detailed curved-beam transfer-matrix methodology …...𝐝 nodal dof’s/displacements column...

Detailed Curved-Beam Transfer-Matrix Methodology

for Non-Linear Quasi-Static & Dynamic Free-Vibrations

Analysis of Complex Structural Chain-Systems

Design and Optimization of Highly-Flexible Independent Leaf-Spring

Suspension System for a Heavy Vehicle

Ricardo Miguel Tourinho Torres

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Supervisors: Prof. Jing-Shan Zhao

Prof. Luís Filipe Galrão dos Reis

Examination Committee

President: Prof. Paulo Rui Alves Fernandes

Supervisor: Prof. Luís Filipe Galrão dos Reis

Member of the Committee: Prof. Luís Alberto Gonçalves de Sousa

June 2017

Abstract

This work intends to provide a new take on the transfer matrix method. Not only by introducing such

methodology in a deliberately instructive manner, as by dedicating considerable attention to its particular

implementation on MATLAB. Resulting in a peculiarly comprehensive methodology for physically and

geometrically linear static analysis, as well as undamped dynamic free-vibrations analysis. Nonetheless,

the main realm of application is strictly focused on complexly shaped unidimensional discrete structural

chain-like systems. According to which, an extensive adaptive derivation for employing constant

curvature curved beam elements is presented. Additionally, fully compiled stress expressions for

topologically combined straight-curved beam systems, in both radial coordinates and according to the

internal efforts, are formulated. An approximative artifact to incrementally generate non-linear results

from purely linear stiffness matrices is introduced based on Quasi-statics, resulting on a rather pragmatic

but successful approach to improve a statics method’s accuracy. Ultimately, all derived aspects of the

TMM method are effectively put into practice with success, being this demonstrated through a simple,

but consensually insufficient, numerical example. However, despite the methodological success, while

attempting to employ it to contribute towards a new design for a highly-flexible independent leaf-spring

suspension for a heavy vehicle, a purely organizational mistake on the design process, ended up leading

to an inevitable crippling of the procedure, wherefore no optimal solution is presented.

Keywords

Transfer Matrix Method

Elastomechanics

Structural Mechanics

Quasi-static Loading

Curved Beam Stresses

Leaf-spring Suspension

v

Resumo

Este trabalho proporciona uma nova abordagem ao método das matrizes de transferência. Quer através

de uma introdução deliberadamente instrutiva à metodologia, quer pela atenção dedicada à sua

particular implementação em MATLAB. Resultando uma metodologia peculiarmente detalhada no que

respeita a análises física e geometricamente lineares tanto estáticas lineares, como dinâmicas de

vibração livre não amortecida. Todavia, o seu domínio de aplicação é estritamente vocacionado a

sistemas estruturais discretos unidimensionais que tomem formas complexas em corrente. Realizando-

se uma total adaptação desta metodologia ao emprego de elementos de viga curvos. Adicionalmente,

formulam-se expressões para as tensões existentes em sistemas com elementos de viga retos/curvos

topologicamente combinados, isto tanto seguindo coordenadas radiais como em função dos esforços

internos. Depois, uma ferramenta aproximativa para gerar de forma incremental resultados não-lineares,

recorrendo apenas a matrizes de rigidez puramente lineares, é introduzida com base na Quasi-estática,

resultando numa abordagem exitosa no que toca a melhorar pragmaticamente a precisão de um

método estático. Em última análise, todos os aspetos relativos ao método das matrizes de transferência

derivados, são efetivamente postos em prática com sucesso, ficando isto demonstrado através de um

simples, e consensualmente insuficiente, exemplo numérico. No entanto, e embora todo o sucesso

verificado no que toca à metodologia apresentada, aquando de uma tentativa de a utilizar no sentido

de contribuir para o design de uma nova suspensão de lâminas altamente flexível para um veículo

pesado, um erro puramente organizacional no processo de design levou a uma inevitável paralisação

do processo, não sendo apresentada qualquer solução ótima.

Palavras-Chave

Método das Matrizes de Transferência

Elastomecânica

Mecânica estrutural

Carregamento Quasi-estático

Tensões em elementos de viga curvos

Suspensão de lâminas

vii

Table of Contents

Abstract.................................................................................................................................................... iii

Keywords ................................................................................................................................................. iii

Resumo ....................................................................................................................................................v

Palavras-Chave ........................................................................................................................................v

List of Symbols ........................................................................................................................................ xi

List of Figures ........................................................................................................................................ xvi

List of Tables ......................................................................................................................................... xix

List of Acronyms and Abbreviations ....................................................................................................... xx

1. Introduction ....................................................................................................................................... 1

1.1. Background & Motivation ......................................................................................................... 1

1.2. Literature Review ..................................................................................................................... 2

1.3. Thesis Organisation ................................................................................................................. 4

1.4. Novel Aspects of the Work ...................................................................................................... 5

2. Structure Synthesis .......................................................................................................................... 7

3. Transfer Matrix Method .................................................................................................................. 11

3.1. Application ............................................................................................................................. 11

3.2. Methodology .......................................................................................................................... 12

3.2.1. Basic Elements and Rules ............................................................................................. 12

3.2.2. Alternative Programming Styles Comparison ................................................................ 19

3.2.3. TMM Dynamic Undamped Free-Vibrations Analysis (DFVA) ....................................... 21

3.2.4. TMM Linear Static Elastic (LSEA) & In-Plane Stress Analysis (SA) ............................. 24

3.3. Novel Aspects of the Implemented TMM .............................................................................. 28

3.3.1. Transfer Matrix Method vs. Finite Element Method ....................................................... 28

3.3.2. Convention Adjustment & New State Vector Topology ................................................. 28

3.3.3. Derivation of Field Transfer Matrices from an Exact Arch-Element Stiffness Matrix .... 29

3.3.4. Profile Data Input and Pre-processing Treatment ......................................................... 33

3.3.5. Model Update According to Allowed In-Plane Loading ................................................. 34

3.3.6. Coordinate System Transformations & Updated Matrix Process .................................. 36

3.3.7. Distributed Mass Considerations ................................................................................... 40

viii

3.3.8. Floating Point Arithmetic Issues .................................................................................... 43

3.3.9. Additional Consequences to the Adopted Convention .................................................. 45

3.3.10. In-Plane Stresses in Curved Bar-Beam Elements ........................................................ 45

3.3.11. In-Plane Stresses Methodology Chosen for Topologically Combined Curved-Straight

Systems 51

3.3.12. Buckling Analysis ........................................................................................................... 54

3.3.13. Arch Element Versus Straight Element ......................................................................... 55

3.4. Non-Linear Quasi-Static Loading Static Analysis .................................................................. 58

3.5. Numerical Example ............................................................................................................... 59

4. Leaf-Spring Suspension Mechanical Design Process ................................................................... 61

4.1. Briefing ................................................................................................................................... 61

4.1.1. Starting Model ................................................................................................................ 61

4.1.2. Heavy-Vehicle Synthesis ............................................................................................... 61

4.1.3. Critical Load Condition .................................................................................................. 61

4.1.4. Derivation of a Suspension’s Flexibility Related Parameter, 𝚻 ..................................... 63

4.1.5. Imposed Design Requirements ..................................................................................... 65

4.2. Presumptions on Leaf-Spring’s Material – TMM LSEA Optimization .................................... 66

4.2.1. Brief Setup ..................................................................................................................... 66

4.2.2. Iterative Computational Optimization Procedure ........................................................... 67

4.3. Leaf-Spring’s Material Selection – TMM Non-Linear Quasi-Static Loading Optimization ..... 71

4.4. Leaf-Spring’s Geometrical Optimization – TMM NLQSA-OP ................................................ 74

4.4.1. Geometrical Parameters Parallel Optimization Step ..................................................... 74

4.5. Final Remarks on the Results ............................................................................................... 77

5. Conclusions & Future Developments ............................................................................................. 79

5.1. Conclusions ........................................................................................................................... 79

5.2. Future Developments ............................................................................................................ 80

References .............................................................................................................................................. 1

Appendices .............................................................................................................................................. 1

A. Numerical Example .......................................................................................................................i

B. Relation Between Safety Factor and other Major Physical and Geometrical Parameters ........... ii

C. Documental Pictures on Our Innovative Suspension’s First Full-Prototype, from Dongfeng’s

Technical Center, in Wuhan................................................................................................................. iv

ix

D. CAD Representation Showing the Current State of Design of the Unique Heavy-Vehicle Being

Developed ............................................................................................................................................ vi

E. Evaluation of the Vertical Wheel Loads of a Heavy-Vehicle ....................................................... vi

F. Table with First Selection of Materials and their Physical Properties described at Room

Temperature ...................................................................................................................................... viii

G. Diagram on Further Computational Meaning Behind the Plots of Figure 4.2 ............................. ix

H. Operating profiles of the entirety of NLQSA-OP optimized materials ......................................... ix

I. Leaf-Spring’s Geometrical Optimization for two Additional Materials – TMM NLQSA-OP ..........x

I.1. Geometrical Parameters Parallel Optimization Step ...................................................................x

xi

List of Symbols

Every category organised in alphabetic order.

Convention

𝑎, 𝐴, 𝛼 Scalar

𝐚 Vector

𝐀 Matrix

Over Script

�̃�, �̃� Extended vector or matrix

�̂�, �̂� Unit norm normalized scalar or vector

�̅� Average scalar

Superscript

𝐚𝑖𝐿 Vector at the infinitesimal left-side of position 𝑖

𝐚𝑖𝑅 Vector at the infinitesimal right-side of position 𝑖

𝐚𝑇 , 𝐀𝑇 Vector or matrix transpose

Latin Symbols

𝑎 Radial distance from the centre of curvature to the cross-section’s inner fibre

𝑎𝐻−𝑆 Intended ideal 𝑎𝑣 to cause ultimate operational amplitude (at critical loading)

𝑎𝑙 Arch length

𝑎𝑣 Vertical acceleration

𝐚 Acceleration vector

𝐴 Cross-section area

𝐴′ Radial integral of area until layer of the cross-section being studied

𝐴𝑚 Auxiliary area to the calculation of longitudinal stress in curved beams

𝐴𝑚′ Same as 𝐴𝑚 but with its formulation integrated radially until layer of the

cross-section being studied

𝐀, 𝐁, 𝐂, 𝐃 Partition matrices of 𝐊

𝑏 Width at a certain layer of the cross-section under study

𝑏𝑦 Sectional width at distance 𝑦𝑝 from cross-section’s neutral axis

𝑐 Radial distance from the centre of curvature to the cross-section’s outer fibre

𝐝 Nodal DOF’s/displacements column vector

𝑑𝑑 Dimensionless parameter that characterizes the shear effect

𝑑1, 𝑑2, … , 𝑑6 DOF’s/Displacements participant in Euler-Bernoulli’s beam-bar finite

element

𝐸 Young’s/Elasticity Modulus

xii

𝐸𝐴 Axial stiffness

𝑒 Shifted distance of the neutral axis relative to the centroidal axis

𝑒𝑑 Dimensionless parameter that characterizes the membrane effect

𝐸𝐼 Flexural stiffness

𝐹 Vertical Force

𝐅 Field transfer matrix

𝐅𝒅𝒎 Force vector of uniformly distributed masses on curved-beam field element

𝐅𝑒𝑥𝑡 Applied/External loading force vector (on field element)

𝐅𝑞 Force vector of uniformly distributed in-plane forces and moments (on field

element)

𝐅𝑇𝑜𝑡𝑎𝑙𝐸𝑥𝑡 Total applied/external loading force vector

𝑔 Gravitational acceleration (constant and equal to 9.80665 𝑚 𝑠2⁄ )

𝐺 Shear/Kirchhoff’s Modulus

𝐺𝐴 Shear Stiffness

ℎ Height/Thickness (of rectangular cross-section) along the direction of the

radius of curvature (if existent)

𝐼𝑧 Second moment of area/Area moment of inertia of cross-section about its

neutral axis 𝑦

𝐼𝑧 Second moment of area/Area moment of inertia of cross-section about its

neutral axis 𝑧

𝐼∗ Second moment of area/Area moment of inertia about 𝑧-axis modified for

curvature of bar

𝐽 Polar area moment of inertia of the cross-section

𝑘𝑖𝑗 Entry coefficient of stiffness matrix 𝐊

𝐊 or 𝐊0 Traditional Stiffness matrix (used in linear analysis)

𝐊1, 𝐊2 Stiffness matrices whose components are, respectively, linear and quadratic

functions of the nodal DOF’s compiled in 𝐝

𝑙 Length

𝐿 Total length of the chain system

𝑚 Mass

𝑚𝐴𝐵 , 𝑚𝐵𝐶 Slopes of segments AB and BC, respectively

𝑀 Bending moment about 𝑧-axis

𝑛 Number of layers of materials in a cross-section

𝑛𝑠 Project’s safety factor

𝑁 Longitudinal force

𝐩 Nodal forces column vector

𝑃 Vertical force in kgf

𝐏 Point transfer matrix

𝐏𝑒𝑥𝑡 Applied/External loading force vector (on point element)

xiii

𝑞 Uniformly distributed mass

𝑞𝑑𝑚 Element’s distributed mass per unit length

𝑞𝑥 , 𝑞𝑦 Longitudinal and transversal distributed forces, respectively

𝑞𝑚 Distributed moment by unit length

𝑄 First moment of area of cross-section about its neutral axis 𝑥

𝑄′ Slightly different first moment of area. Adjusted to the calculation of the

transversal shear force in curved beams

𝑟 Radial coordinate to locate a point on the cross section, measured from the

center of curvature

𝑟𝑛 Distance from the center of curvature to the neutral axis

𝑅 Distance from the center of curvature to the centroid of the cross section, or,

if loosely speaking, the radius of curvature

𝐑 Rotation transformation matrix

𝑠 End-to-end length

SRP Suspension’s responsiveness related parameter

𝑆𝑦 Material’s yield strength

𝑡 Time

𝑇 Torque or torsion moment

𝐓 Transformation matrix

𝑢 Extension/longitudinal displacement

𝑢𝑖𝑗 Coefficients of transfer matrix 𝐔

𝐮𝑞 , 𝐟𝑞 Auxiliary matrices in computing 𝐅𝑞

𝐔 Encompassing transfer matrix

𝐔𝐂 Complementary encompassing transfer matrix

𝐔∗ Encompassing transfer matrix to consider TMM operations with 𝐔’s and 𝐓’s,

i.e. to encompass coordinate system transformations

𝑉 Transversal shear force

𝑉𝐹 Volume fraction of either matrix or fibres, respectively subscripted, 𝑚 or 𝑓

𝑤 Deflection/transversal displacement

𝑊𝑑 Distortion/Shear energy

𝑥𝐷 , 𝑦𝐷 Coordinates of center of circumference containing an arch element

𝑥, 𝑦, 𝑧 Global coordinates

𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 Local coordinates

𝑦 In SA is the distance in the 𝑦-direction from the neutral axis to the point

where stresses are calculated

𝑦𝑝 Perpendicular distance from point of study to cross-section’s neutral axis

𝐳 State vector

xiv

Greek Symbols

𝛼 Arch curvature angle

𝛼𝑠 Shear correction factor

𝛽 Curved-beam integration auxiliary angle

𝛾 Shear strain

𝛿𝐫 Virtual displacements vector

Δ Frequency determinant

𝜃 Slope angle

𝜅𝑠 Dimensionless shear form factor

𝜆1, 𝜆2, 𝜆3, 𝜆4 Auxiliary expressions, of trigonometric nature, in computing 𝐅𝑞

𝜈 Poisson’s coefficient

𝜉 Random property of a material

𝜌 Volumetric mass density

𝜎 Overall stress (normal + bending)

𝜎𝑀 Bending stress

𝜎√𝑀𝑦

2+𝑀𝑧2

Bi-directional bending stress (on Tresca equivalent stress’s expression)

𝜎𝑁 Normal/Longitudinal stress

𝜎𝑇𝑟𝑒𝑠𝑐𝑎 Equivalent stress according to Tresca yield criterion

𝜎𝑉𝑀 Equivalent stress according to Von-Mises yield criterion

𝜎𝑥 Longitudinal/circumferential/axial/tangential/hoop stress

𝜎𝑦 Radial stress inducing curvature along 𝑥𝑦-plane

𝜎𝑌 Yield stress

𝜎𝑧 Radial stress inducing curvature along 𝑥𝑧-plane

𝜎1, 𝜎2, 𝜎3 Principal stresses

𝜏 Overall shear stress (transversal + torsional) at the centroid of the cross-

section

𝜏𝑎𝑣 Average shear stress on the cross-section

𝜏𝑇 Torsional shear stress

𝜏𝑉 Transversal shear stress

𝜏𝑦 Shear stress at yielding

𝜏𝑉𝑦, 𝜏𝑉𝑧

Components of the transversal shear stress in special coordinates

Τ Suspension’s Flexibility Parameter

𝜙𝑐 Orientation of the imaginary dotted line segment connecting an arch’s caps

𝜙𝑐𝑡 Angle formed at each arch cap, between the tangent with the curvature and

the chord connecting the two arch caps

𝜙𝑑 Difference between right and left infinitesimal sides of a position 𝑖

𝜙𝑖𝐿 , 𝜙𝑖𝑅 Angles ruling nodal local referential on each infinitesimal side of a position 𝑖

𝛟𝑚 𝑚th normal mode shape vector

xv

𝜒 Bending curvature

𝜓 Curvature

𝜔 Circular frequency

𝜔𝑚 𝑚th natural frequency of vibration

xvi

List of Figures

FIGURE 2.1: GENERIC VIEW OF THE PROPOSED INDIVIDUAL SUSPENSION MECHANISM'S GEOMETRY (LEFT).

DISCRETIZATION OF THE DIFFERENT CONSTITUENT BODIES OF THE PROPOSED SUSPENSION SYSTEM

(RIGHT). ............................................................................................................................................. 7

FIGURE 2.2: ANISOTROPIC ELASTIC MODEL OF FLEXIBLE PLATE-FORM BODY, COMPENSATING UNDER

GEOMETRICAL CONSTRAINTS IMPOSED BY THE SUSPENSION DESIGN (LEFT). INFINITE NUMBER OF REVOLUTE

PAIRS DISCRETIZING EACH COMPLIANT CHAIN (CENTRE). THEORETICAL MODEL OF KINEMATIC PAIRS WITHIN

EACH COMPLIANT FLEXIBLE LINK (RIGHT). ALL FROM [76]. ..................................................................... 9

FIGURE 2.3: SINGLE LEAF-SPRING PLANAR DEFORMATION PROFILES WITHIN SUSPENSION'S RANGE OF

OPERATION. ....................................................................................................................................... 9

FIGURE 2.4: POSSIBLE CONSEQUENCES OF CHOOSING A WRONG INITIAL LEAF-SPRING PROFILE. .................. 10

FIGURE 3.1: QUALITATIVE COMPARISON OF THE RESULTS FROM DIFFERENT TYPES OF ELASTIC ANALYSIS: LINEAR

ANALYSIS (LA), LINEAR STABILITY ANALYSIS (LSA) AND NON-LINEAR ANALYSIS (NLA) (ADAPTED FROM

[81]). ............................................................................................................................................... 11

FIGURE 3.2: CANTILEVER SUBJECTED TO FORCE 𝑉 AND MOMENT 𝑀 (FROM [6]). .......................................... 13

FIGURE 3.3: CANTILEVER WITH A CONCENTRATED END-MASS 𝑚 (FROM [6]). ............................................... 13

FIGURE 3.4: BEAM WITH DISCRETE MASSES (LUMPED-MASS TECHNIQUE) (FROM [6]). ................................. 14

FIGURE 3.5: GENERIC LUMPED-MASS BEAM SECTION (FROM [6]). ............................................................... 14

FIGURE 3.6: END FORCES AND DEFLECTIONS FOR BEAM OF LENGTH 𝑙 AND UNIFORMLY DISTRIBUTED MASS 𝑞

(FROM [6]). ....................................................................................................................................... 15

FIGURE 3.7: EQUILIBRIUM OF AN INFINITESIMAL BEAM SECTION IN ITS DEFORMED CONFIGURATION (FROM [81]).

........................................................................................................................................................ 15

FIGURE 3.8: PROCESSING TIMES, EMPLOYING TWO DIFFERENT MASS-TECHNIQUES, WHEN DETERMINING

NATURAL FREQUENCIES FOR AN INCREASING DEGREE OF DISCRETIZATION. IN BOTH THESE CASES NO

EXTENDED VECTORS OR MATRICES WERE USED. ................................................................................. 20

FIGURE 3.9: PROCESSING TIMES, EMPLOYING TWO DIFFERENT STATE VECTOR TOPOLOGIES, WHEN

DETERMINING NATURAL FREQUENCIES FOR AN INCREASING DEGREE OF DISCRETIZATION. IN BOTH CASES A

UNIFORMLY DISTRIBUTED MASS WAS USED. ........................................................................................ 20

FIGURE 3.10: PLOT OF FREQUENCY DETERMINANT AS A FUNCTION OF CIRCULAR FREQUENCY (FROM [6]). .... 24

FIGURE 3.11: BENDING AND TRANSVERSAL SHEAR STRESSES IN A RECTANGULAR BEAM (FROM [85]). .......... 25

FIGURE 3.12: MOHR'S CIRCLE FOR A TYPICAL UNIAXIAL TENSION TEST (FROM [86]). .................................... 26

FIGURE 3.13: REPRESENTATION OF TRESCA AND VON MISES CRITERIA IN BOTH THE PRINCIPAL STRESSES’ 3D

SPACE (LEFT) AND IN THE (𝜎1, 𝜎2) PLANE (RIGHT) (RESPECTIVELY, FROM [87] AND [85]). ...................... 27

FIGURE 3.14: ARCH ELEMENT CONSIDERED BY LITEWKA & RAKOWSKI (LEFT). ELEMENTAL MODEL ADOPTED IN

OUR NEWLY DEVELOPED TMM (RIGHT) (BOTH ADAPTED FROM [89]). ................................................... 30

FIGURE 3.15: SECTION DISCRETIZATION INTO LAYERS (FROM [93]). ............................................................ 32

FIGURE 3.16: INTERSECTION AT D OF BOTH LINE SEGMENTS’ PERPENDICULAR BISECTORS (FROM [94]). ....... 34

FIGURE 3.17: CURVED ARCH-LIKE ELEMENT SUBJECT TO IN-PLANE LOADING (ADAPTED FROM [69]). ............. 35

xvii

FIGURE 3.18: GENERIC CURVILINEAR BEAM SECTION DISCRETIZED INTO CONSTANT CURVATURE ARCH-LIKE

ELEMENTS, WITH RESPECTIVE TRANSFER MATRICES, STATE VECTORS AND TRANSFORMATION MATRICES IN-

BETWEEN ELEMENTS. ........................................................................................................................ 38

FIGURE 3.19: GLOBAL & LOCAL REFERENTIAL SYSTEMS WITH RESPECTIVE GLOBAL CHARACTERISING ANGLES.

........................................................................................................................................................ 39

FIGURE 3.20: SCHEME OF NEGATIVE TRANSVERSAL DISTRIBUTED FORCE ON A GENERIC CURVED-BEAM ELEMENT.

........................................................................................................................................................ 40

FIGURE 3.21: PROPER SCHEME ON HOW A CURVED-BEAM ELEMENT SHOULD HAVE ITS MASS DISTRIBUTED PER

UNIT LENGTH, 𝑞𝑑𝑚. .......................................................................................................................... 41

FIGURE 3.22: IEEE 754 FLOATING POINT FORMAT (FROM [97]). ................................................................. 43

FIGURE 3.23: FREE-BODY DIAGRAM OF LUMPED-MASS 𝑚𝑖 (ADAPTED FROM [6]). ......................................... 45

FIGURE 3.24: NORMAL STRESS DISTRIBUTION OF CURVED BEAM (ADAPTED FROM [69]). .............................. 47

FIGURE 3.25: RECTANGULAR CROSS SECTION AND RESPECTIVE AXIS PASSING THROUGH CENTRE OF

CURVATURE (DOT-DASHED LINED) (FROM [69]). .................................................................................. 48

FIGURE 3.26: RADIAL STRESSES: (A) EQUILIBRIUM OF SEGMENT OF CURVED BEAM; (B) RESULTANTS (ADAPTED

FROM [69]). ...................................................................................................................................... 49

FIGURE 3.27: DEFINITIONS FOR SHEAR STRESS (ADAPTED FROM [69]). ....................................................... 50

FIGURE 3.28: VARIATION OF TRANSVERSAL SHEAR STRESS AT NEUTRAL AXIS (LEFT) AND OF BENDING STRESS

AT OUTER FIBRE (RIGHT) FOR AN INCREASING R/H GEOMETRICAL RATIO. .............................................. 53

FIGURE 3.29: INDIVIDUAL LEAF-SPRING’S BOTH 2D NON-DEFORMED AND DEFORMED PROFILES (ABOVE).

VERTICAL DEFLECTIONS AT LEAF-SPRING'S TIP, REGISTERED FOR DIFFERENT MODELS WITH DIFFERENT

DISCRETIZATION LEVELS (BELOW). ..................................................................................................... 56

FIGURE 3.30: MAXIMUM EQUIVALENT VON-MISES STRESSES (LEFT) AND TOTAL COMPUTATIONAL PROCESSING

TIMES (RIGHT), REGISTERED FOR DIFFERENT MODELS AND DIFFERENT DISCRETIZATION LEVELS. ........... 57

FIGURE 3.31: TOTAL COMPUTATIONAL PROCESSING TIME WITH TMM DFVA (LEFT) AND FIRST THREE NATURAL

FREQUENCIES OF VIBRATION (RIGHT), REGISTERED FOR DIFFERENT MODELS AND DIFFERENT

DISCRETIZATION LEVELS (LEFT). ........................................................................................................ 58

FIGURE 3.32: COMPARATIVE RESULTS BETWEEN DIFFERENT METHODOLOGIES APPLIED TO THE NUMERICAL

EXAMPLE. ......................................................................................................................................... 60

FIGURE 4.1: PROJECT GUIDELINES WITH RESPECT TO POSSIBLE SUSPENSIONS' RANGE OF OPERATION ABILITY.

ACCEPTABLE RANGE FOR ULTIMATE OPERATIONAL POSITION’S CAUSAL ACCELERATION PAINTED GREEN.

PURPLE AND DARK ORANGE, SHOWCASE THE TWO FULL-SPECTRUM OF ACCELERATION RANGE OF

OPERATION, BOUNDING BOTH ACCEPTANCE MARGINS. (ADAPTED FROM [109]). .................................... 64

FIGURE 4.2: FIRST COMPUTATIONAL OPTIMIZATION EXPERIMENT'S DIAGRAM. .............................................. 68

FIGURE 4.3: STATIC SAFETY FACTOR FOR DIFFERENT LEAF-SPRING’S MATERIALS AND THICKNESSES, WHEN AT

THE MOST CRITICAL OPERATIONAL POINT (100±0.1MM AMPLITUDE ABOVE NEUTRAL STAGE). EVERY

CHARTED POINT IS ASSOCIATED TO A UNIQUE LEAF-SPRING PROFILE WHICH GUARANTEES PERFECT

NEUTRALITY UNDER ALVW CONDITIONS. ........................................................................................... 70

FIGURE 4.4: FOR EACH POSSIBLE LS ANALYZED, IT IS ILLUSTRATED ITS TIP POSITION TRAVEL STARTING AT THE

BOTTOM AT ITS INITIAL (AS MANUFACTURED) POSITION, THEN, UPON ALVW, REACHING A PERFECTLY

xviii

NEUTRAL POSITION WITHIN ±1MM, AND, FINALLY, ALSO WITHIN ±1MM A MAXIMUM AMPLITUDE OF 100MM IS

SECURED FOR THE PARTICULAR DESIGN UNDER 𝑃𝐿𝑆𝑀𝐻𝐽. ................................................................... 73

FIGURE 4.5: COMPARATIVE RESULTS BETWEEN A CONCAVE LS DESIGN (TOP-LEFT) AND A CONVEX ONE (TOP-

RIGHT). ............................................................................................................................................ 76

xix

List of Tables

TABLE 4.1: FIRST RESULTS FROM EMPLOYING A SINGLE-THICKNESSED NLQSA-OP ON PREVIOUS BEST

PERFORMING MATERIALS. .................................................................................................................. 71

TABLE 4.2: REPETITION OF SAME SINGLE-THICKNESSED NLQSA-OP ON SECOND BATCH OF SELECTED

MATERIALS. ...................................................................................................................................... 72

TABLE 4.3: FULLY CONSTRAINED NLQSA-OP RESULTS ON THICKNESS VARIATION (HH IM9). ..................... 74

TABLE 4.4: FULLY CONSTRAINED NLQSA-OP RESULTS ON WIDTH VARIATION (HH IM9). ............................ 75

TABLE 4.5: FULLY CONSTRAINED NLQSA-OP RESULTS ON LENGTH VARIATION (HH IM9). .......................... 75

TABLE 4.6: JOINT TOPLOGY OPTIMIZATION. ................................................................................................ 77

xx

List of Acronyms and Abbreviations

In alphabetic order:

ALVW Averagely Loaded Vehicle Weight

ANCF Absolute Nodal Coordinate Formulation

BC Boundary Condition

DFVA Dynamic (Undamped) Free-Vibrations Analysis

DOF Degree of Freedom

EL-EVWR Suspension’s Operational Effectiveness Loss at EVWR

EL-GVWR Suspension’s Operational Effectiveness Loss at GVWR

ELOPC Suspension’s Operational Effectiveness Loss Over Payload Capacity Range

EVWR Empty Vehicle Weight Rating

FEM Finite Element Method

FE-TMM Finite Element-Transfer Matrix Method

GVLS Subscript: (applied to each) LS by evenly distributing GVWR (fully-loaded)

GVWR Gross Vehicle Weight Rating

IS Independent Suspension

LA Linear Analysis

LS Leaf-Spring

LSA Linear Stability Analysis

LSEA Linear Static Elastic Analysis

LSEA-OP LSEA Optimization Program

LSMHJ Subscript: (applied to each) LS in order to deform it up to its SMHJ

LSN Subscript: (applied to each) LS in order to deform it up to its neutral position

max. Maximum

min. Minimum

MSS Maximum-Shear-Stress

NLA Non-Linear Analysis

NLQSA Non-Linear Quasi-Static Analysis

NLQSA-OP NLQSA Optimization Program

OT Optimized Thickness

RRR Revolute-Revolute-Revolute

S Subscript: (applied on a) suspension

SA (In-Plane) Stress Analysis

SF Static safety factor

SFN Subscript: (applied to each) suspension counting from its neutral position

SMHJ Suspension’s Maximum Half-Journey

TMM Transfer Matrix Method

2D Two-dimensional

%PCS Payload Capacity Share in Percentage

1

1. Introduction

1.1. Background & Motivation

The framework for this research product arises from a recent collaborative research project launched

between Tsinghua University, in Beijing, and an important industrial Chinese manufacturer entity, by the

name Dongfeng Motor Corporation (东风汽车公司). This is a state-owned automobile manufacturer

corporation, established in 1969, and headquartered in Wuhan – capital city of Hubei Province, in

Central China. As of 2014, this was the second largest Chinese vehicle maker, by having a production

volume of over 3.5 million whole vehicles in a single year. Thus, presenting the highest commercial

vehicle production volume between all the remainder domestic manufacturers. Greatly contributing to

this elevated national status, were the joint ventures signed with many internationally reputed brands,

namely, Honda, Nissan, Infiniti, PSA Peugeot Citroën, Renault, Kia, etc. [1].

In this regard, this corporation, with support from governmental funding, sponsors an academic research

project aiming to fully design and materialize a next-generation industrial heavy vehicle, whose novelties

dwell upon a set of challenging state-of-the-art features. Hence, it is amongst these propounded set of

hallmarks that this particular thesis fits in, as we were given the concise task of academically contribute

to the design and optimization of an innovative highly-flexible independent suspension mechanism.

It should be noted, though, that this project stands under a collaborative context, going far beyond the

academic spectrum itself, wherefore the independent suspension system results from a team effort,

which is shared between the author of this thesis, two other laboratory colleagues at Tsinghua University

(Shuai-Song Hou and later on Hong-Wei Song), and, last but not least, by a team of professional

engineers at Dongfeng’s Technical Center, in Wuhan. However, truth be told, practically the entirety of

what is presented in this academic document was developed independently, due to the rather subjective

nature of the guidelines of the project, but also, and mostly, due to the organisational scheme carefully

planned by Prof. Dr. Jing-Shan Zhao, in which, while some graduates would have to take a more

pragmatic approach to the challenge in hands, in order to fulfil the particularly tight deadlines imposed

by the industrial stakeholders, and to keep up with the progress at the Technical Center, thus focusing

more on CAD/CAM/CAE/PLM commercial software usage, others, would have to provide the necessary

theoretical and analytical support, in order to properly and academically justify the design decisions

taken throughout the process. This thesis falls under the latter category.

In this respect, while seeking for a pure analytical approach to solve this mechanical design challenge,

inherently emerged the need to build a computational program (on MATLAB) able to run an optimization

routine for planar chain-like structural mechanism, so to structurally and vibrationally analyse each leaf-

spring mechanical behaviour. In trying to do so, we adventurously opted to address this issue by

employing a less explored method literature-wise, so much so that, at attempting it, we would invariably

have to derive and further expand the current literature on the matter. Thus, from an academic

standpoint, we decided to accept the challenge of proposing such a tool for theoretical purposes.

Becoming the development and fully detailed methodological portrayal of such breakthrough the main

purpose of this thesis.

2

This theoretical instrument resorts to an interesting methodology developed in the field of linear

Elastomechanics during the 50’s, its name is Transfer Matrix Method (TMM), and it comes almost as an

underground brother to the widely popular Finite Element Method (FEM), in the sense that both share

most of the same background knowledge, but the former has a few tweaks to its mathematical

formulation, which bestows particular advantages for circumstances as the ones faced in this project.

Being notably captivating when used to solve complex mechanical problems without proceeding to

commercial software.

Apart from this goal, the second objective set for this thesis, is to effectively contribute, with such

originally engineered methodology, to the design and optimization of an innovative highly flexible

independent suspension mechanism.

1.2. Literature Review

The focus of the literature review is strictly turned to the TMM methodology undergone, since the

academic and literary value mostly abides on the novel and original aspects related to its development

and application. In this sense, we would like to thanks He, Rui & Zhang [2] for some of the information

herein detailed.

Leadingly, it ought to be said that the published literature on this instrumented method was found to be

either majorly inapplicable to the structural case in hands, or understandably tawdry both depth-wise

and explanatory-wise.

If, in one hand, was unfortunate the utter state of shamble seemingly installed in this niche research

issue, at least regarding its applications on the field of structural engineering and in English language,

in which there is a clear lack of standardised rules, notations and convention, wherefore any lone

researcher will undoubtedly face a scatter of knowledge upon ambulation. On the other hand, this, was

seen as an opportunity to conjoin in this thesis a carefully arranged compilation of information from the

most useful literature uncovered. Following an adaptation, at times taking distinctively original contours,

to the particular structural optimization challenge herein seized.

Regarding this literary review, the first usage of TMM is commonly credited to Heinrich Holzer (1921),

who initially applied it to torsion vibration of rods [3]. Decades later, Nils Otto Myklestad (1945)

developed a TMM to determine the bending-torsion modes of beams. To which followed Thomson

(1950), with the usage of TMM on more general vibration problems [5], marking the official insurgency

and popularization of TMM in the academic world. About ten years afterwards, Pestel & Leckie (1963)

produced a ground-breaking piece of work, including a list of transfer matrices for elastomechanical

elements up to the twelfth order [6]. The latter work paved the way for TMM’s golden years (1964-1976),

with numerous researchers applying TMM on a wide variety of engineering programs, namely, Rubin

[7][8], Targoff [9], Lin [10], Mercer & Seavey [11], Lin & McDaniel [12], Mead [13][14], Henderson &

McDaniel [15], McDaniel [16][17], Murphy [18][19][20], and Dokanish [21], who in 1972 developed the

finite element-transfer matrix method (FE-TMM). In these series of works, TMM is used to deal with

beams, beam-type periodic structures, skin-stringer panels, rib-skin structures, curved multispan

structures, cylindrical shells, stiffened rings, plate structure vibration analysis and so forth.

3

In the subsequent years, progress continued at a steady rate. Horner & Pilkey (1978) proposed Riccati

TMM in order to circumvent the numerical stability of the boundary value problem [22]. Several

researchers, namely, Loewy, Degen & Shepard (1985), Ohga & Shigematsu (1987), Xue (1994), and

also, Bhutani & Loewy (1999), studied and improved FE-TMM for structure dynamics [23][24][25][26].

