Longitudinal Slip

For the case of pure longitudinal slip κ_coof the contact patch, the point of transition from adhesion to sliding is located according to Eqn (3.44) with (3.34, 3.40) at a distance x_t from the contact center:

(9.1)

The composite parameter θ has been defined in Chapter 3, Eqn (3.46). The length of the adhesion range 2am = a − x_t that begins at the leading edge is characterized by the fraction m that in the present chapter replaces the symbol λ to avoid confusion with the notation for the wavelength. We have

(9.2)

An important observation is that at small variations of slip we may assume that only in the adhesion range changes in deflection occur.

For the development of the transient model we start out from the basic rolling contact differential Eqns (2.55, 2.56). In the adhesion range we find for the longitudinal tread element deflection u_c in the case that only longitudinal slip is considered:

(9.3)

where the tilde designates the variation with respect to the steady-state level. The average linear speed of rolling of the contact patch is equal to that of the wheel rim and reads

(9.4)

The Fourier transform of the deflection becomes with U and K denoting the transformed quantities of u and κ, respectively, and ω_s the path frequency:

(9.5)

Here, the boundary condition that says that the deflection vanishes at the leading edge, is satisfied. By integrating over the range of adhesion and multiplying with the tread element stiffness c_p, the frequency response function of the variation of F_x to the variation of κ_c is obtained. With the reduced frequency

(9.6)

and the local derivative of F_x to κ_c

(9.7)

the expression for the response function becomes, when still adhesion occurs in the contact patch (first condition of (9.2)),

(9.8)

In Figure 9.2 the resulting amplitude and phase characteristics have been shown together with those of an approximate first-order system. Especially the phase curve appears to exhibit a wavy pattern in the higher frequency range. These waves are considerably attenuated when the contact model is incorporated in a more complete tire model including carcass compliance, as will be shown below.

FIGURE 9.2 Frequency response function of the longitudinal force variation to longitudinal slip variation of the brush model versus reduced path frequency according to the exact analytical solution and that of an approximate substitute first-order system at a given level of longitudinal slip.

The first-order substitute model has a frequency response function that reads

(9.9)

The approximation shows the same high frequency asymptote and steady-state level as the exact model. Apparently, the actual cutoff path frequency reads

(9.10)

that, obviously, reduces to 1/a when the average slip κ_co vanishes and, as a consequence, m (9.2) becomes equal to unity. When the average slip is chosen larger, the length of the adhesion range decreases and the cutoff frequency becomes higher. The relaxation length of the approximate contact model reads, according to Eqn (9.9) with (9.6),

(9.11)

which reduces to zero when total sliding occurs.

The performance of the first-order system is reasonable but appears to improve when the filtering action of the carcass compliance is taken into account. In that configuration, we have the wheel slip velocity V_sx that acts as the input quantity. Adding the time rate of change of the carcass deflection produces the slip velocity of the brush model. The carcass deflection u equals (with fore-and-aft carcass stiffness c_x introduced)

(9.12)

The feedback loop in the augmented system apparently contains a gain equal to iω_s/c_x. The resulting complete frequency response function reads

(9.13)

When the approximate first-order system is employed for the contact model, the frequency response function for the complete model becomes, using (9.13) with (9.9),

(9.14)

where the total relaxation length has been introduced which apparently reads

(9.15)

The local longitudinal slip stiffness of the contact patch is here equal to the local longitudinal slip stiffness of the complete model, which obviously is due to the assumed inextensibility of the base line of the brush model. So we have at steady state:

(9.16)

At vanishing slip we will add a subscript 0. When total sliding occurs, both σ_c and C_Fκ reduce to zero and the total relaxation length σ vanishes.

Introduction of the longitudinal carcass compliance may be accomplished by considering the feedback loop similar to the one (c_y) for the more complex configuration for lateral slip shown in Figure 9.6. Figure 9.3 presents the frequency response function for this total model at vanishing slip, κ₀ = 0.

FIGURE 9.3 Frequency response function of the longitudinal force variation to longitudinal slip variation of the brush model attached to a flexible carcass versus path frequency according to the exact analytical solution and that of the approximate first-order system at zero longitudinal slip.

It is noted that compared with the curves of Figure 9.2 the wavy pattern is considerably reduced. The approximate system performs very well. A wavelength of 10 cm occurs at the path frequency ω_s = ca. 60 rad/m.

The differential equation that governs the transient slip response of the contact patch and through that the longitudinal force response becomes for the approximate system

(9.17)

The variation of the force becomes

(9.18)

The structure of Eqn (9.17) corresponds with that of Eqn (7.37) of Chapter 7. The response, however, is insensitive to load variations but shows a nice behavior. The transient response to load variations is sufficiently taken care of through the effect of carcass compliance. As a result, the relaxation length for the response to load variations becomes equal to C_Fκc/c_x, which is somewhat smaller than σ (9.15) that holds for the response to slip variations.

When the average steady-state relation for the longitudinal slip

(9.19)

is added to Eqn (9.17), the equation for the total transient slip is obtained:

(9.20)

that completely corresponds to Eqn (7.54) of the enhanced nonlinear transient tire model. The transient slip is subsequently used as input into the steady-state longitudinal force characteristic, as will be explained later on, and has already been indicated in Chapter 7, Section 7.3.

Lateral Slip

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 the lateral direction but also about the vertical axis. This is in accordance with the ultimate SWIFT configuration. The connected turn slip behavior of the contact patch will be dealt with further on.

As with the longitudinal model development, first, the analytic response functions of the brush model will be assessed, in this case to side slip variations. Eqn (2.56) gives rise to the following equation for the lateral tread deflection variations:

(9.21)

Note that for the sake of simplification in the present chapter the notation tanα is replaced by α, with or without a subscript. As for the longitudinal deflection we find for the Fourier transform V_c of the lateral deflection responding to slip angle variations, A_c denoting the brush model slip angle’s transform (or actually of tan α_c):

(9.22)

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

(9.23)

and the corresponding adhesion fraction becomes, cf. Eqn (3.8),

(9.24)

As the slip angle of the contact patch remains small in the range where adhesion still occurs, cos α_co has been replaced by unity. By integration of the transformed deflection over the range of adhesion the frequency response functions of the force and the moment variations are established. They read, at pure lateral slip (V_rco = V_x, cf. (9.4)),

(9.25)

and

(9.26)

The local slope of the steady-state side force characteristic of the brush model is given by

(9.27)

The normalized response function of the side force is identical to that of the longitudinal force if the factor 1 + κ_co is omitted in (9.6). The approximate first-order description with response function

(9.28)

where

(9.29)

shows the same very good agreement with the exact result, as assessed with the longitudinal force response. Similarly we have the differential equation for the variation of the transient side slip:

(9.30)

The variation of the side force becomes

(9.31)

After adding the steady-state equation

(9.32)

to Eqn (9.30), the equation for the total transient side slip is obtained:

(9.33)

As before, the resulting is used as the input of the steady-state side force function.

