Tire Dynamics - Fundamentals

1. Calculate the tire stiffness. Tire size – 275/40R18, Free radius is 13.33 inches and loaded radius is 12.36 inches.

- Deflection = Free radius - Loaded radius = 13.33 - 12.36 = 0.97 inches

Tire stiffness = Force * Deflection

(Force is not given)

2. Explain in your own words what slip ratio and slip angle are.

- Slip ratio is a measure of tire slipping with respect to the road along the longitudinal direction.

Mathematically,

`s = frac(v_(vehicl e) - v_ (tire))(v_(vehicl e)) = frac(v_(vehicl e) - r omega)(v_(vehicl e))`

Ideally, the tire points straight but due to the distortion arising in the contact patch, it travels away from the expected trajectory. This angle made by both the paths is called as the slip angle.

3. What is the risk of having too much mechanical trail? (think in terms of driving at the limit)

- The lateral force peak arises in between 5-6 degrees of slip angle. The percentage reduction of steering wheel rim force occurs during this 1 degree of span.

The % reduction is more when only Aligning Torque is considered solely due to the Pneumatic Trail (depends on slip angle/loading).

As Mechanical Trail (depends on steering geometry) is progressively added, the percentage reduction is substantially diluted.

The feedback while driving should come from the tires and not the suspension as the force is generated first in the tires.

Thus, a skid warning (significanct % reduction) is essential when you are on the limit of traction else you will have false sense of security while driving thinking that the tires have good cornering capability.

4. You know lateral force is proportional to slip angle. Can a tire produce lateral force at zero slip angle?

- Yes, a tire can produce lateral force at zero slip angle. Because during cornering (leaning on corner), a camber thrust is generated which results in some amount of lateral force.

`F_y = F_(y1) + F_(y2)`

`F_(y1) = C_(alpha) . alpha`

`F_(y2) = C_(gamma) . gamma`

where;

`C_(alpha)` = Stiffness

`alpha` = Slip angle

`C_(gamma)` = Camber stiffness

`gamma` = Camber angle

5. Explain tire load sensitivity and how increasing the rear roll bar size can induce oversteer.

- Lateral force coefficient vs Slip angle is not a constant (mu) curve for different loads. It varies according to the relation `F_y prop F_z ^ (0.7 -0.8)`

So how the curve responds to different loads is Tire Load Sensitivity.

Even though the total vertical load remains constant, the anti-roll bar has compliant characteristics and it sustains the load. So depending on the anti-roll bar size, the weight on the tire will increase and it saturates early during cornering. This will result in the loss of lateral force.

If the lateral force reduces in the rear, we get an oversteer condition.

If the lateral force reduces in the front, we get an understeer condition.