Kumar and Sankar (1986) constructed the discrete time TMM to analyse the dynamical response of the

vibration system [27]. Rui et al., over the years (1998, 2005, 2010) have developed the discrete time

TMM for multibody system dynamics [28][29][30]. Liu (1999) adopted TMM to analyse the plane frame

with variable section and branch [31]. Lee (2000) analysed one-dimensional structural problems using

spectral TMM [32]. Choi & Man (2001) dealt with the dynamic analysis of a geared rotor-bearing system

by TMM [33]. Huang & Horng (2001) applied an extended TMM with complex numbers to analyse

branched torsional systems [34]. Zu & Ji (2002) proposed an improved TMM for steady-state analysis

of nonlinear rotor-bearing systems [35]. Yu et al. (2002) dealt with furcated structural systems by TMM

[36]. Gu et al. (2003) analysed the transient response analysis of large-scale rotor-bearing systems [37].

Zou et al. (2003) analysed the torsional vibration of complicated multibranched shafting systems by the

modal synthesis method [38]. Liew et al. (2004) used the TMM for transient analysis of nonlinear rotor-

bearing systems [39]. Ellakany et al. (2004) dealt with free vibration analysis of composite beams by a

combined transfer matrix and analogue beam method [29][40]. Yeh & Chen (2006) analysed wave

propagations problems of a periodic sandwich beam by FEM and TMM [41]. Hsieha et al. (2006)

proposed a modified TMM to analyse the coupling lateral and torsional vibrations of symmetric rotor-

bearing systems [42].

Summarily, after proceeding to analyse the current state of the art on this matter (2013-2017), we

essentially verified that TMM is experiencing a considerable new rise in popularity. Nonetheless, its

range of application still predominantly revolves upon the same handful of research topics. The most

overwhelmingly common ones relate to eigenvalue TMM algorithms for vibration control and natural

modes & frequencies of diverse mechanical systems [43][44][45][46][47][48][49]. In this particular area

of study, being indeed the stronghold of TMM as it stands nowadays, there hasn’t been any significant

innovation other than, application-wise, the increasing popularisation, since the beginning of the century,

of TMM on dynamic crack propagation evaluations [50][51][52]. Also, we verify an expansion of its usage

on a series of other vibroacoustic elastodynamic problems, such as vehicle exhaust system acoustic

simulations, noise control treatment, acoustic resonators design, omnidirectional acoustic cloaking and

other transformation-acoustical novelties [53][54][55][56]. Additionally, the latter often relate to another

particularly common research topic on TMM amongst academics, which is to use it as a tool to analyse

mechanical properties of composite multi-layered laminated parts, having a wide range of application,

e.g. as sound absorbing materials, optical systems, wave propagation in anisotropic materials,

investigation on multi-layered complex molecular structures, laminated rubber bearings, and many more

[57][58][59][60][61]. On the opposite side of the spectrum, TMM is now starting to be used in robotics

and flexible manipulators too, essentially as a multibody dynamics modelling tool [62][63][64].

However, if we move the spotlight seeking for areas closer to the one we approach in this particular

thesis, we find that it is remarkably more difficult to come across any state-of-the-art research works. In

fact, we have Liu, Zhao & Feng (2016), who worked on a discrete time TMM to study compliant dynamics

4

of a rectilinear rear independent suspension system [65]. Plus, Cramer et al. (2015, 2016), who

employed a slightly different discrete time FE-TMM, built a predictive model able to capture the physical

properties of airplane wings, offering a simple, stable, fast, and, most importantly, implementable

aeroelastic model for embedded controllers to efficiently control wings deformations and shape during

flight, since finite element models have the drawback of being too large to be conveniently implemented

[66][67]. And, finally, we stress the work of Rong (2014) at developing a combined absolute nodal

coordinate formulation (ANCF) and TMM, to produce efficient dynamic analysis of large-deformation

flexible beams [68].

To conclude, despite decades of TMM academic knowledge pile up, we are left with a sense of

tremendous potential still out there to be exploited. Thus, we hope with this thesis to a give a good

contribution towards the same common academical purpose shared with all the researchers above

mentioned.

1.3. Thesis Organisation

Very straightforwardly, this thesis’s outline can be detailed as follows.

Chapter 1, provides a proper introduction to the research project being presented, starting with the

background and motivations behind it, as well as the objectives stipulated. To which follows, a

satisfyingly descriptive literature review, able to provide the reader a clear panorama over the current

stance and reach of TMM in the academic world. This is done chronologically, since its origins up until

the current state-of-the-art. The chapter finishes up by glancing over the novel aspects to be featured

hereinafter.

In Chapter 2, a structural synthetisation, of the mechanism to be mechanically designed, is presented.

Encompassing an overview of the generic model for the suspension and its topology, as well as, the

first guidelines over some of the project’s design directions to be addressed, which will ultimately lead

to the project’s impositions, only presented on the following chapter. On doing this, key aspects from

other influential academic works are introduced, followed by a briefing over some of the most relevant

qualitative and quantitative approaches to the design procedure.

On Chapter 3, a unique introduction to the Transfer Matrix Method (TMM) is made. The extent of such

introduction goes at least as far as into the necessary detail to implement and apply it to any complexly

shaped unidimensional discrete structural chain-like systems. Wherefore, after having introduced all the

basic elements, rules and analysis procedures, a sequential formulation of a series of underlying new

features is made, ultimately enabling the methodology presented to encompass the previously

described range of application.

As to the final chapter, Chapter 4, it showcases the application of the methodology in a properly

contextualized real-world engineering problem. In this case, it demonstrates the applicability of the TMM

methodology developed, by employing it in an attempt to contribute towards the mechanical design of a

highly flexible leaf-spring suspension for a heavy vehicle.

5

1.4. Novel Aspects of the Work

Regarding the proposed suspension design, novelty abides in almost all aspects of it, given its originally

intended innovative purpose. Such as its uncanny independent layout, which includes carefully designed

leaf-springs, whose own structural elasticity is harnessed to leverage the mechanism’s performance.

Not only that, but they are mounted so that each independent system constitutes a

redundant/overconstrained light-weight mechanism, whose technological linchpin is its one-off ability to

guarantee elastic invariableness to the wheels’ alignment parameters. Something regular suspensions

are far from offering, and whose advantages are described further on in this work. Additionally, the

design of the suspension was initially bounded in such a meticulous way that it bolsters the ability of the

heavy-vehicle as a whole, so to allow it to excel in a series of important performance parameters,

definitely imparting it a prominent position at the edge of the market for the same class of trucks.

Methodologically speaking, this thesis was developed mainly from the background knowledge passed

down by the brilliantly written works of Pestel & Leckie [6] and Pilkey [69], who complement each other

in the sense that, while one properly and instructively introduces the methodology, the other

pragmatically and descriptively provides some advanced tools, which, together with the author’s acumen,

allowed for the presentation of an algorithm which offers a new approach to TMM with a bunch of newly

derived aspects to it. Being the developed method in many levels more advanced than either of the

suggested TMM’s on any of the main background references consulted.

Regarding the methodological novelties, we will have an additional set of bold features which we believe

to be quite interesting. In this regard, there is a particularly special elemental convention adopted, which

we think is much more intuitive than the ones found in the two most important books integrally read on

the matter [73][77]. Then, we also employed an exact stiffness matrix for curved beam elements,

adapted from Litewka and Rakowski [74], which takes into consideration the combined effects of flexural,

axial and shear deformations, leaving the discretizing elements completely free of shear and membrane

locking effects. As a consequence, all of the methodology had not only to be formulated for the straight

beam case, as, followingly, it had to be adapted in its entirety to be able to handle the considerably more

complex, but also more accurate, curved beam elements. In this grievous process, a huge number of

complex considerations are encompassed. Additionally to this, an updated stiffness matrix form for

handling multi-layered beams in arch-like elements [78] is also formulated, although ultimately not

implement on MATLAB (in fact, from all detailed issues throughout the thesis, this is the only case where

such transpires). An introduction on how to easily transform any conveniently supplied stiffness matrix,

as they are widely available on the literature, into its homologous field matrix, necessary to undergo a

TMM [73], is fully detailed. A complex derivation of the final form for the external forces vector, so to be

able to handle a wide range of different in-plane loadings, in fact, to be able to simulate any kind of in-

plane loading at the elemental level, is shown. Yet included in the previous point, but worthy singular

mention, is the laborious derivation of a way to properly consider distributed mass along any system

within the realm of planar chain-like systems, meaning it may be composed of multiple elements with

any global inclinations relative to each other, by methodically employing sequential coordinate system

transformations. Aside from these, some care is given to the supply of particular computational

implementation tips, such as floating-point arithmetic issues, which are addressed in this thesis with the

6

intent of helping the reader to successfully implement a similar program to the one developed by the

author for this work. Last but definitely not least, are the compiled and adapted considerations on how

to properly compute all in-plane stresses present in curved bar-beam elements, regarding which, a fully

detailed instructive summary of each participating expressions is presented. Such stresses’ expressions

were carefully built for different degrees of curvature and following a direct correspondence of the final

stresses with the causal internal forces or moments (normal stress, bending stress, transversal shear

stress).

To the previously densely introduced aspects, a few more, no less important, but rather less heavier

formulation-wise, were also either derived or compiled, and ultimately presented and implemented on

this work. For organization purposes, they are followingly listed:

- Formulation, implementation and comparative analysis of different programming styles;

- Full TMM dynamic undamped free-vibrations analysis (DFVA) implementable formulation;

- TMM implementation of unit norm normalization & orthogonality property of the normal modes;

- Full TMM linear static elastic analysis (LSEA) & in-plane stress analysis (SA) procedures;

- Equivalent stress according to Von-Mises failure criterion;

- Transfer matrix method vs. finite element method;

- Detailed profile data input & pre-processing treatment for curved-beam elements on TMM;

- Fully compiled radial coordinate in-plane stresses in curved bar-beam elements;

- Qualitative continuity analysis of in-plane stresses formulation to deal with topologically curved-

straight systems;

- Arch element vs. straight element comparative analysis;

- TMM non-linear quasi-static loading static analysis.

As stated, all of these small details are then uniquely put together into a single methodological effort,

which will be detailed throughout this thesis.

7

2. Structure Synthesis

Prof. Dr. Jing-Shan Zhao, in charge of supervising this academic venture, draw a notably unfettered

academic blueprint for this particular research work. From a creative standpoint, there were only two

major highlightable limitations imposed. One, was that the suspension ought to be composed of thin

leaf-springs, whose structural elasticity was suggested to be used in order to leverage the mechanism’s

performance. The other, relates to the methodology to be undergone, and was that we should follow an

as analytical as possible path in order to provide theoretical support to the overall project.

Moreover, the most basic principles for using a suspension system must not be overlooked. According

to [70], the summarized purposes for a suspension system are to:

• locate the wheels while allowing them to move up & down, and steer;

• maintain the wheels in contact with the road and minimize road noise;

• distribute the weight of the vehicle to the wheels;

• reduce vehicle weight as much as possible – in particular the unsprung mass;

• resist the effects of steering, braking, and acceleration;

• work in conjunction with the tires and seat springs to give acceptable ride comfort.

Therefore, a vehicle’s suspension should effectively attenuate vibrations caused by various road

conditions [71][72][73], which is obviously the main cause for ride comfort deterioration [74]. However,

these are difficult to achieve simultaneously, hence a conventional vehicle suspension mechanical

design procedure generally involves trade-offs between handling stability and ride comfort [75].

Going straight to the point, innumerable layout combinations were proposed, drafted and discussed,

but, a first-design similar to the ones shown in Figure 2.1 ended up being the one chosen. This is an

independent rectilinear/straight-line multi-leaf-spring suspension.

Figure 2.1: Generic view of the proposed individual suspension mechanism's geometry (left). Discretization of the

different constituent bodies of the proposed suspension system (right).

This proposed concept was strongly influenced by the co-author, Prof. Dr. Jing-Shan Zhao, who for

several years has been developing important work on the field of redundant/overconstrained light-weight

mechanism design. Wherefore, four of his research works were particularly influential - [71][75][76] and

[65] as co-author with Dr. Liu – all of them focusing on independent suspensions, but, most importantly,

if it wasn’t for its elasticity, this design would share with these researched designs the fact that they are

exact planar straight-line linkages, consequently bestowing our suspension mechanism a capacity to

elastically guarantee invariableness to the wheels’ alignment parameters. This is definitely a

technological edge, unreachable to virtually all currently production installed suspensions, since, and

8

quoting Zhao et al. on [71], “(…) any changes of the wheel orientation parameters, such as kingpin,

caster, camber, toe change, axes distance and the wheel track, surely bring undesirable dynamic

phenomena such as shimmy and caster wobble [77]”.

In this regard, by having an arrangement inspired on Zhao’s previous designs, the suspension shares

those unique kinematic advantages described. However, the morphology of the main kinematic

branches is completely novel, as they are now composed of non-rigid structural bodies, which obviously

implies different theoretical methods to justify and analyse this suspension’s behaviour. Nonetheless,

such challenge, belongs to the next chapters of this thesis.

In synthesizing the proposed mechanism, the focus is strictly directed to only one of the vehicle’s

independent suspensions composing the overall system. Since, even though they might be subject to

different loading conditions depending on their position along the chassis, locally speaking, all of them

offer the same static and dynamic response under equal loading conditions.

Knowing this, let’s first look at the generic model on Figure 2.1 (right). Each system is composed of six

bodies, two rigid and four elastic ones, whose locations are below discretized. Basically, we have a

network of four symmetrically disposed thin leaf-springs composing the compliant connection between

the steering knuckle and the chassis frame, the latter here represented by four supports located

rectangularly on the three-dimensional ground. This symmetry is an essential part of the geometrical

design developed for this suspension, as it is fundamental in guaranteeing the suspension’s elastic

rectilinearity in respect to the wheel’s only elastically allowable vertical degree of freedom. We always

disclaim ourselves, by referring the term “elastic”, because, due to the flexible nature of this

mechanism’s main constitutive links, upon a considerable impact loading, the mechanism’s supposedly

rectilinear trajectory may suffer a punctual deviation, which is immediately corrected by the inherent

stiffness of these same links.

Essentially, the geometry of these four branches provides rectilinear guidance, and constitute indeed

an overconstrained parallel mechanism with only one translational DOF under normal conditions, which

assures the wheels’ alignment parameters to be invariable in mechanism theory during a wheel’s jounce

and rebound, where, once again, the capacity to conserve a straight-line motion depends on the

resultant stiffness of the suspension [75][76]. This, was already demonstrated by Zhao et al. in two

occasions. First, for a similar layout, but, where branches are composed by stretchable 3-RRR rigid

multilink kinematic chains [75]. And second, and ultimately most relevant, for flexible plate-form

branches of anisotropic elasticity [76], which, should be understood, was the predecessor concept for

the current thesis.

Thus, to ascertain this so-called rectilinear motion of the end effector, and study the overall kinematic

synthesis of the suspension system, Zhao et. al applied a very interesting methodology, designated

Screw Theory. An introduction to this theory can be found on [78]. Figure 2.2 (centre), shows that all

flexible links, as 𝐴1𝐵1, were simplified into theoretical models composed of an infinite number of identical

revolute pairs 𝑃11, … , 𝑃1𝑛 (𝑛 ≥ 1), being also revolute jointed at both ends of the plate-form. The results

applying Screw Theory were conclusive, hence [76] states that the free motion of the end effector

allowed by two compliant chains, Figure 2.2 (right), is exclusively translational in 𝑧-axis direction. And,

this result will always hold as long as the two planes specified by the kinematic chains 𝐴1𝐵1 and 𝐴2𝐵2

9

are neither parallel nor coplanar [79][80]. Following Zhao et. al, the angle between these planes was

adopted as 90° degrees for this thesis, nonetheless, it may obviously be defined within ]0°, 180°[. This

particular design parameter was not herein approached, as it is innocuous considering the conditions

studied in this work. A proper justification for this is detailed in chapter 5.2.

Figure 2.2: Anisotropic elastic model of flexible plate-form body, compensating under geometrical constraints

imposed by the suspension design (left). Infinite number of revolute pairs discretizing each compliant chain

(centre). Theoretical model of kinematic pairs within each compliant flexible link (right). All from [76].

On the contrary, a crucial aspect to design is the initial profile shape to follow on the manufacturing of

these elastic plates. This issue will now be qualitatively approached, in order to supply important guiding

rules according to which the final profile should comply to. However, throughout this thesis a progressive

refinement process can be counted on.

Our first intuition would be to opt for an initially curved leaf-spring (LS), as proposed on Figure 2.2, since,

it is obvious that this enables us a more “comfortable” elastic deformation up until the suspension’s

maximum half-journey (SMHJ) amplitude, exemplified in Figure 2.2 (left).

Using a linear static TMM algorithm, applying eight straight-beam elemental stiffness matrixes with

added shear deformation effects and referentially described in a local manner followed by a global

continuous insertion, whose methodology will be fully introduced on Chapter 3, we built the following

visually appealing chart (Figure 2.3), which shows, for a qualitative LS, the progression of the planar

deformed profiles towards both critical pre-defined SMHJ vertical amplitudes required for a single LS,

namely, [−100, 100] millimetres. To do so, we progressively applied vertical forces at the LS’s tip, which

were carefully calculated so to produce equal deflection steps in each direction, wherefore resulting in:

Figure 2.3: Single Leaf-spring planar deformation profiles within suspension's range of operation.

10

Obviously, the purpose of these rudimentary TMM results are just to give us a rough idea over the actual

LS non-linear profiles happening during operation. By further observing Figure 2.3, it would be an

obvious first choice to go for an initial LS with the green layout, i.e. symmetrically curvilinear. However,

one thing we immediately realised, is that the neutral position during operation (in green) would

necessarily have to be the one occurring when subjecting the suspension system to the vehicle’s

“normal” weight. In sum, taking in consideration the weight associated to an averagely loaded heavy

vehicle, we should manufacture a LS whose wheel-side extremity is lower in relation to the frame-side

one, so to obtain a reasonable static neutral position for the global vehicle system. This reasoning is

illustrated on Figure 2.4, whose first row of pictures illustrates the case where what has been discussed

is not taken into consideration, resulting in a clear waste of suspension operative range of effectiveness,

being a quite obvious but highly limitative mistake performance-wise.

Figure 2.4: Possible consequences of choosing a wrong initial leaf-spring profile.

Thus, this will force us to completely change many design considerations. Not only will the suspension

have to assume a more carefully geometrized initial profile, in order to achieve this requirement, implying

computational optimization processes, but, this also means the system is subject to considerably higher

stresses during operation, particularly during a wheel’s jounce motion, such as when going through a

bump. On the flip side, the advantage is that we will have a more responsive/active suspension, since

we are dealing with more critical elastic states, hence the natural behaviour will be to bring the LS back

to the initial profile more vehemently, consequently guaranteeing a tighter grip to the floor under any

condition. Equally consequent, is the inherent necessity for a better and stronger damping dynamic

control, shock absorbers being a must (strictly damper – no spring needed), as structural damping is by

all means insufficient. Such dynamic challenge is not approached on this analytical thesis.

Additionally, and perhaps already on the radar to the most critical reader, is the possibility for a convex

LS, instead of the so far illustratively considered concave one. Or, by giving another example, the

possibility for different kinds of jointed connections between the LS’s and the frame or steering knuckle.

These, and many other design propositions, will be thoroughly approached and resolved on Chapter 4,

using for that purpose a specially conceived discrete quasi-static non-linear curved-beam TMM

methodology, which is detailed on Chapter 3.

11

3. Transfer Matrix Method

Transfer Matrix Method began to establish itself in the field of linear Elastomechanics during the 50’s,

following the developments on electronic computers, which opened new possibilities for more complex

matrix calculus. Since then, this methodology quickly became widely used, with its success bolstering

the degree of its development in great extent. Its usage mostly comprises static and dynamic treatment

of complicated elastic systems. Read Subchapter 1.2 for a more in-depth historical and literary review.

The herein developed and disclosed work, focuses on a modified TMM methodology, whose

conventions and notations tend to follow Matrix Methods in Elastomechanics, by Pestel & Leckie [6], as

such played an important foundational role, knowledge-wise, to the author of this thesis. Yet, the

established intent here is to provide a detailed, and instructionally-mannered, vade-mécum work on

TMM, at an introductory to medium level, to which are then additionally added a handful of bold unique

features.

3.1. Application

TMM stands out from the rest in its extreme versatility and ability to be analytically programmable, such

as using MATLAB, without any immediate breakdown of expediency, efficiency or avail. This perfectly

complied with our academic project necessities. And so, MATLAB was indeed the conducive medium for

a tool able to promptly assist throughout the design process towards the intended highly flexible thin

plate leaf-spring independent suspension.

In this respect, the particular TMM developed, strictly applies to the realm of linear and non-linear (the

latter upon a quasi-static approximation) unidimensional discrete structural systems. At which, this work

provides guidance on how to execute non-linear quasi-static, linear static, and dynamic free-vibrations

analysis. It should be known that, the term “linear” classifies any structure where there is a proportional

relation between solicitations (e.g. external forces) and the consequent structural response (e.g.

displacements). However, such an artifact at predicting a structural behaviour rapidly loses its accuracy

for increased deformations (see Figure 3.1).

Since, as synthetized, the projected LS is

expected to undergo considerable

deformations, it is then justified the effort put

into providing a non-linear analysis in the mix.

On a side note, every time linearity is

considered in this work, it refers to both physical

(constitutive relations are linear – linear elastic

materials) and geometrical linearity (equilibrium

equations are written in the non-deformed

configurations of the structure and kinematic

relations are linear - this last hypothesis is also

known as “small displacements hypothesis”).

As to the non-linear approximative model,

Figure 3.1: Qualitative comparison of the results

from different types of elastic analysis: Linear

Analysis (LA), Linear Stability Analysis (LSA) and

Non-linear Analysis (NLA) (adapted from [81]).

12

obtained through a quasi-static loading proposition, this is an approximation to a geometrical non-linear

case.

Concerning the term “unidimensional”, it applies to systems where one dimension is way bigger than all

others. Bars, beams, shafts and other structures composed of them, such as trusses and gantries, fall

into this category. In opposition, two-dimensional systems possess one dimension much smaller than

all others, e.g. plates and shells. Since this project comprises thin-plate leaf-springs, a two-dimensional

model should ideally be considered, however, knowing only in-plane loadings are hereinafter studied, a

classical planar unidimensional Cartesian model offers a more reasonable formulative challenge.

Finally, the structural systems to be studied are obviously classified as “discrete”, since their deformed

configurations are characterised by a finite number of parameters (DOF’s - degrees of freedom). Our

TMM covers flexible chain systems, which are discretized into multiple beam sections endowed with

local physical and geometrical properties. Each section is bounded by two nodes, which in turn are ruled

by three DOF’s each: two displacements (horizontal and vertical) and one rotation. Thus, the number of

DOF’s in a system depends on the discretization adopted. In reality, every structural system is

continuous, having infinite DOF’s, nonetheless, engineers artfully adopt practical approximations by

resorting to discretization methods, as Galerkin’s Method, FEM, or, such as in this case, TMM.

Throughout the following chapters, is assumed the reader is already acquainted with elementary

mechanics and strength of materials, as well as elementary algebra and calculus.

3.2. Methodology

The methodological philosophy behind this matrix calculus technique, focuses on studying a complex

system by breaking it apart into less complicated building blocks, each, seeing its own elastic and

dynamic properties carefully embedded in an ID-like matrix. Then, by having learned the rules on how

to “glue” these blocks together, virtually any system can be analysed. Such matrix formulation results

particularly efficient, being easily adaptable to be ran on modern digital computers.

However, although TMM is known to be suitable for the treatment of branched and coupled systems, its

application is inadvisable for systems that lack a predominant chain topology, so much so, that the

methodology developed by the author for this thesis only allows chain systems. But, fortunately, on the

particular case-study we are focusing on, not only the mechanism’s specially conceived symmetrical

layout and redundancy allow for a perfectly synchronised deformation of all four LS’s upon vertical

motion of the steering knuckle (demonstrated on [75][76]), as, by strictly considering in-plane loadings,

together enable the overall system to be studied as a whole only by singularly analysing one of its LS’s

planar behaviour. With this in mind, we stress the importance of paying special care when selecting

formulations, as they need to adequately fit the particular engineering challenge in hands.

3.2.1. Basic Elements and Rules

The most basic element of TMM is the state vector, 𝐳𝒊. Each discretized point of an elastic system has

an associated state vector. These, supply valuable information about some of the most important

parameters on that particular position on the system. Each mechanical system is different, and there

are different sets of differential equations that might rule it. Therefore, the components comprising the

13

state vectors of each system are determined and built accordingly to necessity. Nonetheless, usually all

of them follow the same scheme, where the top half of the state column vector relates to different nodal

degrees of freedom, while the bottom half to the corresponding causal parameters. Using an example

might be more intuitive, wherefore, if looking at a typical state vector for ruling plane flexural vibrations

of a straight beam, which will be very similar to the one we will define for our suspension system, we will

have the following column state vector configuration:

𝐳𝑖 = [

−𝑤𝑖

𝜓𝑖

𝑀𝑖

𝑉𝑖

] (3.1)

In which, 𝑤 stands for deflection (in 𝑚𝑚), 𝜓 for curvature/slope (in radians), 𝑀 for bending moment (in

𝑁.𝑚𝑚) and 𝑉 for transversal shear force (in 𝑁). All of these at the particular node 𝑖 , indicated as

subscript. A physical understanding of these parameters is easily attained by visualizing the following

example for a cantilevered beam.

Figure 3.2: Cantilever subjected to force 𝑉 and moment 𝑀 (from [6]).

Attention should be paid to the minus signal put behind the deflection on the state vector, equation (3.1).

This is strictly conventional, meaning that, henceforth any downward displacement is assumed positive.

Such inconsequential adoption complies to [6], however, throughout this thesis a different convention

on this issue might be preferred, which, in due course would clearly be indicated.

Having used 𝐳𝑖 to fully define a system’s node 𝑖, it is to wonder how the latter relates with the remainder

nodes, so to elaborate a more complex chain system. Well, being this an incredibly versatile matrix

method, it is possible to compute any state vector of a chain system only by employing simple matrix

multiplications. However, we first need to introduce the rules on how to build such matrices. And, even

before that, it is important recall how to properly discretize a system for TMM usage.

As stated before, each discrete beam section, bounded by two nodes, has associated a unique matrix

embedded with its physical and geometrical properties. We call these, Field Matrices (𝐅). At the nodes,

in turn, elastic or rigid supports, as well as external forces and moments might be applied. Thus, although

being lengthless, they also have their own unique descriptive matrix, called, Point Matrices (𝐏).

TMM’s unique discretizational style is exemplified below, for a simple case, on Figure 3.3.

Figure 3.3: Cantilever with a concentrated end-mass 𝑚 (from [6]).

14

It is common practise to assign superscripts 𝐿 or 𝑅 to 𝐳𝑖, respectively, before or after every point element

at a corresponding node 𝑖, as a way to pinpoint its specific domain of action.

Another interesting aspect of the TMM’s discretization process ought to be introduced. Well, mass-wise,

while for more accurate results all field matrices should be built considering adequate distributed mass

functions, it is often advisable to follow the technique of concentrating the mass on the nodes, while

replacing the actual beam for a massless one with the same flexural stiffness 𝐸𝐼, since such a system

is more easily computed by TMM [6]. This, is called lumped-parameter or lumped-mass technique, being

illustrated on both Figures 3.3 and 3.4, and, its effectiveness and practicality in matrix calculus becomes

even more imperative when undergoing dynamic free-vibrations analysis, since, if considering circular

frequencies amidst the field matrices, the so-called “consistent mass matrices” or distributed-parameter

technique, it would exponentially increase the computational effort involved in the process, as it would

increase both the complexity and degree of the resultant frequency determinant equation [6][69]. Thus,

by considering this lumped-mass “approximation” at modelling the structure, the computational burden

of identifying the frequencies can be highly reduced [69]. For this reason, henceforth all vibration

analysis strictly use lumped-masses, while proper distributed masses are kept for the regular analysis.

Figure 3.4: Beam with discrete masses (Lumped-mass Technique) (from [6]).

At last, we can introduce how the computational matrix calculus is processed. By looking at the example

on Figure 3.3, and knowing there is a field transfer matrix 𝐅 between 𝐳0 and 𝐳1𝐿, and a point transfer

matrix 𝐏 between 𝐳1𝐿 and 𝐳1

𝑅. Then, the process simply ensues the following purely sequential scheme:

𝐳1𝐿 = 𝐅𝐳0 = 𝐔1

𝐿𝐳0 (3.2)

𝐳1𝑅 = 𝐏𝐅𝐳0 = 𝐔1

𝑅𝐳0 (3.3)

Where 𝐔 are the encompassing transfer matrices resulting from the sequential multiplication of any

existent field and point transfer matrices in between two state vectors being studied. This same

procedure can be indefinitely reproduced throughout any chain system’s length. Take the example,

Figure 3.5: Generic lumped-mass beam section (from [6]).

𝐳𝑖−1𝑅 = 𝐏𝑖−1𝐳𝑖−1

𝐿 = 𝐔𝑖−1𝑅 𝐳𝑖−1

𝐿 (3.4)

𝐳𝑖𝐿 = 𝐅𝑖𝐏𝑖−1𝐳𝑖−1

𝐿 = 𝐔𝑖𝐿𝐳𝑖−1

𝐿 (3.5)

𝐳𝑖R = 𝐏𝑖𝐅𝑖𝐏𝑖−1𝐳𝑖−1

𝐿 = 𝐔𝑖𝑅𝐳𝑖−1

𝐿 (3.6)

𝐳𝑖+1𝐿 = 𝐅𝑖+1𝐏𝑖𝐅𝑖𝐏𝑖−1𝐳𝑖−1

𝐿 = 𝐔𝑖+1𝐿 𝐳𝑖−1

𝐿 (3.7)

𝐳𝑖+1𝑅 = 𝐏𝑖+1𝐅𝑖+1𝐏𝑖𝐅𝑖𝐏𝑖−1𝐳𝑖−1

𝐿 = 𝐔𝑖+1𝑅 𝐳𝑖−1

𝐿 (3.8)

15

In order to portray a more understandable and organised method, the above calculus always stems from

the first state vector. Yet, given the vital info, calculations could in truth be set between any two points.

In TMM, such vital information comes in the form of transfer matrices. Thus, we will now focus on how

to build these. By carrying on with the same planar straight beam case (state vector on equation (3.1)),

we convention the end forces and deflections, for a uniformly distributed mass element, to be as shown

in Figure 3.6. Based on which, we can write down the immediately following two equilibrium equations.

Figure 3.6: End forces and deflections for beam of length 𝑙 and uniformly distributed mass 𝑞 (from [6]).

𝑉𝑖𝐿 = 𝑉𝑖−1

𝑅 − 𝑞𝑖𝑙𝑖 (3.9)

𝑀𝑖𝐿 = 𝑀𝑖−1

𝑅 + 𝑉𝑖−1𝑅 𝑙𝑖 −

𝑞𝑖𝑙𝑖2

2 (3.10)

Complementarily, by resorting to elementary beam theory, we obtain the two missing equations for both

end deflection and slope of a beam section with uniformly distributed mass and flexural stiffness (𝐸𝐼)𝑖.

For robustness sake, such equations’ full derivation is herein showcased. The difficulty lies in deriving

Euler-Bernoulli’s classic equation, which describes the relationship between a beam’s deflection upon

an applied load, and goes as follows:

𝑑2

𝑑𝑥2 (𝐸𝐼

𝑑2𝑤

𝑑𝑥2) = 𝑞(𝑥) (3.11)

To do so, first, an infinitesimal 𝑑𝑥 section in its deformed configuration is considered, Figure 3.7. Then,

the system of equilibrium equations is written by employing the Virtual Work Principle, which is in fact a

special case of the D’Alembert Principle for the dynamic case (∑(𝐅𝑇𝑜𝑡𝑎𝑙𝐸𝑥𝑡− 𝑚𝐚) . 𝛿𝐫 = 𝟎) , but

considering the system to be still instead (𝐚 = 𝟎), wherefore inertial forces are null [81]. A posteriori, any

higher order infinitesimals were dismissed from the two resultant equilibrium equations, and, after having

played around with such two expressions, we reach the following final one:

−𝑞 + 𝑅𝐴𝑑2𝑤

𝑑𝑡2 + 𝑁𝑑2𝑤

𝑑𝑥2 +𝑑𝑁

𝑑𝑥

𝑑𝑤

𝑑𝑥−

𝑑2𝑀

𝑑𝑥2 = 𝑅𝐼𝑑2

𝑑𝑡2 (𝑑2𝑤

𝑑𝑥2) (3.12)

Figure 3.7: Equilibrium of an infinitesimal beam section in its deformed configuration (from [81]).

16

Hitherto equilibrium equations were examined, however, any structural problem requires compatibility

and constitutive equations to be considered too. Thus, let’s start by taking Euler-Bernoulli’s beam theory

classical elastic constitutive relation, which simplistically relates moment and bending curvature.

𝑀 = 𝐸𝐼𝜒 (3.13)

The bending curvature 𝜒 can also be deduced from the previously established equilibrium, already

shown in Figure 3.7, resulting (particular attention to the adopted referential convention is called for):

𝜒 =𝑑𝜃

𝑑𝑠=

𝑑𝜃

𝑑𝑥

𝑑𝑥

𝑑𝑠=

𝑑

𝑑𝑥(tan−1 (

𝑑𝑤

𝑑𝑥))

1

√(𝑑𝑤

𝑑𝑥)2+1

=𝑑2𝑤

𝑑𝑥2

(𝑑𝑤

𝑑𝑥)2+1

1

√(𝑑𝑤

𝑑𝑥)2+1

=𝑑2𝑤

𝑑𝑥2

[(𝑑𝑤

𝑑𝑥)2+1]

32

(3.14)

𝑀 = −𝐸𝐼𝑑2𝑤

𝑑𝑥2

[(𝑑𝑤

𝑑𝑥)2+1]

32

(3.15)

The latter, is a second order non-linear differential equation, having a hardly attainable solution, thus

critical simplifications had to be admitted. Flexural stiffness is assumed constant along the beam section.

And, we consciously opted to limit this method’s accuracy range by assuming small deformations (Figure

3.1), which is achieved by expanding the denominator in equation (3.15) using Taylor series, so that

𝟏

[(𝒘′)𝟐+𝟏]𝟑𝟐

= 𝟏 −𝟑

𝟐(𝒘′)𝟐 +

𝟏𝟓

𝟖(𝒘′)𝟒 + 𝑶((𝒘′)𝟓) (3.16)

Now, if we are considering small deformations on the beam, then we must analyse the values around

𝜃 = 0° ⇒ 𝑤′ = 0. For example, for 𝑤′ = 0.1 or a rotation of 𝜃 = tan−1(0.1) = 5.71°, we get the following

exact value

𝟏

[(𝒘′)𝟐+𝟏]𝟑𝟐

=𝟏

[(𝟎.𝟏)𝟐+𝟏]𝟑𝟐

= 𝟎. 𝟗𝟖𝟓𝟐 (3.17)

Thus, by using the Taylor series approximation with only the first two terms, we verify we get a value

quite close to the exact one, being this

𝟏 −𝟑

𝟐(𝒘′)𝟐 = 𝟏 −

𝟑

𝟐(𝟎. 𝟏)𝟐 = 𝟎. 𝟗𝟖𝟓𝟎 (3.18)

With only 1.48% of difference associated, this ended up being a surprisingly reasonable approximation.

