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THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT UNIVERSAL JOINT
A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF MIDDLE EAST TECHNICAL UNIVERSITY
BY
ÇAĞIL MERVE TANIK
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF MASTER OF SCIENCE IN
MECHANICAL ENGINEERING
JANUARY 2014
Approval of the thesis:
THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT UNIVERSAL JOINT
submitted by ÇAĞIL MERVE TANIK in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Department, Middle East Technical University by, Prof. Dr. Canan Özgen Dean, Graduate School of Natural and Applied Sciences _____________________ Prof. Dr. Süha Oral Head of Department, Mechanical Engineering _____________________ Prof. Dr. F. Suat Kadıoğlu Supervisor, Mechanical Engineering Dept., METU _____________________ Assoc. Prof. Dr. Volkan Parlaktaş Co-Supervisor, Mechanical Engineering Dept., HU _____________________ Examining Committee Members: Prof. Dr. Orhan Yıldırım Mechanical Engineering Dept., METU _____________________ Prof. Dr. F. Suat Kadıoğlu Mechanical Engineering Dept., METU _____________________ Assoc. Prof. Dr. Volkan Parlaktaş Mechanical Engineering Dept., HU _____________________ Prof. Dr. Metin Akkök Mechanical Engineering Dept., METU _____________________ Assist. Prof. Dr. Ergin Tönük Mechanical Engineering Dept., METU _____________________
Date: 30.01.2014
iv
I hereby declare that all information in this document has been obtained and
presented in accordance with academic rules and ethical conduct. I also declare
that, as required by these rules and conduct, I have fully cited and referenced
all material and results that are not original to this work.
Name, Last name : Çağıl Merve TANIK
Signature :
v
ABSTRACT
THEORETICAL AND EXPERIMENTAL ANALYSES OF COMPLIANT
UNIVERSAL JOINT
Tanık, Çağıl Merve
M.S., Department of Mechanical Engineering
Supervisor: Prof. Dr. Fevzi Suat Kadıoğlu
Co-Supervisor: Assoc. Prof. Dr. Volkan Parlaktaş
February 2014, 115 pages
In this study, a compliant version of the cardan universal joint whose compliant parts
are made of blue polished spring steel is considered. The original design consist of
two identical parts assembled at right angles with respect to each other. Identical
parts can be produced from planar materials; thus, it has the advantage of easiness in
manufacturing. As a design example, two mechanisms are dimensioned with
different plate thicknesses. The resultant stresses at flexural hinges of these samples
are determined via analytical and finite element analysis method. Torque capacity of
these mechanisms are determined. Also fatigue analysis of these mechanisms are
performed. Further, one of these samples is manufactured and operated under three
different conditions. It is verified that results of experiments are consistent with
theoretical approaches.
Keywords: Compliant Mechanisms, Universal Joint, Finite Element Analysis,
Fatigue Analysis
vi
ÖZ
ESNEK KARDAN MAFSALININ TEORİK VE DENEYSEL ANALİZLERİ
Tanık, Çağıl Merve
Yüksek Lisans, Makine Mühendisliği Bölümü
Tez Yöneticisi: Prof. Dr. Fevzi Suat Kadıoğlu
Ortak Tez Yöneticisi: Doç. Dr. Volkan Parlaktaş
February 2014, 115 sayfa
Bu çalışmada esnek kısımları yay çeliğinden yapılan kardan milinin esnek bir
versiyonu incelenmiştir. Özgün tasarım birbirlerine dik olarak monte edilmiş iki
özdeş parçadan oluşmaktadır. Özdeş parçalar düzlemsel malzemelerden imal
edilebilir, bu nedenle mekanizmanın üretimi kolaydır ve bu önemli bir avantajdır. Bu
çalışmada iki farklı plaka kalınlığında mekanizma tasarım örnekleri
boyutlandırılmıştır. Bu örneklerin bükülen mafsallarındaki gerilmeler analitik
metotlar ve sonlu elemanlar analizi ile incelenmiştir. Böylece mekanizmaların tork
kapasiteleri belirlenmiştir. Ayrıca mekanizmaların yorulma analizleri de yapılmıştır.
Yapılan analitik ve sayısal çalışmaları doğrulamak amacıyla, tasarlanan örneklerin
bir tanesi üretilmiş ve üç farklı koşulda çalıştırılmıştır. Bu deneylerin sonuçlarının
kuramsal yaklaşımlar ile tutarlı olduğu gözlemlenmiştir.
Anahtar Kelimeler: Esnek Mekanizmalar, Üniversal Mafsal, Sonlu Eleman Analizi,
Yorulma Analizi
vii
To My Husband
viii
ACKNOWLEDGMENTS
The author wishes to express her deepest gratitude to his supervisor Prof. Dr. F. Suat
Kadıoğlu and co supervisor Assoc. Prof. Dr. Volkan Parlaktaş for their guidance,
advice, criticism, encouragements and insight throughout the research.
The author would also like to thank her husband Assoc. Prof. Dr. Engin Tanık for his
support and patience during the thesis study.
ix
TABLE OF CONTENTS
ABSTRACT ................................................................................................................. v
ÖZ ............................................................................................................................... vi
ACKNOWLEDGMENTS ....................................................................................... viii
LIST OF TABLES ..................................................................................................... xii
LIST OF FIGURES ................................................................................................. xiii
LIST OF SYMBOLS ................................................................................................ xvi
CHAPTERS ................................................................................................................. 1
1. INTRODUCTION ................................................................................................... 1
1.1 Literature Review ............................................................................................... 1
1.2 Objective and Scope of the Thesis ..................................................................... 4
2. COMPLIANT MECHANISMS ............................................................................... 7
2.1 Introduction to Compliant Mechanisms ............................................................. 7
2.2 Pseudo-Rigid-Body Model ................................................................................. 8
2.2.1 Small-Length Flexural Pivots ...................................................................... 9
2.2.2 Active and Passive Forces ......................................................................... 11
2.2.3 Cantilever Beam with a Force at the Free End .......................................... 13
2.2.4 Moment at the Free End............................................................................. 17
2.3 Nomenclature ................................................................................................... 17
2.4 Diagrams .......................................................................................................... 19
2.5 Pseudo-Rigid-Body Replacement ................................................................... 20
2.6 Material Considerations ................................................................................... 22
x
3. CONVENTIONAL UNIVERSAL JOINT ............................................................ 25
3.1 Introduction to Cardan Universal Joint ............................................................ 25
3.2 Kinematic Analysis of the Cardan Universal Joint .......................................... 26
4. DESIGN OF THE COMPLIANT UNIVERSAL JOINT ...................................... 31
4.1 Dimension Synthesis of the Compliant Universal Joint................................... 31
4.2 Material Selection of the Compliant Universal Joint ....................................... 35
5. THEORETICAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT .... 39
5.1 Stress Analysis of the Compliant Universal Joint with FEA ........................... 39
5.1.1 Boundary Conditions and Meshing ........................................................... 46
5.2 Static Analysis for Deflection-Only Case ....................................................... 48
5.3 Study of Mesh Refinement for Bending-Only Case ........................................ 58
5.4 Path Analysis .................................................................................................... 60
5.5 Static Analysis to Determine Torque Limits ................................................... 66
5.6 Study of Mesh Refinement for Combined Loading Case ................................ 73
5.7 Static Analysis of the Mechanism at Zero Degree Shaft Angle ..................... 76
5.7 Fatigue Life Estimations ................................................................................. 82
6. EXPERIMENTAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT . 97
6.1 Manufacturing of the Prototype .................................................................... 97
6.2 Components of the Experimental Setup ....................................................... 99
6.3 Experimental Verification .......................................................................... 101
7. RESULTS, CONCLUSION AND FUTURE STUDY ........................................ 103
7.1 RESULTS AND CONCLUSION .................................................................. 103
7.2 FUTURE STUDY .......................................................................................... 104
REFERENCES ........................................................................................................ 105
APPENDICES ......................................................................................................... 109
xi
A: PROPERTIES OF AISI 1080 ............................................................................. 109
B: TECHNICAL DRAWINGS ................................................................................ 110
C: SPECIFICIONS OF BEARINGS ...................................................................... 112
D: SPECIFICIONS OF DC ELECTRIC MOTOR .................................................. 113
E: ISOMETRIC VIEW OF THE TEST SETUP ..................................................... 115
xii
LIST OF TABLES
TABLES
Table 2.1 The Coefficients of the Compliant Mechanisms ....................................... 16
Table 2.2 Ratio of Yield Strength to Young's Modulus for Some Materials ............. 24
Table 4.1 Properties of AISI 1080 (Spring Steel) ...................................................... 35
Table 5.1 Analytical and Numerical Normal Stress .................................................. 51
Table 5.2 Analytical and Numerical Normal Stress .................................................. 54
Table 5.3 Number of Elements through the Thickness vs. Maximum von-Mises
Stress for Bending-Only Case .................................................................................... 60
Table 5.4 Steps of the Loading .................................................................................. 67
Table 5.5 Stress Values vs. Time Increments ............................................................ 68
Table 5.6 Interpolation of the Torque Values and Stress Values for �ℎ = 0.75 70
Table 5.7 Torque Limits for Different Shaft Angles (�ℎ =0.75 mm) ....................... 70
Table 5.8 Interpolation of the Torque Values and Stress Values for �ℎ =0.5 mm .... 71
Table 5.9 Torque Limits for Different Shaft Angles (�ℎ =0.5 mm) ......................... 72
Table 5.10 Number of Elements through the Thickness vs. Maximum von-Mises
Stress for Combined Loading Case............................................................................ 75
Table 5.11 Torque vs. Shear Stress for �ℎ = 0.75 mm .............................................. 79
Table 5.12 Torque vs. Shear Stress Values for �ℎ = 0.5 mm .................................... 81
Table 5.13 Values of a and b for Surface Factor ....................................................... 83
Table 5.14 Reliability Factors, � ............................................................................. 85
Table 5.15 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�ℎ = 0.75 mm) ......................................................................................................... 90
Table 5.16 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�ℎ = 0.5 mm) ........................................................................................................... 91
Table 5.17 Maximum and Minimum Normal Stress Values for Combined Loading 94
Table 6. 1 Experiments for Different Conditions .................................................... 101
xiii
LIST OF FIGURES
FIGURES
Figure 1.1 Compliant Spatial Four Bar (RSSR) Mechanism (Tanık E. and Parlaktaş
V. (2011)) ..................................................................................................................... 2
Figure 1.2 Compliant Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012)) ..... 4
Figure 1.3 Flow Chart of the Stress Analysis .............................................................. 6
Figure 2.1 Cantilever Beam (a) and its Pseudo-Rigid Body Model (b) ..................... 10
Figure 2.2 Cantilever Beam with a Force at the Free End and its Pseudo-Rigid Body
Model with a Torsional Spring .................................................................................. 12
Figure 2.3 Flexible Segment and its Pseudo-Rigid-Body Model .............................. 13
Figure 2.4 Plot of Characteristic Radius Factor, �, versus n...................................... 14
Figure 2.5 Flexible Beam with a Moment at the Free End ........................................ 17
Figure 2.6 Component Characteristics of Links ........................................................ 18
Figure 2.7 Component Characteristics of Segments .................................................. 18
Figure 2.8 Symbol Convention for Compliant Mechanism Diagrams ...................... 19
Figure 2. 9 An Example of a Compliant Mechanism Diagram ................................. 20
Figure 2.10 A Complaint Four Bar and Slider-Crank Mechanism and Pseudo Rigid
Body Models .............................................................................................................. 22
Figure 2.11 Flexible Cantilever Beam ....................................................................... 23
Figure 3.1 Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012)) ..................... 25
Figure 3.2 Variables of Cardan Universal Joint ......................................................... 26
Figure 3.3 Input Shaft Angle versus Output Shaft Speed for Different Shaft Angles
.................................................................................................................................... 28
Figure 3.4 Input Shaft Angle versus Output Shaft Angle for Different Shaft Angles
.................................................................................................................................... 29
Figure 4.1 Spherical 4R Linkage (Tanık E. and Parlaktaş V. (2012)) ....................... 31
xiv
Figure 4.2 Original Compliant Universal Joint Design (Tanık E. and Parlaktaş V.
(2012))........................................................................................................................ 32
Figure 4.3 Dimensions of the Compliant Universal Joint (Tanık E. and Parlaktaş V.