We may follow the same procedure to assess the aligning torque equations as was done in Section 7.3. The resulting first-order response, however, does not always agree with the analytically found tendency. The phase lag and the slope of the high-frequency amplitude asymptote indicate that a second-order behavior prevails at zero average slip angle while at larger slip angles the response gradually changes into a first-order nature. Apparently, the mechanism is more complex and we should account for the transient response of the pneumatic trail. The variation of the aligning torque may be written as follows:

(9.34)

The analysis conducted by Maurice (2000) shows that the analytically assessed response function of the pneumatic trail variation to slip angle variations can be approximated by the first-order system (9.33) and a so-called phase leading network in series. The frequency response function of the latter reads

(9.35)

The factor in this formula represents the local slope of pneumatic trail characteristic:

(9.36)

which, apparently, is a negative quantity. According to Maurice, adequate values for the parameters σ₁ and σ₂ can be obtained through the formulas

(9.37)

or alternatively

(9.38)

and

(9.39)

The block diagram of the current system governed by Eqns (9.28, 9.34, 9.35) is presented in the upper diagram of Figure 9.4. The lower diagram shows an alternative structure of the same system, thereby displaying the extra moment ΔM_z, which is governed by the ratio of parameters (9.39). This ratio tends to infinity when full adhesion occurs. Then, m = 1 and α_co = 0, as becomes clear from Eqn (9.24). The singularity involved has been circumvented by Maurice through the introduction of a function that limits the value of (9.39) around zero lateral slip, α_co = 0. This, however, will slightly disturb the proper response at zero slip angle and thus degrades the linear analysis around zero slip. An alternative way of avoiding the singularity, which obviously is caused by the fact that we actually may have a moment without the simultaneous presence of a force, is the consideration of the extra transient moment ΔM_z. This moment is obtained by multiplication of the difference of the transient slip quantities for the force and for the pneumatic trail with three factors the combination of which may be designated with C_ΔM:

(9.40)

FIGURE 9.4 Block diagram of the contact patch model to generate short wavelength transient responses of the side force and the aligning torque to small slip angle variations. The original upper diagram can be replaced by the lower diagram, thereby avoiding singularity at zero slip.

It turns out that now the singularity does not show up because both F_yo and 1/σ₁ become zero at the same time. This indicates that indeed a moment may arise although at that instant of time the side force is zero. After writing out the factors in (9.40) by using expressions for the side force (3.11), the pneumatic trail (3.13) and further (9.36, 9.39) while in (3.11) λ is replaced by m and θ_yσ_y by θα_co = z we find, for C_ΔM,

(9.41)

where C_Mαco denotes the aligning stiffness of the brush model at zero slip angle and the nondimensional factor ξ; is introduced:

(9.42)

with

(9.43)

Evaluation of (9.42) reveals that for z < 1 the factor ξ; may possibly be approximated by

(9.44)

This approximation will be introduced later on when we will deal with the application of the Magic Formula.

The differential equations that apply for the contact patch model subjected to small side slip variations with respect to a given side slip level now read

(9.45)

(9.46)

from which the variation of the force and moment result:

(9.47)

(9.48)

It is of importance to check the step responses to side slip starting from zero slip angle, especially right after the start of the step change. Initially we have all variables equal to zero while the slip angle input has reached the new value α_co. The various time derivatives become

(9.49)

These results are correct. The last equation holds indeed because we have at vanishing slip angle according to Eqn (9.42): ξ; = 1 and consequently C_ΔM = t_coC_Fαco, where t_co = a/3. Evidently, the extra moment is responsible for the proper start of the course of the aligning torque showing zero slope. This property has been ascertained to occur both in reality and with models, cf. Figures 5.10 and 5.11. Also the response to a lateral wheel displacement y = ∫V_sydt at zero forward speed develops correctly. We find with Eqn (7.6) for the force: F_y = −C_Fy y and for the moment: M_z = 0.

The equations may now further be appraised by first introducing tire carcass lateral and torsional compliance, as depicted in Figure 9.5 and in the corresponding block diagram of Figure 9.6, and subsequently comparing the results with analytical solutions obtained by using Eqns (9.25, 9.26).

FIGURE 9.5 The brush model attached to a carcass possessing lateral and torsional compliance.

FIGURE 9.6 Block diagram of the augmented system including carcass lateral and torsional compliance.

As a reference, the steady-state characteristics of the model have been presented in Figure 9.7. The diagram contains the curves for both the complete model and for the brush model alone. It is of interest to note the lower cornering stiffness due to the introduction of carcass compliance. The expression for the lower side slip stiffness can be found to read

(9.50)

FIGURE 9.7 Steady-state side slip force and moment characteristics and the relaxation length of the brush model (the contact patch) and of the model including the flexible carcass through which the brush model is attached to the wheel plane.

The relaxation length of the complete model found from the cutoff frequency of the side force response function turns out to become

(9.51)

The way the relaxation length changes with slip angle is depicted in the right-hand diagram of Figure 9.7. As one might expect, we find that this length multiplied by the increment in wheel slip angle is equal to the increase of the sum of carcass lateral deflection and the average deflection of the adhering tread elements.

At three different levels of side slip, α_o = 0, 0.08, and 0.16 rad, the comparison of the simulation model with the analytical model has been conducted in terms of the path frequency response functions. Figure 9.8 shows the results. The general conclusion is that, at least for wavelengths larger than ca. 15 cm, the correspondence can be judged to be very good. The upper pair of diagrams that refer to the side force response shows the sideways ‘shift’ of the phase lag curves that is caused by the drop in relaxation length with increasing average slip angle. The diagram for the aligning torque response clearly shows the transition from second- to first-order behavior when the average slip angle changes from zero to larger values.

FIGURE 9.8 Frequency response functions of a linearized system including carcass flexibility at different average slip angle levels according to the analytical solution and to the approximate simulation model. The path frequency at a wavelength of the input wheel plane motion equal to 20 cm has been indicated.

The Eqns (9.45–9.48) can be made applicable to the general case of large slip angle variations by adding to (9.45, 9.46) the steady-state relations and by rewriting the Eqns (9.47, 9.48) in complete nonlinear form so that, when considering a small variation, the linearized equations are recovered. We obtain, for the transient slip quantities,

(9.52)

(9.53)

and, for the force and moment,

(9.54)

(9.55)

The last term representing the extra moment is left in linearized form. This term may be replaced by the difference of two equal functions the derivative of which equals C_ΔM, one with argument and the other with . Because of the fact that the difference of these two arguments remains small, the linearized version is expected to be sufficiently accurate. The block diagram of the nonlinear system displayed in Figure 9.9 may further clarify the structure of the model.

