3.11 Vehicle Model for Handling and Performance
3.11.4 Principles of Any Differential Mechanism
The non-driven wheels of a car can spin independently since there is no connec- tion between them. But the driven wheels must be linked together so that a single engine and transmission can turn both wheels. The mechanism that links the two driven wheels of the same axle is called the differential. Basically, it is a device that splits the engine power two ways, allowing each output wheel to spin at a dif- ferent speed [2]. But this is too loose an explanation. In this section the equations governing any differential are discussed in detail.
Regardless of the specific mechanical design, a differential is essentially a hous- ing with two aligned shafts that must fulfill one very specific requirement: the two shafts must have opposite angular speeds with respect to the housing, as shown in Fig.3.21, bottom.
Letωl,ωhandωr be the absolute angular speeds of the left shaft, of the housing and of the right shaft, respectively (see Fig.3.21although the subscripts are dif- ferent. Their meaning will be given shortly). Moreover, letMl,MhandMr be the corresponding moments (torques) applied to them. All angular speeds are always positive, while the moments must be such that
MlMr≥0 and (MlMr)Mh≤0 (3.126) as will be shown in a while.
Every differential is governed by the following set of three equations:
ωl−ωh
ωr−ωh = ωˆl
ˆ ωr = −1 Mh+Ml+Mr =0
Mhωh+Mlωl+Mrωr =Wd
(3.127)
whereWd>0 is the power lost inside the housing and
Fig. 3.21 Angular speeds and moments in any differential mechanism. Top:
absolute speeds, bottom:
relative speeds
ˆ
ωl=ωl−ωh and ωˆr =ωr−ωh (3.128) are the relative angular speeds of the shafts with respect to the housing, usually very small. As already stated, under any working conditions we always haveωˆl= − ˆωr. The first equation in (3.127) is the Willis formula, the second equation is the torque balance and the third equation is the power balance.
The first equation can be better rewritten as
ωl+ωr=2ωh (3.129)
which shows a very important kinematic feature: if one wheel rotates faster than the housing, the other wheel must rotate slower. Let us callωf andωs these angular speeds, respectively, andMf andMs the corresponding torques (Fig.3.21). Since the differential has two degrees of freedom, the Willis formula alone cannot say how big is the difference between the two rotation speedsωf andωs.
Combining the last two equations in (3.127), we get the internal power balance of the housing
Ml(ωl−ωh)+Mr(ωr−ωh)=Mlωˆl+Mrωˆr
=Mf(ωf −ωh)+Ms(ωs−ωh)=Mfωˆf +Msωˆs
=Wi−Wo=Wd
(3.130) whereWi is the input power andWois the output power, both assumed as positive.
Obviously, by definition
(ωˆf =ωf−ωh) >0 and (ωˆs=ωs−ωh) <0 (3.131) withωˆf = − ˆωs.
92 3 Vehicle Model for Handling and Performance Fig. 3.22 Longitudinal
forces during power-on in a vehicle equipped with a limited slip differential:
(a) low lateral acceleration, (b) high lateral acceleration
The knowledge of the internal efficiencyηhof the housing is very helpful, since ηh=Wo
Wi =Wi−Wd
Wi ≤1 (3.132)
Instead ofηh, it is common practice to use the Torque Bias Ratio (TBR) which is exactly equal to 1/ηh
TBR= 1 ηh
(3.133) There are two possible working conditions:
(1) positive torqueMhfrom the engine (power-on), which means that bothMf and Ms are negative (for the differential, but positive for the wheels). Therefore,
−Mfωˆf =WoandMsωˆs=Wi (Fig.3.21);
(2) negative torqueMh from the engine (power-off), which means that both Mf andMsare positive (for the differential, but negative for the wheel). Therefore,
−Msωˆs=WoandMfωˆf =Wi. Inserting these results in (3.132) we get
power-on ηh=−Mfωˆf Msωˆs =Mf
Ms ≤1 power-off ηh=−Msωˆs
Mfωˆf = Ms
Mf ≤1
(3.134)
As shown in Fig.3.22, the outcome of a power-on condition strongly depends on the vertical loads acting on the two wheels. As a rule of thumb, the slower wheel has always the higher torque.7The power-off condition, instead, is more predictable, as shown in Fig.3.23.
7Just consider that, sinceωh> ωs, bothMsandωˆsare negative and hence their product is positive, meaning input power for the differential mechanism inside the housing. Consistently,Mfωˆf<0, which is an output power for the differential mechanism.
Fig. 3.23 Longitudinal forces during power-off (coasting mode) in a vehicle equipped with a limited slip differential (not locked)
For the purpose of practical implementation, it is useful to rewrite (3.134) in a more compact way. Let ς =sign(Mh); therefore ς =1 during power-on and ς= −1 during power-off. We have that
Mf =ηςhMs (3.135)
Moreover, letϕ=sign(ωr−ωl). We obtain Mr =
ηςhϕ
Ml (3.136)
which covers all cases.
Most road cars are equipped with a so-called open differential, which hasηh1 and henceMl=Mr under all working conditions.
On the other hand, off-road vehicles, race cars and other sports cars have a limited slip differential (also called self-locking). A low efficiencyηh (and hence a high TBR) can be achieved in several ways, but all rely on friction inside the housing.
There are differentials with constant ηh (typically ηh 1/4), others which have some sort of clutches and haveηhsensitive to torque.
Many limited slip differentials which employ clutches behave like a totally locked differential whenever the differenceΔM= |Ml−Mr|is below a threshold value.
In that case, there is no differential effect and both wheels rotate at the same angular speedωl=ωr=ωh, the moments being any, provided they fulfill the equationMh+ Ml+Mr=0. For instance, it is even possible to have the two longitudinal forces pointing in opposite directions, as shown in Fig.3.24.
Summing up, the type of differential mechanism affects the handling behavior through the following equations:
open differential: Mi1=Mi2, and henceΔXi=0;
locked differential: ωi1=ωi2, and hence, in general,ΔXi =0;
limited slip differential: Mi1=Mi2 andωi1=ωi2(more precisely, during power- on|Mf| =ηh|Ms|, withηh<1, that is the slower wheel receives by the engine the higher torque).
94 3 Vehicle Model for Handling and Performance Fig. 3.24 Possible
longitudinal forces in a vehicle equipped with a locked differential