Cover
Foreword
Series Preface
List of Symbols and Abbreviations
1. Abbreviations
Introduction
1. Composites in Automotive Chassis and Drivetrain
2. Physical Properties of Composite Materials
3. Structure of the Book
4. Target Audience of the Book
5. References
About the Companion Website
1 Elastic Anisotropic Behavior of Composite Materials
1. 1.1 Anisotropic Elasticity of Composite Materials
2. 1.2 Unidirectional Fiber Bundle
3. 1.3 Rotational Transformations of Material Laws, Stress and Strain
4. 1.4 Elasticity Matrices for Laminated Plates
5. 1.5 Coupling Effects of Anisotropic Laminates
6. 1.6 Conclusions
7. References
2 Phenomenological Failure Criteria of Composites
1. 2.1 Phenomenological Failure Criteria
2. 2.2 Differentiating Criteria
3. 2.3 Physically Based Failure Criteria
4. 2.4 Rotational Transformation of Anisotropic Failure Criteria
5. 2.5 Conclusions
6. References
3 Micromechanical Failure Criteria of Composites
1. 3.1 Pullout of Fibers from the Elastic‐Plastic Matrix
2. 3.2 Crack Bridging in Elastic‐Plastic Unidirectional Composites
3. 3.3 Debonding of Fibers in Unidirectional Composites
4. 3.4 Conclusions
5. References
4 Optimization Principles for Structural Elements Made of Composites
1. 4.1 Stiffness Optimization of Anisotropic Structural Elements
2. 4.2 Optimization of Strength and Loading Capacity of Anisotropic Elements
3. 4.3 Optimization of Accumulated Elastic Energy in Flexible Anisotropic Elements
4. 4.4 Optimal Anisotropy in a Twisted Rod
5. 4.5 Optimal Anisotropy of Bending Console
6. 4.6 Optimization of Plates in Bending
7. 4.7 Conclusions
8. References
5 Optimization of Composite Driveshaft
1. 5.1 Torsion of Anisotropic Shafts With Solid Cross‐Sections
2. 5.2 Thin‐Walled Anisotropic Driveshaft with Closed Profile
3. 5.3 Deformation of a Composite Thin‐Walled Rod
4. 5.4 Buckling of Composite Driveshafts Under a Twist Moment
5. 5.5 Patents for Composite Driveshafts
6. 5.6 Conclusions
7. References
6 Dynamics of a Vehicle with Rigid Structural Elements of Chassis
1. 6.1 Classification of Wheel Suspensions
2. 6.2 Fundamental Models in Vehicle Dynamics
3. 6.3 Forces Between Tires and Road
4. 6.4 Dynamic Equations of a Single‐Track Model
5. 6.5 Conclusions
6. References
7 Dynamics of a Vehicle With Flexible, Anisotropic Structural Elements of Chassis
1. 7.1 Effects of Body and Chassis Elasticity on Vehicle Dynamics
2. 7.2 Self‐Steering Behavior of a Vehicle With Coupling of Bending and Torsion
3. 7.3 Steady Cornering of a Flexible Vehicle
4. 7.4 Estimation of Coupling Constant for a Twist Member
5. 7.5 Design of the Countersteering Twist‐Beam Axle
6. 7.6 Patents on Twist‐Beam Axles
7. 7.7 Conclusions
8. References
8 Design and Optimization of Composite Springs
1. 8.1 Design and Optimization of Anisotropic Helical Springs
2. 8.2 Conical Springs Made of Composite Material
3. 8.3 Alternative Concepts for Chassis Springs Made of Composites
4. 8.4 Conclusions
5. References
9 Equivalent Beams of Helical Anisotropic Springs
1. 9.1 Helical Compression Springs Made of Composite Materials
2. 9.2 Transverse Vibrations of a Composite Spring
3. 9.3 Side Buckling of a Helical Composite Spring
4. 9.4 Conclusions
5. References
10 Composite Leaf Springs
1. 10.1 Longitudinally Mounted Leaf Springs for Solid Axles
2. 10.2 Leaf‐Tension Springs
3. 10.3 Transversally Mounted Leaf Springs
4. 10.4 Conclusions
5. References
11 Meander‐Shaped Springs
1. 11.1 Meander‐Shaped Compression Springs for Automotive Suspensions
2. 11.2 Multiarc‐Profiled Spring Under Axial Compressive Load
3. 11.3 Sinusoidal Spring Under Compressive Axial Load
4. 11.4 Bending Stiffness of Meander Spring With a Constant Cross‐Section
5. 11.5 Stability of Corrugated Springs
6. 11.6 Patents for Chassis Springs Made of Composites in Meandering Form
7. 11.7 Conclusions
8. References
12 Hereditary Mechanics of Composite Springs and Driveshafts
1. 12.1 Elements of Hereditary Mechanics of Composite Materials
2. 12.2 Creep and Relaxation of Twisted Composite Shafts
3. 12.3 Creep and Relaxation of Composite Helical Coiled Springs
4. 12.4 Creep and Relaxation of Composite Springs in a State of Pure Bending
5. 12.5 Conclusions
6. References
Appendix A: Mechanical Properties of Composites
1. A.1 Fibers
2. A.2 Physical Properties of Resin
3. A.3 Laminates
4. References
Appendix B: Anisotropic Elasticity
1. B.1 Elastic Orthotropic Body
2. B.2 Distortion Energy and Supplementary Energy
3. B.3 Plane Elasticity Problems
4. B.4 Generalized Airy Stress Function
Appendix C: Integral Transforms in Elasticity
1. C.1 One‐Dimensional Integral Transform
2. C.2 Two‐Dimensional Fourier Transform
3. C.3 Potential Functions for Plane Elasticity Problems
4. C.4 Rotationally Symmetric, Spatial Elasticity Problems
5. C.5 Application of the Fourier Transformation to Plane Elasticity Problems
6. C.6 Application of the Hankel Transformation to Spatial, Rotation‐Symmetric Elasticity Problems
Index
End User License Agreement

