ASC Paper Notes
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The distance of the NSP from the front axle is determined by the following equation   Assuming the tires in the front and back are the same model, and thus the cornering stiffness values are the same, the equation simplifies down to 1/3WB. |
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
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The Static Margin (SM) is defined as “the distance from the CG rearward to the NSP divided by the wheelbase wheelbase” and is expressed with the following equation (somehow…?). It is the distance from the front axle to the NSP as defined earlier, minus the distance from the front axle to the CG (denoted as LG), all divided by WB. Notice that for the first term since WB is in the definition, dividing by WB cancels it out.  In our case, with 3 wheels of the same cornering stiffness, this can be simplified down to  |
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When turning, the side load from before becomes the centrifugal force. With this model, the steering angle can be described with the following equation
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This, using lateral force vs slip angle data such as figure 3 (in the first expandable section), this equation can be re-written as 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.
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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.
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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).
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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.
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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)
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The paper also explores how this lateral acceleration value, Fc, can be determined using the tipping table angle outlined in the regulations.
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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
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Braking Weight Transfer
Under braking, the following side view model can be used. In this scenario, the car may have 3 or 4 wheels.
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Using this model, the following equation can be found.
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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
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Note: Underlined terms have definitions in expandable sections above. WB - Wheelbase, the distance between the rear and front axle CG - Center of Gravity HG - The height of the CG TR - Track, the distance between the tires on the same axle NSP - Neutral Steer Point, the point at which a lateral load can be applied to the vehicle without inducing rotation (yaw) SM - Static Margin, the distance from the CG rearward to the NSP divided by the wheelbase K - Understeer gradient, performance indicator indicating under and oversteer. LG - Distance rearward from the front axle to the CG LNSP - Distance rearward from the front axle to the NSP C_f - Front TIRE cornering stiffness C_r - Rear TIRE cornering stiffness C_f bar - Sum of C_f for all tires on the axle C_r bar - Sum of C_r for all tires on the axle W_f - Weight on the front axle WF_i - Weight on front inside wheel WF_o - Weight on the front outside wheel W_r - Weight on the rear axle a_y - Lateral acceleration; can be present through an applied force OR the centrifugal force δ - Delta, steering angle Slip_F - front tire slip angle; the angular difference between the front tire’s direction and the actual velocity Slip_R - rear tire slip angle; the angular difference between the rear tire’s direction and the actual velocity |
Reading List
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.633.5587&rep=rep1&type=pdf
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