## Chapter

NON-STEADY-STATE OUT-OF-PLANE STRING-BASED TYRE MODELS The transient and oscillatory dynamic behaviour of the tyre will be discussed in this and two ensuing chapters. The present chapter is devoted to the model development of the tyre as an integral component. The stretched string model is chosen as the basis for the physical description of the out-of-plane (antisymmetric) behaviour. This model exhibits a finite contact length that allows the study of short path wavelength phenomena. The model...

## Kx Va yboW siny dconxy Vr d7xyyboQsmy 3102

V -V k , V r Q V -V , V -V tan a (3.104) sx cx ' r e cx sx ' sy cx v ' The lateral displacement yb of the belt at the contact centre is attributed to camber, conicity and the lateral external force (through the lateral compliance of the carcass). The gradient dyb dxb may be approximately assessed by assuming a parabolic base line yb(xb) exhibiting an average slope cs influenced by the aligning torque (through the yaw compliance) and ply-steer, and a curvature cc influenced by the side force...

## Info

Because, obviously, the derivative of Fyo with respect to its argument aeq at aeq 0 is equal to the original cornering stiffness CFm, we find for the derivative of Fy with respect to a the same value for the slope at a 0 which proves that the slope at the origin of the characteristic is not affected by the successive multiplications. The second step in the manipulation of the Fig.4.4. Cornering and aligning stiffness versus wheel load. Fig.4.3. Using the similarity method to adapt F and Mz...

## M M I ur

When lifted from the ground, the equivalent ground to tyre forces vanish if the frequency of excitation is not too high (low with respect to the first natural frequency of the tyre). For the set of parameter values measured on a radial ply steel-belted tyre listed in Table 5.3, the amplitude and phase of the equivalent moment as a response to yaw angle have been computed and are presented in Fig.5.40 as a function of the excitation frequency n coHk. Experimentally obtained curves up to a...

## 31 Introduction

This chapter is devoted to the analysis of the properties of a relatively simple theoretical tyre model belonging to the third category of Fig.2.11. The mathematical modelling of the physical model shown in Fig.2.13 has been a challenge to various investigators. Four fundamental factors play a role frictional properties in the road-tyre interface, distribution of the normal contact pressure, compliance of the tread rubber and compliance of the belt carcass. Models of the carcass with belt and...

## 24 Fundamental Differential Equations for a Rolling and Slipping Body

A wheel with tyre that rolls over a smooth level surface and at the same time performs longitudinal and lateral slipping motions, will develop horizontal deformations as a result of the presence of frictional forces which attempt to prevent the tyre particles, that have entered the contact area, from sliding over the road. Besides areas of adhesion, areas of sliding may occur in the contact patch. The latter condition will arise when the deflection generated in the range of adhesion would have...

## 82 Cornering on Undulated Roads

When a car runs along a circular path over an uneven road surface the wheels move up and down and may even jump from the road while still the centripetal forces are to be generated. Under these conditions, the wheels run at slip angles which may become considerably larger than on a smooth road. Consequently, on average, the cornering stiffness diminishes. This phenomenon was examined in Chapter 5, Section 5.6, where the stretched string model was found to be suitable to explain the decrease in...

## 92 The Contact Patch Slip Model

In this section, we will first represent the contact patch with tread elements by the brush model. Because of its relative complexity, the analytical model that describes the non-steady-state response to slip variations is approximated by a set of first-order differential equations. This contact model is tested by attaching the base line of the brush model to the wheel plane through a compliant carcass. For reasons of practical use, we finally introduce the Magic Formula to handle the...

## 23 Assessment of Tyre Input Motion Components

The location of the contact centre C and the magnitude of the wheel radius r result from the road geometry and the position of the wheel axle. We consider the approximate assumption that the road plane is defined by the plane touching the surface at point Q located vertically below the wheel centre A. The position of point Q with respect to the inertial frame (Ox , '. z ) is given by vector q. The normal to the road plane is defined by unit vector n. The location of a reference point B of the...

## 94 Dynamic Tyre Model Performance

A number of experiments has been conducted at the Delft University of Technology to assess the parameters of the dynamic model and to judge its performance. The steady-state side slip, longitudinal slip and camber force and moment characteristics have been typically assessed from over the road experiments with the Delft Tyre Test Trailer. For the model performance evaluation, steady-state characteristics have been used obtained from tests carried out on the drum with the strain gauge equipped...

