madymo theory manual0eca4fe331aaaa0387ab-39017777f15f755539d3047328d4a990.r16.cf3...coupling manual...

Design, Simulation and Virtual Testing

madymo

Theory Manual | VERSION 7.7

www.tassinternational.com

c© Copyright 2017 by TASS InternationalAll rights reserved.

MADYMO R© has been developed at TASS International Software BV.

This document contains proprietary and confidential information of TASS International. Thecontents of this document may not be disclosed to third parties, copied or duplicated in anyform, in whole or in part, without prior written permission of TASS International.

The terms and conditions governing the license of MADYMO R© software consist solely of thoseset forth in the written contracts between TASS International or TASS International authorisedthird parties and its customers. The software may only be used or copied in accordance withthe terms of these contracts.

www.tassinternational.com

MADYMO Theory Manual

MADYMO Manuals

An overview of the MADYMO solver related manuals is given below. From Acrobat Reader,these manuals can be accessed directly by clicking the manual in the table below. Manualsmarked with a star (⋆) are also provided in hard-copy (major releases only).

Theory Manual The theoretical concepts of the MADYMO solver.Reference Manual⋆ Detailed information on how to use the MADYMO solver

and how to specify the input.Model Manual⋆ Dummy, Dummy Subsystem and Barrier Models with

simple examples.Human Model Manual Human Models and applications that make use of Human

Models.Tyre Model Manual Documentation about Tyre Models.Utilities Manual User’s guide for MADYMO/Optimiser, MADYMO/Scaler,

MADYMO/Dummy Generator, MADYMO/Tank TestAnalysis

Folder Manual Describes the use of MADYMO/Folder.Programmer’s Manual Information about user-defined routines.Release Notes Describes the new features, modifications and bug fixes

with respect to the previous release.Installation Instructions Description for the system administrator to install

MADYMO.Coupling Manual Description of coupling with ABAQUS, LS-DYNA, PAM

CRASH/SAFE and Radioss and the TCP/IP coupling withMATLAB/Simulink.

TASS International provides extensive and high quality support for its products to help youin utilizing the software most efficiently. TASS International offers extensive hotline supportfor our software products, MADYMO, PreScan and Delft-Tyre. Our hotline support can bereached over phone as well as via email and will assist you with your questions regarding ourdifferent software products. Your requests will be dealt with in a fast and effective manner tosupport you in the continuation of your work in progress. On the website you will find yourlocal representative with the accompanying support contact details.

iii

MADYMO Theory Manual

iv

MADYMO Theory Manual CONTENTS

Table of contents

MADYMO Manuals iii

Table of contents iv

1 Introduction to MADYMO 1

1.1 Overview of the Theory Manual chapters . . . . . . . . . . . . . . . . . . . . . . 2

2 Reference space 5

3 Multi-body systems 7

3.1 System structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Systems with closed chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Rigid bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Kinematics of a rigid body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.5 Kinematics of bodies connected by a joint . . . . . . . . . . . . . . . . . . . . . . 14

3.6 Flexible bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.7 Flexible beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.8 Kinematic joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.8.1 Revolute joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.8.2 Translational joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.8.3 Spherical joint with Euler parameters . . . . . . . . . . . . . . . . . . . . 28

3.8.4 Spherical joint with Euler angles . . . . . . . . . . . . . . . . . . . . . . . 29

3.8.5 Spherical joint with Bryant angles . . . . . . . . . . . . . . . . . . . . . . 30

3.8.6 Universal joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.8.7 Cylindrical joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.8.8 Planar joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.8.9 Bracket joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.8.10 Free joint with Euler parameters . . . . . . . . . . . . . . . . . . . . . . . 35

3.8.11 Free joint with Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.8.12 Free joint with Bryant angles . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8.13 Free joint with rotation and translation . . . . . . . . . . . . . . . . . . . 39

3.8.14 Translational-revolute joint . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.8.15 Revolute-translational joint . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.8.16 Translational-universal joint . . . . . . . . . . . . . . . . . . . . . . . . . . 41

v

CONTENTS MADYMO Theory Manual

3.8.17 Universal-translational joint . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.9 Body surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.10 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.11 Numerical integration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Sensor, signal, operator and control elements 55

4.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Finite element model 65

5.1 Finite element concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Time integration method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Stresses and strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4 Principal and effective stresses and strains . . . . . . . . . . . . . . . . . . . . . 76

5.5 Material models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5.1 Elastic material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5.2 Elasto-plastic material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5.3 Damage models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5.4 Woven fabric material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5.5 Interface material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5.6 Hyperelastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.5.7 Visco elastic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.5.8 Sandwich material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5.9 Honeycomb material behaviour . . . . . . . . . . . . . . . . . . . . . . . 109

5.5.10 Solid foam material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.5.11 Fu-Chang foam material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.5.12 Spotweld material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.5.13 Other material models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.6 Element types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.7 Initial conditions, prescribed motion and supports . . . . . . . . . . . . . . . . 138

5.8 Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.9 Linear constraint equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.10 Spotwelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.11 Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

vi


6 Acceleration field model 147

7 Restraints 151

7.1 Kelvin restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.2 Maxwell restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.3 Point restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7.4 Joint restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.5 Cardan and flexion-torsion restraints . . . . . . . . . . . . . . . . . . . . . . . . 161

8 Belt model 169

8.1 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

8.2 Belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.3 Slip models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8.3.2 Belt forces and friction forces . . . . . . . . . . . . . . . . . . . . . . . . . 172

8.3.3 Quasi-static belt slip model . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.3.4 Dynamic slip model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

8.4 Retractor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8.5 Pretensioner models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.6 Load Limiter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.7 Fuse belts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.8 Hybrid belt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9 Contact interaction models 185

9.1 Contact between ellipsoids, cylinders and planes . . . . . . . . . . . . . . . . . 185

9.2 Contact between FE-surfaces and MB-surfaces . . . . . . . . . . . . . . . . . . . 198

9.3 Contacts between finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.3.1 Contact algorithm based on intersections . . . . . . . . . . . . . . . . . . 203

9.3.2 Contact algorithm based on penetrations . . . . . . . . . . . . . . . . . . 206

9.3.3 Contact force models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9.3.4 Initial intersection check . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.3.5 Extra options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

10 Airbag models 223

10.1 Finite element airbag model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

10.2 Airbag finite element mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

10.3 Material behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

vii


10.4 Airbag inflation process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

10.5 Ideal gas law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

10.6 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

10.7 Overall bag leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

10.8 Overall energy leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

10.9 Straps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

10.10 Suggestions and warnings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

10.11 Gasflow Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

10.12 Multiple Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

11 Muscle model 263

12 Tyre model 269

13 Energy preservation 279

14 Filters 281

14.1 Filtering time-history signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

14.2 Why filter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

14.3 Selecting a filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

14.4 Choosing the sampling frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 282

14.5 Pre-event and post-event data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

15 Injury parameters 287

15.1 Gadd Severity Index (GSI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

15.2 Head Injury Criterion (HIC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

15.3 Neck Injury Criteria - Forward (NIC_FORWARD) . . . . . . . . . . . . . . . . . 292

15.4 Biomechanical neck injury predictor Nij . . . . . . . . . . . . . . . . . . . . . . . 293

15.5 3MS and XMS (CONTIGUOUS and CUMULATIVE) . . . . . . . . . . . . . . . 294

15.6 Thoracic Trauma Index (TTI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

15.7 Viscous Injury Response (VC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

15.8 Combined Thoracic Index (CTI) . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

15.9 Femur Force Criterion (FFC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

15.10 Tibia Index (TI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

15.11 Tibia Compressive Force Criterion (TCFC) . . . . . . . . . . . . . . . . . . . . . 298

15.12 Abdominal Peak Force (APF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

15.13 Combined Iliac and Acetabulum Peak Force (CIAPF) . . . . . . . . . . . . . . . 298

viii


15.14 Head Contact Duration (HCD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

15.15 Weighted Head Injury Criterion (HICd) . . . . . . . . . . . . . . . . . . . . . . . 299

15.16 Total Moment about Occipital Condyle (MOC) . . . . . . . . . . . . . . . . . . . 299

15.17 Neck Injury Criteria - Rearward (NIC_REARWARD) . . . . . . . . . . . . . . . 299

15.18 Neck injury predictor Nkm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

15.19 Lower Neck Load Index (LNL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

15.20 Brain Injury Criterion (BrIC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

16 Dynamic relaxation 303

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

16.2 Description of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

16.3 Convergence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

16.4 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

16.5 Extra options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

A Functions 307

B Hysteresis 309

B.1 Hysteresis model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312



C Dynamic amplification factor 325

D Coordinate system orientation 327

D.1 Rotation angle method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

D.2 Screw axis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

D.3 Vector method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

Index 334

ix


x

MADYMO Theory Manual Introduction to MADYMO

11 Introduction to MADYMO

MADYMO (MAthematical DYnamic MOdel) is a computer program that sim-ulates the dynamic behaviour of physical systems emphasizing the analysis ofvehicle collisions and assessing injuries sustained by passengers. Althoughoriginally developed for studying occupant behaviour during car crashes,MADYMO is sufficiently flexible to analyse collisions involving other meansof transport such as trains, aeroplanes, motorcycles and bicycles. It also al-lows assessments to be made of the suitability of various restraint systems,including seat belts and airbags.

MADYMO combines in one simulation program the capabilities offered bymulti-body (for the simulation of the gross motion of systems of bodies con-nected by kinematical joints) and finite element techniques (for the simulationof structural behaviour), Figure 1.1. A model can be created with only finiteelement models, or only multi-bodies, or both.

systems ofbodies

finiteelementmodels

belt systems strapssprings/damperstyres

actuators

controllers

user routines acceleration fields airbag loading

supports

contacts

point, edge,

surface loads

Figure 1.1: MADYMO structure.

Within the airbag module MADYMO offers a set of standard force modelsfor belts, airbag and contact between bodies or with the surroundings (seeFigure 1.1 and Figure 1.2). To create a MADYMO input data file, the user firstselects the number of multi-body systems and finite element structures to beincluded in the simulation model. For example, a simulation model can con-sist of one multi-body system for a dummy, one for a deformable steeringcolumn and one for a child restraint system.

Release 7.7 1

1

Introduction to MADYMO MADYMO Theory Manual

user defined forces

acceleration

gravity

plane contact

ellipsoid contact

Figure 1.2: Examples of systems of bodies with force interactions.

Finite element structures can be used for the driver, passenger side airbag andthe knee bolster. For crash dummy models, standard models are availablethat have been already created by TNO (see the Database Manual). An inputdata file is then set up which specifies the mass distribution of the bodies, theconnections between the bodies and the joint properties, and for finite elementstructures.

1.1 Overview of the Theory Manual chapters

The multi-body algorithm in MADYMO yields the second time derivatives ofMulti-bodysystems the degrees of freedom in an explicit form. The number of computer opera-

tions is linear in the number of bodies if all joints have the same number ofdegrees of freedom. This leads to an efficient algorithm for large systems ofbodies. At the beginning of the integration, the initial state of the systems ofbodies has to be specified in terms of joint positions and velocities. Severalkinematic joint types are available with dynamic restraints to account for jointstiffness, damping and friction. Joints can be locked, unlocked or removedbased on user-defined conditions. (Chapter 3)

The control module applies loads to bodies, which depend on the motionSensor andControlelements

of bodies. Sensors, transformers, controllers and actuators can be used.(Chapter 4)

The finite element module provides a method for dividing the actual contin-Finite elementmodels uum into finite volumes, surfaces or line segments. The continuum is then

analysed as a complex system, composed of relatively simple elements wherecontinuity should be ensured along the interface between elements. These el-ements are interconnected at a discrete number of points, the nodes. The ini-tial nodal positions and velocities, the nodes corresponding to each element,

2 Release 7.7

MADYMO Theory Manual Introduction to MADYMO

1the connectivity, as well as the element properties such as material behaviour,must be specified at the start of the simulation. Truss, beam, membrane, shelland solid elements can be applied. Material models are available for metals,fabrics, foams, composites, rubbers and honeycomb.

The manner that the interaction between bodies and finite elements is mod-elled allows the use of different time integration methods for the equationsof motion of the finite element part and the multi-body part. The integra-tion methods used are conditionally stable and therefore put limitations onthe maximum time step that can be used. To increase the efficiency of the en-tire analysis, the finite element module can be sub-cycled with respect to themulti-body module using different time steps for each module. (Chapter 5)

The acceleration field model calculates the forces at the centres of gravity ofAccelerationfield model bodies or finite elements due to a homogeneous acceleration field. This model

is particularly useful for the simulation of the acceleration forces on a vehicleoccupant during an impact. Acceleration fields can be applied to any groupof bodies a user chooses. (Chapter 6)

Five types of restraints are available. (Chapter 7)Restraints

• The Kelvin restraint is a uni-axial element that simulates a spring paral-lel with a damper.

• The Maxwell restraint is a uni-axial element that simulates a spring anddamper in series. Non-linear spring characteristics as well as velocitydependent damping can be defined.

• A Point restraint is a combination of three spring-damper elements eachparallel to one of the axes of an orthogonal coordinate system.

• Joint restraint.

• Cardan and flexion-torsion restraint.

The belt model accounts for initial belt slack or pre-tension and rupture ofBelt modelbelt segments. Elastic characteristics can be specified separately for each beltsegment and slip of belt material from one segment to another is accountedfor. A special option is available for fuse belts. For each belt one retractor withan optional webbing grabber, one pretensioner and one load limiter can bemodelled (Chapter 8).

Planes, ellipsoids and cylinders can be attached to a body to represent itsContactmodels shape. These surfaces are also used to model contact with other bodies or

with finite element models. The contact surfaces are of major importance inthe description of the interaction of the occupant with the vehicle interior. Theelastic contact forces, including hysteresis, are a function of the penetration ofthe contact surfaces. In addition to elastic contact forces, damping and frictioncan be specified. (Chapter 9)

The finite element model contains special functionality for airbag modelling,Airbag modelsfor example, straps, inflators and gas jets. (Chapter 10)

Release 7.7 3

1

Introduction to MADYMO MADYMO Theory Manual

The muscle model implemented in MADYMO is a uni-axial element describ-Muscle modeling active and passive muscle forces using nonlinear Hill type equations. Thetwo ends can be attached to arbitrary points on (flexible) bodies or the refer-ence space. The model describes the tension force F, as a function of the dis-tance l between the two attachment points (the muscle length), the speed ofchange in l (the lengthening velocity v) and the active state (A). (Chapter 11)

A tyre model is available for studies involving vehicle handling. Standard andTyre modeluser defined road elevation profiles can be applied. (Chapter 12)

Analog signals measured during experiments usually are filtered. To com-Filteringpare MADYMO simulations with those experiments, filters from the SAE J211Draft (CFC) and a NHTSA document (FIR100) can be specified for filteringMADYMO time-history signals. (Chapter 14)

A large number of standard output parameters are available, such as acceler-Injuryparameters ation, forces, torques and kinematic data. MADYMO (Chapter 15) also offers

in addition the possibility to calculate standard injury parameters such as:

• Head injuries (HIC, GSI),

• Neck injuries (NIC, NIJ),

• Thoracic injuries (3MS, CTI, TTI, VC),

• Femur injuries (FFC, TCFC) and

• Tibia injuries (TI).

4 Release 7.7

MADYMO Theory Manual Reference space

22 Reference space

A coordinate system (X, Y, Z) is connected to the reference space (Figure 2.1).The origin and orientation of this reference space coordinate system can be se-lected arbitrarily. Usually the positive Z axis is chosen pointing upwards, thatis, opposite to the direction of gravity. The motion of all systems is describedrelative to this coordinate system.

X

Z

Y

Figure 2.1: Reference space coordinate system.

Surfaces such as planes, ellipsoids and restraints as well as nodes of finiteelement models can be attached to the reference space.

Release 7.7 5

2

Reference space MADYMO Theory Manual

6 Release 7.7

MADYMO Theory Manual Multi-body systems

3

3 Multi-body systems

A multi-body system is a system of bodies. Any pair of bodies in the samesystem can be interconnected by one kinematic joint. The MADYMO multi-body formalism for generating the equations of motion is suitable for systemsof bodies with a tree structure (Figure 3.1) and systems with closed chains.

Systems with closed chains are reduced to systems with a tree structure byremoving a kinematic joint in every chain. Removed joints are subsequentlyconsidered as "closing" joints. For each (reduced) system with a tree struc-ture, one body can be connected to the reference space by a kinematic joint, orthe motion relative to the reference space of one body can be prescribed as afunction of time.

Figure 3.1: Examples of single and multi-body systems with tree structure.

A kinematic joint restricts the relative motion of the two bodies it connects.Kinematicjoints A specific type of kinematic joint is characterised by the way the relative mo-

tion of two bodies is constrained. The relative motion allowed by a joint isdescribed by quantities called joint degrees of freedom. The number dependson the type of joint. In MADYMO, the most common joint types are availablesuch as spherical joints, translational joints, revolute joints, cylindrical joints,planar joints and universal joints.

A system of bodies is defined by:

• the bodies: the mass, the inertia matrix and the location of the centre ofgravity,

Release 7.7 7

3

Multi-body systems MADYMO Theory Manual

• the kinematic joints: the bodies they connect, the type, and the locationand the orientation, and

• the initial conditions.

In addition, the shape of bodies may be needed for contact calculations orpostprocessing (graphics) purposes. Applied loads on bodies can be modelledwith the force models described in the following chapters.

3.1 System structure

Several multi-body systems can be defined in MADYMO. A multi-body sys-tem is a set of bodies interconnected by kinematic joints. Two sets of bodies areseparate multi-body systems when there are no kinematic joints between the two sets.The interconnection structure of a multi-body system is implicitly defined inthe input file. Kinematic joints connect the bodies and define the structure ofthe multi-body system.

A multi-body system has a tree structure if it is possible to move from onearbitrary body i to another arbitrary body j along a unique sequence of bodiesand joints (Figure 3.2). The travelled route is called the path between bodiesi and j. If another path exists between a pair of bodies, the bodies in thesetwo paths form a closed chain (Figure 3.3). MADYMO reduces a system withclosed chains to a system with a tree structure by removing appropriate kine-matic joints (Section 3.2). Such removed joints are taken into account as "clos-ing" joints.

The interconnection structure of a multi-body system follows from the defini-tion of the kinematic joints. The definition of a kinematic joint consists of thespecification of the two bodies connected by the joint, and the location andorientation of the joints. The first (second) referenced body is called the par-ent (child) body. A kinematic joint can also connect a body with the referencespace. The reference space must always be the first referenced "body". Theremust be at least one kinematic joint between a multi-body system and the referencespace.

8 Release 7.7


3i

j

Figure 3.2: The path between body i and body j.

j

i

Figure 3.3: Example of a system with a closed chain.

Every body has to be used as child body at least once. (This does not apply to thereference space since it is not considered as a body.) When this is not strictlynecessary from a modelling point of view, a joint must be added with such abody as child body and another body or the reference space.

The order in which the two bodies are specified should be done with care. Itis best to use a particular body only once as child body. (This is only possiblefor multi-body systems with a tree structure.) Only then the system will beanalysed as a system with a tree structure. When a body is referenced morethan once as a child body, MADYMO analyses the multi-body system as asystem with closed chains. This is more time consuming than an equivalentsystem with a tree structure.

Figure 3.4 shows a multi-body system. Body 1 is connected by a joint withthe reference space. When for each kinematic joint, the lower numbered body

Release 7.7 9

3


is used as parent body, this system will be analysed as a system with a treestructure; otherwise, it will be analysed as a system with closed chains.

Figure 3.4: Example of a multi-body system.

3.2 Systems with closed chains

In MADYMO, systems of bodies may have closed chains. The theoreticalnumber of degrees of freedom ncc of a closed chain equals the sum of the num-ber of joint velocity degrees of freedom (Table 3.2) of joints in the chain minus6. The value of ncc is less than or equal to the actual number of degrees offreedom na,cc of the closed chain. When ncc < na,cc or ncc < 0 , the chain clos-ing constraint equations are dependent. MADYMO does not allow dependentconstraint equations. For calculating ncc , degrees of freedom of joints that arein more than one closed chain may be counted just once. Locked joints andjoints with a prescribed motion have zero degrees of freedom.

It may be necessary to use different joint types in the model of a system withclosed chains as compared to the physical mechanism in order to obtain closedchains with na,cc = ncc . This is illustrated with the planar four-bar linkageshown in Figure 3.5. The bars are connected by revolute joints with their rota-tion axis perpendicular to the plane of the mechanism. For this model we havencc = 4 × 1 − 6 = −2 and na,cc = 1 which is not allowed because ncc < na,cc .

Some joints are not really revolute joints because they also allow some addi-tional rotations/translations. This eliminates malfunctioning of the mecha-nism due to manufacturing inaccuracies that may lead to revolute joints withtheir rotation axis not perpendicular to the plane of the mechanism. In factthey behave in a real mechanism as if they are spherical or cylindrical joints.

In order to obtain a closed chain with ncc = 1 , revolute joints have to bereplaced by other joint types such that three extra joint velocity degrees offreedom are gained. For example, joint B can be replaced by a spherical jointand joint C by a cylindrical joint. Replacing joints B and C by a spherical anda cylindrical joint, respectively, does not change the behaviour of this four-bar linkage. Note that replacing cylindrical joint C also by a spherical joint

10 Release 7.7


3

does not lead to one extra degree of freedom for the chain; it only gives theconnecting rod BC the freedom to rotate around its axis.

B

A

C

D

Figure 3.5: Four-bar linkage.

For systems with closed chains, it may be difficult to specify initial valuesof the joint position and velocity degrees of freedom such that the system isinitially assembled. MADYMO automatically assembles systems with closedchains. For every closed chain, the user must set STATUS = INITIAL for thosejoints that have in total exactly 6 velocity degrees of freedom. During theassembly analysis, the specified initial degrees of freedom of these joints arechanged by MADYMO. In general, it is best to select joints for which the de-grees of freedom are bounded. For the four-bar linkage shown in Figure 3.5,the degrees of freedom of revolute joint A can have arbitrary values; the de-grees of freedom of the other joints vary between specific limits. For that rea-son, it is best to set STATUS = INITIAL for joints B, C and D. Note that whenjoint B is replaced by a spherical joint and joint C by a cylindrical joint, theirtotal number of degrees of freedom equals 5.

The system and the initial values of the joint position degrees of freedomwhich are kept constant during the assembly analysis, must be such that thesystem can be assembled. It is, for example, not possible to assemble the four-bar linkage of Figure 3.5 when the distance between joints A and D is largerthan the sum of the lengths of the three bars.

The specified initial values of the degrees of freedom of the joints with STA-TUS = INITIAL are used as starting values for the assembly analysis. Whenthe initial configuration of the mechanism is not unique, the initial values ofthese joint degrees of freedom must be close enough to the actual initial valuesin order to avoid convergence to an incorrect initial configuration. Figure 3.6shows two possible solutions for the assembly analysis of a four-bar linkagewhen the degree of freedom of revolute joint D is kept constant during theassembly analysis (STATUS = FREE).

The number of degrees of freedom of joints with STATUS = INITIAL may notchange during the analysis. Therefore these joints may not be locked or broken(Section 3.8) and stick due to Coulomb friction (Section 7.4) may not occur.

Release 7.7 11

3


B

A

C

D

B'

Figure 3.6: Different initial configurations of a four-bar linkage.

3.3 Rigid bodies

A rigid body is defined by:

• the mass,

• the location of the centre of gravity, and

• the moments of inertia and the products of inertia.

The shape of a body is not relevant to the equations of motion except whena body contacts other bodies or its environment. Only in the latter case it isnecessary to define the shape of the bodies.

In order to quantify body data, the user must choose a body local coordinatesystem. The user can choose the origin and orientation of this coordinate sys-tem depending on the user’s needs. Normally the axes are chosen such thatthey can be easily recognised in order to facilitate measurement of geometricdata. The local coordinate system of a body i will be denoted by (xi, yi, zi) .Data corresponding to a specific body is defined with respect to this body lo-cal coordinate system.

The location of the centre of gravity is denoted by the vector gi . The com-ponents gi of this vector must be expressed in the local coordinate system ofbody i. The three mass moments of inertia and three products of inertia mustbe specified relative to a coordinate system, the inertia coordinate system, withits origin in the centre of gravity of the body. The orientation of this coordinatesystem must be specified when it differs from the orientation of the body localcoordinate system. The products of inertia are equal to zero when the axes ofthe inertia coordinate system are parallel to the principal axes of inertia.

3.4 Kinematics of a rigid body

Consider the rigid body i shown in Figure 3.7. The motion of this body is com-pletely defined by the location of the origin and the orientation of the body

12 Release 7.7


3

local coordinate system (xi, yi, zi) relative to the reference space coordinatesystem (X, Y, Z) .

Figure 3.7: Specification of the motion of a rigid body.

The position of the origin of the body local coordinate system relative to theorigin of the reference space coordinate system is given by the vector ri . Theorientation of the body local coordinate system relative to the reference spacecoordinate system is defined by the direction cosine matrix Ai .

Let the vector from the origin of the body local coordinate system to an arbi-trary point P on body i be given by xi . The position vector Xi of this point inthe reference space is given by

Xi = ri + xi (3.1)

This equation can be written in the matrix form

Xi = ri + Aixi (3.2)

or

X1

X2

X3

i

=

r1

r2

r3

i

+

A11 A12 A13

A21 A22 A23

A31 A32 A33

i

x1

x2

x3

i

(3.3)

Release 7.7 13

3


where and ri are column matrices containing the components relative to thereference space coordinate system of the vectors Xi and ri , respectively. xi is acolumn matrix containing the components of the vector xi relative to the bodylocal coordinate system. The direction cosine matrix Ai relates xi to Xi .

The first time derivative of (3.1) equals

Xi = ri + ωi × xi (3.4)

where ωi is the angular velocity vector of body i.

The second time derivative of (3.1) equals

Xi = ri + ωi × xi + ωi × (ωi × xi) (3.5)

3.5 Kinematics of bodies connected by a joint

Two bodies interconnected by an arbitrary kinematic joint are shown inFigure 3.8. A kinematic joint can connect only two bodies. The bodies aredenoted by i and j. In MADYMO, the motion of a body j is described rela-tive to the corresponding parent body i. The joint position degrees of freedomdefine the motion within the joint. Their number, nij , equals the number ofdegrees of freedom of the joint. They will be contained in the column matrixq

ij.

14 Release 7.7


3

body j

body i

ηj

ηi

ζi

ζjzi

yi

xi

cij

ξi

dij

ξj

cji

yjxj

zj

z

xy

ri rj

Figure 3.8: Two interconnected bodies.

On each body, a body-fixed joint coordinate system (ξ, η, ζ) is introduced inorder to describe the relative motion of body j relative to body i. Section 3.8explains how the joint coordinate systems are defined.

Let the orientation of the joint coordinate system on body i (j) relative tothe body local coordinate system on body i (j) be specified by the time-independent direction cosine matrix Cij (Cji) .

Let the orientation of the joint coordinate system on body j relative to the jointcoordinate system on body i be specified by the direction cosine matrix Dij .This matrix is a function of the joint position degrees of freedom q

ijonly.

Let the orientation of the local coordinate system of body i and body j bespecified by the direction cosine matrix Ai and Aj, respectively. Using thesedirection cosine matrices, Aj can be written in terms of Ai as

Aj = AiCijDijCTij (3.6)

Let cij and cji be the position vectors of the origins of the joint coordinate sys-tems on body i and j, respectively, relative to the origin of the local coordinatesystem of the corresponding body. For a rigid body, the components of thesevectors relative to the corresponding body local coordinate system are con-stant. The vector from the origin of the joint coordinate system on body i to

Release 7.7 15

3


the joint coordinate system on body j is given by the vector dij . The compo-nents of this vector relative to the joint coordinate system on body i, dij , area function of the joint position degrees of freedom q

ijonly. The position vec-

tor of the origin of the local coordinate system on body j, rj , can be writtenin terms of the position vector of the origin of the local coordinate system onbody i, ri , as

rj = ri + cij + dij − cji (3.7)

Applying equations (3.6) and (3.7) successively for a body connected to thereference space until the peripheral body of each branch yields the positionsand orientations of all the body local coordinate systems relative to the refer-ence space coordinate system.

Taking the first time derivative of equations (3.6) and (3.7) yields the followingexpressions for the angular and linear velocity (Wittenburg1).

ωj = ωi + ωij (3.8)

rj = ri + ωi × cij + dij − ωj × cji (3.9)

ωij is the angular velocity vector of body j relative to body i. Taking the timederivative of equations (3.8) and (3.9) yields similar expressions for the angu-lar and linear acceleration (Wittenburg).

1Wittenburg, J., "Dynamics of Systems of Rigid Bodies", Teubner, Stuttgart, Germany, 1977.

16 Release 7.7


3

3.6 Flexible bodies

Bodies or structural parts, which experience small deformations, but largeKinematicstranslations and/or rotations, can be modelled as flexible bodies. For a flexi-ble body, the motion of a point on the body is considered to be composed of arigid body motion and a motion due to deformation.The rigid body motion isdetermined by the motion of the body local coordinate system (the position ofits origin) defined by the position vector r, and its orientation, defined by thedirection cosine matrix A, (see Section 3.4). The motion due to deformation issuperimposed on this rigid body motion.

Let the displacement due to deformation be given by the vector u. This vectordepends on the point on the body, which is defined by position vector x in theundeformed state of the body, and time, u = u(x, t) . The position vector X ofan arbitrary point on the body relative to the origin of the reference coordinatesystem is then given by (Figure 3.9).

Figure 3.9: Specification of the motion of a flexible body.

X = r + x + u (3.10)

This equation can be written in component form as

X = r + A(x + u) (3.11)

Release 7.7 17

3


where X and r are column matrices containing the components with respect tothe reference space coordinate system of the vectors X and r, respectively, andx and u are the components with respect to the body local coordinate systemof the vectors x and u, respectively. The origin and orientation of the bodylocal coordinate system of a flexible body can be chosen at will.

It is assumed that the rotations due to deformation (not the rigid body ro-tations) are sufficiently small, such that they can be represented by a vectorquantity denoted by φ. The direction of this vector defines the rotation axisand the rotation angle is defined by its magnitude. The rotation vector de-pends on the point on the body, which is defined by the position vector x inthe undeformed state of the body, and time, i.e. φ = φ(x, t) . The compo-nents of this vector field with respect to the body local coordinate system aredenoted by φ(x, t) . The direction cosine matrix due to deformation corre-sponding to this small rotation is given by the matrix B. This matrix can bedetermined from φ(x, t) . The direction cosine matrix D of an arbitrary bodyfixed coordinate system with respect to the reference space coordinate systemis then given by

D = A B C (3.12)

where C is the direction cosine matrix of the body fixed coordinate systemwith respect to the body local coordinate system in the undeformed state ofthe body.

The displacement due to deformation is approximated by a linear combina-tion of n predefined displacement fields

u(x, t) =n

∑k=1

αk(t)Φtk(x) (3.13)

Similarly, rotations due to deformation are approximated by a linear combi-nation of n predefined rotation fields

φ(x, t) =n

∑k=1

αk(t)Φrk(x) (3.14)

If a displacement field involves a rotation field, the vector fields Φtk(x) and

Φrk(x) must agree, otherwise one of the two vector fields must be 0. The com-

bination of a displacement field and a rotation field makes up a deformationmode. Deformation fields must be independent and are not allowed to de-scribe a rigid body motion.

The weight factors αk(t) are the modal degrees of freedom. Their values fol-low from time integration of the equations of motion. The vector fields Φt

k(x)and Φr

k(x) must be chosen such that they approximate the deformation of thebody well. For a body with a simple shape, analytical functions can be used

18 Release 7.7


3

(Section 3.7). For a body with a complex shape, the value of the functionsmust be specified at a limited number of points on the body, the nodes. Thedisplacements and rotations at these nodes can be obtained for instance froma finite element analysis.

Interactions of a flexible body with its surroundings can take place only atthe nodes. Therefore nodes must be defined at locations at which interactionswith the surroundings take place. Not all interactions of a body with its sur-roundings have been fully adapted to the option of taking the flexibility intoaccount, e.g. ellipsoids, cylinders, planes and point restraints.

The kinematic joints and the corresponding joint restraints have been adaptedsuch that the flexibility of bodies is taken into account. However, for kinematicjoints for which the origins of the joint coordinate systems do not coincidedue to a relative translation of the interconnected bodies, i.e. for translational,cylindrical and planar joints, the deformation is not fully accounted for. Incalculating the kinematics of a pair of bodies interconnected by a kinematicjoint, the deformation of the parent body at the origin of the correspondingjoint coordinate system is used for the above mentioned joint types.

3.7 Flexible beams

In MADYMO, analytic deformation modes have been implemented that ap-proximate the deformation of a straight prismatic slender beam. This is basedon the assumption that the centroid, the centre of mass and the shear centreof the cross-section coincide. A beam coordinate system is used to simplifythe expressions for the deformation modes (Figure 3.10). The origin of thiscoordinate system is in one of the ends of the beam, at the centroid of thecross-section. The xb axis of the beam coordinate system passes through thecentroid of the cross-section at the other end of the beam. The yb and zb axisare along the principal axes of the cross-section of the beam.

Figure 3.10: Definition of coordinate systems for flexible beam.

Let L be the length of the undeformed beam and ξ the distance between the

Release 7.7 19

3


origin of the beam coordinate system and an arbitrary point on the beamwhich is normalized with respect to L. In order to obtain a good approxi-mation of the deformation of a beam due to various loadings, six deformationmodes are defined:

• A linearly varying rotation around the xb axis corresponding to a con-Deformationmodes stant torque. This deformation has no displacements. The correspond-

ing modal degree of freedom represents the relative angle of rotation ofthe ends of the beam due to torsion.

Φrx = −1

2+ ξ (3.15)

• A linearly varying axial displacement corresponding to a constant ax-ial load. This deformation has no rotations. The corresponding modaldegree of freedom represents the axial strain in the beam.

Φtx = L(−1

2+ ξ) (3.16)

• Two deformation modes representing deflection in the yb direction cor-responding to a balanced set of transverse forces and bending momentsat the ends of the beam; the corresponding modal degrees of freedomrepresent the angles of rotation around the zb axis at the ends of thebeam due to bending.

Φty = L(ξ − 2ξ2 + ξ3) Φr

z = 1 − 4ξ + 3ξ2 (3.17)

Φty = L(−ξ2 + ξ3) Φr

z = −2ξ + 3ξ2 (3.18)

• Two deformation modes representing deflection in the zb direction cor-responding to a balanced set of transverse forces and bending momentsat the ends of the beam; the corresponding modal degrees of freedomrepresent the angles of rotation around the yb axis at the ends of thebeam due to bending.

Φtz = −L(ξ − 2ξ2 + ξ3) Φr

y = 1 − 4ξ + 3ξ2 (3.19)

Φtz = −L(−ξ2 + ξ3) Φr

y = −2ξ + 3ξ2 (3.20)

The bending deformation modes are normalised such that the rotation at oneof the ends equals 1.0. This causes the difference in sign of the displacementfields for bending in the yb and zb direction.

20 Release 7.7


3

The input of a flexible beam makes it possible to specify that one or more ofthe above six deformation patterns must be omitted. Then the correspondingdeformation mode(s) will not be included, reducing the number of deforma-tion modes. This may be worthwhile if a high eigenfrequency corresponds tosuch a deformation pattern, such as may be the case for the axial deformationof a beam.

The deformation modes are calculated only at the nodes. The nodes are gen-erated at constant intervals along the beam axis, starting with the node at theorigin of the beam coordinate system. Increasing the number of nodes doesnot increase the accuracy of the flexible beam model, but does lead to an in-crease of CPU time. The nodal coordinates are written to the KIN3 file in thesame way as a finite element model.

The elastic loads in the beam corresponding to the above displacement fieldsSlender beamtheory are based on slender beam theory. The beam is assumed to be linear elastic

with Young’s modulus E and shear modulus G = E/[2(1 + ν)] , where ν isPoisson’s ratio. This leads to the following modal stiffness matrix:

GIxxL

EAL

4EIzzL

2EIzzL

2EIzzL

4EIzzL

4EIyy

L2EIyy

L2EIyy

L4EIyy

L

(3.21)

where A, Ixx , Iyy and Izz are the area, the torsion constant, and the two areamoments of inertia of the cross-section, respectively.

The damping matrix is assumed to be proportional to the stiffness matrix.

3.8 Kinematic joints

Two bodies can be connected by one kinematic joint. A kinematic joint con-strains the relative motion of this pair of bodies, so a translational joint allowsonly relative translation. In this section, the parent body will be denoted by iand the child body by j.

The constraints imposed by a kinematic joint cause a load on the pair of in-terconnected bodies, the constraint load. This load is such that the relativemotion of the pair of bodies is restricted to a motion that does not violate theconstraints imposed by the kinematic joint. The constraint loads on the sepa-rate bodies are equal but opposite loads (Figure 3.11). Constraint loads can beused to assess the strength of the joint.

Release 7.7 21

3


j

i

Fj

Fi

Figure 3.11: Constraint load in a spherical joint.

A kinematic joint can connect only two bodies. A joint which connects threebodies, like the one illustrated in Figure 3.12, is not allowed. It can be mod-elled, however, by using two joints, such as a joint that connects bodies 1 and2 and another joint that connects bodies 1 and 3. Both joints have the samelocation on body 1.

1

2

1

3

1

2 3

Figure 3.12: Three bodies linked at one point.

A kinematic joint is defined by the type of the joint, and the joint coordinatesystems (ξi, ηi, ζi) and (ξ j, ηj, ζ j), one rigidly fixed to each of the two bodiesconnected by the joint (Figure 3.13). The origin and the orientation of the jointcoordinate systems on the two bodies have to be specified in accordance withthe way they have been chosen in MADYMO. For example, for a sphericaljoint, the common pivot points are defined as the location of the joint; for atranslational joint one of the axes of the joint coordinate systems must coincideand the other two pairs of axes must be parallel. The possible relative motionof the joint coordinate systems depends on the joint type. Initial conditionsspecify the relative position and orientation of the joint coordinate systems.

22 Release 7.7


3

body j

body i

ηj

ηi

ζi

ζjzi

yi

xi

cij

ξi

dij

ξj

cji

yjxj

zj

Figure 3.13: Kinematic joint connecting bodies i and j.

The origins of the joint coordinate systems have been chosen such that themathematical expression for the relative translation of these points is as sim-ple as possible. For example, because the origins for a spherical joint havebeen chosen to coincide with the articulation points on the bodies, the relativetranslation vector is identically 0. The orientations of the joint coordinate sys-tems have been selected in the same way. An example, the ξ axis of each jointcoordinate system for a revolute joint has been chosen to coincide with therotation axis. In the input, the orientation and the location of the origin of thejoint coordinate systems have to be specified in accordance with this choice.Section 3.8.1 - Section 3.8.17 describe the possible relative motion of the jointcoordinate systems for the standard types of joints.

The location of the origin of the joint coordinate system on the parent bodyis defined by the components of the vector cij with respect to the local coor-dinate system of the parent body. Similarly, the location of the origin of thejoint coordinate system on the child body is defined by the components of thevector cji with respect to the local coordinate system of that body.

The origin of a joint coordinate system which is fixed to a flexible body(Section 3.6) must be coincident with a node. Consequently a node must bedefined at this location. Due to deformation of the body the location of thenode, and consequently the location of the origin of the joint coordinate sys-tem, may change.

The orientation of a joint coordinate system relative to the corresponding bodylocal coordinate system in the undeformed state of the body can be defined bymeans of one of the methods described in Appendix D. A joint coordinate sys-tem will be parallel to the body local coordinate system when no orientationis specified.

The relative motion of a pair of joint coordinate systems is described byJoint degreesof freedom variables, the joint position, velocity and acceleration degrees of freedom.

Table 3.1 summarises for all standard joint types the joint position and velocitydegrees of freedom. Their physical meaning will be explained in subsequentsections. The joint acceleration degrees of freedom are equal to the first timederivatives of the joint velocity degrees of freedom. The number of joint de-grees of freedom depends on the joint type. The joint degrees of freedom are

Release 7.7 23

3


used as the basic variables to describe the motion of systems of bodies.

Table 3.1: Joint position degrees of freedom for standard joint types.

joint position degrees of freedom

joint type Figure Q1 Q2 Q3 Q4 Q5 Q6 Q7

revolute Figure 3.14 φ

translational Figure 3.15 s

spherical, Euler param(a) Figure 3.16 q0 q1 q2 q3

spherical, Euler angles Figure 3.17 φ1 φ2 φ3

spherical, Bryant angles Figure 3.18 φ1 φ2 φ3

universal Figure 3.19 φ1 φ2

cylindrical Figure 3.20 φ s

planar Figure 3.21 φ sη sζ

bracket Figure 3.22

free, Euler param(a) Figure 3.23 q0 q1 q2 q3 sξ sη sζ

free, Euler angles Figure 3.24 φ1 φ2 φ3 sξ sη sζ

free, Bryant angles Figure 3.25 φ1 φ2 φ3 sξ sη sζ

free, rotation/translation(a) q0 q1 q2 q3 sξ sη sζ

translational/revolute Figure 3.26 φ s

revolute/translational Figure 3.27 φ s

translational/universal Figure 3.28 φ1 φ2 s

universal/translational Figure 3.29 φ1 φ2 s

(a) The Euler parameters (q0, q1, q2, q3) are multiplied by√

2 as compared with the defi-nition given in Appendix D.2.

Table 3.2: Joint velocity degrees of freedom for standard joint types.

joint velocity degrees of freedom

joint type Figure QD1 QD2 QD3 QD4 QD5 QD6

revolute Figure 3.14 φ

translational Figure 3.15 s

spherical, Euler param Figure 3.16 ωξ ωη ωζ

spherical, Euler angles Figure 3.17 φ1 φ2 φ3

spherical, Bryant angles Figure 3.18 φ1 φ2 φ3

Continued on the next page

24 Release 7.7


3

Table 3.2 cont.

joint velocity degrees of freedom

joint type Figure QD1 QD2 QD3 QD4 QD5 QD6

universal Figure 3.19 φ1 φ2

cylindrical Figure 3.20 φ s

planar Figure 3.21 φ sη sζ

bracket Figure 3.22

free, Euler param Figure 3.23 ωξ ωη ωζ sξ sη sζ

free, Euler angles Figure 3.24 φ1 φ2 φ3 sξ sη sζ

free, Bryant angles Figure 3.25 φ1 φ2 φ3 sξ sη sζ

free, rotation/translation ωξ ωη ωζ sξ sη sζ

translational/revolute Figure 3.26 φ s

revolute/translational Figure 3.27 φ s

translational/universal Figure 3.28 φ1 φ2 s

universal/translational Figure 3.29 φ1 φ2 s

The default initial condition is such that for every joint, the joint coordinateInitialconditions systems coincide. The initial motion of a system of bodies is defined by the ini-

tial values of the joint position and velocity degrees of freedom. These definethe initial relative motion of the joint coordinate systems. The initial motionof the body local coordinate systems relative to the reference space coordinatesystem is obtained from the kinematic relations (3.6)–(3.9). The initial val-ues of joint position and velocity degrees of freedom are specified under theINITIAL.JOINT_POS and INITIAL.JOINT_VEL elements.

Joint position degrees of freedom (see Table 3.1) can be prescribed at discretePrescribedjoint positiondegrees offreedom

time points under the MOTION.JOINT_POS element. A spline interpolation isused to obtain the prescribed value at an arbitrary point of time; the corre-sponding joint velocity and acceleration degrees of freedom are determinedfrom this spline approximation.

For the spherical and free joints that use Euler parameters, the relative angu-lar acceleration of the joint coordinate systems is set to zero when the relativeorientation is prescribed using joint position degrees of freedom. The relativeorientation of the joint coordinate systems of spherical and free joints can alsobe prescribed using successive rotations. Then the spherical joint or free jointthat uses Euler or Bryant angles may be selected. These angles have the ad-vantage that they have a direct physical meaning and that the correspondingrelative angular acceleration is interpolated properly.

Joint acceleration degrees of freedom, i.e the first time derivative of the jointPrescribedjointaccelerationdegrees offreedom

velocity degrees of freedom (see Table 3.2) can be prescribed at discrete timepoints under the MOTION.JOINT_ACC element. A linear or spline interpolation

Release 7.7 25

3


is used to obtain the prescribed value at an arbitrary point of time; the corre-sponding joint position and velocity degrees of freedom are obtained from anumerical time integration of the joint acceleration degrees of freedom. Whenjoint position and/or velocity degrees of freedom are non-zero at the start ofthe simulation, the initial conditions must be specified.

Unlocked joints may be locked. Then the corresponding joint position degreesLocking/unlocking andremoval ofjoints

of freedom remain constant and consequently the joint velocity and accelera-tion degrees of freedom are identically zero. This results in a rigid connectionbetween the bodies connected by the joint. A joint should only be locked whenthe corresponding joint velocity degrees of freedom are small, because the im-pact required to make the joint velocity degrees of freedom equal to zero afterlocking, is not taken into account. Locked joints may be unlocked. After un-locking the joint degrees of freedom follow from time integration of the equa-tions of motion. Joints can be locked and unlocked depending on specifiedconditions. This option is useful for modelling a buckle pretensioner.

To model a buckle pretensioner, the body to which the stalk is attached mustbe defined and connected by a translational joint to its supporting body. Thisjoint must be initially locked. A pretensioned spring must be defined betweenthe two bodies. Conditions can be defined for unlocking the translational jointand consequently releasing the spring. The connection between the stalk bodyand the supporting body is generally such that the stalk body can slide in justone direction. This can be modelled by locking the joint when the relativevelocity changes sign using the STATE.JOINT element.

To model a breaking body, it can be modelled as two bodies representing theparts after breaking. Using the STATE.JOINT_REMOVE element, the joint is re-placed by a free joint resulting in two disconnected bodies.

3.8.1 Revolute joint

A revolute joint (Figure 3.14) constrains the relative motion of the intercon-nected bodies to a rotation around the ξ axes of the joint coordinate systems.The origins of the joint coordinate systems remain coincident.

The number of degrees of freedom of a revolute joint equals one; the joint posi-tion degree of freedom φ is the angle between the ζ axes of the joint coordinatesystems. The joint velocity degree of freedom is its first time derivative.

26 Release 7.7


3

i

j

ξi,ξj

ζi

ζj

ηi ηj

φ

Figure 3.14: Revolute joint.

joint position degrees

of freedom

Q1R1

φ

joint velocity degrees

of freedom

QD1W1

φ

3.8.2 Translational joint

A translational joint (Figure 3.15) constrains the relative motion of the inter-connected bodies to a translation along the ξ axes of the joint coordinate sys-tems. The ξ axes are coincident and the η (ζ) axes are parallel.

The number of degrees of freedom of a translational joint equals one; the jointposition degree of freedom s is the ξ coordinate of the origin of the joint coor-dinate system on the child body in the joint coordinate system on the parentbody. The joint velocity degree of freedom is its first time derivative.

Release 7.7 27

3


Figure 3.15: Translational joint.

joint position degrees

of freedom

Q1D1

s

joint velocity degrees

of freedom

QD1V1

s

3.8.3 Spherical joint with Euler parameters

A spherical or ball and socket joint (Figure 3.16) constrains the relative motionof the interconnected bodies to a rotation around the origins of the joint coor-dinate systems. The number of joint position degrees of freedom of a sphericaljoint equals four, i.e., the Euler parameters q0 , q1 , q2 , q3 (see Appendix D.2)which describe the relative rientation of the joint coordinate systems. Thenumber of joint velocity degrees of freedom equals three, these are the com-ponents of the angular velocity vector of the child body relative to the parentbody with respect to the joint coordinate system on the parent body.

28 Release 7.7


3

i

j

ζj

ζi

ξj

ξi

ηi

ηj

Figure 3.16: Spherical joint.

joint position degreesof freedom

Q1 Q2 Q3 Q4

q0 q1 q2 q3

joint velocity degreesof freedom

QD1W1

QD2W2

QD3W3

ωξ ωη ωζ

3.8.4 Spherical joint with Euler angles

A spherical or ball and socket joint (Figure 3.17) constrains the relative motionof the interconnected bodies to a rotation around the origins of the joint coor-dinate systems. The number of joint position degrees of freedom equals three,i.e., the Euler angles φ1 , φ2 , φ3 , which describe the relative orientation of thejoint coordinate systems. The Euler angle φ1 is the rotation around the jointξi axis on the parent body, φ2 is the rotation around the resulting ηi axis andφ3 is the rotation around the joint ζ j axis on the child body. The joint velocity

Release 7.7 29

3


degrees of freedom are the first time derivatives of the Euler angles. This jointshould only be used in combination with prescribed joint position degrees offreedom because Euler angles suffer from singularity when the second Eulerangle is approximately 0 ± nπ .

φ1

φ2φ1

φ2

ζi

ζj

ξj

ξi ηi

j

i

φ2φ3

ηj

Figure 3.17: Spherical joint with Euler angles.


Q1R1

Q2R2

Q3R3

φ1 φ2 φ3


QD1W1

QD2W2

QD3W3

φ1 φ2 φ3

3.8.5 Spherical joint with Bryant angles

A spherical or ball and socket joint (Figure 3.18) constrains the relative mo-tion of the interconnected bodies to a rotation around the origins of the jointcoordinate systems.

30 Release 7.7


3

The number of joint position degrees of freedom equals three, i.e., the Bryantangles φ1 , φ2 , φ3 , which describe the relative orientation of the joint coordi-nate systems. The Bryant angle φ1 is the rotation around the joint ξi axis onthe parent body, φ2 is the rotation of the resulting ηi axis and φ3 is the rotationaround the joint ζ j on the child body. The joint velocity degrees of freedom arethe first time derivatives of the Bryant angles. This joint should only be used incombination with prescribed joint position degrees of freedom because Bryantangles suffer from singularity when the second Bryant angle is approximatelyπ/2 ± nπ .

φ1φ2

φ2

φ3 φ1

φ3

ζiζj

ξj

ξi ηi

ηj

j

i

Figure 3.18: Spherical joint with Bryant angles.


Q1R1

Q2R2

Q3R3

φ1 φ2 φ3


QD1W1

QD2W2

QD3W3

φ1 φ2 φ3

Release 7.7 31

3


3.8.6 Universal joint

A universal joint (Figure 3.19) constrains the relative motion of the intercon-nected bodies, starting from the orientation for which the joint coordinate sys-tems coincide, to a rotation around ξi axis followed by a rotation around theηj axis. The origins of the joint coordinate systems remain coincident.

The number of degrees of freedom of a universal joint equals two; the jointposition degrees of freedom are the angle φ1 between the η axes, and the angleφ2 between the ξ axes. The joint velocity degrees of freedom are their first timederivatives.

Figure 3.19: Universal joint.


Q1R1

Q2R2

φ1 φ2


QD1W1

QD2W2

φ1 φ2

32 Release 7.7


3

3.8.7 Cylindrical joint

A cylindrical joint (Figure 3.20) constrains the relative motion of the intercon-nected bodies to a rotation around and translation along the ξ axes of the jointcoordinate systems.

The number of degrees of freedom of a cylindrical joint equals two; the jointposition degrees of freedom are the angle φ between the η axes, and the ξcoordinate s of the origin of the joint coordinate system on the child body inthe joint coordinate system on the parent body. The joint velocity degrees offreedom are their first time derivatives.

sζi

ξj

ξi

i jζj

ηi

ηj

φ

Figure 3.20: Cylindrical joint.


Q1R1

Q2D1

φ s

Release 7.7 33

3



QD1W1

QD2V1

φ s

3.8.8 Planar joint

A planar joint (Figure 3.21) constrains the relative motion of the intercon-nected bodies to motions for which the ηζ planes remain coincident.

The number of degrees of freedom of a planar joint equals three; the jointposition degrees of freedom are the angle φ between the η axes and the coor-dinates sη and sζ of the origin of the joint coordinate system on the child bodyin the joint coordinate system on the parent body. The joint velocity degreesof freedom are their first time derivatives.

j

i

ξi

ξjsζsη

ζiζj

φηjηi

Figure 3.21: Planar joint.


Q1R1

Q2D2

Q3D3

φ sη sζ

34 Release 7.7


3


QD1W1

QD2V2

QD3V3

φ sη sζ

3.8.9 Bracket joint

A bracket joint (Figure 3.22) does not allow a relative motion of the intercon-nected bodies. The joint coordinate systems remain coincident.

The number of degrees of freedom of a bracket joint equals zero; a bracketjoint has no joint position and velocity degrees of freedom.

i

j

ξi,ξj

ζi,ζj

ηi,ηj

Figure 3.22: Bracket joint.

3.8.10 Free joint with Euler parameters

A free joint with Euler parameters (Figure 3.23) does not constrain the relativemotion of the interconnected bodies.

The number of position degrees of freedom of a free joint equals seven, theseare the Euler parameters q0 , q1 , q2 , q3 (cf. Appendix D.2) which define therelative orientation of the joint coordinate systems and the coordinates of theorigin of the joint coordinate system on the child body in the joint coordinatesystem on the parent body: sξ , sη , sζ . The number of joint velocity degrees offreedom equals six, these are the components of the angular velocity vector ofthe child body relative to the parent body with respect to the joint coordinatesystem on the parent body and the first time derivatives of sξ , sη , sζ .

Release 7.7 35

3


i

j

ξi

ξj

ζ i

ζj

ηi

η j

s

s

sξ

η

ζ

Figure 3.23: Free joint with Euler parameters.


Q1 Q2 Q3 Q4 Q5D1

Q6D2

Q7D3

q0 q1 q2 q3 sξ sη sζ


QD1W1

QD2W2

QD3W3

QD4V1

QD5V2

QD6V3

ωξ ωη ωζ sξ sη sζ

3.8.11 Free joint with Euler angles

A free joint with Euler angles does not constrain the relative motion of the in-terconnected bodies. The number of position degrees of freedom equals six.The Euler angles φ1 , φ2 and φ3 (cf. Figure 3.24) describe the relative orienta-tion of the joint coordinate systems. The Euler angle φ1 is the rotation aroundthe joint ξi axis on the parent body, φ2 is the rotation around the resultingηi axis and φ3 is the rotation around the joint ξ j axis on the child body. Therelative position is described with the coordinates of the origin of the joint co-ordinate system on the child body in the joint coordinate system on the parentbody: sξ , sη , sζ .

The number of joint velocity degrees of freedom equals six, these are the firsttime derivatives of the joint position degrees of freedom.

This joint should only be used in combination with prescribed joint position

36 Release 7.7


3

degrees of freedom because Euler angles suffer from singularity when the sec-ond Euler angle is approximately 0 ± nπ .

φ1

φ2φ1

φ2

ζi

ζj

ξjξi ηi

j

i

φ2φ3

ηj

ζi

ξi ηi

sξ sη sζ

’

’

’

Figure 3.24: Free joint with Euler angles.


Q1R1

Q2R2

Q3R3

Q4D1

Q5D2

Q6D3

φ1 φ2 φ3 sξ sη sζ


QD1W1

QD2W2

QD3W3

QD4V1

QD5V2

QD6V3


3.8.12 Free joint with Bryant angles

A free joint with Bryant angles (Figure 3.25) does not constrain the relativemotion of the interconnected bodies.

The number of position degrees of freedom equals six. The Bryant anglesφ1 , φ2 and φ3 (cf. Figure 3.25) describe the relative orientation of the joint

Release 7.7 37

3


coordinate systems. The Bryant angle φ1 is the rotation around the joint ξi

axis on the parent body, φ2 is the rotation around the resulting ηi axis and φ3

is the rotation around the joint ζ j on the child body. The relative position isdescribed with the coordinates of the origin of the joint coordinate system onthe child body in the joint coordinate system on the parent body: sξ , sη , sζ .

The number of joint velocity degrees of freedom equals six, these are the firsttime derivatives of the joint position degrees of freedom.

This joint should only be used in combination with prescribed joint positiondegrees of freedom because Bryant angles suffer from singularity when thesecond Bryant angle is approximately π/2 ± nπ .

φ1φ2

φ2φ3

φ1

φ3

ζ iζ j

ξj

ξiηi

ηj

j

i

ξi ηi

ζ i

sξ sη

sζ

’

’’

Figure 3.25: Free jointr with Bryant angles.


Q1R1

Q2R2

Q3R3

Q4D1

Q5D2

Q6D3



QD1W1

QD2W2

QD3W3

QD4V1

QD5V2

QD6V3


38 Release 7.7


3

3.8.13 Free joint with rotation and translation

The components of the acceleration of a body relative to its parent body w.r.t.the joint coordinate system on the child body must be prescribed when thebody is connected by a joint of type FREE_ROT_DISP to the parent body. Thisjoint type uses as joint position degrees of freedom the FREE joint Euler pa-rameters and the components of the vector from the origin of the joint coordi-nate system on the parent body to the origin of the joint coordinate system onthe child body w.r.t. the joint coordinate system on the child body (for a FREEjoint the components of this vector are w.r.t. the joint coordinate system onthe parent body!). The joint velocity degrees of freedom are the componentsof the velocity vector w.r.t. the joint coordinate system on the child body andthe first time derivative of the relative position vector. The latter does notinclude velocity due to the relative angular velocity and may therefore be dif-ficult to use. The relative angular velocity does not result in a contribution tothe linear velocity when the origins of the joint coordinate systems coincide.It is important to realize this in specifiying the initial conditions for that joint.

3.8.14 Translational-revolute joint

A translational-revolute joint (Figure 3.26) is a combination of a translationaljoint and a revolute joint. The translational joint is fixed to the parent body;the revolute joint is fixed to the child body. The translation direction and therotation axis are perpendicular. The translation takes place along the ξi axis,the rotation around the translated ηj axis.

The number of degrees of freedom of a translational-revolute joint equals two;the joint position degrees of freedom are the angle φ between the ζ axes, andthe ξ coordinate s of the origin of the joint coordinate system on the child bodyin the joint coordinate system on the parent body. The joint velocity degreesof freedom are their first time derivatives.

This joint type can also be built up from a translational joint, a nearly masslessbody and a revolute joint. The advantage of a translational-revolute joint isthat the nearly massless body needs not to be defined and is not included inthe model.

Release 7.7 39

3


j

ηi

ηjξj

φζj

s

ζi

ξi

i

Figure 3.26: Translational-revolue joint.


Q1R2

Q2D1

φ s


QD1W2

QD2V1

φ s

3.8.15 Revolute-translational joint

A revolute-translational joint (Figure 3.27) is a combination of a translationaljoint and a revolute joint. It is identical to a translational-revolute joint withthe exception that the parent and child body are interchanged. The transla-tional joint is fixed to the child body; the revolute joint is fixed to the parentbody. The translation direction and the rotation axis are perpendicular. Therotation takes place around the ηi axis, the translation along the ξ j axis.

The number of degrees of freedom of a revolute-translational joint equals two;the joint position degrees of freedom are the angle φ between the ζ axes, andthe ξ coordinate s of the origin of the joint coordinate system on the child bodyin the joint coordinate system on the parent body. The joint velocity degreesof freedom are their first time derivatives.