List of Tables

Chapter 1
1. Table 1.1 Coefficients of Voigt's and Kelvin's matrices for orthotropic material...
2. Table 1.2 Effective modules of unidirectional composite material (Schürmann 2007...
3. Table 1.3 Input values for calculation of effective modules.
4. Table 1.4 Elastic constants and densities of matrix and fibers of UD fiberglass.
Chapter 2
1. Table 2.1 Overview of strength criteria for fiber‐plastic composites.
2. Table 2.2 Coefficients of the matrix and tensor Mises–Hill criterion.
3. Table 2.3 Eigenvalues of the matrix and tensor Mises–Hill criterion.
4. Table 2.4 Eigenvalues of the matrix for the pressure‐sensitive Mises–Hill crite...
5. Table 2.5 Material parameters for failure criteria (Vasilev et al. 1990).
6. Table 2.6 Coefficients of quadratic form of the Hashin strength criterion in a g...
7. Table 2.7 Coefficients of quadratic form of the Hashin strength criterion in a p...
8. Table 2.8 Coefficients of quadratic form of the Hashin strength criterion in cas...
9. Table 2.9 Coefficients of the pressure‐sensitive Mises–Hill criterion in the inc...
Chapter 3
1. Table 3.1 Coefficients of Eqs. ( 3.47 ) and ( 3.48 ).
2. Table 3.2 Properties of fiber and matrix for calculations.
3. Table 3.3 Expressions for integralχ_n(t) and solutions for coefficients C_n.
4. Table 3.4 Kernel of the integral equation for elastic cylinder, Eq. ( 3.145 ).
5. Table 3.5 Kernel of the integral equation for elastic cavity, Eq. ( 3.147 ).
6. Table 3.6 Kernel of the integral equation for contact problem of elastic cylinde...
7. Table 3.7 Material data for debonding problem (Barbero 1999 ).
Chapter 5
1. Table 5.1 Areas and area moments for solid triangular and circular cross‐section...
2. Table 5.2 Areas and area moments for hollow triangular and circular cross‐sectio...
Chapter 6
1. Table 6.1 Angle definitions in vehicle dynamics.
2. Table 6.2 Angle definitions in geometry of chassis.
3. Table 6.3 Components of forces and moments.
4. Table 6.4 Under‐, over‐ or neutral‐driving vehicle.
Chapter 8
1. Table 8.1 Spring rates and masses of the linear springs with a non‐circular wire...
2. Table 8.3 Stiffness of wires with different cross‐sections of wire.
3. Table 8.4 Section modules of wires with different cross‐section.
Chapter 9
1. Table 9.1 Effective stiffnesses for shear, bending and compression and mass per ...
2. Table 9.2 Conditions at the ends of the spring in terms of shear force, moment a...
3. Table 9.3 Conditions at the ends of the spring for different loading.
Chapter 10
1. Table 10.1 Comparison of different concepts for leaf springs (Kobelev et al. 201...
Chapter 11
1. Table 11.1 Effective spring constants of meander and coil springs for bending an...
2. Table 11.2 Design formulas for axial and bending stiffness of multiarc and sinus...
3. Table 11.3 Effective axial stiffness and spring rates of multiarc and sinusoidal...
4. Table 11.4 Dimensionless functions for multiarc and sinusoidal meander spring wi...

