5.3.1 Model Development

To obtain an expression for the lateral deflection v of the string, we consider the lateral equilibrium of an element of the tread band shown in Figure 5.2. The element is of full tread width and contains elements of the parallel strings with the rubber in between. In the lateral direction (y), the equilibrium of forces acting on the element of length dx and width 2b results in the following equation:

(5.1)

where c_c denotes the lateral carcass stiffness per unit length, S₁ the circumferential (in the x-direction) component of the total tension force acting on the set of strings, and D the shear force in the cross section of the tread band acting on the rubber matrix. The shear force is assumed to be a linear function of the shear angle according to formula

(5.2)

FIGURE 5.2 Lateral equilibrium of deflected tread band element.

With the introduction of the effective total tension S = S₁ + S₂, we deduce from Eqn (5.1)

(5.3)

In the part of the tire not making contact with the road, the contact pressure vanishes so that q_y = 0 and

(5.4)

The lateral behavior of the model with several parallel strings and of the model with a single string will be identical if parameter S is the same. In Figure 5.3, a top view of the single string model is depicted. The length σ, designated as the relaxation length, has been indicated in the figure. The relaxation length equals:

(5.5)

FIGURE 5.3 Top view of the single string model and its position with respect to the fixed frame.

With this quantity introduced, Eqn (5.4) for the free portion of the string becomes

(5.6)

If we consider the circumference of the tire to be much longer than the contact length, we may assume that the deflection v₂ at the trailing edge has a negligible effect on the deflection v₁ at the leading edge. The deflections of the free string near the contact region may then be considered to be the result of the deflections of the string at the leading edge and at the trailing edge respectively and not of a combination of both. The deflections in these respective regions now read approximately:

(5.7)

which constitute the solution of Eqn (5.6) considering the simplifying boundary condition that for large |x| the deflection tends to zero. At the edges x = ±a where v = v₁ and v = v₂ respectively, we have, for the slope,

(5.8)

Because we do not consider the possibility of sliding in the contact zone, a kink may show up in the shape of the deflected string at the transition points from the free range to the contact zone. It seems a logical assumption that through the rolling process, the string forms a continuously varying slope around the leading edge while at the rear, because of the absence of bending stiffness, a discontinuity in slope may occur. An elegant proof of this statement follows by considering the observation that in vanishing regions of sliding at the transition points, cf. Figure 5.4, the directions of sliding speed of a point of the string with respect to the road and the friction force exerted by the road on the string that is needed to maintain a possible kink are compatible with each other at the trailing edge but incompatible at the leading edge. Therefore, it must be concluded that a kink may only arise at the trailing edge of the contact line. Consequently, the equation for the slope at the leading edge (first of (5.8)) can be rewritten as

(5.9)

FIGURE 5.4 Two successive positions of the string model. Vanishing regions of sliding at the leading and trailing edges and the compatibility of sliding speed V_g and frictional forces F required to maintain a possible kink which apparently can only exist at the trailing edge (2).

This equation constitutes an important relationship for the development of the ultimate expression for the deflection of the string as we will see a little later.

In Chapter 2, we have derived the general differential equations for the longitudinal and lateral sliding velocity of a rolling body which is subjected to lateral slip and spin. These equations (2.58, 2.59) become, if the sliding speed equals zero as is our assumption here,

(5.10)

(5.11)

In these equations, s denotes the distance traveled by the wheel center (or better: the contact center) and x and y the coordinates the considered particle would have with respect to the moving axes system (C, x, y) in the horizontally undeformed state.

These partial differential equations will be solved by using Laplace transformation. The transforms will be written in capitals. The transformation will not be conducted with respect to time but with respect to the distance traveled s = Vt where the speed V is assumed to be a constant. The Laplace transform of a variable quantity, generally indicated by q, is defined through

(5.12)

where p is the Laplace variable.

With initial conditions u(x, 0) = v(x, 0) = 0 at s = 0, we obtain, from (5.10, 5.11) the transformed equations,

(5.13)

(5.14)

The solutions of these ordinary first-order differential equations read

(5.15)

(5.16)

In Eqns (5.15, 5.16), the coefficients C_u and C_v are constants of integration. They are functions of p and depend on the tire construction, which is the structure of the model. For our string model with tread elements which can be deformed in the longitudinal direction only, we have the boundary conditions at the leading edge x = a:

(5.17)

and

(5.18)

with the latter Eqn (5.18) corresponding to Eqn (5.9).

The constant C_u now obviously becomes

(5.19)

For the determination of C_v, we have to differentiate Eqn (5.16) with respect to x:

(5.20)

With (5.18), we obtain

(5.21)

which with (5.16) yields for the deflection at the leading edge

(5.22)

and for the deflection in the contact zone

(5.23)

The terms containing e^p(x−a) point to a retardation behavior, which corresponds to delay terms in the original expressions. Note that a memory effect exists due to the fact that the nonsliding contact points retain information about their location with respect to the inertial system of axes as long as they are in the contact zone.

Eqn (5.22) transformed back yields the first-order differential equation for the deflection of the string at the leading edge:

(5.24)

This equation which is of fundamental importance for the transient behavior of the tire model (note the presence of the relaxation length σ in the left-hand member) might have been found more easily by considering a simple trailing wheel system with trail equal to σ and swivel axis located in the wheel plane a distance σ + a in front of the wheel axis, cf. Figure 5.3. The equation may also be found immediately from Eqn (5.14) by taking x = a and considering condition (5.9). Eqn (5.16) may also be transformed back which produces the delay terms mentioned before. However, we prefer to maintain the expression in the transformed state since we would like to obtain the result, that is: the force and moment response, in the form of transfer functions.

