modelagem full car

*Corresponding author.1Formerly with the Ford Motor Co.

Finite Elements in Analysis and Design 31 (1999) 295—315

Vehicle dynamics simulations with coupled multibodyand finite element models

C.W. Mousseau!,*,1,, T.A. Laursen", M. Lidberg#, R.L. Taylor$! Transportation Research Institute, University of Michigan, 2901 Baxter Road, Ann Arbor, MI 48109, USA

" Department of Civil and Environmental Engineering, Duke University, USA# Mechanical Dynamics Inc., USA

$ Department of Civil and Environmental Engineering, University of California, Berkeley, USA

Abstract

With the advent of multibody system simulations (MSS) programs, it has become common practice to use computermodeling to evaluate vehicle dynamics performance. This approach has proved to be very effective for predicting thehandling performance of vehicles; however, it has proved less successful for predicting the vehicle response at frequenciesthat are of interest in ride harshness and durability applications. The lack of correlation between theory and experimentcan be partially traced back to tire models that are inadequate for rough road simulation. This paper presentsa comprehensive vehicle dynamics model for simulating the dynamic response of ground vehicles on rough surfaces. Thisapproach uses a MSS program to simulate the vehicle and a nonlinear FE program for the tires. Parallel processing of thetire models improves the efficiency of the overall simulation. Applications for this technology include vehicle ride andharshness analysis and durability loads simulation. This paper describes the MSS vehicle model, the tire FE model, andthe interface which transfers data between the two simulations. Simulation and experiment results for a single tirewithout a vehicle encountering an obstacle and for a vehicle with four tires driving across a pot hole are presented.Conclusions and opportunities for further research end the paper. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Vehicle dynamics; Tire mechanics; Durability; Friction; Surface contact

1. Introduction

With the advent of multibody system simulations (MSS) programs, it has become commonpractice to use computer modeling to evaluate vehicle dynamics performance. This approach hasproved to be very effective for predicting the handling performance of vehicles; however, it hasproved less successful for predicting the vehicle response at frequencies that are of interest in ride

0168-874X/99/$ — see front matter ( 1999 Elsevier Science B.V. All rights reservedPII: S 0 1 6 8 - 8 7 4 X ( 9 8 ) 0 0 0 7 0 - 5

harshness and durability applications. For instance, a rigid-body vehicle model may proveadequate for predicting the handling response on flat surfaces. However, it may prove inadequatefor simulating a vehicle traveling over irregular terrain, since high-frequency modes may be excitedin the vehicle and tire structure. Also, the vast majority of vehicle handling simulations areconducted on flat surfaces, and the tire is supported at the ground by only a single point. In reality,contact between a tire and ground occurs over a distributed area, requiring a more sophisticatedmodeling approach.

Because of the continuous nature of the deformation and contact, experience has shown thattires are best modeled using the nonlinear finite element (FE) method. MSS programs tends workwell for simulating systems of interconnected rigid elements, but become problematic when contactand large structural deformations occur. On the other hand, it is more difficult to build rigid-bodymodels with nonlinear FE programs than with MSS codes. Libraries of joint constraints, discreteforce elements, and built in subroutines make the MSS approach more straightforward to use. Onesolution to this dilemma is to incorporate both types of methodologies into a single simulation. Inthis approach we simulate the vehicle with MSS program and tires with nonlinear FE programs.This approach was demonstrated with a simple two-degree-of-freedom (DOF) vehicle dynamicsmodel and a single-tire finite element (FE) model [1]. However, the vehicle simulated in that studywas very generic; therefore, the results could not be directly applied to a production engineeringapplication. Also no experimental data were available to validate the combined vehicle and tiresimulation.

This paper presents a comprehensive vehicle dynamics model for simulating the dynamicresponse of ground vehicles on rough surfaces. This approach uses a MSS program to simulate thevehicle and a nonlinear FE program for the tires. Parallel processing of the tire models improvesthe efficiency of the overall simulation. The paper is organized as follows. The multibody vehicledynamics model, tire model, and interface, which controls the program execution and the flow ofdata, are described. Simulation and experiment results for a single tire without a vehicle encounter-ing an obstacle and for a vehicle with four tires driving across a pot hole are presented. Conclusionsand suggestions for future work close the paper.

