theoretical and experimental analyses of compliant

THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT UNIVERSAL JOINT

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

ÇAĞIL MERVE TANIK

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE IN

MECHANICAL ENGINEERING

JANUARY 2014

Approval of the thesis:

THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT UNIVERSAL JOINT

submitted by ÇAĞIL MERVE TANIK in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department, Middle East Technical University by, Prof. Dr. Canan Özgen Dean, Graduate School of Natural and Applied Sciences _____________________ Prof. Dr. Süha Oral Head of Department, Mechanical Engineering _____________________ Prof. Dr. F. Suat Kadıoğlu Supervisor, Mechanical Engineering Dept., METU _____________________ Assoc. Prof. Dr. Volkan Parlaktaş Co-Supervisor, Mechanical Engineering Dept., HU _____________________ Examining Committee Members: Prof. Dr. Orhan Yıldırım Mechanical Engineering Dept., METU _____________________ Prof. Dr. F. Suat Kadıoğlu Mechanical Engineering Dept., METU _____________________ Assoc. Prof. Dr. Volkan Parlaktaş Mechanical Engineering Dept., HU _____________________ Prof. Dr. Metin Akkök Mechanical Engineering Dept., METU _____________________ Assist. Prof. Dr. Ergin Tönük Mechanical Engineering Dept., METU _____________________

Date: 30.01.2014

iv

I hereby declare that all information in this document has been obtained and

presented in accordance with academic rules and ethical conduct. I also declare

that, as required by these rules and conduct, I have fully cited and referenced

all material and results that are not original to this work.

Name, Last name : Çağıl Merve TANIK

Signature :

v

ABSTRACT

THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT

UNIVERSAL JOINT

Tanık, Çağıl Merve

M.S., Department of Mechanical Engineering

Supervisor: Prof. Dr. Fevzi Suat Kadıoğlu

Co-Supervisor: Assoc. Prof. Dr. Volkan Parlaktaş

February 2014, 115 pages

In this study, a compliant version of the cardan universal joint whose compliant parts

are made of blue polished spring steel is considered. The original design consist of

two identical parts assembled at right angles with respect to each other. Identical

parts can be produced from planar materials; thus, it has the advantage of easiness in

manufacturing. As a design example, two mechanisms are dimensioned with

different plate thicknesses. The resultant stresses at flexural hinges of these samples

are determined via analytical and finite element analysis method. Torque capacity of

these mechanisms are determined. Also fatigue analysis of these mechanisms are

performed. Further, one of these samples is manufactured and operated under three

different conditions. It is verified that results of experiments are consistent with

theoretical approaches.

Keywords: Compliant Mechanisms, Universal Joint, Finite Element Analysis,

Fatigue Analysis

vi

ÖZ

ESNEK KARDAN MAFSALININ TEORİK VE DENEYSEL ANALİZLERİ

Tanık, Çağıl Merve

Yüksek Lisans, Makine Mühendisliği Bölümü

Tez Yöneticisi: Prof. Dr. Fevzi Suat Kadıoğlu

Ortak Tez Yöneticisi: Doç. Dr. Volkan Parlaktaş

February 2014, 115 sayfa

Bu çalışmada esnek kısımları yay çeliğinden yapılan kardan milinin esnek bir

versiyonu incelenmiştir. Özgün tasarım birbirlerine dik olarak monte edilmiş iki

özdeş parçadan oluşmaktadır. Özdeş parçalar düzlemsel malzemelerden imal

edilebilir, bu nedenle mekanizmanın üretimi kolaydır ve bu önemli bir avantajdır. Bu

çalışmada iki farklı plaka kalınlığında mekanizma tasarım örnekleri

boyutlandırılmıştır. Bu örneklerin bükülen mafsallarındaki gerilmeler analitik

metotlar ve sonlu elemanlar analizi ile incelenmiştir. Böylece mekanizmaların tork

kapasiteleri belirlenmiştir. Ayrıca mekanizmaların yorulma analizleri de yapılmıştır.

Yapılan analitik ve sayısal çalışmaları doğrulamak amacıyla, tasarlanan örneklerin

bir tanesi üretilmiş ve üç farklı koşulda çalıştırılmıştır. Bu deneylerin sonuçlarının

kuramsal yaklaşımlar ile tutarlı olduğu gözlemlenmiştir.

Anahtar Kelimeler: Esnek Mekanizmalar, Üniversal Mafsal, Sonlu Eleman Analizi,

Yorulma Analizi

vii

To My Husband

viii

ACKNOWLEDGMENTS

The author wishes to express her deepest gratitude to his supervisor Prof. Dr. F. Suat

Kadıoğlu and co supervisor Assoc. Prof. Dr. Volkan Parlaktaş for their guidance,

advice, criticism, encouragements and insight throughout the research.

The author would also like to thank her husband Assoc. Prof. Dr. Engin Tanık for his

support and patience during the thesis study.

ix

TABLE OF CONTENTS

ABSTRACT ................................................................................................................. v

ÖZ ............................................................................................................................... vi

ACKNOWLEDGMENTS ....................................................................................... viii

LIST OF TABLES ..................................................................................................... xii

LIST OF FIGURES ................................................................................................. xiii

LIST OF SYMBOLS ................................................................................................ xvi

CHAPTERS ................................................................................................................. 1

1. INTRODUCTION ................................................................................................... 1

1.1 Literature Review ............................................................................................... 1

1.2 Objective and Scope of the Thesis ..................................................................... 4

2. COMPLIANT MECHANISMS ............................................................................... 7

2.1 Introduction to Compliant Mechanisms ............................................................. 7

2.2 Pseudo-Rigid-Body Model ................................................................................. 8

2.2.1 Small-Length Flexural Pivots ...................................................................... 9

2.2.2 Active and Passive Forces ......................................................................... 11

2.2.3 Cantilever Beam with a Force at the Free End .......................................... 13

2.2.4 Moment at the Free End............................................................................. 17

2.3 Nomenclature ................................................................................................... 17

2.4 Diagrams .......................................................................................................... 19

2.5 Pseudo-Rigid-Body Replacement ................................................................... 20

2.6 Material Considerations ................................................................................... 22

x

3. CONVENTIONAL UNIVERSAL JOINT ............................................................ 25

3.1 Introduction to Cardan Universal Joint ............................................................ 25

3.2 Kinematic Analysis of the Cardan Universal Joint .......................................... 26

4. DESIGN OF THE COMPLIANT UNIVERSAL JOINT ...................................... 31

4.1 Dimension Synthesis of the Compliant Universal Joint................................... 31

4.2 Material Selection of the Compliant Universal Joint ....................................... 35

5. THEORETICAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT .... 39

5.1 Stress Analysis of the Compliant Universal Joint with FEA ........................... 39

5.1.1 Boundary Conditions and Meshing ........................................................... 46

5.2 Static Analysis for Deflection-Only Case ....................................................... 48

5.3 Study of Mesh Refinement for Bending-Only Case ........................................ 58

5.4 Path Analysis .................................................................................................... 60

5.5 Static Analysis to Determine Torque Limits ................................................... 66

5.6 Study of Mesh Refinement for Combined Loading Case ................................ 73

5.7 Static Analysis of the Mechanism at Zero Degree Shaft Angle ..................... 76

5.7 Fatigue Life Estimations ................................................................................. 82

6. EXPERIMENTAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT . 97

6.1 Manufacturing of the Prototype .................................................................... 97

6.2 Components of the Experimental Setup ....................................................... 99

6.3 Experimental Verification .......................................................................... 101

7. RESULTS, CONCLUSION AND FUTURE STUDY ........................................ 103

7.1 RESULTS AND CONCLUSION .................................................................. 103

7.2 FUTURE STUDY .......................................................................................... 104

REFERENCES ........................................................................................................ 105

APPENDICES ......................................................................................................... 109

xi

A: PROPERTIES OF AISI 1080 ............................................................................. 109

B: TECHNICAL DRAWINGS ................................................................................ 110

C: SPECIFICIONS OF BEARINGS ...................................................................... 112

D: SPECIFICIONS OF DC ELECTRIC MOTOR .................................................. 113

E: ISOMETRIC VIEW OF THE TEST SETUP ..................................................... 115

xii

LIST OF TABLES

TABLES

Table 2.1 The Coefficients of the Compliant Mechanisms ....................................... 16

Table 2.2 Ratio of Yield Strength to Young's Modulus for Some Materials ............. 24

Table 4.1 Properties of AISI 1080 (Spring Steel) ...................................................... 35

Table 5.1 Analytical and Numerical Normal Stress .................................................. 51

Table 5.2 Analytical and Numerical Normal Stress .................................................. 54

Table 5.3 Number of Elements through the Thickness vs. Maximum von-Mises

Stress for Bending-Only Case .................................................................................... 60

Table 5.4 Steps of the Loading .................................................................................. 67

Table 5.5 Stress Values vs. Time Increments ............................................................ 68

Table 5.6 Interpolation of the Torque Values and Stress Values for �ℎ = 0.75 70

Table 5.7 Torque Limits for Different Shaft Angles (�ℎ =0.75 mm) ....................... 70

Table 5.8 Interpolation of the Torque Values and Stress Values for �ℎ =0.5 mm .... 71

Table 5.9 Torque Limits for Different Shaft Angles (�ℎ =0.5 mm) ......................... 72

Table 5.10 Number of Elements through the Thickness vs. Maximum von-Mises

Stress for Combined Loading Case............................................................................ 75

Table 5.11 Torque vs. Shear Stress for �ℎ = 0.75 mm .............................................. 79

Table 5.12 Torque vs. Shear Stress Values for �ℎ = 0.5 mm .................................... 81

Table 5.13 Values of a and b for Surface Factor ....................................................... 83

Table 5.14 Reliability Factors, � ............................................................................. 85

Table 5.15 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�ℎ = 0.75 mm) ......................................................................................................... 90

Table 5.16 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�ℎ = 0.5 mm) ........................................................................................................... 91

Table 5.17 Maximum and Minimum Normal Stress Values for Combined Loading 94

Table 6. 1 Experiments for Different Conditions .................................................... 101

xiii

LIST OF FIGURES

FIGURES

Figure 1.1 Compliant Spatial Four Bar (RSSR) Mechanism (Tanık E. and Parlaktaş

V. (2011)) ..................................................................................................................... 2

Figure 1.2 Compliant Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012)) ..... 4

Figure 1.3 Flow Chart of the Stress Analysis .............................................................. 6

Figure 2.1 Cantilever Beam (a) and its Pseudo-Rigid Body Model (b) ..................... 10

Figure 2.2 Cantilever Beam with a Force at the Free End and its Pseudo-Rigid Body

Model with a Torsional Spring .................................................................................. 12

Figure 2.3 Flexible Segment and its Pseudo-Rigid-Body Model .............................. 13

Figure 2.4 Plot of Characteristic Radius Factor, �, versus n...................................... 14

Figure 2.5 Flexible Beam with a Moment at the Free End ........................................ 17

Figure 2.6 Component Characteristics of Links ........................................................ 18

Figure 2.7 Component Characteristics of Segments .................................................. 18

Figure 2.8 Symbol Convention for Compliant Mechanism Diagrams ...................... 19

Figure 2. 9 An Example of a Compliant Mechanism Diagram ................................. 20

Figure 2.10 A Complaint Four Bar and Slider-Crank Mechanism and Pseudo Rigid

Body Models .............................................................................................................. 22

Figure 2.11 Flexible Cantilever Beam ....................................................................... 23

Figure 3.1 Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012)) ..................... 25

Figure 3.2 Variables of Cardan Universal Joint ......................................................... 26

Figure 3.3 Input Shaft Angle versus Output Shaft Speed for Different Shaft Angles

.................................................................................................................................... 28

Figure 3.4 Input Shaft Angle versus Output Shaft Angle for Different Shaft Angles

.................................................................................................................................... 29

Figure 4.1 Spherical 4R Linkage (Tanık E. and Parlaktaş V. (2012)) ....................... 31

xiv

Figure 4.2 Original Compliant Universal Joint Design (Tanık E. and Parlaktaş V.

(2012))........................................................................................................................ 32

Figure 4.3 Dimensions of the Compliant Universal Joint (Tanık E. and Parlaktaş V.