This joint type can also be built up from a revolute joint, a nearly massless

40 Release 7.7


3

body and a translational joint. The advantage of a revolute-translational jointis that the nearly massless body needs not to be defined and is not included inthe model.

i

ηj

ξi

φζi

s

ζj

j

ηi

ξj

Figure 3.27: Revolute-translational joint.


Q1R2

Q2D1

φ s


QD1W2

QD2V1

φ s

3.8.16 Translational-universal joint

A translational-universal joint (Figure 3.28) is a combination of a translationaljoint and a universal joint. The translational joint is fixed to the parent body;one of the rotation axes of the universal joint is fixed to the child body. Thetranslation direction and one rotation axis are perpendicular; the rotation axesare also perpendicular. The translation takes place along the ξi axis, the rota-tions around a translated ηi axis and the ζ j axis.

The number of degrees of freedom of a translational-universal joint equals

Release 7.7 41

3


three; the joint position degrees of freedom are the angle φ1 between the ζaxes, the angle φ2 between the η axes, and the ξ coordinate s of the origin ofthe joint coordinate system on the child body in the joint coordinate systemon the parent body. The joint velocity degrees of freedom are their first timederivatives.

This joint type can also be build up from a translational joint, a nearly masslessbody and a universal joint. The advantage of a translational-universal joint isthat the nearly massless body needs not to be defined and is not included inthe model.

j

ξj

φ1ζjs

ζi

i

ηiξj

φ2ηj

Figure 3.28: Translational-universal joint.


Q1R2

Q2R3

Q3D1

φ1 φ2 s


QD1W2

QD2W3

QD3V1

φ1 φ2 s

3.8.17 Universal-translational joint

A universal-translational joint (Figure 3.29) is a combination of a translationaljoint and a universal joint. It is identical to a translational-universal joint with

42 Release 7.7


3

the exception that the parent and child body are interchanged. The transla-tional joint is fixed to the child body; one of the rotation axes of the universaljoint is fixed to the parent body. The translation direction and one rotation axisare perpendicular; the rotation axes are also perpendicular. The rotations takeplace around the ζi axis and the ηj axis, the translation along the ξ j axis.

The number of degrees of freedom of a universal-translational joint equalsthree; the joint position degrees of freedom are the angle φ1 between the ζaxes, the angle φ2 between the η axes, and the ξ coordinate s of the origin ofthe joint coordinate system on the parent body in the negative ξ direction ofthe joint coordinate system on the child body. The joint velocity degrees offreedom are their first time derivatives.

This joint is similar to the translational-universal joint, except that the parentand the child are interchanged.

i

φ1 ζi

s

ζj

j

ηj

φ2<0ηi

ξiξj

Figure 3.29: Universal-translational joint.


Q1R2

Q2R3

Q3D1

φ1 φ2 s


QD1W2

QD2W3

QD3V1

φ1 φ2 s

Release 7.7 43

3


3.9 Body surfaces

In MADYMO, rectangular planes, (hyper)ellipsoids, (hyper)elliptical cylin-ders and finite element surfaces can be attached to any body of any system.These can be combined to represent the surface of a specific body. Contact in-teractions between the different surfaces and finite element structures can bespecified (see Chapter 9).

A rectangular plane is defined by the coordinates of three points. The firstPlanesand second point are vertices on one edge, A and B, of the rectangle; the thirdpoint is on the opposite edge of the rectangle (Figure 3.30). The vertices on theopposite edge, C and D, are calculated by the program.

Figure 3.30: Points defining a plane.

The direction of the outward normal of the plane determines the "materialside" of the plane:

• the outside normal is defined by the direction of a right handed screwwhen the rotation is from point 1 to 2 to 3 (Figure 3.31). In terms of avector product: the direction of the outward normal is determined byd12 × d23 , where d12 is the vector from point 1 to point 2 and d23 thevector from 2 to 3.

44 Release 7.7


3

Figure 3.31: Specification of the outside normal of a plane.

Ellipsoids and hyper-ellipsoids can be attached to bodies or to the referenceEllipsoidsspace. A (hyper)ellipsoid is given by:

|x|a

n

+

|y|b

n

+

|z|c

n

= 1 (3.22)

where a, b and c are the semi-axes of the (hyper)ellipsoid and n is the de-gree. If n = 2 , this equation describes an ellipsoid. If the degree n in-creases the (hyper)ellipsoid will approximate more and more a rectangularshape (Figure 3.32). Higher degree ellipsoids can be especially useful for themodelling of edge contacts.

Release 7.7 45

3


bc

n = 2n = 8

y

z

z

y

e

e

x y

z

x

y

z

ab

c

e

e

e

Figure 3.32: Ellipses and ellipsoids for several degrees n.

The (hyper)ellipsoids are defined by the semi-axes a, b and c, the degree n ofthe (hyper)ellipsoid, the coordinates of the ellipsoid centre in the local coordi-nate system of the object to which it is attached and the orientation.

The semi-axes do not need to be parallel to the axes of the local coordinatesystem of the object to which the ellipsoid is attached. The direction of thesemi-axes can be specified by defining the orientation of the ellipsoid coordi-nate system (xe, ye, ze). The origin of this coordinate system coincides withthe centre of the ellipsoid; the coordinate axes are parallel to the axes of theellipsoid. In MADYMO, the orientation of the ellipsoid coordinate system canbe specified by of one of the standard methods (Appendix D).

An elliptical cylinder can be attached to a body or to the reference space. ThisEllipticalcylinders surface can be used for visualisation purposes or for defining contact with

(hyper)ellipsoids. An elliptical cylinder is the surface of an object that is pris-matic in one direction, the axis of the cylinder, and has a hyper-elliptical cross-section perpendicular to this direction (Figure 3.33). The end faces are not in-

46 Release 7.7


3

cluded in an elliptical cylinder.

xy

z

x

a

y

b

z

c

c c

c

Figure 3.33: Elliptical cylinder for n = 4.

An elliptical cylinder is defined with respect to its cylinder coordinate system(xc, yc, zc). The centre of the cylinder is in the origin of this coordinate system.The axis of the cylinder coincides with the xc axis. The semi-axes of a cross-section are parallel to the yc and zc axes. The equation for an elliptical cylinderis given by

−a ≤ x ≤ a

|y|b

n

+

|z|c

n

= 1 (3.23)

where a is one-half the length of the cylinder, b and c are the semi-axes of the(hyper) elliptical cross-section in the yc and zc direction, respectively, and n isthe degree. x, y, and z are the coordinates of an arbitrary point on the surfaceof the elliptical cylinder with respect to the cylinder coordinate system. Ifn = 2 the cross-section is an ellipse. For increasing values of the degree n, thecross-section gets a more rectangular shape.

An elliptical cylinder is defined by one-half the length a, the semi-axes b andc, the degree n of the (hyper)elliptical cross-section, the coordinates of the cen-tre of the cylinder in the local coordinate system of the object to which it isattached and the orientation of the cylinder coordinate system.

The orientation of the cylinder coordinate system w.r.t. the local coordinatesystem of the object to which it is attached can be specified by one of the stan-dard methods (Appendix D). When this orientation is not specified, it will be

Release 7.7 47

3


parallel to the axes of the local coordinate system of the object to which thecylinder is attached.

The standard surfaces, planes, (hyper)ellipsoids and (hyper)elliptical cylin-Finite elementsurfaces ders, offer the possibility to model a large variety of surfaces with minimum

input. For a more detailed or a more general description of surfaces, a finiteelement surface can be used. With a finite element surface, a surface is approx-imated by triangular or quadrangular elements (Figure 3.34). A finite elementsurface is not necessarily a closed surface, in fact it may consist of differentparts. A finite element surface can be attached to the reference space or multi-ple bodies.

Figure 3.34: Finite element surface.

A finite element surface is defined by the coordinates of the nodes and foreach element the connectivity of the nodes, these are the numbers of the nodesthat define the element. The coordinates of the nodes must be with respect toa finite element model coordinate system. The position and orientation ofthis coordinate system relative to the coordinate system of the object to whichthe finite element surface is attached can be specified, offering the possibilityto use a finite element model coordinate system that is most convenient fordetermining the nodal coordinates.

3.10 Equations of motion

The equations of motion (Newton-Euler) of a rigid body i referred to its centreof gravity are (Wittenburg1).

1Wittenburg, J., "Dynamics of Systems of Rigid Bodies", Teubner, Stuttgart, Germany, 1977.

48 Release 7.7


3miri = Fi (3.24)

Ji · ωi + ωi × Ji · ωi = Ti (3.25)

where mi is the mass, Ji is the inertia tensor with respect to the centre of grav-ity, ωi is the angular velocity vector, Fi is the resultant force vector, and Ti

is the resultant torque vector relative to the centre of gravity. For a body ina system of bodies, Fi and Ti include the constraint forces and torques dueto joints. These cannot be determined until the acceleration of the system isknown. This in contrast with all other forces and torques which depend onlyon position and velocity quantities.

The unknown joint constraint forces and torques can be eliminated using theprinciple of virtual work. First equations (3.24) and (3.25) are multiplied bya variation of the position vector, δri , and a variation of the orientation, δπi ,and the resulting equations are summed for all bodies of the system:

∑ δri · miri − Fi+ δπi · Ji · ωi + ωi × Ji · ωi − Ti = 0 (3.26)

When the variations δri and δπi of connected bodies are such that the con-straints caused by the joint are not violated, the constraint forces and torquesin joints will cancel (principle of virtual work). Such variations can be ob-tained from equations (3.6) and (3.7) by varying the joint degrees of freedom.These expressions can be substituted in equation (3.26). Starting with periph-eral bodies, expressions for the second time derivative of the joint degrees offreedom are obtained:

qij

= MijYi + Qij

(3.27)

Yi is a 6 × 1 column matrix which contains the components of the linear andangular acceleration of the coordinate system of the parent body i. The nij × 6matrix Mij and the nij × 1 column matrix Q

ijdepend on the inertia of the

bodies and the instantaneous geometry of the system. Qij

depends addition-

ally on the instantaneous velocity of the system and the applied loads. Thematrices Mij and Q

ijare calculated successively starting with peripheral bod-

ies. Then starting with the joint between the reference space and the referencebody the second time derivatives of the joint degrees of freedom can be cal-culated from equation (3.27). Note that for this joint, i equals 0 and j equals 1and the acceleration of the reference space, Y0 = 0.

This algorithm yields the second time derivatives of the joint degrees of free-dom in explicit form. The number of computer operations is linear in thenumber of bodies if all joints have the same number of degrees of freedom.This leads to an efficient algorithm for large systems of bodies.

Time integration of the second time derivatives of the joint coordinates givesthe joint coordinates and their first time derivatives at a new point of time.

Release 7.7 49

3


These are used to calculate the motion of the body coordinate systems relativeto the reference space coordinate system using equations (3.6) and (3.7) andtheir first time derivatives (3.8) and (3.9). At the start of the integration, thejoint coordinates and their first time derivatives have to be specified (initialconditions).

In addition to data specifying systems of bodies and the applied loads, theInitialconditions initial state must be set. In MADYMO, the initial values of the joint position

and velocity degrees of freedom must be specified.

3.11 Numerical integration methods

The equations of motion form a system of coupled non-linear second orderdifferential equations (Section 3.10). These equations can be written as:

q = h(q, q, t) (3.28)

with initial values q0 and q0 .

h defines the equations of motion; q is a column matrix with the generalisedcoordinates, the joint position degrees of freedom; q and q are the joint ve-locity degrees of freedom and their first time derivatives. The column matrixq contains m elements, corresponding with the m degrees of freedom of themodel.

The equations of motion are solved numerically. Three methods are available:

• a modified Euler method with a fixed or variable time step;

• a Runge-Kutta method with a fixed or variable time step;

• a Runge-Kutta Merson method with a variable time step.

These are one-step explicit methods, which means the solution at a time pointtn+1 can be written explicitly in terms of the solution at the preceding timepoint tn . For most problems, the error in the solution will reduce when thetime step is decreased. If one of the fixed time or variable time step methodsis used, the user is recommended to run a problem with different time stepsin order to estimate the accuracy of the solution.

If one of the Runge-Kutta methods is used, the system of m second order dif-ferential equations is reduced to 2m first order differential equations. Intro-duce the column matrix x defined by:

x =

q

q

(3.29)

Using this substitution, equation (3.28) becomes

x =

q

q

=

h(q, q, t)

q

= f (x, t) (3.30)

50 Release 7.7


3

with initial condition

x(t0) =

q0

q0

(3.31)

Equation (3.30) is integrated, resulting in solutions for q and q in the next timepoint.

The modified Euler method is a one-step method with a fixed time step ts .Euler methodStarting point for the time integration of the equations of motion are the sec-ond order differential equations given by equation (3.28). These accelerationequations are integrated using the explicit Euler method which gives the so-lution for the velocity variables at time point tn+1 = tn + ts

qn+1

= qn+ tsq

n(3.32)

The velocity variables are integrated using the implicit Euler method whichgives the solution for the position variables at time point tn+1

qn+1

= qn+ tsq

n+1(3.33)

The fourth order Runge-Kutta method with fixed time step is a simpleRunge-Kuttamethod and straight forward method. The fourth order Runge-Kutta solution of

equation (3.30) at time point tn+1 = tn + ts can be written as

xn+1 = xn +1

6ts(k1 + 2k2 + 2k3 + k4) (3.34)

where ts = fixed integration time step and

k1 = f (tn, xn)

k2 = f (tn +1

2ts, xn +

1

2tsk1)

k3 = f (tn +1

2ts, xn +

1

2tsk2)

k4 = f (tn + ts, xn + tsk3) (3.35)

The fifth order Runge-Kutta Merson solution of equation (3.30) at time pointRunge-KuttaMersonmethod

tn+1 = tn + ts can be written as

xn+1 = xn +1

6ts(k1 + 4k4 + k5) (3.36)

Release 7.7 51

3


where ts = current integration time step and

k1 = f (tn, xn)

k2 = f (tn +1

3ts, xn +

1

3tsk1) (3.37)

k3 = f (tn +1

3ts, xn +

1

6tsk1 +

1

6tsk2)

k4 = f (tn +1

2ts, xn +

1

8tsk1 +

3

8tsk3)

k5 = f (tn + ts, xn +1

2tsk1 +

3

2tsk3 + 2tsk4)

An estimation of the local truncation error TEn+1 is given by:

TEn+1 =1

30ts max

i| − 2k1 + 9k3 − 8k4 + k5|i (3.38)

If the truncation error exceeds the condition

TEn+1 > INT_TOL max(1, |xn+1|i) (3.39)

the integration step is repeated with the step size ts halved.

If the truncation error satisfies the condition

TEn+1 < 0.1INT_TOL max(1, |xn+1|i) (3.40)

the step size ts is doubled.

The error tolerance INT_TOL is an input parameter. A recommended value is0.001, but for very stiff systems, a lower value (such as 0.0001) may be neces-sary.

In MADYMO, a lower limit for the time step has been set. The time stepcannot be less than 0.001 · TIME_STEPwhere TIME_STEP is the initial time step.The time step can be prevented from becoming too large by specifying anupper limit MAX_STEP.

The Runge-Kutta Merson method cannot be used for applications with fi-nite element models because these do not allow the repeated time integra-tion over the same time interval, which occurs when the step size is reduced.

For Euler method and the fourth order Runge Kutta method, the smallest timeUsing FE timestep step of the selected finite element models can be used as the multibody inte-

gration time step, depending on whether a multibody integration time stepis specified. It is recommended to set the output time step to zero if the userwants output at every multibody time step, otherwise the computation effi-ciency may deteriorate due to synchronisation. When the prescribed outputtime step is smaller than the initial multibody integration time step, the actualoutput time step will be equal to the multibody integration time.

52 Release 7.7


3

For a given time step, the modified Euler method is less accurate than theStability andaccuracy Runge-Kutta methods. In order to obtain the same accuracy, the time step

used for the modified Euler method should be theoretically about 1/8 of theRunge-Kutta time step and 1/16 of the Runge-Kutta-Merson time step. How-ever, this holds true only if for the Runge-Kutta method the right side ofequation (3.28) is four times continuously differentiable and five times con-tinuously differentiable for the Runge-Kutta-Merson method. For practicalapplications, the right side of equation (3.28) is not even continuous, it is dif-ficult to predict in advance which method is most efficient.

Explicit numerical integration methods are conditionally stable. When thetime step is too large, an error in the solution will grow. The maximum timestep that leads to a stable solution depends on the largest eigenvalue in themodel. Because the eigenvalues for non-linear differential equations dependon the solution, it is not possible to estimate the maximum time step in ad-vance. However, an unstable solution can be easily recognised from the factthat the solution goes to infinity. The maximum time step for the modifiedEuler method allowed for stable time integration is about 2/3 of the maxi-mum time step for the fourth order Runge-Kutta method. When stability de-termines the step size, the modified Euler method is more efficient than theRunge-Kutta method. The Runge-Kutta method requires four function evalu-ations, the calculation of h in equation (3.28), for one time step in comparisonto the modified Euler method, which requires only one function evaluation.When a finite element model is supported on a rigid body, the Runge-Kuttamethod may become unstable. For this reason, the user is advised to use thenthe Euler method.

Release 7.7 53

3


54 Release 7.7

MADYMO Theory Manual Sensor, signal, operator and control elements

4

4 Sensor, signal, operator and control elements

The sensor, signal, operator and control elements are described in this chapter.These elements allow control of the system (as opposed to sensors for outputpurposes, see Reference Manual). A desired time-dependent motion or loadcan be defined with a function signal element. Sensor, external input andfunction signals can be manipulated with operators and PID controllers.

The output signals of all sensor, operator and control elements can be used as:

• input signals for operators and controllers,

• input for actuators that apply forces or torques to bodies,

• input for switches that define state conditions,

• input for external output signals that define interaction with externalprograms.

4.1 Sensors

Sensors can be used to extract the quantities of multi-body systems and airbagchambers. The output value of a sensor can be used as input for signals, oper-ators, controllers and actuators. The types of sensors available are:

• airbag sensors.

• belt sensors.

• body sensors.

• joint sensors.

• surface distance sensors.

• restraint sensors.

• contact sensors.

• switch sensors.

• node distance sensors.

• injury sensors.

These sensors detect the value of an airbag chamber property (volume, tem-Airbag sensorsperature, gas mass, pressure, outflow to ambient or another chamber).

The sensors detect the belt force in a belt segment, the friction force in a beltBelt sensorstying, the belt feed velocity in a belt retractor, the belt payout of a load limiteror the belt payout velocity of a pretensioner.

When the sensor relates to one body, the sensor signal is:Body sensors

Release 7.7 55

4

Sensor, signal, operator and control elements MADYMO Theory Manual

• the resultant linear acceleration of a point on a body.

• the component of the position, linear velocity, angular velocity or linearacceleration. The component is expressed w.r.t. the reference space inthe sensing direction (unit vector n, Figure 4.1). The direction is fixed ineither the reference coordinate system or the local coordinate system ofthe body for which the motion is to be determined. The component canbe extracted in either a reference space-fixed direction or a body-fixeddirection.

In calculating the motion at the mounting point of the sensor, the contributiondue to the deformation of the body may be taken into account.

z1

n

Y

X1

y1

x1

X

Z

X1• ω1

Figure 4.1: Definition of absolute body sensor signals.

In the case of relative motion, the motion of the point on the body is deter-mined relative to a reference point (Figure 4.2). The reference point can alsobe on the same body, which may be of interest if the body is flexible.

• The relative motion is projected onto the line connecting the two points.

• The relative position indicates the distance between the two points.

• The relative linear velocity/acceleration gives the speed/accelerationthat the relative distance changes between the two points.

• The relative position is always positive and the relative veloc-ity/acceleration is positive when the two points move apart.

56 Release 7.7


4

z1

Y

X1

y1

x1

X

Z

X1• ω1

ω2

X2•

X2

z2

y2 x2

Figure 4.2: Definition of relative body sensor signals.

Table 4.1 summarises the quantities that can be extracted with a body sensor.The subscript 1 refers to the point for which the (relative) motion is deter-mined. The subscript 2 refers to the reference point if the relative motion isrequested.

noteFor a relative (angular) velocity signal, the projection axis is not definedwhen the two points coincide. The relative (angular) velocity is discontinuousin this position. The value of the body sensor in this position is then set equalto the magnitude of the relative (angular) velocity vector.

Contact sensors extract the resultant or a component of a force due to the con-Contactsensors tact between two multibody surfaces, a multibody surface and a finite element

model, between two finite element models or between a tyre and a road.

Table 4.1: Body sensor signals

Body Body relative

Orientation ∠(n × x1)t0, (n × x1)t ∠(x1 × x2)t0, (x1 × x2)t

Position x1 · n ‖x1 − x2‖

Linear velocity x1 · n (x1 − x2) · (x1 − x2)/ ‖x1 − x2‖

Linear acceleration x1 · nx1 · −x1 · nx1

(x1 − x2) · (x1 − x2)/ ‖x1 − x2‖

Angular velocity ω1 · n (ω1 − ω2) · (x1 − x2)/ ‖x1 − x2‖


Release 7.7 57

4


Table 4.1 cont.

Body Body relative

Angular acceleration ω1 · n (ω1 − ω2) · (x1 − x2)/ ‖x1 − x2‖

These sensors extract the value of a relative motion quantity in a kinematicJoint sensorsjoint. The output value of a joint sensor is the value of a joint degree of free-dom, or the first or the second time derivative (velocity, acceleration) of a jointdegree of freedom. The physical meaning of the joint degrees of freedom forall the available types of kinematic joints is described in Section 3.8.

Joint constraint sensors extract the resultant or a component of a joint con-straint load on the parent or the child body.

These sensors extract the value of a restraint load.Restraintsensors

The output signal of a surface distance sensors is the distance between theSurfacedistancesensors

attachment point of the sensor and the closest point of intersection of a linein the sensing direction with sets of ellipsoids, cylinders and planes. Hyper-ellipsoids and hyper-elliptical cylinders (degree > 2) are treated as ellipsoidsand cylinders (degree = 2) in calculating this point of intersection.

A cylinder has a finite length and the end faces are not taken into considera-tion. This means a cylinder consists only of the wall and the sensor can "look"through the end faces. A point of intersection is also determined when theattachment point of the sensor is inside an ellipsoid or a cylinder.

A plane is finite and there is no inside or outside. When there is no point ofintersection with one of the ellipsoids, cylinders or planes, the sensor signalis equal to the user specified distance MAX_DIST. For example, the solid lineshown in Figure 4.3 corresponds to the surface distance sensor signal whenthe attachment point moves along the x-axis and the sensing direction is they-direction. The vertical dashed lines correspond to a transition from one sur-face to another or to no surface.

sens

ing

dire

ctio

n

x

y

MAXDIS

Figure 4.3: Surface distance sensor signal when attachment point moves along the x-axis.

58 Release 7.7


4

These sensors extract the value of the state of the switch. Where TRUE isSwitch sensorsrepresented by the value 1 and FALSE by the value 0.

These sensors detect the actual distance between two nodes, where the nodesNode distancesensors are part of one Finite Element model.

These sensors detect the actual injury value of the injury criterion to whichInjury sensorsthey refer, based on unfiltered source signals detected at each integration timepoint up to actual time.

4.2 Signals

External input signals are quantities that can be used as input for operators,Externalsignals controllers and actuators. They can be used to interact with an output of an

external program. External output signals are quantities that can be used asoutput for operators and controllers and sensors. They can be used to interactwith an input of an external program.

Function signals are time-dependent quantities that can be used as input forFunctionsignals operators, controllers and actuators. They can be used to define, for example,

a desired motion. This motion can be subtracted from the actual motion witha sum operator. The resulting error signal can be supplied to a controller,which will generate an actuator input signal that in turn will reduce the error.Also, a function signal can be supplied directly to an actuator, which appliestime-dependent loads to bodies.

Constant value signals are signals with a constant value.Constant valuesignals

4.3 Operators

Operators can be classified into operators that use only one input signal, andoperators that use one or more input signals.

Operators that use one input signal return with:Operators thatuse one inputsignal • the absolute value of the input signal,

• the cosine of the input signal,

• the input signal with a time delay,

• the value of a specified function with the input signal as independentvariable,

• a polynome a0 + a1x + a2x2 + . . . + anxn with x as input signal,

• xa with x as input signal and a specified exponent,

• the sine of the input signal,

Release 7.7 59

4


• the tangent of the input signal.

Operators that can use one or more input signals return with:Operators thatuse one oremore inputsignals

• the weighted average value of the selected input signals,

• the maximum value of a set of input signals, each of them may be mul-tiplied by a constant,

• the minimum value of a set of input signals, each of them may be multi-plied by a constant,

• the product of a set of input signals, each of them may be multiplied bya constant,

• the reciprocal value of the product of a set of input signals, each of themmay be multiplied by a constant,

• the sum of a set of input signals, each of them may be multiplied by aconstant.

4.4 Controllers

Controllers can be used to affect the behaviour of bodies’ systems. A single in-put or single output linear Proportional Integrating and Differentiating (PID)controller model is available. This type of controller generates an output sig-nal, usually an actuator load, which is proportional to the input signal andusually represents the control error. The ratio of the output signal and the in-put signal is given by the transfer function H of the controller. In the literature,two different notations are used for this transfer function. MADYMO uses the’true three term’ notation (4.1). The constants Km , τim and τdm are controllerparameters.

Hcm = Km(1 +1

τims+ τdms) (4.1)

The other common notation is given by equation (4.2), which describes thePID controller as a PI and PD controller in series.

Hc = K(1 +1

τis)(1 + τds) (4.2)

The relationship between the parameters of the true three term controller andthe serial notation is given by:

Km = K(τi + τd)/τi (4.3)

τim = τi + τd (4.4)

τdm = τiτd/(τi + τd) (4.5)

60 Release 7.7


4

By setting τim equal to infinity, the integrating action is removed and by settingτdm equal to zero, the differentiating action is removed. This way the PIDcontroller can be reduced to a P(τim = ∞, τdm = 0) , PI(τdm = 0) or PD(τim =∞) controller.

To determine the controller output signal, a time integration is required, whichis done simultaneously with the time integration of the equations of motion.This corresponds to an analog controller.

The output signal of a P-controller is proportional to the input signal. TheChoice ofcontroller type constant of proportionality, the control gain Km , determines the controlled

system’s speed of response. An increase of the control gain leads to an increasein the response speed. A control gain value that is too large will lead to anunstable response.

Generally a P-controller is not appropriate for position control of a system.This is because a P-controller does not eliminate a constant error when a con-stant load is applied that differs from the nominal load for which the controllerhas been designed. Such a constant error can be eliminated by a control loadwhich is proportional to the time integral of the error. This results in a PI-controller. Usually the control gain Km must be reduced when an integratingcomponent is added to the controller because, otherwise, the controller maybecome unstable. However, this will reduce the response speed.

The speed of the response can be increased by including a differentiating ac-tion in the controller. The PD controller reacts to the time rate of change of thecontrol error. Also a higher maximum control gain can be used when com-pared with a proportional controller.

A PID controller is obtained when both the integrating and differentiatingparts are included in the controller. This controller combines the advantagesof both the PI and PD controllers.

4.5 Actuators

Actuators apply concentrated forces or torques on bodies. The magnitude ofthe actuator load equals the value of the input signal of the actuator. Twotypes of actuators are available: body actuators and joint actuators. In thissection, these two types of actuators are explained.

A body actuator load can be either a force or a torque. Two kinds of bodyBody actuatorsactuators can be distinguished:

• type 1: actuators that apply a load at a point on only one body,

• type 2: actuators that apply a load at a point on one body and the corre-sponding reaction load at a point on another body.

The direction of the load of this type of actuator is determined by a vectortype 1n (Figure 4.4). This direction is fixed in either the reference space coordinatesystem or the local coordinate system of the body to which the load is applied.

Release 7.7 61

4


This type of actuator can be used to model the reaction force of a reaction jet,for example.

Figure 4.4: Body actuator torque in fixed reference direction.

For this type of actuator, the line of action of the load passes through the twotype 2points of application (Figure 4.5). The load is positive when it is directed fromthe first point of application to the second point of application. This corre-sponds to a tensile force in the actuator when the load is a force. The load isset equal to zero when the points coincide because the line of action cannot bedetermined in this position. This type of actuator can be used to model, forexample, a hydraulic cylinder with negligible mass.

62 Release 7.7


4

Figure 4.5: Definition of a positive body actuator force.

This actuator is specified for a specific degree of freedom of a kinematic joint.Joint actuatorsIt represents a force (torque) when the corresponding joint degree of freedomis a displacement (rotation). For a joint that has a joint actuator, the joint ac-tuator applies a load to the parent body and the reaction load to the childbody. The direction of the load is determined by the kinematic joint axis on theparent body that corresponds to the specified joint degree of freedom. Jointactuators cannot be specified for free joints and spherical joints.

Joints actuators can be used to model, for example, the torque due to an elec-tric servomotor if the rotor and stator are modelled as bodies that are inter-connected by a revolute joint.

Release 7.7 63

4


64 Release 7.7

MADYMO Theory Manual Finite element model

5

5 Finite element model

MADYMO features full FE capabilities for structural impact analysis. In thefinite element module, truss, beam, membrane, shell and brick elements areimplemented. Several material models such as elastic, elasto-plastic, Mooney-Rivlin and hysteresis can be used. Several finite element models can be usedwithin one simulation. A MADYMO model can consist of only multi-bodysystems, only finite element models or both.

The interaction between the multi-body model and the finite element modelis explained in Figure 5.1. Two kinds of interactions, supports and contacts,generate forces between finite element models and the multi-body systems.

Figure 5.1: Interaction between multi-body and finite element module.

This approach allows the use of different time integration methods for theTimeintegrationmethods

equations of motion for the finite element and the multi-body modules. Forshort duration crash analysis, explicit integration methods are preferred. Fora MADYMO analysis with a finite element model, the 4th order Runge-Kuttaor Euler method must be used for the time integration of the equations of mo-tion of the multi-body module. The central difference method is used for thetime integration of the equations of motion of the finite element models. Ac-tual positions and velocities at each time step of the central difference methoddetermine the support and contact forces. The forces acting on the multi-bodysystem are accounted for in each main time point of the 4th order Runge-Kuttaand each time step of the Euler method.

Explicit methods are conditionally stable and therefore put limitations onTime stepselection which time step can be used. Due to the fine spatial discretization often re-

quired, a much smaller time step is needed for finite element models than formulti-body models. To increase the efficiency of the entire analysis, the finiteelement analysis is sub-cycled with respect to the multi-body analysis using adifferent constant time step for each finite element model. If contacts betweendifferent finite element models are specified, the time step is identical for allthe finite element models that are in contact. MADYMO automatically selectsthe smallest time step used in any of the finite element models defined.

In order to be able to model parts of belts with membrane or truss elements,a node can be tied to a belt segment (See "Belt model" on page 169). All cur-rent belt model options can be used, including retractors, pretensioners andload limiters, so the finite element belt model can slide over dummy modelsurfaces.

Release 7.7 65

5

Finite element model MADYMO Theory Manual

This chapter provides the background information needed to set up a suitablefinite element model using the MADYMO finite element module.

Reference: A more thorough introduction to finite element techniques is pro-vided by Bathe1,2, and Hughes3.

1Bathe, K.J., & Wilson, E.L., Numerical Methods in Finite Element Analysis, Prentice-Hall Inc.,New Jersey, England, 1976.

2Bathe, K.J., Finite Element Procedures in Engineering Analysis, Prentice-Hall Inc., New Jersey,England, 1982.

3Hughes, T.J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Anal-ysis, Prentice-Hall Inc., New Jersey, England, 1982.

66 Release 7.7


5

5.1 Finite element concepts

The response of solid structures is governed by a set of partial differentialequations for the unknown state variables. A direct method or a variationalmethod can be used to generate these equations. In the absence of thermo-mechanical effects, the basic equations are:

• the momentum equation,

• the constitutive equation (the material behaviour: a relationship be-tween stresses and strains), and

• the strain-displacement relationship (deformation as a function of thedisplacements).

For given initial and boundary conditions, the system response is embeddedin these equations. However, an analytic solution of the resulting partial dif-ferential equations satisfying the initial and boundary conditions exists onlyfor very simple cases, and thus a numerical procedure must generally be used.

This is the method used to reduce a continuum to a discrete numerical model.Finite elementmethod The continuum is divided into relatively simple finite elements representing

its shape. These elements can be volumes, surfaces, lines or combinations ofthese (see Figure 5.2). The elements are interconnected at a discrete number ofpoints, the nodes. MADYMO uses a Lagrangian description. This means thatthe nodes, and therefore also the elements, are fixed to the material and thusmove through space with the material.

The system is discretized by interpolating the displacement, velocity and ac-celeration of any point in an element in terms of the same quantities at thenodes connected to this element. The interpolation functions, the elementshape functions, must be such that rigid body motions, motions for whichthe strains are zero, can be described. Almost all major general purpose, finiteelement programs for structural analyses use this displacement-based finiteelement formulation because of its simplicity and generality.

Figure 5.2: Examples of commonly used element types.

Release 7.7 67

5


The state of stress results from the strains and the constitutive equation of theTheConstitutiveEquation

material. In quasi-static analyses the equilibrium requirements for each ele-ment, see Figure 5.3, are then established in terms of the motion of the nodeswhich can be solved for successive points in the deformation process. There-fore the displacements of the nodes will be the basic unknown parameters forquasi-static analyses.

In dynamic analyses, the equilibrium requirements for each element are estab-lished in terms of the motion of these nodes which can be solved for successivepoints in time. Therefore, the displacements, the velocities and the accelera-tions of the nodes are the basic unknowns for a dynamic analysis.

Figure 5.3: Static equilibrium between elements.

A well-defined problem includes the definition of the geometry in terms ofnodal coordinates and element connectivities. In addition to that, materialand geometrical properties of the elements need to be specified. Initial con-ditions, such as initial displacements or velocities, as well as boundary condi-tions, such as prescribed motions, contacts and loads can be specified to com-plete the model definition. The generation of the input data, especially thegeometry, can be a rather laborious task. Therefore a general finite elementpreprocessor is often used, or special programs that are available for certainfrequently studied applications like the airbag mesh generator.

5.2 Time integration method

The finite element method is used to reduce a solid structure to a discreteEquations ofmotion numerical model. The equations of motion of a finite element model can be

written as:

M a + D v = Fext − Fint(u, v) (5.1)

68 Release 7.7


5

where M and D are the mass and damping matrices; Fext is the vector of ex-ternally applied loads; Fint is the vector of internal nodal forces and a, v andu are the nodal acceleration, velocity and displacement vectors, respectively.The matrix D generally is not assembled from element damping matrices, butinstead is constructed from the mass and stiffness matrix of the complete ele-ment assemblage, i.e., Rayleigh damping is assumed:

D = αM + βK (5.2)

where α and β are damping coefficients. The coefficient α may be a function oftime. In explicit finite element analysis the stiffness matrix is not available andtherefore the damping proportional to the stiffness is included in the internalforce vector Fint .The equilibrium equations can now be transformed into:

M(a + αv) = Fext − Fint(u, v). (5.3)

Generally, the mass matrix is not diagonal (consistent mass matrix) and a ma-trix inversion is needed to solve the equations of motion. For explicit timeintegration a diagonal mass matrix and hence an uncoupled set of equationsof motion for all degrees of freedom is highly advantageous. A diagonal massmatrix is generally obtained through mass lumping. Note that then each de-gree of freedom must have a non-zero contribution to the mass matrix.

In addition to the spatial discretization, a time discretization is necessary. ThisTimediscretization means the quantities describing the structural behaviour are calculated at a

discrete number of points in time. A direct time integration method is used sothat the equations of motion are satisfied only at discrete time points. The con-cept for the direct integration method is based on the assumption that thereis a variation of displacements, velocities, and accelerations within each timeinterval. The form of this variation determines the accuracy, stability, and thecosts of the solution procedure. In the MADYMO finite element module, acentral difference method with a constant time step and a variable time step isavailable. The relations for the central difference method with a constant timestep are:

vn+ 12

= vn− 12+ ∆tan (5.4)

un+1 = un + ∆tvn+ 12

Subscripts n − 12 , n , n + 1

2 and n + 1 correspond with time points t − ∆t/2 , t ,t + ∆t/2 and t + ∆t respectively, where t is the current time point. Incorporat-ing the central difference equations in the equations of motion leads to:

vn+ 12

= A1vn− 12+ A2 M−1(Fext − Fint) (5.5)

where

A1 = (1 − α∆t)/(1 + α∆t) (5.6)

A2 = ∆t/(1 + α∆t)

Release 7.7 69

5


A lumped mass distribution is used so that the mass matrix M is diagonal andthe determination of the inverse of the mass matrix M − 1 becomes insignifi-cant. By default, no damping is used (a = 0).

In the central difference method, the displacements and velocities are calcu-Explicit timeintegrationmethod

lated from quantities at previous points in time only. This method is called anexplicit time integration method. This method is conditionally stable, mean-ing that the time step must be small enough to ensure that the solution doesnot grow without bound. For undamped linear systems, the time step is lim-ited by:

∆t ≤ 2/ω (5.7)

where ω is the maximum frequency appearing in the numerical model. ThisCourantstabilitycondition

is the Courant stability condition (Courant1). For a linear truss element withtwo nodes, it can be shown that:

∆t ≤ L/c (5.8)

where c is the dilatational wave speed and L is the characteristic length of theelement. This condition requires that the time step is small enough to ensurethat a sound wave may not cross the smallest element during one time step.The speed of sound for linear elastic material is a function of the elasticity anddensity of the material:

c =

√E

ρ, (5.9)

where E is the Young’s modulus and ρ the density of the material. Thus thecritical time step for stability depends on the size of the smallest element aswell as the elasticity and density of the material modelled. For nonlinear sys-tems, a similar stability criterion cannot be derived. However, for most prac-tical nonlinear problems an extra 10% reduction on the Courant condition issufficient.

When the constant time step integration is used, the time step is based onCalculatingelement length the Courant criterion calculated for the initial geometry. This time step can

be reduced by specifying a smaller time step. Large element distortions canmake this time integration unstable because the characteristic lengths of theelements become smaller and/or the stiffness of the material increases. Thenthe variable time step integration is more suitable because it is based on theCourant criterion calculated for the current geometry.

1Courant, R., Friedrichs, K. and Lewy, H., On the Partial Difference Equations of MathematicalPhysics. Math. Ann, 100, pp. 32-74, 1928.

70 Release 7.7


5

Element Characteristic element lengths

Beam elements the critical time step size is determined by thelongitudinal sound speed unless thebending-related time step size is smaller:

∆t =L

c· min

l,L√

12r2g

(12r2g+4L2)

(12r2g+L2)

where rg is the radius of gyration of the

cross-section given by r2g = I

A with I the maximummoment of inertia and A the cross-section area.

Three-nodemembrane elements

the equation for the element length Lmin is:

Lmin =2A

max(li)

where A is the element surface area and li thelength of element side i (i = 1, 2, 3).

Four-node membraneand shell elements;Three-node shellelements

the default equation for shell elements the elementlength Lmin is: ( TIME_INT_MTH = NORMAL):

Lmin = max(min(li),A

max(dj))

the default equation for membrane elements theelement length Lmin is: ( TIME_INT_MTH = NORMAL):

Lmin = max(min(li),A

min(dj))

where A is the element surface area,li the length of element side i (i = 1, 2, 3, 4) anddj the length of the element diagonal j (j = 1, 2).For degenerated four-node and three-node shellelements Lmin is :

Lmin = min(l1, l2, l3)

For degenerated four-node membrane elementsLmin is :

Lmin =2A

max(li)


Release 7.7 71

5


Table cont.

Element Characteristic element lengths

Four-node membraneand shell element;Three-node shellelementsmethod=1

When method 1 (TIME_INT_MTH = ACCURATE) isselected an extra condition is used:

L∗min = min(Lmin,

2A

max(l1 + l3, l2 + l4))

This condition may be needed when there areelements for which the angle between adjacentsides is small. For degenerated four-node elementsthe equation used is :

L∗min = Lmin

For three-node elements the equation used is :

L∗min =

2A

max(li)

Four-node membraneand shell element;Three-node shellelementsmethod=2

Method 2 (TIME_INT_MTH = PRECISE) yields themost strict condition and can be applied to someshell elements only. The resulting extra conditioninvolves the thickness of the element:

L∗∗min = L∗

min ·min(1,L∗

min

t√

3)

This condition may be needed when there areelements that are thick compared with the otherdimensions.

Four-node solidelement

the equation for the element length is:

Lmin =3 ·V

max(A1, A2, A3, A4)

where V is the element volume andAi the area of element face i (i = 1, 2, 3, 4)

Eight-node solidelement

the equation for the element length is:

Lmin =V

1.7662 · max(A1, A2, A3, A4, A5, A6)

where V is the element volume andAi the area of element face i (i = 1, 2, 3, 4, 5, 6).

Membrane and shellelements

Young’s modulus is corrected for the plane stresssituation.

Solid elements Young’s modulus is replaced by K + 4G/3 with Kthe bulk modulus and G the shear modulus.

The speed of sound for beam elements is determined in the same manneras for truss elements. For membrane and shell elements Young’s modulus iscorrected for the plane stress situation. For volume elements Young’s modulusis replaced by K + 4G/3 with K, the bulk modulus and G, the shear modulus.

72 Release 7.7


5

5.3 Stresses and strains

Internal forces will be generated within a structure under the action of appliedloads. This is illustrated in Figure 5.4 for a cylindrical bar. Assuming that theinternal forces are uniformly distributed over the cross-section as illustratedin Figure 5.4, the stress can be obtained by dividing the total tensile force Fby the cross-sectional area. The stress acting on the cross-section is a vector.If the area of the cross-section perpendicular to the centre line is A, the cross-sectional area shown in Figure 5.4 will be A/ cos α .

Figure 5.4: Stresses in a bar under tension.

The magnitude of the stress vector P is

P = (F/A) cos α (5.10)

The stress vector can be resolved into a component perpendicular to the cross-section, the normal stress vector σ, and a component parallel to the cross-section, the shearing stress vector τ.

Release 7.7 73

5


The magnitudes of the normal and stress vectors are

σ =P cos α = (F/A) cos2 α (5.11)

τ =P sin α = (F/A) sin α cos α

The normal stress is defined as positive when it produces elongation and neg-ative when it produces compression. In general, the normal stress vector andshearing stress vector are related to the stress vector by:

σ = (P · n)n and τ = P − σn (5.12)

where n is the unit outward normal vector to the cross section. For a givenorthogonal coordinate system the projections on the axes are denoted by nx ,ny , nz and Px , Py , Pz respectively resulting in the magnitude of the normalstress:

σ =Pxnx + Pyny + Pznz (5.13)

τ2 =P2x + P2

y + P2z − σ2

For the special case shown in Figure 5.4:

Px =P Py =0 Pz =0 (5.14)

nx = cos α ny = sin α nz =0

By substituting (5.14) in (5.13) equations (5.11) are easily obtained.

For an infinitesimal cubic element with edges parallel to the coordinate axes,the notation for the stress components acting on the faces of this element andthe positive directions are as indicated in Figure 5.5. Subscripts are used toindicate the direction and the plane on which the stress acts. The first index ofa stress component indicates the direction of the normal to the plane on whichthe stress is acting. The second index indicates the direction of the stress.

74 Release 7.7


5

σyPy

Pz

Px

P τyx

τyz τxy

τxz

σx

τzx

τzy

σzy

x

z

O

Figure 5.5: Definition of the stress components.

Based on equilibrium considerations the following relationships between thecomponents of the stress vector and the normal and shearing stress compo-nents can be found:

Px =σxxnx + τyxny + τzxnz (5.15)

Py =σxynx + σyyny + τzynz (5.16)

Pz =τxznx + τyzny + σzznz (5.17)

Relationships of this form, where for every (unit) vector n a second vector, theStress tensorstress vector P, is represented by a square matrix:

σxx τyx τzx

τxy σyy τzy

τxz τyz σzz

(5.18)

are common in mechanics. The quantity described by the matrix is called thestress tensor. Because the stress tensor is symmetric, six quantities σxx , σyy ,σzz , τxy = τyx , τxz = τzx and τyz = τzy are sufficient to describe the stressesacting on the coordinate planes through a point.

Release 7.7 75

5


A similar symmetric tensor can be defined for the deformations:

εxx12 ψyx

12 ψzx

12 ψxy εyy

12 ψzy

12 ψxz

12 ψyz εzz

=

∂u∂x

12 ( ∂u

∂y + ∂v∂x ) 1

2 ( ∂u∂z + ∂w

∂x )

· ∂v∂y

12 ( ∂v

∂z + ∂w∂y )

· · ∂w∂z

(5.19)

where u, v and w are the displacements in x, y and z direction, respectively.The diagonal components of the strain tensor are normal strains (relativeelongations) whereas the non-diagonal components are shearing strains. Thestrain rates are partial derivatives of the velocities with respect to the actualposition

εxx =∂u

∂xεyy =

∂v

∂yεzz =

∂w

∂z(5.20)

ψxy =∂u

∂y+

∂v

∂xψyz =

∂v

∂z+

∂w

∂yψzx =

∂w

∂x+

∂u

∂z

In large displacement applications, the strains are commonly determined byintegration of the rate equations. For the uni-directional strain case, this re-sults in a logarithmic strain

ε = ln(l

l0) (5.21)

5.4 Principal and effective stresses and strains

If the components of the stress tensor are known, it is always possible to findStressesthree perpendicular planes for which the shearing stresses vanish, the resul-tant stresses are perpendicular to the planes on which they act. These stresses(σ1 , σ2 and σ3) are called the principal stresses, and the corresponding direc-tions are called the principal axes of stress.

For the strains, it is also always possible to find three perpendicular planesStrainsfor which the shearing strains disappear. These directions are called principalaxes of strain, and the corresponding strains are called the principal strains(ε1 , ε2 and ε3). For isotropic and orthotropic material behaviour, the principalaxes of stress and strain coincide. For other types of material behaviour, thisis often only true for very special loading cases.

For arbitrary stress and strain situations, effective stresses and strains can becalculated, which provide scalar representations of structural loading. Com-parison with maximum allowable values gives a safety margin for typicalloading cases. Several effective stress and strain measures are calculated inMADYMO:

The most simple form has been proposed by Tresca1 as a result of a long seriesTrescaof experiments in which the loads required to extrude metals through dies of

1Tresca, H., Comptes Rendus Acad. Science, Paris, 1864.

76 Release 7.7


5

various shapes were measured. Tresca concluded that the elastic limit wasreached when the maximum shear stress reached a certain value. This hasbeen used to define an effective stress as:

σe = σ1 − σ3 where σ1 ≥ σ2 ≥ σ3 (5.22)

and an effective strain as:

εe = ε1 − ε3 where ε1 ≥ ε2 ≥ ε3 (5.23)

Another formulation frequently used has been proposed by Von Mises1. ThisVon Misescriterion is based on the fact that plastic deformations are essentially sheardeformations which entail no volume changes. The resulting effective stresscan be defined in terms of principal stresses as:

σe =1

2

√2√

(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 (5.24)

and the effective strain as:

εe =1

2

√2√

(ε1 − ε2)2 + (ε2 − ε3)2 + (ε3 − ε1)2 (5.25)

For anisotropic plasticity, it is advantageous to apply a Hill equivalent stress.Its definition is given on page 86.

In addition to the stresses and strains with respect to the local element coor-dinate system, the principal stresses and strains as well as equivalent stressesand strains can be obtained for selected elements as time history output of thefinite element model.

1Mises, R. von, Göttinger Nachrichten, Math.–Physics Klasse, 1913.

Release 7.7 77

5


5.5 Material models

All structural materials are elastic to a certain extent, if the applied loads donot exceed a certain limit, the deformation disappears with the removal of theloads. When loaded beyond the elastic limit, plastic deformations remain afterremoval of the loads. Often the material is assumed to be:

• homogeneous = the smallest part cut from the body possesses the samespecific mechanical properties as the body and

• isotropic = the properties are the same in all directions.

Structural materials usually do not satisfy all of the above assumptions. Evenmetals consist of crystals that vary in size and orientation. However, as longas the dimensions of a body are large in comparison to the dimensions of asingle crystal the assumption of homogeneity can be used, and if the crystalsare oriented at random the material can be treated as isotropic.

Materials such as (polymer) foams and biological tissues exhibit time depen-Foams andbiologicaltissues

dent mechanical properties (visco-elasticity) so that time must be consideredin the constitutive equations. Rubber-like materials are incompressible andmay undergo extremely large elastic deformations.

Most biological materials and polymers exhibit some form of material damp-Materialdamping ing. In addition to a linear visco-elastic material model, a similar model can

be used in which the damping is represented by a linear dependence on strainrate

σ = S ε + γε (5.26)

with S the stiffness matrix and γ a rate sensitivity parameter, which is chosenas

γ = Ed(µ + (1 − µ)∆te) (5.27)

with E, d, ∆te denoting the Young’s modulus, a damping constant and theelement time step according to the undamped stability criterion, respectively.The parameter µ can have two discrete values:

µ = 0 ⇒ γ = Ed∆te Damping depends on the element time step and as aresult on element size; small elements show less damping than large elements.

µ = 1 ⇒ γ = Ed Damping is identical for all elements irrespective ofsize.

Material damping also influences the time step criterion. As a result of damp-ing, stability is obtained for

∆t = min

[√(γ/E)2 + ∆t2

e − γ/E|0 < e ≤ Nelem

](5.28)

78 Release 7.7


5

In the absence of damping (d = 0), the finite element time step equals theundamped Courant time step.

ω

Dam

ping

rat

io

ζ

Figure 5.6: Damping ratio versus frequency.

Note that the damping ratio, i.e. ζ = d for µ = 0 and ζ = d/∆te for µ = 1 ,increases linearly with frequency, see Figure 5.6. This implies for µ = 1 thatfor a constant damping coefficient the material may become critically dampedif the element time step decreases. It is advised in this case to choose such thatfor the smallest integration time step the material is not critically damped.

Damping is available for linear elastic isotropic material, linear elastic or-thotropic material, loosely woven fabric material, Mooney-Rivlin hyperelasticisotropic material and honeycomb material models.

5.5.1 Elastic material

The mechanical behaviour of materials is specified as the relationship betweenConstitutiveequation the stresses and strains, constitutive equations. Hooke’s law is the general

constitutive equation for linear elastic material behaviour that shows a linearrelationship between the six strain components and six stress components.This relationship can be written in matrix notation as:

σ = S ε (5.29)

where S is the stiffness matrix with elasticity coefficients and where σ and εare column matrices with the following stress and strain components:

σT = (σxx, σyy, σzz, τxy, τyz, τzx)

εT = (εxx, εyy, εzz, ψxy, ψyz, ψzx)(5.30)

Release 7.7 79

5


The stiffness matrix can be inverted in order to obtain the strains in terms ofthe stresses:

ε = C σ (5.31)

with C denoting the compliance matrix.

For isotropic material, the behaviour in all directions is the same and the com-Isotropicmaterial pliance matrix can be expressed as:

C =

1/E −ν/E −ν/E 0 0 0

1/E −ν/E 0 0 0

1/E 0 0 0

symmetrical 1/G 0 0

1/G 0

1/G

(5.32)

where E is Young’s modulus, ν is Poisson’s ratio (≤ 0.5 for isotropic materials)and G the shear modulus. The relationship between these constants is:

G =1

2

E

1 + ν(5.33)

This means that only two constants, E and ν, have to be specified for linearisotropic material behaviour.

In shells and membranes, a plane stress state is often assumed. If the z direc-Shells andmembranes tion is perpendicular to the surface, then this leads to:

σzz = τyz = τzx = 0 (5.34)

The stiffness matrix reduces to

S =E

1 − ν2

1 ν 0

ν 1 0

0 0 1−ν2

(5.35)

Materials that have three mutually orthogonal planes of elastic symmetry areOrthotropicmaterial called orthotropic. Due to material symmetry, only nine independent elastic

constants have to be determined. Introducing E1 , E2 and E3 as the moduli ofelasticity in the 1st , 2nd and 3rd principal material directions, G12 , G23 and G31

as the shear moduli in the 1-2, 2-3 and 3-1 planes respectively and the Pois-son’s ratios ν12 , ν13 and ν23 , the compliance matrix for the principal material

80 Release 7.7


5

directions becomes:

C =

1/E1 −ν21/E2 −ν31/E3 0 0 0

−ν12/E1 1/E2 −ν32/E3 0 0 0

−ν13/E1 −ν23/E2 1/E3 0 0 0

0 0 0 1/G12 0 0

0 0 0 0 1/G23 0

0 0 0 0 0 1/G31

(5.36)

This means that in the full three-dimensional case, nine elasticity constantshave to be specified. Symmetry of the elastic compliances implies

νij

Ei=

νji

Eji, j = 1, 2, 3 (5.37)

Positive definiteness of the compliance matrix implies

|νij| <

√Ei

Ej(5.38)

For shells and membranes with a plane stress state, Hooke’s law with respectto the principal material directions simplifies to:

ε1

ε2

ψ12

=

1/E1 −ν21/E2 0

−ν12/E1 1/E2 0

0 0 1/G12

σ1

σ2

τ12

(5.39)

Thus only four material constants (E1 , E2 , ν12 and G12) have to be specified ifan orthotropic material model is used. The other input data corresponding tothe three-dimensional case is not used and can be specified as zero.

The compliance matrices specify the elasticity in the principal material di-rections. To determine the material orientation within finite elements for or-thotropic materials, the principal material directions with respect to the localelement coordinate system must be known. In the input deck, this directionmust be specified by a vector parallel to a principal material direction (seeFigure 5.7). The components of this vector with respect to the coordinate sys-tem in which the nodal coordinates are expressed, have to be specified.

Release 7.7 81

5


Figure 5.7: Material direction vector.

For an anisotropic material, Hooke’s law is given by:Anisotropicmaterial

σ1

σ2

τ12

=

S11 S12 S14

S21 S22 S24

S41 S42 S44

ε1

ε2

ψ12

(5.40)

The six independent elastic coefficients, S21 = S12 , S14 = S41 , S42 = S24 , mustbe specified by the user.

The principal stress and strain directions do not coincide for S41 6= 0 or S24 6= 0due to the coupling of tensile and shearing actions.

Layered orthotropic materials can be described by specifying the ply stack-Ply stackingsequence ing sequence in the elements. A material direction vector must be specified

whose projection on the element surface points in the fibre direction of thefirst layer. Either the bottom or top layer can be chosen as the first layer. Theangle between the fibre directions in subsequent layers can be defined in theProperties table. From these angles, the transformation matrices between thelocal element coordinate system and the principal material axes in all pliescan be determined. This material behaviour can be used for four-node shellelements and is applied when linear elastic orthotropic material is specified.

5.5.2 Elasto-plastic material

If metals are loaded beyond the elastic limit, or stresses exceed the yield stress,yielding occurs and often the stiffness of the material will be reduced signifi-cantly. If the metal does not possess strain-hardening, the structure could evenfail without further loading.

Constitutive equations for elasto-plastic material behaviour with isotropicPlasticitytheory hardening for one and two dimensional finite elements are derived from the

82 Release 7.7


5

basic equations for time-independent elasto-plasticity. Before the onset ofplastic deformation, the stress-strain relation is given by

σ = S ε (5.41)

The total deformation consists of an elastic and a plastic part, denoted withsuperscripts e and p, respectively

ε = εeεp (5.42)

The yield criterion determines the stress level at which plastic deformationtakes place. For metal plasticity, the Von Mises yield function is frequentlyused

φ =√

23 σT

DσD − σy(εp),

ρD

= σD = σ − 13 tr(σ)I,

(5.43)

where σD is the deviatoric stress tensor and σy is the yield stress, which isrelated to the value of the equivalent plastic strain. The equivalent plasticstrain rate is defined as:

εp =

√3

2(εp)T εp (5.44)

The purely elastic behaviour of the material is limited to negative values ofthe yield function, φ < 0 , while φ > 0 cannot occur. For associated plasticity,the direction of plastic strain rate is normal to the yield surface. The plasticstrain rate tensor may then be obtained as

εp = λ∂φ

∂σ(5.45)

With λ the ’plastic multiplier’. Substituting the equation above in the defini-tion for the equivalent plastic strain rate, the result is

εp = λ

√2

3

∂φ

∂σ

T ∂φ

∂σ= Bλ (5.46)

For the present von Mises yield criterion, B = 1 , but for other criteria such asthe Hill criterion this may not be true. The stress rate tensor can be specifiedby

σ = S εe = S(ε − εp) = S(ε − λ∂φ

∂σ) (5.47)

Because φ > 0 cannot occur, it holds that during plastic deformation

φ = 0 ,

φ = ∂φ∂σ

Tσ + ∂φ

∂σy

∂σy

∂ε p˙εp = 0 (5.48)

Release 7.7 83

5


Combining the preceding equations, it follows that:

λ =

∂φ∂σ

TSε

∂φ∂σ

TS

∂φ∂σ + ∂φ

∂σy

∂σy

∂ε pB

(5.49)

The plasticity theory presented above can easily be extended to an elasto-Elasto-plasticdamage theory plastic damage theory by replacing Cauchy stresses with effective Cauchy

stresses.

In the case of damage, the concept of effective stress is used, in which an ef-fective stress tensor is defined as

σ =σ

1 − D(5.50)

where D denotes the damage. The damage growth law is of the form of:

D =

P1XP2

(1−D)P3εp ifεp ≥ εc

0 otherwise(5.51)

with

X =1

2εeS0εe (5.52)

The parameters in the damage evolution law must be specified by the user.

As a result of plastic deformation, the stresses in a shell do not vary linearlyover the thickness. Stress resultants can be obtained by numerical integrationof the stresses in discrete points over the thickness

N =

h/2∫

−h/2

σdz, M = −h/2∫

−h/2

σzdz, (5.53)

Elasto-plastic material behaviour can be used for truss, membrane, shell andsolid elements. The user must specify the Young’s modulus, Poisson’s ratioand the yield stress. Additionally the hardening function, the plastic modulusas function of the yield stress, can be specified. If no hardening function isspecified an elasto-ideal plastic material behaviour is obtained.

Strain rate effects can be included in the plasticity formulation by making theStrain ratedependentplasticity

yield stress dependent on strain rate. Then, an increase or decrease in thestrain rate corresponds to an isotropic expansion or contraction of the yieldsurface. In MADYMO, it is assumed that yield-stress is composed of twoterms:

• dependent on effective strain rate only and

84 Release 7.7


5

• dependent on both effective strain rate and plastic strain εp .

The dependence of both yield-stress terms on effective strain rate is differenti-ated from an amplification factor ζ. This means the isotropic strain hardeninglaw is formulated as

σy = σy0 g( ˙ε) + [(1 − ζ) + ζg( ˙ε)] σy1(εp) (5.54)

where σy0 is the initial yield stress.