## K cMy

Where a 0 when the single contact point tyre model is considered. To reduce the number of governing system parameters we will introduce the following non-dimensional quantities with the reference length a0 representing the actual or the nominal half contact length 0. , s , e , t -

## Y Jl e Ml922

Again, the responses to variations of the side slip only occur in the range of adhesion. The transition point from adhesion to sliding now occurs at and the corresponding adhesion fraction becomes, cf. Eq.(3.8) m -9 aco if aj< else m 0 (9.24) As the slip angle of the contact patch remains small in the range where adhesion still occurs, cosaco has been replaced by unity. By integration of the transformed deflection over the range of adhesion the frequency response functions of the force and...

## Expressions Of Steering Angle Required To Negotiate The Given Curve

The system depicted in Fig. 1.4 and described in the preceding subsection performs a motion over a flat level road. Proper coordinates are the Cartesian coordinates X and Y of reference point A. the yaw angle of the moving x axis with respect to the inertialXaxis and finally the roll angle (p about the roll axis. For motions near the Xaxis and thus small yaw angles, Eq.(l .25) is adequate to derive the equations of motion. For cases where may attain large values, e.g. when moving along a...

## 83 Longitudinal Force Response to Tyre Non Uniformity Axle Motions and Road Unevenness

In this section the response of the axle forces Fx and Fz to in-plane axle motions (x, z), road waviness and tyre non-uniformities will be discussed. For a given tyre-wheel combination the response depends on rolling speed and frequency of excitation. This dependency, however, appears to be of much greater significance for the fore and aft force variation Fx than for the vertical load variation ' The vertical force response has an elastic component and a component due to hysteresis. A similar...

## 1 K

If the thus obtained function of Fx is plotted versus ax the resulting curve becomes symmetric. However, if then the abcsissa is transformed into cby using (4.30) the resulting curve turns out to become asymmetric with the braking side identical to the characteristic we started out with. This asymmetry was already found to occur with the brush model discussed in Chapter 3. Next, we should realise that through (4.18) camber has been accounted for and that, as a consequence, ay is to be replaced...

## 85 ABS Braking on Undulated Road

The aim of this section is to investigate the influence of dynamic effects due to vertical and longitudinal wheel vibrations excited by road irregularities upon the braking performance of the tyre and anti-lock braking system. These disturbing factors affect the angular velocity of the wheel and consequently may introduce disinformation in the signals transmitted to the ABS system upon which its proper functioning is based. One may take the simple view that the primary function of the anti-lock...

## TK Kvn cbvn hyQyl9206

Values of inertia parameters normalised with tyre mass m0 and reference moment of inertia m0r20 with r0 the unloaded tyre radius have been listed in App.3. In the study of Zegelaar (1998) important observations have been made regarding contact area dimensions, static and dynamic vertical stiffness and characteristics at different speeds of rolling, static longitudinal stiffness of the standing tyre, tyre radius growth with speed, rolling resistance, effective rolling radius and rolling...

## 81 Vehicle Response to Steer Angle Variations

In Chapter 1, Section 1.3.2 the dynamic response of the two-degree of freedom vehicle model depicted in Figs. 1.9,11 to steer angle input has been analysed. As an extension to this model we will introduce tyres with lagged side force response. The system remains linear and we may use Eq.(7.18). The relaxation length will be denoted by a. We have the new set of equations m(v + Vr) CFala + CFa2a'2 (8.1) Figure 8.1 presents the computed frequency response functions, which hold for the extended...

## T a3

And if a > av (but < ' m) Fig. 3.4. Characteristics of the simple brush model side force, aligning torque and pneumatic trail vs slip angle. Fig. 3.4. Characteristics of the simple brush model side force, aligning torque and pneumatic trail vs slip angle. These relationships have been shown graphically in Fig.3.4. At vanishing slip angle expression (3.13) reduces to This value is smaller than normally encountered in practice. The introduction of an elastic carcass will improve this...

## P Iji Jiel01p

The transfer functions of the responses to y and are obtained by considering the relations between the transformed quantities and inserting these in (5.29). We find in general for the transfer function conversion By transforming back the expressions such as (5.23,29), the deflection, the force and the moment can be found as a function of distance travelled 5 for a given variation of a and (p or of y and . An interesting observation may be made when considering the situation depicted in Fig.5.6....

## 95 Dynamic Tyre Response to Short Road Unevennesses

The actual road surface profile over which the tyre rolls may contain spectral components showing relatively short wavelengths. If the wavelength is smaller than two to three times the contact length a geometric filtering of the profile becomes necessary if the tyre model employed is assumed to contact the road in a single point. For the SWIFT model a special filter has been developed that takes care of the envelopment properties of the tyre and the variation in effective rolling radius that...

