Considering Compliance: Using the OptimumKinematics Forces Module to Determine the True Toe Gain

by | Jun 24, 2019 | OptimumKinematics | 0 comments

The Forces Module in OptimumKinematics is one feature that can not only benefit the performance of the car through optimization of component design, but also optimization of vehicle setup. Using OptimumKinematics Forces Module the user can use the Multiple Iteration Feature to determine the total compliance of an element under steering, heave, pitch or roll. This allows a user to more accurately predict where to setup components and extract a higher performance out of the vehicle.

Kinematic Points

By changing the kinematic pickup points of the suspension, we not only change the relative motion of the suspension link, but also the load path that the suspension link is sustaining. This in turn changes the deflection of the suspension, which can change how the tire is slipping and whether the tire performance is where it should be. Given that a car can be most sensitive to an excessive change in toe, we will measure the bump steer of the car under a full heave event, considering both the kinematic change but also the compliance using the pickup point forces to find the link force and extension. The vehicle that we are using can adjust the kinematic pickup point up to 40mm from its lowest position. You can find the properties of the car on a previous article (here). The simulation being used will be a 4G equivalent heave event, simulating a curb strike or similar event.

 OptimumG Tool for Motion and Force Estimation

Figure 1 – OptimumG Tool for Motion and Force Estimation

Tierod Force Variation per Vertical Movement

The simulation parameters window requires a vehicle setup instance to be input in addition to a motion and a load case. Within the parameter window, the parameter that is being modified can be selected along with the number of iterations to consider. The tools that are available to retain the steering rack pickup point allow for 2.5mm of variation in the pickup point, which now sets the number of iterations for the simulation and the resolution of the iterations. The forces with each iteration of the heave can then be analyzed and a 2D plot can be generated to compare the forces at the steering rack pickup point

Variation in the vertical load

Figure 2: Variation in the vertical load across a 40mm change in the vertical displacement of the inboard steering point

Using the 2D plot, we can see that the vertical load in the tie rod link goes up significantly as the mounting point on the steering rack is increased. This would suggest that the compliance of the tie rod will increase with a greater vertical displacement. The compliance of the link will act as an additional toe gain and induce scrub on the tire that would not otherwise be present in the design of the system. To approximate the compliance of the system, the elongation length of the tie rod can be calculated using the lateral force being applied on the tie rod to steering rack pickup point. While there will be an additional vertical load and an additional longitudinal load applied to the system, the additional loads relative to the lateral load can be considered negligible as the longitudinal loading is just over 2.5% of the lateral load while the vertical loading is 15% of the lateral loading.

Outputs for the force components

Figure 3: Outputs for the force components for one iteration of the batch simulation


To calculate the maximum elongation on the tie rod, we will take the maximum loading on the link from either the right-hand side or the left-hand side.  In this case, the right-hand side has a higher load case such that it will have the higher magnitude of loading. The maximum and minimum value can then be exported from the results window. The elongation of the tie rod can then be calculated using the following equation:

In the equation, the P term refers to the load being applied to the link, L refers to the length of the link, E refers to the elastic modulus of the link, and A refers to the cross-sectional area of the link. The cross-sectional area of the steering link is defined from the worst-case loading of the suspension. We are also assuming no play in the attachment points and no deflection in the upright or the steering rack mounting points. For more information on how to use the OptimumKinematics Forces Module to define the link geometry of the suspension using multiple different load cases, check out our case study going through how to generate different load cases and define the suspension geometry.

Calculation parameters

Figure 4: Calculation parameters for the extension of the tie rod

The length for the tie rod link has been defined as a constant value with minimal adjustment needed in order to return the car to zero toe. There will be minor variation in the length of the link due to adjusting to return to zero toe, however the effect that will have on the overall system can be considered negligible, as the change in load is only 3.3% for a 10% change in the vertical displacement of the inboard point, which in turn yields no significant change in the toe length, as seen in figure 5.

Compliance Results

Figure 5: Compliance results of the system in terms of toe displacement and toe angle

Once we have calculated the elongation of the link from the forces applied, we can determine the total toe change under heave with the different options available for the vertical toe link. Based on the total toe change, an engineer now has a value closer to the true toe variation of the front suspension. Additionally, the ride gradient of the toe link can be determined. This can provide a gauge when empirical testing a car on a K&C rig to prove that the vehicle is behaving as predicted in simulation.  Additionally, while testing different suspension designs in simulation, the ride gradient can provide a gauge to show which designs minimize the compliance the most.

Total Compliances

Figure 6: Total toe angle change for each kinematic point and the toe gradient of the system

In figure 6, the leftmost value refers to a steering pickup point 200mm above the ground, with each column being a 2.5mm increase in the vertical component of the suspension pickup point.  Based on the combination of the kinematic and the compliance toe change, we see that the least amount of toe change occurs with the vertical point set to the 202.5mm setting. The toe change that does occur is toe out, which means that there will be additional slip angle induced on the tire as the car is turning in, which will tend to be more stable than toe in. We can additionally see that the toe compliance will induce a more toe out vehicle by 0.012 degrees, which is not significant enough to be observed by most toe gauges.  Therefore, for this vehicle, we have proven that the system compliance is minimal based on preliminary calculations of the links.

Key Points

The OptimumKinematics Forces Module can be used for much more than determining the pickup points and performing structural analysis. Using the geometric properties of the suspension links and the forces being applied to the system, we can determine the true toe gradient of a system including the compliances of the system. This can help a team make sure that their tires are being scrubbed as expected and the car is reacting to suspension movement as would be expected.  This is just another way that the OptimumKinematics Forces Module can aid in the process of understanding a vehicle’s suspension.


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