3.1 Introduction

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

Models of the carcass with belt and side walls with encapsulated pressurized air that are commonly encountered in the tire modeling literature are based either on an elastic beam or on a stretched string both suspended on an elastic foundation with respect to the wheel rim. The representation of the belt by a beam instead of by a stretched string is more difficult because of the fact that the differential equation that governs the lateral deflection of the belt under the action of a lateral force becomes of the fourth instead of the second order. For the study of steady-state tire behavior, most authors approximate the more or less exact expressions for the lateral deflection of the beam or string.

As an extension to the original ‘brush’ model of Fromm and of Julien (cf. Hadekel (1952) for references) who did not consider carcass compliance, Fiala (1954) and Freudenstein (1961) developed theories in which the carcass deflection is approximated by a symmetric parabola. Böhm (1963) and Borgmann (1963), the latter without the introduction of tread elements, used asymmetric approximate shapes determined by both the lateral force and the aligning torque. Pacejka (1966, 1981) established the steady-state side-slip characteristics for a stretched-string-tire model without and with the inclusion of tread elements attached to the string. The lateral stiffness distribution as measured on a slowly rolling tire in terms of influence or Green’s functions (cf. Savkoor 1970) may be employed in a model for the side-slipping tire possibly in connection with the tread-element-following method that was briefly discussed in the preceding section (cf. Pacejka 1972, 1974) and will be demonstrated later on in Section 3.3 of the present chapter.

Frank (1965a) has carried out a thorough comparative investigation of the various one-dimensional models. He employed a general fourth-order differential equation with which tire models can be examined that feature a stretched string, a beam, or a stretched beam provided with elastic tread elements. Frank obtained a solution of the steady-state slip problem with the aid of a special analog computer circuit. A correlation with Fourier components of the measured deformation of real tires reveals that the stretched string-type model seems to be more suitable for the simulation of a bias ply tire, whereas the beam model is probably more appropriate for representing the radial ply tire.

Figure 3.1 (from Frank 1965b) presents the calculated characteristics of several types of carcass models provided with tread elements. The curves represent: a. stretched string model, b. beam model, c. approximation based on Fiala’s model (symmetric parabolic carcass deflection), d. model of Fromm (brush model with rigid carcass). The tread element stiffness is the same for each model. The parameters in the cases a, b, and c have been chosen in such a way as to give a best fit to experimental data for the peak side force and the cornering force at small slip angles (that is, same cornering stiffness). It appears then that model c shows close correspondence with curve a for the side force. Curves d show the result when the carcass elasticity is neglected and only the flexibility of the tread elements is taken into account. When the tread element stiffness of model d is adapted (i.e., lowered) in such a way that the cornering stiffness becomes equal to that of the other three models, no difference between the side force characteristics according to Fromm’s and Fiala’s models appears to occur. Due to approximations introduced by Fiala, the coefficients in the expression for the side force versus slip angle (if parabolic pressure distribution is adopted) become equal to those obtained directly by Fromm.

FIGURE 3.1 Comparison of calculated characteristics for four different tire models with tread elements and nonsymmetrical pressure distribution at a given wheel vertical load (a: string, b: beam, c: Fiala, d: brush i.e., with rigid carcass) (from Frank 1965a,b).

In the calculations for Figure 3.1, Frank employed a constant coefficient of friction μ and a slightly asymmetric vertical pressure distribution q_z(x), found from measurements. The positive aligning torque obtained at larger values of the slip angle α arises as a result of this asymmetry. The phenomenon that in practice the aligning torque indeed varies in this way is due to a combination of several effects. The main cause is probably connected with the asymmetric pressure distribution of the rolling tire (due to hysteresis of the tire compound) resulting in a small forward shift of the point of application of the normal load (giving rise to rolling resistance) and, consequently, at full sliding also of the resulting side force. Another important factor causing the moment to become positive is the fact that the coefficient of friction is not a constant but tends to decrease with sliding velocity. As may be derived from e.g., Eqn (2.59), the sliding velocity attains its largest values in the rear portion of the contact area where the slope becomes largest. Consequently, we expect to have larger side forces acting in the front half of the contact area at full sliding conditions than in the rear half. The rolling resistance force that due to the lateral distortion acts slightly beside the wheel plane may also contribute to the sign change of M_z. A μ that decreases with the sliding velocity (not considered by Frank) causes the creation of a peak in the F_y(α) curves and a further slight decay. This has often been observed to occur in practice, especially on wet and icy roads. In the longitudinal force characteristic the peak is usually more pronounced.