The rate dependency amplification factor ζ indicates how strongly the strainrate effect works on the hardening yield-stress σy1 . For ζ = 0 , the strain rateonly affects the initial yield-stress, whereas for ζ = 1 , the actual yield-stress isscaled bye the rate dependency function g:

σy =σy0 g( ˙ε) + σy1(εp), ζ =0 (5.55)

σy =[σy0 + σy1(εp)

]g( ˙ε), ζ =1

For linear hardening, can be written as

σy1 = Epεp (5.56)

with Ep the plastic modulus. Softening can be modelled using a negativevalue for Ep . Depending on the material model, the effective strain rate isdefined as

˙ε =

√2

3εT ε (5.57)

or as

˙ε =

√2

3εT

D εD; εD = ε − 1

3tr(ε)I (5.58)

where εD is the deviatoric strain tensor. The strain rate dependency functiong in equation (5.54) can be specified according to one of two empirical laws

Cowper-Symonds: g( ˙ε) = 1 +

(˙ε

c1

) 1c2

(5.59)

Johnson-Cook: g( ˙ε) = 1 + c2 ln

(max

(˙ε

c1, 1

))(5.60)

Both strain rate dependency factors can have a user-defined upper limit. It isalso possible to define a strain rate dependency function g as a user specifiedfunction.

The Johnson-Cook model is preferable for problems showing large variationsin strain rate.When little variation in strain rate is expected, the Cowper-Symonds law is preferred. Rate dependent plasticity can be combined with

Release 7.7 85

5


damage analysis and is suitable for failure analysis of metal parts and plas-tics.

For specific applications, it may be necessary to include the effects ofElasto-plasticanalysis usingthe Hillcriterion

anisotropy, which is usually done by using the Hill criterion. Assuming that alocal coordinate system coincides with the axes of orthotropy, the Hill criterioncan be expressed as

φ = K2 f (σ) − 1; K =σy0

σy(5.61)

where K is a hardening parameter that is defined as the ratio of the initial andcurrent yield stresses. For plane stress conditions, the stress function f can bewritten as

f (σ) = p1σ21 − p2σ1σ2 + p3σ2

2 + p4σ24 + p5σ2

5 + p6σ26 (5.62)

The coefficients p1, . . . , p6 are given by

p1 =1

XT2; p2 =

1

XT2+

1

YT2− 1

ZT2; p3 =

1

YT2

p4 =1

S212

; p5 =1

S223

; p6 =1

S213

(5.63)

where XT, YT, ZT, S12 , S23 , S13 denote yield stresses in the tensile and shear

tests, respectively. For XT = YT = ZT and S12 = S23 = S13 = XT/√

3 ,the von Mises criterion is obtained. Normally a yield surface would changeits basic shape. When XT and YT are not equal, a smooth yield surfacecan still be achieved by using ZT = min(XT, YT) and S12 = S23 = S13 =

min(XT, YT)/√

3 . However, there is very little known about hardening rulesfor anisotropic plasticity, so the assumption is that a single hardening param-eter is adequate to model the material behaviour.

A Hill equivalent stress is computed in MADYMO, which is defined as

σH = σy0

√f (σ) (5.64)

This material behaviour can be used in combination with damage and strainrate dependency. It is also suitable to analyse the failure of composite materi-als.

For the analysis of metal foams and other complex plasticity phenomena, theElasto-plasticanalysis usingtheDeshpande-Fleckmodel

plasticity model developed by V. S. Deshpande and N. A. Fleck is available forsolid elements. The Deshpande-Fleck model is based on an incrementally ob-jective rate formulation and, similarly to the previous plasticity models, usesan additive decomposition of the elastic and plastic part of the strain.

The model uses different criteria for the onset of plasticity and the direction ofplastic flow, i.e. the plastic flow is allowed to be non-associative. The onset of

86 Release 7.7


5

plastic flow is determined by

φ(σ, κ) = σ − σy(κ) (5.65)

in which the uniaxial yield strength σy is a function of the internal state vari-able κ and where the equivalent stress σ is given by

σ2 =1

1 +(

α3

)2(σ2

e + α2σ2m). (5.66)

The Von Mises effective stress is given by σe =√

32 σd

ijσdij , the mean stress

equals σm = 13 σkk and α is a parameter that determines the shape of the yield

surface.

The direction of plastic flow is determined by a similar equation, i.e.,

ψ(σ, κ) = σ − σy(κ) (5.67)

with

σ2 =1

1 +(

β3

)2(σ2

e + β2σ2m) (5.68)

where β determines the shape of the plastic flow surface. The plastic strainrate is now obtained from

εp = λ∂ψ

∂σ(5.69)

where λ is the non-negative plastic multiplier.

Within the Deshpande-Fleck material model only work hardening is included.The rate of plastic work if obtained from Wp = σy ˙ε , where ˙ε ≡ κ is an equiv-alent strain rate work conjugate to the yield stress. The tangent modulusH = dσy/dε to the σy − ε diagram is obtained from

H =dσy

dεe

dεe

dε+

dσy

dεm

dεm

dε(5.70)

where εe is the Von Mises deviatoric plastic strain rate which is work conju-gate to the Von Mises effective stress σe and εm is the volumetric plastic strainrate which is work conjugate to the mean stress σm . The tangent modulus foruniaxial loading hσ = dσy/dεe and the tangent modulus for hydrostatic load-ing hp = dσy/dεm may either be specified as constants or as functions of thedeviatoric and volumetric plastic strain respectively.

Release 7.7 87

5


5.5.3 Damage models

Degradation of material properties is the result of initiation, growth and co-alescence of microdefects, such as microvoids and microcracks. In circum-stances where the defects are distributed in a statistically homogeneous man-ner, it is advantageous to model the mechanisms associated with materialdegradation within the context of continuum damage mechanics (CDM). InCDM internal state, variables (also called damage variables) are introduced,which represent the local distribution of microdefects in an averaged sense1.

On a macroscale, relationships between material response characteristics andthe damage variable are formulated. The evolution of the damage variablethrough the loading process results in a continuous deterioration of the mate-rial stiffness. Macrocrack initiation and propagation can be modelled by con-sidering the crack tip as a process zone in which the damage state increases.Crack growth is identified with the evolution of a completely damaged zone,without any reference to concepts taking into account the material discontinu-ity of the cracks. The CDM approach to fracture can be applied when classicalfracture mechanics cannot cope with, thus providing a viable tool for failureanalysis.

The damage state can be defined by the existing distribution and type of mi-crodefects. The local damage can be defined as the ratio of damaged and un-damaged area in a plane through a volume element. Its value ranges between0 for undamaged and 1 for completely damaged. In the case of isotropic dam-age, the damage state can be characterized by a scalar quantity D = D(x, t) .

Generally, a complete damage theory requires:Genericdamage model

• the assessment of a stress-strain relationship for damaged materials,

• a damage evolution law, and

• a criterion for damage growth.

1Krajcinovic, D., Damage Mechanics, Mech. Mater., Vol. 8, pp. 117-197, 1989.

88 Release 7.7


5

For brittle damage states, the only source of energy dissipation is in damagegrowth. Using the concept of effective stress, the stress-strain relationship forisotropic damage can be written as

σ = (1 − D)S0ε (5.71)

where S0 is the initial stiffness matrix. Damage growth is assumed to takeplace when a critical threshold X in the thermodynamic force (the free energydensity of the undamaged material) is exceeded.

X =1

2εTS0ε (5.72)

The damage evolution law that is used isDamageevolution law

D =

P1XP2

(1−D)P3X , X > X

0 , otherwise

(5.73)

where the current damage threshold is determined by

X = maxτ≤t

[X(τ), X0] (5.74)

where X0 is the initial damage threshold thermodynamic force, which followsfrom the critical strain for the onset of damage evolution ε0 provided by theuser,

X0 =1

2E(ε0)

2 (5.75)

It is noted that P1 > 0 , since damage evolution must be an irreversible process.The parameters P1 , P2 and P3 must be determined experimentally from uni-axial tests.

Anisotropic damage states can be observed in laminated fibre-reinforced com-posites (layered orthotropic materials). For these materials, three mutually or-thogonal damage modes can be distinguished: transverse matrix cracks, fibrebreaking and local delaminations. These damage modes can be character-ized by a symmetric second order tensor. In MADYMO, two in-plane damagemodes are taken into account and the damaged stiffness matrix S is given by

S =

(1 − D1)S011 (1 − D1)(1 − D2)S0

21 0

· (1 − D2)S022 0

· · (1 − D1)(1− D2)S044

(5.76)

Release 7.7 89

5


where D1, D2 are the damage variables associated with fibre cracking andtransverse matrix cracking, respectively. The damage evolution law associ-ated with the kth damage mode is chosen as:

Dk =

Pk1XPk2k Xk if Xk ≥ Xk (k = 1, 2)

0 otherwise

(5.77)

where the thermodynamic force Xk conjugate to the kth damage evolutionvariable is given by

Xk = −1

2εT ∂S

∂Dkε (5.78)

Material damping can also be modelled by introducing a viscous (rate depen-dent) term

σ = Sε + γS11ε (5.79)

where the parameter γ is a damping constant.

Damage is available for linear elastic isotropic material, isotropic elasticplasticmaterial and linear elastic orthotropic material models for four-node shell andfour-node solid elements.

A simple damage model is available for defining the failure behavior ofPlastic-strainbased damagemodel

isotropic plastic material models where the failure state and growth of dam-age is defined in terms of the magnitude of the effective plastic strain only.Material damage is initiated at the defined effective plastic failure strain εp f

and grows in proportion to the effective plastic strain until the effective plas-tic rupture strain εpr is reached, see Figure 5.8. These parameters are easilyobtained by inspection of experimental results as a first approximation to thematerial damage process.

90 Release 7.7


5

σ

εεεpf pr

(D=0) (D=1)

Figure 5.8: Illustration of a stress-strain relation for an isotropic plastic material com-bined with the plastic-strain based damage model.

The scalar damage state is defined by

D = 0.0 for εp ≤ εp f

D =ε p−ε p f

ε pr−ε p ffor εp f < εp < εpr

D = 1.0 for εp ≥ εpr

where d = 0.0 signifies no damage and d = 1.0 is complete damage.

If the rupture strain is not specified, brittle fracture is assumed and the rupturestrain is assumed to equal the failure strain (εpr = εp f ). The present imple-mentation does not include a damage process zone, hence the finite elementsize sets the length scale in the failure process and determines the amount ofenergy dissipated upon failure.

Many engineering materials, including metals, polymers, soil, concrete andMeshsensitivity rock, are classified as softening materials. These materials exhibit a reduction

of the load-carrying capacity accompanied by increasing localized deforma-tions after reaching the limit load1. If strain softening occurs also over a smallportion of the length, which would be correct for a perfectly homogeneousmedium, instability always occurs at the peak stress point. In finite elementanalysis, strain softening causes spurious mesh sensitivity. Spurious meshsensitivity can not only occur for the post peak response, but also for the valueof the maximum load.

To correct this situation, a localization limiter must be introduced, which pre-vents the structure from localization in an infinitely small region. An effective

1Bazant, Z.P., Mechanics of distributed cracking, Appl. Mech. Rev., Vol. 39, pp 675–705, 1986.

Release 7.7 91

5


method is to assume that damage processes occur in a zone of dimensions de-pending on the given type of material. In this zone, it is assumed that crackingis uniformly smeared.

Consider a damaged element of length L. Let the damage process zone in theelement equal length h. The corresponding global damage state in the elementand local damage state in the process zone are denoted as D and Ω, respec-tively. It can be shown that the local element damage Ω can be expressed interms of the damage in the damage process zone D, yielding

Ω =hL D

1 +(

hL − 1

)D

0 <h

L≤ 1 (5.80)

Thus if the damage process zone is specified, the element damage state can becomputed. Damage growth in the process zone takes place if the local ther-modynamic force X exceeds the maximum value reached during its loadinghistory. Because all quantities are computed at the element level, a transfor-mation between the global and local quantities must be made. The globalthermodynamic force Xg can be written as

Xg = X

(1 + D

(h

L− 1

))2

(5.81)

In MADYMO, mesh insensitive results are obtained if the damage processzone is several times smaller than the element size. If the process zone islarger than the local element size, local and global damage states coincideand classical continuum mechanics solutions are obtained. If the value of thedamage zone has not been specified, no corrections are made.

5.5.4 Woven fabric material

Airbag fabrics are typically of a plain-woven construction and may include alayer of coating material. Plain woven fabrics consist of two sets of interlacedyarns, the lengthways set of yarns is called the warp, and the crosswise set ofyarns is called the weft, see Figure 5.9.

During manufacture the warp yarns are typically drawn out in straight lines,whilst the weft yarns are woven across (above and below) the warp yarnsresulting in the zigzag construction of crimping.

Both ‘tightly’ and ‘loosely’ woven fabrics are available with differences in yarnspacing and crimping effects, both of which will have an effect on the overallmechanical properties of the fabric under load.

92 Release 7.7


5

WEFT

WARP

Figure 5.9: Typical loosely woven fabric construction.

Woven fabrics are not able to resist out-of-plane loading (bending moments)due to the typically small overall thickness and the woven construction. Thefabric yarns are able to resist in-plane elongations (tensile), but not compres-sive loads, due to the buckling of the yarns and build up of wrinkles in thefabric. Woven fabrics are also able to resist in-plane shear loads once theyarn compaction or ’lock-up’ angle has been reached, see Figure 5.10 andFigure 5.11. The ‘lock-up’ angle is much lower for tightly woven than looselywoven fabrics.

Y

X’

X

Y’

angle between thread 1 and element frame

angle btween thread 1 and thread2

Figure 5.10: Shear deformation of tightly woven fabric.

Release 7.7 93

5


α

θ

η

X´Y´

Figure 5.11: Shear deformation of loosely woven fabric.

Two materials are available in MADYMO to model fabrics, i.e. a modelthat only includes two thread stiffness properties and ignores the in-planeshear stiffness and a model that incorporates, in addition to the two threadproperties, an in-plane shear stiffness property and a coating layer. Theformer model is suitable for loosely woven fabrics, whereas the latter issuitable for a typical airbag tightly woven fabric.

The initial orientation of the warp and weft yarns are specified relative to theInitialorientation ofyarns

global reference frame. They are then stored internally relative to the localelement frame (see Figure 5.12). If a reference mesh is supplied for the airbag,then the yarn orientations for the reference mesh are supplied (relative to theglobal reference frame).

94 Release 7.7


5Y

GLOBAL REFERENCE SYSTEM

4

1 2

X

Y

X

FIBRE 2 DIRECTION

FIBRE 1 DIRECTION

LOCAL ELEMENT FRAME

3

Figure 5.12: Yarn orientation.

The orientation of the fibres is updated at each time step, allowing relativerotation to occur between the two yarns. This effect is shown in a uni-axialtest with two yarn orientations (see Figure 5.13). In both cases (a and b) theyarns are initially orthogonal. In case (a) the yarns are aligned to the loadingdirection and no relative rotation of the yarns takes place under the applieduni-axial load. In case (b) the yarns are inclined at 45 to the loading axis(‘bias’ test), and under load will rotate relative to each other.

initialstate

(b)(a)

finalstate

Figure 5.13: Yarn rotation.

The fabric material response can be divided into three basic structural parts:yarn stiffness (warp and weft), shear stiffness between the two layers of yarnsand a coating stiffness. Each of these three basic parts has distinct stiffnessand load carrying characteristics.

Release 7.7 95

5


Under tensile loading the yarns typically have a high tensile stiffness alongYarn stiffness(Warp & weft) their length enabling high tensile loads to be sustained. Under compressive

loading the geometry of the individual yarns results in a negligible axial com-pressive stiffness, due to individual yarn buckling and overall fabric wrin-kling.

The warp and weft yarn axial stiffness may be defined by either a constantelastic modulus or a user defined stress-strain characteristic. Tension-only be-haviour may be used to provide a smeared wrinkling effect in the fabric. Atension-only behaviour may be invoked for use with a defined constant elasticmodulus. For the Yarn stiffness defined by a characteristic, the tension-only ef-fects should be incorporated into the loading functions itself (see Figure 5.14).

function

definedincluded in userTension only behaviour

stress−strain

Figure 5.14: Tension-only behaviour defined in load function.

The geometry of the weave pattern has a large influence on the in-plane shearShear stiffnessstiffness of the fabric. Under shear loading, the warp and weft yarns typicallydisplace in a trellis-like manner with little initial resistance until the yarns be-gin to lock-up against each other (Figure 5.10 and Figure 5.11). This lockupangle is dependent upon the geometry of the weave pattern.

The fabric shear stiffness is a function of the angle change between the twoyarns. The stiffness may be described using either a constant shear modulusor a user defined function.

With the user-defined function it is possible to simulate the effects of the lockup angle using data derived from either a bias45 or picture frame test (seeFigure 5.15). It is not possible to simulate the lock up angle effect with a con-stant shear modulus defined.

96 Release 7.7


5

shear lock−up angle

shea

r fo

rce

angle(radians)

shear lock−up angle

Figure 5.15: Shear stiffness function with lock-up angle effect included.

The coating stiffness consists of an isotropic coating layer. The coating stiffnessCoatingstiffness is calculated using an isotropic elastic material model. The thickness of the

coating material is specified relative to the total fabric thickness.

Strain rate dependency may be introduced for fabric yarn axial stresses (notRate effects,materialdamping

to the shear or coating stresses) by using a stress scaling function (CowperSymonds, Johnson-Cook or User-Defined). The continuum element strain rateis used for the material damping calculation.

The element mass is calculated using the product of the element in-plane area,Element massthe element thickness, and the material density. The element timestep calcula-tion is based upon the maximum elastic modulus of the warp and weft yarns.If a load function is used, the maximum slope (modulus) of the load func-tion is used. The fabric material models are available for both triangular andquadrilateral membrane element formulations.

Two methods are commonly used to describe the mechanical properties for aMechanicalbehaviour ofthreads andfabrics

woven fabric, i.e.,

(a) Describe the individual yarn properties and yarn density (yarns per unitlength).

(b) Describe the overall properties of the fabric (smeared properties).

The benefit of the second approach (b) is that the final state of the fabric includ-ing the effects of the weaving process (including crimping) is incorporated inthe material description. Yarn properties alone may be used when the effectsof the weaving process are not large (for example: some loosely woven fabricswith negligible crimping).

Release 7.7 97

5


Let the mechanical behaviour of a thread be described by a linear relationshipMechanicalbehaviour ofthreads

between the thread force and the logarithmic strain

Fth = kεlog (5.82)

where k is the thread stiffness. The internal force per unit of length, f f , ina patch of fabric is proportional to the number of threads per unit of lengthperpendicular to the thread direction, or the weaving density ζ

f f = ζFth (5.83)

When t is the thickness of the fabric, the stress in the fabric in the thread di-rection is

σ =f f

t=

ζk

tεlog = Eεlog (5.84)

This means that E = (ζk)/t can be considered as the modulus of elasticityof the fabric in the thread direction. When the characteristic of the thread isknown as a (non-linear) function of the logarithmic strain, the stress-strainrelation for the fabric is given by

σ =ζ

tFth(εlog) (5.85)

The fabric characteristic is equal to the thread characteristic multiplied by thefactor ζ/t . The thickness of the fabric is a redundant parameter, because themechanical behaviour is controlled by the stiffness parameter Et = ζK andthe mass per unit area ρt , where ρ denotes the mass density.

98 Release 7.7


5

The material tensile properties are typically measured with a uni-axial test.Smearedmechanicalbehaviour offabric

The mechanical behaviour of the fabric is described by a relationship betweenthe force and displacement recorded:

Ffabric = fa(dfabric) (5.86)

For MADYMO input, the force-displacement relationship may be convertedto a nominal stress versus nominal strain function,

σfabric(nominal) = fb(εfabric(nominal)) (5.87)

or a true stress versus true strain (log) function

σfabric(true) = fb(εfabric(true)) (5.88)

This data may be used directly as the input or a constant modulus E may becalculated and used instead. The use of properties measured directly from thefabric automatically incorporates effects such as crimping. In some cases, a bi-axial test may provide more appropriate data if the loading of the fabric is bi-axial. This will reduce the crimp effect and provides a stiffer initial response.

The stress and strain measures are calculated from the force and displacementdata:

σfabric(nominal) =Ffabric

Ainitial cross-sectionεfabric(nominal) =

(lfabric − l0)

l0(5.89)

σfabric(true) =Ffabric

Acurrent cross-sectionεfabric(true) = ln

(lfabric

l0

)(5.90)

The shear stiffness of the fabric is derived from a picture frame (or trellis) test,which measures shear force versus shear angle change between fibres:

Ffabric(shear) = fa(γfabric)

The fabric shear stiffness versus shear angle change is then normalised perunit area to provide a shear stress versus shear angle function. Either a func-tion, a characteristic or a constant shear modulus may be input. If a functionis used, it is possible to include the effects of the ‘lock-up’angle, where theshear stiffness rises considerably due to the compaction between fibres. If acharacteristic is used, it is possible to include the loading-unloading behavior.The stiffness constants for the fabric coating are input directly in the form ofthe elastic constants E (elastic modulus) and ν (Poisson’s ratio).

Release 7.7 99

5


Table 5.1: Proposed tests to measure woven fabric material parameters.

TestConfigura-tions

Test Type Uni-axial tensile Uni-axial tensile Bi-axial tensile Picture framePure shear

Specimen Single yarn Woven fabric Woven fabric Woven fabric

RequiredOrientation

Yarn axis 0 degrees (warp)90 degress (weft)45 degrees (BIAS)

Warp & weftparallel toloading axes

Warp & weftparallel to shearframe sides

Typical testresults

(a) (b) (c) (d)

(e) Comparison of the results of (a)through (c)

Four distinct tests are proposed to measure the various material stiffness prop-Measurementof materialproperties

erties required for MADYMO input, depending on the material chosen, seeTable 5.1.

In the figures in Table 5.1, the solid line indicates the uni-axial fabric, thedashed line indicates the bi-axial fabric, the dash-dotted line corresponds tothe fabric with shear response and the dotted line indicates the fabric withuni-axial response.

There are three common methods available to measure the fabric elongationMeasuringfabric materialproperties

(tensile) stiffness, each of which will include different effects due to the fabricweaving process and the loading conditions. Two methods are available to

100 Release 7.7


5

determine the fabric shear stiffness. For the coating layer, it is assumed theelastic modulus E and Poisson’s ratio are known.

The single yarn uni-axial tensile test measures the yarn properties in isola-Single yarn/Uni-axialtensile

tion of the mechanical interaction that takes place in a woven fabric (crimpingeffects) as shown in Figure (a) of Table 5.1.

The woven fabric uni-axial tensile test determines the yarn properties includ-Woven fabric/Uni-axialtensile

ing the mechanical interaction that takes place in a woven fabric. Tests maybe undertaken to measure both the warp and weft response, in order to deter-mine the distribution of crimping between warp and weft yarns.

The difference between the woven fabric and single yarn test indicates thedegree of crimping that has occurred during weaving. The uni-axial responsefor the woven fabric typically shows a low initial stiffness as the crimpingstraightens out, before attaining a similar stiffness to a single yarn (see Figure(b) of Table 5.1).

The bi-axial tensile test will determine the yarn properties including the me-Wovenfabric/Bi-axialtensile

chanical interaction that takes place in a woven fabric under the influence ofdifferent strain ratios between the warp and weft yarns.

The results from the bi-axial response will illustrate the effect of the mechani-cal weaving process on the yarn response. The response for a typical bi-axialtest is shown in Figure (c) of Table 5.1. Due to the straining of perpendicularyarns, the crimp effect is less pronounced, and an initial stiffness between asingle yarn and a uni-axial test is typically attained, as shown in Figure (e) ofTable 5.1.

The picture frame test determines the structural response of the fabric underWoven fabric/Picture frame(pure shear)

pure shear loading (Figure (d) of Table 5.1), in effect the in-plane resistance toa ’trellis’-like deformation mode. A low initial shear stiffness is expected, in-creasing to a well-defined ‘lock-up angle’ after which there is a large increasein shear stiffness as the angle change between warp and weft is severly con-strained and the fabric wrinkles.

A common alternative to the picture frame test is the bias45 uni-axial tensileBias 45Uni-axial test test that has the fabric yarns oriented at +/− 45 degrees to the loading axis.

However, the results are more difficult to interpret than the picture frame test,due to the combination of shear and tensile loads.

5.5.5 Interface material

The interface material model provides a simple representation of a tear seamincluding failure and post-failure energy dissipation as shown in Figure 5.16.The pre-failure behaviour of the material is modelled as rigid and is achievedby imposing kinematic constraints across each pair of nodes that bridge thetear seam to prevent any relative movement across the interface elements. Thetwo integration points at the ends of each interface element are then associatedwith the respective kinematic constraints, see Figure 5.17.

Release 7.7 101

5


Displacement

Failure load

Ultimate relativedisplacement

Post−failure energy dissipation

(area under line)

Force

Figure 5.16: Interface material model representation for tear seam failure.

Interface elementInterface element

Figure 5.17: Association of interface element integration point with kinematic con-straint.

The kinematic constraints are only removed when the failure criteria for eachintegration point associated with a particular constraint has been met simul-taneously during the specified accumulated time window. The failure crite-rion evaluated is similar to that for spotwelds as defined in (5.91), where FN ,FN,max , FS and FS,max denote the interface element integration point actingnormal force, maximum allowable normal force, acting shear force and max-imum allowable shear force respectively. The coefficients A and B determinethe shape of the failure criterion. The normal and shear directions of each in-terface element are defined by the local coordinate system (described in theelement section).

[FN

FN,max

]A

+

[FS

FS,max

]B

≥ 1 (5.91)

The user defined post-failure energy dissipation controls the rate of removal

102 Release 7.7


5

of the failure load and provides a material model to simulate the energy dissi-pation occurring when material failure and tearing occur.

The failure force and post-failure energy dissipation for the in-plane normaldirection of an interface element are required to be specified by the user. Ifin-plane shear failure is also to be included, the respective shear failure forceand energy dissipation are also required. The failure force and post-failureenergy dissipation are defined per unit area, where the area of the tear seamis defined as the length of the element edge multiplied by the thickness.

The post-failure behaviour controls the rate of load removal. This is speci-fied as the energy dissipated per unit area following failure. The reduction infailure load versus the relative displacement of the tear seam is calculated toprovide the correct energy dissipation. The ultimate normal displacement un

of the tear seam may be calculated from the ultimate normal traction force tn

and the mode I fracture energy GI , i.e.,

un =2GI

tn(5.92)

The same procedure can be applied for the calculation of the relative shear dis-placement us . After failure, the kinematic constraints are not re-imposed onunloading, this results in different behaviour for the shear and normal direc-tions. For the normal direction, energy is only disspated for positive relativedisplacement increments. For the shear direction energy dissipation can oc-cur in either direction once the maximum relative displacement magnitudehas been exceeded.

5.5.6 Hyperelastic materials

To model rubber-like material behaviour, a Mooney-Rivlin material model hasbeen implemented in MADYMO based on the strain energy function

W = A(J1 − 3) + B(J2 − 3) + C(J−23 − 1) + D(J3 − 1)2 (5.93)

with J1 , J2 and J3 the invariants of the right Cauchy-Green strain tensor e,which are defined as

J1 = trace(e)

J2 = 12 (trace2(e) − trace(e2))

J3 = det(e)

(5.94)

The material parameters C and D are functions of the coefficients A and B

C =1

2A + B (5.95)

D =A(5ν − 2) + B(11ν − 5)

2(1 − 2ν)(5.96)

Release 7.7 103

5


The incompressibility of the material can be taken into account by setting Pois-son’s ratio to nearly 0.5. The resulting large value for the penalty factor Dforces the third invariant J3 to 1, resulting in nearly incompressible materialbehaviour.

The 2nd Piola-Kirchhoff stress sensor is obtained by differentiating the strainenergy function W with respect to the right Cauchy-Green strain tensor e

S = 2∂W

∂e(5.97)

This material behaviour can be used for solid elements only. The user mustspecify the material constants A, B and ν. For this material model the recom-mended value of Poisson’s ratio is: 0.460 < ν < 0.499 .

5.5.7 Visco elastic materials

The mechanical response of most materials depends on the applied strain rate.In particular certain foams and biological materials exhibit a typical visco-elastic behaviour, i.e. the resistance against sudden changes of shape is rela-tively high compared to the quasi-static resistance against deformation. Anobjective, linear visco-elastic material model has been developed based ona multi-mode Maxwell model. A simple, one-dimensional representation ofthe model can be given by the combination of a spring in parallel with up tofour Maxwell elements. The volumetric response is taken to be purely elastic,whereas the visco-elastic behaviour is accounted for in the deviatoric part ofthe stress response only. Especially for nearly incompressible materials thisassumption simplifies the model without loss of accuracy.

The Green-Lagrangian strain tensor E follows from the deformation gradienttensor F

E =1

2(FTF − I) (5.98)

The second Piola-Kirchhoff stress tensor P is related to the Green-Lagrangianstrain tensor and rate of deformation by means of the relation

P = Ktr(E)I + 2

t∫

−∞

G(t − τ)EDdτ (5.99)

Where the subscript D denotes the deviatoric part, defined as

ED = E − 1

3tr(E)I (5.100)

It should be noted that the constitutive relation (5.99) is fully objective, since inthe convolution integral the Green-Lagrangian rate of deformation has beenused1.

1Some formulation use the rate of deformation D, defined as the symmetric part of the velocitygradient L = D + W . Such an approach does not render an objective formulation.

104 Release 7.7


5

The rate-dependent, instantaneous shear modulus G(t) is

G(t) = γ∞ + ∑i

γie−t/τi (5.101)

Where t is the time scale of the deformation process and γ∞ is the static shearmodulus. The dynamic shear modulus of mode i is denoted γi and has anassociated relaxation time constant τi . The time step is based on the upperlimit of the material stiffness, which is applicable at infinite rate of deforma-tion (t → 0). To ensure that the rate of change of the strain energy is positivedefinite, the following condition has to be satisfied

G ≤ 3

2K (5.102)

For nearly incompressible materials K/G ≫ 1 . It is recommended not tochoose this ratio too high because of the unfavourable effect on the integrationtime step. A value of K/G between 30 and 100 is usually high enough toeffectively suppress volumetric deformations.

The mechanical behaviour of biological tissue and most rubber like materi-als can be described as nearly incompressible non-linear viscoelastic. A non-linear viscoelastic constitutive model is developed on the basis of a secondorder Mooney-Rivlin model. A unique feature of the model is that it is formu-lated in a differential form with special attention paid to accurate predictionof nearly incompressible material behaviour at large deformations and rota-tions at acceptable time steps. A brief description of the constitutive modelis provided here. A more comprehensive description is presented elsewhere(Brands et al., 2002; Brands, 2002).

For accurate modelling of nearly incompressible materials the Cauchy stress,σ, is additively composed of a volumetric part σv and multiple deviatoric con-tributions σd

i which depend on change of shape only (i.e. not on change ofvolume).

σ = σv +n

∑i=0

σdi (5.103)

The hydrostatic stress component uses the linear relation

σv = K

(V

V0− 1

)I, (5.104)

where K is the bulkmodulus and I is the unit tensor. The deviatoric part of thestress is modelled nonlinearly viscoelastic. It is constructed from a numberof contributions (modes) σd

i , each with its own time constant, τi . In a one-dimensional analogue, these modes can be regarded as a number of spring-dashpot elements in parallel (generalized Maxwell model) with non-linearsprings and linear dashpots. In the three-dimensional model, the deforma-tion gradient tensor F, is decomposed multiplicatively into an elastic part (c.f.

Release 7.7 105

5


elongation of spring), Fp,i , and an inelastic part (c.f. elongation of dashpot)for each mode i.

F = Fe,iFp,i (5.105)

The elastic behaviour is taken to be independent of volumetric changes and ismodelled by a second order non-linear Mooney-Rivlin strain energy densityfunction,

Wi = C10,i( I1,i − 3) + C01,i( I2,i − 3)+

C20,i( I1,i − 3)2 + C02,i( I2,i − 3)2(5.106)

in which I1,i and I2,i are the first and second invariants acting on the isochoricelastic Finger tensor in mode i,

Be,i = I−1/33,i Fe,iF

ce,i (5.107)

and Cjk,i are Mooney Rivlin material parameters of mode i.

The inelastic rate of deformation Dp,i , related to the deviatoric part of thestress, obtained from equation (5.106), by a simple Newtonian fluid descrip-tion (or linear damper) with viscosity (damping constant), ηi

Dp,i =σd

2ηi(5.108)

This inelastic rate of deformation is used to update both elastic and inelasticparts of the deformation using a strain evolution equation stated in rotation in-variant strain(rate) quantities [Brands, 2002]. This provides accurate solutionswhen modelling nearly incompressible materials undergoing large rotationsand deformations, even at large time steps.

The MADYMO input parameters are chosen such as to be easily determinablefrom experiments. For nearly incompressible materials the bulk modulus K,can be determined directly from ultrasonic data, using K = c2

pρ (with massdensity ρ and sound velocity cp). Also hydrostatic compression experimentscan be used. Linear material parameters C10,i , C01,i and ηi are related to thelinear viscoelastic shear modulus Gi and time constant τi via

(C10 + C01)i = Gi and ηi = Giτi (5.109)

These can be found by fitting a generalized Maxwell model to small strainsimple shear experiments or oscillatory shear experiments (e.g. in Brands etal. 2002). Subsequently the second order parameters can be found from stressrelaxation experiments or quasi-static simple shear experiments. Elaborationof equation (5.106) for quasi static simple shear provides

σ = 2(C10,0 + C01,0)γ + 4(C20,0 + C02,0)γ3 (5.110)

106 Release 7.7


5

To reduce the number of unknowns it is assumed that the nature of the non-linear behaviour is the same in every mode i. This is done by defining thefollowing ratio’s between the Mooney-Rivlin parameters,

fNLS = (C20 + C02)i/(C10 + C01)i Relation linear vs nonlinear

Mooney-Rivlin parameters

f1 = C10,i/C01,i Ratio between first order Mooney-

Rivlin parameters

f2 = C20,i/C02,i Ratio between second order

Mooney-Rivlin parameters

f3 Fraction of small strain shear stiff-

ness providing minimal stiffness

when shear softening is modelled

( fNLS < 0)

Non-linear shear parameter fNLS determines the overall nonlinear shear be-haviour. If fNLS > 0 the material will display shear hardening, i.e. increasingstiffness with increasing strain (as for many rubbers). When fNLS < 0 the ma-terial displays shear softening, i.e. decreasing stiffness with increasing strain(as for brain tissue). However, at large strains negative incremental shear stiff-ness might occur. For this reason a small stress contribution is added thathas negligible effect at small strains but overrules the negative stiffness in-duced by the negative second order parameters at large strains. This is doneby adding third order Mooney-Rivlin parameters. A parameter f3 can be pro-vided that provides the minimal remaining (positive) shear stiffness writtenas a fraction of the small strain stiffness: Gmin = f3Gi . Using these parametersequation (5.109) can be rewritten as function of small strain shear stiffness as,

σi = Gi

γe + 2 fNLSγ3

e,i +9 f 2

NLS

5 − 5 f3γ5

e,i

(5.111)

Note that f1 and f2 are absent in this equation as the ratio of the Mooney-Rivlin parameters cannot be determined from simple shear experiments. In-stead unconfined compression experiments can be used to fully characterizethe material.

5.5.8 Sandwich material

Sandwich plates are layered structural elements in which a relatively thicklayer, the core, is placed in between two relatively thin layers, the facings. Thecore is characterised by a relative low density and a low stiffness in shear, ten-sion and compression, while the facings have a relatively high density and ahigh stiffness in shear, tension and compression. The function of the facings ina sandwich element is to carry in-plane stresses resulting from in-plane shear,tension and compression forces and from bending moments. The function ofthe core is to take up out-of-plane shear forces and to stabilize the facings.

Release 7.7 107

5


Sandwich plates and layered composite material plates behave differentlywhen experiencing out-of-plane loads. The assumption that cross-sectionalareas remain flat does not hold for sandwich plates. Often a sandwich plateexperiences a transverse shear deflection. Sandwich material can only be ap-plied for the four-node shell elements because transverse shear is included inthis element type only. The out-of-plane shear stiffness of a sandwich plate isdetermined by the shear modulus of the core which is much lower than thatof the facings. To account for this behaviour in lamination theory, differentconstitutive equations are used for the material of the facings and the materialof the core.

For the material of the facings, the terms in the constitutive equations corre-sponding to the out-of-plane shear stiffness are disregarded. The material isassumed to behave differently in layers belonging to a core than in layers be-longing to a facing, this means the function of each layer, either core or facing,must be specified.

For the constitutive equations describing the material behaviour, the follow-ing assumptions have been made:

• The core material is orthotropic and much thicker than the facings.Stresses that can occur in the core are in-plane normal and shear stressesand out-of-plane shear stresses.

• The facings can consist of a number of integration points and a state ofplane stress is assumed to exist in each integration point. Each integra-tion point is assumed to be orthotropic.

The stress-strain relation for a layer of a facing with respect to the principalmaterial directions of this layer is given by

σ1

σ2

τ12

=

S11 S12 0

S12 S22 0

0 0 S44

·

ε1

ε2

ψ12

(5.112)

Equation 5.112 is also used for the core. Here two extra equations are addedto describe the relationships between out-of-plane shear stresses and strains

τ23

τ13

=

S55 0

0 S66

·

ψ23

ψ13

(5.113)

The stiffness components in (5.112) are given by

S11 = E11(1−ν2

12·E22E11

) , S12 = ν12E22(1−ν2

12·E22E11

) ,

S22 = E22(1−ν2

12·E22E11

) , S44 = G12

(5.114)

108 Release 7.7


5

Thus only four engineering constants are needed to specify in-planeorthotropic material behaviour, namely E11 , E22 , G12 and ν12 (seeEquation (5.39)). For the shear in the core material, isotropy is assumed, re-sulting in S44 = S55 = S66 = G12 .

5.5.9 Honeycomb material behaviour

A semi-empirical model is used to represent crushable honeycomb mate-rial. The material is orthotropic with uncoupled relationships between thestress and strain tensor components, or the tangent stiffness matrix S (seeEquation (5.29)) is diagonal

S = diag(E1, E2, E3, G12, G23, G13) (5.115)

These moduli correspond to user-defined material directions. The materialdirections rotate with the material based on the rotation component of thedeformation gradient tensor in the sense of polar decomposition. This meansthat this material model can only be used in with the default (ADV_STRAIN)strain-rate formulation (page 134).

As is shown for material direction DIRAA in Figure 5.18, the moduli maychange during compression. This occurs when the logarithmic strain becomessmaller than specified values. These values are denoted compaction factors.

The compaction factors must be negative (compression) and CMPFC_B <NoteCMPFC_A .

σ

εCMPFC_ACMPFC_B

ln

Ei1

Ei2

Ei3

Figure 5.18: Loading curve of honeycomb material.

Unloading from compaction greater then CMPFC_A occurs along a slope of

Release 7.7 109

5


Ei1. When unloading takes place from compaction smaller than CMPFC_B(larger compressions!), the unloading slope is max (Ei1, Ei3).

The time step is based on the steepest slope in the stress-strain curve. Whena fixed time step is used, the time step is calculated from the Courant stabil-ity condition applied to elements compressed to the level corresponding toCMPFC_B. This maintains stability at least until the second compaction factoris reached. When a variable time step is used, the Courant stability conditionis automatically satisfied for compressive deformations.

5.5.10 Solid foam material

The behaviour of solid foams can be described as highly non-linear and strain-rate dependent with high energy dissipation characteristics and hysteresis incyclic loading. Low density combined with high energy dissipation capacitymake foams attractive for energy absorbing functions in automotive applica-tions. However, the three-dimensional mechanical response of foam materialsis quite difficult to capture in a mathematical model.

Foams are typically used under compression (Figure 5.19). At small strains,Foam’smechanicalbehaviour

the mechanical behaviour is close to linear elastic, followed by a large order ofmagnitude reduction in slope. Then, there is a long region in which the slopechanges gradually. This stage corresponds to the collapse of cells. After thecells have collapsed, the final stage of densification is reached in which thecells come in contact with one another causing a sharp increase in the stress.

collapse densification

strain

linear elastic stage

stre

ss

Figure 5.19: Typical stress-strain curve for solid foam material.

Strain-rate dependency is a very important factor that must be consideredwhen modelling the characteristics of foam. If the dynamic stress-strain re-lationship from experiments is used directly in a simulation without consid-ering strain rate effects, the foam model will almost certainly be either stifferor softer than the real foam.

110 Release 7.7


5

The Foam material model uses an experimental stress-strain curve rather thanStress-straincurve a material law. It is available for solid elements only. The model is based on

the following two assumptions

• There is no coupling between stresses and strains of different principaldirections, which means Poisson effects are neglected.

• Strain rate effects can be characterized by a strain rate dependent scalingfactor.

As a result, the stress-strain curve can be determined from uni-axial compres-sion and tension tests for different loading rates. Mathematically, the stress-strain relationship has the following form:

σ = g( ˙ε)σr (5.116)

where g is a scaling factor that depends on the effective strain rate, and σr isa user specified reference stress curve. This curve represents the quasi-staticuni-axial behaviour of foam under both compression and tension. A piecewiselinear interpolation is used. Currently, both hysteresis models 1 and 2 canbe applied for this curve (see Appendix B). However, hysteresis model 2 ispreferred because it works better with the physical behaviour of foams.

Two analytical laws are available for scaling up the user-defined stress-straincurve, the Cowper-Symonds and the Johnson-Cook formulations.

Cowper-Symonds g( ˙εn) = 1 +

(˙εn

c1

) 1c2

(5.117)

Johnson-Cookg( ˙εn) = 1 + c2 ln

(max

(˙εn

c1, 1

))(5.118)

where c1 and c2 are user specified positive constants. Besides, it is also possibleto define a strain rate dependency function g as a user specified function. Boththis user specified function and these empirical laws are based on nominalstrain rates so a transformation of logarithmic strain rates to nominal strainrates must be carried out. Taking the derivative of the effective strain withrespect to time and using the relationship between nominal and logarithmicstrains, the nominal strain rate is obtained as

εn = exp(sε)ε−1tr(ε · εT); s = sign[min(ε) + max(ε)] (5.119)

The nominal strain rate is adjusted depending on the sign of the largest abso-Notelute value of the principal strains. As a result, the stresses are scaled up muchmore for loading than for unloading.

Release 7.7 111

5


Material damping can be taken into account by a similar model as used forMaterialdamping isotropic and orthotropic materials, in which the damping is represented by a

linear dependency on the strain rate:

σ = σ + γε (5.120)

where σ is the stress tensor derived from the specified stress-strain curve in-cluding strain-rate dependency effects. γ is the damping coefficient, which isdefined as

γ = Kd(µ + (1 − µ)dte) (5.121)

where K, d and dte denote, respectively, the maximum slope of the definedstress-strain curve, a damping constant and the critical element time-step ac-cording to the undamped stability criterion. The parameter can have two dis-crete values:

µ = 0 ⇒ γ = Kd∆te Damping depends on the element time step and asa result on element size; small elements show less damping than large ele-ments.

µ = 1 ⇒ γ = Kd Damping is identical for all elements irrespective ofsize.

5.5.11 Fu-Chang foam material

Foams are typically used under compression (Figure 5.19). At small strains,the mechanical behaviour is close to linear elastic, followed by a large order ofmagnitude reduction in slope. Then, there is a long region in which the slopechanges gradually. This stage corresponds to the collapse of cells. After thecells have collapsed, the final stage of densification is reached in which thecells come in contact with one another causing a sharp increase in the stifness.

The behaviour of foams can be described as highly non-linear and strain-ratedependent with high energy dissipation characteristics and hysteresis in cyclicloading (Figure 5.20). Low density combined with high energy dissipation ca-pacity make foams attractive for energy absorbing purposes in automotiveapplications. To capture the three-dimensional mechanical response of foammaterials, the stress-strain relation in the Fu-Chang foam material model isdefined by a tabulated formulation. The tabulated stress-strain relations arebased on uniaxial static and dynamic tensile tests at different strain rates, thatare used directly as input. An extension of the model with elastic damageis included that is capable of identifying unloading in a natural way, i.e. bya decrease of the stored hyperelastic energy of the system. With the presentFu-Chang foam model, hysteresis effect can be simulated and energy is dissi-pated.

The Fu-Chang foam material model uses experimental stress-strain curvesrather than a material law. It is available for solid elements only (i.e. using anobjective strain and stress rates formulation for finite increments). The modelis based on the following three assumptions

112 Release 7.7


5

• There is no coupling between stresses and strains of different principaldirections, which means Poisson effects are neglected. As a result, thestress-strain curve can be determined from uni-axial tests.

• Strain rate effects can be characterized by tabulated stress-strain rela-tions at different strain rates.

• The curve with the lowest strain rate value corresponds to the load-ing path of the material as measured in a quasistatic test. Unloadingresponse will be computed internally by a damage formulation, usingthe hysteretic unloading factor HU and the shape factor for unloadingSHAPE .

Figure 5.20: Typical stress-strain curves at different strain rates for Fu-Chang foammaterial.

The Fu-Chang foam material model is based on a tabulated approach of hy-Stress-strainalgorithm perelasticity formulated in the principal (true) stress space. The stress-strain

evaluation algorithm can be described as follows:

- Compute the square of the left stretch tensor V from the deformation gradi-ent F

V2 = FFT (5.122)

- Diagonalize the left stretch tensor by computing the eigenvectors and com-pute the principal stretch ratios λi .

- Compute the strain rates using the velocity gradient L in the principal direc-tions of the left tensor

ε =1

2(L + LT) (5.123)

Release 7.7 113

5


- Filter the principal strain rate values, using the last twelve values or aweighted running average

λni =

n

∑m=n−11

λmi

12or λn

i =λn

i

12+

n−1

∑m=n−11

λmi

11(5.124)

- Compute principal engineering strains ε i = λi − 1 and determine principalengineering stresses σi by a table lookup. The table lookup uses strain andstrain rate in each principal direction. For tensile regime, the failure behaviouris considered when the stress is larger than the tension cut-off stress.

- Then compute Cauchy stresses in the global system using the fact that ina hyperelastic material the eigenvectors of the true stress tensor and the leftstretch tensor are identical.

- Compute the damage parameter d based on the hyperelastic energy per unitundeformed volume of the material W

W =3

∑i=1

Wu(λi), Wmax = max(W, Wmax) (5.125)

d = (1 − HU)

[1 −

(Wcur

Wmax

)SHAPE]

(5.126)

where HU is the hysteretic unloading factor, SHAPE is the shape factor forunloading and Wcur is the current value of the hyperelastic energy per unitundeformed volume.

- Compute the stress incorporating damage

σi = (1 − d)σi (5.127)

Showing that the rate effects are also applied during unloading and hysteresisis a consequance of the damage mechanism rather than the viscosity.

5.5.12 Spotweld material

This material model can be applied to the spotweld beam element type. Thesespotweld beam elements, based on a degeneration of an isoparametric 8-nodesolid element (the Hughes-Liu formulation), may be placed between two de-formable surfaces that have to be tied together. The spotweld material be-haviour can be defined as:

• linear elastic isotropic behaviour combined with a stress-resultant fail-ure model

• elasto-plastic behaviour with linear hardening combined with an equiv-alent plastic strain failure model.

114 Release 7.7


5

A plane stress state is assumed. If the local x-axis is aligned with the beam axisIsotropicbehaviour and the local y- and z-axes define the cross-section plane, then the σxx = σyy

stress components of the local 3D stress-tensor have to be zero. The relationbetween the stresses and strains can now be written as

σxx

σxy

σxz

= E

∣∣∣∣∣∣∣∣∣

1 0 0

0 12(1+ν)

0

0 0 12(1+ν)

∣∣∣∣∣∣∣∣∣

εxx

γxy

γxz

(5.128)

The entire spotweld beam fails if the stress-resultants in a cross-section areoutside of the failure-surface defined by

max(Nxx,0)

Nxxmax

2

+ Qxy

Qxymax

2

+ Qzx

Qzxmax

2

+

Mxx

Mxxmax

2

+ Myy

Myymax

2

+ Mzz

Mzzmax

2

− 1 = 0

(5.129)

where Nxxmax , Qxymax , Qzxmax , Mxxmax , Myymax and Mzzmax are the pre-definedfailure parameters for the failure surface and Nxx , Qxy , Qxz , Mxx , Myy , Mzz

are the stress-resultants calculated in the local coordinate system of the cross-section via

Nxx =∫

σxxdA

Qxy = s fy

∫σxydA

Qxz = s fz

∫σxydA

Mxx =∫

σyzrdA

Myy =∫

σxxydA

Mzz =∫

σxxzdA

(5.130)

where s fy and s fz are shear correction factors in the y- and z-direction respec-tively.

The material shows a linear elastic isotropic behaviour until the Von MisesElasto-plasticbehaviour equivalent stress exceeds the initial yield stress σy0 . The Von Mises yield func-

tion is defined as

φ =

√3

2σT

DσD − σy(ep) (5.131)

where σD is the deviatoric stress tensor and σy is the yield stress which de-pends on the equivalent plastic strain ep . The equivalent plastic strain rate isdefined as

εp = λ∂ϕ

∂σ(5.132)

Release 7.7 115

5


with λ the ‘plastic’ multiplier. The incremental stress tensor ∆σ can now bedetermined by integrating the stress rate tensor

σ = Sεe = S(ε − εp) (5.133)

When linear hardening is used, the actual yield stress is determined as

σy(ep) = σy0 + Epep (5.134)

where Ep is the elastic-plastic modulus. An internal iteration loop is per-formed to satisfy the plane stress condition, i.e. σyy = σzz = 0 .

Each integration point in the beam cross section can fail independently; theentire spotweld fails if all integration points have failed. Failure of an inte-gration point occurs if the corresponding equivalent plastic strain exceeds aspecified threshold strain parameter ePF .

The post-failure behaviour can be controlled by a specified rupture strain mea-sure ePR . The concept of effective stress is used to reduce the stress statel theequivalent Von Mises stress Σeq is reduced according to

Σeq = (1 − D)ΣeqF (5.135)

where ΣeqF is the equivalent Von Mises stress at failure and D is a failure pa-rameter defined by

D =ep − ePF

ePR − ePF(5.136)

Afterwards, the stress tensor σ is reduced according to the ratio between Σeq

and ΣeqF

σ = σFΣeq

ΣeqF

(5.137)

where σF is the stress tensor at failure.

5.5.13 Other material models

The material properties for Tension ONly Elastic material models with Rup-Tension-onlyelastic materialwith rupture

ture (TONER) are defined similarly to the Strap in Airbag chamber. The ma-terial model TONER can be used for describing elastic connections betweenfinite element nodes that have an optional rupture capability. In additionto modelling straps, this option can be useful in modelling, for example, ripstitch in seat belts or tearing of fabrics.

The user must specify an untensioned length, a stiffness, a mass and a relativeelongation at which rupture occurs. After rupture has occurred in one of theelements, no forces will be generated by any of the elements of the same part.

The Hysteresis model is a general non-linear material characteristic using theHysteresismodel same syntax as hysteresis models 1 and 2 (Appendix B). The user must specify

116 Release 7.7


5

a stress-strain relationship that includes a loading curve, an unloading curveand a hysteresis slope. Poisson’s ratio can not be specified and equals zero.The user should be aware that the steepest slope specified is used to determinethe stable integration time step. By choosing a hysteresis slope that is not toosteep, a considerable amount of computational time can often be saved.

This material behaviour can be applied to truss, membrane and solid ele-ments. In combination with truss or membrane elements, this material modelis suitable for the modelling of belt webbing or airbag fabric. In combina-tion with solids, the material model is suitable for the characterisation of (lowdensity) high hysteresis materials such as polypropylene and polyurethanefoams, which exhibit very high hysteresis and have a very low Poisson’s ratio.For belt applications using membrane elements, the belt force is calculatedfrom the specified belt characteristic and damping coefficient and/or damp-ing function, i.e.,

Fi = Fei + Fdi (5.138)

in which Fei is the elastic belt force and Fdi is the damping force. In combina-tion with strain-rate model using membrane elements, the elastic belt force iscomputed as:

Fei = g(ε i) f (ε i) (5.139)

with g(ε i) a strain rate sensitivity scale factor (Cowper-Symonds, Johnson-Cook or a user-defined function of the strain rate) and f is the elastic beltcharacteristic. Scaling using g(ε i) is only applied for positive values of thestrain rate and if a function is specified for g(ε i) it should be a non-decreasingfunction with only positive function values and g(0) = 1 (so no scaling if ε i =0). The damping force calculation is similar to the damping force calculationin other FE material models.

For fabrics, it is recommended to use a material model that can only carrytension. During compression, wrinkling prevents the build-up of stressesto counteract this deformation. To account for this nonlinear behaviour, thestress components are calculated first and transformed into principal stresses.Negative principal stresses are discarded during the reverse transformation.The resulting stresses are subsequently used to calculate the internal nodalforces.

Rigid behaviour can be applied to any part of a finite element model and isRigidbehaviour represented by a rigid element. A number of defined rigid elements can be

constrained together to form one integral rigid element. A rigid element istreated as an independent rigid body. The advantage of using rigid behaviourover supporting the corresponding nodes to a rigid body is that the calculationof the internal forces of the corresponding elements is eliminated, which savescomputation time.

The kinematic contact model (Section 9.2) should not be applied for contactbetween nodes of a rigid finite element model, and planes, ellipsoids or cylin-ders. With this contact model it is assumed that the surfaces do not deform by

Release 7.7 117

5


placing the nodes of the finite element model onto the surface of the ellipsoid(or plane, cylinder). For a rigid model this is not possible because the modelwould have to deform locally, so the nodes are removed from the contact. Ifa rigid finite element model comes into contact with another finite elementmodel, it will be necessary to specify a modulus of elasticity and a Poisson’sratio for the calculation of the compliance of the rigid finite element model.

For some applications, a finite element model can be considered unde-Rigid switchformable for only a part of the simulation. For example, the deformation ofan airbag can be neglected before the inflator is triggered. This can be mod-elled with the rigid time switch. This switch can be turned on at the start of thesimulation. At the point in time when deformations are expected, for examplewhen the inflator is triggered, the switch must be turned off. From that pointin time, the internal nodal forces are calculated using the specified materialbehaviour. For the entire analysis either the calculated or specified, the finiteelement time step is used.

5.6 Element types

As stated in page 67, the finite element method is used to reduce a continuoussystem to a discrete numerical model. The actual continuum is separated byimaginary points, lines or surfaces into a collection of finite elements. Theelements are assumed to be interconnected at a discrete number of points, thenodes of the elements. Each node can have up to six degrees of freedom, threetranslational components and three rotational components, depending on thetype of elements to which the node is connected.

There are many types of elements available within MADYMO: trusses, beams,membranes, shells and solids. Elements are available which are particularlysuitable for the analysis of highly nonlinear dynamic behaviour of three-dimensional structures. Most elements are based on linear displacement in-terpolation and are integrated at a single point at the centroid of the element.For some elements, this leads to zero-energy or hourglass modes which aresuppressed in MADYMO by an effective hourglass control algorithm.

Most elements are based on a co-rotational velocity strain (or rate of defor-mation) formulation leading to linear and frame-invariant kinematic relation-ships. Rigid body rotations are treated by embedding a local element coor-dinate system that moves with the element allowing arbitrarily large elementrotations. Fictitious mass moments of inertia are used for all elements withrotational degrees of freedom. This requires much less computational timethan using the actual mass moments of inertia because these would normallydetermine the integration time step. If only one or two elements are used, themass moment of inertia of the structure will be over estimated. For normalcrash applications, however, there is no significant influence on the results.

All element types can be joined together by connecting them to commonnodes. The user should be aware, however, that there is only a transfer ofloads corresponding to the common degrees of freedom at a common node oftwo elements. For example, if a beam element (three translational and three

118 Release 7.7


5

rotational degrees of freedom for each node) and a solid element (three trans-lational degrees of freedom) are connected, the solid element does not applya torque on the beam element at the common node. This means that the con-nection between the two elements behaves as a spherical joint.

This element is a prismatic one dimensional element that connects two nodesTwo-nodetruss and can carry only axial tension and compression. The mass of this type of

element is lumped and equally distributed over the two nodes. An updatedLagrange description, based on a linear displacement interpolation, is used.Each node has three translational degrees of freedom (see Figure 5.21).

Figure 5.21: Two-node truss element.

The only geometrical property needed is the cross-sectional area of the truss.This parameter is not updated during the simulation. Loading of the truss ischaracterised by the axial force.

The two-node beam element is a prismatic, one-dimensional element that con-Two-nodebeam element nects two nodes N1 and N2 and can carry tension, bending and torsion. Each

node has three translational degrees of freedom, denoted with u, v and w, andthree rotational degrees of freedom, denoted with α, β and γ (see Figure 5.22).

Figure 5.22: Two-node beam element.

For beams with an asymmetric cross section, the orientation of the local coor-dinate system in the cross section must be specified by a third node N3. Thenodes N1, N2 and N3 form the element (x, y)-plane. The local x-axis pointsfrom node N1 to node N2, i.e.,

e˜x =

x˜2 − x

˜1∥∥x˜2 − x

˜1

∥∥ (5.140)

Release 7.7 119

5


The local z-axis is perpendicular to the (x,y)-plane and points outwards, i.e.,

e˜z =

e˜x × (x

˜3 − x˜1)∥∥e

˜x × (x˜3 − x

˜1)∥∥ (5.141)

The local y-axis is perpendicular to the (z, x)-plane and points into the direc-tion of node N3, i.e.,

e˜y = e

˜z × e˜x (5.142)

For beams with a symmetric cross-section (e.g. a circular or pipe cross-section), or with symmetric cross-sectional properties (e.g., Iyy = Izz), the localy- and z-axes can be chosen arbitrarily and the third node N3 may be omitted.In that case, the local x-axis e

˜x points from node N1 to node N2 as defined in(5.140) and a local z′-axis is defined in the direction of the smallest componentof e˜x . The local y-axis is defined to be perpendicular to the (z′, x)-plane, i.e.,

e˜y = e

˜z′ × e˜x (5.143)

and the z-axis is defined to be perpendicular to the x-axis and y-axis

e˜z = e

˜x × e˜y (5.144)

Two types of beam elements are available:

• A classical beam element-type based on the Euler-Bernoulli beam theory.

• A degenerated, continuum-based element-type based on the Hughes-Liu beam formulation.

In this element formulation, it is assumed that the beam cross-section remainsEuler-Bernoullibeam plane and perpendicular to the beam-axis, which implies that the angle be-

tween the cross-section and the beam-axis does not change. The definition ofthe beam is completed by its area and its area moments of inertia for elasticmaterial behaviour. For plastic material behaviour also the section modulusmay be needed. Depending on the integration method chosen for the analysis,the geometrical properties are either also directly given by the user or they arecomputed from the integration points in the cross section. The cross-sectionalproperties are not updated during the simulation.

Note that many predefined beam shapes are nonsymmetric and a coupling ispresent between bending in y and z direction. For plastic material behaviorthis implies that the orientation of the neutral axis depends on the history ofthe deformation. For this reason, plastic material behavior in non-symmetriccross-sectional shapes is only allowed in combination with one of the numeri-cal cross-sectional integration methods. The section moduli are ignored in thiscase.

Not withstanding the possible coupling between bending and torsional defor-mation of the beam, it is assumed, with a small loss of accuracy, that the centre

120 Release 7.7


5

of mass, the shear centre and the centroid of the cross-section all coincide. Themass of this type of element is lumped and equally distributed over the twonodes. Fictive mass moments of inertia are used. Finally, the loading of thebeam is characterised by an axial force, two bending moments and a torsionalmoment.

