ASC Paper Notes

https://www.americansolarchallenge.org/ASC/wp-content/uploads/2013/01/Dr_Starr_Stability_Paper_-_Rev_20060811.pdf

General Concepts

Placement of the CG is very important when it comes to a side load being applied, which primarily occurs during turning. The following shows the scenarios that can occur when the CG is moved fore and aft of the Neutral Steer Point (NSP)

Open

The location of the CG relative to the NSP determines the characteristic of the yaw response. This can be summarized in variables known as the static margin and the understeer coefficient

Neutral steer is when SM = 0, which means LG = WB/3 in our case.

Understeer is achieved when SM > 0, and the CG is ahead of the NSP which in turn means LG/WB < 1/3. This is considered the more stable setup.

Oversteer occurs when SM < 0 and the CG is behind the NSP. This means LG/WB > 1/3 and is considered unstable.

For more info on over and understeer; see this.

This will be elaborated on further. Typical American passenger cars have an SM of ~0.06 while sports cars typically come closer to neutral and oversteer characteristics (Ferrari Monza hits an SM of around 0.003).

When turning, the side load from before becomes the centrifugal force. With this model, the steering angle can be described with the following equation

Re-writing this equation by replacing slip angles alpha yields the following equation.

where W_f is the weight on the front axle and W_r is on the rear. This can be further simplified by introducing the understeer gradient and simplifying the lateral acceleration.

The final equation can (somehow) be written, to include static margin, as

Given this equation, and the one with the understeer gradient (K) are equivalent equations, it can be seen that SM and K are the same in signs. That is, if SM = 0, K = 0; SM > 0, K > 0 etc.

This means we can further define our under/oversteer criteria. A key step is observing what occurs to the steering angle, δ, as the lateral acceleration term a_y increases.

Neutral Steer → K = 0 = SM, Slip_F = Slip_R

Here, the steering angle stays at a constant 57.3WB/R. This, however, implies that more lateral force is needed at each end of the vehicle, which means the slip angles would need to increase (or so the paper says; I don’t really follow this logic based on the given equations but it does make sense logically)

Understeer → K > 0, SM > 0, Slip_F > Slip_R

It can be seen that if these variables are positive, then the steering angle would need to increase relative to 57.3WB/R as the lateral acceleration increases. This indicates the understeer-ey behaviour of the vehicle.

The benefit of this is that the vehicle becomes self-correcting. That is to say, if the steering angle follows neutral steer, then the lateral acceleration must change. Given the following equation for lateral acceleration

it can be seen that either the radius must increase, leading to the understeer characteristic OR the velocity must decrease in order to maintain turning radius R.

Oversteer → K<0, SM<0, Slip_F < Slip_R

Opposite to the above, to achieve a given lateral acceleration, the steering angle must decrease, relative to neutral steer. Due to the rear slipping more than the front, if the driver were to make no correction to the steering relative to neutral steer, the back of the vehicle would “step out” and slip more than the front, causing the radius to decrease. This decrease in turning radius means more lateral acceleration which in turn requires EVEN MORE steering correction. This is why oversteer is widely considered as “unstable”.

3-Wheel Design Considerations

When looking at 3 wheels, a slight modification needs to be made to the understeer gradient and static margin equations. Instead of using C_f bar and C_r bar, they are instead replaced with C_f and C_r.

This is simply modifying the values present in the bicycle model and does not change the underlying performance metrics. This means that the same conditions for under/oversteer are the same for 3 wheels and 4 wheels; with the only difference being in the formula.

With only 2 wheels at the front and 1 wheel in the back, it is important to note that only lateral weight transfer between tires can occur at the front. This means, that due to load sensitivity, the front stiffness would decrease, which can increase the K value as lateral acceleration increases (assuming steering angle stays constant).

This means that the K value CAN be negative at the start but quickly become positive given increasing lateral acceleration.

This is still indicative of a stable vehicle.

This means that, in theory, the maximum weight on the rear axle can be increased from 33.33% up to 36% while maintaining the same stability.

Tipping of 3-Wheel Vehicle - Tipping Test

The diagram that will be used for reference here is as follows.

The vehicle pictured above is in a right-hand turn, with subscript “i” denoting inside and “o” denoting outside. The height of the CG is HG and the distance from the front axle to the CG is LG.

Although slipping can occur before tipping (somehow, the paper says the largest value of a_y that the vehicle will experience is equal to the coefficient of lateral friction at the tires), it is still wise to design the vehicle to a higher a_y since tires can bump while sliding which can cause tipping.

In the analysis done in the paper, it was assumed the tires were not sliding, therefore allowing us to isolate and examine the lateral forces needed to tip the car. When this happens, WFi will be 0. This lateral acceleration can be expressed as Fc, measured in g’s (note this could be gravity using imperial units)

The paper also explores how this lateral acceleration value, Fc, can be determined using the tipping table angle outlined in the regulations.

For ASC regulations, the car must be able to withstand a 45-degree tipping table. This gives us an Fc of at least 1.

This is the equation for a 4 wheeler

Braking Weight Transfer

Under braking, the following side view model can be used. In this scenario, the car may have 3 or 4 wheels.

Using this model, the following equation can be found.

F_t, which is ambiguously described in the paper, is the target % of weight transfer. That is to say, the percentage of the total weight on the rear axle that would be desired to leave the rear axle under braking. UMinnesota used F_t of 30%, which indicated that 30% of the default rear weight was transferred to the front axle.

F_b is the prescribed max braking deceleration, which in our case, is 1g minimum.

Glossary

Reading List

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.633.5587&rep=rep1&type=pdf