Static Tyre-Road Interaction Modelling

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Meccanica 32: 473479, 1997. 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Static Tyre/Road Interaction Modelling

A.M. BURKE and O.A. OLATUNBOSUN

University of Birmingham, Automotive Engineering Centre, School of Manufacturing and MechanicalEngineering; Edgbaston, Birmingham, B15 2TT, England

(Received: 15 May 1997)

Abstract. The understanding, modelling and predicting of tyre behavioural characteristics, for both static anddynamic applications, requires the consideration of many detailed aspects of this seemingly simple component.In order to investigate the problem more fully, computer analysis techniques are becoming more common thanthe simplifications associated with analytical methods. The finite element method is one such technique thatenables engineers to examine tyre behaviour comprehensively and to predict tyre performance at the designstage. In this paper, attention is drawn to the problem of tyre/road interaction modelling. A purely theoretical

approach is presented which gives the analyst more flexibility in changing parameters such as inflation pressure,hub load, and material properties than previously developed experimental/numerical techniques. A gap elementformulation is used to model the interaction so that contact patch area, shape and deflection are automaticallyaccounted for under a given load and inflation pressure. Modelling and experimental results are also presentedto illustrate the accuracy of the technique.

Sommario. La comprensione, la simulazione e la valutazione del comportamento di uno pneumatico, sia nelcaso statico che dinamico, richiede di tenere conto di numerosi aspetti che riguardano questo componenteapparentemente semplice. Allo scopo di studiare piu approfonditamente il problema stanno diventando semprepiu diffuse tecniche di analisi computazionale, in luogo di metodi analitici semplificati. Il metodo degli elementifiniti rappresenta una di queste tecniche, la quale permette allingegnere di esaminare piu approfonditamente ilcomportamento di uno pneumatico nonche di valutarne le prestazioni sin dalla fase della progettazione. Questo

articolo e focalizzato sullo studio della interazione pneumatico/strada. Viene presentato un approccio puramenteteorico che, rispetto ad altre tecniche sperimentali/numeriche sviluppate in precedenza, fornisce al progettistamaggiore flessibilita nella variazione di alcuni parametri, quali la pressione di gonfiaggio, il carico sul mozzo,e le proprieta dei materiali. Linterazione pneumatico/strada viene modellata mediante una formulazione aglielementi interstiziali (gap elements) in modo tale da tenere automaticamente conto della superficie di contatto,della forma e della flessione dello pneumatico per un assegnato carico e per una data pressione di gonfiaggio. Alloscopo di illustrare la precisione del modello vengono anche presenati i risultati sperimentali e teorici ottenuti.

Key words: Tyre/road interaction, Finite element method, Tyre modelling, Automotive applications.

Nomenclature

Fx axial compressive force,Fy friction force lateral direction 1,

Fz friction force lateral direction 2,

u , v , w displacements in the element co-ordinate system,

Kb arbitrary open gap stiffness,

Ka penalty value for closed gap stiffness,

s static coefficient of friction,

Kt gap transverse shear stiffness,

k kinetic coefficient of friction.

1. Introduction

The normal operating conditions of a tyre require it to be in contact with the ground plane. This

contact in conjunction with the vehicle mass causes the tyre to deflect, and due to the flexible


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474 A.M. Burke and O.A. Olatunbosun

nature of the tyre, this deflection is quite considerable. Predictions by Szilard [1] suggest that the

resulting deflection of a tyre under normal loading conditions does not follow linear mechanical

laws, and therefore, for modelling applications, non-linear techniques have to be employed. The

amount of deflection is obviously related to the load; the greater the load the greater will be the

deflection. The contact area itself is also load related. An imprint of the contact patch at various

loads would reveal a change in area and possibly a change in shape.

Researchers in the past [24] developing finite element tyre models have approached thecontact problem from an empirical standpoint, by deriving force and displacement data exper-

imentally, and then applying the information to the model. However, this approach, although

successful, does have significant disadvantages:

1. Due to the requirement of experimental testing of the finished tyre, purely theoretical design

stage modelling is not possible.

2. Experimental/numerical techniques do not allow the analyst to monitor naturally occurring

load/deflection behaviour, preventing static stiffness evaluation.

3. Experimental load/deflection data are derived at a constant inflation pressure loading. There-

fore, if further investigations are to be carried out involving other inflation pressures, addi-tional experimental testing is required.

4. Experimental testing often only reveals the resulting deflection for given loading conditions.

The shape and area of the contact patch then has to be derived by the analyst using trial and

error solution methods of the finite element model.

In order to overcome the disadvantages associated with previous techniques, this paper

describes a purely theoretical approach that allows the analyst to change hub load and infla-

tion pressure whenever required and the contact patch deflection, area and shape are accounted

for automatically. In the analysis MSC/NASTRAN V67 finite element solver was used to model

the contact, by using a gap element formulation, as well as the geometric non-linear effects that

occur due to large displacements.

