Finite Element Analysis of Rolling Tires A State of the Art Review

Transcription

1 Finite Element Analysis of Rolling Tires A State of the Art Review Maik Brinkmeier, Udo Nackenhorst and Matthias Ziefle Institut für Baumechanik und Numerische Mechanik Gottfried Wilhelm Leibniz Universität Hannover, Germany Finite Element methods are well established for the analysis of stationary rolling contact problems especially in tire industries. However, the available methods often lack on numerical stability and some open questions regarding the physical reliability remain. The most recent progress made in this field is shown here, covering tractive rolling under steady state conditions, high frequency transient dynamics of rolling tires and the treatment of inelastic constitutive properties within the chosen relative kinematics ALE description. 1 Introduction Computational methods for rolling contact analysis, e.g. for tire road contact, wheel rail contact or roller bearings, are of great interest, because in general it is not possible to investigate the mechanisms within the contact region experimentally under usual environmental conditions. Therefore, reliable and efficient numerical tools based on a careful modeling of the physical effects are needed. Analytical models available for rather simple problems can serve to verify the computational methods. First analytical approaches for simple geometries based on the assumption of elastic half spaces have been published by Carter (1926), which have been extended for rolling of discs with dissimilar material properties by Desoyer (1957) and Bufler (1959). Based on this fundamental theoretical work efforts on the development of computational methods started in the late 60s. Pioneering work has been done by Kalker (1967), who developed computational schemes for discretized version of the underlying integral equations, which is interpreted as an early version of Boundary Element Methods (BEM) for contact problems. Because this approaches are based on analytical, so called fundamental solutions, which are available for linear mechanical response of the contacting bodies only, their regime of applications is limited. The development of more general computational techniques by Finite Element Methods (FEM) started in the 80s. Padovan (1987) proposed the Moving Lagrange Observer theory. Disadvantages of this approach are caused from the additive decomposition of motion with restrictions to kinematically linear problems, and the rather heuristical approach suggested for the treatment of second order gradients based on a C 0 smooth finite element approximation. A more general kinematic formulation has been presented by Oden and Lin (1986), a three dimensional generalization has been published by Bass (1987). In these early contributions already viscoelastic material properties have been taken into account, see e.g. Oden and Lin (1986), which have been treated numerically by an engineering approach. A numerically sound approach for the treatment of inelastic material properties within the finite element analysis of rolling has been discussed by Tallec and Rahier (1994). A more general kinematic formulation has been introduced by Faria et al. (1992) and Nackenhorst (1993) by placing it into the framework of Arbitrary Lagrangian Eulerian (ALE) methods. At that time ALE methods where under the development for the finite element computation of fluid structure interaction problems, cf. Donea (1983); Hughes et al. (1981), nowadays a great potential is seen in tackling metal forming processes etc, see Gadala (2004) for example. Actually we 1

