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More About This Title Design and Analysis of Composite Structures forAutomotive Applications - Chassis and Drivetrain
English
A design reference for engineers developing composite components for automotive chassis, suspension, and drivetrain applications
This book provides a theoretical background for the development of elements of car suspensions. It begins with a description of the elastic-kinematics of the vehicle and closed form solutions for the vertical and lateral dynamics. It evaluates the vertical, lateral, and roll stiffness of the vehicle, and explains the necessity of the modelling of the vehicle stiffness. The composite materials for the suspension and powertrain design are discussed and their mechanical properties are provided. The book also looks at the basic principles for the design optimization using composite materials and mass reduction principles. Additionally, references and conclusions are presented in each chapter.
Design and Analysis of Composite Structures for Automotive Applications: Chassis and Drivetrain offers complete coverage of chassis components made of composite materials and covers elastokinematics and component compliances of vehicles. It looks at parts made of composite materials such as stabilizer bars, wheels, half-axes, springs, and semi-trail axles. The book also provides information on leaf spring assembly for motor vehicles and motor vehicle springs comprising composite materials.
- Covers the basic principles for the design optimization using composite materials and mass reduction principles
- Evaluates the vertical, lateral, and roll stiffness of the vehicle, and explains the modelling of the vehicle stiffness
- Discusses the composite materials for the suspension and powertrain design
- Features closed form solutions of problems for car dynamics explained in details and illustrated pictorially
Design and Analysis of Composite Structures for Automotive Applications: Chassis and Drivetrain is recommended primarily for engineers dealing with suspension design and development, and those who graduated from automotive or mechanical engineering courses in technical high school, or in other higher engineering schools.
English
Vladimir Kobelev, PhD, is a professor of mechanical engineering at the University of Siegen in Germany. He is a member of the International Society for Structural and Multidisciplinary Optimization and EUROMECH. He has authored three other books, including Durability of Springs, and has been published in journals more than 60 times.
English
List of symbols 1
Introduction 1
Composites in automotive chassis and drivetrain 1
Physical properties of composite materials 4
Structure of the book 9
Target audience of the book 13
References 13
Chapter 1 Elastic anisotropic behavior of composite materials 21
1.1 Anisotropic elasticity of composite materials 21
1.1.1 4th rank tensor notation of Hooke’s law 21
1.1.2 Voigt’s matrix notation of Hooke’s law 22
1.1.3 Kelvin’s matrix notation of Hooke’s law 25
1.2 Unidirectional fiber bundle 26
1.2.1 Components of unidirectional fiber bundle 26
1.2.2 Elastic properties of unidirectional fiber bundle 27
1.2.3 Effective elastic constants of unidirectional composites 30
1.3 Rotational transformations of material laws, stress and strain 32
1.3.1 Rotation of 4th rank elasticity tensors 33
1.3.2 Rotation of elasticity matrices in Voigt’s notation 34
1.3.3 Rotation of elasticity matrices in Kelvin’s notation 36
1.4 Elasticity matrices for laminated plates 37
1.4.1 Voigt’s matrix notation for anisotropic plates 37
1.4.2 Rotation of matrices in Voigt’s notation 39
1.4.3 Kelvin’s matrix notation for anisotropic plates 40
1.4.4 Rotation of matrices in Kelvin’s notation 42
1.5 Coupling effects of anisotropic laminates 43
1.5.1 Orthotropic laminate without coupling 43
1.5.2 Anisotropic laminate without coupling 43
