1.2.1 Introduction to Tire Characteristics

The upright wheel rolling freely, that is without applying a driving torque, over a flat level road surface along a straight line at zero side slip, may be defined as the starting situation with all components of slip equal to zero. A relatively small pulling force is needed to overcome the tire-rolling resistance, and a side force and (self)-aligning torque may occur as a result of the not completely symmetric structure of the tire. When the wheel moves in a way that the condition of zero slip is no longer fulfilled, wheel slip occurs that is accompanied by a buildup of additional tire deformation and possibly partial sliding in the contact patch. As a result, (additional) horizontal forces and the aligning torque are generated. The mechanism responsible for this is treated in detail in the subsequent chapters. For now, we will suffice with some important experimental observations and define the various slip quantities that serve as inputs into the tire system and the moment and forces that are the output quantities (positive directions according to Figure 1.1). Several alternative definitions are in use as well. In Appendix 1, various sign conventions of slip, camber, and output forces and moments together with relevant characteristics have been presented.

For the freely rolling wheel, the forward speed V_x (longitudinal component of the total velocity vector V of the wheel center) and the angular speed of revolution Ω_o can be taken from measurements. By dividing these two quantities, the so-called effective rolling radius r_e is obtained:

(1.1)

Although the effective radius may be defined also for a braked or driven wheel, we restrict the definition to the case of free rolling. When a torque is applied about the wheel spin axis, a longitudinal slip arises that is defined as follows:

(1.2)

The sign is taken such that for a positive κ, a positive longitudinal force F_x arises, that is, a driving force. In that case, the wheel angular velocity Ω is increased with respect to Ω_o and consequently Ω > Ω_o = V_x/r_e. During braking, the fore-and-aft slip becomes negative. At wheel lock, obviously, κ = −1. At driving on slippery roads, κ may attain very large values. To limit the longitudinal slip K to a maximum equal to one, in some texts the longitudinal slip is defined differently in the driving range of slip: in the denominator of (1.2), Ω_o is replaced by Ω. This will not be done in the present text.

Lateral wheel slip is defined as the ratio of the lateral and the forward velocity of the wheel. This corresponds to minus the tangent of the slip angle α (Figure 1.1). Again, the sign of α has been chosen such that the side force becomes positive at positive slip angle.

(1.3)

The third and last slip quantity is the so-called spin which is due to rotation of the wheel about an axis normal to the road. Both the yaw rate resulting in path curvature when α remains zero and the wheel camber or inclination angle γ of the wheel plane about the x axis contribute to the spin. The camber angle is defined to be positive when, if looking from behind the wheel, the wheel is tilted to the right. In Chapter 2, more precise definitions of the three components of wheel slip will be given. The forces F_x and F_y and the aligning torque M_z are results of the input slip. They are functions of the slip components and the wheel load. For steady-state rectilinear motions, we have, in general,

(1.4)

The vertical load F_z may be considered as a given quantity that results from the normal deflection of the tire. The functions can be obtained from measurements for a given speed of travel and road and environmental conditions.

Figure 1.1 shows the adopted system of axes (x, y, z) with associated positive directions of velocities and forces and moments. The exception is the vertical force F_z acting from road to tire. For practical reasons, this force is defined to be positive in the upward direction and thus equal to the normal load of the tire. Also, Ω (not provided with a y subscript) is defined positive with respect to the negative y-axis. Note that the axes system is in accordance with SAE standards (SAE J670e 1976). The sign of the slip angle, however, is chosen opposite with respect to the SAE definition, cf. Appendix 1.

In Figure 1.2 typical pure lateral (κ = 0) and longitudinal (α = 0) slip characteristics have been depicted together with a number of combined slip curves. The camber angle γ was kept equal to zero. We define pure slip to be the situation when either longitudinal or lateral slip occurs in isolation. The figure indicates that a drop in force arises when the other slip component is added. The resulting situation is designated as combined slip. The decrease in force can be simply explained by realizing that the total horizontal frictional force F cannot exceed the maximum value (radius of ‘friction circle’) which is dictated by the current friction coefficient and normal load. Later, in Chapter 3 this becomes clear when considering the behavior of a simple physical tire model. The diagrams include the situation when the brake slip ratio has finally attained the value of 100% (κ = −1) which corresponds to wheel lock.

FIGURE 1.2 Combined side force and brake force characteristics.