List of Illustrations

Chapter 1
1. Figure 1.1 Coordinate system and elastic symmetry.
2. Figure 1.2 Coordinate system associated with fiber direction and rotated coordi...
3. Figure 1.3 (a) Circumferentially asymmetric stiffness configuration (CAS), (b) ...
Chapter 2
1. Figure 2.1 The surface of the safety factors for the Tsai–Wu criterion (α = 1, ...
2. Figure 2.2 The surface of the safety factors for the Goldenblat–Kopnov criterio...
3. Figure 2.3 Transformation of tensor‐polynomial failure criteria for a unidirect...
4. Figure 2.4Figure 2.4 Transformation of tensor‐polynomial failure criteria for a...
5. Figure 2.5 Transformation of tensor‐polynomial failure criteria for a unidirect...
Chapter 3
1. Figure 3.1 Types of failure for fiber‐reinforced composite materials.
2. Figure 3.2 Common types of fracture mechanism of fiber‐reinforced composite mat...
3. Figure 3.3 Pullout of fibers and rupture surfaces of fibers and matrix on the p...
4. Figure 3.4 Zones along the pullout fibers. (a) Pulling the fibers from the elas...
5. Figure 3.5 Bridging of crack surfaces by fibers (a) and by matrix (b).
6. Figure 3.6 Dependence of the average stress upon the displacement of the matrix...
7. Figure 3.7 Crack in the unidirectional material.
8. Figure 3.8 Dependence of the stress to unlimited break in fibers as function of...
9. Figure 3.9 Dependence of the stress to unlimited break in matrix as function of...
10. Figure 3.10 Dependence of the stress to unlimited break in fibers and matrix as...
11. Figure 3.11 Dependence of the stress to unlimited break in fibers and matrix as...
12. Figure 3.12 Crack section and zone structure.
13. Figure 3.13 Zones at penny‐shaped crack.
14. Figure 3.14 Stress intensity factor K_I as the function of .
15. Figure 3.15 Plane crack problem.
16. Figure 3.16 Elastic‐plastic pullout of the fiber from the matrix (a) and the sp...
17. Figure 3.17 Fatigue failure of unidirectional fiberglass‐epoxy composite. Visib...
18. Figure 3.18 Fiber‐matrix debonding model. (a) Undeformed stress state, perfect ...
19. Figure 3.19 Fiber‐matrix debonding model. (C) Deformed stress state, perfect ad...
20. Figure 3.20 Fiber‐matrix debonding model. (A) Undeformed stress state, perfect ...
21. Figure 3.21 Exact and approximated kernels K_c(η), k_c(η) of the integr...
22. Figure 3.22 Relation of exact to the approximate kernel K_c(η)/k_c(η).
23. Figure 3.23 Normalized solution g₁(η)/g₁(0) ( 3.159 ) for the dimensionle...
24. Figure 3.24 Right side of the equation f(η) for different dimensionless le...
25. Figure 3.25 The coefficient c_g(a)/ exp(a/2) as a function of debonding le...
26. Figure 3.26 Graphs of the exact and asymptotical expressions of the solution ne...
27. Figure 3.27 Solution g₂(t) ( 3.167 ) for a semi‐infinite contact region a = ∞.
28. Figure 3.28 Solutions for a finite (D, shaded lines) and semi‐infinite contact ...
Chapter 4
1. Figure 4.1 Elastic energy for uniaxial tension as a function of angle Φ.
2. Figure 4.2 Specific elastic energy for uniaxial tension as a function of angle
3. Figure 4.3 Elastic energy for pure shear as a function of angle Φ.
4. Figure 4.4 Specific elastic energy for pure shear as a function of angle Φ
5. Figure 4.5 Direction‐dependent strength criterion g(Φ) for a uniaxial load...
6. Figure 4.6 Specific, direction‐dependent strength criterion g(Φ) for a uni...
7. Figure 4.7 Direction‐dependent strength criterion g(Φ) for a pure shear lo...
8. Figure 4.8 Specific, direction‐dependent strength g(Φ)/ρ for a pure s...
9. Figure 4.9 Specific direction‐dependent ultimate stored energy for uniaxial ten...
10. Figure 4.10 Specific direction‐dependent ultimate stored energy for a pure shea...
Chapter 5
1. Figure 5.1 Driveshafts of a common passenger car.
2. Figure 5.2 Composite propshaft in a Renault Espace Quadra (Pollard, 1989 ).
3. Figure 5.3 Composite driveshaft (Keys et al., 2018 ).
4. Figure 5.4 Torsion of a composite driveshaft by the terminal torque M_T.
5. Figure 5.5 Geometry of the cross‐section of the rod.
6. Figure 5.6 Types of fastening of the ends of the rod.
7. Figure 5.7 Driveshaft on the periodically spaced momentless supports.
8. Figure 5.8 Driveshaft on the periodically spaced rotation‐free supports.
9. Figure 5.9 Solid triangular and circular cross‐sections of the driveshaft of th...
10. Figure 5.10 Hollow triangular and circular cross‐sections of the driveshaft.
11. Figure 5.11 US3651661(A), 1972: “Composite shaft with integral end flange.”
12. Figure 5.12 US4171626(A): “Carbon fiber‐reinforced composite drive shaft.”
13. Figure 5.13 US4173128(A): Composite drive shaft.
14. Figure 5.14 US4551116(A): Drive shaft assembly.
15. Figure 5.15 US5725434(A): Shaft of fiber‐reinforced material.
16. Figure 5.16 US2017082149(A1): Composite driveshaft for a rotary system.
Chapter 6
1. Figure 6.1 Basic types of suspension.
2. Figure 6.2 Semi‐independent axles with Watt linkage wheel guide correction.
3. Figure 6.3 Semi‐independent axles.
4. Figure 6.4 Vehicle dynamics effects.
5. Figure 6.5 Roll center of semi‐independent axles.
6. Figure 6.6 Coordinate systems of vehicle (DIN ISO 8855).
7. Figure 6.7 Suspension geometry.
8. Figure 6.8Figure 6.8 Suspension geometry, projection on the yz‐plane.
9. Figure 6.9Figure 6.9 Suspension geometry, projection on the xz‐plane.
10. Figure 6.10 Suspension geometry, projection on the xy‐plane.
11. Figure 6.11 Bicycle model of the vehicle in an earth‐fixed coordinate system.
12. Figure 6.12 Side slip curve and lateral force properties.
13. Figure 6.13 Wheel slip angles in a single‐track model.
14. Figure 6.14 Slip angle and lateral force on the front wheel.