2. Simulation models

2.1. Vehicle dynamics model

A MSS model of a mid-size automobile with a unitized body, a MacPherson strut [2] frontsuspension, and a multi-link [2] rear suspension was constructed for this study. Fig. 1 illustratesthe front and rear suspensions. This figure shows that the suspension components (e.g., controlarms and spindle) are connected together with a combination of ball joints (i.e., rigid metal to metalconnections), and bushings (i.e., relatively complaint elastomeric elements). Bushings and a sub-frame help to isolate the body from vibrations that originate in the front suspension. The engine,which is connected to the body via elastomeric mounts, is also an important contributor to thedynamic response of the vehicle.

The steps involved in constructing the MSS model include: (1) identifying the vehicle compo-nents that have significant mass (e.g., car body, spindle, and control arms), (2) identifying how the

296 C.¼. Mousseau et al. /Finite Elements in Analysis and Design 31 (1999) 295—315

Fig. 1. (a) Front and (b) rear vehicle suspensions.

components connect together (i.e., system topology), and (3) determining the nature of forcesexerted on the components. We assume that components with significant masses can be adequatelyrepresented by rigid bodies [3,4]. For instance, the vehicle body, sub frame, engine, controls arms,and spindle are modeled as rigid bodies. Note this approach works well for components that are

C.¼. Mousseau et al. /Finite Elements in Analysis and Design 31 (1999) 295—315 297

Fig. 2. Vehicle dynamics model left front suspension topology.

fairly rigid (e.g., control arms), and consequently, have natural frequencies that are far above theexcitation frequencies. However, errors may result if significant structural deformation occurs orcritical modes of vibration become excited [5].

Fig. 2 illustrates the left front suspension portion of the MSS vehicle dynamics model. Eachelement in the figure represents a particular dynamic process and corresponds directly to a suspen-sion element. This figure shows that constraint equations, represented pictorially by joints [3,4],are used to model rigid connections between components. For example, a ball joint is modeled byimposing a constraint requiring two points on two bodies to be coincident during the analysis.Likewise, bushing elements [3] are used to model compliant connections between components.These elements use forces that are solely function of displacement and velocity to link bodies, tomaintain the connection, allowing for deformations to occur during the analysis. Shock absorbers,springs, and jounce stops are represented with forces that depend nonlinearly on the relativedisplacement and velocity between two points [3,4].

The MSS’s described in this paper were generated with the ADAMS [3,4] program. The data filethat describes the simulation model, is constructed from the system topology; mass and inertiaproperties of vehicle body and suspension components; the force-deflection characteristics of thebushings and springs; the force-velocity characteristics of shock absorbers. The complete MSSvehicle model has 32 rigid bodies, 32 joints, 26 bushing elements and 17 spring-damper elements.ADAMS uses this information to numerically assemble and solve the dynamic equations of motionof the system.


Fig. 3. Tire finite element model.

2.2. Tire model

The deformation of the tire structure and contact between the tire and ground is modeled usingthe FE method. The tire FE described below is implemented as user element in FEAP [6,7],a research FE program. Since large deformations occur in the tire structure and discontinuities arepresent in contact model, the resulting equations of motion are nonlinear. Thus the quasi-static anddynamic solutions are generated by the Newton Raphson [6], Newmark [6], and HHT [6]nonlinear, implicit solution procedures in FEAP.

A brief description of the tire FE model is provided below. The tire structure model, illustrated inFig. 3, consists of 50 to 100 wedge-shaped finite elements equally spaced along the circumference of


Fig. 4. Deformation of tire finite element.

the tire. The conductivity of each element is defined by three nodes. The center node defines thespindle coordinates and the other two describe tread coordinates. Each node has 3 degrees offreedom (DOF) corresponding to vertical displacement along the ½-axis, horizontal displacementalong the X-axis, and rotation about the Z-axis. The tire rim rotation is described by the rotationof the spindle node. A geometrically nonlinear beam element that accommodates large deforma-tion [8] models the tread, while the circular arch model approximates the sidewall behavior.Membrane forces from the arch provide both radial and circumferential support continuouslyalong the tread. The circular arch introduces the effect of sidewall bulging into the model toimprove accuracy under large deformations.

