Stability of Trailers
VIII. DYNAMIC STABILITY IN A STEADY TURN
The considerations of the stability of an automobile as well as the study of the steady turn behavior have so far considered only linearized relations between lateral tire forces and slip angles. This is logical for the consider- ation of the stability of straight-line motion because the basic motion required no lateral forces and the perturbed motion required only small forces and slip angles. This assumption is more restrictive for steady turns, because it means that the steady lateral forces must remain fairly small if the linear assumption is to remain valid. This restriction implies that the lateral acceleration must also be limited if the conclusions are to be valid. In normal driving, most maneuvers are actually accomplished at rather low values of lateral acceleration. Most turns are taken at speeds resulting in accelerations of no more than 10% or 20% of the acceleration of gravity. Under these conditions, for most cars and tires, the linear tire force assumption is a fairly good approximation.
On the other hand, if we consider emergency maneuvers or limit speeds in turns, the tire force-slip angle becomes significantly nonlinear. As we have seen in Chapter 4, at high slip angles, the tire force reaches a maximum value and further increases in slip angle typically cause to tire force to decline as the tire begins to slide along the roadway. Under these conditions, the understeer and oversteer properties of the automobile model discussed in the previous sections are no longer constant properties of the car itself but rather change with the severity of the maneuver.
The characterization of the dynamics of automobiles in the nonlinear range can be a complex task. To gain some insight into this area and yet to keep the discussion reasonably simple, we will restrict the discussion to the stability analysis of cars while they are in a steady turn. The new feature is that we will no longer assume that the lateral acceleration is sufficiently small such that a linear tire force model is adequate. The bicycle model will be retained, which means that each axle will be represented by a single force-slip angle law including the effects of both tires.
Fig. 6.8,which was used previously to define terms for a negotiating a steady turn, can still be used, but now the slip angles will not be assumed to
be small enough that a nearly constant cornering coefficient can be used. For typical tires on dry pavement, the linear range extends to slip angles of about 5j or so, and the lateral forces at the top end of this range correspond to accelerations of more than one-half the acceleration of gravity. This means that for normal driving, the linear approximation has validity. We now consider larger slip angles and steer angles, but even as the tires begin to behave in a nonlinear way, the angles are assumed to be small enough that the approximations used for the trigonometric functions in the analysis remain reasonable.
A. Analysis of the Basic Motion
The basic motion for a stability analysis is now a steady turn and the variable values necessary to accomplish the turn will be denoted with the subscript ‘‘s.’’ We still assume that the forward speed U=constant. The remaining variables for the basic motion are defined as follows:
V¼ Vs; r ¼ rs¼ U=R;
Vs¼ rs¼ 0; Yf ¼ Yfs Yr¼ Yrs; af ¼ afs; ar¼ ars; d ¼ ds
The steady steer angle is determined by a version of Eq. (6.52), because we still assume that the angles are small enough to use small angle approx- imations for trigonometric functions. Note that in the tire force plots in
Fig. 4.1,significant nonlinear behavior is exhibited for tire slip angles of only 10j or 15j. From Eq. (6.52), the steer angle for the basic motion is approximately
ds¼ afsþ arsþ ða þ bÞ=R ð6:81Þ
The dynamic equations remain as before [see Eqs. (6.18) and (6.19)] Now, however, there are the so-called trim conditions, which con- strain the steady values of variables of the basic motion so that the car can execute a steady turn.
mrsU¼ Yfsþ Yrs
0¼ aYfs bYrs rs¼ U=R
Fig. 6.12shows generally how the lateral tire forces are related to slip angles when the slip angles are no longer assumed to be very small. As discussed previously, positive slip angles yield negative forces when the notation ofChapter 2is used, so it is convenient to plot negative forces vs. slip angles.
ð6:80Þ
The previous stability analysis treated the case of straight-line motion. Then the basic motion corresponded to zero values for both lateral force and slip angle. The cornering coefficients used, Cfand Cr, are the slopes of the force curves at the origin. In the present case, there are steady values for the forces and slip angles at the operating points shown in Fig. 6.12. These steady values are determined by the trim conditions, Eq. (6.82). The slopes of the force curves at these operating points will be designated CfV and CrV. As Fig. 6.12 shows, these slopes can be significantly different form the cornering coefficients when the steady forces in the turn become large.
B. Analysis of the Perturbed Motion
The perturbed motion is described by variables representing small devia- tions from the constant steady values associated with the basic motion. These time-varying variables will be denoted using the symbolD. In every case, the actual variables will be written as the steady variables plus the deviation variables. For example, the yaw rate, r(t), will be expressed as r=rs+Dr, and the front lateral force, Yf(t), is written as Yf=Yfs+DYf.
