10.1.1 Tire Envelopment Properties

In the literature one finds numerous publications on tire envelopment behavior. We refer to the study of Zegelaar (1998) for an extensive list of references. A number of these will be mentioned here. Important experimental observations have been made by Gough (1963). He indicated that the tire that is slowly rolled at constant velocity and axle height, over a cleat with length much smaller than the contact length, exhibits three distinct responses: (1) variations in the vertical force, (2) variations in the (horizontal) longitudinal force, and (3) variations in the angular velocity of the wheel. Lippmann et al. (1965,1967) studied the responses of both truck and passenger car tires rolling over short sharp unevennesses like cleats and steps of several heights. From the experimental observations it has been concluded that an almost linear relationship exists between tire force variation and step height. The superposition principle may therefore be employed to assess the response to an arbitrarily shaped unevenness by taking the sum of responses to a series of step changes in road surface height. These observations are expected to be approximately true if the obstacle height is not too large.

For three typical road unevennesses, depicted in Figure 10.1, Zegelaar has measured the responses of the tire at three different constant axle heights. To avoid dynamic effects, the velocity of the drum on which the cleat is attached (Figure 9.37) was maintained at the very low level of 0.2 km/h.

FIGURE 10.1 Three typical obstacles used in Zegelaar's research.

Figure 10.2 presents the measured vertical load F_V, horizontal longitudinal force F_H, and the effective rolling radius r_e as derived from the measurements. The latter quantity is obtained from the variation of the wheel rotation rate dθ/ds, which is defined as: the difference of the incremental wheel angular displacement and the constant (undisturbed) incremental rotation, as a ratio to the increment of the traveled distance of the wheel axle. The following equations apply:

(10.1)

FIGURE 10.2 Rolling over a trapezium cleat, an upward, and a downward step (Figure 10.1) at very low speed and for three axle heights. Diagrams show measured variations of the vertical force F_V (upward +), the longitudinal horizontal force F_H (forward +), and the effective rolling radius r_e, Eqn (10.2). Tire dimensions: 205/60R15.

and hence

(10.2)

The peculiar shapes of the various response curves correspond very well with results found in the literature. Several tire models have been used to simulate the experimental observations.

Davis (1974) has developed a model featuring independent radial springs distributed along the circumference. By giving the individual springs a non-linear degressive characteristic the model is able to generate a vertical force response curve with the typical dip that shows up when running at relatively high initial loads. Badalamenti and Doyle (1988) developed a model also consisting of radial springs but now with additional interradial spring elements that connect the end points of adjacent radial springs in the radial direction. When the deflections of neighboring radial springs are not equal to each other, the interradial spring generates a radial ‘shear force’ that acts on the end points of the radial springs. Mousseau et al. (1994) and Oertel (1997) simulate the tire rolling over a positive step by means of (different types of) finite element models; also cf. Gipser (1987, 1999). Zegelaar uses the flexible ring (belt) model of Gong (1993) that was developed with the aid of the modal expansion method, as a reference model in his research. The addition of tread elements with radial and tangential compliances to Gong's model did enable Zegelaar to employ the model for the study of traversing obstacles. Also this model shows responses very similar to the measured behavior. Schmeitz and Pauwellussen (2001) employ the radial interradial model as a possible basis for the pragmatic model running over an arbitrary road surface.

10.1.2 The Effective Road Plane Using Basic Functions

To arrive at a geometrically filtered road profile, Bandel and Monguzzi (1988) follow the idea of Davis and introduce the effective road plane. The effective plane height and slope variation may be established by conducting an experiment where the wheel is rolled at a very low velocity over an uneven road surface at constant axle height (with respect to a horizontal reference plane) and the forces are measured, cf. Figure 10.3. It is argued that the resulting force (with the rolling resistance force omitted) that acts upon the wheel axle is directed perpendicularly to the effective road plane. By dividing the variation of the measured vertical force, which is approximately equal to the vertical component of the normal force F_N, by the radial stiffness of the tire, the effective height variation, −w, is obtained. The effective slope, tan β_y, is found by dividing the longitudinal horizontal force (after having subtracted the relatively small rolling resistance force) by the vertical force. Both effective quantities are functions of the longitudinal position s of the wheel center. The following formulas apply for the effective height w:

(10.3)

FIGURE 10.3 The wheel rolled over a road irregularity at constant axle height to establish the effective road plane variation.

In Section 10.1.5 a more precise definition and expression for the effective road plane height is given.