5.3.2 Step and Steady-State Response of the String Model

An important characteristic aspect of transient tire behavior is the response of the lateral force to a stepwise variation of the slip angle α. The initial condition at s = 0 reads: v(x) = 0; for s > 0, the slip angle becomes α = α_o. From Eqn (5.23), we obtain, for small slip angles by inverse transformation for the lateral deflection of the string in the contact region,

(5.45)

while for the old points which are still on the straight contact line, the simple expression holds:

(5.46)

Expression (5.45) is composed of a part (a − x)α_o which is the lateral displacement of the wheel plane during the distance rolled a − x and an exponential part. The point which at the distance rolled s considered is located at coordinate x was the point at the leading edge when the wheel was rolled a distance a − x ago that is at s − a + x. At that instant, we had a deflection at the leading edge v_1(s−a+x). The new v = v_(s) equals the old v₁ = v_1(s−a+x) plus the subsequent lateral displacement of the wheel (a − x)α_o. Obviously, the exponential part of (5.45) is the v₁ at the distance rolled s − a + x. This can easily be verified by solving Eqn (5.24) for v₁.

Finally, with (5.25) the expression for the force can be calculated for the two intervals, with and without the old contact points:

(5.47)

(5.48)

For the aligning torque, we obtain, by using (5.26),

(5.49)

(5.50)

where the quantities designate the unit step response functions. These functions correspond to the integral of the inverse Laplace transforms of the transfer functions (5.30, 5.31) given above.

The graph of Figure 5.7 shows the resulting variation of F_y and vs traveled distance s. As has been indicated, the curves are composed of a parabola (of the second and third degree respectively) and an exponential function. The step responses have been presented as a ratio to their respective steady-state values.

FIGURE 5.7 The response of the lateral force F_y and the aligning torque M_z to a step input of the slip angle α, calculated for a relaxation length σ = 3a.

The steady-state values of F_y and (or M_z) are directly obtained from (5.25) and (5.26) by considering the shape of the deflected string at steady-state side slip (Figure 5.8), i.e. a straight contact line at an angle α with the wheel plane and a deflection at the leading edge v₁ = σα through which the condition to avoid a kink in the string at that point is obeyed, or from Eqns (5.48, 5.50) by letting s approach infinity. We have

(5.51)

(5.52)

FIGURE 5.8 Steady-state deflection of a side slipping string tire model (complete adhesion).

with the cornering and aligning stiffnesses

(5.53)

(5.54)

and the pneumatic trail

(5.55)

The variation of these quantities and of the pneumatic trail t = C_Mα/C_Fα and also of the relaxation length with wheel load F_z, the latter being assumed to vary proportionally with a², turns out to be quite unrealistic when compared with experimental evidence. A variation much closer to reality would be obtained if tread elements were attached to the string. For more details, also concerning the nonlinear characteristics and the nonsteady analysis of that enhanced but much more complicated model, we refer to Section 5.4.3.

The unit step response functions to the other input variable φ and the associated variables and ψ are of interest as well. They can be derived by integration of the unit impulse response functions Π(s), which are found by inverse transformation of the transfer functions (5.30, 5.31, 5.36, 5.37). Alternatively, the other step response functions may be directly established by considering the associated string deflections similar to (5.45, 5.46) or from the unit step responses to the slip angle, corresponding to the coefficients of α₀ shown in (5.47–5.50), by making use of the following relationships analogous to Eqns (5.39, 5.36, 5.37):

(5.56)

(5.57)

(5.58)

Figure 5.9 illustrates the manner in which the deflection of the string model reacts to a step change of each of the four wheel motion variables (slip angle, lateral displacement of the wheel plane, turn slip, and yaw angle). Figure 5.10 presents the associated responses of the side force and the aligning torque. The responses have been divided by either the ultimate steady-state value of the transient response or the initial value, if relevant. For the moment response to turn slip and lateral displacement, both the initial and the final values vanish and a different reference value had to be chosen to make them nondimensional. The various steady-state coefficients and the lateral and torsional stiffnesses read in terms of the model parameters:

FIGURE 5.9 Transient response of string deflection to step change in motion variables.

FIGURE 5.10 Unit step response of side force F_y, aligning torque and tread width moment to slip angle α, lateral displacement , turn slip φ, and yaw angle ψ. Computed for string model with relaxation length σ = 3a.

Lateral stiffness of the standing tire:

(5.59)

Cornering or lateral slip stiffness

(5.60)

Aligning stiffness

(5.61)

Torsional stiffness of (thin) standing tire

(5.62)

Turn slip stiffness for the force

(5.63)

Note that the steady-state response of to φ equals zero. The responses to side slip have already been presented in Figure 5.7. It now appears that the response of the side force F_y to turn slip φ is identical to the response of the aligning torque due to lateral deflections to α. This reciprocity property is also reflected by the equality of the slip stiffnesses given by (5.61, 5.63). It furthermore appears that the responses of F_y to ψ and φ and of to α tend to approach the steady-state condition at the same rate. This will be supported by the later finding that the corresponding frequency responses at low frequencies (large wavelengths) are similar. The frequency responses at short wavelengths are mainly governed by the step response behavior shortly after the step change has commenced. As appears from the graphs, at distances rolled smaller than the contact length large differences in transient behavior occur. As expected, the initial responses of F_y to and of to ψ are immediate and associated with the respective stiffnesses (5.59, 5.62).

The response of the moment due to tread width modeled with the brush model that deflects only in the longitudinal direction may be derived by considering the Laplace transform of the longitudinal deflection u according to Eqn (5.15) with (5.19). Through inverse transformation or simply by inspection of the development of this deflection while the element moves through the contact range, the following expressions are obtained:

(5.64)

(5.65)

By using Eqn (5.27), the following expressions for the step response of to φ result:

(5.66a)

(5.66b)

The graphical representation of these formulas is given in Figure 5.10. The slip stiffness and stiffness coefficients employed read

Turn slip stiffness for the moment, cf. (5.66b):

(5.67a)

Torsional stiffness of standing tire due to tread width, cf. (5.32) with p → ∞:

(5.67b)

Note that the steady-state response of to φ equals zero.

As a result of a step change in turn slip, longitudinal slip at both sides of the contact patch occurs. The transient response extends only over a distance rolled equal to the contact length, at the end of which the steady-state response has been reached. As indicated in the graph, the approach curve has a parabolic shape. The response to a step change in yaw angle is immediate (like that of ) after which a decline occurs which for is linear (derivative of response to φ).