(2012))........................................................................................................................ 33
Figure 4.4 Critical Position of the Compliant Universal Joint (Tanık E. and Parlaktaş
V. (2012)) ................................................................................................................... 34
Figure 4.5 Isometric View of the Design ................................................................... 36
Figure 4.6 For a 10° Shaft Angle and Different Input Shaft Angles Views of the
Mechanism ................................................................................................................. 37
Figure 5.1 Flow Chart of the Analysis ....................................................................... 39
Figure 5.2 Definitions of the Parts ............................................................................. 47
Figure 5.3 Connections and Newly Formed Parts ..................................................... 47
Figure 5.4 The Meshing of the Model ....................................................................... 48
Figure 5.5 Representation of the Boundary Condition and Shaft Angle ................... 50
Figure 5.6 ANSYS Simulation for �ℎ =0.75 mm and 1° Shaft Angle ...................... 52
Figure 5.7 ANSYS Simulation for �ℎ =0.75 mm and 5° Shaft Angle ...................... 52
Figure 5.8 ANSYS Simulation for �ℎ =0.75 mm and 10° Shaft Angle .................... 53
Figure 5.9 Plot of Shaft Angle vs. Analytical and ..................................................... 54
Figure 5.10 ANSYS Simulation for �ℎ=0.5 mm and 1° Shaft Angle ........................ 55
Figure 5.11 ANSYS Simulation for �ℎ=0.5 mm and 5° Shaft Angle ........................ 56
Figure 5.12 ANSYS Simulation for �ℎ =0.5 mm and 10° Shaft Angle .................... 56
Figure 5.13 ANSYS Simulation for �ℎ =0.5 mm and 15° Shaft Angle .................... 56
Figure 5. 14 Plot of Shaft Angle vs. Analytical and .................................................. 57
Figure 5.15 Plot of Shaft Angle vs. Analytical and ................................................... 58
Figure 5.16 Simulation Results for Four Elements through the Thickness for
Bending-Only Case .................................................................................................... 59
Figure 5.17 Simulation Results for Five Elements through the Thickness for
Bending-Only Case .................................................................................................... 59
Figure 5.18 Number of Elements through the Thickness vs. Maximum von-Mises
Stress for Bending-Only Case .................................................................................... 60
xv
Figure 5.19 Paths for One of the Deflected Flexural Hinge ...................................... 61
Figure 5.20 Path Analysis in ANSYS ........................................................................ 61
Figure 5.21 Length vs. Equivalent Stress for Path 1 .................................................. 62
Figure 5.22 Length vs. Equivalent Stress for Path 2 .................................................. 62
Figure 5.23 Length vs. Equivalent Stress for Path 3 .................................................. 63
Figure 5.24 Length vs. Equivalent Stress for Path 4 .................................................. 63
Figure 5.25 Length vs. Equivalent Stress for Path 5 .................................................. 64
Figure 5.26 Length vs. Equivalent Stress for Path 6 .................................................. 64
Figure 5.27 Length vs. Equivalent Stress for Path 7 .................................................. 65
Figure 5.28 Length vs. Equivalent Stress for Path 8 .................................................. 65
Figure 5.29 Length vs. Equivalent Stress for Path 9 .................................................. 66
Figure 5.30 Analysis settings ..................................................................................... 67
Figure 5.31 Simple Model with Applied Torque and Reactions at Flexural Hinges . 69
Figure 5.32 Finite Element Analysis and Results for Simple Model ......................... 69
Figure 5.33 Shaft Angle vs. Maximum Torque Values for �ℎ = 0.75 .............. 71
Figure 5.34 Shaft Angle vs. Maximum Torque Values for �ℎ =0.5 mm .................. 72
Figure 5.35 Shaft Angle vs. Maximum Torque Values for Both Models .................. 73
Figure 5.36 Simulation Results for Three Elements through the Thickness for
Combined Loading Case ............................................................................................ 74
Figure 5.37 Simulation Results for Four Elements through the Thickness for
Combined Loading Case ............................................................................................ 74
Figure 5.38 Simulation Results for Five Elements through the Thickness for
Combined Loading Case ............................................................................................ 75
Figure 5.39 Number of Elements through the Thickness vs. Maximum von-Mises
Stress for Combined Loading Case ............................................................................ 76
Figure 5.40 Shear Stress Values for 1 N.m Torque (�ℎ = 0.75 mm) ........................ 77
Figure 5.41 Shear Stress Values for 2.5 N.m Torque (�ℎ = 0.75 mm) ..................... 77
Figure 5.42 Shear Stress Values for 5 N.m Torque (�ℎ = 0.75 mm) ........................ 77
Figure 5.43 Shear Stress Values for 7.5 N.m Torque (�ℎ =0.75 mm) ...................... 78
Figure 5.44 Shear Stress Values for 10 N.m Torque (�ℎ = 0.75 mm) ...................... 78
Figure 5.45 Shear Stress Values for 12.5 N.m Torque (�ℎ = 0.75 mm) ................... 78
xvi
Figure 5.46 Shear Stress for 15 N.m Torque (�ℎ = 0.75 mm) .................................. 79
Figure 5.47 Torque vs. Shear Stress for �ℎ = 0.75 mm ............................................. 79
Figure 5.48 Shear Stress for 1 N.m Torque (�ℎ= 0.5 mm) ........................................ 80
Figure 5.49 Shear Stress for 2.5 N.m Torque (�ℎ = 0.5 mm) ................................... 80
Figure 5.50 Shear Stress Values for 5 N.m Torque (�ℎ= 0.5 mm) ............................ 80
Figure 5.51 Shear Stress for 7.5 N.m Torque (�ℎ = 0.5 mm) ................................... 81
Figure 5.52 Torque vs. Shear Stress for �ℎ = 0.5 mm ............................................... 81
Figure 5.53 Torque vs. Shear Stress for Both Models ............................................... 82
Figure 5.54 Notch-Sensitivity Charts for Steels and UNS A92024-T Wrought
Aluminum Alloys Subjected to Reversed Bending or Reversed Axial Loads
(Budynass and Nisbett (2011)) .................................................................................. 86
Figure 5.55 Rectangular Fillet Bar in Bending .......................................................... 87
Figure 5.56 Estimation of Theoretical Stress Concentration factor with ANSYS .... 87
Figure 5.57 Loading Condition of the Model ............................................................ 89
Figure 5.58 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�ℎ =
0.75 mm) .................................................................................................................... 90
Figure 5.59 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�ℎ = 0.5 mm) ..................................................................................................................... 91
Figure 5.60 Von-Mises Stress Distribution, Most Critical Point on the Flexural
Hinge, its Infinitesimal Cube and ��� Stress Distribution ........................................ 92
Figure 5.61 Infinitesimal Cube of Critical Point ....................................................... 93
Figure 5.62. Normal Stress Variations for Different Shaft Angles............................ 94
Figure 5.63 Performance Graph of the Compliant Universal Joint ........................... 95
Figure 6.1 Assembly of the Prototype ....................................................................... 97
Figure 6.2 Components of the Prototype ................................................................... 98
Figure 6. 3 Different Ways of the Connections ......................................................... 99
Figure 6.4 Components of the Experimental Setup ................................................. 100
Figure 6.5 A Failed Compliant Universal Joint ....................................................... 102
Figure 6.6 Shaft Angle vs. Torque Output and Experimentally Verified Data ....... 102
Figure A.1 Properties of AISI 1080 ......................................................................... 109
Figure B.1 Technical Drawing of the Model with 0.75 mm Plate Thickness ......... 110
xvii
Figure B.2 Technical Drawing of the Model with 0.5 mm Plate Thickness ............ 111
Figure C.1Specifications of UCP 200 Bearing ........................................................ 112
Figure D.1 Specifications of the DC Electric Motor................................................ 114
Figure E.1 Isometric View of the Fatigue Test Setup .............................................. 115
xviii
LIST OF SYMBOLS
� Horizontal component of free end, length of rigid link, angular acceleration,
curve fitting parameter, constant
� Vertical component of free end, distance, width, curve fitting parameter,
constant
� Distance
�� Parametric angle coefficient
�� Constant
� Width of connection slot, diameter
� Modulus of elasticity
� Force
� Fatigue strength factor
� Width of compliant universal joint
� Second moment of area
� Spring constant
� Marin endurance limit modifying factor
�� Geometric stress concentration factor
�� Stiffness coefficient
� Length
� Length
Moment
! Number of cycles
" Horizontal coefficient of applied force, factor of safety
xix
"# Horizontal component of force
# Vertical component of force
$ Notch sensitivity
% Radius
& Contact surface of compliant universal joint
'( Endurance limit
'() Unmodified endurance limit
'*� Ultimate tensile strength
'+ Yield strength
, Torque
�- Thickness of flexural hinge
�. Thickness of compliant universal joint
/ Width of flexural hinge
01 Initial position vector
23 Engineering normal strain for fracture
4 Pseudo rigid body angle, angular deflection
� Characteristic radius factor, angle of rotation for shafts
��5 Pseudo rigid body constant
6 Deflection
7 Bend angle, shaft angle
8 Angle
� Normal stress
9 Angular velocity
: Direction angle of force
1
CHAPTER 1
INTRODUCTION
1.1 Literature Review
Universal joints are common mechanical devices which are used for transmitting
rotary motion between misaligned intersecting shafts. That is, a universal joint is a
joint or coupling that is capable of transmitting rotary motion from one shaft to
another which are not in line with each other. Classical analysis of the universal joint
involves the determination of angular displacements, velocities, accelerations and
torque ratios of the shafts.
In this study a novel universal joint design is proposed. A compliant universal joint
whose compliant parts are made of blue polished spring steel is taken into
consideration. This chapter presents a survey of the related literature for compliant
mechanisms and universal joint mechanisms.
In literature universal joint became an attractive topic because of its capabilities of
easy mounting, resisting high loads and commercial availability. Early articles on
universal joints made of rigid links address various aspects of these mechanisms.
Basically a universal joint is a spherical four bar linkage. In literature there are lots
of studies about this type of mechanism addressing its analysis, synthesis,
applications and type determination. For example Mohan et al. (1973) introduced
closed form synthesis of a spatial function generation mechanism which consists of a
spherical four bar linkage. Freudenstein (1965) proposed a new type of a spherical
2
mechanism. Yang (1965) worked on static force and torque analysis of a spherical
four bar mechanism. Dynamic analysis of a universal joint and its manufacturing
tolerances are introduced by Chen and Freudenstein (1986). Freudenstein and Macey
(1990) worked on the inertia torques of the Hooke joint. Moment transmission by a
universal joint is studied by Porat (1980). Homokinetic joint allows to transmit
power through a variable angle, at constant rotational speed. For a double cardan
homokinetic joint Wagner and Cooney (1979) developed a new approach to increase
its dynamic mechanical efficiency.
Universal joint has the advantage of easiness in manufacturing. On the other hand
traditional universal joints consist of many parts which are assembled and therefore
manufacturing tolerances on these parts must be complied with. Tolerances of a
universal joint are studied by Fischer and Freudenstein (1984).
Figure 1.1 Compliant Spatial Four Bar (RSSR) Mechanism (Tanık E. and Parlaktaş
V. (2011))
Compliant mechanisms are flexible mechanisms, which gain some or all of their
motion through the deflection of members. They can be fully or partially compliant.
Generally compliant mechanisms have lower number of parts which reduce
manufacturing and assembly time. Some of them may even be made of a single
3
piece. They are lighter and they have fewer number of movable joints, which cause
wear and need lubrication. The main disadvantage of compliant mechanisms is that,
their analyzes and design is difficult to accomplish. The pseudo rigid body model is
used to simplify the analysis and design of compliant mechanisms. In Figure 1.1 first
compliant spatial four bar mechanism is shown which is designed by Tanık and
Parlaktaş (2011).
Salamon (1989) introduced a methodology which uses a pseudo rigid body model of
the compliant mechanisms with compliance modelled as torsional and linear springs.
Howell and Midha (1994) and (1998) used closed form elliptic integral solutions to
develop deflection approximations for an initially straight flexible segment subjected
to bending.
A spherical four bar mechanism which is a special case of spatial four bar
mechanism that possesses out of plane motions is studied by Tanık and Parlaktaş
(2012). Another spatial four link mechanism studied by Parlaktaş and Tanık (2011)
is the compliant spatial slider crank mechanism.
In the literature a compliant universal joint is previously considered by Trease et al.
(2005) who proposed a design for a compliant universal joint. There is also a
prototype of a compliant universal joint in the library of Cornell University that can
be found in http://kmoddl.library.cornell.edu website. In that prototype the compliant
section is made of leather.
Recently, Tanık and Parlaktaş (2012) proposed a new design for a compliant cardan
universal joint which is shown in Figure 1.2. The design consists of two identical
parts assembled at right angles with respect to each other. In that study, dimensions
of the mechanism are designed in order to satisfy the Cardan joint theory and to
avoid an undesired contact between the identical parts for proper functioning of the
mechanism. This prototype is made of polypropylene and manufactured and
operated under specified loading conditions to verify the theoretical approaches.
4
Figure 1.2 Compliant Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012))
1.2 Objective and Scope of the Thesis
The successful implementation of a compliant universal joint in real life applications
depends not only on its kinematic design but its strength as well. The purpose of this
thesis is therefore to analyze the stresses and fatigue strength of a compliant
universal joint, whose flexible parts are made of blue polished spring steel by
analytical, numerical and experimental methods. Hence theoretical approaches will
be experimentally verified. To the best of Author's knowledge there are not any
studies in the literature which address the strength issues of compliant universal
joints except the study of Tanık and Parlaktaş (2012) , where only a preliminary
5
finite element analysis has been done to determine the torque capacity of the
mechanism.
Here, a design is proposed according to the dimensional constraints that satisfy the
theory of universal joints and thereby avoid undesired contact between the parts. The
stress analysis is done analytically and numerically by finite element method using
ANSYS software. Fatigue analysis is done and it is experimentally verified with the
prototypes that are manufactured. The flow chart of the strength analysis can be seen
in Figure 1.3 schematically.
