Now that we have a basic understanding of how a very basic car model with specific features (that we summarized in our previous entry as “neutral steering”) handles, we can go on with our analysis and focus on more generic cases, where the CG is not at the same distance from the front and rear wheel center.
Using again our simplified (bicycle) vehicle model, with the same tyres at the front and at the rear, we can imagine it having its CG seating closer to the front wheel than to the rear.
We will still suppose that our vehicle moves on a circular path with radius R, moving with constant forward velocity V or, in other terms, in steady state.
If we proceed, as for the neutral car, assuming all angles to be small, we can see how in this case the front tyre will have to experience a force with a bigger magnitude than the rear one, in order to have both force and moments equilibrium, because the CG is now located in a much closer position to the front axle. This means, if we consider the following equilibrium equations:
being a is smaller than b, to have a net yaw moment N equal to zero, Fyf will have to be bigger than Fyr.
Since front and rear tyres are the same, in order to have the front tyres producing a bigger cornering force, they will have to also experience a bigger slip angle than the rear ones.
A similar condition is normally defined as “understeer” and the corresponding vehicle is said to be “understeering”. This term is used because, in order for the vehicle to travel on a path of radius R, with a forward velocity V, where both R and V are the same as for the neutral steering example, the steering angle δ must be bigger than the Ackermann angle δack. In other terms, if the driver would apply the same steering angle δack as in the neutral handling case, this particular vehicle would move on a path with a bigger radius: this means it would steer less effectively than the neutral one or understeers. This comes because the front slip angle, being bigger, will rotate the car anticlockwise more, compared to the previous case, hence pushing the driver to steer even more to travel on a path with the same radius.
As a consequence, also the body slip angle β changes and, as we will see in more detail, its value depends on several other parameters, like the corner radius or front and rear slip angles.
In order for the total vehicle cornering force to remain constant (as it must be, because the CG of the car still moves on a circular path with the same radius R and with the same velocity V as the one of the neutral car), if the front slip angle (and front cornering force) increases, the rear must decrease, as dictated by the lateral direction equilibrium equation:
Body slip angle β, front and rear slip angles and steering angle δ are linked together. In particular, it can be shown that front and rear slip angles are functions of β and of the rotational velocity of the car, often called r in technical literature.
Considering the car as a rigid body that rotates around a fixed point (as we are doing in our description), we know that r depends on the corner radius R and on the velocity V.
Since β is normally small, we can consider V being pretty much equal to u:
Given these assumptions, it can be derived that:
These two equations are derived basically applying some trigonometry to the simplified car model. As a part of the rigid body, the rear wheel must move forward with a velocity u ≈ V, and laterally with a velocity v (toward the center of the corner). Anyway, because of body rotational speed r, rear wheel center will also move toward the outside of the corner with a velocity given by “-br”. This means, wheel center’s net velocity will be given by:
The rear slip angle will be then given by:
But, since we also know that:
A similar approach can be used for the front, but here we also have the steering angle δ playing a role.
At the front wheel, we will have once again the same lateral velocity v as the CG (pointing toward corner center) and another lateral velocity component depending, as for the rear, on car rotation with angular velocity r and given by “ar”.
Anyway, this time the two components will add together, because they both point toward corner center. Steering angle will play against them though:
This description is most probably not the most fun part we dealt with until now and we purposely didn’t go into too much details, leaving to more interested readers the complete proofs and spotting any mistake, that could well be there.
Anyway these topics are very useful to understand which parameters influence each other in terms of cornering handling. Moreover, they will serve always as a basis to understand the principal features of an understeering or oversteering car.
With this in mind, we will now consider another generic case, that will close our analysis of how a basic vehicle model handles, depending on its features. Let’s again focus on our bicycle model, anyway now assuming that its CG seats closer to the rear wheel, than to the front one.
We will assume the car moving again on a circular path with radius R, at constant speed V, hence in steady state conditions.
Following the same logic we used for the “understeering car”, in order for our vehicle to keep moving on the same path as in the other two cases and with the same velocity (and hence lateral acceleration), the sum of front and rear tyre forces must still be the same and equal to Fc. Anyway, since now the CG seats much closer to the rear wheel, in order to have a moment equilibrium, the rear tyre force must be bigger than the front, because b is now shorter.
Again, since front and rear tyres are the same and experience the same vertical load, in order to have the rear tyre producing a bigger cornering force, it will also have to experience a bigger slip angle than the front one.
A similar condition is normally identified as “oversteer” and a car behaving this way as “oversteering”. If the car keeps moving on a circular trajectory with the same radius as in the two previous cases, the steering angle δ required to keep the car on its path will be smaller than the Ackermann angle δack. Turning this around, if the driver would still use a steering angle equal to δack, the car would move on a circular path with a smaller radius compared to the neutral car, this meaning it would steer more, or “oversteer”.
We need to stress once again, that this description of a neutral, understeering and oversteering behavior is strictly valid only if the tyres operate in their linear range. Nonetheless, it helps to give a physical meaning to one of the most used terminology in the racecar industry.
It is also worth to mention that our discussion has been focused on a car cornering in steady state. In this condition, we have seen how moving the CG forward would cause the car to understeer (more) while moving it rearward would cause the car to oversteer (or to understeer less). Anyway, a racecar operates most of the times in a condition that the engineers calls “transient”, which means that the acceleration vector (hence, also his two planar components, the longitudinal and lateral one) change in time. Moreover, to further complicate things, most of the times a driver tries to use the available grip in a combined direction (using for example at the same time throttle and/or brakes and steering) and puts the car in something close to a steady state condition only very rarely (sometimes maybe at the apex of a corner, for a very short time).
Dealing with transients would make our description too complex and would be out of the scope of these articles. Some of the basic principles we described relatively to steady conditions can give the reader a feeling about the effects of certain parameters on car handling.
One of the most important foundations, also when dealing with transient maneuvers, is that the two fundamental variable to define how a car handle are always the steering angle δ and the body slip angle β because, as we saw, all the other variables can be calculated basing on them (for example front and rear slip angle).
Steering angle δ is also the only variable among the one we discussed that is directly under driver’s control.