In Figures 5.11 and 5.12, experimentally obtained responses to small step changes of the input have been presented. The diagrams show very well the exponential nature of the force response to side slip. Especially the aircraft tire exhibits the ‘delayed’ response of the aligning torque to side slip as predicted by the theory (Figure 5.10). Figure 5.12 shows a similar delay in the responses of F_y to turn slip also found in the theoretical results. The peculiar response of the moment to yaw and turn slip is clearly formed by the sum of the responses of and of , although their ratio differs from the assumption adopted in Figure 5.10.

FIGURE 5.11 Step response of side force F_y and aligning torque M_z to slip angle α as measured on an aircraft tire with vertical load F_z = 35 kN.

(From Besselink 2000; test data is provided by Michelin Aircraft Tire Corporation).

FIGURE 5.12 Step response of side force F_y and aligning torque M_z to slip angle α, yaw angle ψ, and turn slip φ (α = 0) as measured on a passenger car tire at load F_z = 4 kN. Tests were conducted on the flat plank machine of TU-Delft, cf. Figure 12.6 (Higuchi 1997, p. 44).

The response to a small step change in turn slip φ_o = −1/R_o has been obtained by integration of the response to a small turn slip impulse = − step yaw angle ψ_o (while α remains zero) and division by R_o ψ_o:

(5.68a)

(5.68b)

For ψ_o = −0.5° = −1/115 rad and by choosing R_o = 115 m, we divide by unity. For more details, we refer to Pacejka (2004).

Graphs of step response functions may serve to compare the performance of different models and approximations with each other. This will be done in Section 5.4. First we will discuss the frequency response functions.

5.3.3 Frequency Response Functions of the String Model

The frequency response functions for the force and the moment constitute the response to sinusoidal motions of the wheel and can be easily obtained by replacing in the transfer functions (5.30, 5.31, 5.32) the Laplace variable p byiω_s. The path frequency ω_s (rad/m) is equal to 2π/λ, where λ denotes the wavelength of the sinusoidal motion of the wheel. Figure 5.13 illustrates the manner in which the string deflection varies with traveled distance when the model is subjected to a yaw oscillation with different wavelengths.

FIGURE 5.13 The string model at steady-state side slip and subjected to yaw oscillations with two different wavelengths λ.

FIGURE 5.14 String model frequency response functions for the side force with respect to various motion input variables. The nondimensional path frequency is equal to half the contact length divided by the wavelength of the motion: a/λ = ω_s/2π. For the response to α a first-order approximation with the same cutoff frequency, ω_s = 1/(σ + a), has been added (Fα₁). Three ways of presentation have been used: log–log, lin–log, and lin–lin. The model parameter σ = 3a, which leads to t = 0.77a.

The frequency response functions such as H_F,ψ(i_ωs) are the complex ratios of output, e.g. F_y, and input, e.g. ψ. In Figures 5.14 and 5.15, the various frequency response functions have been plotted as a function of the nondimensional path frequency a/λ = ½ω_sa/π. The functions are represented by their absolute value |H| and the phase angle ϕ of the output with respect to the input (if negative then output lags behind input), e.g.

(5.69)

If the variables are considered as real quantities, one gets α = |α| cos(ω_ss) and, for its response, F_y = |F_y| cos(ω_ss + ϕ). In the figure, the absolute values have been made nondimensional by showing the ratio to their values at ω_s = 0, the steady-state condition. Three different ways of presentation have been used, each with its own advantage.

The force response to slip angle very much resembles a first-order system behavior, as can be seen in the upper graph with a log–log scale. The cutoff frequency that is found by considering the steady-state response and the asymptotic behavior at large path frequencies appears to be equal to

(5.70)

However, the phase lag at frequencies tending to zero is not equal to ω_s(σ + a), as one would expect for a first-order system, but somewhat smaller. Analysis reveals that the phase lag tends to

(5.71)

with the relaxation length for the side force with respect to the slip angle

(5.72)

which with (5.55) becomes equal to 3.23a if σ = 3a. The phase lag does approach 90° for frequencies going to infinity. The first-order approximation with the same cutoff frequency has been added in the graph for comparison. The corresponding frequency response function reads

(5.73)

The frequency response of the force to yaw shows a wavy curve for the amplitude at higher frequencies (at wavelengths smaller than ca. two times the contact length). The decline of the peaks occurs according to the same asymptote as found for the slip angle response. Consequently, the same cutoff frequency applies:

(5.74)

When analyzing the behavior at small frequencies, it appears that here the phase lag does tend to

(5.75)

with the relaxation length for the side force with respect to the yaw angle:

(5.76)

Further analysis reveals that, when developing the frequency response function H_F,ψ in a series up to the second degree in iω_s,

(5.77)

and subsequently employing the fundamental relationship (5.42) between α and φ responses, the frequency response function H_F,α up to the first degree in iω_s becomes

(5.78)

which shows that

(5.79)

which is an important result in view of assessing σ_Fα from yaw oscillation measurement data and checking the correspondence with (5.72).

The aligning torque (, Figure 5.15) shows a response to the slip angle which is closer to a second-order system with a phase lag tending to a variation around 180° and a 2:1 asymptotic slope of the amplitude with a cutoff frequency equal to

(5.80)

where t denotes the pneumatic trail, cf. (5.55). Again, the response of F_y to φ turns out to be the same as the response of to α. As the graph of Figure 5.15 shows, the amplitude of as a response to yaw oscillations ψ exhibits a clear dip at (with parameter σ = 3a) a wavelength λ = ∼12a. This condition corresponds to the situation depicted in Figure 5.13 (third case) and is referred to as the meandering phenomenon or as kinematic shimmy which occurs in practice when the wheel is allowed to swivel freely about the vertical axis through the wheel center and the system is slowly moved forward. The nearly symmetric string deformation explains why the amplitude of the aligning torque almost vanishes at this wavelength. At higher frequencies, the amplitude remains finite and approaches at ω_s → ∞ the same value as it had at ω_s = 0 that is: . The phase angle approaches −360°. It is interesting that analysis at frequencies approaching zero shows that the phase lag both for the response of to α and to ψ (and thus also for F_y to φ) approaches the value ω_s (σ + a) that also appeared to be true for the response of F_y to ψ. So we have

(5.81)

FIGURE 5.15 String model frequency response functions for the aligning torque with respect to various motion input variables. Same conditions as in Figure 5.14. The response functions of the moment due to tread width have been added.