The influence of different but symmetric shapes for the vertical force distribution along the x-axis has been theoretically investigated by Borgmann (1963). He finds that, especially for tires exhibiting a relatively large carcass compliance, the influence of the pressure distribution is of importance. Nonsymmetric more general distributions were studied by Guo (1994). Many authors adopt, for the purpose of mathematical simplicity, the parabolic distribution (Fiala 1954, Freudenstein 1961, Bergman 1965, Pacejka 1958, Sakai (also n-th degree parabola, 1989), Dugoff et al. 1970 (uniform, rectangular distribution) and Bernard et al. 1977 (trapezium shape)). Models of Fiala and Freudenstein feature a flexible carcass while the remaining authors have restricted themselves to a rigid carcass or a uniformly deflected belt. That, however, enabled them to include the description of the more difficult case of combined slip. The introduction of a nonconstant friction coefficient has been treated by others: Böhm (1963), Borgmann (1963), Dugoff et al. (1970), Sakai (1981), and Bernard et al. (1977).

Figure 3.1 shows that, when the model parameters are chosen properly, the choice of the type of carcass model (beam, string, or rigid) has only a limited effect. Qualitatively, the resulting curves are identical. The rigid carcass model with elastic tread elements is often referred to as the brush tire model. Because of its simplicity and qualitative correspondence with experimental tire behavior, we will give a full treatment of its properties, mainly to provide understanding of steady-state tire slipping properties which may also be helpful in the development of more complex models. A uniform carcass deflection will be considered to improve the aligning torque representation at combined slip.

In Section 3.3, we deal with the effect of nonuniform carcass deflection, nonconstant friction coefficient, and the inclusion of camber and turning (path curvature) combined with side slip and braking/driving. For this purpose, the tread simulation model will be employed.

3.2 Tire Brush Model

The brush model consists of a row of elastic bristles that touches the road plane and can deflect in a direction parallel to the road surface. These bristles may be called tread elements. Their compliance represents the elasticity of the combination of carcass, belt, and actual tread elements of the real tire. As the tire rolls, the first element that enters the contact zone is assumed to stand perpendicularly with respect to the road surface. When the tire rolls freely (that is without the action of a driving or braking torque) and without side slip, camber, or turning, the wheel moves along a straight line parallel to the road and in the direction of the wheel plane. In that situation, the tread elements remain vertical and move from the leading edge to the trailing edge without developing a horizontal deflection and consequently without generating a fore-and-aft or side force. A possible presence of rolling resistance is disregarded. When the wheel speed vector V shows an angle with respect to the wheel plane, side slip occurs. When the wheel velocity of revolution Ω multiplied with the effective rolling radius r_e is not equal to the forward component of the wheel speed V_x = Vcos α, we have fore-and-aft slip. Under these conditions, depicted in Figure 3.2, horizontal deflections are developed and corresponding forces and moment arise. The tread elements move from the leading edge (on the right-hand side of the pictures) to the trailing edge. The tip of the element will, as long as the available friction allows, adhere to the ground (that is, it will not slide over the road surface). Simultaneously, the base point of the element remains in the wheel plane and moves backward with the linear speed of rolling V_r (that is equal to r_eΩ) with respect to wheel axis or better: with respect to the contact center C. With respect to the road, the base point of the element moves with a velocity that is designated as the slip speed V_s of the wheel.

FIGURE 3.2 The brush tire model. Top left: view of driven and side-slipping tire. Top right: the tire at different slip conditions. Bottom left: the tire at pure side slip, from small to large slip angle. Bottom right: the resulting side force and aligning torque characteristics.

