Perfect 4WD

In this work we address the problem of designing a perfect 4WD (four wheel drive) car. Since the main problem is caused by incorrect distribution of torque to the wheels, proposed method suggests to use computational devices to keep the certain computed ratios of angular velocities of the wheels. This concept is suitable for the modern electric cars and allows to bypass the usual mechanical differential. The main result is more stability and efficiency on the road.

Why do we need this?

To understand why we need a better 4WD, let us consider several situations. All these situations happen to us every day: when we drive a car in the rain or in the sunny weather, when we drive around a corner or drive on a strait road. We experience the problem so often that we think that it is the way it should be.

The first situation is when a car is on a solid ground. It is well known that the angular velocity of each wheel of the car may be different: when a car makes a left turn, right wheels spin faster than left wheels. To provide the ability for the wheels to have different velocities, a differential is installed in the car. And it seems that it solves the problem, but...

Here is the second situation. Suppose that we are on some slippery surface now (it may be just ice or mud, or may be one wheel is hanging in the air). Now, the differential works against us: it transmits all torque to the loose wheels. The loose wheels spin, but the car does not move. (For example, see here.)

Probably, the problem is now clear: we need a better way to distribute torque to the wheels. The usual mechanical differential cannot do this properly in all situations. Although the differentials in our days are much "smarter", than they were before, they are not able to eliminate the problem completely.

The Concept

Proposed is a simple concept of an electric car with 4WD without a differential. This car has 4 independent electric motors (one motor per each wheel). A computational device distributes torque for each wheel by adjusting the current in the circuit of the corresponding electric motor. This concept gives the following:

stability on the slippery roads;
efficiency on the turns;
efficiency on the slippery surfaces;
reliability of the four electric motors.

Mathematical Part of the Problem

In this section we address the problem of the correct relative angular velocities of the wheels of a car. By the end of the section the explicit formulas for them will be obtained.

Figure 1: positions of the wheels

Suppose that we have a car with four wheels numbered counter-clockwise starting with the left rear wheel. We want the velocity vectors A_i to be strictly parallel to the orientation of the corresponding wheels. (This condition simply requires that the wheels roll, but not slide.)

Now, let us denote the ratio of the length and width of the car by h: h=length/width. Also, suppose that the orientation of the wheels is the following: wheels 1 and 2 are directed strictly forward, wheels 3 and 4 have the angles α+β and α-β, respectively (see figure 1).

It turns out that the value of β is a function of α and h:

and for any other value of β the velocity of the wheels is not strictly parallel to the orientation of the wheels. The graph of tan β versus α is shown below:

One can see that the values of tan β (and, therefore, the values of β) are close to 0, but, nevertheless, β is important for the car to move properly.

Next, we can write out the relative velocities s_i of each wheel. Since (usually) all wheels have the same radius, s_i can be considered also as relative angular velocities of the corresponding wheels. Taking the velocity of the first wheel as the basis (that is the relative velocity of the first wheel is s₁=1) we get the following values for the remaining three wheels:

Conclusion

Now, since we know the proper relative velocities of the wheels, we just need a computer that will increase or decrease the current in the circuits of the motors to maintain the right ratio. Looking at the achievements of the modern electronics, it seems that this part will be relatively simple. And, hopefully, soon we will see real 4WD cars on the road.