June 29, 2023

# Bolide suspension optimization

*Industry: Automotive | Product: pSeven | Company: Formula Student*

## Introduction

We already told you about one project for Formula Student when pSeven helped to calibrate car’s tire dynamics model. You can read more about it in our article "Tire Dynamics Model Identification."

Today we present another use case. In this project, a group of students from the Formula Student optimized the attachment points of the rear suspension of the new version of their racing bolide using pSeven. The kinematics of the suspension directly depends on the attachment points of the suspension. And in turn, the cornering speed of the bolide depends on the kinematics.

Correct suspension kinematics were already found by the team for the previous version of their bolide. That was achieved due to successful layout, which made it possible to implement long suspension arms, and an increase in the arm length reduced the range of the wheel settings. Due to the design changes, the rear frame of the new bolide version turned out to be too wide, so the rear suspension arms became significantly shorter, which affected the kinematics for the worse (Figure 1).

*Figure 1. Rear suspension of the new bolide*

The team made a mathematical model of the new bolide’s suspension in Amesim (Figure 2) and by simulating the movement of the wheel, they got graphs of changes of the wheel’s settings.

*Figure 2. Mathematical model of the new suspension in Amesim*

This mathematical model allowed obtaining the following toe and cambering graphs (Figure 3):

*a)*

*b)*

*Figure 3. a) Toe plot of the new suspension; b) Camber plot of the new suspension*

The dependences of the wheel alignment angles on its movement coincided with those obtained in the Lotus Kinematic Suspension software, which indicates that the mathematical model is correct.

## Problem Statement

In the task, it was necessary to determine attachment points of the rear suspension that would allow the new bolide’s kinematics to be as close as possible to the kinematics of the previous version. The following input parameters were considered:

- Coordinates of the point of the front attachment of the upper arm to the supporting structure.
- Coordinates of the point of the rear attachment of the upper arm to the supporting structure.
- Coordinates of the point of the front attachment of the lower arm to the supporting structure.
- Coordinates of the point of rear attachment of the lower arm to the supporting structure.
- Coordinates of the attachment point of the jet thrust to the wheel-hub assembly.
- Coordinate of the attachment point of the jet thrust to the supporting structure.

There are two methods of suspension optimization:

- Synthesizing suspension configuration with the necessary parameters from scratch.
- Improvement of existing suspension attachment points.

The first approach is computationally expensive and incredibly difficult to implement, so the team decided to choose the second one and used effective optimization algorithms implemented in pSeven.

Thus, these optimization goals were defined:

- Minimize the difference between the toe curves of the new and the previous version.
- Minimize the difference between the camber curves of the new and the previous version.

## Challenges

- Complex dependency of the suspension kinematics on its attachment points, which is non-linear and non-obvious.
- Countless possible options for attaching the suspension to the supporting structure, thus it is not possible to consider each of them.

## Solution

The first optimization run was unsuccessful. Obtained camber and toe curves were very similar to the desired ones, but the resulting attachment points were impossible to implement on the car, and all other parameters indicated that this suspension did not meet the requirements.

To obtain the required suspension kinematics with the capability of being assembled on a supporting structure, it is not enough just to achieve the required wheel alignment graphs. Moreover, certain restrictions must be satisfied.

The first constraint is the attachment of three suspension points (2, 4, 6) to the plate, which is a part of the supporting structure (Figure 4).

*Figure 4. The new rear suspension in Amesim: blue body – plate*

Knowing the location of the plate in space, its mathematical equation in the form of Ax+By+Cz+D=0 is obtained. In Amesim, a “super component” is implemented, which substituted the coordinates of points 2, 4, 6 into this equation. For each such equality, in pSeven the following constraints were added (Figure 5):

- Upperlimit – rear upper arm attachment point.
- Lowerlimit – rear lower arm attachment point.
- Rodlimit – jet thrust attachment point.

In order for the sum Ax+By+Cz+D to be equal to zero, the point must meet a very strict condition of belonging to the plane, which is almost impossible to fulfill, so an error of ± 10^{-4} m was introduced.

*Figure 5. Optimization configuration*

The second constraint is the roll center coordinate, which directly depends on the location of the arms. The roll center of the rear axle must be above the roll center of the front axle, but at the same time below the center of gravity of the vehicle. The height coordinate of the roll center of the front axle in the reference system is -0.4545m and the center of gravity of the car is 0m. Therefore, this range of values is set for the “rollcentre” variable (Figure 5).

Since there were too many variable parameters for optimization, and it was necessary to set a small step of their change, in order to reduce the calculation time, the difference in the toe and camber graphs was reduced to one variable “toecamber”.

To solve the optimization problem, the Amesim model was integrated into pSeven and connected to the Design space exploration block, in which Surrogate-based optimization algorithm was selected (Figure 6).

*a)*

*b)*

*Figure 6. a) Optimization workflow in pSeven; b) Content of the Composite block – integration with the Amesim simulation model*

## Results

As the result of optimization, kinematics that exceeded the original model in terms of the range of wheel alignment angles, while satisfying all the necessary conditions, was obtained.

*Figure 7. Toe curves. The blue line is the desired graph; the red line is the result*

The area under the toe curve was 2.1172 arbitrary units, and after the optimization, it became 0.731234. The area under the graph **has decreased by more than 65%**.

*Figure 8. Camber curves. The blue line is the desired graph; the red line is the result*

The area under the camber curve was 1.38555 conventional units, and after the optimization, it became 1.45181. The area under the graph **has decreased by almost 5%**. Single-criteria optimization took **less than 10 minutes**, and its budget was 472 calls to the calculation model.