FIGURE 9.9 Block diagram of the nonlinear model of the contact patch (tread) also valid for larger slip angle variations.

To ensure that the above equations correctly describe the response to large slip angle variations, the simulation model results have been compared with the response of a physical model. That model features a finite number of tread elements attached to a straight base line. The deflections of carcass and elements are computed at each time step in which the wheel is rolled further over a distance equal to the interval between two successive tread elements and is moved sideways according to the current value of the input slip angle. The actual model employed contains 20 elements. The parameter values used in both models have been listed in Table 9.1.

TABLE 9.1 Parameter Values Used for Brush Model with Flexible Carcass

The slip angle variation is sinusoidal around a given average level. To cover a broad range of operations, the computations have been conducted at three wavelengths: λ = 0.2, 1, and 5 m, two average slip angle levels: α_o = 0 and 0.08 rad, and one amplitude: .

Figure 9.10 presents the results for the two models. The range of the abscissa has been chosen such that precisely two wavelengths are covered. The distance rolled is large enough to have a situation close to the periodic state. Again, the agreement is quite good and we may have confidence in the model. The bottom diagram presents the variation of the slip angle. The top pair of diagrams shows the responses at the relatively long wavelength of 5 m so that a condition closer to steady state has been reached. From Figure 9.7 it can be seen that at the maximum α of 0.16 rad or almost 9° the aligning torque peak has been surpassed by far. This explains the two dips per wavelength. At a maximum of 0.08 rad the peak has just been surpassed.

FIGURE 9.10 Side force and aligning torque responses of the nonlinear brush simulation model with flexible carcass to sinusoidal slip angle input of the wheel plane with a slip angle amplitude of 0.08 rad and average levels of 0 (thin curves) and 0.08 rad (fat curves), compared with results of the physical model (broken curves).

The less deep dips occurring in the upper curves of diagrams b and d are delayed with respect to the slip angle when this has reached its minimum value equal to zero. At steady state, when λ → ∞, the moment (and the force) would have become equal to zero at that instant. The deeper dips belong to the maxima of the slip angle variation. The considerable reduction in amplitude of the response at the shorter wavelengths agrees nicely with the findings of Figure 9.8.

The force responses correspond almost perfectly with the outcome of the physical model. Evidently, a similar correspondence is expected to occur with the response of the longitudinal force to longitudinal slip variations.

Turn Slip

The last item to be studied in the development of the contact model is the response to variations in path curvature while the slip angle remains zero. We will not attempt to develop a background analytical model but will take a more heuristic route. The results will be checked by comparing these with the computed responses of the physical model. The responses derived for the string model with tread elements as presented in Chapter 5, especially the step responses to φ as depicted in Figure 5.21 (ex.tr.el.) and Figure 5.10 may be helpful. We observe that the force response is similar to that of the aligning torque to a step change in slip angle. In both cases the slope at the start is zero.

The side force relates to the transient turn slip (if linear) as: . The uncorrected further approach of the force to its steady-state level is assumed to occur according to the first-order equation, with σ_c as relaxation length. The initial slope of the uncorrected response curve of the normalized transient turn slip of the contact patch versus s becomes: 1/σ_c. The zero slope at the start may be modeled by subtracting a correction response curve that starts with the same slope but dies out after having reached its peak. Such a short term response may be obtained by taking the difference of two responses, , each leading to the same level but starting at different slopes the difference of which should correspond to the initial slope of the uncorrected force response curve. For simplicity we take for one of the two responses the uncorrected force response, so that σ_F1 = σ_c. The relaxation length of the second response should then be equal to σ_F2 = σ_c/2, so that its initial slope becomes: 2/σ_c. Subtracting these two slopes results in the desired initial slope of the correction curve: 1/σ_c. The resulting equations for the force response to the turn slip velocity read

(9.56)

(9.57)

with σ_F2 = σ_c/2. The transient turn slip for the force finally becomes

(9.58)

The side force at pure path curvature is obtained from the nonlinear steady-state response function:

(9.59)

For the range of turn slip a|φ| < 1/θ the relation remains linear and equals for the brush model:

(9.60)

The moment response may be divided into the response due to the contact patch length that involves lateral tread deflections, and the response due to tread width giving rise to longitudinal deflections. First, we will address the moment generated by the brush model with zero width. Figures 5.9, 5.10, and 5.21 indicate that we are dealing here with a response that after having reached a peak tends to zero. Again we may model this behavior by subtracting two first-order responses. For this, we introduce two transient turn slip quantities and with respective relaxation lengths: σ_φ1 and σ_φ2. The two differential equations become

(9.61)

(9.62)

At zero speed the response of the difference would become

(9.63)

The deflection angle of the tread due to transient spin is defined as

(9.64)

which at zero speed becomes

(9.65)

The condition to be satisfied is that the deflection angle is equal to the yaw angle: . Hence, we have

(9.66)

In the case of small angles, we may write for the moment

(9.67)

From Eqns (9.61 and 9.62), it can be assessed that the initial slope of the response of the moment to a step change in spin from zero to φ_c0 turns out to be

(9.68)

For the second condition to assess the ratios of the σ_φ’s to half the contact length a, the best fit of the remaining course of the step response may serve, cf. Eqns (9.114 and 9.115).

When the angle of rotation ψ continues to grow, the state of total sliding will be attained and the moment can be calculated to become

(9.69)

It has not been tried to derive the functional relationship between moment and increasing steer angle for the standing (nonrolling) brush model. The following nonlinear function to describe the moment response in between the two extremes has been chosen:

(9.70)

The factor ε_φ has been introduced to have a parameter available to better approach the response shown by the physical model. The value 1.15 appeared to be appropriate. Similar to the relaxation length σ_c, Eqns (9.29 and 9.24), the lengths σ_φ1,2 are reduced in proportion with the magnitudes of transient turn slip quantities and .