1.5.3 Anisotropic laminate with coupling 44
1.5.4 Coupling effects in laminated thin-walled sections 44
Conclusions 45
References 45
Chapter 2 Phenomenological failure criteria of composites 45
2.1 Phenomenological failure criteria 45
2.1.1 Criteria for static failure behavior 45
2.1.2 Stress failure criteria for isotropic homogenous materials 45
2.1.3 Phenomenological failure criteria for composites 46
2.1.4 Phenomenological criteria without stress coupling 47
2.1.5 Phenomenological criteria with stress coupling 49
2.2 Differentiating criteria 61
2.2.1 Fiber and intermediate break criteria 61
2.2.2 Hashin strength criterion 62
2.2.3 Delamination criteria 64
2.3 Physically based failure criteria 65
2.3.1 Puck criterion 65
2.3.2 Cuntze criterion 66
2.4 Rotational transformation of anisotropic failure criteria 67
Conclusions 72
References 72
Chapter 3 Micromechanical failure criteria of composites 81
3.1. Pull-Out of fibers from elastic-plastic matrix 81
3.1.1 Axial tension of fiber and matrix 81
3.1.2 Shear stresses in matrix cylinders 91
3.1.3 Coupled elongation of fibers and matrix 94
3.1.4 Failures in matrix and fibers 95
3.1.6 Rupture of fibers, matrix joints crack edges 104
3.2 Crack bridging in elastic-plastic unidirectional composites 106
3.2.1 Crack bridging in unidirectional fiber reinforced composites 106
3.2.2 Matrix crack growth 107
3.2.3 Fiber crack growth 108
3.2.4 Penny-shaped crack 114
3.2.5 Plane crack problem 125
3.3. Debonding of fibers in unidirectional composites 131
3.3.1 Axial deformation of unidirectional fiber composites 131
3.3.2 Stresses in unidirectional composite in cases of ideal debonding or adhesion 138
3.3.3 Stresses in unidirectional composite in case of partial debonding 143
3.3.4 Contact problem for finite adhesion region 152
3.3.5 Debonding of semi-infinite adhesion region 160
3.3.6 Debonding of fibers from matrix under cyclic deformation 164
Conclusions 168
References 168
Chapter 4 Optimization principles for structural elements made of composites 151
4.1 Stiffness optimization of anisotropic structural elements 151
4.1.1 Optimization problem 151
4.1.2 Optimality conditions 153
4.1.3 Optimal solutions in anti-plane elasticity 157
4.1.4 Optimal solutions in plane elasticity 157
4.2 Optimization of strength and loading capacity of anisotropic elements 160
4.2.1 Optimization problem 160
4.2.2 Optimality conditions 162
4.2.3 Optimal solutions in anti-plane elasticity 163
4.2.4 Optimal solutions in plane elasticity 164
4.3 Optimization of accumulated elastic energy in flexible anisotropic elements 167
4.3.1 Optimization problem 167
4.3.2 Optimality conditions 168
4.3.3 Optimal solutions in anti-plane elasticity 169
4.3.4 Optimal solutions in plane elasticity 170
4.4 Optimal anisotropy in twisted rod 172
4.5 Optimal anisotropy of bending console 175
4.6 Optimization of plates in bending 177
Conclusions 179
References 180
Chapter 5 Optimization of composite driveshaft 181
5.1 Torsion of anisotropic shafts with solid cross-section 181
5.2 Thin-walled anisotropic driveshaft with closed profile 186
5.2.1 Geometry of cross-section 186
5.2.2 Main kinematic hypothesis 188
5.3 Deformation of composite thin-walled rod 191
5.3.1 Equations of deformation of anisotropic thin-walled rod 191
5.3.2. Boundary conditions 196
5.3.4 Symmetry of section 200
5.4 Buckling of composite driveshafts under twist moment 201
5.4.1 Greenhill’s buckling of driveshafts 201
5.4.2 Optimal shape of the solid cross-section for driveshaft 204
5.4.3 Hollow circular and triangle cross-sections 206
5.5 Patents for composite driveshafts 208
Conclusions 214
References 214
Chapter 6 Dynamics of vehicle with rigid structural elements of chassis 213
6.1. Classification of wheel suspensions 213
6.1.2 Common design of suspensions 213
6.1.3 Types of twist-beam axles 215
6.1.4 Kinematics of wheel suspension 216
6.2 Fundamental models in vehicle dynamics 218