2 are on the way to merge these techniques developed independently with the aim of more stable, reliable and efficient numerical methods for rolling contact analysis. This presentation is aimed as survey on the current stage of ALE methods for rolling contact problems. We start with a short introduction into the ALE description of rolling bodies, where hints on details of implementation will be given. The rolling contact problem is discussed in a next section in detail, where the focus is laid onto a physically correct formulation and numerically robust implementation of the tangential contact conditions including frictional sliding. In section 4 the computation of the high frequency response of rolling tires with emphasis to rolling noise prediction is discussed, where a modal superposition approach is suggested. A computational consistent method for treatment of inelastic material properties is presented in the final part of this contribution, where a Time Discontinuous Galerkin (TDG) scheme is introduced for the numerically stable advection of the material history along path lines within the spatially fixed ALE mesh. 2 ALE Description of Rotating Bodies Rolling contact problems are described preferably within a spatial (Eulerian) framework. Imagine a laboratory test setup, where a tire is rolling on a drum, then the contact area appears spatially fixed. However, when large deformations have to be taken into account a more general kinematic framework is needed, for which the Arbitrary Lagrangian Eulerian (ALE) description has been established. The advantages are summarized as follows: stationary rolling is described independent of time, a spatially fixed discretization is introduced, which enables local refinement in the contact zone for more accurate analysis, error control and adaptive mesh refinement can be performed with respect to the spatial discretization only, superimposed transient dynamics is immediately described in space domain, which is required for example for rolling noise analysis. This is underlined by the following arguments: Within a purely Lagrangian description (body fixed observer) the whole circumference of the wheel has to be discretized as fine as needed for a detailed contact analysis. The number of unknowns is drastically reduced when the rolling process is observed in a spatial observer framework. In addition, for the treatment of the explicit time dependency time discretization schemes have to be involved. A stationary operating point has to be computed starting from the resting state. The theoretical basis of the ALE description of rolling is well understood, cf. Nackenhorst (2004), in the following only the basic results will be repeated. The general idea of this special kind of ALE formulation is the decomposition of motion into a pure rigid body motion, denoted by the mapping χ, and the superimposed deformation, denoted by φ. Illustratively this is the introduction of a intermediate configuration as depicted in fig. 1, which is used as stress free reference configuration. Mathematically this scenario is described by a multiplicative decomposition of motion. The material deformation gradient reads F = ˆF R, (1) where the tensor R describes the pure rigid body motion and therefore, the orthogonality property R 1 = R T holds. The tensor ˆF is a measure for the deformation of the rolling body and serves as basic kinematic description for the derivation of objective strain measures. As well known from Eulerian mechanics the material time derivative is decomposed into a relative and convective part, which reads dφ dt = φ t + Gradφ w, (2) χ 2

3 initial configuration X φ current configuration B x φ (B) φ = ˆφ χ χ χ χ (B) ˆφ reference configuration Figure 1: ALE-decomposition of motion. where the gradient Grad is formulated with respect to the coordinates of the reference configuration χ, and w describes the rigid body velocity of the material point. With this basics a continuum description of rolling bodies including objective tensors and thermodynamically consistent constitutive laws is developed straight forward. Starting point of our investigations is the momentum balance equation written with respect to the reference configuration, Div ˆP + ˆ b = ˆ dv dt. (3) Herein the caret ˆ indicates quantities measured in the reference configuration. Thus ˆP is the First Piola Kirchhoff stress tensor, b describes the body force density, ˆ is the mass density and v the velocity of material particles. This elliptic partial differential equation has to be solved under consideration of the boundary conditions, i.e. φ = φ on φ (B), (4) ˆP ˆN = T on T (B). (5) The final ingredient is the formulation of the constitutive equations, where here the hyperelastic concept is assumed, i.e. the stress tensor is derived from a free energy density Additionally, the contact conditions have to be satisfied. 2.1 Weak Formulation and Finite Element Representation χ(b) ˆP = ˆ ψ. (6) ˆF For approximate solutions using the finite element method the balance law is re written in a weak form, by multiplying eq. (3) with a test function η and integrating over the bodies volume, ( Div ˆP + ˆ b ˆ dv ) η dˆv. (7) dt After mathematical transformation taking into account the specialties of the ALE description and discretizing with standard C 0 smooth displacement based finite elements, consistent linearization leads to the incremental finite element representation of the equations of motion, M ϕ + G ϕ + [K W ϕ = f ext + f inertia f int, (8) 3