Note that most predefined beams require the specification of material prop-erties and geometrical properties. For the concept analysis of spatial framestructures a separate beam is available for which the axial, torsional and bend-ing response may be specified through user-defined functions. These func-tions may be non-linear and the bending moment functions are allowed todepend on the axial force. Note that the user-defined functions lump togetherthe material and geometrical properties of the beam.

An updated Lagrange description, based on a cubic bending and linear tensileand torsion displacement interpolation, is used. The differential equationsgoverning the bending deformation of the beam are

dMy

dx = Vz My = EIyydϕy

dx − EIyzdϕz

dx

dMzdx = −Vy Mz = −EIyz

dϕy

dx + EIzzdϕz

dx

(5.145)

where My is the moment around the y-axis, Vy is the shear force in y-direction,Iyy is the moment of inertia around the y-axis, ϕy is the rotation around the y-axis and similar quatities are defined for the z-axis. Effects of shear deforma-tion are taken into account through an effective shear area of the cross-section,i.e.,

Vz

χA= G

(dw

dx+ ϕy

)Vy

ψA= G

(dv

dx+ ϕy

)(5.146)

where χ and ψ are shear area correction factors. For the stress calculationsin beam elements essentially two methods are available. One approach is adirect evaluation in terms of stress resultants, i.e., axial force, axial torque andbending moments. This approach is called the global formulation and givesaccurate results for a linear material response. With the global formulation,a non-linear distribution of stress over the cross-section due to, for example,plasticity can only be approximated. A more accurate solution is then ob-tained using the integration point formulation, which employs one of the nu-merical integration procedures.

The numerical integration procedures that are available in MADYMO areGaussian, Lobatto, Trapezoidal quadrature, as well as a user defined quadra-ture. Gaussian quadrature provides a high accuracy for the number of integra-tion points used. The Lobatto rule is very similar, and it always has integra-tion points at the domain boundaries. The Trapezoidal rule uses equidistantintegration points.

The accuracy of the integration rule can be assessed by comparing the re-sults of numerical integration with exact values for area, moments of iner-tia and section moduli, if available. In this, the absolute section modulus

Release 7.7 121

5


Qyy =∫|z|dA is a measure of the ultimate bending moment, the moment of

inertia is a measure of the elastic bending stiffness and the area is a measure ofthe axial stiffness. An example of this is given for 7 by 7 Gaussian integrationof a solid rectangular section with width w = 0.12 and height h = 0.2:

CROSS SECTION TYPE SOLID RECTANGULAR

EXACT VALUE NUMERICAL VALUE

AREA 0.24000E-01 0.24000E-01

IXX 0.10880E-03 0.10880E-03

IYY 0.80000E-04 0.80000E-04

IZZ 0.28800E-04 0.28800E-04

QYY 0.12000E-02 0.11646E-02

QZZ 0.72000E-03 0.69879E-03

Both the Gaussian and Lobatto rules exactly recover the moments of inertia,but the section modulus Qyy originates from a piecewise linear function, sothe result is not exact. The weight factors for the trapezoidal rule are based onlinear interpolation. Hence, the elastic bending stiffness is not recovered withthis integration rule, but the section modulus can be given exactly.

For the user-defined integration rule, the integration point locations and theassociated weight factors must be defined by the user. A simple way toachieve this is to divide the cross-section into a sufficient number of subdo-mains. For each subdomain, an integration point is defined by the location ofthe center of gravity of the subdomain. The weight factor equals the area ofthe subdomain.

For elastic behaviour, it is recommended to use the GLOBAL formulation (us-ing stress resultants) because it is much more efficient without a loss of ac-curacy. For plastic behaviour, it can be beneficial or even necessary to use theintegration point formulation. For efficiency and simplicity, the torque-torsionrelations are taken to be independent of the axial stress and are assumed to befully elastic (both in global and integration points formulation).

The user may specify an offset for the beam axis, for example to more ac-curately position the beam with respect to other elements. The offset p

˜=

(0, y0, zT)

is specified within the local element frame defined by (5.140) to

(5.144) and does not affect the orientation of the local element frame, i.e. onlythe origin of the local element frame is shifted. An offset along the beam axisis not allowed for.

The nodal linear velocities u˜s in the shifted element frame are obtained from

the nodal linear velocities u˜

and angular velocities ω˜

in the unshifted elementframe by

u˜s = u

˜+ (ω˜× p˜)∆t (5.147)

The angular velocities in both element frames are identical.

The element response, i.e. the forces F˜s and moments M

˜ s in the shifted ele-ment frame, are subsequently mapped back to the unshifted element frameby

M˜

= M˜

s + p˜× F˜s (5.148)

122 Release 7.7


5

The nodal forces in both element frames are identical.

This beam element is based on a degeneration of the isoparametric 8-nodeHughes-Liubeam solid element. The concept of an isoparametric element is based on the usage

of shape functions given in terms of the local coordinates. The correspon-dence between the Cartesian and the local coordinates can be established bymapping the solid element on a biunit cube:

x(ξ, η, ζ) = Nk(ξ, η, ζ)xk (5.149)

in which x is an arbitrary point in the element, ξ, η, ζ are the local coordinates,xk are the global Cartesian coordinates of node k and Nk are the element shape-functions at node k, which are given by

Nk(ξ, η, ζ) =1

8(1 + ξkξ)(1 + ηkη)(1 + ζkζ) (5.150)

In this description of beam geometry, ξ determines the location along the beamaxis (x-axis) and the (η, ζ) coordinate pair defines a point on the cross-section.Orthogonal inextensible fibers are defined at the nodal cross-section for defin-ing the kinematics of the element. The following assumptions about the mo-tion and stress-state of the element are made:

• The fibers remain both inextensible and straight.

• The transverse normal stresses (σyy, σzz) are forced to be zero, whichsatisfies the normal plane-stress condition.

• The element is incrementally objective, no strains are generated underrigid body rotations.

• Shear deformation effects are taken into account; reduction of the effec-tive shear force can be obtained by specifying a reduction factor for theshear area.

The integration of the beam element is performed with a one-point integrationNumericalintegration along the beam axis at ξ = 0 and multiple points in the cross-section, both in

η− and ζ−direction. Also for this beam formulation, the user may specify anoffset p

˜= (0, y0, z0) for the beam axis. Contrary to the Bernoulli formulation,

the Hughes-Liu element formulation allows for the offset to be added to thelocation of the integration points.

At each node, a fiber coordinate system Vη , Vζ is set-up for treating the nodalFibercoordinatesystem

rotations. This fiber coordinate system is defined by two orthogonal, in-extensible fibers pointing in the local y- and z-direction. The nodal fibersare updated using a second-order accurate update formulation following theHughes and Winget procedure.

A local coordinate system T = (e1, e2, e3) is necessary to enforce the zero nor-Localcoordinatesystem

mal stress condition perpendicalar to the beam axis. The orthogonal basis

Release 7.7 123

5


with 〈e1〉 normal to the deformed cross-section plane is defined as follows:

e1 =tVη×tVζ

|Vη×Vζ| ,

ey =tVη+tVζ

|Vη+Vζ| ,

ez =e1×ey

|e1×ey| ,

e2 = 12√

2(ey − ez) ,

e3 = 12√

2(ey + ez) .

(5.151)

where Vη and Vζ are the fiber directional vectors in η− and ζ−direction at themidpoint of the beam axis (ξ = 0).

The incremental strain and spin tensor are obtained by integration of the ve-Incrementalstrain andstress update

locity gradient

Gij =∆ui

t+∆txj(5.152)

where ∆ui are the incremental displacements from t to t + ∆t and t+∆tx thecoordinates in deformed configuration:

∆ε ij =1

2(Gij + Gji), ∆ωij =

1

2(Gij − Gji) (5.153)

The incremental spin tensor is used to the contribution of the Jaumann rate ofstress tensor that is approximated as

σij = tσij +tσip∆ωpj +

tσjp∆ωpi (5.154)

To evaluate the stress-strain relation, the updated stress tensor and the incre-mental strain tensor are rotated from the global coordinate system to the localelement coordinate system using the transformation as defined in the above:

tσ1ij =TipσpqTjq

∆ε1ij =Tip∆εpqTjq (5.155)

After the evaluation of the constitutive law, the stress tensor is updated incre-mentally:

t+∆tσ1ij = tσ1

ij + ∆σ1ij (5.156)

and afterwards transformed back to the global coordinate system:

t+∆tσij = Tpit+∆tσ1

pqTqj (5.157)

124 Release 7.7


5

For the membrane elements a corotational finite element formulation is used.Membraneelements:Strainmeasures andcorotationalformulation

In such a formulation the element co-ordinate system is attached to an ele-ment and rotated with the element. The element strains and element stressesare related to this co-rotating element co-ordinate system If a velocity field rep-resents a pure rigid body rotation, then the rotation of the element co-ordinatesystem is obviously equal to this rigid body rotation.

Three different strain measures can be used for membrane elements: the engi-neering or linear strain denoted as Elin , the Green-Lagrangian strain denotedas EGL , and the natural or logarithmic strain denoted as Elog .

In the linear strain formulation the classical engineering strains and theCauchy stresses are used and are defined with respect to the initial element co-ordinate system The engineering strain tensor can directly expressed in termsof the deformation gradient F by

Elin = F − I (5.158)

This formulation is in general not objective and the results obtained can de-pend on the element nodal topology. Objectivity can be achieved by forcingthat the local spin at the centre of the element is zero, i.e.

ω =∂u

∂Y− ∂v

∂X= 0 (5.159)

In the Green strain formulation the right Green strains and the 2nd Piola Kirch-hoff stresses are used and are defined with respect to the initial element co-ordinate system The Green strain tensor can directly expressed in terms of thedeformation gradient F by

EGL =1

2(FTF − I) (5.160)

Because the Green strains will vanish in any rigid body motion, this formula-tion is objective and the time integration is objective for finite time increments.

In the Logarithmic strain formulation the logarithmic or natural strains andthe Cauchy stresses are used and defined with respect to the updated elementco-ordinate system An objective strain rate formulation is used and the timeintegration is objective for finite time increments.

The incremental strain tensor ∆Elog is computed by performing a polar de-composition of the incremental deformation gradient Λ

Λ =(F(n))−1F(n+1) (5.161)

Λ =VR

∆Elog = ln(V)

Release 7.7 125

5


where V is the left stretch tensor obtained from polar decomposition of theincremental deformation gradient.

This is a flat, two-dimensional, triangular element (constant strain triangle)Three-nodemembraneelement

that connects three nodes and carries in-plane loads only. Due to the absenceof bending, the deformations are fully determined by the three translationaldegrees of freedom of these nodes (see Figure 5.23). A linear interpolation ofthe nodal displacements is used for the displacements within an element. Themass of this type of element is lumped and distributed over the three nodes byusing element distribution factors. These factors are proportional to the angleenclosed by the two element edges joining in the node.

Figure 5.23: The three-node membrane element.

Each membrane element has its own right-handed, orthogonal, local elementcoordinate system. The element plane is defined as the plane through thethree nodes of the element. The ξ and η axis lie in the element plane. Theζ axis is perpendicular to the element plane. The direction of the ξ axis isfrom the first to the second node specified in the Elements Table. The η axis isperpendicular to the ξ axis pointing into the direction of the third node. The ζaxis follows from the outer product of vectors along the positive ξ and η axis(see Figure 5.23). The order in which the node numbers are given thereforedefines the direction of the ζ axis. The positive direction of the ζ axis canbe easily determined as it corresponds with the direction of a right-handedscrew if rotating from node 1 past node 2 towards node 3. This direction isimportant for pressure loads and for determining of the gas volume inside anairbag chamber.

The only geometrical property needed is the thickness of the membrane. Theloading of the membrane is characterised by the Cauchy stresses, which areconstant within each membrane element. The three non-zero components ofthe Cauchy stress are determined with respect to the element coordinate sys-tem, Figure 5.24.

126 Release 7.7


5

y

1 2x

3

σy

σx σx

σy

τxy

τxy

Figure 5.24: Stresses in membrane element.

The elements can be distorted significantly during the analysis. The code doesnot carry out any checks on element shape. The performance is best when theyare close to an equilateral triangle.

This is a quadrilateral element with four nodes (see Figure 5.25). The stan-Four-nodemembraneelement

dard isoparametric formulation with bi-linear interpolation functions is used.Each node has three translational degrees of freedom. The mass of this typeof element is lumped in a work equivalent manner that results in an equaldistribution over the nodes.

Figure 5.25: Four-node membrane element.

The local coordinate system of a four-node membrane element is defined asfollows. The diagonals 1-3 and 2-4 form the ξη-plane. The ζ axis is perpen-dicular to the ξη-plane pointing outwards; if a right handed screw is rotated

Release 7.7 127

5


from node 1 paste node 2 to node 3 the translation of the screw is in the pos-itive direction of the ζ axis. The ξ and η axis lie in the ξη-plane. The ξ axis isdirected from the middle of edge 1-4 to the middle of edge 2-3 and the η axisis perpendicular to the ξ axis pointing into the direction of edge 3-4.

A thickness has to be assigned to an element. Also a choice can be madebetween reduced integration and full integration over the element area. Re-duced integration decreases the computing time, however, sometimes hour-glass modes can be activated which may lead to inaccurate results. Four-nodemembrane element has three hourglass modes: two in plane modes and onewarp mode, see Figure 5.26.

Figure 5.26: Hourglass modes for four-node membrane element.

These hourglass modes are suppressed by an effective hourglass control algo-rithm using a default dimensionless stiffness hourglass stabilisation parame-ter of 0.1. The user can tune the hourglass stabilisation parameter by changingit to a value between 0.0 and 0.5 As an alternative, full integration can be used,offering the advantage of robustness.

Four-node membrane element degenerates into a triangular element when thethird node and the fourth node are equal. This degeneration has a negativeeffect on the efficiency and should be avoided. If a mesh is available for acomplex geometry with quadrilateral elements, it can be useful to use a fewdegenerated triangular elements also. For a full mesh of triangular elements,however, a three-node membrane element is preferred.

This is a six-node triangular shell element that carries both in-plane loadsSix-node shellelement (membrane behaviour) as well as perpendicular loads (bending behaviour).

The sequence of the node-numbering is depicted in Figure 5.27: first the threecorner nodes N1-N3 must be numbered, followed by the mid-side nodes N4-N6.

128 Release 7.7


5

φ4u2

v2

w2

u3

v3

w3

φ5φ6

uv

w11

1

ξηζ

Figure 5.27: Six-node triangular shell element.

The geometry of the element is only described by the three corner nodal co-ordinates via a linear interpolation. The three corner nodes N1-N3 have threetranslational degrees of freedom; the mid-side nodes have only one rotationaldegree of freedom about the element side. A mixed formulation with a lin-ear displacement interpolation of the corner nodal displacements and a linearrotation interpolation of the mid-side nodal rotation is used.

Coupling between membrane and bending stiffness can be taken into account,but this coupling can introduce the so-called membrane-locking phenomenon.

The local coordinate system of the element is set up as follows. The elementxy-plane is defined by the three corner nodes; the local x-axis points fromnode N1 to node N2. The local y-axis is perpendicular to the x-axis in thexy-plane and points into the direction of node N3. Finally, the local z-axis isperpendicular to the xy-plane (from the vector product of x and y).

The initial curvature of the element can be specified by the coordinates of themid-side nodes. The distance out of the xy-plane from the mid-side nodes isused for defining the initial curvature of the element. If the coordinates of themid-side nodes are specified as (0, 0, 0), the element is assumed to have a flatinitial configuration.

The strain relationships are based on the Kirchhoff-Love hypothesis thatstates:

points on a straight line perpendicular to the mid-surface in theinitial state stay on a straight line perpendicular to the mid-surface.

The geometrical properties needed are the thickness of the shell and the num-ber of integration points along the thickness. The integration points alongthe thickness may be interpreted as layers, consisting of the same material,but not necessarily with the same properties. The two outer layers of a threelayer shell element with plastic material behaviour may be yielded whereasthe mid-layer still behaves as a linear elastic material. The bending modes

Release 7.7 129

5


are described by curvature deformation parameters so that for isotropic elas-tic material behaviour one integration point along the shell thickness is suffi-cient. For an elasto-plastic material model, the number of integration pointsalong the thickness must be at least two. The loading of the shell elements ischaracterised by the components of the Cauchy stress tensor with respect tothe element coordinate system.

The elements can be distorted significantly during the analysis. The code doesnot carry out any checks on element shape. The performance is best when theyare close to an equilateral triangle.

This is a two-dimensional quadrilateral element that connects four nodes andFour-nodeshell element can carry in-plane loads as well as bending loads. A bi-linear displacement

and rotation interpolation is used, i.e.,

Ni =1

4(1 + ξiξ)(1 + ηiη) (5.162)

where i ∈ [1, 2, 3, 4] is the node number, (ξ, η) are isoparametric coordinatesand (ξi, ηi) are the coordinates of node i. Each of the nodes has three transla-tional and three rotational degrees of freedom (see Figure 5.28). The mass ofthis type of element is lumped and equally distributed over the four nodes.Fictitious mass moments of inertia are used.

Each element has its own right-handed orthogonal local element coordinatesystem. The diagonals 1-3 and 2-4 form the ξη-plane. The ζ axis is perpen-dicular to the ξη-plane pointing outwards as shown in Figure 5.28; if a righthanded screw is rotated from node 1 past node 2 to node 3 the translation ofthe screw is in the positive direction of the ζ axis. The ξ and η axis lie in the ξη-plane. The ξ axis is directed from the middle of edge 1-4 to the middle of edge2-3 and the η axis is perpendicular to the ξ axis pointing into the direction ofedge 3-4.

Figure 5.28: Four-node shell element.

The rotation about the ζ axis of the local element coordinate system is notused. This means that a moment or prescribed rotation about the element ζaxis has no influence on the element. Also if two shell elements are connected,

130 Release 7.7


5

the component of a moment of one element about the ζ axis of the other ele-ment are not transferred.

The basic strain relationships are based on the Mindlin-Reissner hypothesisthat states:

points on a straight line perpendicular to the mid-surface in theinitial state stay on a straight line not necessarily perpendicular tothe mid-surface.

Deformations due to transverse shear are taken into account; no reduction ofthe shear stiffness is applied. Detailed information on the formulation of thiselement can be found in the work of Belytschko and coworkers1,2.

The geometrical properties that need to be specified are the thickness of theshell and the number of integration points through the thickness. A normalintegration of this element would require four integration points over the ele-ment area. With regard to efficiency, only one integration point is used result-ing in five zero-energy or hourglass modes : two in plane modes, one warpmode and two bending modes (see Figure 5.29).

Figure 5.29: Hourglass modes for four-node shell element.

In order to suppress the hourglass modes, an effective hourglass control al-gorithm is used with a default dimensionless stiffness hourglass stabilisationparameter of 0.1. The user can tune the hourglass stabilisation parameter bychanging it to a value between 0.0 and 0.5.

The elements can be distorted significantly during the analysis. The code doesnot carry out any checks on element shape. The performance is best when theyare close to a square.

1Belytschko, T. and Tsay, C. S., 1981, "Explicit Algorithms for Nonlinear Dynamics of Shells."in Nonlinear Finite Elements Analysis of Plates and Shells, ed. by Hughes, T. J. R., ASME, NewYork, pp. 209-231.

2Belytschko, T., Lin, J. I. and Tsay, C. S., 1984, "Explicit Algorithms for Nonlinear Dynamics ofShells." Computer Methods in Applied Mechanics and Engineering, 42, 225-251.

Release 7.7 131

5


The four-node shell element degenerates into a triangular element when thethird and fourth node are equal. However, this degeneration results in a lossof efficiency and needs to be avoided.

The standard Belytschko-Tsay element is known to perform poorly when theelement is subjected to a warping deformation. The original element formula-tion was improved by Belytschko and coworkers1,2,3 and these improvementsare available in madymo through the warping enhancement option. Briefly,the effect of warping is taken into account through two improvements, i.e.

1. The terms in the strain-displacement relation (B-matrix) involving cur-vatures and translations are coupled.

2. The non-zero contribution to the energy due to rigid body motion anddrilling rotations when the element is warped is removed through a pro-jection that extracts the pure deformation from the displacement field.

1Belytschko, T., Leviathan, I., 1993, "Projection schemes for one-point quadrature shell ele-ments." Computer Methods in Applied Mechanics and Engineering, 115, 277-286.

2Belytschko, T., Wong, B. L. and Chiang, H.-Y., 1989, "Advances in onepoint quadrature shellelements." Computer Methods in Applied Mechanics and Engineering, 96, 93-107.

3Belytschko, T., Bachrach, W. E., 1986, "Efficient implementation of quadrilaterals with highcoarse-mesh accuracy." Computer Methods in Applied Mechanics and Engineering, 54, 279-301.

132 Release 7.7


5

Note that the original formulation of the element does not have a non-zeroenergy contribution due to rigid body motion or drilling degree of freedom.This contribution enters the element formulation due to the coupling betweennodal curvatures and translations.

This is a two-dimensional triangular C0-element that connects three nodesThree-nodeshell element and can carry in-plane loads as well as bending loads. Similarly to the four-

node shell element, the strain relationships are based on the Mindlin-Reissnerhypothesis and deformations due to transverse shear are taken into account.

Each of the nodes has three translational and two rotational degrees of free-dom (see Figure 5.30) in the local reference frame, i.e., the drilling rotation isnot used. Linear interpolation for the displacement and rotation fields is used.The mass of this type of element is lumped and equally distributed over thethree nodes. Fictitious mass moments of inertia are used.

ηφφξ

uζ

uξ

uζ

uζ

uη

φξ

ηφ

φξ

uξ uη

ηφ

uη

N1

N2

N3uξ

ζ

ξ η

Figure 5.30: Three-node shell element.

Each element has its own right-handed orthogonal local element coordinatesystem. The ξ-axis is defined to run from node 1 to node 2 and the ζ-axis isperpendicular to the plane spanned by the three nodes. The η-axis is perpen-dicular to the ξ-axis and the ζ-axis, i.e.,

eξ =x2 − x1

‖x2 − x1‖eζ =

x2 − x1

‖x2 − x1‖× x3 − x1

‖x3 − x1‖eη = eζ × eξ

The four noded interface element is a two dimensional quadrilateral ele-Four-nodeinterfaceelement

ment used to define the geometry and the orientation of a tear seam (seeFigure 5.31).

Release 7.7 133

5


Ν1

Ν2

Ν3

Ν4

Figure 5.31: Four-node interface element.

The geometry of the tear seam is defined by the line connecting the midpointof the element edge from node 1 to node 4 (N1-N4) to the midpoint of the ele-ment edge N2-N3. This is equivalent to the average of the two edges definedby N1-N2 and by N4-N3.

The orientation of the tear seam is defined with respect to the element localcoordinate system. The local z-axis (out-of-plane direction) is defined as thecross product of the local vectors N1-N3 and N2-N4. The local s-axis (in-planeshear direction) is defined as the average of edges N1-N2 and N4-N3 or de-rived from a preferred normal direction. The local n-axis (in-plane normaldirection) is calculated from the cross product of the local z-axis and s-axis.The local in-plane shear and normal directions of the tear seam are used forboth the evaluation of the failure load and the calculation of the post-failureenergy dissipation.

The geometrical property required for the interface element is the thickness ofthe tear seam. The cross sectional area of the tear seam (the product of the tearseam thickness times length) is used to calculate the failure and post-failurecharacteristics.

The failure load and post-failure energy dissipation are defined per unit areafor the local normal and shear directions. Both the failure load and post-failure behaviour are evaluated and processed at the element nodal integra-tion points.

There is no density, and no mass associated with the interface element. Inthe current implementation the pre-failure behaviour is rigid, implementedby kinematic constraints. There is no minimum stable timestep requirementfor the element.

Solid elements are three dimensional elements that can carry tensile, compres-Solid elementssion and shear loads. The stress components for a solid element are definedwith respect to the inertial coordinate system. An element local coordinatesystem is defined for all element types except the solid elements. The motionof element local coordinate system is used as rigid body motion of the elementin calculating the rate of deformation. For solid elements, an alternative choiceis made to obtain an objective strain and stress rate, that is, the same strain andstress rate are obtained with respect to coordinate systems that have a relative

134 Release 7.7


5

motion. A well-known objective stress rate is the Jaumann stress rate σ givenby

σ = σ − W σ + σ W (5.163)

where the spin tensor W is the skew-symmetric part of the velocity gradientand represents the angular velocity corresponding to the rigid body rotation.

The time integration of the strain and stress rate must also be objective. Forlarge deformations, special measures must be taken to assure that the timeintegration is also objective for finite time steps. Otherwise the solution maybe inaccurate or become unstable. Formulations have been put forward thatare objective for large rotation increments based on rotation-neutralised stressmeasures1. In the spirit of the Hughes-Winget formulation and more recentwork on the subject2, an incrementally objective formulation has been im-plemented in MADYMO. The strain increments are based on the incrementaldeformation gradient between the configurations corresponding to the beginand end time of a finite element time step. This formulation not only givescorrect integration of the stress rate equations, but also the total strains areintegrated properly.

Alternatively, an advanced formulation may be selected which uses objectivestress and strain rates, and the time integration is objective for finite incre-ments. This formulation is more accurate than the Hughes-Winget formula-tion and requires just about 10% more computation time.

This element is based on a tri-linear displacement interpolation. Each of theEight-nodesolid element element’s nodes has three translational degrees of freedom (u, v, w) as illus-

trated in Figure 5.32. The mass of this type of element is lumped and equallydistributed over the eight nodes.

1Hughes, T.J.R., and Winget, J.,1980, "Finite Rotation Effects in Numerical Integration or RateConstitutive Equations Arising in Large-Deformation Analysis", Int. J. Meth. Eng., Vol. 15, pp.1862-1867.

2Paas, M.H.J.W., and Steenbrink, A.C., 1999, "An Objective Integration Procedure for RateEquations in Explicit Structural Dynamics", European Conference on Computational Mechanics,August 31–September 3, Munich, Germany.

Release 7.7 135

5


Figure 5.32: Eight-node solid element.

The node numbers of a hexahedron must be specified in the order in whichthe nodes are numbered in Figure 5.32. The first and fifth node will alwaysbe connected. Degenerated solid elements like wedges or pentahedrons canbe created by repeating two or more consecutive nodes in the Elements Table.For obtaining a wedge node numbers five and six must coincide with respec-tively with node numbers eight and seven, whereas for obtaining a penta-hedron node numbers six, seven and eight must coincide with node numberfive. However, this degeneration results in a loss of efficiency and should beavoided whenever possible.

The reduced integration of the eight-node solid element will reduce the com-putational effort, but results in twelve zero-energy or hourglass modes, whichcan be characterised by four principal modes in three different directions (seeFigure 5.33).

136 Release 7.7


5

Figure 5.33: Hourglass modes for eight-node hexahedral element in one direction.

In order to suppress these hourglass modes, an effective hourglass controlalgorithm is used with a default dimensionless stiffness hourglass stabilisationparameter of 0.1. The user can tune the hourglass stabilisation parameter bychanging it to a value between 0.0 and 0.5.

Elements can be distorted significantly during an analysis. The code does notcarry out any checks on element shape. The best accuracy is achieved whenthey form a cube.

This is a four-node tetrahedral element and it has a single integration point.Four-nodesolid element Unlike the eight-node hexahedral element, the one-point integration of the

four-node tetrahedral element does not result in any zero energy or hourglassmode.

Due to the simple geometry, the element is often more suitable for modellingcomplex bodies and surfaces than the eight-node hexahedral elements.

The node numbers of a tetrahedron must be specified counter-clockwise asshown in Figure 5.34. Each of the element’s nodes has three translational de-grees of freedom (u, v, w) as illustrated in Figure 5.34. The element’s displace-ment field is based on a tri-linear displacement interpolation. The mass of thistype of element is lumped and equally distributed across the four nodes.

Release 7.7 137

5


w1v1

u1 w2

v2

u2

w3v3

u3

w4v4

u4

Figure 5.34: Four-node tetrahedral element.

The four-node tetrahedral element is less sensitive to element shape distor-tion than the eight-node solid elements due to its simple geometry. Modellingcomplex bodies and surfaces including headform models, honeycomb barri-ers, tyres or foam objects such as seats and padding often involves automaticmesh generation. When using eight-node solid elements, the shape of the ele-ments is often quite poor, resulting in a loss of accuracy or leading to numeri-cal instability. These problems can be avoided by using four-node tetrahedralelements.

5.7 Initial conditions, prescribed motion and supports

Initial values can be specified for nodal displacements and orientations rela-tive to the inertial space, and their first time derivatives. Also the motion orvelocity of nodes relative to the reference space may be prescribed as a func-tion of time.

When the influence of the environment on the deformation of (a part of) afinite element model is not significant, the motion or velocity of the corre-sponding nodes can be prescribed as a function of time. For example, theinfluence of occupants on the deformation of the vehicle during a collision isusualy negligible. This means the finite element analysis of the vehicle can bedone without occupants. The resulting nodal motion can be saved and pre-scribed in subsequent analyses for optimizing the occupants’ restraint system.The vehicle finite element model has many nodes, so it is more convenient tostore the nodal motion in a separate file and use the structural motion inputoption. The analysis is faster, without affecting the final results, because allcalculations corresponding to elements which have prescribed nodal motionare eliminated.

A support is a rigid connection between a degree of freedom of a node (theSupportednodes components with respect to the inertial coordinate system) and the reference

138 Release 7.7


5

space or system, or a body. The position of the node, relative to the object towhich it is connected, is fixed during the simulation. Therefore the position ofa node is determined by the motion of this object only. The inertia propertiesof the supported nodes can be taken into account in the equations of motionof the supporting body (as a user-defined option).

The kinematic contact model (Section 9.2) should not be applied for contactbetween nodes with a prescribed motion or supported nodes, and planes, el-lipsoids or cylinders. This is because that contact model is based on placingnodes in contact with a surface that violates the prescribed motion or the mo-tion introduced by the support.

5.8 Loads

Loads can be defined for elements and nodes by specifying a load versus timefunction. There are four different types of loading options:

• concentrated forces and moments at nodes,

• loads on element edges,

• loads on element surfaces,

• inertia forces due to the mass of elements.

A concentrated force or moment is defined by the components with respect tothe inertial coordinate system as a function of time.

Components of edge loads with respect to the inertial coordinate system canbe specified for a number of nodes. The direction of the load does not changeduring the simulation. A set of nodes is transferred into a collection of pairs(1, 2, 3 → 1 − 2, 2 − 3), defining edges. The edge load is multiplied by the dis-tance between each pair of nodes. The resulting force for each pair is equallydistributed over the two nodes. If only one node is specified, an error mes-sage is generated. Edge forces can be specified for all element types. Edgemoments can only be specified for the SHELL6 element.

A surface pressure can be specified for membrane and shell elements. Thepressure is perpendicular to the element; a positive pressure corresponds witha pressure in the local element zeta-direction.

The components of acceleration fields with respect to the inertial coordinatesystem can be specified for all element types. The acceleration is multiplied bythe mass of the element and the resulting force is distributed over the nodesof the element using the mass distribution factors.

5.9 Linear constraint equations

Linear constraints are commonly used in finite element analysis to representvarious boundary conditions, such as skewed supports, rigid links and for

Release 7.7 139

5


ensuring interelement continuity between different order elements. Linearconstraint equations can be written as:

nj

∑i=1

cjiui = 0 j = 1, 2, . . . , neq (5.164)

where ui are nodal displacement components and cji are user defined coeffi-cients. This form can be rewritten as

ue = Lur (5.165)

where ue is a vector of eliminated degrees of freedom and ur is a vector of re-tained degrees of freedom. The components of ue must be unique. In addition,vectors ue and ur must have distinct (non overlapping) components. Matrix Lcontains scaling factors.

When the nodal damping is omitted, the remaining set of equations for thetotal set of constrained degrees of freedom becomes

[LTMeL + Mr

]ur = LT f

e+ f

r(5.166)

The matrix at the left-hand side is generally a full symmetric matrix. Thismeans an implicit solution procedure for the constrained degrees of freedommust be used.

The kinematic contact model (Section 9.2) should not be applied for contactbetween nodes used in linear constraints, and planes, ellipsoids or cylinders.

5.10 Spotwelds

A node-node spotweld and a three-node spotweld define a rigid connectionNode-node and3-nodespotwelds

between two and three spot-welded nodes, respectively. Connected elementsare attached rigidly to such a spotweld. Consequently these spotwelds trans-fer forces (when the element has translational degrees of freedom) and mo-ments (when the element has rotational degrees of freedom).

The distance between the two or three nodes of a spotweld must be larger thanzero. In addition, nodes in node-node spotwelds cannot be used in supports,rigid elements, constraint equations, kinematic ellipsoid/cylinder/plane-node contact, prescribed motion and prescribed structural motion.

Spotweld failure occurs if the following criterion is violated during an accu-mulated time interval:

[FN

FNmax

]A

+

[FS

FSmax

]B

< 1 (5.167)

140 Release 7.7


5

where FN , FNmax , FS , and FSmax denote the spotweld normal force, the max-imum allowable normal force, the spotweld shear force and the maximumallowable shear force, respectively. The coefficients A and B determine theshape of the rupture criterion. The normal direction of the spotweld is de-termined by the line between the two nodes (or two lines for three-nodespotwelds, each starting at the middle node). A user-specified time windowdetermines the time required for violation of the failure criterion before initi-ating failure. If the time window has not been specified, failure is initiated assoon as the criterion has been violated for the first time.

5.11 Joints

Certain parts of FE models can be made rigid through the use of rigid ele-ments, rigid materials or through supported nodes. These rigid parts can beconnected by joints and take advantage of other Multi-Body functionality likerestraints, sensors, actuators, etc.

To make use of this functionality, coordinate system objects and/or point ob-jects have to be created for these rigid FE parts. The definition of these objectsis done through nodes; please note that all the nodes in an object have to berelated to the same rigid element, rigid material or support for which the ob-ject is created. MADYMO will internally create an MB body on these relatedparts. All the related nodes will be internally supported on that MB body.The mass properties of the rigid elements, rigid materials or supports will betransferred to the MB body. Note that since these MB bodies are integratedwith the multi-body time step, this time step should be chosen equal to the FEtime step if the resulting rigid bodies have small mass properties.

The definition of joints between FE objects is different to joint definitions be-tween MB objects. Joints in MB objects position the MB objects. This is notthe case for FE objects. FE objects are positioned by the user as normal andthe joints are defined between these FE objects. The parent FE object of a jointis defined with a positioning node and a coordinate system which is definedthrough nodes (see Figure 5.35).

Release 7.7 141

5


x

y

z

1

3

4

2

x

y

z

Figure 5.35: Coordinate system defined with nodes for a FE object.

Again: all these nodes should be related to the same rigid element/rigid ma-terial/support. The child FE object of a joint is defined as a positioning nodeonly. This node identifies the part that will be connected to the parent. Thecoordinate system of the child object does not need to be specified as it will becalculated from the parent object and the initial joint degrees of freedom. Thechild position node also specifies the approximate position of the child jointcoordinate system, however as the actual position is calculated by MADYMO,the user may specify a tolerance (or gap) that the created child coordinate sys-tem must lie within. MADYMO will check the user defined position (in thechild FE object node) with the calculated child position point. If the user de-fined position lies within the tolerance MADYMO will move the point to thecalculated point. The position of the specified node itself will not be moved,only the joint coordinate system.

For configuration tree branches with MB objects, a branch must always startwith a joint connected to the reference space which in fact positions the branch(Figure 5.36). For branches with FE objects, the branch does not start with ajoint connected to the reference space since the FE objects are positioned bythe nodes (Figure 5.37).

142 Release 7.7


5

MBMB

MB

MB

Figure 5.36: A MB branch of bodies and joints always start with the reference space.

FEFE

FE

FE

rigid

rigid

Figure 5.37: A FE branch of bodies and joints does not start with the reference space.Note that joints can only be connected to rigid parts of the FE model. Note that forevery joint the left body is the parent and the right body is the child.

For example, a joint can be defined between rigid elements and no other jointsare needed. Joints may not have an MB parent object and an FE child ob-ject. Conversely joints may have an FE parent object and an MB child object.Therefore, in a mixed FE-MB configuration the FE object must always be theparent. Finally, the bodies and related surfaces used in a branch that containsFE objects cannot be used in any FE model for supports, contact, jets, etc. (seeFigure 5.38, 5.39 and 5.40).

Release 7.7 143

5


FEFE

FEMB

MB

MB

Figure 5.38: A parent FE object can have a MB child object in a joint. Note that forevery joint the left body is the parent and the right body is the child.

MB FEMB

FEMBFE

Figure 5.39: A parent MB object cannot have a FE child object in a joint. Note that forevery joint the left body is the parent and the right body is the child.

144 Release 7.7


5

FEFE

FEMB

MB support

FE

MB

Figure 5.40: MB objects in a child chain of FE parent objects cannot be used in supports,contact, etc of any FE model. Note that for every joint the left body is the parent andthe right body is the child.

Release 7.7 145

5


146 Release 7.7

MADYMO Theory Manual Acceleration field model

6

6 Acceleration field model

This force model calculates the forces at the centres of gravity in bodies with anuniform acceleration field a(t) (Figure 6.1). The acceleration field is defined asa function of time by means of function pairs (Appendix A). The componentsof a(t) must be expressed relative to the reference space coordinate system(X, Y, Z). An acceleration field does not need to be applied to all bodies or allsystems.

Figure 6.1: A system of bodies in an uniform acceleration field.

The positive Z-axis of the reference space coordinate system usually selectedis opposite of the acceleration due to gravity. Then, gravity is described by anegative acceleration-time function in Z-direction (Figure 6.2).

Release 7.7 147

6

Acceleration field model MADYMO Theory Manual

Figure 6.2: Description of a gravity field against a rigid barrier.

The acceleration forces on a vehicle occupant during an impact can be simu-Accelerationforces lated using this model. As an example, the impact of a vehicle against a rigid

barrier is considered here.

One way to model this test is to describe the motion of the vehicle. If thevehicle is represented by one body, the displacement and orientation of thisbody must be prescribed. These can be obtained from film data or from thedouble integration of the acceleration signals. The vehicle can be modelledas a body with a prescribed acceleration. Because the relative motion of theoccupant to the vehicle is most relevant, an alternative method is to considerthe vehicle deceleration as a fictitious acceleration field on the occupant (seeFigure 6.3). This is only possible if vehicle rotation can be eliminated. Thevehicle can then be fixed to the reference space.

148 Release 7.7

MADYMO Theory Manual Acceleration field model

6

X

Z

Y

time (s)

measured vehicleacceleration

a(m/s2)

acceleration field

time (s)

fictitious accelerationfield in MADYMO

ax(m/s2)

laboratory + MADYMOinertial coordinate system

Figure 6.3: Simulation of vehicle deceleration by means of an acceleration field.

A deceleration measured at the vehicle is prescribed as a fictitious accelerationNotefield on the occupant.

If a fictitious acceleration field is prescribed, the calculated acceleration mustbe corrected for this acceleration field in order to receive the correct values.Consider a one-body system located in an impacting vehicle. In the simula-tion, this vehicle is considered to be connected to the reference space. A fic-titious acceleration field based on the vehicle deceleration pulse is prescribedon the one-body system. The resulting acceleration in the model for the one-body system is identical to the vehicle deceleration as long as no other forcesare acting on the body. By subtracting the prescribed acceleration field, thecorrect acceleration is obtained.

Release 7.7 149

6

Acceleration field model MADYMO Theory Manual

150 Release 7.7

MADYMO Theory Manual Restraints

7

7 Restraints

Several types of restraints and the associated features will be described in thisTypes ofrestraints chapter:

• Kelvin,

• Maxwell,

• Point,

• Joint,

• Cardan and flexion-torsion.

7.1 Kelvin restraints

This restraint calculates the forces produced by a spring parallel with adamper (Figure 7.1). A Kelvin restraint is a massless, uni-axial element with-out bending or torsion stiffness. The two ends can be attached to arbitrarypoints on any two bodies. These bodies may be in the same system or in dif-ferent systems. A Kelvin restraint may also be attached to the reference space.The spring and damper forces act on the bodies at the Kelvin restraint’s at-tachment points.

Figure 7.1: A Kelvin restraint.

The following features are available:Kelvin restraintfeatures

Release 7.7 151

7

Restraints MADYMO Theory Manual

• Non-linear force-(relative) elongation characteristics of the spring. Thespring force must be given as a function of either the relative elonga-tion or the elongation of the spring and is defined in tabular form (SeeAppendix A). A positive force corresponds to tension and a negativeforce to compression.

• Hysteresis can be defined using the hysteresis models (See Appendix B).Separate characteristics for loading and unloading are entered.

• A dynamic amplification factor can be defined to account for the de-pendency of the force-(relative) elongation characteristic on the rate ofdeformation (See Appendix C).

• Damping is specified using a constant viscous damping coefficient Cd

and a damping force function Fd(v) . The damping force is calculatedfrom:

Fd = Cd · v + Fd(v) (7.1)

where v is the rate of change in the distance between the attachmentpoints over time. A positive coefficient Cd results in resistive damping.The damping force can be set to zero if the spring force is equal to zero.

• Initial strain or initial length (to account for slack or pre-tension). Theinitial state can be defined by the untensioned spring length L0 or theinitial strain dL(t0) which is defined as:

dL(t0) =L(t0) − L0

L0(7.2)

where L(t0) is the distance between the attachment points at the start ofthe simulation. L0 is calculated by the program from the factor dL(t0) .A negative value of dL(t0) represents compression and a positive valuetension. Note that dL(t0) > −1 .

The relative elongation follows from:

dL(t) =L(t) − L0

L0(7.3)

where L(t) is the distance between the attachment points at time t.

7.2 Maxwell restraints

This restraint calculates the forces produced by a spring in series with adamper (Figure 7.2). A Maxwell restraint is a massless, uni-axial elementwithout bending or torsion stiffness. The two ends can be attached to arbi-trary points of any two bodies. These bodies may be in the same system orin different systems. A Maxwell restraint may also be attached to a refer-ence space. The spring and damper forces act on the bodies at the attachmentpoints of the Maxwell restraint.

152 Release 7.7


7

Figure 7.2: A Maxwell restraint.

The following features are available:Maxwellrestraintfeatures • Non-linear force-relative elongation characteristics of the spring. The

spring characteristic is defined in tabular form (See Appendix A). A pos-itive force corresponds to tension and a negative force to compression.

• Hysteresis can be defined using the hysteresis models (See Appendix B).Separate characteristics for loading and unloading are entered.

• Damping is defined by the damping force as a function of the velocityat which the damper ends move apart. The damping force is defined by:

Fd = Fd(v) (7.4)

The damping force function must be strictly increasing.

• The untensioned length and the initial length of the spring must bespecified. They define the initial state of the Maxwell restraint.

7.3 Point restraints

This restraint calculates elastic and damping forces on a fixed point P (at-tached to a body j) as a function of its position relative to the point re-straint coordinate system (xp, yp, zp), as well as the reaction forces on bodyi (Figure 7.3). This coordinate system can be located arbitrarily within anybody i or within the reference space. The orientation of this coordinate sys-tem is specified using the standard methods available in MADYMO (SeeAppendix D).

Release 7.7 153

7


Figure 7.3: A point restraint.

A point restraint is a combination of three orthogonal Kelvin restraints withconstant damping coefficients parallel to the coordinate axes xp , yp and zp ,respectively. At one end the Kelvin restraints are connected to point P and atthe other end to slider joints in three orthogonal planes parallel to the pointrestraint coordinate system (Figure 7.4).

154 Release 7.7


7

Figure 7.4: A point restraint is a combination of three Kelvin restraints.

The following features are available:Point restraintfeatures

The elastic, hysteresis and damping properties can be specified independentlyfor each of the coordinate axis directions.

• The elastic spring properties are defined as force-displacement charac-teristics. The displacements are defined as the coordinates of P in thepoint restraint coordinate system. A spring force on point P in the corre-sponding negative point restraint coordinate axis direction is a positiveforce.

• Hysteresis is defined using the hysteresis models available inMADYMO (See Appendix B).

• The dependency of the force-displacement characteristic on the rate ofdeformation can be accounted for by a dynamic amplification factor(See Appendix C). If the dependency differs for different directions, twoor three point restraints can be super-imposed on each other.

• Damping is defined using a viscous damping coefficient and a dampingforce function similar to the damping in a Kelvin restraint.

Release 7.7 155

7


MADYMO can calculate the location of the origin of the point restraint coor-dinate system such that it initially coincides with the restrained point P. Thisis a useful option if the point restraint is used to restrain the distance betweentwo points of different bodies (for example to model a closed chain).

The model of a flexible steering column shown in Figure 7.5 is an applicationSteering wheeland column of point restraints. The steering wheel and steering column are represented

by a system made up of two bodies. The steering column is connected to thevehicle with two point restraints. For each point restraint, a separate pointrestraint coordinate system is defined in the vehicle. The deformation charac-teristics of the steering column in the axial direction can be assigned to one ofthe point restraints.

Figure 7.5: Application of point restraints in a flexible steering column.

7.4 Joint restraints

Most types of kinematic joints have a corresponding joint restraint. It defineselastic, damping and friction loads which depend on the relative motion in thekinematic joint. For all kinematic joints, except spherical and free joints, theloads are an explicit function of the joint degrees of freedom. The load is eithera force or a torque depending on whether the joint degree of freedom repre-sents a translation or a rotation, respectively. Spherical joints and free jointsare treated differently because the joint degrees of freedom that are used fordefining the relative orientation, Euler parameters, are not quantities whichcan be correlated to elastic, damping and friction torques. For these joints, an-gles of rotation are introduced that define the relative orientation of the jointcoordinate systems.

An elastic load is specified as a non-linear function of the corresponding jointElastic loadposition degree of freedom. It is positive when the load tends to reduce the

156 Release 7.7


7

joint degree of freedom. A resistive load corresponds with a strictly increas-ing function. Hysteresis (Appendix B) can also be defined. A dependency ofthe elastic load on the rate of deformation can be taken into account with adynamic amplification factor (Appendix C). The rates of deformation are thejoint velocity degrees of freedom.

Damping is specified by a constant damping coefficient and a damping loadDampingas a function of the corresponding joint velocity degree of freedom. A posi-tive damping coefficient results in a damping load that counteracts the jointvelocity degree of freedom.

Friction is specified by a constant friction load. It counteracts the joint velocityFrictiondegree of freedom. This friction load is multiplied with a ramp function thatdepends on the joint velocity degree of freedom leading to no friction load forsmall values of the joint velocity degree of freedom and the specified frictionload for higher values of the joint velocity degree of freedom.

The remaining part of this section discusses how applied joint loads dependon a joint degree of freedom. Although only a one degree of freedom joint isconsidered, the description applies also to joints with more degrees of freedombecause the applied loads corresponding to different degrees of freedom arenot coupled.

The applied loads calculated by a joint restraint consist of:

• a non-linear elastic load Qe

• a viscous damping load Qd

• a Coulomb friction load Q f

The total load on the parent body is the sum of the elastic, damping and fric-tion loads:

Qt = Qe + Qd + Q f (7.5)

The reaction load is applied to the corresponding child body.

The non-linear elastic load Qe is a function of the joint position degree of free-dom q:

Qe = Qe(q) (7.6)

This is usually a resistive load limiting the motion in the joint. Joint stops canbe modelled in this way. An elastic load Qe(q) is resistive if it is defined as astrictly increasing function of q. Typical examples of resistive elastic joint loadfunctions are shown in Figure 7.6.

Release 7.7 157

7


Qe

q

Qe

q

Figure 7.6: Examples of resistive elastic load functions.

An elastic load does not necessarily have to be resistive. It is also possibleto specify an active load. The load function is then a decreasing function(Figure 7.7).

Optionally, hysteresis can be defined. In that case separate functions for load-ing and unloading must be specified. Details on the hysteresis option aregiven in Appendix B.

Qe

q

Figure 7.7: Example of an active elastic load function.

In the input file, a function is defined by a number of coordinate pairs, seeAppendix A. The data for these functions can be obtained, for example, from

158 Release 7.7


7

experiments or from the literature. The accuracy of the approximation usuallyincreases if more coordinate pairs are selected.

Damping can be specified by either a viscous damping coefficient Cd , or adamping load function Qd of the corresponding joint velocity of degree offreedom q. In the former case, the resulting damping load is computed bymultiplying the damping coefficient with the corresponding joint velocity de-gree of freedom q. The resulting damping load is given by:

Qd = Cd q + Qd(q) (7.7)

The resulting damping load Qd counteracts the relative velocity when it ispositive for positive values of the corresponding joint velocity degree of free-dom.

A constant dry friction (Coulomb friction) load C f can be specified. To avoidvibrations induced by a nearly zero relative velocity, a linear ramp functionis applied which makes the friction load velocity dependent for small relativejoint velocities. Figure 7.8 illustrates a typical set of ramp values. If the ab-solute value of the relative joint velocity is less than ramp1 , no friction loadwill act on the joint. For velocities larger than ramp2 , the full friction loadwill be applied. For velocities between ramp1 and ramp2 , the friction load iscalculated from:

Q f = sign(q)|q| − ramp1

ramp2 − ramp1

C f (7.8)

where ramp1 < |q| < ramp2 .

Qf

ramp 1

Cf

ramp 20

q•

typical ramp values: ramp 1 = 0.1 (rad/s or m/s)

ramp 2 = 1 (rad/s or m/s)

Figure 7.8: Dry friction load with typical ramp values.

Release 7.7 159

7


A friction load counteracts the relative velocity provided that C f is positive. IfC f is negative, the friction load is an active load.

The standard friction model does not describe slip-stick because the rampfunction leads to zero friction when the relative velocity equals zero. For rev-olute joints, translational joints and spherical joints, a special friction model isavailable which includes slip-stick and a dependency of the friction load onthe resultant force transferred by the joint.

When there is a relative velocity in the joint, the friction load Q f is given by

Q f = Qk + fkN (7.9)

where Qk is a constant kinetic friction load, fk is the kinetic friction coefficient,and N is the resultant force transferred by the joint. (Note that the friction coef-ficients for revolute and spherical joints have the dimension of length.) Whenthe relative velocity in the joint changes sign, the joint is locked (stick). Whenthe load required to keep the joint locked exceeds the maximum possible fric-tion load (which is given by the above equation, but now with a constant staticfriction load and a static friction coefficient), the joint is unlocked again.

It is not possible to define this special friction model in joints for which the mo-tion is prescribed (element MOTION.JOINT_POS and MOTION.JOINT_ACC) andjoints in a closed chain.

This special friction model is only applicable for the constructions of the jointtypes for which it is implemented. For example, consider the piston and cylin-der shown in Figure 7.9. The connection between these two bodies can bemodelled as a translational joint if the rotation around the piston axis is pre-vented. Friction may be present in the contact areas between the cylinder andthe piston, and between the cylinder and the piston rod. These friction forcesdepend on the normal forces on the contacting surfaces. However, these nor-mal forces do not only depend on the resultant force, but also on the torquetransferred by this joint. Such a torque is transferred as a couple of contactforces between the surfaces and therefore does not give a contribution to theresultant force transferred by the joint. As a result, the friction force will beinaccurate if there is a significant contribution of the transferred torque.

Figure 7.9: Piston and cylinder.

160 Release 7.7


7

7.5 Cardan and flexion-torsion restraints

Cardan and flexion-torsion restraints apply opposite torques on the connectedobjects that depend on angles that define the relative orientation of restraintcoordinate systems that can be defined for these restraints. A restraint can bedefined between two bodies or between a body and the reference space. In thesequel, the case that a restraint interconnects two bodies is considered only fornotational convenience. Several restraints can be defined between two bodiesproviding the possibility to model a complicated dependency of torques onthe relative orientation of the restraint coordinate systems.

A restraint coordinate system must be defined on each body of the two bod-ies that are connected by a restraint. These coordinate systems are denotedby (x∗i , y∗i , z∗i ) and (x∗j , y∗j , z∗j ), rigidly connected to the parent body i and the

child body j, respectively. The origin of the restraint coordinate systems isnot of interest for rigid bodies. The orientation is defined relative to the lo-cal coordinate systems of each of the bodies to which the coordinate systemsare connected using one of the methods described in Appendix D. A restraintcoordinate system will be parallel to the body local coordinate system if noorientation has been specified.

The difference between the cardan restraint and the flexion-torsion restraintTwo restraintmodels is the angles that are used to define the relative orientation of the restraint

coordinate systems.

For a Cardan restraint, the relative orientation of the restraint coordinate sys-Cardanrestraint model tems is described by means of three successive rotations (Figure 7.10). These

three rotations define the orientation of restraint coordinate system j relativeto restraint coordinate system i.

Release 7.7 161

7


zi* z

j*

xi*

xj*

yj*

yi*

zi*

xi*

ϕ (+) yi*

zi*

xi*

yj'

yi*

θ(+)

zi*

xi* x

j*

yj*

yi*

ψ(+)

zj*

Figure 7.10: Relative orientation of coordinate systems using Bryant angles.

Initially, both coordinate systems are parallel. The first rotation ϕ is carriedout about the x∗i -axis of coordinate system i. The second rotation θ is aboutthe resulting y′j-axis and the third rotation ψ about the z∗j -axis. This set of

rotations is known as Bryant or Cardan angles.

When θ = π/2 , the first rotation axis is parallel to the third rotation axis.Then the first and third rotation angle cannot be discriminated (gimbal lock).In order to avoid this, the second rotation angle θ is limited to the followingrange:

− π

2< θ <

π

2(7.10)

The total torque in the restraint is the sum of the elastic, damping and frictiontorque contributions corresponding to the rotation angles ϕ, θ and ψ:

Mt = M1e(ϕ) +M2e(θ) +M3e(ψ) +

M1d(ϕ) +M2d(θ) +M3d(ψ) +

M1 f +M2 f +M3 f

(7.11)

162 Release 7.7


7

where M1 is a torque vector parallel to the x∗i -axis, M2 is a torque vector par-allel to the floating y′j-axis and M3 is a torque vector parallel to the z∗j -axis.

Hysteresis and a dynamic amplification factor can be defined for the elastictorques.

Application of this restraint model is recommended if the relative rotations areindeed achieved by rotations about one or more physical axes. This appearsto be the case in many joints of the present crash dummies.

The rotations between the restraint coordinate systems as described above canbe interpreted also as follows (see Figure 7.11):

• a first rotation ϕ about an axis fixed to body i;

• a second rotation θ about a floating axis;

• a third rotation ψ is about an axis fixed to body j.

Figure 7.11: Rotation axes in the cardan restraint model.

In order to relate the restraint coordinate systems to the rotation axes, at leastone orientation should exist in which the rotation axes are perpendicular toeach other.

The following guidelines can be given for the selection of the orientation ofSelectingcardanrestraintsystemorientation

the cardan restraint coordinate systems:

• First a reference restraint position should be defined. When two or threephysical rotation axes exist, they must be perpendicular to each other inthis position.

Release 7.7 163

7


• Both restraint coordinate systems are defined in such a way that they areparallel in this reference position and, as a consequence, all three relativerotation angles ϕ, θ and ψ are equal to zero.

• If there are physical rotation axes, the x∗-axes of the restraint coordinatesystems should be selected along the physical rotation axis in body i andthe z∗-axes along the physical rotation axis of body j.

• The rotation axis which corresponds to the highest resistive torqueshould be selected as y∗-axis to avoid gimbal lock.

For a flexion-torsion restraint, the relative orientation is the result of two suc-Flexion-torsionrestraint model cessive rotations of a restraint coordinate system j relative to a restraint coordi-

nate system i (Figure 7.12). Initially, both coordinate systems are parallel. Noelastic torques are allowed in this position. The first rotation is called "bend-ing" and is defined by the bending angle α between the z∗-axes. The secondrotation is called "torsion" defined by the torsion angle β about the z∗j -axis.

zi*

zj*

yi*

xi*

i

β (+)

γ

α

j

Figure 7.12: Flexion-torsion restraint model with flexion angle α, torsion angle β anddirectional dependency angle γ.

A set of unit vectors is defined along the axes of both restraint coordinatesystems. These vectors are indicated by exi , eyi and ezi for the restraint coor-dinate system (x∗i , y∗i , z∗i ) connected to body i and by exj , eyj and ezj for thecoordinate system (x∗j , y∗j , z∗j ) connected to body j.

164 Release 7.7


7

The bending angle α is then given by

α = arccos(ezi · ezj) (7.12)

The angle α is defined in the interval

0 ≤ α ≤ π (7.13)

γ is the angle between the projection of ezj on the x∗i y∗i plane and the x∗i -axis.This angle is defined in the interval

− π ≤ γ ≤ π (7.14)

The unit vector u is defined by

u =ezi × ezj∣∣ezi × ezj

∣∣ (7.15)

In order to define the torsion angle, consider the unit vector exj . Before flexionthis vector coincides with exi . Due to the rotation α this vector is mappedto the intermediate image vector e′xj (Figure 7.13). This intermediate image

vector is given by:

Mte

ezi

ezj

u

exi

eyi

exj

e'xj

α

β

βα

Figure 7.13: Torsion angle β.

e′xj = (exi · u)u + (exi − (exi · u)u) cos α + (u × exi) sin α (7.16)

Release 7.7 165

7


The torsion angle β is defined as the angle between e′xj and exj:

β = arccos(e′xj · exj) (7.17)

The angle β is positive if

(e′xj × exj) · ezj > 0 (7.18)

If α = 0 , the vector u is not defined. In that case the torsion angle is definedas the angle between exj and exi .

Elastic (resistive) torques can be specified which are functions of these rotationangles α and β. In addition, the bending torque can be defined as a functionof the angle γ (Figure 7.12).

The flexion or bending torque in the restraint is defined by (Figure 7.14):Flexion

M f e = −M f e(α)C(γ)u (7.19)

where u is the unit vector perpendicular to the two z∗-axes, a is the bendingangle, and M f e and C are functions of α and γ, respectively.

zi*

zj*

Mfe

eziezj

u

exi

eyi

xi*

yi*

α

γ

Figure 7.14: Flexion-torsion model: the flexion torque is directed along vector u.

The torque-angle function M f e(α) is defined analogous to other elastic jointcharacteristics (Equation (7.6)). Figure 7.15 shows a typical loading function.

166 Release 7.7


7

This function is only defined for:

0 ≤ α ≤ π

and that for α = 0 the corresponding torque should be equal to zero. A resis-tive torque is defined as a strictly increasing function.

Mfe(α)

+ πα

Figure 7.15: Resistive elastic flexion torque – angle function.

The function C(γ) is an optional dimensionless multiplication factor for thetorque-angle function M f e(α) . If not specified in the input file, C(γ) automat-ically defaults to 1.

The function C(γ) must be defined in the interval

− π ≤ γ ≤ π (7.20)

In order to avoid a discontinuity in the torque, the function values of C(γ)should be equal at γ = −π and γ = π . The tabular function for C(γ) can beapproximated by piece-wise linear interpolation or by a cubic spline. If splineinterpolation is selected, the user should in specify the function pairs such thatthe derivatives of the spline functions are equal at γ = −π and γ = π .