Thereafter, equation (3.15) is rewritten, and substituted into equation (3.12), finally providing the actual

general Euler-Bernoulli theory-based equation ruling our model, which is

𝐸𝐼𝑑4𝑤

𝑑𝑥4 + 𝑁𝑑2𝑤

𝑑𝑥2 +𝑑𝑁

𝑑𝑥

𝑑𝑤

𝑑𝑥+ 𝑅𝐴

𝑑2𝑤

𝑑𝑡2 − 𝑅𝐼𝑑2

𝑑𝑡2 (𝑑2𝑤

𝑑𝑥2) = 𝑞 (3.19)

From where equation (3.11), also known as “equation of the elastic curve”, is deduced by simply

considering we are dealing with linear analysis, i.e. by disregarding terms originated from inertia forces,

this is all terms dependent on time 𝑡. Apart from this, longitudinal forces 𝑁 are also ignored, as this

simplistic model doesn’t consider their effect (equation (3.1)). Equation (3.11) exact solution, 𝑤(𝑥),

corresponds to a forth degree polynomial, shown on equation (3.20), whose four constants of integration

depend on the type of boundary conditions governing the beam. The following equations can be written:

𝑤(𝑥) =𝑞

24𝐸𝐼𝑥4 + 𝐶4𝑥

3 + 𝐶3𝑥2 + 𝐶2𝑥 + 𝐶1 (3.20)

𝜓(𝑥) =𝑑𝑤

𝑑𝑥=

𝑞

6𝐸𝐼𝑥3 + 3𝐶4𝑥

2 + 2𝐶3𝑥 + 𝐶2 (3.21)

𝑀(𝑥) = −𝐸𝐼𝑑2𝑤

𝑑𝑥2 = −𝑞

2𝑥2 − 6𝐸𝐼𝐶4𝑥 − 2𝐸𝐼𝐶3 (3.22)

17

𝑉(𝑥) = −𝐸𝐼𝑑3𝑤

𝑑𝑥3 = −𝑞𝑥 − 6𝐸𝐼𝐶4 (3.23)

Not forgetting we are evaluating the example represented on Figure 3.6, let’s then consider the following

two boundary conditions (BC’s) affecting the degrees of freedom (DOF’s):

𝑤(0) = 𝑤𝑖−1𝑅 ⇒ 𝐶1 = 𝑤𝑖−1

𝑅 (3.24)

𝜓(0) = 𝜓𝑖−1𝑅 ⇒ 𝐶2 = 𝜓𝑖−1

𝑅 (3.25)

To which follows the sequential deduction of the remainder two necessary constants of integration from

equations (3.23) and (3.22), as

𝐶4 = −𝑉(𝑥)

6𝐸𝐼−

𝑞𝑥

6𝐸𝐼 (3.26)

𝐶3 = −𝑀(𝑥)

2E𝐼+

𝑞𝑥2

4𝐸𝐼+

𝑉(𝑥)𝑥

2𝐸𝐼 (3.27)

Resulting, after substitution, in

𝑤(𝑥) = 𝑤𝑖−1𝑅 + 𝜓𝑖−1

𝑅 𝑥 −𝑀(𝑥)𝑥2

2𝐸𝐼+

𝑉(𝑥)𝑥3

3𝐸𝐼+

𝑞𝑥4

8𝐸𝐼 (3.28)

𝜓(𝑥) = 𝜓𝑖−1𝑅 −

𝑀(𝑥)𝑥

𝐸𝐼+

𝑉(𝑥)𝑥2

2𝐸𝐼+

𝑞𝑥3

6𝐸𝐼 (3.29)

Which can be rewritten for the condition on the right side of Figure 3.6, i.e. for 𝑥 = 𝑙, as

𝑤(𝑥 = 𝑙) = 𝑤𝑖𝐿 = 𝑤𝑖−1

𝑅 + 𝜓𝑖−1𝑅 𝑙 −

𝑀𝑖𝐿𝑙2

2𝐸𝐼+

𝑉𝑖𝐿𝑙3

3𝐸𝐼+

𝑞𝑙4

8𝐸𝐼 (3.30)

𝜓(𝑥 = 𝑙) = 𝜓𝑖𝐿 = 𝜓𝑖−1

𝑅 −𝑀𝑖

𝐿l

𝐸𝐼+

𝑉𝑖𝐿𝑙2

2𝐸𝐼+

𝑞𝑙3

6𝐸𝐼 (3.31)

However, Figure 3.7, on which was based the previous derivation, doesn’t follow the same convention

as Figure 3.6, which governs our TMM, therefore, noting there is a difference of adopted sign convention

for either sides’ curvatures 𝜓, this should easily be corrected by multiplying these by (−1). Additionally,

the vertical deflection’s sign should also be inverted due to a convention assumption which was already

explained before. After these modifications, results:

−𝑤𝑖𝐿 = −𝑤𝑖−1

𝑅 + 𝜓𝑖−1𝑅 𝑙𝑖 +

𝑀𝑖𝐿𝑙𝑖

2

2(𝐸𝐼)𝑖−

𝑉𝑖𝐿𝑙𝑖

3

3(𝐸𝐼)𝑖−

𝑞𝑖𝑙𝑖4

8(E𝐼)𝑖 (3.32)

𝜓𝑖𝐿 = 𝜓𝑖−1

𝑅 +𝑀𝑖

𝐿𝑙𝑖

(𝐸𝐼)𝑖−

𝑉𝑖𝐿𝑙𝑖

2

2(𝐸𝐼)𝑖−

𝑞𝑖𝑙𝑖3

6(𝐸𝐼)𝑖 (3.33)

Last but not least, by substituting into these the previous two equilibrium equations, (3.9) and (3.10), we

are able to rewrite the equations such that all state vector elements at point 𝑖𝐿 can be expressed in

terms of those at point (𝑖 − 1)𝑅, which is what we want to attain when building a TMM field transfer

matrix. After doing so, we finally get the set of four ruling equations for this particular field:

−𝑤𝑖𝐿 = −𝑤𝑖−1

𝑅 + 𝜓𝑖−1𝑅 𝑙𝑖 +

𝑀𝑖−1𝑅 𝑙𝑖

2

2(𝐸𝐼)𝑖+

𝑉𝑖−1𝑅 𝑙𝑖

3

6(𝐸𝐼)𝑖−

𝑞𝑖𝑙𝑖4

24(𝐸𝐼)𝑖 (3.34)

𝜓𝑖𝐿 = 𝜓𝑖−1

𝑅 +𝑀𝑖−1

𝑅 𝑙𝑖

(𝐸𝐼)𝑖+

𝑉𝑖−1𝑅 𝑙𝑖

2

2(𝐸𝐼)𝑖−

𝑞𝑖𝑙𝑖3

6(𝐸𝐼)𝑖 (3.35)

𝑀𝑖𝐿 = 𝑀𝑖−1

𝑅 + 𝑉𝑖−1𝑅 𝑙𝑖 −

𝑞𝑖𝑙𝑖2

2 (3.36)

𝑉𝑖𝐿 = 𝑉𝑖−1

𝑅 − 𝑞𝑖𝑙𝑖 (3.37)

Which ought to be put into matrix notation, following

18

[

−𝑤𝜓𝑀𝑉

]

𝑖

𝐿

=

[ 1 𝑙

𝑙2

2𝐸𝐼

𝑙3

6𝐸𝐼

0 1𝑙

𝐸𝐼

𝑙2

2𝐸𝐼

0 0 1 𝑙0 0 0 1 ]

. [

−𝑤𝜓𝑀𝑉

]

𝑖−1

R

+

[ −

𝑞𝑙4

24𝐸𝐼

−𝑞𝑙3

6𝐸𝐼

−𝑞𝑙2

2

−𝑞𝑙 ]

(3.38)

Or, according to TMM’s notation,

𝐳𝑖𝐿 = 𝐅𝑖𝐳𝑖−1

𝑅 + 𝐅𝑞 (3.39)

Where, 𝐅𝑞 represents an “appendix” vector carrying information on the effect of the uniformly distributed

mass 𝑞, although it may include contributions from virtually any kind of external phenomena.

Similarly, when going through a point instead of a field element, we have

𝐳𝑖𝑅 = 𝐏𝑖𝐳𝑖

𝐿 + 𝐏𝑒𝑥𝑡 (3.40)

Homologously, the latter case has his own “appendix” vector, 𝐏𝐸𝑥𝑡, which in turn encompasses external

punctual phenomena, as applied forces, moments, or even pre-defined impositions to the DOF’s of a

specific node. As to the point transfer matrix definition, 𝐏𝑖 , this may, respectively, take any of the

following two forms, depending if distributed mass or lumped mass technique is being employed. In this

work, the latter case is only used on dynamic free-vibrations analysis (DFVA), as already explained.

[

1 0 0 00 1 0 00 0 1 00 0 0 1

] and [

1 0 0 00 1 0 00 0 1 0

𝑚𝜔2 0 0 1

] (3.41)

The emphasis here goes to the second matrix in (3.41), whose distinct entry is justified by the

concentrated vibrating mass’ inherent inertia force, causing a discontinuity in the shear, which can be

yielded from simple equilibrium considerations as

𝑉𝑖𝑅 = 𝑉𝑖

𝐿 − 𝑚𝑖𝜔2𝑤𝑖 (3.42)

Here, the free parameter 𝜔2 represents the squared circular frequency associated to the particular mass

(if subscripted 𝑖) or entire system. Lumped-mass technique can also be applied on regular analysis with

no modifications to the point matrix on the left, by having the mass contribution introduced through 𝐏𝑒𝑥𝑡.

As these matrices on (3.41) were defined, numerous others, with unique features able to defined all

sorts of nodal conditions, could have been. From abrupt changes in slope or in beam axis, to linear or

rotary hinges, to an almost limitless variety of flexible supports, all are possible to be computed and

applied using this methodology. And the exact same happens with field transfer matrices. Throughout

this work, are only presented solutions applicable to systems similar to the suspension case-study in-

hands. For further information, Walter D. Pilkey’s Formulas for Stress, Strain, and Structural Matrices

[69] is seriously recommended to the reader, for the invaluable and extensive work written on this issue.

The justification for the, at the time seemingly innocuous, decision of assigning a negative sign to 𝑤 on

the equation (3.1), is now clear by analysing equations (3.38) and (3.41), as this allows cross-symmetric

matrices with only positive entries. Nonetheless, this is purely an aesthetic option.

All pieces of the puzzle have been explained, however, if proceeding as thoroughly detailed, we are not

able to conveniently and straight-forwardly implement a sequential multiplication TMM, as first promised.

The problem resides on the “appendix” vectors, 𝐅𝑞 and 𝐏𝐸𝑥𝑡. In sum, while before each matrix 𝐔 carried

all necessary info to allow us to transfer from a state vector to another, now an additional complementary

19

matrix 𝐔𝐂 is required to work, in parallel with 𝐔, to carry in it info relative to the appendix vectors’

contributions. In this situation, a generic operation between two state vectors 𝑗 and 𝑖 goes as follows:

𝐳𝑖 = 𝐔𝑗→𝑖𝐳𝑗 − 𝐔𝐂𝑗→𝑖=

𝐔0→𝑖

𝐔0→𝑗𝐳𝑗 +

𝐔𝐂0→𝑖

𝐔𝐂0→𝑗

(3.43)

In which,

𝐔𝐂0→𝑖= ∑

𝐔𝑖

𝐔𝑘𝐅𝑇𝑜𝑡𝑎𝑙𝐸𝑥𝑡𝑘

𝑖𝑘=1 (3.44)

𝐅𝑇𝑜𝑡𝑎𝑙𝐸𝑥𝑡 𝑖= 𝐅𝑞𝑖

− 𝐏𝐸𝑥𝑡𝑖 (3.45)

𝐏𝐸𝑥𝑡𝑖= [

−𝑤𝑒𝑥𝑡

𝜓𝑒𝑥𝑡

𝑀𝑒𝑥𝑡

𝑉𝑒xt

]

𝑖

(3.46)

However, in order to provide a so-called sequential TMM implementation, we should seek for an

alternative procedure. Starting from equations (3.38) and (3.39), such alternative is obtained by adapting

the matrix notation used to one comprising “extended” state vectors, which consist of state vectors as

in equation (3.1) added a simple unitary extra line as follows,

[ −𝑤𝜓𝑀𝑉1 ]

𝑖

𝐿

=

[ 1 𝑙

𝑙2

2𝐸𝐼

𝑙3

6𝐸𝐼−

𝑞𝑙4

24𝐸𝐼

0 1𝑙

𝐸𝐼

𝑙2

2𝐸𝐼−

𝑞𝑙3

6𝐸𝐼

0 0 1 𝑙 −𝑞𝑙2

2

0 0 0 1 −𝑞𝑙0 0 0 0 1 ]

.


𝑖−1

𝑅

(3.47)

Or, using the adequate representation for both extended state vectors and extended field transfer matrix,

�̃�𝑖𝐿 = �̃�𝑖�̃�𝑖−1

𝑅 (3.48)

Basically, the is no longer need for any “appendix” vectors or complementary matrices, since all their

information is instead included in the matrices’ last column. This is a neat and convenient way to obtain

the intended straightforward sequential multiplication TMM implementation.

3.2.2. Alternative Programming Styles Comparison

In fact, almost unnoticeably, we ended-up presenting four different ways to build a TMM. It can either

be done using distributed masses, or, we might instead adopt a lumped-mass technique. On the other

hand, we may either use extended vectors and matrices, or, if the intent is to reduce the size of the

elements being operated we may opt for a methodology involving “appendix” vectors and

complementary matrices. All of these possibilities have their own advantages and disadvantages.

However, the two things we should be careful when dealing with different programming styles are, the

user-friendliness or promptness of usage, and their effect on the total computation processing time.

Up until a certain point in time, in order to comply to all combinatory possibilities, four different versions

were parallelly and carefully programmed. Then, a detailed stage-by-stage performance analysis was

developed to study each of the programming style’s processing times, charted on figures 3.8 and 3.9.

20

Figure 3.8: Processing times, employing two different mass-techniques, when determining natural frequencies for

an increasing degree of discretization. In both these cases no extended vectors or matrices were used.

Figure 3.9: Processing times, employing two different state vector topologies, when determining natural

frequencies for an increasing degree of discretization. In both cases a uniformly distributed mass was used.

Such analysis, brought up two big conclusions. First, was that, for whatever programming style

employed, there won’t be any significant performance differences when running linear static elastic

analysis (LSEA). So much so, that neither of the charts above illustrates these situations. In absolute

contrary tide came the results relative to DFVA, where, for the random model studied, there was a clear

and sudden increase in processing time when employing a distributed mass comparatively to lumped-

mass technique (Figure 3.8). Interestingly enough, in [6] the reader is early on advised on this same

issue, and bluntly guided to apply lumped-mass technique for vibrations analysis, which is herein

corroborated. The reasoning for this phenomenon was explained on subchapter 3.2.1.

Has to the usage of different state vectors’ topologies, we verified that, within the range of discretization

which is going to be used in this project, there was absolutely no need to opt for a faster program in

detriment of the appalling convenience conveyed by “extended” elements (Figure 3.9).

In sum, lumped-parameter technique is employed on DFVA, and distributed-parameter technique on

LSEA. Furthermore, all kinds of analysis use extended elements.

To conclude, a proper disclaimer must be made, as the previous charts were built from a very early

implementation of the program, by all means rudimentary in complexity when compared to the current

version. Regardless of this, at the time all four possible programming styles were tested under equal

conditions, thus, despite now only holding qualitative value, they still present absolutely valid arguments.

21

3.2.3. TMM Dynamic Undamped Free-Vibrations Analysis (DFVA)

3.2.3.1. Methodology

Following the presentation of all basic elements and rules, the TMM DFVA analysis may now be

introduced. In doing so, we will continue to assist ourselves with the same simplistic example of a planar

straight beam with two DOF per node. As justified in subchapter 3.2.2, lumped-mass parameters and

extended elements are employed.

The procedure, as well as the transfer matrices for both a uniform massless beam and a concentrated

mass, have already been derived on sub-chapter 3.2.1. Hence, the relation between any state vector

along the chain system and the initial one is symbolically written as:

�̃�𝑖 = �̃�0→𝑖�̃�0 ⇔

[ −𝑤𝜓M𝑉1 ]

𝑖

=

[ 𝑢11 𝑢12 𝑢13 𝑢14 𝑢15

𝑢21 𝑢22 𝑢23 𝑢24 𝑢25

𝑢31 𝑢32 𝑢33 𝑢34 𝑢35

𝑢41 𝑢42 𝑢43 𝑢44 𝑢45

𝑢51 𝑢52 𝑢53 𝑢54 𝑢55]

0→𝑖

.


0

(3.49)

By removing the last column as well as the last line from �̃�0→𝑖, we are left with the free equations relating

the DOF’s with the corresponding internal forces and moments, i.e. free of external influences, which is

how to correctly proceed when dealing with DFVA. Being all active coefficients, 𝑢11 to 𝑢44, functions of

the circular frequency 𝜔. By expanding such matrix product result the following four equations:

−𝑤𝑖 = −𝑢11𝑤0 + 𝑢12𝜓0 + 𝑢13𝑀0 + 𝑢14𝑉0 (3.50a)

𝜓𝑖 = −𝑢21𝑤0 + 𝑢22𝜓0 + 𝑢23𝑀0 + 𝑢24𝑉0 (3.47b)

𝑀𝑖 = −𝑢31𝑤0 + 𝑢32𝜓0 + 𝑢33𝑀0 + 𝑢34𝑉0 (3.47c)

𝑉𝑖 = −𝑢41𝑤0 + 𝑢42𝜓0 + 𝑢43𝑀0 + 𝑢44𝑉0 (3.47d)

Imposing the BC’s on these, leads eventually to the formulation of the frequency condition/determinant.

Exempli gratia, let’s assume the beam from Figure 3.2, with a built-in left-side and a released right-side.

Thus, its boundary conditions are:

𝑀𝑖 = 0 𝑉𝑖 = 0 𝑤0 = 0 𝜓𝑖 = 0 (3.51)

By substituting these into equations (3.50c) and (3.50d),

0 = 𝑢33𝑀0 + 𝑢34𝑉0 0 = 𝑢43𝑀0 + 𝑢44𝑉0 (3.52)

For a nontrivial solution of these equations the determinant of the coefficients must be zero, i.e. the

following frequency condition/determinant is obtained

Δ = |𝑢33 𝑢34

𝑢43 𝑢44| = 0 (3.53)

All of the system’s natural frequencies of vibration, 𝜔𝑚, are then computed as roots of the determinant

equation above. This being a classical eigenvalue problem of matrix algebra. Such frequencies, might

result real, imaginary or a combination of both. Real solutions represent frequencies that induce

oscillatory motions, or resonance, in the system. On the other hand, imaginary frequencies produce

exponentially diverging or unstable motions [82]. The latter, however, may be neglected in this work,

since our focus is on undamped systems [82].

As demonstrated before, the “lumped-mass matrix” method is much more conveniently employed than

its alternative, the “consistent-mass matrix” method, but, interestingly enough, the first natural frequency

resulting from a singularly discretized beam is, in general, around 10% below the true value with lumped-

mass, and 10% higher with the consistent-mass [83][69], once more corroborating our previous decision.

22

However, the Lumped-parameter, more than the homologous, leads to results of within a wide degree

of accuracy. Nonetheless, equation (3.53), as a polynomial, offers a highly tractable form for roots

computing, especially when compared to the unwieldy transcendental functions incurred with

distributed-parameter models [69]. Since, discretization of parameters limits the degrees of freedom of

motion, usually, there are only as many frequencies as existing “lumps”. Typically, for a reasonable

number of lumps, the leading frequencies can be found with adequate accuracy, which is fortunate,

given that these are normally those of greatest engineering concern [69].

With the natural frequencies of an elastic system having been found, by means of transfer matrices, it

is an easy matter to compute and visualize the corresponding normal modes’ shapes, 𝛟𝑚. For instance,

the first natural vibration mode 𝛟1 is obtained by substituting in equations (3.50), having 𝑖 as last node

of the system, both, the BC’s in equation (3.51), and, in this case, the first natural frequency of vibration

𝜔1 resultant from (3.53). Having done this, we are left with four equations and four unknowns: 𝑤𝑖, 𝜓𝑖,

𝑀0 and 𝑉0. After solving them, the entire state vectors, at positions 0 and 𝑖, are known. From this point

on, all of the remainder state vectors can be obtained, using either one of the already known ones. For

that, the exact same process just needs to be repeated, this time supressing the BC step, but still

substituting the current natural frequency on all transfer matrices. This is a classical eigenvectors

problem, after which we can build two-dimensional profiles for the corresponding natural vibration modes.

3.2.3.2. Unit Norm Normalization & Orthogonality Property of the Normal Modes

Although commonsensical, it should be noted that, like all vibration modal analysis, the resultant modes

are calculated relative to a prescribed value on one of the system’s DOF, so that all resultant vibration

modes may be “scalarly” relativized to this imposed value, wherefore modal shapes representations are

qualitative in nature and should be normalized beforehand. Ideally, the prescribed value should be

unitary or small enough to avoid disproportionate, but yet visually clear, modes.

The resultant normal modes’ shapes 𝛟1 , 𝛟2 ,…, 𝛟𝑚 , can be checked using a property know as

orthogonality of the normal modes between two modes [6][82][84]. Yet again, for our 2D straight beam

model, let −�̂�𝑖1 and �̂�𝑖1 respectively refer to the normalized DOF’s at node 𝑖 due to the first natural

frequency, and, those due to the second natural frequency, be −�̂�𝑖2 and �̂�𝑖2. Then, the property states:

∑ 𝑚𝑖[(−�̂�𝑖1)(−�̂�𝑖2) + �̂�𝑖1�̂�𝑖2]𝑛𝑖=0 𝜔1

2 = 0 (3.54a)

Where, each normal mode shape vector is,

𝛟𝑚 = [−𝑤0𝑚 𝜓0𝑚 −𝑤1𝑚 𝜓1𝑚 ⋯ −𝑤𝑛𝑚 𝜓𝑛𝑚]𝑇 (3.51b)

The norm (or length) of any mode shape is defined as

‖𝛟𝑚‖ = √(−w0𝑚)2 + (𝜓0𝑚)2 + ⋯+ (−𝑤𝑛𝑚)2 + (𝜓𝑛𝑚)2 (3.55)

Because each mode shape has an arbitrary scaling factor, it can always be normalized towards an

universal unitary norm, this is designated as unit norm normalization or length normalization [84].

Denoting the normalized mode shapes that have a unit norm as �̂�1, �̂�2,…, �̂�𝑚, follows

�̂�𝑚 =𝛟𝑚

‖𝛟𝑚‖= [−

𝑤1𝑚

‖𝛟𝑚‖

𝜓1𝑚

‖𝛟𝑚‖⋯ −

𝑤𝑛𝑚

‖𝛟𝑚‖

𝜓𝑛𝑚

‖𝛟𝑚‖]𝑇

= [−�̂�1𝑚 �̂�1𝑚 ⋯ −�̂�𝑛𝑚 �̂�𝑛𝑚]𝑇 (3.56)

23

This normalization procedure always precedes the orthogonality proof on (3.54a). Moreover, the latter

proof can be efficiently improved by omitting 𝜔12, and by using a proper matrix notation. Therefore, the

new form for an orthogonality proof between the 1st and 2nd normal modes, takes the following shape:

�̂�1𝑇𝐌�̂�2 = 0 ⇔

[ −�̂�01

�̂�01

⋮−�̂�𝑛1

�̂�𝑛1 ] 𝑇

[ 𝑚0 0 0 0 00 𝑚0 0 0 00 0 ⋱ 0 00 0 0 𝑚𝑛 00 0 0 0 𝑚𝑛]

[ −�̂�02

�̂�02

⋮−�̂�12

�̂�12 ]

= 0 (3.57)

Note that, on equation (3.54a), the sum of 𝑚𝑖(−�̂�𝑖1)𝜔12 with 𝑚𝑖�̂�𝑖1𝜔1

2 comprises the normalized

instantaneous inertia force of mass 𝑖 for the first natural mode, therefore this property physically

expresses the fact that the work done by the inertia forces occurring in the first normal mode and moving

through the displacements of the second normal mode is zero [6]. For a valid vice-versa statement,

equation (3.54a) just needs to be multiplied by 𝜔22 instead. This property explains why normal modes

are often described as orthogonal to one another. It goes without saying that this procedure can be done

between any two normal modes, however, our particular MATLAB program only runs a single

orthogonality check between the first two normal modes’ results.

3.2.3.3. Frequency Determinant Function

On a separate note, by looking at the procedure from equations (3.49) to (3.53), the matrix

multiplications carry 𝜔 as a free parameter, and, by applying the BC’s, 𝜔 is determined from the

resulting frequency equation. However, for models with slightly more refined discretization the amount

of algebra grows quite rapidly, and it is easy to imagine that with complicated systems the algebraic

labour may eventually become prohibitive. This is mostly because the frequency equation becomes very

complicated and it is particularly cumbersome to extract the roots out from it. It is then advisable to

replace this algebraic computation by a numerical one [6].

Assuming all field and point matrices are known, after applying the BC’s, we have already seen that the

frequency condition on equation (3.53) is obtained. As explained, had the matrix multiplication been

carried out algebraically, then, coefficients 𝑢33, 𝑢34, 𝑢43, 𝑢44 and the frequency condition itself would

result as complicated functions of 𝜔 . In opposition, the numerical procedure presently proposed

indicates that certain values of 𝜔 should be selected and the corresponding frequency determinants Δ

computed. Then, a discrete representation of function Δ(𝜔) is obtained, from which, it is known that

getting closer to zero implies dangerously subjecting our system to one of its resonating circular natural

frequencies (Figure 3.10a). In engineering, oftentimes the problem is not to accurately find all the natural

frequencies of a system, but to discover whether or not it has any natural frequencies within a certain

frequency range (Figure 3.10b), hence the purpose and convenience of this straight-forward method.

24

Figure 3.10: Plot of frequency determinant as a function of circular frequency (from [6]).

3.2.4. TMM Linear Static Elastic (LSEA) & In-Plane Stress Analysis (SA)

3.2.4.1. TMM Linear Static Elastic Analysis (LSEA)

In the aftermath of having understood the procedure behind a TMM DVFA, it will now be introduced the

slightly simpler TMM LSEA. Which we know, according to subchapter 3.2.1, employs extended elements

and a distributed-parameter discretization technique.

At this point all transfer matrices 𝐔 are assumed to be known, therefore, the first step is to take equation

(3.49) and calculate the state vectors at the LS’s end sides, which are the locations where we have

information available on the BC’s. Thus, by directly introducing these known BC’s on equation (3.49) we

end up with a linear system of equations which allows us to obtain all the parameters at these two

positions. Knowing the initial state vector 𝐳0, all remainder state vectors of the system can be obtained

in an absolutely straightforward fashion, by simply substituting and applying the respective data on

equation (3.49). This two-stepped iterative process literally englobes this entire analysis, and is all that

needs to be done in order to obtain all the state vectors. The results must, however, undergo a cunning

set of post-treatment procedures in order to adequately compose intuitive visualizations/plots.

This methodology’s outputted results, apart from many other things, are used to build efforts’ distribution

diagrams, where forces and moments are directly presented, instead of reactions, as often seen in other

literature. We thought it was important to absolutely clarify this aspect.

As to the TMM SA, it ends up being a kind of freeloader to the LSEA. Since, once all nodal forces and

moments are known, the procedure to be described on the following chapter can immediately be run.

3.2.4.2. In-Plane Stress Analysis (SA)

In this work, since we only approach in-plane stresses, i.e. stresses caused by in-plane loadings, no

torsional shear stresses were considered. It should also be said that, in the following formulas we will

consider one extra nodal degree of freedom, the horizontal displacement 𝐮𝑖. This aspect was supposed

to only be introduced from chapter 3.3 onwards, however it would make no sense to make ourselves go

later on through such monotonous methodological update.

Firstly, at each state vector 𝑖, the following stresses are computed according to [85]: normal stress 𝜎𝑁𝑖

due to longitudinal force 𝑁𝑖 (equation (3.58)), bending stress 𝜎𝑀𝑖 due to bending moment 𝑀𝑖 (equation

(3.59)), and transversal shear stress 𝜏𝑉𝑖 due to transversal shear force 𝑉𝑖 (equation (3.60)).

25

𝜎𝑁𝑖=

𝑁𝑖

𝐴𝑖 (3.58)

𝜎𝑀𝑖=

𝑀𝑖𝑦𝑝𝑖

𝐼𝑧𝑖

(3.59)

𝜏𝑉𝑖=

𝑉𝑖𝑄𝑖

𝐼𝑧𝑖𝑏𝑦𝑖

(3.60)

In which, 𝐴𝑖 is the cross-section area at 𝑖; 𝑦𝑝𝑖 the perpendicular distance, at the cross-section plane,

from the point of study to the neutral axis 𝑧; 𝐼𝑧𝑖 the second moment of area of the cross section about

the neutral axis 𝑧; 𝑄 the first moment of area of the cross section about the neutral axis 𝑥; and 𝑏𝑦𝑖 the

sectional width at the distance 𝑦𝑝 from the neutral axis.

When analysing a rectangularly cross-sectioned LS, as it will effectively happen, all the cross-section’s

critical points must be first identified. In this respect, it should be known that normal stresses have an

evenly distributed stress across the entire section, as to the remainder two types of stresses, given that

their distribution patterns are not as simple, these are educatively illustrated on Figure 3.11.

Figure 3.11: Bending and transversal shear stresses in a rectangular beam (from [85]).

In sum, as one moves away from the neutral axis, the shear stress decreases parabolically until it

reaches zero at the outer surfaces, where 𝑦1 = ±𝑐 (Figure 3.11). It is particularly interesting to observe

that maximum shear stress occurs at the neutral axis, i.e. where bending stress is null, and vice-versa

at the outer surfaces. Most importantly is to know that the bending stress is by far the greatest stress,

hence all critical points may be situated in either one of the outer surfaces. In fact, there is a popular

rule of thumb in engineering stating that if a beam is slender, i.e. when the beam’s length divided by the

biggest of its two other dimensions is higher than 10, the transverse shear is negligible.

However, for a rectangular cross-section, the following expression may be deduced:

𝐴𝑖 = 𝑏𝑖ℎ𝑖 𝑦𝑝𝑖=

ℎ𝑖

2 𝐼𝑧𝑖

=𝑏𝑖ℎ𝑖

3

12 𝑄𝑖 = ∫ 𝑦 𝑑𝐴 = 𝑏𝑖 ∫ 𝑦 𝑑𝑦

𝑐𝑖

𝑦1𝑖

=𝑏𝑖

2(𝑐𝑖

2 − 𝑦1𝑖

2 )𝑐𝑖

𝑦1𝑖

(3.61)

𝜏𝑉𝑖=

𝑉𝑖

2𝐼𝑥𝑖

(𝑐𝑖2 − 𝑦1𝑖

2 ) =3𝑉𝑖

2𝐴𝑖(1 −

𝑦1𝑖2

𝑐𝑖2 ) ⇒ 𝜏𝑉𝑖𝑚𝑎𝑥

= 𝜏𝑉𝑖|𝑦1𝑖

=0=

3𝑉𝑖

2𝐴𝑖 (3.62)

Next, an appropriate failure criterion is selected, according to which the equivalent stress 𝜎 is computed.

There are two major theories to choose from: Maximum-Shear-Stress Theory for Ductile Materials (MSS

or Tresca Theory) and Distortion-Energy Theory for Ductile Materials (or Von-Mises Theory).

For the first one, yielding/failure is assumed to occur when the material’s maximum shear stress (MSS)

𝜏𝑚𝑎𝑥 equals the MSS at yielding in a simple uniaxial tension test 𝜏𝑦, where 𝜏𝑚𝑎𝑥 = (𝜎𝑚𝑎𝑥 − 𝜎𝑚𝑖𝑛) 2⁄ .

Thus, knowing Plane Stress (𝜎𝑧 = 0) is assumed, and the principal stresses are 𝜎1 ≥ 𝜎2 ≥ 𝜎3, we get

26

Therefore, the general form of the Tresca Criterion for yielding is written as,

𝜎1 − 𝜎3 ≥ 𝑆𝑦 or 𝜏𝑚𝑎𝑥 =𝜎1−𝜎3

2≥

𝑆𝑦

2 (3.64)

𝑆𝑦 denoting the material’s yield strength. Then, by considering the project’s safety factor 𝑛𝑠, results

𝜏𝑚𝑎𝑥 =𝜎1−𝜎3

2𝑛𝑠≥

𝑆𝑦

2𝑛𝑠 ⇒ 𝜎𝑇𝑟𝑒𝑠𝑐𝑎 = 2𝜏𝑚𝑎𝑥 = 2√(

𝜎

2)

2

+ 𝜏2 = √𝜎2 + 4𝜏2 ≥𝑆𝑦

𝑛𝑠 ⟹

Which, upon rearrangement, provides the general formula for the equivalent stress according to Tresca

Yield Criterion. For academical purposes, all possible directional stresses are there included, inclusively,

the torsional shear stress 𝜏𝑇 . Also, 𝜎√𝑀𝑦

2+𝑀𝑧2

designates a bi-directional bending stress, whose

applicability is strictly recommended on Tresca equivalent stress’ expression.

⟹ 𝜎𝑇𝑟𝑒𝑠𝑐𝑎 = √𝜎2 + 4𝜏2 = √(𝜎𝑁 + 𝜎√𝑀𝑦

2+𝑀𝑧2)

2

+ 4(𝜏𝑉𝑦

2 + 𝜏𝑉𝑧

2 + 𝜏𝑇2) ≥

𝑆𝑦

𝑛𝑠 (3.65)

Where

𝜎√𝑀𝑦

2+𝑀𝑧2

=√𝑀𝑦

2+𝑀𝑧2.𝑦𝑝

𝐼𝑧 and 𝜏𝑇 =

𝑇𝑐

𝐽 (3.66)

𝑇 being the applied torque, 𝑐 the distance from the center of the cross-section until the outer fiber, and

𝐽 the polar area moment of inertia of the cross-section (𝐽 = 𝐼𝑧 + 𝐼𝑦).

Now going to the second theory above presented, this states that yielding or failure occurs when the

maximum distortion/shear energy in the material 𝑊𝑑,𝑚𝑎𝑥 equals the maximum distortion/shear energy

at yielding in a simple uniaxial tension test 𝑊𝑑,𝑦 [85].

𝑊𝑑,𝑚𝑎𝑥 =1

12𝐺[(𝜎𝑥𝑥 − 𝜎𝑦𝑦)

2+ (𝜎𝑦𝑦 − 𝜎𝑧𝑧)

2+ (𝜎𝑧𝑧 − 𝜎𝑥𝑥)

2 + 6(𝜏𝑥𝑦2 + 𝜏𝑦𝑧

2 + 𝜏𝑧𝑥2 )] (3.67)

𝑊𝑑,𝑦 =1

6𝐺𝑆𝑦

2 (3.68)

Where, 𝐺 is the material’s shear modulus. From these, the general form of the Von Mises Criterion for

yielding is:

1

√2[(𝜎𝑥𝑥 − 𝜎𝑦𝑦)

2+ (𝜎𝑦𝑦 − 𝜎𝑧𝑧)

2+ (𝜎𝑧𝑧 − 𝜎𝑥𝑥)

2 + 6(𝜏𝑥𝑦2 + 𝜏𝑦𝑧

2 + 𝜏𝑧𝑥2 )]

1 2⁄

= 𝑆𝑦 (3.69)

⇒ {𝜎1 = 𝜎𝑌

𝜎3 = 𝜎2 = 0 (3.63)

Figure 3.12: Mohr's circle for a typical uniaxial

tension test (from [86]).

27

From where we derive the actual formula of the equivalent Von Mises stress, 𝜎𝑉𝑀, which, after already

including the respective project’s safety factor 𝑛𝑠, goes as follows

�̅�𝑉𝑀 = √𝜎2 + 3𝜏2 = √1

2[(𝜎𝑁 − 𝜎𝑀𝑦

)2

+ (𝜎𝑀𝑦− 𝜎𝑀𝑧

)2

+ (𝜎𝑀𝑧− 𝜎𝑁)

2] + 3 (𝜏𝑉𝑦

2 + 𝜏𝑉𝑧2 + 𝜏𝑇

2) ≥𝑆𝑦

𝑛𝑠 (3.70)

Regarding both alternative equivalent stresses, it is known that the latter gives us a more precise

estimation of the necessary stress level for yielding to occur on ductile materials. However, although it

is true that the first is less accurate, this has the advantage of always supplying conservative

estimations. Hence, if followed, it assures safer conditions. Therefore, while draft-designing, it may be

a good choice to opt for the first theory, as it will provide an easier and safer solution. Nonetheless, if

some level of precision is required, then Von Mises Theory is recommended. In this work, the latter was

the one computationally implemented. Every comparative statement discussed can be traced down on

Figure 3.13 (left), although it is more easily understood if we consider planar stress conditions, so to

have a simpler 2D stress diagram, Figure 3.13 (right). In this, Tresca theory is represented by a dotted

line, while Von Mises by a solid one. Each of these lines delimit the corresponding criteria’s theoretically

safe areas, i.e. any combination of stresses there located does not induce yielding. Through such figure,

being Von-Mises criterion more accurate, it is easy to verify that it lets us be more dangerously closer

to the risk, leading to less conservative design decisions. Nonetheless, given a proper project’s safety

factor, there is no real risk in employing either of the criteria.

