## Y yDyssin cssycicos 0ssys 2 sin 2 815

y ,a> coso> s- y ,co sina> s + 2y ,a> cos2co s- (8.16) J , S si s s J cl s s J s2 s s v z zo + z sina s with a> s (8.17) Substitution of (8.15,16,17) in (8.14) and subsequently making the coefficients of corresponding terms in the left and right members equal to each other yields for the average output The average side force Fyave expressed in terms of the original quantities now reads For large Fz amplitudes and short wavelengths X 2k oos the formula shows that the dynamic loss...

## S

A parabolic vertical pressure distribution is assumed according to Eq.(3.4). At the trailing edge a sliding range starts to built up when side slip increases. Also at the leading edge sliding may occur when the tread element stiffness is relatively large. The bare string model does show a relatively short sliding range that grows and ultimately meets the rear sliding range at increasing side slip. We will assume that the tread element stiffness is sufficiently small to disregard front sliding....

## O y

Which quantity is a function of both the wheel load Fz and the transient slip angle or deformation gradient a'. It reduces to the relaxation length a0 ( < r at a' 0) when a' tends to zero and Fy a' to CFa. Obviously, offsets of the side force versus slip angle characteristic near the origin have been disregarded here. The transient slip angle follows from v The side force and moment are obtained by using the steady-state characteristics Fy Fy a',Fz) (8.26) With the gyroscopic couple, cf....

## E te a a 0620

Apparently, in the (e, V) parameter plane the boundaries of (6.19) reduce to two parallel lines at e a + a and e -1. When the caster lies in between these two values the yaw angle performs an oscillation with exponentially increasing amplitude at any speed of travel. Apparently, when the damping is zero, the speed and the torsional stiffness do not influence the extent of the unstable area. They will, however, change the degree of instability and the natural frequency (eigenvalues). If we have...

## 214

The 'spin' slip (p is defined as the component -to. of the absolute speed of rotation vector (o of the wheel body along the normal to the road plane n divided by the forward running speed. We obtain the expression in terms of yaw rate y and camber angle y (cf. Fig.2.6) The minus sign is introduced again to remain consistent with the definitions of longitudinal and lateral slip (2.11, 2.12). Then, we will have as a result of a positive (p a positive moment M . It turns out that then also the...

## A

The equations (2.55,56) apply in general. Their solutions contain constants of integration which depend on the selected physical model description of the tyre. Applications of the differential equations derived above will be demonstrated in subsequent chapters both for steady-state and non-steady-state conditions. In the case of steady-state motion of the wheel, the partial derivatives with respect to time t or travelled distance 5 become equal to zero (dv ds du ds 0). Then, the deformation...

## Cw F

In the ensuing theory we will make use of the theoretical slip quantities (4.27a,b), which does produce the slope in the constant a curves in the Fy vs Fx diagram at ',, 0. It should be noted, however, that experience has shown that the use of practical slip quantities .v., and s* (that is k and tana* which may be obtained again by omitting the denominators in (4.27a,b)) may already give very good results as will be indicated later on in Fig.4.8. Note that when approaching wheel lock This calls...

## Y

Integration yields the following expressions for the horizontal deformations in the contact area The second expression is, of course, an approximation of the actual variation which is according to a circle. The approximation is due to the assumption made that the deflections v are much smaller than the path radius R. The constants of integration follow from boundary conditions which depend on the tyre model employed and on the slip level. As an example, consider the simple brush model with...

## Fy cFyy0 y

Using the conversion formulae (5.36,37) the transfer functions with respect to a and (pcan be obtained. For the order n of (5.99) equal to 2, the model becomes of the second order and one finds the following transfer functions in vector form a(a + V2a)p + a+a, a (a + Via) , a+a) a (o + ' 2 a) p2 + (< j+a) p + 1 (1 , -a(a + V2a)p , a(a + V2d)p2 + l) a (o + ' 2 a) p2 + (a+a) p + 1 It may be noted that the side force turn slip stiffness for this approximate model is somewhat larger than...

## 610

Which includes the non-dimensional pneumatic trail t t a0. After elimination of the time and all the variables except and v, in Eqs.(6.1-9) and using the conversions (6.10) we obtain the non-dimensional differential equations + (e-a)- - + y ds o_ ds The system is linear and of the third order. It is assumed that the tyre deflection relaxation length a 3a or non-dimensionally a 3. In case of the single contact point tyre model a must be taken equal to zero and a must be given the value of a + a...

## 1

In practice, points C and C* lie closely together and making distinction between the longitudinal or the lateral velocities of these points is only of academic interest and may be neglected. Instead of V in the denominator we may write Vc . and if we wish, instead of Vcy in the numerator the lateral speed of point S (parallel to road plane) which is Vsy. This may even be a better choice if large values of camber are considered while a vertical tyre deflection rate occurs. The definitions of the...

## 64 Shimmy and Energy Flow

To sustain the unstable shimmy oscillation, energy must be transmitted 'from the road to the wheel'. We realise that, ultimately, the power can only originate from the vehicle's propulsion system. The relation between unstable modes and the self-excitation energy generated through the road to tyre side force and aligning moment is discussed in detail in subsection 6.4.1. The transition of driving energy (or the vehicle forward motion kinetic energy) to self-excitation energy is analysed in...