15. Figure 6.15 Slip angle and lateral force on the rear wheel.
16. Figure 6.16 Neutral steer in the bicycle model.
17. Figure 6.17Figure 6.17 Steer angle for low speed turning. “Ackermann” cornering...
18. Figure 6.18 Explanation of the understeer effect from the viewpoint of the bicy...
19. Figure 6.19 Explanation of the oversteer effect from the viewpoint of the bicyc...
20. Figure 6.20 US2002000703 (A1) Wheel suspension system with an integrated link, ...
21. Figure 6.21 EP2423012 (A2) Fiber composite anti‐roll bar, https://www.epo.org.
Chapter 7
1. Figure 7.1 Cornering of a car with stiff and with flexible rear axles.
2. Figure 7.2 Steering of vehicle with (a) absolutely rigid axle and (b) induced s...
3. Figure 7.3 Induced camber angle for a flexible rear axle (roll motion is not di...
4. Figure 7.4 Relationship between the rolling motion of the vehicle and twisting ...
5. Figure 7.5 Lateral shift of the wheel and additional steer angle due to centrif...
6. Figure 7.6 Relation between the torque M_TB in the twist‐beam and the roll momen...
7. Figure 7.7 Suspension roll stiffness of a twist‐beam axle.
8. Figure 7.8 Lateral rigidity of a twist‐beam axle.
9. Figure 7.9 Camber stiffness of a twist‐beam axle.
10. Figure 7.10 Bending‐torsion and tensile‐torsion coupling for a composite fiber ...
11. Figure 7.11 Laminate structure in the cross‐beam for the countersteering compos...
12. Figure 7.12 H‐shaped and Π‐shaped cross‐sections.
13. Figure 7.13 Bucket design.
14. Figure 7.14 Representation of the cross‐beam in two views.
15. Figure 7.15 Display of the bucket design in two views.
16. Figure 7.16 Step‐by‐step assembly of the countersteering twist‐beam axle.
17. Figure 7.17 Finite‐element model of the countersteering twist‐beam axle.
18. Figure 7.18 All‐synthetic plastic concepts: twist‐beam rear suspension, Volkswa...
19. Figure 7.19 EP3174744B1, Motor vehicle suspension with glued composite torsion ...
20. Figure 7.20 DE202013004035U1 (2013): Assembly for a vehicle.
21. Figure 7.21 US6382649B1 (1999‐07‐16): Wheel suspension in a motor vehicle.
22. Figure 7.22 DE202016105937U1, “Chassis axle and vehicle.”
Chapter 8
1. Figure 8.1 Glass fiber reinforced plastic (GFRP) springs: (a) fiberglass vol...
2. Figure 8.2 Helical composite spring, subjected to axial load and axial torque.
3. Figure 8.3 Helical spring loaded by torque and axial force.
4. Figure 8.4 Helical spring with variable wire diameter and non‐cylindrical form.
5. Figure 8.5 Middle surface ϖ of a free spring.
6. Figure 8.6 Middle surface Ω of a deformed spring.
7. Figure 8.7 Orientation of reinforcement fibers in the anisotropic conical sprin...
8. Figure 8.8 Orientation of two families of reinforcement fibers in the orthotrop...
9. Figure 8.9 Dependence effective circumferential modules upon meridian angle Φ...
10. Figure 8.10 US20040256829A1 (2003) “Suspension system having a composite beam.”
11. Figure 8.11 EP0459220B1 (1993): “Ring‐shaped spring made out of fiber composite...
12. Figure 8.12 US4801019 (1989): “Shock Absorbing Unit.”
13. Figure 8.13 DE102009029300A1 (2009): “Plastic spring for a motor vehicle underc...
14. Figure 8.14 DE102012202625A1 (2013): “Plastic composite spring.”
15. Figure 8.15 US20030222385 (2003) “Composite wave ring spring.”
Chapter 9
1. Figure 9.1 Lateral surface and equivalent beam of the composite spring.
2. Figure 9.2 Composite spring (right) and its equivalent beam (left).
3. Figure 9.3 Contour plot of first fundamental frequency and critical loads as fu...
4. Figure 9.4 Critical loads as function of the degree of slenderness ξ durin...
Chapter 10
1. Figure 10.1 Leaf‐tension composite spring (a, c) and conventional leaf spring (...
2. Figure 10.2 Leaf‐tension composite spring.
3. Figure 10.3 ZF lightweight axle CLA. Source: Courtesy of ZF Friedrichshafen AG.
4. Figure 10.4 ZF CLA system (ZF Friedrichshafen AG, Press Information 2015).
5. Figure 10.5 Ratio of roll spring rate to vertical spring rate.
6. Figure 10.6 EP0660005A1 ( 1993 ) “Composite material leaf spring for motor veh...
7. Figure 10.7 DE102013003958A1 ( 2013 ) Storage arrangement of a transverse leaf...
8. Figure 10.8 DE102014215871 (2014) Vehicle axle with an axis extending in the in...
9. Figure 10.9 US2016/0347139A1 ( 2016 ) Transverse leaf spring for a motor vehic...
Chapter 11
1. Figure 11.1 Elements of a meander spring.
2. Figure 11.2 Different concepts of meander springs and a helical spring.
3. Figure 11.3 Anisotropic behavior of a meander compression spring: different sti...
4. Figure 11.4 Representation of the continuous meander spring and its replacement...
5. Figure 11.5 Directrix of a quarter of the neutral surface of a multiarc‐profile...
6. Figure 11.6 Moments and forces in the meander.
7. Figure 11.7 Mass ratio of an optimal spring to the mass of a spring with a cons...
8. Figure 11.8 Mass ratio of constant to optimal sinusoidal meander springs, see E...
9. Figure 11.9 WO1994018019A1 ( 1993 ‐02‐15) Suspension for the wheel of a vehicl...
10. Figure 11.10 WO8500207(A1) Springs for high specific energy storage.
11. Figure 11.11 WO8700252: Spring assemblies.
12. Figure 11.12 DE3641108C2: Spring element.
13. Figure 11.13 US4927124 Spring assemblies.
14. Figure 11.14 DE19962026A1 ( 2001 ): Spring/suspension device.
15. Figure 11.15 DE102012111252A1 (2012) Spring of suspension for a vehicle.
16. Figure 11.16 US2007267792A1 ( 2007 ) Sigma‐springs for suspension systems.
17. Figure 11.17 DE102008006411 ( 2009 ) Vehicle spring made from composite materi...
18. Figure 11.18 DE102008057463 ( 2008 ) Spring, particularly drawing and pressure...