Figure 3.13: Representation of Tresca and Von Mises Criteria in both the principal stresses’ 3D space (left) and in

the (𝜎1, 𝜎2) plane (right) (respectively, from [87] and [85]).

After having calculated the three kinds of stresses considered for every single discretized element on

the system, plus a forth one which is the equivalent stress value, we are then able to plot a wide variety

charts on these stresses’ distributions, consequently, identifying the mechanism’s critical point.

Ultimately, it’s for this particular critical point that, the safety factor under static loading conditions is

obtained, by applying equations (3.65) or (3.70), in accordance to whichever failure theory is adopted.

Our MATLAB program, is only able to output the value of a safety factor under static conditions, which

gives us a rather loose quantitative measure of safety. In order to evaluate the true safety, we should

instead look at the fatigue safety factor, since our mechanism operates under continuous cyclic loading

in time. Unfortunately, this parameter cannot be computationally outputted due to intricacies associated

to the necessary collection of data from a series of literature tables, being these parameters highly

dependent on geometrical and physical factors, hence it is not implementable on our TMM.

28

3.3. Novel Aspects of the Implemented TMM

Transparently speaking, the so far explained methodology represented essential, but also admittedly

perfunctory, knowledge on TMM. All based on rather solid literature, notably, [6], [69], [84] and [85].

However, in seeking to achieve a hand-tailored methodology to help design our suspension, some

breaching adjustments to the conventional TMM methodology, previously presented in subchapter 3.2,

need to be craftily forged. It is on these small, but quite hard to implement, tweaks to the system, that

relies a significant share of this thesis’ academic value.

In the following subchapters, all issues addressed are particularities of our originally developed

implementation, i.e. either added improvements or considerable methodological deviations to the

procedures earlier presented. Most of these accruing from the author’s self-intuition on shallowly

referenced issues, which, upon onerous trial and error, were successfully merged into the methodology.

In this respect, the author’s academic claims proudly dwell on this set of novel implemented features.

This “new” TMM methodology is indeed, as expected, considerably more complex.

3.3.1. Transfer Matrix Method vs. Finite Element Method

Although algebraically similar, TMM is completely different than other commonly taught and applied

matrix methods, such as the hugely popular FEM. TMM opts for a non-federal method, where, a vector

with one node’s displacements and internal forces, is then multiplied by the elemental transfer matrices

comprising the chain-like bridge between the initial point and the final point to be computed. After such

operation, a topologically equal column vector portraying the final point is obtained. In opposition, FEM

stacks together all nodal displacements into a (n x 1) column vector, which is then multiplied by a huge

(n x n) global stiffness matrix, ensuing from a carefully orchestrated assemblage of the different

elemental stiffness matrices. The result of such multiplication, is a (n x 1) column vector of all

corresponding nodal internal forces (see equation (3.82)).

Regarding efficiency, this methodology ideally fits our purposes, since only successive matrix

multiplications are necessary to fit the elements together, while intermediate conditions and the number

of degrees of freedom present no difficulty since they have no effect on the order of the transfer matrices

required; in fact, their size is dependent only on the order of the differential equations governing the

behaviour of the elements of the system [6].

To conclude, assuming systems become more complex, while FEM deals with increasingly complex

systems by equally increasing the size of the elements under operation, the other commits on keeping

the same original-sized matrix making instead more sequential operations until all discretized elements

have been traversed. The TMM methodology, although being a much less studied and inhospitable

research matter, computationally speaking is tremendously more efficient and has a larger range of

application. Being particularly apt for studying vibration of structures and both rigid and rigid-flexible

multibody systems, since differential equations don’t need to be established [65][88].

3.3.2. Convention Adjustment & New State Vector Topology

Therefore, as shown from equations (3.9) to (3.39), any field or point transfer matrix in TMM might be

deduced only recurring to equilibrium and elementary beam theory equations, which are then disposed

29

in order to fit whichever TMM’s matrix notation we opt to define. The state vector’s form now chosen to

rule the “new” TMM implementation, as well as our particular MATLAB program. This is slightly more

complex than the one used to explain the methodology in subchapter 3.2 (equation (3.1)), since one

more degree of freedom and its corresponding internal force was added to the mix (equation (3.71)),

obviously affecting the overall methodology’s complexity level. Moreover, the state vector itself will also

be improved to become more intuitive, namely, by dropping the negative sign on the vertical

displacement and by re-organising the vector’s internal forces and moment in order to follow the same

order as their homologous displacements and rotational degrees of freedom. Resulting in the following

generic state vector 𝑖,

𝐳𝑖 =

[ 𝑢𝑖

𝑤𝑖

𝜓𝑖

𝑁𝑖

𝑉𝑖

𝑀𝑖]

(3.71)

Where 𝑢 stands for extension/longitudinal displacement in millimetres, 𝑤 for deflection/transversal

displacement in millimetres, 𝜓 for curvature/slope in radians, 𝑁 for normal/longitudinal force in newtons,

𝑉 for shear/transversal force in newtons, and 𝑀 for bending moment in newton-metres.

On the other hand, for intuitive purposes as well, the convention ruling each kind of displacement and

internal force’s positive directions on a generic two node element, shown in Figure 3.6, will also be

changed, as, actually, any kind of convention can be used as long as it is indeed thoroughly followed.

In this respect, the stipulated positive directions for the displacements will be the same on both left and

right nodes, then, the direction for both transversal displacements, longitudinal displacements and

curvatures are, respectively, upwards, rightwards and anti-clockwise. However, for the internal forces,

the same convention as in Figure 3.6 is kept. This is done for equilibrium purposes. Wherefore, the

positive direction of the internal forces associated to each one of the degrees of freedom is the same as

the later on the right node, but opposite for the left one (which is convenient for aesthetic reasons, for

example, in this way all simple point matrices are just identity matrices).

3.3.3. Derivation of Field Transfer Matrices from an Exact Arch-Element

Stiffness Matrix

Concurrently to the previous modifications, one more change will also be made, but, much less

inconsequent, since, with it we will take a huge leap forward in designing a better fit elemental geometry.

Knowing from the mechanism synthetization that this will be composed by concave-shaped thin leaf-

springs, then, we will audaciously formulate all TMM analysis using discretized arch field elements. This

will bring tremendous advantages in terms of processing time and accuracy, as we can get better

accuracy using a less refined discretization. The new TMM element is represented on Figure 3.14 (right).

Additionally, since we live in an open source engineering world where information on stiffness matrices

of all sorts are constantly being researched about, updated and are readily available, then, we

capacitated our MATLAB program to be able to receive as an input not only transfer matrices according

to TMM rules, but also any stiffness matrix, on which it applies a shifting procedure so to transform it

into its corresponding equivalent transfer matrix, already properly adapted to our particular convention.

30

Yet again, knowing that the synthesised design of our suspension’s most important structural element

resembles an arch-shaped leaf-spring, with this methodology we can discretized it into several better-fit

curved beam elements of constant curvature. In this regard, quite interesting literature was found on the

development of curved beam elements, some of them based on exact strain energy or nature shape

functions, others, based on assumed displacement fields.

Thus, we carefully chose the outstanding work from Litewka & Rakowski [89], who derived the exact

stiffness matrix of a curved beam element with constant curvature. A planar two-node with six degrees

of freedom element was considered, Figure 3.14 (left), in which the effect of flexural, axial and shear

deformations were taken into account. Therein, the analytical shape functions describing radial and

tangential displacements as well as cross section rotations are given in algebraic trigonometric form.

They include the coupled influences of shear and membrane effects, being an element completely free

of shear and membrane locking effects. Although shearing effects are known to only start to be

significant for thicker and short beams [69], while membrane effects for arches with small heights [89],

it is rather reassuring to consider them in our work.

Figure 3.14: Arch element considered by Litewka & Rakowski (left). Elemental model adopted in our newly

developed TMM (right) (both adapted from [89]).

Based on the shape functions developed in [89], the strain energy formula was then used to obtain the

stiffness matrix 𝐊 for shear flexible and compressible arch element is formulated, being expressed as

shown immediately below (surprisingly, the original references had more than a few typos, which we

corrected). This formulation guarantees results that coincide exactly with the analytical ones obtained

for continuous arches [89][90].

𝐊 =𝐸𝐼

𝑎𝑙3 [𝑘𝑖𝑗], 𝑘𝑖𝑗 = 𝑘𝑗𝑖 (3.72)

𝑘11 = 𝛼4(𝐷′ + 𝐷′′ cos 𝛼); 𝑘12 = 𝛼4𝐷′′ sin 𝛼 ; 𝑘13 = 𝑎𝑙 [−𝛼3(𝐷′ + 𝐷′′ cos 𝛼) +2

𝐷1𝛼2 sin 𝛼] ; (3.73a)

𝑘14 = −𝛼4(𝐷′′ + 𝐷′ cos 𝛼); 𝑘15 = 𝛼4𝐷′ sin 𝛼 ; 𝑘16 = 𝑎𝑙 [𝛼3(𝐷′′ + 𝐷′ cos 𝛼) −

2

𝐷1𝛼2 sin 𝛼] ; (3.70b)

𝑘22 = 𝛼4(𝐷′ − 𝐷′′ cos 𝛼); 𝑘23 = 𝑎𝑙 [−𝛼3𝐷′′ sin 𝛼 +2

𝐷1𝛼2(1 − cos 𝛼)] ; 𝑘24 = −𝑘15; (3.70c)

𝑘25 = −𝛼4(𝐷′ cos 𝛼 − 𝐷′′); 𝑘26 = 𝑎𝑙 [𝛼3𝐷′ sin 𝛼 −

2

𝐷1𝛼2(1 − cos 𝛼)] ; (3.70d)

𝑘33 = 𝑎𝑙2 [𝛼2(𝐷′ + 𝐷′′ cos 𝛼) + 1 +

4

𝐷1(1 − cos 𝛼 − 𝛼 sin 𝛼)] ; 𝑘34 = 𝑘16; 𝑘35 = −𝑘26; (3.70e)

𝑘36 = 𝑎𝑙2 [−𝛼2(𝐷′ cos 𝛼 + 𝐷′′) − 1 −

4

𝐷1(1 − cos 𝛼 − 𝛼 sin 𝛼)] ; 𝑘44 = 𝑘11; 𝑘45 = −𝑘12; (3.70f)

𝑘46 = 𝑘13; 𝑘55 = 𝑘22; 𝑘56 = −𝑘23; 𝑘66 = 𝑘33 (3.70g)

Where,

31

𝐷1 = 𝛼(𝛼 + sin 𝛼)(1 + 𝑑1 + 𝑒1) − 2[2(1 − cos𝛼) + 𝛼𝑑1 sin 𝛼] (3.74)

𝐷2 = 𝛼(𝛼 + sin 𝛼)(1 + 𝑑1 + 𝑒1) − 2𝛼 sin 𝛼 (1 + 𝑒1) (3.75)

𝐷′ =1

𝐷1+

1

𝐷2 𝐷′′ =

1

𝐷1−

1

𝐷2 (3.76)

𝑑1 = 𝛼2𝑑𝑑 𝑒1 = 𝛼2𝑒𝑑 (3.77)

𝑑𝑑 =12𝐸𝐼

𝜅𝑠𝐺𝐴𝑎𝑙2 =

2(1+𝜈)ℎ2

𝜅𝑠𝑎𝑙2 𝑒𝑑 =

𝐸𝐼

𝐸𝐴

1

𝑎𝑙2 (3.78)

𝜅𝑠 =𝜏𝑎𝑣

𝜏=

𝑉 𝐴⁄

𝐺𝛾=

1

𝛼𝑠 (3.79)

Most of these symbols are either clearly illustrated on Figure 3.14, or were already characterized before,

or are just an artifact for simplification purposes, having no physical meaning. Knowing this: 𝑅 is the

elemental curvature radius in millimetres, 𝛼 the elemental curvature angle in radians, 𝑎𝑙 is the elemental

length (𝑎𝑙 = 𝑅𝛼) , 𝐸𝐼 the flexural stiffness, 𝐺𝐴 the shear stiffness, 𝐸𝐴 the axial stiffness, E the

elastic/Young’s modulus in MPa, 𝐺 the shear/Kirchhoff’s modulus in MPa, 𝐼 the second moment of

area/area moment of inertia in mm4, 𝐴 the cross-section area in mm2, 𝑑𝑑 and 𝑒𝑑 the dimensionless

parameters that characterize, respectively, the shear and membrane effects in the element (see [91]),

𝜅𝑠 the dimensionless shear form factor that depends on the cross-sectional area shape, 𝜏 the shear

stress at the centroid of the cross section, 𝜏𝑎𝑣 the average 𝜏, 𝑉 the shear force, 𝛾 the shear strain, and,

finally, 𝛼𝑠 the shear correction factor, which, according to Walter D. Pilkey’s amazing book [69], is more

accurately determined when applying the theory of elasticity, leading to the following equation for a solid

rectangular cross section, where 𝜈 is the Poisson’s coefficient,

𝛼𝑠 =12+11𝜈

10(1+𝜈) (3.80)

It is worth noting that setting 𝑑𝑑 = 0 means neglecting of the shear locking effect, while setting 𝑒𝑑 = 0

neglects the compressibility (membrane effect). For more information on the shear locking phenomenon

in beam elements and its interpretation the reader is suggested to read reference [91].

Purely due to organization purposes, we will now address one aspect of our newly derived stiffness

matrix’s formulation, which, under normal circumstances, should only be referred later on in this work.

This is because, further on, one of the design solutions we intended to propose for our suspension’s

leaf-springs, is to use more than one material per leaf-spring in a multi-layered manner. However,

ultimately, such proposal is not undergone. Nonetheless, we would like to demonstrate how to

compute/update our stiffness matrix to conceive such manoeuvre. This was studied by Muhaisin M. H.

[92], who derived a new form of the stiffness matrix for a reinforced concrete plane frame element

including the effect of shear deformations, but in doing so he took a layered element approach in order

to consider variation of material properties through the thickness, obtaining with this, quite good results

when compared with other authors. Sadjad A. Hemzah [93], in turn took Muhaisin’s methodology and

adapted it to arch beam finite elements. In sum, we consider three assumptions: the element is

subdivided into a set number of layers 𝑛𝑖 along its depth as shown in Figure 3.15, all plane sections

before bending remain plane after bending, and the stress in each layer is assumed to be uniform along

the depth of the layer.

32

Figure 3.15: Section discretization into layers (from [93]).

Having done this, we may now introduce the updated multi-layered stiffness matrix 𝐊 for each element

according to [92][93], where 𝑏𝑖 and 𝑡𝑖 are, respectively, each layer width and thickness.

𝐊 = ∫ ∑ 𝑏𝑖ℎ𝑖𝑛𝑖𝑖=1

[ 𝑘11 𝑘12 𝑘13 𝑘14 𝑘15 𝑘16

𝑘22 𝑘23 𝑘24 𝑘25 𝑘26

𝑘33 𝑘34 𝑘35 𝑘36

𝑘44 𝑘45 𝑘46

𝑘55 𝑘56

𝑠𝑦𝑚. 𝑘66]

𝑑𝑥𝐿

0 (3.81)

The following step on the overall methodology is now to take the stiffness matrix 𝐊 we formulated up

until this point, and convert it into a massless field transfer matrix 𝐅. To do so, first one needs to seek

for any disagreement between adopted conventions, which in our case do exist between ours (Figure

3.14 (right)) and the convention adopted by Litewaka & Rakowski [89]. Using as an example a single

element’s regular procedure, between two nodes 1 and 2, such differences clearly stands out. We incite

the reader to compare both equivalent situations herein presented on equation (3.82), where Litewaka

and Rakowski’s [89] finite element convention is defined on the left, while our TMM one is on the right.

The negative signals used on the internal forces of node 1, are there to indicate the difference in terms

of positive direction stipulated at [89] relative to the one adopted for this TMM implementation.

[ (−)𝑁1

(−)𝑉1

(−)𝑀1

𝑁2

𝑉2

𝑀2 ]

= 𝐊𝑖

[ 𝑢1

𝑤1

𝜓1

𝑢2

𝑤2

𝜓2]

⟺⏞?

[ 𝑢2

𝑤2

𝜓2

𝑁2

𝑉2

𝑀2]

= 𝐅1→2

[ 𝑢1

𝑤1

𝜓1

𝑁1

𝑉1

𝑀1]

(3.82)

In order for the transformation procedure herein presented to be successful, the first three rows of the

stiffness matrix 𝐊 need to be multiplied by (−1) to be in accordance to the intended new convention.

Actually, we were accidently lucky that all the remaining stipulated positive directions and vector

positions of either of the node’s displacements and internal forces all came out already in the correct

way after we symmetrized the element in Figure 3.14 (left) from [89] into the concave shape needed

(Figure 3.14 (right)), otherwise the procedure would involve changing rows, columns or inversions of

signals. In sum, for the situation in hands, the new non-symmetrical stiffness matrix’s is

𝐊 = ∫ ∑ 𝑏𝑖𝑡𝑖𝑛𝑖𝑖=1

[ −𝑘11 −𝑘12 −𝑘13 −𝑘14 −𝑘15 −𝑘16

−𝑘21 −𝑘22 −𝑘23 −𝑘24 −𝑘25 −𝑘26

−𝑘31 −𝑘32 −𝑘33 −𝑘34 −𝑘35 −𝑘36

𝑘41 𝑘42 𝑘43 𝑘44 𝑘45 𝑘46

𝑘51 𝑘52 𝑘53 𝑘54 𝑘55 𝑘56

𝑘61 𝑘62 𝑘63 𝑘64 𝑘65 k66 ]

𝑑𝑥𝐿

0 (3.83)

33

The final steps in deriving 𝐅 from 𝐊 consist of a matrix partitioning followed by a re-assemblage. This

methodology is found on Pestel and Leckie [6], where the square matrix on equation (3.83) (with all

operations already included on the 𝑘𝑖𝑗 coefficients), is first partitioned as shown below in equation (3.84),

which is also done simultaneously for both displacements and nodal forces column vectors, 𝐝 and 𝐩,

uniquely for a better understanding of the line of thought behind this procedure. Taking equation (3.82)

(left) proceed as follows

[

𝐩𝑖−1

⋯𝐩𝑖

] = [𝐀 ⋮ 𝐁⋯ ⋮ ⋯𝐂 ⋮ 𝐃

]

𝑖

[𝐝𝑖−1

⋯𝐝𝑖

] (3.84)

Expanding equation (3.84) yields

𝐩𝑖−1 = 𝐀𝑖𝐝𝑖−1 + 𝐁𝑖𝐝𝑖 and 𝐩𝑖 = 𝐂𝑖𝐝𝑖−1 + 𝐃𝑖𝐝𝑖 (3.85)

Solving for 𝐝𝑖 from the first of these equations, we obtain

𝐝𝑖 = −𝐁𝑖−1𝐀i𝐝𝑖−1 + 𝐁𝑖

−1𝐩𝑖−1 (3.86)

This expression is substituted in the second equation to give

𝐩𝑖 = (𝐂 − 𝐃𝐁−1𝐀)𝑖𝐝𝑖−1 + 𝐃𝑖𝐁𝑖−1𝐩𝑖−1 (3.87)

Equations (3.86) and (3.87) are combined into a single equation to give

[𝐝⋯𝐩

]

𝑖

= [−𝐁−1𝐀 ⋮ 𝐁−1

⋯ ⋮ ⋯𝐂 − 𝐃𝐁−1𝐀 ⋮ 𝐃𝐁−1

]

i

[𝐝⋯𝐩

]

𝑖−1

⇔ 𝐳𝑖 = 𝐅(𝑖−1)→𝑖𝐳𝑖−1 (3.88)

From the process just shown, results the final massless and un-extended field transfer matrix used on

our MATLAB program to define each discretized field of the leaf-springs.

3.3.4. Profile Data Input and Pre-processing Treatment

The particular data relative to the profile shape of the system being studied goes through some crucial

steps, which are relevant to be known. In fact, regarding the shape of the intended LS to be studied, it

can literally follow any kind of function at all. In the MATLAB program developed, the input information

can be any number of points’ 𝑥𝑦 coordinates, which will then be interpolated using a piecewise cubic

Hermite polynomial numeric function. In this, the first and last points given will always be the initial and

final extremities of the system. Right there, we have a routine computing the total length of the leaf-

spring too. It goes without saying that, the more points are supplied the better will fit the interpolation to

the intended shape for the leaf-spring.

Then, another input is needed, which is the number of discrete sections in which the leaf-spring will be

discretized into. This value will immediately trigger a run-down along the previously interpolated function,

from where are defined all of the nodes’ initial 𝑥𝑦 coordinates. This is done so that the discretization is

made with equally lengthened sections. Actually, a little more than this is done. This is explained by the

fact that every single field was defined as an arch element, therefore, we also need information relative

to the radius and angle of curvature of every single discretized element, hence further explanation needs

to be added.

34

To compute the different radii of curvature, we look at each case individually and determine the position

and size of a circle that goes through the field’s two extreme points in such a way that it approximately

superimposes with the intended curvature for that field according to the piecewise cubic Hermite

interpolating polynomial function. This procedure requires three points per field, thus, when doing the

previously described “run-down”, one more point, sensibly located between the two discretizing ones, is

also taken out from the interpolating data for every section. After having three points 𝐴, 𝐵 and 𝐶, and

the resultant slopes 𝑚𝐴𝐵 and 𝑚𝐵𝐶, the following two formulas for the exact position of the circle’s center

𝐷, based on finding the intersection of the perpendicular bisectors of the line segments 𝐴𝐵 and 𝐵𝐶

(Figure 3.16), can be written as

𝑥𝐷 =𝑚𝐴𝐵𝑚𝐵𝐶(𝑦𝐴−𝑦𝐶)+𝑚𝐵𝐶(𝑥𝐴+𝑥𝐵)−𝑚𝐴𝐵(𝑥𝐵+𝑥𝐶)

2(𝑚𝐵𝐶−𝑚𝐴𝐵) (3.89a)

𝑦𝐷 = −1

𝑚𝐴𝐵[𝑥𝐷 − (

𝑥𝐴+𝑥𝐵

2) + (

𝑦𝐴+𝑦𝐵

2)] (3.86b)

Where,

𝐴(𝑥𝐴, 𝑦𝐴) 𝐵(𝑥𝐵 , 𝑦𝐵) 𝐶(𝑥𝐶 , 𝑦𝐶) 𝑚𝐼𝐽 =𝑦𝐽−𝑦𝐼

𝑥𝐽−𝑥𝐼 (3.86c)

Being the radius of curvature, for example from point 𝐴,

𝑅 = √(𝑥𝐷 − 𝑥𝐴)2 + (𝑦𝐷 − 𝑦𝐴)2 (3.90)

Since each arch element’s length 𝑎𝑙 is already known at this point, it is

easy to calculate the corresponding angle of curvature 𝛼, using

𝛼 =𝑎𝑙

R (3.91)

3.3.5. Model Update According to Allowed In-Plane Loading

Two important things need to be restated, about the approach we’ve previously decided to follow on our

MATLAB implementation. One, is that we chose to formulate massless field and point transfer matrices.

This is an inconsequent decision, because the mass will still be considered, the only difference is that it

is added a posteriori to the transfer matrices formulation. This is perhaps counterintuitive, but in fact this

approach proves to be extremely versatile and useful organization-wise. Why? Yet again, because,

when running undamped free vibrations dynamic analysis, we are then able to directly use the already

derived massless field transfer matrices, taking in this case a lumped-mass approach simply by adding

a discontinuity in the shear to the massless point transfer matrix, which is induced by the intrinsic inertia

force associated to the concentrated vibrating masses, as in equation (3.42).

As to the second, it is also related to the previous one in the sense that, it is related to the way this “a

posteriori” mass addition is made. We opted to go for an approach using extended elements. The

possible alternatives, implications and motivations behind this decision were all already thoroughly

explained in sub-chapter 3.2.

Regarding this matter, we will also take this chance to enhance our MATLAB implemented model’s range

of allowed in-plane loading. This improvement will be derived from Walter D. Pilkey’s work [69], whose

computed formulas enable us to consider both longitudinal and transversal distributed forces 𝑞𝑥 and 𝑞𝑦,

Figure 3.16: Intersection at D

of both line segments’

perpendicular bisectors (from

[94]).

35

as well as a distributed moment per unit length 𝑞𝑚 , this for every single discretized arch element

composing the system. This derivation, however, required a considerable number of modifications due

to differing conventions adopted, which won’t be detailed. The final newly introduced extended

distributed in-plane loading column vector �̃�𝑞 results from the operation on equation (3.92). This vector

final expression was not described in equation (3.92) though, for the simple reason that such expression

would occupy a lot of space and it is not worthy since we will still perform further improvements to it later

on, at such time it will then be properly presented. The new TMM element representation is illustrated

on Figure 3.17. Obviously, punctual forces or moments can also easily be added at any discretized node

as shown already in equation (3.40).

Figure 3.17: Curved arch-like element subject to in-plane loading (adapted from [69]).

The new extended distributed in-plane loading column vector �̃�𝑞 will be positioned in the extended field

transfer matrix �̃� as proceeded in equation (3.47), or, imagining a generic single-transfer extended

transfer matrix �̃� with coefficients 𝑢𝑖𝑗 (𝑖, 𝑗 = 1,… ,7) as in equation (3.49), representing a transfer over a

field element, then it would be equivalent to state that

�̃�𝑞 =

[ 𝑢17

𝑢27

𝑢37

𝑢47

𝑢57

𝑢67

𝑢77]

=

[ 𝐹𝑢

𝐹𝑤

𝐹𝜓

𝐹𝑁

𝐹𝑉

𝐹𝑀

1 ]

= �̃�𝑞𝐟𝑞 (3.92)

Where,

�̃�𝑞 =

[ cos 𝛼 sin 𝛼 −𝑅(1 − cos 𝛼)

𝑅

𝐸𝐴𝜆4 +

𝑅3

𝐸𝐼∗𝜆3

𝑅

2𝐴𝐸𝛼 sin 𝛼 −

𝑅3

𝐸𝐼∗2𝜆2

𝑅2

𝐸𝐼∗(𝛼 − sin 𝛼) 0

− sin 𝛼 cos 𝛼 −𝑅𝑠𝑖𝑛 𝛼 −𝑅

2𝐴𝐸𝛼 sin 𝛼 +

𝑅3

𝐸𝐼∗2𝜆2 (

1

𝐸𝐴+

𝑅2

𝐸𝐼∗) 𝑅𝜆1

𝑅2

𝐸𝐼∗(1 − cos 𝛼) 0

0 0 1 −𝑅2

𝐸𝐼∗(𝛼 − sin 𝛼)

𝑅2

𝐸𝐼∗(1 − cos 𝛼) −

𝑅

𝐸𝐼∗𝛼 0

0 0 0 cos 𝛼 sin 𝛼 0 00 0 0 − sin 𝛼 cos𝛼 0 00 0 0 𝑅(1 − cos𝛼) −𝑅 sin 𝛼 1 00 0 0 0 0 0 1]

(3.93a)

36

𝐟𝑞 = 𝑅 ∫

(

[ −𝑅 [𝑞𝑥 (

1

𝐸𝐴𝜆4 +

1

𝐸𝐼∗ 𝑅2𝜆3) −𝑞𝑚𝑅

𝐸𝐼∗(𝛽 − sin 𝛽) + 𝑞𝑦 (

𝛽

2𝐸𝐴sin 𝛽 −

1

𝐸𝐼∗ 𝑅2𝜆2)]

𝑅 [𝑞𝑥 (𝛽


1

𝐸𝐼∗ 𝑅2𝜆2) +𝑞𝑚𝑅

𝐸𝐼∗(1 − cos𝛽) + 𝑞𝑦𝜆1 (

𝛽

𝐸𝐴+

𝑅2

𝐸𝐼∗)]

𝑅 [𝑞𝑥𝑅

𝐸𝐼∗(𝛽 − sin 𝛽) −

𝑞𝑚𝛽

𝐸𝐼∗−

𝑞𝑦𝑅

𝐸𝐼∗(1 − cos 𝛽)]

−𝑞𝑥 cos𝛽 − 𝑞𝑦 sin 𝛽

𝑞𝑥 sin 𝛽 − 𝑞𝑦 cos 𝛽

𝑞𝑥𝑅(1 − cos 𝛽) − 𝑞𝑚 − 𝑞𝑦𝑅 sin 𝛽 ]

)

𝑑𝛽𝛼

0

=

=

[ −

𝑅2

2𝐸𝐴𝐼∗[2𝐴𝛼𝑅2(𝑞𝑥𝜆3 − 𝑞𝑦𝜆2) − 𝐴𝑞𝑚𝑅(2 cos 𝛼 + 𝛼2 − 2) − 𝐼∗(𝛼𝑞𝑦 cos𝛼 − 𝑞𝑦 sin 𝛼 − 2𝛼𝑞𝑥𝜆4)]

−𝑅2

2𝐸𝐴𝐼∗[2𝐴𝛼𝑅2(𝑞𝑥𝜆2 − 𝑞𝑦𝜆1) + 2𝐴𝑞𝑚𝑅(sin 𝛼 − 𝛼) + 𝐼∗(𝛼𝑞𝑥 cos𝛼 − 𝑞𝑥 sin 𝛼 − 𝛼2𝑞𝑦𝜆1)]

𝑅2

2𝐸𝐼∗[𝑅(2𝑞𝑥 cos𝛼 + 2𝑞𝑦 sin 𝛼 + 𝑞𝑥𝛼

2 − 2𝑞𝑦𝛼 − 2𝑞𝑥) − 𝑞𝑚𝛼2]

−𝑅[𝑞𝑦(1 − cos𝛼) + 𝑞𝑥 sin 𝛼]

𝑅[𝑞𝑥(1 − cos 𝛼) − 𝑞𝑦 sin 𝛼]

−𝑅2[𝑞𝑥(sin 𝛼 − 𝛼) − 𝑞𝑦(cos 𝛼 − 1)] − 𝑅𝑞𝑚𝛼

1 ]

(3.90b)

𝜆1 =1

2(sin 𝛼 − 𝛼 cos𝛼) (3.90c)

𝜆2 =1

2(2 − 2 cos 𝛼 − 𝛼 sin 𝛼) (3.90d)

𝜆3 =1

2(2𝛼 + 𝛼 cos 𝛼 − 3 sin 𝛼) (3.90e)

𝜆4 =1

2(𝛼 cos 𝛼 + sin 𝛼) (3.90f)

And, 𝐼∗ is the area moment of inertia/second moment of area about 𝑧-axis modified for curvature of bar

in mm4, being expressed as (from [69])

𝐼∗ = ∫ [𝑦2 (1 −𝑦

𝑅)⁄ ] 𝑑𝐴

𝐴 (3.94a)

Thus, for our rectangular cross-section as in Figure 3.15, we derive

𝐼∗ = ∫ ∫ [y2 (1 −𝑦

𝑅)⁄ ] 𝑑𝑦 𝑑𝑧

ℎ 2⁄

−ℎ 2⁄

𝑏 2⁄

−𝑏 2⁄= −𝑏𝑅2[𝑅 𝑙𝑛(|2𝑅 − ℎ|) − 𝑅 ln(|2𝑅 + ℎ|) + ℎ] (3.91b)

For the previous derivation, we considered that all three kinds of distributed loads are continuously

spread along the entire length of the field element. If for some reason that’s not the case, a new vector

�̃�𝑞 might also be derived by modifying the integral in equation (3.93b), from the formulas on [69], in order

to fit such condition. However, it is our belief that this is absolutely unnecessary, because it would be

easier to simply discretize more elements and proceed with an element-by-element parameter

modification, in this case simply modifying each localized element’s distributed loads at will.

(Important additional information is added to this sub-chapter on sub-chapter 3.3.7).

3.3.6. Coordinate System Transformations & Updated Matrix Process

Up until this point, the methodology presented in all previous sub-chapters, since sub-chapter 3.2, has

proved to be extremely versatile, in the sense that virtually any unit length, of a discretized system, may

have its very own geometrical and physical properties. All of this allied with what we believe to be an

outstanding convenience of implementation. However, to the most watchful readers there has been a

permanent elephant in the room, and this is the fact that the methodology can only process straight

open-chain systems. In an attempt to improve this limitation, we will dedicate some time to give this

methodology the necessary tools to be able to handle non-straight systems. Indeed, following the

37

synthetisation on chapter 2, it’s clear that the leaf-spring is not straight. As to the openness of the

systems being studied, this particular issue will not be addressed on this work, since, when applying the

methodology to the leaf-spring, they are not actually needed, as this is also an open-chain system.

Therefore, by being aware that each elemental matrix, used in a typical TMM straightforward sequential

multiplication implementation (equations (3.4)-(3.8)), follows their own locally adapted convention, which

are intrinsically defined at the time of their field or point transfer matrices derivation, then, it is obvious

that facing any DOF’s direction discontinuity, between two elements in a row, definitely will generate a

problem (Figure 3.16). This situation absolutely precludes the usage of this method on non-straight

systems, hence being a major let down in terms of freedom of implementation by vastly limiting the

possible range of its applications when studying real systems. In addition, there is also another problem

arising from this issue, which is related to the mass distribution. The consequent contribution to

equations (3.14)-(3.20) by the distributed mass, is actually not in accordance with its definition previously

illustrated in Figure 3.6, however, since at the time small displacements were assumed, the

approximation was seen as acceptable and a reasonable. But this is an incorrect procedure to continue

to follow if dealing with non-straight systems, leading to absolutely wrong results.

Fortunately, both these problems can be solved by using transformation matrices 𝐓. Basically, since all

transfer matrices are defined in their own elemental/local coordinate system, all that needs to be done

is to guarantee that any previous results arising to a new element should be transformed in accordance

to this new element’s local coordinate system in order to be correctly processed by it.

Now, this directly applies to straight beam elements, however, for curved beam elements, although the

implementation process is pretty much the same, there is a slight twist which needs to be acknowledged.

What happens in the latter case is that each element doesn’t have a single local coordinate system

ruling the entire element. By looking at Figures 3.14 (right) and 3.17, it can be seen that the orthogonal

directions stipulated for the displacements and internal forces, at either side of the curved beam, are

aligned with the curvature of the arch-like element itself, resulting in two unique local referential systems

for each extremity of the beam. In fact, this is not a problem, because, as long as the field transfer matrix

receives the data from its left extremity in accordance to its respective local referential, then, this matrix

will process it and output an already “transformed” homologous set of data for the right side in

accordance to this side’s local referential. Hence, the only twist is that, for a planar application like the

one we have, two angles (relative to the global orientation) per element need to be stored in order to be

promptly used to build all the necessary transformation matrices 𝐓 of the in-between-elements

referential transformation process.

For any strangely shaped chain system, if the level of refinement employed is extremely high, then, any

two contiguous terminal local referential systems will tend to be equal, indeed, in this situation only one

angle by element might be needed to be stored. Hence, upon a very refined discretization, one would

think that this phenomenon could be neglected, being in fact just as a regular straight-beam’s

transformation procedure. However, we already derived an exact stiffness matrix formulation for a

curved beam element (equation (3.57)), so we think it’s important to keep some congruence and

maintain the same level of rigor. Besides, the value on this methodology is that it requires a less refined

discretization to obtain accurate results.

38

In sum, such kind of operation can now be generically represented as follows:

Figure 3.18: Generic curvilinear beam section discretized into constant curvature arch-like elements, with

respective transfer matrices, state vectors and transformation matrices in-between elements.

�̃�𝑖−2 = �̃�(𝑖−3)→(𝑖−2)�̃�𝑖−3 (3.95a)

�̃�𝑖−1 = �̃�(𝑖−2)→(𝑖−1)�̃�(𝑖−2)𝐿→(𝑖−2)𝑅�̃�(𝑖−3)→(𝑖−2)�̃�𝑖−3 (3.92b)

�̃�𝑖 = �̃�(𝑖−1)→𝑖�̃�(𝑖−1)𝐿→(𝑖−1)𝑅�̃�(𝑖−2)→(𝑖−1)�̃�(𝑖−2)𝐿→(𝑖−2)𝑅�̃�(𝑖−3)→(𝑖−2)�̃�𝑖−3 (3.92c)

We call attention to the fact that matrix multiplication is and associative mathematical operation, meaning

that if any combination of parenthesis was to be used in the above equations the final result would still

be the same regardless of the order by which multiplications are made, as in (𝐀𝐁)𝐂 = 𝐀(𝐁𝐂) = 𝐀𝐁𝐂.

However, the purpose of this attention call is because this property should be handled with carefulness,

since the order by which matrixes are disposed does in fact matter, as in 𝐀𝐁𝐂 ≠ 𝐀𝐂𝐁.

