Durability of Springs (eBook)

Vladimir Kobelev (Autor)

eBook Download: PDF

2017 | 1st ed. 2018
XXXI, 270 Seiten
Springer International Publishing (Verlag)
978-3-319-58478-2 (ISBN)

Infos

This book offers an advanced treatise of the mechanics of springs with focus on the springs for automotive industry. It demonstrates new and original results for the optimization of helical springs as well the design of disk springs and thin-walled springs and presents the new results for creep and relaxation of springs made of steel under high static loads. The fatigue of springs and weak link concept for cyclically loaded springs are enlightened. The closed form solutions of advanced problems allow the deeper understanding of spring mechanics and optimization of energy harvesters.

Inhalt

Foreword 5
Preface 7
Introduction 8
References 13
Contents 14
List of Symbols 20
Chapter 1 20
Chapter 2 21
Chapter 3 22
Chapter 4 23
Chapter 5 24
Chapter 6 26
Chapter 7 27
Chapter 8 27
Chapter 9 28
Chapter 10 29
Chapter 11 29
Chapter 1: Principles of Spring Design 31
1.1 Design Formulas Cylindrical Springs 31
1.1.1 Cylindrical Springs with Circular Wire 31
1.2 Forces and Moments in Helical Springs 32
1.2.1 Stiffness and Stored Energy of Cylindrical Helical Springs 35
1.2.2 Fatigue Life and Damage Accumulation Criteria 37
1.3 Compression and Torque of Cylindrical Helical Springs 38
1.3.1 Spring Rates of Non-Cylindrical Helical Springs 38
1.3.2 Diameter Alteration Due to Simultaneous Compression and Torque 40
1.4 Helical Springs of Minimal Mass 42
1.4.1 Restricted Optimization Problem 42
1.4.2 Optimization of Helical Springs for Maximal Stress 43
1.4.3 Design for Fatigue Life 46
1.4.4 Spring Quality Parameter for Helical Springs 47
1.5 Semi-elliptic Longitudinal and Transverse Leaf Springs of Minimal Mass 47
1.6 Multi-material Design of Springs 52
1.7 Conclusions 54
References 55
Chapter 2: Stress Distributions Over Cross-Section of Wires 56
2.1 Warping Function 56
2.2 Prandtl Stress Function 58
2.3 Shear Stresses on Surface of Elliptic and Circular Wires 61
2.4 Shear Stresses on Surface of Ovate Wire 64
2.5 Quasi-elliptical Cross-Section 67
2.6 Hollow Ovate Wire 69
2.7 Conclusions 71
References 72
Chapter 3: ``Equivalent Columns´´ for Helical Springs 73
3.1 Static Stability Criteria of Helical Springs 73
3.2 Static ``Equivalent Column´´ Equations 75
3.3 Dynamic ``Equivalent Column´´ Equations 77
3.4 Natural Frequency of Transverse Vibrations 81
3.5 Stability Conditions and Buckling of Spring 85
3.6 Instability of Twisted and Tensioned Helical Spring 89
3.6.1 Buckling of Twisted Helical Spring 89
3.6.2 Instability of Tensioned Helical Spring 93
3.7 Spatial Models for Dynamic Behavior of Helical Springs 94
3.8 Conclusions 98
References 99
Chapter 4: Coiling Process for Helical Springs 102
4.1 Elastic-Plastic Bending and Torsion of Wire 102
4.2 Modified Ramberg-Osgood´s Law 104
4.3 Plastic Deformation of Wire During Coiling 106
4.4 Behavior of Wire in Manufacturing Process 107
4.5 Elastic Spring-Back and Appearance of Residual Stresses 111
4.6 Post-coiling Shape of Helical Spring 112
4.7 Conclusions 119
References 119
Chapter 5: Disk Springs 120
5.1 Thick Shell Model for Disk Springs 120
5.1.1 Mechanical Models of Elastic Disk Springs 120
5.1.2 Geometry of Disk Spring in Undeformed State 122
5.1.3 Load-Caused Alteration of Strain and Curvature 123
5.1.4 Disk Springs of Moderate Material Thickness 125