(2012))........................................................................................................................ 33

Figure 4.4 Critical Position of the Compliant Universal Joint (Tanık E. and Parlaktaş

V. (2012)) ................................................................................................................... 34

Figure 4.5 Isometric View of the Design ................................................................... 36

Figure 4.6 For a 10° Shaft Angle and Different Input Shaft Angles Views of the

Mechanism ................................................................................................................. 37

Figure 5.1 Flow Chart of the Analysis ....................................................................... 39

Figure 5.2 Definitions of the Parts ............................................................................. 47

Figure 5.3 Connections and Newly Formed Parts ..................................................... 47

Figure 5.4 The Meshing of the Model ....................................................................... 48

Figure 5.5 Representation of the Boundary Condition and Shaft Angle ................... 50

Figure 5.6 ANSYS Simulation for �ℎ =0.75 mm and 1° Shaft Angle ...................... 52

Figure 5.7 ANSYS Simulation for �ℎ =0.75 mm and 5° Shaft Angle ...................... 52

Figure 5.8 ANSYS Simulation for �ℎ =0.75 mm and 10° Shaft Angle .................... 53

Figure 5.9 Plot of Shaft Angle vs. Analytical and ..................................................... 54

Figure 5.10 ANSYS Simulation for �ℎ=0.5 mm and 1° Shaft Angle ........................ 55

Figure 5.11 ANSYS Simulation for �ℎ=0.5 mm and 5° Shaft Angle ........................ 56



Figure 5. 14 Plot of Shaft Angle vs. Analytical and .................................................. 57

Figure 5.15 Plot of Shaft Angle vs. Analytical and ................................................... 58

Figure 5.16 Simulation Results for Four Elements through the Thickness for

Bending-Only Case .................................................................................................... 59

Figure 5.17 Simulation Results for Five Elements through the Thickness for

Bending-Only Case .................................................................................................... 59

Figure 5.18 Number of Elements through the Thickness vs. Maximum von-Mises

Stress for Bending-Only Case .................................................................................... 60

xv

Figure 5.19 Paths for One of the Deflected Flexural Hinge ...................................... 61

Figure 5.20 Path Analysis in ANSYS ........................................................................ 61

Figure 5.21 Length vs. Equivalent Stress for Path 1 .................................................. 62









Figure 5.30 Analysis settings ..................................................................................... 67

Figure 5.31 Simple Model with Applied Torque and Reactions at Flexural Hinges . 69

Figure 5.32 Finite Element Analysis and Results for Simple Model ......................... 69

Figure 5.33 Shaft Angle vs. Maximum Torque Values for �ℎ = 0.75 .............. 71

Figure 5.34 Shaft Angle vs. Maximum Torque Values for �ℎ =0.5 mm .................. 72

Figure 5.35 Shaft Angle vs. Maximum Torque Values for Both Models .................. 73

Figure 5.36 Simulation Results for Three Elements through the Thickness for

Combined Loading Case ............................................................................................ 74






Stress for Combined Loading Case ............................................................................ 76

Figure 5.40 Shear Stress Values for 1 N.m Torque (�ℎ = 0.75 mm) ........................ 77

Figure 5.41 Shear Stress Values for 2.5 N.m Torque (�ℎ = 0.75 mm) ..................... 77

Figure 5.42 Shear Stress Values for 5 N.m Torque (�ℎ = 0.75 mm) ........................ 77

Figure 5.43 Shear Stress Values for 7.5 N.m Torque (�ℎ =0.75 mm) ...................... 78

Figure 5.44 Shear Stress Values for 10 N.m Torque (�ℎ = 0.75 mm) ...................... 78

Figure 5.45 Shear Stress Values for 12.5 N.m Torque (�ℎ = 0.75 mm) ................... 78

xvi

Figure 5.46 Shear Stress for 15 N.m Torque (�ℎ = 0.75 mm) .................................. 79

Figure 5.47 Torque vs. Shear Stress for �ℎ = 0.75 mm ............................................. 79

Figure 5.48 Shear Stress for 1 N.m Torque (�ℎ= 0.5 mm) ........................................ 80

Figure 5.49 Shear Stress for 2.5 N.m Torque (�ℎ = 0.5 mm) ................................... 80

Figure 5.50 Shear Stress Values for 5 N.m Torque (�ℎ= 0.5 mm) ............................ 80

Figure 5.51 Shear Stress for 7.5 N.m Torque (�ℎ = 0.5 mm) ................................... 81

Figure 5.52 Torque vs. Shear Stress for �ℎ = 0.5 mm ............................................... 81

Figure 5.53 Torque vs. Shear Stress for Both Models ............................................... 82

Figure 5.54 Notch-Sensitivity Charts for Steels and UNS A92024-T Wrought

Aluminum Alloys Subjected to Reversed Bending or Reversed Axial Loads

(Budynass and Nisbett (2011)) .................................................................................. 86

Figure 5.55 Rectangular Fillet Bar in Bending .......................................................... 87

Figure 5.56 Estimation of Theoretical Stress Concentration factor with ANSYS .... 87

Figure 5.57 Loading Condition of the Model ............................................................ 89

Figure 5.58 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�ℎ =

0.75 mm) .................................................................................................................... 90

Figure 5.59 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�ℎ = 0.5 mm) ..................................................................................................................... 91

Figure 5.60 Von-Mises Stress Distribution, Most Critical Point on the Flexural

Hinge, its Infinitesimal Cube and �� Stress Distribution ........................................ 92

Figure 5.61 Infinitesimal Cube of Critical Point ....................................................... 93

Figure 5.62. Normal Stress Variations for Different Shaft Angles............................ 94

Figure 5.63 Performance Graph of the Compliant Universal Joint ........................... 95

Figure 6.1 Assembly of the Prototype ....................................................................... 97

Figure 6.2 Components of the Prototype ................................................................... 98

Figure 6. 3 Different Ways of the Connections ......................................................... 99

Figure 6.4 Components of the Experimental Setup ................................................. 100

Figure 6.5 A Failed Compliant Universal Joint ....................................................... 102

Figure 6.6 Shaft Angle vs. Torque Output and Experimentally Verified Data ....... 102

Figure A.1 Properties of AISI 1080 ......................................................................... 109

Figure B.1 Technical Drawing of the Model with 0.75 mm Plate Thickness ......... 110

xvii

Figure B.2 Technical Drawing of the Model with 0.5 mm Plate Thickness ............ 111

Figure C.1Specifications of UCP 200 Bearing ........................................................ 112

Figure D.1 Specifications of the DC Electric Motor................................................ 114

Figure E.1 Isometric View of the Fatigue Test Setup .............................................. 115

xviii

LIST OF SYMBOLS

� Horizontal component of free end, length of rigid link, angular acceleration,

curve fitting parameter, constant

� Vertical component of free end, distance, width, curve fitting parameter,

constant

� Distance

�� Parametric angle coefficient

�� Constant

� Width of connection slot, diameter

� Modulus of elasticity

� Force

� Fatigue strength factor

� Width of compliant universal joint

� Second moment of area

� Spring constant

� Marin endurance limit modifying factor

�� Geometric stress concentration factor

�� Stiffness coefficient

� Length

� Length

Moment

! Number of cycles

" Horizontal coefficient of applied force, factor of safety

xix

"# Horizontal component of force

# Vertical component of force

$ Notch sensitivity

% Radius

& Contact surface of compliant universal joint

'( Endurance limit

'() Unmodified endurance limit

'*� Ultimate tensile strength

'+ Yield strength

, Torque

�- Thickness of flexural hinge

�. Thickness of compliant universal joint

/ Width of flexural hinge

01 Initial position vector

23 Engineering normal strain for fracture

4 Pseudo rigid body angle, angular deflection

� Characteristic radius factor, angle of rotation for shafts

��5 Pseudo rigid body constant

6 Deflection

7 Bend angle, shaft angle

8 Angle

� Normal stress

9 Angular velocity

: Direction angle of force

1

CHAPTER 1

INTRODUCTION

1.1 Literature Review

Universal joints are common mechanical devices which are used for transmitting

rotary motion between misaligned intersecting shafts. That is, a universal joint is a

joint or coupling that is capable of transmitting rotary motion from one shaft to

another which are not in line with each other. Classical analysis of the universal joint

involves the determination of angular displacements, velocities, accelerations and

torque ratios of the shafts.

In this study a novel universal joint design is proposed. A compliant universal joint

whose compliant parts are made of blue polished spring steel is taken into

consideration. This chapter presents a survey of the related literature for compliant

mechanisms and universal joint mechanisms.

In literature universal joint became an attractive topic because of its capabilities of

easy mounting, resisting high loads and commercial availability. Early articles on

universal joints made of rigid links address various aspects of these mechanisms.

Basically a universal joint is a spherical four bar linkage. In literature there are lots

of studies about this type of mechanism addressing its analysis, synthesis,

applications and type determination. For example Mohan et al. (1973) introduced

closed form synthesis of a spatial function generation mechanism which consists of a

spherical four bar linkage. Freudenstein (1965) proposed a new type of a spherical

2

mechanism. Yang (1965) worked on static force and torque analysis of a spherical

four bar mechanism. Dynamic analysis of a universal joint and its manufacturing

tolerances are introduced by Chen and Freudenstein (1986). Freudenstein and Macey

(1990) worked on the inertia torques of the Hooke joint. Moment transmission by a

universal joint is studied by Porat (1980). Homokinetic joint allows to transmit

power through a variable angle, at constant rotational speed. For a double cardan

homokinetic joint Wagner and Cooney (1979) developed a new approach to increase

its dynamic mechanical efficiency.

Universal joint has the advantage of easiness in manufacturing. On the other hand

traditional universal joints consist of many parts which are assembled and therefore

manufacturing tolerances on these parts must be complied with. Tolerances of a

universal joint are studied by Fischer and Freudenstein (1984).

Figure 1.1 Compliant Spatial Four Bar (RSSR) Mechanism (Tanık E. and Parlaktaş

V. (2011))

Compliant mechanisms are flexible mechanisms, which gain some or all of their

motion through the deflection of members. They can be fully or partially compliant.

Generally compliant mechanisms have lower number of parts which reduce

manufacturing and assembly time. Some of them may even be made of a single

3

piece. They are lighter and they have fewer number of movable joints, which cause

wear and need lubrication. The main disadvantage of compliant mechanisms is that,

their analyzes and design is difficult to accomplish. The pseudo rigid body model is

used to simplify the analysis and design of compliant mechanisms. In Figure 1.1 first

compliant spatial four bar mechanism is shown which is designed by Tanık and

Parlaktaş (2011).

Salamon (1989) introduced a methodology which uses a pseudo rigid body model of

the compliant mechanisms with compliance modelled as torsional and linear springs.

Howell and Midha (1994) and (1998) used closed form elliptic integral solutions to

develop deflection approximations for an initially straight flexible segment subjected

to bending.

A spherical four bar mechanism which is a special case of spatial four bar

mechanism that possesses out of plane motions is studied by Tanık and Parlaktaş

(2012). Another spatial four link mechanism studied by Parlaktaş and Tanık (2011)

is the compliant spatial slider crank mechanism.

In the literature a compliant universal joint is previously considered by Trease et al.

(2005) who proposed a design for a compliant universal joint. There is also a

prototype of a compliant universal joint in the library of Cornell University that can

be found in http://kmoddl.library.cornell.edu website. In that prototype the compliant

section is made of leather.

Recently, Tanık and Parlaktaş (2012) proposed a new design for a compliant cardan

universal joint which is shown in Figure 1.2. The design consists of two identical

parts assembled at right angles with respect to each other. In that study, dimensions

of the mechanism are designed in order to satisfy the Cardan joint theory and to

avoid an undesired contact between the identical parts for proper functioning of the

mechanism. This prototype is made of polypropylene and manufactured and

operated under specified loading conditions to verify the theoretical approaches.

4

Figure 1.2 Compliant Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012))

1.2 Objective and Scope of the Thesis

The successful implementation of a compliant universal joint in real life applications

depends not only on its kinematic design but its strength as well. The purpose of this

thesis is therefore to analyze the stresses and fatigue strength of a compliant

universal joint, whose flexible parts are made of blue polished spring steel by

analytical, numerical and experimental methods. Hence theoretical approaches will

be experimentally verified. To the best of Author's knowledge there are not any

studies in the literature which address the strength issues of compliant universal

joints except the study of Tanık and Parlaktaş (2012) , where only a preliminary

5

finite element analysis has been done to determine the torque capacity of the

mechanism.

