In the last section, only the effects of steady roll angle on the vehicle dynamics were considered for simple understanding. Therefore, the vehicle equations of motion still remain in a form of two-degree-of-freedom of side slip and yaw, even when a roll is considered, and the analytical descriptions of such parameters as stability factor, natural frequency, damping ratio, etc., are available.
Next, the transient effects of body roll on vehicle dynamics, especially on response stability of the vehicle to steering input, will be examined. For a basic understanding, the characteristic equation of vehicle motion with roll is regarded to be given again byEqn (6.35)under the same premise for simplification taken inSection 6.4.4. As it is possible to understand fromEqns (6.32) and (6.33)that the effects of roll steer and camber change are identical, here only roll steer is considered with no camber change.
Generally, the roll steer is used for the adjustment of the vehicle steer characteristics.
It is obvious from Eqn (6.46)that the roll steer has no effects on the steer characteristics if both front and rear axles have the same roll steer. Therefore, the same amount of roll steer at the front and rear axles with counter direction of each other are taken into consideration, as shown byEqn (6.52):
vaf
vf ẳ var
vf ẳva
vf (6.52)
PuttingEqn (6.52)intoEqns (6.32) and (6.33), the following are obtained as the camber change is zero:
Yfẳ0 (6.53)
Nfẳva
vflfKfþva
vflrKrẳva
vflK (6.54)
As is discussed in Section 6.5, ifva/vf is positive, the vehicle becomes OS, and then negative, and then US. In addition to the previous, using the assumptionCfẳ0, the characteristic equation becomes the following:
A4s4ỵA3s3ỵA2s2ỵA1sỵA0ẳ0 where
A0ẳ16K2Kf
mV 32mShsK2 ml
va vfV A1ẳ8KKf
A2ẳmKfVỵ16K2If mV A3ẳ4K 2Ifm2Sh2s
m
!
A4ẳm Ifm2Sh2s m
! V
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;
(6.35)0
AsA1,A2,A3, andA4are always positive, this is one of the stability conditions:
A0ẳ16K2Kf mV
12mShs
lKf va vfV2
>0 (6.55)
In addition, the following condition must be satisfied for the vehicle stability:
A1A2A3A0A23A21A4>0 UsingEqn (6.35)0, the previous condition is rewritten as follows:
2 66 66 41þ
16K2
2Ifmm2Sh2s 2 mmShslKf2
va vf 3 77 77 5V2þ
8K2
2Ifmm2Sh2s
m2Kf >0 (6.56)
Now at first, from the stability condition,Eqn (6.55), ifva/vfis positive, which means the vehicle is OS, then the vehicle has a critical speed,VC1, described as follows:
VC1ẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lKf 2mShsvavf s
(6.57)
The previous critical speed has the same implication as is discussed in Section 3.3.2 that if the vehicle is oversteer, there is the critical speed for stability limit, which has nothing to do with whether the vehicle motion has a roll mode or not. The vehicle is non-oscillatory unstable at the vehicle speed greater than that described byEqn (6.57).
The next stability condition isEqn (6.56). Because the independent term to the vehicle speed inEqn (6.56)is always positive, the following condition must be satisfied for the vehicle stability at any speed:
1þ 16K2
2Ifm2Smh2s 2 mmShslKf2
va
vf0 (6.58)
This is rewritten as follows:
va vf
mmShslKf2 16K2
2Ifmm2Sh2s
2 (6.59)
and, the critical value of the roll steer is defined as follows:
va vf
C
ẳ mmShslKf2 16K2
2Ifmm2Sh2s
2 (6.60)
As (va/vf)Cis negative, the stability condition described byEqn (6.56)is rewritten as the following:
1 va=vf ðva=vfịC
V2þ
8K2
2Ifm2Smh2s
m2Kf >0 (6.56)0
This is always satisfied at any vehicle speed whenva/vfis positive, which means the roll steer makes the vehicle OS. On the other hand, ifva/vfis negative, which means the roll steer makes the vehicle US andva/vf<(va/vf)C, then there is a critical vehicle speed due toEqns (6.56) or (6.56)0, described as follows:
VC2ẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m8K2K2f
2Ifm2Smh2s 1 ðva=vfịva=vf
C
vu uu
ut (6.61)
The vehicle is oscillatory unstable at the vehicle speed greater than that described by Eqn (6.61).