Figure 7.16 shows a typical example of a C(γ) function. γ = 0 correspondsto forward flexion (in a neck), γ = −π and γ = π correspond to backwardflexion. In the example, the multiplication factor is equal to 1 for both forwardand backward flexion. For lateral flexion (γ = −π/2 and γ = π/2) thisfactor is 0.5. The example shows the piece-wise linear and the cubic splineapproximation.

Release 7.7 167

7


0

C( )

0.5

1.0

−π π

γ

γ

Figure 7.16: Example of the multiplication factor (γ) function.

The torsion torque is defined by (Figure 7.13)Torsion

Mte = −Mte(β)ezj (7.21)

where β is the torsion angle.

The torque-angle function Mte(β) is defined analogous to other elastic jointcharacteristics (Equation (7.6)). The function is only defined for

− π ≤ β ≤ π (7.22)

For β = 0 the torque value should be equal to zero. A resistive torsion torqueis defined as a strictly increasing function.

The relative angular velocity in a flexion-torsion restraint can be representedDamping andfriction torques by a vector ∆ω. A (positive) viscous damping coefficient can be defined in the

flexion-torsion restraint model. The corresponding (resistive) damping torqueis given by:

Md = −Cd∆ω (7.23)

Finally, a Coulomb friction torque M f with corresponding ramp values canbe defined analogous to the friction torque for the other joint restraint models(see Figure 7.8). This friction torque opposes the relative angular velocity.

This restraint allows, for example, the simulation of the rubber neck and spinecylinders of a dummy. The z∗-axes are then selected along the centre line of thecylinder body. Due to the rotational symmetry of the neck and spine bodiesof many dummies, the bending torque has to be defined only as a function ofthe bending angle α. In the case of non-symmetrical structures, the bendingtorque becomes directionally dependent by means of the angle γ.

168 Release 7.7

MADYMO Theory Manual Belt model

8

8 Belt model

8.1 General Information

Simple belts can be described with Kelvin restraints. For more complicatedbelt systems, the standard belt model is available which describes the realspatial belt geometry. The standard belt model can model belts, belt slip andspecial seat belt components such as a retractor with an optional webbinggrabber, a pretensioner and a load limiter. In MADYMO, the standard beltmodel has fixed attachment points. The hybrid belt model is a combination ofa standard belt model with finite element options in order to achieve the ef-fects of slip in different directions across the dummy model such as occurs inbelt roll-out and submarining. Several examples of the standard belt model’suses are illustrated in Figure 8.1.

Figure 8.1: Examples of applications of the belt model.

Release 7.7 169

8

Belt model MADYMO Theory Manual

8.2 Belt

A belt consists of a chain of straight belt segments interconnected by tyings(Figure 8.2). The ends of a belt segment are called the attachment points. At-tachment points are fixed points on bodies, on the reference space or on FE-nodes. These FE-nodes can also be part of an external FE-model in a coupledsimulation. The model accounts for slip of belt material from one segment toan adjacent segment. At an attachment point, the belt can only slide in thedirection of the belt segment.

segment 1

segment 2segment 3

segment n

b

optional retractor

e

eb b

e

be

Figure 8.2: Basic set-up of a belt.

Initial belt slack (or pre-tension) can be specified. If there is slack in the belt,the forces in all of the belt segments will be zero. The belt stiffness character-istic is defined as a force-relative elongation function. Hysteresis of the beltmaterial can be defined in the belt model. This option accounts for the energydissipation and permanent belt deformation. Hysteresis is defined by meansof the hysteresis models available in MADYMO (See Appendix B).

For each belt segment, a relative elongation at which rupture occurs can bedefined. After rupture occurs in one of the belt segments in the consideredbelt, the entire belt is disabled and belt forces are no longer calculated for thisbelt.

An option is available to model rip stitch (fuse belts). Each belt can be pro-vided with one retractor with an optional webbing grabber (see Section 8.4),one pretensioner (see Section 8.5) and one load limiter (see Section 8.6), allpresent at the same end point of the belt. In addition, a buckle pretensionercan be modelled with standard MADYMO options.

170 Release 7.7


8

A simulation can contain more than one belt. This allows the simulation ofseveral restrained occupants or several independent belts on one occupant.Parts of a belt can be modelled by truss or membrane elements. This is done bytying the end of a belt to a node of a finite element mesh representing part of abelt system and the end of another belt to a node on the other side of the finiteelement mesh (hybrid belt). If finite elements are used for specific segments,the nodes can slide over the dummy surface in an arbitrary direction so thatsubmarining and belt roll-out can be modelled.

The belt model allows slip between two adjacent belt segments. The slipSlipmodel is described in Section 8.3.

To model slip between two adjacent belt segments, the adjacent ends of thetwo belt segments should be connected to the same body or FE-model. Ata slip ring, the two ends are attached to the same point on the body or tothe same FE-node. For attachments to bodies this is generally not the case(Figure 8.3).

An additional belt length LA needs to be entered to account for the belt lengthAdditionalLength L between the two attachment points. For example, in Figure 8.3, an addi-

tional length of 0.5L can be added to the length of segment 2 and another ad-ditional length of 0.5L to the length of segment 3. Also additional belt lengthcan be used to account for the extra length of a belt segment because it is notstraight.

Figure 8.3: Connection between two belt segments.

The slip models depend on a Coulomb friction coefficient . At each belt tyingFrictioncoefficient different friction coefficients can be defined to control the slip of belt material

between the two adjacent belt segments.

If the friction coefficient is zero, slip will take place until the forces in the two

Release 7.7 171

8


segments are the same. If no slip is possible between two belt segments, thebelt must be divided into two separate belts.

Different elastic force-relative elongation characteristics can be specified foreach belt segment. When slip occurs between two adjacent belt segments, iand j, belt material that belongs to segment i moves to segment j and adoptsthe material properties of belt segment j. Belt segments that are part of thesame belt should therefore have identical material properties.

8.3 Slip models

8.3.1 Introduction

No slip calculations are performed and the forces in all belt segments are setto zero when the retractor is not locked, when there is slack in an entire beltsystem or after rupture of a belt segment.

The amount of slip of belt material from a belt segment to a neigbouring beltsegment depends on the difference of the belt forces and the friction force atthe common tying. Two belt slip models are available; the quasi-static belt slipmodel and the dynamic belt slip model. For the quasi-static belt slip model,the slip is based on quasi-static equilibrium: in an iterative process the amountof slip is determined such that the belt forces and the friction forces are inequilibrium. The slip velocity is needed for belt damping and for slip velocitydependent friction. In principle, it can be obtained by differentiating the slipnumerically. However this appears to give a rather noisy slip velocity. Thedynamic belt slip model does not have this problem. For this belt slip model,the resultant of neighbouring belt forces and friction force results in a beltslip acceleration. Time integration of this acceleration yields the belt slip andbelt slip velocity. It can be expected that this belt model will give only moreaccurate results when belt damping is taken into account. The same belt slipmodel will be applied for all tyings of the same belt system.

8.3.2 Belt forces and friction forces

The actual length Li of a belt segment i equals the distance between the at-Belt forcestachment points added to the user specified additional belt length. The un-tensioned belt segment length will be denoted as Lu

i . The initial untensioned

belt length Lu,0i is determined from the initial actual belt length and the spec-

ified initial strain or initial elongation. When slack in the entire belt systemdisappears, the untensioned belt segment lengths are updated to the values ofthe actual belt segment lengths. The untensioned belt segment lengths changedue to slip of belt material from one segment to another and due to the actionof a retractor, a pretensioner, a load limiter and/or a belt fuse.

The strain in a belt segment is given by

ε i =Li

Lui

− 1 (8.1)

172 Release 7.7


8

The strain rate of a belt segment is given by

ε i =Li − Li

Lui

Lui

Lui

(8.2)

where the time derivative of Li is calculated from the velocities of the beltattachment points and the time derivative of Lu

i is calculated from the belt slipvelocities. The "slip" velocity between a belt segment and a retractor/loadlimiter/pretensioner is obtained from a numerical time differentiation.

The belt force is calculated from the specified belt characteristic and dampingcoefficient and/or damping function, i.e.,

Fi = Fei + Fdi (8.3)

in which Fei is the elastic belt force:

Fei = g(ε i) f (ε i) (8.4)

with g(ε i) a strain rate sensitivity scale factor (Cowper-Symonds, Johnson-Cook or a user-defined function of the strain rate) and f is the elastic beltcharacteristic. Scaling using g(ε i) is only applied for positive values of thestrain rate and if a function is specified for g(ε i) it should be a non-decreasingfunction with only positive function values and g(0) = 1 (so no scaling if ε i =0). Fdi is the damping force in belt segment i. The scale factor g(ε i) as wellas the damping force Fdi can be applied only when the dynamic slip model(Section 8.3.4) is used. The belt segment applies two collinear forces with op-posite direction. They work along the belt segment direction.

The damping force in belt segment i can be calculated in different ways, de-pending on material damping switch µ.

µ = 0 or µ = 1

Fdi = γi ε i (8.5)

µ = 0 ⇒ γi = Kd∆tsi where K, d and ∆tsi denote, respectively, the max-imum slope of the defined force-relative elongation characteristic, a user-specified damping constant and the critical belt segment time-step in the un-damped situation:

∆tsi =(Lu

i )√dρ

(8.6)

With ρ the specific mass of the belt. This damping model can be applied to addcritical damping to each belt segment (d = 1.0, independent of its untensionedlength Lu

i ).

Release 7.7 173

8


µ = 1 ⇒ γi = Kd Damping is identical for all belt segments irrespec-tive of length.

µ = 2

Fdi = dε i + fd(ε i) (8.7)

where d is the damping coefficient and fd is a function relating the dampingforce and the relative elongation rate defined in (8.2). This damping functionshould be positive for positive relative elongation rates and negative for neg-ative relative elongation rates. In order to obtain an order of magnitude forthe belt damping coefficient, one may consider the order of magnitude of thecritical damping of a belt segment, i.e.,

O(d) = 2O(Lui )√

ρO(K) (8.8)

where O(Lui ) is the order of magnitude of a belt segment length and O(K) is

the order of magnitude of the slope of a representative tangent line to the beltsegment elastic characteristic.

For all three damping models (µ = 0, 1, 2): When the belt segment elongation isincreasing, the damping force and elastic force are both tensile forces (positive)and the total tensile force in the belt segment is given by

Fi = Fei + Fdi (8.9)

When the belt segment elongation is decreasing, both forces act in oppositedirection and the damping force then counteracts the elastic force. The sum ofthe two forces is forced to result in a tensile force, i.e. a lower bound for thesegment damping force is set equal to minus the segment elastic force, i.e.,

Fdi = max(−Fei, Fdi) (8.10)

if a negative damping force is calculated from (8.7). The sum of the elastic anddamping force thus has a minimum value zero.

The slip algorithm in the belt model is based on the slip between the rim of aFriction forcespulley and a belt (Figure 8.4). The tension forces in the belt segments are F1

and F2 , with F1 representing the largest tension force.

174 Release 7.7


8Figure 8.4: Slip between a belt and a pulley.

If slipping is about to occur and during slip, the friction force W between thepulley and belt is given by

W = F1 − F2 = F2(eβ(µ+ f (t)+g(Fn)+ fv(vs)) − 1) (8.11)

where µ is a constant friction coefficient, f is a friction coefficient-time func-tion, g is a friction coefficient-normal force function, fv is a friction coefficient-slip velocity function, Fn is the component of the belt forces normal to thepulley at the base of vector W in Figure 8.4, vs is the slip velocity and β is theangle of belt contact. The functions g and fv can only be applied when thedynamic slip model is used. The friction coefficients for slip and stick may bedifferent.

8.3.3 Quasi-static belt slip model

Two possible cases can occur: Only a Coulomb friction coefficient in all tyingsof a belt or a static Coulomb friction coefficient in at least one tying of a belt.

For each combination of adjacent belt segments, the resulting slip forceOnly aCoulombfrictioncoefficient Fslip = F1 − F2 − Wd = F1 − F2eβ(µd+ fd(t)) (8.12)

is calculated.

Next, that pair of adjacent belt segments is found where the maximum Fslip

occurs. Belt material slips in the direction of the segment with the highesttension force F1 if

Fslip > 1N (8.13)

Release 7.7 175

8


After this slip step, the expressions for Fslip are calculated again for all pairs ofbelt segments. The following slip step is carried out for the belt pair with thehighest Fslip at that point. The process of material exchange is repeated untilFslip < 1N for all pairs of adjacent belt segments. There is no slip at a specifictying if Fslip < 1N is satisfied for all slip steps. For proper slip functioning,the belt characteristics must be strictly increasing.

A static friction coefficient can be specified for a tying by means of a functionAlso a staticCoulombfrictioncoefficient

of time, a constant or both. The resulting static friction coefficient must alwaysbe larger than the friction coefficient used in the model previously described.

If a static friction coefficient is defined for a tying, it can be either in the slipstate or in the stick state. The state of all tyings for which static friction isdefined is initialized to be stick at the start of the simulation. Tyings for whichno static friction coefficient is defined can only be in the slip state. For tyingswhich are in the slip state, the slip process is exactly the same as describedfor the previous slip model. The other tyings are skipped during the belt slipprocess, i.e., there is no slip.

After the belt slip process, it is checked if the state of any tying has to bechanged. If the state of a tying is stick, it can only switch to slip at time if thefollowing condition is satisfied:

Fslip = F1 − F2 −Ws = F1 − F2eβ(µs+ fs(t))> 1N (8.14)

in which Ws is the maximum static friction force, µs is the constant static fric-tion coefficient, fs(t) is the static friction coefficient-time function and Fslip ,F1 , F2 and β as in (8.11) and (8.12).

If the state of any other tying is slip, it is checked if it should switch from theslip to the stick state. This is done when either the slip velocity at a tying inthe slip state has remained zero during a user-specified time interval or whenit has changed sign.

8.3.4 Dynamic slip model

If a retractor and/or load limiter is present in the belt system, first belt materialexchange is carried out between retractor/load limiter and first belt segment,to get a force balance between retractor/load limiter and first belt segment. Inthis iteration process damping in the first belt segment is taken into account.Once there is a force balance between retractor/load limiter and first belt seg-ment, the resulting belt force in the first belt segment is used in the calculationof the slip acceleration at the neighbouring tying. The dynamic slip modelmay result in oscillatory slip when there is not sufficient dissipation in the beltmodel. This may be suppressed by applying sufficient friction and dampingfor the belt model.

For the dynamic slip model, two cases must be considered for every tying:stick at a tying and slip at a tying.

The slip model is in the stick phase with slip velocity vs = 0 initially, whenStick at a tyingthe slip velocity is zero and when the slip velocity reverses direction. The

176 Release 7.7


8

magnitude of the maximum friction force between the belt and the tying iscalculated from

Wmax = F2(eβ(µs+ fs(t)+gs(Fn)+ fv(0)) − 1) (8.15)

where F2 is the smaller of the belt forces in the two connected belt segmentsand where the static friction coefficients are used. Fn, the component of thebelt forces normal to the pulley at the base of vector W, is calculated as:

Fn = (F1 + F2)cos(π/2 − β/2) = (F1 + F2)sin(β/2) (8.16)

There is stick as long as the maximum friction force

Wmax > F1 − F2 (8.17)

with given by (8.15). The actual friction force is then equal to

W = F1 − F2 (8.18)

Thus during stick

W = min(Wmax, F1 − F2) (8.19)

where is in the direction of F2 . When F1 − F2 > Wmax stick is switched to slip.

The model is in slip phase when it is switched from stick to slip or when theSlip at a tyingslip velocity is non-zero. The friction force equals

W = F2(eβ(µd+ fd(t)+gd(Fn))+ fv(vs)) − 1) (8.20)

The dynamic and static friction coefficients must satisfy:

µd + fd(t) + gd(Fn) ≤ µs + fs(t) + gs(Fn) (8.21)

The direction of the friction force is opposite to the belt slip velocity vs . Theslip acceleration equals

as =1

mb

(±(F1 − F2) −Wsgn(vs) −

1

2ρvs(v−s − v+

s )

)(8.22)

In ±, the + (−) sign applies when the segment with the larger (smaller) beltsegment force is closer to POINT_REF_1. The last term in (8.22) accounts forthe change of the mass corresponding to the tying, where v−s and v+

s are theslip velocities at the neighbouring tyings. The belt segment mass is given by

mb =1

2ρ(Lu

1 + Lu2 ) (8.23)

Release 7.7 177

8


where ρ is the belt mass per unit undeformed length.

The slip acceleration is integrated twice, yielding the slip velocity vs and theamount of slipped belt material ss since the start of the simulation. With ss ,the untensioned belt segments are updated according to

Lu1 = Lu,0

1 + (±ss) and Lu2 = Lu,0

2 − (±ss) (8.24)

where Lu,0i is the initial untensioned length of belt segment i.

When the direction of the slip velocity reverses, slip is switched to stick.

8.4 Retractor model

If a retractor is modelled, it must be located at one of the ends of the chain ofbelt segments. If the pretensioner and/or load limiter model are also used forthe current belt, they must both be located at the same location as the retractor.The retractor model features include reel locking and film spool effect. Oncea retractor is locked, it will remain locked for the remainder of the simulation.The retractor reel is either vehicle sensitive or webbing sensitive.

Different conditions can cause this type of reel to lock by modelling the properVehiclesensitive reel switch, for example:

• at a specified time (modelled by a time switch);

• if the acceleration signal of a specified acceleration field exceeds a pre-scribed level for a minimum duration. This option can be used if theretractor is connected to the reference space and the vehicle decelerationis modelled as an acceleration field acting on the occupant(s);

• if the norm of the projection ap of the calculated linear acceleration a ofthe retractor on the user-defined plane exceeds a prescribed level for aminimum duration. The plane is defined by its normal (N1, N2, N3).This is illustrated in Figure 8.5. This type of locking can be used in asimulation with a moving vehicle. The desired projected accelerationsignal can be modelled by a body sensor, and selecting the appropriatesignal type (PLAN_ACC).

178 Release 7.7


8

n

O

a

ap

Figure 8.5: The sensitivity of the retractor (the location of the retractor is O).

This type of reel locks if the belt feed rate (the relative belt velocity) exceeds aWebbingsensitive reel specified limit.

In a belt equipped with a retractor:Retractor

• pretension in any segment of the belt is removed;

• slack in the adjacent segment is removed.

Force calculations are carried out only if there is no net slack in the belt andthe retractor is locked. Additionally, a pretensioner and/or load limiter canonly start working after the retractor is locked.

Before the locking mechanism of a retractor is activated, a free supply of beltmaterial is provided by the retractor such that pretension in all segments orslack in the segment adjacent to the retractor is removed. As a result, theuntensioned belt length of the belt segment attached to the retractor is equalto the distance between the attachment points at the time of locking.

After locking, the retractor can still supply some belt material due to thetightening of the belt material on the reel, or the film spool effect. The usercan define a force-belt outlet characteristic including hysteresis as shown inFigure 8.6. This characteristic must be non-decreasing. If also an active loadlimiter is present in the same belt, the retractor characteristic must be an in-creasing characteristic. For a retractor, the film spool effect will remain activeuntil a webbing grabber is activated.

Release 7.7 179

8


retractorforce(N)

(m)belt outlet of retractor

(after locking)

Figure 8.6: Example of film spool characteristic.

8.5 Pretensioner models

Two different pretensioner models are available. The pretensioner is activatedand de-activated by a switch. Activating the pretensioner will not cause aretractor in the same belt to be locked. However, the pretensioner will notwork during the period that there is a retractor which is still unlocked andwhich is not disabled by a webbing grabber.

This type of pretensioner winds in the belt material as a prescribed function ofPretensionermodel workingon a payin-timefunction

time. This is specified as the amount of untensioned belt length as a functionof the time relative to the activation of the pretensioner. This must be a mono-tonically increasing function that ends in a constant value, the total belt lengthtaken in by the pretensioner. The last constant part of this function must bespecified because extrapolation might otherwise lead to a further retraction ofthe belt material.

This type of pretensioner takes in belt material as a function of the momentPretensionermodel workingon aforce-payoutfunction

on the pretensioner spool. This moment equals the spool radius multiplied bythe sum of the force in the attached belt segment and the force derived fromthe user-specified force-payout function. It results in an angular accelerationof the spool. The corresponding rotation of the spool leads to a payout (payin)of belt material. Increase of the spool radius due to the thickness of the beltmaterial which winds around the spool is not taken into account in this model.The user specifies a force-payout function to be compatible with e.g. the way aretractor characteristic is specified. To define the pretensioner force as a func-tion of belt payin, a positive force for negative belt payout should be specified

180 Release 7.7


8

(the function is specified in the second quadrant as shown in Figure 8.7)

0.0−0.5−1.0

1000

3000

2000

Belt payout (m)

PretensionerForce (N)

Figure 8.7: Example of a pretensioner force-payout function.

For both types of pretensioners there are two cases in which the amount of beltmaterial wound in by the pretensioner in one multibody integration time stepcan not exceed 90% of the untensioned length of the belt segment attached tothe pretensioner. These are:

• if there is a node of a finite element model attached to the belt

• if the belt only consists of one segment and there is no load limiter andno retractor or the webbing grabber is active.

A buckle pretensioner can be modelled in MADYMO as a separate body andNotea pretensioned spring.

8.6 Load Limiter

If a load limiter is modelled, it must be located at one of the ends of the chainof belt segments. If the pretensioner and/or retractor model are also used forthe current belt, they must both be located at the same location as the loadlimiter. A load limiter limits the load level in the connected belt segment orretractor by giving out belt material. A load limiter is activated when at leastone of its load levels is active. The switch specified for that specific load levelhandles activation and de-activation of a load level. If more than one loadlevel is active at a time, the minimum of the active load levels is used.

The load limiter will not work during the period that there is a retractor whichis still unlocked and which is not disabled by a webbing grabber.

Release 7.7 181

8


If at least one of the load levels of the load limiter has been activated, the forceon the load limiter (belt force or retractor force) must exceed the minimumactive load level of the load limiter. When this level is reached the load limiterwill start to give out belt material.

The transition from one active load level to the next will follow a user-specified slope. If the transition is from a higher level to a lower level, thesign of the slope will be reversed.

To realise a transition from a lower load level to a higher one, the switch of thelower load level must de-activate the lower load level, while the higher loadlevel becomes active simultaneously or is active already.

When belt payout decreases, unloading will take place along a user-definedhysteresis slope, until the zero force level is reached. Further unloading willresult in a belt segment force equal to zero. Subsequent reloading will followthe same path in opposite direction.

8.7 Fuse belts

The fuse belt option can be used to model the tearing of belt stitches. Severalfuse belts can be defined for one belt segment.

Tearing takes place in n steps (equal to the number of stitches). Each sepa-Tearing of beltstitches rate tearing step in the fuse belt requires the belt force in the belt segment to

exceed the specified tearing force after belt slip has been taken into account.After each tearing step, the total extra untensioned belt length divided by n isadded to the untensioned length of the belt segment. Within one integrationtime step, several tearing steps can take place within the same fuse belt. Ineach tearing step, untensioned belt material is added to the belt segment andthe belt forces at this time point are recalculated, taking belt slip into account.If the tearing force is still exceeded with the new untensioned belt length, thenext stitch will tear. If the tearing force is not exceeded any longer, the pro-gram proceeds to the next time point and the same process is repeated untilall tearing steps are completed.

More than one fuse belt can be defined for a belt segment. The order in whichthese fuse belts should tear can be defined.

If belt rupture occurs before the stitches of a fuse belt within the same beltstart tearing, the entire belt is disabled and the stitches will not tear apart.

8.8 Hybrid belt

Belt parts can be modelled with truss or membrane elements in order to beable to model complex behaviour such as multi-directional belt slip, sub-marining and roll-out (Figure 8.8). A belt can be tied to a finite element beltsegment. All of the belt model options can be used, including retractors, pre-tensioners and load limiters. The finite element belt segment can slide over thedummy surfaces subjected to Coulomb friction. Sub-cycling of the finite ele-ment time integration with respect to the multibody time integration (where

182 Release 7.7


8

the multibody time step is a multiple of the finite element time step) is imple-mented in order to reduce CPU time.

Time: 0. ms

Figure 8.8: A hybrid belt system using membrane elements.

A belt modelled with both finite elements and the standard belt model shouldsatisfy the following condition:

• A finite element belt segment should not be tied to the end of a beltsegment when this end is tied to a retractor, pretensioner or load limiter.

If a part of a belt is modelled with finite elements, the position of the initialmesh on the dummy can be obtained with the Belt Fitting Wizard of XMADgicor by a pre-simulation. In such a pre-simulation, the end points of the FEMbelt are connected to bodies with prescribed motion that are moved to thecorrect position. To prevent dummy motions due to the contact forces, alljoints can be locked and the nodal degrees of freedom of flexible bodies canbe fixed during the pre-simulation. The nodal positions in MADYMO inputformat are obtained from the FEMESH file. The nodal positions at a time pointwith the most suitable mesh to be used as the initial mesh should be includedin the input file for the actual analyses.

Release 7.7 183

8


Figure 8.9: Examples of combinations of finite element belt segments and standard beltsegments.

For modelling the effects of belt penetration (pocketing) on friction, or-thotropic friction and penetration dependent friction are available. Or-thotropic friction provides separate friction coefficients for longitudinal andlateral friction. Both types of friction can be found under the CON-TACT.FE_FE and CONTACT.MB_FE options (see the Reference Manual).

184 Release 7.7

MADYMO Theory Manual Contact interaction models

9

9 Contact interaction models

Contact interaction can be defined by a master surface against a slave surface. Inthese contacts, planes, (hyper-)ellipsoids, (hyper-)elliptical cylinders and finite ele-ment models can be specified. A possible contact between two surfaces is only evalu-ated when the user has specified that the two surfaces can be in contact.

There are two contact models available: the elastic and the kinematic contactmodels (Table 9.1).

This model is available for all possible contacts, and the contacting surfacesElastic contactmodel can penetrate each other. The corresponding elastic contact force depends on

the penetration. This dependency must be specified as a force-penetrationcharacteristic, a stress-penetration characteristic or a penalty factor.

This model does not allow the contacting surfaces to penetrate each other.Kinematiccontact model This model is only available for the contact between nodes of a finite element

model and ellipsoids, cylinders and planes. For this model, the contact forceis based on an inelastic impact of the node with the contact surface.

Contacts are evaluated independently. This means that when a surface is be-hind or inside another surface, it can be still contacted through that other sur-face.

Table 9.1: Overview of available contacts. Elastic contacts are denoted by "e" (withpenalty contacts as a special case denoted by "p"), kinematic contacts aredenoted by "k".

ellipsoid node

plane e e, k

cylinder e e, k

ellipsoid e e, k

element e, p

9.1 Contact between ellipsoids, cylinders and planes

In these contacts, the slave surface is defined as ellipsoids and the mastersurface is defined as planes, cylinders and ellipsoids. Only the elastic contactmodel can be applied. The resulting contact force is a point force that is madeup of elastic, damping and friction parts.

The elastic contact force (including hysteresis) is a user-defined func-tion of the surface penetration. The penetration corresponding to plane-ellipsoid, ellipsoid-ellipsoid and cylinder-ellipsoid contacts will be explainedon page 186, page 188 and page 190 respectively.

Release 7.7 185

9

Contact interaction models MADYMO Theory Manual

A force-penetration characteristic can be defined for each surface separatelyor for a specific contact interaction.

A dynamic amplification factor (see Appendix C) can account for the depen-dency of the elastic force on the rate of penetration.

The damping and friction forces depend on the relative velocity of the con-tacting surfaces.

The penetration (λ in Figure 9.1) is the distance between the contact plane,Penetration inplane -ellipsoidcontacts

and a plane that is parallel to the contact plane and tangent to the ellipsoid atthe penetrated side of the ellipsoid. P1 , the tangent point on the ellipsoid, isthe contact point on the ellipsoid. P2 , the projection of P1 on the contact plane,is the contact point on the contact plane.

The elastic force is perpendicular to the contact plane. The example inFigure 9.1 shows the elastic contact force on the body to which the ellipsoid isattached. (The reaction force on the plane is not shown.)

ellipse (ellipsoid)

elastic forceon ellipse(ellipsoid)

P2

P

plane

1

P1

Fe

2

λ

Figure 9.1: Penetration in ellipsoid-plane contact with elastic contact load.

The size of a plane for contact analysis is determined by the specified planeBoundarycontact area dimensions and the boundary width w (BOUNDARY_WIDTH). For w ≥ 0 , the

specified plane is extended with a boundary of width w (Figure 9.2). Theboundary on the plane’s edge with a width of 2 · w is called the boundarycontact area. If the contact point on the plane lies within the boundary contactarea, the contact force (the sum of elastic, damping and friction force) is mul-tiplied by a correction factor. This factor varies between 0 and 1. Figure 9.3illustrates this for a two-dimensional section of the plane. The contact force isset to zero when the contact point on the plane is outside the boundary contactarea (Figure 9.4). For w = ∞ , the plane is infinite and the contact force will beapplied also when the contact point is outside the specified plane.

186 Release 7.7


9

FIN FIN

MADYMO 3D

A B

CD

Boundary contact area

Figure 9.2: The boundary contact area.

Figure 9.3: Correction factor for contact loads.

Release 7.7 187

9


P1

PP2plane

FIN FIN

ellipse (ellipsoid)

Figure 9.4: Point of application of contact force outside plane.

Ellipsoids with a degree larger than 10 are treated as 10th degree ellipsoidsPenetration inellipsoid -ellipsoidcontacts

for these contacts. The definition of the penetration is illustrated in Figure 9.5.Let l1 be a plane tangent to ellipsoid 1. The plane tangent to ellipsoid 2 andparallel to l1 is denoted by l2 . The distance between l1 and l2 is denotedby λ. The penetration between both ellipsoids is defined as the minimum ofdistance λ. An iteration process is used to find this penetration by modifyingthe tangent planes. P1 is the contact point on ellipsoid 1, P2 is the contact pointon ellipsoid 2.

P1P

P2

ellipse 1(ellipsoid 1)

ellipse 2(ellipsoid 2) I1

I2

λ

Figure 9.5: Penetration in ellipsoid-ellipsoid contacts.

The elastic force is perpendicular to the tangent planes. The point of appli-cation P of the contact force depends on how the contact stiffnesses of theellipsoids have been specified (page 192). The four cases are the same as those

188 Release 7.7


9

of the plane-ellipsoid contact, except that now two ellipsoids are interacting.It can occur that the contact points P1 and P2 do not lie within the intersectionof the ellipsoids. The contact force is then set to zero.

In the example shown in Figure 9.6, where point P is located on ellipsoid 1,the contact forces acting on the body to which ellipsoid 1 is connected areillustrated. (The reaction load on the other body is not shown.)

Fe

λ

ellipse (ellipsoid) 1

point ofapplicationof forces

ellipse (ellipsoid) 2

Fd

FfP

Figure 9.6: Ellipsoid-ellipsoid contact (only forces acting on ellipsoid 1 are shown).

The iteration process for the penetration calculation finds a minimum for thedistance λ. If the penetration is large, the program may iterate to a minimumthat is not the global minimum (Figure 9.7). The penetration algorithm deter-mines only one minimum based on initial values obtained in the precedingtime step. At the start of the simulation or if there is no penetration in thepreceding time step, the initial value for the tangent planes is chosen perpen-dicular to the line connecting the ellipsoid centres.

Release 7.7 189

9


λ1

λ2

Figure 9.7: Situation with two local minima for the penetration λ.

A transition from one local minimum to another may occur in some casescausing a sudden change in the magnitude and direction of the contact force.This undesirable behaviour can be avoided by defining the ellipsoids suffi-ciently large enough in comparison with the expected penetrations.

Ellipsoids and elliptical cylinders with a degree larger than 10 are treated asPenetration incylinder -ellipsoidcontacts

10th degree ellipsoids and elliptical cylinders for these contacts. The definitionof the penetration is illustrated in Figure 9.8. Let l1 be a plane tangent to theelliptical cylinder. The plane tangent to the (hyper-)ellipsoid and parallel tol1 is denoted by l2 . The distance between l1 and l2 is denoted by λ. Thepenetration of the cylinder and the ellipsoid is defined as the minimum of thedistance λ. An iteration process is used to find this penetration by modifyingthe tangent planes.

190 Release 7.7


9

elliptical cylinder

hyper-ellipsoidP

P

l

l

1

2

2

1

λ

Figure 9.8: Penetration in cylinder-ellipsoid contacts.

Only tangent planes parallel to the cylinder’s centre line are considered. Asa result, contact with the end faces is not considered. If the cylinder is ex-pected to be contacted by an ellipsoid at an end face, this end face should beclosed by specifying ellipsoids with the same orientation, semi-axes b and c,and degree n as the cylinder, but with their centres in the centre of the endfaces (Figure 9.9). Also if the contact point on the cylinder P1 is outside thecylinder, no contact forces will be generated, even when the ellipsoid inter-sects the cylinder (Figure 9.10). This limitation is also solved when the endface is closed by an ellipsoid.

Release 7.7 191

9


Figure 9.9: Cylinder with end faces closed by ellipsoids.

elliptical cylinder

hyper-ellipsoidP

P

l

l

1

2

2

1

λ

Figure 9.10: Contact point falls outside cylinder.

The evaluation option should be used in order to prevent that there are forcesNotegenerated from contact between the ellipsoid and both the cylinder and theellipsoid that closes the end face.

The elastic force is perpendicular to the tangent planes. The contact force’spoint of application P depends on how the contact stiffness of the cylinder andellipsoids has been specified. The four cases are the same as those of the plane-ellipsoid contact (page 192), except that now an (hyper-)elliptical cylinder andan (hyper-)ellipsoid are interacting.

An elastic contact force Fe is generated if an ellipsoid penetrates a plane, aThe elasticforce cylinder or another ellipsoid, provided that the interaction is defined as a pos-

192 Release 7.7


9

sible contact. This force is perpendicular to the (tangent) plane. The elasticforce depends on the penetration and the force-penetration characteristic(s).

Force-penetration characteristics are defined by means of the function option.A positive value of the force corresponds to a resistive contact force. In addi-tion, hysteresis and dynamic amplification can be defined (Appendix B andAppendix C, respectively). Separate characteristics for loading and unload-ing must be entered.

The contact force’s point of application P depends on the manner the contactContactcharacteristics characteristic has been specified. In the following, the situation in which an

ellipsoid contacts a plane is described. If an ellipsoid contacts an ellipsoid orcylinder, the "plane" must be replaced by "ellipsoid" or "cylinder", respectively.The following four options are available:

• The contact characteristic of the plane is used. Point P coincides withthe contact point on the ellipsoid P1 because the ellipsoid is consideredto be infinitely stiff (Figure 9.11).

• The contact characteristic of the ellipsoid is used. Then point P coin-cides with the contact point on the plane P2 because now the plane isconsidered to be infinitely stiff (Figure 9.12).

• The contact characteristic is combined from the characteristic definedseparately for the plane and the ellipsoid (Figure 9.13). The combinedcontact characteristic is obtained by adding the penetrations of the planeand ellipsoid for each value of the contact force.

All three hysteresis models can be applied for both surfaces. The loadingcurves must always strictly increase with a function value 0.0 for x =0.0 . For hysteresis models 1 and 2, the unloading curves must be eitherstrictly increasing with a function value 0.0 for x = 0.0 , or they can bothbe zero. If hysteresis model 3 is applied, the unloading curves muststrictly increasing. Different hysteresis models can be applied for thetwo surfaces.

The elastic force is used to calculate the penetration into the ellipsoidxe,el from the ellipsoid characteristic and the penetration into plane xe,pl

from the plane characteristic. Next point P is calculated from:

P = P1 +xe,el

xe,el + xe,pl· (P2 − P1) (9.1)

where

P1 contact point on ellipsoid

P2 contact point on plane

• The combined contact characteristic (USER_MID_POINT) of the inter-acting plane and ellipsoid is specified in the input file (Figure 9.14).Point P is in the middle of the line segment P1 – P2 .

Release 7.7 193

9


Figure 9.11: Contact characteristic of plane is used.

Figure 9.12: Contact characteristic of ellipsoid is used.

Figure 9.13: Both contact characteristics are used.

Figure 9.14: Combined contact characteristic (USER_MID_POINT) is used.

In order to define damping and friction forces, a reference plane is introduced.Damping andfriction forces This plane contains the point P and is parallel to the contact plane if there

194 Release 7.7


9

is plane-ellipsoid contact (Figure 9.15). For ellipsoid-ellipsoid and cylinder-ellipsoid contact, the reference plane is parallel to the tangent planes corre-sponding to a minimum value of λ (page 188 and page 190).

The relative velocity v between the interacting contact surfaces is defined asthe relative velocity at the point P of the two contacting objects. This velocityvector is resolved into two components: a component vplane in the referenceplane and a component vnorm normal to this plane.

Figure 9.15: The relative velocity v resolved into two components.

The damping force Fd is defined as:Damping force

Fd =[Cd · |vnorm|+ Fdamp

]· Fampl(Fe) (9.2)

where Cd is the (positive) damping coefficient, Fdamp is the damping velocityfunction and Fampl is the amplification function dependent of the elastic forceFe .

A damping coefficient cannot be defined separately for each contact surface.Note

If the penetration increases(loading), the damping force is added to the elas-tic force (Figure 9.16a). If the penetration decreases (unloading), the dampingforce counteracts the elastic force (Figure 9.16b). If the damping force exceedsthe elastic force, contact forces are not applied during unloading because con-tact forces are resistive forces.

Release 7.7 195

9


Figure 9.16: Damping in loading (a) and unloading (b).

In addition to the damping force, a dry friction force Ff can be specified. ThisFriction forcefriction force acts in the reference plane in the direction opposite to the relativevelocity component vplane:

Ff = C · f (|Fe + Fd|) · |Fe + Fd| (9.3)

where f (|Fe + Fd|) is the Coulomb friction coefficient and C a ramp function.This ramp function varies between 0 and 1 as a function of the relative velocityvplane (Figure 9.17). The ramp function has been introduced in order to avoidvibrations induced by dry friction. The friction coefficient can be defined as afunction of the magnitude of the normal force.

196 Release 7.7


9

Figure 9.17: Definition of the ramp function C.

The same ramp function is used for all contacting surfaces.Note

Simultaneous multiple interactions between an ellipsoid and several other el-Evaluationlipsoids, cylinders or planes can occur. The user can specify the type of evalu-ation. Two types are possible:

• Discrete: the force corresponding to the contact with the largest elasticforce is applied.

• Continuous: a scaling factor equal to the maximum elastic force dividedby the summed elastic force is calculated and limited to the interval(0, 1] . For each interaction in the list the force is scaled with this fac-tor.

Consider an ellipsoid connected to the upper leg of a pedestrian and a chainExampleof finite planes attached to a vehicle representing the front end of the bonnet(Figure 9.18). The bonnet’s stiffness can be measured by means of tests witha leg-shaped impactor. If this stiffness is assigned to each of the interactionsseparately and the upper leg ellipsoid penetrates several of the planes at thesame time, the effective total bonnet stiffness in the model will be much toohigh. If the evaluation option of type discrete is used, only the forces, thatare the result of contact with the largest elastic force occurring in a series ofspecified interactions, will be taken into account. They will be applied to thepedestrian upper leg body and to the vehicle body.

Release 7.7 197

9


Figure 9.18: Example of multiple contact interactions.

Instabilities can develop because the contact changes from one plane to an-Noteother.

9.2 Contact between FE-surfaces and MB-surfaces

Contact can be defined between finite element surfaces and multi-body sur-Introductionfaces. In these contacts, the slave surface is defined by the nodes of the finiteelement surface and the master surface is defined as planes, cylinders andellipsoids. Two models are available to calculate the contact force.

The kinematic contact model does not allow a finite element node to pen-etrate the master contact surface. The contact force is based on an inelasticimpact between the node and the surface.

The elastic contact model allows a finite element node to penetrate the mastercontact surface. The contact force is based on a specified force-penetration ora stress-penetration characteristic.

A cylinder is handled as if it has an infinite length (the a-axis is infinite).Note

In the kinematic contact model a plane is handled as a rectangular volumeKinematiccontact model with one of the faces made up by the plane (Figure 9.19). The opposite face

is at a user-defined distance. This thickness is determined by the user or adefault value of 5 mm is used. The thickness should be larger than the largest

198 Release 7.7


9

displacement of a penetrating node in one finite element time step. The en-closed volume is rectangular and positioned directly behind the plane.

Figure 9.19: Connecting contact planes.

For the contact algorithm, the finite element nodes are treated as point massesand master contact surfaces as impenetrable. During each finite element timestep, the relative position of the node and the relevant contact surface are eval-uated. If a node is inside an ellipsoid, cylinder or the rectangular volume cor-responding to a plane, the contact algorithm is activated for this node. If anode enters a plane volume through a side or the rear face, the node will betreated as if it entered the volume through the frontal face, the specified plane.

Contact forces are calculated based on the relative velocity of the node andthe contact surface. A normal impulse is applied to the node and the contactsurface so that the relative velocity component perpendicular to the contactsurface becomes zero. This corresponds to an inelastic impact. The penetratednode is placed on the contact surface. However, a node can become trapped,for example, between two penetrating ellipsoids for which contacts with thisnode are specified. Then the node will be placed on the surface of one of theellipsoids and inside the other ellipsoid. The contact algorithm assumes thatnodes always penetrate ellipsoids and cylinders from the outside and planesfrom the front.

A tangential impulse based on Coulomb friction is applied. The tangentialimpulse equals the product of the normal impulse and the friction coefficient.If this results in a reversion of the relative tangential velocity, the tangentialimpulse is reduced such that the resulting relative velocity of the node andthe contact surface equals zero.

The impulses due to all contacts of nodes with contact surfaces attached to abody are used to calculate the linear and angular impulse in the body’s cen-tre of gravity. The force and torque acting on the body’s centre of gravity arecalculated based on this linear and angular impulse. For contact surfaces at-tached to the inertial space, the force and torque are calculated with respect tothe inertial coordinate system.

Release 7.7 199

9


At the start of a simulation, avoid penetrating a plane, cylinder or ellipsoidwith a node because a large contact force will result and the node will be shotout of the contact surface. A list of initially penetrating nodes is given in theREPRINT file for each contact.

A surface can be represented by several planes. If two or more planes form acorner with an inside angle of less than 180 degrees, a node could be forcedthrough, unless there is absolutely no space between the two. Avoid this bydefining the planes with a small (about 5 mm) overlap (Figure 9.19). If a two-sided plane is required, the user should define two planes with the fronts fac-ing in opposite outward directions with a space equal to at least the combinedthicknesses.

Often only a fraction of the nodes can make contact with a certain contact sur-face. Carefully select the relevant nodes to significantly reduce computationtime needed compared to defining contact for all of the nodes.

Nodes selected in the slave surface should not be supported to a body or beNotepart of another constraint because the contact algorithm modifies the positionof the nodes.

This contact model allows a finite element node to penetrate the master con-Elastic contactmodel tact surface. A node can penetrate a plane and it will be found on the opposite

side of the plane. A node can penetrate an ellipsoid (cylinder) and it will befound inside the ellipsoid (cylinder). The penetration λ is equal to the mini-mum distance between the node and the contact surface.

Because the penetration equals the minimum distance between the contactNotesurface and the node, the penetration and contact force in an ellipsoid is lim-ited. This can result in incorrect contact forces if an ellipsoid penetrates afinite element surface more than the minimum of the semi-axes. In that casethe ellipsoid should be covered by elements and the finite element contactalgorithm should be used.

The elastic contact model treats one of the surfaces as ‘rigid’ and one of thesurfaces as ‘deformed’. Dependent on the contact type the master or slavesurface is treated as deformed. If the contact algorithm treats a surface asdeformed this means that the hysteresis in the contact forces is stored for thatsurface.

In the elastic contact model two models are available for calculating the con-tact force: contact model force and contact model stress.

In contact model stress, the contact force is given by:

Fi = Ai

[σe

(λi

ti

)+

[Cd

λi

ti+ σd

(λi

ti

)]fd(σe)

](9.4)

The elastic contact stress σe is taken from the specified stress-penetration/thickness characteristic σe(λ/t) . t is the thickness of themaster or slave contact surface. By dividing the penetration by t, thecharacteristic is independent of the thickness of the surface’s padding. Thethickness of the deformable surface is used. Which surface is treated as

200 Release 7.7


9

deformable is defined by the contact type as mentioned before. When this isthe master surface, the thickness is specified in the contact interaction (defaultis 1.0 m). When this is the slave surface, the average thickness of the elementsconnected to the penetrating node is used.

The second term in the above equation represents the damping contact stress.Cd is a constant damping coefficient. σd is a damping stress function σd(λ/t) ,which can account for non-linear damping. fd is the damping amplificationfactor that is a function of the elastic stress σe . This can prevent a large damp-ing stress when a node impacts the surface with a large velocity and there isno elastic contact stress, or the surface is deformed.

The Coulomb friction stress σf is

σf ≤ f (vplane) · σ (9.5)

where f (vplane) is a user-defined friction function and vplane is the node’s ve-locity component relative to the surface parallel to the tangent plane of thesurface.

The contact force is the result of multiplying the contact stress by the node’sarea, or the element’s area that is attached to the nodes.

If the slave surface is selected as deformable, it is possible to have multiplecontact stress characteristics in one contact. In that case these characteristicsshould be defined with the finite element groups and the contact type shouldbe selected from the slave surface.

In contact model force, the contact force is given by:

Fi =Ftotal

∑ncj=1 λj Aj

λi Ai (9.6)

where Fi is the contact force of the penetrating contact node i,nc is the number of penetrating contact nodes,λj is the penetration of the contact node j,Aj is the contact area associated with the contact node j,Ftotal is the total contact force which is composed of the elastic force Fe and thedamping force Fd:

Ftotal = Fe + Fd (9.7)

The elastic force Fe is derived from a specified contact characteristic which isa function of the maximum penetration across all penetrating contact nodes:

λmax = max(λ1, λ2, ..., λnc) (9.8)

The damping force is defined as:

Fd = (Cdvn + Fdamp(vn))Fampl(Fe) (9.9)

Release 7.7 201

9


where Cd is the (positive) damping coefficient,vn is the normal velocity of the contact node with the maximum penetration,Fdamp is the damping velocity function,Fampl is the amplification function dependent on the elastic force.

If the penetration increases(loading), the damping force is added to the elasticforce. If the penetration decreases (unloading), the damping force counteractsthe elastic force. If the damping force exceeds the elastic force, contact forcesare not applied during unloading because contact forces are resistive forces.

It is not always possible to have hysteresis in the contact stress characteris-tics. Table 9.2 shows an overview of the available hysteresis in the contactalgorithm.

Table 9.2: Overview of available hysteresis models in contacts.

contact model/contact type

force/master

force/slave

stress/master

stress/slave

hysteresis model none yes yes yes yes

hysteresis model 1 or 2 yes yes no yes

hysteresis model 3 yes yes no no

Initial penetrations will generate initial contact forces which can cause insta-Initialpenetrations bilities. In order to avoid initial contact forces, the INITIAL_TYPE option

should be used. The penetration of the contact nodes relative to their posi-tions at the start of the simulation are stored and remain unchanged duringthe whole simulation. Contact forces are generated during the simulation onlyif the actual penetrations exceed the initial penetrations.

9.3 Contacts between finite elements

The finite element contact algorithm searches for contact between a masterIntroductionsurface and a slave surface.

The master surface is defined as a group of contact segments that are formedby one or more finite element groups.

The slave surface is defined as a group of contact nodes that are formed byone or more finite element groups.

In MADYMO there are two algorithm’s to search for contact between ele-ments:

• Contact is found based on intersections of slave surface nodes throughthe master surface contact segments.

• Contact is found based on penetration of slave surface nodes in the con-tact thickness (gap) of the master surface contact segments.

202 Release 7.7


9

Intersection based contact is mainly used for contact between rigid surfaceswhere the surface stiffness is defined in the contact force characteristic. Pene-tration based contact is mainly used for deformable FE structures where pene-tration between the surfaces has to be kept as low as possible. In the followingsections both algorithm’s are explained.

9.3.1 Contact algorithm based on intersections

In the intersection based contact algorithm, three different phases can be dis-Concepttinguished:

• the search phase,

• the detailed search phase and

• the force calculation phase.

In the search phase for each contact node on the slave surface is checked forintersection of the master surface’s contact segment during the current timestep. For these calculations, the positions of both the contact node and contactsegment of the current and previous time step are used. In this manner allnew contacts are detected. All the contacts (contact node intersects a contactsegment) are stored.

Figure 9.20 shows a contact node penetrating a master surface. In the searchphase, a new contact node at t = 1 is found. The search phase is the mosttime-consuming part of the computation of the contact algorithm.

t = 2

t = 1

t = 0

t = 3

contact node

penetration

master surface

321

Figure 9.20: Penetration of contact nodes in contact surface.

Release 7.7 203

9


1

t = 1

t = 0

2

t = 2

penetration

master surface

t = 3

contact node

3

Figure 9.21: Penetration of contact nodes in master surface.

In the detailed search phase, all old contacts (contacts that occurred ear-lier) and new contacts (contacts found in the search phase) are checked. InFigure 9.20, the old contact is the node at t = 2 and t = 3 . Each contact ischecked to see if the contact node is still penetrating the contact segment. Ifthe contact node is on the wrong side of the segment (node at t = 3), the con-tact node has moved back to the surface and is not penetrating any longer.These contacts are removed from storage.

In addition, during the detailed search phase, the contact node’s projection ischecked to see if it lies within the contact segment. The projection is calculatedusing the normal of the contact segment.

If the projection does not lie within the contact segment (node moving fromt = 1 to t = 2), the connecting contact segments are checked. This process con-tinues until a contact node is found with a projection lying within the contactsegment.

If a contact segment does not have a connecting segment (node at t = 3 inFigure 9.21), the contact node remains with the closest contact segment. Usingthe option RELEDG, this contact can be released (the contact is removed fromstorage).

In the force calculation phase for all contacts, the contact nodes’ penetrationsof the contact segment are calculated. The penetrations are calculated as theminimum distance between the node and the segment (see Figure 9.20 andFigure 9.21). Using the penetration, the contact forces are calculated depend-ing on which contact force model is used. The contact force has the same di-rection as the penetration and is applied to both the contact node and mastersurface.

It is possible to specify also a contact thickness in the contact. Using thisThickness incontact feature, the contact algorithm can handle a master surface with a thickness,

i.e. contact forces are generated for contact nodes entering the gap (seeFigure 9.22).

204 Release 7.7


9

penetration

master surface

t = 3

t = 0contact node

gap

gap

1

3

2t = 1

t = 2

Figure 9.22: Contact with gap.

The contact algorithm works exactly the same and the only difference is thatNotecontact forces are generated earlier.

The gap can be specified in three ways:

• The gap can be specified as a function of time. The gap is the same forthe entire master surface.

• The master surface’s gap is equal to half the element thickness of theelements that make contact. For volume elements, the thickness is cal-culated as 20th part of the initial volume divided by the initial contactsurface of the volume element. Optionally this thickness can be scaledwith a scale function dependent on time.

• The master surface’s gap is equal to half the node thickness of the contactnodes that make contact (of the slave surface). The node thickness isthe average thickness of connecting element thicknesses. Optionally thisthickness can be scaled with a scale function dependent on time.

If the gap feature is used, initially there can be contact nodes in the gap. Thismeans that initially contact forces are generated for those nodes. If these con-tact forces are relatively too high, instability can occur. MADYMO alwayschecks which nodes lie initially in the contact gap and reports them in theREPRINT file.

In other software codes this check is often called an "initial penetration check".NoteHowever, initial intersections can still occur if no contact nodes lie initially inthe gap (see Figure 9.23).

Release 7.7 205

9


gapgap

slave surface

penetrations

master surface

initial intersectionsinitial

Figure 9.23: Initial interactions not found using gap feature.

Using the scale functions or gap function, the gap can be made initially verysmall in order to prevent contact nodes from being in the contact gap. How-ever, if the functions do scale up the gap too quickly, instabilities can occur.

If the element thickness of the master surface is used for the gap and the mas-ter surface has multiple thicknesses, discontinuance can exist which can causestability problems.

Keep the gap steady during the simulation and work with models that doRecommenda-tion not have contact nodes initially in the contact gap. If the master surface has

multiple thicknesses, do not specify the gap using the element thickness of themaster surface. In this case, the other gap options should be used.

9.3.2 Contact algorithm based on penetrations

The penetration based contact algorithm consists of two phases:Concept

• Contact search phase

• Force calculation phase

The force calculation phase works in the same way as for the intersectionbased contact. In the contact search phase, all nodes are found penetratingthe gap of the contact segments. The gap, i.e. the contact thickness, is speci-fied by the user. Nodes penetrating the gap are pushed back outside the gap

206 Release 7.7


9

(Figure 9.24). This means that if the nodes penetrate through the contact seg-ment mid line, they will be pushed away on the back of the contact segment.This means that if the gap is defined too small or the contact forces are toolow, nodes can go through the surface.

t = 3

master surface

penetration

contact node

t = 1 2

3

1

gap

gap

t = 2

t = 0

t = 0

t = 2

t = 3

Figure 9.24: Penetration based contact.

The gap can be specified in two ways:

• The gap can be specified as a function of time. The gap is the same forthe entire master/slave surface.

• The gap is the surface’s thickness, i.e. half the thickness of the masterelement + half the thickness of the slave element. For volume elements,the thickness is calculated as 20th part of the initial volume divided bythe initial contact surface of the volume element. Optional this thicknesscan be scaled with a scale function dependent on time.

Again here, initial penetrations of the nodes in the gap are reported in theREPRINT file. Also if the gap is scaled up to quickly in the simulation, insta-bilities can occur.

Initial penetrations will generate initial contact forces which can cause insta-INITIAL_

PEN_TRACK bilities. In order to avoid initial contact forces for initial penetrations, theINITIAL_PEN_TRACKoption should be used. If this option is used, all the initialcontact forces will be zero. If during the simulation the penetration increasesa contact force is generated based on the "extra" penetration compared to thelowest penetration of that node has had during that simulation. In this way stillcontact is detected during the simulation but no initial forces are calculated. Itis recommended to use this option.

There are two ways of penetration based contact: "node to surface" and "sur-"Node tosurface" and"surface tosurface"

face to surface". In the "node to surface" the slave nodes are compared with

Release 7.7 207

9


the contact gap around the master contact segments. In the "surface to sur-face" contact, not only the slave nodes are compared with the master contactsegments but also the master nodes are compared with the slave contact seg-ments.

In the "surface to surface" contact it is possible to add additional edge-edgeEdge-edgecontact contact. In that case on top of the "surface to surface" algorithm also edge-

edge contacts of the segments of the master and slave surface are detected.(Every triangle segment has 3 edges; every quad segment is divided into twotriangle segments). In this case also 2-node elements like trusses and beamsare taken into account in the contact.

Also if this option is used, intersections of edges with segments are detectedevery INTERSECT_CHECK_INTERVAL cycles. For the intersections found, thecontacts are temporarily turned off until these intersections have disappeared.Intersections of edges with segments indicate that somewhere in the modelthe penetrations are too large (gone through the surface). The intersection al-gorithm detects these "fault" cases and removes temporary these contacts sothat there is no energy loss of contacts working in the wrong direction. Noticethat if the time step is too large or the contact stiffness is too low, too manyintersections are found and too many contacts are turned off resulting in notrealistic behaviour of the model. If this is seen, the contact stiffness should beincreased or better, the time step of the model should be reduced.

The intersection algorithm is CPU expensive and that is why the default forINTERSECT_CHECK_INTERVAL is set to 20. This is a good value for airbag simu-lations.

When edge-edge contact is used, the CPU times of the model will increasewith a factor 3 to 4 depending on the model.

It is recommended to use edge-edge contact for all airbag single surface calcu-Recommen-dation lations.

9.3.3 Contact force models

Three contact force models are available: penalty based contact, adaptiveIntroductionbased contact and elastic characteristic based contact. Elastic characteristicbased contact is only available for the intersection based contact algorithm.

In the penalty based model, the bulk modulus of the contact segment (masterPenaltycontact surface) is used to calculate the contact force:

Fi = (K/V0)A2i ψλi (9.10)

where:

• K is the bulk modulus of the contact segments penetrated,

• V0 is the initial volume of the contact segment,

• Ai is the area of the contact segment,

208 Release 7.7


9

• ψ is the penalty factor and

• λi is the penetration of the contact node.

The penalty based model is designed for non rigid finite element surfaces andshould keep the penetrations as low as possible.

Elements with hole material do not have a bulk modulus so contact forces areNotenot generated for these elements. Also line elements do not have a surfaceso these elements will be rejected from the master surface (not from the slavesurface!) if no edge contact is specified.

The penalty based model limits the contact forces based on the time step inorder to keep the simulation stable. The maximum contact force depends onthe mass of the contacting node, the mass of the contacting segment and thetime step (the eigenfrequency of the contact). The lower the time step, thehigher the maximum contact force. This means that sometimes it has no useto increase the penalty factor because the contact forces are already limited bythe time step. In that case the time step should be decreased. Also this meansthat if a variable time step is used in a model, the contact forces can scale up ordown as the time step varies. This can cause peaks in the results. In that caseit is recommended to use CONTACT_FORCE.ADAPTIVE with a static TIME_STEPdefined.

When CONTACT_FORCE.ADAPTIVE is specified, the contact stiffness is calculatedAdaptivecontact based on the current time step of the model or the TIME_STEP value. Again the

contact forces are limited by the eigenfrequency of the contacts (i.e. the currenttime step). It is recommended to use the TIME_STEP option if a variable FEtime step is used in the model because otherwise the contact forces can changedrastically if the time step changes in the model resulting in instability of themodel or unrealistic peaks on the signals.

In CONTACT_FORCE.PENALTY and CONTACT_FORCE.ADAPTIVE, damping can beintroduced in the contact by the option DAMP_COEF. DAMP_COEF is the percent-age of critical damping based on the eigenfrequency of a contact (DAMP_COEF=1.0 is critical damping). Contact damping can reduce the oscillations in thecontact forces.

In the penetration based contact the option VAR_TIME_STEP can be defined. InVAR_TIME_

STEP that case the contact algorithm controls the time step i.e. based on the contactstiffness a critical time step is calculated and used in the simulation for therelated FE models.

Furthermore, additional contact forces are calculated for contactingnodes/edges, which have reached a certain level of penetration. This level canbe specified using the attribute CRITICAL_PEN (default 0.8) and is a percentageof the gap size (maximal 1.0). These additional contact forces are calculatedin order to keep the relative velocity between the node and the segment ortwo edges zero in the next time step. In this way nodes cannot go through themaster surface. Because the individual contacts are dependent of each other,these extra contact forces have to be calculated in an iterative way until theyare converged:

Release 7.7 209

9


1. Calculate contact forces based on specified contact stiffness.

2. Calculate critical time step for contact stiffness.

3. Calculate critical time step so that contacting nodes/edges cannot gothrough the surface in the next time step.

4. Calculate additional contact forces for critical penetration contacts.

Do step 3 and 4 again until the additional contact forces are converged.

