7.2.1 Linear Model

Figure 7.1 depicts the model in top view. At the instant considered, the wheel slip point S (attached to the wheel rim at a level near the road surface) and the contact point are defined to be located in the plane through the wheel axis and normal to the road. These points (which may be thought to lie on two parallel slip circles) move over the road surface with the wheel and contact patch slip velocities, respectively. In the figure, the x and y components of the slip velocities have been shown as negative quantities. The difference of the velocities of the two points causes the carcass springs to deflect. Consequently, the time rates of change of the longitudinal and lateral deflections u and v read

(7.1)

and

(7.2)

FIGURE 7.1 Single-contact-point tire model showing lateral and longitudinal carcass deflections, u and v (top view).

If we assume small values of slip, we may write, for the side force acting from road to contact patch with C_Fα denoting the cornering or side slip stiffness,

(7.3)

It is assumed here that the difference between the wheel center longitudinal velocity V_x and the longitudinal velocity V_cx of the contact center is negligible:

(7.4)

Consequently, we may employ V_x in the present chapter. With the lateral tire stiffness at road level C_Fy, we have, for the elastic internal force that balances the side slip force,

(7.5)

and we can write, for Eqn (7.2) with (7.3, 7.5) after having introduced the relaxation length for side slip σ_α,

(7.6)

the differential equation for the lateral deflection due to side slip v_α (later on we will also have a lateral deflection due to camber):

(7.7)

where α is the wheel slip angle: . The side force is obtained by multiplying v_α with C_Fy.

In a similar way, we can deal with the longitudinal force response. With the longitudinal tire stiffness C_Fx at road level and the longitudinal slip stiffness C_Fκ, we obtain, for the relaxation length σ_κ,

(7.8)

and we can derive with Eqn (7.1) the differential equation for the fore and aft deflection u:

(7.9)

with κ the longitudinal wheel slip ratio: κ ≈ −V_sx/|V_x|. The longitudinal force may be obtained by multiplying u with the stiffness C_Fx.

Next, we consider wheel camber as input. For a suddenly applied camber angle γ (about the line of intersection at ground level!), we assume that a contact line curvature and thus the camber thrust C_Fγγ are immediately felt at the contact patch. As a reaction, a contact patch side-slip angle is developed that builds up the lateral carcass deflection v = v_γ. Again Eqn (7.5) applies. The side force that acts on the wheel now becomes

(7.10)

With wheel side slip kept equal to zero, , and , Eqn (7.2) can be written in the form

(7.11)

This equation shows that according to this simple model the camber force relaxation length σ_γ is equal to the relaxation length for side slip σ_α. This theoretical result is substantiated by careful step response experiments, performed by Higuchi (1997) on a flat plank test rig (cf. Section 7.2.3).

A similar equation results for the total spin φ including turn slip and camber:

(7.12)

with, according to (4.76),

(7.13)

that shows that the turn slip velocity can be converted into an equivalent camber angle.

The forces and moment are obtained from the deflections u and v by first assessing the transient slip quantities and from these with the slip stiffnesses the forces and moment.

According to the steady-state model employed here, different from Eqn (4.E71,72), the moment response to camber (and turn slip) is the sum of the residual torque, M_zr, supposedly mainly due to finite tread width, and −t_α F_y, supposedly caused by camber induced side slip, cf. discussion later on (below Eqn (7.40)). A first-order approximation for the response of M_zr with the short relaxation length equal to half the contact length a may be employed. This, however, will be saved for Chapter 9 where short wavelength responses are considered. Here we suffice with the assumption that the moment due to tread width responds instantaneously to camber and turn slip.

For the linear, small-slip condition, we find

(7.14)

(7.15)

(7.16)

and similar for φ. The pneumatic trail due to side slip is denoted here by t_α. The total aligning torque becomes (cf. Eqn (4.E71) and Figure 4.21)

(7.17)

In an alternative model, used for motorcycle dynamics studies, the terms with t_αC_Fγ or t_αF_yγ in (7.16, 7.17) are omitted, cf. discussion below Eqn (7.40).