2. Theoretical Background

MSC/NASTRAN development of the gap, or contact element has allowed the possibility of

modelling a wide variety of contact problems. It simulates a unidirectional point-to-point con-

tact between two degrees of freedom by adding a large value of stiffness so that the two have

approximately the same displacements. It does this by using the penalty method which simu-

lates the rigidity between the two degrees of freedom. Penalty values are introduced to avoid

penetration and to enforce the sticking condition (static friction) between two contact points.

The gap element changes its status when load is applied. Consider the internal forces of

a gap element with an isotropic friction. Force components in the gap element are the axial

compressive force and the friction forces in lateral directions in terms of displacements in the

element co-ordinate system. These internal forces can be computed based on the gap status as

follows:

When the gap is open (no contact, no lateral stiffness)

Fx = Kbu 0 and Fy = Fz = 0. (1)

When the gap is closed and sliding (no friction)

Fx = Kau > 0 and Fy = Fz = 0. (2)


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Static Tyre/Road Interaction Modelling 475

When the gap is closed and sticking (static friction)

F2y + F2z (sFx )2, (3)

where Fx = Kau, Fy = Ktv and Fz = Ktw. It is noted that the sticking condition is dictatedby the static coefficient of friction and the transverse shear stiffness.

When the gap is closed and slipping (kinetic friction)

F2y + F2z > (kFx )2, (4)

where

Fx = Kau, Fy =vkFxv2 + w2

and Fz =wkFxv2 + w2

with the friction dictated by the kinetic coefficient of friction.

The element stiffness matrix can be formed as

[K] =

K KK K

with K = {F}

{u} , (5)

where the stiffness components are derived for each gap status as follows:

When the gap is open

K =

Kb 0 0

0 0

sym 0

. (6)

When the gap is closed and sliding

K = Ka 0 00 0

sym 0

. (7)

When the gap is closed and sticking

K = Ka 0 0Kt 0

sym Kt

. (8)When the gap is closed and slipping

K = Ka(v2 + w2)3/2

(v

2 + w2)3/2 0 0kv(v

2 + w2) kuw2 kuvwkw(v

2 + w2) kuvw kuv2

. (9)

It is noted that slipping due to friction introduces an unsymmetric stiffness matrix. There aresome inherent difficulties in the penalty gap element. Since the Newtons method is sensitive to

abrupt changes in stiffness during iterations, it could cause a divergent or oscillatory solution

when the gap changes its status. The timing of the stiffness update is crucial to the convergence,

which necessitates an adaptive stiffness update criterion based on the gap status. Furthermore,


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if the gap status undergoes two or more consecutive changes, adaptive stiffness update alone

cannot stabilise the solution and the increment size should be reduced.

The stick-slip behaviour of the gap element is path-dependent in a manner similar to the

plasticity. The lateral forces due to friction should be considered for equilibrium. If the internal

forces in the gap element are computed in a single step, the accuracy deteriorates when the

increment produces large changes in displacement with friction. In order to trace the path-

dependent solution, a sub-incremental scheme within an incremental load step is desired.Optimal penalty values (Ka and Kt) are difficult to assess. Furthermore, they have to be

updated continuously during the deformation process, because the stiffness of the adjacent

structure changes during the incremental process. It becomes necessary to adjust penalty values

adaptively by the program throughout the analysis.

In this study the adaptive gap element formulation was used to represent the ground plane at

the tyre/road interface. Several geometrically non-linear solution sequences were run to inves-

tigate the effect of changes in load and inflation pressure. The resulting deflection was then

compared to experimental results for the purposes of correlation. The method developed is

described and the correlation results are also presented.

3. Model Development

The 195/65R15 tyre was modelled using 780 MSC/NASTRAN CQUAD4 quadrilateral com-

posite shell elements to represent the composite material nature of the tyre cross section. An

area larger than the estimated contact area (taking into account the possibility of low inflation

pressures or excessive loads) was selected, encompassing the nodes to which the gap elementswere to be attached. One of the advantages of the gap element formulation is its ability to model

the contact without prior knowledge of the exact contact area. The open gap element has very

little stiffness and so contributes almost no structural effect to the model, therefore, allowing

more gap elements than are required to be specified. This ensures that the whole of the contact

area is represented.

Ground plane nodes were created that shared the same lateral and longitudinal co-ordinates

as the nodes identified in the contact patch, providing the required secondary attachment point

for the gap elements. The vertical position of the ground plane nodes were common to the centre

node of the contact patch. The clearance setting of the gap element at this node was set to zeroto mimic the tyre resting on the ground, but with no load applied. The remaining gap elements

were specified with clearance settings equal to the vertical distance between the contact patch

node and the corresponding ground plane node, so that under loading conditions the tyre could

deflect vertical only up to the impenetrable barrier set up by the ground plane. The tyre model

with the ground plane nodes in position is shown in Figure 1.