Outline of the thesis is as follows. In Chapter 2, a review of compliant mechanisms
is done. Then pseudo rigid body model is briefly explained and deflection and stress
equations are derived. Rigid body replacement synthesis is presented. The kinematic
equations of the universal joint are given in Chapter 3. In Chapter 4 dimensioning of
the mechanism and selection of the material is covered. In Chapter 5 the analytical
and numerical solution procedure and approaches are discussed. Experimental setup
is introduced in Chapter 6. Finally, the results are discussed in Chapter 7.
Figure 1.3
Determine stresses under static loading
Perform stress analysis using FEA
Evaluate fatigue life of the mechanism
Manufacture a real model
Compare the fatigue life estimates with
Discuss the results by various methods
and explain discrepancies if any
6
3 Flow Chart of the Stress Analysis
Determine stresses under static loading
analytically
Perform stress analysis using FEA
Evaluate fatigue life of the mechanism
by analytical methods
Manufacture a real model
Compare the fatigue life estimates with
experiments
Discuss the results by various methods
and explain discrepancies if any
Conclusions
7
CHAPTER 2
COMPLIANT MECHANISMS
2.1 Introduction to Compliant Mechanisms
According to Shigley and Uicker (1980) a mechanism is a mechanical device used to
transfer or transform motion, force, or energy. Rigid-link mechanisms gain their
mobility from the movable joints.
Compliant mechanisms are flexible mechanisms, that gain some or all of their
motion through the deflection of flexible members rather than movable joints.
Compliant mechanisms can be fully compliant or partially compliant. Fully and
partially compliant mechanism definitions are given by Howell (2001). Fully
compliant mechanisms obtain all their motion from the deflection of compliant
members and partially compliant mechanisms contain one or more traditional
kinematic pairs along with compliant members.
Required input output relationship is obtained by the combination of the rigid and
compliant parts or fully compliant elements. The strength of the deflecting members
limits deflection of compliant link therefore a compliant link cannot produce a
continuous rotational motion.
The advantages of compliant mechanisms can be divided into two subgroups: cost
reduction and increased performance. Compliant mechanisms require fewer parts to
accomplish a certain task. A reduction in the number of parts reduces manufacturing
and assembly time, and cost. Some compliant mechanisms can be manufactured as a
8
single piece by injection molding process. Compliant mechanisms also have fewer
movable joints. That results in reduced wear and need for lubrication.
Using compliant mechanisms reduces the number of movable joints which increases
mechanism precision since backlash may be reduced or eliminated. Vibration and
noise caused by the revolute and sliding joints of rigid-body mechanisms may also
be reduced by using compliant mechanisms.
In compliant mechanisms energy is stored in the form of strain energy in the flexible
members. This property can be an advantage for some cases and a disadvantage for
some other cases. As an advantage the stored or transformed energy can be released
at a later time or in a different manner. A bow and arrow system is a good example.
All of the energy is not transferred, but some is stored in the mechanism.
Compliance becomes a disadvantage if function of a mechanism is to transfer energy
from input to output.
A major disadvantage of compliant mechanisms is the lack of knowledge regarding
analysis and synthesis methods for such mechanisms and the requirement to
determine the deflections of flexible members. Therefore analysis and design of
compliant mechanisms has difficulties compared to conventional mechanisms.
Fatigue analysis is another vital issue. Some compliant members are loaded in a
cyclic manner. To perform prescribed functions it is important to design those
compliant members with sufficient fatigue life.
Compliant links that remain under stress for long periods of time or subject to high
temperatures may experience stress relaxation and creep.
2.2 Pseudo-Rigid-Body Model
The purpose of the pseudo-rigid-body model is modelling the deflection of flexible
members by using rigid-body components which have equivalent force-deflection
9
characteristics. This method of modelling allows well-known rigid-body analysis
methods to be used in the analysis of compliant mechanisms. Howell (2001) says
that the pseudo-rigid-body model is a bridge that connects rigid-body mechanism
theory and compliant mechanism theory.
Salamon (1989) introduced a methodology for compliant mechanism design that
used a pseudo-rigid-body model of the compliant mechanism with compliance
modelled by torsional and linear springs. These models are much easier to analyze
than idealized models that require finite element or elliptic integral solutions. The
most important attribute of the pseudo-rigid-body model is that it significantly
simplifies the design process.
Closed-formed elliptic-integral solutions are used by Howell and Midha (1994 and
1998) to develop deflection approximations for an initially straight, flexible segment
with linear material properties.
2.2.1 Small-Length Flexural Pivots
An important component that exist in compliant mechanisms is the so called ''small
length flexural pivot''. The beam shown in Figure 2.1 has two segments. The small
segment is shorter and more flexible than the large segment. This small segment is
called small-length-flexural pivot. Usually large segment is at least 10 times larger
than small segment. The large segment is also much stiffer.
� ≫ � (2.1)
(��)< ≫ (��)= (2.2)
10
Figure 2.1 Cantilever Beam (a) and its Pseudo-Rigid Body Model (b)
For the flexible segment with end moment loading the deflection equations are
derived by Howell (2001) as follows:
8> = ��� (2.3)
6+� = 1 − �A&8>8> (2.4)
6B� = 1 − &C"8>8> (2.5)
These equations could be used to model small-length flexural pivots with pseudo-
rigid-body model. Figure 2.1 shows a member and its pseudo-rigid-body model . The
model consist of two rigid equal links, connected by a characteristic pivot.
Characteristic pivot represents the displacement and torsional spring models the
beam stiffness or resistance to deflection. This model gives an accurate solution for
the deflection path of the beam end for a given end load. The percentage error
between this model and the closed-form elliptic integral solutions is 0.5 for large
11
deflections. The angle of pseudo rigid link is the pseudo rigid body angle, 4, that is
equal to the beam end angle for small-length flexural pivots.
Θ= 8>(small-length flexural pivots) (2.6)
A torsional spring with spring constant � is used to model the beam's resistance to
deflection. The required torque to deflect the torsional spring at an angle 4 is
, = �Θ (2.7)
From the elementary beam theory the spring constant � could be found. For a beam
with an end moment, the end angle is
8> = �(��)= (2.8)
Since = , and 4 = 8>, the spring constant can be found as
� = (��)=� (2.9)
This model is more accurate for bending dominant cases than transverse and axial
loading dominant cases.
2.2.2 Active and Passive Forces
Figure 2.2 shows a cantilever beam with a force at the free end. The force, F, must
be defined by its magnitude and direction. The direction may be defined by the angle : or by the horizontal and vertical components of the force. In Figure 2.2 horizontal
component is shown as "# and the vertical component as #. F is a nonfollower
12
force which means it remains at the same angle regardless of the deflection of the
beam. The nonfollower force's magnitude and direction is
� = #D"E + 1 (2.10)
: = atan 1−" (2.11)
Figure 2.2 Cantilever Beam with a Force at the Free End and its Pseudo-Rigid Body
Model with a Torsional Spring
The force can also be resolved into its normal and tangential components. The
tangential component to the path which is also normal to the pseudo-rigid-link, ��,
causes a moment at the torsional spring
, = ��(� + �2) (2.12)
The tangential component, which causes the deflection of the pseudo-rigid-link, is
called an active force. The normal component is called a passive force and it has no
contribution to the deflection of the beam.
13
When the deflection changes the active and passive components change because F is
a non-follower force. Then the active force is,
�� = �&C"(: − 4) (2. 13)
2.2.3 Cantilever Beam with a Force at the Free End
In Figure 2.3 the flexible beam with a force applied at its free end is shown. If
deflections are large, the linear beam deflection equations may not give accurate
solutions. To perform the analysis, elliptic integral solutions or nonlinear finite
element analysis could be used. Instead of these methods, pseudo-rigid-body model
which is a simpler but accurate method of analysis may also be used.
Figure 2.3 Flexible Segment and its Pseudo-Rigid-Body Model
14
The location of the characteristic pivot is expressed in terms of the characteristic
radius factor, �, which represents the fraction of the beam length at which the pivot
is located. Once � is determined the deflection path may be parameterized in terms
of 4, the pseudo-rigid-body angle.
The characteristic radius factor is a function of ", that is horizontal coefficient of the
applied force. Howell (2001) introduced a formulation of � in terms of " as:
� = K0.841655 − 0.0067807" + 0,000438"E (0.5 < " < 10.0)0.852144 − 0.0182867" (−1.8316 < " < 0.5)0.912364 + 0.0145928" (−5 < " < −1.8316) S (2.14)
Figure 2.4 Plot of Characteristic Radius Factor, �, versus n
The �-" graph that is seen in Figure 2.4 shows that the characteristic radius factor �
does not vary much. Thus � is approximated by Howell (2001) as:
�TU( = V ��" WXWYV �" WXWY (2.15)
0.81
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
-5 -3 -1 1 3 5 7 9
Ch
ara
cte
rist
ic r
ad
ius
fact
or
(γ)
Coefficient of horizontal force component (n)
15
For −0.5 ≤ " ≤ 1
�TU( ≅ 0.85 (2.16)
The relationship between 8> and 4 is approximated by Howell (2001) as:
8> = ��4 (2.17)
where �� is constant and called the parametric angle coefficient.
The torsional spring constant that is used to model the beam's resistance to deflection
is, �Θ, the stiffness coefficient which is nondimensionalized torsional spring
constant.
Let
��E�� = �Θ4 (2.18)
�Θ =
\]]]]]]̂]]]]]]_ 3.024112 + 0.121290" +0.003169"E(−5 < " ≤ −2.5)
1.967647 − 2.616021" −−3.738166"E−2.649437"` − 0.891906"a−0.113063"b(−2.5 ≤ " ≤ −1)2.654855 − 0.509896 × 10de"+0.126749 × 10de"E−0.142039 × 10dE"`+0.584525 × 10da"a(−1 < " ≤ 10)
S (2.19)
where 4 < 4fTB ,
16
�ΘTU( = V �Θ�"WXWYV �"WXWY �A% g "e = 5"E = 10S (2.20)
�ΘTU( = 2.61 (2.21)
considering the loads are in a range of 63h < : < 135h or −0.5 < " < 1.0,
�ΘTU( = 2.65 A% �Θ ≅ i� (2.22)
The torque applied at the pin joint is,
, = �4 (2.23)
where � is the torsional spring constant [!/%��] and 4 is the angular deflection.
Torsional spring constant is defined by Howell (2001) as,
� ≅ i�E ���
(2.24)
where ��Θ is pseudo-rigid body constant.
For different values of n, value of the coefficients are shown in Table 2.1,
Table 2.1 The Coefficients of the Compliant Mechanisms
n m nopq(m) rs ts nopq(ts)
0 0.8517 64.3o 1.2385 2.677 58.5o
2 0.8276 109.0o 1.2511 2.597 69.0o
5 0.8192 121.0o 1.2557 2.562 67.5o
-0.5 0.8612 47.7o 1.2358 2.693 44.4o
-3 0.8669 16.0o 1.2119 2.688 12.9o
17
2.2.4 Moment at the Free End
The flexible beam with an end moment at its free end is shown in Figure 2.5.
Figure 2.5 Flexible Beam with a Moment at the Free End
The coordinates of the free end, maximum normal stress value and angle of the
beam end are determined by Howell (2001) as,
� = �[1 − 0.7346(1 − �A&4)] (2.25)
� = 0.7346�&C"4 (2.26)
8> = 1.51644 (2.27)
�fTB = >�� (2.28)
2.3 Nomenclature
In rigid-body mechanisms motion is transferred or transformed by rigid links and
traditional joints. However the working principle of the compliant mechanisms are
different. The deflection of the flexible members gives the motion. Therefore
identification of the compliant mechanism's parts are more difficult than the rigid
body mechanisms.
A link is defined by Howell (2001) as the continuum connecting the mating surfaces
of one or more kinematic pairs.
Figure 2.6
For a rigid-body mechanism the distance between the joints is constant and the shape
of the link is kinematically unimportant. However the motion of a compliant link is
dependent on link geometry and the forces. As seen in Figure
rigid or compliant and a compliant link can be simple or compound.
Figure 2.7 Component Characteristics of Segments
Rigid Link
18
of the compliant mechanism's parts are more difficult than the rigid
A link is defined by Howell (2001) as the continuum connecting the mating surfaces
of one or more kinematic pairs.
Component Characteristics of Links
body mechanism the distance between the joints is constant and the shape
of the link is kinematically unimportant. However the motion of a compliant link is
dependent on link geometry and the forces. As seen in Figure 2.6 the links may be
rigid or compliant and a compliant link can be simple or compound.
Component Characteristics of Segments
Link
Rigid Link Compliant Link
Simple Compound
Segment
Rigid Compliant
of the compliant mechanism's parts are more difficult than the rigid-
A link is defined by Howell (2001) as the continuum connecting the mating surfaces
body mechanism the distance between the joints is constant and the shape
of the link is kinematically unimportant. However the motion of a compliant link is
the links may be
19
When a compliant link is analyzed, it could be observed that the compliant segment
may be composed of rigid or compliant segments. Material discontinuities or
geometric changes are often the starting points of a new segment. Segments can be
either rigid or compliant as shown in Figure 2.7.