## M M Ml M

With this simple addition to the enhanced stretched string tyre model (provided with tread elements distributed over the assumedly rectangular contact area) calculations have been performed. Figure 5.38 shows the Nyquist plot of the responses with along the curves cosa lKna V 2ka It should be noted that the vector of the gyroscopic couple is directed perpendicularly to the vector of the lateral force. For the range of frequencies considered, the simple approximate dynamic extension theory gives...

## F

Slip velocities of contact patch forming input to transient slip equations. Fig. 9.29. Slip velocities of contact patch forming input to transient slip equations. With the transient response variables computed with the aid of Eqs.(9.78-86) plus Eq.(9.20) the contact forces and moments may be found by using these variables as argument in the steady-state equations presented in Sections 4.3.2 and 4.3.3. We should, however, properly account for the response to a varying camber angle of...

## Vsy vo

Parameter values of wheel suspension system and tyre considered. Table 8.2. Parameter values of wheel suspension system and tyre considered. The tyre side force differential equation (7.20) has been used and similar for the longitudinal force. The longitudinal slip speed follows from (8.45) with (8.52). Note that mechanical caster has not been considered so that the lateral slip speed is simply expressed by (8.80). The aligning torque is based on Eq.(7.49) with for Mzr the spin...

## V

In the contact area regions of adhesion may occur as well as regions where sliding takes place. In the region of adhesion the tyre particles touching the road do not move and we have V,,s V,,y 0. In this part of the contact area, the frictional shear forces acting from road to tyre on a unit area (with components denoted by qx and qy not to be confused with the components of the position vector (2.44) used earlier) do not exceed the maximum available frictional force per unit area. The maximum...

## 65 Non Linear Shimmy Oscillations

Non-linear shimmy behaviour may be investigated by using analytical and computer simulation methods. The present section first gives a brief description of the analytical method employed by Pacejka (1966) that is based on the theory of the harmonic balance of Krylov and Bogoljubov (1947). The procedure that is given by Magnus (1955) permits a relatively simple treatment of the oscillatory behaviour of weekly non-linear systems. Further on in the section results obtained through computer...

## McF F F v3

Fy - Fx characteristics for 6.00-13 tyre measured on dry internal drum with diameter of 3.8m (from Henker 1968). Fig. 3.17. Fy - Fx characteristics for 6.00-13 tyre measured on dry internal drum with diameter of 3.8m (from Henker 1968). Fig. 3.18. Combined slip side force and moment versus longitudinal force characteristics measured at two values of the slip angle for a 7.60-15 tyre on a dry flat surface (from Nordeen and Cortese (1963)). where the compliance coefficient c has been...

## 1 ae2co2

Equations (6.89-91) provide sufficient information to compute the amplitude and the frequency of the limit-cycle. The stability of the limit-cycle of the weakly non-linear system may be assessed by taking the derivative of the Hurwitz determinant with respect to the amplitude. If d 2 d > () the limit-cycle is stable and attracts the trajectories, if negative, the limit-cycle is unstable and the oscillations deviate more and more from this periodic solution. In the original publication more...

## Ac vK WKvsxc920

That completely corresponds to Eq.(7.54) of the enhanced non-linear transient tyre model. The transient slip k'c is subsequently used as input into the steady-state longitudinal force characteristic as will be explained later on and has already been indicated in Chap.7, Sec.7.3. The lateral slip condition is more complex to handle because we have to deal with both the side force and the aligning torque. In addition, in the test condition, the carcass is allowed to not only deflect in lateral...

## Ds ds ds2 2 dr

After substitution of these series in Eq.(5.96) the following generic formula is obtained The side force and the aligning torque can be found by multiplying the lateral deflection v0 y0 - y and the torsion angle fi dPM d.v - with the respective stiffnesses CFy and CMy, cf. Eqs.(5.59,62). We obtain

## J a2 c2

This expression holds because we have assumed that the brush model is isotropic. Using the magnitude of combined slip according to (9.76) the factor m 1 - OQ if Q< else m 0 (9.77) Maurice (2000) found excellent agreement with the physical model for the combined slip cases a0 0.08rad and 0.08rad and k0 0.06 and fc 0.06 with a phase difference of 45 and wavelengths of 0.2, 1 and 5m. Adding turn slip will influence the combined slip response further. The next section approaches this matter in a...

## BAF2z

The static loss can be simply found to become Fig. 8.4. The static loss in average side force due to the curved force vs load relationship. Fig. 8.4. The static loss in average side force due to the curved force vs load relationship. The occurrence of the dynamic loss may be explained by assuming a linear variation of both the cornering stiffness and the relaxation length with load with z cFz, y Fy, 3 ea c. Consider a truncated Fourier series approximation of the periodic solution

## 1 i 1 s

The last term represents the curvature -1 Ry of the tyre peripheral line touching a frictionless surface at a cambered position. When disregarding a possible uniform offset of this line we obtain when integrating (2.52) by using (3.54) and approximating (2.51) by taking xo x rJJ - which reduces expressions (2.55, 56) for the sliding velocities with (3.53-55) to In the range of adhesion where the sliding velocities vanish, the deflection gradients become

## 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...

## 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...