Fig. 6.13shows how the change of the front lateral force is related to the change of the slip angle away from the steady value by the local slope of the force law CfV.
The force-slip angle is now linearized about the operating point determined by the basic motion.
DðYfÞi d daf
YfðafÞ
½ Daf¼ CfVDaf ð6:83Þ
A similar expression is used at the rear.
The dynamic equations for the perturbed motion can be now written by substitution into the general dynamic equations, Eqs. (6.18) and (6.19). Because the steady, basic motion variables are constant in time,
V ¼ DV; r ¼ Dr ð6:84Þ
The dynamic equations are then
mDV þ mðrsUþ DrUÞ ¼ Yfsþ DYfþ Yrsþ DYr ð6:85Þ
IzDr ¼ aðYfsþ DYfÞ bðYrsþ DYrÞ ð6:86Þ
Returning to the trim conditions, Eq. (6.82), we find that all the steady variables cancel out of Eqs. (6.85) and (6.86), leaving dynamic equations that resemble the equations which apply to perturbations from straight-line motion [Eqs. (6.18) and (6.19)]. Now, however, the dynamic equations apply to perturbations from a steady cornering situation.
mDV þ mDrU ¼ DYfþ DYr ð6:87Þ
IzDr ¼ aDYf bDYr ð6:88Þ
The slip angles for the basic motion are related to the motion variables as derived previously, Eq. (6.20).
afs¼ ðVsþ arsÞ=U ds; ars¼ ðVs brsÞ=U ð6:89Þ The perturbed variables obey similar equations.
Daf ¼ ðDV þ aDrÞ=U Dd; Dar¼ ðDV bDrÞ=U ð6:90Þ
Now, using linearized lateral force-slip angle laws such as Eq. (6.83), the final equations for the perturbed variables may be written.
mðDVþ DrUÞ ¼ ðCfV þ CrVÞDV=U ðaCf bCrÞ
ð6:91Þ Dr=U þ CfVDd
IzDr ¼ ðaCfV bCrVÞDV=U ða2CfV þ b2CrVÞDr=U þ aCfVDd ð6:92Þ These equations closely resemble the equations previously derived using body-centered coordinates for the stability analysis of straight-ahead running [Eqs. (6.22) and (6.23)]. The lateral velocity and the raw rate are now replaced by the corresponding perturbation variables and local slopes of the force-slip angle curves replace the cornering coefficients.
Because of the similarity between the dynamic perturbation equa- tions for straight line running and for steady turns, the previous stability criterion, Eq. (6.14), can be used with the new variables. The car in a steady turn will be stable if the expression
mðbCrV aCfVÞ þ CfVCrVða þ bÞ2=U2> 0 ð6:93Þ is satisfied.
For straight-ahead running, it was possible to classify cars as understeer, oversteer, or neutral steer depending on the sign of bCraCf independent of speed. It was also clear that the criterion [Eq. (6.14)] would always be satisfied for small enough U, and therefore all cars are stable at low speeds. The situation is less clear in the case of a steady turn because CfV and CrV are the local slopes of the force-slip angle curves that change if either the turn radius or the speed is changed.
It is clear that the cornering coefficients Cfand Cr are positive as defined for straight line running, but CfV and CrV can approach zero or even become negative at very large slip angles. Thus, although the criterion for stability [Eq. (6.93)] will always be satisfied for low speeds, because then CfV and CrV approach Cfand Cr, the situation in a turn at high speeds is not clear. The term involving 1/U2diminishes with increasing speed, but the quantity bCrVaCfV may change sign as the parameters U and R of the basic motion change. The car can therefore behave quite differently in curves than it does in a straight line, and it is not possible to define critical speeds or characteristic speeds for cars in steady turns unless the tire forces remain in the linear region.
C. Relating Stability to a Change in Curvature
If the tire characteristics of an automobile are assumed to be strictly linear, then there is no difference in the stability analysis between a basic motion in a straight line and in a steady curve. In the discussion of steady turns for a linear car model, it was noted that instability occurred at a steady lateral acceleration level for an oversteer vehicle at which the steer angle in the turn became zero. When nonlinear tire characteristics are assumed, the situation is more complex, but it turns out to be possible to relate dynamic stability to the change in steady steer angle with a change in turn radius at constant speed. This allows one to extend the definition of understeer and oversteer to the nonlinear case. The terms do not apply just to the car but to the car and the particular turn it is negotiating.
