Complex Tire-Ground Interaction Simulation: Recent Developments Of An Advanced Shell Theory Based Tire Model

Transcription

1 . ozdog and W. W. Olson Complex Tire-Grond Interaction Simlation: ecent eelopments Of n danced Shell Theory ased Tire odel EFEECE: ozdog. and Olson W. W. Complex Tire-Grond Interaction Simlation: ecent eelopments Of n danced Shell Theory ased Tire odel sbmitted for presentation at the 5 Tire Society meeting and for consideration for pblication in the jornal Tire Science and Technology. STCT: The formlation of an adanced shell theory based tire model (STT) proides the fondation for tire-grond interaction analysis. icro-mechanics composite laminate and shell theories are integrated in a consistent tire model soled by a dal symbolic-nmeric algorithm. The complex tire-grond interaction is careflly examined by a sensitie nmerical procedre and adaptie iteration step. eformations and stress-strains of the entire tire strctre are described as fnction of applied forces at the contact interface. The accracy of soltions is fond to be highly depended on mesh size bt consistent oer the iterations. The elaborate theoretical model is efficiently balanced by the symbolic comptation redction techniqe and the system nmerical optimization. Finite difference method is sed extensiely. Simlation proides a stepby-step insight into force generation and macroscopic behaior of the tire. The code deeloped proides an alternatie to existing tire model implementations in ehicle dynamics simlation softare. Table of omenclatre p p p z srface load components displacements components Lamé parameters ij ij ij stiffness coefficients K K K - change of cratre force resltants - moment resltants - shearing forces radii of cratre ε ε γ - strain components z crilinear coordinate system σ σ τ stress components

2 Introdction The complexity of anisotropic shells of arbitrary shapes ith non-symmetrical loads has generated an intense field of research for many decades. lthogh an exact soltion for the general case does not exist arios mathematical models hae broght their contribtion to a general formlation of adanced tire models. This paper contines the deelopment of an adanced shell theory based tire model []. Tire odel The general system of eqations defining the tire behaior is composed of compatibility constittie and eqilibrim eqations. y redcing the three dimensional carcass of the tire to to dimensions modeling all loadings at middle srface applying bondary conditions and performing ftre simplifications the system can be ritten as [345]: Compatibility eqations: K K K γ ε ε () Constittie eqations: γ ε ε K K K ()

3 Eqilibrim eqations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) p p p z (3) ondary conditions: ) ( ) ( ) ( (4) There are to techniqes to sole the general system of eqations in terms of the displacements or in terms of the internal force and moment resltants. oth techniqes reqire a redction of the nmber of eqations by sbstittions and by increasing the order of the differential eqations. The first method is preferred becase the final system can be expressed as a system of three goerning partial differential eqations for displacements and. The first sbstittion replaces the strains in constittie eqations () ith the compatibility eqations (). The force and moment resltants are no expressed as fnctions of displacements and their deriaties. {6} {6} } { ) ( j i F for f F ij ij ij (5) The inter-ply shear forces are determined from the eqilibrim eqations (3):

4 ( ) ( ) ( ) ( ) (6) In the final sbstittion the force and moment resltants and the inter-ply shear forces are replaced in the first three eqilibrim eqations (3). The reslting forth-order differential system of eqations relates the applied forces to the displacements. Together ith the bondary conditions the system is flly determined and finite difference method is applied. s m Eqation p s ij ij ij n for m { 3 } { 34 } { 6} { 6} { } t s i j n { } p p z t (7) The finite difference method is applied sing the central difference scheme. The comptations ere performed sing athematica T Softare. lgorithm For each iteration step corresponding to an increment in the applied internal and external load the radii of the cratre and the deformed profile are compted. The oerall algorithm has the folloing steps: Inpt profile Comptation of shells position Find fitting fnctions for profile Comptation of cratres radii of cratre and Lamé parameters Inpt stiffness matrix for composite strctre Symbolic comptation and redction techniqe for system of eqations Generation of complete system of eqations and F scheme implementation Inpt C s and applied force matrix Sole system and otpt and for all shell elements Compte all other 5 ariables Compte deformed profile

5 The expression for the deformed shape compted after each iteration: x z k k x z k k k y k cos( )sin( ) k y k k cos( )cos( ) k sin( ) k cos( ) k cos( ) sin( ) k k sin( )sin( ) sin( )cos( ) (8) Figre : isplacement ectors and eslts There are to effectie ays to determine the tire behaior one is to apply external loads on the srface of the tire and determine the resltant deformations and the other one is to se contact constrains for displacement and ealate the force reactions. The pressre distribtion in the contact area is strongly nonlinear haing small ales in the center of area and high ales close to the limits of the contact area. n experimental set of data is reqired for the loads inpt case hile the second method is easier to apply and more efficient for this tire model. The contact constrains are a ne set of eqations that are added to the general system of eqations (7) by defining a ale of displacement in Z direction. Z cos( )sin( ) cos( ) sin( )sin( ) (9) def k k k here Z def is the iteration displacement and k k and k are the corresponding deformations in the and z direction.