For the evaluation of the model a comparison with the physical brush model has been executed. First, the flexible carcass is attached to the tread model, cf. Figure 9.5. To simulate this more complex situation, the approach of the enhanced transient model of Section 7.3 has been adopted. The additional dynamic equations for the contact patch with mass m_c and moment of inertia I_c and carcass stiffnesses c_y,ψ and damping coefficients k_y,ψ read, cf. Figure 9.5:

(9.71)

(9.72)

(9.73)

(9.74)

(9.75)

This extension of the model may, of course, also be used for the previously treated response to pure side slip. In fact, it is to be noted that the model for side slip, Eqns (9.52–9.55), should be added to the spin model, Eqns (9.61, 9.62, 9.64, and 9.70), to correctly account for their interaction in the complete model with carcass compliance included. In the physical model the brush model automatically responds to both the side slip and spin. When the wheel plane is subjected to only side slip, the spin of the base line of the tread model remains very small and may be neglected. On the other hand, when the wheel is being steered with wheel side slip remaining zero (path curvature), the base line does show non-negligible side slip, especially at shorter wavelengths, where the moment becomes considerable and as a result the base line is yawed and thus induces side slip. This effect vanishes at steady-state turning. However, if we would add the effect of tread width, the spin torque also acts at steady state and thereby contributes largely to the side force response to spin of the complete model.

Due to the complexity that arises when adding tread width to the brush model, cf. Chapter 3, it has been decided to consider this aspect when dealing with the ultimate model adapted to the use of the Magic Formula in the next section.

The diagrams of Figure 9.11 present the computed responses to varying turn slip. It is seen that the correspondence with the physical model, again with 20 tread elements, is quite good. The deformation of the moment response curve occurring at shorter wavelengths is caused by the extra yaw moment generated through the base line slip angle variation.

FIGURE 9.11 Side force and aligning torque responses of the nonlinear brush simulation model with flexible carcass to sinusoidal path curvature variations (turn slip at α = 0) of the wheel with an amplitude of aφ equal to 0.08 and average levels of 0 (thin curves) and 0.08 (fat curves), compared with results of the physical model (broken curves).

As a reference, the steady-state characteristics of the force and moment response to turn slip, as computed for the single row brush model with and without carcass compliance, have been shown in Figure 9.12. Up to aφ = 1/θ the aligning torque remains zero which causes the characteristics for the cases without and with flexible carcass to become identical. The remaining course of the curves for the system including carcass compliance has not been computed as that part lies outside the range of evaluation.

FIGURE 9.12 Steady-state turn slip force and moment characteristics of the brush model both with and without flexible carcass. The effect of tread width has not been included.

Combined Slip

To cover the case of combined slip, also including longitudinal slip, the steady-state brush model characteristics are to be adapted as formulated in Chapter 3, Section 3.2.3. In addition, the factor m that indicates the fraction of the contact length where adhesion occurs and is used to reduce the relaxation length σ_c is to be adapted by using the composite magnitude of slip:

(9.76)

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 can be assessed:

(9.77)

Maurice (2000) found excellent agreement between the simulation model and the physical model for the combined slip cases: α_o = 0.08 rad and = 0.08 rad and κ_o = 0.06 and = 0.06 with a phase difference of 45° and wavelengths of 0.2, 1, and 5 m. For a more precise treatment with the interaction in the sliding range taken into account as well, we refer to the work of Berzeri et al. (1996). Adding turn slip will influence the combined slip response further. The next section approaches this matter in a pragmatic way.

Steady-State, Step Response, and Frequency Response Characteristics

To demonstrate the performance of the model, a number of typical characteristics will be presented. The hypothetical steady-state pure side slip and pure turn slip characteristics of the model have been given in Figure 9.14.

FIGURE 9.14 Steady-state side slip and turn slip force and moment characteristics of the overall tire model as defined by the Magic Formula. The effect of tread width has been included.

The step response graphs of Figure 9.15 show the proper shapes of the various curves, notably the initial horizontal tangent of the response curves of the side force to turn slip and of the moment to side slip, also shown in Figure 5.21 (no tread width). Also, the peak of the moment response curve to spin and the dip of the curve of the moment response to steer angle are exhibited, as expected.

FIGURE 9.15 Step response curves of the side force and of the aligning torque to side slip α, turn slip (path curvature) φ, and steer angle ψ.

Figure 9.16 presents the frequency response functions of the linearized system at zero average side and turn slip with tread width effect included. The curves may be compared with those of the string model with tread elements, cf. Figure 5.23 (with zero tread width). In Figure 9.17 the Nyquist plots of the moment response to steer angle have been depicted. The upper diagram shows the influence of tread width by changing the parameter . If equal to zero, the thin tire is represented. The value 1 corresponds with the base line configuration. The lower graph provides insight into the influence of the parameter ε_φ12 that governs the magnitude of the effect of the transient yaw deflection angle α_M (9.64, 9.86). The value 4 is used in the base line configuration. When compared with the plots of Figures 5.27 and 5.35, it may be concluded that the model is capable of approaching the responses of more complex infinite order (string) models and of actual tires.

FIGURE 9.16 Frequency response function of system with tread width. Curves for the force and moment responses to side slip α, turn slip (path curvature) φ, and steer angle ψ have been indicated.

FIGURE 9.17 Nyquist frequency response plots of the aligning torque to steer angle. Upper diagram: influence of tread width (4 is base line value), lower diagram: influence of the transient yaw deflection response to turn slip (4 is base line value).

Large Slip Angle and Turn Slip Response Simulations

The same sinusoidal maneuvers have been simulated as was done before with the brush-based contact model. The complete nonlinear set of Eqns (9.78)–(9.120) has been used with parameters listed in Table 9.2.

TABLE 9.2 Parameter Values for Tire Model with Magic Formula Including Quantities Introduced Later on.

The slip angle variation is sinusoidal around average levels α_o = 0 and 0.08 rad with one amplitude: = 0.08 rad at three wavelengths: λ = 0.2, 1, and 5 m. Similarly, the turn slip has two average levels: aφ_o = 0 and 0.08 and one amplitude: , also at wavelengths: λ = 0.2, 1, and 5 m.

The diagrams of Figures 9.18 and 9.19 present the results. We observe that the curves are quite similar to those depicted in the Figures 9.10 and 9.11. Only, as expected, the moment response to turn slip is very much affected by the now introduced effect of the tire tread width.

FIGURE 9.18 Side force and aligning torque responses of the Magic Formula-based simulation model with flexible carcass and finite tread width to sinusoidal slip angle variation of the wheel plane with an amplitude of 0.08 rad and average levels of 0 (thin curves) and 0.08 rad (fat curves).

FIGURE 9.19 Side force and aligning torque responses of the Magic Formula-based simulation model with flexible carcass and finite tread width to sinusoidal path curvature variations (turn slip at α = 0) of the wheel plane with an amplitude of aφ of 0.08 and average levels of 0 (thin curves) and 0.08 (fat curves).

Maurice has conducted extensive experiments with a 205/60R15 91V tire at 2.2 bar inflation pressure on a 2.5 m drum using the pendulum test rig of Figure 9.38. The diagrams of Figure 9.20a present the computed results compared with experimental data for the case of pure side slip. The experiments have been carried out at very low speed to avoid inertia effects of the moving tire.