For the effective forward slope tan β_y we have

(10.4)

From Eqn (10.3) the approximate value (effect of small f_r disregarded) of the radial deflection ρ_z can be obtained. If needed, the actual effective road plane height, w′ defined below the wheel spin axis, may be assessed, cf. Section 10.1.5. For the description of the effective road input the pragmatic modeling approach initiated by Bandel et al. (1988) and further developed by Zegelaar and extended by Schmeitz is most useful and will be discussed below.

10.1.3 The Effective Road Plane Using the ‘Cam’ Road Feeler Concept

Instead of using the basic profile and running over that with the two-point follower, we may more closely consider the actual tire shape that moves over the road surface profile. Schmeitz discovered that the principle of the circle moving over the surface (Figure 10.8) may be adopted but with an ellipse instead of the circle. This super ellipse that takes the shape of a standing egg has a height approximately equal to that of the tire but a radius of curvature at the lowest point smaller than that of the free tire. In that way, the ‘cam’ touches the step later than the circle would. By choosing an optimal shape of the ellipse, the role of the offset l_f (that changes with step height) of the sine-based basic curve (Figure 10.9) can be taken care of automatically. Figure 10.10 depicts the cam moving over a step. The dimensions of the cam are defined by the super ellipse parameters. In terms of the coordinates x_e and z_e the ellipse equation reads

(10.6)

FIGURE 10.10 ‘Cam' moving over step road profile, producing a basic curve.

The effective road height and slope can be assessed by using two cams following each other at a distance equal to the shift length l_s. The change in height of the midpoint of the connecting line and the inclination of this line represent the effective height and effective slope, respectively.

Figure 10.11 illustrates this ‘tandem cam’ configuration. Fitting the tandem cam parameters follows from assessing the best approximation of low speed responses of a tire running over steps of different heights at a number of constant vertical loads. Once the ellipse parameters have been established, the cam dimensions can be approximately considered to be independent of step height and vertical load. The tandem base length l_s, however, does depend on the vertical load. It is interesting that analysis shows that the lower part of the ellipse turns out to be practically identical to the contour of the tire in side view just in front of the contact zone up to the height of the highest step considered in the fitting process (Schmeitz 2004).

FIGURE 10.11 The tandem cam configuration that generates the (actual) effective height and slope corresponding to the use of a basic curve and two-point follower. The total model including residual spring and rigid belt ring running over an arbitrary road profile.

In a vehicle simulation, it may be more efficient to assess the basic profile first, that is: before the actual wheel rolls over the road section considered. This is achieved by sending one cam ahead over a given section of the road and having that determine the basic profile. The two-point follower is subsequently moved, concurrently with the actual wheel forward motion, over the basic profile, thereby generating the (actual) effective road inputs, and β_y, which are fed into the tire model. Figure 10.12 illustrates the procedure. When traversing a single step it is, of course, more efficient to use an analytic expression for the basic curve based on Eqn (10.6), cf. Figure 10.11 left.

FIGURE 10.12 The cam generating the basic road profile and two-point follower moving over it.

10.1.4 The Effective Rolling Radius When Rolling Over a Cleat

The third effective input is constituted by the effective road forward curvature that significantly changes the effective rolling radius when a road unevenness is traversed. In Figure 10.2 these variations have been shown in the lower diagrams. The curves are derived from measurements by using the Eqns (10.1) and (10.2).

In the effort to model the aspect of rolling over an obstacle, it is important to realize that we have three elements that contribute to the variation of the rolling radius:

Figure 10.16 illustrates the matter. The first item has been dealt with before, cf. Section 8.3.1, Figure 8.12, and Eqn (8.38) and Eqn (9.232). According to the latter equation the effective radius is a function of vertical load and speed of rolling. In Figure 10.17, model considerations and the graph resulting from experiments have been repeated. For small variations in radial deflection we may employ the equation

(10.7)

FIGURE 10.16 Three contributions to the change in apparent effective rolling radius.

FIGURE 10.17 The effective rolling radius varying with a change in normal load.

The second contribution accounts for the fact that at a slope and unchanged normal load, the axle speed parallel to the road surface is larger than the horizontal component V_x. We have for the change in the apparent effective rolling radius

(10.8)

The third contribution comes from the road surface curvature. The analysis that attempts to model the relation between curvature and effective rolling radius is more difficult and requires special attention. Figure 10.18 unravels the process of rolling over a curved obstacle and indicates the connection with rolling over a drum surface with same curvature. In contrast to the drum, the obstacle does not rotate. Consequently, to compare the process of rolling over a curved obstacle with that of rolling over a rotating drum surface, we must add the effect of the tire supported by a counter-rotating surface that does not move forward. The left-hand diagram of Figure 10.19 depicts a possible test configuration with a plank that can be tilted about a transverse line in the contact surface. When being tilted, point S, that is attached to the wheel rim, must move along with the plank in the longitudinal direction since the wheel slip is zero (V_sx = 0), as brake or drive torque is not applied. Consequently, the wheel rotates slightly and the following equation applies:

(10.9)

FIGURE 10.18 Unraveling the process of rolling over a curved obstacle.