2.4 Diagrams
Skeleton diagrams are used to represent rigid-body mechanisms easily. Similar
diagrams are also used for compliant mechanisms. Symbols which represent joints
and segments are shown in Figure 2.8.
Figure 2.8 Symbol Convention for Compliant Mechanism Diagrams
By using symbols an example of a compliant mechanism diagram is shown in Figure
2.9. The mechanism has two compliant links. First link is compliant and it is a
Rigid Segment
�
Axially Compliant
Segment
Pin or Revolute
Joint Flexural Pivot
Slider (Prismatic)
Joint
or
Fixed Connection
or
20
composition of a compliant and a rigid segment. Second link is also compliant and
has 3 segments that are rigid and compliant. There are rigid traditional kinematic
pairs therefore it is a partially compliant mechanism.
Figure 2. 9 An Example of a Compliant Mechanism Diagram
2.5 Pseudo-Rigid-Body Replacement
In a compliant mechanism analysis or synthesis, sometimes transformation of the
mechanisms is required. This transformation can be compliant mechanism to rigid-
body mechanism or vice versa. In compliant mechanism analysis, a pseudo rigid
segment 1(rigid)
link 1
(compliant)
segment 2 (compliant)
segment 5(rigid)
link 2
segment 4 (compliant)
6 variable
21
body model is obtained from the compliant mechanism. However, in rigid body
replacement synthesis, a pseudo rigid model is equivalent to a rigid-body mechanism
model and the resulting mechanism is determined from these models.
In rigid-body mechanisms the distance between the joints is the kinematically most
important parameter. During all the replacement processes the joints must be fixed.
Joint is on the midpoint of the compliant segment in small length flexural hinges if
transformation is compliant mechanism to rigid-body mechanism or vice versa.
In flexible beams there is a relationship between the rigid and compliant link which
is defined by characteristic radius factor, �.
� = �� (2.29)
Where � is length of the rigid link and � is the length of the compliant link. The
characteristic radius factor value according to " values can be seen in Table 2.1.
Another important issue during pseudo-rigid-body replacement is taking the torque
,which comes from the deflection of complaint link, into account. Therefore torsional
springs must be attached to the mechanism.
In Figure 2.10 a four bar and slider crank mechanism's pseudo-rigid-body
replacements are shown. In both mechanisms the black mechanism represents
compliant mechanism and grey one is its pseudo-rigid-body model.
22
Figure 2.10 A Complaint Four Bar and Slider-Crank Mechanism and Pseudo Rigid
Body Models
2.6 Material Considerations
For a beam that is shown in Figure 2.11 the deflection at the free end is,
6 = 4��`��ℎ` (2.30)
23
Figure 2.11 Flexible Cantilever Beam
The maximum stress occurs at the fixed end and equals to,
�fTB = 6���ℎE (2.31)
The failure occurs when the �fTB equals to the yield strength, '+, then,
'+ = 6���ℎE (2.32)
From Equation 2.32 � can be expressed in terms of yield strength as,
� = '+�ℎE6� (2.33)
24
Substituting Equation 2.33 into Equation 2.30 results in the maximum deflection, 6fTB (assuming geometrically linear load vs. deflection relationship is valid),
6fTB = 23 '+� �Eℎ (2.34)
According to maximum deflection equation, ratio of the strength to modulus of
elasticity shows us how much does a beam deflect without permanent deformation.
Thus the material with the highest value of '+/� will allow larger deflection. In
Table 2.2 ratio of yield strength to modulus of elasticity for several materials are
shown.
Table 2.2 Ratio of Yield Strength to Young's Modulus for Some Materials
Material u (vwp) xy (zwp) (xy/u) × {|||
Steel (1010 hot rolled) 207 179 0.87
Steel (4140 Q&T@400) 207 1641 7.9
Blue Polished Spring Steel
(AISI 1080)
200 880 4.4
Aluminium (1100 annealed) 71.7 34 0.48
Aluminium (7075 heat treated) 71.7 503 7.0
Titanium (Ti-13 heat treated) 114 1170 10
Polyethylene (HDPE) 1.4 28 20
Polypropylene 1.4 34 25
As seen in Table 2.2 elastic modulus does not change much with addition of alloying
elements or heat treatments. However yield strength value could be increased by heat
treatment that also makes the material more brittle.
Polypropylene has a very high yield strength to young's modulus ratio which allows
large deflections. It is available in the market, inexpensive, very ductile, easy to
process and has a low density. It can yield thousands of time without fracturing.
25
CHAPTER 3
CONVENTIONAL UNIVERSAL JOINT
3.1 Introduction to Cardan Universal Joint
A universal joint is a joint or coupling that is capable of transmitting rotary motion
from one shaft to another which are not in line with each other. It consists of a pair
of hinges, oriented at 90o to each other that are connected by a cross shaft as seen in
Figure 3.1. Kinematically a universal joint is equivalent to a slotted sphere type of a
joint which has two degrees of rotational freedom.
Figure 3.1 Cardan Universal Joint (Tanık E. and Parlaktaş V. (2012))
7
26
3.2 Kinematic Analysis of the Cardan Universal Joint
Even when the input shaft rotates at a constant speed, output shaft could rotate at a
variable speed. For this reason the universal joint suffers from vibration and wear.
The speed of the output shaft varies and this variation depends on the configuration
of the joint that is specified by the variables given below,
� �e, the angle of rotation for axle 1
� �E, the angle of rotation for axle 2
� 7, the bend angle of the joint or the input shaft angle of the joint
Figure 3.2 Variables of Cardan Universal Joint
These variables can be seen in Figure 3.2. The red plane and axle 1 are perpendicular
to each other and axle 2 is always perpendicular to the blue plane. These planes are
at an angle 7. �e and �E are the angular displacement of each axle. �e, �E are the
angles between the 01e and 01E and initial positions along the 0 and � axes. The 01e
and 01E vectors are fixed with the cross shaft that connects the two axles therefore
they remain perpendicular to each other.
01E �1
01
01e
�e
�E
7
27
01e draws the border of the red plane and related to �eby,
q}{ = [�A&�e, &C"�e, 0] (3.1)
01E draws the border of the blue plane and is the result of the unit vector 01 = [1,0,0] being rotated through Euler angles [i/2, 7, 0],
q}~ = [−�A&�&C"�e, �A&�E, &C"�&C"�E] (3.2)
The 01e and 01E vectors are fixed with the cross shaft therefore they must remain at
right angles,
q}{. q}~ = 0 (3.3)
Thus the equation relating the rotations of axles is,
tan�e = cos7 tan�E (3.4)
and the solution for �Ein terms of �eand 7 is,
�E = ���"2(&C"�e, �A&7 �A&�e) (3.5)
The angles �e and �E are the functions of the time. Differentating the rotation angles
with respect to time gives us the angular velocities of the axles,
9e = ��e/�� (3.6)
9E = ��E/�� (3.7)
The relationship between the angular velocities of axle 1 and 2 is,
28
9E = 9e�A&71 − &C"7E�A&�eE (3.8)
The relationship between the angular accelerations can be derived by differentiating
the angular velocity equation,
�E = �e�A&71 − &C"7E�A&�eE − 9eE�A&7&C"7E&C"�eE(1 − &C"7E�A&�eE)E (3.9)
The angular velocity of the output shaft versus rotation angle of the input shaft for
different shaft angles are plotted by using Equation 3.8 in Figure 3.3. �eis in degrees
and for a complete rotation of input shaft.
Figure 3.3 Input Shaft Angle versus Output Shaft Speed for Different Shaft Angles
29
The output shafts's rotation angle versus input shaft rotation angle for different shaft
angles are plotted by using Equation 3.5 in Figure 3.4. �eis in degrees and for a
rotation of input shaft 0° to 180°.
Figure 3.4 Input Shaft Angle versus Output Shaft Angle for Different Shaft Angles
30
31
CHAPTER 4
DESIGN OF THE COMPLIANT UNIVERSAL JOINT
4.1 Dimension Synthesis of the Compliant Universal Joint
Tanık and Parlaktaş (2012) described the universal joint as a spherical four bar
linkage which is a special case of spatial four bar mechanism. In the universal joint
the arc lengths of the moving links are exact right angles and the connected shafts
intersect at an angle as seen in Figure 4.1.
Figure 4.1 Spherical 4R Linkage (Tanık E. and Parlaktaş V. (2012))
Tanık and Parlaktaş (2012) proposed a novel design for a compliant universal joint
whose compliant components are flexural pivots as seen in Figure 4.2. In Figure 4.3
dimensions of the compliant universal joint which is made of polypropylene are
32
presented. For different applications some of the dimensions are free parameters
however three constraints must be satisfied.
Figure 4.2 Original Compliant Universal Joint Design (Tanık E. and Parlaktaş V.
(2012))
The first constraint comes from the theory of the spherical four bar mechanism.
According to this constraint centerlines of all single axis flexural hinges must
intersect at the center of the sphere. The centerline axis of the flexural hinges and the
inside contact surface must be aligned as seen in Figure 4.3.
33
Figure 4.3 Dimensions of the Compliant Universal Joint (Tanık E. and Parlaktaş V.
(2012))
To satisfy this constraint,
� = � + �/2 (4.1)
The second constraint is to obtain a form closed structure. The thickness of the
identical parts and the width of the connection slot must be equal as shown in Figure
4.3.
� = �. (4.2)
34
Figure 4.4 Critical Position of the Compliant Universal Joint (Tanık E. and Parlaktaş
V. (2012))
The third constraint is to avoid a contact between &e and &E surfaces during operation
as seen in Figure 4.4. For the compliant universal joint Tanık and Parlaktaş (2012)
defined the deflection of single axis flexural hinges. When the input angle 8 equals
0 , 90°, 180° and, 270° one set of the hinges will be at maximum deflected position
and the other set will not be deflected at all. The position in Figure 4.4 most
probably is the critical position when the contact occurs between &e and &E.
However, the mechanism is rotating and just before and after this position 7 will
decrease. On the other hand the sharp corners of the surface &E may contact to &e due
to compliance of the structure and plate thickness. However if compliance factor is
taken into account, further analysis would be challenging. Therefore the relationship
between dimensions can be determined from the geometry as,
0&C"(90 − 7) ≈ 0.5�. + �&C"7 (4.3)
0�A&(90 − 7) + ��A&7 ≈ � − � (4.4)
Eliminating 0 from Equations 4.3 and 4.4 , the lower limit of � is determined by
Tanık & Parlaktaş (2012) as,
35
� ≈ 0.5�.(��"7) + �(��"7&C"7 + �A&7 + 1) (4.5)
H value seen in Figure 4.3 is a free parameter that defines the whole size of the
mechanism.
4.2 Material Selection of the Compliant Universal Joint
In this thesis, first of all blue polished spring steel (AISI 1080) is chosen as the
material of the compliant part of the mechanism. This material has high yield
strength and relatively ductile behavior. Since, fatigue characteristics of polymers are
not definite as in steels in the literature, estimating the life of the mechanism
analytically would be very hard if not impossible. In Appendix A, properties of the
material can be found and some of the properties are given in Table 4.1.
Table 4.1 Properties of AISI 1080 (Spring Steel)
Density 7800-7900 kg/m3
Young's modulus 200-215 GPa Shear modulus 77-84 GPa Bulk modulus 155-175 GPa Poisson's ratio 0.285-0.295 Yield strength 880-1080 MPa Tensile strength 1170-1440 MPa
According to the three constraint equations given above, the dimensions of the
design are determined. Two different designs are proposed where thicknesses of the
small length flexural hinges are 0.5 and 0.75 mm, respectively in the experiments
performed in this study, the geometry with 0.5 mm thickness is used thus the
detailed design of this geometry will be explained.
Free parameters are chosen as follows: � =20 mm, �- =0.5 mm, � =15 mm and �. =5.68 mm. For the ease of manufacturing three plates will be assembled therefore �. is the summation of blue polished spring steel, two steel plates and tolerances of
the plates. According to the Equation 4.1, 4.2 and 4.5 the dimensions are determined
36
as, � =25 mm, � =5.68 mm and the minimum value of � =50.34 mm. All of the
dimensions of the design can be seen in Appendix B. The isometric view can be seen
in Figure 4.5 and deflected position of the mechanism for different input shaft angles
is shown in Figure 4.6.
Figure 4.5 Isometric View of the Design
Flexural Hinges
Shaft 1
Shaft 2
37
Figure 4.6 For a 10° Shaft Angle and Different Input Shaft Angles Views of the
Mechanism
38
THEORETICAL ANALYSIS
5.1 Stress Analysis of the
Preliminary stress analyses of the design are
then finite element analysis is u
ANSYS 13.0 Workbench software is used.
Firstly, both of the geometries are sketched with Catia V5.
analyses are performed
the analysis can be seen in Figure
following parts.