Copyright

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Library of Congress Cataloging‐in‐Publication Data

Names: Kobelev, Vladimir, 1959‐ author.

Title: Design and analysis of composite structures for automotive

applications : chassis and drivetrain / Vladimir Kobelev, Department of

Natural Sciences, University of Siegen, Germany.

Description: First edition. | Hoboken, NJ : Wiley, 2019. | Series: Automotive

series | Includes bibliographical references and index. |

Identifiers: LCCN 2019005286 (print) | LCCN 2019011866 (ebook) | ISBN

9781119513841 (Adobe PDF) | ISBN 9781119513865 (ePub) | ISBN 9781119513858

(hardback)

Subjects: LCSH: Automobiles–Chassis. | Automobiles–Power trains. |

Automobiles–Design and construction.

Classification: LCC TL255 (ebook) | LCC TL255 .K635 2019 (print) | DDC

629.2/4–dc23

LC record available at https://lccn.loc.gov/2019005286

Cover Design: Wiley

Cover Images: © Vladimir Kobelev, Background: © solarseven/ShuWerstock

Foreword

From a materials science point of view, composite materials of glass and carbon fibers have a specific potential and already some practical importance in several applications under high dynamic loads. Comparing the fibers, glass fibers are the better material for spring applications because their lower modulus of elasticity compared to carbon fibers. This is favorable in terms of high strokes and deformation requirements. Due to their high specific strength and the stiffness of composite materials, it is in principle possible to achieve weight savings of 30 – 70% of the weight of a steel spring depending on application. In addition to reduce the unsprung masses for suspension, it is also possible to improve driving dynamics as well as noise, vibration and hardness behavior (NVH), since the material properties are better in some significant areas. Furthermore, due to the high corrosion resistance and resistance against other environmental influences, surface protection is not necessary in most of the applications.

However, the usage of composite materials for springs have not reached high quantities due to some limitations. Load transmission requires special designs. Considering suspension coil springs, high loads transverse to the main load direction occur. Therefore, the load transmission does not follow ideally to the fiber direction and only medium loads can act on the matrix. In addition, in the case of large‐scale production and the available manufacturing processes, value adjustments must be made in comparison with units made of steel. These are currently the focus of research and development efforts throughout the world. Endless, unidirectional fiber materials, such as those used for structural elements in automotive engineering, exhibit strong anisotropic, i.e. direction‐dependent, properties. The fibers used are oriented with respect to the loads that occur. Therefore, the leaf spring, where loading results almost in tension stresses of the fibers is the perfect match with composite materials. Huge weight reduction up to 75% is possible to achieve by using the material properties and the design flexibility of glass fiber reinforced composite in the best way. A single composite tension leaf spring can substitute a steel multi‐leaf spring with a progressive spring load characteristic. The special design leads to a very homogenous, progressive spring characteristic and therefore, a better driving performance. Furthermore, we know already some designs for suspension steel coil springs substitution such as one‐by‐one substitution by composite coil spring and a meander spring design. In both case these springs do need special tools for the design and did not reach the market breakthrough due to huge different load‐rate requirements within the platforms.

There are some processes existing for the production of glass fiber composite springs. Nevertheless, the prepreg process (pre‐impregnated fibers) has proven itself as the best due to the realizable good properties under dynamic loads. Prepreg processes result in an optimal adhesive strength due to low porosity and allows flexibility in design, such as geometry, width and height of the spring. It is also possible to produce the elements of chassis in general and suspension particular using the resin injection process. For this resin injection process, a fiber structure is first produced from the dry reinforcement fibers, which follows the desired component geometry. If required, structural cohesion can be achieved using textile methods, such as sewing or bonding, which bond the fibers together. Such fiber structures are called preforms. The injection of the resin influences the orientation of the fibers and therefore, those springs do not reach the performance of prepreg composites due to potential ondulation.

Automotive manufacturers' requirements for carbon dioxide reduction, lower vehicle weight, the reduction of unsprung masses and the robustness of the springs, especially in the event of corrosion, will further increase in the future. The optimal application of the materials used plays a decisive role, supported by material properties, best technology and processes as well as an efficient design. Therefore, alternative materials, such as composites, may become higher importance for dynamic loaded suspension applications.

