Stability of Trailers
III. STABILITY ANALYSIS USING INERTIAL COORDINATES
Fig. 6.1 shows a plane motion model of an automobile nominally moving along the x-axis, but with a small heading angle W(t). The variables x(t) and y(t) describe the position of the center of mass. In this model, all motion is assumed to occur in the x–y plane, so there is no consideration of vertical (heave) motion or roll, or pitch angular motion.
The parameters of this model are a and b, the distances between the axles and the center of mass; m, the mass; Iz, the polar moment of inertia referred to the center of mass and the z-axis; and the two half-tracks, tfand tr, which will subsequently be shown to be unimportant for the following stability analysis.
The basic motion is described by x=U=constant, y=W=W=0. For the perturbed motion, we assume that the x-motion remains approximate- ly the same as for the basic motion but that y, W, and W can have small values. This means that no equation of motion for x is required because x=Ut+constant.
The main difficulty in setting up the dynamic equations for this model lies in the computation of the slip angles for the four tires.Fig. 6.2
shows the velocity components of the wheel motion at the left front. As with the vehicle models studied in Chapters 1and5,the wheel velocity again involves the summation of the velocity of a particular point (the center of mass) and the velocity components induced by the angular velocity, W.
The velocity components shown in Fig. 6.2 bear a good deal of resemblance to those shown inFig. 5.3for the trailer model. For the trailer, the velocity at the wheel was the velocity at the hitch, plus the velocity induced by the rotation. In Fig. 6.2, the velocity at the wheel is the velocity of the center of mass plus the vector components aW and tfW. These components can be recognized as components of the general term x! r!AB, which was introduced in Eq. (1.15) and applied inFig. 5.3.In the present case, r!AB is the vector distance from the center of mass to the center point of the wheel.
By resolving the velocity components shown in Fig. 6.2 in the direction the tire is pointing, the rolling velocity of the tire, using a small angle approximation and the fact that all the perturbed variables are assumed to be small, is given by the expression
Ucos W tfW þ y sin WiU ð6:1Þ
Similarly, the lateral velocity perpendicular to the tire pointing direction is found to be
ycos Wþ aW U sin Wiyþ aW UW ð6:2Þ
The front slip angle is then, af¼
Lateral Velocity Rolling Velocity¼
yþ aW
U W ð6:3Þ
Note that after the small perturbed variable assumption has been used, the influence of the front half-track, tf, disappeared. This means that the slip angle expression applies just as well for the right front wheel. As both rear wheels also have approximately the same slip angles for the perturbed motion, this leads to the common term bicycle model for this four-wheel car, because it is possible to consider a single equivalent wheel for the front axle and another for the rear axle.
Fig. 6.3shows the velocity components at the left rear wheel. The computation of the rear slip angle, ar, follows closely the procedure for the front wheels.
The rolling velocity is
Ucos W trW þ y sin WiU ð6:4Þ
The lateral velocity is
ycos W bW U sin Wiy bW UW ð6:5Þ
The slip angle at the rear is ar¼
Lateral Velocity Rolling Velocity¼
y bW
U W ð6:5aÞ
Once again, because of the small angle assumptions, the track has no influence on the slip angle at the rear so the slip angle expression in Eq. (6.5a) applies to both rear wheels.
Fig. 6.4shows the forces exerted on the automobile by the tires. The forces Yfand Yrare shown as positive in the nominal y-direction, and the slip angles are shown as though they were positive (as though the lateral velocities were positive in the nominal y-direction). This is a perfectly logical way to determine the signs of the forces and velocities. However, when the slip angles are positive as shown, the forces exerted on the tires from the ground would actually be in the negative y-direction.
This is a common problem in describing tire–roadway interactions, which leads either to negative cornering coefficients in the linearized case, or to a special way of writing the force laws to have positive coefficients. Later on, when stability criteria are being applied to the equations of motion, it is inconvenient to have some parameters in the equations that are inherently negative. Therefore the laws relating lateral tire forces to slip angles will be written in a special form that will avoid the problem of negative cornering coefficients.
In this chapter, a negative sign will be written in to the force laws so that the cornering coefficients at the front and rear axles, Cfand Cr, are positive. Just as in the case in Chapter 5, the cornering coefficients are assumed to account for the sum of the forces on both of the two tires at each axle.
Yf ¼ Cfaf Yr¼ Crar ð6:6Þ
It should be remembered that the use of linearized force-slip angle relations is only valid for small slip angles. This is fine for a stability analysis that dealing with only small perturbations from the basic motion, but if the vehicle is unstable, the slip angles will build up until the approximation is no longer appropriate.