Using the trim conditions, Eq. (6.82), the steady forces are Yfs¼ ðmU2=RÞb=ða þ bÞ ¼ ðmU2bÞ ða þ bÞ2 aþ b R ð6:94Þ Yrs¼ ðmU2=RÞa=ða þ bÞ ¼ ðmU2aÞ ða þ bÞ2 aþ b R ð6:95Þ and the local slopes of the force laws are
CfV ¼ dYfs
dafs; CrV ¼ dYrs
dars ð6:96Þ
The steady steer angle obeys Eq. (6.81). Now, using a simplified notation for the wheelbase,
luða þ bÞ ð6:97Þ
we can calculate the change in steer angle when the turn radius is changed or, more precisely, when l/R is changed by differentiating Eq. (6.81).
Bds Bðl=RÞ¼ B afs Bðl=RÞþ B ars Bðl=RÞþ 1 ¼ dafs dYfs BYfs Bðl=RÞþ dars dYrs BYrs Bðl=RÞþ 1 ¼ 1 CfV mU2b l2 1 CrV mU2a l2 þ 1 If Eq. (6.98) is rearranged into Eq. (6.99)
l2CfVCrV U2
Bds
Bðl=RÞ¼ mðbCrV aCfVÞ þ l2CfVCrV=U2 ð6:99Þ ð6:98Þ
one sees that the stability expression, Eq. (6.93), appears on the right-hand side of the equation. If the expressions on the two sides of Eq. (6.99) are positive, the car is stable. If they are negative, the car is unstable.
Let us suppose that the two slopes CfV and CrV are positive. (This is generally the case, except for very large slip angles.) Then the stability of the vehicle depends on the sign ofBds/B(l/R), because this determines the sign of the stability expression on the left-hand side of Eq. (6.99). IfBds/B(l/ R) is positive, the car is stable; ifBds/B(l/R) is negative, the car is unstable for the particular speed and curve radius.
To interpret this fact, it is worthwhile to return for a moment to the strictly linear case in which a constant understeer coefficient, K, can be defined. In the section on steady turns, an expression was derived for the steer angle required given a turn radius, R, and a constant speed, U [Eq. (6.65)]. Using the notation of this section, it is
ds¼ KU2=R þ l=R ¼ ðKU2=lÞðl=RÞ þ l=R
¼ ðKU 2=lÞ þ 1ðl=RÞ ð6:100Þ
In the linear case then, Bds
Bðl=RÞ¼ KU2
l þ 1 ð6:101Þ
This result confirms what we already know. All cars are stable when U is sufficiently small. No matter what the sign of K happens to be, both sides of Eq. (6.101) will be positive if U is small enough. Understeer cars, for which K is positive, or neutral steer cars with K=0, are stable at all speeds. Oversteer cars for which K is negative become unstable only when U exceeds the critical speed and the terms on both sides of Eq. (6.101) become negative.
In the section on steady turns, the steer angle was plotted vs. speed for a constant radius turn. It is useful now to plot steer angle vs. l/R for constant speed. The plot is shown inFig. 6.14.
For the linear tire force assumption, the slopes of the plot of dsas a function of (l/R) are constant because K and U are both constant. Note that an oversteer situation means that the slope of d vs. l/R is less than unity, but only if the slope is negative is the car unstable. This only happens for the oversteer case and at a speed higher than the critical speed.
For nonlinear tire characteristics, the slopes are not constant and vary with the turn radius when the speed is constant. The reason, of course, is that sharper turns require greater lateral acceleration and thus
the tires operate at different points on their tire characteristics when the radius changes. In the nonlinear case, the terms understeer and oversteer can only be defined for particular turn radii and speeds, but one can show regions of oversteer, understeer, stability and instability on a plot of dsas a function of (l/R), as shown inFig. 6.15.
As Fig. 6.15 shows, understeer and oversteer relate to the local slopes of the steer angle curve. Instability sets in only when the slope becomes negative. Under the assumption that CfV and CrV remain positive, the vehicle becomes unstable only from an oversteer condition.
Fig. 6.15 is somewhat difficult to understand, and one should remember that it is just one possible example of how the steer angle could vary with turn radius at a constant speed. However, it does illustrate the fact that the nonlinearity of tire (or, more correctly, axle) characteristics mean that cars cannot be classified as understeer, neutral, or oversteer except for maneuvers during which the axle forces and slip angles remain essentially in their linear range.