Here, a design is proposed according to the dimensional constraints that satisfy the

theory of universal joints and thereby avoid undesired contact between the parts. The

stress analysis is done analytically and numerically by finite element method using

ANSYS software. Fatigue analysis is done and it is experimentally verified with the

prototypes that are manufactured. The flow chart of the strength analysis can be seen

in Figure 1.3 schematically.

Outline of the thesis is as follows. In Chapter 2, a review of compliant mechanisms

is done. Then pseudo rigid body model is briefly explained and deflection and stress

equations are derived. Rigid body replacement synthesis is presented. The kinematic

equations of the universal joint are given in Chapter 3. In Chapter 4 dimensioning of

the mechanism and selection of the material is covered. In Chapter 5 the analytical

and numerical solution procedure and approaches are discussed. Experimental setup

is introduced in Chapter 6. Finally, the results are discussed in Chapter 7.

Figure 1.3

Determine stresses under static loading

Perform stress analysis using FEA

Evaluate fatigue life of the mechanism

Manufacture a real model

Compare the fatigue life estimates with

Discuss the results by various methods

and explain discrepancies if any

6

3 Flow Chart of the Stress Analysis

Determine stresses under static loading

analytically

Perform stress analysis using FEA

Evaluate fatigue life of the mechanism

by analytical methods

Manufacture a real model

Compare the fatigue life estimates with

experiments

Discuss the results by various methods

and explain discrepancies if any

Conclusions

7

CHAPTER 2

COMPLIANT MECHANISMS

2.1 Introduction to Compliant Mechanisms

According to Shigley and Uicker (1980) a mechanism is a mechanical device used to

transfer or transform motion, force, or energy. Rigid-link mechanisms gain their

mobility from the movable joints.

Compliant mechanisms are flexible mechanisms, that gain some or all of their

motion through the deflection of flexible members rather than movable joints.

Compliant mechanisms can be fully compliant or partially compliant. Fully and

partially compliant mechanism definitions are given by Howell (2001). Fully

compliant mechanisms obtain all their motion from the deflection of compliant

members and partially compliant mechanisms contain one or more traditional

kinematic pairs along with compliant members.

Required input output relationship is obtained by the combination of the rigid and

compliant parts or fully compliant elements. The strength of the deflecting members

limits deflection of compliant link therefore a compliant link cannot produce a

continuous rotational motion.

The advantages of compliant mechanisms can be divided into two subgroups: cost

reduction and increased performance. Compliant mechanisms require fewer parts to

accomplish a certain task. A reduction in the number of parts reduces manufacturing

and assembly time, and cost. Some compliant mechanisms can be manufactured as a

8

single piece by injection molding process. Compliant mechanisms also have fewer

movable joints. That results in reduced wear and need for lubrication.

Using compliant mechanisms reduces the number of movable joints which increases

mechanism precision since backlash may be reduced or eliminated. Vibration and

noise caused by the revolute and sliding joints of rigid-body mechanisms may also

be reduced by using compliant mechanisms.

In compliant mechanisms energy is stored in the form of strain energy in the flexible

members. This property can be an advantage for some cases and a disadvantage for

some other cases. As an advantage the stored or transformed energy can be released

at a later time or in a different manner. A bow and arrow system is a good example.

All of the energy is not transferred, but some is stored in the mechanism.

Compliance becomes a disadvantage if function of a mechanism is to transfer energy

from input to output.

A major disadvantage of compliant mechanisms is the lack of knowledge regarding

analysis and synthesis methods for such mechanisms and the requirement to

determine the deflections of flexible members. Therefore analysis and design of

compliant mechanisms has difficulties compared to conventional mechanisms.

Fatigue analysis is another vital issue. Some compliant members are loaded in a

cyclic manner. To perform prescribed functions it is important to design those

compliant members with sufficient fatigue life.

Compliant links that remain under stress for long periods of time or subject to high

temperatures may experience stress relaxation and creep.

2.2 Pseudo-Rigid-Body Model

The purpose of the pseudo-rigid-body model is modelling the deflection of flexible

members by using rigid-body components which have equivalent force-deflection

9

characteristics. This method of modelling allows well-known rigid-body analysis

methods to be used in the analysis of compliant mechanisms. Howell (2001) says

that the pseudo-rigid-body model is a bridge that connects rigid-body mechanism

theory and compliant mechanism theory.

Salamon (1989) introduced a methodology for compliant mechanism design that

used a pseudo-rigid-body model of the compliant mechanism with compliance

modelled by torsional and linear springs. These models are much easier to analyze

than idealized models that require finite element or elliptic integral solutions. The

most important attribute of the pseudo-rigid-body model is that it significantly

simplifies the design process.

Closed-formed elliptic-integral solutions are used by Howell and Midha (1994 and

1998) to develop deflection approximations for an initially straight, flexible segment

with linear material properties.

2.2.1 Small-Length Flexural Pivots

An important component that exist in compliant mechanisms is the so called ''small

length flexural pivot''. The beam shown in Figure 2.1 has two segments. The small

segment is shorter and more flexible than the large segment. This small segment is

called small-length-flexural pivot. Usually large segment is at least 10 times larger

than small segment. The large segment is also much stiffer.

� ≫ � (2.1)

(��)< ≫ (��)= (2.2)

10

Figure 2.1 Cantilever Beam (a) and its Pseudo-Rigid Body Model (b)

For the flexible segment with end moment loading the deflection equations are

derived by Howell (2001) as follows:

8> = �� (2.3)

6+� = 1 − �A&8>8> (2.4)

6B� = 1 − &C"8>8> (2.5)

These equations could be used to model small-length flexural pivots with pseudo-

rigid-body model. Figure 2.1 shows a member and its pseudo-rigid-body model . The

model consist of two rigid equal links, connected by a characteristic pivot.

Characteristic pivot represents the displacement and torsional spring models the

beam stiffness or resistance to deflection. This model gives an accurate solution for

the deflection path of the beam end for a given end load. The percentage error

between this model and the closed-form elliptic integral solutions is 0.5 for large

11

deflections. The angle of pseudo rigid link is the pseudo rigid body angle, 4, that is

equal to the beam end angle for small-length flexural pivots.

Θ= 8>(small-length flexural pivots) (2.6)

A torsional spring with spring constant � is used to model the beam's resistance to

deflection. The required torque to deflect the torsional spring at an angle 4 is

, = �Θ (2.7)

From the elementary beam theory the spring constant � could be found. For a beam

with an end moment, the end angle is

8> = �(��)= (2.8)

Since = , and 4 = 8>, the spring constant can be found as

� = (��)=� (2.9)

This model is more accurate for bending dominant cases than transverse and axial

loading dominant cases.

2.2.2 Active and Passive Forces

Figure 2.2 shows a cantilever beam with a force at the free end. The force, F, must

be defined by its magnitude and direction. The direction may be defined by the angle : or by the horizontal and vertical components of the force. In Figure 2.2 horizontal

component is shown as "# and the vertical component as #. F is a nonfollower

12

force which means it remains at the same angle regardless of the deflection of the

beam. The nonfollower force's magnitude and direction is

� = #D"E + 1 (2.10)

: = atan 1−" (2.11)

Figure 2.2 Cantilever Beam with a Force at the Free End and its Pseudo-Rigid Body

Model with a Torsional Spring

The force can also be resolved into its normal and tangential components. The

tangential component to the path which is also normal to the pseudo-rigid-link, ��,

causes a moment at the torsional spring

, = ��(� + �2) (2.12)

The tangential component, which causes the deflection of the pseudo-rigid-link, is

called an active force. The normal component is called a passive force and it has no

contribution to the deflection of the beam.

13

When the deflection changes the active and passive components change because F is

a non-follower force. Then the active force is,

�� = �&C"(: − 4) (2. 13)

2.2.3 Cantilever Beam with a Force at the Free End

In Figure 2.3 the flexible beam with a force applied at its free end is shown. If

deflections are large, the linear beam deflection equations may not give accurate

solutions. To perform the analysis, elliptic integral solutions or nonlinear finite

element analysis could be used. Instead of these methods, pseudo-rigid-body model

which is a simpler but accurate method of analysis may also be used.

Figure 2.3 Flexible Segment and its Pseudo-Rigid-Body Model

14

The location of the characteristic pivot is expressed in terms of the characteristic

radius factor, �, which represents the fraction of the beam length at which the pivot

is located. Once � is determined the deflection path may be parameterized in terms

of 4, the pseudo-rigid-body angle.

The characteristic radius factor is a function of ", that is horizontal coefficient of the

applied force. Howell (2001) introduced a formulation of � in terms of " as:

� = K0.841655 − 0.0067807" + 0,000438"E (0.5 < " < 10.0)0.852144 − 0.0182867" (−1.8316 < " < 0.5)0.912364 + 0.0145928" (−5 < " < −1.8316) S (2.14)

Figure 2.4 Plot of Characteristic Radius Factor, �, versus n

The �-" graph that is seen in Figure 2.4 shows that the characteristic radius factor �

does not vary much. Thus � is approximated by Howell (2001) as:

�TU( = V ��" WXWYV �" WXWY (2.15)

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0.88

0.89

-5 -3 -1 1 3 5 7 9

Ch

ara

cte

rist

ic r

ad

ius

fact

or

(γ)

Coefficient of horizontal force component (n)

15

For −0.5 ≤ " ≤ 1

�TU( ≅ 0.85 (2.16)

The relationship between 8> and 4 is approximated by Howell (2001) as:

8> = ��4 (2.17)

where �� is constant and called the parametric angle coefficient.

The torsional spring constant that is used to model the beam's resistance to deflection

is, �Θ, the stiffness coefficient which is nondimensionalized torsional spring

constant.

Let

��E�� = �Θ4 (2.18)

�Θ =

\]]]]]]̂]]]]]]_ 3.024112 + 0.121290" +0.003169"E(−5 < " ≤ −2.5)

1.967647 − 2.616021" −−3.738166"E−2.649437"` − 0.891906"a−0.113063"b(−2.5 ≤ " ≤ −1)2.654855 − 0.509896 × 10de"+0.126749 × 10de"E−0.142039 × 10dE"`+0.584525 × 10da"a(−1 < " ≤ 10)

S (2.19)

where 4 < 4fTB ,

16

�ΘTU( = V �Θ�"WXWYV �"WXWY �A% g "e = 5"E = 10S (2.20)

�ΘTU( = 2.61 (2.21)

considering the loads are in a range of 63h < : < 135h or −0.5 < " < 1.0,

�ΘTU( = 2.65 A% �Θ ≅ i� (2.22)

The torque applied at the pin joint is,

, = �4 (2.23)

where � is the torsional spring constant [!/%��] and 4 is the angular deflection.

Torsional spring constant is defined by Howell (2001) as,

� ≅ i�E ��

(2.24)

where ��Θ is pseudo-rigid body constant.

For different values of n, value of the coefficients are shown in Table 2.1,

Table 2.1 The Coefficients of the Compliant Mechanisms

n m nopq(m) rs ts nopq(ts)

0 0.8517 64.3o 1.2385 2.677 58.5o

2 0.8276 109.0o 1.2511 2.597 69.0o

5 0.8192 121.0o 1.2557 2.562 67.5o

-0.5 0.8612 47.7o 1.2358 2.693 44.4o

-3 0.8669 16.0o 1.2119 2.688 12.9o

17

2.2.4 Moment at the Free End

The flexible beam with an end moment at its free end is shown in Figure 2.5.

Figure 2.5 Flexible Beam with a Moment at the Free End

The coordinates of the free end, maximum normal stress value and angle of the

beam end are determined by Howell (2001) as,

� = �[1 − 0.7346(1 − �A&4)] (2.25)

� = 0.7346�&C"4 (2.26)

8> = 1.51644 (2.27)

�fTB = >�� (2.28)

2.3 Nomenclature

In rigid-body mechanisms motion is transferred or transformed by rigid links and

traditional joints. However the working principle of the compliant mechanisms are

different. The deflection of the flexible members gives the motion. Therefore

identification of the compliant mechanism's parts are more difficult than the rigid

body mechanisms.