Because the axle cornering coefficients relate to a summation of the forces on both tires at each axle, they depend partly on the normal forces supported at the individual wheels, the characteristics of the road surface, and of the tires themselves. Although there are many suspension design factors for a vehicle that also contribute to the determination of the axle cornering coefficients beside the characteristics of the tires themselves (Bundorf, 1967b), for now we regard the effective front and rear axle cornering coefficients simply as given quantities.
The equations of motion relate the acceleration of the center of mass to the applied forces and the angular acceleration to the applied moments. Keeping in mind that W is assumed to be a small angle, one can write Newton’s law for linear acceleration of the center of mass in the form
m ¨y¼ Yfþ Yr¼ Cfaf Crar ð6:7Þ
The angular acceleration law is
IzW¨ ¼ aYf bYr¼ aCfafþ bCrar ð6:8Þ
After substituting expressions for the slip angles, Eqs. (6.3) and (6.5), the final equations may be put into the following matrix form:
m 0 0 Iz 2 4 3 5 ¨y ¨ W 2 4 3
5 þ ðCfþ CrÞ=U ðaCf bCrÞ=U ðaCf bCrÞ=U ða2Cfþ b2CrÞ=U 2 4 3 5 ð6:9Þ þ 0 ðCfþ CrÞ 0 ðaCf bCrÞ 2 4 3 5 y W 2 4 3 5 ¼ 0 0 2 4 3 5
Equation (6.9) is in the form of Eq. (3.21), so the characteristic equation is of the form of Eq. (3.23). The same procedure for finding the char- acteristic equation was used previously for the two degree-of-freedom
y W 2 4 3 5
trailer model. See Eqs. (5.31–5.33). In the case at hand, the expression for the characteristic equation is
det
ms2þ ððCfþ CrÞ=UÞs ððaCf bCrÞ=UÞs ðCfþ CrÞ ððaCf bCrÞ=UÞs Izs2þ ðða2Cfþ b2CrÞ=UÞs ðaCf bCrÞ 2 4 3 5 ¼ 0 0 2 4 3 5 ð6:10Þ
When this expression is evaluated, the fourth-order characteristic expres- sion emerges explicitly.
ðmIzÞs4þ mða2Cfb2CrÞ=U þ IzðCfþ CrÞ=U
s3
þ ½ðCfþ CrÞða2Cfþ b2CrÞ=U2 ðaCf bCrÞ2=U2 ð6:11Þ mðaCf bCrÞs2þ 0s1þ 0s0¼ 0
Because the last two coefficients vanish, the equation can be written in the following form:
ðmIzÞs2þ mða2Cfb2CrÞ=U þ IzðCfþ CrÞ=U
s
ð6:12Þ þ½ðCfþ CrÞða2Cfþ b2CrÞ=U2 ðaCf bCrÞ2=U
mðaCf bCrÞgs2¼ 0
This form of the characteristic equation makes it clear that two of the eigenvalues are zero. The stability of the automobile will depend on the eigenvalues associated with the remaining quadratic part of the equation when the factor s2is removed. The two zero eigenvalues are associated with solution components that have the time behavior of e0t=constant. These solutions have no particular significance for the analysis of stability. Obviously, adding a constant distance to the y-coordinate or adding a constant heading angle to W merely means that location and direction of the basic motion path have been changed. The stability of the automobile actually has nothing to do with where in the x–y plane the x-axis is located or with the direction of the x-axis. Thus the use of inertial coordinates has produced a fourth-order system that is more complicated then necessary for stability analysis.
If we now consider the three coefficients of the quadratic part of the characteristic equation when s2in Eq. (6.12) has been eliminated, we see that the coefficients of s2 and s are inherently positive, but that the complicated last term is not necessarily positive.
As we have seen in Chapter 3, for a second-order characteristic equation, the eigenvalues will represent a stable system only if all coef- ficients are positive. Therefore the criterion for stability in this case has to do with the last coefficient in the quadratic part of Eq. (6.12) that must be positive if the car model is to be stable.
ðCfþ CrÞða2Cfþ b2CrÞ=U2 ðaCf bCrÞ2=U2
ð6:13Þ mðaCf bCrÞ > 0
After some algebraic manipulation, this criterion for stability can be simplified to
ða þ bÞ2
CfCr=U2þ mðbCr aCfÞ > 0 ð6:14Þ
If the inequality in Eq. (6.14) is true, the car is stable; if not, it is unstable.