Importing the CAD
model of the design into the ANSYS 13.0
Programme's design
modeller module
Meshing the model
Defining the boundary
conditions and loadings
39
CHAPTER 5
CAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT
Stress Analysis of the Compliant Universal Joint with FEA
stress analyses of the design are performed by analytical methods and
hen finite element analysis is used for comparison. For finite element analysis
ANSYS 13.0 Workbench software is used.
both of the geometries are sketched with Catia V5. Then
performed for different shaft angles and torque values. The flow chart of
the analysis can be seen in Figure 5.1. Details of the analysis will be explained in the
Figure 5.1 Flow Chart of the Analysis
Importing the CAD
model of the design into the ANSYS 13.0
Programme's design
modeller module
Pre processing of the
model (defining rigid and flexible parts, simplfying
the geometry etc.)
Importing the model to
the analyser module
Assigning the material
properties to the parts
Defining the contacts
and joints between partsMeshing the model
Defining the boundary
conditions and loadings
Analysis settings and
solving the problem
Obtaning the results and
making comments
COMPLIANT UNIVERSAL JOINT
Compliant Universal Joint with FEA
by analytical methods and
sed for comparison. For finite element analysis
hen, finite element
angles and torque values. The flow chart of
etails of the analysis will be explained in the
Importing the model to
the analyser module
Assigning the material
properties to the parts
Obtaning the results and
making comments
40
The steps shown in Figure 5.1 will be explained schematically for ANSYS 13.0
Workbench.
Defining the type of the analysis (Modal, Static, Transient Structural, etc.)
Entering the engineering data for AISI 1080
41
By using ANSYS Workbench Design Modeller prepare the model so that it is ready
for the analysis (Deleting the idle parts, improving the surfaces, etc.)
Analysis part
42
Defining the behaviors of the parts (Flexible or rigid)
Assigning the material properties that are defined in Engineering Data
43
Creating new coordinate systems to identify motions or forces
Defining contacts and joints between parts
44
Meshing the parts
Defining the analysis settings
45
Defining the boundary conditions and loadings
46
Getting the solutions and making comments
5.1.1 Boundary Conditions and Meshing
The spring steel universal joint is divided into subgroups whose characteristics will
be different in the analysis. Small length flexural hinges are the most critical parts of
the design therefore these hinges are meshed finely which means the number of
elements for unit area has the highest value compared to the other parts. And the
other parts are meshed coarsely which behave as relatively rigid and not as much
critical as flexible hinges. Hence the analysis took less time with a good accuracy of
results. The parts of the joint could be seen in Figure 5.2.
47
Figure 5.2 Definitions of the Parts
The connections of the parts are shown in Figure 5.3. Shaft 1 and clutch 1 are
modelled as a unique part by using Form New Part command. And also compliant
joint 1 is bonded to clutch 1 and body 1. Bonding creates a multi point constraint so
that the bonded surfaces behave like a single surface. Same procedure is done for the
other compliant joints. The connections and the parts that have been formed, are
shown in Figure 5.3.
Figure 5.3 Connections and Newly Formed Parts
As seen in Figure 5.4 the model is divided into two main parts whose meshes are
finer than other part. Part A is meshed with body sizing of 0.65 mm and part B is
48
meshed with 2 mm hex-dominant method which uses an unstructured meshing
approach to generate a quad-dominant surface mesh and then fill it with a hex-
dominant mesh. The total number of the elements is 19053 and there are 27428
nodes for the design with 0.75 mm thickness. The model with 0.5 mm thickness has
21705 elements 31896 nodes.
Figure 5.4 The Meshing of the Model
5.2 Static Analysis for Deflection-Only Case
Firstly the bending capacity of the compliant universal joint without loading of a
torque should be identified. In the design thicker parts are modelled as rigid and are
not bent compared to the flexural hinges.
For different shaft angles stress values are calculated analytically and numerically.
The analysis of the design with thickness of 0.75 mm is done for shaft angles
between 1° to 13°. Same procedure is done for the design with 0.5 mm thickness for
shaft angles 1° to 18°.
49
Analytical calculations that can be seen in Table 5.1 are done with the procedure
given below,
Width, thickness and length of the hinge are,
/ = 20 × 10d` (5.1)
�- = 0.75 × 10d` (5.2)
� = 20 × 10d` (5.3)
Second moment of area is,
� = /�-̀12 = 7.031 × 10de`a (5.4)
Modulus of elasticity for blue polished spring steel is,
� = 200 × 10�#� (5.5)
Taking shaft angle, 7, in radians, moment and maximum stress values could be
found with the equations given below,
= ��7� (5.6)
�fTB = �-/2� (5.7)
50
Figure 5.5 Representation of the Boundary Condition and Shaft Angle
For the numerical solution the model which is shown in Figure 5.5 is used. One end
is fixed and the other end is bent from analysis settings by inserting supports, fixed
support and remote displacement, in FEA. Remote displacement command is used
by rotating the shaft in the desired direction and for the desired shaft angle. The
equivalent stress values are obtained. Numerical solutions can be seen in Table 5.1
for the material with thickness of 0.75 mm and in Table 5.2 for the other model for
shaft angles as limited by yielding. The average of the chosen data is calculated.
During the analysis for the angles between 1° to 4° small deflection analysis is used
and for the other angles large deflection analysis is chosen from analysis settings.
Stress concentration is a highly localized effect. In some cases the reason may be
surface scratches. Engineering normal strain for fracture, 23, is an important
parameter for defining a material as ductile or brittle. If 23 ≥ 0.05, material is
ductile, otherwise material is brittle. In ductile materials the stress concentration
factor is not usually applied to predict the critical stress, because plastic strain in the
region of the stress is localized and has a strengthening effect. In other words, if the
material is ductile and the load is static, load may cause yielding in the critical
loading near the notch. This yielding can involve strain hardening of the material and
increases yield strength at the small critical notch location. In brittle materials the
geometric stress concentration factor, ��, is applied to the nominal stress.
51
AISI 1080 material has an elongation of 10-14 % which means 23 changes between
0.1 to 0.14. Therefore blue polished spring steel is a ductile material. Considering
that loads are static and the material is ductile, this part can withstand the loads with
no general yielding. Budynass and Nisbett (2011) points that in these cases the
designer sets the geometric (theoretical) stress concentration factor, ��, to unity.
Therefore, in this study all the following static analyses are performed by
disregarding stress concentration effects.
Table 5.1 Analytical and Numerical Normal Stress
Values for Different Shaft Angles (�- =0.75 mm)
Deflection-only for th=0.75 mm Shaft Angle
(deg) σ-max-
numerical*(MPa) σ-max-
analytical**(MPa) % Error 1 66.6 65.5 1.7 2 133.2 130.9 1.7 3 199.7 196.3 1.7 4 266.3 261.8 1.7
5 (large) 337.6 327.2 3.1 6 (large) 406.3 392.7 3.3 7 (large) 475.2 458.1 3.6 8 (large) 544.4 523.6 3.8 9 (large) 613.7 589 4.0
10 (large) 683.2 654.5 4.2 11 (large) 752.8 719.9 4.4 12 (large) 822.4 785.4 4.5 13 (large) 892.0 850.8 4.6
* The max stress represents the maximum value of the stresses available on the
hinges disregarding the stress concentration regions
** The related stress does not include stress concentrations
The numerical solutions for 1° , 5° , 10° shaft angles are shown in Figures 5.6 to 5.8.
The data that is chosen to calculate the average values are shown in Table 5.1.
During the selection of the data the hinge with the maximum stress value is found
and then the equivalent stress values for this hinge are chosen by using probe. The
average values of the stress values can be seen in Table 5.1 for the model with the
thickness of 0.75 mm.
52
Figure 5.6 ANSYS Simulation for �- =0.75 mm and 1° Shaft Angle
Figure 5.7 ANSYS Simulation for �- =0.75 mm and 5° Shaft Angle
53
Figure 5.8 ANSYS Simulation for �- =0.75 mm and 10° Shaft Angle
To compare the analytical and numerical solutions shaft angle vs. normal stress
values are plotted seen in Figure 5.9. As the shaft angle increases the normal stress
values increase which is expected. For the 0.75 mm thickness the maximum shaft
angle without yielding is found as 13°. This value increases for the model with 0.5
mm thickness. It is observed that the difference between the solutions increases with
the increasing shaft angle values. From the percentage error it is obvious that the
analytical approach is more suitable for the angles that are not large. The large
deflection analysis rises the percentage error for larger shaft angles, since the
numerical solution deviates from the analytical one.
54
Figure 5.9 Plot of Shaft Angle vs. Analytical and
Numerical Normal Stress Values (�- = 0.75 )
Table 5.2 Analytical and Numerical Normal Stress
Values for Different Shaft Angles (�- = 0.5 )
Deflection-only th=0.5 Shaft Angle
(deg) σ-max-
numerical*(MPa) σ-max-
analytical**(MPa) % Error
1 41 43.6 1.3
2 89.6 87.3 2.6
3 134.6 130.9 2.7
4 180.2 174.5 3.2
5 (large) 223.4 218.2 2.3
6 (large) 282.0 261.8 7.2
7 (large) 331.7 305.4 7.9
8 (large) 374.2 349.1 6.7
9 (large) 449.8 392.7 12.7
10 (large) 477.3 436.3 8.6
11 (large) 528.9 480.0 9.2
12 (large) 569.6 523.6 8.1
13 (large) 604.1 567.2 6.1
14(large) 680.7 610.9 10.3
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
0 2 4 6 8 10 12 14
σ_
ma
x(M
Pa
)
Shaft Angle (deg)
Numerical
Analytical
55
15 (large) 717.6 654.5 8.8
16 (large) 755.7 698.1 7.6
17 (large) 822.3 741.8 9.8
18 (large) 869.5 785.4 9.7
19 (large) - 829.0 -
20 (large) - 872.7 -
* The max stress represents the maximum value of the stresses available on the
hinges disregarding the stress concentration regions
** The related stress does not include stress concentrations
The numerical solutions for 1°, 5°, 10° and 15° shaft angles are shown in Figures
5.10 to 5.13. Same procedure is done for the model with 0.5 mm thickness. The
average values of the stress values can be seen in Table 5.2.
Figure 5.10 ANSYS Simulation for �-=0.5 mm and 1° Shaft Angle
56
Figure 5.11 ANSYS Simulation for �-=0.5 mm and 5° Shaft Angle
Figure 5.12 ANSYS Simulation for �- =0.5 mm and 10° Shaft Angle
Figure 5.13 ANSYS Simulation for �- =0.5 mm and 15° Shaft Angle
Shaft angle vs. normal stress values for the second model are plotted which can be
seen in Figure 5. 14. For the 0.5 mm thickness the maximum shaft angle without
57
yielding is found as 18° numerically and 20° analytically. Analytical approach is a
good estimation of the simulations however simulation values will be used to
determine critical shaft angle values.
Figure 5. 14 Plot of Shaft Angle vs. Analytical and
Numerical Normal Stress Values (�- = 0.5 )
To make a comparison shaft angle vs. normal stress values for both of the models are
shown in Figure 5.15. Normal stress values in terms of �-, � and can be written
as,
� = 6 /��-E (5.8)
As thickness increases, normal stress value decreases because �- is in the
denominator in Equation 5.8. However value increases as the flexibility decreases
and this increase is much more than the increase with the thickness change.
Therefore stress values are larger for the model with 0.75 mm thickness for the same
shaft angles.
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
σ(M
Pa
)
Degree (deg)
Numerical
Analytical
58
Figure 5.15 Plot of Shaft Angle vs. Analytical and
Numerical Normal Stress Values for Both Models
5.3 Study of Mesh Refinement for Bending-Only Case
Mesh refinement is an important tool for editing finite element meshes in order to
increase the accuracy of the solution. However, as a mesh is made finer, the
computation time increases. The density of mesh must satisfactorily balance
accuracy and computing resources. Refinement is performed in an iterative manner
in which a solution is found, error estimates are calculated, and elements in regions
of high error are refined. This process is repeated until the desired accuracy is
obtained.
For the model with the thickness of 0.5 mm finite element analyses are done for
different mesh densities. The analysis are done for the deflection-only case with a
shaft angle of 15°. The most critical part of the mechanism is one set of the
compliant flexural hinges. The most critical set of the hinges are meshed with three
different mesh densities. During the refinement study three, four and five elements
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
800.0
900.0
1000.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
σ_
ma
x(M
Pa
)
Bend Angle (deg)
Analytical 0.5
Numerical 0.5
Analytical 0.75
Numerical 0.75
59
are used through the thickness and the results are shown in Table 5.3. In Figure 5.13
the simulation results for three elements are shown and in Figure 5.16 and 5.17 other
simulation results can be seen.
Figure 5. 16 Simulation Results for Four Elements through the Thickness for
Bending-Only Case
Figure 5. 17 Simulation Results for Five Elements through the Thickness for
Bending-Only Case
At least three convergence runs are required to plot a curve which can then be used
to indicate when convergence is achieved or, how far away the most refined mesh is
from full convergence. Therefore convergence curve is plotted and shown in Figure
60
5.18. Three runs of different mesh density give the nearly same result, therefore
convergence is already achieved and no more refinement is necessary. After this
study of refinement 3 elements through the thickness are used.