Fig. 4, shows a cross section of the tire element during deformation and demonstrates itoperation. Initially, the element is unloaded, and the tread and sidewall are at their relaxedconfigurations. The equilibrium configuration after pressurization depends on these stiffnesses andthe tread width, and the sidewall radius. Application of a vertical load on the tread nodes causes itto move vertically, and the sidewall radius to decrease until a new equilibrium point is reached. The


Fig. 5. Forces acting on ideal sidewall geometry.

figure demonstrates that even though the tire finite element is two dimensional, it does incorporatethe effect of sidewall bulging.

2.2.1. Sidewall deformationThe sidewall provides support for the tread in both the circumferential and radial directions.

One way to visualize how the sidewall works, is to assume that the tread is linked to the rim bya bundle of inextensible fibers, a concept very similar to the cords of a radial ply tire. Circumferen-tial forces are added to model to account for the shearing resistance of the sidewall rubber-cordcomposite. Since the model is 2D, the two inextensible circular membranes act in tandem, so nolateral motion is allowed. For a given cross section, the kinematics of the two sidewall cords aredefined by an upper point which is attached to the rim, and the lower point which is attached to thetread. The upper point displacement is determined by the displacement and rotation of the rimnode. The lower point displacement is determined by the displacement of the two nodes that definethe tread.

We make the following assumptions about the behavior of the sidewall: (1) the elastic character-istics of the arch are assumed to be independent of h; (2) the membrane analogy still applies; (3) theadditional force that arises from elastic deformation of the sidewall can be approximated usinga circular arch (i.e., beam) model; and (4) the direction of the additional force is determined by themembrane model. With these assumptions, the forces acting at the sidewall/tread interface containstwo components: a membrane force and a force resulting from the elastic deformation of the arch.The sidewall force relationship that results is both sufficiently accurate and easy to compute.

Since bending alters the nature of the deformation field, the arch will no longer maintain a truecircular shape. However, the sidewall bending stiffness is relatively low and membrane effectsdominate. The resulting shape is very close to a circular arc and the arch radius is independent ofthe meridian angle. Given that the arch remains circular, from Fig. 5, the meridian angle h can be


determined from the length of the sidewall (l48

) and the sidewall section height (¸) as

l482

cos h#¸

2 Ah!p2B"0. (1)

This relation is needed for computing the sidewall’s contribution to the tangent stiffness matrix.Also of use is the relation between the deformed radius r, h and ¸:

r"¸

2 cos h. (2)

The following expression for vertical projection of the sidewall support force acting on the treadwas developed in [15],

¹5"CA

1r2!

1rr

0Bkr#rp

tD sin h. (3)

Eq. (3) depends on the empirically derived effective sidewall bending stiffness (k4r) and the unde-

formed sidewall radius (r0).

The model as it stands, relies on horizontal projection of the membrane force to provideresistance to shear. The additional resistance to shear that sidewall rubber-cord compositeprovides needs to be accounted for in the model. We use an empirical model to approximate theadditional shear forces that act on the rim and the tread. We assume that a force to be proportionalthe shear angle c, and it acts on the tread and rim in circumferential direction in an equal andopposite manner.

Fx"kcc, (4)

where, kc is the effective sidewall shear stiffness.

2.2.2. Element kinematicsTo derive the FE tangent stiffness matrix and force residual, we need to determine the force

acting on an arbitrary cord. The force per unit length acting on the tread and rim is determined bythe cord length, cord orientation, and inflation pressure. Fig. 6 shows a view in the X—½ plane ofthe element in undeformed position and in the deformed position, (i.e., with the rim rotated byh and with the tread nodes deformed). The direction of the distributed force that acts between therim and the tread is assumed to lie along the direction of the cord (e

y). Since the sidewall must

maintain a circular shape, the membrane radius will change as the two points move closer together,causing the force acting on the tread to also change. This simulates the effect of sidewall bulging.

To determine the cord force, we need to determine the length and direction of a sidewall cord interms of the displacements (u

x, u

y, and uh) and reference coordinates (X, ½). Referring to Fig. 6, the

vector that defines the fiber in the deformed condition is given by

P"{@5{

"P4{#P

"{@4{!P

t{"(X

4#u

4x)i#(½

4#u

4y) j#r

4sin(/!/

0#h)i

!r4sin(/!/

0#h) j!(X

5#u

5x)i!(½

5#u

5y) j. (5)

The horizontal projection of the sidewall cord is given by

¸"DDP"{@5{

DD"JP"{@5{

· P"{@5{

(6)


Fig. 6. Deformation of a single tire finite element.