Expressions equivalent to (5.70, 5.72, 5.74, 5.79–5.81) appear to hold for the enhanced model with tread elements attached to the string, cf. Section 5.4.3.

The torque due to tread width shows a response to yaw angle ψ that increases in amplitude with path frequency and starts out with a phase lead of 90° with respect to ψ. At low frequencies, the moment acts like the torque of a viscous rotary damper with damping rate inversely proportional with the speed of travel V. We find, with Eqn (5.67),

(5.82)

At high frequencies ω_s → ∞, that is at vanishing wavelength λ → 0 where the tire is standing still, the tire acts like a torsional spring and the moment approaches , cf. (5.68). The cutoff frequency appears to become

(5.83)

The total moment about the vertical axis is obtained by adding the components due to lateral and longitudinal deformations:

(5.84)

For the standing tire, one finds from a yaw test the total torsional stiffness C_Mψ which relates to the aligning stiffness and the stiffness due to tread width as follows:

(5.85)

Obviously, this relationship offers a possibility to assess the turn slip stiffness for the moment C_Mφ.

Due to the action of , the phase lag of will be reduced. This appears to be important for the stabilization of wheel shimmy oscillations (cf. Chapter 6). The total moment and its components can best be presented in a Nyquist plot where the moment components and the resulting total moment appear as vectors in a polar diagram. Figure 5.16 depicts the vector diagram at a low value of the path frequency ω_s.

FIGURE 5.16 Complex representation of side force and moment response to yaw oscillations ψ.

By considering the various phase angles at low frequencies, we may be able to extract the moment response due to tread width from the total (measured) response and find the response of the moment for a ‘thin’ tire. Since at low frequencies the moment vector tends to point upward, we find, while considering (5.75, 5.81) and (5.82),

(5.86)

With known σ_Fψ and ϕ_−Mψ to be determined from the measurement at low frequency, the moment turn slip stiffness C_Mφ = κ^∗ may be assessed in this way.

In the diagrams of Figures 5.17 and 5.18, the nondimensional frequency response functions H_F,ψ(iω_s)/C_Fα and −H_M,ψ(iω_s)/C_Mα with its components and have been presented as a function of the nondimensional path frequency a/λ. The parameter values are σ = 3a and C_Mφ = aC_Mα.

The diagram of Figure 5.17 clearly shows the increase in phase lag and decrease of the amplitude of F_y with decreasing wavelength λ. The wavy behavior and the associated jumps from 180° to 0 of the phase angle displayed in Figure 5.14 become clear when viewing the loops that appear to occur when the wavelength becomes smaller than about two times the contact length.

FIGURE 5.17 Nyquist plot of the nondimensional frequency response function of the side force F_y with respect to the wheel yaw angle ψ. Parameter: σ = 3a.

The aligning torque vector, Figure 5.18, for the ‘thin’ tire turns over 360° before from a wavelength of about the contact length the loops begin to show up. At about λ = 12a, the curve gets closest to the origin. This corresponds to the frequency where the dip occurs in Figure 5.15 and is illustrated as the last case of Figure 5.13.

FIGURE 5.18 Nyquist plot of the nondimensional frequency response function of the aligning moment M_z with respect to the wheel yaw angle ψ. The contributing components due to lateral deformations and due to tread width and associated longitudinal deformations. Parameters: σ = 3a and C_Mφ = aC_Mα .

For values of ^∗ sufficiently large, the total moment curve does not circle around the origin anymore. The curve stretches more to the right and ends where the wheel does not roll anymore and the tire acts as a torsional spring with stiffness expressed by (5.85). In reality, the tire will exhibit some damping due to hysteresis. That will result in an end point located somewhat above the horizontal axis.

The calculated behavior of the linear tire model has unmistakable points of agreement with results found experimentally at low values of the yaw frequency. At higher frequencies and higher speeds of rolling, the influence of the tire inertia and especially the gyroscopic couple due to tire lateral deformation rates is no longer negligible. In Section 5.5 and Chapter 9, these matters will be addressed.

5.4.1 Approximate Models

In the literature, several simpler models have been proposed. Not all of these are based on the string model but many are. Figure 5.20 depicts a number of approximated contact lines as proposed by several authors. The most well-known and accurate approximation is that of Von Schlippe (1941) who approximated the contact line by forming a straight connection between the leading and trailing edges of the exact contact line. Kluiters (1969) gave a further approximation by introducing a Padé filter to approximately determine the location of the trailing edge. Smiley (1957) proposes an alternative approximation by considering a straight contact line that touches the exact contact line in its center in a more or less approximate way. Pacejka (1966) considered a linear or quadratic approximation of the contact line touching the exact one at the leading edge; the first and simplest approximation is referred to as the straight tangent approximation. A further simplification, completely disregarding the influence of the length of the contact line, results in the first-order approximation referred to as the point contact approximation.

FIGURE 5.20 Several approximative shapes of the contact line of the string model.

In the sequel, we will discuss Von Schlippe's and Smiley's second-order approximation as well as the straight tangent and point contact approximations. The performance of these models will be shown in comparison with the exact ‘bare’ string model and with the enhanced model with laterally compliant tread elements. Figures 5.21–5.27 give the results in terms of step response, frequency response Bode plots, and Nyquist diagrams.

FIGURE 5.21 Step responses of side force and aligning torque to several inputs for exact and approximate models: ex.: exact ‘bare’ string model; vSch.: Von Schlippe; Sm2: Smiley second order; s.t.: straight tangent; a0: point contact; ex.tr.el.: exact with tread elements.