4 to be solved for the evolution of the displacement field t+ t ϕ = t ϕ + ϕ. (9) In (8) K is the tangential stiffness matrix, M is the standard mass matrix, G = ˆ (N T A A T N)dˆV (10) χ(b) is referred to as gyroscopic matrix and W = ˆ A T AdˆV (11) χ(b) is called ALE inertia matrix obtained from the linearization of the centrifugal forces. The matrix A connects the convective velocity with the nodal displacements, i.e. c = Gradφ w = Aϕ. (12) The right hand side of eq. (8) describes the equivalent nodal forces due to applied external loads, centrifugal forces and internal forces due to the divergence of the stress tensor, respectively. 3 Stationary Rolling In stationary cases all time derivatives in eq. (8) vanish and a rotating body underlying large deformations is described by a non linear problem which is solved incrementally using Newton Raphson equilibrium iteration as follows: Within an incremental load step t solve the linearized problem [ t K W ϕ = t+ t f ext + f inertia ( t ϕ n ) f int ( t ϕ n ) (13) and iteratively (internal counter n) update the displacement field ϕ n+1 = ϕ n + ϕ. (14) Repeat the iteration until convergence criteria are fulfilled, i.e. the equilibrium of nodal point forces is obtained within suitable error bounds, and increase loads incrementally for the next step t + t. For the analysis of rolling contact problems now, in addition the contact conditions have to be taken into account. Whereas the normal contact restraints can be modeled directly by efficient and well established methods known from Lagrangian mechanics of solid body contact as far as the history of contacting particles can be neglected, for which it is referred to the recent monographs Laursen (2002) and Wriggers (2002), in the ALE framework of rolling special treatment of the tangential contact constraints including slip stick distinction in the finite contact area has to be considered. Despite the fact, that today commercial software offers options on these computations, the scientific literature on this detail is quite limited, cf. Hu and Wriggers (2002), Nackenhorst (2004) and Laursen and Stanciulescu (2006). The following section is aimed for a careful analysis of this delicate topic with respect to physical consistent and numerically reliable solutions. 3.1 Treatment of the Contact Conditions. A well accepted outcome of continuums contact mechanics is the assumption, that normal and tangential contact can be treated locally decoupled. The treatment of the normal contact conditions, i.e. the enforcement of the Signorini conditions d 0, p 0, pd = 0, (15) 4

5 is straight forward, because it can be evaluated within the spatial configuration. Well established algorithms for contact computation can be applied directly to enforce the normal contact constraints. The penalty method for example leads to the contact force contribution f (cn) = N T α t nd da (16) cφ(b) and a contribution to the tangent matrix K (cn) = cφ(b) α n NT an a T n Nda. (17) The tangential contact constraints are defined by the Kuhn Tucker conditions where R 0 s 0 R s = 0, (18) R = τ τ max (19) describes the friction law. However, well established techniques developed within a pure Lagrangian framework can not be applied directly to enforce the tangential contact constraints within the ALE picture, because the history of the contacting particles is not computed implicitly. Therefore, additional effort is necessary for the description of tractive tangential rolling contact problems. This outline is based on classical predictor corrector schemes, where in a first (trial) step sticking is assumed and the contact traction is evaluated from reaction forces. These are used for the evaluation of the friction law in a second step, and if violated, it is switched from enforced kinematic constraints to enforced traction derived from the constitutive friction law. It has to be concluded, that the physics of tractive rolling has to be explained in a material picture, where the relative tangential movement between two contacting particles has to be traced during their movement. A general path lines tracking algorithm will be presented in section 5, where the treatment of inelastic material behavior will be discussed. However, this approach ends up in a staggered explicit scheme, and therefore, for the numerical treatment of frictional rolling contact here an alternative method is suggested. The slip velocity can be expressed as ṡ p = Grads w, (20) within the ALE description, leading to a C 0 smooth approximation for the nodal slip distance and a coupled system of equations, cf. Ziefle (2007), K c ϕϕ K c sϕ K c ϕs K c ss ϕ = s 4 High Frequency Dynamics of Rolling Tires f c t r c t. (21) In this section the modal superposition approach for the solution of the transient dynamics of rolling tires will be presented. This provides more basic insight into the physics of rolling bodies than a straight forward non linear analysis in advance. The basic modeling assumptions for this approach are summarized as follows: Small vibrational amplitudes are superimposed onto the large deflection of the stationary rolling problem, the contact conditions do not change transiently. 5