## 73 Enhanced Non Linear Transient Tyre Model

A totally different approach to model the transient rolling properties of the tyre is based on the separation of contact patch slip properties and carcass compliance not through the use of relaxation lengths but by incorporating the carcass springs in the model explicitly. The contact patch is given some inertia to facilitate the computational process (computational causality). This has the drawback that a relatively high natural frequency is introduced, possibly making the computation slower....

## Vsx8119

With the k' dependent relaxation length The model is not sensitive to wheel load variations which constitutes the restriction of the model. For the problem at hand this restriction is not relevant and the model can be used. The great advantage of the model is the fact that an algebraic loop does not occur. And again a u limitation is not needed. A straight forward simulation can be conducted. For the relation (8.120) the following approximate function is used where < rmm represents the...

## C c

With CMaaccording to Eq.(4.E48) and the initial residual torque from (4.E47) Finally,the initial side force and torque are to be removed from the equations by putting the parameters pHyh pHy2, pVyh pVy2, qme and qi, - or the scaling factors . , .,-,. and ., , equal to zero and by replacing in Eqs.(4.E20,37) the original side slip input variable a* tana -sgn V, by its effective value and in Eqs.(4.E28,29,47) the original camber y* sin by the effective total Fig. 4.24. The final diagram with...

## P qpp qa

Surprisingly, it turns out that these transfer functions (except the one with respect to y) correspond to the functions of Rogers (with ()) (Eqs.5.102,103 with Via omitted) if, as Besselink indicated, the following equivalence conditions hold q -a> qr > cy CFy a> < V c(5147) Furthermore, comparison with the Von Schlippe approximation at steady state, shows that where, as before, CFy designates the camber force stiffness. Mor eland's model This model, first published in 1954 in the paper...

## A3ancc

The calculated variation of half the contact length and the relaxation length as a ratio Fig. 5.48. The calculated variation of half the contact length and the relaxation length as a ratio to the static half contact length versus the vertical load ratio for the model with tread elements using Eq.(5.171). In Fig.5.49 a comparison is made between the results of the three models without tread elements, with tread elements (exact) and according to the approximation with varying a a*(a)....

## F yFs2Sgn

At < p < ps the force reduces to ul' sgn(p and the moment to zero. The same can be obtained from the expressions (3.62) holding for the case of full adhesion when 2c',y b is replaced by cpy and the width 2b is taken equal to zero. peripheral line on frictionless surface Fig. 3.24. The tyre brush model with zero width rolling while turning or at a camber angle (a) at full adhesion. Turning at large spin showing sliding at the front half and at the rear end (b,c), with parabolic approximation...

## 25 Tyre Models Introductory Discussion

Several types of mathematical models of the tyre have been developed during the last half century. Each type for a specific purpose. Different levels of accuracy and complexity may be introduced in the various categories of utilization. This often involves entirely different ways of approach. Figure 2.11 roughly illustrates how the intensity of various consequences associated with different ways of attacking the problem tend to vary. From left to right the model is based less on full scale tyre...

## 2 y l xi xtana383

And in second sliding range (-a < x< xv2) 2 2 a range for -Jr -y p to be 'large' s2, Fig. 3.30. The model running at large spin (turning and equivalent camber) at a relatively small (a,b) or large positive or negative slip angle (c). Fig. 3.30. The model running at large spin (turning and equivalent camber) at a relatively small (a,b) or large positive or negative slip angle (c). Integration over the contact length after addition of vlfl and multiplication with the stiffness per unit length...

## Wcx

If the forward speed Va becomes or is equal to zero one might add a small quantity s in the denominator of (4.E3, E5) to avoid singularity, or, when transient slip situations occur, one should use the transient slip quantities (or deformation gradients) tana' and k' as defined and used in Chapters 7 and 8. To avoid the occurrence of similar singularities in the ensuing equations due to e.g. zero velocity or zero vertical load, a small additional quantity s (with same sign as its neighbouring...

## 84 Forced Steering Vibrations

A steering suspension system of an automobile exhibits a rather complex configuration and possesses many degrees of freedom. A simplification is necessary to conduct a sensible analysis to gain insight into its general dynamic behaviour and into the influence of important parameters of the system. Investigation of the steering mode of vibration requires at least the steering degree of freedom of the front wheel, possibly extended with the rotation degree of freedom of the steering wheel. For...