Prof. Dr. Vladimir Kobelev was born in Rostow‐na Donu, Russian Federation. He studied Physical Engineering at the Moscow Institute of Physics and Technology. After his PhD at the Department of Aerophysics and Space Research (FAKI), he habilitated at the University of Siegen, Scientific‐Technical Faculty. Today, Prof. Kobelev is lecturer and APL professor at the University of Siegen in the subject area of Mechanical Engineering.

In his industrial career, Prof. Kobelev is an employee at Mubea, a successful automotive supplier located near Cologne/Germany. In the Corporate Engineering Department, Prof. Kobelev is responsible for the development of calculation methods and physical modeling of Mubea components.

Joerg Neubrand

CTO, Managing Director and

Member of the Executice Board of the Mubea Group

Series Preface

Fuel efficiency continues to “drive” significant research and development in the automotive sector. In many instances, this is propelled by regulations that target reduced emissions as well as reduced fuel consumption. Even with more efficient vehicles and electric hybrid or purely electric driven systems, the need for reduced energy consumption is demanded by the market. This is due to the fact that the customer base is demanding increased efficiency as this brings better performance, lower costs and extended range of the vehicle. One clear means by which fuel efficiency can be enhanced is by reducing the weight of the vehicle. Lightweighting can be accomplished by a number of means, one of which is lighter weight material substitution. That is to say, one may substitute a lighter material for a heavier one on a vehicle component. Composites have been used to replace metal components in efforts to lightweight aircraft for decades. More recently, advances in materials, manufacture, and design have made composites cost effective and viable in the automotive sector. Two major stumbling blocks that have hindered composite use in the automotive sector are the cost of the composite components, and the ability to rapidly and economically produce such components in quantities that are needed by the automotive sector. Recently, these stumbling blocks have been overcome. However, for most commercial automotive applications, composites remain relegated to less critical elements of the vehicle system such as body panels. The use of composite for more critical vehicle applications such as suspension and drive train elements have been left to extremely demanding automotive scenarios such as Formula One. However, this scenario is about to change.

Design and Analysis of Composite Structures for Automotive Applications, provides an in‐depth technical analysis of critical suspension and drive train elements with a focus on composite materials. This includes basic principles for the design and optimization of critical vehicle elements using composite materials, as well as classical concepts related to mass reduction in automotive systems. The author, Professor Kobelev, skillfully integrates concepts related to vehicle parameters such as stiffness into overall vehicle dynamics using closed form solutions that are described in exquisite detail. The discussions focus on key elements of the vehicle including suspension and powertrain. These discussions are both comprehensive as well as the first of their kind in a text book, making this text an important reference for any automotive engineer on the leading edge.

Design and Analysis of Composite Structures for Automotive Applications is part of the Automotive Series that addresses new and emerging technologies in automotive engineering, supporting the development of next generation vehicles using next generation technologies, as well as new design and manufacturing methodologies. The series provides technical insight into a wide range of topics that is of interest and benefit to people working in the advanced automotive engineering sector. Design and Analysis of Composite Structures for Automotive Applications is a welcome addition to the Automotive Series as it primary objective is to supply pragmatic and thematic reference and educational materials for researchers and practitioners in industry, and postgraduate/advanced undergraduates in automotive engineering. The text is a state‐of‐the art book written by a leading world expert in composites and its application to suspensions and is a welcome addition to the Automotive Series.