A. Stability, Critical Speed, Understeer, and Oversteer
Several interesting facts can be easily observed from the simplified stability criterion, Eq. (6.14). First, if the speed is sufficiently low, the first term involving 1/U2will be a large enough positive number that the criterion will be satisfied for any parameter set. This confirms the obvious idea that all automobiles are stable at very low speeds.
Second, if the second term in Eq. (6.14) is positive, the car will surely be stable at any speed. The second term is positive when
bCr> aCf ð6:15Þ
This condition is described by the term understeer. Clearly, this term relates to steering properties of the car. For now, all we know is that an understeer vehicle is stable even at high speeds, where the 1/U2term in the stability criterion becomes small, but later we will discuss how understeer affects the steady state cornering behavior.
Finally, if
bCr< aCf ð6:16Þ
we see that the second term in the stability criterion, Eq. (6.14), is negative. This means that if the speed U is gradually increased from zero, the positive first term will decrease to a point at which the first term will just balance the negative second term. This speed is called the critical speed, Ucrit, and above this speed the stability criterion will no longer be satisfied and the car will
be unstable. This speed is determined by equating the positive and negative terms in Eq. (6.14) and solving for the speed.
U2crit¼ ða þ bÞ2CfCr=mðbCr aCfÞ ð6:17Þ
Note that this expression yields a positive number for Ucrit2because we assume that bCr<aCf.
This situation is described by the term oversteer, and it does not only imply that the car is unstable for speeds greater than the critical speed, but also that the speed drastically affects the steady state cornering behavior of the vehicle. The steady state cornering of automobiles with understeer and oversteer characteristics will be discussed in a later section.
B. Body-Fixed Coordinate Formulation
Most modern vehicle dynamic studies use a coordinate system attached to the vehicle as described inChapter 2to describe the vehicle motion. This indicates that the coordinate system rotates with the vehicle, which somewhat complicates the acceleration terms in the basic dynamic equa- tions. On the other hand, the use of body-centered coordinates often results in simpler final equations. In the case at hand, this formulation will directly result in a second-order characteristic equation, and will eliminate the bother of the two zero eigenvalues, which were found in the previous analysis to result from fourth-order equation of motion.
Fig. 6.5 shows the so-called bicycle model of an automobile, in which only a single equivalent front wheel and a single equivalent rear wheel are shown together with the body-centered coordinate system discussed in Chapter 2. A new feature is the steer angle, d, shown at the front. This feature is not needed for stability analysis at present, but is included for later use when steering behavior is discussed. For the stability analysis, the steer angle d will simply be assumed to be zero.
Now U and V are velocity components of the center of mass, but with respect to the body coordinates x and y that are attached to the vehicle body and moving with it. Once again, it will be assumed that the velocity in the forward x-direction, U, is strictly constant for the basic motion and is essentially constant for the perturbed motion, so it is a parameter rather than a variable. The two variables needed to describe the motion are V, the velocity of the center of mass in the y-direction, and r, the angular velocity or the yaw rate about the z-axis.
The lateral tire forces are denoted Yr and Yf, although the front lateral tire force does not point strictly in the y-direction if the front wheel is steered as shown. These definitions are in conformity with the general notation introduced in Chapter 2. The parameters of the vehicle are the same as those used previously in the inertial coordinate system analysis.
The equations of motion are specialized from the general equations of Chapter 2. It is assumed that the steer angle d is a small angle, so that cos di1. The equation
mðVþ rUÞ ¼ Yfþ Yr ð6:18Þ
determines the lateral acceleration of the center of mass and the equation
Izr¼ aYf bYr ð6:19Þ
determines the angular acceleration.
Calculation of the slip angles is accomplished with the help of the sketches shown in Fig. 6.6. Note that b is a sort of slip angle for the center of gravity of the vehicle itself. We will later see that when cornering, the car does not generally go exactly in the direction it is pointed, and thus biV/U is not zero. Again, the velocities at the wheels are found by adding the velocity of the center of mass to components induced by the angular velocity, r. In this coordinate system, the extra components, ar and br are easily observed to add in the fashion shown in Fig. 6.6.
Using the sketches and assuming small angles, the slip angles are readily expressed as ratios of the lateral velocities (with respect to the wheel pointing direction) to the rolling velocities.
ar¼ ðV brÞ=U; af ¼ ðV þ arÞ=U d ð6:20Þ
Note how the steer angle d affects the front slip angle.