Table 5. 3 Number of Elements through the Thickness vs. Maximum von-Mises
Stress for Bending-Only Case
Number of Elements through the Thickness
Maximum von-Mises Stress (MPa)
3 759.95
4 759.94
5 759.54
Figure 5. 18 Number of Elements through the Thickness vs. Maximum von-Mises
Stress for Bending-Only Case
5.4 Path Analysis
In addition to the results given in section 5.2 9 paths are defined to determine the
equivalent stresses along these path. These are shown in Figure 5.19. For this
analysis the model with the thickness of 0.5 mm is chosen and the shaft angle is set
750
752
754
756
758
760
762
3 3.5 4 4.5 5
Ma
xim
um
vo
n-M
şsse
s S
tre
ss (
MP
a)
Number of Elements
61
as 10°. The analysis shown in Figure 5.19, is done by finite element method using
ANSYS. Post processing results along a path is part of the Workbench Mechanical .
In the analysis a path is defined as construction geometry on which to map the
results. Then a linearized stress result is obtained. Finally the desired results along
the path using the Linearizad Stress item are calculated. The path linearized stress
results can be seen in Figure 5.20 to Figure 5.29.
Figure 5.19 Paths for One of the Deflected Flexural Hinge
Figure 5.20 Path Analysis in ANSYS
Path 1 Path 2 Path 3
Path 4
Path 5
Path 6
Path 7 Path 8 Path 9
62
Figure 5.21 Length vs. Equivalent Stress for Path 1
Figure 5.22 Length vs. Equivalent Stress for Path 2
Length (m)
Equ
ival
ent
Str
ess
(Pa)
Length (m)
Equ
ival
ent
Str
ess
(Pa)
63
Figure 5.23 Length vs. Equivalent Stress for Path 3
Figure 5.24 Length vs. Equivalent Stress for Path 4
Length (m)
Equ
ival
ent
Str
ess
(Pa)
Length (m)
Equ
ival
ent
Str
ess
(Pa)
64
Figure 5.25 Length vs. Equivalent Stress for Path 5
Figure 5.26 Length vs. Equivalent Stress for Path 6
Length (m)
Equ
ival
ent
Str
ess
(Pa)
Length (m)
Equ
ival
ent
Str
ess
(Pa)
65
Figure 5.27 Length vs. Equivalent Stress for Path 7
Figure 5. 28 Length vs. Equivalent Stress for Path 8
Length (m)
Equ
ival
ent
Str
ess
(Pa)
E
quiv
alen
t S
tres
s (P
a)
Length (m)
66
Figure 5. 29 Length vs. Equivalent Stress for Path 9
5.5 Static Analysis to Determine Torque Limits
In the second part of the analysis torque limits are calculated for some specific shaft
angles that were analyzed in the deflection-only case. In this part analyses are done
in two steps. In the first step one of the shaft is bent by the desired shaft angle, in the
second step torque is applied to the same shaft. In order to apply the torque to the
shaft which is bent in the first step, a new coordinate system overlapping with the
centerline of the shaft is created. In the analysis large deflection assumption is used
and also nonlinear solution option is chosen.
Initial time step is taken as 0.1 second, minimum time step is 0.01 and maximum
time step is 1 second. These settings can be seen in Figure 5.30.
Length (m)
Equ
ival
ent
Str
ess
(Pa)
67
Figure 5.30 Analysis settings
The configuration with 1° shaft angle will be explained briefly, other shaft angle
configurations can be analyzed with the same procedure.
In the first step of the loading normal stress has the maximum value of 68.5 MPa for
the first model. In the second step this position is protected and torque is applied.
This torque causes a maximum of 880 MPa stress value on the flexural hinges. The
steps of the loading and the stress values vs. time increments are shown in Table 5.4
and Table 5.5.
Table 5.4 Steps of the Loading
68
Table 5.5 Stress Values vs. Time Increments
To determine the exact torque value that causes yielding, torque values are applied
iteratively. Two suitable torque values are applied to the joint and the exact value of
the torque is found by interpolation. In Table 5.6 and 5.7 the interpolated data can be
seen for both models.
Torque is applied to the universal joint at its free end. However the torque on the
flexural hinges are indeterminate. Applied torque could increase bending in the
flexural hinges and the reactions could be a combination of torques carried by the
flexural hinges and the couple moment of shear forces. Therefore a simple model
shown in Figure 5.31 is proposed to determine the percentage effects of the couple
and torques on the hinges. Using ANSYS 5000 N.mm torque is applied and the
reactions of the flexural hinges are calculated. Torques are determined as 112.2
N.mm and forces as 55.1 N. The ANSYS model and some of the results are shown in
Figure 5.32.
From statics applied torque is defined as,
, = �� + ,e + ,E (5.8)
, = (55.1! × 87) + 112.2! + 112.2!= 5018!
(5.9)
69
Therefore 96% of the torque comes from the couple moment and the rest comes from
the ,e and ,E torques. This confirms that the compliant parts in this mechanism are
predominantly flexural hinges.
Figure 5.31 Simple Model with Applied Torque and Reactions at Flexural Hinges
Figure 5.32 Finite Element Analysis and Results for Simple Model
,
� = 87 � = 87
�
�
,e ,E 20
20
0.5
70
Table 5.6 Interpolation of the Torque Values and Stress Values for �- = 0.75
Interpolation values th=0.75 mm Shaft Angle (deg)
Upper torque
Upper stress
Lower torque
Lower stress
Torque value for 880 MPa
1 12 955.01 11 876.1 11.0 2 11 925.81 10 848.16 10.4 3 11 974.64 9 818.72 9.8 4 10 945.33 8 789.64 9.2 5 10 994.60 8 849.39 8.4 6 8 911.20 7 838.46 7.6 7 8 974.46 6 828.46 6.7 8 6 891.64 5 824.94 5.8 9 6 955.59 4.5 855.52 4.9
10 4 885.99 3 853.2 3.8 11 3 885.8 2 827.5 2.9 12 3 946.85 2 893.02 1.8 13 1 919.02 0.5 896.65 0.1
In Table 5.6 torque limits for different shaft angles are shown. In Table 5.7 torques
that causes yielding are tabulated. According to this data Figure 5.33 is plotted.
When the shaft angle increases the maximum torque capacity decreases as expected.
Table 5.7 Torque Limits for Different Shaft Angles (�- =0.75 mm)
Torque limits for σ-yield=880 MPa th=0.75 mm Shaft Angle (deg) T-max (N.m)
1 11 2 10.4 3 9.8 4 9.2 5 8.4 6 7.6 7 6.7 8 5.8 9 4.9
10 3.8 11 2.9 12 1.8 13 0.1
71
Figure 5.33 Shaft Angle vs. Maximum Torque Values for �- = 0.75
The same procedure is done for other model. The interpolation table and the results
can be seen in Table 5.8 and Table 5.9. The Figure 5.34 is plotted and gives the
similar trend with the first model.
Table 5.8 Interpolation of the Torque Values and Stress Values for �- =0.5 mm
Interpolation values th=0. 5 mm Shaft Angle (deg)
Upper torque
Upper stress
Lower torque
Lower stress
Torque value for 880 MPa
1 5.8 881.99 5 727.48 5.8 2 5.6 905.80 5 801.80 5.5 3 6 1019.67 5 829.17 5.3 4 6 1059.83 5 869.47 5.1 5 5 927.21 4 742.96 4.7 6 5 955.18 4 784.00 4.6 7 5 981.66 4 803.70 4.4 8 5 1090.19 4 845.49 4.1
10 4 937.14 3 792.90 3.6 12 4 1007.86 3 862.24 3.1 14 3 963.49 2 826.09 2.4
0
2
4
6
8
10
12
0 2 4 6 8 10 12 14
T m
ax (
Nm
)
Bend angle (deg)
72
Table 5.9 Torque Limits for Different Shaft Angles (�- =0.5 mm)
Torque limits for σ-yield=880 MPa th=0.5 mm Shaft Angle (deg) T-max (N.m)
1 5.8 2 5.5 3 5.3 4 5.1 5 4.7 6 4.6 7 4.4 8 4.1 9 3.9
10 3.6 11 3.3 12 3.1 13 2.8 14 2.4 15 2 16 1.6 17 1 18 0.6 19 0
Figure 5.34 Shaft Angle vs. Maximum Torque Values for �- =0.5 mm
0
1
2
3
4
5
6
7
0 5 10 15 20
T m
ax (
Nm
)
Shaft Angle (deg)
73
Torques can be applied for the model with the thickness of 0.5 mm are lower for
small shaft angles as shown in Figure 5.35. At 10° shaft angle curves intersect and
torque capacity of the thicker model decreases. The model with the thickness of 0.5
mm can operate at larger shaft angles where thicker one cannot transmit torque at
13° shaft angle.
The reason can be explained as follows. For lower shaft angles, the thicker model
carries more torque as expected. Since the cross section is larger, for a given torque
stresses are lower. As the bend angle increases, however, bending stresses for the
thicker model becomes dominant so its torque carrying capacity diminishes.
Figure 5.35 Shaft Angle vs. Maximum Torque Values for Both Models
5.6 Study of Mesh Refinement for Combined Loading Case
For the model with the thickness of 0.5 mm finite element analyses are done for
different mesh densities. The analysis are done for the combined loading case with a
shaft angle of 14°. The most critical part of the mechanism is one set of the
0
2
4
6
8
10
12
0 5 10 15 20
T m
ax (
Nm
)
Bend Angle (deg)
t_h = 0.5 mm
t_h = 0.75 mm
74
compliant flexural hinges. The most critical set of the hinges are meshed with three
different mesh densities. During the refinement study three, four and five elements
are used through the thickness and the results are shown in Table 5.10. In Figure
5.36 the simulation results for three elements are shown and in Figure 5.37 and 5.37
other simulation results can be seen.
Figure 5.36 Simulation Results for Three Elements through the Thickness for
Combined Loading Case
Figure 5.37 Simulation Results for Four Elements through the Thickness for
Combined Loading Case
75
Figure 5.38 Simulation Results for Five Elements through the Thickness for
Combined Loading Case
The convergence curve is plotted which can be seen in Figure 5.39.Three runs of
different mesh density give the nearly same result, therefore convergence is already
achieved and no more refinement is necessary. After this study of refinement 3
elements through the thickness are used.
Table 5. 10 Number of Elements through the Thickness vs. Maximum von-Mises Stress for Combined Loading Case
Number of Elements through the Thickness
Maximum von-Mises Stress (MPa)
3 903.54
4 904.53
5 937.59
76
Figure 5.39 Number of Elements through the Thickness vs. Maximum von-Mises
Stress for Combined Loading Case
5.7 Static Analysis of the Mechanism at Zero Degree Shaft Angle
During this step of the analysis maximum shear stress values for zero shaft angle are
determined. One end is fixed from analysis settings by inserting fixed support in
ANSYS other end is not bent as previous steps. Only increasing torque values are
applied to the joint by using loads from analysis settings. The desired direction and
desired value of torque is applied to the shaft without fixed support. Firstly 1 N.m
torque is applied to the shaft then 2.5 N.m is applied and this value is increased by
2.5 N.m increments. The stress values are obtained by maximum shear stress
solution. Numerical solutions can be seen in Table 5.11 for the material with
thickness of 0.75 mm and in Table 5.12 for the other model.
The simulations for the first model are shown in Figure 5.40 to 5.46. Average of the
higher shear stress values which are represented by red color in simulations are taken
and tabulated. In Table 5.11 the average value of the shear stresses on the hinge are
shown.
900
905
910
915
920
925
930
935
940
3 3.5 4 4.5 5
Ma
xim
um
vo
n-M
isse
s S
tre
ss (
MP
a)
Number of Elements
77
Figure 5.40 Shear Stress Values for 1 N.m Torque (�- = 0.75 mm)
Figure 5.41 Shear Stress Values for 2.5 N.m Torque (�- = 0.75 mm)
Figure 5.42 Shear Stress Values for 5 N.m Torque (�- = 0.75 mm)
78
Figure 5.43 Shear Stress Values for 7.5 N.m Torque (�- =0.75 mm)
Figure 5.44 Shear Stress Values for 10 N.m Torque (�- = 0.75 mm)
Figure 5.45 Shear Stress Values for 12.5 N.m Torque (�- = 0.75 mm)
Figure 5.
Table 5.
Only torque (Torque (
The results are plotted in Figure
stress increases in an almost linear manner.
Figure 5.
0.0
100.0
200.0
300.0
400.0
500.0
600.0
0 1
Sh
ear
Str
ess
(MP
a)
79
Figure 5.46 Shear Stress for 15 N.m Torque (�- = 0.75 mm)
Table 5.11 Torque vs. Shear Stress for �- = 0.75 mm
Only torque (N.m) for zero shaft angle th=0.75 Torque (N.m) �_�0 (MPa)
1 35.3 2.5 89.3 5 178.5
7.5 247.7
10 331.5 12.5 344.9
15 490.5
The results are plotted in Figure 5.47. As expected when the torque increases
in an almost linear manner.
Figure 5.47 Torque vs. Shear Stress for �- = 0.75 mm
2 3 4 5 6 7 8 9 10 11
Torque (Nm)
0.75 mm)
0.75 mm
torque increases, shear
0.75 mm
11 12 13 14
The simulations for the second model are shown in Figure
procedure is used and shear stress values are taken and tabulated in Table
Figure 5.48 Shear Stress for 1
Figure 5.49 Shear Stress for 2.5
Figure 5.50 Shear Stress Values for 5
80
The simulations for the second model are shown in Figure 5.48 to 5
procedure is used and shear stress values are taken and tabulated in Table 5
Shear Stress for 1 N.m Torque (�-= 0.5 mm)
Shear Stress for 2.5 N.m Torque (�- = 0.5 mm)
Shear Stress Values for 5 N.m Torque (�-= 0.5 mm)
5.51. Same
5.12.