## CO J UJsl

Similarly, the formula for the response of lateral acceleration a can be derived . ( f amv - dl.2i j D c2 By considering Eq.(1.77) it can now be explained for instance that at higher frequencies the system shows features of a first-order system because of the jco term in the numerator the yaw rate amplitude response tends to a decay at a 6dB per octave rate (if plotted in log-log scale) and the phase lag approaches yaw velocity response to steer angle yaw velocity response to steer angle...

## MnQ FnV M Qf V

This suggests, at least for the model employed, that the moment arm equals the Fig. 9.33. Power flow diagram (bond graph) of driven tyre wheel combination in steady state. Fig. 9.33. Power flow diagram (bond graph) of driven tyre wheel combination in steady state. Fig. 9.34. Tyre radii as function of the normal load measured at zero or very low speed. Fig. 9.34. Tyre radii as function of the normal load measured at zero or very low speed. effective rolling radius (defined at zero driving or...

## Drilling Torque Van Der Jagt

BHyq gt CP a R0 CHy p DHy p CPa DDrv, Mz sin 0.57T CDr p KzRtprQ CM a t0 CP a R0 B rq, KzR pro l CDr p DDr f, Vera, RofFz Table 9.2. Parameter values for tyre model with Magic Formula including quantities introduced later on. Table 9.2. Parameter values for tyre model with Magic Formula including quantities introduced later on. Steady-State, Step Response and Frequency Response Characteristics To demonstrate the performance of the model a number typical characteristics will be presented. The...

## Semiempirical Tyre Models

In the preceding chapter the theory of the tyre force and moment generating properties have been dealt with based on physical tyre models. The present chapter treats models that have been specifically designed to represent the tyre as a vehicle component in a vehicle simulation environment. The modelling approach is termed 'semi-empirical' because the models are based on measured data but may contain structures that find their origin in physical models like those treated in the preceding...

## 22 Definition of Tyre Input Quantities

If the problem which is going to be investigated involves road irregularities, then the location and the orientation of the stub axle spindle axis must be known with respect to the specific irregularity met on the road. The road surface is defined with respect to a coordinate system of axes attached to the road. If the position and orientation of the axle is known with respect to the fixed triad then the exact position of the wheel with respect of the possibly irregular road surface can be...

## Motorcycle Dynamics

The single track vehicle is more difficult to study than the double track automobile and poses a challenge to the vehicle dynamicist. Stability of motion is an important issue and it turns out that the stabilising actions of the human rider are essential to properly handle the vehicle. Steady-state cornering behaviour can be analysed in a straightforward manner together with the examination of the stability of the equilibrium motion. While for an automobile only the lateral and yaw degrees of...

## 32 Tyre Brush Model

The brush model consists of a row of elastic bristles that touches the road plane and can deflect in a direction parallel to the road surface. These bristles may be called tread elements. Their compliance represents the elasticity of the combination of carcass, belt and actual tread elements of the real tyre. As the tyre rolls, the first element that enters the contact zone is assumed to stand perpendicularly with respect to the road surface. When the tyre rolls freely that is without the...

## Tyre Characteristics And Vehicle Handling And Stability

This chapter is meant to serve as an introduction to vehicle dynamics with emphasis on the influence of tyre properties. Steady-state cornering behaviour of simple automobile models and the transient motion after small and large steering inputs and other disturbances will be discussed. The effects of various shape factors of tyre characteristics cf. Fig. 1.1 on vehicle handling properties will be analysed. The slope of the side force Fy vs slip angle a near the origin the cornering or side slip...

## Dal a

For the sake of simplicity we have assumed m k2 ab. By using Eq. 1.94 the traj ectories solution curves can be constructed in the au a2 plane. The isocline method turns out to be straightforward and simple to employ. The pattern of the trajectories is strongly influenced by the so-called singular points. In these points the motion finds an equilibrium. In the singular points the motion is stationary and consequently, the differentials of the state variables vanish. From the handling diagram K...

## Cxc2 I Cf2

With g denoting the acceleration due to gravity. After having defined the lateral acceleration which in the present linear analysis equals the centripetal acceleration Eq. l .53 can be written in the more convenient form The meaning of understeer versus oversteer becomes clear when the steer angle is plotted against the centripetal acceleration while the radius R is kept constant. In Fig. 1.10 left-hand diagram this is done for three types of vehicles showing understeer, neutral steer and...