The unit vector ey

is aligned along the deformed sidewall cord.

ey"

P"{@5{

DDP"{@5{

DD. (7)

The unit vector ex

lies in the X—½ plane and is normal to the deformed sidewall fiber.

ex"k]e

y. (8)

We are also interested in determining the forces that the cord act on the tread and react at thespindle point. Recall that the cord illustrated in Fig. 6 is actually a projection of the circular arch inthe X, ½ plane. The force acting between the sidewall and tread consists of a membrane componentand an elastic component due bending that acts along the e

ydirection, and a shear component that

acts along the ex

direction. Eqs. (3) and (4) are used to express the force in vector form.

T5"Cp5r#k

rA1r2

!

1rr

0BD sin he

y#kcce

x. (9)

Referring again to Fig. 6, the forces and moment that act at the spindle point S are given by

T4"!Cp5r#k

rA1r2!

1rr

0BD sin he

y#kcce

x, (10)

M4"!Cp5r#k

rA1r2!

1rr

0BD sin hp

4{@"{]e

y!jkc p

4{@"{]e

x. (11)

Note that we are assuming that the sidewall fiber cannot support a moment about the Z-axis.


2.2.3. Sidewall element internal forces and tangent matrixOur intention is to use implicit methods to solve for the displacements during both the

quasi-static and dynamic analysis. This requires the formation of the element tangent stiffnessmatrix and force residual. The tangent stiffness matrices for the implicit solution procedures arefound by constructing and linearizing the weak form [6,7]. At this point it is convenient to useEqs. (9)—(11) to form the following residual vector

u(u)"MT5· i T

5· j 0 T

4· i T

4· j M

4· kN5. (12)

The weak form for the distributed force is formed by integrating over the element interval, theproduct of the residual and a vector of weighting functions (w) [6].

G48

(u)"Ph

0

u(u) · w dX. (13)

The weak form is linearized about a displacement u and the finite element matrices are formed withthe Galerkin method [7]. In this step, the tread, spindle nodal point positions, and nodaldisplacements are interpolated isoparametricly [7]. The element tangent stiffness matrix is foundby substituting in the interpolated variables and the stiffness matrix is obtained by factoring out theweighting coefficients and nodal displacements. This process is described in more detail in [9].

2.2.4. Material dampingThe primary source of energy dissipation in a rolling tire is hysteresis in the tread and the

sidewall. However, hysteresis can be difficult to implement and significantly increase the amount ofcomputation required in a finite element simulation. Therefore, the energy dissipation is calculatedin the model with equivalent viscous damping. The damping coefficient is obtained by equating theenergy dissipated during a sinusoidal deformation of a material specimen to the energy dissipatedby a viscous damper, as described in [10]. There are three sources of damping in the simulationmodel, all of which use a linear damping law. In the tread, the damping moment is assumed to beproportional to the rate of change of the curvature. We assume that the sidewall damping acts inthe radial and circumferential directions.

2.2.5. Contact modelThe mathematical constraints involved in defining the road/tire mechanical interaction can be

generally understood by referring to Fig. 7. In the figure, X refers to the reference position of a nodeon the periphery of the tire, and the contact kinematic quantities associated with X serve toquantify its motion relative to the opposing roadway surface. The first of these is the gap function g,which can be written as:

g(X)"([X#u(X)]![YM #u(YM )]) ) l(YM ), (14)

where YM is the closest point projection of X on the surface and l is the surface normal at this location.Impenetrability of tire and roadway is then mathematically expressed as

g(X))0 (15)

for all nodes X on the periphery of the tire. It is further assumed that all mechanical interactionnormal to the interface is compressive, so that no adhesive forces form between the tire and


Fig. 7. Penetration calculation for contact model.

roadway. In view of this fact, the relationship between the contact pressure tN

(assumed positive incompression) and the gap function g is expressed via the following classical Kuhn—Tuckerconditions:

g(X))0, tN(X)*0, t

N(X )g(X)"0. (16)

Expressed in this form, we can think of tN

as being the Lagrange multiplier needed to enforceinequality condition (15) for node X.