Thomas Kurfess

March 2019

List of Symbols and Abbreviations

E_L: Longitudinal modulus of elasticity of composite parallel to fiber direction
E_T: Transverse modulus of elasticity of composite perpendicular to the fiber direction
G_TL: Transverse‐longitudinal shear modulus of composite
G_TT: Transverse shear modulus of composite
r_f: Radius of the fiber
: External radii of hypothetical matrix cylinders
V_f: Fiber volume content
V_m: Matrix volume content, V_m = 1 − V_f
ν_TL: Transverse‐longitudinal Poisson ratio of a composite
ν_TT: Transverse Poisson's ratio of a composite
E_f: Longitudinal modulus of elasticity of fibers
E_{f. T}: Transverse modulus of elasticity of fibers
E_m: Modulus of elasticity of matrix (resin)
ν_m: Poisson's ratio of matrix (resin)
G_m: Shear modulus of matrix (resin)
G_{f. TL}: Transverse‐longitudinal shear modulus of fibers
ν_{f. TL}: Transverse‐longitudinal Poisson's ratio of fibers
ν_{f. TT}: Transverse Poisson's ratio of fibers
S = [S_ijpq]: Compliance tensor of rank four, i, j, p, q = 1, 2, 3
C = [C_ijpq]: Elasticity tensor of rank four
C⁽⁰⁾ = [c_ijpq]: Elasticity tensor of rank four, in the layer coordinate system
S⁽⁰⁾ = [s_ijpq]: Compliance tensor of rank four, in the layer coordinate system
I = [I_ijpq]: Fourth rank identity tensor
σ: Voigt's stress vector
ɛ: Voigt's strain vector
C: Voigt's elasticity matrix
: Kelvin's stress vector
: Kelvin's strain vector
: Kelvin's elasticity tensor
S⁽⁰⁾: Compliance matrix in Voigt's notation in intrinsic coordinates
: Compliance matrix in Kelvin's notation in intrinsic coordinates
t = [t_lk]: Transformation (rotation) square 3×3 or 2×2 matrix
T_σ: σ‐transformation (rotation) square 6×6 matrix, in Voigt's notation
T_ε: ε‐transformation (rotation) square 6×6 matrix, in Voigt's notation
: Transformation (rotation) square 6×6 matrix, in Kelvin's notation
A: Plane modulus quadrant, square 3×3 matrix (entries for the membrane elasticity tensor) in Voigt's notation
B: Coupling quadrant, square 3×3 matrix (entries for the coupling elasticity tensor) in Voigt's notation
D: Plate quadrant, square 3×3 matrix (entries for the bending elasticity tensor) in Voigt's notation
ɛ^T = [ε₁₁, ε₂₂, γ₁₂ = 2ε₁₂]: Strain vector in Voigt's notation
κ^T = [κ₁₁, κ₂₂, 2κ₁₂]: Curvature vector in Voigt's notation
N^T = [N₁₁, N₂₂, N₁₂]: In‐plane forces vector in Voigt's notation
M^T = [M₁₁, M₂₂, M₁₂]: Bending moments vector in Voigt's notation
Q: Reduced stiffness matrix in Voigt's notation
: σ‐transformation (rotation), square 3×3 matrix in Voigt's notation
: ε‐Transformation (rotation) square 3×3 matrix in Voigt's notation
: Plane modulus quadrant, square 3×3 matrix (entries for the membrane elasticity tensor) in Kelvin's notation
: Coupling quadrant, square 3×3 matrix (entries for the coupling elasticity tensor) in Kelvin's notation
: Plate quadrant, square 3×3 matrix (entries for the bending elasticity tensor in Kelvin's notation
: Strain vector in Kelvin's notation
: Curvature vector in Kelvin's notation
: In‐plane reaction forces, vector in Kelvin's notation
: Bending moments, vector in Kelvin's notation
: Reduced stiffness matrix in Kelvin's notation
: Transformation matrix for rotation (3×3) in Kelvin's notation
X_t, X_c: Tensile or compressive strengths in the fiber direction
Y_t, Y_c: Tensile or compressive strengths in the transverse direction
: Matrix of the Mises–Hill criterion, Voigt's notation in intrinsic coordinates
F, G, H, L, M, N: Characteristic values of the Mises–Hill criterion
Λ_i: Eigenvalues of the Mises–Hill criterion, i = 1..6
: Ultimate normal stresses in the Mises–Hill criterion
: Ultimate shear stresses in the Mises–Hill criterion
: Matrix of the Mises–Hill criterion, Kelvin's notation in intrinsic coordinates
Φ: Rotation angle of fibers in the plane “1–2”
: Matrix of the Mises–Hill criterion in the rotated axes, Kelvin's notation
: Matrix of the pressure‐sensitive Mises–Hill criterion, Voigt's notation in intrinsic coordinates
: Matrix of the pressure‐sensitive Mises–Hill criterion, Kelvin's notation in intrinsic coordinates
F⁽²⁾, F⁽⁴⁾, F⁽⁶⁾: Tensors of the 2nd, 4th and 6th ranks of the Goldenblat–Kopnov tensor fracture criterion
σ_f: Axial stresses in cylindrical fibers
σ_m: Axial stresses in hollow matrix cylinders
τ: Shear stress at the fiber surface
τ_p: Yield point of the matrix
p(z), q(z): Auxiliary functions, p = u_m − u_f, q = u_m + u_f.
u_m,u_f: Axial displacements of matrix and fiber cylinders
λ, μ,: Parameters of the length dimension
: Parameters of the inverse length dimension
l_p: Length of the plastic zone
R_f, R_f: Crack extension resistance of fibers and matrix
: Auxiliary modules
K_f, K_m: Fracture toughness of fibers and matrix
dU_e/da: Energy release rate per thickness unit
dU_f/da: Crack extension resistance per thickness unit
ϕ_i(ρ, η): Potential functions,i = 1, 2
a = l_c/2r_f: Dimensionless length of adhesive or debonding region
K_max, K_max: Maximum and minimum stress intensity factor
Y(a): Dimensionless parameter that reflects the geometry
c_f = c_f(R_σ): Material constant of matrix or resin for a given stress ratio R_σ
R_σ = K_min/K_max: Stress ratio of cyclic load
p_c(K) = K^−p: Paris–Erdogan crack propagation function
C_T: Torsional rigidity of bar (driveshaft)
I_b1, I_b2: Moment of inertia of cross section with respect to both bending axes
: Critical torque in Greenhill's problem
W_e(Φ): Density of elastic energy
W_e^*(Φ): Elastic energy per mass unit (specific elastic energy)
: Ultimate elastic energy per mass unit (specific ultimate elastic energy)
: Ultimate elastic energy
X_E,Y_E,Z_E: Axes of the earth‐fixed coordinate system
X_V,Y_V,Y_V: Axes of the vehicle‐fixed coordinate system
X, Y, Z: Axes of the horizontal coordinate system
X_W, Y_W,Z_W: Axes of the wheel coordinate system
ϕ: Roll angle
θ: Pitch angle
ψ: Yaw angle
β: Sideslip angle of a vehicle
ς = ψ − β: Course angle
S_X;V: Circumferential slip
ω: Rotational speed of a wheel
ω₀: The rotational speed of a straight and freely rolling wheel
α: Sideslip angle of a wheel
μ_X,W: Coefficients of circumferential force
μ_Y,W: Coefficients of lateral force
C_α: Cornering stiffness
C_αf,C_αr: Cornering stiffness of front and rear tire
I_z: Mass moment of inertia of the vehicle around the vertical axis
m: Mass of a vehicle
L_f: Horizontal distance from front axle to center of mass
L_r: Horizontal distance from rear axle to center of mass
L_w = L_f + L_r: Wheel base
δ: Steer angle at the wheel (part of steer angle due to steer)
: Steer angle gradient
δ_a: Steer angle according to Ackermann
Δ_V: External excitation in the course of steering
ω_V: Yaw circular frequency of a vehicle
D_V: Damping factor of a vehicle's yaw oscillation
L₀: Free length of a helical spring
L_rel: Released length of a helical spring
L_comp: Compressed length of a helical spring
L_c: Close up length of a compressed spring
s = L_rel − L_comp: Spring travel
c: Axial compression or extension spring constant
c_θF: Compression‐twist spring rate
c_θ: Twist spring rate
U_e: Elastic energy
U_f: Work of applied forces
F: Axial force on a helical spring
M_θ: Axial torque on a helical spring
c^*: Design value for a spring constant
τ_w: Ideal stress at solid height
d_opt: Optimal diameter of a wire
m_opt: Lower boundary for spring mass
ϖ: Middle surface of an undeformed conical spring
Ω: Middle surface of a deformed conical spring
t_c: Thickness of an anisotropic conical spring
r_a_,r_b: Inner and outer radius of the middle surface of a free conical spring
R_a,R_b: Outer radius of the middle surface of a deformed conical spring
Δ = r_b/r_a: Ratio of the outer radius to inner radius of a conical spring
: Inversion point of a conical spring
z_a, z_b: Heights of the inner and outer edges of a free spring
Z_a, Z_b: Heights of the inner and outer edges of a deformed spring
ε₁: Circumferential mid‐surface strain of a conical spring
κ₁: Circumferential curvature changes of a conical spring
: Effective elastic modulus
: Effective orthotropic elastic modulus
s_Q = s_b + s_s: Total transversal displacement of a helical spring
Q: Shear force of equivalent column for a helical spring
M_B: Bending moment of a helical spring
m_B: External torque per unit length of a helical spring
f_Q: External load in the transverse direction of a helical spring
C₄₄I_T: Twist stiffness of a wire with respect to its axis
C₃₃I_b: Stiffness of a spring wire in the case of bending in a binormal direction
C₃₃I_n: Stiffness of a spring wire in the case of bending in a normal direction
s_o(z): Initial transverse deflection of a helical spring
v_o(z): Initial transverse velocity of a helical spring
ω_k: Circular natural frequency in the order of k of a helical spring
ξ = L₀/D: Slenderness ratio of a helical spring
μ = L/L₀: Dimensionless length of a helical spring
Ω_k: Dimensionless frequency of transverse oscillations in the order of k of a helical spring
: Critical deflection during compression from the free length of a helical spring
: Critical deflection during expansion from the flattened state of a helical spring
f(σ_eff, t): Anisotropic stress function for creep
t: Time for creep
: Deviatoric component of creep strain
s_ij: Deviatoric component for creep stress
σ_eq: Mises equivalent stress
c_τ: Creep constant for shear strain
c_σ: Creep constant for uniaxial strain
γ_e: Elastic component of shear strain
γ_c: Creep component of shear strain
: Torque at the moment t = 0
M_T(t): Torque as a function of time
: Spring force at the moment t = 0
F_z(t): Spring force as a function of time
: Bending moment at the moment t = 0
M_B(t): Bending moment as a function of time
Φ_T(t),Φ_B(t),Φ_H(t): Relaxation functions for twisting, bending and helical spring
f^′ = ∂f/∂z, or ∂f/∂x₁, or ∂f/∂ξ₁: “Prime” denotes a derivative with respect to a coordinate: z, or x₁, or ξ₁
: “Dot” denotes a time derivative