6 Figre : Contact constraint Z displ The nmber of contact constraint eqations introdced is eqal to the nmber of shells in contact ith the grond. dditionally an eqal nmber of ariables representing the normal load ector force on the shell elements are added to the matrix of ariables. The mesh size for this analysis is chosen to shell elements and the system of eqations is composed of 439 eqations. The additional contact constraints ill increase the nmber of eqations to an aerage of 44. Figre 3: Tire esh of 464 elements oth the mesh size and the iteration step ere chosen sch that the accracy of soltions is balanced by a reasonable comptational time. The analysis has been performed on a Windos based machine ith a Pentim 4.4 GHz processor. The aerage time for a complete rn of one iteration step ith athematica T softare as abot 6 min and 38 alid iterations ere reqired to load the tire on a flat srface. The algorithm for this simlation takes into consideration an important aspect of loading of the tire on a srface; in order to define a alid constraint the resltant applied load has

7 to act as an external force and the resltant forces need to hae the same sign [6]. If the constraints are not chosen properly the iteration step has to be repeated at a different Z def ale. The total nmber of iterations performed as 6 ith only 38 iterations alid. Figre 4 represents the displacements in the direction. The ales are both positie and negatie and sho symmetry ith respect to the cron of the tire. Figre 4: isplacement for ertical loading isplacement in direction has small ales ith maximm magnitde of. [in] as presented in Figre 5. The shape of the displacement plot and its ales are as expected. t the contact region the displacements hae higher ales and sloly decay as the crosssections are located at significant distance from the area. The maximm ales compted for both ends of the contact cross section are the reslt of local rotations. The main reason for haing rotations of the elements is the anisotropic properties of the strctre. The longitdinal displacement has small inflence on the cross-section profile.

8 Figre 5: isplacement for ertical loading The normal displacement to the srface of the tire is represented in Figre 6. eformations are concentrated arond the center of the contact area. aximm ale recorded along the profile is at the cron. The positie ales beteen node 4-9 and 5-57 are a reslt of the expansion of the sideall nder deformation. Figre 6: isplacement for ertical loading Using Eqation (8) the deformed profile is compted after iteration. Figre 7 shos seeral iterations performed on the tire and gie a combined ie of the profile after inflation and the profile after ertical load is applied. The graphs represents iteration nmber and the last iteration 38. While the cron follos the constraints the

9 sideall shos more deformations in the sholder region as expected. The sensitiity of the code to iteration step became significant after the th iteration and a loer z deformation as reqired. Figre 7: Cross-section deformed profile for ertical loading The folloing table presents the loaded radis the ertical deformation and the maximm pressre applied on the shell elements in contact ith the grond.

10 Iteration z displacement ax p z [psi] The actal load distribtion in the contact area is presented in Figre 8. s expected the distribtion is ariable ith increased ales toards the end limit of the area. ring the iterations and progressie loading the ales of external load hae a nonlinear distribtion. lthogh the resoltion of the otpt data is limited by the nmber of nodes chosen the shape of the graphs are close to data presented in experimental reports [7]. Figre 8: Pressre distribtion in contact area

11 Conclsions The recent deelopments of an adanced shell theory based tire model proide a frameork for comptation and simlation of tire-road interaction. icro-mechanics composite laminate theory and shell theory are integrated in a consistent tire model soled by dal symbolic-nmeric algorithm. eformations strains and moment resltants of the entire strctre are described as fnction of strctral material properties ndeformed tire profile and applied forces. The analysis as performed by applying local loads on the contact region beteen the tire and grond. The main adantage of performing a fll analysis of the tire is that beside deformations the internal forces and moments are determined more accrately. The only significant limitation in the analysis as the comptational poer aailable at the time of the simlation. The code deeloped proides a conenient ay to adjst the eqations generation modle based on the mesh size. Oerall the tire model adance the theoretical and comptational deelopments to a ne leel and contine the research of analytical tire models in a continm groing field of interest. From a design point of ie the compter code is bild ith seeral modles ell defined representing the compatibility the constittie and the eqilibrim eqations modles finite difference scheme implementation generation of eqations soler and post-processing data modle. One of the main adantages reslting from this tire model is the possibility of cstomizing for a ide ariety of tire constrction designs. oth radial and bias-ply types of tires are spported and detailed strctral parameters are easily integrable in the tire model. eferences [] ozdog. Olson W. W. n danced Shell Theory ased Tire odel sbmitted for pblication in Tire Science and Technology Jornal 4. [] reer H. K. Stresses and eformations in lti-ply ircraft Tires Sbject to Inflation Pressre Loading ir Force Flight ynamics Laboratory Wright-Patterson ir Force ase Ohio 97. [3] Vasilie V. V. orozo E. V. echanics and nalysis of Composite aterials Elseier etherlands. [4] Ventzel E. Krathammer T. Thin Plates and Shells: Theory nalysis and pplications arcel ekker e York. [5] Walter J.. Cord einforced bber Chapter 3 of echanics of Pnematic Tires U.S. epartment of Transportation ational Highay Traffic Safety dministration.c. 98.

12 [6] X Y. Jia L. Zhang J. odeling Tire/oad Contact Using Pieceise itz Procedre Jornal of Terramechanics 4(5) [7] El-Gindy. Leis H. eelopment of a Tire/Paement Contact-Stress odel ased on rtificial eral etorks Pb. FHW Federal Highay dministration.