In the model we propose, frictional interaction is described using a Coulomb law. For a nodeundergoing persistent sliding, where gR "0, a frame indifferent measure of relative sliding is given bya projection of the relative material velocity into the plane tangent to the roadway surface

VT(X )"(I!l?l) (V(X)!V(YM )). (17)

A Coulomb description governing the tangential stress tT

is given in terms of a slip function U via

U(tT, t

N) :"DDt

TDD!kt

N)0, V

T(X)"1

tT

DDtTDD, 1*0, 1U"0. (18)

Eq. (18) imply a law of constrained evolution for the frictional stress tT, in a manner largely

analogous to the way rate independent plasticity equations are often formulated. The familiarphysical assumptions of Coulomb are readily confirmed: the magnitude of frictional stress DDt

TDD may

not exceed the coefficient of friction k times the pressure tN, all sliding occurs in a direction

opposing the applied frictional stress, and no sliding occurs unless f'0, which is only allowed ifDDt

TDD is exactly equal to kt

N.

With respect to the global finite element equations, contact effects appear as

Md$ (t)#F*(d)#F#"0, (19)

where the first term represents the dynamic effects, F* represents the internal forces generated bytire deformation as discussed above, and F# is the contact force vector, assembled from the contacttractions satisfying conditions (16) and (18). Of course, F# is an extremely nonlinear and discontinu-ous function of the tire deformation. In particular, contact conditions are widely recognized to be


subject to ill-conditioning if penalty methods are used. Accordingly, we use the augmentedLagrangian method as documented in [11] to enforce the aforementioned constraints.

2.2.6. Tire model parametersThe model parameters are organized into a group related to tire geometry and another

associated with elastic properties. The parameters that describe the tire geometry, including, theuninflated tire diameter, effective rim diameter, tread width, and sidewall length are measureddirectly from the tire cross section. Other parameters such as inflation pressure and surface frictionare specified by the user. Since the circular membrane model only approximates the actual tirecross-sectional shape, the user must exercise some judgment when interpreting the sidewall lengthand tread width parameters. The axial stiffness is best measured directly with a uniaxial test. Thebending stiffness can be estimated either with a three point bending test on a specimen cut from thetire, or from load deflection tests on a flat surface and knife edge with the complete tire.

To enable the model to better simulate the response of the tire during large deformations, anempirical term that approximates bending in the sidewall is added. Since the elastic sidewallbending stiffness (k

48) parameter is difficult to measure directly, an indirect approach was used. For

our studies, it was determined by matching simulations to force—deflection experimental data ata single inflation pressure. These simulations took less than a minute to run, and a suitable valuewas determined after only five simulations. The parameter is independent of inflation pressure, andthus data are required only for a single inflation pressure to characterize the model.

2.3. Interfacing the nonlinear finite element multibody dynamics simulations

To conduct a full vehicle simulation, the MSS vehicle dynamics model requires forces for each ofthe four spindle locations and each FE tire model requires the corresponding spindle displace-ments. These data needs to be passed between the simulation during every simulation time step.The interface needs to perform this task both reliably and efficiently.

2.3.1. MSS simulation data requirementsUsing information from a data file, the ADAMS program assembles and solves the Eu-

ler—Lagrange dynamic equations in first order form. A predictor—corrector algorithm with step sizecontrol integrates the resulting set of differential algebraic equations (DAEs). A backward differ-ence formula (BDF) approximates the solution over an interval and transforms the DAEs into a setof coupled implicit algebraic equations [12,13]. The following BDF formula is used:

qi(tl)"hb

0qRi(tl~j

)#k+j/1

(ajqi(tl~j

)#hbjqRi(tl~j

)). (20)

The ith generalized coordinate (q) at time step l is estimated from a linear combination of thegeneralized coordinate at the previous kth time steps and its derivative. Where, a and b areweighting coefficients and h is the time step size. The predictor provides an initial estimate of thesolution, while the corrector uses the BDF and a Newton-Raphson like algorithm to iterativelysolve for the final value of q

i(tl) [13]. At each iteration of the solver, this procedure requires

evaluation of the forces and formation of the Jacobian matrix (i.e., the partial derivative of theforces with respect to the generalized coordinates).