In the lower part of the figure, the model is shown at pure side slip. The slip changes from very small to relatively large. We observe that the deflection increases while the element moves further through the contact patch. The deflection rate is equal to the supposedly constant slip speed. The resulting deflection varies linearly with the distance to the leading edge and the tips form a straight contact line that lies in a direction parallel to the wheel speed vector V. The figure also shows the maximum possible deflection that can be reached by the element depending on its position in the contact region. This maximum is governed by the (constant) coefficient of friction μ, the vertical force distribution q_z, and the stiffness of the element c_py. The pressure distribution and consequently also the maximum deflection v_max have been assumed to vary according to a parabola. As soon as the straight contact line intersects the parabola, sliding will start. The remaining part of the contact line will coincide with the parabola for the maximum possible deflection. At increasing slip angle, the side force that is generated will increase. The distance of its line of action behind the contact center is termed the pneumatic trail t. The aligning torque arises through the nonsymmetric shape of the deflection distribution and will be found by multiplying the side force with the pneumatic trail. As the slip increases, the deformation shape becomes more symmetric and, as a result, the trail gets smaller. This is because the point of intersection moves forward, thereby increasing the sliding range and decreasing the range of adhesion. This continues until the wheel speed vector runs parallel to the tangent to the parabola at the foremost point. Then, the point of intersection has reached the leading edge and full sliding starts to occur. The shape has now become fully symmetric. The side force attains its maximum and acts in the middle so that the moment vanishes. That situation remains unchanged when the slip angle increases further. The resulting characteristics for the side force and the aligning torque have been depicted in the same figure. In the part to follow next, the mathematical expressions for these relationships will be derived, first for the case of pure side slip.

3.3 The Tread Simulation Model

In this section, a methodology is developed that enables us to investigate effects of elements in the tire model which were impossible to include in the analytical brush model dealt with in the preceding Section 3.2. Examples of such complicating features are: arbitrary pressure distribution; velocity and pressure dependent friction coefficient; isotropic stiffness properties; combined lateral, longitudinal and camber or turn slip; lateral, bending and yaw compliance of the carcass and belt; and finite tread width at turn slip or camber.

The method is based on the time simulation of the deformation history of one or more tread elements while moving through the contact zone. The method is very powerful and can be used either under steady-state or time-varying conditions. In the latter nonsteady situation, we may divide the contact length into a number of zones of equal length in each of which a tread element is followed. In the case of turning or camber, the contact patch should be divided into several parallel rows of elements. While moving through the zones, the forces acting on the elements are calculated and integrated. After having moved completely through a zone, the integration produces the zone forces. These forces act on the belt and the corresponding distortion is calculated. With the updated belt deflection, the next passage through the zones is performed and the calculation is repeated.

Here, we will restrict the discussion to steady-state slip conditions and take a single zone with length equal to the contact length. In Section 2.5, an introductory discussion has been given and reference has been made to a number of sources in the literature. The complete listing of the simulation program TreadSim written in Matlab code can be found online (cf. App. 2). For details, we may refer to this program.

Figure 3.33 depicts the model with deflected belt and the tread element that has moved from the leading edge to a certain position in the contact zone. In Figure 3.34, the tread element deflection vector e has been shown. The tread element is assumed to be isotropic, thus with equal stiffnesses in the x- and y-direction. Then, when the element is sliding, the sliding speed vector V_g, that has a sense opposite to the friction force vector q, is directed opposite to the deflection vector e. The figure depicts the deflected element at the ends of two successive time steps i − 1 and i. The first objective is now to find an expression for the displacement g of the tip of the element while sliding over the ground.

FIGURE 3.33 Enhanced model with deflected carcass and tread element that is followed from front to rear.

FIGURE 3.34 The isotropic tread element with deflection e in two successive positions i − 1 and i. Its base point B moves with speed V_b and its tip P slides with speed V_g.

The contact length is divided into n intervals. Over each time step Δt, the base point B moves over an interval length toward the rear. With given length

(3.99)

and the linear speed of rolling V_r = r_eΩ, the time step Δt is obtained

(3.100)

with the velocity vector V_b of point B, the displacement vector Δs of this point over the time step becomes

(3.101)

The base point B moves along the belt peripheral line or a line parallel to this line. With the known lateral coordinate y_b of this line of base points with respect to the wheel center plane, the local slope can be assessed. Then, with the slip velocity V_s of the slip point S, the yaw rate of the line of intersection , and the rolling speed V_r, the components of V_b can be found.