The slopes of the pure slip curves at vanishing slip are defined as the longitudinal and lateral slip stiffnesses, respectively. The longitudinal slip stiffness is designated as C_Fκ. The lateral slip or cornering stiffness of the tire, denoted by C_Fα, is one of the most important property parameters of the tire and is crucial for the vehicle’s handling and stability performance. The slope of minus the aligning torque versus slip angle curve (Figure 1.1) at zero slip angle is termed as the aligning stiffness and is denoted by C_Mα. The ratio of minus the aligning torque and the side force is the pneumatic trail t (if we neglect the so-called residual torque to be dealt with in Chapter 4). This length is the distance behind the contact center (projection of wheel center onto the ground in wheel plane direction) to the point where the resulting lateral force acts. The linearized force and moment characteristics (valid at small levels of slip) can be represented by the following expressions in which the effect of camber has been included:

(1.5)

These equations have been arranged in such a way that all the coefficients (the force, moment slip, and camber stiffnesses) become positive quantities.

It is of interest to note that the order of magnitude of the tire cornering stiffness ranges from about 6 to about 30 times the vertical wheel load when the cornering stiffness is expressed as force per radian. The lower value holds for the older bias-ply tire construction and the larger value for modern racing tires. The longitudinal slip stiffness has been typically found to be about 50% larger than the cornering stiffness. The pneumatic trail is approximately equal to a quarter of the contact patch length. The dry friction coefficient usually equals ca. 0.9 on very sharp surfaces and ca. 1.6 on clean glass; racing tires may reach 1.5–2.

For the side force which is the more important quantity in connection with automobile handling properties, a number of interesting diagrams have been presented in Figure 1.3. These characteristics are typical for truck and car tires and are based on experiments conducted at the University of Michigan Transportation Research Institute (UMTRI, formerly HSRI), cf. Ref. (Segel et al. 1981). The car tire cornering stiffness data stem from newer findings. It is observed that the cornering stiffness changes in a less than proportional fashion with the normal wheel load. The maximum normalized side force F_y,peak/F_z appears to decrease with increasing wheel load. Marked differences in level and slope occur for the car and truck tire curves also when normalized with respect to the rated or nominal load. The cornering force vs slip angle characteristic shown at different speeds and road conditions indicates that the slope at zero slip angle is not or hardly affected by the level of speed and by the wet or dry condition. The peak force level shows only little variation if the road is dry. On a wet road, a more pronounced peak occurs and the peak level drops significantly with increasing speed.

FIGURE 1.3 Typical characteristics for the normalized cornering stiffness, peak side force, and side force vs normalized vertical load and slip angle respectively. F_zo is the rated load.

Curves which exhibit a shape like the side force characteristics of Figure 1.3 can be represented by a mathematical formula called ‘magic formula’. A full treatment of the empirical tire model associated with this formula is given in Chapter 4. For now we can suffice with showing the basic expressions for the side force and the cornering stiffness:

(1.6)

The shape factors C and E as well as the parameters c₁ and c₂ and the friction coefficient μ (possibly depending on the vertical load and speed) may be estimated or determined through regression techniques.

1.2.2 Effective Axle Cornering Characteristics

For the basic analysis of (quasi)-steady-state turning behavior, a simple two-wheel vehicle model may be used successfully. Effects of suspension and steering system kinematics and compliances such as steer compliance, body roll, and also load transfer may be taken into account by using effective axle characteristics. The restriction to (quasi)-steady-state becomes clear when we realize that for transient or oscillatory motions, exhibiting yaw and roll accelerations and differences in phase, variables like roll angle and load transfer can no longer be written as direct algebraic functions of one of the lateral axle forces (front or rear). Consequently, we should drop the simple method of incorporating the effects of a finite center of gravity height if the frequency of input signals such as the steering wheel angle cannot be considered small relative to the body roll natural frequency. Since the natural frequency of the wheel suspension and steering systems is relatively high, the restriction to steady-state motions becomes less critical in the case of the inclusion of e.g., steering compliance in the effective characteristic. Chiesa and Rinonapoli (1967) were among the first to employ effective axle characteristics or ‘working curves’ as these were referred to by them. Vågstedt (1995) determined these curves experimentally.

Before assessing the complete nonlinear effective axle characteristics, we will first direct our attention to the derivation of the effective cornering stiffnesses which are used in the simple linear two-wheel model. For these to be determined, a more comprehensive vehicle model has to be defined.

Figure 1.4 depicts a vehicle model with three degrees of freedom. The forward velocity u may be kept constant. As motion variables, we define the lateral velocity v of reference point A, the yaw velocity r, and the roll angle φ. A moving axes system (A, x, y, z) has been introduced. The x-axis points forwards and lies both in the ground plane and in the plane normal to the ground that passes through the so-called roll axis. The y-axis points to the right and the z-axis points downward. This latter axis passes through the center of gravity when the roll angle is equal to zero. In this way, the location of the point of reference A has been defined. The longitudinal distance to the front axle is a and the distance to the rear axle is b. The sum of the two distances is the wheel base l. For convenience, we may write a = a₁ and b = a₂.