This iterative loop is stopped if the additional contact forces are con-verged (tolerance can be specified under CONTROL_ANALYSIS.TIME using theCONTACT_TOL attribute (default 1E-4)) or the maximum number of iterationshas exceeded (can be specified under CONTROL_ANALYSIS.TIME using theCONTACT_MAX_ITER attribute (default 20)).

The CPU -usage of this calculating depends on the tolerance, the maximumnumber of iterations, the contact stiffness, gap size. If the tolerance is too highor the maximum number of iterations is too low, still nodes will go throughthe surface. If the contact stiffness is too low, many contacts will lie in thecritical penetration zone, which means that much iteration for calculating theadditional contact forces will be needed which slows down the calculations.

It is recommended to control the contact stiffness using the TIME_STEP at-Recommen-dation tribute in CONTACT_FORCE.ADAPTIVE. Using this value the nominal time step

for the calculation can be set.

The elastic characteristic contact model provides three ways to treat the con-Elasticcharacteristiccontact

tact dependent on the contact type:

1. slave: the slave surface is treated as ‘deformed’ and the master surfaceis assumed to be ‘rigid’ (Figure 9.25(a)). The contact area is calculatedusing the segment area connected to the contacting node of the slave set.In addition, the contact history is stored for the slave surface’s contactingnode.

2. master: the master surface is treated as ‘deformed’ and the slave surfaceis assumed to be ‘rigid’ (Figure 9.25(b)). The contact area is calculated inthe same way as for ‘slave’. The contact history is stored for the mastersurface’s contacting segments. The maximum penetration of all nodescontacting one segment is used to calculate the contact force’s hysteresisbecause more than one node can penetrate a segment.

3. combined: the slave and master surface are treated as ‘deformed’. Thismeans that an intermediate surface is calculated between the master andslave surface and that these penetrations are used for calculation of thecontact forces. The contact area is calculated in the same way as for‘slave’.The hysteresis in the contact forces are stored in the slave surfacefor the slave contact forces and in the master surface for the master con-tact forces. These forces are the same (action=reaction) but the contactcharacteristics of the master and slave surface can be different which

210 Release 7.7


9

means that also the hysteresis can be different. The contact forces arecalculated in an iterative way in which also the intermediate surface iscalculated. The above algorithms for slave and master are combined inthis calculation. The algorithm uses the derivatives of the contact forcecharacteristics. If these characteristics are highly non linear, oscillationscan occur in the contact forces. In this case the time step of the simu-lation should be lowered and if this does not help, the characteristicsshould be modified.

master surface

slave surface

(a) (b)

Figure 9.25: Contact type: (a) slave surface deformed, (b) master surface deformed.

In the elastic characteristic contact model two models are available for calcu-lating the contact force: textbfcontact model force and contact model stress.Contact model force does not work in combination with contact type com-bined.

In contact model stress, the contact force is given by:

Fi = Ai

[σe

(λi

ti

)+

[Cd

λi

ti+ σd

(λi

ti

)]fd(σe)

](9.11)

The elastic contact stress se is taken from the specified stress-penetration/thickness characteristic σe(λ/t) . t is the thickness of themaster or slave contact surface. By dividing the penetration by t, the charac-teristic is independent of the thickness of the surface’s padding. The thicknessof the deformed surface is used. Which surface is treated as deformed isdefined by the contact type as mentioned before. When this is the mastersurface, the thickness of the elements of the master surface is used. Whenthis is the slave surface, the maximum thickness of the elements connectedto the penetrating node is used. For combined contact type, both thicknessesfor the slave and master surface are used. There are two methods to calculatethe master surface thickness. One method just uses the element thickness,i.e. the thickness is constant als long as the slave node is penetrating thatelement. In the other method the thickness is calculated based on the positionof the penetrating slave node. The thickness is interpolated from the masternode thickness of the element. The master node thickness is calculated as themaximum element thickness of the elements connected to the node.

The second term in the above equation represents the damping contact stress.Cd is a constant damping coefficient. σd is a damping stress function σd(λ/t) ,

Release 7.7 211

9


which can account for non-linear damping. fd is the damping amplificationfactor that is a function of the elastic stress σe . This can prevent a large damp-ing stress when the slave surface impacts the master surface with a large ve-locity and there is no elastic contact stress, or the surface is deformed.

The contact force is the result of multiplying the contact stress by the contactarea.

It is possible to have multiple contact stress characteristics in one contact. Inthat case these characteristics should be defined with the finite element groupsand the contact type should be selected from the slave surface or master sur-face.

In the stress model, the total contact force Ftotal =nc

∑j=1

Fj depends on the contactNote

area.

In contact model force, the force is defined as a function of the penetration.The total contact force Ftotal of a contact is calculated using the maximum pen-etration λmax of all contacting nodes:

Ftotal = Γ(λmax) (9.12)

where

λmax = max(λ1, λ2, . . . , λnc) (9.13)

The contact forces Fi per contact node is calculated using:

Fi =

Ftotalnc

∑j=1

λj Aj

λi Ai (9.14)

where nc is the total number of contact nodes,

λj is the penetration of contact node j and

Aj is the contact area.

The contact area of the ‘deformed‘ surface is taken. If the slave surface is takento be ’deformed’, the average area of elements surrounding the node is taken.If the master surface is taken to be ’deformed’, the contact area is defined asthe area of the element which is penetrated by the node.

The total contact force depends only on the maximum penetration of all con-Notetact nodes, which means it is independent of the number of nodes in contactand the total contact area. Because of this the contact model stress is preferredto the contact model force in all cases.

A friction coefficient is a function of the relative velocity between the contact-Frictioning node and segment:

Ff ≤ µ(vplane)Fc (9.15)

where:

212 Release 7.7


9

• µ is the friction coefficient,

• vplane is the relative velocity (see Figure 9.26) and

• Fc is the normal contact force without friction.

The friction force Ff is smaller than µ(0)Fc for stick and is equal to µ(vplane)Fc

for slip.

v

v

v

vsegment

fF

rel

plane

nodeFigure 9.26: Direction of friction force.

The relative velocity is defined as the relative velocity between the velocityof the projection of the node on the contact segment and the velocity of thecontact segment. The tangential relative velocity vplane is the component ofthe relative velocity parallel to the contact segment. The friction force acts inthe opposite direction as this velocity.

No stick friction is obtained if the static coefficient of friction µ(0) is equal tozero.

Stick-slip friction can introduce vibrations in the model.Note

9.3.4 Initial intersection check

The contact algorithm assumes that initially there is no contact. Only for nodesinitially in the gap a contact force will be calculated (if INITIAL_PEN_TRACKis OFF). Using the initial intersection check, initial intersections are detectedif they occur. Initial intersections are defined as cross-sections of the mastersurface and the slave surface.

It is possible to only check for initial intersections. In that case for both themaster and slave surfaces, the contact segments are calculated (normally only

Release 7.7 213

9


the contact segments of the master surface are needed because only the con-tact nodes are used for the slave surface). Then it is determined if a mas-ter surface’s contact segment crosses the slave surface’s contact segment (seeFigure 9.27). The initial penetrations are reported in the REPRINT file. In thiscase initial contact forces are not generated. This option is useful for checkingfor initial penetrations in folded airbags.

For the intersection based contact, it is also possible to generate initial con-tact forces. If this option is used, the results from initial intersection check areused to determine which contact nodes initially penetrate within contact seg-ments. In order to achieve this, it is required that the master surface segments’faces are opposite of the slave surface segments’ faces. Also, the contact seg-ments in one contact set which cross each other initially must form a closedchain in the master surface and the slave surface. In this way, the contact seg-ments inside the chain can be selected for the slave and master surface (seeFigure 9.27). These segments are used for calculating the initial penetrationsfor each contact node.

214 Release 7.7


9detected initial penetrations

selected contact segments

selected contact nodes

slave surface

master surface

element normal

element normalslave surface

master surface

Figure 9.27: Initial penetration check.

To find for each initial penetrating node its contact segment 3 options are avail-able:

• For each initial penetrating node in the slave surface, the segment inthe master surface is searched to which the distance is minimal if forcalculation of the distance the master surface’s face is used.

• For each initial penetrating node in the slave surface, the segment in

Release 7.7 215

9


the master surface is searched to which the distance is minimal if forcalculation of the distance the slave surface’s face is used.

• For each initially penetrating node in the slave surface, the segment inthe master surface is searched to which the distance is minimal if forcalculation of the distance a user-defined normal is used. This user-specified normal and the slave surface’s face must point in the samedirection.

Initial forces should only be generated for rigid finite element models becauseNoteinitial forces can make a deformed finite element model unstable due to thehigh initial contact forces.

9.3.5 Extra options

The penalty factor is a tuning parameter for calculating the contact force if aMAX_FORCE_

PAR penalty based contact is used. The default value is 1.0. If stability problemsarise, it is better to adjust the MAX_FORCE_PAR instead of the penalty factor. Thisparameter limits the contact force for each node to a contact force equal to thecontact force which occurs if the penetration equals the penetrated elementthickness te scaled with this parameter:

Fc = (K/V0)A2ψ min(λ, MAX_FORCE_PARte) (9.16)

In MADYMO, the time step is not adjusted for the contact forces, only the el-ements determine the stable time step using the Courant criterion. The corre-sponding contact forces can be high and may cause (local) instabilities becausevery large penetrations in the contact can occur. By limiting the contact force,these (locally) high contact forces can be prevented and instability is avoided.MAX_FORCE_PAR is a very powerful way to avoid instabilities.

Use a MAX_FORCE_PAR of 1.0. If instabilities occur due to the contact, scale thisRecommen-dation parameter to 0.1 − 0.001 . Do not modify the penalty factor in these cases.

As stated above, the intersection based contact algorithm checks for contactsChoosingmaster andslave surfacefor intersectionbased contactalgorithm.

between a master and slave surface. If a slave surface’s contact node contactsthe master surface, the segment of the master surface is taken for which theprojection of the contact node lies within that segment (using the normal ofthe segment surface).

Figure 9.28 shows an example in which the choice of the master surface andthe slave surface is very important. It shows a small sphere that penetrates aplate very deeply. In this case, if the slave surface is the sphere, no problemswill arise. However if the sphere is chosen as the master surface, incorrectresults are obtained. The contact nodes will be the nodes located at the pointwhere the sphere penetrated the plate. For these nodes, a segment on thesphere is searched for which the projection of the node lies within the segment.These segments will be on the upper part of the sphere. Also, if the sphere

216 Release 7.7


9

moves to the right or left, the same nodes will stay in contact resulting inhigher contact forces, or in other words, the results will be incorrect.

velocity

velocity

1

2

3

master surface

slave surface

all nodes in contact

elements in contact

all nodes in contact

penetrations

penetrations

velocity

velocity

master surface

slave surface

1

2

3

penetrations

penetrations

nodes in contact

nodes in contact

element in contact

element in contact

elements in contact

Figure 9.28: Choice of master and slave surface.

Another example is shown in Figure 9.29. Again the choice of master andslave surface has a great effect on the results because the projection of thecontact nodes on the master surface are used to calculate the penetrations.

penetrations

velocity

penetrations

master surface

slave surfacemaster surface

slave surface

velocity

Figure 9.29: Choice of master and slave surface.

Choose the contact surface with the most curvature as the slave surface.Recommen-dation

Release 7.7 217

9


In the intersection based contact, if no contact segment can be found for a con-RELEDG

tact node, or there is no segment for which the projection of the node lieswithin the segment, the contact node will stay in contact with the segmenton the edge of the master contact surface (see Figure 9.30). With the optionRELEDG, these contact nodes can be released from the contact. However, thismeans that if the node moves back to the same path, no contact forces willbe generated. If the node moves through the surface again, a contact force inanother direction will be generated (see Figure 9.31).

2

RELEDG OFF RELEDG ON

penetration

node in contact

11

3

node in contact

2

3

4

5 5

4

node in contact

slave surface

master surface

slave surface

master surface

node in contact

Figure 9.30: Option RELEDG.

Generally, it is better to extend the master surface in the way that these casescannot occur during the simulation (see Figure 9.31).

218 Release 7.7


9

node in contact

penetration

6

5

node in contact

slave surface

master surface

nodes in contact

master surface

slave surface

4 4

5

6

node in contact

33

2

1 1

2

Figure 9.31: How to deal with master surface edges.

If the elastic characteristic contact models force or stress are used, the contactCONTACT_ AREA

nodes in the slave surface should have a contact area if the contact force his-tory is stored for the slave surface (the slave surface is chosen as ‘deformed‘).Normally, these contact areas are calculated using the connected segments tothese contact nodes. For line elements, such as truss and beam elements, thesegments have no surface which results in a zero contact area for nodes con-nected to these elements. This means no contact forces are generated for thesenodes. This can be avoided using CONTACT_AREA. If this option used, the spec-ified area is used for nodes which have no contact area due to the connectingelements. The default value for CONTACT_AREA is 0.

If a set containing volume elements is selected for the master surface or slaveCONTACT_

SURFACE surface, the contact algorithm will select the volume’s outer surface for the

Release 7.7 219

9


master surface or slave surface. Using the CONTACT_SURFACE=OFFalso the con-tact segments/nodes inside the volume will be selected in the contact.

To achieve realistic results, the mesh of the slave surface and the master sur-General rec-ommendation face should have approximately similar fineness. The mesh of the slave sur-

face should always be as fine or finer than the master surface. If a elastic char-acteristic force model is used and the master surface is chosen as deformed,the master surface should have a similar mesh density as the slave surface.

The performance of the contact algorithm highly depends on the number ofelements and nodes in the master and slave surface. To reduce computationaltime, these surfaces should be as small as possible. However, these surfacesshould be selected with care. If these surfaces are to small, problems can de-velop.

In Figure 9.32, an example is shown in which a sphere penetrates a surfaceSurface withtwo differentelasticcharacteristicforce models

with two different characteristics. In the example on the left, the two contactsare defined where the sphere is the slave surface and two master surfaces (onefor each contact) are used for modelling the two different characteristics. Theinput deck is:

<CONTACT.FE_FE ID="1" NAME="contact_1"

MASTER_SURFACE = "MASTER_SURFACE_1"

SLAVE_SURFACE = "SLAVE_SURFACE">

<CONTACT_METHOD.NODE_TO_SURFACE_CHAR>

<CONTACT_FORCE.CHAR CONTACT_TYPE = "MASTER"/>

</CONTACT_METHOD.NODE_TO_SURFACE_CHAR>

</CONTACT.FE_FE>


MASTER_SURFACE = "MASTER_SURFACE_2"





</CONTACT.FE_FE>

As shown in Figure 9.32, the results will be not realistic because the first con-tact will detect the contact and stay in contact if the sphere moves to mastersurface 2. The second contact will not detect contact at all because no nodesare penetrating.

On the right side of Figure 9.32, the correct method of modelling this problemis shown. Only one contact is defined by selecting more than one facet set inthe contact. The input deck should be:


MASTER_SURFACE="MASTER_SURFACE_1 MASTER_SURFACE_2"



220 Release 7.7


9



</CONTACT.FE_FE>

This example shows that it is always important to define the master surface asonly one surface in a contact.

slave surface

master surface 1 master surface 2

node in contact

node in contact

1

2

3

penetration

master surface

master surface 1 master surface 2

penetration

slave surface

node in contact

1

2

3

Figure 9.32: Selection of master surface and slave surface.

Release 7.7 221

9


222 Release 7.7

MADYMO Theory Manual Airbag models

10

10 Airbag models

10.1 Finite element airbag model

The finite element model, described in Chapter 5, has been expanded with gasthermodynamics to make it suitable for airbag applications. For each airbagin a simulation model, a separate finite element model should be specified.A multi-chamber airbag must be specified as one finite element model. Bythe discretization of the fabric skin in finite elements, the motion of the fabricand the contacts with objects allows for a more accurate description. Effectsof inertia, bag slap, and pressure forces on objects that come into contact areaccounted for during airbag deployment.

The gas in an airbag chamber is treated as a mixture of ideal gases and the statevariables, pressure and temperature, are assumed to be uniform throughoutthe chamber. As an option, the pressure distribution in the airbag can be mod-elled with a CFD method (see Section 10.11).

This chapter provides more detailed information concerning the gas thermo-dynamics implemented in the program. Suggestions and warnings for the useof the finite element model for airbag applications are presented. A more de-tailed introduction to thermodynamics can be found in Huang1 and Abbott &Van Ness2.

1Huang, F.F., Engineering thermodynamics, Collier MacMillan Publishers, London, 1976.2Abbott, M.M. and Ness, H.C. van, Thermodynamics, Schaum’s outline series, Second Edition,

McGraw-Hill Book Company, New York, 1989.

Release 7.7 223

10

Airbag models MADYMO Theory Manual

10.2 Airbag finite element mesh

The fabric skin of an airbag must be modelled with membrane or shell ele-ments. Other element types can also be used in the same FE model, as long asthey are not part of the surface of the airbag skin. For example, truss elementscan be used to model extra stiffness caused by seams.

For airbag analyses, the elements used to model an airbag chamber must formAirbagchambervolume

a closed surface so that the volume inside the chamber can be defined. Thevolume of an airbag chamber is the sum of volume contributions of the mem-brane and shell elements that make up the skin of the chamber. The volumecontribution of an element Ve equals:

• the product of the inertial Z-coordinate of the centre of the element Zc

and the area of the element projected on the XY-plane (Figure 10.1);

• the projected element area equals the scalar product of the base vectorin the inertial Z-direction e3 and

• the element unit normal n, multiplied by the element area Ae:

Ve = Zc Aen · e3 (10.1)

The volume calculation defines the volume inside a chamber only if all ele-ment normals point out of the airbag chamber and the mesh forms a closedsurface. MADYMO detects whether the element normals point out of theairbag chamber and elements form a closed surface, and then adjusts elementnormals and closes the holes when needed by adding massless triangular ele-ments that connect the nodes on the edge of the mesh.

The automatic adjustment of the airbag mesh can be eliminated if the elementmesh of an airbag chamber forms a closed surface and all element normalspoint out of the airbag. This makes it necessary to cover exhaust orifices withmembrane elements and specify gas outflow properties for these elements.MADYMO checks if the airbag mesh is properly defined by calculating thechamber volume by projecting elements on the inertial XY, YZ and ZX planes(see Figure 10.1). If the differences between these three volumes are largerthan might be expected from numerical inaccuracies, a warning will be writ-ten to the REPRINT file.

224 Release 7.7


10Figure 10.1: Contribution of triangular element to airbag volume.

The finite element mesh of the airbag fabric must represent the airbag in theInitial MetricMethod initial configuration, this means that each node’s initial coordinates must be

specified. A second, separate mesh can be specified to represent the unde-formed, design configuration.

The design configuration can be transformed into a folded configuration usingutility software (e.g. MADYMO/Folder). In the folding process, the elementswill always be distorted to some extent. In some cases, the folded configura-tion is simply represented by scaling parts, or the entire airbag. To account forthe distortions introduced in the folding or scaling process, the solver musthave knowledge of both the reference (or design) configuration, and the ini-tial (or folded) configuration. A mapping between the two configuration isestablished to calculate the strains and stresses with respect to the referenceconfiguration. This mapping, or method, is called the Initial Metric Method(IMM). Hence, the IMM is used when the initial and an undeformed configu-ration (the reference or design configuration) of the airbag are specified.

The IMM can be used for airbags modelled with membrane elements com-bined with all valid airbag materials. The material direction vector, used forthe orthotropic and anisotropic material model, is related to the referencemesh. Two IMM procedures are available: a spring-damper model and astrain-based model.

This IMM procedure is based on a discrete spring-damper model and can beSpring-dampermodel used for both geometrically linear and non-linear membrane elements. The

method uses a special tension-only state of the elements: if an element issmaller compared to its size in the reference configuration, denoted as the un-tensed state, no element strains and stresses are introduced. In the untensedstate fictitious element forces are generated to prevent instability; these ficti-tious forces are calculated from the elongation and/or relative elongation ve-

Release 7.7 225

10


locity of the element sides and diagonals (quadrilateral element). The springstiffness to resist the side elongation is derived directly from the element ma-terial properties. The damping constant of the damper,which produces a forceagainst the side elongation rate, is related to a user-defined damping parame-ter. If the initial state of the element is tensed, i.e. all elementsides in the initialconfiguration are larger compared to its size in the reference configuration, theelement will have stresses due to the initial, positive strains. These stresseswill cause the airbag to move, even before the airbag is triggered. These posi-tive stresses can be partially eliminated by specifying a threshold-strain levelparameter, which causes an increase of the reference size of the initial tensedelements with strains smaller than this threshold value.

When during the airbag simulation the state of the elements changes from un-tensed to tensed, these elements switch from the specific IMM formulation tothe standard stress-strain formulation and a discontinuity in the nodal forcesdue to this transition may occur. Moreover, continuity of nodal forces over thetransition, and therefore a constant total energy, cannot be guaranteed.

In an airbag deployment simulation using the spring-damper model, there isno guarantee that all spring-damper systems will be transformed into FE ele-ments. For instance for the elements nearby the boundary or nearby a wrin-kled surface, the transformation may never take place, since those elementsmay never reach their reference geometry that is necessary for the transition.When not all elements are fully transformed, the airbag suffers from inaccu-rate area, volume and shape which adversely affects the simulation results.

Since MADYMO R7.4, an improved transition algorithm is available basedon the global area ratio between the current airbag area and reference airbagarea. The current and reference area are determined based on the total airbagarea, i.e. the sum of area of all airbag chambers in an FE-model. If the globalarea ratio exceeds a certain predefined threshold value (the default value is0.7) and at least one element is out of IMM then the enforced transition willbe applied. During the enforced transition, all untensioned spring-damperswill be forced to transfer into FE-elements within a certain predefined time-window (the default is 0.02 s). The number of transformed elements per time-step vtrans f er can be roughly computed as vtrans f er = Γ/(tw/∆t), with Γ is thetotal number of untensed elements, tw is the user-predefined time-window,and ∆t is the current time-step. Note that during the imposed transition pro-cess some spring-dampers might be transfered to FE-state in an natural wayand the time-step also changes during the simulation. Additionally, only theelements that satisfy a certain minimum area requirement will be involved inthe enforced transition process. Hence, the above relation for vtrans f er is onlya rough estimation of the actual transition progress.

During imposed transition the elements are actually being "pushed" to assumetheir reference geometry in an FE-state. During this process some artificialexternal force will be involved while moving the nodes from their untensedstate onto their reference state. After the imposed transition the elements willhave a new equilibrium state, different from the one before the imposed tran-sition. Hence, the new energy balance ∆E can be computed as ∆E = ∆E∗ -Wtrans f er, with ∆E∗ is the total energy balance after imposement and Wtrans f er

is the amount of energy being induced during imposed transition (i.e. the so-

226 Release 7.7


10

called Transition Energy). Wtrans f er is computed based on the sum of transitionenergy of all elements undergoing imposed transition.

This IMM procedure is a strain-based method, making use of the standardStrain-basedmodel strain and stress calculations; this method is only valid for geometrical linear

membrane elements.

Before the airbag simulation is started, i.e. the airbag is triggered, the IMMtransition must be performed during a pre-simulation. This IMM transition,i.e. the movement from the initial configuration to the reference configura-tion is performed by applying an initial displacement field (which is given bythe mapping between the reference and initial configuration) during a pre-defined time duration. After the initial displacement field has been applied, arelaxation phase is needed for obtaining an equilibrium state before the airbagis triggered. In this way, the starting point of the airbag simulation is a quasi-static equilibrium state, but not necessary free of strains and stresses.

Because there is no transition from a specific IMM formulation to the standardstress-strain formulation (as in the spring-damper model), a discontinuity inthe nodal forces due to this transition will not occur.

Rayleigh damping is used to obtain a quasi-static equilibrium state at the endof the pre-simulation. An optimal convergence can be obtained by using thecritical damping constant for the system. This critical damping constant canbe estimated by choosing a proper value for the alpha Rayleigh dampingcoefficient during the pre-simulation. The estimated range for the Rayleighdamping coefficient is α = 1000 − 10000 , with a recommended value forα = 10000 . At the end of the pre-simulation the alpha damping coefficienthas to be smoothly decreased to the normal alpha damping coefficient usedduring the unfolding process.

The time step calculation and the computation of the element masses areElement meshtime stepcalculation

based on the reference mesh. This means a very distorted element mesh inthe initial configuration can also yield a "normal" time step.

10.3 Material behaviour

The stress distribution in the airbag fabric can be calculated and comparedwith critical values for the rupture of fibres. The material properties of techni-cal fabrics that are used for airbag manufacturing are directional dependent.Usually two perpendicular principal material directions can be recognisedthat coincide with the warp and weft directions of the fabric. Due to the kine-matic interaction between the warp and weft threads and their undulation inthe unstressed state, the stiffness increases with the loading for uniaxial ten-sion tests. However, this effect is small under bi-axial tension. Therefore alinear orthotropic material behaviour is sufficient for most applications.

Most current driver-side airbags consist of two flat, circular pieces of fabricWrinklingsewed together. During inflation, the airbag is filled and will experience in-creasing stressing and wrinkling of the fabric. This wrinkling is due to thereduction of the circumference of the driver-side airbag during deployment.

Release 7.7 227

10


Using membrane elements with linear material behaviour would require sev-eral elements per wrinkle to achieve a realistic model, which would lead tomeshes with thousands of elements. Tension-only models have been devel-oped to account for the stiffness reduction due to wrinkling with a small num-ber of finite elements.

10.4 Airbag inflation process

The airbag chamber inflation process is modelled as an expanding volumewith mass flowing into and out of the volume enclosed by the airbag skin.The fabric skin of the chamber is loaded by the internal pressure, which resultsfrom the gas blown into the volume by the inflator(s). Gas can flow into andout of the chamber through exhaust orifices and pores in the fabric.

The gas mass in the chamber m is the result of the inflator-supplied gas massms , the inflowing gas mass mi and the exhausted gas mass mex . The changeof mass in a chamber is:

m = ms + mi − mex (10.2)

where the m , ms , miand mex are the time derivatives of m, ms , mi and mex

respectively.

Several inflators can be defined for each chamber. Each inflator supplies gas,Inflatormass and heat into the chamber and must be defined separately, its supplymechanism is activated by a switch. The heat supply, or enthalpy, is deter-mined by the temperature and the gas species in the gas mixture coming out ofthe inflator. The mass outflow of each inflator must be specified by means of amass flow rate versus time characteristic. Also the supplied (time-dependent)gas mixture and the temperature at the exit plane of the inflator Texit as a func-tion of time must be defined. If the jet option is selected, this temperature willbe used in the gas jet calculations (page 229).

• If the expansion of the gas from the inflator is isothermal, the tempera-ture at the exit plane Texit will be used as the supply temperature of theinflator Ts in the thermodynamics calculations for the gas in the chamber(Section 10.6).

• If polytropic expansion takes place at the inflator exit plane, the supplytemperature of the inflator will be calculated as:

Ts = Texit

(P

Pexit

) n−1n

(10.3)

• With pexit the pressure at the exit plane of the inflator, p the uniformpressure in the chamber and n the polytropic constant, 1 ≤ n ≤ γ .Parameter γ represents the ratio of the constant pressure heat capacitycp and the constant volume heat capacity cv of the inflator gas mixture(page 235).

228 Release 7.7


10

• Isothermal expansion from the inflator gas takes place if n equals 1, andisentropic expansion takes place if n equals γ.

• By default, a value 1 is assumed for n (isothermal expansion). If a differ-ent value is specified, the inflator exit pressure time history must also bespecified.

The inflator input data is often obtained from a tank test of the actual infla-tor. The MTA or CGI program can be used to obtain the desired mass flowrate, gas temperature, exit pressure and polytropic constant from tank testdata. The gas characteristics, heat capacities and molecular weight that mustbe specified will be explained on page 235.

Different analytical models can be applied to account for the inertia effectsGas jetof a gas jet coming out of an inflator. The advantage of this analytical ap-proach, compared to a separate discretization for the gas in the airbag, is thatthe necessary computational time will approximately only double (due to thecomplex search algorithm to determine the elements effected by the jet). Asopposed to using a separate discretization that would increase the computa-tional time by at least an order of magnitude.

By default, all airbag elements, except the hole elements, in the airbag cham-ber in which the jet is defined and that enter the jet zone are influenced by thejet. A reduction in computer time can be achieved by limiting the number ofelements that are used in the jet calculation.

Jet forces will only be applied on those nodes that are selected and "visible"from that part of the inflator outflow area that is situated within the controlvolume of the airbag. The part of the inflator area that is situated outside thecontrol volume does not contribute in determining on which nodes jet forcesneed to be applied.

The flow is assumed to be supersonic in the inflator outlet. Combining thiswith the ideal gas law and the energy equation for an adiabatic flow processleads to the following equation for the gas velocity in the outlet vo , the densityof the gas in the outlet ρo and the critical pressure po:

vo =√

(γ − 1)cpTexit (10.4)

ρo =ms

Aivo(10.5)

po =ms

Ai

√γ(γ − 1)cvTexit (10.6)

where Ai is the inflator outlet area, ms is the supplied gas mass, Texit the tem-perature at the exit plane of the inflator, cv and cp the heat capacities and γ theratio of heat capacities.

The shape of the jet depends on the shape of the inflator outlet defined inJet shapethe input. The centre line velocity in each jet’s cross-section is a non-linearfunction of:

Release 7.7 229

10


• the gas velocity in the outlet,

• the inflator’s dimensions and

• the cross-sectional distance z to the inflator outlet vmax = vo f (z, Do)where Do is the radius of the circular outlet Ro or the half-length of therectangular outlet opening’s Bo shorter side.

1. For a circular outlet, use a cone-shaped jet (Figure 10.2).

Figure 10.2: Circular outlet: cone-shaped jet.

2. For a rectangular outlet, use a wedge-shaped jet (Figure 10.3). The widthof the wedge-shaped jet is constant and equal to the length of the longerside of the inflator outlet.

230 Release 7.7


10Figure 10.3: Rectangular outlet: wedge-shaped jet.

Figure 10.4: Velocity distribution in the jet.

A jet consist of two zones (Figure 10.4):

• Within the initial zone (z < zo), a constant velocity core is present withInitial zoneits maximum width at the inflator outlet opening and contracting to zero

Release 7.7 231

10


width at the end of the initial zone. Outside the constant velocity core, aGaussian velocity profile is assumed.

• In the main zone, the centre line velocity decreases with an increasingMain zonecross-sectional distance to the inflator outlet. For the velocity distribu-tion, a Gaussian profile is assumed (for a rectangular outlet only) in thedirection parallel to the inflator outlet’s shorter side. The velocity profilein the main zone at distance z to the outlet opening has the form:

v

vmax= e

−r2

2S2 (10.7)

where r is the distance to the centre line for a cone-shaped jet or the distance tothe centre plane for a wedge-shaped jet. S, the standard deviation, is a linearfunction of z. S is calculated as:

S = cD (10.8)

where D is the local jet radius R for a cone-shaped jet:

D = R = Ro + z tan α (cone) (10.9)

or the local half-width B of a wedge-shaped jet:

D = B = Bo + z tan α (wedge) (10.10)

Three models are available to simulate cone-shaped jets, and two models toThree jetmodels: simulate wedge-shaped jets:

This model is based on the theory of Idelchik1 for cone-shaped and wedge-Idelchik jetmodel shaped jets:

The angle of the jet divergence 2α is obtained from tan α = 3.4a′ for a cone-shaped jet. Where a′ represents the coefficient of turbulence for the jet, whichis equal to:

• 0.08 for a cone-shaped jet resulting in α = 0.262rad(15) .

1Idelchik, I.E., Handbook of Hydraulic Resistance, Hemisphere Publishing Corp., Washington,U.S.A., 1986.

232 Release 7.7


10

Within the main zone, the centre line velocity for a cone-shaped jet is calcu-lated as:

vmax =0.96vo

a′zRo

+ 0.29(10.11)

The angle of the jet divergence 2α is obtained from tan α = 2.4a′ for a wedge-shaped jet. Where a′ represents the coefficient of turbulence for the jet, whichis equal to:

• 0.105 for a wedge-shaped jet resulting in α = 0.247rad(14.14) .

Within the main zone, the centre line velocity for a wedge-shaped jet as:

vmax =1.2vo√

a′zBo

+ 0.41(10.12)

The Constant momentum jet model is based on conservation of momentum inConstantmomentum jetmodel

cone-shaped jets. The user can specify the half angle of jet divergence α andparameter c, which defines the standard deviation S of the Gaussian distribu-tion (10.7). Because of conservation of momentum and using (10.8) and (10.9),it follows that in the main zone of a cone-shaped jet:

vmax =voRo

S(z)=

vo

c(

1 + z tan αRo

) (10.13)

The length of the initial zone follows from:

S(zo) = Ro (10.14)

so

zo =Ro(

1c − 1)

tan α(10.15)

From (10.8) and (10.9) and tan α > 0 , it follows:

S = cR = c tan α

(Ro

tan α+ z

)(10.16)

With Ro/ tan α + z representing the distance of a cross-section to the imaginary"source" of the jet.

Good agreement with experimental data was found with c tan α between 0.100and 0.077 for values of the Reynolds number between 1.0E4 and 1.0E5. Arecommended value for inflator gas jets is c tan α = 0.081 .

Release 7.7 233

10


If α is defined as zero, the initial zone will have an infinite length and anuniform velocity vo will be used in the jet.

The user-defined jet model allows the centre line velocity vmax to be specifiedUser-definedjet model for cone-shaped as well as wedge-shaped jets. The user can also specify the

half angle of jet divergence α and c which defines the standard deviation S.The end of the initial zone is located at that distance to the outlet for whichthe prescribed gas centre line velocity vmax starts deviating from vo . If thefollowing conditions are met, no momentum will be gained at the end of theinitial zone.

For a cone-shaped jet:(

1

c− 1

)Ro ≥ zo tan α (10.17)

For a wedge-shaped jet:

(2

c√

π− 1

)Bo ≥ zo tan α (10.18)

To guarantee that no momentum is gained in the main zone, the centre linevelocity must meet the following conditions for z > zo for a cone-shaped jet:

dvmax

dz≤ −vo tan α(Ro + zo tan α)

(Ro + z tan α)2(10.19)

and for a wedge-shaped jet:

dvmax

dz≤ −0.5vo tan α

√Bo + zo tan α√

(Bo + zo tan α)3(10.20)

For all three models, the velocity of every point in the jet is known. The onlyGeneral pointselements affected directly by the jet are those:

• that do not represent holes with a centre inside the jet defined by thecone or the wedge and

• which are not hidden behind other elements.

The extra pressure pjet applied to these elements depends on the gas velocityv at the centre of the element, the density of the gas in the jet ρo and the jetefficiency factor η:

pjet = ηρov2 (10.21)

In the Idelchik jet model, only the outlet geometry has to be specified.

In the constant momentum jet model, the half angle of jet divergence α andparameter c defining the standard deviation S of the Gaussian distributionalso have to be specified.

Finally, the user-defined jet model needs to include:

234 Release 7.7


10

• the outlet geometry,

• the angle α,

• the ratio of centre line velocity and gas velocity in the outlet as a functionof the distance to the outlet opening z and the

• parameter c defining the standard deviation S of the Gaussian distribu-tion.

The definition of an ideal gas requires that (at all temperatures and pressures)Thermalproperties ofgases

the internal energy is only a function of the temperature and that the ideal gaslaw is valid.

Often the amount of substance contained in a system is given as a number ofmoles. A mole is defined as the amount of substance made up of 6.02252 · 1023

molecules. The mass of one mole of a certain gas is called its molar weight(MW). The molar weights of the predefined gases have been summarised inTable 10.1.

The amount of heat which must be added to a closed system to achieve a givenchange of state depends on how the process is carried out. The constant pres-sure heat capacity cp and the constant volume heat capacity cv are often usedparameters. For ideal gases, these are related to the universal gas constant R:

R = cp − cv (10.22)

with R = 8.314J mol−1K−1 for all gases. The ratio of heat capacities is oftendenoted by γ:

γ =cp

cv(10.23)

Two models can be used to model the temperature dependency of the heatcapacity at constant pressure. The first model is described by the NIST Chem-istry WebBook, NIST Standard Reference Database Number 69 – March, 2003Release (http://webbook.nist.gov) and is applicable for gases with relavilyhigh temperatures. It is described by the following equation:

cp = a0 + a1T + a2T2 + a3T3 +a4

T2(10.24)

where a0, a1, a2, a3 and a4 are constants and are given in Table 10.2 and T isthe absolute temperature of the gas.

Another possible model for the heat capacity at constant pressure is based onThe Properties of Gases and Liquids, 5th edition by B.E. Poling, J.M. Prausnitzand J.P. O’Connell. The heat capacity at constant pressure is given by:

cp = b0 + b1T + b2T2 + b3T3 + b4T4 (10.25)

Release 7.7 235

http://webbook.nist.gov

10


This model is more suitable for low temperature gases and it is recommendedto use this when stored gas inflators are used, because in this case low tem-peratures will be encountered.

The effect of temperature is minimal for monatomic gases such as helium,neon and argon. For diatomic gases such as oxygen and nitrogen, the heatcapacities change slowly with temperature. For poly-atomic gases, the heatcapacities vary considerably with temperature and differ significantly fromgas to gas. The coefficients a0 through a4 and b0 through b4 with their temper-ature range have been summarised in Table 10.2 and Table 10.3, respectively,for several gases that can be used for airbag applications. All of these gasesare predefined in the code using the values listed in Table 10.1 and 10.2.

The molar fractions of the species in the gas mixture flowing out of the inflatormust be defined. The user can, however, redefine these values or specify newgases by using one of the user-defined gases. This last option is useful only ifproperties of the gas mixture are available.

Table 10.1: Molar weights.

Gas MW[kg/mol ]

Gas MW[kg/mol ]

Nitrogen (N2) 0.02801 Hydrogen (H2) 0.00202

Oxygen (O2) 0.03200 Water vapour(H2O) 0.01802

Carbon dioxide(CO2)

0.04401 Ammonia(NH3) 0.01703

Carbon monoxide(CO)

0.02801 Hydrogensulphide(H2S)

0.03408

Argon (Ar) 0.03995 Benzene (C6H6) 0.07811

Neon (Ne) 0.02018 Nitrous oxide (N2O) 0.04401

Helium (He) 0.00400

Table 10.2: Heat capacity coefficients: NIST model

Gas a0

[J/(mol K) ]a1

[J/(mol K 2)]a2

[J/(mol K 3)]a3

[J/(mol K 4)]a4

[J K/mol ]Temp.range(K)

N2 26.092 8.218801×10−3 -1.976141×10−6 1.59274×10−10 4.4434×104 298 –6000

O2 29.659 6.137261×10−3 -1.186521×10−6 9.578×10−11 -2.19663×105 298 –6000


236 Release 7.7


10

Table 10.2 cont.

Gas a0

[J/(mol K) ]a1

[J/(mol K 2)]a2

[J/(mol K 3)]a3

[J/(mol K 4)]a4

[J K/mol ]Temp.range(K)

CO2 24.99735 5.518696×10−2 -3.369137×10−5 7.948387×10−9 -1.36638×105 298 –1200

CO 25.56759 6.09613×10−3 4.054656×10−6 -2.671301×10−9 1.31021×105 298 –1300

Ar 20.786 2.825911×10−10 -1.464191×10−13 1.092131×10−17 -3.661371×10−2 298 –6000

Ne 20.78603 4.850638×10−13 -1.582916×10−16 1.525102×10−20 3.196347×10−5 298 –6000

He 20.78603 4.850638×10−13 -1.592916×10−16 1.525102×10−20 3.196347×10−5 298 –6000

H2 33.066178 -1.136342×10−2 1.1432816×10−5 -2.772874×10−9 -1.58558×105 298 –1000

H2O 30.092 6.832514×10−3 6.793435×10−6 -2.534480×10−9 8.2139×104 500 –1700

NH3 19.99563 4.977119×10−2 -1.537599×10−5 1.921168×10−9 1.89174×105 298 –1400

H2S 26.88412 1.867809×10−2 3.434203×10−6 -3.378702×10−9 1.35882×105 298 –1400

C6H6 -36.22 4.8475×10−1 -3.157×10−4 7.762×10−8 0.0 298 –1500

N2O 27.67988 5.114898×10−2 -3.064454×10−5 6.847911×10−9 -1.57906×105 298 –1400

Table 10.3: Heat capacity coefficients: Poling model

Gas b0

[J/(mol K) ]b1

[J/(mol K 2)]b2

[J/(mol K 3)]b3

[J/(mol K 4)]b4

[J/(mol K 5)]Temp.range(K)

N2 29.4249 -2.17008×10−3 5.82013×10−7 1.30537×10−8 -8.23133×10−12 50 –1000

O2 30.1815 -1.49462×10−2 5.47092×10−5 -4.99700×10−8 1.48829×10−11 50 –1000

CO2 27.0969 1.12744×10−2 1.24883×10−4 -1.97386×10−7 8.78008×10−11 50 –1000

CO 32.5262 -3.25345×10−2 9.82771×10−5 -1.08254×10−7 4.28195×10−11 50 –1000

Ar 20.7862 0.000 0.000 0.000 0.000 0 – ∞


Release 7.7 237

10


Table 10.3 cont.

Gas b0

[J/(mol K) ]b1

[J/(mol K 2)]b2

[J/(mol K 3)]b3

[J/(mol K 4)]b4

[J/(mol K 5)]Temp.range(K)

Ne 20.7862 0.000 0.000 0.000 0.000 0 – ∞

He 20.7862 0.000 0.000 0.000 0.000 0 – ∞

H2 23.9706 3.06056×10−2 -6.41877×10−5 5.75361×10−8 -1.77098×10−11 50 –1000

H2O 36.5421 -3.48044×10−2 1.16818×10−4 -1.30038×10−7 5.25475×10−11 50 –1000

NH3 35.2367 -3.50455×10−2 1.69698×10−4 -1.76766×10−7 6.32731×10−11 50 –1000

H2S 35.4695 -2.85852×10−2 1.09668×10−4 -1.10666×10−7 4.05746×10−11 50–1000

C6H6 29.5247 -5.14167×10−4 1.19437×10−3 -1.64685×10−6 6.84614×10−10 50 –1000

N2O 26.3153 2.82775×10−2 8.22301×10−5 -1.56312×10−7 7.39988×10−11 50 –1000

The gas in an airbag is treated as a mixture of ideal gases. The average heatcapacities of a mixture of gases are determined by the Amagat-Leduc rule ofpartial volumes. This rule states that any thermal property of a gas mixturecan be approximated by the molar average value, for the NIST model:

cp,mix = ∑ a0ixi + T ∑ a1i

xi + T2 ∑ a2ixi+

T3 ∑ a3ixi +

1

T2 ∑ a4ixi (10.26)

And for the Poling model this is:

cp,mix = ∑ b0ixi + T ∑ b1i

xi + T2 ∑ b2ixi+

T3 ∑ b3ixi + T4 ∑ b4i

xi (10.27)

where xi is the molar fraction of gas species i.

10.5 Ideal gas law

Based on moles, the ideal gas law for a certain volume V occupied by the gascan be written as:

pV = nRT (10.28)

where p is the pressure and n is amount of gas expressed in moles. Often theideal gas law is formulated based on mass:

pV = mRT (10.29)

238 Release 7.7


10

The constant R used in this equation is different for different gases, whereas

R = 8.314J mol −1K−1 for all gases. The relationship between R and R is givenby:

R =R

MW(10.30)

where MW is the molar weight of the gas.

10.6 Conservation of energy

The first law of thermodynamics for a homogeneous gas system may be writ-ten as:

d(mU) = dQ − dW (10.31)

where m is the mass of the system, U the specific internal energy, dQ the heatflow and dW the work done by the gas. This equation is the mathematicalformulation of the first principle of energy conservation as applied to a processoccurring in a closed system.

This equation can also be applied to steady-state flow. In unsteady-state flow,Unsteady-stateflow as is the case for an airbag chamber (Figure 10.5), the mass in the system no

longer needs to be constant and energy may be accumulated or depleted. Ne-glecting the kinetic and potential energy, the net energy transport of the gasflowing into and out of a chamber equals the enthalpy difference, ∆(Hdm) , ofthese flows. The energy equation then becomes:

d(mU) + ∆(Hdm) + dW − dQ = 0 (10.32)

If a significant amount of heat flow through the airbag fabric occurs, thiswill be accounted for by defining a non-zero coefficient of heat transfer k (seepage 241):

dQ = kA(T1 − T2)dt (10.33)

where T1 is the temperature of the gas mixture inside one airbag chamber,T2 the temperature in the other chamber or the ambient temperature, A thesurface of the fabric where the heat flow occurs, and t the elapsed time.

Release 7.7 239

10


(mex, Hex)1

inflatorTs

(mex, Hex)1•

ms, Hs•

pa, Ta (environment)

Chamber 1p1 V1 m1 T1

Chamber 2p2 V2 m2 T2

• (mex, Hex)2 (vents)•

(mex, Hex)2 (pores)•

Figure 10.5: Mass and heat flow into and out of the airbag chambers.

The gas mixture in the chamber is treated as an ideal gas:Gas mixture inairbagchamber dU =d(cvT) (10.34)

∆H =∆(cpT) (10.35)

where cp and cv are the constant pressure and constant volume heat capacityper unit of mass of the gas mixture respectively.

The ideal gas heat capacities are different for different gases and functions oftemperature only. For chamber i substitution in the energy equation leads to:

micviTi + mi cvi

Ti + micviTi+

mexicpi

Ti + kA ∑chamj 6=i

(Ti − Tj) + piVi + Qged =

= ∑inflchami

mscpTs + ∑chamj 6=i

minjcp jTj

(10.36)

where:

• mexiis the mass flow rate through exhaust orifices and pores,

• ms is the mass flow rate coming out of an inflator,

240 Release 7.7


10

• Ts is the temperature of the gas coming out of the inflator and

• minj is the mass flow rate from chamber j into chamber i,

• Qged represents the global energy discharge.

This equation is used for the calculation of the temperature change of the gasTemperaturechange of gas inside the airbag chamber.

For an airbag chamber, the pressure and temperature of the gas mixture arecalculated based on the volume enclosed by the finite elements used to modelthe chamber fabric at the end of the preceding time step. This volume, how-ever, is often very small before the inflators are activated.

For a stable integration of the thermodynamic equations, a minimum initialgas volume is required. MADYMO estimates the necessary minimum initialvolume so that the Courant condition determines a time step for the airbag’selement mesh. This leads to a stable time integration of the thermodynamicequations. The required extra volume, V0, is added to the volume enclosedby the finite elements to give the total gas volume:

p =mRT

V + V0(10.37)

For most airbag meshes, the extra volume of a chamber V0 is so small that ithas only a significant influence during a few finite element time steps aftertriggering of the inflators. The user can specify the number of subincrementsto be integrated in the differential equations that describe the gas thermody-namics.

Due to this alteration, the user does not have to include a small initial volumefor every chamber in the mesh. The volume V0 and any initial volume en-closed by the finite elements is filled with air at ambient pressure and temper-ature. Air is modelled as a gas mixture with molar fractions nitrogen 0.78084,oxygen 0.20946, carbon-dioxide 0.00033 and argon 0.00937. The ambient tem-perature and pressure can be redefined. If the value for the ambient pressureis altered, this value will also be used to determine the relative pressure (thepressure difference between the airbag and the environment). The relativepressure determines the forces acting on the airbag skin’s elements and theoutflow of gases to the surroundings through exhaust orifices and pores.

In equation (10.36), heat flow through the airbag material is characterised byHeat flowthrough airbag a coefficient of heat transfer k. Heat flow without mass flow can be accounted

for by with this option. Heat flows from one chamber to the another or from achamber to the ambient air. Due to the very complex phenomena that causesthis heat flow, the coefficient k will mainly be used as a tuning parameter forthis heat flow.

For steady-state heat flow through a flat wall, a physical relationship for kConvectionandconduction

can be derived. Coefficient k then accounts for two types of heat transport(convection and conduction) which act simultaneously. On both sides of the

Release 7.7 241

10


wall, convection takes place (in the heat transport from gas to wall and fromwall to gas). Each form of convection is defined by a coefficient of heat transferαi, i = 1, 2 . Also conduction of heat through the wall takes place, defined bythe conductivity factor λ, which is material specific.

λ

Tambient

α2

α1

h

Q•

T

Figure 10.6: Heat flow through airbag fabric.

k is related to α1 , α2 , λ and h as:

1

k=

1

α1+

h

λ+

1

α2(10.38)

where h is the thickness of the wall. The parameter k can be defined with thematerial properties.

For efficiency reasons and due to the limited effect of the area change, the un-deformed element surface is used instead of the actual area in the heat transfercalculations.

242 Release 7.7


10

During the inflation process of an airbag chamber, gas can flow out of and intoGas flowthroughexhaustorifices andfabric

the volume bounded by the airbag fabric through exhaust orifices and poresin the airbag fabric. This gas flow represents a flow of mass and heat. Not onlyseparate models for exhaust orifices and porous fabrics are available, but alsooverall outflow to the ambient air for the chamber as a whole can be defined.The resulting mass flow rate is calculated as the sum of the contributions re-sulting from the different outflow models.

If a flow between two airbag chambers is modelled, the gas can flow in eitherdirection depending on the pressure values in both chambers. Flow betweena chamber and the ambient air can only take place into the ambient air. Anorifice can also be modelled using the model for porous fabrics. Parameterscan be chosen such that the models for the orifices and porous fabrics result inalmost the same flow of gas. However, some differences still remain. Thepressure works on the elements used to model the fabric, but no pressureworks on exhaust orifices. The material stiffness of the fabric is also takeninto account, but holes have no resistance against deformations because theyhave almost no stiffness. Since this may introduce instabilities, it is essentialthat hole elements always consist of nodes which are already used by fabricelements.

Outflow from any of the models to the ambient air will take place only if theAspirationpressure in the airbag exceeds the ambient pressure. Aspiration can be ac-counted for by specifying an extra inflator that supplies air at the ambienttemperature. Air can be modelled as a gas mixture with molar fractions nitro-gen 0.78084, oxygen 0.20946, carbon-dioxide 0.00033 and argon 0.00937. Themass flow for this inflator can be approximated by the difference of the massflows obtained from two similar experiments, one with and one without aspi-ration.

The gas flow through the airbag fabric and exhaust orifices which are in con-Block flowtact is reduced by multiplying the outflow by a blocking factor. Gas flowthrough elements attached to a node, which is specified in a contact inter-action, is reduced when that node is in contact. This reduction is also appliedto hole elements that are created by MADYMO if the chamber mesh is nota closed surface. The blocking factor can be specified on global level for allelements, but can be overwritten on material level, for the elements of thatmaterial.

To model an exhaust orifice, a hole material must be used. This material typeExhaustorifices implies that the element considered has no resistance against deformations.

This means that stress concentrations around exhaust orifices can be calcu-lated. Also the airbag pressure is not applied on hole elements. This resultsin a pressure deficit that can lead to more realistic motion of the entire airbag.The area of an exhaust orifice is equal to the total area of the elements usedto model the exhaust orifice; depending on the chosen area type, the area iseither fixed in time or updated according to the actual deformation.

Two different models can be applied to account for the mass flow through ex-haust orifices. For flow between a chamber and the ambient air, these modelsare only active at time points when the pressure in the airbag chamber ex-

Release 7.7 243

10


ceeds the ambient pressure. For flow between two chambers, the gas exhausttakes place from the chamber with the higher pressure. A bursting cover canbe simulated by specifying a relative pressure level ∆p0 and a time intervalduring which this level must be exceeded to start the flow.

In the first model, the mass flow through exhaust orifices is approximated byGas outflowmodel 1 a one-dimensional, quasi-steady, isentropic flow model. Poisson’s equation

states:

ρ = ρex(p/pex)1/γ (10.39)

where:

• p is the pressure in the airbag chamber,

• pex is the pressure in the exhaust orifice,

• ρ is the density of the gas in the airbag chamber,

• ρex is the density of the gas in the exhaust orifice and

• γ is the ratio of the heat capacities of the gas in the chamber.

Conservation of energy for a compressible gas is formulated by the Bernoulliequation:

v2 = 2γ/(γ − 1)(p/ρ − pex/ρex) (10.40)

where v is the average gas velocity in the exhaust orifice. The mass outflowrate through the exhaust orifice is given by:

mex = ρexAeffv (10.41)

where Aeff is the effective area of the exhaust orifice. This area is smaller thanthe actual area of the exhaust orifice A because of non-isentropic flow effects.These can be accounted for by the discharge coefficient CDex defined by:

Aeff = CDexA (10.42)

The discharge coefficient can be specified by the user. If the total area of theelements used to model the exhaust orifice is not equal to the effective outflowarea, this will be corrected by altering the discharge coefficient of the finiteelements. For this specific reason, values even larger than 1 are allowed asinput.

A time and relative pressure dependency in the discharge can be defined by:

Aeff = CDexCDpCDT A (10.43)

in which CDP and CDT represent a pressure difference and time-dependentdischarge factor, to be specified as functions.

244 Release 7.7


10

Combining these equations and the ideal gas law (see equation (10.29)) resultsin:

mex = CDexCDpCDT A

(p

R

)√

2(cp

T)

[(

pex

p)

2γ − (

pex

p)

(γ+1)γ

](10.44)

This mass flow equation is used to calculate the mass outflow. For subsonicflow, pex is equal to the ambient pressure p2 for flow between a chamber andits surrounding. For flow between chambers, pex is equal to the pressure p2 inthe chamber to which the gas flows. The flow in the orifice is sonic or chokedwhen the pressure inside the chamber exceeds the critical level:

p > p2

(1

2γ +

1

2

)γ/(γ−1)

(10.45)

Then the pressure in the exhaust orifice is equal to

pex = p [2/(γ + 1)]γ/(γ−1) (10.46)

In the second model, the mass flow rate through exhaust orifices is calculatedGas outflowmodel 2 as:

mex = ρCDex fpf(∆p) ftf(t)A (10.47)

The product CDex fpf(∆p) ftf(t) defines a specific leakage rate with dimensionsm/s . A is the deformed area of the orifice, CDex an area scale factor. fpf is afunction that can be defined for a relative pressure dependent specific leakagerate factor. ftf is a function prescribing a time dependent specific leakage ratefactor. If the total area of the finite elements representing the exhaust orifice isnot equal to the actual outflow area, this will be compensated by defining anarea scale factor CDex unequal to 1.

The third model is especially designed for Gasflow chamber-to-chamber ap-Gas outflowmodel 3 plications. The massflow m, the momentum mv and the energy flux u are

determined based on the local state conditions of the gas on both sides of thehole, using a FCT-calculation. For this model, the transport can be scaled us-ing CDex and time dependent function ftf . This model also incorporates theblock-flow effect (outflow is reduced when the element is in contact). Notethat Gasflow grids may be orientated arbitrary with respect to each other, be-cause the local FCT-calculation for this hole creates its own local grid to deter-mine flow conditions.

Four different models can be applied to account for the leakage throughPorous fabricsporous fabric. This leakage can take place between two airbag chambers oran airbag chamber and its surroundings.

Mass flow between two chambers will take place from the chamber with thehigher pressure to the chamber with the lower pressure. Mass flow from a

Release 7.7 245

10


chamber to the ambient air will only take place when the pressure in the cham-ber exceeds the ambient pressure.

The first model calculates leakage through porous fabrics by using an equationPermeabilitymodel 1 for the gas flow through a flat screen placed in a channel perpendicular to the

flow. The pressure drop over the screen is calculated based on a resistancecoefficient ζ defined by:

∆p = ζ1

2ρv2 (10.48)

with ∆p the pressure drop over the screen, ρ the gas density and v the gasvelocity in the channel.

The resistance coefficient depends on the free area coefficient η, such as theratio of the total area of the pores in the screen, the free area Afree , and thescreen area A:

η = Afree/A (10.49)

Afree can depend on the pressure drop, undeformed screen area A0 and screenarea increase ∆A:

Afree = C [ f1(∆p)A0 + f2(∆p)∆A] (10.50)

If both f1 and f2 are constant functions with value 1, the free area coefficient ηwill equal C.

For airbag applications, the free area is usually very small compared to thescreen area, η ≪ 1 . In this case, a good approximation for the resistancecoefficient is given by:

ζ = 1/η2 (10.51)

The mass outflow rate through airbag fabric with free area Afree is given by:

mex = ρ f v f Afree (10.52)

where ρ f and v f are the density and the velocity of the gas flowing throughthe fabric, respectively.

If ∆p is interpreted as the pressure difference between the two sides of thefabric, the mass outflow rate will be found by combining the above equations(10.48) and (10.51), resulting in:

mex =Afree

√2ρ∆p (10.53)

Afree =η2p ft(t) [ f1(∆p)A0 + f2(∆p)∆A]

with ρ the gas density in the chamber with the higher pressure (or the gasdensity in the chamber for flow between a chamber and the ambient air) andη2

p ft(t) = η .

246 Release 7.7


10

The user only has to specify the value for the permeability factor ηp and thefunctions ft , f1 and f2 for the relevant elements. Only positive values of ∆pare used for f1 and f2 .

As is the case for the exhaust orifices, the area of the airbag fabric varies intime due to deformation resulting in a more realistic mass outflow.

In the second model, the mass flow rate is calculated from:Permeabilitymodel 2

mex = ρ f3(∆p) f4(t)A (10.54)

The product f3(∆p) f4(t) defines a specific leakage rate with dimensions m/s .A is the deformed area of the porous airbag fabric. f3 is a function prescribinga pressure difference dependent specific leakage rate factor. Only positivevalues of ∆p are used for f3 . f4 is a function prescribing a time dependentspecific leakage rate factor. This function can be used, for example, if outflowthrough some of the airbag fabric stops because the fabric comes into contactwith surrounding surfaces.

The global permeability model is compatible with the global permeabilityPermeabilityglobal model of LS-Dyna. The mass outflow rate is calculated as:

mex = ∑elements

ρ fp(∆p) ft(t)Aelement

Atotal(10.55)

in which Atotal is the total area of the chamber, excluding the holes, and ft(t)is the time-scaling function (similar to that of permeability models 1 and 2).The product fp(∆p) ft(t) defines a leakage rate (in units of m3/s). Note thatthe summation involves only the airbag elements.

For uniform pressure, this outflow model uses chamber-average data for tem-perature, pressure and density; for Gasflow local values are used. This modelis very similar to the overall bag leakage, but also incorporates the block-floweffect.

The global isentropic permeability model is compatible with the global per-Permeabilityglobalisentropic

meability model of PamCrash. The mass outflow rate is calculated as:

mex = ∑elements

ρ fp(∆p) ft(t)

√Telement

Tneighbour

Aelement

Atotal(10.56)

in which Atotal is the total area of the chamber, excluding the holes, ft(t) isthe time-scaling function (similar to that of permeability models 1 and 2) andTelement is the temperature near the segment at the side where the flow comesfrom. For outflow to ambient, Tneighbour is the ambient temperature; for out-flow to another chamber it is equal to the temperature at the side of the seg-ment where the gas is flowing to. The product fp(∆p) ft(t) defines a leakage

rate (in units of m3/s). Note that the summation involves only the airbag ele-ments.

Release 7.7 247

10


For uniform pressure, this outflow model uses chamber-average data for tem-perature, pressure and density; for Gasflow local values are used. This modelalso incorporates the block-flow effect.