The velocity of point B may be considered as the sliding velocity of this point with respect to the ground and we may employ equations (2.55, 2.56) for its assessment. In these equations, at steady state, the time derivatives of u and v vanish, the slope is replaced by zero, and for we take the gradient of the belt deflection caused by the external force and moment. We have, with average x position x_b = x + 0.5Δx,

(3.102)

and

(3.103)

with the slip and roll velocities

(3.104)

The lateral displacement y_b of the belt at the contact center is attributed to camber, conicity, and the lateral external force (through the lateral compliance of the carcass). The gradient may be approximately assessed by assuming a parabolic base line y_b(x_b) exhibiting an average slope c_s influenced by the aligning torque (through the yaw compliance) and ply-steer, and a curvature c_c influenced by the side force (through the bending stiffness) and camber and conicity (cf. (3.56)). We have, for the lateral coordinate,

(3.105)

and, for its approximation used in (3.102),

(3.106)

and the slope:

(3.107)

Conicity and ply-steer will be interpreted here to be caused by ‘built-in’ camber and slip angles. These equivalent camber and slip angles γ_con and α_ply are introduced in the expressions of the quantities θ. We define

(3.108)

and

(3.109)

The coefficients ε_γx and ε_γy may be taken equal to each other. The first term of the displacement (3.105) is just a guess. It constitutes the lateral displacement of the base line at the contact center when the tire is pressed on a frictionless surface in the presence of conicity and camber. The displacements at the contact leading and trailing edges are assumed to be zero under these conditions. The approximation y_bo(3.106) is used in (3.102) to avoid apparent changes in the effective rolling radius at camber. The actual lateral coordinate y_b (3.105) plus a term y_rγ is used to calculate the aligning torque. With this additional term, the lateral shift of F_x due to sideways rolling when the tire is being cambered is accounted for. We have y_b,eff = y_b + y_rγ with y_rγ = ε_yrγ b sin γ with an upper limit of its magnitude equal to b. The moment causes the torsion of the contact patch and is assumed to act around a point closer to its center as depicted in Figure 3.19. A reduction parameter is used for this purpose. More refinements may be introduced to better approximate the shape of the base line, especially near the leading and trailing edges (bending back).

With the displacement vector Δs (3.101) established, we can derive the change in deflection e over one time step. By keeping the directions of motion of the points B and P in Figure 3.34 constant during the time step, an approximate expression for the new deflection vector is obtained. After the base point B has moved according to the vector Δs, we have

(3.110)

Here, e_i−1 denotes the absolute value of the deflection and g_i, the distance P, has slided in the direction of −e_i−1. In the case of adhesion, the sliding distance g_i = 0. When the tip slides, the deflection becomes

(3.111)

From (3.110), an approximate expression for g_i can be established, considering that g is small with respect to the deflection e. We obtain

(3.112)

in which expression (3.111) is to be substituted. If g_i is positive, sliding remains. If not, adhesion commences. In the case of sliding, the deflection components can be found from (3.110) using (3.112). Then, the force vector per unit length is

(3.113)

If adhesion occurs, g_i = 0 and with (3.110) the deflection is determined again. The force per unit length now reads

(3.114)

As soon as the condition for adhesion

(3.115)

is violated, sliding begins. The sliding distance g_i is calculated again and its sign checked until adhesion may show up again.

In (3.113), the friction coefficient appears. This quantity may be expressed as a function of the sliding velocity of the tip of the element over the ground. However, this velocity is not available at this stage of the calculation. Through iterations, we may be able to assess the sliding speed at the position considered. Instead, we will adopt an approximation and use the velocity of the base point (3.102, 3.103) to determine the current value of the friction coefficient. The following functional relationship may be used for the friction coefficient versus the magnitude of the approximated sliding speed V_b:

(3.116)

During the passage of the element through the contact zone, the forces ΔF_i are calculated by multiplying q_i with the part of the contact length Δx covered over the time step Δt.

Subsequently, the total force components and the aligning moment are found by adding together all the contributions ΔF_xi, ΔF_yi, and x_biΔF_yi−y_bi,effΔF_xi, respectively. A correction to the moment arm may be introduced to account for the side ways rolling of the tire cross section while being cambered and deflected (causing lateral shift of point of action of the resulting normal load F_z and similarly of the longitudinal force F_x). Also, the counter-effect of the longitudinal deflection u_c may contribute to this correction factor (cf. Eqn (3.51)).

For details and possible application of the model, we refer to the complete listing of the Matlab program TreadSim presented online.

In the sequel, a number of example results of using the tread simulation model have been presented. The following cases have been investigated:

The computations have been conducted with the set of parameter values listed in Table 3.1.

TABLE 3.1 Parameter Values used in the Tread Simulation Model (Figures 3.35–3.37)

In the case of a ‘rigid’ carcass, parameter c (= 0.01 m/kN), Eqn (3.52), is used while c_lat, c_bend, c_yaw → ∞. As indicated, quantities c_p and θ follow from the model parameters.

