ASC Paper Notes
Aspects of stability
rear end stability in turning and crosswinds
high-speed straight-line ability; reducing steering corrections
resistance to tipping in turns and sudden changes in road surfaces (from slippery to grippy)
“swapping ends” under hard braking? not quite sure what this means
CG must be a design specification
Slip angle stuff
Slip angle is a function of the lateral load, as well as the vertical load (on that tire)
generally, automotive tires have a max slip angle of around 10 degrees. further than that, and the vehicle will lose grip and slide
The plot of bike tire slip angles, showcasing how increasing load increases the lateral load for an X slip angle
this increase in grip is not proportional, however, with doubling the load from 67-135lb only yielding a 66% increase in lateral load
this is known as “load sensitivity”
coefficient of lateral load is the ratio of lateral to the vertical load
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)
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
The Static Margin (SM) is defined as “the distance from the CG rearward to the NSP divided by the wheelbase and is expressed with the following equation (somehow…?)
In our case, with 3 wheels of the same cornering stiffness, this can be simplified down to
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
This, using lateral force vs slip angle data such as figure 3 (in the first expandable section), this equation can be re-written as
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 understeer gradient is a performance metric that also dictates under/oversteer. It is defined as the following
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
Tipping of 3-Wheel Vehicle
Braking Weight Transfer
Reading List
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.633.5587&rep=rep1&type=pdf