10.7 Overall bag leakage

In addition to the outflow models discussed in the previous two sections, anoverall discharge of gas (mass and heat) to the surroundings only can be de-fined for an airbag chamber as a whole.

The mass outflow rate is calculated as:

mex = ρ fp(∆p) ft(t) (10.57)

In which ρ is the gas density in the airbag chamber, and ∆p is the relativepressure in the airbag chamber. The product fp(∆p) ft(t) defines a leakage rate

(m3/s) for the gas flowing out of the airbag chamber. For example, outflowthrough seams can be simulated in this way. If the pressure does not exceedthe ambient pressure (∆p ≤ 0), this outflow model will not be active.

10.8 Overall energy leakage

For tuning purposes, it is also possible to extract energy from the thermody-namic system. This can be done by using global energy discharge. For thechamber, an additional heatloss flux term is introduced:

Q = fged(t) (10.58)

This term extracts energy from the chamber, but the mass is conserved. Over-all energy leakage is independent of all other mass and energy outflow func-tions.

10.9 Straps

In order to limit the deployment range of the airbag, straps or tethers are oftenused to connect the front and the back of the airbag. This delays the momentthat the occupant comes into contact with the airbag. A strap is modelled asa massless, linear tension-only spring between two nodes. The untensionedlength and the stiffness must be defined. Forces are only calculated in thestraps when the actual length is longer than the untensioned length. A relativeelongation at which rupture occurs can be specified. The mass of the strap canbe accounted for by specifying an additional mass for the nodes to which thestrap is tied. Several straps can be modelled within each finite element airbag.

Non-linear material behaviour for straps can be modelled by specifying morestraps tied to the same two nodes with different untensioned lengths. Thesestraps will only act as parallel springs when the distance between the nodesexceeds the untensioned lengths which result in a non-linear characteristic.

248 Release 7.7


10

An alternative approach is to model the strap with one or more truss elements.Modelling straps with membrane elements is possible but discouraged whenthe bag is folded. The elements used to model the straps have to be placed sothat a folded strap is created which will have the correct length after unfold-ing. In addition, the strap elements should not be defined as being part of anyairbag chamber volume.

10.10 Suggestions and warnings

This section presents best-practice modelling techniques for FE airbag applica-tions and also provides information on input mistakes, in order to help usersdesign robust, accurate models. Examples of airbag analyses that follow thesemodelling guidelines can be found in the Application Manual.

It is recommended that the airbag is modelled using MEM3 elements basedElement choiceon GREEN strain formulation. The element behaviour is more stable underlarge distortions, such as are introduced during folding or deployment. Idealelement size is 10 mm or less, particularly in OOP studies, as the deploymentis more like a continuum.

If a coarse mesh is used, the wrinkling effect can be described by the mate-rial REDUCTION_FACTOR. This parameter reduces the element stiffness inthe compressive direction, simulating a wrinkling compressive mode; coarsermeshes require lower values (10−4 – 10−1) To prevent total element collapseto zero length, the original stiffness can be restored at a given compressionusing the REDUCTION_LIMIT_STRAIN parameter. If the GREEN strain for-mulation is used, a value of −0.49995 will restore the original stiffness oncean element shrinks to 1% of its original length.

IMM1 is usually used in OOP simulations, in which accurate folding of theIMM1airbag is important. Ideally, the ratio of the airbag reference (actual) area tothe initial folded area should be between 90% and 110%, and there should beno significant local element distortions.

During pre-simulation, an initial stress field is applied over a user-definedperiod, capturing the stretching that occurs during the folding process. Toreach static equilibrium, a user defined alpha damping function must be de-fined. This damping parameter should have a value of between 103 and 104

during the pre-simulation analysis, and this value should then be reduced tobetween 0 and 10 at the moment of triggering; otherwise, high alpha damp-ing will damp out the low frequencies. This pre-simulation process can beautomated using DYNAMIC_RELAX.

Equilibrium is more easily achieved the more the airbag is constrained, e.g. acontainer with a cover will generally allow easier equilibration than an opencontainer open.

In-Position simulation generally makes use of scaled airbags; in such casesIMM2IMM2 may be used. In order to prevent element collapse, the elements inIMM can be given a stiffness by defining the IMM_STIF_REDUC parameterat a low value, e.g. 0.0001.

Release 7.7 249

10


The default damping type is IMM_DAMP_MTH=0, and typical damping val-ues used with this option (IMM_DAMP) lie between 0.1 and 0.2. This damp-ing method is independent of the FE time step.

If the user chooses IMM_DAMP_MTH=2, the IMM_DAMP parameter shouldhave a value in the range of (10−5 – 10−4). If the damping coefficient is toohigh, too much energy will be dissipated and the elements may not come outof IMM. Conversely, too low a damping coefficient will lead to instability, es-pecially if no IMM_STIF_REDUC value is defined.

If the elements are significantly stretched, there may be stability problems.The IMM_STRAIN option can be invoked to prevent this instability; however,the disadvantage of this parameter is that it will also adjust the area of thereference mesh, potentially introducing inaccuracies. Therefore instead of us-ing IMM_STRAIN, it is better to adjust the initial mesh (so that fewer or noseverely stretched elements are introduced) in combination with high alphadamping during the pre-simulation; alternatively IMM1, which can handlestretched elements very well, can be used.

It is recommended that material damping be used in order to damp out high-Materialdamping frequency oscillations in the airbag. Useful material damping values are typ-

ically around 0.1 – 0.2, for MU=0. Low-frequency motions are not affected bymaterial damping.

For contact guidelines, please refer to the Appendix "Contact ModellingContactdefinition Guidelines" in the Reference manual.

As discussed in Section 10.6, a small initial volume is added to the airbag vol-Extra initialvolume ume enclosed by the finite elements in order to achieve a stable integration of

the thermodynamic equations. This only applies to Uniform Pressure simu-lations, as for Gasflow simulations this extra initial volume is not taken intoaccount. However, if during inflation the nodes have extra velocity due to a(prescribed) motion of the airbag support, a small extra initial volume may beneeded. The user can achieve this by enclosing a small volume with the finiteelements at the start of the simulation.

For Uniform Pressure simulations, the thermodynamic calculationsThermo-dynamicsub-cycling

(THERMC) should sub-cycle at least 25 times per FE time step. Since itis a straightforward Euler integration calculation it consumes very little CPUtime, but it makes the thermodynamic calculations more accurate.

Always check the reprint file for extrapolation warnings. Ideally, functionsExtrapolationwarning should be defined over the complete range of operation. If a function is not

defined over the whole range, MADYMO will linearly extrapolate this func-tion from the last two points at the end of the function exceeded. This has thepotential to affect the results enormously, depending on the gradient of thefunction and the amount by which the limits are exceeded.

250 Release 7.7


10

10.11 Gasflow Module

The Gasflow module implemented in MADYMO uses a combined CFD (Com-CFD/FEcouplingmethodology

putational Fluid Dynamics) FE coupling methodology. The standard FE capa-bilities of the MADYMO code are used to model the airbag components andtheir interaction with the vehicle or occupants. The interior volume of theairbag is modelled using a CFD approach. In the coupling algorithm, the FEairbag mesh forms a closed surface that defines the volume enclosed by theairbag. This surface acts as a boundary condition applied to the Gasflow meshthat constrains the motion of the inflator gases (see Figure 10.7). The pressurein the adjacent cell determines the pressure on an element. Integrating thepressure over the element surface results in nodal forces.

AIRBAG

INFLATOR

ACTIVE CELLS

INACTIVE CELLS

Figure 10.7: Airbag simulation using Gasflow: the airbag volume is divided into smallcells, each with a distinct physical state.

The volume enclosed by the airbag at the start of the simulation is automati-CalculationMethodology cally discretised into a user-defined number of computational cells. For com-

putational efficiency, a regular mesh of hexahedral cells aligned with a fixedlocal co-ordinate system is used with a uniform cell size in each direction. Themotion of the gas within the Gasflow mesh is calculated from first principlesusing the Euler conservation equations for inviscid compressible hydrody-namic flow. These equations are solved simultaneously in all three spatial

Release 7.7 251

10


dimensions using the high resolution Flux Corrected Transport method1,2. Asecond order accurate explicit method calculates the time integration in whichthe time step is controlled by the Courant stability criterion. The stability cri-terion is that a disturbance should not be able to travel across more than a sin-gle cell dimension in on single integration step. Small cell dimensions or highflow velocities in the Gasflow mesh results in small time integration steps. Theintegration time step for each cycle is the minimum of that required by the FEand Gasflow modules of the simulation.

The portions of the Gasflow mesh inside and outside the airbag are updatedas the airbag surfaces move. An efficient, approximate geometric calculationis used to determine the fraction of each cell volume and face area that is opento gas flow. If a cell face is completely blocked, gas cannot flow through thisface. If a cell volume is completely covered, the cell does not take part in theGasflow calculation. The calculation of cell and face blockage fractions is nu-merically expensive so it is only performed if the airbag surface has movedsignificantly since the last geometry update. So, for unfolding airbags the up-date will be very frequent but for tank tests there will be no update at all.Unless treated in a special manner, Gasflow cells that are almost covered willneed a small Courant stability time step. A special technique is used to over-come this problem by blending nearly covered cells with an adjacent cell.

Inflator gases are added into the Gasflow mesh where the jet outflow orifice ofthe inflator(s) intersect with the Gasflow mesh. The state of the inflator gasesis specified in the same way as for the MADYMO isobaric airbag model usingstandard data that is usually derived from tank test experiments. This processadds mass, energy and momentum into the Gasflow mesh. For efficiency, thegas within the airbag is modelled as a single ideal gas material. The ideal gasparameter, gamma, varies as the gas composition within the airbag changesduring the inflation process, and the average pressure and temperature withinthe airbag change. The inflator gas is mixed homogeneously with the gas thatis already present inside the airbag.

In order to successfully model the inflation process within a complex foldedGasflowAlgorithm airbag, the cells used in the Gasflow mesh must be small enough to resolve the

features of the interior volume. For example, a folded airbag with the foldsinitially separated by 1 mm would require an initial cell size no larger than1 mm. Enclosing the entire volume of the inflated airbag with small sizedcells would require a prohibitively large number of numerical cells, a largeamount of memory and long computation times. For this reason, the numberof cells within the Gasflow mesh is kept constant throughout the simulation,but the cells are allowed to grow or shrink when the overall airbag dimensionschange. The cell sizes are changed at discrete times and during this processthe state of the gas is mapped from the old Gasflow mesh to the new mesh,under the condition of mass, momentum and energy conservation.

1JP Boris, DL Book, ’Solution of Continuity Equations by the Method of Flux-Corrected Trans-port’, Vol. 16, Methods in Computational Physics, Academic Press, 1976

2ST Zalesak, ’Fully Multidimensional Flux-Corrected Transport Algorithms for Fluids’, Jour-nal of Computational Physics Vol. 31, pp335-362, 1979

252 Release 7.7


10

If the airbag surface is tangled, the tangled region is considered as "outside theairbag". As a consequence the physical state of the airbag (such as chambervolume) is incorrect. It is recommended to prevent contact tangling e.g. byusing a smaller FE time step. For the same reason it is also advised to have noinitially intersected airbag elements.

For folded airbags this would require a large amount of Gasflow cells in orderAnti ThroughFlow to prevent flows through layers initially but also during the unfolding process

when the volume starts to grow and as a consequence, the Gasflow cells willalso grow. In order to prevent this flow without using an excessive amountof Gasflow cells, the anti through flow option can be used. Using this op-tion, no flow through layers of fabric will occur. Punch-out effects are welldescribed with the new algorithm, just like the general unfolding behaviour.If this option is used in combination with a Gasflow mesh which is too coarse,the Gasflow cells in small volume regions might become inactive. For exam-ple: connection tunnels in curtain airbags or inflator houses. As a consequenceno gas can flow through this region. These phenomena can be relatively easilychecked by output data like kinematics and time history files. To circumventinactive cells in those small, smaller cells need to be defined or the geometryof the inflator has to be adjusted. Using of a multiple chamber approach couldbe another alternative solution. Gasflow cells that contain a jet area or part ofa jet area are in principle always active.

As a rule of thumb an Gasflow cell will be active if 50% or more of its volumeis inside the FE airbag volume (the cells with the dark borders in Figure 10.7),although exceptions remain. The pressure of an active Gasflow cell is appliedon the internal airbag side of those FE-elements intersecting this Gasflow cell.If an airbag FE-element is in a non-active cell and also no direct active neigh-bourhood cell can be found, a zero pressure element warning will be gener-ated if the anti through flow option is not used and the number of zero pres-sure elements is above 5% of the total number of FE-elements of the airbagchamber.

When the anti-through flow option is not used those FE-elements that areflagged as zero pressure FE-elements will be pressurized with the averagepressure of the airbag chamber. We would therefore recommend that the num-ber of zero pressure elements should be low compared to the total number ofFE airbag elements (not more than a few percent).

The anti-through flow option applies some additional tests to close extra Gas-flow cells to prevent gas flow through tightly packed airbag layers. Thismeans that an increased number of airbag elements are likely to be flagged aszero pressure elements. No zero pressure warnings are written to the reprintfile. Gasflow will apply the ambient pressure on both sides of these flaggedFE-elements (both internal as well as external to the airbag, i.e. effectively zeropressure difference).

For Gasflow calculations it is possible to extend the closed surface by an ad-Tethersditional element list, forming a tether. These tether elements can prevent gasflowing through it, using the same technique as the anti-through flow algo-rithm. As a result of this, tether elements can have a pressure difference. Note

Release 7.7 253

10


that permeability model 1 and 2 are more suitable for tether permeability thanthe global and global_isentropic permeability models (which are more suit-able for chamber to ambient permeability). Tether permeability only works ifthere are active Gasflow cells on both sides of the tether.

The start of the Gasflow calculation is initiated at the moment the first inflatorGasflowTriggering is triggered. This is implemented to save CPU time during the pre-simulation

when no Gasflow calculations are needed. However, for synchronisation ofoutput data, Gasflow output files will contain "empty data" output time stepsduring this period.

For complex folded multiple chamber airbags, it can be difficult to find a goodGasflowInitialisationDelay

grid resolution, such that there are always active cells present in each chamber.To make this easier, it is possible to start a chamber using uniform pressure cal-culations, and switch to Gasflow after some period (controlled by a switch).In the time period you run the Gasflow chamber with uniform pressure calcu-lations, jets are not supported (but gas mass can be added by the inflator) andHOLE.MODEL3 does not work. Instead, this is replaced by HOLE.MODEL1.

For performance reasons, it can make sense to switch from a Gasflow calcu-Isobaric Switchlation to a uniform pressure calculation, for example, at the moment that theairbag is almost fully deployed. This switching process can only be done forall chambers at once. To prevent instabilities, a small time window is usedfor the transfer. After the transfer, jets are not supported (but gas mass canbe added by the inflator) and HOLE.MODEL3 does not work. Instead, this isreplaced by HOLE.MODEL1.

For uniform pressure and Gasflow, the effects of airbag porosity, holes andInflow andOutflowmodels

heat conduction through the airbag material can be modelled. Porosity of theairbag fabric has the effect of allowing gas to escape from the cells adjacentto the airbag material. Heat conduction through the airbag material reducesthe temperature of the gas in cells adjacent to the airbag material. Reductionof gas outflow due to permeability and/or holes because of (self)contact isdefined by the block flow option. Gasflow outflow due to global dischargeis based on the average pressure inside the airbag chamber. All other outflowtypes and heat conduction are calculated per FE-element. The physical state inthe Gasflow cell related to this particular FE-element is used for those outflowmodels or the heat conduction model.

In comparison to the standard isobaric inflation model available withinGrid DefinitionMADYMO,very little additional data is required to conduct a detailed infla-tion simulation. The user must specify:

• the number of numerical cells to be used in each local co-ordinate direc-tion of the Gasflow mesh,

• the orientation of the Gasflow mesh.

• the minimum cell size to be used (optional)

• the orientation and dimensions of the jet orifice of the inflator(s). For

254 Release 7.7


10

uniform pressure models, no jet is required. An existing uniform pres-sure jet is easily modified into a Gasflow jet.

Two methods of injecting gas are available for the Gasflow module, a mo-Jetsmentum based method and a sonic based method. Both take the sonic inflowcondition into account. However, the momentum-based method will in gen-eral not have the sonic gas velocity, but a lower velocity in those cells that areintersected by the inflator jet area. When the sonic method is used, the gasvelocity in those cells that are intersected by the inflator jet area will be set tothe local sonic velocity. When the inflator jet area is small with respect to theGasflow cell size the amount of kinetic energy brought into the system couldbe too large. When more jets are defined in an airbag chamber, they will be allof the same type.

When a part of the inlet jet area is outside any active cell a warning will begenerated, reporting the jet is partly blocked. The simulation will continue,but the supplied gas is now divided over the other Gasflow cells cut by thisinflator. If an inflator is situated in non-active cells only an error will be gener-ated: Jet fully blocked. In situations that the jet inlet area is large or positionedin such a way that at least a part of the area is outside the Gasflow meshboundary an error will be generated as well. A warning will be generatedat initialisation or at the first occurrence if more than one jet is located in asingle Gasflow cell. This warning indicates that the grid density is too coarseto resolve an accurate solution, especially if the jets are pointing in differentdirections.

Initially, the boundary of the Gasflow mesh is created such that at each sideGasflow Meshof each direction a quarter of a Gasflow cell size is outside with respect to theairbag mesh (Figure 10.8). In general an inflated airbag is growing in size, so ata certain moment the airbag mesh will reach on of the boundary sides. At thatmoment the Gasflow mesh will regrid in those directions needed, in such away, that in those directions there will be a distance between Gasflow bound-ary surface and the airbag surface of 10% of the Gasflow boundary lengthin that direction. Shrinkage of the Gasflow grid is handled in a similar way(Figure 10.9).

Release 7.7 255

10


1/4 Euler cell size

Airbag boundary

Gasflow boundary

Figure 10.8: Gasflow mesh at initialisation. The boundary of the Gasflow mesh is at adistance of a quarter of a Gasflow cell from the airbag boundary.

10%

100%

10%

Figure 10.9: Growth of a Gasflow mesh boundary after regridding. The new meshborders are at a distance of 10% of the grid length.

In principle, the cell sizes are derived from the amount of cells specified ineach direction and the length of the needed Gasflow boundaries. As a result,

256 Release 7.7


10

it might happen that the cell size in a specific direction is very small and there-fore causes a small time step in the simulation. If such a fine Gasflow meshresolution is not needed, an option can be used to specify a minimum cell sizein each Gasflow direction. As a consequence the boundary length in thosedirections will increase and the calculation timestep will be increased.

If a lot of unused Gasflow cells are located in an area that the airbag will notenter, e.g. the instrument panel or the back plate of a airbag test set-up, use canbe made of the offset functionality. This can be used to reposition the Gasflowmesh with the restriction that the Gasflow grid must always engulf the entireairbag chamber.

Gas transportation through holes at high velocities becomes less accurate ifHolesubsegments the Gasflow cells are small compared to the hole segments. These hole seg-

ments can either consist of user-specified elements with material type hole, orthey consist of automatically generated segments closing a hole in the airbagmesh when using the AUTO_VOLUME option under AIRBAG_CHAMBER.The hole subsegment functionality improves the accuracy of the hole outflowcalculation by dividing each side of a triangular hole segment into N equalparts, thus creating N − 1 extra equidistant nodes for each side of the triangle.

First, a quadrangular hole segment is divided into two triangles. In the nextstep, the sides of these triangles are divided into N equal parts. Finally, foreach triangle a grid is generated, connecting these new nodes in such a waythat N2 new subtriangles are created. These subtriangles exactly cover theoriginal triangle. The new nodes and subtriangles are updated at each timepoint in the simulation, based on the actual geometry of the hole segments.They are used exclusively for the calculation of gas transportation throughholes. N, the number of subdivisions of each hole segment side, can be speci-fied by the user or it can be calculated by MADYMO. N is constant during thesimulation. The following expression is used by MADYMO to calculate N:

N ≥ lavg3√

0.5ncells/Vref (10.59)

in which lavg is this hole’s average undeformed triangular segment sidelength, 0.5 ncells is an estimate for the number of active Gasflow cells once theairbag chamber is fully inflated (the estimation is that as a rule of thumb 50%of the cells are active, i.e. situated within the airbag chamber mesh then), andVref is the volume of the fully inflated airbag chamber. The result of (10.59) isrounded off upwards to the next integer number.

If a hole between 2 chambers is modelled, then (10.59) is calculated for bothchambers and the maximum N resulting from both calculations is used.

10.12 Multiple Chamber

Uniform pressure and Gasflow airbag models can be split into multiple cham-bers. For uniform pressure models this can be useful to introduce a time delayin the inflating/deployment process and therefore a more realistic unfoldingpattern. For Gasflow it can be useful to define a multiple chamber model in

Release 7.7 257

10


order to be more efficient with the amount of used Gasflow cells or to studymore details in specific regions by refining the mesh in this region.

The flow of the gas through the exchange area between chambers can be mod-elled by hole elements or elements that contain permeability. The same out-flow models are available for outflow to ambient and flow between chambers.However, instead of the ambient pressure, the pressure difference between therelevant chambers is used for uniform pressure simulations. For Gasflow sim-ulations, the physical state of the active cells at both side of the exchange areaare used. Each chamber must have a closed volume. Elements and parts onthe exchange surface will be part of the relevant chambers. See Figure 10.10and 10.11 for an example of the principle set-up of a multiple chamber model,in which chamber 1 consists of part 1 through 4. while chamber 2 consists ofthe parts 4 through 7.

3 5

6

71

4

Exchange area

2 Chamber 2Chamber 1

Figure 10.10: A uniform-pressure two-chamber model in which part four is the ex-change area.

3 5

6

71

4

Exchange area

2 Chamber 2Chamber 1

Figure 10.11: A two-chamber model with Gasflow in which part four is the exchangearea.

258 Release 7.7


10

In principle the exchange areas can be defined anywhere in the airbag model.However, not all locations are suitable for defining an exchange area. When amultiple chamber model is defined in a wrong way, negative volumes can beencountered, which is reported as a warning in the reprint file. An exampleof this is given in Figure 10.12. In particular, if the ratio between the heightand width is small, the shape of a chamber may become non-realistic due todeployment, see Figure 10.13. In those situations a warning is listed of anincorrect, negative volume.

Figure 10.12: A correct chamber volume can be derived, because the ratioheight/width is large enough.

Figure 10.13: The volume is not correct or even negative. This is due to unfolding andmay occur when the ratio of height and width is small.

Typical examples to position an exchange area are between the inflator house

Release 7.7 259

10


and the airbag or curtain airbags. Figure 10.14 shows an example of an un-folded curtain airbag that is divided into five chambers. For simplicity, theunfolded situation is shown but it will also work in folded situations.

Inflator

Chamber 2 Chamber 5Chamber 4Chamber 3

Chamber 1

Figure 10.14: Example of using the multiple chamber functionality. The arrows indicatethe regions where gas can flow from chamber to chamber.

Two modelling aspects should be taken into account when creating exchangearea(s). The first issue is related to the triple or more points. Figure 10.15shows an example of a 3-chamber model containing a triple point. Automatichole generation will fail because the boundary of an exchange area betweentwo chambers needs to be identical for both chambers (top right of the figure).Therefore the hole elements must be generated by hand (bottom right).

The second issue is related to single closed boundary. Automatic segmentgeneration can not be used for an exchange area between two chambers, ifmore than one boundary exist. An example is given in Figure 10.16, in which adonut shaped airbag is split into two parts. Automatic hole generation will failbecause automatic segment generation requires one closed boundary. There-fore the hole elements must be generated by hand.

260 Release 7.7


10

Figure 10.15: Modelling issues related to multiple chamber models. The boundarybetween two chambers must be identical. The arrow indicates the ‘triple point’.

Release 7.7 261

10


2 1

Figure 10.16: The exchange area between the two chambers contains more than oneclosed boundary.

262 Release 7.7

MADYMO Theory Manual Muscle model

11

11 Muscle model

Usually Hill-type muscle models are applied in biomechanical studies of mus-Hill-typemuscle model cular coordination. These empirical models do not use a large amount of

computing power. In addition, parameters for the human muscles modelscan be found in the literature1,2. A Hill-type muscle model is illustrated inFigure 11.1. It consists of:

• a Contractile Element (CE) which describes the active force generated bythe muscle,

• a Parallel Elastic element (PE) which describes the elastic properties ofmainly muscle fibres and the surrounding tissue,

• two Series Elastic elements (SE1 and SE2) which describe the elasticproperties of mainly the tendons and aponeurosis,

• two masses (M1 and M2) which describe muscle mass as lumpedmasses.

In many applications, muscle mass can be ignored and the series elastic ele-ments SE1 and SE2 are combined into one element. However, for applicationin impact conditions, muscle mass needs to be included. The masses can alsobe used to define interactions with other model elements such as contact withbony structures.

The muscle model implemented in MADYMO is based on a description orig-inally formulated by Hill3, which is illustrated in Figure 11.1.

1Yamaguchi G.T., Sawa A.G.U., Moran D.W., Fessler M.J., Winters J.M. (1990). A survey ofhuman musculotendon actuator parameters. In: Winters J.M., Woo S.L.Y. (eds) Multiple musclesystems: Biomechanics and movement organization, Springer Berlin Heidelberg New York.

2Winters J.M., Stark L. (1988). Estimated properties of synergistic muscles involved in move-ments of a variety of human joints. J. Biomechanics, 21-12:1027:1041.

3Hill, A.V. (1938). The heat of shortening and the dynamic constants of muscle. Proc. R. Soc.Lond. 126: 136-195.

Release 7.7 263

11

Muscle model MADYMO Theory Manual

Figure 11.1: A Hill type muscle model.

A muscle model of varying complexity can be constructed from this basicmuscle model and additional MADYMO elements:

• Tendons can be modelled using Kelvin restraints, conventional belt sys-tems or finite elements.

• Muscle mass can be modelled with standard MADYMO bodies.

• Control options can be used to model neural feedback.

• Neural control signals can also be specified as a function of time.

The muscle model implemented in MADYMO describes the behaviour of theMADYMO’smuscle model contractile element (CE) and the parallel elastic element (PE) defined above

and shown in Figure 11.1. The two ends can be attached to arbitrary pointson (flexible) bodies or the reference space. The model describes the tensionforce F, as a function of the distance l between the two attachment points (themuscle length), the speed of change in l (the lengthening velocity v) and theactive state (A).

The active state A is a measure of the muscle’s neural excitation. A maximalActive state A

activation is usually represented by A = 1 . For a minimal activation or "reststate," a value of A = 0.005 has been proposed by Hatze1. The active statemay depend on time or on the (relative) motion of bodies. Due to dynamicprocesses, the active state does not change instantaneously: it takes about 25 –100ms for the active state to transfer from the rest state to maximal activation.This is not included in the muscle model. It is advisable to take this into ac-count when defining the active state. The muscle model can be used with anyactive state (also negative).

1Hatze H. (1981). Myocybernetic control models of skeletal muscle, characteristics and appli-cations. University of South Africa, ISBN 0 86981 216 5.

264 Release 7.7


11

The muscular tension force is calculated as:

F =Fce + Fpe (11.1)

where

Fce =AFmax fH(vr) fL(lr) (11.2)

Fpe =Fmax fp(lr) (11.3)

The dimensionless muscle length lr and lengthening velocity vr are defined by

lr =l/(lref) (11.4)

vr =v/Vmaxa (11.5)

Vmaxa = fv(A)Vmax (11.6)

The parameter lref is a reference length used for normalizing the muscle lengthOptimumlength in the active and passive force-length relationships. This reference length is

commonly defined as the optimum length, at which active force generationis most efficient. Vmax is a parameter describing the maximum shorteningvelocity. The function fv defines the dependency of the effective maximumshortening velocity Vmaxa on the active state A. The parameter Fmax is definedas the force exerted with maximal activation in isometric conditions (v = 0)and at the reference length.

The function fH describes the normalized active force-velocity relationship(or Hill curve) of the contractile element CE. Usually the force-velocity rela-tionship has a positive slope; force increases during lengthening and reducesduring shortening (Figure 11.2). The function fL describes the normalised ac-tive force-length relationship of the CE. The function fP describes the passiveforce-length relationship of the parallel elastic element PE. These active andpassive force-length relationships are shown in Figure 11.3, which illustratesrelative force contributions of the CE and PE elements with full activation inisometric conditions.

Release 7.7 265

11


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

relative lengthening velocity (vr)

rela

tive

forc

e (f

h)

Figure 11.2: Standard force-velocity relationship ( fH) for contractile element CE.CEsh = 0.25 , CEml = 1.5 , CEshl = 0.05 .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

relative length (lr)

rela

tive

forc

e (f

l and

fp)

PEsh=5

PEsh=10

PEsh=20Sk=.40

Sk=.54

Sk=.68

Figure 11.3: Standard force-length relationships for several parameter values. Activefunctions fL (middle) and passive functions fP (right). For passive curves PExm = 0.8 .

For a submaximal active state, the active force will be reduced proportionally.Note

266 Release 7.7


11

The slope of a force-length relationship determines a stiffness coefficient. Asshown in Figure 11.3, the slope of the active force-length relationship is nega-tive for a relative length larger than one. In this range, the active force-lengthrelationship introduces a negative stiffness. This will be partially compen-sated by the passive force-length relationship. However, when the resultingstiffness is negative, feedback may be required in order to stabilise the model.

The user can either specify functions fV , fH , fL and fP in tabular form, or usestandard functions which are described below.

The standard function for fV is the unity function in which the effective max-Standardfunctions imum shortening velocity is independent of the active state.

The standard force-velocity relation or Hill curve fH has been defined as fol-lows. Separate equations are defined for lengthening and shortening of theCE. When the maximum effective shortening velocity is exceeded (vr ≤ −1),no force is delivered:

fH(vr) = 0 (11.7)

During normal shortening (−1 < vr ≤ 0):

fH(vr) =1 + vr

1 − vr/CEsh(11.8)

During lengthening (vr > 0):

fH(vr) =1 + vrCEml/CEshl

1 + (vr/CEshl)(11.9)

The parameters CEsh and CEshl determine the shape of the force-velocity rela-tionship. The parameter CEml represents the ultimate tension during length-ening relative to the isometric force with maximal activation.

The standard active force-length relationship fL of the contractile element (CE)is given by:

fL(lr) = exp

(−[

lr − 1

Sk

]2)

(11.10)

The parameter Sk determines the width of the active force length curve (seeFigure 11.3).

The standard passive force-length relationship fP of the Parallel Elastic ele-ment (PE) is given by:

fp(lr) =

k1exp [k2(lr − 1)]− 1 for lr > 1

0 for lr < 1(11.11)

with constants

k1 = 1/exp(PEsh) − 1k2 = PEsh/PExm

(11.12)

Release 7.7 267

11


The parameter PExm describes the relative elongation (l − lref)/lref inducing apassive muscle force equal to Fmax . The parameter PEsh determines the shapeof the PE force-length curve (see Figure 11.3).

268 Release 7.7

MADYMO Theory Manual Tyre model

12

12 Tyre model

A road vehicle’s tyres are an important factor in the dynamic behaviour of theTyre-roadcontact loads vehicle. The major disturbance loads and the control loads on a vehicle are a

result of the contact between the tyres and the road. The vertical loads transferthe weight of the vehicle to the road. Due to the compliance of the tyres, avehicle is cushioned against disturbances due to small road irregularities. Thelongitudinal tyre forces are required for traction and braking to take place.Lateral tyre forces are required for controlling the direction of the vehicle’stravel. The lateral behaviour of tyres is an important factor for vehicle stability.The tyre-road contact loads depend on:

• the characteristics of the tyre (stiffness, damping),

• the road condition (friction between tyre and road), and

• the motion of the tyre relative to the road (the amount of slip).

A good model of the tyre-road contact loads is needed to describe the dynamicbehaviour of a vehicle.

For each pneumatic tyre and road condition, the tyre loads due to slip be-Magic formulahave in a typical manner (Figure 12.1). A special mathematical function, the"magic formula", approximates this behaviour quite well. The parameters inthe magic formula depend on the tyre type and the road conditions. Theseparameters are taken from experimental data obtained from tests in whichthe tyre is dragged over a road in various positions and orientations. Themagic formula and the experimentally determined parameters describe thetyre loads accurately for the specific tyre type and road conditions used in theexperiments.

Release 7.7 269

12

Tyre model MADYMO Theory Manual

longitudinal slip

long

itudi

nal f

orce

F

1.5 F

2 F

lateral slip

late

ral f

orce

F

1.5 F

2 F

lateral slip

alig

ning

torq

ue

2 F

1.5 F

F

Figure 12.1: Measured tyre loads due to slip for different values of the normal load.

The tyre model implemented in MADYMO is based on the magic formula1.Dynamiceffects Also dynamic effects are considered such as the lagging of tyre loads with re-

spect to the slip due to the time needed by the tyre to build up the deformationwhich it requires to transfer the loads. In addition, the loads due to combined

1Here, we only present selected equations from the magic formula theory. For further detailssee: Tyre Model Manual.

270 Release 7.7


12

lateral and longitudinal slip are taken into account. The behaviour of a tyrecan be modified by scaling parameters.

Standard road elevation profiles are available that describe a flat road andStandard roadelevationprofiles

roads with a sinusoidal wave, a sinusoidal bump or a sinusoidal ramp, eitherparallel or perpendicular to the axis of the road. Other profiles can be definedwith a meshed road surface or a user-defined routine USRRD3. For details,please refer to the Programmer’s Manual.

A road profile must satisfy the following condition:Road profile

The radius of the road profile’s curvature is large when compared to the radiusof the tyre. It is assumed that there is only a single contact between the tyreand the road. In addition, for calculating the motion of the tyre relative to theroad, the road is approximated by a plane which is tangent to the road at thepoint that is vertically below the wheel centre. When the radius of curvatureof the road is too small, this tangent plane gives an inaccurate approximationof the road.

The road condition, or the friction between tyre and road, cannot be speci-fied separately. The road condition is included in the magic formula parame-ters. This means a set of parameters is only valid for a given road condition.Each road condition requires a different set of parameters. The magic formulaparameters corresponding to a different road condition can be approximatedwith scaling parameters.

The road is fixed to the reference space. This coordinate system must be cho-sen such that the xy-plane is more or less parallel to the road and the Z-axis isoutward of the road.

In MADYMO, a tyre must be connected to a rigid body that represents theThe motion ofa tyre relativeto a road

wheel. In calculating the motion of the tyre relative to the road, the tyre isrepresented by a disk that is rigidly connected to the wheel. This disk is co-incident with the central plane of the tyre, the wheel plane (Figure 12.2). Thecentre of this disk is coincident with the wheel centre. The wheel axis, the axisthrough the wheel centre and perpendicular to the wheel plane, is defined bya unit vector nw that is fixed to the wheel. The wheel plane is taken coinci-dent with the xz-plane of the body local coordinate system of the wheel. Thewheel centre is taken coincident with the origin of this coordinate system. Asa result, the wheel axis coincides with the y-axis of the body local coordinatesystem of the wheel.

Release 7.7 271

12


z x

ynw

vx

yr

zr

xr

Figure 12.2: Kinematics of wheel with tyre.

The wheel axis is not necessarily parallel to the axis of the (revolute) jointNotebetween the wheel carrier and the wheel. The road is approximated by a planethat is tangent to the road at the point that is vertically below the wheel centre.It is assumed that the road is stationary.

The motion of the wheel is defined by:Motion of thewheel

• the position and linear velocity of the wheel centre,

• the direction of the wheel axis, and

• the angular velocity of the wheel.

The motion of the tyre relative to the road is described with respect to a co-ordinate system which is fixed to the road, the road coordinate system. Thepositive yr-axis of this coordinate system is coincident with the projection ofthe vector nw on the road tangent plane. The zr-axis is perpendicular to theroad, positive for the outward direction of the road. The xr-axis follows fromthe vector product of these two directions and is parallel to the line of intersec-tion of the road plane and the wheel plane. The origin of the road coordinatesystem is located at the intersection of the wheel plane and the projection ofthe wheel axis onto the road plane, the centre of tyre contact C.

For a given tyre and road, the tyre loads depend only on the motion of the tyrerelative to the road. This motion is characterised by the loaded tyre radius, thecamber angle, the slip angle and the longitudinal slip.

The loaded tyre radius rl is defined by the distance of the wheel centre andLoaded tyreradius the centre of tyre contact (see Figure 12.3). rw is the unloaded tyre radius.

272 Release 7.7


12

γ

rw

yr

C

zrrl

nw

Figure 12.3: Back view of wheel.

The camber angle γ is defined by the angle between the zr-axis and the lineCamber anglethrough the wheel centre, and the centre of tyre contact. The camber angle hasa value between −π/2 < γ < π/2 .

vx

F

C

Fxα

vcy

v

yr Fy < 0Mz

My < 0

xr

Figure 12.4: Top view of wheel.

The slip angle α is defined by the angle between the forward velocity of theSlip anglewheel centre vx , the component of the velocity of the wheel centre in the di-rection of the xr-axis, and the lateral velocity of the centre of tyre contact vcy

(see Figure 12.4):

α = arctanvcy

|vx|(12.1)

vcy is the lateral velocity of the point on the wheel body that is momentarilythe centre of tyre contact. The slip angle has a value between −π/2 < α <

Release 7.7 273

12


π/2 . A backward velocity is allowed, however, it is not possible to specifydifferent tyre characteristics for forward and backward velocities.

The longitudinal slip κ is defined by the ratio of the longitudinal velocity vsxLongitudinalslip and the forward velocity of the wheel centre vx:

κ = −vsx

vx(12.2)

The longitudinal velocity vsx requires some explanation. Consider a tyrerolling without slip at a constant velocity with a wheel centre vx and a con-stant spin velocity ωn . Under this condition, the point on the wheel for whichthe longitudinal velocity is zero is usually not the centre of tyre contact C, buta point which is somewhat below the road plane.

This can be made feasible by considering a tyre with a tread that does notelongate. For such a tyre, the longitudinal slip will be zero if the displacementof the centre of the wheel due to one revolution equals the length of the cir-cumference of the tread, or vx(2π/ωn) = 2πrw . Since the undeformed tyreradius, rw , is larger than the loaded tyre radius rl , the instantaneous centreof rotation (this is the point on the wheel for which the velocity relative to theroad is zero) is below the road plane. In general, the instantaneous centre ofrotation is at a distance re , the effective rolling radius of the tyre, from thecentre of the wheel. re depends on a given tyre on the loaded tyre radius. Forcalculating the longitudinal slip, the effective rolling radius is approximatedby

re = rw − F(rw − rl) − D(rw − rln) arctan [B(rw − rl)/(rw − rln)] (12.3)

where rln is the tyre radius loaded at nominal load. The values of the constantsB, D and F, which make this function fit the effective rolling radius, must bespecified in the input data file. vsx is then the longitudinal velocity of thepoint on the wheel body which is instantaneously at a distance re from thewheel centre.

Due to the minus signs in the definitions of the slip quantities α and κ, a posi-Notetive slip corresponds to a positive force.

The magic formula is a mathematical function that can produce characteristicsThe magicformula which closely match measured characteristics for:

• the longitudinal tyre force,

• the lateral tyre force and

• the aligning torque

as functions of the corresponding slip quantities κ and α. The basic form ofthis formula is given by

y(x) = D sin C arctan[Bx − E(Bx − arctan(Bx))] (12.4)

274 Release 7.7


12

y

y

s

m

xm

y

x

Figure 12.5: Curve of the magic formula.

Figure 12.5 shows the curve of this formula. The parameters in the formula,the stiffness factor B, the shape factor C, the peak factor D and the curvaturefactor E, are directly related to this curve:

B =(CD)−1(dy/dx)x=0 (12.5)

C =2 − (2/π) arcsin(ys/ym) (> 0) (12.6)

D =ym (12.7)

E =Bxm − tan(π/2C)

Bxm − arctan(Bxm)(≤ 1) (12.8)

Equation (12.4) represents an odd function, y(x) = −y(−x) . Generally a tyreload is not an odd function of the corresponding slip quantity (Figure 12.6).To account for the difference in magnitude of the peak loads for positive andnegative slip, add to the load that results from equation (12.4), the mean valueSv of the measured maximum and minimum peak loads.

x = X + Sh (12.9)

Y(X) = y(x) + Sv (12.10)

If the tyre load is non-zero for zero slip (X = 0), the horizontal offset Sh mustbe added to the slip X before it is substituted into the magic formula.

Release 7.7 275

12


x

slip X

y(x) lo

ad Y

(X)

S > 0v

S > 0h

Figure 12.6: Magic formula curve with offsets.

The parameters B, C, D, E, Sv and Sh depend on the normal tyre load and thecamber angle. The tyre load-slip characteristic can be made asymmetric byintroducing into E a dependency on the corresponding slip quantity, which isnot an odd function.

The road applies the normal force Fz on the wheel. To calculate this force, theThe normaltyre force tyre is treated as a linear spring and damper with a stiffness coefficient kz and

a damping coefficient cz (see Figure 12.7).

Fz = kzεz + cz εz cos γ (12.11)

where the deformation of the tyre is defined by

εz = rw − rl (12.12)

rw is the undeformed tyre radius. The normal tyre force is set to zero if thewheel disk does not intersect the road plane, that is rw < rl .

276 Release 7.7


12

γ

rw

yr

C

zrrl

Fz

z

Figure 12.7: Normal tyre force and deformation.

The normal tyre force is applied on the wheel body at the centre of tyre contact.The rolling torque includes the fact that the normal force’s line of action is infront of the tyre contact’s centre.

The longitudinal tyre force Fx0 depends basically on the longitudinal slip κ. ItThelongitudinaltyre force

is calculated from the basic magic formula with

κx =κ + SHx (12.13)

Fx0 =Dx sin[Cx arctanBxκx − Ex(Bxκx − arctan(Bxκx))]+ SVx (12.14)

The rolling torque is approximated by

My = (SVx + KxSHx) · R0 (12.15)

The tyre’s longitudinal stiffness is high in the region of zero longitudinal slip.For low values of the forward velocity of the wheel centre vx , a small changein the wheel’s spin velocity causes a large change in the longitudinal force.Because the wheel’s moment of inertia is relatively small, this introduces alarge change in the wheel’s spin acceleration. As a result, a small time step forthe numerical integration may be necessary. This can be avoided in two ways:

• If the longitudinal force and rolling torque can be neglected or if theseare not important, they can be eliminated by specifying no longitudinalforce data.

• If the wheel’s spin velocity is approximately constant, the joint accel-eration degree of freedom of the revolute joint between the wheel andthe wheel carrier can be prescribed equal to zero. This corresponds to

Release 7.7 277

12


a wheel that is driven by a torque which equals the rolling torque. Theadvantage of this approximation is that there is still a longitudinal forcethat may be required to sustain the aerodynamic drag on the vehicle.

The lateral tyre force Fy0 depends basically on the slip angle α. It is calculatedThe lateral tyreforce from the basic magic formula with

αy =α + SHy (12.16)

Fy0 =Dy sin[Cy arctanByαy − Ey(Byαy − arctan(Byαy))]+ SVy (12.17)

For combined longitudinal and lateral slip, the lateral force Fy is approxi-mated by

Fy = Fy0 · Gyκ(α, κ, γ, Fz) + SVyκ (12.18)

with Gyκ a weight function and SVyκ the "κ-induced" side force.

The aligning torque Mz0 depends basically on the lateral slip. It is calculatedThe aligningtorque from the basic magic formula with

αt =α + SHt (12.19)

Mz0 = − t · Fy0 + Mzr (12.20)

The pneumatic trail t and the residual torque Mzr are approximated by

t(αt) = Dt cos[Ct arctanBtαt − Et(Btαt − arctan(Btαt))] (12.21)

For combined longitudinal and lateral slip, the aligning torque is approxi-mated by

M′z = −t′ · F′

y + M′zr + s · Fx (12.22)

with

t′(αt,eq) =

Dt cos[Ct arctanBtαt,eq − Et(Btαt,eq − arctan(Btαt,eq))] (12.23)

F′y = Fy − SVyκ (12.24)

M′zr(αr,eq) = Dr cos[arctan(Brαr,eq)] (12.25)

s = ssz1 + ssz2(Fy/Fz0) + (ssz3 + ssz4d fz)γR0 · λs (12.26)

278 Release 7.7

MADYMO Theory Manual Energy preservation

13

13 Energy preservation

An analysis of the energy of a Madymo model can give a better understandingof the model’s response. Monitoring the energy gives insight in the behaviourof the model and the reliability of the results. An appropriate breakdown ofthe energy of the model simplifies this analysis. A natural choice is to usethe four energy parameters corresponding with the four types of forces in theequations of motion:

• the kinetic energy related to the inertia forces,

• energy loss (dissipation) related to the damping and friction forces,

• energy storage (internal energy) related to the elastic terms,

• external work related to the external loading.

The force balance of the equations of motion implies an energy balance: thetotal energy, which is the sum of kinetic + dissipation + internal − externalenergy, must remain constant. The results of an analysis may be unreliablewhen the total energy is not constant. This may be caused by an instability ora loss of a contact.

A better understanding can be obtained by a further breakdown of the energyto distinguish between various sources of energy dissipation.

The total internal energy is the energy corresponding to reversible deforma-Total internalenergy tion of elements and bodies. This includes:

1. the elastic strain energy

2. the energy due to hourglassing of reduced integrated elements (reducedintegrated four-node membrane, four-node shell and eight-node solidelements),

3. the elastic energy in membrane elements in Initial Metric Method(IMM)mode,

4. the energy due to deformation of flexible bodies.

The total dissipation energy is the energy corresponding to irreversible defor-Totaldissipationenergy

mations of elements and bodies. This includes:

1. the plastic deformation energy,

2. the energy due to damage,

3. the dissipation energy of flexible bodies, restraints and belts.

Release 7.7 279

13

Energy preservation MADYMO Theory Manual

For materials and objects that have a loading and an unloading curve, the dis-sipative part of the strain energy is not calculated separately, but is includedin the elastic strain energy.

The following four sources of energy dissipation can be identified:

1. Rayleigh damping,

2. material damping,

3. plastic deformation and/or damage,

4. IMM damping and energy loss in IMM mode transition,

The work done by loads that are related to the boundary conditions is in-Total externalenergy cluded in the external energy:

1. work done by external loading, such as applied loads, prescribed mo-tions, restraints, belts, supports and spotwelds,

2. work done by contact forces.

The forces that are applied to a model due to contact are regarded as externalforces, including forces due to self contact. Contact interactions are not part ofa model, which implies that the energy stored in the "contact springs" or dissi-pated in the "contact dampers" or by friction cannot be monitored separately.Energy dissipated in contacts is included in the work done by external contactforces.

The gas inside an airbag chamber is regarded as a separate entity and thepressure forces from the gas are considered as external loading on the model,doing external work on the structure. Because the gas pressure is calculatedbased on the first law of thermodynamics, the energy conservation principleis automatically satisfied for the gas.

280 Release 7.7

MADYMO Theory Manual Filters

14

14 Filters

14.1 Filtering time-history signals

Several filters can be specified for filtering of MADYMO time-history signals.First, the solver calculations are carried out, producing time-history files withunfiltered signals. Next for each time-history file that requires filtering, filter-ing is carried out. After signals from a time-history file have been filtered, thefiltered file replaces the original.

14.2 Why filter?

MADYMO simulation results should resemble and predict the behaviour ob-served during experiments. Therefore, MADYMO offers the ability to applythe filters commonly used in impact tests.

Analog signals measured during experiments, usually are low-pass filtered(as opposed to band-pass or high-pass filtering) before being digitized andrecorded. There are several reasons for filtering:

• to prevent aliasing errors during subsequent sampling (see Section 14.4),

• to reduce high frequency environmental noise, or

• to remove high frequencies that are considered not important for thephenomena being studied.

14.3 Selecting a filter

To enable comparison of data from different sources, it is important to conformto an accepted standard. The filters available in MADYMO are defined by theSAE J211-1 (CFC filters), and by a NHTSA document (FIR100 Filter Program,Version 1.0, July 16,1990). These are all low-pass filters.

Among the specifications by the SAE J211-1 are the specifications for four fil-ters denoted:

CFC60, CFC180, CFC600, and CFC1000

The lower the CFC number, the lower the cutoff frequency of the filter. InMADYMO, the cut-off frequency divided by the CFC number varies from 1.67to 1.98 . (The SAE J211-1 Draft contains some errors in the formulas for the co-efficients of a two pole Butterworth filter; in MADYMO the corrected formulashave been used.)

The SAE J211-1 provides recommendations on which filter to use for each par-ticular signal, see Table 14.1.

The software for the FIR100 filter is defined by NHTSA. This filter should beselected for signals used in calculation of the TTI injury criterion.

Release 7.7 281

14

Filters MADYMO Theory Manual

14.4 Choosing the sampling frequency

This section describes how to choose a correct sampling frequency for time-history files. In MADYMO, the user is responsible for choosing a correct sam-pling frequency. Its reciprocal, the sampling period, is equal to the time stepof the time history file.

When representing a signal by its samples, it is important not to loose relevantNoteinformation, or worse, misinterpret information.

The Sampling Theorem states: A band-limited analog signal can be recon-Nyquistfrequency structed from its samples, if fs > 2 fm , where fs is the sampling frequency and

fm the maximum frequency present in the analog signal. The frequency 2 fm

is commonly referred to as Nyquist frequency, see for example "Signals andSystems", Oppenheim, Willsky, and Young, Prentice-Hall.

When the sampling frequency is smaller than the Nyquist frequency, distor-Aliasingtion will be introduced when converting the samples back to a continuous-time signal. This is aliasing, or also known as foldover or mixing. Moreover,if the samples are digitally filtered even more distortion is added.

Table 14.1: Recommended CFC filters.

Typical Test Measurements CFC number

Vehicle structural accelerations for use in:

Total vehicle comparison 60(1)

Collision simulation input 60

Component analysis 600

Integration for velocity or displacement 180

Barrier face force 60

Belt restraint system loads 60

Anthropomorphic Test Device:

Head accelerations (linear and angular) 1000

Neck:

Forces 1000(2)

Moments 600

Arm:

Forces, Moments and Accelerations 600

Thorax:

Spine accelerations 180

Rib and Sternum accelerations 1000

Deflections 600

Lumbar:


282 Release 7.7


14

Table 14.1 cont.

Typical Test Measurements CFC number

Forces and Moments 600

Pelvis:

Accelerations 1000


Femur/Knee/Tibia/Ankle:

Accelerations 1000


Displacements 180

Sled acceleration 60

Steering column loads 600

Headform acceleration 1000

(1) When overall acceleration of the frame or body in a given direction is desired and a higherfrequency response class is used, readability of the data may be improved by averaging outputsof two or more transducers at different locations.(2) These classifications are needed to calculate head impact forces based on neck forces and headaccelerations when using an ATD. When Force channels are multiplied by a moment arm, a CFC600 filter shall be used.

A high sampling frequency will result in large time-history files. A samplingfrequency lower than the Nyquist frequency could lead to signals which con-tain aliasing errors.

To avoid distortion on a filtered signal which is caused by aliasing, the signalscan be temporarily stored with a sampling frequency which is equal to theintegration time step. These extended sampled signals are now filtered bythe selected filter and subsequently, the filtered signals are written to the timehistory file with a sampling frequency which is equal to the output time stepwhich has been defined for the time history file.

This extended sampling option can be activated for specific time history out-put sections; when no available filter type is selected under the correspondingtime history output section, the signals are filtered by a suitable low-pass filter.

If this extended sampling option is switched on, the performance will slightlydecrease and the memory usage will slightly increase

Each CFC filter implementation has a limited range of sample frequencies forRange ofsamplefrequencies

which it can be used; outside this range the implementation will not conformto the standard and can become unstable (an input signal with small am-plitude, produces an output signal with an amplitude that increases withinbound). If the sampling frequency is not acceptable for a filter, an error mes-sage is written to REPRINT file, and the request for the filter is ignored.

Table 14.2 shows the range of sample frequencies that can be used for the var-ious filters.

Release 7.7 283

14


Table 14.2: Filter names and sampling frequency ranges.

Filter name Sampling frequency Range (kHz)

CFC60 2 – 1000

CFC180 2 – 1000

CFC600 8 – 1000

CFC1000 10 – 1000

FIR100 2 – min(

1PTI , 10

TE−T0+PTI

)

MADYMO will not allow a user to filter a signal more than once. In general, ifFiltered signalcan not befiltered again

a signal is filtered twice, once with cutoff frequency flow and once with fhigh ,where flow ≪ fhigh , then the total effect will be approximately, but not exactly,the same as filtering with cutoff frequency flow .

14.5 Pre-event and post-event data

To avoid "start-up effects" in a filter output signal, part of the input signalPre-event datais added before the actual signal. The added part is referred to as pre-eventdata. The time length of the added interval is referred to as PTI (Padding TimeInterval; PADDING_TIME).

The start-up effect is caused by a filter’s memory. At the start of a signalsupplied to the filter, the filter’s memory will contain values not related to theinput signal. This will give a component in the output signal that is unrelatedto the input signal. For practical filters, this component will go to zero as timeprogresses, hence the name "start-up effect".

By reflecting part of the input signal with respect to its first sample, the start-up effect will appear at the start of the reflected signal part. This reflected partis called the "pre-event data".

For example:

The original signal starts at t = T0 , and lasts until t = TE . Pre-event data with length PTI is added. The start-up effect in the out-put signal will begin at t = T0− PTI . At t = T0 , the start-up effectin the output signal will be negligible.

After filtering, the pre-event part of the output signal is discarded. The defaultvalue for PTI is 0.01 s. This value applies to all signals being filtered.

Several choices for constructing pre-event data are possible, but the choicedepends on the history available about the input signal. MADYMO constructspre-event data by reflecting the input signal with respect to its first sample;this usually gives a good initial value and first derivative of the output signal.

284 Release 7.7


14

For the filters also post-event data are necessary; part of the input signal isPost-eventdata reflected with respect to its last sample. MADYMO applies the same value of

PTI for both pre-event and post-event data.

Release 7.7 285

14


286 Release 7.7

MADYMO Theory Manual Injury parameters

15

15 Injury parameters

Injury biomechanics describes the effect mechanical loads have on the hu-Injurybiomechanics man body, particularly impact loads. Due to a mechanical load, a body re-

gion will experience mechanical and physiological changes, the biomechani-cal response. Injury occurs if the biomechanical response is so severe that thebiological system deforms beyond a recoverable limit, resulting in damageto anatomical structures and altering the normal function. The mechanisminvolved is called injury mechanism, the severity of the resulting injury is in-dicated by the expression "injury severity".

An injury parameter is a physical parameter or a function of several physi-Injuryparameter cal parameters that correlates well with the injury severity of the body region

being examined. Many schemes have been proposed for ranking and quanti-fying injuries. Anatomical scales describe the injury in terms of its anatomicallocation, the type of injury and its relative severity. These scales rate the in-juries instead of the results of the injuries. The most well-known, widely ac-cepted anatomical scale is the Abbreviated Injury Scale (AIS). Although orig-inally intended for impact injuries in motor vehicle accidents, the updates ofthe AIS allow its application also for other injuries such as burns and pene-trating injuries.

The AIS distinguishes the following levels of injury:AbbreviatedInjury Scale

0 no injury

1 minor

2 moderate

3 serious

4 severe

5 critical

6 maximum injury (causes death)

9 unknown

The AIS is a "threat to life" ranking. The numerical values have no significanceother than to designate order.

Biomechanical tolerance is the magnitude of a biomechanical response of thehuman body due to an impact that causes a defined level of injury, often givenby an AIS level. Biomechanical tolerance is not the same for each individualin a population and varies from low to high values within the population.This means the average tolerance level represents the protection required by acertain percentage of the population.

An injury criterion can be defined as a biomechanical index of exposure sever-Injury criterionity, which indicates the potential for impact induced injury by its magnitude.

There are several reasons why injury criteria are developed. The search for a

Release 7.7 287

15

Injury parameters MADYMO Theory Manual

valid criterion improves the understanding of injury mechanisms and the sit-uations in which they occur. An injury criterion also relates loading conditionsduring impacts on human bodies to certain levels of injury scales as the AISscale. Another practical reason is that experiments with cadavers, animals,mechanical surrogates (dummies) of occupants provide only measurementsof forces, displacements, velocities and accelerations and no injuries. The in-jury criteria, based on data of these experiments or mathematical simulations,can be used for an efficient analysis of car safety design and optimization.

Most injury criteria are based on accelerations, relative velocities or displace-ments, or joint constraint forces. These quantities must be requested with stan-dard output options. Most injury criteria need some mathematical evaluationof a time history signal.

Filtering applied to signals used to calculate the injury parameters, may influ-Filteringence the results significantly. Recommendations on how to filter signals aregiven below. MADYMO does not enforce these recommendations, warningsare given only; it is the user’s responsibility to use correctly filtered signals foreach injury parameter.

Table 15.1: Recommendations for filters.

Injury parameter Filter Related documents

GSI CFC1000

HIC CFC1000 FMVSS 214, 96/79/EC

NIC_FORWARD CFC600 and CFC1000 SAE J221/1, 96/79/EC

Nij CFC600 or CFC1000

3MS, XMS

TTI FIR100 FMVSS 214

VC CFC180 and CFC600 SAE J1727, 96/79/EC

CTI CFC180 and CFC600

FFC CFC600 96/79/EC

TCFC CFC600 96/79/EC

TI CFC600 96/79/EC

APF CFC600 96/27/EC

HCD CFC1000 SAE J2052

HICd CFC1000 FMVSS 201

MOC CFC600 SAE J1727, SAE J211

NIC_REARWARD CFC180

NKM CFC600 SAE J221, SAE J1733

LNL CFC600

288 Release 7.7


15

15.1 Gadd Severity Index (GSI)

The first extensive quantification of head tolerance to impact was the WayneState Tolerance Curve (WSTC). This still remains the basis for the most cur-rently accepted injury criteria. The WSTC shows a relationship between a lin-ear acceleration level and pulse duration that gives similar head injury sever-ity in head contact impact. The experimental data used to develop this curvewas:

• acceleration levels for short pulse duration (1–6 ms), necessary to pro-duce linear skull fracture (known to be highly associated with brain con-cussion) in embalmed cadaver heads;

• acceleration levels for intermediate pulse duration (6–10 ms) in experi-ments in which cadaver and animal brain pressure responses were com-pared;

• acceleration levels for long pulse duration obtained in human volunteertests, not producing brain injury and thus to be considered as asymptoticvalue of the curve. This asymptotic value was originally proposed to be42 g but later proposed to be raised to 80 g.

Figure 15.1 shows the now accepted form of the WSTC. The ordinate repre-sents the "effective" or average acceleration (measured at the rear of the head)and the abscissa represents the time duration of this acceleration.

Figure 15.1: The Wayne State Tolerance Curve.

Combinations of acceleration and time which lie above the curve are likely toresult in considerable brain damage (AIS 3 or higher) and combinations thatlie below this curve stay below human tolerance.

