This case study shows how to optimize the motion ratio values of a Push/Pull Rod system. The main objective is to achieve a targeted value of motion ratio for a heave motion by changing the rocker pickup points and the coil-over mount on the chassis. Imagine a situation where there is the need to change the motion ratio of a car and you can only manufacture the rocker. This case study will show how an optimized motion ratio can be achieved quickly.
The Baseline System
The baseline suspension we picked for this case study is a common Double A-Arm suspension with a simple Push/Pull Rod actuation. The baseline can be seen in the picture below.
Design Variables and Boundaries
As mentioned in the overview, the points that we’re changing on this case study are the two rocker points (pivot and push rod link). The design variables are the points that we want to change to optimize our system. The boundaries are packaging constraints that define the search space and possibly limit the point coordinates.
We’re only using spherical boundaries on this case study, but box or cylindrical boundaries could be used as well. The geometries’ sizes were chosen based on experience and packaging: we don’t want components colliding and we also want our solution to be inside the search volume. The spheres shown in the image below represent the search space.
Both boundaries are set to hard, which means that the resulting points can’t be located outside their region.
The motion used to analyze the suspension in this case study is a pure heave with linear variation, from -30mm to +30mm, as shown below. We will use the default 20 steps to measure the motion ratio.
This motion is what will determine how we evaluate the suspension system, thus, every candidate result will be simulated using this motion. Note that the line starts at -30mm, when the % Completion is zero and goes all the way up to +30mm when the % Completion is 100. The function passes through the zero heave, which is assumed to be the static deflection position.
In this case study we will work with multiple objectives. The main goal is to optimize the coil over motion ratio in heave assuming that we can only manufacture a new rocker.
Thus, the coil over length must be similar to the original one when the system is in droop, ideally identical. The push rod must also have equal length, so the current one can be re-used in the new configuration. Additionally, we want to minimize the moment loads in the rocker pivot. A total of 5 objectives will be used in this case study, where four of them work like constraints.
Objective #01 – Motion Ratio
The motion ratio is chosen as to provide a progressive stiffness to the suspension as the car goes down. The desired motion ratio lies far from the baseline, as shown by the chart. Since the stiffness is less relevant when the car is being lifted, the weight function – which measures the degree of importance of the objective at that specific step – is progressively set to zero. This will prioritize the function on the compression region, which is more critical than rebound.
Objective #02 – Angle between the Rocker and the Push Rod
In order to minimize the loads on the rocker pivot, the ideal angle between the rocker plane and the push rod is zero (considering that the rocker plane is parallel to the pivot axis) to minimize the bending moments in the pivot. For that reason, the objective is defined as a constant line at zero. The same weight function used in Objective #01 is applied to this objective, since minimizing the loads is not as important in the rebound region as the suspension is unloaded.
Objective #03 – Angle between the Rocker and the Coil Over
Similarly to Objective #02, the angle between the rocker and coil over must be reduced for the loads in the pivot to be minimized as well. Notice that the weight function once again being used to prioritize the motion range that is more important to us, where it induces higher loads on the suspension.
Objective #04 – Push Rod Length
As for our described conditions above, we want to keep the same push rod, so the push rod length must remain constant to comply with that restriction. Since its length is not changed through the simulation, the weight function was set as constant.
Objective #05 – Coil Over Length
Even though it is possible to change the coil over length by setting the spring preload, the coil over working range is desired to remain similar, specially when fully extended, where the damper is physically limited. Notice how we can use the weight function to only give importance to the coil over length at full droop, in the 100% completion region.
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. OptimumKinematics Help File helps the user to pick the proper optimization settings for each optimization scenario.
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.
The time taken to run the 200 generations was 3 minutes and 54 seconds, even though the solution converged much sooner than that. From generation 100 to 200 there was very little improvement on the objectives’ fitness. The convergence graph is shown below.
The final (overall) fitness found was 10.645 and the greatest variation found was on the X coordinate of the push rod connection to the rocker, as shown in the table:
The objectives are considered satisfactory, The figures below show the results for each objective defined. They display the original baseline curve, in light green; the defined objective curve, in dark green; and the optimized curve, in blue.
The optimized suspension is shown next. Notice how the coil over is better aligned with the push rod pivot than the original system. Also, the lever arm difference is quite big when compared to the baseline system.
It is known that getting the right motion ratio can be quite challenging without proper tools. For a simple optimization, the time we took to get a new and improved solution is indubitably lower than trial-and-error approaches or by running several batch simulations. This proves that a new actuation system can be optimized using the Optimization Module in under 15 minutes (considering the setup time). If needed, it is possible to take it further and use the Forces Module to extract the load cases to input in a FEA before manufacturing the new rocker.