FIGURE 9.20a Comparison of theoretical model calculations and experimental results performed at 0.6 m/s on a 2.5 m drum (Maurice). The curves cover one wavelength of the periodic responses to sinusoidal slip angle variations. Force response to slip angle variations at two levels of side slip.

The wavelength ranges from 0.3 to 2.4 m. The upper diagram refers to the case of zero average slip angle and 4° amplitude. The lower diagram shows the responses around an average slip angle of 4°, which causes the curves to deviate considerably from the input sinusoidal shape.

The curves clearly show that a shorter wavelength causes the response amplitude to decrease and the phase lag to increase. It can also be observed that the responses occur more quickly at larger levels of slip, which is due to the sharp decrease of the relaxation length with increasing slip. The Magic Formula was used to model the steady-state characteristics. The parameters were obtained from separate tests performed on the drum at the much higher speed of 60 km/h. The different conditions may explain the deviation in level of the calculated responses with respect to the measured ones.

Figure 9.20b presents the comparison with the results of a second series of experiments under the same conditions. Here, the vertical load is changed sinusoidally while the slip angle is kept at a low level and at a higher level.

FIGURE 9.20b Comparison of theoretical model calculations and experimental results performed at 0.6 m/s on a 2.5m drum (Maurice). The curves cover one wavelength of the periodic responses to sinusoidal load variations. Force and moment response to vertical load variations at two values of slip angle.

For these experiments the measuring tower of Figure 9.37 (left) has been employed. The results are similar to those discussed in Chapter 8, Figure 8.9. The moment response seems to be improved with the more complex tire model.

The agreement of the computed results with experimental data is quite good. As has been reported by Maurice, also for the moment response a rather good agreement has been established. Since at the maximum slip angle of 8° the peak of the moment characteristic has been surpassed, the result becomes quite sensitive to small differences between actual and model steady-state characteristics of the aligning torque.

Dimensions of the Contact Area

Prints of the contact patch may be obtained by using ink or carbon paper. The shape appears to change from an oval shape at very low normal loads to a more rectangular shape at higher values of the load. An effective rectangular contact area may be defined with an area equal to that of the envelope of the actual print. The ratio of the width and length of the rectangle is taken equal to that of the actual contact area. The effective half length and half width are denoted as a and b. The dimensions depend on the normal load F_N (=|F_z|) and the following formulas have been found to give a good approximation:

(9.207)

and

(9.208)

with r_o denoting the free tire radius. Figure 9.30 presents the curves compared with the measured effective quantities for the tire pressed on a flat surface and on a curved drum surface with 2.5 m diameter. Obviously, the results are satisfactory. The dimensions of the tire were again: 205/60R15 91V at 2.2 bar inflation pressure. The nondimensional parameter values can be found in Table 9.3. In App. 3(3) an alternative expression for a is presented based on the radial deflection ρ_z instead of on the normal load F_N. The resulting value is much less dependent on the possibly changed inflation pressure.

FIGURE 9.30 Measured and calculated half length a and half width b of the contact patch vs wheel load, for the cases: loading on a flat plate and on a drum surface.

TABLE 9.3 Parameter Values for Contact Patch Dimensions (205/60R15 91V at 2.2 bar)

The Sidewall Stiffnesses and Damping

The rigid ring model of the tire freely rolling and loaded on the road shows three in-plane modes of vibration: the vertical mode and two angular modes. One of these rotational modes vibrates in phase with rim angular vibration while the other moves in anti-phase. The natural frequencies have been estimated with the aid of experiments conducted on the drum test stand where the wheel, at fixed axle position, rolls over a short cleat or is excited by brake torque fluctuations, cf. Section 9.4.2.

The experiments indicate that the natural frequencies lying in the range of 0–100 Hz decrease with velocity. Other researchers found the same tendency, notably Bruni, Cheli and Resta (1996). Since the sidewall stiffnesses are much larger than the residual stiffnesses it is decided to make the in-plane sidewall stiffnesses dependent on the speed of rolling. As to the out-of-plane vibrations, Maurice did not ascertain the necessity to make the lateral, yaw, and camber stiffnesses speed dependent.

Zegelaar introduces a variable quantity Q_V that is a measure of the time rate of change of the loaded tire deformation due to rolling. We have the nondimensional quantity (V_o representing the reference velocity, cf. Section 4.3.2):

(9.209)

The following expressions for the sidewall stiffnesses have been found to be appropriate:

(9.210)

(9.211)

(9.212)

The additional subscript 0 designates the situation of the loaded nonrotating tire. The vertical and longitudinal stiffnesses have been assumed equal to each other. The parameters q_bVx,z,θ govern the speed dependency of the stiffnesses.

The sidewall damping coefficients k_bi = k_{bx,y,z,γ,θ,ψ} are considered to be constant quantities. The interaction terms appearing in Eqns (9.131–9.136) containing the coefficients k_biΩ are omitted since these terms affect the rolling resistance and the aligning torque (also in steady state) and would make these speed dependent. The introduction of material damping being inversely proportional with frequency would be closer to reality. Further on, the rolling resistance will be introduced in an alternative, better controlled way.

For the constant stiffnesses, nondimensional parameters may be introduced. We define with F_No, r_o (=R_o) and m_o (the reference load, free tire radius, and tire mass) the nondimensional parameters q:

(9.213)

(9.214)

(9.215)

(9.216)

To provide more damping when the wheel speed gets close to zero, we may follow the theory of Chapter 7 and introduce k_V,low as demonstrated in Chapter 8, Eqns (8.127, 8.128), where the slip speed V_sx may be replaced by V_sxc. In a similar way the residual stiffness and damping parameters c_c and k_c have been normalized.

The Normal Force

The spring with residual stiffness c_cz indicated in Figure 9.27 hides a structure that is a lot more complex than a spring with constant stiffness. Experiments reveal that the force deflection characteristics are nonlinear: the force develops after contact has been made and increases slightly more than proportionally with the overall normal deflection ρ_z. Also, the tire grows with speed due to centrifugal action. Figure 9.31 illustrates both phenomena. Furthermore, it has been found useful to introduce F_x and F_y interaction terms in the vertical stiffness, cf. Reimpell et al. (1986). The following formula is proposed for the normal force including interaction and overall stiffness and growth functions (also see Eqn (4.E68) for the influence of inflation pressure):

(9.217)

FIGURE 9.31 Vertical load vs normal deflection characteristics at various forward velocities and tire radius growth with speed.