6.2.1 Basic variables of vehicle dynamics 218
6.2.2 Coordinate systems of vehicle and local coordinate systems 221
6.2.3 Angle definitions 223
6.2.4 Components of force and moments in car dynamics 228
6.2.5 Degrees of freedom of vehicle 230
6.3 Forces between tires and road 231
6.3.1 Tire slip 231
6.3.2 Side slip curve and lateral force properties 233
6.4 Dynamic equations of single-track model 235
6.4.1 Hypotheses of single-track model 235
6.4.2 Moments and forces in single-track model 237
6.4.3 Balance of forces and moments in single-track model 241
6.4.4 Steady cornering 242
6.4.5 Non-steady cornering 248
6.4.6 Anti-roll bars made of composite materials 251
Conclusions 253
References 253
Chapter 7 Dynamics of vehicle with flexible, anisotropic structural elements of chassis 255
7.1 Effects of body and chassis elasticity on vehicle dynamics 255
7.1.1 Influence of body stiffness on vehicle dynamics 255
7.1.2 Lateral dynamics of vehicles with stiff rear axle 256
7.1.3 Induced effects on wheel orientation and positioning of vehicles with flexible rear axle 258
7.2 Self-steering behavior of the vehicle with coupling of bending and torsion 264
7.2.1 Countersteering for vehicles with twist-beam axes 264
7.2.2 Bending-twist coupling of countersteering twist-beam axle 267
7.2.3 Roll angle of vehicle 269
7.3 Steady cornering of flexible vehicle 273
7.3.1 Stationary cornering of car with flexible chassis 273
7.3.2 Necessary steer angles for coupling and flexibility of chassis 274
7.4 Estimation of coupling constant for twist member 279
7.4.1 Coupling between vehicle roll angle and twist of cross-member 279
7.4.2 Stiffness parameters of twist-beam axle 280
7.5 Design of the countersteering twist-beam axle 287
7.5.1 Requirements for countersteering twist-beam axle 287
7.5.2 Selection and calculation of cross-section for the cross-member 292
7.5.3 Elements of countersteering twist-beam axle 293
7.6 Patents on twist-beam axes 301
Conclusions 305
References 306
Chapter 8 Design and optimization of composite springs 309
8.1 Design and optimization of anisotropic helical springs 309
8.1.1 Forces and moments in helical composite springs 309
8.1.2 Symmetrically designed solid bar with circular cross-section 317
8.1.3 Stiffness and stored energy of helical composite springs 319
8.1.4 Spring rates of helical composite springs 323
8.1.5 Helical composite springs of minimal mass 326
8.1.6 Axial and twist vibrations of helical springs 330
8.2 Conical springs made of composite material 333
8.2.1 Geometry of anisotropic conical spring in undeformed state 333
8.2.2 Curvature and strain deviations 336
8.2.3 Thin-walled conical shells made of anisotropic materials 337
8.2.4. Variation principle 339
8.2.5. Structural optimization of conical spring due to ply orientation 342
8.2.6 Conical spring made of orthotropic material 344
8.2.7. Bounds for stiffness of spring made of orthotropic material 346
8.3 Alternative concepts for chassis springs made of composites 355
Conclusions 363
References 367
Chapter 9 “Equivalent Beams” of helical anisotropic springs 365
9.1 Helical compression springs made of composite materials 365
9.1.1 Statics of “equivalent beam” for anisotropic spring 365
9.1.2 Dynamics of “equivalent beam” for anisotropic spring 368
9.2 Transverse vibrations of composite spring 371
9.2.1 Separation of variables 371
9.2.2 Fundamental frequencies of transversal vibrations 374
9.2.3 Transverse vibrations of symmetrically stacked helical spring 376
9.3 Side buckling of helical composite spring 377
9.3.1 Buckling under axial force 377
9.3.2 Simplified formulas for buckling of symmetric stacked helical spring.. 379
Conclusions 379
References 380
Chapter 10 Composite leaf springs 393
10.1 Longitudinally mounted leaf springs for solid axes 393
10.1.1 Predominantly bending-loaded leaf springs 393
10.1.2 Moments and forces of leaf springs in pure bending state 395