Eqns (7.7, 7.9, 7.11) may be written in terms of the transient slip quantities. For example, we may express v_α in terms of by using the first equation of (7.14). Insertion in (7.7) gives

(7.18)

If we recognize the fact that the relaxation length is a function of the vertical load and if the average slip angle is unequal to zero, an additional term shows up in the linearized equation (variation of and of F_z(t) are small!) which results from the differentiation of v_α = σ_αα′ with respect to time. Then, Eqn (7.7) becomes

(7.19)

Obviously, when using Eqn (7.18) a response to a variation of the vertical load cannot be expected. If the load varies, Eqn (7.18) is inadequate and the original Eqn (7.7) should be used or the corresponding Eqn (7.19).

With (7.5), Eqn (7.7) may be written directly in terms of F_y. If we may consider the carcass lateral stiffness C_Fy virtually independent of the wheel load F_z, we obtain, by using Eqn (7.6),

(7.20)

Since we have the same relaxation length for both the responses to side slip and camber, this equation appears to hold for the combined linear response to the inputs α and γ or φ. In the right-hand member, F_yss denotes the steady-state response to these inputs and possibly a changing vertical load at a given slip condition. Multiplication with the pneumatic trail produces the moment (if γ = φ = 0). A similar differential equation may be written for the F_x response to κ.

Eqns (7.7, 7.9) have been written in the form (V_x not in denominator and V_sx,y used as right-hand member) that makes them applicable for simulations of stopping and starting from zero speed occurrences. At speed V_x = 0, Eqn (7.9) turns into an integrator: . With Eqn (7.15), the longitudinal force becomes which with (7.8) is equal to C_Fxu. This is the correct expression for the tire that at standstill acts like a longitudinal or tangential spring. When the wheel starts rolling, the tire gradually changes into a damper with rate: C_Fκ/|V_x|. Figure 7.2 depicts a corresponding mechanical model with spring and damper in series. It shows that at low speed the damper becomes very stiff and the spring dominates. At higher forward velocities, the spring becomes relatively stiff and the damper part dominates the behavior of the tire. A similar model may be drawn for the transient lateral behavior. It may be noted that Eqn (7.20) is not suited for moving near or at V_x = 0. In Section 8.6, the use of the transient models for the response to lateral and longitudinal wheel slip speed at and near zero speed will be demonstrated.

FIGURE 7.2 Mechanical model of transient tangential tire behavior.

Apparently, Eqn (7.12) with (7.13) fails to describe the response to variations in wheel yaw angle ψ at vanishing speed. Then, the lateral deflection becomes . With , we have indicating an instantaneous response of F_y which, however, should remain zero! We refer to Chapter 9, Section 9.2.1, Eqn (9.56 etc.), that suggests a further developed model that can handle this situation correctly. Consequently, it must be concluded that the present transient model cannot be employed to simulate parking maneuvers unless F_y is suppressed artificially in the lower speed range.

7.2.2 Semi-Non-Linear Model

For the extension of the linear theory to cover the non-linear range of the slip characteristics, it may be tempting to employ Eqn (7.20) and use the instantaneous nonlinear force response as input in the differential equation. The input steady-state side force is calculated, e.g. with the Magic Formula, using the current wheel slip angle α. This method, however, may lead to incorrect results as, due to the phase lag in side force response, the current (varying) wheel load may not correspond to the calculated magnitude attained by the side force. In limit conditions, the tire may then be predicted to be still in adhesion while in reality full sliding occurs. A better approach is to use the original Eqn (7.7) and to calculate the side force afterward by using the resulting transient slip angle as input in the Magic Formula.

In general, we have the three Eqns (7.7, 7.9, 7.11) and possibly (7.12) and the first equations of (7.14–7.16) producing , and or which are used as input in the nonlinear force and moment functions (γ or φ directly in the expressions for M_r), e.g. the equations of the Magic Formula tire model (Chapter 4):

(7.21)

(7.22)

(7.23)

(7.24)

(4.E71)

where, if required, γ may be replaced by φ as the spin argument.