The graphs of the lower half of Figure 3.35a relative to the upper half show the most prominent effect of a with sliding velocity decreasing friction coefficient. At a_μ = 0.03, the side force F_y exhibits a clear peak at a slip angle of about seven degrees when the wheel is rolling freely (κ = 0). Also, the fore-and-aft force F_x tends to decrease after having reached its peak value. The inward endings of the curves at constant values of slip angle shown in the bottom diagram are typical especially for a tire running on wet road surfaces. The peak values themselves will decrease when the speed of travel is increased (not shown) while the initial slopes (slip stiffnesses) remain unchanged. The change in sign of the aligning torque in the braking range of its diagram occurs because of the retained parameter c that produces the effect of the flexible carcass according to Eqns (3.51, 3.52). The expected change in sign of the aligning torque curve at pure side slip at higher levels of side slip does not occur due to the limitation of the estimation of the sliding speed for which the speed of a point of the belt is taken (here rigid). In Figure 3.35b with the carcass considered flexible, this sign change does show up as illustrated in the upper right diagram.

A more important effect of the flexible carcass presented in Figure 3.35b is the decreased cornering stiffness while the fore-and-aft slip stiffness remains the same. The lower value of F_y at α = 2° in the F_y vs F_x diagram clearly demonstrates this effect (the curves become less curved). The aligning torque, however, is not so much affected. This is due to the larger pneumatic trail which is a result of the curved deflection line of the belt. Figure 3.36 shows the pneumatic trail diagram for the model with and without carcass compliance (the number of rows has no influence in case of zero spin). As expected, the simple brush model with rigid carcass has a trail of 0.33a when the slip angle approaches zero. The flexible carcass model considered features a pneumatic trail of about 0.46a. See Figure 3.38 for the deflection pattern.

FIGURE 3.38 Examples of deflection patterns at various combinations of slip for the single row model with rigid and flexible carcass.

The influence on the curves of the introduction of the relatively large camber angle of 10° is indicated in the lower half of Figure 3.35b. The effect is most clearly demonstrated in the lower-left diagram. The camber thrust is accompanied by a lateral deflection that causes a shift of the line of action of the longitudinal force. The resulting torque tends to rotate the lower part of the belt about the vertical axis which now can be accomplished through the yaw compliance of the carcass. At braking, the rotation is such that an apparent slip angle arises that increases the camber side force. At driving, the opposite occurs. As a result, the constant slip angle curves plotted in the diagram show an inclination. The corresponding influence diagram of κ on F_y shows distorted curves when compared with those of Figure 3.11. The inversed S shape of the curves at small slip angle is a feature that is commonly encountered in measured characteristics. The aligning torque diagram (lower-right picture) is considerably changed as a result of the action of the torque mentioned above that originates from F_x.

In Figure 3.35c, the effect of a finite width of the contact patch is demonstrated. Two rows of tread elements have been considered. At the camber angle of 10°, a spin torque is generated that appears to rotate the lower part of the belt in such a way that an apparent slip angle arises that increases the camber thrust. The upper-left-hand diagram shows the increase in side force. The right-hand diagram indicates the considerable rise in the aligning torque as a result of the spin torque in the range of small longitudinal force F_x.

The two diagrams in the second row demonstrate the effects of tread width, carcass flexibility, and friction decay with velocity on the side force and moment vs slip angle characteristics for a series of camber angles. The plots may be compared with those of Figure 3.28. The lower half of Figure 3.35c refers to the case without camber and may be compared with the plots of the upper half of Figure 3.35b that does not include the effects of tread width.

Figure 3.35d presents the force and moment characteristics for a series of levels of turn slip, including large values of spin corresponding to a radius of curvature equal to a/0.8. The figures of the second row refer to the case of constant friction (a_μ = 0) and may be compared with Figures 3.31, 3.32. An interesting effect of tread width is the further decrease of the side force at higher levels of turn slip due to longitudinal slip that occurs on both sides of the contact patch which consumes a lot of the available frictional forces. Of course, the aligning torque is now considerably larger. The third row of diagrams represents the spin force and moment characteristics at various levels of side slip. In Figure 3.37, the pure spin characteristics have been drawn for the model with a rigid carcass provided with one row of tread elements (same as Figure 3.25) and with two rows computed with the tread simulation model. Comparison clearly shows the considerable reduction of the peak side force and the much larger level of the aligning torque caused by the finite tread width. The influence of carcass flexibility appears to be very small both for the force and for the moment.