In the following expression, (3.96c), it is presented the final generic equation for the computing of any

state vector 𝑖, belonging to element (𝑖 − 1)-𝑖, from the very first extended state vector �̃�0 of the system.

Actually, equations (3.96a)-(3.96c) show exactly how our MATLAB implementation proceeds. Since our

entire model is built sequentially from the first element to the last, meaning with this that all matrix data

is stored in a cumulative way starting at the first state vector �̃�0. Therefore, apart from each element’s

local transfer matrix �̃�(𝑖−1)→𝑖, we also need to keep record of another kind of matrix �̃�∗0→i, this matrix

results from the correct cumulative multiplication of all the previous matrices involved in both transfer

and transformation processes up until point 𝑖, giving us a direct tool to establish the equations between

any state vector succeeding this matrix, 𝒛𝑖, and 𝒛0. As a side note, equation (3.96c) could easily be

extrapolated to encompass the calculation made from any other state vector 𝑘 located on the system

instead of the initial node, just by changing 0 for 𝑘 , although it’s not as intuitive for the reader to

understand.

�̃�𝑖−2 = �̃�(𝑖−3)→(𝑖−2)�̃�(𝑖−3)𝐿→(𝑖−3)𝑅�̃�0→(𝑖−3)∗ �̃�0 = �̃�0→(𝑖−2)

∗ �̃�0 (3.96a)

�̃�𝑖−1 = �̃�(𝑖−2)→(𝑖−1)�̃�(𝑖−2)𝐿→(𝑖−2)𝑅�̃�0→(𝑖−2)

∗ �̃�0 = �̃�0→(𝑖−1)∗ �̃�0 (3.93b)

�̃�𝑖 = �̃�(𝑖−1)→𝑖�̃�(𝑖−1)𝐿→(𝑖−1)𝑅�̃�0→(𝑖−1)

∗ �̃�0 = �̃�0→𝑖∗ �̃�0 (3.93c)

Obviously, for the point elements only one angle in relation to the global referential orientation should

be stored, as the element doesn’t have any curvature in and of itself. Knowing this, it is advisable to

39

follow a default zero-radian strategy, by using always the same orientation as the global referential for

points. This is immediately justifiable due to obvious advantages in terms of boundary condition

implementation and external concentrated forces or moments application.

During our MATLAB implementation, there needs to be an extra referential system transformation of

every state vector results. This is a transformation from their local referential systems to the global one

(equation (3.97)), which is done in order to have a full mapping of the system’s state vectors progression

according to a global referential, 𝐳𝑔𝑏𝑖. This is essential to output intuitive results and plots.

�̃�𝑔𝑏𝑖= �̃�𝑙𝑐𝑖→𝑔𝑏�̃�𝑖 (3.97)

To finalize this sub-chapter, we ought to supply the formula, derived by the author in accordance to the

adopted convention, which allows us to build any necessary extended transformation matrix �̃�𝑖𝐿→𝑖𝑅 at

any position 𝑖. Knowing this is always a planar referential system transformation, the only two required

inputs are the two disparate angles 𝜙𝑖𝐿 and 𝜙𝑖𝑅

on each infinitesimal side of a position 𝑖. The difference

between those angles, 𝜙𝑑, represents the existent discontinuity in the orientations between two spatially

coincident referential systems. These angles are measured in the anti-clockwise direction relative to the

original global referential orientation, in radians. The formula to build �̃�𝑖𝐿→𝑖𝑅 is:

�̃�𝑖𝐿→𝑖𝑅 =

[ cos 𝜙𝑑 sin 𝜙𝑑 0 0 0 0 0−sin 𝜙𝑑 cos 𝜙𝑑 0 0 0 0 0

0 0 1 0 0 0 00 0 0 cos𝜙𝑑 sin𝜙𝑑 0 00 0 0 − sin𝜙𝑑 cos𝜙𝑑 0 00 0 0 0 0 1 00 0 0 0 0 0 1]

(3.98)

Where,

𝜙𝑑 = 𝜙𝑖𝑅 − 𝜙𝑖𝐿 (3.99)

As stated only a few paragraphs before, the newly introduced twist came from having now two angles

(relative to the global orientation) per element that need to be stored in order to be promptly used to

build all the necessary transformation matrices 𝐓 of the in-between-elements referential transformation

process. Hence, every element (𝑖 − 1)-𝑖 should have readily available information relative to two angles,

𝜙(𝑖−1)𝑅 and 𝜙𝑖𝐿

. These two angles directly characterize the orientation of the local referential systems

on each element’s tip. Computing such angles may only be difficult for the arch-like constant curvature

field elements, thus, this situation will be detailed for a generic case illustrated in Figure 3.19 below.

Figure 3.19: Global & Local referential systems with respective global characterising angles.

40

𝜙(𝑖−1)𝑅= 𝜙𝑐 − 𝜙𝑐𝑡 (3.100)

𝜙𝑖𝐿 = 𝜙𝑐 + 𝜙𝑐𝑡 (3.101)

Where 𝜙(𝑖−1)𝑅 and 𝜙𝑖𝐿 give the orientation, relative to the fixed global orientation, of the two local

referential coordinate systems at the element’s extremities. 𝜙𝑐 is the orientation of the imaginary dotted

line segment connecting the two discretizing points, and 𝜙𝑐𝑡 is the angle formed at each curvilinear

element’s tip between the tangent at that point and the chord connecting the two discretizing points. The

exact 𝑥𝑦 positions of the two discretizing points are known at this point in the implementation, therefore

𝜙𝑐 is easily computed trigonometrically. As to 𝜙𝑐𝑡, this can be computed by knowing some basics of

circle theorems [95], which state this angle equals exactly half of the central angle intercepting the arch

comprised within the two points 𝛼 (𝜙𝑐𝑡 = 𝛼 2⁄ ). Since 𝛼 is known, then all data may be collected.

3.3.7. Distributed Mass Considerations

Figure 3.17 shows a curved arch-like field element, with a wide range of in-plane loading, which we have

previously fully defined. However, unfortunately, this cannot yet represent our final elemental model,

since there is still an important problem which perhaps a less attentive reader might not have been able

to spot. One would think that the formulation on equation (3.92) would be able to render a distributed

mass loading on each of the discretized elements of the model, simply by using an adequate transversal

distributed force 𝑞𝑦, as illustrated on Figure 3.20.

Figure 3.20: Scheme of negative transversal distributed force on a generic curved-beam element.

However, just by looking at the previous illustration, we realise that this is absolutely not true. According

to the Newton’s law of universal gravitation applied to our planetary system, the force vector on the

mass being studied, resulting from the attractive gravitational field of planet Earth, will be directed from

the studied system’s mass to the Earth’s mass center. Thus, any constituent vector of a beam’s mass

distribution can never be radially directed to any other point in the Euclidean space other than planet

Earth’s center. Then, by common-sensibly converting earth-centred polar coordinates to “flat-earth

coordinates”, i.e. by treating radial lines as vertical and lines of constant radial distance as horizontal,

we can equivalently state that the distributed mass along any beam should always be downwardly

positive in the global coordinates referential vertical direction. A proper mass distribution loading for the

beam on Figure 3.20 is now shown in Figure 3.21.

41

Figure 3.21: Proper scheme on how a curved-beam element should have its mass distributed per unit length, 𝑞𝑑𝑚.

We have now to derive an added contribution to the extended loading column vector �̃�𝑞, on equation

(3.92), which we will designate 𝐅𝑑𝑚. However, in order to include the effect of this new distribution on

the elemental field, such has to be done relative to the tangential-radial directed referential ruling the

element itself. Therefore, such implementation is not that straight-forward. The solution herein presented

by the author undergoes taking the generic integral formulas for in-plane loading, on [69], only

considering a transversal distributed force 𝑞𝑦 different than zero (which is changed to 𝑞𝑑𝑚 in order to

stand for the new mass distribution per unit length), whose integral calculation is done not without first

multiplying the vector by an adequate rotation transformation matrix 𝐑, which not only transforms a

distribution as in Figure 3.20 into one as in Figure 3.21, but it also inverts the positive direction of the

applied distribution to properly fit a common gravitational mass distribution (compare Figure 3.17 with

3.21). Resulting:

𝐅𝑑𝑚 =

[ 𝐹𝑑𝑚𝑢

𝐹𝑑𝑚𝑤

𝐹𝑑𝑚𝜓

𝐹𝑑𝑚𝑁

𝐹𝑑𝑚𝑉

𝐹𝑑𝑚𝑀]

= 𝐮𝑞𝐟𝑑𝑚 = 𝐮𝑞 ∫

(

𝐑

[ −𝑅2 [𝑞𝑥 (

1

𝐸𝐴𝜆4 +

1

𝐸𝐼∗𝑅2𝜆3) −

𝑞𝑚𝑅

𝐸𝐼∗(𝛽 − sin 𝛽) + 𝑞𝑦 (

𝛽

2E𝐴sin 𝛽 −

1

𝐸𝐼∗𝑅2𝜆2)]

𝑅2 [𝑞𝑥 (𝛽


1

𝐸𝐼∗𝑅2𝜆2) +

𝑞𝑚𝑅

𝐸𝐼∗(1 − cos 𝛽) + 𝑞𝑦𝜆1 (

𝛽

𝐸𝐴+

𝑅2

𝐸𝐼∗)]

𝑅2 [𝑞𝑥𝑅

𝐸𝐼∗(𝛽 − sin 𝛽) −

𝑞𝑚𝛽

𝐸𝐼∗−

𝑞𝑦𝑅

𝐸𝐼∗(1 − cos 𝛽)]

−𝑅[𝑞𝑥 cos 𝛽 + 𝑞𝑦 sin 𝛽]

𝑅[𝑞𝑥 sin 𝛽 − 𝑞𝑦 cos 𝛽]

𝑅[𝑞𝑥𝑅(1 − cos 𝛽) − 𝑞𝑚 − 𝑞𝑦𝑅 sin 𝛽] ]

|

|

|

𝑞𝑥=𝑞𝑚=0𝑞𝑦=𝑞𝑑𝑚 )

𝑑𝛽𝛼

0= 𝐮𝑞

[ 𝑓𝑑𝑚𝑢

𝑓𝑑𝑚𝑤

𝑓𝑑𝑚𝜓

𝑓𝑑𝑚𝑁

𝑓𝑑𝑚𝑉

𝑓𝑑𝑚𝑀]

(3.102a)

𝑓𝑑𝑚𝑢=

𝑞𝑑𝑚𝑅2

16𝐸𝐴𝐼∗[16𝐴𝑅2 (𝜆1 cos (

𝛼

2− 𝜙𝑐) − 𝜆2 sin (

𝛼

2− 𝜙𝑐) − 𝜆1 cos(

𝛼

2+ 𝜙𝑐) − 𝜆2 sin (

𝛼

2+ 𝜙𝑐)) − 𝐼∗ (2𝛼 cos (

3𝛼

2+

𝜙𝑐) − sin (3𝛼

2+ 𝜙𝑐) − sin(

𝛼

2− 𝜙𝑐) (2𝛼2 + 16𝜆1 + 1) + 16𝜆1 (𝛼 cos (

𝛼

2+ 𝜙𝑐) − sin(

𝛼

2+ 𝜙𝑐)))]

(3.99b)

𝑓𝑑𝑚𝑤= −

𝑞𝑑𝑚𝑅2

16𝐸𝐴𝐼∗[16𝐴𝑅2 (𝜆2 cos (

𝛼

2− 𝜙𝑐) + 𝜆1 sin (

𝛼

2− 𝜙𝑐) − 𝜆2 cos (

𝛼

2+ 𝜙𝑐) + 𝜆1 sin (

𝛼

2+ 𝜙𝑐)) + 𝐼∗ (cos (

3𝛼

2+

𝜙𝑐) +2𝛼 sin(3𝛼

2+ 𝜙𝑐) − cos (

𝛼

2− 𝜙𝑐) (2𝛼2 + 16𝜆1 + 1) + 16𝜆1 (cos (

𝛼

2+ 𝜙𝑐) + 𝛼 sin (

𝛼

2+ 𝜙𝑐)))]

(3.99c)

𝑓𝑑𝑚𝜓=

𝑞𝑑𝑚𝑅3

𝐸𝐼∗(sin𝛼 − 𝛼) (3.99d)

42

𝑓𝑑𝑚𝑁= 𝑞𝑑𝑚𝑅𝛼 sin (

𝛼

2− 𝜙𝑐) (3.99e)

𝑓𝑑𝑚𝑉= 𝑞𝑑𝑚𝑅𝛼 cos (

𝛼

2− 𝜙𝑐) (3.99f)

𝑓𝑑𝑚𝑀= 𝑞𝑑𝑚𝑅2(cos𝛼 − 1) (3.99g)

Where,

𝐑 =

[ cos(

𝛼

2− 𝛽 − 𝜙𝑐 + 𝜋) sin (

𝛼

2− 𝛽 − 𝜙𝑐 + 𝜋) 0 0 0 0

−sin(𝛼

2− 𝛽 − 𝜙𝑐 + 𝜋) cos(

𝛼

2− 𝛽 − 𝜙𝑐 + 𝜋) 0 0 0 0

0 0 1 0 0 0

0 0 0 cos (𝛼

2− 𝛽 − 𝜙𝑐 + 𝜋) sin (

𝛼

2− 𝛽 − 𝜙𝑐 + 𝜋) 0

0 0 0 −sin (𝛼

2− 𝛽 − 𝜙𝑐 + 𝜋) cos(

𝛼

2− 𝛽 − 𝜙𝑐 + 𝜋) 0

0 0 0 0 0 1]

(3.103)

Thus, the updated and final extended in-plane loading column vector �̃�𝑞 to be implemented on MATLAB

should now be the one below instead of the one in equation (3.92).

�̃�𝑞 =

[ 𝑢17

𝑢27

𝑢37

𝑢47

𝑢57

𝑢67

𝑢77]

= �̃�𝑞 + [𝐅𝑑𝑚

0] =

[ 𝐹𝑢 + 𝐹𝑑𝑚𝑢

𝐹𝑤 + 𝐹𝑑𝑚𝑤

𝐹𝜓 + 𝐹𝑑𝑚𝜓

𝐹𝑁 + 𝐹𝑑𝑚𝑁

𝐹𝑉 + 𝐹𝑑𝑚𝑉

𝐹𝑀 + 𝐹𝑑𝑚𝑀

1 ]

(3.104a)

𝐹𝑢 + 𝐹𝑑𝑚𝑢= −

𝑅2

16𝐸𝐴𝐼∗2 [8𝐴𝑅2 (−2𝑞𝑑𝑚 cos (𝛼

2− 𝜙𝑐) (𝐼∗𝜆1 cos 𝛼 − 𝜆2(𝐼

∗ sin 𝛼 + 𝛼)) + 2𝑞𝑑𝑚𝐼∗ sin (𝛼

2− 𝜙𝑐) (𝜆2 cos 𝛼 + 𝜆1 sin 𝛼 − 𝛼𝜆3) +

2𝑞𝑑𝑚𝐼∗𝑐𝑜𝑠 (𝛼

2+ 𝜙𝑐) (𝜆1 cos 𝛼 − 𝜆2 sin 𝛼) + 2𝑞𝑑𝑚𝐼∗ sin (

𝛼

2+ 𝜙𝑐) (𝜆2 cos 𝛼 + 𝜆1 sin 𝛼) − cos𝛼 (𝛼2𝐼∗𝑞𝑥 − 2𝛼𝐼∗(𝑞𝑥𝜆3 − 𝑞𝑦𝜆2) −

2(𝐼∗(2𝑞𝑥 − 𝑞𝑦𝜆3) − 𝑞𝑥𝜆2)) + 2 sin 𝛼 (𝛼𝐼∗(𝑞𝑥(𝜆2 + 2) − 𝑞𝑦𝜆1) + 𝐼∗𝑞𝑥𝜆3 − 𝑞𝑦𝜆2) − 𝛼2𝐼∗𝑞𝑥 − 2(𝐼∗(2𝑞𝑥 − 𝑞𝑦𝜆3) − 𝑞𝑥𝜆2)) −

8𝐴𝑅𝑞𝑚𝐼∗(4 cos2 𝛼 − 2 cos𝛼 + 4𝛼 sin 𝛼 − 𝛼2 − 2) − (−𝑞𝑑𝑚 cos (3𝛼

2+ 𝜙𝑐) (2𝛼 cos 𝛼 + sin 𝛼) + 𝑞𝑑𝑚 sin (

3

2𝛼 + 𝜙𝑐) (cos 𝛼 −

2𝛼 sin 𝛼) + 𝑞𝑑𝑚 sin 𝛼 cos (𝛼

2− 𝜙𝑐) (10𝛼2 + 16𝜆1 + 1) + 𝑞𝑑𝑚 sin (

𝛼

2− 𝜙𝑐) (cos 𝛼 (2𝛼2 + 16𝜆1 + 1) + 16𝛼𝜆4) + 8(−2(𝛼 cos 𝛼 +

sin 𝛼) cos (𝛼

2+ 𝜙𝑐) 𝑞𝑑𝑚𝜆1 + 2𝑞𝑑𝑚𝜆1 sin (

𝛼

2+ 𝜙𝑐) (cos 𝛼 − 𝛼 sin 𝛼) − (−2𝛼𝑞𝑦 + 𝑞𝑥) cos2 𝛼 − ((2𝛼𝑞𝑥 + 𝑞𝑦) sin 𝛼 + 2(𝛼𝑞𝑥 −

𝑞𝑦)𝜆4) cos 𝛼 − (−𝛼2𝑞𝑦𝜆1 − 𝛼𝑞𝑥 + 2𝑞𝑥𝜆4) sin 𝛼 − 𝛼𝑞𝑦 + 𝑞𝑥 − 2𝑞𝑦𝜆4)) I∗2 ] (3.101b)

𝐹𝑤 + 𝐹𝑑𝑚𝑤 =𝑅2

16𝐸𝐴𝐼∗2 [8𝐴𝑅2 (−2(𝜆2 cos 𝛼 + 𝜆1(sin 𝛼 − 𝛼)) cos (𝛼

2− 𝜙𝑐) 𝐼∗𝑞𝑑𝑚 − 2(𝐼∗𝜆1 cos 𝛼 − (𝐼∗ sin 𝛼 + 𝛼)𝜆2) sin (

𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 +

2(𝜆2 cos 𝛼 + 𝜆1 sin 𝛼) cos (𝛼

2+ 𝜙𝑐) 𝐼∗𝑞𝑑𝑚 − 2(𝜆1 cos 𝛼 − 𝜆2 sin 𝛼) sin (

𝛼

2+ 𝜙𝑐) 𝐼∗𝑞𝑑𝑚 − 2(𝛼𝐼∗(𝑞𝑥(𝜆2 + 1) − 𝑞𝑦𝜆1) + 𝐼∗(−2𝑞𝑑𝑚 +

𝑞𝑥𝜆1 − 2𝑞𝑦) − 𝑞𝑦𝜆2) cos 𝛼 − (𝛼2𝐼∗𝑞𝑥 − 2𝛼𝐼∗ (𝑞𝑑𝑚 + 𝑞𝑥𝜆3 − 𝑞𝑦(𝜆2 − 1)) + 2(𝐼∗𝑞𝑦𝜆1 + 𝑞𝑥𝜆2)) sin 𝛼 + 2(𝛼𝐼∗𝑞𝑥 − 𝐼∗(2𝑞𝑑𝑚 − 𝑞𝑥𝜆1 +

2𝑞𝑦) − 𝑞𝑦𝜆2)) − 16𝐴(sin 𝛼 − 𝛼)(2 cos𝛼 − 1)𝐼∗𝑞𝑚𝑅 + (−(cos 𝛼 − 2𝛼 sin 𝛼) cos (3𝛼

2+ 𝜙𝑐) 𝑞𝑑𝑚 − (2𝛼 cos 𝛼 + sin 𝛼) sin (

3𝛼

2+

𝜙𝑐)𝑞𝑑𝑚 + ((2𝛼2 + 16𝜆1 + 1) cos𝛼 + 16𝛼𝜆1) cos (𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 − (10𝛼2 + 16𝜆1 + 1) sin 𝛼 sin (

𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 + 8(−2(cos𝛼 −

𝛼 sin 𝛼) cos (𝛼

2+ 𝜙𝑐) 𝑞𝑑𝑚𝜆1 − 2(𝛼 cos 𝛼 + sin 𝛼) sin (

𝛼

2+ 𝜙𝑐) 𝑞𝑑𝑚𝜆1 − (2𝛼𝑞𝑥 + 𝑞𝑦) cos2 𝛼 + ((−2𝛼𝑞𝑦 + 𝑞𝑥) sin 𝛼 + (𝛼2𝑞𝑦 −

2𝑞𝑥)𝜆1) cos 𝛼 + (𝛼(2𝑞𝑥𝜆4 + 𝑞𝑦) − 2𝑞𝑦𝜆1) sin 𝛼 + 𝛼𝑞𝑥 + 2𝑞𝑥𝜆1 + 𝑞𝑦)) 𝐼∗2] (3.101c)

𝐹𝜓 + 𝐹𝑑𝑚𝜓 =𝑅2

2𝐸𝐼∗[𝑅 (−2𝛼(cos𝛼 − 1) cos (

𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 + 2𝛼(sin 𝛼 − 𝛼) sin (

𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 + 4𝑞𝑥 cos2 𝛼 − 2(−2𝑞𝑦 sin 𝛼 + 𝛼(𝑞𝑑𝑚 + 2𝑞𝑦) +

𝑞𝑥) 𝑐𝑜𝑠 𝛼 + 2(2𝛼𝑞𝑥 + 𝑞𝑑𝑚 − 𝑞𝑦) 𝑠𝑖𝑛 𝛼 − 𝛼2𝑞𝑥 + 2𝛼𝑞𝑦 − 2𝑞𝑥) + 𝛼2𝑞𝑚] (3.101d)

𝐹𝑁 + 𝐹𝑑𝑚𝑁 = 𝑅 [𝛼 sin 𝛼 cos (𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 + 𝛼 cos 𝛼 sin (

𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 + 𝑞𝑦 cos2 𝛼 − (𝑞𝑦 + 2𝑞𝑥 sin 𝛼) cos 𝛼 − sin 𝛼 (qy sin 𝛼 − 𝑞𝑥)] (3.101e)

𝐹𝑉 + 𝐹𝑑𝑚𝑉 = 𝑅 [𝛼 cos𝛼 cos (𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 − 𝛼 sin 𝛼 sin (

𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 − 𝑞𝑥 cos2 𝛼 − (2𝑞𝑦 sin 𝛼 − 𝑞𝑥) cos 𝛼 + sin 𝛼 (𝑞𝑥 sin 𝛼 + 𝑞𝑦)] (3.101f)

𝐹𝑀 + 𝐹𝑑𝑚𝑀 = 𝑅2 [−𝛼 sin 𝛼 cos (𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 − 𝛼(cos 𝛼 − 1) sin (

𝛼

2− 𝜙𝑐) 𝑞𝑑𝑚 − 2𝑞𝑦 cos2 𝛼 + (2𝑞𝑥 sin 𝛼 + 𝑞𝑑𝑚 + 3𝑞𝑦) cos 𝛼 − 3𝑞𝑥 sin 𝛼 +

𝛼𝑞𝑥 − 𝑞𝑑𝑚 − 𝑞𝑦] − 𝛼𝑞𝑚𝑅 (3.101g)

The derivation process just described is particularly complex, however this can be implemented in a

pretty straightforward way by letting MATLAB do all the necessary operations, by simply following the

steps on equations (3.92), (3.102a) and (3.104a), for which we only need to hand-type once matrices

(3.93a), (3.93b) and (3.103). However, there is another reason for the intricate expressions of the final

43

in-plane loading vector to be detailed above, this is because, implementation-wise, it is insurmountably

faster for the computer go through this process if the final vector is hand-typed directly into the code in

the form shown on equations (3.89b)-(3.89g). Obviously, this is due to the fact that in doing so we omit

some operations that otherwise would have to be computed, but more than that, what makes this an

issue worth mentioning is the fact that the integration on (3.102a) is particularly heavy to compute,

representing in average 96.1% of the processing time within this small set of described operations. In

opposition, the suggested alternative will represent a cut in average of -95.2% relative to the previous

time. Anyhow, the results obtained using the formulation derived from the beginning of this sub-chapter,

performed exactly as expected.

3.3.8. Floating Point Arithmetic Issues

In order to represent any real number in a fixed amount of memory MATLAB uses floating point numbers

that conform to the IEEE Standard 754. There are actually several mechanisms by which this can be

done, but all in all the most important aspect to retain is that a floating-point number consist of two basic

parts, a string of digits called significand or mantissa (1 + 𝑓), where 𝑓 is the fraction, and an integer

exponent which modifies the magnitude of the number called characteristic or exponent (𝑒). The number

is then saved in the memory as a significand multiplied by the base raised to the power of the exponent.

Meaning they can be expressed in its normalized way as ([96])

𝑥 = ±(1 + 𝑓). 2𝑒 (3.105)

Where the quantity 𝑓 is the fraction and satisfies

0 ≤ 𝑓 < 1 (3.106)

And the exponent 𝑒 is an integer in the interval

−1022 ≤ 𝑒 < 1023 (3.107)

By default, MATLAB’s numerical data is stored as a double precision type of number (binary64), which,

according to the standard used, is represented in 64 bits (8 bytes), namely, one sign bit, 11 exponent

bits, and 52 bits for 𝑓 [97][96]. The entire significand part of a floating-point number is not 𝑓 but 1 + 𝑓,

which has 53 bits. However, the leading 1 doesn’t need to be stored. In effect, the IEEE format pack 65

bits of information into a 64-bit word [96].

Figure 3.22: IEEE 754 Floating point format (from [97]).

Knowing this, we realize that the smallest positive normalized floating-point number has 𝑓 = 0 and 𝑒 =

−1022, while the largest floating-point number has 𝑓 a little less than 1 and 𝑒 = 1023. MATLAB calls

these numbers “realmin” and “realmax”, which characterize the limits to the standard system. These

limits when converted into decimal numbers are 2.2251e-308 and 1.7977e+308. Any number above the

maximum threshold are considered as infinity or Inf, while any number smaller than the minimum

threshold are set to 0.

44

Now, when using transfer-matrix method (TMM), depending on the level of discretization, we may have

to be dealing with a considerable number of sequential matrix multiplications in order to transfer the

information from one side of the system to the other, even more so when introducing transformation

matrices along the way (see equation (3.96c)). Well, this situation will lead to transfer matrices �̃�∗0→i

often times carrying numbers below realmin. These numbers, despite being so small, are essential in

transferring the proper information across a highly-discretized system, in this way playing an important

role in assuring the precision of the method as a whole.

The concern here is that, due to floating point arithmetic issues, inherent to the normalizing rules

stipulated by IEEE Standard 754, we may have serious disruptions in the information transferring

process, leading to degraded results.

An easy way to identify the degree of this problem’s repercussions on the obtained results, is to look

directly into the most vulnerable element of the system, i.e. the transfer matrix from the initial state vector

𝐳0 until the last state vector of the chain 𝐳𝑓, �̃�∗0→𝑓. When this matrix is divided by the homologous one

for the state vector positioned immediately before, as in equation (3.108), the outcome should ideally

coincide with the very last discretized element’s transfer matrix, �̃�(𝑓−1)→𝑓, which had to be pre-defined

prior to this matrix calculus in its exact form. Unfortunately, we verify that differences between this

division’s output and the original transfer matrix happen to exist, and happen more often than not, even

for relatively simple systems. Therefore, this is in fact a big problem associated to this method’s

implementation. This problem will also greatly affect the computation of the natural frequencies since it

results from an eigenvalue problem applied to matrix �̃�∗0→𝑓.

�̃�∗

0→𝑓

�̃�∗0→(𝑓−1)

=⏞?

�̃�(𝑓−1)→𝑓 (3.108)

A solution must be to increase the overall precision. This is done by modifying MATLAB’s source code,

so that it is enhanced from its defaulted 64-bit double precision into being capable of handling quadruple

precision floating numbers instead, while still keeping its compliance to IEEE 754. However, such 128-

bit floating point arithmetic can only be attained by installing specialized toolboxes for MATLAB, for

instance, Advanpix’s Multiprecision Computing Toolbox [98], which come at an unviable cost for the

author.

Fortunately, there is a cheaper alternative to solve this problem. We can use the Symbolic Math Toolbox

[99] to explicitly set the number of significant digits used in our computations and maintain that accuracy

throughout the computations. In addition, with this method we can control the precision of computations

as well as the trade-off between accuracy and performance. The MATLAB function able to perform this

is called vpa, or variable precision arithmetic, and can be used by simply setting the decimal digit

accuracy as high as needed to maintain the accuracy of symbolic functions and operations. In

comparison to the previous alternative, there is the caveat of having a higher processing time, but this

comes at a compromisable rate given the convenience and accessibility provided.

Thus, our program will at all times be working simultaneously with both high-precision symbolic values

and double-precision default values, and, since double-precision values inherently contain round-off

errors which can’t be restored by a late introduction of vpa, then, it’s imperative to consciously and

efficiently dispose of vpa for key variables and operations along the code, in order to guarantee good

45

results at an as lower computational cost as possible. The same level of care is taken at defining each

vpa’s number of significant decimal digits. What ends up happening is that we have a mix of defaulted

16-digit precision numbers mixed up with different digit precisions imposed by vpa, which are carefully

adapted to the needs of each different routine or section of the program.

3.3.9. Additional Consequences to the Adopted Convention

The major differences to the methodology were already approached, and, as stated before, if any

remaining derivation of process are omitted is because they can be easily extrapolated from chapter 3.2

to fit this new approach. However, some very small disparities are still worthy to be highlighted, since

there is a higher risk of implementation failure in case of their neglection. These, will now be approached.

Taking in consideration that the elemental convention shown in Figure 3.6 is now slightly different, see

Figure 3.14 (right), whenever external contributions such as forces or moments are now applied in a

point element, their input into the system continues to be made through vector 𝐏𝑒𝑥𝑡, however this vector’s

components instead of being added to the respective elemental point matrix should now be subtracted

into it. This comes purely from simple equilibrium considerations. In fact, this same reason also rules

the next alteration, which is first introduced through the following free-body diagram

Figure 3.23: Free-body diagram of lumped-mass 𝑚𝑖 (adapted from [6]).

In this case, we refer to an alteration being made to equation (3.42), which described the introduction of

a discontinuity in the shear due to the inertia force of a concentrated vibrating mass. This same equation

should now be written as,

𝑉𝑖𝑅 = 𝑉𝑖

𝐿 + 𝑚𝑖𝜔2𝑤𝑖 (3.109)

However, at that time we were only dealing with a reduced number of degrees of freedom for

simplification purposes. Now, in order to fit the updated model being introduced, there won’t just be

discontinuities in the shear. In this regard, the new simple point transfer matrix to be used in DFVA with

lumped-mass technique is, according to [6], and in place of equation (3.41),

𝐏𝑖 =

[

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 0

−𝑚𝜔2 0 0 1 0 00 𝑚𝜔2 0 0 1 00 0 −𝐼𝜔2 0 0 1]

(3.110)

3.3.10. In-Plane Stresses in Curved Bar-Beam Elements

In sub-chapter 3.2.4, within the scope of TMM linear static elastic and stress analysis, an introduction

on how to compute all three considered kinds of in-plane stresses was made. These stresses were: the

normal stress (𝜎𝑁), the bending stress (𝜎𝑀), and the transversal shear stress (𝜏𝑉). However, at the time

46

we were considering the simpler straight-beam model, hence, now, with a more advanced constant

curvature arch-like beam element, we ought to take more detailed considerations in these stresses’ re-

definitions.

For the expressions derived in this sub-chapter, we only consider beam-bar elements with a cross

section symmetric about the plane of curvature, and loadings and deformations which lie in the same

plane as the element itself. Such elements may be subject to extension and bending. The expressions

also include the effects of shear deformation and rotary inertia. The conditions described define what

we popularly designate as in-plane stresses, however, out-of-plane stresses could as well be considered

by proceeding in a similar fashion and even by taking the very same references as the ones herein

considered, however deeper changes to the MATLAB implementations developed would be required (for

instance, three-dimensional modelling would be required), which were not made on this particular work,

not due to any unsurpassable difficulty but rather due to simple lack of time. Nonetheless, it should be

stated that, in order to properly simulate all the critical operation states of the suspension we intend to

analyse in this work, particularly to analyse the suspension’s behaviour when the vehicle is accelerating,

decelerating or braking and/or curving, such out-of-plane stresses should in fact be implemented in any

further improvements done over this work.

Before proceeding, it should first be said that, instead of the three kinds of stresses detailed before, we

will now encompass these same kinds of in-plane stresses through a radially adjusted coordinate

system, resulting in two of the three kinds of stresses being defined differently. The types of stresses

are now: circumferential stress (𝜎𝑥), shear stress (𝜏𝑉) and radial stress (𝜎𝑦). While the transversal shear

stress continues to be a pure outcome of the transversal shear force, the remainder two stresses will

result from the combined efforts of the normal force and bending moment, with these specific

contributions going to be analysed forwardly.

3.3.10.1. Longitudinal Stress

Therefore, let us start by the new definition of the longitudinal/circumferential/axial/tangential/hoop

stress, 𝜎𝑥. In fact, it wouldn’t be wrong to continue designating it has normal stress, as this stress is

normal to every section of the curved beam, but we won’t ever use such designation to avoid any

possible mixing of adopted conventions.

During bending, the cross sections of a straight beam are assumed to remain plane, and the strains of

compression and extension fibres equidistant from the centroid of the cross section are equal in

magnitude so that the longitudinal stress is distributed linearly on the cross section. However, for a

curved beam, the cross sections are also assumed to remain plane when the beam is bent, but the

strains at two points on opposite sides of and equidistant from the centroid are no longer equal in

magnitude. Although the magnitude of the extension and compression of the fibres at these points are

the same, the “original lengths” of the fibres are different. The longitudinal stress distribution is no longer

linear. Figure 3.24 shows the “new” longitudinal stress distribution on a cross section, where it can be

seen that the neutral axis of the cross section does not coincide with the centroid but is rather shifted of

a distance 𝑒, which for pure bending is ([69]):

𝑒 = 𝑅 −𝐴

𝐴𝑚= 𝑅 − 𝑟𝑛 0.6 <

𝑅

ℎ< 8 (3.111a)

47

With

𝑅 =1

𝐴∫ 𝑟 𝑑𝐴𝐴

𝐴𝑚 = ∫𝑑𝐴

𝑟𝐴 (3.108b)

Figure 3.24: Normal stress distribution of curved beam (adapted from [69]).

Where 𝐴 is the cross-sectional area, 𝑅 is the distance from the center of curvature to the centroid of the

cross section, ℎ is the height of the cross section along the direction of 𝑅, 𝑟𝑛 is the distance from the

center of curvature to the neutral axis, and 𝑟 locates a point on the section measured from the center of

curvature (resulting in 𝑟 = 𝑎 + ℎ 2⁄ − 𝑦, see Figure 3.25). Note that 𝑒 is a cross-sectional property and

not related to the applied forces. Equations (3.111) are applicable for the range 0.6 < 𝑅 ℎ⁄ < 8. When

𝑅 ℎ⁄ > 8, i.e. for slender curved beams, round-off errors or small inconsistencies in treating a cross

section of complicated shape may have a large effect on the calculated value of 𝑒. To avoid this, 𝑒 can

be computed using ([69])

𝑒 ≅𝐼𝑧

𝑅𝐴

𝑅

ℎ≥ 8 (3.108c)

Where 𝐼𝑧 is the moment of inertia or second moment of area about the centroidal axis.

Knowing 𝑀 is the bending moment about the 𝑧-axis (Figure 3.24), the circumferential stress on the cross

section is (modified from [69])

𝜎𝑥 = −𝑀𝑦

𝐴𝑒𝑟=

𝑀(𝑟−𝑟𝑛)

𝐴𝑒𝑟 (3.112a)

Where 𝑦 is the distance in 𝑦-direction from the neutral axis to the point where stresses are calculated.