This nonlinear model is straightforward and is often used in transient or low-frequency vehicle motion simulation applications. Starting from zero speed or stopping to standstill is possible. However, as has been mentioned before, at V_x equal or close to zero, Eqns (7.7, 7.9) act as integrators of the slip speed components V_sx,y, which may give rise to possibly very large deflections. The limitation of these deflections may be accomplished by making the derivatives of the deflections u and v equal to zero, if (1) the forward wheel velocity has become very small (<V_low) and (2) the deflections take values larger than physically possible. This can be seen to correspond with the combined equivalent side slip value exceeding the level α_sl where the peak horizontal force occurs. Approximately, we may adopt the following limiting algorithm with the equivalent slip angle according to Eqn (4.E78):

(7.25)

with roughly

Experience in applying the model has indicated that starting from standstill gives rise to oscillations which are practically undamped. Damping increases when speed is built up. To artificially introduce some damping at very low speed, which with the actual tire is established through material damping, one might employ the following expression for the transient slip , as suggested by Besselink, instead of Eqn (7.15):

(7.26)

The damping coefficient k_Vlow should be gradually suppressed to zero when the speed of travel V_x approaches a selected low value V_low. Beyond that value the model should operate as usual. In Chapter 8, Section 8.6, an application will be given. A similar equation may be employed for the lateral transient slip.

Another extreme situation is the condition at wheel lock. At steady state, Eqn (7.9) reduces to

(7.27)

which indicates that the deflection u according to the semilinear theory becomes as large as the relaxation length σ_κ. Avoiding the deflections from becoming too large, which is of importance at e.g. repetitive braking, calls for an enhanced nonlinear model. Another shortcoming of the model that is to be tackled concerns the experimentally observed property of the tire that its relaxation length depends on the level of slip. At higher levels of side slip, the tire shows a quicker response to additional changes in side slip. This indicates that the relaxation lengths decrease with increasing slip.

7.2.3 Fully Nonlinear Model

Figure 5.29, repeated here as Figure 7.3, shows the deflected string model provided with tread elements at various levels of steady-state side slip. Clearly, this model predicts that the ‘intersection’ length σ^∗ decreases with increasing α that is: when the sliding range grows. In the single-point-contact model, we may introduce a similar reduction of the corresponding length. The intersection length is defined here as the ratio of the lateral deflection v_α and the transient lateral slip tan :

(7.28)

FIGURE 7.3 The string tire model with tread elements at increasing slip angles showing growing sliding range and decreasing ‘intersection’ length σ^∗.

In Eqn (7.7), the relaxation length is replaced by and we get

(7.29)

with apparently

(7.30)

with the initial relaxation length (at ):

(7.31)

to which approaches when . To avoid singularity, one may use the last expression of (7.30) with small ε_F and add to the shift Δα to arrive at as indicated in Figure 4.22. Figure 7.4 presents the characteristic of the intersection length together with the side force characteristic from which it is derived. Also in the equation for the camber deflection response (7.11), the relaxation length σ_α is replaced by .

FIGURE 7.4 Characteristics of tire side force, ‘intersection length’ , and relaxation length σ_α.

A more direct way to write Eqn (7.29) is yielded by eliminating with the use of Eqn (7.28):

(7.32)

The transient slip angle is obtained from the deflection v_α by using the inverse possibly adapted characteristic, cf. Higuchi (1997):

(7.33)

To avoid double valued solutions for this purpose an adapted characteristic for in (7.30) or (7.33) may be required showing a positive slope in the slip range of interest. The value for or for , when Eqn (7.29) is used, is obtained through iterations or by using information from the previous time step, cf. Higuchi (1997), Pacejka and Takahashi (1992), and Takahashi and Hoshino (1996). The ultimate value of the force F_y is finally obtained from Eqn (7.22) by using the computed .