Abbreviations

AF: Aramid fiber
AFRP: Aramid fiber reinforced plastic
Autoclave: Heated pressure tank
CE: Cyanate ester resin
CF: Carbon fiber
CFRP: Carbon fiber reinforced plastic
DSA: Driveshaft axis
EP: Epoxy resin
Fabric: Biaxially woven textile
FRP: Fiber reinforced plastic
GF: Glass fiber
GFRP: Glass fiber reinforced plastic
HM: High modulus
HT: High tensile strength
Laminate: Layer construction of cured, individual plies
Matrix: Resin in which the fibers are embedded
NVH: Noise, vibration and hardness
PA: Polyamide
PEEK: Polyetheretherketone
PF: Phenolic resin
PMMA: Polymethyl‐methylacrylate
PPS: Polyphenyl sulfide
Prepreg: Preimpregnated fibres – fibers or textiles pre‐saturated with resin
PU: Polyurethane resin
Roving: Soft strand of twisted, attenuated, freed of external matter fiber ready to conversion into yarn
RTM: Resin transfer molding – resin is injected into an enclosed mold in which fibers have been placed
Tg: Glass transition temperature
UD: Unidirectional fibers are oriented in only one direction
UP: Unsaturated polyester
VE: Vinyl ester resin

Chassis and Drivetrain

Abbreviations