FIGURE 3.37 Pure turn slip characteristics according to the model (left, rigid carcass with one or two rows of elements) and results from experiments with a bias-ply truck tire (right, 9.00–20 eHD, p_i = 5.5 bar, V = 1–3 km/h on dry road, from Freudenstein 1961). Model parameters: θ = 5, a_μ = 0, and b_row = 0.5 a.

These results indicate that the single row theory developed in Section 3.2 also for high spin in combination with side slip has only limited practical significance. The influence of small spin, i.e., camber, on the side force vs slip angle characteristic as calculated with the aid of the simple single row, rigid carcass brush model may be considered as reasonable.

Freudenstein (1961) has conducted side-slip and turning experiments with a bias-ply truck tire on a dry road surface. The side-slip measurement results of this tire were already depicted in Figure 3.6. In the right-hand diagram of Figure 3.37, the results from the turning experiments have been presented. As abscissa, both the path radius and the path curvature have been used. Obviously, the calculated two-row model characteristics show good qualitative agreement with the experimental curves. For the values a = 0.1 m, μF_z = 20000 N, and θ = 8, a very reasonable also quantitative correspondence for both the force and the moment characteristics of Figure 3.37 can be obtained. Freudenstein suggests the following formula for the peak moment generated at pure turning at wheel speed V = 0:

(3.117)

As Freudenstein did not give the camber characteristics of the truck tires on which the turning behavior was measured, we are not able to compare the responses to camber and turning. According to Hadekel (1952), for aircraft tires the lateral force due to turning is about four times higher than the camber force at equal values of spin. From experiments performed by Higuchi (1997), a factor of about two can be deduced for a radial ply car tire (cf. Chap. 7 discussion above Figure 7.11). This supports the theory of the reduced curvature of the peripheral line of the cambered tire pressed on a frictionless surface due to the high lateral bending stiffness of the tread band, as illustrated in Figure 2.10, and the associated coefficient ε_γ in Eqn (3.55). The program TreadSim also provides information on the distribution of contact forces and deflections of belt and tread elements. Some examples of the deformation pattern have been depicted in Figures 3.38 and 3.39. In Figure 3.38, the deflections of the single row brush model have been depicted and in addition of the model with carcass lateral, yaw and bending compliance. As indicated, the scale of the drawing has been chosen larger in lateral direction. In the middle two diagrams, the influence of camber and of turning on the deflections at side and brake slip has been shown. Turning was considered with an exceptionally small turn radius R equal to a which is half the contact length. One may note the central section where adhesion occurs. The latter situation also occurs with the middle row of tread elements of the three row brush model depicted in Figure 3.39. In this drawing, the scales are the same. Clearly, the tire generates longitudinal deflections of the outer rows of elements which contribute to the torque.

FIGURE 3.39 Deflections of the tread elements and of the belt for the three row model at side slip and turning.

3.4 Application: Vehicle Stability at Braking up to Wheel Lock

When the vehicle is being braked forcefully and a possible downward slope does not compensate for the reduction in speed of travel, the situation can no longer be considered as steady state. The influence of the rate of change of the longitudinal speed and of the effect of combined slip on the lateral stability may no longer be neglected. Instead of using the two first-order differential equations (1.42) for the lateral and yaw motions, we must now consider the complete set, including the equation for the longitudinal motion which is also of the first order.

Because of the complexity involved, the influence of the height of the center of gravity on the vehicle motion will be disregarded. With a finite height, fore-and-aft but also lateral load transfer would occur, the latter causing unequal braking forces on the left- and right-locked wheels that give rise to a stabilizing torque counteracting the effect of the fore and aft load transfer.