FIGURE 1.4 Vehicle model showing three degrees of freedom: lateral, yaw, and roll.

In a curve, the vehicle body rolls about the roll axis. The location and attitude of this virtual axis are defined by the heights h_1,2 of the front and rear roll centers. The roll axis is assessed by considering the body motion with respect to the four contact centers of the wheels on the ground under the action of an external lateral force that acts on the center of gravity. Due to the symmetry of the vehicle configuration and the linearization of the model, these locations can be considered as fixed. The roll center locations are governed by suspension kinematics and possibly suspension lateral compliances. The torsional springs depicted in the figure represent the front and rear roll stiffnesses c_φ1,2 which result from suspension springs and antiroll bars.

The fore-and-aft position of the center of gravity of the body is defined by a and b; its height follows from the distance to the roll axis. The body mass is denoted by m and the moments of inertia with respect to the center of mass and horizontal and vertical axes by I_x, I_z, and I_xz. These latter quantities will be needed in a later phase when the differential equations of motion are established. The unsprung masses will be neglected or they may be included as point masses attached to the roll axis and thus make them part of the sprung mass, that is, the vehicle body.

Furthermore, the model features torsional springs around the steering axes. The king-pin is positioned at a small caster angle that gives rise to the caster length e as indicated in the drawing. The total steering torsional stiffness, left plus right, is denoted by c_ψ1.

1.3.1 Differential Equations for Plane Vehicle Motions

In this section, the differential equations for the three degree of freedom vehicle model of Figure 1.4 will be derived. In first instance, the fore-and-aft motion will also be left free to vary. The resulting set of equations of motion may be of interest for the reader to further study the vehicle’s dynamic response at somewhat higher frequencies where the roll dynamics of the vehicle body may become of importance, cf. App. 2. From these equations, the equations for the simple two-degree-of-freedom model of Figure 1.9 used in the subsequent section can be easily assessed. In Subsection 1.3.6, the equations for the car with trailer will be established. The possible instability of the motion will be studied.

We will employ Lagrange’s equations to derive the equations of motion. For a system with n degrees of freedom n (generalized) coordinates, q_i are selected which are sufficient to completely describe the motion while possible kinematic constraints remain satisfied. The moving system possesses kinetic energy T and potential energy U. External generalized forces Q_i associated with the generalized coordinates q_i may act on the system. Internal forces acting from dampers to the system structure may be regarded as external forces taking part in the total work W. The equation of Lagrange for coordinate q_i reads

(1.25)

The system depicted in Figure 1.4 and described in the preceding subsection performs a motion over a flat level road. Proper coordinates are the Cartesian coordinates X and Y of reference point A, the yaw angle ψ of the moving x-axis with respect to the inertial X-axis and finally the roll angle φ about the roll axis. For motions near the X-axis and thus small yaw angles, Eqn (1.25) is adequate to derive the equations of motion. For cases where ψ may attain large values, e.g., when moving along a circular path, it is preferred to use modified equations where the velocities u, v, and r of the moving axes system are used as generalized motion variables in addition to the coordinate φ. The relations between the two sets of variables are (the dots referring to differentiation with respect to time)

(1.26)

The kinetic energy can be expressed in terms of u, v, and r. Preparation of the first terms of Eqn (1.25) for the coordinates X, Y, and ψ yields

(1.27)

The yaw angle ψ may now be eliminated by multiplying the final equations for X and Y successively with cos ψ and sin ψ and subsequently adding and subtracting them. The resulting equations represent the equilibrium in the x- and y- (or u and v) direction, respectively.

We obtain the following set of modified Lagrangian equations for the first three variables u, v, and r and subsequently for the remaining real coordinates (for our system only φ):

(1.28)

The generalized forces are found from the virtual work:

(1.29)

with q_j referring to the quasi-coordinates x and y and the coordinates ψ and φ. Note that x and y cannot be found from integrating u and v. For that reason, the term ‘quasi’ coordinate is used. For the vehicle model, we find for the virtual work as a result of the virtual displacements δx, δy, δψ, and δφ:

(1.30)

where apparently

(1.31)

The longitudinal forces are assumed to be the same at the left and right wheels, and the effect of additional steer angles ψ_i is neglected here. Shock absorbers in the wheel suspensions are represented by the resulting linear moments about the roll axes with damping coefficients k_φi at the front and rear axles.