5.2 Isotropic Disk Springs of Moderate Thickness 125
5.2.1 Deformation of Thick Conical Shell 125
5.2.2 Variation Method for Thick Shell Models of Isotropic Disk Springs 126
5.2.3 Comparison of Calculation Techniques 129
5.3 Isotropic, Thin Disk Springs 130
5.3.1 Forces and Moments in Isotropic Disk Springs 130
5.3.2 The Strain Energy of Isotropic Thin Disk Springs 131
5.3.3 Almen and Laszlo Method for Thin, Isotropic Disk Springs 133
5.3.4 Stresses in Disk Springs Made of Isotropic Materials 136
5.4 Anisotropic Disk Springs 137
5.4.1 Model of Anisotropic Disk Spring 137
5.4.2 Optimal Ply Orientation for Anisotropic Disk Springs 140
5.4.3 Model of Orthotropic Disk Spring 141
5.5 Disk Wave Springs 145
5.5.1 Application Fields of Disk Wave Springs 145
5.5.2 Design Formulas for Linear Disk Wave Springs 147
5.5.3 Design Formulas for Non-Linear Disk Wave Springs 149
5.6 Conclusions 152
References 153
Chapter 6: Creep and Relaxation of Springs 155
6.1 Constitutive Equations for Creep of Spring Elements 155
6.2 Common Creep Laws 156
6.2.1 Norton-Bailey Law 157
6.2.2 Garofalo Creep Law 159
6.2.3 Naumenko-Altenbach-Gorash Law 159
6.3 Creep and Relaxation of Twisted Rods 160
6.3.1 Constitutive Equations for Relaxation in Torsion 160
6.3.2 Torque Relaxation for Norton-Bailey Law 161
6.3.3 Torque Relaxation for Garofalo Law 162
6.3.4 Torque Relaxation for Naumenko-Altenbach-Gorash Law 163
6.4 Creep and Relaxation of Helical Coiled Springs 163
6.4.1 Relaxation of Helical Springs 164
6.5 Creep of Helical Compression Springs 166
6.6 Creep and Relaxation of Beams in State of Pure Bending 167
6.6.1 Constitutive Equations for Relaxation in Bending 167
6.6.2 Relaxation of Bending Moment for Norton-Bailey Law 168
6.6.3 Relaxation of Bending Moment for Garofalo Law 169
6.6.4 Relaxation of Bending Moment for Naumenko-Altenbach-Gorash Law 170
6.6.5 Creep in State of Bending 171
6.7 Creep and Relaxation of Disk Springs 172
6.7.1 Creep of Disk Springs 172
6.7.2 Relaxation of Disk Springs 178
6.8 Experimental Acquisition of Creep Laws 181
6.9 Conclusions 183
References 183
Chapter 7: Generalizations of Creep Laws for Spring Materials 185
7.1 Constitutive Equations for Fractional Creep 185
7.1.1 Fractional Generalization of Creep Laws 185
7.1.2 Fractional Norton-Bailey Law 186
7.2 Fractional Creep and Relaxation of Twisted Rods 187
7.2.1 Constitutive Equations for Relaxation in Torsion 187
7.2.2 Torque Relaxation for Fractional Norton-Bailey Law 188
7.3 Fractional Creep and Relaxation of Beams in Bending 189
7.3.1 Constitutive Equations for Relaxation in Bending 189
7.3.2 Bending Moment Relaxation for Fractional Norton-Bailey Law 190
7.3.3 Constitutive Equations for Creep in Bending 191
7.4 Unification of Primary and Secondary Creep Laws 192
7.5 Unified Relaxation Equations of Twisted Rods 194
7.5.1 Unified Constitutive Equations for Relaxation in Torsion 194
7.6 Unified Relaxation Equations of Beams in Bending 195
7.6.1 Unified Constitutive Equations for Relaxation in Bending 195
7.7 Solutions for Common Creep Laws 196
7.8 Conclusions 196
References 196
Chapter 8: Fatigue of Spring Materials 198
8.1 Fatigue Life Estimation Based on Empirical Damage Models 198
8.1.1 Phenomenon of Fatigue 198
8.1.2 Evaluation of Fatigue Life with Goodman Diagrams 200
8.1.3 Stress-Life and Strain-Life Approaches 203
8.1.4 Fatigue Analysis at Very High Number of Cycles 209
8.2 Fatigue Estimation Based on Crack Propagation Laws 210
8.2.1 Crack Propagation Laws of Paris-Erdogan Type 210