Writing Eqn (7.32) entirely in terms of the transient slip angle by using (7.5) with v = v_α and remembering that F_y = F_y(α′, F_z) directly yields

(7.34)

The additional input dF_z/dt requires information of the slope at given values of the slip angle. If the vertical load remains constant, the last term vanishes and we have the often used equation of the restricted fully nonlinear model:

(7.35)

with

(7.36)

If we consider an average slip angle α_o and a small variation of tan α and the corresponding lateral slip velocities, Eqn (7.35) becomes, after having subtracted the average part,

(7.37)

which indicates that the structure of (7.35) is retained and that σ_α (7.36) represents the actual relaxation length of the linearized system at a given load and slip angle. Its characteristic has been depicted in Figure 7.4 as well. Obviously, the relaxation length is associated with the slope of the side force characteristic. It also shows that the relaxation length becomes negative beyond the peak of the side force characteristic which makes the solution of (7.35) but also of the original Eqn (7.29) or (7.32) unstable if the point of operation lies in that range of side slip. We may, however, limit σ_α downward to avoid both instability and excessive computation time: σ_α = max(σ_α, σ_min). The transient response of the variation of the force proceeds in proportion with the variation of as .

When Eqn (7.35) is used, an algebraic loop does not occur as is the case when Eqn (7.29) is employed. The relaxation length σ_α can be directly determined from the already available . However, since the last term of (7.34) has been omitted, Eqn (7.35) has become insensitive to F_z variations.

Similar functions and differential equations can be derived for the transient response to longitudinal slip. We obtain, for the distance factors,

(7.38)

and

(7.39)

It is of interest to note that the extreme case of wheel lock can now be handled correctly. As we have seen below Eqn (7.27), the deflection then becomes equal to minus the relaxation length which according to the fully nonlinear model becomes . With (7.38) and , the deflection takes the value that would actually occur at wheel lock: u = F_x/C_Fx.

The use of Eqns (7.29, 7.32) is attractive for simulation purposes because the solution contains the transient effect of changing vertical load. However, we may have computational difficulties to be reckoned with which may require some preparations. Using Eqn (7.35) proceeds in a straightforward manner if the derivative of the force vs slip characteristic is known beforehand. This requires some preparation and may become quite complex if the general combined slip situation is to be considered. Approximations using equivalent total slip according to Eqn (4.E78) may be realized. A drawback, of course, is the fact that Eqn (7.35) does not respond to changes in F_z. In Chapter 9 this equation is used to handle the transient response of the tread deflections in the contact patch.

Figures 7.5–7.12 present results obtained by Higuchi (1997) for a passenger car 205/60R15 tire tested on a flat plank machine (TU-Delft, cf. Figure 12.6). Figure 7.5 shows the response of the side force to step changes in slip angle at the nominal load of 4000 N. The diagram clearly indicates the difference in behavior at the different levels of side slip with a very rapid response occurring at α = 10°. The fully nonlinear model according to Eqns (7.32) or (7.29) or as in this case, with constant vertical load, Eqn (7.35), in conjunction with Eqns (7.22, 7.23) gives satisfactory agreement. The measured response curves have been corrected for the side force variation that arises already at zero slip angle. The test is performed by loading the tire after the slip angle (steer angle) has been applied. Subsequently, the plank is moved. The aligning torque behaves in a similar manner although a little initial delay in response occurs as correctly predicted by the string model and shown in Figure 5.12. At larger slip angles (beyond the level where the M_z peak occurs), the aligning torque first shows a peak in the response after which the moment decays to its steady-state value. This behavior is nicely followed by the model. Figure 7.6 shows the side force response to small increments in slip angle. The experiment is conducted by, after having stopped the plank motion, steering the wheel half a degree further and continue the forward motion. It was first ascertained that the response of the side force to a step steer input was hardly influenced by having first lifted the tire from the road surface or not. Of course, the aligning moment response, in the latter case, is quite different because of initial torsion about the vertical axis. The resulting side force responses as depicted in Figure 7.6 have been compared with the exponential response of the side force increment as predicted by the solution of Eqn (7.37) with (7.36). Fitting with a least-square procedure gave the relaxation length σ_α as presented in Figure 7.7. The resulting variation of σ_α with slip angle level is compared with the slope of the steady-state side force characteristic, both normalized making their values equal to unity at vanishing slip angle.