With the roll angle φ and the roll axis inclination angle θ_r ≈(h₂ − h₁)/l assumed small, the kinetic energy becomes

(1.32)

The potential energy U is built up in the suspension springs (including the radial tire compliances) and through the height of the center of gravity. We have, again for small angles,

(1.33)

The equations of motion are finally established by using expressions (1.31), (1.32), and (1.33) in the Eqn (1.28). The equations will be linearized in the supposedly small angles φ and δ. For the variables u, v, r, and φ, we obtain successively

(1.34a)

(1.34b)

(1.34c)

(1.34d)

Note that the small additional roll and compliance steer angles ψ_i have been neglected in the assessment of the force components. The tire side forces depend on the slip and camber angles front and rear and on the tire vertical loads. We may need to take the effect of combined slip into account. The longitudinal forces are either given as a result of brake effort or imposed propulsion torque or they depend on the wheel longitudinal slip which follows from the wheel speed of revolution requiring four additional wheel rotational degrees of freedom. The first Eqn (1.34a) may be used to compute the propulsion force needed to keep the forward speed constant.

The vertical loads and more specifically the load transfer can be obtained by considering the moment equilibrium of the front and rear axle about the respective roll centers. For this, the roll moments M_φi (cf. Figure 1.4) resulting from suspension springs and dampers as appeared in Eqn (1.34d) through the terms with subscript 1 and 2 respectively, and the axle side forces appearing in Eqn (1.34b) are to be regarded. For a linear model the load transfer can be neglected if initial (left/right opposite) wheel angles are disregarded. We have at steady state (effect of damping vanishes)

(1.35)

The front and rear slip angles follow from the lateral velocities of the wheel axles and the wheel steer angles with respect to the moving longitudinal x-axis. The longitudinal velocities of the wheel axles may be regarded the same left and right and equal to the vehicle longitudinal speed u. This is allowed when s_i|r| u. Then the expressions for the assumedly small slip angles read

(1.36)

The additional roll and compliance steer angles ψ_i and the wheel camber angles γ_i are obtained from Eqn (1.21) with (1.13–15) or corresponding nonlinear expressions. Initial wheel angles are assumed to be equal to zero. The influence of the steer angle velocity appearing in the expression for the front slip angle is relatively small and may be disregarded. The small products of the caster length e and the time rate of change of ψ_i have been neglected in the above expressions.

Equations (1.34) may be further linearized by assuming that all the deviations from the rectilinear motion are small. This allows neglecting all products of variable quantities which vanish when the vehicle moves straight ahead. The side forces and moments are then written as in Eqn (1.5) with the subscripts i = 1 or 2 provided. If the moment due to camber is neglected and the pneumatic trail is introduced in the aligning torque, we have

(1.37)

The three linear equations of motion for the system of Figure 1.4 with the forward speed u kept constant finally turn out to read, if expressed solely in terms of the three motion variables v, r, and φ:

(1.38a)

(1.38b)

(1.38c)

In these equations, the additional steer angles ψ_i have been eliminated by using expressions (1.21) with (1.13–15). Furthermore, the resulting compliance steer and roll steer coefficients for i = 1 or 2 have been introduced:

(1.39)

with

where the steer stiffness at the rear c_ψ2 may be taken equal to infinity. Furthermore, we have the roll axis inclination angle:

(1.40)

In Chapters 7 and 8, the transient properties of the tire will be addressed. The relaxation length denoted by σ_i is an important parameter that controls the lag of the response of the side force to the input slip angle. For the Laplace transformed version of Eqn (1.38) with the Laplace variable s representing differentiation with respect to time, we may introduce tire lag by replacing the slip angle α_i with the filtered transient slip angle. This may be accomplished by replacing the cornering stiffnesses C_Fαi appearing in (1.38) and (1.39) with the ‘transient stiffnesses’:

(1.41)

A similar procedure may be followed to include the tire transient response to wheel camber variations. The relaxation length concerned is about equal to the one used for the response to side slip variations. At nominal vertical load, the relaxation length is of the order of magnitude of the wheel radius. A more precise model of the aligning torque may be introduced by using a transient pneumatic trail with a similar replacement as indicated by (1.41) but with a much smaller relaxation length approximately equal to half the contact length of the tire. For more details, we refer to Chapter 9 that is dedicated to short wavelength force and moment response.

1.3.2 Linear Analysis of the Two-Degree-of-Freedom Model

From Eqns (1.34b and c), the reduced set of equations for the two-degree-of-freedom model can be derived immediately. The roll angle φ and its derivative are set equal to zero and, furthermore, we will assume the forward speed u (≈V) to remain constant and neglect the influence of the lateral component of the longitudinal forces F_xi. The equations of motion of the simple model of Figure 1.9 for v and r now read

(1.42a)

(1.42b)

with v denoting the lateral velocity of the center of gravity and r the yaw velocity. The symbol m stands for the vehicle mass and I (= I_z) denotes the moment of inertia about the vertical axis through the center of gravity. For the matter of simplicity, the rearward shifts of the points of application of the forces F_y1 and F_y2 over a length equal to the pneumatic trail t₁ and t₂ respectively (that is the aligning torques) have been disregarded. Later, we come back to this. The side forces are functions of the respective slip angles:

(1.43)

and the slip angles are expressed by

(1.44)

neglecting the effect of the time rate of change of the steer angle appearing in Eqn (1.36). For relatively low-frequency motions, the effective axle characteristics or effective cornering stiffnesses according to Eqns (1.17, 1.22) may be employed.

