Table Of ContentF l e x u r e s
Flexures
Elements of Elastic Mechanisms
Stuart T. Smith
University of North Carolina, USA
CRC PRESS
Boca Raton London New York Washington, D.C.
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Contents
Preface xiii
Chapter 1: Introduction 1
1.0 Origins 1
l.l Objective 2
1.2 Advantages of flexures 2
1.3 Disadvantages of flexures 3
1.4 Goals of flexure design 3
1.5 Design considerations 5
References 13
Chapter 2: Essentials 15
2.0 Overview 15
2.1 Basic elasticity 15
2.2 Behavior of materials 17
2.2.1 Metals 17
2.2.2 Non-metals 20
2.3 Principal stresses and strains 22
2.3.1 Biaxial stress 24
2.3.2 Triaxial stresses 25
2.4 Non-principal stresses 26
2.4.1 Plane stress 26
2.4.2 Three dimensional stresses 31
2.4.3 Shear stresses and shear strain 32
2.5 Yield criteria 35
2.5.1 Ductile materials failure criteria 36
2.6 Fatigue 37
2.6.1 SIN curves 37
2.6.2 Effects of notches 40
2.6.3 Effects of mean stress 42
2.6.4 Damage assessment 42
v
CONTENTS
2.7 Bending of symmetric beams 43
2.7.1 Thebendingequation 43
2.7.2 Deflection of beams 45
2.7.2.1 Sign convention for bending moments 46
2.7.3 Moment, shear force and rate ofloading relationships 48
2.7.4 Singularity functions 49
2.8 Torsion 51
2.8.1 Torsion of a prismatic beam of circular cross section 52
2.8.2 Torsion of a prismatic beam of rectangular cross section 53
2.9 Mobility 54
References 56
Chapter 3: Rigid body dynamics 59
3.0 Overview 59
3.1 Generalized coordinates 60
3.2 Properties ofvariational operators 64
3.2.1 Commutation 64
3.2.2 Minima of a function 64
3.3 Hamilton's principle 66
Example I: A simple spring mass system 68
Example 2: Lateral vibration of a bar 69
3.4 Lagrange's equation 71
3.4.1 Rayleigh's dissipation function 74
3.4.2 General use of Lagrange's equation 75
3.5 Linear systems theory 75
3.5.1 The simple, single degree of freedom, linear, spring-mass-damper
system 75
3.5.2 Some equivalent definitions of a linear system 78
3.5.3 Frequency response functions 79
3.6 Measuring the critical damping ratio 82
3.7 General linear systems revisited 87
3.8 Multi-degree of freedom linear systems 92
vi
CONTENTS
3.8.1 Note on the graphical representation of frequency response
functions 98
3.9 General response function short cuts 100
3.9.1 Rayleigh's approach to the problem of computing generalized
frequency response functions 103
3.9.1.1 Example of a two degree of freedom system 105
3.10 Eigen analysis 106
3.10.1 Conservative systems 108
3.10.2 Systems with damping 112
3.1 0.3 Interpretation of complex eigenvalues and eigenvectors 113
3.10.4 Summary of primary steps in the derivation of the response
function of a linear multi-degree of freedom system 114
3.10.5 Example: A series, six mass vibration isolator 115
3.10.5.1 Results 118
3.11 Vibrations and natural frequencies of continuous systems 120
3.11.1 Strings 121
3.11.2 Longitudinal vibrations of a rod 124
3.11.2.1 Longitudinal vibrations of a clamped-free rod 126
3.11.2.2 Longitudinal vibrations of a free-free or fixed-fixed rod 126
3.11.2.3 Longitudinal vibrations of a clamped-free rod with a rigid mass
at the free end 127
3 .11.3 Lateral vibration of a bar 129
3. 11.3 .1 Hinged-hinged beam 130
3.11.3.2 The free-free or clamped-clamped bar 131
3.11.4 Lateral vibration ofbars with a rigid mass attached 134
3. 11.4.1 Cantilever beam with a rigid mass attached at the free end 134
3.11.4.2 Hinged beam with a central mass, M 135
3.11.5 Vibration of plates 136
3.11.5.1 Vibrations of a circular plate 136
3.11.5.1.1 Free vibration of a circular plate clamped at the perimeter 137
vii
CONTENTS
3.11.5.2 Rayleigh's method applied to a circular pl:lte with a
central mass 140
3.11.6 Vibrations of a rectangular plate 145
3.11.6.1 Free vibrations of a rectangular plate si1r.ply supported at the
edges 145
3.11.6.2 Fundamental frequency's of rectangular plates with other
boundary conditions 146
3.12 Case study I: A simple two degree offreedom flexure mechanism 146
References I 51
Chapter 4: Flexure elements 153
4.0 Overview 153
4.1 Leaf type springs 153
4.1.1 The cantilever as a rotary hinge 154
4.1.2 The cantilever hinge subject to an axial compressive force 158
4.1.3 Combined axial and tangential loads 159
4.1.4 Combined axial and tangential loads plus a moment applied to
the free end of a simple cantilever 167
4.1.5 Leaf type flexures for parallelogram flexure applications 173
4.2 Notch hinge 177
4.2.1 Theoretical considerations 179
4.2.1.1 The leaf type flexure reconsidered 179
4.2.2 The circular notch hinge 180
4.2.3 Accuracy of stiffness estimates for a notch type hinge 183
4.2.4 The notch hinge of elliptic cross section 185
4.2.5 Compliance's of elliptic hinges in other axes 188
4.2.6 Results 191
4.2.6. I Finite element results 191
4.3 Other hinge elements 192
4.3.1 The cross strip pivot 193
4.3.1.1 Center shift of the pivot 198
4.3.2 The cattwheel hinge 199
viii 4.3.2.1 Torsional stiffness of the cartwheel hinge 201
CONTENTS
4.3.2.2 Center shift of the pivot 202
4.3.2.3 Stresses m the hinge 203
4.3.3 The cruciform hinge 204
4.4 Two axis hinges 206
4.4.1 A simple two axis hinge {y, B) 206
4.4.2 The two beam, two degree offreedom flexure (y. B) 208
4.4.3 The two axis toroidal hinge (e.,, e,) 211
4.5 Case study 1: Force sensor for contact probe characterization 213
4.5.1 Toroidal notch type flexure 215
4.5.2 Two, stacked, notch type flexmes 217
References 218
Chapter 5: Flexure systems 221
5.0 Overview 221
5.1 The four bar link 222
5 .1.1 The simple leaf type rectilinear spring 222
5.1.2 The simple notch type linear spring 226
5.1.3 The vittual center 231
5.1.4 Effect of the drive on the natural fi·equency 237
5.2 Optimal geometry for the rectilinear motion of components on a
simple linear spring 237
5.2.1 Simple linear spring mechanisms 238
5.2.2 Effect of axial strains in the flexure supports 242
5.2.3 Combined effects 243
5.3 Planar mechanisms 244
5.3.1 Discussion of the simplifying assumptions for reduction of
mobility analysis to planar mechanisms 245
5.4 Dynamics of ideal planar flexures (some common mechanisms) 251
5.4.1 The compound rectilinear spring 252
5.4.2 The double compound linear spring (including the lever driven
spring) 256
5.4.3 A coupled two-axis flexure 259
ix