With radius r_c of assumed circular cross-sectional contour, the deflection becomes at wheel camber angle γ (relative to normal to road plane), also cf. Eqn (7.46):

(9.218)

The tire radial growth changes quadratically with rotational velocity:

(9.219)

Here, r_o denotes the radius of the free nonrotating tire, r_l the loaded radius (distance between wheel center and contact center), and Δr the increase in free tire radius due to wheel rotation velocity. The nondimensional parameter q_V1 governs the influence of tire growth, q_V2 the stiffness variation with speed, q_Fcx,y1 the interaction with horizontal forces, and q_Fz1,2,3 the stiffness and nonlinearity of the force deflection characteristic at zero speed and zero horizontal forces. Appendix 3 presents the parameter values fitted to experimental data. To radically simplify (9.218), we may for small γ, replace r_c by r_o−r_l + Δr which is the calculated tire radial deflection.

From the overall characteristics the properties of the residual spring are to be derived. An exact functional relationship may be established but it can be found that the residual normal spring characteristic can be approximated by the third degree polynomial function:

(9.220)

with the F_N-related normal residual spring deflection (taking into account geometrical interaction terms with horizontal deflections)

(9.221)

Here, is the radial displacement of the center of the belt ring with respect to the wheel center and ρ_x,y represent the longitudinal and lateral tire contact deflections. The actual loaded radius r_l results from the calculated deflections (cf. Figure 4.30 for measured evidence) with camber influence included:

(9.222)

The coefficients appearing in (9.220) can be expressed in terms of sidewall stiffness c_bz and wheel speed of revolution Ω:

(9.223)

(9.224)

(9.225)

where, with Reimpell's terms in (9.217) omitted, we obtain

(9.226)

(9.227)

where the nondimensional parameters q_V2 and q_Fz1,2,3 of Eqn (9.217) appear. The horizontal tire deflections at road surface level with respect to the wheel rim are (at small camber):

(9.228)

(9.229)

Appendix 3 provides the relevant parameter values for the passenger car tire that has been tested.

Free Rolling Resistance

Experiments show that the rolling resistance force F_r (pointing backward) is proportional to the tire normal load F_N. A history on this subject can be found in the publication of Clark (1982). We have

(9.230)

The rolling resistance coefficient f_r depends on the forward speed and may be expressed in terms of powers of the speed, cf. Mitschke (1982):

(9.231)

Parameter q_sy1 governs the initial level of the rolling resistance force and typically lies in between 1 and 2%. Parameter q_sy3 controls the slight slope of the resistance with speed. The last parameter q_sy4 represents the sharp rise of the resistance that occurs after a relatively high critical speed is surpassed. Then, the so-called standing waves show up as a result of instability, cf. Pacejka (1981), or according to an alternative theory due to resonance, cf. Brockman and Braisted (1994). The formation of standing waves gives rise to large deflection variations and considerable energy loss. The phenomenon may result in failure of the tire and poses an upper limit to the safe range of operation of the tire.

Below, we will see that the rolling resistance will be introduced in the tire model through the rolling resistance moment that is imposed on the tire belt ring as an external torque about the y axis, cf. Eqn (9.236).

Effective Rolling Radius, Brake Lever Arm, Rolling Resistance Moment

In Chapter 8 the notion of the effective rolling radius has been introduced. Figure 8.12 shows the results of experiments of tires running over a drum surface. In Subsection 8.3.1 the theory is restricted to a linearized representation of the variation of the effective rolling radius with radial defection. The complete nonlinear variation versus normal load F_N may be described by the expression:

(9.232)

with

(9.232a)

and the vertical stiffness of the standing tire at nominal load F_No, as derived from Eqn (9.217):

(9.232b)

In App. 3, Eq. (A3.4), an alternative expression has been given. The longitudinal slip velocity is defined with r_e introduced as the slip radius, cf. e.g. Eqn (8.32).

In the present model with the belt ring and contact patch modeled as separate bodies, the longitudinal slip velocity V_sxc of the contact patch is used as input in the transient slip differential Eqn (9.174). In Eqn (9.192) with (9.155), the effective rolling radius r_e accomplishes the transmission of the rotational speed of the belt to the residual deflection rate of change. At steady-state condition, the deflection rates vanish and we have the following relation for the longitudinal slip speed:

(9.233)

We may consider the power balance of a wheel subjected to a propulsion torque M_D and a drag force F_D acting backward on the wheel in its center. Figure 9.32 depicts the situation. The connected power flow diagram is presented in Figure 9.33. The S represents a power source (the engine) and the Rs are resistors where energy is dissipated. The balance of power requires that the equation holds:

(9.234)

or with (9.233) and F_x = F_D

(9.235)

where represents the energy dissipating moment due to rolling. This equation suggests, at least for the model employed, that the moment arm equals the effective rolling radius (defined at zero driving or braking torque: free rolling). Consequently, the block named ‘tire' in the diagram of Figure 9.33 represents, when unfolding the bond graph, a junction structure containing a transformer with modulus r_e that transforms the angular speed into (a part of) the slip speed and, in the opposite direction, the slip force into the drive torque.

FIGURE 9.32 The driven tire–wheel combination with deflected belt and residual spring.

FIGURE 9.33 Power flow diagram (bond graph) of driven tire–wheel combination in steady state.

Experiments have been carried out by Zegelaar on both the flat plank machine and the drum test stand to establish the effective rolling radius and the moment arm. In these tests, a (brake) torque was applied to the wheel. The moment arm may be termed as the brake lever arm. The diagrams of Figure 9.34 have been obtained from tests performed at zero speed (for assessing the brake lever arm) and at very low speed of travel (for finding the effective rolling radius). Especially in case of the flat surface, an excellent agreement has been found to occur. It is assumed that the growth of the effective rolling radius with speed is equal to that of the free tire radius, cf. Eqn (9.219).

FIGURE 9.34 Tire radii as function of the normal load measured at zero or very low speed.

The diagrams of Figure 9.35 present the influence of speed on the two radii. The loaded radius has been kept fixed so that the vertical load rises when the speed is increased. Three different axle heights have been selected corresponding to the indicated initial vertical loads at zero speed F_No. The left-hand diagram shows the degree of fit for the effective rolling radius. The right-hand diagram shows the correspondence with the brake lever arm. The tests from which the brake lever arm can be assessed have been conducted at low levels of the average and the standard deviation of the brake torque random input (120 and 22 Nm, respectively). The brake lever arm results from the longitudinal force response to the imposed brake torque variation at zero frequency. The influence of the average brake torque on the ratio of the torque amplitude and the force amplitude at zero frequency, –dM_B/dF_x, is given in Figure 9.36. This ratio does not appear to be a constant. Especially at low loads and relatively large braking forces, large deviations arise from the value of the effective rolling radius. Note that, due to its definition, the effective radius is not affected by the magnitude of the brake force.

