Please enjoy the 2-minute summary gallery below, or scroll further to explore my work in greater detail
Suspension Kinematics
The ultimate job of a suspension system is to control the behavior of all four tires in a way that maximizes the performance, drive-ability, and stability of the vehicle. In order to do this, the tires must be kept in an optimal combination of operating conditions, and the kinematic motions of the suspension linkages play a key role in controlling that window. A well designed suspension will maximize the time spent in the tires' "performance envelope" and manage that subsequent tire performance effectively.
In this page, I will discuss the process I used to develop the kinematic design for Clemson FSAE's 2020 vehicle, Tiger22.
Design Methodology
Tiger20 on the autocross course at FSAE Michigan 2019
In many FSAE vehicles, suspension geometry is often one of the first designs to get finalized. This is because the geometry and orientation of several downstream components are derived from the locations of suspension pickup points. For critical components such as the frame, this is critical for integration and ensuring that suspension points are placed on nodes to maximize installation stiffness. Switching the order of that process, and using chassis nodes to determine suspension points, is ill-advised.
For Tiger22, this remains the case. However, in past vehicle when suspension geometry is finalized first, other components such as the frame have to compromise their design in order to align with kinematic pickup points. These compromised designs are often heavier and more compliant than they need to be. In order to avoid this, I decided to include chassis and outboard design considerations to inform my design from the very beginning of the project.
With this in mind, my approach to the suspension design was to start by defining all of my kinematic behavior targets abstractly. This exploits the fact that there are infinite variations of a suspension that can achieve near-identical kinematic characteristics. Once these targets have been fully identified, I can focus on creating design iterations that meet all of the requirements while simultaneously pursuing secondary design objectives. By including these interactions in the early stages of design, I can help avoid design compromises further downstream.
The non-nodal pickup point on the front upper control arm shown here (c. 2018) is an example of design compromise caused by finalizing suspension geometry without consideration of the downstream subsystems
Vehicle Modelling
The starting point for my vehicle model was the two track steady state vehicle model, using my non-linear CSAPS tire model that can capture load, slip and camper effects in both longitudinal and lateral forces. The model has 7 degrees of freedom: yaw and side slip velocity, longitudinal acceleration, front/rear roll, ride height and pitch angle. Suspension stiffness has an effect on total roll and pitch motions (and by extension kinematics), but no direct effect on mechanical grip is captured. Tire compression is also neglected. This describes the basic model foundation, and as the design process moved forward, kinematic details such as roll and steer camber were added in as required. Final design evaluations take all variables in effect, but by increasing complexity in stages I can better evaluate the effects of individual design parameters without worrying about coupling with others.
In most cases, this vehicle model was used to generate GGV models of the car, which were then input to the lap time simulation to predict performance at FSAE dynamic events. You can learn more about the development of the lap sim here.
Camber Kinematics
The first objective in the design process was to select camber change targets for the vehicle. I defined camber change as "camper compensation", meaning the percentage of camber lost during body roll that is gained back. For example, 50% camber compensation means that for the amount of positive camber generated by body roll, half of it is gained back as negative camber from suspension travel. With the two variables of static camber and camber compensation (front and rear), I am able to abstractly define the full camber behavior of the vehicle.
The general design trade-off here is the compromise between cornering and longitudinal performance. Having small amounts of camber gain means that the camber changes very little under braking and acceleration, helping improve grip in those situations. However, that subsequently means that the positive camber gain in cornering is left unaddressed, hurting lateral acceleration performance. For high camber gain, the opposite is true.
To make sure these interactions were being captured, I started with a gut check design exploration:
This braking example demonstrates that increasing camber gain or static camber will decrease overall braking performance
This cornering example demonstrates that the more camber gain is present, the less static camber is required to achieve maximum cornering performance
With a gut check complete, it was time to move on to actual optimization. The goal here was not to select a final camber configuration just yet, but rather identify a target range of camber compensation to work with. This allowed flexibility for other downstream considerations such as steer-camber and desired vehicle balance.
A series of surface response plots were generated to evaluate the effects of static camber and camber compensation on FSAE dynamic event performance. To simplify analysis, front and rear values were kept equal to each other, and steer camber effects were neglected. Again, the goal was simply to find a target range as a starting point.
The Accel results track pretty well with the design exploration above. For maximum ideal rear end grip in a straight line, one would want zero static camber and zero camber change in squat.