When only small deviations with respect to the undisturbed straight-ahead motion are considered, the slip angles may be assumed to remain small enough to allow linearization of the cornering characteristics. For the side force, the relationship with the slip angle reduces to the linear equation

(1.45)

where C_i denotes the cornering stiffness. This can be replaced by the symbol C_Fαi which may be preferred in more general cases where also camber and aligning stiffnesses play a role.

The two linear first-order differential equations now read:

(1.46)

After elimination of the lateral velocity v, we obtain the second-order differential equation for the yaw rate r:

(1.47)

Here, as before, the dots refer to differentiation with respect to time, δ is the steer angle of the front wheel, and l (= a + b) represents the wheel base. The equations may be simplified by introducing the following quantities:

(1.48)

Here C denotes the total cornering stiffness of the vehicle, s is the distance from the center of gravity to the so-called neutral steer point S (Figure 1.11), q is a length corresponding to an average moment arm, and k is the radius of gyration. Equations (1.46) and (1.47) now reduce to

(1.49)

and, with v eliminated,

(1.50)

The neutral steer point S is defined as the point on the longitudinal axis of the vehicle where an external side force can be applied without changing the vehicle’s yaw angle. If the force acts in front of the neutral steer point, the vehicle is expected to yaw in the direction of the force and, if behind, then against the force. The point is of interest when discussing the steering characteristics and stability.

1.3.3 Nonlinear Steady-State Cornering Solutions

From Eqns (1.42) and (1.59) with the same restrictions as stated below Eqn (1.60), the following force balance equations can be derived (follows also from Eqn (1.61)). The effect of the pneumatic trails will be dealt with later on.

(1.80)

where K = ma_y represents the centrifugal force. The kinematic relationship

(1.81)

follows from Eqns (1.44) and (1.51). In Figure 1.11 the vehicle model has been depicted in a steady-state cornering maneuver. It can easily be observed from this diagram that relation (1.81) holds approximately when the angles are small.

The ratio of the side force and vertical load as shown in (1.80) plotted as a function of the slip angle may be termed as the normalized tire or axle characteristic. These characteristics subtracted horizontally from each other produce the ‘handling curve’. Considering the equalities (1.80), the ordinate may be replaced by a_y/g. The resulting diagram with abscissa α₁−α₂ is the nonlinear version of the right-hand diagram of Figure 1.10 (rotated 90° anticlockwise). The diagram may be completed by attaching the graph that shows, for a series of speeds V, the relationship between lateral acceleration (in g units) a_y /g and the relative path curvature l/R according to Eqn (1.55).

Figure 1.17 shows the normalized axle characteristics and the completed handling diagram. The handling curve consists of a main branch and two side lobes. The different portions of the curves have been coded to indicate the corresponding parts of the original normalized axle characteristics they originate from. Near the origin, the system may be approximated by a linear model. Consequently, the slope of the handling curve in the origin with respect to the vertical axis is equal to the understeer coefficient η. In contrast to the straight handling line of the linear system (Figure 1.10), the nonlinear system shows a curved line. The slope changes along the curve which means that the degree of understeer changes with increasing lateral acceleration. The diagram of Figure 1.17 shows that the vehicle considered changes from understeer to oversteer. We define

(1.82)

FIGURE 1.17 Handling diagram resulting from normalized tire characteristics. Equilibrium points I, II, and III (steady turns), of which only I is stable, arise for speed V = 50 km/h and steer angle δ = 0.04 rad. From the different line types, the manner in which the curves are obtained from the upper diagram may be retrieved.

The family of straight lines represents the relationship between acceleration and curvature at different levels of speed. The speed line belonging to V = 50 km/h has been indicated (wheel base l = 3 m). This line is shifted to the left over a distance equal to the steer angle δ = 0.04 rad and three points of intersection with the handling curve arise. These points I, II, and III indicate the possible equilibrium conditions at the chosen speed and steer angle. The connected values of the relative path curvature l/R can be found by going back to the speed line. As will be shown further on, only point I refers to a stable cornering motion. In points II and III (R < 0!), the motion is unstable.

