How do understeer and oversteer work? (the math and the physics)

[hari00968]

I am reading 'Fundamentals of Vehicle Dynamics' by Gillespie. In the chapter 'Steady State Cornering', it explains how under-steer and over-steer works, using a formula, which is also derived. I can understand the derivation of the formula reasonably well. The formula is:

Steer Angle = 57.3 L/R + K*(lateral acceleration)

where:
L = wheelbase
R = radius of curvature of curve
K = understeer gradient (+ve if the car has understeer, -ve if oversteer, 0 if neutral)

There's a formula for K as well, depending on the front and rear curb weights of the car, in addition to tire properties, which is easy, no problems with that.

This is what the book says (page number 203):

"In the understeer case the lateral acceleration at the CG causes the front wheels to slip sideways to a greater extent than the rear wheels. Thus to develop the lateral force at the front wheels necessary to maintain the radius of turn, the front wheels must be steered to a greater angle."

"In the oversteer case, the lateral acceleration at the CG causes the slip angle on the rear wheels to increase more than at the front, thus diminishing the radius of the turn. The increase in lateral acceleration that follows causes the rear to drift out even further and the process continues unless the steer angle is reduced."

It's obvious from the formula that you would need to increase the steering angle with increase in speed, in the case of understeer, and vice versa. But I don't understand what's happening physically, with respect to the force due to the tire slip angles. I'm very confused with their explanations. Please help.

 

[256bits]

When making a corner with a 4 wheel car, the inner front and back wheels have less of a radius to follow than the outer front and back wheels.
Ideally, the inner wheel will turn just a little bit more than the outer, so that both follow the arc prescibed by the above mentioned radii.

For understeer:
If both front wheels turn the same amount, and the outer wheel follows its arc, the inner wheel will subsequently not be following its arc and will skid a bit forward or sideways if you like, while trying to turn. You will have to turn the steering wheel a bit more to follow the curve.

Also, the car has inertia with a CG and wants to move in a straight line.
The wheels want to roll along a path that they point.
The two are in conflict with each other when making turn.

Its not a very thorough explanation, but I hope it clarifies some aspects of your inquiry.

 

[mender]

Hard to say exactly what you're after with this:
"But I don't understand what's happening physically, with respect to the force due to the tire slip angles."

The force generated by the tire varies with the slip angle. To generate more lateral force, more slip angle is required from the tire (front or rear). The amount of force generated by the slip angle is fairly linear until the tire is getting close to its peak, then the line curves over and starts back down as more of the tire's contact patch slides.

If the car is understeering at the limit, the front tires will be at a higher slip angle than the rear tires and are the limiting factor that determines the maximum cornering rate. Adding more steering results in more slip angle but if the tire is already at the peak, the force generated will be less than at a lower steering angle.

Here's an excerpt from one of the books I have:
http://insideracingtechnology.com/tirebkexerpt2.htm

That hopefully explains what you're asking about in more detail.

 

[hari00968]

@mender,

"If the car is understeering at the limit, the front tires will be at a higher slip angle than the rear tires"

And like you said, up to a certain point, lateral force increases with slip angle. Assuming we're still within that range, if the slip angle in front is higher, the lateral force in front should also be higher, which should cancel the understeer right? I mean, if you further increase the steer angle (and hence the slip angle), you're only making the difference in slip angle between the front and rear higher right?

Besides, why is this difference a bad thing in the first place? Because it looks like it will cancel the understeer!

Sorry if my questions seem very trivial, but I'm really confused here!