FIGURE 7.5 Side force response to small and large step change in slip angle as assessed by flat plank experiments and computed with the model defined by Eqns (7.32) or (7.29).

FIGURE 7.6 Measured side force response to small increment of slip angle Δα at different levels of slip angle indicating the decrease of relaxation length at increasing side slip level which is in agreement with expression (7.36).

FIGURE 7.7 Correspondence between F_y vs α slope and relaxation length σ_α as assessed by experiments.

FIGURE 7.8 Relaxation length measured at various wheel loads, decaying with increasing slip angle.

FIGURE 7.9 The slope of F_y vs F_z as a function of slip angle, appearing in Eqn (7.34), as assessed by tests.

FIGURE 7.10 Relaxation lengths resulting from small step changes in slip angle or wheel load, as a function of slip angle level.

FIGURE 7.11 (a) Side force response to step change in camber angle. (b) Moment response to step change in camber angle.

FIGURE 7.12 Relaxation length for step responses to different camber angles (abscissa) at various wheel loads (camber after loading).

The excellent agreement supports the theoretical findings. Additional experiments have been conducted by Higuchi investigating the response to small increments in vertical load at constant slip angle. As shown by Eqn (7.34) the partial derivative plays the role as input parameter. Figure 7.9 shows its variation with slip angle level. Figure 7.10 compares the resulting relaxation length values with those associated with changes in slip angle at constant load. Apparently, the relaxation length for slip angle change is larger than the one belonging to load change. However, due to the finite magnitude of the increments (0.5° and 400 N respectively), the actual differences are expected to be smaller. In the diagram, the curve for Δα may be better shifted to the left over 0.25° while the curve for ΔF_z may be reduced a little in height using information from Figure 7.8 accounting for a decrease of the average load level of 200 N.

Higuchi also investigated the responses to changes in camber angle, and to a limited extent also to the related turn slip. The change in camber was correctly established by rotating the road surface (the plank) about the line of intersection of wheel center plane and plank surface (Figure 12.5, steer angle is kept equal to zero). The step response to turn slip φ_t = −1/R is obtained by integrating the response to a pulse change in turn slip, that is: load tire, twist (ψ) and then roll in the new wheel plane direction, and multiply the result with 1/(Rψ). The responses to camber and turn slip show quantitatively similar responses, cf. Pacejka (2004). The small initial delay of the response of the car tire side force to camber and turn slip (cf. Figures 5.12, 5.10) will effectively increase the relaxation length a little. Figures 7.11a and 7.11b show the responses of side force and moment to step changes in camber angle. The side force behaves in a manner similar to the response to side slip, Figure 7.5. Since we have a camber stiffness much smaller than the cornering stiffness, the camber force characteristic remains almost linear over a larger range of the camber angle (meaning: there is less sliding in the contact patch). Therefore, the difference in step responses remains relatively small. This is reflected by the diagram of Figure 7.12 showing the variation of the resulting relaxation length. Comparison with Figure 7.8 reveals that the relaxation lengths assessed for the responses to side slip (at small side slip) and camber have comparable magnitudes. This supports the theory of Eqn (7.11).

Comparison of the Figures 5.12 (lower left diagram) and 7.11a may reveal the quantitative difference in steady-state side force response to turn slip and camber angle. According to Eqn (3.55), the relationship between turn slip stiffness and camber stiffness will be: C_Fγ = (1 − ε_γ)C_Fφ/r_e. With the tire effective rolling radius equal to approximately 0.3 m and the estimated steady-state levels reached in Figures 5.12 and 7.11a for R = 115 m and γ = 2° respectively, we find, for the reduction factor approximately, ε_γ = 0.5.

The response of the aligning torque to camber is similar to the response to turn slip (cf. Figure 5.12). The moment quickly reaches a maximum after which a slower decay to the steady-state level occurs. The string model with tread width effect predicts the same for the response to turn slip as can be concluded by adding the curves for and (response to φ, Figure 5.10) in an appropriate proportion. The model developed in the present chapter generates a similar response but through a different mechanism. Eqn (4.E71) that for F_x = 0 takes the form

(7.40)

is used after having computed the transient slip and camber angles and .