At a given speed V, a certain steer angle δ is needed to negotiate a circular path with given radius R. The steer angle required can be read directly from the handling diagram. The steer angle needed to negotiate the same curve at very low speed (V → 0) tends to l/R. This steer angle is denoted by δ_o. Consequently, the abscissa of the handling curve α₁ − α₂ may as well be replaced by δ − δ_o. This opens the possibility to determine the handling curve with the aid of simple experimental means, i.e., measuring the steering wheel input (reduced to equivalent road wheel steer angle by means of the steering ratio, which method automatically includes steering compliance effects) at various speeds running over the same circular path.

Subtracting normalized characteristics may give rise to very differently shaped handling curves only by slightly modifying the original characteristics. As Figure 1.17 shows, apart from the main branch passing through the origin, isolated branches may occur. These are associated with at least one of the decaying ends of the pair of normalized tire characteristics.

In Figure 1.18 a set of four possible combinations of axle characteristics have been depicted together with the resulting handling curves. This collection of characteristics shows that the nature of steering behavior is entirely governed by the normalized axle characteristics and in particular their relative shape with respect to each other.

FIGURE 1.18 A number of handling curves arising from the pairs of normalized tire characteristics shown in left. Only the main branch of the handling curve has been drawn (1: front, 2: rear).

The way in which we can use the handling diagram is presented in Figure 1.19. The speed of travel may be kept constant and the lateral acceleration is increased by running over a spiral path with decreasing radius. The required variation of the steer angle follows from the distance between the handling curve and the speed line. Similarly, we can observe what happens when the path curvature is kept constant and the speed is increased. Also, the resulting variation of the curvature at a constant steer angle and increasing speed can be found. More general cases of quasi-steady-state motions may be studied as well.

FIGURE 1.19 Types of quasi-steady-state maneuvers.

1.3.4 The Vehicle at Braking or Driving

When the vehicle is subjected to longitudinal forces that may result from braking or driving actions possibly to compensate for longitudinal wind drag forces or down or upward slopes, fore-and-aft load transfer will arise (Figure 1.26). The resulting change in tire normal loads causes the cornering stiffnesses and the peak side forces of the front and rear axles to change. Since, as we assume here, the fore-and-aft position of the center of gravity is not affected (no relative car body motion), we may expect a change in handling behavior indicated by a rise or drop of the understeer gradient. In addition, the longitudinal driving or braking forces give rise to a state of combined slip, thereby affecting the side force in a way as shown in Figure 1.2.

FIGURE 1.26 The automobile subjected to longitudinal forces and the resulting load transfer.

For moderate driving or braking forces, the influence of these forces on the side force F_y is relatively small and may be neglected for this occasion. This means that, for now, the cornering stiffness may be considered to be dependent on the normal load only. The upper-left diagram of Figure 1.3 depicts typical variations of the cornering stiffness with vertical load.

The load transfer from the rear axle to the front axle that results from a forward longitudinal force F_L acting at the center of gravity at a height h above the road surface (F_L possibly corresponding to the inertial force at braking) becomes

(1.100)

The understeer gradient reads according to Eqn (1.60):

(1.101)

The static axle loads F_zio (i = 1 or 2) are calculated according to Eqn (1.59), while the actual loads F_zi front and rear become

(1.102)

At moderate braking with deceleration −a_x = F_L/m, the load transfer remains small and we may use the linearized approximation of the variation of cornering stiffness with vertical load:

(1.103)

The understeer gradient (1.101) can now be expressed in terms of the longitudinal acceleration a_x (which might be minus the forward component of the acceleration due to gravity parallel to the road). We obtain

(1.104)

with the determining factor λ approximately expressed as

(1.105)

and η_o denoting the original value not including the effect of longitudinal forces. Obviously, since ζ_α1,2 is usually positive, negative longitudinal accelerations a_x, corresponding to braking, will result in a decrease of the degree of understeer.

To illustrate the magnitude of the effect, we use the parameter values given in Table 1.1 (p. 34) and add the c.g. height h = 0.6 m and the cornering stiffness versus load gradients ζ_αi = 0.5C_io/F_zio. The resulting factor appears to take the value λ = 0.052. This constitutes an increase of η equal to 0.052a_x/g. Apparently, the effect of a_x on the understeer gradient is considerable when regarding the original value η_o = 0.0174.

As illustrated by Figure 1.9, the peak side force will be diminished if a longitudinal driving or braking force is transmitted by the tire. This will have an impact on the resulting handling diagram in the higher range of lateral acceleration. The resulting situation may be represented by the second and third diagrams of Figure 1.18 corresponding to braking (or driving) at the front or rear respectively. The problem becomes considerably more complex when we realize that at the front wheels, the components of the longitudinal forces perpendicular to the x-axis of the vehicle are to be taken into account. Obviously, we find that at braking of the front wheels, these components will counteract the cornering effect of the side forces and thus will make the car more understeer. The opposite occurs when these wheels are driven (more oversteer). For a more elaborate discussion on this item, we may refer to Pacejka (1973b).

