Slip Angle, Part 2

I promised force diagrams.  Force diagrams you shall have.  As previously discussed, slip angle is the angular difference between the direction a tire is moving and the direction a tire is rolling.  If the slip angle is zero, the tire is rolling straight ahead.

If a slip angle is introduced, the tire will develop a friction force perpendicular to the direction the tire is rolling.  The magnitude of that friction force (and how the car feels it) is dependent upon the velocity of the tire and the slip angle.

Lesson #1:  Turn the wheel as little as possible

First, let’s take a diagram for an undriven tire.  Static friction can only oppose motion an impending sliding motion, so in this case the friction will be applied perpendicular to the direction the tire is rolling.  The magnitude of this force is determined by the velocity of the tire and the slip angle.  The diagram shows a tire that isn’t being stressed to the edge of its friction circle.

If I increase the length of the velocity vector, the lateral grip vector will increase in length until it touches the edge of the friction circle.

When I construct the friction force vector, the friction circle is mirrored because the forces I have drawn are those being imposed on the pavement.  In the larger friction circle, I have flipped them over to the conventional orientation.  Note that the tire is only experiencing a lateral force – no braking or acceleration.

Next, I move the friction vector to the center of the tire and flip it to reflect the action on the tire.  This is blown up on the right.  The friction vector breaks down into two components – lateral acceleration (cornering force) and drag (which slows the car down).

The drag will slow the car down even without a braking force!  You can also see from the diagram that the larger the slip angle, the higher the drag will be.


Lesson #2:  Drive through the corner on throttle

These two diagrams show a tire with slip angle that is braking (left) and accelerating (right).  These forces act along the direction the tire is rolling and alter the direction static friction vector.  The braking (or accelerating) action alters the direction of impending sliding motion.

Both diagrams have the same lateral grip component and the longitudinal action takes the tire to the limit of adhesion – to the edge of the friction circle.

Let’s take the braking vector construction first.  The bottom diagram shows the lateral and longitudinal components for a wheel that is turning and braking.  The lateral grip line is the same as before, but the longitudinal line is added.

When the forces are rotated to the direction the car is moving (as opposed to the direction the tire is rolling), the net corning and net braking components are revealed.  Note that the net cornering force is reduced.

Under acceleration, the action is similar.  The outcome is different, however.  When the acceleration is added, it is fighting against the drag caused by the lateral grip vector.  Actual acceleration is reduced from what was being delivered to the tire.  Now, look at the cornering force… the net cornering force is increased from the undriven case!


The above discussion is simplified (some might say it is simplified a lot) and can be somewhat misleading.  In the cases shown, the tire was never approaching the limit of lateral grip in the undriven case.

As we push the tire closer and closer to that limit, the ability to accelerate or brake diminishes.  Maximum cornering force will be achieved under acceleration, but this acceleration will be very small.  I chose a case that illustrated the point well.

In so doing, my diagrams are applicable to corner entry (trail braking) and corner exit (accelerating off the corner).  Mid-corner is much less dramatic.

Next time we’ll remove some of the simplifications.

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