Fig. 8. Simulation model interface.

2.3.2. Interface operationFig. 8 illustrates the major computations that take place during the simulation and the data flow

between processes. During each iteration of the MSS corrector, the vehicle dynamics modelprovides the spindle position to tire model simulation which implicitly solves the nonlinear tire FEproblem for the spindle force. Since both simulations are using implicit methods to solve theirrespective problems, many iterations of nonlinear FE solver are required to generate a simulation.For example, a vehicle dynamics simulation may last 10 or more second, and the each time step isabout 1 ms, the simulation will require about 100 000 iterations of the FE solver.

Data transfer between processes is accomplished with library calls to the transport layerinterface (TLI) [14]. This software allows for data to be transferred reliably between processes thatmay exist on a single computer, or on a group of networked computers. At the start and end of eachiteration of the MSS corrector, only spindle forces and displacements pass back and forth betweenthe MSS and the FE programs. When compared to the simulation as a whole, this is a very smallamount of data. With the exception of this data, the calculations in a given tire model areindependent of the other simulations. Thus, two or more tire simulations can execute concurrently,thereby, further reducing the simulation run time.


2.3.3. Force Jacobian calculationThe convergence of the ADAMS solver can accelerated if, in addition to the spindle force, the

Jacobian matrix is provided. The Jacobian matrix can be calculated either directly from the FEprogram, or by numerically perturbing the spindle position. The former approach generates theJacobian matrix as a by product of the tangent stiffness matrix calculation, the latter requiresadditional evaluations of FE program. Since both evaluating the tire finite element model andmoving data between simulations is very time consuming, and can lead to longer simulation runtimes, the direct method is preferred. The other benefit of the direct method is that the resultingmatrix is consistent [6]. This property increases the rate of convergence; therefore, furtherimproves the efficiently of the simulation.

The procedure we used to extract the spindle force Jacobian from FE tangent matrix is describedbelow. For every iteration of the ADAMS corrector we need to calculate the spindle force, whichdepends on the acceleration of tread mass and damping of the tire structure. The Jacobian matrixfound by differentiating the spindle force with respect to the relevant spindle displacements:

Fi,j"

LFi(u, uR , u( )Lu

j

. (21)

Note that we are dealing with a force that is a function of displacement, velocity, and accelerationof all the nodes in the tire structure. Using the chain rule, we further carry out the differentiation:

Fi, j

"

LFi

Luj

#

LFi

LuRj

LuRj

Luj

#

LFi

Lu(j

Lu(j

Luj

. (22)

Using Newmark predictors [6] to extrapolate the velocity and acceleration in the above equation,the following expression results:

Fi, j

"

LFi

Luj

#AchbB

LFi

LuRj

#A1

h2bBLF

iLu(

j

. (23)

Defining the following quantities:

Kij"

LFi

Luj

, Cij"

LFi

LuRj

and Mij"

LFi

Lu(j

,

we can restate Eq. (23) as

Fi,j"K

ij#A

chbBCij

#A1

h2bBMij, (24)

where h is the time step size and b and c are the Newmark parameters. In Eq. (24) Kij, C

ij, and M

ij,

are loosely described, respectively, as the “effective” stiffness, damping, and mass matrices. Eq. (24)also shows that the force Jacobian (F

i,j) depends on the duration over which the perturbation

occurs. The matrix on the right-hand side of Eq. (24) is identical in form to the stiffness matrix inthe Newmark integration algorithm [6].

To calculate the spindle force Jacobian we sub-divide the tire model FE tangent matrix and forceresidual into parts containing the spindle node (i.e., the subscript s) and not containing the spindle


node (i.e., the subscript u).

CK66

K64

K46

K44D G

Du6

DusH"G

R6

R4H . (25)

The tangent matrix can be partially factored into

CK66

K64

K46

K44D"C

L66

0

L46

ID CD

660

0 H44D C

U66

U64

0 I D . (26)

The matrix H44, the reduced tangent matrix for the spindle node, is found by substitution to be:

H44"K

44!L

46D

66U

64. (27)

The computation of the spindle force that is exported to the MSS program is shown below:

f4"R

4!L

46L~166

R6, (28)

where the inverse is done as a forward solution of following the equations.