A link is defined by Howell (2001) as the continuum connecting the mating surfaces

of one or more kinematic pairs.

Figure 2.6

For a rigid-body mechanism the distance between the joints is constant and the shape

of the link is kinematically unimportant. However the motion of a compliant link is

dependent on link geometry and the forces. As seen in Figure

rigid or compliant and a compliant link can be simple or compound.

Figure 2.7 Component Characteristics of Segments

Rigid Link

18

of the compliant mechanism's parts are more difficult than the rigid


of one or more kinematic pairs.

Component Characteristics of Links

body mechanism the distance between the joints is constant and the shape


dependent on link geometry and the forces. As seen in Figure 2.6 the links may be

rigid or compliant and a compliant link can be simple or compound.

Component Characteristics of Segments

Link

Rigid Link Compliant Link

Simple Compound

Segment

Rigid Compliant

of the compliant mechanism's parts are more difficult than the rigid-


body mechanism the distance between the joints is constant and the shape


the links may be

19

When a compliant link is analyzed, it could be observed that the compliant segment

may be composed of rigid or compliant segments. Material discontinuities or

geometric changes are often the starting points of a new segment. Segments can be

either rigid or compliant as shown in Figure 2.7.

2.4 Diagrams

Skeleton diagrams are used to represent rigid-body mechanisms easily. Similar

diagrams are also used for compliant mechanisms. Symbols which represent joints

and segments are shown in Figure 2.8.

Figure 2.8 Symbol Convention for Compliant Mechanism Diagrams

By using symbols an example of a compliant mechanism diagram is shown in Figure

2.9. The mechanism has two compliant links. First link is compliant and it is a

Rigid Segment

�

Axially Compliant

Segment

Pin or Revolute

Joint Flexural Pivot

Slider (Prismatic)

Joint

or

Fixed Connection

or

20

composition of a compliant and a rigid segment. Second link is also compliant and

has 3 segments that are rigid and compliant. There are rigid traditional kinematic

pairs therefore it is a partially compliant mechanism.

Figure 2. 9 An Example of a Compliant Mechanism Diagram

2.5 Pseudo-Rigid-Body Replacement

In a compliant mechanism analysis or synthesis, sometimes transformation of the

mechanisms is required. This transformation can be compliant mechanism to rigid-

body mechanism or vice versa. In compliant mechanism analysis, a pseudo rigid

segment 1(rigid)

link 1

(compliant)

segment 2 (compliant)

segment 5(rigid)

link 2

segment 4 (compliant)

6 variable

21

body model is obtained from the compliant mechanism. However, in rigid body

replacement synthesis, a pseudo rigid model is equivalent to a rigid-body mechanism

model and the resulting mechanism is determined from these models.

In rigid-body mechanisms the distance between the joints is the kinematically most

important parameter. During all the replacement processes the joints must be fixed.

Joint is on the midpoint of the compliant segment in small length flexural hinges if

transformation is compliant mechanism to rigid-body mechanism or vice versa.

In flexible beams there is a relationship between the rigid and compliant link which

is defined by characteristic radius factor, �.

� = �� (2.29)

Where � is length of the rigid link and � is the length of the compliant link. The

characteristic radius factor value according to " values can be seen in Table 2.1.

Another important issue during pseudo-rigid-body replacement is taking the torque

,which comes from the deflection of complaint link, into account. Therefore torsional

springs must be attached to the mechanism.

In Figure 2.10 a four bar and slider crank mechanism's pseudo-rigid-body

replacements are shown. In both mechanisms the black mechanism represents

compliant mechanism and grey one is its pseudo-rigid-body model.

22

Figure 2.10 A Complaint Four Bar and Slider-Crank Mechanism and Pseudo Rigid

Body Models

2.6 Material Considerations

For a beam that is shown in Figure 2.11 the deflection at the free end is,

6 = 4��`��ℎ` (2.30)

23

Figure 2.11 Flexible Cantilever Beam

The maximum stress occurs at the fixed end and equals to,

�fTB = 6��ℎE (2.31)

The failure occurs when the �fTB equals to the yield strength, '+, then,

'+ = 6��ℎE (2.32)

From Equation 2.32 � can be expressed in terms of yield strength as,

� = '+�ℎE6� (2.33)

24

Substituting Equation 2.33 into Equation 2.30 results in the maximum deflection, 6fTB (assuming geometrically linear load vs. deflection relationship is valid),

6fTB = 23 '+� �Eℎ (2.34)

According to maximum deflection equation, ratio of the strength to modulus of

elasticity shows us how much does a beam deflect without permanent deformation.

Thus the material with the highest value of '+/� will allow larger deflection. In

Table 2.2 ratio of yield strength to modulus of elasticity for several materials are

shown.

Table 2.2 Ratio of Yield Strength to Young's Modulus for Some Materials

Material u (vwp) xy (zwp) (xy/u) × {|||

Steel (1010 hot rolled) 207 179 0.87

Steel (4140 Q&T@400) 207 1641 7.9

Blue Polished Spring Steel

(AISI 1080)

200 880 4.4

Aluminium (1100 annealed) 71.7 34 0.48

Aluminium (7075 heat treated) 71.7 503 7.0

Titanium (Ti-13 heat treated) 114 1170 10

Polyethylene (HDPE) 1.4 28 20

Polypropylene 1.4 34 25

As seen in Table 2.2 elastic modulus does not change much with addition of alloying

elements or heat treatments. However yield strength value could be increased by heat

treatment that also makes the material more brittle.

Polypropylene has a very high yield strength to young's modulus ratio which allows

large deflections. It is available in the market, inexpensive, very ductile, easy to

process and has a low density. It can yield thousands of time without fracturing.

25

CHAPTER 3

CONVENTIONAL UNIVERSAL JOINT

3.1 Introduction to Cardan Universal Joint

A universal joint is a joint or coupling that is capable of transmitting rotary motion

from one shaft to another which are not in line with each other. It consists of a pair

of hinges, oriented at 90o to each other that are connected by a cross shaft as seen in

Figure 3.1. Kinematically a universal joint is equivalent to a slotted sphere type of a

joint which has two degrees of rotational freedom.

Figure 3.1 Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012))

7

26

3.2 Kinematic Analysis of the Cardan Universal Joint

Even when the input shaft rotates at a constant speed, output shaft could rotate at a

variable speed. For this reason the universal joint suffers from vibration and wear.

The speed of the output shaft varies and this variation depends on the configuration

of the joint that is specified by the variables given below,

� �e, the angle of rotation for axle 1

� �E, the angle of rotation for axle 2

� 7, the bend angle of the joint or the input shaft angle of the joint

Figure 3.2 Variables of Cardan Universal Joint

These variables can be seen in Figure 3.2. The red plane and axle 1 are perpendicular

to each other and axle 2 is always perpendicular to the blue plane. These planes are

at an angle 7. �e and �E are the angular displacement of each axle. �e, �E are the

angles between the 01e and 01E and initial positions along the 0 and � axes. The 01e

and 01E vectors are fixed with the cross shaft that connects the two axles therefore

they remain perpendicular to each other.

01E �1

01

01e

�e

�E

7

27

01e draws the border of the red plane and related to �eby,

q}{ = [�A&�e, &C"�e, 0] (3.1)

01E draws the border of the blue plane and is the result of the unit vector 01 = [1,0,0] being rotated through Euler angles [i/2, 7, 0],

q}~ = [−�A&�&C"�e, �A&�E, &C"�&C"�E] (3.2)

The 01e and 01E vectors are fixed with the cross shaft therefore they must remain at

right angles,

q}{. q}~ = 0 (3.3)

Thus the equation relating the rotations of axles is,

tan�e = cos7 tan�E (3.4)

and the solution for �Ein terms of �eand 7 is,

�E = ��"2(&C"�e, �A&7 �A&�e) (3.5)

The angles �e and �E are the functions of the time. Differentating the rotation angles

with respect to time gives us the angular velocities of the axles,

9e = ��e/�� (3.6)

9E = ��E/�� (3.7)

The relationship between the angular velocities of axle 1 and 2 is,

28

9E = 9e�A&71 − &C"7E�A&�eE (3.8)

The relationship between the angular accelerations can be derived by differentiating

the angular velocity equation,

�E = �e�A&71 − &C"7E�A&�eE − 9eE�A&7&C"7E&C"�eE(1 − &C"7E�A&�eE)E (3.9)

The angular velocity of the output shaft versus rotation angle of the input shaft for

different shaft angles are plotted by using Equation 3.8 in Figure 3.3. �eis in degrees

and for a complete rotation of input shaft.

Figure 3.3 Input Shaft Angle versus Output Shaft Speed for Different Shaft Angles

29

The output shafts's rotation angle versus input shaft rotation angle for different shaft

angles are plotted by using Equation 3.5 in Figure 3.4. �eis in degrees and for a

rotation of input shaft 0° to 180°.

Figure 3.4 Input Shaft Angle versus Output Shaft Angle for Different Shaft Angles

31

CHAPTER 4

DESIGN OF THE COMPLIANT UNIVERSAL JOINT

4.1 Dimension Synthesis of the Compliant Universal Joint

Tanık and Parlaktaş (2012) described the universal joint as a spherical four bar

linkage which is a special case of spatial four bar mechanism. In the universal joint

the arc lengths of the moving links are exact right angles and the connected shafts

intersect at an angle as seen in Figure 4.1.

Figure 4.1 Spherical 4R Linkage (Tanık E. and Parlaktaş V. (2012))

Tanık and Parlaktaş (2012) proposed a novel design for a compliant universal joint

whose compliant components are flexural pivots as seen in Figure 4.2. In Figure 4.3

dimensions of the compliant universal joint which is made of polypropylene are

32

presented. For different applications some of the dimensions are free parameters

however three constraints must be satisfied.

Figure 4.2 Original Compliant Universal Joint Design (Tanık E. and Parlaktaş V.

(2012))

The first constraint comes from the theory of the spherical four bar mechanism.

According to this constraint centerlines of all single axis flexural hinges must

intersect at the center of the sphere. The centerline axis of the flexural hinges and the

inside contact surface must be aligned as seen in Figure 4.3.

33

Figure 4.3 Dimensions of the Compliant Universal Joint (Tanık E. and Parlaktaş V.

(2012))

To satisfy this constraint,

� = � + �/2 (4.1)

The second constraint is to obtain a form closed structure. The thickness of the

identical parts and the width of the connection slot must be equal as shown in Figure

4.3.

� = �. (4.2)

34

Figure 4.4 Critical Position of the Compliant Universal Joint (Tanık E. and Parlaktaş

V. (2012))

The third constraint is to avoid a contact between &e and &E surfaces during operation

as seen in Figure 4.4. For the compliant universal joint Tanık and Parlaktaş (2012)

defined the deflection of single axis flexural hinges. When the input angle 8 equals

0 , 90°, 180° and, 270° one set of the hinges will be at maximum deflected position

and the other set will not be deflected at all. The position in Figure 4.4 most

probably is the critical position when the contact occurs between &e and &E.

However, the mechanism is rotating and just before and after this position 7 will

decrease. On the other hand the sharp corners of the surface &E may contact to &e due

to compliance of the structure and plate thickness. However if compliance factor is

taken into account, further analysis would be challenging. Therefore the relationship

between dimensions can be determined from the geometry as,

0&C"(90 − 7) ≈ 0.5�. + �&C"7 (4.3)

0�A&(90 − 7) + ��A&7 ≈ � − � (4.4)

Eliminating 0 from Equations 4.3 and 4.4 , the lower limit of � is determined by

Tanık & Parlaktaş (2012) as,

35

� ≈ 0.5�.(��"7) + �(��"7&C"7 + �A&7 + 1) (4.5)

H value seen in Figure 4.3 is a free parameter that defines the whole size of the

mechanism.

4.2 Material Selection of the Compliant Universal Joint

In this thesis, first of all blue polished spring steel (AISI 1080) is chosen as the

material of the compliant part of the mechanism. This material has high yield

strength and relatively ductile behavior. Since, fatigue characteristics of polymers are

not definite as in steels in the literature, estimating the life of the mechanism

analytically would be very hard if not impossible. In Appendix A, properties of the

material can be found and some of the properties are given in Table 4.1.