The steer angle is kept equal to zero. This two-wheel, single-track, rigid vehicle with zero c.g. height has been depicted in Figure 3.40 where the wheel on axle 2 is considered to be locked. For the three states, we obtain the equations

(3.118)

with

(3.119)

and

(3.120)

For a proper simulation of the motion, the slip ratio’s κ_i should result from the wheel speeds of revolution Ω_i which would require additional degrees of freedom. For this occasion, we will employ an alternative approach that involves the introduction of functions for the side force in which its direct dependence on the braking effort (or brake pressure) is included. This is possible when ‘dry’ friction is assumed to occur between tire and road and the curves of the right-most diagram of Figure 1.2 do not show inward endings and thus double valued functions are avoided. The characteristics of Figure 3.16 will be used for our analysis. In Figure 3.41, the characteristics have been reproduced: one curve for the freely rolling wheel, another one for the wheel that is being braked with a braking effort corresponding to a brake force B₁ = −F_x1 at straight ahead rolling (α small) and finally for a wheel that is locked completely. The second curve shows that at higher slip angles, a lower horizontal plateau is followed until the straight inclined line is reached where also this wheel gets locked (note that sin α has been used as abscissa which makes the relationship linear). A full discussion of this behavior has been given in Subsection 3.2.3 in connection with the treatment of the tire brush model.

FIGURE 3.40 ‘Bicycle’ model with one wheel locked.

FIGURE 3.41 Tire side force characteristics for freely rolling and braked wheels.

As shown in Figure 3.40, we assume that the wheel on axle 2 is locked and that on axle 1 may be braked. The characteristics of Figure 3.41 apply. First, the situation near the undisturbed straight ahead motion will be studied. Linearization with both slip angles assumed small yields for the horizontal wheel forces:

(3.121)

For the locked wheel, we have now an effective side-slip stiffness:

(3.122)

As a result, Eqns (3.118–3.120) reduce to

(3.123)

Furthermore, we have introduced the absolute value of the forward speed u in the denominators to allow for the consideration of negative values of u, while at the same time the sign of α remains unchanged (cf. the corresponding equations (1.46) for the nondriven or braked vehicle). Negative values of u correspond to the case of locked front wheels.

Elimination of the lateral velocity v from Eqn (3.123) yields

(3.124)

When compared with Eqn (1.50), considering (1.48), it is noted that in the second coefficient of (3.124), the term k²C₂ has disappeared and that −k²B₁ has been added. This is due to the differentiation of u in the elimination process of v. We had, originally in the second coefficient of (3.124), the term

(3.125)

which explains the changes observed. In other respects, Eqn (3.124) is similar to the homogeneous version of Eqn (1.50). However, an important difference appears in the coefficients which are now dependent on the time because of the presence of the linearly with time decreasing speed u. We have the additional equation for u (by integration of the first of Eqn (3.123)):

(3.126)

The exact solution of Eqn (3.124) can be found because of the fortunate fact that the equation can be reduced to the differential equation of Bessel, the solution of which is known in tabular form. Before giving an example of such a complete solution, we will analyze the motion just after the application of a slight disturbance which allows us to approximate the equation to one with constant parameters in which u is replaced by its initial value u_o. For this substitutive equation, the solution can be found easily. We obtain

(3.127)

with D_1,2 denoting the constants of integration (governed by the initial values of the state variables v and r) and λ_1,2 representing the eigenvalues, that is, the roots of the characteristic equation of the substitutive differential equation:

(3.128)

Comparison with the coefficients of Eqn (3.124) reveals the expressions for the quantities A, B, C, and D. Since, apparently, the first three are always positive and in the case considered C₂ is much smaller than C₁ which makes D also a positive quantity, we expect that the substitutive system can only become unstable because of a possibly negative third coefficient. Then, one of the roots (lets say λ₁) becomes positive (while remaining real). Obviously, this can only occur when u is positive which means: when the wheels of the rear axle are locked. Locked front wheels which occur when in our system description u < 0 will not destabilize the system but makes the vehicle unsteerable. This is because of the fact that changing the steer angle of a locked wheel cannot effect the orientation of its frictional force vector. The car with locked rear wheels corresponds in behavior with the case of excessive oversteer while locked front wheels would give rise to extreme understeer. The critical speed for the case of locked rear wheels is derived by making the third coefficient of Eqn (3.128), or of Eqn (3.124), equal to zero. We find, by using (3.122),

(3.129)

From this expression, it can be seen that when a rigid front tire is considered so that C₁ → ∞, we obtain the simple form: . The influence of an elastic front tire is not very great as the front cornering stiffness may range from 6 to 30 times the tire normal load. Considering a dry road with μ = 1, we find a critical speed of about 20 km/h.