6 An engineering applications of this approach is the analysis of tire noise radiation with the aim to optimize the tire road interaction with respect to silent traffic. In fact the sound radiated from rolling wheels nowadays is the most dominant source of traffic noise which annoys people in urban areas and, therefore, is judged a risk factor for health. In the following the attention is focussed onto the computational aspects on the dynamics of rolling wheels. The eigenvalue analysis for the gyroscopic system will be discussed in detail, continued by the description of excitation functions derived from the surface texture of roads and the modal superposition approach for the computation of operational modes. 4.1 Eigenvalue Analysis of Rolling Wheels Related to eq. (8) the homogeneous equation of motion is considered, M φ + G φ + Kφ = 0, with K = t K W, G = G T, (22) where now φ indicates the small amplitude vibrations which are superimposed to the large deflections ϕ from the stationary rolling analysis. With the time harmonic ansatz φ = z e λt (23) a quadratic eigenvalue problem Q(λ)z = ( λ 2 M + λg + K ) z = 0 (24) is derived which in its linearized form reads, ([ ig K K 0 [ M 0 ω 0 K ) [ ωz ẑ = 0, ẑ = z, (25) with iω = λ. This presentation remains non symmetric and therefore, leads to complex valued eigenvectors. However, due to the Hermitian structure of the matrices the spectral properties are preserved and therefore, the left and right eigenvectors are equal, i.e. Q(λ)z = 0, z H Q(λ) = 0, Q H = Q. (26) Implicitly Restarted Arnoldy Method (IRAM) in shift invert mode is used for the numerical solution of the eigenvalue problem, for details it is referred to Lehoucq and Yang (1998). The automatic restart requires the solution of a linear system [ [ [ ig σm K x1 Mb1 K σ K = (27) x 2 Kb 2 several times. Unfortunately the structure of the matrices is such bad that the factorization requires a huge amount of storage. This problem can be circumvented by a back projection onto the quadratic form. By this approach the solution of the linear system (27) is substituted by the sequential scheme which is solved numerically much cheaper. 4.2 Modal Superposition x 2 = Q 1 [ Mb 1 (ig σm)k 1 b 2, (28) x 1 = K 1 b 2 + σx 2. (29) The excitation of rolling wheels is approximated by Fourier series expansion, which appears to be natural within the modal superposition approach. A series of time harmonic displacements acting at the contact nodes of the rolling wheel is derived from geometrical irregularities, such as tread pattern or road surface 6

7 textures. This spatial information is easily transformed into time harmonic excitation function taking the rolling speed into account, i.e. u(t) = j û j e i(ϕj+ωex j t). (30) With these preparations, the non homogeneous finite element equation of motion M φ + G φ + Kφ = f(t), f(t) = K c u(t) (31) is considered, where now the excitation forces are computed from the excitation spectrum and the related stiffness of the contact nodes. Again, the equivalent linearized form [ ig K K 0 φ [ M 0 0 K φ = [ f(t) 0, φ = [ φ φ, (32) is considered, which decouples applying a standard modal transformation, i.e. φ = [ ZΩ Z q j, Z =, Ω = diag(ω Z i ), (33) j with the generalized coordinates q and the diagonal matrix of eigenvalues Ω, yielding a set of ordinary first order differential equations, ω i q ij q ij = f ij, (34) for which analytical solutions are well known. A time harmonic ansatz is chosen for the generalized coordinates in order of stationary solutions. With the properties of orthogonality of the linear eigenproblem (25) one obtains the solution q j = ( iω iω ex j I) 1 ZH ˆ f e i(ϕ j+ω ex j t) (35) for the generalized coordinates which is transformed back for the operational vibration of the tire model on rough road surfaces by φ(t) = j Z ( iω iω ex j I ) 1 ΩZ H ˆf e i(ϕ j+ω ex j t). (36) To overcome the resonance case, a reasonable small value δ can be used as virtual damping, avoiding numerical problems when inverting the diagonal matrix ( iω iω ex j I ) ( δi + iω iωj ex I ). (37) 4.3 Sound radiation For the simulation of the sound of the tire the Helmholtz equation is considered, while the natural boundary conditions are the surface velocities of the tire model from the eigenvectors and the mixed boundary conditions are impedance and admittance properties of the road surface. The radiation to infinity is modeled with infinite elements. The sound pressure is simulated for every excitation frequency and finally the pressure field is the result of superposition. 5 Treatment of Inelastic Material Properties The numerical treatment of inelastic material for the simulation of rolling bodies has been discussed in literature already in the early stage of the development of these techniques, see e.g. Oden and Lin (1986). These approaches are dated before the overall ALE framework has been recognized, and therefore, they are based on more or less engineering approaches. The general idea in common is, to trace the motion of 7