Table 4.1 Properties of AISI 1080 (Spring Steel)

Density 7800-7900 kg/m3

Young's modulus 200-215 GPa Shear modulus 77-84 GPa Bulk modulus 155-175 GPa Poisson's ratio 0.285-0.295 Yield strength 880-1080 MPa Tensile strength 1170-1440 MPa

According to the three constraint equations given above, the dimensions of the

design are determined. Two different designs are proposed where thicknesses of the

small length flexural hinges are 0.5 and 0.75 mm, respectively in the experiments

performed in this study, the geometry with 0.5 mm thickness is used thus the

detailed design of this geometry will be explained.

Free parameters are chosen as follows: � =20 mm, �- =0.5 mm, � =15 mm and �. =5.68 mm. For the ease of manufacturing three plates will be assembled therefore �. is the summation of blue polished spring steel, two steel plates and tolerances of

the plates. According to the Equation 4.1, 4.2 and 4.5 the dimensions are determined

36

as, � =25 mm, � =5.68 mm and the minimum value of � =50.34 mm. All of the

dimensions of the design can be seen in Appendix B. The isometric view can be seen

in Figure 4.5 and deflected position of the mechanism for different input shaft angles

is shown in Figure 4.6.

Figure 4.5 Isometric View of the Design

Flexural Hinges

Shaft 1

Shaft 2

37

Figure 4.6 For a 10° Shaft Angle and Different Input Shaft Angles Views of the

Mechanism

THEORETICAL ANALYSIS

5.1 Stress Analysis of the

Preliminary stress analyses of the design are

then finite element analysis is u

ANSYS 13.0 Workbench software is used.

Firstly, both of the geometries are sketched with Catia V5.

analyses are performed

the analysis can be seen in Figure

following parts.

Importing the CAD

model of the design into the ANSYS 13.0

Programme's design

modeller module

Meshing the model

Defining the boundary

conditions and loadings

39

CHAPTER 5

CAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT

Stress Analysis of the Compliant Universal Joint with FEA

stress analyses of the design are performed by analytical methods and

hen finite element analysis is used for comparison. For finite element analysis

ANSYS 13.0 Workbench software is used.

both of the geometries are sketched with Catia V5. Then

performed for different shaft angles and torque values. The flow chart of

the analysis can be seen in Figure 5.1. Details of the analysis will be explained in the

Figure 5.1 Flow Chart of the Analysis

Importing the CAD

model of the design into the ANSYS 13.0

Programme's design

modeller module

Pre processing of the

model (defining rigid and flexible parts, simplfying

the geometry etc.)

Importing the model to

the analyser module

Assigning the material

properties to the parts

Defining the contacts

and joints between partsMeshing the model

Defining the boundary

conditions and loadings

Analysis settings and

solving the problem

Obtaning the results and

making comments

COMPLIANT UNIVERSAL JOINT

Compliant Universal Joint with FEA

by analytical methods and

sed for comparison. For finite element analysis

hen, finite element

angles and torque values. The flow chart of

etails of the analysis will be explained in the

Importing the model to

the analyser module

Assigning the material

properties to the parts

Obtaning the results and

making comments

40

The steps shown in Figure 5.1 will be explained schematically for ANSYS 13.0

Workbench.

Defining the type of the analysis (Modal, Static, Transient Structural, etc.)

Entering the engineering data for AISI 1080

41

By using ANSYS Workbench Design Modeller prepare the model so that it is ready

for the analysis (Deleting the idle parts, improving the surfaces, etc.)

Analysis part

42

Defining the behaviors of the parts (Flexible or rigid)

Assigning the material properties that are defined in Engineering Data

43

Creating new coordinate systems to identify motions or forces

Defining contacts and joints between parts

44

Meshing the parts

Defining the analysis settings

45

Defining the boundary conditions and loadings

46

Getting the solutions and making comments

5.1.1 Boundary Conditions and Meshing

The spring steel universal joint is divided into subgroups whose characteristics will

be different in the analysis. Small length flexural hinges are the most critical parts of

the design therefore these hinges are meshed finely which means the number of

elements for unit area has the highest value compared to the other parts. And the

other parts are meshed coarsely which behave as relatively rigid and not as much

critical as flexible hinges. Hence the analysis took less time with a good accuracy of

results. The parts of the joint could be seen in Figure 5.2.

47

Figure 5.2 Definitions of the Parts

The connections of the parts are shown in Figure 5.3. Shaft 1 and clutch 1 are

modelled as a unique part by using Form New Part command. And also compliant

joint 1 is bonded to clutch 1 and body 1. Bonding creates a multi point constraint so

that the bonded surfaces behave like a single surface. Same procedure is done for the

other compliant joints. The connections and the parts that have been formed, are

shown in Figure 5.3.

Figure 5.3 Connections and Newly Formed Parts

As seen in Figure 5.4 the model is divided into two main parts whose meshes are

finer than other part. Part A is meshed with body sizing of 0.65 mm and part B is

48

meshed with 2 mm hex-dominant method which uses an unstructured meshing

approach to generate a quad-dominant surface mesh and then fill it with a hex-

dominant mesh. The total number of the elements is 19053 and there are 27428

nodes for the design with 0.75 mm thickness. The model with 0.5 mm thickness has

21705 elements 31896 nodes.

Figure 5.4 The Meshing of the Model

5.2 Static Analysis for Deflection-Only Case

Firstly the bending capacity of the compliant universal joint without loading of a

torque should be identified. In the design thicker parts are modelled as rigid and are

not bent compared to the flexural hinges.

For different shaft angles stress values are calculated analytically and numerically.

The analysis of the design with thickness of 0.75 mm is done for shaft angles

between 1° to 13°. Same procedure is done for the design with 0.5 mm thickness for

shaft angles 1° to 18°.

49

Analytical calculations that can be seen in Table 5.1 are done with the procedure

given below,

Width, thickness and length of the hinge are,

/ = 20 × 10d` (5.1)

�- = 0.75 × 10d` (5.2)

� = 20 × 10d` (5.3)

Second moment of area is,

� = /�-̀12 = 7.031 × 10de`a (5.4)

Modulus of elasticity for blue polished spring steel is,

� = 200 × 10�#� (5.5)

Taking shaft angle, 7, in radians, moment and maximum stress values could be

found with the equations given below,

= ��7� (5.6)

�fTB = �-/2� (5.7)

50

Figure 5.5 Representation of the Boundary Condition and Shaft Angle

For the numerical solution the model which is shown in Figure 5.5 is used. One end

is fixed and the other end is bent from analysis settings by inserting supports, fixed

support and remote displacement, in FEA. Remote displacement command is used

by rotating the shaft in the desired direction and for the desired shaft angle. The

equivalent stress values are obtained. Numerical solutions can be seen in Table 5.1

for the material with thickness of 0.75 mm and in Table 5.2 for the other model for

shaft angles as limited by yielding. The average of the chosen data is calculated.

During the analysis for the angles between 1° to 4° small deflection analysis is used

and for the other angles large deflection analysis is chosen from analysis settings.

Stress concentration is a highly localized effect. In some cases the reason may be

surface scratches. Engineering normal strain for fracture, 23, is an important

parameter for defining a material as ductile or brittle. If 23 ≥ 0.05, material is

ductile, otherwise material is brittle. In ductile materials the stress concentration

factor is not usually applied to predict the critical stress, because plastic strain in the

region of the stress is localized and has a strengthening effect. In other words, if the

material is ductile and the load is static, load may cause yielding in the critical

loading near the notch. This yielding can involve strain hardening of the material and

increases yield strength at the small critical notch location. In brittle materials the

geometric stress concentration factor, ��, is applied to the nominal stress.

51

AISI 1080 material has an elongation of 10-14 % which means 23 changes between

0.1 to 0.14. Therefore blue polished spring steel is a ductile material. Considering

that loads are static and the material is ductile, this part can withstand the loads with

no general yielding. Budynass and Nisbett (2011) points that in these cases the

designer sets the geometric (theoretical) stress concentration factor, ��, to unity.

Therefore, in this study all the following static analyses are performed by

disregarding stress concentration effects.

Table 5.1 Analytical and Numerical Normal Stress

Values for Different Shaft Angles (�- =0.75 mm)

Deflection-only for th=0.75 mm Shaft Angle

(deg) σ-max-

numerical*(MPa) σ-max-

analytical**(MPa) % Error 1 66.6 65.5 1.7 2 133.2 130.9 1.7 3 199.7 196.3 1.7 4 266.3 261.8 1.7

5 (large) 337.6 327.2 3.1 6 (large) 406.3 392.7 3.3 7 (large) 475.2 458.1 3.6 8 (large) 544.4 523.6 3.8 9 (large) 613.7 589 4.0

10 (large) 683.2 654.5 4.2 11 (large) 752.8 719.9 4.4 12 (large) 822.4 785.4 4.5 13 (large) 892.0 850.8 4.6

* The max stress represents the maximum value of the stresses available on the

hinges disregarding the stress concentration regions

** The related stress does not include stress concentrations

The numerical solutions for 1° , 5° , 10° shaft angles are shown in Figures 5.6 to 5.8.

The data that is chosen to calculate the average values are shown in Table 5.1.

During the selection of the data the hinge with the maximum stress value is found

and then the equivalent stress values for this hinge are chosen by using probe. The

average values of the stress values can be seen in Table 5.1 for the model with the

thickness of 0.75 mm.

52

Figure 5.6 ANSYS Simulation for �- =0.75 mm and 1° Shaft Angle


53


To compare the analytical and numerical solutions shaft angle vs. normal stress

values are plotted seen in Figure 5.9. As the shaft angle increases the normal stress

values increase which is expected. For the 0.75 mm thickness the maximum shaft

angle without yielding is found as 13°. This value increases for the model with 0.5

mm thickness. It is observed that the difference between the solutions increases with

the increasing shaft angle values. From the percentage error it is obvious that the

analytical approach is more suitable for the angles that are not large. The large

deflection analysis rises the percentage error for larger shaft angles, since the

numerical solution deviates from the analytical one.

54

Figure 5.9 Plot of Shaft Angle vs. Analytical and

Numerical Normal Stress Values (�- = 0.75 )

Table 5.2 Analytical and Numerical Normal Stress

Values for Different Shaft Angles (�- = 0.5 )

Deflection-only th=0.5 Shaft Angle

(deg) σ-max-

numerical*(MPa) σ-max-

analytical**(MPa) % Error

1 41 43.6 1.3

2 89.6 87.3 2.6

3 134.6 130.9 2.7

4 180.2 174.5 3.2

5 (large) 223.4 218.2 2.3

6 (large) 282.0 261.8 7.2

7 (large) 331.7 305.4 7.9

8 (large) 374.2 349.1 6.7

9 (large) 449.8 392.7 12.7

10 (large) 477.3 436.3 8.6

11 (large) 528.9 480.0 9.2

12 (large) 569.6 523.6 8.1

13 (large) 604.1 567.2 6.1

14(large) 680.7 610.9 10.3

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

900.0

1000.0

0 2 4 6 8 10 12 14

σ_

ma

x(M

Pa

)

Shaft Angle (deg)

Numerical

Analytical

55

15 (large) 717.6 654.5 8.8

16 (large) 755.7 698.1 7.6

17 (large) 822.3 741.8 9.8

18 (large) 869.5 785.4 9.7

19 (large) - 829.0 -

20 (large) - 872.7 -

* The max stress represents the maximum value of the stresses available on the

hinges disregarding the stress concentration regions

** The related stress does not include stress concentrations

The numerical solutions for 1°, 5°, 10° and 15° shaft angles are shown in Figures

5.10 to 5.13. Same procedure is done for the model with 0.5 mm thickness. The

average values of the stress values can be seen in Table 5.2.

Figure 5.10 ANSYS Simulation for �-=0.5 mm and 1° Shaft Angle

56

Figure 5.11 ANSYS Simulation for �-=0.5 mm and 5° Shaft Angle



Shaft angle vs. normal stress values for the second model are plotted which can be

seen in Figure 5. 14. For the 0.5 mm thickness the maximum shaft angle without

57

yielding is found as 18° numerically and 20° analytically. Analytical approach is a

good estimation of the simulations however simulation values will be used to

determine critical shaft angle values.

