This case study shows how to find the tie rod inner and outer ball joints coordinates that provide the best compromise between Ackerman angle in steering, the steering ratio and a custom bump steer curve while still keeping the angle between the rods and the steering rack roughly parallel, which reduce the efforts in the steering rack.
The baseline system
To illustrate this case study, we’re using a generic Double Wishbone suspension system with Rack and Pinion steering, as shown below:
As noted by the two pictures below, the steering system of this vehicle is extremely detrimental to the steering rack, once it creates a lot of bending stress on the steering rack. The Optimization Module is able to provide an optimized solution that is far from the initial configuration, both in terms of pickup point coordinates and objectives, which will be explained in this article.
Design variables (boundaries)
The design variables – also called the boundaries or the search space – comprise the tie rod points (inner and outer). We hereby demonstrate a situation where the constraint is that the suspension designer must keep the same steering rack between the systems (the baseline and the optimized). But the situation allows the repositioning of the steering rack and change the tie rod link in the upright. Thus, the Y coordinates of the tie rods inner connection must be kept the same while the X and Z are free to be changed, only limited by the packaging. On the outboard tie rod link, the link can also assume any coordinate that the packaging allows.
To mimic the situation described above, we can create a box boundary with coincident points on the XZ plane (sharing the Y coordinate). The outer boundary is shaped as a sphere, which grants a good search space on the outboard position of the tie rod.
The spherical boundary has its center at (80; 780; 270) and a radius of 90 mm. It is set to hard – which means that the resulting system will always be inside that region – and symmetrical – the opposite side will be mirrored.
The box boundary is turned into a plane by setting the lower point to (-100; 348.229; 180) and the upper point to (150; 348.299; 260). All directions of this boundary are set to hard and it is also symmetric.
For the purposes of this case study we’ll be analyzing only the front suspension, in heave and steering simultaneously. We’re assuming that the steering system works from -270 to +270 degrees of steering angle and the heave range we’re interested in is from -50 to +50 mm.
Objective #01 – Toe Angle [Left] in Heave
For the toe angle in heave (also known as bump steer), instead of minimizing the parameter, we’re setting a target that has a custom curve. This way, the steering in heave can behave differently in bump than it does in rebound. This setup could be used to control the vehicle’s stability in high speed corners, where the downforce is significant.
We’ve chosen to have a toe-out setting when the car is compressed: toe angle is set to be +0.4 degrees when the car goes down 50 mm and -0.2 degrees when it is lifted 50mm. Notice that we’re analyzing the front axle only. The weight function decreases when the car is lifting because it is an unusual situation at the track, even though it is possible to have the car lifting up sometimes.
Objective #02 – Tie Rod Top View Angle [Left] in Heave
To reduce steering efforts and bending moments on the steering rack, it is desired that the tie rods and the steering rack remain roughly parallel. Thus, on top view, it is set to be minimized.
Objective #03 – Tie Rod Front View Angle [Left] in Heave
Similarly, on the front view, the angle between the steering rack and the tie rod is set to be minimized. However, it is more relevant to have them parallel where the vehicle spends most of its time – considering a heave motion. Therefore, the weight is set to maximum in the center (equilibrium position) and fades away sideways.
Objective #04 – Steering Ratio in Steering
The steering ratio influences heavily on the driver perception. Since the target of this case study is to keep the same steering rack between the baseline and the optimized car, the steering ratio should be optimized through kinematic exploitation. A common race car has a steering ratio around 15, which is our target. We want to prioritize the central position, so the weight function follows the pattern shown above as well.
Objective #05 – Ackerman Angle in Steering
On a race car, the Ackerman angle is an important factor for maximizing the lateral acceleration of a vehicle. Specially in low speed corners, it can be beneficial as it would “match” the peak slip angles of the inner and outer tires. Considering that the tire this car is using does not have a considerable amount of slip angle at the peak FY shift as the load increases, the ideal Ackerman is zero (parallel Ackerman).
The Ackerman becomes more important in slow corners where the mechanical grip is the biggest factor of the lateral acceleration on a mid to high downforce car and also has the most steering angle displacement. The weight is zero where the steering angle is zero and as the steering angle increases (either way), the weight increases as well.
The optimization settings can have a great influence on the optimization convergence and results quality. Since this case study uses multiple objectives that are conflicting with each other, we will use a Ranked selection for reproduction, combined with a Truncation selection for replacement. The crossover method is the Voluminal with alpha of 2. The mutation will use a Gaussian distribution with a standard deviation of 1.
Given the number of boundaries and objectives, it is estimated that a population size of 100 individuals is enough. The maximum number of generations was set to 200, based on experience, and 8 threads were used to run the optimization. No stop criteria were defined in this case study.
All the settings described above are explained in detail in OptimumKinematics Help File. The Help File also suggests the best settings for each situation. However, sometimes, there is a need for a little bit of empirical input from the user.
The time it took to run 200 generations was 3 minutes and 58 seconds. Since there was no stopping criteria set to the optimization, the process stops only when requested by the user or the optimization reaches the final generation.
The best solution converged quite quickly to a minimum, as shown by the convergence chart below:
We can see that after the 30th iteration, the decrease in overall fitness is almost zero, even though the monitors show that the algorithm kept finding improved systems constantly until the final generation has been reached.
The fitness (or cost) value of each objective was well distributed except for one objective, the Steering Ratio.
|Evaluation: Objective||Scaling||Fitness||Contribution (%)|
|Heave: Toe Angle [Left]||100.0||2.059||17.1|
|Heave: Tie Rod Front View Angle [Left]||1.0||1.354||11.3|
|Heave: Tie Rod Top View Angle [Left]||1.0||0.455||3.8|
|Steering: Ackerman Angle||20.0||0.121||1.0|
|Steering: Steering Ratio||1.0||8.037||66.8|
This is also shown on the charts below. We have a good correlation between some targets and the optimized behaviors. However, some targets were not as close as expected, as shown in the next several charts.
The final solution is a satisfying system. Since this case study does not take into account the packaging of the vehicle, we suppose that the boundaries we’ve set comprise the packaging limitations.
The front and top view respectively can be seen below. As we notice, the angle between the tie rods and the steering rack are much better compared to the initial solution.
The pickup points comparison can be seen below. This is a good example to show that the Optimization Module in OptimumKinematics is able to find feasible solutions that are quite distant from the initial (baseline) one.
|Point Coordinate||Original System [mm]||Optimized System [mm]||Variation [mm]|
|Tie Rod Chassis X [Left] [Front]||-47.86||150.00||197.86|
|Tie Rod Chassis Y [Left] [Front]||348.23||348.23||0.00|
|Tie Rod Chassis Z [Left] [Front]||210.97||260.00||49.03|
|Tie Rod Upright X [Left] [Front]||68.69||153.94||85.25|
|Tie Rod Upright Y [Left] [Front]||778.66||826.10||47.45|
|Tie Rod Upright Z [Left] [Front]||302.92||247.48||-55.44|
This case study shows that with about 15 minutes – considering boundaries and objectives setup – we’re able to come up with a decent suspension for our goals. Following a legacy procedure – i.e. batch runs and sensitivity studies – the outcome could take days and never be as satisfying as the result we obtained here.
Complex objectives can also be set using the Optimization Module, such as the custom bump steer curve we used in this study. Combining this with an exceptional computation time of just above 3 minutes, the Optimization Module proves to be a powerful tool for any suspension designer.