At hard braking, possibly up to wheel lock, stability and steerability may deteriorate severely. This more complex situation will be discussed in Chapter 3 where more information on the behavior of tires at combined slip is given.

1.3.5 The Moment Method

Possible steady-state cornering conditions, stable or unstable, have been portrayed in the handling diagram of Figure 1.17. In Figure 1.22, motions tending to or departing from these steady-state conditions have been depicted. These motions are considered to occur after a sudden change in steer angle. The potential available to deviate from the steady turn depends on the margin of the front and rear side forces to increase in magnitude. For each point on the handling curve, it is possible to assess the degree of maneuverability in terms of the moment that can be generated by the tire side forces about the vehicle’s center of gravity. Note that at the steady-state equilibrium condition, the tire side forces are balanced with the centrifugal force and the moment equals zero.

In general, the handling curve holds for a given speed of travel. That is so, when e.g., the aerodynamic down forces are essential in the analysis. In Figure 1.27 a diagram has been presented that is designated as the MMM diagram (the Milliken moment method diagram) and is computed for a speed of 60 mph. The force–moment concept was originally proposed by W.F. Milliken in 1952 and thereafter continuously further developed by the Cornell Aeronautical Laboratory staff and by Milliken research associates. A detailed description is given in Milliken’s book (1995).

FIGURE 1.27 The MMM diagram portraying the car’s potential maneuvering capacity.

The graph shows curves of the resulting tire moment N vs the resulting tire side force Y in nondimensional form. The resulting force and moment result from the individual side forces and act from ground to vehicle. For greater accuracy, one may take the effect of the pneumatic trails into consideration. Two sets of curves have been plotted: one set for constant values of the vehicle side slip angle β with the steering wheel angle δ_stw as parameter and the other set for constant steer angle and varying slip angle. Along the horizontal axis, the moment is zero and we have the steady-state equilibrium cornering situation that corresponds to the handling curve. It is observed that for the constant speed considered in the diagram, the steer angle increases when the total side force Y or lateral acceleration a_y is chosen larger which indicates that the motion remains stable. At the limit (near number 2), the maximum steady-state lateral acceleration is attained. At that point, the ability to generate a positive moment is exhausted. Only a negative moment may still be developed by the car that tends to straighten the curve that is being negotiated. As we have seen in Figure 1.18, second diagram, there is still some side force margin at the rear tire which can be used to increase the lateral acceleration in a transient fashion. At the same time, however, the car yaws outward because the associated moment is negative (cf. Figure 1.27, the curve near number 8). How to get at points below the equilibrium point near the number 2 is a problem. Rear wheel steering is an obvious theoretical option. In that way, the vehicle slip angle β and front steer angle δ can remain unchanged while the rear steer angle produces the desired rear tire slip angle.

Of course, the diagram needs to be adapted in the case of rear wheel steering. Another more practical solution would be to bring the vehicle in the desired attitude (β → 8°) by briefly inducing large brake or drive slip at the rear that lowers the cornering force and lets the car swing to the desired slip angle while at the same time the steering wheel is turned backward to even negative values.

The MMM diagram, which is actually a Gough plot (for a single tire, cf. Figures 3.5 and 3.29) established for the whole car at different steer angles, may be assessed experimentally through either outdoor or indoor experiments. On the proving ground, a vehicle may be attached at the side of a heavy truck or railway vehicle and set at different slip angles while the force and moment are being measured (tethered testing), cf. Milliken (1995). Figure 1.28 depicts the remarkable laboratory MMM test machine. This MTS Flat-Trac Roadway Simulator™ uses four flat belts which can be steered and driven independently. The car is constrained in its center of gravity but is free to roll and pitch.

FIGURE 1.28 The MTS Flat-Trac Roadway Simulator™, Milliken (1995).

1.3.6 The Car-Trailer Combination

In this section we will discuss the role of the tire in connection with the dynamic behavior of a car that tows a trailer. More specifically, we will study the possible unstable motions that may show up with such a combination. Linear differential equations are sufficient to analyze the stability of the straight-ahead motion. We will again employ Lagrange’s equations to set up the equations of motion. The original Eqn (1.25) may be employed because the yaw angle is assumed to remain small. The generalized coordinates Y, ψ, and θ are used to describe the car’s lateral position and the yaw angles of car and trailer respectively. The forward speed dX/dt (≈V ≈ u) is considered to be constant. Figure 1.29 gives a top view of the system with three degrees of freedom. The alternative set of three variables v, r, the articulation angle φ, and the vehicle velocity V (a parameter) which are not connected to the inertial axes system (O, X, Y) has been indicated as well and will be employed later on. The kinetic energy for this system becomes, if we neglect all the terms of the second order of magnitude (products of variables),

(1.106)

FIGURE 1.29 Single track model of car-trailer combination.

