The determination of vehicle parameters was split into three steps:
- Finding acceptable wishbone geometry
- Using a pseudo steady-state MATLAB model to find weight and wheel rate distributions
- Using a five mass, five degree-of-freedom MATLAB model to determine damping rates
Step 1.
I used vsusp.com to find suspension geometries that minimize lateral roll center movement. Camber gain was only lightly considered since the Hoosier LC0 tires we planned to run are fairly insensitive to any gain. Here is a screenshot of the front suspension geometry used.
Step 2.
To characterize the vehicle's cornering characteristics, a pseudo steady-state model was derived. Only lateral acceleration is considered so this amounts to summing all the external forces and setting them equal to the vehicle mass times the lateral acceleration. The external forces acting at the tires induce a yawing moment on the vehicle which is generally not the same as that required to take the constant radius corner at constant speed, which are assumed at the beginning. Hence, the model is only a pseudo-state model. Tire data for the Hoosier 6.0/18.0-10 LC0 is parsed and interpolated to get lateral forces from the tires given the load, slip angle, pressure and inclination.
The process went as follows:
Dynamic model
Bump response
Chassis step response
The high damping rates were fiddled with to lower the peaks of the Bode response plots and the low damping rates were adjusted to minimize the settling time. The wheel damping rates were converted to spring damping rates so that they could be compared with the dyno plots produced by Ohlins, the manufacturer of the shocks were were using.
The process went as follows:
- Import car parameters
- Assume body angle, steering angle, and cornering scenario
- Guess input lateral acceleration
- Compute tire forces to compute output lateral acceleration
- Analyze output
- Return to step 2 or 3 unless an acceptable solution is found
If the output lateral acceleration matches the input acceleration and nothing weird has happened (like a tire lifting off), then the following parameters are checked:
- Normalized yawing moment
- The stability and control derivatives, dN/dB and dN/dδ
- Tire saturation
- Ride rates
In general, the input tire spring rates must be close to the output rates, but they are not particularly sensitive to changes in other parameters. (1) is used to characterize the ''under/oversteering" of the vehicle. (2) is used to determine whether changes in body or steering angle that pull the car into a tighter corner correspond to decreases in the yawing moment. (3) is used to check that the front tires saturate before the rears so that the driver can feel the front tires loose grip first and therefore have an easier time making corrections. (4) is used to make sure the car does not get too close to bottoming out.
In general, the most important parameters to tweak were the weight distribution and wheel rate distribution. The final weight distribution was around 50/50 rear-to-front and the wheel rates we ran were 23000/18000 N/m front-to-rear.
Step 3.
The final thing to do was to estimate the damping rates. This was done by determining the dynamics of a five mass, five degree-of-freedom representation of the car then simulating the response of the car to bumps and cornering. The following is a graphic showing the five masses and their connections.
Dynamic model
The linearized 2nd-order ODE system of five equations is put into state-space form and the MATLAB command lsim() is used to pass bump and cornering inputs.
Constant high speed damping was used in bump simulations, while constant low speed damping was used for cornering simulations. The following is a plot of the front wheel's response to being bumped (y-axis in meters). The Bode plots generated were also used to determine the damping rates by trying to minimize their peaks.
Bump response
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Chassis step response