FIGURE 10.19 Two components of rolling over a curved obstacle.

The apparent variation in effective rolling radius derives from the equation

(10.10)

so that we obtain for the contribution from the slope rate

(10.11)

The relationship (10.9) has been confirmed to hold through elaborate experiments conducted by Zegelaar (1998) on the tilting plank of the Delft flat plank machine, cf. Figure 12.5.

The other contribution that comes from the drum analog is found by considering the simple model shown in the right-hand diagram of Figure 10.19. The tread elements with length d_t are assumed to stand perpendicularly on the drum surface. The drum has a curvature with radius R_dr. The belt with radius r_b is considered inextensible. As a consequence, we find the following relation between the wheel and drum velocities:

(10.12)

and, for the effective rolling radius for the tire rolling freely over the drum surface,

(10.13)

For the model, the effective rolling radius of the tire rolling over a flat surface is equal to the radius of the belt:

(10.14)

Now, the tread depth is

(10.15)

Hence, the expression for the effective rolling radius on the drum can be rewritten as

(10.16)

The variation of the radius becomes

(10.17)

The drum radius is equal to the radius of curvature of the (effective) road surface profile. This curvature corresponds to change in slope β_y with traveled distance s. Note that for the convex drum surface the β_y rate of change is negative. So, we have

(10.18)

The slope rate of change may be written as

(10.19)

The drum contribution is now expressed as

(10.20)

Adding up the two contributions (10.11) and (10.20) yields the variation in effective rolling radius due to obstacle forward curvature:

(10.21)

where ρ_z is the radial compression of the tire. By adding up all the contributions, we finally obtain for the variation of the effective rolling radius with respect to the initial condition, where β_y = dβ_y/ds and ρ_z = ρ_zo:

(10.22)

The term with the effective forward curvature dβ_y/ds constitutes by far the most important contribution to the effective rolling radius variation and thus to the wheel rotational acceleration that can only be brought about by a variation in the longitudinal force F_x. This force also often outweighs the part of the horizontal longitudinal force F_H that directly results from the slope itself, cf. Eqn (10.4).

10.1.5 The Location of the Effective Road Plane

The relationship that exists between the effective height w and the actual height w′ of the effective road plane has not been addressed so far. Below, a more precise account will be given of the notion of the effective road plane with a clear definition of the effective road plane height.

In Figures 10.3 and 10.5, the effective height −w is considered as being assessed at a constant axle height above the reference plane that coincides with the initial flat level road surface. In Figure 10.20, the alternative case is illustrated where the effective height is assessed at constant vertical load F_V. The following formula covers both cases. The effective height is defined as

(10.23)

where z_a denotes the vertical axle displacement and ρ_zV the vertical tire deflection when loaded on a flat level road with load F_V. With an assumed linear tire spring characteristic we get

(10.24)

FIGURE 10.20 The effective road plane assessed at constant vertical load.

For the case that the effective height is found at a constant vertical load F_V, the initial vertical deflection is z_ao = ρ_zo = ρ_zV. Consequently by considering (10.24), the variation in effective height equals the change in axle height.

If the axle height is kept constant, we have z_a = z_ao = ρ_zo and the formula becomes: w = ρ_zo − ρ_zV, which corresponds with Eqn (10.3).

In Figure 10.20, the actual position of the effective road plane is defined as the location of the point of intersection of effective road plane and the vertical line through the center of the vertical wheel. Its height below the horizontal reference plane is designated as , also indicated in Figure 10.11.

With the small effect of the rolling resistance force F_r = f_r F_N first included and then neglected, the normal force becomes in terms of the vertical load

(10.25)

Consequently, the normal deflection becomes

(10.26)

The expression for the effective road plane height w′ follows from Figure 10.20 by inspection. With the vertical distance

(10.27)

we obtain, using (10.23) and (10.26)

(10.28)

For vanishing forward slope, we correctly find

(10.29)

For small values of forward slope, the approximation (10.29) is perfectly acceptable.

10.2.1 Two-Dimensional Unevennesses

Figure 10.9 shows the obstacle parameter values for the quarter sine basic curve of Zegelaar. Table 10.1 gives the parameters used by Schmeitz, which are based on the ellipse concept (tandem cam technique). The table shows that the height and length of the ellipse are slightly larger than the dimensions of the free tire. It is the exponent that gives rise to the larger curvature of the ellipse near the ground.