= 0.5 mm)
Figure 5.
Table 5.12
Only torque (Torque (
The results are plotted in Figure
trend with the first model. To make comparison between two models Figure
plotted.
Figure 5.
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0
Sh
ear
Str
ess
(MP
a)
81
Figure 5.51 Shear Stress for 7.5 N.m Torque (�- = 0.5
12 Torque vs. Shear Stress Values for �- = 0.5 mm
Only torque (N.m) for zero shaft angle th=0.5 Torque (N.m) �_�0 (MPa)
1 68.7 2.5 166.9 5 338.3
7.5 516.3
The results are plotted in Figure 5.52. The plot for the second model has a similar
trend with the first model. To make comparison between two models Figure
Figure 5.52 Torque vs. Shear Stress for �- = 0.5 mm
1 2 3 4 5 6
Torque (Nm)
0.5 mm)
0.5 mm
. The plot for the second model has a similar
trend with the first model. To make comparison between two models Figure 5.53 is
0.5 mm
7 8
82
Figure 5.53 Torque vs. Shear Stress for Both Models
The line belongs to the model with the thickness of 0.5 mm is above the other line.
Torque for the same shear stress value is larger for the thicker model. Therefore the
maximum torque value for 440 MPa shear stress is 6.4 N.m for the thinner one and
the thicker one has 14.1 N.m.
5.7 Fatigue Life Estimations
Fatigue life estimation is one of the most important steps of a compliant mechanism.
Machine elements can catastrophically fail even if the maximum stresses are well
below the ultimate strength of material because of fatigue. In this study fatigue life
calculations are performed according to the modified Goodman failure theory,
calculation steps and Marin factors for both models are presented below. Initially,
calculations are performed for deflection-only case and then for the combined
loading case.
For the blue polished spring steel yield strength value is,
'+ = 880 #� (5.10)
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sh
ear
Str
ess
(MP
a)
Torque (Nm)
t_h = 0.75 mm
t_h =0.5 mm
83
Ultimate strength of the material is,
'*� = 1.1 × 10` #� (5.11)
Unmodified endurance limit is estimated as,
'(′ = g0.504'*� C� '*� ≤ 1460740 C� '*� > 1460S (5.12)
'(′ = 554.4 #� (5.13)
Marin factors are determined for the model with the thickness of 0.5mm. In fatigue
life calculations, surface finish is one of the most critical parameters. Shigley and
Mischke (1996) suggest an expression for the surface factor,
�T = �'*�� (5.14)
where � and � values can be found in Table 5.13.
Table 5.13 Values of a and b for Surface Factor
a b
Surface Finish Kpsi MPa
Ground 1.34 1.58 -0.085
Machined, cold rolled 2.67 4.45 -0.265
Hot rolled 14.5 58.1 -0.719
As forged 39.8 271 -0.995
Surface factor is calculated from Equation 5.15 and Table 5.13. The material is blue
polished spring steel. Generally for polished machine parts, surface factor is taken as
unity. During the experiments, for the supplied spring steels from the market, it is
84
observed that there are generally some small surface scratches available. Therefore,
to stay on the safe side, surface finish is taken as ground.
�T = 1.58 × 1100d>.>�b = 0.871 (5.15)
Size of the fatigue specimen can be different than the part being analyzed. As the
cross section becomes bigger, there will be more probability of a surface
imperfection. Shigley and Mischke (2001) approximations for the size factor is as
follows:
For bending and torsion of steel,
�� = g1.24�d>.e>� 2.79 ≤ � ≤ 51 1.51�d>.eb� 51 < � ≤ 254 S (5.16)
For axial loading there is no size effect,
�� = 1 (5.17)
All of the equations given above are for the circular cross section in rotating bending
or torsion. For other conditions an equivalent diameter, �(, must be found.
For a nonrotating part with rectangular cross section whose dimensions are � and ℎ,
�( = 0.808 √�ℎ (5.18)
After calculations of the equivalent diameter, this value can be used in Equation 5.16
as d.
Size factor for the rectangular cross section can be found by using Equation 5.16 and
5.18 for the model with the thickness of 0.5 mm,
�( = 0.808√20 × 0.5 = 2.555 (5.19)
85
�� = � �(7.62 �d>.e>� = 1.124 (5.20)
Another endurance limit modification factor is load factor. The fatigue test specimen
has a rotating bending loading. The loadings different than that causes a reduction in
fatigue strength or endurance limit. Norton (2000) introduced the load factors as,
�� = � 1 � "�C"�0.85 �0C��0.59 �A%&CA"S (5.21)
Load factor is taken as �� = 1 by using Equation 5. 21.
Tensile strengths versus endurance limit plot has a scattered data. Most of the
endurance strength data are mean values. Haugen and Wirching (1975) presented
data with standard deviations of endurance strengths of less than 8%. Thus,
reliability modification factors are shown in Table 5.14.
Table 5.14 Reliability Factors, �(
Reliability, % 50 90 95 99 99.9 99.99 99.999 99.9999 �� 1.000 0.897 0.868 0.814 0.753 0.702 0.659 0.620
For 99.9 % reliability �( is found as 0.753 from Table 5. 13.
There are other factors such as stress concentration, temperature, corrosion etc.
which can affect fatigue life. The mechanism is assumed to be working at room
temperature and non-humid condition. Therefore, miscellaneous effects factor �3 is
considered to be stress concentration only and it is estimated as,
�3 = 1 + $(�� − 1) (5.22)
86
�3 = 1�3 (5.23)
Figure 5.54 Notch-Sensitivity Charts for Steels and UNS A92024-T Wrought
Aluminum Alloys Subjected to Reversed Bending or Reversed Axial Loads
(Budynass and Nisbett (2011))
where $ is determined from Figure 5.54 as 0.9 since % = 2 mm and '*� = 1.1 GPa.
Theoretical stress concentration factor of this study is not available in the literature.
Therefore, the theoretical concentration factor, �� is determined via FEA. The
procedure is as follows: A cantilever beam with 0.5 mm thickness is placed in the
middle of the two rigid supports whose edges have 2 mm fillets as shown in Figure
5.55. The loading is only moment since hinges of compliant mechanisms are
dominantly subjected to moment type of loading. To determine ��, ratio of the stress
at the tip of the cantilever beam and the stress away from the edges are considered.
87
Stress distribution is presented in Figure 5.56 and stress concentration is determined
as 1.27.
Thus miscellaneous factor �3, is calculated as,
Figure 5.55 Rectangular Fillet Bar in Bending
Figure 5.56 Estimation of Theoretical Stress Concentration factor with ANSYS
� = 4.5 � = 0.5
% = 2
88
�3 = 1 + 0.9(1.27 − 1) = 1.24 (5.24)
�3 = 0.81 (5.25)
Fatigue life modifying factors are,
��h�T= = �T�����(�3 = 0.595 (5.26)
Modified endurance strength limit is found as,
'( = ��h�T='(′ = 329.69 #� (5.27)
Fatigue life calculations are also done for both of the models but the calculations of
the second model that has a thickness of 0.5 mm will be briefly explained. For the
bending-only loading case,
Fatigue strength fraction from Budynass and Nisbett (2011) approximation,
� = 0.9 (5.28)
Curve fitting parameters for blue polished spring steel are,
� = − 13 log ��'*�'( � = −0.174 (5.29)
� = (�'*�)E'( = 3.285 × 10` (5.30)
for numerical �fTB values life can be found with,
89
! = ��fTB� �e� (5.31)
The loading condition for the bending-only loading case can be seen in Figure 5.57.
Bending moment is taken as completely reversed loading because the bending
moment is constant and the mechanism rotates continuously.
Figure 5.57 Loading Condition of the Model
Factor of safety according to Goodman approach is,
" = 1
� �T'(fh��3�(� + �f'*�� (5.32)
Normal stress for the completely reversed loading is,
��(U = �T1 − ��f'*�� (5.33)
Life can be found by using,
! = ���(U"� e/� (5.34)
�T Time
�fTB �f�W
0
Nor
mal
Str
ess
90
After all calculations fatigue life estimations are done and tabulated for the first
model in Table 5.15 and for the second model 5.16. The tabulated data also plotted
in Figure 5.58 and 5.61.The finite life limit for the first model is 4.5° and for the
second model 6.9°.
Table 5.15 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�- = 0.75 mm)
Only bending th=0.75 mm
Shaft Angle (deg) σ-max-numerical
(MPa) Life (cycles) 1 66.6 ∞ 2 133.2 ∞ 3 199.7 ∞ 4 266.3 ∞ 5 337.6 756100 6 406.3 241500 7 475.2 92010 8 544.4 39820 9 613.7 19030
10 683.2 9827 11 752.8 5406 12 822.4 3135 13 892.0 1901
Figure 5.58 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�- =
0.75 mm)
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
1000000
4 5 6 7 8 9 10 11 12 13
Life
Shaft Angle (deg)
91
Table 5.16 Fatigue Life with respect to Shaft Angle for Bending-Only Loading Case (�- = 0.5 mm)
Only bending th=0.5
Shaft Angle (deg) σ-max-numerical
(MPa) Life (cycles) 1 44.2 ∞ 2 89.6 ∞ 3 134.6 ∞ 4 180.2 ∞ 5 223.4 ∞ 6 282.0 ∞ 7 331.7 962600 8 374.2 451400 9 449.8 142100
10 503.0 97850 11 528.9 51340 12 569.6 32220 13 604.1 22270 14 680.7 10520 15 717.6 7551 16 755.7 5456 17 822.3 3209 18 869.5 2260
Figure 5.59 Shaft Angle versus Fatigue Life for Bending-Only Loading Case (�- = 0.5 mm)
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
1000000
6 7 8 9 10 11 12 13 14 15 16 17
Life
Shaft Angle (deg)
92
Under bending-only loading case, the flexural hinges are subjected to pure bending,
therefore there is almost a uniaxial stress state. Pure bending arises from the given
shaft angle when the mechanism is subjected to torque as well. The loading on the
flexural hinges is still predominantly bending with also a limited amount of torsional
shear stress. The bending stresses are produced because of the shaft angle as before
and also because of the transverse forces that develop to resist the torque.
For the combined loading case, the loading on the flexural hinge is still
predominantly bending, and torsional shear stress is smaller. Bending stresses are
produced because of the shaft angle as in the bending-only case and also because of
the transverse forces that develop to resist the torque that are shown in Figure 5.45.
The combined loading case is different from the bending-only case. In the bending-
only case, the stresses are completely reversed however in the combined loading
case for the critical elements, mean values of stresses are observed to be different
than zero during FEA. Therefore, maximum and minimum normal stress values are
taken from FEA results and used in fatigue life estimation approach.
Figure 5.60 Von-Mises Stress Distribution, Most Critical Point on the Flexural
Hinge, its Infinitesimal Cube and �++ Stress Distribution
93
From the finite element analysis results, for all cases, it is observed that the most
critical region is near the fillets and in the edge of the flexural hinges as shown in
Figure 5.60. The element that is chosen from this point has two free surfaces thus
there cannot be shear stresses on this element. Therefore, for this case, a uniaxial
normal stress is the unique source of fatigue life as presented in Figure 5.61. The
following fatigue life calculations are based on the stress variation of this point.
Normal stresses of the infinitesimal cube are determined for a full rotation of the
mechanism from finite element analysis. Shear stresses of the infinitesimal cube are
zero or close to zero as expected.
Figure 5.61 Infinitesimal Cube of Critical Point
Thus for the combined loading case is alternating and mean stresses are,
�T) = �++¡¢£ − �++¡¤¥2 (5.35)
�f) = �++¡¢£ + �++¡¤¥2 (5.36)
Maximum and minimum normal stress values in y direction are evaluated from finite
element analysis and presented in Table 5.17. Fluctuations of this normal stress for
different shaft angles are plotted in Figure 5.62.
�++
�++
0
�
¦
94
Table 5.17 Maximum and Minimum Normal Stress Values for Combined Loading
Shaft Angle (deg)
§yyopq
(MPa)
§yyo¨© (MPa)
Torque (N.m) Life (cycles)
3 832.2 521.3 5.53 1000000+ 4 799.8 470.8 4.65 1000000+ 5 722.1 273.5 3.60 1000000+ 6 599.2 12.0 2.30 1000000+
Figure 5.62 Normal Stress Variations for Different Shaft Angles
In order to indicate performance of the compliant universal joint it is beneficial to
plot both static and fatigue resistance in the same figure. With one degree increments
of the bending angle, fatigue life of the mechanism is calculated from Equation 5.32.
It should be noted that different from the bending only case, stress concentration
factor is not taken into account in this case since critical stresses are directly
transferred from FEA results In Figure 5.63, the red line represents the static failure
limit and below this curve, the region is safe against static yielding. The mechanism
has infinite life below the blue curve. In between the blue and red curves, the
mechanism is expected to have a finite life.