MSC/NASTRANs rigid format unstructured solution sequence SOL 66 was used in order to

carry out the analysis. This solution sequence allows geometric non-linear effects to be modelled

by employing incremental and iterative NewtonRaphson methods. Several subcases were used

to apply internal inflation pressure and hub loading in stages so that a converged solution could beobtained. Boundary conditions were such that the wheel centre had only one degree of freedom

in the vertical plane. Attachment of the bead area to the wheel centre by rigid elements ensured

that all nodes associated with the bead were also confined to a vertical only displacement, in the

direction of the applied hub force.


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Figure 1. 195/65R15 tyre model and ground plane nodes.

Figure 2. Force/displacement equipment set-up.

4. Results

Experimental validation was carried by applying force to the wheel hub and monitoring the

resulting vertical displacement as the tyre deflected against a flat surface. The force was applied

in a controlled manner by an electro/hydraulic actuator and displacement was monitored usingthe actuator controller and an externally mounted dial test indicator (d.t.i.) for greater accuracy.

Testing was carried out for a range of hub loads and inflation pressures to correspond to the

Table 1. Hub load variation comparison at 193 kN/m2

Force Measured Model % Diff.

(N) displacement (mm) displacement (mm)

1667.60 9.54 9.26 2.942501.55 14.31 14.30 0.073335.40 19.07 18.44 3.304169.25 23.85 22.52 5.585000.0 28.60 26.93 5.84


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Figure 3. Force/displacement (193 kN/m2).

Figure 4. Force/displacement results, 3335.4 N load, 193 kN/m2.

variety of solution sequences selected. A schematic illustration of the experimental set-up is

shown in Figure 2. Table 1 shows force/displacement results obtained at an inflation pressure of

193 kN/m2, from experimental testing and the tyre model re-run at various selected hub loads.

These results plotted, as shown in Figure 3, show that a straight line is produced for both themeasured and model results, indicating a linear vertical stiffness characteristic.

The validity of the technique over a range of inflation pressures was also investigated, the

results of which are shown in Table 2. Figure 4 illustrates the post processed results for the

3335.4 N loading, 193 kN/m2 case from the lateral and longitudinal, profile, viewpoints.


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Table 2. Inflation pressure variation comparison at 3335.4 N

kN/m2 Measured Model % Diff.

(N) displacement (mm) displacement (mm)

97 25.82 26.16 1.3

145 19.65 21.55 9.7

193 19.07 18.44

3.3

241 14.27 16.00 12.1

290 12.81 14.09 10.0

5. Discussion and Conclusions

A new procedure for tyre/ground interaction analysis has been developed. The method addresses

some of the limitations of previous techniques, whereby experimentally derived data were used

as the basis for displacing the contact patch. The old approach of constraining the wheel centre

while the contact patch is displaced upwards depends on the use of experimental data in themodelling procedure. The new method forces the wheel centre downwards against a ground

plane, thus modelling what actually happens in reality. The gap element formulation is a purely

theoretical approach which allows the analyst to be more flexible in changing parameters. The

effect of inflation pressure and wheel load changes on the contact patch displacement can all be

accounted for.

The numerical stability of the gap element formulation was found to be a problem in the

initial stages. Several trial and error runs were required to obtain an acceptable level of accuracy

and numerical efficiency. Once achieved, however, the model was found to be stable.

It can be seen from Figure 3 and Tables 1 and 2 that excellent correlation was achieved for arange of inflation pressures and hub loads. It is, therefore, concluded that the purely theoretical

gap element formulation approach to tyre/road interaction modelling has been shown to be both

viable and accurate.

References

1. Szilard, R., Theory and Analysis of Plates, Prentice-Hall, Englewood cliffs, NJ, 1974.2. Richards, T.R., Brown, J.E., Hohman R.L., and Sundaram, S.V., Modal analysis of tires relevant to vehicle

system dynamics, Presented at the Third Int. Modal Analysis Conf., Orlando, January 1985.

3. Sundaram, S.V., Hohman, R.L., and Richards T. R., Vibrational Modes of a Tire Using MSC/NASTRAN,Goodyear Tire and Rubber Company Technical Report, Akron, Ohio, 1992.

4. Kao, B.G., et al., Ford Motor Co. and Sundaram, S.V. et al. Goodyear Tire and Rubber Co., A new tiremodel for vehicle NVH analysis, SAE Transactions, Paper no. 870424, 1987.

Static Tyre-Road Interaction Modelling

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