When a tensile axial/longitudinal/normal/circumferential force 𝑁 through the centroidal axis occurs on

the cross section, the term 𝑁 𝐴⁄ (as in equation (3.58)) should be added to equation (3.112a):

𝜎𝑥 =𝑁

𝐴−

𝑀𝑦

𝐴𝑒𝑟 (3.109b)

The expression 𝑁 𝐴⁄ implies that the longitudinal stress due to 𝑁 is taken to be constant over the cross

section, an assumption that tis usually reasonable considering that the stress due to 𝑁 is normally much

smaller than the stress due to 𝑀 [69]. When the first term (𝑁) is comparable in magnitude to the second

term (𝑀) or 𝑅 ℎ⁄ is small, the error of using equation (3.112b) increases significantly. Thus, Cook in [100]

introduced two formula for the circumferential stress 𝜎𝑥 which is able to cope with these inaccuracies

([69]):

𝜎𝑥 =𝑀(𝑟−𝑟𝑛)

𝐴𝑒𝑟+

𝑁

𝐴[𝑟𝑛

𝑟+

𝐴𝑒

𝐼𝑧(𝑟 − 𝑅)] 0.6 <

𝑅

ℎ< 8 (3.113a)

48

𝜎𝑥 =𝑀(𝑟−𝑟𝑛)

𝐴𝑒𝑟+

𝑁

𝐴[𝑟𝑛

𝑟+

𝑟

𝑅− 1]

𝑅

ℎ≥ 8 (3.110b)

The previous equations are derived based on the assumptions that the shear and radial stresses vanish,

so they are best suited for those parts of the cross section where these stresses are not significant.

Nonetheless, this is a considerably accurate approximation, especially when comparing to our previous

assumption regarding in-plane stresses, in which the straight-beam flexure formula (equation (3.59))

clearly indicated that the straight-beam solution was significantly in fault when faced with small values

of 𝑅 ℎ⁄ > 5, and, to make it worse, such error would fall into the not conservative side [69]. However,

according to [101], generally, for curved beams with 𝑅 ℎ⁄ > 5, the straight-beam flexure formula could

still be used, as in fact, it can be proved that when 𝑅 ℎ⁄ → ∞, equations (3.113) become 𝜎𝑁 = 𝑀𝑦 𝐼𝑧⁄

[69].

In this work, 𝑎 and 𝑐 will be the radial distances from the centre of curvature to the inner and outer fibres,

which is common practice when working with rectangular cross-sections (Figure 3.25). If for some

reason this is not the case extra homologous parameters might be needed to properly handle any kind

of cross-section. However, all information regarding a wide variety of cross section kinds, can be found

tabulated in [69], together with the analytical expressions for 𝐴𝑚 and 𝐴 . For instance, regarding a

rectangular cross section the latter are:

𝐴 = 𝑏(𝑐 − 𝑎) (3.114)

𝐴𝑚 = 𝑏 ln (𝑐

𝑎) (3.115)

Similarly, by following the definition shown on Figure 3.25, we are also led to validate the only expression

left to define, i.e. the radius of curvature (equation (3.111b)) when we have a rectangular cross section,

being this

𝑅 =1

𝐴∫ 𝑟 𝑑𝐴𝐴

=1

𝑏(𝑐−𝑎)∫ ∫ 𝑟 𝑑𝑟 𝑑𝑧

𝑐

𝑎

𝑏

2

−𝑏

2

=𝑎+𝑐

2 (3.116)

Knowing that in this work we intend to capacitate our MATLAB program to be able to deal with a multi-

layered leaf-spring design (Figure 3.15), and, since each of the layers may in fact have its own cross

section shape, then we ought to find a solution to handle such a composite cross section. A solution to

this issue is also proposed by Pilkey in [69], simply by

𝐴 = ∑ 𝐴𝑗𝑛𝑗=1 (3.117a)

𝐴𝑚 = ∑ 𝐴𝑚𝑗𝑛𝑗=1 (3.114b)

𝑅 =∑ 𝑅𝑗𝐴𝑗

𝑛𝑗=1

∑ 𝐴𝑗𝑛𝑗=1

(3.114c)

From which the remainder 𝑒 and 𝑟𝑛 may then be subsequently deduced. In these expressions 𝑛 is the

number of regular shapes that form the composite section (see Figure 3.15).

Figure 3.25: Rectangular cross section and

respective axis passing through centre of curvature

(dot-dashed lined) (from [69]).

49

3.3.10.2. Radial Stress

The radial stress, 𝜎𝑦, is usually not a major consideration for the design of curved beams with solid cross

sections, because the magnitude of the radial stress is small compared to the circumferential stress.

But, for curved beams that have flanged cross sections with thin webs, for example, the radial as well

as circumferential stresses may be large at the junctions of the flanges and webs. As a consequence,

the shear stress may also be large, and hence yielding may occur. A large radial stress in a thin web

may also cause the web to buckle. In such cases, the radial stress cannot be neglected [69]. Therefore,

this stress as to be analysed in a careful and case by case manner in whatever project we have in

hands.

According to [101], and following the symbol convention in Figure 3.26, the radial stress is expressed

by

𝜎𝑦 =1

𝑏𝑟[𝐴′

𝐴𝑁 +

𝐴𝐴𝑚′ −𝐴′𝐴𝑚

𝐴(𝑅𝐴𝑚−𝐴)𝑀] (3.118a)

Where

𝐴𝑚′ = ∫

𝑑𝐴

𝑟

𝑟

𝑟0 and 𝐴′ = ∫ 𝑑𝐴

𝑟

𝑟0 (3.115b)

With 𝑏, 𝐴′, 𝑟0 and 𝑟 shown in Figure 3.26 (a). A moment 𝑀 that tends to straighten the curved beam

generates a tensile radial stress. This expression is obtained from the equilibrium of the beam segment

in Figure 3.26 (a), where the resultants 𝐹 of 𝜎𝑥, which takes the form of equation (3.113) and is assumed

to be constant along the beam segment, and 𝑇 of 𝜎𝑦 form an equilibrium system [69].

For a rectangular cross section the newly introduced parameters are defined as

𝐴𝑚′ = ∫ ∫

1

𝑟

𝑟

𝑎

𝑏

2

−𝑏

2

𝑑𝑟 𝑑𝑧 = 𝑏 ln (𝑟

𝑎) (3.115c)

𝐴′ = ∫ ∫ 𝑑𝑟 𝑑𝑧𝑟

𝑎

𝑏

2

−𝑏

2

= 𝑏(𝑟 − 𝑎) (3.115d)

Figure 3.26: Radial stresses: (a) equilibrium of segment of curved beam; (b) resultants (adapted from [69]).

We will now increase the overall complexity by including in the assumptions the existence of shear

stresses 𝜏𝑉 (equation (3.118)) on the cross sections, having now 𝜎𝑥 and 𝜏𝑉 varying along the beam

segment (Figure 3.26 (b)). The conditions of equilibrium for the beam segment result in the following

expression for the radial stress [102][103][69]:

𝜎𝑦 =𝑟𝑛

𝐴𝑒𝑏𝑟[(−𝑀 − 𝑁𝑅) (𝐴𝑚

′ −𝐴′

𝑟𝑛) +

𝑁

𝑟(𝑅𝐴′ − 𝑄)] (3.115e)

50

In comparison to the expression originally referenced, we had to change the signal of the bending

moment, since we adopted a conversely positive convention in this particular effort. And now the first

moment of area is

𝑄 = ∫ 𝑟 𝑑𝐴𝐴′

(3.119a)

Whose integration is taken over the area 𝐴′ that lies between the position at which the stress is desired

and the inner fibre of the cross section nearest to the center of curvature (Figure 3.26).

Looking at the particular case of a rectangular cross section (𝑏xℎ), while in a way updating the equation

for the first moment of area previously presented in equation (3.61) for the straight beam case, but not

considering the case represented in Figures 3.24-3.26, results

𝑄 = ∫ ∫ 𝑟𝑟

𝑎

𝑏

2

−𝑏

2

𝑑𝑟 𝑑𝑧 = ∫ ∫ 𝑟𝑟𝑛−𝑦

𝑎

𝑏

2

−𝑏

2

𝑑𝑟 𝑑𝑧 =𝑏

2[(𝑟𝑛 − 𝑦)2 − 𝑎2] (3.116b)

Equation (3.118e) is more accurate than equation (3.118a). Equations (3.118) are reasonably accurate

approximations for the radial stress 𝜎𝑦 in curved beams although 𝜎𝑥 is derived with the assumption that

the shear and radial stresses vanish. This is similar to the case of a straight beam where the normal

stress is based on the assumption that a plane cross section remains plane, which implies that the shear

stresses vanish. Then the straight-beam shear stresses are obtained from the equilibrium of the

resultants of the normal and shear stresses. A comparison of equation (3.118a) [101], for rectangular

cross-sectional beams subjected to shear loading, with a corresponding theory of elasticity solution

indicates that equation (3.118a) is conservative, and it remains conservative to within 6% for values of

𝑅 ℎ⁄ > 1.0 even without considering the 𝑁 term [69]. For further details on other specific situations we

suggest [102].

3.3.10.3. Transversal Shear Stress

In opposition to the expression we had previously in this work defined for the shear stress 𝜏𝑉, equation

(3.60), now the average shear stress across a width (since this is a planar model), i.e. across line 1-2 in

Figure 3.27, of a cross section in a curved beam may be computed using ([69])

𝜏𝑉 =𝑉𝑟𝑛𝑄′

𝑏𝑦𝐴𝑒(𝑅−𝑦′)2 (3.120)

Figure 3.27: Definitions for shear stress (adapted from [69]).

51

Distance 𝑦 (or 𝑦′ on the figure above, both notations used in this work for the same purpose) is

measured from the neutral axis, and 𝑏𝑦 is the sectional width at the layer being analysed. Equation

(3.120) is derived from the assumption that the shear stress is parallel to the shear force, and the shear

stresses at points on an element perpendicular to the shear force are equal to the parallel shear stresses

where the shear force is applied. Although there are some cases in practice that do not coincide with

these assumptions, the assumptions tend to be good approximations and equation (3.120) can often be

used without serious error [69]. A comparison of the results of 𝜏𝑉𝑚𝑎𝑥𝜏𝑉𝑎𝑣𝑒𝑟𝑎𝑔𝑒

⁄ , with the exact elasticity

solution for a rectangular cross section, shows that when 1.25 ≤ 𝑅 ℎ⁄ ≤ 5.5, the error of the value

𝜏𝑉𝑚𝑎𝑥𝜏𝑉𝑎𝑣𝑒𝑟𝑎𝑔𝑒

⁄ does not exceed 1% [104][69].

Notice also that the first moment of area is slightly different, hence the notation 𝑄′. For a rectangular

cross section (𝑏xℎ), we may write the following two expressions, ending with the new form for the

transversal shear stress 𝜏𝑉 (rectangular cross-section only).

𝑄′ = ∫ ∫ 𝑦 𝑑𝑦 𝑑𝑧−𝑒+

ℎ

2𝑦

𝑏

2

−𝑏

2

=𝑏

8[(2𝑒 − ℎ)2 − 4𝑦2] (3.121)

𝜏𝑉 =𝑉𝑟𝑛

8𝐴𝑒

[(2𝑒−ℎ)2−4𝑦2]

(𝑅−𝑦)2 (3.122)

3.3.10.4. Von-Mises Equivalent Stress

To finalize, should be noted that, by using the three types of stresses just introduced, we are inducing a

different multiaxial stress condition, which cannot be studied with the exact same methodology as the

one we had introduced way before in this work with equation (3.70). In adjusting to this new situation,

while in the midst of it considering out-of-plane stresses too (for academic sake), we re-write the

expression for the equivalent stress according to the Von-Mises Criterion. Then, six stresses participate

in this purposively generic expression, namely, the longitudinal/circumferential/tangential stress (𝜎𝑥), the

transversal shear stress in both directions (𝜏𝑉𝑦, 𝜏𝑉𝑧

), the radial stress inducing curvature along 𝑥𝑦-plane

(𝜎𝑦) and 𝑥𝑧-plane (𝜎𝑧), and the torsional shear stress (𝜏𝑇). The end result is:

𝜎𝑉𝑀 = √1

2[(𝜎𝑥 − 𝜎𝑦)

2+ (𝜎𝑦 − 𝜎𝑧)

2+ (𝜎𝑧 − 𝜎𝑥)

2] + 3 (𝜏𝑉𝑦

2 + 𝜏𝑉𝑧2 + 𝜏𝑇

2) (3.123)

3.3.11. In-Plane Stresses Methodology Chosen for Topologically Combined

Curved-Straight Systems

The truth is that, when facing any problem ahead we may in fact chose freely between a normal-bending-

shear stress approach or a circumferential-radial-shear one. It is pure convention, hence pure choice.

We opted for the first way. The justification is mainly because it feels as a more integrated way to

proceed, since we keep a “purer” look over the results. We say this because in this way we only keep

stresses resulting purely from each discretized internal force application, and not due to difficult

combinations of carefully added contributions coming from different internal efforts. On the other hand,

seems to us that this way is more positive research-wise, since in the previous sub-chapter we had

already detailed a highly accurate stress model able to fully describe a wide spectrum of systems

topologically composed by both straight or curved structural beams, but, such homologous expressions

52

weren’t yet set for this normal-bending-shear convention. This is highly limitative, especially when

considering that this yet downgraded convention is arguably the most commonly used academically,

thus leading to more interesting results, as they may be more accessibly compared. All in all, by taking

this choice it feels we are adding one more valuable tool which, at least upon a quite long research,

seemed inexistent in the literature to us available, for the same sort of model herein presented.

In this respect, to reach the intended definitions we must go through a simple term-by-term segregation

process, where we separate each kind of internal effort’s contributions coming from the different curved-

beam expressions and combine them into “pure” stress ones. Only in the case of the transversal shear

force this process is not needed, as this already only results from internal transversal forces. The highest

adjacent difficulty is obviously not in the previously described process, but in checking the expressions

and making all the necessary alterations, especially due to notation differences, to make sure not only

that the outputted results are correct, but that they are continuous along all cross sections, along all

layers constituting the beam and across the full range of possible radius of curvature variation, in this

way assuring that the expressions’ level of accuracy is kept for the entire range of structural systems

encompassed, i.e. from straight beams to highly curved ones.

These in-plane stresses may be compiled as follows (for rectangular cross sections):

Normal stress

𝜎𝑁 =𝑁

𝐴[𝑟𝑛

𝑟+

𝐴𝑒

𝐼𝑧(𝑟 − 𝑅)] −

𝑁𝑟𝑛

𝐴𝑒𝑏𝑟[1

𝑟(𝑅𝐴′ − 𝑄) − 𝑅 (𝐴𝑚

′ −𝐴′

𝑟𝑛)] 𝑓𝑜𝑟 0.6 <

𝑅

ℎ< 8 (3.124a)

𝜎𝑁 =𝑁

𝐴[𝑟𝑛

𝑟+

𝑟

𝑅− 1] −

𝑁r𝑛

𝐴𝑒𝑏𝑟[1

𝑟(𝑅𝐴′ − 𝑄) − 𝑅 (𝐴𝑚

′ −𝐴′

𝑟𝑛)] 𝑓𝑜𝑟

𝑅

ℎ≥ 8 (3.121b)

𝜎𝑁 =𝑁

𝐴 𝑖𝑓 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝐵𝑒𝑎𝑚 (3.125)

Bending Stress

𝜎𝑀 =𝑀(𝑟−𝑟𝑛)

𝐴𝑒𝑟+

𝑀𝑟𝑛

𝐴𝑒𝑏𝑟(𝐴𝑚

′ −𝐴′

𝑟𝑛) 𝑓𝑜𝑟 0.6 <

𝑅

ℎ< 8 (3.126a)

𝜎𝑀 =𝑀(𝑟−𝑟𝑛)

𝐴𝑒𝑟+

𝑀𝑟𝑛

𝐴𝑒𝑏𝑟(𝐴𝑚

′ −𝐴′

𝑟𝑛) 𝑓𝑜𝑟

𝑅

ℎ≥ 8 (3.123b)

𝜎M = −𝑀𝑦

𝐼𝑧 𝑖𝑓 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝐵𝑒𝑎𝑚 (3.127)

Transversal Shear Stress


𝑏𝐴𝑒(𝑅−𝑦)2 𝑓𝑜𝑟 0.6 <

𝑅

ℎ< 8 (3.128a)


𝑏𝐴𝑒(𝑅−𝑦)2 𝑓𝑜𝑟

𝑅

ℎ≥ 8 (3.125b)

𝜏𝑉 =𝑉𝑄

𝐼𝑧𝑏 𝑖𝑓 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝐵𝑒𝑎𝑚 (3.129)

Von-Mises Equivalent Stress

𝜎𝑉𝑀 = √1

2[𝜎𝑁

2 + 𝜎𝑀2 + (𝜎𝑀 − 𝜎𝑁)2] + 3𝜏𝑉

2 (3.130)

With constant attention to the fact that, in all the above, the following applies

𝑒 = 𝑅 − 𝑟𝑛 = 𝑅 −𝐴

𝐴𝑚 𝑓𝑜𝑟 0.6 <

𝑅

ℎ< 8 (3.131a)

𝑒 =𝐼𝑧

𝑅𝐴 𝑓𝑜𝑟

𝑅

ℎ≥ 8 (3.128b)

𝑒 = 0 𝑖𝑓 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝐵𝑒𝑎𝑚 (3.132)

53

𝑄 = ∫ ∫ 𝑟𝑟𝑛−𝑦

𝑎

𝑏

2

−𝑏

2

𝑑𝑟 𝑑𝑧 =𝑏

2[(𝑟𝑛 − 𝑦)2 − 𝑎2] 𝑖𝑓 𝐶𝑢𝑟𝑣𝑒𝑑 𝐵𝑒𝑎𝑚 (3.133a)

𝑄 = ∫ ∫ 𝑦 𝑑𝑦 𝑑𝑧𝑦

−ℎ

2

𝑏

2

−𝑏

2

=𝑏

2[(

ℎ

2)

2

− 𝑦2] 𝑖𝑓 𝑆𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝐵𝑒𝑎𝑚 (3.130b)

𝑄′ = ∫ ∫ 𝑦 𝑑𝑦 𝑑𝑧−𝑒+

ℎ

2𝑦

𝑏

2

−b

2

=𝑏

8[(2𝑒 − ℎ)2 − 4𝑦2] 𝑜𝑛𝑙𝑦 𝑓𝑜𝑟 𝑆ℎ𝑒𝑎𝑟 (𝑐𝑢𝑟𝑣𝑒𝑑) (3.134)

As a note, it must be said that all parameters participating in the expressions above were already

previously introduced and defined in this work. In order to check the so-called continuity across different

𝑅 ℎ⁄ ratios, countless charts were built. Just to exemplify, let’s arbitrarily choose and show two of these,

for instance, the variation with ratio 𝑅 ℎ⁄ of the shear stress at the neutral layer (Figure 3.28 (left)), and

the bending moment at the outer layer (Figure 3.28 (right)). Both of these charts were computed for the

same particular cross-section of the leaf-spring, which is located somewhere near the middle of this

qualitative leaf-spring model provided with common geometrical and physical properties, and whose

shape is similar to the leaf-springs shown in Figure 2.1 (right). Although it is not important for the

qualitative continuity analysis we intend to produce, the loading conditions at this cross section are:

positive bending moment, negative transversal force and negative normal force (compressive). The way

this is done is by keeping all properties constant (including thickness), while varying not only 𝑅, but also

all other dependent parameters.

Figure 3.28: Variation of transversal shear stress at neutral axis (left) and of bending stress at outer fibre (right)

for an increasing R/h geometrical ratio.

Later on, similar variations across an entire cross-section of all participating stresses, for a particular

case of study, are also shown, complementing this previous visualization. Anyhow, the usage of all the

expressions compiled above was carefully validated and represents indeed an important tool for the

overall stress analysis provided by our MATLAB program.

In this regard, the defaulted way our program follows, looks into every discretised state vector and

defines, for that particular cross-section, four layers of major interest, which were pre-defined as being:

the inner fibre layer (𝑟 = 𝑎, 𝑦 = −𝑒 + ℎ 2⁄ ), outer fibre layer (𝑟 = 𝑐, 𝑦 = −𝑒 − ℎ 2⁄ ), centroidal axis layer

(𝑟 = 𝑅, 𝑦 = −𝑒) and neutral axis layer (𝑟 = 𝑟𝑛 , 𝑦 = 0). With this methodology, we then are able to analyse

our suspension, or any other system, in a local manner. Meaning, it allows us to clearly identify and

54

compute not only the critical cross section of the system, but also the particular critical point (in this case

not a “point” but a “layer”, to be precise, since we are working with planar models) within the section.

This is programmed as a default routine procedure so that every time a calculation of the equivalent

local stresses at these four points is undergone, from which the highest value between them is then

chosen to represent the critical equivalent stress for the entire section being studied. This is also a

conservative approach, but it just follows what is considered common practise in mechanical design.

We could easily proceed in a more exhaustive way by computing all stresses for every single layer of

every single cross section, following a comparison between them all, in fact, such a procedure is also

included in the program but only for the critical section, where we plot the distribution all stresses over

it, but, as an engineer, we already know from our experience that the highest equivalent stress will

always be located on either one of those four major layers of interest previously detailed, therefore we

save some processing time by proceeding as urged in the last paragraph.

3.3.12. Buckling Analysis

In this chapter, we justify the absence of a buckling analysis. For the particular operation synthetized

and adjacent to the innovative design proposed for our leaf-spring suspension, shown in Figure 2.1

(right), such kind of failure mode is not existent, thus this analysis is not needed. However, as a

mechanical engineer, in order back so vehemently such statement we need to properly explain it.

In this respect, we need to understand that buckling analysis, as a mechanical failure mode, consists

on the study of a particular kind of instability prone to happen in structural members under specific

loading conditions. Usually it is due to compressive stress, but it always leads to sudden loss of load-

carrying capacity of the beam, accompanied by roughly predictable deformations, which may lead to

failure or prohibitively condition the operation ability.

Closely linked to the modulus of elasticity and the second moment of area, being related to both physical

and geometrical properties of an element, the truth is that, on the other hand, this phenomenon’s

existence is strictly related to the loading and boundary conditions during operation. Knowing this,

buckling is most often discussed in slender columnar subjects, where, although they might be easily

deflected even under considerably low transversal loads, they might offer high initial inertia to deform

under compression loads, hence existing a threshold from which the column gives out and becomes

suddenly unstable. For this critical threshold, it is important to compute the associated elastic buckling

load, so to avoid it when under mechanical operation.

The previous situation, of members under axial loading, together with a couple other situations comprise

most of the cases where buckling design may be preponderant. One of these are columns with cross

sections that offer low torsional stiffness, such as channels, structural tees, etc., who are then subject

to a less common but equally important flexural-torsional buckling. On the same note, thin walled beams,

such as our leaf-spring, may also undergo pure torsional instability, however our research work is not

comprehensive enough to include out-of-plane loading conditions. The last major case of buckling may

occur, for instance, in restricted (boundary-wise) curved structural elements where the radius of gyration

of the cross section is negligible compared to the radius of curvature of the bar, when these are subject

to radial loading in the centripetal direction relative to the curvature of the beam, opposite to the positive

55

radial direction. The same happens during in-plane loading of rings subject to uniform external pressure,

such as pipes or pressure vessels, being this pressure initially directed to the center of the ring [103][69].

Most important of all, and the major point we intend to reach from all those previous descriptions, is that

the buckling phenomenon always occurs in situations of structural stability (while restricted either

structurally, due for example to closed cross-sections, or due to imposed boundary conditions, as

supports) which are being “contradicted” by the applied loading. Hence, when at the primary stage of

synthesis, we opted to choose an elastic thin leaf-spring whose profile was already initially manufactured

in a seemingly deformed shape, we immediately got ourselves out any buckling problems, because, in

this way we are guaranteed to have a shape which, when upon the initial operational load resultant from

the release of the vehicle’s self-weight over the suspension, is already “virtually” unstable considering

the direction of loading, resulting in an immediate and correct deformation of the leaf spring. Of course,

under any operational condition, the instability state is always continuously maintained. Additionally, the

leaf-spring’s degree of stiffness is also carefully designed to guarantee that at either position of

maximum amplitude there isn’t a chance of profile stretching to occur, which would otherwise lead to

other possible instabilities, such as the profile’s curvature direction to invert. Anyway, this is actually

almost common-sense, being similar to the equally common phenomenon shadowing rigid multi-link

mechanism design, where linkage “lock up” is an issue [105][106].

3.3.13. Arch Element Versus Straight Element

It goes without saying that the model thoroughly explained up until this point, and developed as a tool

to analyse our innovative leaf-spring design, is obviously the latest version we were able to fully

conceive. Nonetheless, as one might expect, the hard process of research to reach this point is

constituted of very small steps and obstacles, therefore, one can imagine that for every step of the way

a model with unique characteristics in and of itself was built. In between such levels of acquired and

applied knowledge, we highlight the upgrade made from a MATLAB program able to handle the inputted

data into a model uniquely constituted by multiple straight beam elements, into one able to make a more

accurate approximation of the real system by employing arch-like curved beam elements instead. The

increase in complexity was major, but, fortunately, the quality of the results was equally escalated. Thus,

we will in this chapter proceed with a quick browse over these improvements.

Following this “complexification, we ended up with two programs, which, although very different, share

all the same ramifications and basic structure. In order to build the version which focuses solely in

straight beam field elements for the model approximation, we used a very interesting formulation derived

by Onyeyili et al. [107], where a stiffness matrix, of a plane element with six degrees of freedom, was

developed taking into consideration the effects of shear deformation. Regardless of this core difference,

each of the developed models are always programmed to undergo the same set of TMM analysis,

namely, LSEA, DFVA and a stress analysis, from where all results are collected and used to build the

most varied kinds of charts.

Therefore, regarding the anticipated improvements on the results relative to the TMM linear static

analysis and stress analysis, it must be said that these were clearly verified at all levels. Such differences

are registered in all displacements and internal efforts distributions without exception. In the following

56

paragraphs we will be able to verify this, demonstrating with some illustrative examples, which were all

taken for the exact same situation, where only the number of discretized field elements used to discretize

each leaf-spring planar chain-system is varied. The latter’s initial geometry, following the case in Figure

2.4, can be seen in Figure 3.29 (above) (herein discretized into 10 field sections), therein together with

its final deformed profile, which is caused by a positive vertical force applied at the tip where the steering

knuckle is connected (rightmost-side), simulating the initial evenly distributed release of an averagely

weighted loaded heavy-duty eight-wheeled vehicle (20 ton) on top of a set of individual suspension

systems, each of them composed by four 4mm thick, 105mm wide, and 571.5102mm long, constant

cross-section AISI 1005 Steel leaf-springs per wheel.

Figure 3.29: Individual leaf-spring’s both 2D non-deformed and deformed profiles (above). Vertical deflections at

leaf-spring's tip, registered for different models with different discretization levels (below).

Since, for the synthesized suspension system developed, the most intuitive parameter to analyse is in

fact the vertical deflection of the leaf-spring at its tip, i.e. at the steering-knuckle connection, because it

gives us a direct and intuitive reading about the movement of the wheel itself, then, let’s use this

particular punctual displacement of the system to exemplify the differences in methodology registered

when proceeding with a TMM linear static analysis (Figure 3.29 (below)).

From the previous plot, it is easily observable that the curved-beam model is much more accurate for

the same level of discretization, being also much more convergent. In opposition, the straight-beam

model would require a much higher degree of discretization to achieve similar results. For instance,

when using as comparative value the lastly computed “asymptotic” result outputted from the curved-

57

beam case, the error associated to using 8 curved-beam elements (0.455%) is only comparable to one

made up from 19 straight-beam elements (0.418%). This is considerable.

A similarly built chart showing the evolution of the maximum equivalent Von-Mises stresses, with the

increase in number of discretized elements (Figure 3.30 (left)), does not present any surprising or

highlightable result, wherefore following the previous one, where the curved-beam model has a

considerably higher converging rate, and both lines tend to stabilize for sensibly the same degree of

discretization. However, although the error between both curves for 30 elements is as small as 0.803%,

the straight-beam model will only be within ±10 𝑀𝑃𝑎 to the curved-beam result when both are computed

for 182 elements, which is a lot considering the curved-beam model can guarantee that same condition

from 20 elements onwards. This is a major advantage, and, note that it is expectable that such results

may only become more disparate the more curved is the system being studied or the tougher the loading

conditions applied.

Figure 3.30: Maximum equivalent Von-Mises stresses (left) and total computational processing times (right),

registered for different models and different discretization levels.

Although we already left a strong suggestion towards choosing to proceed with the curved-beam model,

one must, even if only for awareness purposes, look at the possible increase in computational

processing time associated to the higher degree of complexity inherent to the usage of curved-beam

elements, which can be seen in Figure 3.30 (right). From the latter chart, it is immediate that, for the

same number of elements, the curved-beam model takes up a much higher CPU time to compute,

2.5359 times in average to be precise. Nonetheless, if instead we compare the computational times

measured only between similar results, such as between eight curved elements and nineteen straight

ones, the increase in time stands at 22.2%, which is a more reasonable way to analyse, nonetheless,

it’s still a fair few.

Despite this seemingly demoralising increase in processing time, this is actually deceptive and it will end

up being almost neglectable, because until now we were only reading the times to realise both TMM

linear static and stress analysis, but, when we command the program to undergo its embedded TMM

DFVA too, then, the total times increases emphatically (see Figure 3.31 (left)). The total time function

considering dynamics is now almost exponential for both cases, becoming now our first priority to seek

for high converging solutions that require as low of a discretization level as possible. Hence the curved-

beam model’s tremendous importance in this regard.

58

Figure 3.31: Total computational processing time with TMM DFVA (left) and first three natural frequencies of

vibration (right), registered for different models and different discretization levels (left).

The chart relative to the first three natural frequencies of vibration (Figure 3.31 (right)) develops as

expected, in the sense that the curved-beam model continues to be more convergent and less erratic.

The results between the two models tend to be relatively similar from ten discretized elements onwards,

following an asymptotic stabilization towards higher discretization levels (not fully shown). Before, we

had stated that TMM programs can calculate natural frequencies of very complex systems with particular

ease, however the results outputted tend to become more and more degraded the higher the number

of the natural frequency being computed. This is now fully corroborated.

3.4. Non-Linear Quasi-Static Loading Static Analysis

While seeking for a pragmatic approach to further improve our linear elastic static results’ accuracy, and

considering the following chapter of this work will immediately delve into the design of a highly flexible

and intently innovative suspension, it is obvious that, despite the enormous effort put in all the introduced

novelties, as well as the feat of having taken so far down the road an own-developed TMM methodology,

it is predictable that, if we continue using the method as it is, it will not result in successful optimization

solutions for our suspension.

Therefore, in due course, we were introduced by Prof. Jing-Shan Zhao to an astute way of getting exactly

what we were seeking for, which is to find a cost-effective (especially time-wise) way to get considerably

more accurate results, and, ultimately, more applicable to the suspension design process. The technique

we are addressing is called quasi-static loading static analysis. From where the second “static” is usually

omitted, but it was herein, and only once, represented as a reminder of this method’s true nature.

59

In this respect, all that needs to be done is, instead of regularly running the methodology and getting its

linear static prediction under the static load considered, now, each and every static load applied to the

system needs to be democratically split into “small fractions”. Then, the procedure consists in repetitively

running the analysis for the same number of times as the chosen number of load fractions, while, at the

beginning of each iteration, one must always use as initial system, the one resulting from the previous

application of force, being the applied forces to every iteration strictly the so called small fractions we

initially obtained.

Perhaps it is difficultly understandable, but, it is basically like building a house one brick at the time. In

essence, now the outputted result is no longer like a still picture, it is more like a motion picture movie.

Regardless of the metaphors used to describe it, what effectively happens is that, by proceeding in this

way, we are somehow bestowing a certain geometrical non-linearity to the model. Because, despite

every load fraction being the same, in every “frame” the initial system is different, therefore the occurring

displacements will not follow a linear distribution.

This is questionably a reasonable method according to scientific standards, since there is no apparent

formulation governing such non-linearity, at least in this work we didn’t consider any. However, we

personally verified that this method presents a good consistency, or at least apparent convergence, of

results for an increasing degree of quasi-staticity. Therefore, perhaps we are simply not well informed

about its true scientific background. Nonetheless, the fact is, that this method absolutely provides for

considerably more realistic results, hence, we are eager to demonstrate its potential during the

suspension’s mechanical design process. And, it is oddly interesting to realise that a TMM LSEA ends

up being no more than a sub-case within quasi-static TMM, where a single increment of force is used.

Having already introduced all models of this thesis, the following chapter compares all results with

literarily accepted ones. For such, a simple, but very interesting, demonstrative numerical example is

followed.

3.5. Numerical Example

In order to benchmark the so far proposed TMM methodologies, we will consider a particularly

comprehensive example, which is hugely popular for its simplicity but also for its wide usage in academic

literature for benchmarking purposes.

The example is one of a straight cantilever beam, as in Figure A.1 (left) of Appendix A. The beam is

anchored on extremity B, while A is released. The beam is subject to a vertical for 𝐹. We considered a

constant rectangular cross-section of 105mm in width and 7mm in thickness, the length of the beam is

500mm, and it is composed of an typical AISI 4340 Steel.

Both, TMM LSEA and TMM Non-Linear Quasi-Static Analysis (NLQSA), are performed side-by-side with

their analytically formulated FEM counterparts. Regarding the later, results relative to a linear analysis,

as well as, to a true (given that quasi-static is not) non-linear analysis, were collected based on the direct

application of the analytical FEM methodology derived in detail on [81]. In this respect, and shortly, the

geometrically non-linear analysis formulation, which is ultimately executed following an FEM procedure,

is characterised for having been analytically formulated through the potential energy (this time not taking

a small rotations assumption, but rather a moderate one by using up to the second term of Taylor’s

60

series expansion), where Rayleigh-Ritz’s Method, with the transversal displacement field approximated

by means of cubic Hermite’s functions, is followingly applied with the purpose of obtaining the stiffness

matrix and the vector of forces for the finite element formulation. After doing so, the solution of the

followingly presented non-linear system of equilibrium equations, whose expanded components can be

found on Appendix A, is numerically obtained resorting to Newton-Raphson’s Method, for a singularly

discretized Euler-Bernoulli’s beam-bar 6-DOF finite-element, in order to avoid elevated complexity:

𝐊0𝐝 +𝐸𝐴

2𝐊1𝐝 +

𝐸𝐴

3𝐊2𝐝 = 𝐅 (3.135)

In which, 𝐊0 is the “traditional” stiffness matrix, while 𝐊1 and 𝐊2 are the stiffness matrices whose

components are, respectively, linear and quadratic functions of the nodal DOF’s, or local displacements,

compiled into the column vector 𝐝. To finalize, it is known that the linear analysis results from the

application of the system in equation (3.135), but only considering the linear matrix K𝟎.

On our TMM methodology’s behalf, given that complexity is not increased in any way by employing a

better discretization (only processing time is), we assumed a 5-element per beam discretization of the

beam. As to the NLQSA, it should be added that different degrees of quasi-staticity (or number of force

increments) were employed for academic purposes (1, 5, 10, 20, 50, 80).

The results obtained are synthesized on Table A.1 of Appendix A, and plotted in the following chart:

Figure 3.32: Comparative results between different methodologies applied to the numerical example.

As closing note, this numerical example, not only came out to be an interesting and significant

demonstration, as it corroborated everything what had been discussed beforehand in subchapter 3.4,

but, also, the final figure obtained is inexplicably gratifying to look at, as it absolutely complies to the

typical literary comparative representation used to teach the differences between linear and non-linear

analysis, Figure 3.1, amongst which, Quasi-Static’s spectrum may now be localised. Obviously, if higher

degree of quasi-staticity was employed the results would be better and better, however, we believe it

quickly starts to lose its cost-effectiveness. For this reason, and it can be already stated, all following

quasi-statics analysis will be employed for 80 increments of quasi-static loading.

61

4. Leaf-Spring Suspension Mechanical Design Process

4.1. Briefing

4.1.1. Starting Model

Refer to model previously presented on Chapter 2. However, one of the rigid-rigid set of joints bounding

each LS - see Figure 2.1 (right) - more precisely the ones making the connection with the steering

knuckle, are instead replaced by revolute pairs. At this point, there was no justification for such aside

the author’s initial intuition that the previous model would result too stiff for the purpose.

4.1.2. Heavy-Vehicle Synthesis

Dongfeng’s team of engineers always kept themselves one step ahead of our academic team, so much

so, that they took the chance to prototype and fully install, on a tractor unit’s front axle, a suspension

mechanism following our first guidelines. Their purpose was to immediately start with some fatigue

testing. Pictures on this milestone, are documented in Appendix C, among which can be seen that the

tractor unit has 10 wheels/contact points with the ground. However, based on the assumption that all

axles are to be equipped with a pair of independent suspension systems, and, since, for simplification

purposes, an evenly distributed load across all points of contact with the road is assumed, we are not

going to accept any model with four-wheeled axles. Moreover, we suspect that a tractor unit was used

on the prototype’s demonstration simply due to its convenience, since, as far as we know, the unique

heavy vehicle is still in process of design, and all schemes handed around clearly regard a 4-wheeled

2-axled heavy-vehicle (example on Appendix D). Thus, we also seek to contribute towards that direction.