10.1.3 Optimization of leaf springs for anisotropic Mises-Hill criterion 396
10.2. Leaf-tension springs 401
10.2.1 Combined bending and tension of spring 401
10.2.2 Forces and rates of leaf-tension springs 403
10.3 Transversally mounted leaf springs 407
10.3.1 Axle concepts of transverse leaf spring 407
10.3.2 Analysis of transverse leaf spring 409
10.3.3 Examples and patents for transversely mounted leaf springs 412
Conclusions 415
References 416
Chapter 11 Meander-shaped springs 421
11.1 Meander-shaped compression springs for automotive suspensions 421
11.1.1 Bending stress state of corrugated springs 421
11.1.2 “Equivalent beam” of meander spring 426
11.1.3 Axial and lateral stiffness of corrugated springs 426
11.1.4 Effective spring constants of meander and coil springs for bending and compression 429
11.2 Multiarc-profiled spring under axial compressive load 429
11.2.1 Multiarc meander spring with constant cross-section 429
11.2.2 Multiarc meander spring with optimal cross-section 433
11.2.3 Comparison of masses for fixed spring rate and stress 435
11.3 Sinusoidal spring under compressive axial load 438
11.3.1 Sinusoidal meander spring with constant cross-section 438
11.3.2 Sinusoidal meander spring with optimal cross-section 440
11.3.3 Comparison of masses for fixed spring rate and stress 442
11.4 Bending stiffness of meander spring with constant cross-section 443
11.4.1 Bending stiffness of multiarc meander spring with constant cross-section 443
11.4.2 Bending stiffness of sinusoidal meander spring with constant cross-section 444
11.5 Stability of corrugated springs 448
11.5.1 Euler’s buckling of axially compressed rod 448
11.5.2 Side buckling of meander springs 448
11.6 Patents for chassis springs made of composites in meandering form 450
Conclusions 460
References 460
Chapter 12 Hereditary mechanics of composite springs and driveshafts 460
12.1 Elements of hereditary mechanics of composite materials 460
12.1.1 Mechanisms of time-dependent deformation of composites 460
12.1.2 Linear viscoelasticity of composites 462
12.1.3 Nonlinear creep mechanics of anisotropic materials 463
12.1.4 Anisotropic Norton-Bailey law 466
12.2 Creep and relaxation of twisted composite shafts 467
12.2.1 Constitutive equations for relaxation in torsion of anisotropic shafts 467
12.2.2 Torque relaxation for anisotropic Norton-Bailey law 468
12.3 Creep and relaxation of composite helical coiled springs 470
12.3.1 Compression and tension composite spring 470
12.3.2 Relaxation of helical composite springs 471
12.3.3 Creep of helical composite compression springs 472
12.4 Creep and relaxation of composite springs in state of pure bending 473
12.4.1 Constitutive equations for bending relaxation 473
12.4.1 Relaxation of bending moment for anisotropic Norton-Bailey law 474
12.4.3 Creep in state of bending 475
Conclusions 476
References 476
Appendices i
Appendix A Mechanical properties of composites i
A.1 Fibers i
A.1.1 Glass Fibers i
A.1.2 Carbon fibers i
A.1.3 Aramid Fibers i
A.2 Physical properties of resin iii
A.3 Laminates iv
A.3.2 Unidirectional fiber-reinforced composite material iv
A.3.1 Fabric iv
A.3.3 Non-Woven Fabric iv
Appendix B Anisotropic elasticity v
B.1 Elastic Orthotropic Body v
B.2 Distortion energy and supplementary energy viii
B.3 Plane elasticity problems ix
B.3.2 Plane strain state ix
B.3.3 Plane stress state ix
B.4 Generalized Airy stress function x
B.4.1 Plane stress state x
B.4.2 Plane strain state x
B.4.3 Rotationally symmetric elasticity problems x
Appendix C Integral transforms in elasticity xi
C.1 One-dimensional integral transform xi
C.2 Two-dimensional Fourier-Transform xiii
C.3 Potential functions for plane elasticity problems xiii
C.4 Rotationally symmetric, spatial elasticity problems xvii
C.5 Application of the Fourier transformation to plane elasticity problems xix
C.6 Application of the Hankel transformation to spatial, rotation-symmetrical elasticity problems xxi
Index xxiv