As in the previous analysis, it is once again found that positive slip angles as shown result in forces on the tires in the negative Y direction. Because it is preferable to use positive cornering coefficients rather than negative ones, the linearized force laws will be again be written thus:
Yf ¼ Cfaf; Yr¼ Crar ð6:21Þ
Combining Eqs. (6.18–6.21) the equations of motion then become mðVþ rUÞ ¼ ðCfþ CrÞV=U ðaCf bCrÞr=U þ Cfd ð6:22Þ Izr¼ ðaCf bCrÞV=U ða2Cfþ b2CrÞr=U þ aCfd ð6:23Þ For the basic motion, many variables vanish, d=V=r=Yf=Yr=af =ar=0, and U is constant. For the perturbed motion, we assume d=0, Uis still considered to be constant, and that variables V(t) and r(t) take on small enough values so that the slip angles remain small. The dynamic equations for the perturbed motion then assume the following vector- matrix form: m 0 0 Iz V r
þ ðCfþ CrÞ=U ðaCf bCrÞ=U þ mU ðaCf bCrÞ=U ða2Cfþ b2CrÞ=U
ð6:24Þ V r ¼ 0 0
This equation set is essentially in the form of Eq. (3.5), and the character- istic equation is found following the general procedure in Eq. (3.15).
det
msþ ðCfþ CrÞ=U ðaCf bCrÞ=U þ mU ðaCf bCrÞ=U Izsþ ða2Cfþ b2CrÞ=U 2
4
3
5 ¼ 0 ð6:25Þ
When the determinant is written out, the characteristic equation emerges explicitly.
mIzs2þ mða2Cfþ b2CrÞ=U þ IzðCfþ CrÞ=U
s
þ ðCfþ CrÞða2Cfþ b2CrÞ=U2 ðaCf bCrÞ2=U2 mðaCf bCrÞ ¼ 0
If the last term is algebraically simplified, the characteristic equation becomes
mIzs2þ mða2Cfþ b2CrÞ=U þ IzðCfþ CrÞ=U
s
ð6:27Þ þ ða þ bÞ2
CfCr=U2þ mðbCr aCfÞ ¼ 0
This characteristic equation is exactly the same as the quadratic part of the fourth-order characteristic equation that was previously obtained from the model described in inertial coordinates when the zero eigenvalues were eliminated. Here, we see an example of the fact mentioned inChapter 3that all correct dynamic equations describing a system will yield the same eigenvalues, apart from possible zero eigenvalues, which are of no partic- ular significance for stability analysis.
The body-centered coordinate formulation yields a second-order characteristic equation directly and avoids the more complex fourth-order characteristic equation that, in the end, has two zero eigenvalues of no particular interest for stability analysis. It should be clear that the formu- lation using the body-fixed coordinate system is, in some ways, simpler for this problem than the analysis using an inertial coordinate system.
The discussion of understeer, oversteer, and critical speed are, of course, independent of the particular differential equations used to de- scribe the vehicle as long as they are correct, so they need not be repeated here.
The ‘‘bicycle model’’ has a long history of use in automobile steering and stability studies. For those interested in bond graph representations, the Appendix develops a bond graph for the bicycle model of an automo- bile using the rigid body bond graphs in Chapter 2. This bond graph could be used to construct more complex system models. For example, it would be possible to append an actuator bond graph model for active steering studies.
IV. TRANSFER FUNCTIONS FOR FRONT AND REAR
WHEEL STEERING
In the preceding section, the stability of an automobile traveling in the straight path with zero steering angle was studied. We now extend our study to include steering dynamics using the same vehicle model. The body-centered coordinate system approach is particularly useful for this purpose. When linear tire characteristics are assumed, transfer functions relating steering inputs to various response quantities are a convenient way to represent the steering dynamics. Transfer functions relate input and
output variables for linear systems and are described in automatic control texts such as Ogata (1970).
For generality, we will extend our considerations to both front wheel and rear wheel steering. This will create an opportunity to discuss the significant differences in the dynamic behavior between vehicles steered from the front and from the rear.
It may seem odd to consider rear wheel steering systems because only low-speed vehicles such as lawn mowers, forklift trucks, and street sweepers are commonly found to steer the rear wheels. At low enough speeds, stability considerations are not normally considered to be impor- tant for ground vehicles. On the other hand, there have been a number of studies dealing with a combination of front and rear steering (Furukawa, 1989; Sharp and Crolla, 1988; Tran, 1991).
In principle at least, there are several good reasons for considering the steering the wheels both at the front and at the back of a vehicle. In a conventional front-steered automobile, there is a coupling of angular motion in yaw and lateral motion, and only a single control element— the steering wheel—simultaneously influences both aspects of motion.
For example, a driver wishing to change lanes on a straight road needs to generate lateral forces at the front and rear axles. With front wheel