TABLE 10.1 Parameter Values Used for Envelopment Calculations. Tire: 205/60R15, 2.2bar

Figure 10.24 presents the measured and calculated variations of the effective height w (equal to the variations of the vertical axle displacement z_a at constant vertical load F_V), the horizontal fore-and-aft force F_H, and the effective rolling radius r_e, with the vertical load kept constant while slowly rolling over the trapezium cleat. The effective road plane slope tan β_y follows from the ratio of the horizontal force variation and the vertical load. The calculations are based on the two-cam tandem concept of Figure 10.11. The tandem is moved over the original road profile and the effective height and slope are obtained. From the derivative of β_y, the effective rolling radius is found by using Eqn (10.23), which also contains the two very small additional contributions. It is observed that a good agreement between test and calculation results can be achieved. The use of the quarter sine basic curve function gives very similar results.

FIGURE 10.24 Rolling over a trapezium cleat (length: 50 mm, height: 10 mm). Measured and calculated variation of vertical axle displacement (Δz_a = w), horizontal longitudinal force, and effective rolling radius at three different constant vertical loads. Measurements carried out on the Delft flat plank machine (very low speed) and calculations conducted with the use of the tandem cam technique (Figure 10.11) and Eqn (10.22). Tandem cam parameters according to Table 10.1.

Figures 10.25–10.29 present the results of rolling over the same cleat at different speeds while the axle location is kept fixed. The responses of the vertical, fore, and aft forces and the wheel angular speed have been indicated. In addition, the power spectra of these quantities have been shown.

FIGURE 10.25 Time traces and power spectra of vertical, horizontal, and longitudinal force and angular velocity for wheel running at given speed of 39 km/h and initial vertical load of 2000 N over a trapezium cleat at constant axle height using SWIFT tire parameters and obstacle parameters of Table 10.1. (Schmeitz).

FIGURE 10.26 Same as Figure 10.25 but at different speed and initial vertical load.

FIGURE 10.27 Same as Figure 10.26 but at higher speed.

FIGURE 10.28 Same as Figure 10.27 but at higher speed.

FIGURE 10.29 Same as Figure 10.27 but at higher initial vertical load.

Especially at the higher loads the calculated responses appear to follow the measured characteristics quite well up to frequencies around 50 Hz or higher. Figure 10.30 demonstrates the application to a more general road surface profile. It shows the responses of the tire when moving over a series of different types of cleats that resembles an uneven stretch of road.

FIGURE 10.30 Running over a series of cleats mounted on drum surface, at constant axle height.

Finally, in Figure 10.31 the test and simulation results conducted by Zegelaar (1998) have been depicted, representing a more complex condition where the tire is subjected to a given brake torque (brake pressure) while the wheel rolls over a pothole at a fixed axle location. The complex longitudinal force response conditions that are brought about by load and slip variations induced by tire modal vibrations and road unevennesses are simulated quite satisfactorily using the SWIFT model including obstacle geometric filtering.

FIGURE 10.31 The horizontal force variation when traversing a pothole while the wheel is being braked at three different levels of brake torque (M_B = ca. 4, 375, 850 Nm respectively, Zegelaar 1998). Experiments on 2.5 m drum, calculations using parameters according to Figure 10.9.

10.2.2 Three-Dimensional Unevennesses

In Figure 10.32 example results are presented of experimental and model simulation results of the tire moving at constant axle height at a speed of 59 km/h over an oblique strip (height 10 mm, width 50 mm) mounted at an angle of 43 degrees with respect to the drum axis (wheel slip angle and wheel camber angle remain zero). The variations of the wheel spindle forces K_x,y,z and moments T_x,z and the wheel angular velocity Ω are presented as functions of time and as power spectral densities versus frequency. Note: The output forces and moments K_x,y,z and T_x,z are the same as the quantities K_aξ;,η,ζ and T_aξ;,ζ expressed by the Eqns (9.195–9.200), also cf. Figure 9.25.

FIGURE 10.32 Running over an oblique cleat (10 × 50 mm strip) mounted on the drum surface at an angle of 43 degrees, at constant axle height. Measured and computed variations of wheel spindle forces K and moments T and wheel angular speed Ω versus time and as power spectral densities S.

Again, it is seen that a reasonable or good correspondence can be achieved. The various tire–wheel resonance frequencies can be observed to show up: vertical mode at 78 Hz (K_z), rotational mode at 35 Hz (K_x and Ω), camber mode at 46 Hz (K_y, T_z, and T_x), and yaw mode at 54 Hz (T_z).