FIGURE 5.22 Step response of the moment due to tread width to turn slip (path curvature) and yaw angle for exact and approximations: ex.: exact (brush model); s.c.: straight connection (linear interpolation); 2nd: second-order approximation; 1st: first-order approximation.

FIGURE 5.23 a,b,c,d,e,f. Frequency response functions of side force and aligning torque to several inputs for exact and approximate models: ex.: exact ‘bare’ string model; vSch.: Von Schlippe; Sm2: Smiley second order; s.t.: straight tangent; a0: point contact; ex.tr.el.: exact with tread elements.

FIGURE 5.24 Nyquist plot of the frequency response function of the side force F_y with respect to the wheel yaw angle ψ. Parameter: σ = 3a. ex.: exact ‘bare’ string model; vSch.: Von Schlippe; Sm2: Smiley second order; s.t.: straight tangent; a0: point contact; ex.tr.el.: exact with tread elements.

FIGURE 5.25 Nyquist plot of the frequency response function of the aligning torque − with respect to the wheel yaw angle ψ. Curve a0 is hidden by Sm2. Same conditions as in Figure 5.24.

FIGURE 5.26 Nyquist plot of the frequency response function of the torque due to tread width with respect to the wheel yaw angle ψ. ex.: exact (brush); s.c.: straight connection (linear interpolation); 2nd: second-order app.; 1st: first-order approx.

FIGURE 5.27 Nyquist plot of the nondimensional frequency response function of the aligning moment M_z with respect to the wheel yaw angle ψ. Contributing components due to lateral deformations and due to tread width and associated longitudinal deformations. Parameters: σ = 3a and C_Mφ = aC_Mα. Exact and approximate models: ex.: exact ‘bare’ string model; vSch.: Von Schlippe; Sm2: Smiley second order; s.t.: straight tangent; a0: point contact (hidden by Sm2); ex.tr.el.: exact with tread elements; ex.: exact (brush ); s.c.: straight connection (linear interpolation); 2nd : second-order approximation; 1st : first-order approx. (also C_Mφ = 0.6aC_Mα).

For some of the other approximate models (Rogers 1972, Kluiters 1969, Keldysh 1945 and Moreland 1954), only the governing equations will be provided with some comments on their behavior. For more information, we refer to the original publications or to the extensive comparative study of Besselink (2000).

The simplest models – straight tangent and point contact – can easily be extended into the nonlinear regime. In Chapter 6, this will be demonstrated for the straight tangent model in connection with the nonlinear analysis of the shimmy phenomenon. In Chapter 7, the nonlinear single-point contact model will be fully exploited. These models only show an acceptable accuracy for wheel oscillations at wavelengths which are relatively large with respect to the contact length. Chapter 9 is especially devoted to the development of a model that can operate at smaller wavelengths and nonlinear (combined slip) conditions which requires the inclusion of the effect of the length of the contact zone.

5.4.2 Other Models

Two more models will be briefly discussed. These are the models of Keldysh, cf. Goncharenko et al. (1981) and Besselink (2000), and of Moreland (1954). The models are not based on the stretched string concept. They feature two degrees of freedom – the lateral deflection and the torsion deflection – and disregard the finite length of the contact zone. The single-point or straight tangent approximations of the string model have a single degree of freedom where the lateral and the angular deflections at the leading edge are linked through an algebraic relationship. The additional degree of freedom of Keldysh's and Moreland's single-point models with a nonholonomic constraint (first-order differential equation) raises the order of the description from one to two.

5.4.3 Enhanced String Model with Tread Elements

Although the string model with its simplicity and essential features served us well in gathering insight and establishing proper mathematical descriptions and useful approximations, the string model shows a behavior that in several views differs considerably from the properties of the real tire. The relaxation length σ for the tire deflection does not change with vertical load, and thus the relaxation length for the side force to yaw σ_Fψ = σ + a does not vary sufficiently and does not approach zero when F_z → 0 as would occur with the real tire. The same discrepancy holds for the cornering stiffness. The pneumatic trail of the string model is often too large. When considering larger values of slip, that is, when sliding is allowed to occur in the contact zone, the calculated steady-state characteristics for the bare string model do not appear to be adequate as they exhibit a kink at the slip angle where the force has reached its maximum and the moment becomes equal to zero where full sliding commences. Also, when at side slip the lateral force distribution along the contact line is analyzed, it appears that a discontinuity occurs at the point of transition from adhesion to sliding. This is caused by the fact that in the range of adhesion the contact line (that is: the string) is straight and the external frictional forces are equal to the internal elastic forces and remain relatively small. From the transition point onward, the string slides over the ground and the side forces are governed by the vertical force distribution and the friction coefficient. Here the contact line gets curved to let the string tension force S contribute to the transmission of the ground forces to the tire. Finally, it can be easily assessed that the relaxation length of the bare string model does not change with increasing slip angle as is observed to occur in practice. These reasons lead to the desire to develop a model that shows a better behavior and when too complex may serve as a reference for understanding and for the development of simpler models. In the sequel, we will briefly discuss the development of the string model provided with elastic tread elements; for more information, cf. Pacejka (1966, 1981). The results concerning this model's transient and oscillatory out-of-plane behavior have already been included in the various graphs of the preceding section.

5.5.1 First Approximation of Dynamic Influence (Gyroscopic Couple)

In the low frequency range, n < 10 Hz, the aligning torque appears to be affected most by the inertia of the tire. A simple addition to the kinematic model may approximately describe the dynamic changes in the response of the moment. In this theory, the gyroscopic couple due to the lateral deformation rate of the lower portion of the tire has been introduced as the only dynamic influence. It is assumed here that the dynamic forces have a negligible effect on the distortion of the tire.

The gyroscopic couple about the vertical axis is proportional to the wheel rotational speed and the time rate of change of the tilt (camber) distortion of the tire, cf. Figure 5.37. Since this distortion is connected with the lateral deflection of the lower part of the tire and the side force F_y is proportional to this deflection, we have, for the gyroscopic couple,

(5.178)

in which t denotes the time. The tire constant C_gyr is proportional to the tire mass m_t and inversely proportional to the lateral stiffness C_Fy of the standing tire. We have, with the nondimensional quantity c_gyr introduced,

(5.179)

FIGURE 5.37 The average tilt angle of the peripheral line of the deflected tire.