Release 7.7 289

15


For evaluating complex acceleration-time pulses to the WSTC, difficultiesarise in determining the effective acceleration. To overcome this problem,Gadd developed a weighted acceleration criterion for establishing the GaddSeverity Index (GSI), which for the head is:

GSI =

TE∫

T0

R(t)2.5dt (15.1)

with R(t) = resultant linear acceleration in g’s in the cen-tre of gravity of the head

T0 = starting time of the simulation in seconds

TE = end time of the simulation in seconds

t = time in seconds

The expo-

nent 2.5 applies only to the head and is primarily based on a straight-lineapproximation of the WSTC plotted on log-log paper between 2.5 and 50 ms.The threshold - tolerance level - proposed by Gadd for concussion in the caseof frontal impact is 1000 (thus GSI should not exceed 1000). For non-contactimpact, Gadd proposed a tolerance of 1500 for concussion.

The GSI value is printed in the PEAK output file. The GSI can be calculatedfor other segments than the head, but the usefulness of this is limited.

15.2 Head Injury Criterion (HIC)

In response to a comparison of the WSTC and the GSI, a new injury criterionfor the head was defined by the U.S. government, the Head Injury Criterion(HIC):

HIC = maxT0≤t1≤t2≤TE

1

t2 − t1

t2∫

t1

R(t)dt

2.5

(t2 − t1) (15.2)

where T0 is the starting time of the simulation, TE is the end time of the sim-ulation, R(t) is the resultant head acceleration in g’s (measured at the head’scentre of gravity) over the time interval T0 ≤ t ≤ TE , t1 and t2 are the initialand final times (in s) of the interval during which the HIC attains a maximumvalue.

As for the GSI, a value of 1000 is specified for the HIC as concussion tolerancelevel in frontal (contact) impact. For practical reasons, the maximum timeinterval (t2 − t1) that is considered to give appropriate HIC values was setto 36 ms. The length of the time interval greatly affects the HIC calculation.In order to restrict the use of the HIC to hard head contact impacts this timeinterval has been proposed to be further reduced to 16 ms.

290 Release 7.7


15

An injury criterion and associated tolerance level should relate to the injuryHIC limitationsseverity. Limitations of the HIC are:

• HIC only considers linear acceleration, while biomechanical response ofthe head also includes angular motion which is believed to cause headinjury,

• HIC is only valid for a hard contact, thus the time duration of impact islimited,

• HIC is based on the WSTC, which is derived from subjects loaded inanterior-posterior direction only.

Despite these drawbacks, HIC is the most commonly used criterion for headinjury in automotive research and is believed to be an appropriate discrimina-tor between contact and non-contact impact response.

The HIC algorithm in MADYMO is based on the algorithm developed byMentzer1. If a window ∆t is defined for the HIC calculation, only those com-binations of t1 and t2 are evaluated for which (t2 − t1) ≤ ∆t .

The results of the HIC calculation: HIC, t1 and t2 are printed in the PEAKoutput file. The HIC can be calculated for other segments than the head, butthe usefulness of this is limited.

1Mentzer, S.G., "Efficient Computation of Head Injury Criterion (HIC) Values", NHTSA ReportDOT-HS-806-681, November 1984.

Release 7.7 291

15


15.3 Neck Injury Criteria - Forward (NIC_FORWARD)

Neck injury is often assessed by peak forces and moments in the upper andlower neck. EEVC Working Group 111 has proposed a set of injury criteria,the Neck Injury Criteria. (The element NIC_FORWARD is used in order todiscriminate with the NIC criterion for rear impact neck injury assessmentthat is based on the motion of the head relative to the thorax.)

The NIC_FORWARD is a measure of the injury due to the load transferredNIC_FORWARDcomponents

through the head/neck interface. The NIC_FORWARD consists of three com-ponents:

• the neck axial force,

• the fore/aft neck shear force and

• the neck bending moment about a lateral axis at the head/neck interface.

1Lowne, R.W., "The Validation of the EEVC Frontal Impact Test Procedure", 15th InternationalTechnical Conference on the Enhanced of Safety of Vehicles, 1996, Melbourne, Australia, PaperNumber 96-S3- O-28, pp. 401-413.

292 Release 7.7


15

The NIC_FORWARD injury calculation is applied to joint constraint load sig-nals. It is assumed that the coordinate systems of this (bracket) joint are ori-ented in agreement with SAE J221/1 because as axial force, the componentof the constraint force in the joint ζ-direction is used, as fore/aft shear forceFx , the component of the constraint force in the joint ξ-direction is used and asbending moment, the component of the constraint moment My′ about the jointη-axis is used. The bending moment is, in general, not about the lateral axisof the head/neck interface. The moment My is obtained from the followingequation

My = My′ − eFx (15.3)

where e is the distance between the occipital condyle and the joint in the pos-itive joint ζ-direction.

Duration curves of these time history signals are made. The neck axial forceand neck shear force duration curves must not exceed the values shown inFigure 15.2; the neck bending moment must not exceed 57 Nm in extension.

Figure 15.2: Neck tension and shear force performance criteria.

15.4 Biomechanical neck injury predictor ( Nij)

The biomechanical neck injury predictor, Nij , is a measure of the injury dueto the load transferred through the occipital condyles. This injury parametercombines the neck axial force Fz and the flexion/extension moment about theoccipital condyles My . The Nij injury calculation is applied to joint constraintload signals. It is assumed that the coordinate systems of this (bracket) jointare oriented in agreement with SAE J221/1 because as axial force, the com-ponent of the constraint force in the joint ζ-direction is used, as shear forceFx , the component of the constraint force in the joint ξ-direction is used andas bending moment, the component of the constraint moment My′ about thejoint η-axis is used. This bending moment is, in general, not about the occip-ital condyles. The moment about the occipital condyles My is obtained fromthe following equation

My = My′ − eFx (15.4)

Release 7.7 293

15


where e is the distance between the occipital condyle and the joint in the pos-itive joint ζ-direction. Appropriate values for selected dummies are given inthe Reference Manual, under INJURY.NIJ.

Nij is the collective name of four injury predictors corresponding to differentFour injurypredictors combinations of axial force and bending moment:

• tension-extension (NTE),

• tension-flexion (NTF),

• compression-extension (NCE), and

• compression-flexion (NCF).

The equation for the calculation of Nij is given by

Nij = | Fz

Fzc|+ | My

Myc| (15.5)

where Fzc and Myc are constants that depend on the dummy and on the neckloading condition (compression/tension and flexion/extension). Only the in-jury predictor of the applicable loading condition can be greater than zero.Each predictor may not exceed a value of one.

15.5 3MS and XMS (CONTIGUOUS and CUMULATIVE)

The CONTIGUOUS_3MS injury criterion is computed by tracing a linear ac-CONTIGUOUSinjury criterion celeration or joint constraint load signal using a contiguous time window with

a width of 3 ms. The highest level with a duration of at least 3 ms is called theCONTIGUOUS_3MS injury criterion. The CONTIGUOUS_XMS injury crite-rion is a generalisation of the CONTIGUOUS_3MS injury criterion with a userdefined time window in stead of a 3 ms time window.

The CUMULATIVE_3MS is the highest acceleration or a joint constraint loadCUMULATIVEinjury criterion level that is exceeded during at least 3 ms. The time interval of 3 ms is not

necessarily contiguous. The CUMULATIVE_XMS injury criterion is a gener-alisation of the CUMULATIVE_3MS injury criterion with a user defined timewindow in stead of a 3 ms time window.

The results are written to the PEAK output file.

15.6 Thoracic Trauma Index (TTI)

To predict the probability of serious injury to the "hard" or bony thorax as aresult of blunt lateral impact, the Thoracic Trauma Index (TTI) was proposedin 1984. The TTI is an acceleration criterion based on the accelerations of thelower thoracic spine and the ribs. It also incorporates the weight and the age

294 Release 7.7


15

of the human model. The formulation was derived from a large biomechanicaldatabase consisting of 84 cadaver tests. These tests showed that the occurrenceof injuries to the hard thorax, including the ribs and the internal organs pro-tected by the ribs, is strongly related to the average of the peak lateral acceler-ation experienced by the impacted side of the rib cage and the lower thoracicspine. The TTI can be used as an indicator for the side impact performance ofpassenger cars. The specific benefit of the TTI is that it can be used to addressthe entire population of vehicle occupants because the age and the weight ofthe cadaver is included. The TTI as defined by Morgan:

TTI = 1.4 · AGE + 0.5 · (RIBg + T12g) · MASS/MSTD (15.6)

where:

AGE = age of the test subject in years

RIBg = maximum absolute value of acceleration in g’s ofthe 4th and 8th rib on struck side, in lateral direction

T12g = maximum absolute acceleration value in g’s of the12th thoracic vertebra, in lateral direction

MASS = test subject mass in kg

MSTD = standard reference mass of 75 kg

There is also a definition for the TTI that could be used for dummies withouta specific age, called the TTI(d). It is defined for 50th percentile dummies witha mass of 75 kg:

TTI(d) = 0.5 · (RIBg + T12g) (15.7)

For the US-SID, it is indicated where the instrumentation must be placed tomeasure the lateral accelerations at the level of the 4th and the 8th rib and theT12 spinal vertebra.

The dynamic performance requirement, as stated in the FMVSS 214 regula-tions of 1990, is that the TTI(d) level shall not exceed 85 g for passenger carswith four side doors and 90 g for two side doors.

The TTI(d) value is printed in the PEAK output file.

15.7 Viscous Injury Response (VC)

The vital organs of the chest, the heart and large blood vessels, and the lungsare built of soft tissues. The acceleration of bony structures such as the ribsand the spine, as used for the TTI criterion, the chest deflection and the ap-plied force do not address the injury mechanism at high velocity rates of softtissues and can therefore not be used as an injury criteria. Therefore the un-derstanding of the mechanism of soft tissue injury is critical for improvementof vehicle occupant protection systems.

Release 7.7 295

15


In the past, research has lead to the knowledge that soft tissue injury is in-duced by rate sensitive deformation of the chest. It was found that some occa-sions of pulmonary and cardiac injuries occurred in conditions of high impactvelocities with very little chest deformation. This fact is also reported frominjuries caused by the impact of a bullet on a bullet-proof vest, or a baseballhitting the chest directly. It was found that some of these impacts were fatal,even without any visible chest damage.

The Viscous response, denoted as VC, is the maximum value of a time func-tion formed by the product of the velocity of deformation (V) and the instan-taneous compression function (C):

VC = SF · max

(dD(t)

dt

D(t)

SZ

)(15.8)

where D(t) is a deflection and SZ is a prescribed size (the initial torso thicknessfor frontal impacts or half the torso width for side impacts). SF is a dummy-dependent scale factor; SZ and SF values for selected dummies are providedin the Reference Manual, under INJURY.VC.

Analyses of data from experiments on human cadavers show that a frontalimpact which produces a VC value of 1.3 m/s has a 50% chance of causingsevere thoracic injuries (AIS ≥ 4). A value of 1 m/s may be used as a referencevalue for human tolerance in blunt frontal impact to the chest. Specific detailsmay be found in the dummy hardware regulations and SAE J1727.

The VC value is printed in the PEAK output file.

15.8 Combined Thoracic Index (CTI)

The Combined Thoracic Index (CTI) is a measure of the injuries of the thorax.It is a combination of the maximum chest deflection Dmax and the 3 ms clipmaximum value of the resultant upper spine acceleration Amax . The equationfor the calculation of the CTI is given by

CTI = (Amax/Aint) + (Dmax/Dint) (15.9)

where Aint and Dint are constants that depend on the dummy.

15.9 Femur Force Criterion (FFC)

The Femur Force Criterion (FFC) is a measure of injury to the femur. It isthe compression force transmitted axially on each femur of the dummy as itis measured by the femur load cell. The FFC injury calculation is applied tothe joint constraint force in the bracket joint located at a femur load cell. It isassumed that the coordinate systems of this joint are oriented in agreementwith SAE J221/1 because as axial force, the component of the constraint forcein the joint ζ-direction is used. A duration curve of this time history signal

296 Release 7.7


15

is made. The resulting femur axial force duration curve must not exceed thevalues shown in Figure 15.3.

Figure 15.3: Femur force performance criterion.

15.10 Tibia Index (TI)

The Tibia Index (TI) is a measure of injury to the tibia. The TI injury calcu-lation is applied to the joint constraint load in the bracket joint located at atibia load cell. It is assumed that the coordinate systems of this joint are ori-ented in agreement with SAE J221/1 because as axial force, the component ofthe constraint force in the joint ζ-direction is used and as bending moment,the component of the constraint moment about the joint η-axis is used. Theequation for the calculation of TI is given by

TI = |FZ/(FC)Z| + |MR/(MC)R| (15.10)

where

FZ = compressive axial force in joint ζ-direction

(FC)Z = critical compressive force

MX = bending moment about the joint ξ-axis

MY = bending moment about the joint η-axis

MR =√

(MX)2 + (MY)2

(MC)R = critical bending moment

For the IIHS Frontal Offset Crashworthiness Evaluation test protocol MY

should be corrected as follows:

MY,adjusted = MY - e FZ

with e the value of the ECCENTRITY attribute under INJURY.TI. For the50th percentile Hybrid III and when the tibia load cell is specified accord-

Release 7.7 297

15


ing to the SAE J1733 sign convention : e=0.02832 for the upper tibia loadcell and e=−0.006398 for the lower tibia load cell. See for more informationhttp://www.iihs.org/ratings/protocols/pdf/test_protocol_high.pdf

The Tibia Index can be calculated for the top and bottom of each tibia. Foreach joint, the corresponding axial force FZ is used.

Critical compressive force and bending moment values can be found in theINJURY.TI section of the Reference Manual.

15.11 Tibia Compressive Force Criterion (TCFC)

The Tibia Compressive Force Criterion (TCFC) is a measure of injury to thetibia. It is the compressive force Fz expressed in kN, transmitted axially by atibia load cell. The TCFC injury calculation is applied to the joint constraintforce in the bracket joint located at a tibia load cell. It is assumed that thecoordinate systems of this joint are oriented in agreement with SAE J221/1because as axial force, the component of the constraint force in the joint ζ-direction is used.

15.12 Abdominal Peak Force (APF)

The Abdominal Peak Force (APF) is a measure of injury to the abdomen. Thisis a criterion for European and US side impact regulations. APF is the maxi-mum side abdominal strain criterion and is expressed as the highest value ofthe sum of the three forces [N] measured at each abdominal load cell (front,middle and rear) on the impact side, i.e.,

APF = MAX|Fy, front + Fy, middle + Fy, rear| (15.11)

15.13 Combined Iliac and Acetabulum Peak Force (CIAPF)

The Combined Iliac and Acetabulum Peak Force (CIAPF) is a measure of in-jury to the pelvis. The criterion is used in US regulations. CIAPF is the max-imum side pelvis strain criterion and is expressed as the highest value of thesum of iliac and acetabulum lateral forces [N] measured at the correspondingload cells on the impact side, i.e.,

CIAPF = MAX|Flateral, iliac + Flateral, acetabulum| (15.12)

15.14 Head Contact Duration (HCD)

The HCD value is the maximum of the HIC values during contact intervals.The intervals are determined using the resultant contact force (calculated fromneck force of the upper neck transducer, head acceleration and head mass).

Contact intervals are all the intervals in which a lower threshold value (thresh-old level = 200 N) is constantly exceeded and a lower search level (search level= 500 N) is exceeded at least once.

298 Release 7.7


15

15.15 Weighted Head Injury Criterion (HICd)

The HIC(d) value is the weighted standardized maximum integral value ofthe head acceleration and is calculated from the HIC36 value. The calculationof the HIC(d) is based on the following equation:

HIC(d) = 0.75446 · HIC36 + 166.4 (15.13)

with HIC36 the calculated HIC value using a time window of 36 ms.

15.16 Total Moment about Occipital Condyle (MOC)

The Total Moment about Occipital Condyle injury calculation is applied tojoint constraint load signals. It is assumed that the coordinate systems of this(bracket) joint are oriented in agreement with SAE J221/1 because as shearforce Fx , the component of the constraint force in the joint ξ-direction is usedand as bending moment, the component of the constraint moment My′ aboutthe joint η-axis is used. This bending moment is, in general, not about theoccipital condyles. The moment about the occipital condyles My is obtainedfrom the following equation:

My = My′ − eFx (15.14)

The moment about the occipital condyles Mx is obtained from the followingequation:

Mx = Mx′ + eFy (15.15)

where e is the distance between the occipital condyle and the joint in the pos-itive joint ζ-direction.

15.17 Neck Injury Criteria - Rearward (NIC_REARWARD)

The criterion for the neck injury with a rear impact is expressed by the relativeacceleration between the upper and lower neck acceleration and the relativevelocity.

15.18 Neck injury predictor Nkm

The Nkm1,2 is a criterion for neck injury in rear-end collision, which takes intoaccount a linear combination of the neck shearing force and the Total Momentabout the Occipital Condyle (MOC) (see Section 15.16).

1"Crash Analysis Criteria Description, Version 1.6.2", Arbeitskreis MessdatenverarbeitungFahrzeugsicherheit, April 2005 edition.

2Schmitt, K.-U., Muser, M.H., Niederer, P. "A New Neck Injury Criterion Candidate for Rear-end Collisions Taking into Account Shear Forces and Bending Moments", ESV Conference 2001,Amsterdam, The Netherlands.

Release 7.7 299

15


The Nkm injury calculation is applied to joint constraint load signals of the up-per neck load cell, filtered in accordance with CFC 600. It is assumed that thecoordinate systems of this bracket joint are oriented in agreement with SAEJ221 and SAE J1733; the shear force Fx is the component of the joint constraintforce in the joint ξ-direction and the bending moment is the component of theconstraint moment My′ about the joint η-axis. Since the bracket joint does notcoincide with the location of the occipital condyle, a correction is necessary tocalculate the Total Moment:

My = My′ − eFx (15.16)

where My is the Total Moment about Occipital Condyle and e is the eccentric-ity measured as the z(ζ)-coordinate of the occipital condyle in the upper loadcell (bracket joint) coordinate system.

Appropriate values for selected dummies are given in the Reference Manualunder INJURY.NKM.

Nkm is the collective name for four injury predictors, corresponding to differ-Four injurypredictors ent combinations of shear force and bending moment:

• flexion anterior (N f a). Moment flexion (forwards bending) My > 0.0;Forces anterior (head backwards, torso forwards), Fx > 0.0;

• extension anterior (Nea). Moment extension (backwards extension)My < 0.0; Forces anterior (head backwards, torso forwards), Fx > 0.0;

• flexion posterior (N f p). Moment flexion (forwards bending) My > 0.0;Forces posterior (head forwards, torso backwards), Fx < 0.0;

• extension posterior (Nep). Moment extension (backwards extension)My < 0.0; Forces posterior (head forwards, torso backwards), Fx < 0.0 .

The equation for the calculation of Nkm is given by:

Nkm(t) =|Fx|Fint

+|My|Mint

(15.17)

where Fint and Mint are constants that depend on the dummy and on the neckloading condition (anterior/posterior and flexion/extension). Only the injurypredictor of the applicable loading condition can be greater than zero. A valueof 1.0 is the critical value for all four injury predictors.

15.19 Lower Neck Load Index (LNL)

The risk of injury to the lower neck vertebrae in rear-impact is highest whenforces and moments at the centre of the T1 vertebra are acting simultaneously.

300 Release 7.7


15

The LNL1,2 combines three force components and two moment components,which are measured at the base of the neck. The LNL calculation is applied tojoint constraint load signals of the lower neck load cell, filtered in accordancewith CFC 600. It is assumed that the coordinate systems of this bracket jointare oriented in agreement with SAE J211/2 conventions. LNL is calculatedfrom the following equation:

LNL(t) =

∣∣∣∣∣∣

√M2

y(t) + M2x(t)

CM

∣∣∣∣∣∣+

∣∣∣∣∣∣

√F2

x (t) + F2y (t)

CS

∣∣∣∣∣∣+

∣∣∣∣Fz(t)

CT

∣∣∣∣ (15.18)

where My is the component of the constraint moment about the joint η-axis,Mx is the component of the constraint moment about the joint ξ-axis, Fx isthe component of the joint constraint force in the joint ξ-direction, Fy is thecomponent of the joint constraint force in the joint η-direction and Fz is thecomponent of the joint constraint force in the joint ζ-direction. Furthermore,CM is the critical bending torque, CS is the critical shear force and CT is thecritical tension force.

A correction can be applied for the moment signal My accounting for the offsetbetween the actual T1 location and the lower neck load cell (represented by abracket joint). This correction is not used for he RID2 dummy, as the effect isconsidered small. For the Hybrid III however, the offset between the actual T1location and the load cell is large, so the moment signal My is corrected, usingthe following equation:

My, corrected = My − ezFx + exFz (15.19)

In which ex is the x(ξ)-coordinate of the centre of the T1-vertebra in the lowerneck load cell (bracket joint) coordinate system and ez is the z(ζ)-coordinateof the centre of the T1-vertebra in the lower neck load cell (bracket joint) coor-dinate system.

1"Crash Analysis Criteria Description, Version 1.6.2", Arbeitskreis MessdatenverarbeitungFahrzeugsicherheit, April 2005 edition.

2Heitplatz, F. et all, "An Evaluation of Existing and Proposed Injury Criteria with VariousDummies to Determine Their Ability to Predict the Levels of Soft Tissue Neck Injury Seen in RealWorld Accidents", ESV Conference 2003, Nagoya, Japan.

Release 7.7 301

15


15.20 Brain Injury Criterion (BrIC)

The equation for the calculation of BrIC is given by:

BrIC =

√(max|ωx|)2

ω2xc

+(max|ωy|)2

ω2yc

+(max|ωz|)2

ω2zc

(15.20)

where ωx, ωy and ωz are the x, y and z components of the angular velocity ofthe head and ωxc, ωyc and ωzc are critical angular velocity component values.The three maxima can occur at three different time points. Further informationon this criterion can be obtained from.1

1Erik G. Takhounts et all, "Development of Brain Injury Criteria (BrIC)", Stapp Car Crash Jour-nal, Vol. 57 (November 2013), pp. 243-266.

302 Release 7.7

MADYMO Theory Manual Dynamic relaxation

16

16 Dynamic relaxation

16.1 Introduction

Dynamic relaxation can be used to approximate solutions to linear and non-Introductionlinear static or quasi-static simulations. At present, the procedure only affectsthe finite element part of the code, but models with multi-body parts are al-lowed. Convergence is only checked on the variation of the kinetic and inter-nal energy of the FE models. Models with only multi-body parts cannot betreated due to this.

This method can be used to calculate the initial stresses/strains and displace-ment field of an FE-model due to contact interactions (internal or external withmulti-body components), prior to beginning of a dynamic analysis.

16.2 Description of the method

Dynamic relaxation can be viewed as an explicit solution procedure to obtaina static solution. The basic idea is to solve the dynamic system equations withoptimal damping so that the solution will converge to a static solution in aminimal time.

The dynamic equilibrium equation with damping at time n is given by

Man + Cvn + fint(un) = fext(tn) (16.1)

where M and C are the mass and damping matrices, fint and fext are the in-ternal force and external force (applied load) vectors and a, v and u are thenodal acceleration, velocity and displacement vectors, respectively. A centraldifference scheme is used for the calculation of the nodal velocities and dis-placements; for a fixed time increment ∆t the relations are:

vn+ 12 = vn− 1

2 + ∆tM−1(fnext − fn

int − Cvn) (16.2)

and

un+1 = un + ∆tvn+ 12 (16.3)

The mass matrix M is assumed to be diagonal and the damping matrix C hasthe form

C = αM (16.4)

The rate of convergence to obtain the static solution u is controlled by thetime step ∆t and the damping parameter α. For an optimum convergence thedamping parameter has to be the critical damping constant:

αn = 2ωcrit (16.5)

Release 7.7 303

16

Dynamic relaxation MADYMO Theory Manual

where ωcrit is the natural frequency of the fundamental mode of the system.

In MADYMO, the critical damping constant is calculated per element and thecorresponding dynamic relaxation damping forces are applied per element.This critical damping constant per element eαn is calculated for each elementas

eαn = 2

√max(0, ekmodal)

emmodal(16.6)

where ekmodal is the modal stiffness of the element :

ekmodal = eunT eKn eun (16.7)

and emmodal is the modal mass of the element:

emmodal = eunT eMn eun (16.8)

In these equations, eun are the element displacements, eKn is an approxima-tion of the element tangent stiffness matrix and eMn is a fictitious elementmass matrix.

Numerical differentation of the internal element force vector efint is used todetermine the components of eKn

lknij =

[−e fint(eun−1

i )+e fint(eun

i )]

∆t evn− 1

2j

for evn− 1

2j v 6= 0

0 for evn− 1

2j = 0

(16.9)

The elements lmii of this fictitious matrix eMn are computed from

lmii =

(∆t

2µ

)2

∑j

|lmij| (16.10)

where lkij are the elements of the approximated stiffness matrix lKn and µ isthe reduction factor for the time step ∆t .

This fictitious matrix eMn is only used for the integration of the dynamic equi-librium equation; for gravity loads due to an applied acceleration field andcontact evaluation, the specified mass density of the corresponding materialis used.

The damping force per element for dynamic relaxation is now calculated as

efndamp = ζ max(αmin, eαn)eMnevn (16.11)

where evn is the velocity of the nodes of the element, αmin is the minimumdamping coefficient specified by the user and ζ is a reduction coefficient spec-ified by the user.

304 Release 7.7

MADYMO Theory Manual Dynamic relaxation

16

16.3 Convergence criteria

The dynamic relaxation iteration process is terminated after the convergencecriteria are satisfied for the kinetic energy of all FE models and the internalenergy of all FE models.

The convergence criterion for the kinetic energy is defined as

CEin=

Enkin

Ekinmax

< εk (16.12)

where Enkin is the current kinematic energy and Ekinmax

is the maximum kine-matic energy during the dynamic relaxation process.

The convergence criterion for the elastic energy is defined as

CEelas=

∣∣∣∣∣(En

elas − En−1elas )

En−1elas

∣∣∣∣∣ < 0 (16.13)

where Enelas is the internal energy at time point n.

The convergence criterion for the stored energy is defined as

CEstor = Enmat.damp + En

plas + Enimm.damp + En

elas > 0 (16.14)

where Enmat.damp is the energy associated with material (internal) damping,

where Enplas is the energy due to plastic work,

where Enimm.damp is the work of initial metric damping force,

and Enelas is the internal energy at time point n.

If all three criteria are satisfied for each FE model then the dynamic relaxationprocess will scale down the damping factors eαn until αmin is reached. This isneeded because the kinematic energy of the model may not be zero, implyingthat the models still have velocity and the damping factors contribute to the(quasi)-static solution. The scale down factor (alpha relax factor) φ determinesthe speed for which the eα is scaled down.

ζn = ζn−1φ with ζ0 = ζ (16.15)

During the relaxation of the α the kinetic energy and elastic energy can growagain. If CEkin

≥ 100εk or CEelas≥ 100εe the relaxation of the α will be stopped

and ζ will be increased again until CEkin< 100εk and CEelas

< 100εe againusing

ζn = min(1, ζn−1(1 + φ)) (16.16)

with ζ the reduction coefficient for the alpha damping specified by the user.

Release 7.7 305

16

Dynamic relaxation MADYMO Theory Manual

Notice that φ should not be chosen too low because it will be likely that theconvergence criteria are violated and the differences in kinetic/elastic energywill be too high. This can cause instabilities. If φ is chosen too high it will takemany cycles before the damping factor reaches αmin . Default φ is chosen as0.05 and this value is recommended. If instabilities occur during the relaxationthis value should be increased.

After the damping factors have reached αmin , the normal analysis will startat t0 as starting point with the displacement field, stress and strain field ofthe static equilibrium state as obtained at the end of the dynamic relaxationprocess.

16.4 Output

It is possible to write kinematic and time history output during the dynamicrelaxation process. Output files with kinematic and time history output gen-erated during the dynamic relaxation process will get an extra extension _drlx.

16.5 Extra options

All features like contacts, welds, constraints, loads will work normally duringthe dynamic relaxation process. However for some functionality it is possibleto select if it should work during the relaxation phase only, during the normalanalysis only or in both analysis. This is implemented for

• External loads (FE and MB).

• Switches.

• Initial metric method.

306 Release 7.7

MADYMO Theory Manual Functions

A

APPENDIX A Functions

Functions are used in MADYMO to define material properties, joint charac-teristics, acceleration fields.

• the belt force as a function of the relative elongation;Examples

• the belt outlet as a function of the retractor load (film spool effect);

• the inflator mass flow rate as a function of time;

• an acceleration field as a function of time;

• the elastic contact force as a function of the penetration;

• the spring force as a function of the relative elongation;

• stress as a function of strain for finite element material models.

The general form of these functions is:

y = y(x) (A.1)

where x is the independent variable. Functions are defined in the MADYMOinput data file in tabular form. To define a function y(x) the user must specifyn coordinate pairs:

(x1, y1), (x2, y2), . . . , . . . , (xn, yn) (A.2)

The function pairs must be specified in increasing order of x:

x1 < x2 < . . . < . . . < xn (A.3)

Three methods are available in MADYMO to interpolate between coordinateThreeinterpolationmethods

pairs (Figure A.1):

• linear interpolation;

• cubic spline interpolation;

• 5th degree spline interpolation.

Release 7.7 307

A

Functions MADYMO Theory Manual

Figure A.1: Linear and spline interpolation of function table.

A function approximation by means of cubic splines results in a smooth repre-Cubic splineinterpolation sentation (continuity of y(x) and y′(x)). A cubic spline approximation consists

of a series of third degree polynomials:

y = a0 + a1x + a2x2 + a3x3 (A.4)

The coefficients a0 , a1 , a2 and a3 are determined for each interval betweentwo successive coordinate pairs i and j. The spline approximation of y(x) issuch that y(x) and y′(x) are continuous over the entire range of x.

A function approximation by means of 5th degree splines results in a smooth5th degreesplineinterpolation

representation (continuity of y(x), y′(x) and y”(x)). A 5th degree spline inter-polation consists of a series of 5th degree polynomials:

y = a0 + a1x + a2x2 + a3x3 + a4x4 + a5x5 (A.5)

The coefficients a0 , a1 , a2 , a3 , a4 and a5 are determined for each intervalbetween two successive coordinate pairs i and j. The spline approximation ofy(x) is such that y(x), y′(x) and y”(x) are continuous over the entire range ofx.

If necessary MADYMO will extrapolate the function tables. For linear, splineand spline_5 interpolation, a linear extrapolation is used based on y(x) andy′(x) at the end of the interval. Extrapolation of functions should be avoided.A warning message is printed in the REPRINT file if a function table has beenextrapolated. The ranges of extrapolated functions should be expanded toprevent extrapolation.

308 Release 7.7

MADYMO Theory Manual Hysteresis

B

APPENDIX B Hysteresis

Elastic properties for joints, springs, contacts, belts and restraints are definedby means of functions (see Appendix A), with x representing the deformationand y the corresponding load. Energy dissipation in these force models canbe described by means of hysteresis. Three hysteresis models are available.

Require the specification of (Figure B.1):Hysteresismodels 1 and 2

• a loading curve yl(x)

• an unloading curve yu(x)

• a hysteresis slope sl

• an elastic limit xe .

The slope parameter sl, ∆y/∆x in Figure B.1, defines a linear function joiningthe loading and the unloading curve. The same slope sl is used for positiveand negative values of the deformation x.

Figure B.1: Parameters for description of hysteresis models 1 and 2.

Release 7.7 309

B

Hysteresis MADYMO Theory Manual

Although it should be avoided, the hysteresis models 1 and 2 accept a loadingcurve slope at the end of the curve which is greater than the hysteresis slopesl.

• If this situation occurs for a positive xs , the slope of the loading curvemust be greater than slope sl for all x > xs .

• If this situation occurs for a negative xs , this requirement must be satis-fied for all x < xs .

• If the slope of loading curve yl(x) is greater than the hysteresis slope sl,unloading first proceeds along the loading curve yl(x) rather than alongthe slope sl (Figure B.2). A warning message is printed in the REPRINTfile.

Figure B.2: Unloading behaviour if loading slope is greater than hysteresis slope sl.

Requires the specification of:Hysteresismodel 3

• a loading curve yl(x)

• an unloading curve yu(x)

• an elastic limit xe .

Depending on the hysteresis model 3 options chosen and the sign of the defor-mation x, the part of the unloading curve defining the function values in onequadrant is shifted and scaled in such a way that the function values of theloading and the adjusted unloading curve are the same for the actual value ofx as soon as unloading begins (Figure B.3).

310 Release 7.7


B

yl (xmax)

xminxed

-xe

yl (xmin)

xe0 xpd

xed xmax

Y

X

shiftedyuyl

Figure B.3: Hysteresis model 3.

For all three hysteresis models, the loading and unloading curves are definedby means of the function option. The curves are approximated either by splineor by linear interpolation. If the spline interpolation is chosen for the loadingand unloading curves when using hysteresis model 1 or 2, the spline curvesfor loading and unloading must not intersect (except at the origin where in-tersection is allowed).

The major difference between the hysteresis models is in the options for theDifferencesbetweenhysteresismodels

loading and unloading characteristics and the reloading:

• For all three models, the loading curve must go through the origin.

• For hysteresis models 1 and 2, the loading curve is defined only in thefirst and third quadrant and can thus only be used to model passiveelements.

• For hysteresis model 3, also a curve in the second and fourth quadrantcan be defined, thus also active elements can be modelled.

• For models 1 and 3, the unloading curve must also go through the origin,which is not required for hysteresis model 2.

• For model 3, the unloading curve must be a strictly increasing or de-creasing function.

• For models 1 and 2, a fixed hysteresis slope is defined whereas for model3 the unloading curve is shifted and scaled without using a hysteresisslope.

Release 7.7 311

B


• In models 1 and 3, a reloading will follow the unloading curve in op-posite direction until the point where the unloading curve was entered(model 1) or until the point where the loading curve was left (model 3).In model 2, reloading takes place immediately along the hysteresis slopeuntil the loading curve is reached.

In the contact models (such as an ellipsoid-ellipsoid contact), the interpolationmethod for the loading curve must be the same as for the unloading curve.For the other force models (such as a Kelvin element), different interpolationmethods can be selected such as spline interpolation for loading and linearinterpolation for unloading.

B.1 Hysteresis model 1

In Figure B.4, loading and unloading curves are given that satisfy the condi-tions for hysteresis model 1.


The following conditions must be satisfied

312 Release 7.7


B

x < 0 : yl(x) ≤ 0, yu(x) ≤ 0

x = 0 : yl(0) = 0, yu(0) = 0

x > 0 : yl(x) ≥ 0, yu(x) ≥ 0

for all x : |yl(x)| ≥ |yu(x)|

(B.1)

Hysteresis calculations are performed if (unloading or reloading beyond theelastic limit):

0 < x < xmax and xmax ≥ xe

xmin < x < 0 and xmin ≤ −xe

where

xmin = minimum value x has reached so far (xmin ≤ 0),

xmax = maximum value x has reached so far (xmax ≥ 0).

The elastic limit xe is defined as a positive value which is the same for positiveand negative x.

The hysteresis behaviour for x > 0 is as follows (see Figure B.4):Hysteresisbehaviour forx > 0 • If the independent variable x reaches a maximum value (xmax) and the

elastic limit xe has been exceeded, unloading will take place along thehysteresis slope sl until the unloading curve is reached.

• The unloading then proceeds along the unloading curve.

• A reloading will first follow the unloading curve in opposite directionuntil the point where the unloading curve was first entered.

• Reloading then continues upward along the hysteresis slope sl until theloading curve is reached again.

• Further loading beyond xmax will follow the loading curve until a newxmax . A subsequent sequence of loading and unloading will occur in thesame way as just described.

For x < 0 , the hysteresis behaviour is similar, except that hysteresis calcula-Hysteresisbehaviour forx < 0

tions are carried out if x reaches a minimum value (xmin) and the elastic limitxe has been exceeded in the negative x-direction.

Hysteresis for a positive value of the independent variable x is fully indepen-dent from hysteresis for a negative value of x. For instance, if a spring intension unloads along the unloading curve and the spring goes from tension

Release 7.7 313

B


into compression, the compression is described by the loading function (pro-vided that the spring has not been in compression before). If, subsequently,the spring goes into tension again, the reloading will take place along the un-loading curve.

Figure B.5 illustrates this hysteresis behaviour for a series of loading and un-loading steps:

1. loading along loading curve until maximum xmax 1 ,

2. unloading along hysteresis slope sl and subsequent reloading untilxmax 1 ,

3. loading until maximum xmax 2 ,

4. unloading along slope sl,

5. unloading along unloading curve,

6. loading along unloading curve,

7. loading along hysteresis slope sl,

8. further loading along loading curve.

max,2xmax,1xex

1

3

8

7

42

6

5sl

y

x

Figure B.5: Example illustrating various loading and unloading steps.

314 Release 7.7


B


In contrast to hysteresis model 1, hysteresis model 2 does not require the un-loading curve to go through the origin and reloading is immediately along thehysteresis slope. Three cases of unloading behaviour can be distinguished:

yu(0) < 0

yu(0) = 0

yu(0) > 0

There is no need to select explicitly a type of unloading behaviour for hystere-sis model 2. During the input phase, MADYMO detects automatically whichone of these three cases of unloading is used. The following conditions mustbe satisfied for the loading curve:

x < 0 : yl(x) < 0

x = 0 : yl(x) = 0

x > 0 : yl(x) > 0

For yu(0) < 0 , the unloading curve is allowed to pass from the first to theyu(0) < 0

fourth and third quadrants (Figure B.6). The elastic limit xe is only defined forpositive x.

Figure B.6: Hysteresis model 2 for yu(0) < 0 .

Release 7.7 315

B


The loading curve must be above the unloading curve for all x:

yl(x) ≥ yu(x) (B.2)

Hysteresis only occurs if the elastic limit xe has been exceeded. In other words,hysteresis calculations are only carried out in the following case:

x < xmax , xmax ≥ xe ≥ 0 (B.3)

where xmax = maximum value of x reached so far.

If the elastic limit has not been exceeded (xmax < xe), the loading curve is usedfor positive as well as for negative deformation x. The hysteresis loop is onlytraced in clockwise direction as shown in Figure B.6.

The general behaviour of this hysteresis model is as follows (Figure B.6):

• If the independent variable x reaches a maximum value (xmax) and theelastic limit xe has been exceeded, unloading will take place along thehysteresis slope sl until the unloading curve is reached.

• Continued unloading then proceeds downward along the unloadingcurve.

• Reloading takes place immediately along the hysteresis slope sl until theloading curve is reached again.

• Further loading beyond xmax proceeds along the loading curve until anew xmax is reached. Any subsequent loading and unloading will occurin the same way.

For yu(0) = 0 , the unloading curve passes through the origin (Figure B.7).yu(0) = 0

Its hysteresis behaviour differs from that of hysteresis model 1 (Section B.1) inthe sense that reloading takes place immediately along the hysteresis slope sl,rather than first along the unloading curve.

316 Release 7.7


B

Figure B.7: Hysteresis model 2 for yu(0) = 0 .

The following condition must be satisfied for all x:

|yl(x)| ≥ |yu(x)| (B.4)

Hysteresis calculations are only carried out in the following cases:



(B.5)

where



The elastic limit xe is defined as a positive value. For positive and negativevalues of x the same elastic limit applies. The hysteresis loop can only betraced in clockwise direction as shown in Figure B.7.

Hysteresis for a positive value of the deformation x is fully independent fromhysteresis for a negative value of x. For instance, if a spring in tension unloadsalong the unloading curve and the spring goes from tension into compression,the compression is described by the loading function (provided that the spring

Release 7.7 317

B


has not been in compression before). If, subsequently, the spring goes intotension again, the reloading will take place along the loading curve.

For yu(0) > 0 , the unloading curve is allowed to pass from the first to theyu(0) > 0

second and the third quadrants of the graphs (Figure B.8).

Figure B.8: Hysteresis model 2 for yu(0) > 0 .

Hysteresis will only occur if the elastic limit xe has been exceeded in negativex-direction. In other words, hysteresis calculations are only carried out in thefollowing case:

x > xmin , xmin ≤ −xe (B.6)

where xmin = minimum value of x reached so far, xmin ≤ 0 .

The elastic limit xe is defined as a positive value. The hysteresis loop can onlybe traced in the clockwise direction as shown in Figure B.8. The hysteresisloop is the mirror image of the one for yu(0) < 0 . The loading starts along theloading curve in negative x-direction.

The unloading curve must be above the loading curve for all x:

yu(x) ≥ yl(x) (B.7)

318 Release 7.7


B


In contrast to the first two hysteresis models, hysteresis model 3 does not re-quire a slope parameter. A shifted and (optional) scaled unloading curve isused.

The following conditions must be satisfied (loading and unloading curvethrough the origin and only unloading curve strictly increasing or strictly de-creasing):

x = 0 yl(0) = 0, yu(0) = 0

yl(x) ≤ 0 for all x < 0 and

yl(x) ≥ 0 for all x > 0 and

yu(x1) < yu(x2) for all x1, x2 with x1 < x2

or

yl(x) ≥ 0 for all x < 0 and

yl(x) ≤ 0 for all x > 0 and

yu(x1) > yu(x2) for all x1, x2 with x1 < x2

(B.8)

So the loading function must be defined in the first and third quadrant only. Atthe same time, the unloading function must be a strictly increasing function orthe loading function must be defined in the second and fourth quadrant andalso the unloading function must be a strictly decreasing function.

Hysteresis calculations are performed if:



(B.9)

where



The elastic limit xe is defined as a positive number which is the same for posi-tive and negative x.

The hysteresis behaviour for x > 0 is shown in Figure B.9. After the elasticlimit xe has been exceeded, unloading takes place along a shifted and (op-tional) scaled unloading curve first, followed by further unloading along thex-axis.

Release 7.7 319

B


yl (xmax)

xminxed

-xe

yl (xmin)

xe0 xpd

xed xmax

Y

X

shiftedyuyl


Only the part of the unloading curve which is used for positive deformationis scaled and shifted. The amount of shifting and scaling is determined by thefunction value at which unloading starts: yl(xmax) in Figure B.9. The corre-sponding value of the deformation x is composed of an elastic deformationxed and a permanent deformation xpd .

For x < 0 the hysteresis behaviour is similar, except that hysteresis calcula-tions are carried out if x reaches a minimum value xmin and the elastic limitxe has been exceeded in the negative x-direction. In this case, only that partof the unloading curve which is used for negative deformation x is scaled andshifted.

Hysteresis for positive values of x is independent from hysteresis for a nega-tive value of x.

Type A: Only shifting of the unloading curve.Threeadjustmentoptions forunloadingcurve

Type B: Scaling of load data and subsequent shifting of the unloading curve.

Type C: Scaling of both deformation and load data and subsequent shifting ofthe unloading curve.

When unloading starts for x > 0 , the part of the unloading curve that isType Aused for x ≥ 0 is shifted in such a way that it intersects the loading curve foryl(xmax) , where xmax is the largest x-value reached so far (see Figure B.10).

Because the unloading curve is strictly increasing or decreasing, this shiftingis unique. If necessary the unloading curve is extrapolated. For x < 0 a similarshifting is performed when unloading starts.

320 Release 7.7


B

Figure B.10: Hysteresis model 3, type A.

In addition to the general conditions for hysteresis model 3 (B.8), at least oneType Bnon-zero function value must be defined for the unloading curve for a positiveand negative x value:

• If x > 0 during the simulation, the last function value defined must benon-zero: yu(xm) 6= 0 with xm > 0 .

• If x < 0 during the simulation, the first function value defined must benon-zero: yu(x1) 6= 0 with x1 < 0 .

When unloading starts for x > 0 , the part of the unloading curve which isused for x ≥ 0 is scaled in such a way that the function value belonging toxm , the largest value for x used in the function table of the unloading curve,is scaled to yl(xmax) , with xmax the largest value for x reached so far. The

resulting scaling factor equalsy1(xmax)yu(xm)

. Then the scaled unloading curve is

shifted to connect to the loading curve at x = xmax . (Figure B.11). Because theunloading curve is strictly increasing or decreasing, this shifting is unique.For x < 0 a similar scaling and shifting is performed when unloading starts.

Release 7.7 321

B


Figure B.11: Hysteresis model 3 type B.

In addition to the general conditions for hysteresis model 3 (B.8), at least oneType Cnon-zero function value must be defined for the unloading curve for a positiveand negative x value:

• If x > 0 during the simulation, the last function value defined must benon-zero: yu(xm) 6= 0 with xm > 0 .

• If x < 0 during the simulation, the first function value defined must benon-zero: yu(x1) 6= 0 with x1 < 0 .

When unloading starts for x > 0 , that part of the unloading curve whichis defined for x ≥ 0 is scaled in such a way that the function value belong-ing to xm , the largest value for x used in the function table of the unloadingcurve, is scaled to yl(xmax) , with xmax the largest value for x reached so far.

The resulting scaling factor equalsy1(xmax)yu(xm)

. Subsequently the values of the

independent variable x defining the function are scaled by the same factor.Then the scaled unloading curve is shifted to connect to the loading curve atx = xmax , (Figure B.12). Because the unloading curve is strictly increasing ordecreasing, this shifting is unique. For x < 0 a similar scaling and shifting isperformed.

322 Release 7.7


B

Figure B.12: Hysteresis model 3 type C.

Release 7.7 323

B


324 Release 7.7

MADYMO Theory Manual Dynamic amplification factor

C

APPENDIX C Dynamic amplification factor

Load-deformation characteristics that must be provided for the variousforce models are generally obtained from quasi-static tests. However, load-deformation characteristics may depend on the rate of deformation.

In lumped-mass vehicle crash models, it is customary to account for this de-pendency by multiplying static forces by a factor which depends on the rateof deformation. This approach has been adopted in MADYMO for mostdynamic joint models, the contact models, the Kelvin model and the point-restraint model. The rate of deformation ε used for calculating the amplifica-tion factor of a specific force model is identical to the rate used for calculatingdamping forces for that model. The following frequently used amplificationfactors g are available:

1. γ =C1 + C2|ε|+ C3ε2 + C4|ε|3 + C5 ε4

2. γ =

C1 + C2 log(|ε|/C3)

C1

(|ε| > C3, C3 > 0)

(|ε| < C3, C3 > 0)

(C.1)

3. γ =C1 + C2(|ε|/C3)C4 (C3 > 0)

4. γ =

1 + C1ε + C2 ε2 + C3ε3 + C4 ε4

(1 − C1 ε + C2 ε2 − C3 ε+C4 ε4)−1

(ε ≥ 0)

(ε < 0)

When a dynamic amplification factor is specified, the load which is obtainedfrom the static load-deformation characteristic is multiplied by a factor whichdepends on the instantaneous rate of deformation ε.

As an example, Figure C.1 shows the load-deformation characteristic for somedeformation rate values (dashed lines). The load-deformation characteris-tic determined during the simulation depends on the course of the veloc-ity. The solid curve in Figure C.1 shows the qualitative path of such a load-deformation characteristic if the rate of deformation decreases drastically fromv2 at zero deformation to 0 at deformation δ.

Release 7.7 325

C

Dynamic amplification factor MADYMO Theory Manual

Figure C.1: Load-deformation characteristics for different rates of deformation and thecharacteristic determined during the calculation.

This amplification factor is also useful in parameter studies on the influence ofthe stiffness corresponding to a specific force model. Using the amplificationmodel, the level of a force-deflection characteristic can be easily changed bymultiplying it by a constant factor.

326 Release 7.7

MADYMO Theory Manual Coordinate system orientation

D

APPENDIX D Coordinate system orientation

In Section 3.4, a direction cosine matrix was introduced. With this matrix, theThe orientationof coordinatesystems

orientation of a body local coordinate system was expressed relative to thereference space coordinate system (equation (3.3)). In a similar way, a direc-tion cosine matrix is used to define the orientation of a joint coordinate systemrelative to a body local coordinate system (Section 3.5). A direction cosine ma-trix also allows the user to specify the orientation of any auxiliary coordinatesystem, such as an ellipsoid coordinate system (Figure 3.32) or the inertia co-ordinate system, relative to the body local coordinate system.

In order to generalize the mathematical description of the relative orientationof two coordinate systems, a fixed coordinate system i, (xi, yi, zi), and a rotatedcoordinate system j, (xj, yj, zj), are introduced. The relative orientation of twocoordinate systems does not depend on the relative position of their origins.For convenience, it is assumed that the origins coincide (Figure D.1).

Figure D.1: A rotated coordinate system j and a fixed coordinate system i.

Let Pi be a column matrix with the coordinates in the fixed coordinate systemi of a point P. Pj is a column matrix with the coordinates of P in the rotatedcoordinate system j. The two matrices are related by the following expression:

Pi = A Pj (D.1)

Release 7.7 327

D

Coordinate system orientation MADYMO Theory Manual

In other words, the direction cosine matrix A transforms the coordinates of apoint in the rotated coordinate system j to the fixed coordinate system i. Thek-th column of A consists of the components of a unit vector parallel to thek-th axis of coordinate system j relative to coordinate system i (k = 1, 2, 3).

Consider the case in which coordinate system j, (xj, yj, zj), results from arotation of coordinate system i, (xi, yi, zi), of ϕ radians around the xi axis(Figure D.2). The corresponding direction cosine matrix A is given by

A =

1 0 0

0 cos ϕ − sin ϕ

0 sin ϕ cos ϕ

(D.2)

zj

ϕ

zi

xi, xj

yj

yi

Figure D.2: Specification of coordinate system orientation.

Instead of specifying the direction cosine matrix directly, it is often easier touse one of the three other methods available for MADYMO:

The specification of up to three successive rotations around coordinate axes ofRotation anglemethod the coordinate system j relative to the coordinate system i until it reaches its

final orientation (Section D.1).

The specification of a single rotation of the coordinate system j around an axisScrew axismethod in the coordinate system i (Section D.2).

328 Release 7.7


D

The specification of the components relative to the coordinate system iVector methodof a vector parallel to the xj-axis and a vector parallel to the xjyj-plane(Section D.3).

D.1 Rotation angle method

Assume that initially the fixed coordinate system i and the rotated coordinatesystem j coincide. In the rotation angle method, the final orientation of coor-dinate system j relative to i is the result of up to three successive rotations.

The first rotation is carried out around one of the three coordinate axes ofFirst rotationthe fixed system i. The user has to specify the selected rotation axis and therotation angle. The resulting orientation of coordinate system j is given by afirst intermediate coordinate system (x′j, y′j, z′j).

The second rotation is carried out around one of the axes of the coordinateSecondrotation system (x′j, y′j, z′j). This rotation produces a second intermediate coordinate

system (x′′j , y′′j , z′′j ).

The third rotation which is carried out around one of the axes of coordi-Third rotationnate system (x′′j , y′′j , z′′j ) results in the final orientation of coordinate system

(xj, yj, zj).

In the MADYMO input file, the user specifies the order in which the rotationsare to be carried out and the rotation angles. The sign convention for theserotation angles is illustrated in Figure D.3. With this information, MADYMOcomputes the corresponding direction cosine matrix.

Figure D.3: Sign conventions for the angles in the rotation angle method.

Release 7.7 329

D


An example for two successive rotations is shown in Figure D.4:

• the first rotation is through an angle of π/2 around the zi-axis;

• the second rotation is through an angle of π/4 around the x′j-axis.

A sequence of three successive rotations around the xi-axis, the y′j-axis and the

z′′j -axis, respectively, results in three rotation angles often referred to as Bryant

angles or cardan angles. These Bryant angles are used to define the torquecharacteristics in the cardan restraint model (Section 7.5).

Figure D.4: Example of the rotation angle method.

D.2 Screw axis method

The fixed coordinate system i and the rotated coordinate system j are assumedto coincide initially. The final orientation of the rotated coordinate system isthe result of a single rotation around a specific axis (Euler theorem). Thisaxis (the screw axis) will be denoted by the unit vector u. The rotation anglearound this axis will be denoted by χ. A right-handed rotation is defined aspositive (Figure D.5). It can be shown that the components of u are the samein the rotated and in the fixed coordinate system.

330 Release 7.7


D

zi u

+ χ

yi

xi

Figure D.5: Sign convention for the rotation angle in the screw axis method.

To define the orientation of the rotated coordinate system j, the user needs tospecify the components of u and the rotation angle χ. Because u is normalisedby MADYMO, it is not required that the specified components of u form aunit vector. With this information MADYMO computes the direction cosinematrix A according to:

A =

2(q20 + q2

1) − 1 2(q1q2 − q0q3) 2(q1q3 + q0q2)

2(q1q2 + q0q3) 2(q20 + q2

2)− 1 2(q2q3 − q0q1)

2(q1q3 − q0q2) 2(q2q3 + q0q1) 2(q20 + q2

3) − 1

(D.3)

where q1 , q2 , q3 are the components of a vector q which is defined by:

q = u sin(χ/2)

and

q0 = cos(χ/2)

The four scalars q0 , q1 , q2 and q3 are often referred to as Euler parameters.Because u is a unit vector, Euler parameters satisfy the relation:

q20 + q2

1 + q22 + q2

3 = 1 (D.4)

Figure D.6 shows an example of the screw axis method.

Release 7.7 331

D


zi

yj

xj

zj

xi yi

+45° aboutvector (1,1,1)

zi

xj

zj

xiyi

yj

(1,1,1)

+45°

Figure D.6: Example of screw axis method.

D.3 Vector method

Consider the two coordinate systems (xi, yi, zi) and (xj, yj, zj) shown inFigure D.7. The orientation of (xj, yj, zj) relative to (xi, yi, zi) can be definedby the direction of the three coordinate axes relative to the (xi, yi, zi) coordi-nate system. The direction of two coordinate axes are sufficient to define theorientation of the (xj, yj, zj) coordinate system because the third axis can bedetermined from the condition that coordinate axes are mutually perpendicu-lar.

zi

yj

xj

zj

xi

yi

uv

Figure D.7: Vectors u and v defining orientation of (xj , yj, zj).

332 Release 7.7


D

In the vector method, the direction of two axes are defined by two vectors uand v. The first vector, u, must be parallel to the xj-axis; the second vectormust be parallel to the xjyj-plane, but not parallel to the xj-axis. The compo-nents of these vectors with respect to the (xi, yi, zi) coordinate system have tobe specified.

The rotation matrix A is determined from these two vectors as follows:

• the unit vector parallel to the xj-axis, e1 , is given by

e1 =u

|u|

where |u| is the magnitude of the vector u;

• the unit vector parallel to the zj-axis, e3 , is given by

e3 =u × v

|u × v|

where × is the vector product operator;

• the unit vector parallel to the yj-axis, e2 , is given by

e2 = e3 × e1.

The components of these three unit vectors with respect to the (xi, yi, zi) coor-dinate system are the three columns of the rotation matrix A.

Release 7.7 333

I

INDEX MADYMO Theory Manual

Indexacceleration field, 147actuator, 61airbag

heat flow, 241in/outflow, 243

aspiration, 243assemble, 11

beltfilm spool effect, 178, 179pocketing, 184pretensioner

buckle, 26, 181retractor, 180

retractor, 178rupture, 170, 182segment, 170slack, 170slip, 171, 174tying, 170

bodybreaking, 26child, 9flexible, 17rigid, 12

boundary condition, 68prescribed motion, 138prescribed structural motion, 138support, 138

central difference method, 65, 69centre of gravity, 12characteristic element length, 70closed chain, 7, 8, 10combined thoracic index, 296conduction, 241constitutive equation, 67, 68constraint

joint, 21constraint equations, 139contact

characteristic, 193elastic model, 185, 198FE-FE, 202kinematic model, 185, 198MB-FE, 198MB-MB, 185

334 Release 7.7

MADYMO Theory Manual INDEX

I

controller, 60convection, 241coordinate system

beam, 19body local, 12joint, 15, 22orientation, 327

courant criterion, 70

damage, 84, 88deformation modes, 20degrees of freedom

joint, 7, 23modal, 18position, 14

dynamic amplification, 325dynamic equilibrium equation, 303

elementbeam, 119interface, 133membrane, 126shell, 128solid, 134truss, 119

element:rigid, 117energy, 279equations of motion, 48, 65, 68exhaust orifice, 243explicit integration, 65, 70

femur force criterion, 296flexible beam, 19

gadd severity index, 289gas mixture, 228, 236gasflow

CFD-method, 251global discharge, 254

head injury criterion, 290Hill criterion, 86hourglass, 128, 131, 136hysteresis, 309

model 1, 309, 312model 2, 309, 315model 3, 310, 319

inertiamoments of, 12

Release 7.7 335

I


products of, 12inflator, 228initial condition, 50, 68, 138initial metric method, 225

imposed transition (IMM2), 226spring-damper (IMM1), 249spring-damper (IMM2), 225strain-base (IMM1), 227strain-based (IMM2), 249

injury biomechanics, 287injury severity, 287integration method

Euler, 50, 51Runge-Kutta, 50, 51Runge-Kutta Merson, 50, 51

interpolationcubic spline, 307linear, 307

jet, 229cone, 230, 232, 233constant momentum, 233idelchik, 232shape, 229user-defined, 234wedge, 230, 233, 234

jointbracket, 35closing, 8free

Bryant angles, 37Euler angles, 36Euler parameters, 35rotation-translation, 39

lock, 10, 26planar, 34revolute, 26spherical

Bryant angles, 30Euler angles, 29Euler parameters, 28

translational, 27translational-revolute, 39translational-universal, 41universal, 32universal-translational, 42unlock, 26

joint:revolute-translational, 40

Lagrangian description, 67, 119, 121

336 Release 7.7

MADYMO Theory Manual INDEX

I

leakage, 248load

acceleration, 139edge, 139force, 139moment, 139surface, 139

magic formula, 269, 274material

anisotropic, 82elastic, 79elasto-plastic, 82fabric, 92foam, 110hole, 243honeycomb, 109hysteresis, 116interface, 101isotropic, 80mooney-rivlin, 103nonlinear visco-elastic, 105orthotropic, 80sandwich, 107toner, 116visco-elastic, 104

material damping, 78muscle

Hill type, 263optimum length, 265

neck injury criteria, 292neck injury predictor, 293

operator, 59orientation

rotation angle, 329screw axis, 330vector, 332

orifice, 224, 243outflow

model 1, 244, 245outlet

circular, 230rectangular, 230

permeabilitymodel 1, 246model 2, 247

pore, 243

Release 7.7 337

I


Rayleigh damping, 69reference space, 8restraint

cardan, 161flexion-torsion, 161joint, 156kelvin, 151maxwell, 152point, 153

rigid, 118rigid body motion, 17road profile, 271

sampling frequency, 282sensor, 55signal, 59spotwelds, 140strain, 76

principal, 76strap(tether), 248stress, 75

principal, 76surface

cylinder, 46ellipsoid, 45finite element, 48plane, 44

systemmulti-body, 8

thoracic trauma index, 294thread, 94three-ms criterion, 294three-node shell element, 133tibia compressive force criterion, 298tibia index, 297tree structure, 7, 8Tresca, 76tyre, 269

viscous injury response, 295Von Mises, 77

338 Release 7.7

madymo theory manual0eca4fe331aaaa0387ab-39017777f15f755539d3047328d4a990.r16.cf3...coupling manual...

Documents