As can be seen in, Figure 4.13 or 4.21, the aligning torque caused by camber at zero slip angle is attributed to the residual torque M_zr and to the counteracting moment equal to the aligning stiffness C_Mα times the camber induced slip angle Δα_γ which is the same as −t_α F_y at α = 0. The residual torque responds quickly with a relatively short relaxation length (about equal to half the contact length) or as has been suggested above for the present model instantaneously. As the side force responds slowly with relaxation length σ_α and −t_α F_y is opposite in sign with respect to the residual torque, a similar response as depicted in Figure 7.11b will be developed by the model represented by Eqn (7.40).

The mechanism behind this response is supported by the physical reasoning that the moment due to tread width, that responds relatively fast to changes in camber and turn slip, gives rise to a yaw torsion of the carcass/belt in the contact zone which acts as a slip angle. This side slip generates a side force that acts in the same direction as the camber force generated by the camber induced spin. In addition, an aligning moment is generated that acts in a sense opposite to that of the tread-width camber (spin) moment. Ideally, in Figure 4.21, the moment due to camber spin is equal to the residual torque M_zr while −C_Mα Δα_γ = t_α F_y is equal to the moment due to the yaw torsion-induced side slip. However, in reality, only a part of the camber force F_y results from camber spin-induced side slip. It is expected that the above analysis is partly true, perhaps even for an appreciable part. The remaining part that is responsible for the ‘hump’ in the moment response must then be due to the transient asymmetric lateral tire deformation that vanishes when the steady-state condition is reached. Figure 5.9 depicts the nature of this transient deflection. This concept has been followed in Chapter 9.

In the model of Chapter 4 (used in Chapter 9), that was originally introduced for motorcycle dynamics studies dealing with possibly large camber angles, the other extreme is used and the term t_α F_y is replaced by t_αF_yα, that is: with the side force attributed to the (transient) side slip angle alone (cf. Section 4.3.2 where ).

7.2.4 Nonlagging Part

The force response to a change in camber exhibits a peculiar feature. From low-velocity experiments conducted on the flat plank machine (cf. Figure 12.5), it turns out that directly after that the wheel is cambered (about the line of intersection), a side force is developed instantaneously. This, obviously, is caused by the nonsymmetric distortion of the cross section of the lower part of the tire. This initial ‘nonlagging’ side force that occurs at a distance rolled s = 0, as shown in Figure 7.11a, appears, for the tire considered, to act in a direction opposite to the steady-state side force. Equal directions turn out to occur also, e.g., for a motorcycle tire, cf. Segel and Wilson (1976). Also in these reported experiments, the camber angle is applied after the tire has been loaded. Figure 7.13 gives the percentage of the nonlagging part with respect to the steady-state side force for three wheel loads and for three different ways of reaching the loaded and cambered condition before rolling has started. In the first case (Z), the free wheel is first cambered and then loaded by moving it toward the horizontal road surface in vertical direction. This sequence appears to result in a somewhat larger nonlagging part. In the second case (R), the road surface is cambered first after which the tire is loaded vertically. It turns out that now a response arises with the same sign as the steady-state response. The diagram indicates that also in case (C), the sign may remain unchanged if the wheel load is sufficiently high.

FIGURE 7.13 Nonlagging part of side force response for the cases: (Z) loading the tire vertically after having cambered the wheel, (R) loading the tire radially after having cambered the wheel, or loading the tire vertically after having cambered the road, and (C) applying wheel or road camber after having loaded the tire. Upper-right diagram: New additional side-slip point S_y1 with side-slip velocity component V_sy1.