FIGURE 9.35 Tire radii measured on 2.5 m drum as a function of the forward speed at three axle heights corresponding to the indicated initial loads. Average level and amplitude of brake torque are small.

FIGURE 9.36 Ratio of brake torque increment to brake force increment as a function of level of brake force at a velocity of 25 km/h and three different axle heights (loaded radii).

With the effective radius adopted as the brake or driving torque moment arm, we find from Eqn (9.235) that if M_D = 0 the moment .

For the (forward) rolling tire, this moment is assumed to be independent of F_x and M_D. Furthermore, we obtain, at Ω = 0 where the moment due to rolling , that M_D = F_xr_e. Obviously r_e is the brake lever arm corresponding with the experimental result.

By considering the equilibrium about the wheel axis A, the rolling resistance moment M_y (acting about the C-y axis) becomes

(9.236)

with F_r given by Eqn (9.230). Apparently, at free rolling (M_D = 0) where F_x = −F_r, we correctly have: M_y = F_rr_l.

It is clear that theoretical modeling of the mechanism behind the generation of the rolling resistance moment deserves more study. In Chapter 4 the empirical formula Eqn (4.E70) has been presented for M_y that is based on tire test data.

Dynamic Brake and Cleat Test Rig

Experiments have been conducted on the 2.5 m steel drum test stand provided with a specially designed rig equipped with a disc brake installation, cf. Figure 9.37 (right-hand diagram), and for more details: Chapter 12, Figure 12.7. Brake torque fluctuation tests (Section 9.4.2) and dynamic cleat tests (Section 10.2) have been carried out. The test facilities with numerous experimental and simulation results have been described in detail by Zegelaar (1998).

The wheel axle height can be adjusted to select the tire initial load. During the tests, the axle position is held fixed causing the wheel load to rise with increasing speed. The wheel axle bearing supports are equipped with piezoelectric load cells. Steady-state- or average force levels cannot be measured very well with these force transducers. To provide an indication of the actual load increase with speed as measured in the measurement tower equipped with a hub provided with strain gauges, Table 9.4 gives for a series of initial deflections (ρ_z0 at zero speed) the values of the average vertical force derived from measurements at different speeds. The values have been obtained from Eqns (9.217–9.219) after having fitted the parameters involved. The loads shown apply for the cases of nominal loads 2000, 4000, and 6000 N indicated in the graphs presented in the next section.

TABLE 9.4 Vertical Load on 2.5 m Drum at Constant Axle Height and Increasing Speed (tire: 205/60R15 91V at 2.2 bar)

Pendulum and Yaw Oscillation Test Rigs

To assess the lateral and yaw tire dynamic parameters, two test rigs have been developed. One is the trailing arm ‘pendulum’ test stand with at one end a vertical hinge and at the other the steering head with a piezoelectric measuring hub. At that point the arm is excited laterally up to ca. 25 Hz through a hydraulic actuator, cf. Figure 9.38, and for more details: Chapter 12, Figure 12.8. The rig is useful to assess the overall relaxation length and the gyroscopic couple coefficient, both needed for the simpler transient models treated in Chapter 7. The idea of the pendulum concept originates from Bandel et al. (1989). They designed and used an actual pendulum rig. The natural frequency of the freely swinging trailing arm with a tire rolling on the drum was taken to establish the relaxation length σ. This quantity is the parameter of the first-order differential equation such as is used in Chapter 7. Bandel found that σ increases with speed. However, when using a model in which a belt ring with mass is used, it turns out that the parameters can be kept constant, cf. Vries and Pacejka (1998b). Consequently, tire inertia, notably the gyroscopic couples, gives rise to the speed dependency of the effective relaxation length.

FIGURE 9.38 Principal sketch of the trailing arm ‘pendulum’ test rig exciting the tire almost purely laterally. Frequencies up to ca. 25 Hz, adequate for assessing the tire relaxation length and gyroscopic coupling parameter.

Another rig was developed to investigate the response of the tire subjected to yaw oscillations at frequencies up to ca. 65 Hz. The structure depicted in Figure 9.39 is light and very stiff; also see Figure 12.8. The two guiding members with flexible hinges intersect in the vertical virtual steering axis that is positioned in the wheel center plane (center point steering). A hydraulic actuator is mounted to generate the yaw vibration. The wheel axle is provided with a piezoelectric measuring hub. The tire is loaded by adjusting the axle height above the drum surface. During the test the loaded radius remains constant.

FIGURE 9.39 Principal sketch of the yaw oscillation test rig featuring center point steering. Frequencies up to ca. 65 Hz enabling the assessment of tire out-of-plane inertia and stiffness parameters including residual stiffnesses and rigid modes.

The measuring tower, cf. Figure 9.37, provided with a hydraulic vertical axle positioning installation is used to conduct pure braking and pure side slip tests as well as combined slip experiments at axle height oscillations and radial dynamic stiffness tests up to ca. 15 Hz.

A detailed description of the various side slip test facilities together with a full account of the numerous experiments conducted and the simulation results of the model have been given by Maurice (2000) and for motorcycle tires by Vries and Pacejka (1998a,b).

Vibrational Modes

The vibrational modes of the tire may be assessed through modal analysis of the tire wheel system with axle fixed and tire loaded and/or unloaded. When comparing these results with calculated modes using the parameter values assessed by means of the frequency response functions of the rolling tire, it is found that the stiffnesses found from the dynamic rolling experiments are ca. 30% lower than those estimated from experimental modal analysis, cf. Zegelaar. These differences must be due to the different operational conditions and the larger amplitudes of the vibrations and higher temperatures that occur in the realistic rolling experiments.

The calculated vibrational rigid body modes at zero speed using the parameters as established from experiments carried out on the drum have been depicted in Figures 9.40 and 9.41.

FIGURE 9.40 Calculated in-plane vibrational modes of tire/wheel system with axle fixed and tire free or loaded on the drum surface with vertical load F_N = 4000 N and at zero speed.

FIGURE 9.41 Calculated out-of-plane vibrational modes of tire/wheel system with axle fixed and tire free or loaded on the drum surface with vertical load F_N = 4000 N and zero speed.

We have four in-plane degrees of freedom of the belt ring and the wheel rim (two translational and two rotational) and three out-of-plane degrees of freedom (lateral, yaw, and camber). As a consequence, we can distinguish four in-plane rigid modes and three out-of-plane rigid body modes. The mode shapes change considerably when the tire is making contact with the drum surface. The free rotation (0 Hz) mode changes into a mode with the belt and rim rotating in phase with respect to each other.