In view of this very low speed above which the vehicle with rear wheels locked becomes unstable and the fact that normally the speed will already be a lot larger than the critical speed when the brakes are being applied, it is more useful to consider the degree of instability as a function of speed. We will adopt the root λ₁ to represent the degree of instability. Figure 3.42 illustrates the manner in which the roots vary with speed of travel. The diagram includes the case of front wheels locked (u < 0), where obviously the roots remain negative and become complex at higher speeds of travel (real part: dotted curve). As a practical indication for the degree of instability, we may employ the height of the horizontal asymptote to which λ₁ tends when u → ∞. For that we find

(3.130)

which shows that instability decreases when the front tire cornering stiffness is lower and becomes more in balance with the here very low rear ‘cornering stiffness’ (3.122). It also becomes clear that when the front wheels are locked as well (C₁ = μF_z1), the degree of instability reduces to zero representing indifferent stability.

FIGURE 3.42 Real (part of the) eigenvalues of the substitutive linear system with constant coefficients.

The complete solution of Eqn (3.124) can be found by applying the Lommel transformation of the Bessel differential equation and using the modified functions of Bessel in its solution. These functions are available in tabulated form and can be found in the book by Abramowitz and Stegun (1965, cf. Eqn 9.1.53 for the transformed equation and p. 377 for the solution).

In the paper of Koiter and Pacejka (1969), the exact solution has been given together with examples of the numerically computed solution for the complete nonlinear system. For the special case with parameters a = b = k, B₁ = 0, C₁/μF_z = 6, the solution of the linear Eqn (3.124) is presented in Figure 3.43. The exact solution is compared with the approximate one (3.127) of the substitutive equation. The solution in the form of a stable and an unstable branch has been drawn in the (u, r) plane constituting the projection of the complete three-dimensional trajectory in the (u, v, r) space. Along the u axis, we may introduce the time axis considering the relationship (3.126). In the case considered the nondimensional eigenvalues λ_1,2√(l/μg) take the values 1.26 and −3.43 respectively. For the sake of comparison, the initial values have been taken the same for both pairs of solutions. No visible difference can be detected between the two decaying stable branches. The ‘unstable’ branch of the exact solution, however, shows for greater values of time an increasing difference with the corresponding approximate exponential solution. At the start of the motion where the rear axle tends to break away the agreement is very good. It can be shown that when expanded in a series of powers of the time t, the solutions are identical up to and including the third term (with t²). This supports our choice of defining the degree of instability.

FIGURE 3.43 Solutions of the exact and the approximated linear systems representing the variation of the yaw rate r with forward velocity u or with time t.

Along the vertical axis, the yaw rate is plotted. After having attained a maximum, the yaw angle develops at a lesser rate until the motion reaches a complete stop. The area underneath the curves is proportional to the final angle of swing. This finite angle will linearly depend on the initial values of the variables r and v. Consequently, the final deviation with respect to the original rectilinear path can be kept within any chosen limit and, strictly speaking, the actual system is always stable.

Finally, the solutions for the nonlinear system governed by the Eqns (3.118, 3.119) and tire characteristics according to Figure 3.41 have been established by numerical integration of the equations of motion. Now, the character of the motion may change with the level of the initial disturbance. In Figures 3.44 and 3.45, the resulting motions for two cases have been depicted: the first without braking the front wheels (Figure 3.44) and the second with also the front wheels being braked but at a lower effort to make sure that they roll at least initially. The initial speed of travel and disturbances has been kept the same for all cases but the coefficient of friction μ has been varied. At lower friction, the time available to develop the angle of swing is larger and we see that an angle of more than 180 degrees may be reached. Then the locked wheels are moving at the front (u < 0) and we have seen that situation is stable while the motion can become oscillatory (cf. Figure 3.42). For the case of Figure 3.44 with μ = 0.25, we indeed observe that a sign change occurs once for the yaw rate r. When the brakes are applied also at the front axle, the deceleration is larger and we have less time to come to a stop. Consequently, the final angles of Figure 3.45 are smaller than those reached in Figure 3.44. The shaded zones indicate the ranges where also the front wheels get locked. This obviously occurs when the slip angle of the initially rolling wheels 1 becomes sufficiently large and exceeds the value indicated in Figure 3.41 where the inclined straight line is reached. It has been found (not surprisingly) that the influence of raising the initial speed is qualitatively the same as the effect of reducing the coefficient of friction.

FIGURE 3.44 Development of the vehicle motion after the rear wheels get locked while the front wheels remain rolling freely.

FIGURE 3.45 Development of the vehicle motion after the rear wheels get locked while also the front wheels are being braked.