The total moment which acts upon the wheel is now composed of the moment due to lateral tire deformation and that due to longitudinal anti-symmetric tread deformations and finally the gyroscopic couple M_z,gyr:

(5.180)

With this simple addition to the enhanced stretched string tire model (provided with tread elements distributed over the assumedly rectangular contact area), calculations have been performed. Figure 5.38 shows the Nyquist plot of the responses along with the curves ω_s a = 2π na/V = 2π a/λ. It should be noted that the vector of the gyroscopic couple is directed perpendicularly to the vector of the lateral force. For the range of frequencies considered, the simple approximate dynamic extension theory gives a good representation of the effect on the moment response curves. However, the theory does not account for the dynamic changes in the force response.

FIGURE 5.38 The influence of the frequency of the yaw oscillation on the dynamic response according to the simple approximate model extension using the gyroscopic couple due to the lateral tire deflection time rate of change.

5.5.2 Second Approximation of Dynamic Influence (First Harmonic)

More complex theories as developed by Pacejka (1973a) and by Strackerjan (1976) and more recently by Maurice (2000), cf. Chapter 9, yield better results also with respect to the force response and extends to higher frequencies covering the first natural frequencies of the out-of-plane tire dynamic behavior. In these theories, the tire circular belt is considered to move not only about a longitudinal axis but also about a vertical axis and in the lateral direction. The belt rotation about the vertical axis gives rise to a gyroscopic couple about the longitudinal axis which on its turn results in a change in slip angle which influences F_y (and also M_z).

The interesting feature of the approach followed in the first reference mentioned above is that the kinematic theory and response equations as treated in Sections 5.3 and 5.4 with tire inertia equal to zero can be employed. The kinematic part of the model will be considered to be subjected to a different, effective, set of input motion variables. Instead of taking the motion of the actual wheel plane defined by and ψ (or α and φ), an effective input is introduced that is defined as the lateral and the yaw motion of the line of intersection of the ‘dynamic’ belt plane and the road surface . Figure 5.39 depicts the situation. The dynamic belt plane is defined as the virtual center plane of the belt of the tire that is displaced with respect to the rim through the action of lateral inertial forces only. These inertial forces distributed along the circumference of the belt are approximated in such a way that only the forces resulting from the zeroth and first modes of vibration are taken into account, that is: as if the belt would remain circular. Harmonics of second and higher order are neglected in the dynamic part of the model. We have, for the approximate dynamic lateral, camber, and yaw deflections, respectively,

(5.181)

with the total tire deflections composed of dynamic and static parts (the latter resulting from external ground forces):

(5.182)

and stiffnesses of the whole belt with respect to the rim C_y, C_γ, and C_ψ. Furthermore, the equations contain effective tire inertia parameters m_t, I_t, and polar moment of inertia I_tp. The effective inputs to the kinematic model are defined by

(5.183)

FIGURE 5.39 ‘Dynamic’ average plane deflection of belt and associated effective input.

With the aid of these equations plus the kinematic tire equations (e.g. Von Schlippe's approximate Eqns (5.87, 5.88, 5.92, 5.93) with (5.36, 5.37) and (5.32) for with inputs replaced by effective inputs , the frequency response functions of the ground force F_y and moment with respect to the actual inputs may be readily established.

For applications in vibratory problems associated with wheel suspension and steering systems, it is of greater interest to know the response of the equivalent force and moment that would act on the assumedly rigid wheel with tire inertia included at the contact center. The equivalent force and moment are obtained as follows:

(5.184)

This result should correspond to the force and moment measured in the wheel hub, F_y,hub and M_z,hub, after these have been corrected for the inertial force and moment acting on the assumedly rigid wheel plus tire (inertia between load cells and ground surface). With inertia parameters of wheel plus tire, we find, for this correction,

(5.185)

When lifted from the ground, the equivalent ground to tire forces vanish if the frequency of excitation is not too high (low with respect to the first natural frequency of the tire).

For the set of parameter values measured on a radial-ply steel-belted tire listed in Table 5.3, the amplitude and phase of the equivalent moment as a response to yaw angle have been computed and are presented in Figure 5.40 as a function of the excitation frequency . Experimentally obtained curves up to a maximum of 8 Hz (on 2.5 m drum) have been added. The phase lag that occurs in the lower frequency range is largely responsible for the creation of self-excited wheel shimmy. The influence of tire inertia is evident: the phase lag changes into phase lead at higher values of speed (similar as in Figure 5.38). In contrast to the dynamic response depicted in Figure 5.41, the kinematic model response (without the inclusion of tire inertia) would produce the same response at equal wavelengths, λ = V/n, irrespective of the value of the speed of travel V.

TABLE 5.3 Parameters for Car Tire Model with Dynamic Belt Extension (2^nd Approximation)

FIGURE 5.40 Amplitude and phase response of equivalent aligning torque (as if acting from ground on rigid wheel tire) to yaw angle as a function of time frequency of excitation [Hz] computed using Von Schlippe approximation and exact extended with dynamic theory according to second approximation with dynamic belt. Model parameters are listed in Table 5.3 (radial steel belted tire). Curves hold for given speed of travel V [m/s]. Experimental results are given up to a frequency of 8 Hz. The yaw natural frequency of the free non-rotating tire with fixed wheel spindle is n₀₀.

FIGURE 5.41 Amplitude and phase response as a function of frequency with the frequency of excitation equal to that of wheel revolution.

The resonance frequencies, which at speeds close to zero approach a value somewhat larger than the natural frequency of the free non-rotating tire, n₀₀, appear to separate at increasing speed. Vibration experiments performed on a free tire (not contacting the road) for the purpose of obtaining the values of certain tire parameters gave a clear indication that a reduction of the lowest resonance frequency arises when the wheel is rotated around the wheel spin axis at higher speeds of revolution.