The potential energy remains zero:

(1.107)

and the virtual work done by the external road contact forces acting on the three axles reads

(1.108)

With the use of Eqns (1.25) and (1.29), the following equations of motion are established for the generalized coordinates Y, ψ, and θ:

(1.109)

(1.110)

(1.111)

This constitutes a system of the sixth order. By introducing the velocities v and r, the order can be reduced to four. In addition, the angle of articulation φ will be used. We have the relations

(1.112)

And, with these the equations for v, r, and φ,

(1.113)

(1.114)

(1.115)

The right-hand members are still to be expressed in terms of the motion variables. With the axle cornering stiffnesses C₁, C₂, and C₃, we have

(1.116)

From the resulting set of linear differential equations, the characteristic equation may be derived which is of the fourth degree. Its general structure is

(1.117)

The stability of the system can be investigated by considering the real parts of the roots of this equation or we might employ the criterion for stability according to Routh-Hurwitz. According to this criterion, the system of order n is stable when all the coefficients a_i are positive and the Hurwitz determinants H_n−1, H_n−3 etc. are positive. For our fourth-order system, the complete criterion for stability reads

(1.118)

In Figure 1.30, the boundaries of stability have been presented in the caravan axle cornering stiffness vs speed parameter plane. The three curves belong to the three different sets of parameters for the position f of the caravan’s center of gravity and the caravan’s mass m_c as indicated in the figure. An important result is that a lower cornering stiffness promotes oscillatory instability: the critical speed beyond which instability occurs decreases. Furthermore, it appears from the diagram that moving the caravan’s center of gravity forward (f smaller) stabilizes the system which is reflected by the larger critical speed. A heavier caravan (m_c larger) appears to be bad for stability. Furthermore, it has been found that a larger draw bar length g is favorable for stability.

FIGURE 1.30 Stability boundaries for the car caravan combination in the caravan cornering stiffness vs critical speed diagram. Vehicle parameters according to Table 1.1, in addition: h = 2 m, g = 2 m, k_c = 1.5 m (I_c = m_ck_c²), cf. Figure 1.29.

It turns out that a second type of instability may show up. This occurs when the portion of the weight of the caravan supported by the coupling point becomes too large. This extra weight is felt by the towing vehicle and makes it more oversteer. The critical speed associated with this phenomenon is indicated in the diagram by the vertical lines. This divergent instability occurs when (starting out from a stable condition) the last coefficient becomes negative, that is, a_n = a₄ < 0.

The oscillatory instability connected with the ‘snaking’ phenomenon arises as soon as (from a stable condition) the second highest Hurwitz determinant becomes negative, H_n−1 = H₃ < 0 (then also H_n < 0), cf. Klotter (1960) or Leipholz (1987). When the critical speed is surpassed, self-excited oscillations are created which shows an amplitude that, in the actual nonlinear case, does not appear to limit itself. This is in contrast to the case of the wheel shimmy phenomenon to be treated in Chapter 5 where a stable limited oscillation appears to arise. The cause of the unlimited snaking oscillation is that with increasing amplitudes also the slip angle increases which lowers the average cornering stiffness as a consequence of the degressively nonlinear cornering force characteristic. From the diagram we found that this will make the situation increasingly worse. As has been seen from full vehicle/caravan model simulations, the whole combination will finally overturn. Another effect of this reduction of the average cornering stiffness is that when the vehicle moves at a speed lower than the critical speed, the originally stable straight-ahead motion may become unstable if, through the action of an external disturbance (side wind gust), the slip angle of the caravan axle becomes too large (surpassing of the associated unstable limit cycle). This is an unfortunate, possibly dangerous situation! We refer to Troger and Zeman (1984) for further details.

1.3.7 Vehicle Dynamics at More Complex Tire Slip Conditions

So far, relatively simple vehicle dynamics problems have been studied in which the basic steady-state cornering force versus slip angle characteristic plays the dominant role. The situation becomes more complex when matters like combined slip at hard braking, wheel camber, tire transient and vibrational properties, and e.g., obstacle crossings are to be considered.

In the subsequent chapters, tire performance and modeling will be treated in greater detail which enables us to introduce relevant tire properties in the analysis. The following specific subjects will be studied as applications of the tire modeling theory:

Online, cf. App. 2, several types of car handling experiments are discussed and compared with theoretical results of Sec. 1.3.2. MATLAB programs are presented, also for the three degree of freedom model, for the benefit of the interested reader.