L66

w6"R

6(29)

Thus,

f4"R

4!L

46w

6. (30)

Eqs. (27)—(30) were implemented into the FEAP program as a macro command which is calledwhen the spindle force is required by the vehicle MSS.

3. Results

Simulation results from both the standalone tire model and the combined vehicle and tire modelsare presented below. Experimental results are also presented to demonstrate the accuracy of thesimulation models.

3.1. Tire model alone

To demonstrate the accuracy of the tire model for predicting impact forces, simulation results arecompared to experimental data. The tire FE simulations were conducted with the FEAP programusing the Newmark dynamic solution procedure. The FE model consists 100 tire FE s equallyspaced along the tire circumference. Contact between the tire and ground was modeled with 2Dsurface contact elements. Fig. 9 illustrates the simulation model at the start of the dynamicanalysis; in the inflated condition loaded against a simulated rigid drum. The rigid contact surfacehas a 1700 mm diameter with a step that is 25 mm high by 660 mm long, and the “x” in the center ofdrum defines the position of a node to which the rigid surface is constrained. A coefficient of frictionof 0.3 was defined for the surface.

The total duration of the simulation was 8 s and it consisted of two parts. The first part wasquasi-static and it lasted 2 s; the latter part was dynamic. During the first part, the tire was inflated


Fig. 9. Tire model at start of simulation.

to 0.208 MPa and deformed with a load of 3800 N against the surface. During the second part ofthe simulation, the drum was spun from rest to the final angular velocity over 4 s. The combinationof the surface movement and friction caused the tire to start rolling. After 4 seconds the drumvelocity was then held constant and the tire was allowed to reach a steady-state condition before itimpacted the obstacle. The simulation execution time was 1.3 CPU hours on an HP 735workstation.

The experiment was conducted on a 1700 mm diameter road wheel with a 25 mm high by660 mm long obstacle clamped to the outside surface. The tire was mounted to a dead axle whichwas instrumented to measure longitudinal and vertical forces. The drum was spun up to speed withtire rolling against the road wheel surface. A data acquisition system captured the spindle forceresponse when the tire impacted the obstacle. More details of the tire impact test are describedin [15].

Fig. 10 shows the measured and predicted hub forces for the tire rolling over the obstacle at51.3 KPH. The measured and simulated longitudinal force responses agree reasonably well.However, the model underestimates the peak values slightly and predicts a slightly higherfrequency of oscillation at 44 Hz, compared to 35 Hz observed in the test. The 44 Hz oscillationcorresponds to the first natural frequency of the tire model when loaded against a rigid surface[15]. The reason for differences between the model and test frequencies are described in [15] andstem largely from the simplifying assumption used to model the tread. Very similar results wereobtained with the Newmark and HHT integration algorithms.


Fig. 10. Simulated and measured (a) longitudinal and (b) vertical spindle forces for the 145 SR-12 tire impacting a step ona 25 mm high step rotating drum at 51.3 KPH (31.9 MPH).

The vertical spindle force response shows very good agreement between the experimental resultsand simulation. The simulated response shows a 89 Hz oscillation when the tire reaches the top ofthe step, and an 84 Hz oscillation when the tire leaves the step. These oscillations are attributed tothe first vertical mode of the tire that varies slightly with tire deflection [15]. The measuredresponse shows a 88 Hz oscillation when the tire reaches the top of the step, and an 85 Hzoscillation when the tire leaves the step. This exercise was repeated at different speeds and the peakvalues predicted by the simulation show good agreement with the experimental results [1].

3.2. Combined tire and vehicle dynamics models

To explore accuracy of the combined tire and vehicle simulation, a limited validation study wascarried out. An experiment was conducted on a vehicle instrumented with wheel force transducers


Fig. 11. Measured and simulation left front (a) longitudinal and (b) vertical spindle forces for an automobile traveling48 KPH over a 0.76 m long by 0.1 deep chuckhole.

that measured the longitudinal, lateral, and vertical spindle forces. The experiment was conductedin the following manner. The vehicle was initially driven over a flat surface at a constant speed of48 KPH where it then encountered a 760 mm long by 100 mm deep obstacle on the left side of thevehicle and a flat surface on the right side. During the impact with the obstacle, time history datafrom all the wheel transducers were digitized and stored for later analysis.