The lateral and camber modes appear to form combinations: one low frequency mode with a low axis of camber oscillations and one high frequency mode with a rotation axis closer to the top of the tire and relatively large lateral deflections in the contact zone. The yaw mode in the loaded case shows a higher calculated natural frequency since the effect of turn slip has been included, which was not the case in Maurice’s original model. This means that the yaw stiffness of the contact tread has now been accounted for.

The natural frequencies n and damping ratio ζ change with the speed of rolling. In Table 9.5 the values have been presented for the loaded tire running at a velocity of 0 and 30 m/s. Especially the out-of-plane modes show considerable changes in frequency and damping. The camber and yaw mode natural frequencies which are identical in the unloaded zero speed case, exhibit a with-speed growing mutual difference with the camber mode frequency becoming smaller and the yaw mode frequency larger.

TABLE 9.5 Natural Frequencies and Damping of Vibrational Modes of Rigid Ring Tire Model Calculated at a Vertical Load F_N = 4000 N for Two Values of Forward Speed

Frequency Response Functions

A typical example of measured and calculated in-plane frequency response functions has been depicted in Figure 9.42. Coherence functions show that the tests give sensible results up to ca. 80 Hz. Similar response functions have been obtained by Kobiki et al. (1990). The left-hand diagram of the figure represents the response function of the longitudinal force F_x (=K_aξ; in Eqn (9.195)) acting on the wheel axle to the imposed brake torque variation considered to be applied in the torque meter. The right-hand diagram shows the response of the force to wheel slip variations. The wheel slip is derived from the measured wheel and drum speeds.

FIGURE 9.42 Measured and calculated in-plane frequency response functions at an average braking force of 450 N and assessed at a braking force standard deviation of 75 N.

The two peaks occurring in the left-hand diagram belong to the in-phase and the anti-phase modes. The single peak showing up in the right-hand diagram belongs to the mode that would arise if the rim were fixed also in rotation. The natural frequency lies in between the frequencies of the peaks in the left diagram. The natural frequencies contribute to assess the sidewall stiffnesses.

From the right-hand diagram two quantities can be derived: the slip stiffness and the overall relaxation length. Through the latter, additional information is obtained to find the fore-and-aft residual stiffness. From careful interpretation of the frequency response functions assessed at different speeds, the sidewall stiffness dependence on the speed of rolling has been ascertained. Resulting calculated response functions at different loads and brake torque level gave satisfactory agreement with measured behavior. We refer to the original work of Zegelaar (1998) for detailed information.

The out-of-plane frequency response functions of the side force F_y (=K_aη) and the aligning torque M_z (=T_aζ) to yaw oscillations have been presented in Figures 9.43 and 9.44. The parameters have been assessed by minimizing the difference between measured and calculated (complex) response functions. The correspondence achieved between measured and computed curves at different speeds, loads, and side slip level is quite satisfactory; cf. Maurice (2000) for more details. To conduct a proper comparison, the measured data have been corrected for the inertia of the wheel and part of the tire that moves with the wheel.

FIGURE 9.43 Measured and calculated out-of-plane frequency response functions of the side force and aligning torque to steer angle variations at zero average slip angle for three values of forward speed and normal load F_N = 4000 N.

FIGURE 9.44 Measured and calculated out-of-plane frequency response functions of the side force and aligning torque to steer angle variations at one value of speed and at four levels of average side slip and wheel load F_N = 4000 N.

The expected splitting up of the single peak at low velocity into two peaks, one belonging to the camber mode and the other to the yaw mode, and the growing difference of the two natural frequencies with increasing speed is clearly demonstrated. This phenomenon, which is due to gyroscopic action, has already been observed to occur with the stretched string model with inertia included approximately, cf. Figure 5.40. It is noted that the theoretical results of Maurice have been established by using the model that did not include the equations for the response to spin (9.177–9.180). Especially the aligning moment is sensitive to turn slip.

The moment response curves to side slip and yaw of the massless tire model as depicted in Figure 9.16 are quite different. The dip in the moment amplitude response curve to yaw oscillations occurring in the curve for M_ψ in Figure 9.16 does not appear in the curve for the response to side slip M_α. This dip also appears in the curves of Figure 5.40 where spin is included as well. It is surprising to see that a similar dip occurs in Figures 9.43 and 9.44. The minimum arises here due to the tendency of the response to side slip to decrease and tend to zero at increasing path frequency while at the same time the moment amplitude increases due to tire inertia when approaching the resonance peak. The dip in the measured curve of the upper right-hand diagram of Figure 9.43 at 25 km/h is deeper than the one shown by the theoretical curve of the lower diagram. This may be due to the additional action of spin in the actual tire. As shown by both the theoretical and measured curves of Figure 9.44, the dip disappears altogether at larger average side slip.

The experiments conducted with the trailing arm ‘pendulum’ test rig, where the tire is subjected to almost pure side slip and the spin is very small, also show satisfactory correspondence with model behavior (Maurice 2000; also: De Vries and Pacejka, motorcycle tires, 1998b). This also appears to hold for a limited number of conducted combined slip tests and the response to vertical axle motions at side slip and braking carried out with the measuring tower.

Apparently, the rigid ring model provided with the short wavelength transient slip model is very well capable of describing the dynamic tire behavior in the frequency range up to about 60 Hz. Furthermore, it may be concluded that the spin part is only necessary when dealing with short wavelength, especially low-speed phenomena where tire inertia is less important, such as with parking.

Time Domain Responses

To demonstrate the performance of the model, simulations and experiments have been carried out pertaining to successive stepwise increases in brake pressure and steer angle.

The response of the longitudinal force and the associated wheel speed has been presented in Figure 9.45. The lower diagram clearly depicts the oscillatory variation of the force vs slip ratio. The measured response shows a faster decay of the wheel velocity after the highest brake effort has been reached. Apparently, this is due to the friction coefficient being lower in the experiment than assumed in the model. Finally, the brake is released and the wheel spins up again. The oscillations (ca. 28 Hz) correspond to the in-phase vibrational mode of the system, with the brake disc/axle inertia included.

FIGURE 9.45 Brake force response to successive step increments of brake pressure. The upper diagrams depict the variation of force and wheel speed of revolution with time. The lower figure shows the loops in the force vs wheel slip diagram. Apparently, the actual friction coefficient is lower than assumed in the model. (Zegelaar 1998).

Figure 9.46 shows the responses to successive changes in steer angle for both the side force and the aligning torque at a given load and speed. The responses clearly show vibrations attributed to the yaw/camber mode with natural frequency of ca. 40 Hz (cf. Figure 9.43, 25 km/h). Also, the decrease of the overall relaxation length at larger slip angle can be recognized.

FIGURE 9.46 Side force and aligning torque response to successive step changes in steer angle. (Maurice 2000).