Figure 5. 14 Plot of Shaft Angle vs. Analytical and

Numerical Normal Stress Values (�- = 0.5 )

To make a comparison shaft angle vs. normal stress values for both of the models are

shown in Figure 5.15. Normal stress values in terms of �-, � and can be written

as,

� = 6 /��-E (5.8)

As thickness increases, normal stress value decreases because �- is in the

denominator in Equation 5.8. However value increases as the flexibility decreases

and this increase is much more than the increase with the thickness change.

Therefore stress values are larger for the model with 0.75 mm thickness for the same

shaft angles.

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

900.0

1000.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

σ(M

Pa

)

Degree (deg)

Numerical

Analytical

58

Figure 5.15 Plot of Shaft Angle vs. Analytical and

Numerical Normal Stress Values for Both Models

5.3 Study of Mesh Refinement for Bending-Only Case

Mesh refinement is an important tool for editing finite element meshes in order to

increase the accuracy of the solution. However, as a mesh is made finer, the

computation time increases. The density of mesh must satisfactorily balance

accuracy and computing resources. Refinement is performed in an iterative manner

in which a solution is found, error estimates are calculated, and elements in regions

of high error are refined. This process is repeated until the desired accuracy is

obtained.

For the model with the thickness of 0.5 mm finite element analyses are done for

different mesh densities. The analysis are done for the deflection-only case with a

shaft angle of 15°. The most critical part of the mechanism is one set of the

compliant flexural hinges. The most critical set of the hinges are meshed with three

different mesh densities. During the refinement study three, four and five elements

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

900.0

1000.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

σ_

ma

x(M

Pa

)

Bend Angle (deg)

Analytical 0.5

Numerical 0.5

Analytical 0.75

Numerical 0.75

59

are used through the thickness and the results are shown in Table 5.3. In Figure 5.13

the simulation results for three elements are shown and in Figure 5.16 and 5.17 other

simulation results can be seen.

Figure 5. 16 Simulation Results for Four Elements through the Thickness for

Bending-Only Case

Figure 5. 17 Simulation Results for Five Elements through the Thickness for

Bending-Only Case

At least three convergence runs are required to plot a curve which can then be used

to indicate when convergence is achieved or, how far away the most refined mesh is

from full convergence. Therefore convergence curve is plotted and shown in Figure

60

5.18. Three runs of different mesh density give the nearly same result, therefore

convergence is already achieved and no more refinement is necessary. After this

study of refinement 3 elements through the thickness are used.

Table 5. 3 Number of Elements through the Thickness vs. Maximum von-Mises

Stress for Bending-Only Case

Number of Elements through the Thickness

Maximum von-Mises Stress (MPa)

3 759.95

4 759.94

5 759.54

Figure 5. 18 Number of Elements through the Thickness vs. Maximum von-Mises

Stress for Bending-Only Case

5.4 Path Analysis

In addition to the results given in section 5.2 9 paths are defined to determine the

equivalent stresses along these path. These are shown in Figure 5.19. For this

analysis the model with the thickness of 0.5 mm is chosen and the shaft angle is set

750

752

754

756

758

760

762

3 3.5 4 4.5 5

Ma

xim

um

vo

n-M

şsse

s S

tre

ss (

MP

a)

Number of Elements

61

as 10°. The analysis shown in Figure 5.19, is done by finite element method using

ANSYS. Post processing results along a path is part of the Workbench Mechanical .

In the analysis a path is defined as construction geometry on which to map the

results. Then a linearized stress result is obtained. Finally the desired results along

the path using the Linearizad Stress item are calculated. The path linearized stress

results can be seen in Figure 5.20 to Figure 5.29.

Figure 5.19 Paths for One of the Deflected Flexural Hinge

Figure 5.20 Path Analysis in ANSYS

Path 1 Path 2 Path 3

Path 4

Path 5

Path 6

Path 7 Path 8 Path 9

62

Figure 5.21 Length vs. Equivalent Stress for Path 1


Length (m)

Equ

ival

ent

Str

ess

(Pa)

Length (m)

Equ

ival

ent

Str

ess

(Pa)

63



Length (m)

Equ

ival

ent

Str

ess

(Pa)

Length (m)

Equ

ival

ent

Str

ess

(Pa)

64



Length (m)

Equ

ival

ent

Str

ess

(Pa)

Length (m)

Equ

ival

ent

Str

ess

(Pa)

65


Figure 5. 28 Length vs. Equivalent Stress for Path 8

Length (m)

Equ

ival

ent

Str

ess

(Pa)

E

quiv

alen

t S

tres

s (P

a)

Length (m)

66

Figure 5. 29 Length vs. Equivalent Stress for Path 9

5.5 Static Analysis to Determine Torque Limits

In the second part of the analysis torque limits are calculated for some specific shaft

angles that were analyzed in the deflection-only case. In this part analyses are done

in two steps. In the first step one of the shaft is bent by the desired shaft angle, in the

second step torque is applied to the same shaft. In order to apply the torque to the

shaft which is bent in the first step, a new coordinate system overlapping with the

centerline of the shaft is created. In the analysis large deflection assumption is used

and also nonlinear solution option is chosen.

Initial time step is taken as 0.1 second, minimum time step is 0.01 and maximum

time step is 1 second. These settings can be seen in Figure 5.30.

Length (m)

Equ

ival

ent

Str

ess

(Pa)

67

Figure 5.30 Analysis settings

The configuration with 1° shaft angle will be explained briefly, other shaft angle

configurations can be analyzed with the same procedure.

In the first step of the loading normal stress has the maximum value of 68.5 MPa for

the first model. In the second step this position is protected and torque is applied.

This torque causes a maximum of 880 MPa stress value on the flexural hinges. The

steps of the loading and the stress values vs. time increments are shown in Table 5.4

and Table 5.5.

Table 5.4 Steps of the Loading

68

Table 5.5 Stress Values vs. Time Increments

To determine the exact torque value that causes yielding, torque values are applied

iteratively. Two suitable torque values are applied to the joint and the exact value of

the torque is found by interpolation. In Table 5.6 and 5.7 the interpolated data can be

seen for both models.

Torque is applied to the universal joint at its free end. However the torque on the

flexural hinges are indeterminate. Applied torque could increase bending in the

flexural hinges and the reactions could be a combination of torques carried by the

flexural hinges and the couple moment of shear forces. Therefore a simple model

shown in Figure 5.31 is proposed to determine the percentage effects of the couple

and torques on the hinges. Using ANSYS 5000 N.mm torque is applied and the

reactions of the flexural hinges are calculated. Torques are determined as 112.2

N.mm and forces as 55.1 N. The ANSYS model and some of the results are shown in

Figure 5.32.

From statics applied torque is defined as,

, = �� + ,e + ,E (5.8)

, = (55.1! × 87) + 112.2! + 112.2!= 5018!

(5.9)

69

Therefore 96% of the torque comes from the couple moment and the rest comes from

the ,e and ,E torques. This confirms that the compliant parts in this mechanism are

predominantly flexural hinges.

Figure 5.31 Simple Model with Applied Torque and Reactions at Flexural Hinges

Figure 5.32 Finite Element Analysis and Results for Simple Model

,

� = 87 � = 87

�

�

,e ,E 20

20

0.5

70

Table 5.6 Interpolation of the Torque Values and Stress Values for �- = 0.75

Interpolation values th=0.75 mm Shaft Angle (deg)

Upper torque

Upper stress

Lower torque

Lower stress

Torque value for 880 MPa

1 12 955.01 11 876.1 11.0 2 11 925.81 10 848.16 10.4 3 11 974.64 9 818.72 9.8 4 10 945.33 8 789.64 9.2 5 10 994.60 8 849.39 8.4 6 8 911.20 7 838.46 7.6 7 8 974.46 6 828.46 6.7 8 6 891.64 5 824.94 5.8 9 6 955.59 4.5 855.52 4.9

10 4 885.99 3 853.2 3.8 11 3 885.8 2 827.5 2.9 12 3 946.85 2 893.02 1.8 13 1 919.02 0.5 896.65 0.1

In Table 5.6 torque limits for different shaft angles are shown. In Table 5.7 torques

that causes yielding are tabulated. According to this data Figure 5.33 is plotted.

When the shaft angle increases the maximum torque capacity decreases as expected.

Table 5.7 Torque Limits for Different Shaft Angles (�- =0.75 mm)

Torque limits for σ-yield=880 MPa th=0.75 mm Shaft Angle (deg) T-max (N.m)

1 11 2 10.4 3 9.8 4 9.2 5 8.4 6 7.6 7 6.7 8 5.8 9 4.9

10 3.8 11 2.9 12 1.8 13 0.1

71

Figure 5.33 Shaft Angle vs. Maximum Torque Values for �- = 0.75

The same procedure is done for other model. The interpolation table and the results

can be seen in Table 5.8 and Table 5.9. The Figure 5.34 is plotted and gives the

similar trend with the first model.

Table 5.8 Interpolation of the Torque Values and Stress Values for �- =0.5 mm

Interpolation values th=0. 5 mm Shaft Angle (deg)

Upper torque

Upper stress

Lower torque

Lower stress

Torque value for 880 MPa

1 5.8 881.99 5 727.48 5.8 2 5.6 905.80 5 801.80 5.5 3 6 1019.67 5 829.17 5.3 4 6 1059.83 5 869.47 5.1 5 5 927.21 4 742.96 4.7 6 5 955.18 4 784.00 4.6 7 5 981.66 4 803.70 4.4 8 5 1090.19 4 845.49 4.1

10 4 937.14 3 792.90 3.6 12 4 1007.86 3 862.24 3.1 14 3 963.49 2 826.09 2.4

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14

T m

ax (

Nm

)

Bend angle (deg)

72

Table 5.9 Torque Limits for Different Shaft Angles (�- =0.5 mm)

Torque limits for σ-yield=880 MPa th=0.5 mm Shaft Angle (deg) T-max (N.m)

1 5.8 2 5.5 3 5.3 4 5.1 5 4.7 6 4.6 7 4.4 8 4.1 9 3.9

10 3.6 11 3.3 12 3.1 13 2.8 14 2.4 15 2 16 1.6 17 1 18 0.6 19 0

Figure 5.34 Shaft Angle vs. Maximum Torque Values for �- =0.5 mm

0

1

2

3

4

5

6

7

0 5 10 15 20

T m

ax (

Nm

)

Shaft Angle (deg)

73

Torques can be applied for the model with the thickness of 0.5 mm are lower for

small shaft angles as shown in Figure 5.35. At 10° shaft angle curves intersect and

torque capacity of the thicker model decreases. The model with the thickness of 0.5

mm can operate at larger shaft angles where thicker one cannot transmit torque at

13° shaft angle.

The reason can be explained as follows. For lower shaft angles, the thicker model

carries more torque as expected. Since the cross section is larger, for a given torque

stresses are lower. As the bend angle increases, however, bending stresses for the

thicker model becomes dominant so its torque carrying capacity diminishes.

Figure 5.35 Shaft Angle vs. Maximum Torque Values for Both Models

5.6 Study of Mesh Refinement for Combined Loading Case

For the model with the thickness of 0.5 mm finite element analyses are done for

different mesh densities. The analysis are done for the combined loading case with a

shaft angle of 14°. The most critical part of the mechanism is one set of the

0

2

4

6

8

10

12

0 5 10 15 20

T m

ax (

Nm

)

Bend Angle (deg)

t_h = 0.5 mm

t_h = 0.75 mm

74

compliant flexural hinges. The most critical set of the hinges are meshed with three

different mesh densities. During the refinement study three, four and five elements

are used through the thickness and the results are shown in Table 5.10. In Figure

5.36 the simulation results for three elements are shown and in Figure 5.37 and 5.37

other simulation results can be seen.

Figure 5.36 Simulation Results for Three Elements through the Thickness for

Combined Loading Case



75



The convergence curve is plotted which can be seen in Figure 5.39.Three runs of

different mesh density give the nearly same result, therefore convergence is already

achieved and no more refinement is necessary. After this study of refinement 3

elements through the thickness are used.