According to [108], vehicles are commonly categorized within ten weight classes (from 1 to 8b). The

purpose of our vehicle is to be state-of-the-art, so we would like to first categorize it and then make sure

we out-perform all competition on its market. The truck class selected is number 7. First, because it is a

perfect fit with the previous synthetization, as this is a typical four-wheel heavy-duty class, couple

examples being concrete and fire engine trucks. The other reason, is that this class really plays the

frustrating “always-second-place” role, being one class behind in both vehicle’s gross weight rating

(GVWR) and empty weight rating (EVWR). Consequently, the fact that it is lighter, makes for an almost

cut in half payload capacity share in percentage (%PCS) in relation to his two “big brothers”. Knowing

this, let’s embrace the size limitation, but still reach out to equal the highest %PCS, i.e. 200%.

After some instrumental ghost-optimization, we are satisfied with the followingly proposed values.

Therefore, the vehicle has a GVWR of 33069.3393 lbs (class’s max. is 33000 lbs) and an EVWR of

11023.1131 lbs (class’s min. is 11500 lbs), making up for a perfectly rounded %PCS of 200%. This, in

turn, gives us an averagely loaded vehicle weight (ALVW) (at 100% %PCS) of 22046.2262 lbs.

4.1.3. Critical Load Condition

From the assumptions and loading conditions described above, we can, for example, compute the

weight transmitted to a single independent suspension (IS) for an ALVW, which is 5511.5566 lbs. Then,

by assuming the force transmission ramifies perfectly throughout the symmetrical suspension system

designed, it is possible to state that each LS has under these conditions a vertical force applied at its tip

62

of 1377.8891 lbs or, a metrically intuitive, 625.0000 kg. Homologously, we obtain for the GVWR and

EVWR cases, the respective two values, 938.0000 kg and 312.5000 kg.

However, despite these being important case-studies, none of them come even close to represent the

ultimate critical load condition, which, as one would expect, necessarily stems from dynamic “vehicle-

highway interactions”, as, in order for the suspension to operate over obstacles amidst the vehicle’s

journey, it is subjected to numerous peaks in acceleration, which, structurally speaking, may translate

to tremendous applied forces leading to challenging stress settings do deal with.

Therefore, undergoing an as comprehensive as possible analysis of the different interactions between

the wheel and the road becomes imperative for a valid mechanical design procedure. Obviously,

statically and analytically speaking, the best we can do is to seek to evaluate a set of loading conditions

identified as being the most problematic, and then, make sure the model withstands such conditions

within a rather loose margin of safety. Evidently, TMM may be usefully handled on an optimization script,

since, although static and dynamic analysis may lead to shockingly different results, the truth is that, to

a certain extent, the outcomes from a panoply of static analysis can be relativized and expeditiously

compared with each other, efficiently assisting to differentiate options by their potential.

Fortunately, there is enough literature available concerning dynamic tire forces, wherefore we were able

to gather invaluable experimental data on vertical wheel loads for a comprehensive range of scenarios,

namely, while: cornering, accelerating, going on road with slope, going past a step-like obstacle on the

road, and, the icing on the cake, while performing a so-called “complete road test” [109]. All experimental

results can be seen on Appendix E.

However, non-expectably, the academic worth of [109] is not on the experimental points per se, as,

despite having conduced the experiment on a four-wheeled heavy truck-unit, its measured weight only

comes close to our vehicle’s empty condition (Figure E.2), being still a far stretch to the sought critical

condition at full-load. Going straight to the point, Koulocheris, after analysing the experimental points,

concluded that, in all cases, the final linear regression model showed that measured wheel loads follow

Newton’s Second Law applied on the wheels, as all the fitted equations comply with:

𝑭𝑺𝑭𝑵 = 𝑷𝑺. 𝒂𝒗 + 𝒆𝒓𝒓𝒐𝒓̅̅ ̅̅ ̅̅ ̅̅ (4.1)

In which, 𝐹𝑆𝐹𝑁 stands for the vertical force applied on each independent suspension system, while

considering the initial suspension’s profile to be in its neutral position, 𝑃𝑆 is the force in kgf distributed

by the vehicle on each suspension, 𝑎𝑣 is the vertical acceleration, and 𝑒𝑟𝑟𝑜𝑟̅̅ ̅̅ ̅̅ ̅̅ represents the average

error of this assumption, which, henceforth is considered as the value of an arithmetic average operation

between all zero intercept values from the ten available linear regression equations, thus −45.411.

This premise is even more solid for the most important experimental case-study, the complete road test,

compare Figure E.2 (knowing the measurements were made on the rear wheels) to the linear regression

equations obtained at Figure E.3. Nonetheless, in order for one to be able to use equation (4.1) to

effectively extrapolate the vertical forces involved at the critical load condition, the vertical acceleration

at such condition need to first be identified. In this regard, one would wrongly think that we just need to

consider the highest acceleration experimentally registered (30 𝑚 𝑠2⁄ ) and get on with the corresponding

analysis. But, it is not as simple as that. For such high-performance mechanisms, as suspension are,

specialized design becomes the guiding norm at approaching the market. In sum, the critical loading

63

condition should comply to the product’s context of usage, at risk of, otherwise, not only make for an

unfairly difficult and limitative engineering challenge, as also, truth be said, of blatantly prejudicing the

chances of achieving a good design solution. The following subchapter provides an important guiding

tool on this respect.

4.1.4. Derivation of a Suspension’s Flexibility Related Parameter, 𝚻

On a suspension system, flexibility is arguably, but rather commonly, looked up as an important

performance measure, probably, given its visually appealing outcome. In this sense, a suspension is

the more flexible the lower is the peak vertical acceleration necessary to elevate it up to its maximum

amplitude, which, for example, allows for cruising undamagingly over a treacherous obstacle course at

low speed. Additionally, and complementarily making up for comfortableness (given also the compulsory

existence of a proper dynamic damping solution), we introduce another important performance

measure, designated responsiveness, which for sake of the previous example, results, upon

improvement, on a clear boost to the vehicle’s ability to undamagingly cruise over the same course at

slightly higher speeds, covering with the same degree of thoroughness the same range of amplitudes

performed at lower speed. The challenge of reaching a design able to encompass all these aspects, is

seized on this thesis with the very own-carved static optimization tools priorly proposed. As to

considerations towards a dynamically viable solution, we refer to chapter 5.

Going back to the flexibility, being this the main purpose on this subchapter, we did not intend, with the

previous paragraph, to suggest that an as flexible as possible solution should be sought for. This would

be erroneous, as we shall explain next.

First, such flexible suspension may also be described as soft, behaviourally speaking, and, one of its

possible implications, is its higher proneness to large-amplitude vibrations, thus requiring a much better

associated dampening solution. Furthermore, we need to acknowledge the unconditional incompatibility

between having a highly flexible suspension and wanting to comfortably drive in regular high-speed

roads. This is not possible, since, even typically derisory obstacles, would, in this case, cause peaks in

acceleration for which the flexible suspension was not designed to handle, resulting in a wobbly driving.

Summarising, if on one hand, we wish to have a vehicle able to adjustably and properly handle any

driving condition, i.e. within a full range of possible vertical accelerations between [−30, 30] 𝑚 𝑠2⁄ , as

shown in Figure E.3 [109]. On the other hand, there is the battling wish to achieve an extremely flexible

and responsive suspension, able to malleably grip to any obstacle on the road. However, the latter

inevitably implies to have a dangerously low allowance for vertical accelerations. Hence, a

compromising middle ground needs to be found.

This being said, and knowing that we were assigned to design a suspension for a highly categorized

(weight-wise) cargo vehicle, we may assume this will not move at great speeds. Moreover, it was not

provided any information regarding the necessity for an all-terrain vehicle. Therefore, we believe it is

important to propose a suspension which allows, both a comfortable ride under any so-called normal

driving condition, at slightly faster speeds, and, also proper operation ability and flexibility to allow for

safe fully-loaded and low-speeded trips. Hence, in this work we will demonstrate the design procedure

on how to come up with the exact physical and geometrical suspension design to sensibly fit all needs.

64

After much consideration, we came up with an interesting way to approach the design problem

described. In fact, we could start by predefining the critical vertical acceleration, out from [109], and

then, calculate, for the fully-loaded vehicle, what is the most critical load scenario (equation (4.1)), which

is followingly introduced into our TMM model so to visualize the respective results. Alternatively, which

is how we proceeded, we instead start by imposing de ultimate deformation scenario on our suspension,

having then TMM give us the corresponding causal loading. From there, it is immediately establishable

the value of the maximum vertical acceleration associated to the ultimate operational position.

In order to evaluate the results, we will consider that this

vertical acceleration, associated to the ultimate

operational position, and which drags with it the rest of

the suspension accelerations range of operation, must

fall down between 15 ± 4.5 𝑚/𝑠2. Such range, painted

green in Figure 4.1, results from our personal

understanding on which ranges could or could not be

accepted as effectively being reasonably all-

encompassing, while simultaneously equally

reasonable at allowing for as wide as possible elastic

deformations to occur under low accelerations.

Obviously, both purple and dark orange painted areas,

showcase the respective spectrums of operational

accelerations associated to each of the previously

selected bounding conditions. As to the negative

accelerations, these are needless be shown in their full

spectrum, because not only weren’t they experimentally

registered beyond what is represented, as, most

importantly, due to the particular adopted design for the

LS to be manufactured, which is explained in chapter 2,

it always performs much better when flexing

downwardly, largely surpassing the safety coefficient

registered at the upwards-most point of operation,

which is always the critical position of operation.

In this connection, we decided to derive, inspired in equation (4.1), an expression for a dimensionless

parameter able to provide in a single number an evaluation of the suspension’s accelerations range

position within the chart before shown, i.e. able to evaluate in a single number it’s flexibility. This

parameter was called Suspension Flexibility Related Parameter, Τ, and its expression is:

𝚻 =𝟒𝒈(𝑷𝑳𝑺𝑴𝑯𝑱 − 𝑷𝑳𝑺𝑵) − 𝒆𝒓𝒓𝒐𝒓̅̅ ̅̅ ̅̅ ̅̅

𝟒𝑷𝑮𝑽𝑳𝑺𝒂𝑯−𝑺

(4.2)

In which, 𝑔 is the gravitational acceleration, 𝑃𝐿𝑆𝑀𝐻𝐽 the necessary force in kgf applied to each LS in order

to achieve SMHJ, 𝑃𝐿𝑆𝑁 is the kgf applied to each LS so to achieve its neutral position, 𝑒𝑟𝑟𝑜𝑟̅̅ ̅̅ ̅̅ ̅̅ was already

described, and 𝑃𝐺𝑉𝐿𝑆 is the kgf distributed by each LS under the vehicle’s full-load capacity (GVWR).

Figure 4.1: Project guidelines with respect to

possible suspensions' range of operation ability.

Acceptable range for ultimate operational position’s

causal acceleration painted green. Purple and dark

orange, showcase the two full-spectrum of

acceleration range of operation, bounding both

acceptance margins. (adapted from [109]).

65

In this respect, Τ equals 1 when the vertical acceleration, associated to the critical case, exactly

coincides with the center of the “green” range, being therefore the ideal scenario. In this particular case,

because the “green” range chosen was 15 ± 4.5 𝑚/𝑠2, 𝑎𝐻−𝑆 is the intended ideal 𝑎𝑣 to cause ultimate

operational amplitude (at critical loading), in this case predefined as 15. Obviously, as a side note, the

latter variable can be used to study completely different ranges, but, as a disclaimer, if we don’t situate

it near the center of the spectrum, then the name for Τ, as flexibility parameter, is no longer adequate,

as instead it is just a parameter to check on how close to 𝑎𝐻−𝑆 our results are. Proceeding with the

explanation, 4.5 is actually an outcome of having previously expressed the desire for a solution to have

a Τ ∈ [0.7, 1.3]. In sum, we predefined that, if Τ < 0.7, then the suspension is considered too soft/flexible

to have a good performance, while, if Τ > 1.3 , it’s too hard/stiff. Hence, if properly adjusted this

parameter’s purpose, can both expeditiously give a notion on the proposed suspension’s flexibility, or,

simply be handily employed as an important design parameter in mechanical design.

4.1.5. Imposed Design Requirements

First of all, knowing this particular TMM’s formulation limitations, namely, not assuming large

deformations, neither three-dimensional true non-linear dynamic analysis, then, we cannot expect the

results to be as accurate as otherwise they would. Additionally, it is also common sense in Mechanical

Engineering that a static safety factor is far less critical than a fatigue safety factor, which relates to

dynamic loading conditions, such as the ones this suspension has to characteristically be subject to. For

this reason, we take advantage of the developed TMM procedure’s convenience, promptness and

speed, to effectively narrow a huge variety of proposed solutions down to a small lot of highly-potential

ones. This is done in accordance to a total of eight carefully established design requirements, namely:

⟹ TMM Static Linear Elastic Safety Factor ≥ 10 under most critical loading condition

Acceptance is awarded if proposed physical and geometrical properties lead to a final static safety factor

not inferior to 10. Being this value selection solely based on the author’s experience.

⟹ Heavy Vehicle’s EVWR ≥ 5 ton (10000 lbs) & %PCS equal to 200%

Due to their implications, both these impositions had to already be addressed at the vehicle’s

synthetisation, see subchapter 4.1.2.

⟹ Averagely Loaded Vehicle Weight (ALVW) Sustained by the Suspension at a Half-Journey of

0±1mm (Perfectly Neutral Position)

Careful geometrical and physical design of LS components, so to guarantee a perfectly stable

suspension neutral position at the vehicle’s average load condition (%PCS equal to 100%).

⟹ Suspension’s Maximum Half-Journey (SMHJ) equal to 100±1mm

The maximum half-journey of the proposed suspension system under operation must equal 100mm

counting from the neutral position, clearly categorizing this suspension system, amongst all traditional

suspension, as an high-amplitude one, with an operation range between [−100, 100] 𝑚𝑚.

66

⟹ Suspension’s Operational Effectiveness Loss at EVWR (EL-EVWR) ≤ 70mm

⟹ Suspension’s Operational Effectiveness Loss at GVWR (EL-GVWR) ≤ 70mm

(ELOPC ≤ 70mm. Where ELOPC equals the highest value between EL-EVWR and EL-GVWR)

The fifth and sixth performance impositions, devolve upon the Suspension’s Operational Effectiveness

Loss Over Payload Capacity Range (ELOPC). For instance, we would never accept a leaf-spring solely

made of rubber, which obviously would easily reach 100mm of half-amplitude, however, this amplitude

is reached for an extremely low and impractical static load, hence the vehicle’s chassis would never be

able to stand still on top of its suspension. To avoid this problem is the suspension should be able to

sustain an averagely loaded heavy vehicle with ideally little to none amplitude variation, this requires a

certain structural stiffness. This, paves the way to the fifth and sixth limitations, which state that, in the

range of loading imposed by the vehicle at EVWR and GVWR, at least 30% of the half operational

effectiveness range of the suspension, previously defined as 100 mm, must be safeguarded. Basically,

this means that, at either EVWR or GVWR, the absolute value of the variation in amplitude relative to

the neutral position (while subject to ALVW) must never surpass the 70 mm.

⟹ Suspension’s Flexibility Related Parameter, 𝚻, between 𝟎. 𝟕 ≤ 𝚻 ≤ 𝟏. 𝟑

Parameter derivation, thorough explanation, as well as, associated physical and performance

implications, were already thoroughly detailed in subchapter 4.1.4.

⟹ Suspension’s Responsiveness Related Parameter, SRP, as close to 1 as possible

The simplistic way we found to loosely grasp the suspension’s responsiveness, was to intuitively state

that a suspension is the more responsive the smaller is the difference between the maximum force (in

kgf) applicable to each LS during operation (at which it reaches 100mm half-journey), designated by

𝑃𝐿𝑆𝑀𝐻𝐽, and, the force (in kgf) transmitted to each LS, due to an evenly distribution of the vehicle’s weight

when at full payload capacity, 𝑃𝐺𝑉𝐿𝑆. To constitute SRP, this is computed through a simple division,

namely, 𝑃𝐿𝑆𝑀𝐻𝐽 𝑃𝐺𝑉𝐿𝑆⁄ . The justification behind such statement is explained by the following premise: If,

by means of vehicle-road interaction, there needs to be employed a minimal vertical force for the

suspension to go from its current state to achieve its maximum state of operability, then, it must mean

the design used is extremely flexible, however, it may or may not be responsive. But, if we make sure

this occurs for a design that concurrently sustains an EL-GVWR within boundaries, then we can state

that the suspension must have an outstanding performance, being not only flexible but responsive. In

this respect, if the maximum force applicable was equal to a GVWR condition, it would correspond to

an SRP equal to 1, from that point on, any extraordinarily applied force, resulting for example from a

bump on the road, will be as responsively reacted to, as closer to 1 is the actual SRP. By the way, SRP

is always > 1, otherwise other project impositions would already be at fault.

4.2. Presumptions on Leaf-Spring’s Material – TMM LSEA Optimization

4.2.1. Brief Setup

In this chapter, a full study on the safety factor under critical static loading conditions, while involving a

deliberately huge panoply of materials, is undergone. Herein, only a selected few, but rather essential,

67

project requirements are considered. All information is computed and collected by means of a set of

custom-made complex MATLAB programs, which iteratively run an even more complex group of

functions, embedded with the previously introduced TMM at its full extent. Once again, such TMM-based

program, is able to nimbly build any open-chain multi-element structural model, and carry on to compute

natural vibration frequencies, normalized natural modes of vibration, frequency determinant function, for

a better evaluation of the safe ranges of frequency during operation, compute 2D deformed and non-

deformed profiles, including visualizations of the nodal extension, deflection and curvature functions in

respect to both local and global referential, compute resultant profile inclination function, compute

distributions of bi-directional components of force and bending moment distribution, these, arranged

along the system’s deformed longitudinal length, also for both local and global referential, compute

corresponding internal stresses’ distributions for all sectional layers of the model, as well as a fully

detailed distribution over the critical cross-section, and, finally, it computes and supplies a distribution

diagram of the equivalent Von-Mises stress, accompanied by the respective static safety factor.

All these results are encompassed within two chooseable, but non-exclusive, paths, a TMM linear static

elastic analysis (LSEA) and a TMM dynamic undamped free-vibrations analysis (DFVA), having these

already been fully characterised in the previous chapters of this work.

4.2.2. Iterative Computational Optimization Procedure

First and foremost, as one would imagine, came a series of roughly drafter individual static tests, being

unmindfully concerted simply to give us a general idea over any obvious interrelations between some

of the different geometrical parameters being handled, particularly, by seeking to relate them with the

overarching static safety factor (SF) parameter. These, are detailed in Appendix A. In this respect, if, for

instance, we use a thicker LS, one would intuitively expect a considerable gain in SF (as validated in

Figure B. (right)), but, inherent to such enhancement, comes an obvious need to employ higher

magnitude forces in order to achieve similar deformation states. Not only that, but, performance-wise,

there are many adjustments that come with even slightest of changes in design. For context, one could

take the project requirement of having to achieve a perfectly neutral profile when supporting the

averagely loaded vehicle, and notice that an increase in the LS’s thickness, would necessarily trigger

the designer towards a mitigating solution, such as by opting for a less concave profile. However, such

stemming modification, could domino effect onto different levels of design, such as suspension

amplitude range, stiffness or ride comfort, possibly hindering an exceptional performance. In fact, most

of mechanical design entails gambling with intricate trade-off situations, ultimately falling on the

engineer’s hands the responsibility to take on action and find adequate solutions.

Regarding this academic challenge, the overall problem clearly results in a considerably complex non-

linear programming optimization problem. Unfortunately, due to the level of complexity associated to the

combination of numerous governing elemental structural equations, common non-linear programming

techniques, such as the Lagrange Multiplier Method or the Kuhn-Tucker conditions [110][110], cannot

be used, and, even resorting to advanced optimization software’s, such as GAMS IDE, would still lead

to extremely difficult implementations, requiring a veteran-like level of know-how, which is not at our

reach.

68

Therefore, an aliased “fully-constrained parameter-by-parameter” approach is followed, meaning, a

specialized own developed program, able to strategically factor in almost all project impositions, and

ultimately outputting a fully optimized landscape around a determined parameter. Due to the already

processually high-revving execution of numerous intricate, and continuously iterative, calculations, as

stated before, only one parameter can be optimized at the time under these conditions. Thus, there is

indeed need for the process to be repeated several times for different parameters, until we achieve a

close-to-optimal solution which satisfies all the project constraints. For this reason, it is not in truth an

optimization program, as it is more of a computational trial-and-error program with a cunning

convergence scheme behind it, nonetheless, for a lack of better word, we will designate it as optimization

program. This approach efficiency will depend on the user’s choice of path, such as when playing around

with which parametric values to input onto each optimization iteration, thus being highly unpredictable.

On the flip side, unpredictability does not preclude efficacy, at which, relative to other non-computational

approaches, it excels.

By getting back on tracks with the current process of design, since we are dealing with a compliant

flexible mechanism, the LS’s material properties will be extraordinarily influential. Hence, we should start

by selecting a range of reasonable materials to kick off the computational tests. Wherefore, given a

certain present sense of uncertainty, we opted to choose fifteen “common” materials, complemented by

an additional group of seven, whose “special” properties, based on Appendix A, led us to believe might

end up rendering quite promising results. All properties at room temperature are followingly summarized:

Table F. in Appendix F.

With this project’s geometrical and physical impositions in mind (subchapter 4.1.5), and with the intent

to develop a trustworthy set of computational experiments, capable of giving us palpable and robust

results, the very first complex multiple optimization MATLAB script was programmed. In it, each and

every one of the materials presented on Table F., are take through the following process:

Figure 4.2: First computational optimization experiment's diagram.

Yet again, regarding this suspension’s spectrum of vertical motion, the most severe situation, is one

generating positive peaks of vertical acceleration, for instance, immediately upon a bump or immediately

after the bump, during the restitution period, when the vehicle chassis tends to fall below the suspension

level. The reason why negative accelerations are not as critical, reside on our particular LS design, as

it was already explained.

69

Back to the first computational run, thickness is the first parameter statically optimized, as it is arguably

the most impactful parameter to the overall performance. Regardless, as stated, of having considered

all other parameters constant, the process is monumentally heavy in terms of CPU time consumed. This

is mostly, as can be seen on the diagram above, because, not only the whole process is repeated for

twenty-two materials, but, within each of these, it will be iterated an egregious amount of times in order

to study a full range of thicknesses varying from 0.01mm to 30mm. As if this wasn’t enough, for each

thickness, within each material, the program run a particularly complex iterative process, where different

initial profiles are sequentially being supplied and tested, only stopping upon finding one that can

effectively guarantee the imposition for neutral position under ALVW. Lastly, it undergoes an incremental

loading procedure, which only stops at the maximum operational displacement, i.e. at the critical

condition, where vital data is collected. It goes without saying that tremendous effort was put onto this

computational methodology, as well as, onto numerous enhancements, which peremptorily made for an

outstanding converging ability.

On another note, in order to reconstruct both all deformed and also non-deformed profiles participating

on the procedure at multiple iterative stages, a piecewise cubic Hermite polynomial interpolation

technique is employed. This played a major role during the ever-changing profile shaping process, since,

discretization procedures abide to certain rules, hence every time a deformed profile needed to be “re-

used” again in another routine, it would first be properly mapped, then deconstructed and finally

reconstructed again, this time according to the rules, but also, according to some cunning node-

positioning techniques we developed to allow for accurate results at strategic points on the system, as

well, as to efficiently avoid accumulation induced errors too (which is especially important for the non-

linear quasi-static loading analysis).

Yet another point that deserves attention, is, without a doubt, the constant “feeding” of possible initial

profiles, which are then relentlessly tested, following a classic feedback loop control structure, until

“perfection” is reached. Well, since the utopic intent is to guarantee a LS design which “curls up”, upon

625kg (ALVW), onto a constant-curvature profile where both caps stay at the same level (within ±1mm),

then, what he cunning process found does, is to simply start from the intended profile, deform it, remap

it, and provide to the main function as possible initial profile to undergo the tests. Of course, this is yet

again another incremental procedure. By the way, just because will was conveyed to have such a neutral

shape, that doesn’t make it any easier, since the actual geometry is still an unknown, so much so, that

later on routines are ran to study cases with multiple different curvatures, within the full range from

concave to convex, as well as having more spaced or closed together caps. For all of them an ideal

profile was found, otherwise the remaining procedure does not take place

Finally, after having traversed such remarkably exhaustive computational process, many were the

figures, out of which Figure 4.3 perhaps stands out, that can be presented. Its “special touch”, resides

on the fact that drawn every does not represent every analysed point, but, it rather comes on behalf of

a considerable cloud of dispersed points, out of which this one selected is their best representative,

being the so-called optimal solution for a particular scenario. Figure G.4.4, works as a tutorial on the

meaning behind the lines drawn in Figure 4.3. On a rather anticlimactic statement, we would like to add

that, unless otherwise stated, all LS’s are discretized into eight field elements.

70

Figure 4.3: Static safety factor for different leaf-spring’s materials and thicknesses, when at the most critical

operational point (100±0.1mm amplitude above neutral stage). Every charted point is associated to a unique leaf-

spring profile which guarantees perfect neutrality under ALVW conditions.

On Figure 4.3, material Quadrant EPP Proteus® Layflat Natural Co-Polymer Polypropylene was not

even regarded, because it achieved an extremely high safety factor, nonetheless, due to its extremely

low young modulus, the necessary forces to achieve the required levels of elastic deformation were too

low to be considered acceptable throughout the remaining design process, for instance, for the

maximum thickness considered above (30mm), it would only take about 8 N to achieve the maximum

amplitude variation, for which it had a safety factor of 126.40.

If, on one hand, we were stunned by how close to the threshold of failure all results were, mercilessly

giving us an immediate wake-up call to the sheer magnitude of the engineering feat we proposed

ourselves to achieve. Perhaps, knowing we were proposing such an unequivocally innovative

suspension design, we should have had the wisdom not to immediately embrace a design for such a

high-performing vehicle, as we proposed ourselves to successfully design an highly-flexible suspension

for an heavy-vehicle. Nonetheless, we will not stand by, as this is indeed such a great challenge.

In this sense, we immediately decided to change our approach. Instead of following with the plan of first

narrowing the best designs down by employing simple linear static analysis, which, although

considerably less accurate, could still quite rapidly provide a comparative stance between a wide variety

of designs relative to each other. However, in the light of the current events, we realised that we are

already in a stage where every little bit of optimizable performance may invaluably give us the much-

sought-after edge. Therefore, given the LSEA’s inaccurate nature at analysing large deformation

mechanisms (also proven in subchapter 3.5), we don’t intend to run the risk of deviating ourselves from

the best possible design solution, which, given the TMM methodologies we previously derived, may only

be reached by employing a TMM non-linear quasi-static loading analysis.

71

4.3. Leaf-Spring’s Material Selection – TMM Non-Linear Quasi-Static

Loading Optimization

First off, we wish to give another chance to some of the previous materials to shine in quasi-statics, as

we cannot really 100% trust the linear analysis. Therefore, we opt to select, as our new first range of

materials, all those highly-performing ones at LSEA, so-called “special” materials, nonetheless, together

with these, we are bringing back the three best performing ones between the so-called “common”

materials. Although the previous analysis had left us perplexed, it truly was a mesmerizing testimony to

the current state-of-the-art in materials engineering, with all the selected composite materials blatantly

leaving the metallic solutions behind, we still wish to take until the end of this thesis at least one metallic

solution, tough, even if only for comparison purposes, or, who knows, for future prototyping, since metals

are known to be much more accessible.

Just to be clear, now we will not simply run NLSQA on the described materials, no, that would not be

enough. Actually, we are instead running a massive NLSQA optimization program (NLSQA-OP). With

the only (major) simplification residing in the fact that we are adopting the previously optimized thickness

(OT). Just to recall the reader, please check on subchapter 4.2.2, for a fully disclosed explanation on

what is undergone during a so-called “optimization program”. Obviously, as we had deliberated for

Quasi-Statics, there is an added compromise of having to divide and proceed, for every step of the way,

in 80 segregated and posteriorly stacked increments. Additionally, it ought to be said that all through-

out, 8 discretized elements per LS is also the rule-of-thumb employed. The results are then:

Table 4.1: First results from employing a single-thicknessed NLQSA-OP on previous best performing materials.

Material 𝝈𝒚

[MPa]

𝑬

[GPa]

𝑮

[GPa]

𝝂 𝝆

[kg/mm3]

OT

[mm]

𝑷𝑳𝑺𝑵

[kgf]

𝑷𝑳𝑺𝑴𝑯𝑱

[kgf] SRP SF

AISI Steel 4340 1178 193 78 0.284 7.85e-06 10 625.00 2766.40 2.949 0.2229

Titanium Ti-6Al-4V 805 121 44 0.34 4.43e-06 15 625.00 5009.78 5.341 0.1756

AISI Steel 9260 (High

Yield) 1149 200 80 0.29 7.85e-06 10 625.00 2868.38 3.058 0.2079

AISI Grade 18Ni

Maraging Steel (Aged) 2395 200 77 0.30 8.08e-06 10 625.00 2868.38 3.058 0.4335

2800 Maraging Steel 2617 160 70 0.30 6.60e-06 10 625.00 2409.50 2.569 0.6088

Sandvik Nanoflex Steel 2400 180 80 0.30 7.90e-06 10 625.00 2664.43 2.841 0.4847

Hexcel® HexPly® 8551-7

Epoxy Matrix, IM7 [0º,

Dry] Carbon Fibre

2760 159 5.86 0.416 1.49e-06 10 625.00 1950.63 2.080 0.8037

HH F650 Bismaleimide

Resin-IM6 Carbon Fibre 2774 165 67.1 0.35 1.51e-06 10 625.00 2460.49 2.623 0.6240

HH 954-2A Curing

Cyanate Resin-G40-800

Carbon Fibre

3017 159 57.6 0.38 1.52e-06 10 625.00 2358.52 2.514 0.7164

Participating parameters, but non-resultant from computational experiment:

𝑃𝐺𝑉𝐿𝑆 = 938.0 𝑘𝑔𝑓

The results are then, still, quite negative. But, we should not loose motivation, as effectively the process

of design has just begun. And, truth be told, this was already expected to happen, because we know

72

from a fact, that LSEA tends to allow quite unrealistically loose deformations, therefore it is normal to

have even lower SF’s than before, despite having already run one time the NLQSA-OP.

Our next decision, was then to try to further, and for the last time, improve our range of studied materials

based on the experience we already collected so far. Wherefore, after running an “advanced search by

properties” on [113], hand-tailored to our needs, once again, all potentially advantageous results ended

up being composites.

Left with no other choice, then really having to deal with composites, then we better properly do so.

Hence, first, sought literature on how to more accurately define composite materials’ physical properties

were sought for. Provided our constant care for rigor, this issue was troubling us, but, an interesting

work on the matter was, fortunately found available, we refer to [111]. Wherefore, from this point on, all

following materials were updated in accordance to the following expression, in its generic form:

𝜉 = 𝜉𝑓𝑉𝐹𝑓 + ξ𝑚𝑉𝐹𝑚 (4.3)

This expression was developed for simple composites with only one kind of unidirectional fibres.

Considering 𝜉 a random property of the material, and 𝑉𝐹 the volume fraction of either the matrix (𝑚) or

fibres (𝑓). Wherefore, for every material we tried to consult their 𝑉𝐹’s. In case such was not possible,

we opted to assume them equal to others belonging to the same trademarked brand, hence, even if

having some error, at least it is fair to say that materials are then mechanically performed under the

same circumstances.

After this, the exact same step as before was repeated, this time, for the second batch of materials. For

which, since they hadn’t had their thicknesses previously optimized from an erroneous LSEA-OP, as

others did, we decided to adopt the same thickness for every one of the current cases, keeping everyone

rest assured. Note, that the best-performing material from the previous series, got the privilege of being

put at the top of this table. This was done for no any reason other than to ease any comparison.

Table 4.2: Repetition of same single-thicknessed NLQSA-OP on second batch of selected materials.

Material 𝝈𝒚

[MPa]

𝑬

[GPa]

𝑮

[GPa]

𝝂 𝝆

[kg/mm3]

OT

[mm]

𝑷𝑳𝑺𝑵

[kgf]


[kgf] SRP 𝚻 SF

Hexcel® HexPly® 8551-7

Epoxy Matrix, IM7 [0º,

Dry] Carbon Fibre

2760 159 5.86 0.416 1.49e-06 10 625.00 1950.63 2.080 0.925 0.804

Hexcel® HexTow™

AS2C Carbon Fibre (62%

𝑉𝐹𝑓)

3584.6 188.7 5.86 0.416 1.80e-06 10 625.00 2154.57 2.297 1.067 0.883

Hexcel® HexTow™

AS4C Carbon Fibre (62%

𝑉𝐹𝑓)

3594.14 197.56 5.86 0.416 1.78e-06 10 625.00 2205.56 2.351 1.102 0.850

Hexcel® HexTow™ AS4

Carbon Fibre (62% 𝑉𝐹𝑓) 3621.58 196.8 5.86 0.416 1.79e-06 10 625.00 2154.57 2.297 1.067 0.873

Hexcel® HexTow™

AS4D Carbon Fibre (62%

𝑉𝐹𝑓)

3893.7 209.66 5.86 0.416 1.79e-06 10 625.00 2256.55 2.406 1.138 0.879

Hexcel® HexTow™ AS7


Hexcel® HexTow™ IM6


73

Hexcel® HexTow™ IM9


Hexcel® HexTow™

PV42/850 Carbon Fibre

(62% 𝑉𝐹𝑓)

4885.84 248.68 5.86 0.416 1.79e-06 10 625.00 2460.49 2.623 1.280 0.961

Because our program is capacitated with the ability to map and track every node on a leaf-springs design

upon their analysis, we thought of accompanying the table above with a more visually appealing point

of view, showing for instance the optimized LS’s profiles during operation at the three most important

stages of design, namely, initial position (as manufactured), neutral operational position (under ALVW),

and, at the maximum/critical amplitude of operation (at which sustains 𝑃𝐿𝑆𝑀𝐻𝐽).

Such a chart, was effectively done, but, given its unclearness and unappealingness, it was instead

nudged to Appendix H. Alternatively, we found a better way to provide an easily viewable panorama

over the profile conditions during operation. Such representation goes as follows:

Figure 4.4: For each possible LS analyzed, it is illustrated its tip position travel starting at the bottom at its initial

(as manufactured) position, then, upon ALVW, reaching a perfectly neutral position within ±1mm, and, finally, also

within ±1mm a maximum amplitude of 100mm is secured for the particular design under 𝑃𝐿𝑆𝑀𝐻𝐽.

Having finished this analysis, we should now take a decision on what material or materials to bring for

the extremely thorough geometrical analysis, as until now almost all focus was on the physical properties

of the design. In this regard, the decision taken is admittedly the most prone to uproar. Actually, at this

stage, all materials are particularly similar. For that reason, we opted to go ahead with three materials,

one would have the highest SF (HHT IM9 Carbon Fibre), another which offered the best responsiveness

and flexibility (HH 8551-7 Epoxy Matrix, IM7 Carbon Fibre), and a third consisting of the best overall

metallic solution (2800 Maraging Steel).

74

4.4. Leaf-Spring’s Geometrical Optimization – TMM NLQSA-OP

For the three elite materials selected, a proper and personalized geometrical solution is going to be

proposed. However, due to organisation concessions, every step henceforth taken is only fully

committed in detailing “Hexcel® HexTow™ IM9 Carbon Fibre”’s case. The remainder two materials will

have all the same exact work put into them, simply only their final results are herein shown. Nonetheless,

the entire progress can still be accompanied to detail throughout Appendix I.

Back to what matters, the next procedure will be slightly different than before. Although It is common

sense that, any multi-parameter optimal solution can be more accurately found if, given a series of

optimizations, the result of the first is used as part of the next and so forth, to be fully honest, we don’t

have enough time to really find the “one and only” optimal solution, so, we just hope to come up with a

more convergent methodology to ultimately obtain an as optimal solution as possible.