8.2.2 Propagation Laws for Crack Under Cyclic Loading 214
8.3 Fatigue Estimation Based on Unified Propagation Functions 215
8.3.1 Unification of Paris Law 215
8.3.2 Unification of Paris Law Type I 216
8.3.3 Limit Cases of Type I Propagation Function 220
8.3.4 Unification of the Fatigue Law Type II 221
8.3.5 Limit Cases of Type II Propagation Function 224
8.4 Sensitivity of Fatigue Crack Propagation Upon Stress Ratio 229
8.5 Conclusions 234
References 235
Chapter 9: Failure Probability of Helical Spring 239
9.1 Evaluation of Failure Probability of Springs 239
9.2 Weakest Link Concepts for Homogeneously Loaded Elements 240
9.3 Weakest Link Theory for Heterogeneously Loaded Elements 242
9.4 Applications of Weakest Link Concept to Helical Springs 244
9.4.1 Failure Probability of Helical Springs 244
9.4.2 Influence of Spring Index on Instantaneous Failure of Helical Springs 245
9.4.3 Influence of Spring Index on Fatigue Life of Helical Springs 247
9.5 Conclusions 250
References 251
Chapter 10: Thin-Walled Rods with Semi-Opened Profiles 252
10.1 Theory of Thin-Walled Rods with Semi-opened Profiles 252
10.1.1 Open, Closed and Semi-opened Wall Sections 252
10.1.2 Base Line of Semi-opened Cross-Section 254
10.2 Thin-Walled Rods with Semi-opened Profile 255
10.3 Deformation Behavior of Cross-Sections 255
10.4 Deformation of Rods with Semi-opened Profiles 257
10.5 Statics of Semi-opened Profile Bars 259
10.5.1 Normal Stresses in Semi-opened Profile Bars 259
10.5.2 Torque and Bi-Moment 260
10.5.3 Tangential Stresses in Bar Cross-Sections 261
10.6 Tangential Stress in Semi-opened Profiles 261
10.7 Strain Energy of Semi-opened Rod 263
10.8 Conclusions 264
References 265
Chapter 11: Semi-Opened Profiles for Twist-Beam Automotive Axles 267
11.1 Applications of Thin-Walled Rods with Semi-Opened Cross-Sections 267
11.1.1 Semi-Solid Suspension with Twist Beam 267
11.1.2 Mechanical Models of Twist Beam Axle 269
11.2 Elastic Behavior of Twist-Beam Axles Under Load 269
11.2.1 Loads and Displacements of Twist-Beam Axles 269
11.2.2 Roll Stiffness of Twist-Beam Axle 270
11.2.3 Lateral Stiffness of Twist-Beam Axle 271
11.2.4 Camber Stiffness of Twist-Beam Axle 272
11.3 Deformation of Semi-Opened Beam Under Terminal Load 273
11.3.1 Bending of Semi-Opened Profile Beam Due to Terminal Moments 273
11.3.2 Torsion Stiffness of Beam with Constant Section Due to Terminal Torques 274
11.3.3 Stresses in the Beam with Constant Section Due to Terminal Torques 275
11.3.4 Equivalent Tensile Stress Due to Simultaneous Bending and Torsion 277
11.3.5 Stiffness Properties of Semi-Opened Profiles for Automotive Applications 278
11.3.6 Semi-Opened Beams with Variable Cross-Sections 278
11.4 Conclusions 280
References 281
Appendices 282
Appendix A: Integrals with Polylogarithm 282
Appendix B: Integrals with Hypergeometric Function 283
Appendix C: Integrals with Incomplete Beta Function 284
Appendix D: Complete Elliptic Integrals 285
Appendix E: Appell Hypergeometric Function 285
References 285
Index 286

Produkt-Details

• Erscheint lt. Verlag: 7.8.2017
• Zusatzinfo: XXXI, 270 p. 72 illus., 49 illus. in color.
• Verlagsort: Cham
• Sprache: englisch
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• Themenwelt: Wirtschaft
• Schlagworte: automotive springs • fatigue of springs • stability of springs • steel springs • vibration of springs
• ISBN-10: 3-319-58478-2 / 3319584782
• ISBN-13: 978-3-319-58478-2 / 9783319584782