To simulate the development of the nonlagging side force, the following model is suggested. The tire response that arises due to loading and/or tilting of the wheel while V_x = 0 is considered to be the result of the integrated lateral horizontal velocity of the lower part of the wheel. Besides the lateral velocity V_sy of the contact center C, the lateral velocity V_sy1 of a newly introduced slip point S_y1 that is thought to be attached to the wheel at a radius r_sy, is used as an additional component of the effective lateral slip speed V_sy,eff. The upper-right diagram of Figure 7.13 depicts the situation. We define

(7.41)

with ε_c being the participation factor. The effective lateral slip speed replaces V_sy in Eqn (7.7). In the case of a horizontal flat road surface, the velocities of the two points C and S_y1 become

(7.42a)

(7.42b)

The location of S_y1 is defined through the proposed function for the slip radius:

(7.43)

The participation factor ε_c is defined by the proposed function:

(7.44)

It may be seen from Eqns (7.42a, 7.42b) that at constant camber angle and a purely vertical axle motion (case Z), V_sy1 = V_ay = 0, and V_sy = V_az tan γ is the governing component of the effective slip speed (7.41). If the loading is conducted by a radial approach of the road surface (case R), we have V_sy = 0, and the governing part is V_sy1 = V_ay = −V_az tan γ. The third case C is achieved by first vertical loading of the upright tire, V_sy = V_sy1 = 0, and subsequently tilting the wheel about the line of intersection that is: about point C. In the latter phase, the lateral slip velocity components become V_sy = 0 and V_sy1 = (r_l − r_sy)(dγ/dt)cos γ.

For the tire parameter values (C_Fz and C_Fy possibly considered functions of γ): C_Fz = 200 kN/m, C_Fy = 130 kN/m, F_zo = 4 kN, r_o = 0.3 m, r_c = 0.15 m, and for Eqns (4.E19–4.E30) with ζ′s and λ′s = 1: p_Cy1 = 1.3, p_Dy1 = 1, p_Ey1 = −1, p_Ky1 = 15, p_Ky2 = 1.5, p_Ky3 = 6, p_Ky4 = 2, p_Ky6 = 1, p_Vy3 = 1 and remaining p′s = 0, the following values for p_NL1. p_NL8 were assessed through a manual fitting process: p_NL1 = 2.5, p_NL2 = 0.8, p_NL3 = 0, p_NL4 = 3, p_NL5 = 1, p_NL6 = 2, p_NL7 = −2.5, p_NL8 = 10. With these values, the responses computed for the nonlagging part of the side force show reasonable correspondence with the experimental results of Figure 7.13. See Exercise 7.1 for parameters of a motorcycle tire.

The vertical load has been calculated using a tire model with a circular contour of the cross section with radius r_c. For the more general case of an elliptic contour, cf. Figure 7.14, the following equations apply. We have, for the coordinates of the lowest point,

(7.45)

FIGURE 7.14 Cross-section contour with elliptical shape.

The vertical compression ρ_z, which is the distance of the lowest point of the ellipse to the road surface if this distance is non-negative, now reads with r_o the free tire radius and r_l the loaded tire radius:

(7.46)

In the simpler case of a circular contour (r_c = a = b), the formula reduces to

(7.47)

The normal load is now calculated as

(7.48)

For a wheel subjected to oscillations, the observed tire properties will be of importance especially when loss of road contact occurs. In Exercise 7.1 given at the end of this chapter and in Section 8.2, this problem is addressed. The more complex case of moving over short obstacles, exhibiting forward and transverse road slope variations, is treated in Section 10.1.6, Eqns (10.33, 10.34).

7.2.5 The Gyroscopic Couple

In Chapter 5 the gyroscopic couple that arises as a result of the time rate of change of the average tire tilt deflection angle has been introduced. This angle is considered to be proportional with the lateral tire deflection v or the side force F_y. Equation (5.178) may be written in terms of the deflection. With (5.179) and with the F_z/F_zo factor added to give smoother results for a jumping tire, we get

(7.49)

where at free rolling the wheel speed of revolution equals the forward velocity divided by the effective rolling radius of the tire. In general, we have

(7.50)

For a radial ply steel-belted car tire, the nondimensional coefficient c_gyr has been estimated to take the value 0.5. The quantity m_t represents the mass of the tire. Extending the expression (7.40) yields for the total aligning moment:

(7.51)