The analytical results were generated with the combined tire and vehicle simulation model. Thesimulation was conducted with the ground moving underneath the vehicle at a prescribed velocity,the tires rolling freely, and no tractive torques were applied to the tires. The simulation wasconducted in the following manner. During the first second, the tire was quasi-statically inflated to0.243 MPa. During the next second, the vehicle and tires were allowed to settle out dynamically toan equilibrium position. Over the following 5 s, the ground was accelerated at constant rate from0 to 48 KPH. For the last two seconds of the simulation, the ground was moved at constant speedof 48 KPH, during which the vehicle encountered the obstacle. The simulation lasted a total of 9 sand 13.5 CPU h were required to generate the results on a two processor, shared memory SGIworkstation.

Fig. 11 shows the experimental and simulated spindle forces for left front wheel during impactwith the obstacle. This figure shows that in the vertical direction, the simulation agrees well withthe test results. Peak values are within 1% of one another and many of the oscillations exhibited bythe simulation are also present in experiment. The magnitude of the FFT of vertical force in Fig. 13shows that the frequency of oscillation of the first vertical tire mode predicted by the simulationagrees very well with the test results. It also shows that the vehicle dynamics model also overpredicted the frequency of hop mode at 11.5 Hz]2.7 Hz. However, Fig. 11 also shows that thelongitudinal forces predicted by the simulation over estimates the magnitude of the longitudinalforce. It also shows that the simulation under estimates the duration of the force during the initialimpact.


Fig. 12. Measured and simulation left rear (a) longitudinal and (b) vertical spindle forces for an automobile traveling48 KPH over a 0.76 m long by 0.1 deep chuckhole.

Fig. 12 shows the experimental and simulated longitudinal and vertical spindle forces for leftrear wheel during the impact with the obstacle. This figure also shows good agreement between thesimulation and test in the vertical direction. However, the simulation underestimates the magni-tude of the longitudinal force by almost 50%. Although, the duration time of the initial impactforce predicted by the simulation agrees well with the experimental results.


Fig. 13. FFT of the left front (a) vertical and (b) longitudinal spindle forces.

4. Conclusions

An approach for coupling a vehicle dynamics model to tire model to from a comprehensivesimulation was presented. This approach uses a MSS program to simulate the vehicle anda nonlinear FE program for tires. The efficiency of the overall simulation is improved by usinga special purpose FE to model the tire structure and by parallel processing the nonlinear FEprograms.

The results of this study indicates that this approach is a viable technique for simulating vehicledynamic response on rough roads. The tire model alone agrees well with experimental data andexecutes in a reasonable amount of time. Although, the combined tire and vehicle simulationsexecute much more slowly than either simulation alone. direct computation of the force Jacobiancan improve the computational efficiency. For the case of the a vehicle driving over an obstacle, thecombined tire and vehicle simulation was able to predict the vertical spindle forces very accurately.However, the longitudinal spindle forces predictions were less accurate. The sources of thedisagreement between simulation and experiment are currently being investigated.

Future work will include addressing the cause of the discrepancies in longitudinal forcepredictions, increasing the efficiency of the interface between, and conducting a more comprehens-ive validation study. Work is already underway on a 3D version of the tire model [10,16] that isalso capable of predicting lateral loads.

Acknowledgements

The authors wish to acknowledge the support of the Research Staff, Light Truck, and CarChassis Engineering activities of Ford Motor Co. Also we wish to acknowledge the encouragement


and contributions of Jim Avouris, Vikas Chawla, Sam Clark, Al Conle, Ian Darnell, Greg Hulbert,John Hogan, Bill Oates, Steve Riley, and Natarajan Saravanan. Also, Tod Laursen was supportedin part through NSF CAREER award number CMS-9703356; this support is gratefully acknow-ledged.

References

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deformation frictional contact problems, Int. J. Numer. Meth. Eng. 36 3451—3485.[12] C.W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs,

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[14] M. Padovano, Networking Applications on UNIX System V Release 4, Prentice-Hall, Englewood Cliffs, NJ, 1993.[15] C.W., Mousseau, A numerical and experimental investigation into the force response of a tire impacting a large

obstacle, Ph.D. Dissertation, University of Michigan, 1994.[16] J.P. Den Hartog, Mechanical Vibrations, McGraw-Hill, New York, 1934.


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