8 a material particle along concentric rings, which are defined by a list of integration points. This approach works for structured spatial finite element meshes. However, troubles occur, when the spatial discretization is not unique, keeping in mind that local mesh refinement for the contact analysis is an explicit goal at least for stationary analysis. It is emphasized that there is no sound mathematical basis for these approaches and numerical errors can not be controlled at all. This section is aimed to present a theoretical framework for the numerical treatment of inelastic material behavior within the ALE framework of rolling contact, where special emphasis is laid onto a mathematically consistent theory, which enables for error estimation and adaptive strategies. From the computational point of view the treatment of inelastic material properties within the ALE framework of rolling is a challenging task, because the finite element mesh is not attached to the material points like in a Lagrangian picture. Therefore, the history of material particles has to be traced along the path lines on which the points are moving. This class of problems has been intensively investigated in fluid mechanics where the challenge lies on the numerical treatment of advection equations, which till now seems not to be solved satisfactory. A Time Discontinuous Galerkin (TDG) approach described below has been proven for best accuracy in this context. For the description of inelastic material behavior within a thermodynamically consistent constitutive theory a free energy density function of the form ψ = ψ(f,α,...) (38) is stated, from which the stress tensor is derived as sketched in eq.(6). The internal variables α are candidates for the description of the inelastic material history, they can be scalar, vector, or tensor valued. For the mathematical determination of the internal variables under some basic considerations evolution rules are derived from the second law of thermodynamics. In the case of rate dependent constitutive behavior with fading memory a mathematical structure of the evolution rule like α = 1 (C α) (39) τ can be derived, where C is the Right Cauchy Green tensor and τ a constitutive parameter. For the numerical solution of the evolutions rules for the internal state variables within the ALE framework of rolling a fractional step strategy is applied, Benson (1989). In a first (Lagrangian) step the evolution equation (39) is solved locally by standard integration schemes, see Simo and Hughes (1998) for example. In a second step the internal variables are pushed within the spatially fixed ALE mesh along the path lines of the moving particles. This second step is described mathematically by an advection equation, which in one dimensional form reads α t + gradα c = 0, (40) where c = Gradϕ w is the convective velocity. Numerical methods for the stable and accurate solution of advection problems are under investigation for a long time. Nowadays time discontinuous Galerkin (TDG) methods appear to be most promising approaches for numerically stable and accurate solutions. For a comparison of numerical methods for the solution of advection dominated problems it is referred to Ziefle (2007). An overall solution algorithm for the treatment of inelastic material behavior within the ALE framework of rolling bodies is sketched as follows: 1. In each time step t a compute the stationary solution by solving the linear system and update the displacement field within a Newton scheme. [K W u = f ext + f inertia + f cont f int (41) t a u n = ta u n 1 + u. (42) 8

