The purpose of the multi-body modelling
The aim of the simulation is to analyse how the limited slip differential works as well as to compare the behaviour of a vehicle provided with this kind of differential to an automobile furnished with a normal differential and another one with a rigid axle.
The results obtained match the expectations: the car with the limited slip differential shows a behaviour similar to the one with an open differential when its engine torque is low (e.g. curve entrance, low velocity). At high torque (e.g. curve exit) the behaviour is similar to a vehicle with rigid axle to maximize the traction.
When a car turns, the radiuses of curvature of internal and external wheel are different therefore, each tyre covers a different distance rotating at a different speed. If between the two driving wheels there were a rigid constraint one of the two wheels should skid causing not only an under steering effect but also the wearing of the tyres.
The differential is used to avoid these problems and to let the wheels turn at different speeds as well as to distribute the engine torque equally to the left and right wheels.
As figure 1 shows, the spider gear acts as a lever dividing the force coming from the differential box between the two side gears.
The use of the normal differential presents some disadvantages: when one of the two wheels loses traction (e.g. slippery street or during a curve) also the torque transmitted to the other wheel goes down. This can cause considerable difficulties for the car to proceed or even its standstill. Therefore, in some vehicles some mechanisms are employed to block the differential temporarily. Two examples of self-locking differential are the limited slip differential and the Torsen one.
How the limited slip differential works
The limited slip differential exploits two clutches to lock the differential. Compared to a normal differential, the limited slip differential has some more components such as two pressure rings (or sun pinion rings, fig.2, n.7) and a series of friction disks (half joined to the differential box and half joined to the side gear, fig.2 n.3,8). The two pressure rings present two ramps in which the cross pin is inserted.
When the engine transmits torque to the two pressure rings, they are pushed away and press the two clutches, the more the torque the more the clutch pressure is. According to the pressure level, the clutches block the movement between the shaft pinion and the differential box. This permits to limit the speed difference between the two wheels.
One of the advantages of this solution is its flexibility; once we vary the preload of the clutches we also vary the level of intervention of the differential. Another way to set the behaviour of the differential is to change the outline and the angle of the ramps.
On the other hand, this mechanism has also the disadvantage of being subjected to the disk wear that have to be often substituted.
The differential modelling
If compared to the real mechanism we have made some simplifications to model the differential; anyway, such modifications do not prejudice the quality of the simulation.
The teeth of the gear have not been modelled and the relative motion between the wheel gears has been obtained through the feature “three-body relative constraint”.
Another fundamental aspect concerns the contact forces between the pressure ring and the pinion shaft; this has been made through the feature “sphere-to-extruded-surface contact force”. The sphere has been positioned on the end of the shaft, the extruded surface has been created thanks to a pocket feature. In this case, a linear ramp has been used so the axle force is directly proportional to the applied torque.
The clutches are represented by a unique disk that interacts with the pressure ring; in this case, we used the feature “sphere-to-revolved-surface contact” to simulate the contact between the two bodies. The feature permits to set the parameter of friction coefficient and so, simulate the friction between the disks. We have located the sphere on the pressure ring and we have used the disk surface as revolved surface. Varying the friction coefficient we also vary the locking level of the differential.
The other parts of the differential have been assembled through revolute-joint and cylindrical-joint.
The dragging of the two pressure ring by the box takes place thanks to translational-joint, that blocks the rotation but at the same time allows the translation along the rotation axle. The disks and the side gears have been bound thanks to the same procedure; a linear spring presses the disks and the pressure rings together.
The picture shows the model:
The car modelling
We used a very simple model of a car; the chassis has no suspensions and we inserted the inertial parameter manually using data of commercial vehicles.
We used the model “simple tyres” to create the tyres. We used softer tyres on the front axle and more rigid ones on the rear axle to amplify the load transfer between the right and the left wheel during the curve. We added also a steering system which allows the car to follow the road path. The differential previously created has been installed on the car.
In the following pictures the car is displayed:
We created a path road made up with a straight stretch and a long curve with a radius of curvature of 20 meters to study the car behaviour on a bend.
An initial velocity was assigned to the car whereas the torque follows a pre-set spline.
In particular, we examined two different cases: one with a high torque and one with a low one. In the first case, the torque started from 100 Nm and went up till 700 Nm after the entrance on the band; in the second instance, the torque remained constant and equal to 100Nm during the entire path.
The graph shows the torque as a function of the time:
For each case, we analysed the behaviour of the three different kinds of differential.
Analysis of the results
The normal load on the rear tyres is similar in the three configurations studied therefore, we may compare the graphs of the longitudinal force transferred to the ground by the rear gear.
With the normal differential the internal tyre (left) is not able to transfer great force to the ground because of the reduction of normal load; the wheel starts to slip and, consequently, also the external wheel (right) transfers a limited force.
Using the limited slip differential, the force on the internal wheel is approximately the same as described in the previous case; however, the external wheel is able to apply a relevant force. This occurs thanks to the effect of the clutches that prevent an excessive sliding between the two wheels.
The graphs explain the behaviour of a vehicle with rigid axle; the forces applied are similar to the case with the limited slip differential except for the first part of the diagram where the torque is still low. This case will be considered later.
In the following graphs the slip speeds of the tyres are represented. With a normal differential the internal wheel slips very fast causing an anomalous wear and the deterioration of the tyre. Using the limited slip differential the slip speed is about six times lower than with the other type of differential ; this preserves the tyre from an excessive wear.
The efficiency of the limited slip differential is pointed out by the vehicle speed; as shown in the diagram, the car equipped with this differential reaches a higher speed even though in the last part the speed is lower than with the normal differential because the car starts to drift. This effect can be avoided using a more accurate traction control.
Applying a low torque, the behaviour of the limited slip differential is similar to a normal differential although there is a difference between the forces transmitted by the two wheels. Anyway, the difference is negligible if compared to the car with the rigid axle.
Comparison between high and low torque
As we expected, the behaviour of the differential is different in the two cases analysed.
We may understand the feedback on the driver observing the graphs of the steering angle and yaw torque generated by the rear wheels. At low engine torque there is a little under steering effect unimportant if compared with the rigid axle case. At high engine torque the yaw torque is higher than with a normal differential and it helps the vehicle to follow the curve easily.
We can also observe the benefits of the differential in the curve entry; while the torque is low the car can easily enter the curve, because the yaw torque, even if negative, has a low rate. On the contrary, if the car were equipped with a rigid axle the yaw torque effect would backfire, in fact its yaw torque level is negative and in this case very high as well.
To understand better the effect of the yaw torque we can calculate the position of the point of thrust. Such value can be estimated dividing the yaw torque for the whole longitudinal thrust exerted by the rear wheels; this represents a virtual point on which all the forces are centred, and it is located on the transversal axle of the vehicle.
From the picture we notice that such position can also reach values higher than 3 metres; this means that the centre of thrust is ipothetically located outside the car frame.
The limited slip differential works as we expected; its positive effect on the traction is considerable at high torque, so this mechanism is suitable for sportive and race cars.
Moreover, in the future the system may be improved using different models of clutch friction implementing for instance a viscous type. A further analysis can concern the preload of the clutch springs that in our study was considered null. Finally, the differential may be installed on a more developed and realistic car model.