Table 5. 10 Number of Elements through the Thickness vs. Maximum von-Mises Stress for Combined Loading Case

Number of Elements through the Thickness

Maximum von-Mises Stress (MPa)

3 903.54

4 904.53

5 937.59

76


Stress for Combined Loading Case

5.7 Static Analysis of the Mechanism at Zero Degree Shaft Angle

During this step of the analysis maximum shear stress values for zero shaft angle are

determined. One end is fixed from analysis settings by inserting fixed support in

ANSYS other end is not bent as previous steps. Only increasing torque values are

applied to the joint by using loads from analysis settings. The desired direction and

desired value of torque is applied to the shaft without fixed support. Firstly 1 N.m

torque is applied to the shaft then 2.5 N.m is applied and this value is increased by

2.5 N.m increments. The stress values are obtained by maximum shear stress

solution. Numerical solutions can be seen in Table 5.11 for the material with

thickness of 0.75 mm and in Table 5.12 for the other model.

The simulations for the first model are shown in Figure 5.40 to 5.46. Average of the

higher shear stress values which are represented by red color in simulations are taken

and tabulated. In Table 5.11 the average value of the shear stresses on the hinge are

shown.

900

905

910

915

920

925

930

935

940

3 3.5 4 4.5 5

Ma

xim

um

vo

n-M

isse

s S

tre

ss (

MP

a)

Number of Elements

77

Figure 5.40 Shear Stress Values for 1 N.m Torque (�- = 0.75 mm)

Figure 5.41 Shear Stress Values for 2.5 N.m Torque (�- = 0.75 mm)


78

Figure 5.43 Shear Stress Values for 7.5 N.m Torque (�- =0.75 mm)


Figure 5.45 Shear Stress Values for 12.5 N.m Torque (�- = 0.75 mm)

Figure 5.

Table 5.

Only torque (Torque (

The results are plotted in Figure

stress increases in an almost linear manner.

Figure 5.

0.0

100.0

200.0

300.0

400.0

500.0

600.0

0 1

Sh

ear

Str

ess

(MP

a)

79

Figure 5.46 Shear Stress for 15 N.m Torque (�- = 0.75 mm)

Table 5.11 Torque vs. Shear Stress for �- = 0.75 mm

Only torque (N.m) for zero shaft angle th=0.75 Torque (N.m) �_�0 (MPa)

1 35.3 2.5 89.3 5 178.5

7.5 247.7

10 331.5 12.5 344.9

15 490.5

The results are plotted in Figure 5.47. As expected when the torque increases

in an almost linear manner.

Figure 5.47 Torque vs. Shear Stress for �- = 0.75 mm

2 3 4 5 6 7 8 9 10 11

Torque (Nm)

0.75 mm)

0.75 mm

torque increases, shear

0.75 mm

11 12 13 14

The simulations for the second model are shown in Figure

procedure is used and shear stress values are taken and tabulated in Table

Figure 5.48 Shear Stress for 1

Figure 5.49 Shear Stress for 2.5

Figure 5.50 Shear Stress Values for 5

80

The simulations for the second model are shown in Figure 5.48 to 5

procedure is used and shear stress values are taken and tabulated in Table 5

Shear Stress for 1 N.m Torque (�-= 0.5 mm)

Shear Stress for 2.5 N.m Torque (�- = 0.5 mm)

Shear Stress Values for 5 N.m Torque (�-= 0.5 mm)

5.51. Same

5.12.

= 0.5 mm)

Figure 5.

Table 5.12

Only torque (Torque (

The results are plotted in Figure

trend with the first model. To make comparison between two models Figure

plotted.

Figure 5.

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

0

Sh

ear

Str

ess

(MP

a)

81

Figure 5.51 Shear Stress for 7.5 N.m Torque (�- = 0.5

12 Torque vs. Shear Stress Values for �- = 0.5 mm

Only torque (N.m) for zero shaft angle th=0.5 Torque (N.m) �_�0 (MPa)

1 68.7 2.5 166.9 5 338.3

7.5 516.3

The results are plotted in Figure 5.52. The plot for the second model has a similar

trend with the first model. To make comparison between two models Figure

Figure 5.52 Torque vs. Shear Stress for �- = 0.5 mm

1 2 3 4 5 6

Torque (Nm)

0.5 mm)

0.5 mm

. The plot for the second model has a similar

trend with the first model. To make comparison between two models Figure 5.53 is

0.5 mm

7 8

82

Figure 5.53 Torque vs. Shear Stress for Both Models

The line belongs to the model with the thickness of 0.5 mm is above the other line.

Torque for the same shear stress value is larger for the thicker model. Therefore the

maximum torque value for 440 MPa shear stress is 6.4 N.m for the thinner one and

the thicker one has 14.1 N.m.

5.7 Fatigue Life Estimations

Fatigue life estimation is one of the most important steps of a compliant mechanism.

Machine elements can catastrophically fail even if the maximum stresses are well

below the ultimate strength of material because of fatigue. In this study fatigue life

calculations are performed according to the modified Goodman failure theory,

calculation steps and Marin factors for both models are presented below. Initially,

calculations are performed for deflection-only case and then for the combined

loading case.

For the blue polished spring steel yield strength value is,

'+ = 880 #� (5.10)

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sh

ear

Str

ess

(MP

a)

Torque (Nm)

t_h = 0.75 mm

t_h =0.5 mm

83

Ultimate strength of the material is,

'*� = 1.1 × 10` #� (5.11)

Unmodified endurance limit is estimated as,

'(′ = g0.504'*� C� '*� ≤ 1460740 C� '*� > 1460S (5.12)

'(′ = 554.4 #� (5.13)

Marin factors are determined for the model with the thickness of 0.5mm. In fatigue

life calculations, surface finish is one of the most critical parameters. Shigley and

Mischke (1996) suggest an expression for the surface factor,

�T = �'*�� (5.14)

where � and � values can be found in Table 5.13.

Table 5.13 Values of a and b for Surface Factor

a b

Surface Finish Kpsi MPa

Ground 1.34 1.58 -0.085

Machined, cold rolled 2.67 4.45 -0.265

Hot rolled 14.5 58.1 -0.719

As forged 39.8 271 -0.995

Surface factor is calculated from Equation 5.15 and Table 5.13. The material is blue

polished spring steel. Generally for polished machine parts, surface factor is taken as

unity. During the experiments, for the supplied spring steels from the market, it is

84

observed that there are generally some small surface scratches available. Therefore,

to stay on the safe side, surface finish is taken as ground.

�T = 1.58 × 1100d>.>�b = 0.871 (5.15)

Size of the fatigue specimen can be different than the part being analyzed. As the

cross section becomes bigger, there will be more probability of a surface

imperfection. Shigley and Mischke (2001) approximations for the size factor is as

follows:

For bending and torsion of steel,

�� = g1.24�d>.e>� 2.79 ≤ � ≤ 51 1.51�d>.eb� 51 < � ≤ 254 S (5.16)

For axial loading there is no size effect,

�� = 1 (5.17)

All of the equations given above are for the circular cross section in rotating bending

or torsion. For other conditions an equivalent diameter, �(, must be found.

For a nonrotating part with rectangular cross section whose dimensions are � and ℎ,

�( = 0.808 √�ℎ (5.18)

After calculations of the equivalent diameter, this value can be used in Equation 5.16

as d.

Size factor for the rectangular cross section can be found by using Equation 5.16 and

5.18 for the model with the thickness of 0.5 mm,

�( = 0.808√20 × 0.5 = 2.555 (5.19)

85

�� = � �(7.62 �d>.e>� = 1.124 (5.20)

Another endurance limit modification factor is load factor. The fatigue test specimen

has a rotating bending loading. The loadings different than that causes a reduction in

fatigue strength or endurance limit. Norton (2000) introduced the load factors as,

�� = � 1 � "�C"�0.85 �0C��0.59 �A%&CA"S (5.21)

Load factor is taken as �� = 1 by using Equation 5. 21.

Tensile strengths versus endurance limit plot has a scattered data. Most of the

endurance strength data are mean values. Haugen and Wirching (1975) presented

data with standard deviations of endurance strengths of less than 8%. Thus,

reliability modification factors are shown in Table 5.14.

Table 5.14 Reliability Factors, �(

Reliability, % 50 90 95 99 99.9 99.99 99.999 99.9999 �� 1.000 0.897 0.868 0.814 0.753 0.702 0.659 0.620

For 99.9 % reliability �( is found as 0.753 from Table 5. 13.

There are other factors such as stress concentration, temperature, corrosion etc.

which can affect fatigue life. The mechanism is assumed to be working at room

temperature and non-humid condition. Therefore, miscellaneous effects factor �3 is

considered to be stress concentration only and it is estimated as,

�3 = 1 + $(�� − 1) (5.22)

86

�3 = 1�3 (5.23)

Figure 5.54 Notch-Sensitivity Charts for Steels and UNS A92024-T Wrought

Aluminum Alloys Subjected to Reversed Bending or Reversed Axial Loads

(Budynass and Nisbett (2011))

where $ is determined from Figure 5.54 as 0.9 since % = 2 mm and '*� = 1.1 GPa.

Theoretical stress concentration factor of this study is not available in the literature.

Therefore, the theoretical concentration factor, �� is determined via FEA. The

procedure is as follows: A cantilever beam with 0.5 mm thickness is placed in the

middle of the two rigid supports whose edges have 2 mm fillets as shown in Figure

5.55. The loading is only moment since hinges of compliant mechanisms are

dominantly subjected to moment type of loading. To determine ��, ratio of the stress

at the tip of the cantilever beam and the stress away from the edges are considered.

87

Stress distribution is presented in Figure 5.56 and stress concentration is determined

as 1.27.

Thus miscellaneous factor �3, is calculated as,

Figure 5.55 Rectangular Fillet Bar in Bending

Figure 5.56 Estimation of Theoretical Stress Concentration factor with ANSYS

� = 4.5 � = 0.5

% = 2

88

�3 = 1 + 0.9(1.27 − 1) = 1.24 (5.24)

�3 = 0.81 (5.25)

Fatigue life modifying factors are,

��h�T= = �T��(�3 = 0.595 (5.26)

Modified endurance strength limit is found as,

'( = ��h�T='(′ = 329.69 #� (5.27)

Fatigue life calculations are also done for both of the models but the calculations of

the second model that has a thickness of 0.5 mm will be briefly explained. For the

bending-only loading case,

Fatigue strength fraction from Budynass and Nisbett (2011) approximation,

� = 0.9 (5.28)

Curve fitting parameters for blue polished spring steel are,

� = − 13 log ��'*�'( � = −0.174 (5.29)

� = (�'*�)E'( = 3.285 × 10` (5.30)

for numerical �fTB values life can be found with,

89

! = ��fTB� �e� (5.31)

The loading condition for the bending-only loading case can be seen in Figure 5.57.

Bending moment is taken as completely reversed loading because the bending

moment is constant and the mechanism rotates continuously.

Figure 5.57 Loading Condition of the Model

Factor of safety according to Goodman approach is,

" = 1

� �T'(fh��3�(� + �f'*�� (5.32)

Normal stress for the completely reversed loading is,

��(U = �T1 − ��f'*�� (5.33)

Life can be found by using,

! = ��(U"� e/� (5.34)

�T Time

�fTB �f�W

0

Nor

mal

Str

ess

90

After all calculations fatigue life estimations are done and tabulated for the first

model in Table 5.15 and for the second model 5.16. The tabulated data also plotted

in Figure 5.58 and 5.61.The finite life limit for the first model is 4.5° and for the

second model 6.9°.