-400
-200
0
200
400
600
800
1000
0 90 180 270 360
No
rma
l S
tre
ss (
MP
a)
Rotation Angle of Input Shaft (deg)
0 Degree Bend Angle
3 Degrees Bend Angle
4 Degrees Bend Angle
5 Degrees Bend Angle
6 Degrees Bend Angle
7 Degree Bend Angle
95
Figure 5.63 Performance Graph of the Compliant Universal Joint
96
97
CHAPTER 6
EXPERIMENTAL ANALYSIS OF THE COMPLIANT UNIVERSAL JOINT
6.1 Manufacturing of the Prototype
After theoretical calculations are performed the mechanism is designed for
manufacturing. For the experimental setup second model with the thickness of 0.5
mm is used because it has larger torque capacity and is more available in the market.
The most critical parts of the mechanism are the four compliant hinges. For those
parts blue polished spring steel is used which is also available as a sheet metal in the
market. This material which is also called C80 (AISI 1080) has high yield strength.
The properties of the AISI 1080, are found from Ashby (2005), and are given
Appendix A. For the other parts the stainless sheet steel called C35 is used.
Figure 6.1 Assembly of the Prototype
98
By using Catia V5 the drawings of the components are prepared. Because of the
importance of the distances between the connections laser cutting machine is used
for manufacturing. In the assembly of the design, there are two main parts that are
connected to each other as shown in Figure 6.1 and there are five components in a
main component. In a main part four of the parts are made from C35 and other is
made from C75 which can be seen Figure 6.2. The production of the other parts like
brake dynamometer, connecting shafts and the assembly are done in the laboratory
of Hacettepe University. Compliant part is placed in the middle of the relatively rigid
parts and assembled in three different ways. First one is bonding the parts with a
special glue which is for assembling metals. Second one is by welding the parts from
the edges far from the critical hinges. Last one is the connecting by bolts and nuts.
All of these connections are tried to see the differences between them. The
experimental results agree with theoretical results in all connection types. However
welding is a difficult manufacturing process, and during bonding with glue non
homogenous layers could occur. As a result bolt and nut connection is decided to be
the best connection type among them. The different ways of the connections can be
seen in Figure 6.3.
Figure 6.2 Components of the Prototype
Both ends of the mechanism are connected to bearings and fixed to a wooden
platform. To estimate the fatigue life of the mechanism an inductive proximity
sensor detection switch and a counter is assembled. Proximity sensor is a component
widely used in automatic control industry for detecting, controlling, and noncontact
99
switching. When proximity switch is close to some target object, it will send out
control signal. In the design proximity sensor detects the presence of metal without
physical contact, counts the number of cycles and sends out signal to the counter. In
this way fatigue life is found experimentally. And also for the case with torque a
brake dynamometer is designed with a shoe plate. A weight is attached to the
dynamometer to apply torque and the applied torque is measured by a torque meter.
Figure 6. 3 Different Ways of the Connections
6.2 Components of the Experimental Setup
In Figure 6.4 experimental setup is seen also its isometric view is shown in
Appendix E. The specifications of the bearing and the brushed DC electric motor can
be found in Appendix C and D. The components of the setup are,
1. Power supply (max 3A, 30 Volt)
2. Counter
3. Inductive proximity sensor
4. Bearing
5. Compliant universal joint
6. Brake dynamometer
7. Torque meter
8. Dead weight
9. Shoe plate
10. Brushed DC electric motor
100
Figure 6.4 Components of the Experimental Setup
1
2 3
4 5
6
7 8
9
6
10
101
6.3 Experimental Verification
After the manufacturing of the experimental setup is finished some experiments that
are shown in Table 6.1 are done to verify the theoretical approaches.
Table 6. 1 Experiments for Different Conditions
Shaft Angle (deg)
Torque (N.mm)
Theoretical data (cycles)
Experimental data (cycles)
Experiment 1 6.5 - ∞ Survived more than 106 cycles
Experiment 2 8 - 451400 420000
Experiment 3 6 500 ∞ Survived more than 106 cycles
For fatigue life experiments the mechanism is run continuously until an indication of
failure like a crack is observed or the mechanism passes the infinite life limit that is
1000000 cycles for steels. Three different experiments are done for verification. Two
of them are for bending-only mode at shaft angles 6.5° and 8°. The other one is for
6° degrees shaft angle under 500 N.mm torque. In theoretical calculations, it is found
that an infinite life (i.e. 1000000 cycles) can be reached for a maximum shaft angle
of 6.9° in bending-only case. First test is done for 6.5 degrees and the mechanism is
run 1200000 cycles and the design has reached the infinite life. Second test is done
for 8 degrees shaft angle without torque. The mechanism failed as shown in Figure
6.5 after 420000 cycles which is calculated as 451400 theoretically. In other words
test is completed with a percentage error of 7. The source of errors can be surface
imperfections, material or manufacturing defects. Last test is performed with 500
N.mm torque for a 6° shaft angle. The value of the torque is measured by torque
meter and suitable weight is determined for the brake dynamometer. For 6° shaft
angle the theoretical maximum torque value can be sustained for infinite life is 1130
N.mm as seen in Figure 6.5. However 500 N.mm torque is decided to apply to the
mechanism with a factor of safety and to avoid excessive wear of the disc and shoe
pad. This test is completed at 1200000 cycles without any indication of failure.
102
Hence theoretical prediction is experimentally verified. The finite life limits for
fatigue failure, static failure and the experimentally verified data are shown in Figure
6.6.
Figure 6.5 A Failed Compliant Universal Joint
Figure 6.6 Shaft Angle vs. Torque Output and Experimentally Verified Data
103
CHAPTER 7
RESULTS, CONCLUSION AND FUTURE STUDY
7.1 Results and Conclusion
In this study, a compliant universal joint whose compliant parts are made of blue
polished spring steel is considered and its fundamentals are described. This is the
first steel compliant universal joint in the literature and its feasibility is verified with
a real model.
The mechanism consists of two simple identical parts. Identical parts are produced
by connecting the planar parts that are manufactured with laser cutting process.
Compliant parts of the mechanism are made of blue polished spring steel and for
other parts stainless steel is used. Identical parts are assembled at right angles with
respect to each other.
Three constraint equations determined by Tanık and Parlaktaş (2012) are used to
satisfy the Cardan joint theory and to avoid undesired contact between parts.
After dimensioning a compliant cardan mechanism with two different complian part
thicknesses and choosing an appropriate material, static analyses are carried out:
Stress analysis are performed for different conditions to determine the capacity of the
universal joint. For bending-only condition normal stresses are obtained both
analytically and numerically to determine the maximum possible shaft angle without
yielding. On the other extreme, for zero shaft angle case, maximum torque capacity
of the mechanism is obtained. Then shaft angles with corresponding maximum
104
torque capacities are determined. Finally, for one of the mechanisms, a fatigue
analysis is performed. Life of the mechanism is determined for corresponding torque
values and shaft angles.
In order to verify theoretical approaches, a real model is manufactured. Three
different experiments with different shaft angles and torque loadings are done. First
experiment is done with 6.5° degrees shaft angle without torque and it is observed
that there is no indication of failure. In the second experiment shaft angle is set as 8°
without torque loading and failure is observed as expected. It is also verified that
compliant universal joint has an infinite life at a 6° shaft angle with a loading of 500
N.mm torque.
Hence strengthwise feasibility study of a steel compliant universal joint is done
theoretically and experimentally. It is believed that, compliant universal joint may be
a good alternative for the applications where transmitted torque values are not very
high. Also the design has the advantages of having small number of different parts,
ease of manufacturing and compactness.
7.2 Future Study
For the combined loading case shear stress values due to the torque are determined
numerically. One can attempt to develop an analytical model to predict the torque
capacity and try to derive closed form equations for the stresses. Such an attempt
would require large deflection analysis of a non-circular cross section which is also
subjected to torsion. After this study design charts of the compliant universal joint
can be obtained.
105
REFERENCES
Ashby, M.F., 2005, Material Selections in Mechanical Design, Elsevier, 81, 93 s.
Biancolini, M. E., Brutti, C., Pennestr`i, E., and Valentini, P. P., 2003, “Dynamic
Mechanical Efficiency and Fatigue Analysis of the Double Cardan Homokinetic
Joint,” Int. J. Veh. Des., 32, pp. 231–249.
Chen, C. K., and Freudenstein, F., 1986, “Dynamic Analysis of a Universal Joint
With Manufacturing Tolerances,” ASME J. Mech., Transm., Autom. Des., 108, pp.
524–532.
Fischer, I., and Freudenstein, F., 1984, “Internal Force and Moment Transmission in
a Cardan Joint With Manufacturing Tolerances,” ASME J. Mech., Transm., Autom.
Des., 106(9), pp. 301–311.
Freudenstein, F., 1965, “On the Determination of the Type of Spherical
Mechanisms,” Contemporary Problems in the Theory of Machines and Mechanisms,
USSR Academy of Sciences, pp. 193–196.
Freudenstein, F., and Macey, J. P., 1990, “The Inertia Torques of the Hooke Joint,”
Proceedings of the 21st Biennial ASME Mechanisms Conference, Chicago, Sept.
16–19, Vol. 24, pp. 407–413.
Haugen, E. B. and Wirsching, P. H, “Probabilistic Design,” Machine Design, vol.
47, no. 12, 1975, pp. 10–14.
Haugen, E. B. and Wirsching, P. H. “Probabilistic Design,” Machine Design, vol.
47, no. 12, 1975, pp. 10–14
Howell, L.L., 2001, "Compliant Mechanisms", Wiley-Interscience Publications.
106
Howell, L.L., Midha, A., "Parametric deflection approximation for end-loaded,
large-deflection beams in compliant mechanisms", Journal of Mechanical Design
117, p156-165, 1988.
http://kmoddl.library.cornell.edu/model.php?m¼500, Number 099 in the Clark
collection
L.L. Howell, A. Midha, "A Method for The Design of Compliant Mechanism with
Small-Length Flexurl Pivots", Journal of Mechanical Design 116, p280-290, 1994.
Mohan, R. A. V., Sandor, G. N., Kohli, D., and Soni, A. H., 1973, “Closed Form
Synthesis of Spatial Function Generating Mechanism for the Maximum Number of
Precision Points,” ASME J. Eng. Ind., 95, pp. 725–736.
Norton R. L., 2000, Machine Design, 2nd Ed., Prentice Hall, Upper Saddle River,
NJ.
Parlaktas, V., and Tanik, E., 2011, “Partially Compliant Spatial Slider-Crank (RSSP)
Mechanism,” Mech. Mach. Theory, 46(11), pp. 1707–1718.
Porat, I., 1980, “Moment Transmission by a Universal Joint,” Mech. Mach. Theory,
15(4), pp. 245–254.
Salamon, B.A. "Mechanical Advantage Aspects in Compliant Mechanisms Design",
M.S. Thesis, Purdue University, 1989.
Savage, M., and Hall, J., 1970, “Unique Descriptions of All Spherical Four-Bar
Linkages,” ASME J. Eng. Ind., 92, pp. 559–563.
Shigley, J. E. and Uicker, J. J.., Theory of Machines and Mechanisms, McGraw-Hill, New York, 1980, p. 262. Shigley, J. E., and Mischke, C. R., 1996, Standard Handbook of Machine Design,
2nd Ed., McGraw-Hill, New York.
Shigley, J. E., and Mischke, C. R., 2001, Mechanical Engineering Design, 6th Ed.,
McGraw-Hill, New York.
107
Tanik, E. and Parlaktas, V., 2011, “Compliant Cardan Universal Joint,” ASME J.
Mech. Des., 134 (2012) 021011
Tanik, E., and Parlaktas, V., 2011, “A New Type of Compliant Spatial Four- Bar
(RSSR) Mechanism,” Mech. Mach. Theory, 46(5), pp. 593–606.
Trease, B., Moon, Y., and Kota, S., 2005, “Design of Large-Displacement Compliant
Joints,” ASME J. Mech. D., 127, pp. 788–798.
Wagner, E. R., and Cooney, C. E., 1979, “Cardan or Hooke Universal Joint”
Universal Joint and Driveshaft Design Manual, Society of Automotive
Engineers,Warrendale, PA, pp. 39–75.
Yang, A. T., 1965, “Static Force and Torque Analysis of Spherical Four-Bar
Mechanisms,” ASME J. Eng. Ind., 87, pp. 221–227.
108
109
APPENDIX A
PROPERTIES OF AISI 1080
Figure A.1 Properties of AISI 1080
110
APPENDIX B
TECHNICAL DRAWINGS
Figure B.1 Technical Drawing of the Model with 0.75 mm Plate Thickness
111
Figure B.2 Technical Drawing of the Model with 0.5 mm Plate Thickness
112
APPENDIX C
SPECIFICIONS OF BEARINGS
Figure C.1Specifications of UCP 200 Bearing
113
APPENDIX D
SPECIFICIONS OF DC ELECTRIC MOTOR
114
Figure D.1 Specifications of the DC Electric Motor
115
APPENDIX E
ISOMETRIC VIEW OF THE TEST SETUP
Figure E
.1 Isometric V
iew of the F
atigue Test S
etup
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