Nyquist plots for the M_z,eq and F_y,eq responses to ψ have been presented in Figures 5.42 and 5.43. Both for the bias-ply and for the radial steel-belted tire of Figure 5.40, theoretical and experimental results are shown. Parameters are according to Table 5.3. For each of the curves, the excitation frequency remains constant while the speed varies along the curve.

FIGURE 5.42 Theoretical frequency response Nyquist plots of side force and aligning torque to yaw angle for different time frequencies. Same conditions as in Figure 5.40.

FIGURE 5.43 Experimental frequency response Nyquist plots of side force and aligning torque to yaw oscillations for a radial and a bias-ply car tire measured on a 2.5-m drum.

An increase in frequency tends to raise the moment curves while the force response curves appear to expand with the phase angle at a given wavelength remaining not much changed. The typical differences between the two types of tires, notably in the shape of the upward ends of the moment curves where the speed assumes high values, are well represented in the theoretical diagrams.

We may conclude that the simple first approximation appears to give a considerable improvement for the moment response in the lower frequency range, while for the force response and the higher frequency excitations the inclusion of the dynamics of the belt up to the first modes of vibration is required.

Since we have here a dynamic model extension that allows the use of existing kinematic (mass less) models, we have been able to investigate the effect of tire inertia on these models. However, for the sake of simplicity these models have been restricted to a linear description allowing straightforward frequency response analysis. Extension into the nonlinear regime in the time domain is cumbersome for models including a finite contact length (retardation) effect (although an approximate nonlinear Von Schlippe model may be established by attaching nonlinear springs to the wheel plane at the leading and trailing edges the tips of which follow the path as if we had full adhesion) and also because in the dynamic extension, Eqns (5.181–5.183), the function F_y = F_y(α′) would be needed in an inverse form.

To further develop mathematical models that can be used under conditions that include short wavelengths, higher frequencies, and large (combined) slip, a different route appears to be more promising. The adopted principle is: separate modeling of the flexible belt with carcass and the contact patch with its transient slip properties. The theory of the non-steady-state rolling and slipping properties of the brush model helps in this modeling effort. In Chapter 9, this advanced model will be discussed.

For less demanding cases (large wavelength, low frequency), the straight tangent or single-point contact models may be employed. Such models may be used in combination with the approximate dynamic extension that accounts for the gyroscopic couple due to the lateral deflection velocity. Furthermore, these simple models can be easily adapted to cover large slip nonlinear behavior, cf. Eqns (5.125, 5.126, 5.128) or (5.130–5.132) and Eqn (5.178) with F_y replaced by C_Fyv₀. Chapter 7 is devoted to the treatment of the very useful and relatively simple, possibly nonlinear, single contact point transient slip models. With that single-point contact model, the problem of cornering (and braking) on undulated road surfaces can be made more accessible for vehicle simulation studies. First, we will address this difficult subject in the subsequent section using exact string models with and without tread elements which shows the deeper cause of the associated loss in cornering power.

5.6.1 String Model with Tread Elements Subjected to Load Variations

Figure 5.44 shows the tire model in two successive positions while the wheel moves at a constant (small) slip angle over an undulated road surface which causes the contact length to vary with distance traveled s. When the contact center C is moved a distance Δs, the following changes occur simultaneously: lateral shift of the wheel plane, forward rolling, and change in contact length. These changes influence the position of the leading and trailing edges of the string. The coordinates of these points are:

(5.186)

the changes of which become

(5.187)

FIGURE 5.44 Two successive positions of tire model with varying contact length at constant slip angle α (sliding not considered).

The changes in front and rear lateral string deflections v_c1,2 are composed of various contributions. For the increment of the deflection at the leading edge Δv_c1 we obtain

Similar expressions are obtained for the contributions to the change of the lateral deflection at the rear . The contact length-dependent coefficients appearing in the equations are

(5.189)

(5.190)

(5.191)

(5.192)

They are derived from solutions of the differential equations for the shape of the string deflection (Eqns (5.155, 5.156)). For the parameters employed, we refer to Eqn (5.153). From the increments (5.187, 5.188), the following differential equations are established for the unknown string deflections v_c1 and v_c2:

(5.193)

(5.194)

The two remaining unknowns v₁ and v₂ representing the total lateral deflections and consequently the distances between wheel plane and contact line at the leading and trailing edges are found by considering the following. In the rolling process of a side slipping tire with a continuously changing contact length, in general, three intervals can be distinguished during the interval of the loading cycle where contact with the road exists. In Figure 5.45, the load variation together with the corresponding variation of the contact length 2a of a side slipping tire that periodically jumps from the road has been shown in the road plane for the interval that contact exists. Immediately after the first point touches the road, the contact line grows in two directions (interval I). This growth continues until the second interval II is reached, where growth of contact takes place only at the front, and at the rear loss of contact occurs. Finally, in the third interval III, loss of contact at both ends takes place until the tire leaves the road. When the tire does not leave the road, an additional interval II occurs before interval I is reached again and the cycle starts anew. In less severe cases, intervals I and III may not occur. We then have the relatively simple situation of continuous growth of contact at the front and loss of contact at the rear.

FIGURE 5.45 On the development of the contact line of the tire model with tread elements that periodically loses contact with the road.

The unknowns v₁ and v₂ and the coordinate of the contact points in the plane are obtained as follows (cf. Figure 5.45):

(5.195a)

(5.195b)

(5.195c)

By means of numerical integration, Eqns (5.193, 5.194) have been solved with the aid of the Eqns (5.195) and the expressions (5.189–5.192) for the case formulated below in which the vertical tire deflection ρ has been given a periodic variation with amplitude greater than the static deflection ρ_o, so that periodic loss of contact between tire and road occurs.