Table 5.15 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�- = 0.75 mm)

Only bending th=0.75 mm

Shaft Angle (deg) σ-max-numerical

(MPa) Life (cycles) 1 66.6 ∞ 2 133.2 ∞ 3 199.7 ∞ 4 266.3 ∞ 5 337.6 756100 6 406.3 241500 7 475.2 92010 8 544.4 39820 9 613.7 19030

10 683.2 9827 11 752.8 5406 12 822.4 3135 13 892.0 1901

Figure 5.58 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�- =

0.75 mm)

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1000000

4 5 6 7 8 9 10 11 12 13

Life

Shaft Angle (deg)

91

Table 5.16 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�- = 0.5 mm)

Only bending th=0.5

Shaft Angle (deg) σ-max-numerical

(MPa) Life (cycles) 1 44.2 ∞ 2 89.6 ∞ 3 134.6 ∞ 4 180.2 ∞ 5 223.4 ∞ 6 282.0 ∞ 7 331.7 962600 8 374.2 451400 9 449.8 142100

10 503.0 97850 11 528.9 51340 12 569.6 32220 13 604.1 22270 14 680.7 10520 15 717.6 7551 16 755.7 5456 17 822.3 3209 18 869.5 2260

Figure 5.59 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�- = 0.5 mm)

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1000000

6 7 8 9 10 11 12 13 14 15 16 17

Life

Shaft Angle (deg)

92

Under bending-only loading case, the flexural hinges are subjected to pure bending,

therefore there is almost a uniaxial stress state. Pure bending arises from the given

shaft angle when the mechanism is subjected to torque as well. The loading on the

flexural hinges is still predominantly bending with also a limited amount of torsional

shear stress. The bending stresses are produced because of the shaft angle as before

and also because of the transverse forces that develop to resist the torque.

For the combined loading case, the loading on the flexural hinge is still

predominantly bending, and torsional shear stress is smaller. Bending stresses are

produced because of the shaft angle as in the bending-only case and also because of

the transverse forces that develop to resist the torque that are shown in Figure 5.45.

The combined loading case is different from the bending-only case. In the bending-

only case, the stresses are completely reversed however in the combined loading

case for the critical elements, mean values of stresses are observed to be different

than zero during FEA. Therefore, maximum and minimum normal stress values are

taken from FEA results and used in fatigue life estimation approach.

Figure 5.60 Von-Mises Stress Distribution, Most Critical Point on the Flexural

Hinge, its Infinitesimal Cube and �++ Stress Distribution

93

From the finite element analysis results, for all cases, it is observed that the most

critical region is near the fillets and in the edge of the flexural hinges as shown in

Figure 5.60. The element that is chosen from this point has two free surfaces thus

there cannot be shear stresses on this element. Therefore, for this case, a uniaxial

normal stress is the unique source of fatigue life as presented in Figure 5.61. The

following fatigue life calculations are based on the stress variation of this point.

Normal stresses of the infinitesimal cube are determined for a full rotation of the

mechanism from finite element analysis. Shear stresses of the infinitesimal cube are

zero or close to zero as expected.

Figure 5.61 Infinitesimal Cube of Critical Point

Thus for the combined loading case is alternating and mean stresses are,

�T) = �++¡¢£ − �++¡¤¥2 (5.35)

�f) = �++¡¢£ + �++¡¤¥2 (5.36)

Maximum and minimum normal stress values in y direction are evaluated from finite

element analysis and presented in Table 5.17. Fluctuations of this normal stress for

different shaft angles are plotted in Figure 5.62.

�++

�++

0

�

¦

94

Table 5.17 Maximum and Minimum Normal Stress Values for Combined Loading

Shaft Angle (deg)

§yyopq

(MPa)

§yyo¨© (MPa)

Torque (N.m) Life (cycles)

3 832.2 521.3 5.53 1000000+ 4 799.8 470.8 4.65 1000000+ 5 722.1 273.5 3.60 1000000+ 6 599.2 12.0 2.30 1000000+

Figure 5.62 Normal Stress Variations for Different Shaft Angles

In order to indicate performance of the compliant universal joint it is beneficial to

plot both static and fatigue resistance in the same figure. With one degree increments

of the bending angle, fatigue life of the mechanism is calculated from Equation 5.32.

It should be noted that different from the bending only case, stress concentration

factor is not taken into account in this case since critical stresses are directly

transferred from FEA results In Figure 5.63, the red line represents the static failure

limit and below this curve, the region is safe against static yielding. The mechanism

has infinite life below the blue curve. In between the blue and red curves, the

mechanism is expected to have a finite life.

-400

-200

0

200

400

600

800

1000

0 90 180 270 360

No

rma

l S

tre

ss (

MP

a)

Rotation Angle of Input Shaft (deg)

0 Degree Bend Angle

3 Degrees Bend Angle




7 Degree Bend Angle

95

Figure 5.63 Performance Graph of the Compliant Universal Joint

97

CHAPTER 6

EXPERIMENTAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT

6.1 Manufacturing of the Prototype

After theoretical calculations are performed the mechanism is designed for

manufacturing. For the experimental setup second model with the thickness of 0.5

mm is used because it has larger torque capacity and is more available in the market.

The most critical parts of the mechanism are the four compliant hinges. For those

parts blue polished spring steel is used which is also available as a sheet metal in the

market. This material which is also called C80 (AISI 1080) has high yield strength.

The properties of the AISI 1080, are found from Ashby (2005), and are given

Appendix A. For the other parts the stainless sheet steel called C35 is used.

Figure 6.1 Assembly of the Prototype

98

By using Catia V5 the drawings of the components are prepared. Because of the

importance of the distances between the connections laser cutting machine is used

for manufacturing. In the assembly of the design, there are two main parts that are

connected to each other as shown in Figure 6.1 and there are five components in a

main component. In a main part four of the parts are made from C35 and other is

made from C75 which can be seen Figure 6.2. The production of the other parts like

brake dynamometer, connecting shafts and the assembly are done in the laboratory

of Hacettepe University. Compliant part is placed in the middle of the relatively rigid

parts and assembled in three different ways. First one is bonding the parts with a

special glue which is for assembling metals. Second one is by welding the parts from

the edges far from the critical hinges. Last one is the connecting by bolts and nuts.

All of these connections are tried to see the differences between them. The

experimental results agree with theoretical results in all connection types. However

welding is a difficult manufacturing process, and during bonding with glue non

homogenous layers could occur. As a result bolt and nut connection is decided to be

the best connection type among them. The different ways of the connections can be

seen in Figure 6.3.

Figure 6.2 Components of the Prototype

Both ends of the mechanism are connected to bearings and fixed to a wooden

platform. To estimate the fatigue life of the mechanism an inductive proximity

sensor detection switch and a counter is assembled. Proximity sensor is a component

widely used in automatic control industry for detecting, controlling, and noncontact

99

switching. When proximity switch is close to some target object, it will send out

control signal. In the design proximity sensor detects the presence of metal without

physical contact, counts the number of cycles and sends out signal to the counter. In

this way fatigue life is found experimentally. And also for the case with torque a

brake dynamometer is designed with a shoe plate. A weight is attached to the

dynamometer to apply torque and the applied torque is measured by a torque meter.

Figure 6. 3 Different Ways of the Connections

6.2 Components of the Experimental Setup

In Figure 6.4 experimental setup is seen also its isometric view is shown in

Appendix E. The specifications of the bearing and the brushed DC electric motor can

be found in Appendix C and D. The components of the setup are,

1. Power supply (max 3A, 30 Volt)

2. Counter

3. Inductive proximity sensor

4. Bearing

5. Compliant universal joint

6. Brake dynamometer

7. Torque meter

8. Dead weight

9. Shoe plate

10. Brushed DC electric motor

100

Figure 6.4 Components of the Experimental Setup

1

2 3

4 5

6

7 8

9

6

10

101

6.3 Experimental Verification

After the manufacturing of the experimental setup is finished some experiments that

are shown in Table 6.1 are done to verify the theoretical approaches.

Table 6. 1 Experiments for Different Conditions

Shaft Angle (deg)

Torque (N.mm)

Theoretical data (cycles)

Experimental data (cycles)

Experiment 1 6.5 - ∞ Survived more than 106 cycles

Experiment 2 8 - 451400 420000

Experiment 3 6 500 ∞ Survived more than 106 cycles

For fatigue life experiments the mechanism is run continuously until an indication of

failure like a crack is observed or the mechanism passes the infinite life limit that is

1000000 cycles for steels. Three different experiments are done for verification. Two

of them are for bending-only mode at shaft angles 6.5° and 8°. The other one is for

6° degrees shaft angle under 500 N.mm torque. In theoretical calculations, it is found

that an infinite life (i.e. 1000000 cycles) can be reached for a maximum shaft angle

of 6.9° in bending-only case. First test is done for 6.5 degrees and the mechanism is

run 1200000 cycles and the design has reached the infinite life. Second test is done

for 8 degrees shaft angle without torque. The mechanism failed as shown in Figure

6.5 after 420000 cycles which is calculated as 451400 theoretically. In other words

test is completed with a percentage error of 7. The source of errors can be surface

imperfections, material or manufacturing defects. Last test is performed with 500

N.mm torque for a 6° shaft angle. The value of the torque is measured by torque

meter and suitable weight is determined for the brake dynamometer. For 6° shaft

angle the theoretical maximum torque value can be sustained for infinite life is 1130

N.mm as seen in Figure 6.5. However 500 N.mm torque is decided to apply to the

mechanism with a factor of safety and to avoid excessive wear of the disc and shoe

pad. This test is completed at 1200000 cycles without any indication of failure.

102

Hence theoretical prediction is experimentally verified. The finite life limits for

fatigue failure, static failure and the experimentally verified data are shown in Figure

6.6.

Figure 6.5 A Failed Compliant Universal Joint

Figure 6.6 Shaft Angle vs. Torque Output and Experimentally Verified Data

103

CHAPTER 7

RESULTS, CONCLUSION AND FUTURE STUDY

7.1 Results and Conclusion

In this study, a compliant universal joint whose compliant parts are made of blue

polished spring steel is considered and its fundamentals are described. This is the

first steel compliant universal joint in the literature and its feasibility is verified with

a real model.

The mechanism consists of two simple identical parts. Identical parts are produced

by connecting the planar parts that are manufactured with laser cutting process.

Compliant parts of the mechanism are made of blue polished spring steel and for

other parts stainless steel is used. Identical parts are assembled at right angles with

respect to each other.

Three constraint equations determined by Tanık and Parlaktaş (2012) are used to

satisfy the Cardan joint theory and to avoid undesired contact between parts.

After dimensioning a compliant cardan mechanism with two different complian part

thicknesses and choosing an appropriate material, static analyses are carried out:

Stress analysis are performed for different conditions to determine the capacity of the

universal joint. For bending-only condition normal stresses are obtained both

analytically and numerically to determine the maximum possible shaft angle without

yielding. On the other extreme, for zero shaft angle case, maximum torque capacity

of the mechanism is obtained. Then shaft angles with corresponding maximum

104

torque capacities are determined. Finally, for one of the mechanisms, a fatigue

analysis is performed. Life of the mechanism is determined for corresponding torque

values and shaft angles.

In order to verify theoretical approaches, a real model is manufactured. Three

different experiments with different shaft angles and torque loadings are done. First

experiment is done with 6.5° degrees shaft angle without torque and it is observed

that there is no indication of failure. In the second experiment shaft angle is set as 8°

without torque loading and failure is observed as expected. It is also verified that

compliant universal joint has an infinite life at a 6° shaft angle with a loading of 500

N.mm torque.

Hence strengthwise feasibility study of a steel compliant universal joint is done

theoretically and experimentally. It is believed that, compliant universal joint may be

a good alternative for the applications where transmitted torque values are not very

high. Also the design has the advantages of having small number of different parts,

ease of manufacturing and compactness.

7.2 Future Study

For the combined loading case shear stress values due to the torque are determined

numerically. One can attempt to develop an analytical model to predict the torque

capacity and try to derive closed form equations for the stresses. Such an attempt

would require large deflection analysis of a non-circular cross section which is also

subjected to torsion. After this study design charts of the compliant universal joint

can be obtained.

105

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109

APPENDIX A

PROPERTIES OF AISI 1080

Figure A.1 Properties of AISI 1080

110

APPENDIX B

TECHNICAL DRAWINGS

Figure B.1 Technical Drawing of the Model with 0.75 mm Plate Thickness

111

Figure B.2 Technical Drawing of the Model with 0.5 mm Plate Thickness

112

APPENDIX C

SPECIFICIONS OF BEARINGS

Figure C.1Specifications of UCP 200 Bearing

113

APPENDIX D

SPECIFICIONS OF DC ELECTRIC MOTOR

114

Figure D.1 Specifications of the DC Electric Motor

115

APPENDIX E

ISOMETRIC VIEW OF THE TEST SETUP

Figure E

.1 Isometric V

iew of the F

atigue Test S

etup

theoretical and experimental analyses of compliant

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