It has been shown in Chapter 3 that the understeer vehicle shows the oscillatory response to steering input at high vehicle speed; however, there is no condition to make the vehicle unstable at any speed so far as the vehicle motion is considered by the two-degree-of-freedom
equation of motion, side slip, and yaw. Things are different when the roll motion is taken into consideration. There is the critical vehicle speed that makes the vehicle oscillatory unstable when the vehicle is adjusted to become understeer by some roll steer.
The effects of the body roll, especially roll steer, on vehicle stability obtained in the previous are summarized as follows:
Whenva/vf>0 (OS), non-oscillatory unstable atVVC1.
When (va/vf)Cva/vf0 (NS, US), always stable at any vehicle speed.
Whenva/vf<(va/vf)C(US), oscillatory unstable atVVC2.
A schematic image of the stable/unstable region on the Vva/vf plane is shown in Figure 6.25.
It is important to note that the previous analysis is based on the premise of the suspension damper equal to zero, however, there is a possibility of the existence of a practical vehicle speed for oscillatory unstable under some practically reasonable values of suspension damper and roll steer[4].
PROBLEMS
6.1 Sometimes, the roll angle of the vehicle subjected to the 0.5 g steady lateral acceleration is defined as the roll rate. Calculate the roll rate of the vehicle with the mass of the vehicle bodymSẳ1400 kg, the body C.G. height from the roll axishsẳ0.52 m, the front roll stiffnessKffẳ65.0 kNm/rad, and the rear roll stiffnessKfrẳ35.0 kNm/rad.
6.2 Calculate the lateral load transfer at the front and rear suspensions, respectively, for the vehicle in Problem 6.1. Use the following vehicle parameters in addition to those in 6.1: the position of the front wheels from the C.G.lfẳ1.1 m, the position of the rear wheelslrẳ1.6 m, the front treaddfẳ1.5 m, the rear treaddrẳ1.5 m, the front roll center height from the groundhfẳ0.05 m, and the rear roll center height hrẳ0.2 m.
6.3 Calculate what percent of the cornering stiffness is equivalently reduced by the suspension compliance steer if the cornering stiffness of the original tire is 60 kN/rad and the compliance steer to a unit lateral force is 0.00185 rad/kN.
0 )C
/ (∂α ∂φ
Roll steer
Under-steer Over-steer
Oscillatory unstable
Non-oscillatory unstable Stable
Vehicle speed
FIGURE 6.25
Image of vehicle stability limit due to roll steer.
6.4 Investigate the relative value of the camber change rate and the roll steer rate that give us almost the same effects on the steer characteristics of the vehicle.
6.5 Find the roll steer rate at the rear suspension that is needed to make the OS vehicle inFigure 6.24NS. Use the following vehicle parameters,mẳ1500 kg,
mSẳ1400 kg,lfẳ1.5 m,lrẳ1.1 m,Kfẳ55 kN/rad,Krẳ62 kN/rad,hsẳ0.52 m, andKfẳ100 kNm/rad. Neglect all the camber changes and the roll steer in the front suspension.
REFERENCES
[1] Eberan R. ROLL ANGLESdthe calculation of wheel loads and angular movement on curves. Automob Eng October, 1951; 379–84.
[2] Segel L. Theoretical prediction and experimental substantiation of the response of the automobile to steering control. Proc IMechE (AD) 1956–1957; 310–31.
[3] Ellis JR. Vehicle dynamics. London: London Business Book Ltd; 1969.
[4] Ishio J, Abe M. Effect of body roll on vehicle dynamics. Trans Jpn Soc Automot Eng September 2013;44(5).