The static situation will be indicated with subscript _o. The tire radius is denoted with r. The vertical deflection and the contact length are governed by the equations

(5.196)

(5.197)

The numerical values of the parameters of the system under consideration are

(5.198)

In Figure 5.46, the calculated variation of contact length 2a (oval curve), the path of the contact points (lower curve AB), and the course of the points of the string at the leading and trailing edge (AC and BC) have been indicated. Two positions of the tire have been depicted, one in interval II and the other in interval III.

FIGURE 5.46 The deflected tire model in two positions during the interval of contact with the road.

It is necessary that the calculations in the numerical integration process be carried out with great accuracy. A small error gives rise to a rapid build up of deviations from the correct course. In the case considered above, the integration time is limited and accurate results can be obtained. In cases, however, where continuous contact between tire and road exists, a long integration time is needed before a steady-state situation is attained, which is due to the fact that the exact initial conditions of a periodic loading cycle are not known. For this sort of situation, the exact method described above is difficult to apply due to strong drift of v_c2 in particular.

For further investigation of the effect of a time-varying load, we will turn to an approximate method based on the string model without tread elements.

5.6.2 Adapted Bare String Model

The use of this model is much simpler since the deflections at both edges are independent of each other. Drift does not occur in the calculation process. In Figure 5.47, the deflected model has been shown in interval II. For the three intervals, the following sets of equations apply:

FIGURE 5.47 On the development of the contact line of the bare string model.

growth of contact at leading edge:

(5.199a)

(5.199b)

(5.199c)

(5.199d)

growth of contact at trailing edge:

(5.200a)

(5.200b)

(5.200c)

(5.200d)

loss of contact at leading edge:

(5.201a)

(5.201b)

(5.201c)

loss of contact at trailing edge:

(5.202a)

(5.202b)

(5.202c)

Solutions of the above equations show considerable differences from the results obtained using the more advanced model with tread elements. The most important difference is the fact that with the simple model the lateral force does not gradually drop to zero as the tire leaves the ground.

In order to get better agreement, we will introduce a nonconstant relaxation length σ = σ^∗(a) that is a function of the contact length 2a and thus of the vertical load. We take σ^∗ equal to the relaxation length of the more advanced model according to Eqn (5.171).

Figure 5.48 presents the variation of relaxation length and contact length as a function of the vertical load as calculated for the current parameter values (5.198).

FIGURE 5.48 The calculated variation of half the contact length and the relaxation length as a ratio to the static half contact length versus the vertical load ratio for the model with tread elements using Eqn (5.171).

In Figure 5.49, a comparison is made between the results of the three models: without tread elements, with tread elements (exact), and according to the approximation with varying σ = σ^∗(a). The calculations have been carried out for the parameter values (5.198). The approximate path of contact points shows good agreement with the path for the exact model with tread elements. When the tread elements are omitted, we have the bare string model for which the path becomes wider and the lateral deflections greater.

FIGURE 5.49 Paths of contact points and variations of the side force according to three models.

5.6.3 The Force and Moment Response

The lateral force F_y and the moment M_z which act on the tire can be assessed with good approximation through the following simplified formulas. In their derivations, we have replaced the contact line by the straight line connecting the beginning and end points of the contact line (Von Schlippe's approach). For the model with tread rubber elements, we obtain

(5.203)

(5.204)

and, for the model without tread elements,

(5.205)

(5.206)

In these expressions, the following quantities have been introduced:

(5.207)

and

(5.208)

in which A, B, and B^∗ are given by expressions (5.189–5.192).

For the sake of completeness, the formulas for the moment have been added. However, we will restrict ourselves to the discussion of the variation of the force. For the three models considered, the variation of the cornering force has been shown in the same Figure 5.49. As expected, the approximate and exact theories drop to zero at the point of lift-off, whereas the force acting on the simple string model remains finite, at least under the here assumed condition of no sliding. The correspondence between exact and approximate solutions of the path and of the side force is satisfactory, and we will henceforth use the approximate method according to the adapted bare string model for the investigation of the model with tread elements.

For a series of deflection amplitudes and wavelengths λ, the variation of the cornering force, or rather of the cornering stiffness, has been calculated. Since during loss of contact the vertical load remains zero, the period of the total loading cycle must become longer in order to keep the average vertical load unchanged. This change in period has been taken into account in the calculation of the average side force or cornering stiffness C_Fα.

Figures 5.50 and 5.51 show the final result of this investigation, viz. the cornering stiffness averaged over one complete loading cycle as a function of the vertical deflection amplitude and the path wavelength λ for both models with and without tread elements, that is: with the adapted bare string model and with the original bare string model. With the parameter values σ = 3a_o and c_p = 15c_c, the relaxation length of the more advanced model with tread elements at the static load becomes . For the model without tread rubber, the relaxation length has been taken equal to σ = 2a_o.

FIGURE 5.50 Computed variation of average cornering stiffness with amplitude of the vertical tire deflection, ρ, and path frequency, 1/λ, for the bare string model.

FIGURE 5.51 Computed variation of average cornering stiffness with amplitude of the vertical tire deflection (ρ) and path frequency (1/λ) using the adapted bare string model that approximates the string model with tread elements.

The figures clearly illustrate the unfavorable effect of increasing the amplitude and the path frequency (decreasing the wavelength λ). The curve at zero path frequency is purely due to the nonlinear variation of the cornering stiffness C_Fα versus vertical load F_z shown in Figure 5.32 (cf. Figure 1.3 and discussion on the effect of load transfer in Chapter 1, Figure 1.7) and reflects the ‘static’ drop in average cornering stiffness. A pronounced difference between the response of both types of models is the sudden drop in average cornering stiffness that shows up in the curves of the bare string model when the frequency is larger than zero and the deflection amplitude reaches the value of the static deflection. This is in contrast to the gradual variation in average cornering stiffness which is observed to occur with the more advanced model, and no doubt also with the real tire. Furthermore, the more advanced model with tread elements already shows a noticeable decrease in average cornering stiffness before the tire starts to periodically leave the ground. The ‘dynamic’ drop is represented by the dotted curves along which the path frequency changes and the deflection amplitude remains the same. The very low level, to which the average stiffness is reduced, once loss of contact occurs, is of the same order of magnitude for both tire models.