9 2. When the stationary solution is found, update the internal variables by a local solution of the evolution equations, α = F (C, α). (43) This is the so called Lagrangian step within the fractional step approach which can be performed efficiently by standard algorithms established in plasticity or visco elasticity. 3. In a third step the smoothing of the internal variable field is required. The local solution of the evolution equation in step 2 has been performed at integration points of the spatial finite element mesh, resulting in a non smooth representation denoted by α GP. This non smoothness is obvious within the finite element framework, because the internal variables depend on the gradients of the primal variables for which only a C 0 smooth approximation has been chosen. To obtain a C 0 smooth distribution for the gradient fields, a standard super convergent projection scheme is applied, cf. Zienkiewicz and Taylor (2000). With the ansatz α = H ˆα, where α expresses the continuous distribution, H are standard spatial finite element shape functions and ˆα are the nodal values of the field, from a least squares fit a linear system H T H dv ˆα = H T α GP dv (44) φ(b) φ(b) is derived to compute a C 0 smooth gradient field. The solution of this linear system is numerically cheap, because it can be performed locally by so called super convergent patch recovery techniques, Zienkiewicz and Zhu (1992). 4. Finally, in each time step the advection of the inelastic variables is performed by [Υ a (t)m(x) + Υ b (t)q(x) [ˆα(x, t) = [ Υ c (t)m(x) ˆα n 1 (x, t), (45) with time and space dependent matrices. It should be recognized that this so called fractional step approach, which is well established in ALE theory, is fully explicit. The computational effort compared to a pure elastic solution is increased mentionable. However, by the examples shown below its practicability even for large scaled problems is demonstrated. References J. M. Bass. Three dimensional finite deformation, rolling contact of a hyperelastic cylinder: formulation of the problem and computational results. Computers & Structures, 26: , D. Benson. An efficient accurate simple alemethod for nonlinear finite element programs. Computer Methods in Applied Mechanics and Engineering, 72: , H. Bufler. Zur Theorie der rollenden Reibung. Ingenieur Archiv, 27: , F. W. Carter. On the action of a locomotive driving wheel. Proceedings of the Royal Society of London, A112: , K. Desoyer. Zur rollenden Reibung zwischen Scheiben mit verschiedenen Elastizitätskonstanten. Östereichisches Ingenieur Archiv, 11: , J. Donea. Arbitrary Lagrangian Eulerian finite element models. In T. Belytschko and Hughes T. J. R, editors, Computational Methods for Transient Analysis, pages Elsevier, Amsterdam, L. O. Faria, J. T. Oden, B. Yavari, W. W. Tworzydlo, J. M. Bass, and E. B. Becker. Tire modeling by finite elements. Tire Science & Technology, 20(1):33 56, M. S. Gadala. Recent trends in ALE formulation and its applications in solid mechanics. Comp. Meth. Appl. Mech. Eng., 193: ,

10 G. Hu and P. Wriggers. On the adaptive finite element method of steady state rolling contact for hyperelasticity in finite deformations. Computer Methods in Applied Mechanics and Engineering, 191: , T. J. R. Hughes, W. K. Liu, and T. K. Zimmermann. Lagrangian Eulerian finite element formulation for incompressible viscous flows. Computer Methods in Applied Mechanics and Engineering, 29: , J. J. Kalker. On the rolling contact of two elastic bodies in the presence of dry friction. Phd thesis, Technical University Delft, T. A. Laursen. Computational Contact and Impact Mechanics. Springer, T. A. Laursen and I. Stanciulescu. An algorithm for incorporation of frictional sliding conditions within a steady state rolling framework. Communications in Numerical Methods in Engineering, 22: , R. Lehoucq and D. Sorensenand C. Yang. Arpack users guide: Solution of large scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM series in Software, Environments, and Tools, U. Nackenhorst. The ALE-Formulation of Bodies in Rolling Contact - Theoretical Foundations and Finite Element Approach. Computer Methods in Applied Mechanics and Engineering, 193: , U. Nackenhorst. On the finite element analysis of steady state rolling contact. In M. H. Aliabadi and C. A. Brebbia, editors, Contact Mechanics Computational Techniques, pages Computational Mechanics Publication, Southampton, Boston, J. T. Oden and T. L. Lin. On the general rolling contact problem for finite deformations of a viscoelastic cylinder. Computer Methods in Applied Mechanics and Engineering, 57: , J. Padovan. Finite element analysis of steady and transiently moving/rolling nonlinear viscoelastic structure I. theory. Computers & Structures, 27(2): , J. C. Simo and T. J. R. Hughes. Computational Inelasticity. Springer, New York, P. Le Tallec and C. Rahier. Numerical models of steady state rolling for non linear viscoelastic structures in finite deformations. International Journal for Numerical Methods in Engineering, 37: , Peter Wriggers. Computational Contact Mechanics. John Wiley & Sons Ltd. Chichester, M. Ziefle. Numerische Konzepte zur Behandlung inelastischer Effekte beim reibungsbehafteten Rollkontakt. Phd thesis, Leibniz Universität Hannover, O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method, Volume 2: Solid Mechanics. Butterworth Heimann, Oxford, O. C. Zienkiewicz and J. Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. Part I: the recovery technique. International Journal for Numerical Methods in Engineering, 33: ,