In this regard, this will be done in three steps. First, all geometrical parameters we can possibly optimize

will go through a fully-constrained NLQSA-OP centered on them individually, while the rest parameters

are intuitively selected to be close to an expectable final solution. Then, by taking each separately

optimized parameter and put all together, of course we would already get a good solution, but, it is

certainly not one able to fulfill some of the very rigorous performance impositions. For this reason, one

las NLQSA-OP will kind of proceed to the necessary small adjustments in that direction. Ultimately, there

is one last overall analysis, working this simply as celebrative parade to showcase the results obtained.

4.4.1. Geometrical Parameters Parallel Optimization Step

Participating but non-outputted parameters throughout process: 𝑃𝐸𝑉𝐿𝑆 = 312.5 𝑘𝑔𝑓


Thickness Optimization

Table 4.3: Fully constrained NLQSA-OP results on thickness variation (HH IM9).

Hexcel® HexTow™ IM9 Carbon Fibre (62% 𝑽𝑭𝒇)

Thickness

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

4 4.3729 625.0 767.76 0.1003 188.47 143.002 0.8185

6 1.7997 625.0 1134.86 0.3561 155.419 76.4666 1.2098

8 1.1825 625.0 1695.7 0.7470 105.04 45.9597 1.8077

9 1.0554 625.0 2052.6 0.9958 74.005 35.9905 2.1882

10 0.9783 625.0 2460.49 1.2801 49.1902 29.4661 2.6231

11 0.9412 625.0 2970.35 1.6354 35.1983 24.4624 3.1666

12 0.9517 625.0 3480.21 1.9908 26.8639 20.4455 3.7102

14 1.0089 625.0 4499.92 2.7015 17.5325 14.721 4.7973

16 1.0425 625.0 5723.58 3.5544 12.5002 11.0746 6.1019

Since the ideal value

seems to fall in-between

categories, more precisely

somewhere around an EL-

EVWR = 65 mm. Then, we

proceed with a piecewise

cubic Hermite polynomial

interpolation, which is

shown in Figure I.1 (left) of

Appendix I, which allows us

to define the optimal

thickness as 9.3264 mm.

75

However, if taking into consideration the Renard Series, [85], which usually tend to rule mechanical

engineering design, optimal thickness is instead 9.30 mm.

Width Optimization

Table 4.4: Fully constrained NLQSA-OP results on width variation (HH IM9).


Width

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

73.5 1.0016 625.0 1899.65 0.88922 79.5013 39.2197 2.0252

94.5 0.97073 625.0 2307.53 1.1735 56.6980 32.1318 2.4601

105 0.97826 625.0 2460.49 1.2801 49.1902 29.4661 2.6231

115.5 0.97747 625.0 2664.43 1.4223 44.3844 27.0140 2.8405

136.5 0.9851 625.0 3072.32 1.7066 35.3524 23.6494 3.2754

157.5 1.026 625.0 3378.23 1.9198 28.1643 21.6188 3.6015

178.5 1.0449 625.0 3786.12 2.2041 25.9616 18.4285 4.0364

199.5 1.0844 625.0 4092.04 2.4173 22.5521 16.7959 4.3625

220.5 1.1057 625.0 4499.92 2.7016 19.1561 15.9038 4.7974

Length Optimization

Table 4.5: Fully constrained NLQSA-OP results on length variation (HH IM9).


Length

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

386.566 0.63832 625.0 6437.38 4.052 17.6039 13.6887 6.8629

441.669 0.73786 625.0 4397.95 2.6305 25.6632 18.0455 4.6886

496.764 0.85324 625.0 3174.29 1.7776 34.9814 23.8006 3.3841

551.949 0.97927 625.0 2460.49 1.2801 49.6933 29.4898 2.6231

607.16 1.1196 625.0 2001.62 0.9603 66.0119 36.9962 2.1339

662.409 1.3051 625.0 1665.11 0.72575 93.0375 43.6435 1.7752

717.57 1.5013 625.0 1440.77 0.56939 120.3545 52.0833 1.536

772.73 1.7423 625.0 1257.22 0.44146 146.0476 62.1072 1.3403

827.956 2.001 625.0 1134.86 0.35617 172.3946 72.5465 1.2099

Concave vs. Convex Leaf-spring

This issue was, as it should have, initially intuited to generate our first model. However, in order to

guarantee that we are indeed offering a close to optimal solution, we will make a comparative study

between these too alternative, for a random, but quite similar, leaf-spring design.

The results obtained, were the following:

By proceeding exactly in

the same manner, we end

up with an optimal width

value, according to the

preferred series in

mechanical engineering, of

86.16 mm.

As to the Length, it should

be said that lengths were

varied while keeping the

same exact shape,

meaning they were

obtained by means of

magnification. All in all, the

final value for length,

according to the preferred

numbers, 607.16 mm.

76

Figure 4.5: Comparative results between a concave LS design (top-left) and a convex one (top-right).

In this analysis, it is clear that using a convex LS design allows for an outstanding ease of flexibility,

specially accentuate towards the maximum amplitude of the suspension, when in opposition the

concave design tends to become harder to deform. This is huge in terms of overall flexibility and

responsiveness. To be fair, it is also clear that the main advantage of continuing to employ a concave

design (left), resides on its ability to allow for lower Von-Mises stress distributions along the profile

throughout the entire range of operation, which perhaps would allow for a longer suspension life.

Nonetheless, at the critical point, and ultimately most important, the stress level is the same, therefore,

we will for the last optimization adopt a convex design.

Joint Topology

Seems a little odd that only now, already so far down the design route, we would for the first time

question what kind of joint to bound each flexible component. Well, truth be told, this was actually the

very first thing we did when we started developing the first TMM methodology. However, especially now

that the full design is almost complete, we think we should employ the brand new and more accurate

NLQSA to re-check on this important situation.

In this matter, we would like to highlight the fact that our MATLAB is able to receive as an input a vector

of zeros and ones to fully define if each and every degree of freedom on a certain node is released or

fixed. This seems rather simple, but it is one of the many features which in our view make our particular

program so valuable. We always tried to make a program which would be promptly used for any situation

and not only our particular one. Therefore, as it stands the user only needs to change some parameters

confined to a small area called “inputs” and all the remaining code will quite complexly readjust itself if

needed to compute the particular system in hands.

In order to properly analyses the different scenarios a full NLQSA-OP needs to be developed for each

case, as we wish to compare some of these assumptions’ performance, as if they had to really operate.

In this regard all possible combination of joints were tested, as they are not that many, and employing

77

yet again a randomly similar LS as the one we are designing. The results for the best three cases were

conclusive, and go as follows (where the left-side is considered the connection to the frame, and right

to the steering knuckle):

Table 4.6: Joint toplogy optimization.

Joint Topology SF SRP Τ

Rigid – Rigid 1.1991 2.9493 1.4933

Rigid – Revolute 0.9783 2.6231 1.2801

Revolute – Rigid 1.5886 1.2642 0.3917

We are satisfied to see that our initial tests were not wrong, and that, we really did choose the best

possible topology throughout this design procedure.

4.5. Final Remarks

Following the geometrical “optimization”, and given that it involved a lot more parameters, namely, width,

length, thickness, possible joints and leaf-spring’s shape, as had been previously explained, there was

a deliberate decision to proceed with all these individual optimizations in a parallel manner. According

to this decision, such step would be followed by one last optimization to adjust together all parameters

outputted from both the physical and geometrical design, so to fit some of the more rigorous

requirements. The idea behind doing so, was to seek for a more convergent organizational scheme due

to time limitations, however, and despite the methodology developed in itself being working in perfect

conditions, which was ultimately the one thing we intended to showcase, it ended up being a terrible

idea, because the final joint solution, after the parallel optimization step, was far from being acceptable,

as, apparently, all individual optimizations outputted over-compensated parameters, because of the

negative influence of all assumed parameters that remain constant, which were at time given what were

considered to be “reasonable” values. Facing this misfortune, and left with no more time for manoeuvre,

we were inevitably led to a sudden interruption of the design and “optimization” process.

For us, this was not even the biggest misfortune, the true problem was that, due to this, our plan to take

advantage of the final/optimal solution found to finally showcase all of the developed MATLAB program’s

capabilities just broke down right there and then. At the moment we had a program grievously written to

robustly be able to, for an intuitive relatively small set of inputs, build any complexly shaped

unidimensional discrete structural chain-like system, compute natural vibration frequencies, compute

normalized natural modes of vibration, proceed with and orthogonality check on resultant natural modes,

compute frequency determinant function for visualization of safe frequency ranges of operation,

compute 2D deformed and non-deformed profiles, including isolated visualizations of the nodal

extension, deflection and curvature functions in respect to both local and global referential, compute

resultant profile inclination function, compute distributions of bi-direction planar components of force and

bending moment distribution, these, along the system’s deformed longitudinal length and also relative

to both local and global referential, compute corresponding internal stresses’ distributions for all layers

of the 2D model, as well as their distributions over the cross-section automatically identified as critical

78

one, after which the critical point is also identified, finalizing with the computation of the equivalent Von-

Mises stress distribution visualization and resulting static safety factor.

In actuality, there were not supposed to be eight project requirements, but nine, as we ought to have

analyzed the typical ranges of vibration on the wheels due to vehicle-road interactions, which was in

effect ready to be done with what we had already developed so far in respect to DFVA, since we know

that these must not fall within a margin of 15% distance away from any of the suspension’s natural

frequencies of vibration.

Regardless of this, a TMM methodology capacitated with what is believed to be a bunch of unique own

developed formulaic characteristics, was indeed successfully developed. Equally successful was its

capacity in generating results in compliance to all initially established expectations. In this regard, the

methodology excels in its major purpose, known to be at producing linear static analysis for complex

chain systems. Indeed, the numerical example (in Chapter 3.5) proved this method’s capacity to

outperform the popularly used finite element method for the same kind of analysis being run.

Nonetheless, it is also related to this last note that the biggest criticism to this work takes shape, residing

on the number of validations, comparisons and/or numerical examples specified, which are almost

inexistent, especially considering we are introducing a methodology with so many new features. This

would have been critical. The excuse on this matter resides on the size and time limitations imposed to

this work, since we would necessarily have to chose between doing all that benchmarking work or

absolutely turn our backs to all the work developed in laboratory towards the suspension system

proposal. Ultimately, we believe to have a made a wrong choice between the two, as our decision ended

up affecting negatively the quality of the displayed work. This is a clear demonstration that sometimes

less is more, since, by taking the other possible research direction, we would actually have had less

work, but a much more consistent research work in hands.

This purely strategical decision not only disallowed us to better benchmark our work, as it eventually led

us to a disappointing outcome for the concrete design attempt. As if this wasn’t enough, it also led to a

lack of attention given to quite important specific aspects of the MATLAB implementation in itself, since

we cannot forget that this program’s execution was in fact the overwhelming consumer of our research

time, as a lot of hard working hours were put into its programming, but with no apparent translation into

the written work. On another note, there was also a clear disconnection between the complexity of the

method being developed from scratch, which is understandably not the highest, and the particular case-

study, simply resulting on a failed intent, as this particular suspension would necessarily need a much

more complex method behind it, in order to be able to take any realistic results, and, consequently, valid

conclusions. Those are essentially our self-criticisms.

On a personal note there is a sense of lack of guidance throughout this work, which in effect happened,

and could otherwise have avoided most of these situations. The author is also self-critic of his social

abilities in seeking for mentoring from superior academics. Many important life lessons may be taken

out of this experience though.

79

5. Conclusions & Future Developments

5.1. Conclusions

In an attempt to bring back the overall spirit to a brighter one following the critical analysis made on

Chapter 4.5, all those euphemistically less positive aspects described do not in any way necessarily

take merit away from the transfer matrix methodology derivation in itself, and all its originally introduced

aspects, which comprise the greatest portion of the thesis, and which were in truth successfully

completed in accordance to the objectives established. Actually, even after accepting for a fact that the

concrete suspension design was indeed too much of a demanding mechanical challenge for the

methodology’s reach, it was still taken forward in all proper seriousness, with maximum effort put into

employing the tools that we had worked so hard to develop, and there was a clear intent at effectively

demonstrating this method’s applicability. Proof of that, was the expeditious application of a pragmatic

solution to immediately address the accuracy issues experienced, which were successfully tackled with

the introduction of the so-called quasi-static loading methodology, as it truly boosted forward the rest of

the mechanical design procedure, having, inclusively, the results from the numerical example proved its

great usefulness in approximating linear results to more realistic non-linear ones.

In this sense, knowing we had established as major objective for the design of the suspension system,

to employ our newly derived methodology onto a properly contextualized real-world engineering

challenge, we humbly think that this was a success. However, on the path towards an optimal overall

solution, a purely strategical mistake ended up compromising the final result acquisition. This mistake

led to a sudden collapse of the design procedure. Since, at that point as, a new direction would

necessarily imply repetition of a huge number of simulations, and, unfortunately, such an amount of

CPU time was definitely not available at those final stages of research, the current misfortune outcome

was inevitable.

In sum, our personal objective with this work relates to a method’s formulation and its application in

mechanical/structural design procedures. In this respect, regardless of a final solution having been found

for the particularly demanding case study in hands, we truly believe that both goals we initially proposed

ourselves to demonstrate were shiningly done so, as not only methodologically, as result-wise, the

performance proved flawless in its known nature. Ultimately, we are all humans, and the final mistake

is also a testimony to our humanness, nonetheless we should take things lightly and embrace the

opportunity to properly address the mistake in order to avoid committing similar ones in the future.

80

5.2. Future Developments

As a final note, we would like to state that this thesis provided a true sense of borderless unmet potential

for the methodology so far developed, giving the impression that there is much more able to be achieved

right around the corner. And with possible applications in nearly all fields of expertise related to

mechanics, for instance, I see this method being soon applied in the field of biomechanics, given the

complexity and sheer variety of any two neighboring bodies involved in those mechanical systems. Such

problem could easily be addressed with this methodology, given its versatility. Indeed, some of the

advancements we see being grasped in the near horizon comprise:

- Three-dimensional displacements and internal forces;

- Both in-plane and out-of-plane external loading;

- All kinds of buckling analysis (axial, flexural, torsional, combined, etc.);

- Non-linear stiffness matrices;

- Structural damping;

- Steady-state and forced dynamic vibration analysis;

- Dynamic multibody analysis;

- Models with coupled and branched systems;

- Models with closed systems;

- Models also including dampening elements besides elastic ones;

- Models including effect of kinematic joints besides structural connections;

- Fatigue and life-cycle considerations.

Not only do all of these seem possible, as in fact we already found some literature starting to approach

these aspects, unfortunately in a fairly non-explanatory and light way. Therefore, we also hope with this

work to positively impact the next generation of researchers towards this field of elastomechanical matrix

methods, which such a positive mark as left on us.

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Appendices

i

A. Numerical Example

Considering that, the straight beam cantilever model assumed is,

Figure A.1: Geometry and loading pattern governing the cantilever beam numerical example (left). Euler-

Bernoulli’s beam-bar finite element (right).

And, the FEM matrixed behind the analytical formulation are shown below, on which Euler-Bernoulli’s

beam-bar finite elements are employed (Figure A.1 (right)),

𝐊0 =

[

𝐸𝐴

𝑙0 0 −

𝐸𝐴

𝑙0 0

12𝐸𝐼

𝑙36𝐸𝐼

𝑙20 −

12𝐸𝐼

𝑙36𝐸𝐼

𝑙2

4𝐸𝐼

𝑙0 −

6𝐸𝐼

𝑙22𝐸𝐼

𝑙𝐸𝐴

𝑙0 0

12𝐸𝐼

𝑙3−

6𝐸𝐼

𝑙2

𝑠𝑦𝑚𝑚.4𝐸𝐼

𝑙 ]

(A.1)

The non-null components of the (6x6) matrix 𝐊1 are,

𝐊1(1,2) = 𝐊1(4,5) = −𝐊1(2,4) = −𝐊1(1,5) =6

5𝑙2(𝑑5 − 𝑑2) −

1

10𝑙(𝑑3 + 𝑑6)

𝐊1(1,3) = −𝐊1(3,4) =1

10𝑙(𝑑5 − 𝑑2) +

1

30(𝑑6 − 4𝑑3)

𝐊1(1,6) = −𝐊1(4,6) =1

10𝑙(𝑑5 − 𝑑2) +

1

30(𝑑3 − 4𝑑6)

𝐊1(2,2) = 𝐊1(5,5) = −𝐊1(2,5) =6

5𝑙2(𝑑4 − 𝑑1)

𝐊1(3,3) = 𝐊1(6,6) =2

15(𝑑4 − 𝑑1)

𝐊1(3,6) = −1

30(𝑑4 − 𝑑1)

𝐊1(3,5) = 𝐊1(5,6) = −𝐊1(2,3) = −𝐊1(2,6) =1

10𝑙(𝑑4 − 𝑑1)

The non-null components of the (6x6) matrix 𝐊2 are,

𝐊2(2,2) = 𝐊2(5,5) = −𝐊2(2,5) =1

140[18

𝑙(𝑑3

2 + 𝑑62) −

108

𝑙2(𝑑5 − 𝑑2)(𝑑3 + 𝑑6) +

432

𝑙3(𝑑5 − 𝑑2)

2]

(A.2)

𝐊2(2,3) = −𝐊2(3,5) =1

280[3𝑑6

2 − 3𝑑32 + 6𝑑3𝑑6 +

108

𝑙2(𝑑5 − 𝑑2)

2 −72

𝑙𝑑3(𝑑5 − 𝑑2)]

𝐊2(2,6) = 𝐊2(5,6) =1

280[3𝑑3

2 − 3𝑑62 + 6𝑑3𝑑6 +

108

𝑙2(𝑑5 − 𝑑2)

2 −72

𝑙𝑑6(𝑑5 − 𝑑2)]

𝐊2(3,3) =1

280[12𝑙𝑑3

2 + 𝑙𝑑62 − 3𝑙𝑑3𝑑6 + 3(𝑑5 − 𝑑2)(𝑑3 − 𝑑6) +

18

𝑙]

𝐊2(3,6) =1

280[−3𝑙𝑑3

2 − 3𝑙𝑑62 + 4𝑙𝑑3𝑑6 − 6(𝑑5 − 𝑑2)(𝑑3 + 𝑑6)]

(A.3)

The results, relative to the vertical displacements at the tip of the cantilever beam proposed, as well as

the relative errors of all results obtained are then computed in relation to the Non-Linear literary values.

For organisation reasons, we only wrote down NLQSA’s results where a degree of quasi-staticity of 80

increments is applied. All this is synthesized on the next page’s table:

ii

Table A.1: Results of numerical example performed in subchapter 3.5.

Force

𝑭 [𝑵]

Literary FEM Results Proposed TMM Results

Linear

Analysis

[mm]

%error

Non-Linear

Analysis

[mm]

LSEA

[mm] %error

NLQSA

(80 Inc.)

[mm]

%error

0 0 0 0 0 0 0 0

3400 236.013 91.45 20.1825 166.647 87.89 152.361 86.75

6800 472.026 94.25 27.1330 333.119 91.85 247.62 89.04

9350 649.035 95.27 30.7253 457.972 93.29 285.49 89.24

11000 - - 32.6819 538.759 93.93 300.69 89.13

13000 - - 34.7894 636.683 94.54 313.387 88.90

15000 - - 36.6777 734.608 95.01 322.583 88.63

17000 - - 38.396 832.532 95.39 330.012 88.37

19000 - - 39.978 930.456 95.70 336.281 88.11

40000 - - 52.0787 1958.66 97.34 374.253 86.08

80000 - - 66.210 3917.14 98.31 414.495 84.02

B. Relation Between Safety Factor and other Major Physical and

Geometrical Parameters

All of the following charts were built while keeping a set of defaulted parameters constant, then, case by

case, we took only one of these parameters and varied it. All cases were obtained for a leaf-spring with

a constant rectangular cross-section (𝑏𝑥ℎ) and an initial profile shaped as in Figure 3.29 (above). The

remaining parameters go as follows (for AISI Steel 1005 [112][113]):

𝜎𝑦 = 226 𝑀𝑃𝑎 (B.1) 𝐸 = 200 𝐺𝑃𝑎 (B.2)

𝐺 = 80 𝐺𝑃𝑎 (B.3) 𝜈 = 0.25 (B.4)

𝜌 = 7.872 𝑔 𝑐𝑚3⁄ (B.5) ℎ = 4 𝑚𝑚 (B.6)

𝑏 = 105 𝑚𝑚 (B.7) 𝐿𝑡𝑜𝑡𝑎𝑙 = 571.5102 𝑚𝑚 (B.8)

# 𝑜𝑓 𝐹𝑖𝑒𝑙𝑑 𝐸𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑑𝑖𝑠𝑐𝑟. = 8 (B.9) 𝐹𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙𝑟𝑖𝑔ℎ𝑡 𝑡𝑖𝑝= 6129.16 𝑁 (B.10)

(Symbology used has already been previously introduced)

Figure B.1: Relation between safety factor and both material’s yield strength (left) and modulus of elasticity (right).

iii

Figure B.2: Relation between the safety factor and both the material's shear modulus of elasticity (left) and

Poisson coefficient (right).

Figure B.3: Relation between the safety factor and both the material's density (left) and the leaf-spring's cross-

section thickness (right).

Figure B.4: Relation between the safety factor and both the leaf-spring's cross-section width (left) and its initial

profile total longitudinal length (right), the latter being varied while keeping the overall profile shape the same.

iv

C. Documental Pictures on Our Innovative Suspension’s First Full-

Prototype, from Dongfeng’s Technical Center, in Wuhan

Figure C.1: CAD drawing built for first full-prototype.

Figure C.2: Tractor unit already with innovative suspension installed on its front axle.

Figure C.3: Close-up on installed front-axle suspension (left). Overview on setup for hysteresis testing (right).

v

Figure C.4: Close-ups on arranged experimental setup.

For the simple experimental setup above documented, static hysteresis tests were performed to study

the plastic deformation mechanism. Registering the correlation between applied forces to the steering

knuckle and corresponding deformations of a suspension mechanism with its flexible members

manufactured from a common 60Si2Mn spring steel. Since these tests were static, the results are

independent of the time, load speed or deformation speed [114]. Basically, what was done were a series

of repetitive hysteresis single or double loops, setting the control on one of the parameters studied for

cyclical control purposes.

Two groups of tests were undergone, either with vertical or horizontally applied loading. For each

particular case, we will supply one charted example with an adequately explanatory legend.

Figure C.5: Hysteresis single-loop test under horizontal loading conditions. Control executed on horizontal

displacement, and set for cycle between [-2.5, 2.5] 𝑚𝑚.

vi

Figure C.6: Hysteresis double-loop test under vertical loading conditions. Control executed on vertical force, and

set for first cycle between [0, -10000] 𝑁, followed by a second between [0, -40000] 𝑁.

D. CAD Representation Showing the Current State of Design of the

Unique Heavy-Vehicle Being Developed

Figure D.1: 4-wheeled, 2-axled, heavy vehicle currently under academic research development.

E. Evaluation of the Vertical Wheel Loads of a Heavy-Vehicle

The following data was collected by means of a number of sensors, but, most importantly, by

accelerometers on the rear axle, which directly measured the vertical acceleration of either wheel (𝑧, in

this work), and by strain-gauged axle housing transducers carefully mounted so to measure strains due

to axle shear and hence shear forces. Then, while the output of the accelerometers, placed near the

wheels, provide the axle’s roll acceleration, the vertical forces (𝐹𝑠, in this work) are computed using

Newton’s Second Law considering the dynamic equilibrium of vertical forces.

vii

The data was analysed in order to try and fit the resulted forces of each wheel in linear regressionm

odels, with the vertical acceleration as a predictor variable.Then the following charts were obtained,

where the measured forces and the fitted linear models of each wheel are presented for each road test.

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

Figure E.1: Linear regression models for the relations

between the forces and the vertical accelerations of

the wheels, respectively, during acceleration (a),

during a right corner (b), while going on a road with

slope (c) and when going past an obstacle on the

road (d).

Figure E.2: Measured mass distribution of the vehicle

over its four wheels.

All results can be intuitively validated. For sake of

an example, if considering the natural vibration

occuring upon the vehicle’s motion, a negative

aceleration implies the moment when the wheels

are being compressed by the body of the vehicle

against the road. Obviosuly, this corresponds to

a positive force associate has the wheels

reached their maximum compression and

rebound in order to restitute the vehicle to its

stable intermediate oscillatory position.

Figure E.3: Linear regression model for the relation

between the forces and the vertical accelerations of

the wheels, while performing a “complete road test”.

𝑦 = −1122𝑥 + 30.50

𝑦 = −1022𝑥 − 49.92

𝑦 = 1162𝑥 − 188.65

𝑦 = −1049𝑥 − 239.54

𝑦 = −1145𝑥 + 59.89

𝑦 = −1066𝑥 + 226.67 𝑦 = −1137𝑥 − 39.75

𝑦 = −966𝑥 − 136.75

𝑦 = −1122𝑥 + 30.50

𝑦 = −1022𝑥 − 49.92

viii

The final conclusion is presented on the body of work of subchapter 4.1.3. Furthermore, in subchapter

4.1.4, Figure E.3 is employed with its acceleration axis inverted. This is because when we apply a

positive force on the LS’s vehicle tends to “sink” relatively to the wheels’ level, but, at this instant, and

according to this charts, despite the positive force described, the accelerations are instead at their

maximum negative value. So by inverting the axis we just make things more intuitive from the point of

view of the LS’s design.

F. Table with First Selection of Materials and their Physical Properties

described at Room Temperature

Table F.1: Data relative to first selected range of materials' mechanical properties.

Material Yield Strength

[MPa]

Modulus of

Elasticity

[GPa]

Shear Modulus

of Elasticity

[GPa]

Poisson

Coefficient

Density

[kg/mm3]

AISI Steel 1005 226 200 80 0.25 7.872e-06

AISI SS 304-Annealed 276 190 77 0.30 7.9e-06

AISI SS 410 483.1 219.36 80 0.268 7.734e-06

AISI Steel Maraging 1482 186 74 0.31 8e-06

AISI Steel 4340 1178 193 78 0.284 7.85e-06

Tungsten Alloy 750 400 156 0.28 1.93e-05

Waspaloy 793.18 212.46 83 0.314 8.266e-06

Titanium Ti-6Al-4V 805 121 44 0.34 4.43e-06

Titanium -Annealed 275 120 44 0.34 4.5e-06

Aluminium 2014 393.7 73.12 28 0.33 2.794e-06

Aluminium 5086 217 72 26.4 0.33 2.66e-06

Aluminium 6061 241.7 6.898 26 0.33 2.711e-06

Aluminium 7050 455 71.7 26.9 0.33 2.83e-06

AISI Steel 9260 (High Yield) 1149 200 80 0.29 7.85e-06

AISI Steel 9260 (Low Yield) 440 200 80 0.29 7.85e-06

AISI Grade 18Ni Maraging Steel (Aged) 2395 200 77 0.30 8.08e-06

2800 Maraging Steel 2617 160 70 0.30 6.60e-06

Sandvik® Nanoflex Strip Steel 2400 180 80 0.30 7.90e-06

Hexcel® HexPly® 8551-7 Epoxy Matrix, IM7

[0º, Dry] Carbon Fibre 2760 159 5.86 0.416 1.49e-06

Hexcel® HexPly® F650 Bismaleimide Resin

(26%) with [0º] IM6 Carbon Fibre (67%) 2774 165 67.1 0.35 1.51e-06

Quadrant EPP Proteus® Layflat Natural Co-

Polymer Polypropylene 14000 1.31 0.465 0.41 0.91e-06

Hexcel® HexPly® 954-2A Curing Cyanate

Resin, G40-800 Carbon Fibre 3017 159 57.6 0.38 1.52e-06

All properties presented on Table 4.1 were concertedly collected from SIEMENS NX 10’s Library [112]

and from the incredible library available online at MATWEB [113]. For instance, since Waspaloy lacked

information on its shear modulus of elasticity, this was cautiously such value was collected from the

official Haynes website [115], from another similar nickel alloy. The shear modulus of elasticity for the

Sandvik® Nanoflex strip steel was taken from [116]. Regarding the composite of curing cyanate resin

ix

with G40-800 carbon fibre, its Poisson coefficient was assumed to be the same as the one of the pure

resin, while its density was estimated assuming we have a 50-50 matrix-fibre composite. As for the

cases of the Bismaleimide resin matrix with carbon fibre and the LF Co-Polymer Polypropylene, their

Poisson coefficients were respectively found on [117] and [118], after which, by erroneously, but

deliberately, assuming the materials to be linear, homogeneous and isotropic, we were then able to

apply the following expression to estimate the shear modulus of elasticity ([119]):

𝐺 =𝐸

2(1+𝜈) (F.1)

G. Diagram on Further Computational Meaning Behind the Plots of

Figure 4.2

Figure G.1: Comprehensive diagram on each punctually optimized profile’s shape and operational performance.

H. Operating profiles of the entirety of NLQSA-OP optimized materials

Figure H.1: For each material, 3 profiles are drawn. An initial one as manufactured, another under ALVW

conditions (neutralized position), and, at last, its maximum operational position.

x

I. Leaf-Spring’s Geometrical Optimization for two Additional Materials –

TMM NLQSA-OP

I.1. Geometrical Parameters Parallel Optimization Step

Participating but non-outputted parameters throughout process: 𝑃𝐸𝑉𝐿𝑆 = 312.5 𝑘𝑔𝑓


Thickness Optimization

Table I.1: Fully constrained NLQSA-OP results on thickness variation (HH 8551-7).

Hexcel® HexPly® 8551-7 Epoxy Matrix, IM7 [0º, Dry] Carbon Fibre

Thickness

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

4 3.5426 625.0 706.577 0.057665 139.18 172.621 0.75328

6 1.8629 625.0 951.309 0.22824 170.326 98.4596 1.0142

8 1.0755 625.0 1359.2 0.51253 137.457 59.9488 1.449

9 0.8999 625.0 1644.72 0.71154 114.092 48.4215 1.7534

10 0.8037 625.0 1950.63 0.92476 87.8286 38.7403 2.0796

11 0.7337 625.0 2307.53 1.1735 60.5739 32.2527 2.4601

12 0.7008 625.0 2664.43 1.4223 43.2404 27.2178 2.8405

14 0.6904 625.0 3582.18 2.0619 25.9647 19.7896 3.819

16 0.7119 625.0 4601.89 2.7727 17.788 14.7909 4.9061

Table I.2: Fully constrained NLQSA-OP results on thickness variation (2800 M. Steel).

2800 Maraging Steel

Thickness

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

4 3.3336 625.0 710.656 0.060508 197.719 172.905 0.75763

6 1.6253 625.0 992.098 0.25667 168.511 92.1201 1.0577

8 0.8597 625.0 1542.74 0.64047 132.776 53.6832 1.6447

9 0.7004 625.0 1950.63 0.92476 106.146 42.1182 2.0796

10 0.6088 625.0 2409.5 1.2446 74.6179 32.6792 2.5688

11 0.5408 625.0 2970.35 1.6355 47.3715 26.3816 3.1667

12 0.4944 625.0 3684.15 2.133 32.5396 21.5684 3.9277

14 0.4639 625.0 5417.67 3.3412 18.6101 14.4893 5.7758

16 0.4393 625.0 7661.04 4.9049 12.0561 10.0924 8.1674

xi

Figure I.1: Piecewise cubic Hermite polynomial interpolation, followed by identification of each cases’ thought

adequate EL-EVRW value, based on Tables 4.3, I.1 and I.2. Cases: HH IM9 (left), HH 8551-7 (center) and 2800

M. Steel (right).

HH IM9: HH 8551-7: 2800 M. Steel:

Thickness=9.3264 mm Thickness=10.8103 mm Thickness=10.1554 mm

⇒ 9.30 mm ⇒ 10.80 mm ⇒ 10.16 mm

Width Optimization

Table I.3: Fully constrained NLQSA-OP results on width variation (HH 8551-7).


Width

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

73.5 0.85331 625.0 1542.74 0.64047 118.2642 51.6340 1.6447

94.5 0.82421 625.0 1797.67 0.81815 98.5366 42.1498 1.9165

105 0.80108 625.0 1950.63 0.92476 85.6882 39.4931 2.0796

115.5 0.78796 625.0 2103.59 1.0314 77.1240 36.2993 2.2426

136.5 0.77849 625.0 2358.52 1.2091 60.6122 31.9136 2.5144

157.5 0.77525 625.0 2613.45 1.3867 49.3063 28.4846 2.7862

178.5 0.77755 625.0 2868.38 1.5644 41.4930 25.7151 3.058

199.5 0.7772 625.0 3174.29 1.7776 37.3746 22.7736 3.3841

220.5 0.7974 625.0 3378.23 1.9198 30.3181 22.0724 3.6015

Table I.4: Fully constrained NLQSA-OP results on width variation (2800 M. Steel).

2800 Maraging Steel

Width

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

73.5 0.6531 625.0 1848.66 0.85369 111.2596 43.0273 1.9709

94.5 0.61246 625.0 2256.55 1.138 86.3364 35.5337 2.4057

105 0.60879 625.0 2409.5 1.2446 74.6179 32.6792 2.5688

115.5 0.60322 625.0 2562.46 1.3512 61.4638 31.0302 2.7318

xii

136.5 0.58711 625.0 2970.35 1.6355 48.1968 26.9768 3.1667

157.5 0.58001 625.0 3378.23 1.9198 41.6341 23.1542 3.6015

178.5 0.58637 625.0 3684.15 2.133 32.6473 21.8132 3.9277

199.5 0.5886 625.0 4092.04 2.4173 29.9528 19.0904 4.3625

220.5 0.60299 625.0 4397.95 2.6305 26.7991 17.2877 4.6886


adequate EL-EVRW value, based on Tables 4.4, I.3 and I.4. Cases: HH IM9 (left), HH 8551-7 (center) and 2800

M. Steel (right).

HH IM9: HH 8551-7: 2800 M. Steel:

Width=86.1667 mm Width=130.3272 mm Width=108.4888 mm

⇒ 86.16 mm ⇒ 130.30 mm ⇒ 108.50 mm

Length Optimization

Table I.5: Fully constrained NLQSA-OP results on length variation (HH 8551-7).


Length

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

387.013 0.44712 625.0 5111.75 3.128 27.9787 17.5560 5.4496

442.122 0.54175 625.0 3480.21 1.9909 40.9739 24.0286 3.7102

497.166 0.66332 625.0 2511.47 1.3157 60.6910 31.1480 2.6775

552.314 0.80108 625.0 1950.63 0.92476 85.6882 39.4931 2.0796

607.533 0.96546 625.0 1593.73 0.676 115.0883 48.2070 1.6991

662.798 1.1611 625.0 1338.8 0.49832 140.0257 59.1988 1.4273

718.037 1.3965 625.0 1175.65 0.3846 167.1000 69.5746 1.2534

773.29 1.6691 625.0 1053.28 0.29931 190.5601 82.2735 1.1229

828.59 2.0111 625.0 961.506 0.23535 213.8151 95.6845 1.0251

xiii

Table I.6: Fully constrained NLQSA-OP results on width variation (2800 M. Steel).

2800 Maraging Steel

Length

[mm] SF

𝑷𝑳𝑺𝑵

[Kgf]


[Kgf] 𝚻

EL-EVWR

[mm]

EL-GVWR

[mm] SRP

387.472 0.26145 625.0 9292.59 6.042 19.4482 12.9477 9.9068

442.511 0.36354 625.0 5111.75 3.128 29.2099 19.1355 5.4496

497.432 0.47279 625.0 3378.23 1.9198 46.4842 25.6394 3.6015

552.458 0.60879 625.0 2409.5 1.2446 74.6179 32.6792 2.5688

607.589 0.76365 625.0 1848.66 0.85369 103.7463 41.8891 1.9709

662.805 0.95305 625.0 1491.76 0.60493 133.3840 51.7581 1.5904

718.089 1.1563 625.0 1277.62 0.45568 159.6615 63.0191 1.3621

773.305 1.4085 625.0 1124.66 0.34907 185.0739 74.9326 1.199

828.579 1.7215 625.0 1012.49 0.27089 209.2490 87.9032 1.0794


adequate EL-EVRW value, based on Tables I.5 and I.6. Cases: HH 8551-7 (left) and 2800 M. Steel (right).

HH IM9: HH 8551-7: 2800 M. Steel:

Length=607.16 mm Length=507.5982 mm Length=543.7952 mm

⇒ 607.16 mm ⇒ 507.60 mm ⇒ 543.80 mm

detailed curved-beam transfer-matrix methodology …...𝐝 nodal dof’s/displacements column...

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