Similarly, the Skid pad results track nicely as well. It is clear to see here the desire to have high static camber and low compensation, vice versa, or somewhere in between.
The Autocross and Endurance Event results show a slightly more complex relationship, due to the nature of both lateral and longitudinal performance requirements.
Finally, the total points prediction for all 4 dynamic events is shown below:
For the autocross and endurance event, as well as the total score, higher rates of camber compensation yielded higher overall points hauls. This suggests that, in suspension design, favoring cornering performance over braking and acceleration will have a greater effect on lap time. This makes intuitive sense for power-limited FSAE cars running on autocross courses with low speeds and small braking zones. It is interesting to note that the higher the camber compensation, the higher sensitivity to static camber.
Based on these results, I selected a target range of between 60 to 80 percent camber compensation, with low static camber (0.25-.5 deg). Front and rear values would ultimately vary based on desired balance, steer camber, etc.
Steering and Kingpin Axis
The next design section to tackle was the kingpin axis. Unlike camber curves, this one is a little trickier because there are more opposing criteria that you are trying to achieve. KPA affects not only the performance due to weight jacking and steer camber effects, but also the balance and force feedback to the driver. With that in mind, my primary design criteria were the following, in order of highest to lowest priority:
1) Control the limit approaching behavior of self returning steer moment (MZ) in order to consistently communicate to the driver that they are approaching the limit. The baseline target is for the returning moment to peak at around 70 percent of where peak lateral force occurs, to allow the drivers enough bandwidth to anticipate the limit.
2) Manage the steer camber and jacking characteristics so that both vehicle performance (lateral acceleration) and vehicle balance (understeer gradient) is consistent across the full operating range.
3) Achieve a target max steering wheel force of 25 pounds (based on previous driver preference data)
4) Maximize outright lateral acceleration performance.
Visualization of the variables being explored.
Source: Jambukar S., Sujatha C. (2020) Effects of Kingpin and Caster Offset on Braking Stability of Long Wheelbase Bus. In: Biswal B., Sarkar B., Mahanta P. (eds) Advances in Mechanical Engineering. Lecture Notes in Mechanical Engineering. Springer, Singapore
Before moving forward, it's important to know the characteristics of the steering rack used. Our team uses the NARRCO FSAE rack due to it's low cost and low weight
The first step is always to explore the design space, so I started with some simple spreadsheets to visualize the relationship between steering kinematics and resultant driver force
Steering force solver spreadsheet
KPI and Caster trade-off visualization
From there, the next step was to explore the MZ saturation behavior. I did this using a single axle cornering model, applying a total slip angle and solving for quasi-steady state equilibrium.
This enabled me to narrow in on the combination of settings required to achieve my MZ saturation target near the cornering limit (see bolded table entry)
However, with driver steering force there is another important consideration to make. If the majority of returning moment comes from the mechanical (caster) trail, there can be a loss of resolution, and natural MZ fall-off from the tire will pass unnoticed by the driver. So it's important to keep track of this.
After playing around with the variables at my disposal and exploring their interactions, I narrowed down on three primary design candidates to consider further:
To evaluate these three concepts, I used the same vehicle model as for the camber curve analysis. Each concept was evaluated for a range of turn radii and camber compensation values. Outputs of interest were peak lateral acceleration and understeer gradient. For lateral acceleration, I took the average value across all turn radii, with the goal of maximizing that value. For understeer gradient, I took the average slope across turn radiuses, with the objective to minimize the slope (i.e. minimize UG variation).
The final selected concept was option #3. The lower caster angle improved the resolution of the tire self-returning moment, as well as the consistency of the understeer gradient due to reduced steer-camber. In addition, peak MZ was closer to the saturation target than the other 2 configurations. The only downside is that peak steering wheel force was reduced, but this was considered a favorable trade-off for the improvements in drive-ability and consistency.
The final step was to pair this KPA configuration to a final camber configuration. This was largely a manual and iterative process. Ultimately I converged on a camber compensation of 25%/65% front/rear, and static camber of 0.5/0.25 degrees front/rear.
I wish I could say I had a more structured way of doing this, but ultimately this worked well for me.
Before moving on with control arm kinematics, I tackled selecting the target Ackermann. The geometric nature of navigating a corner means that the inside front wheel takes a shorter path radius than the outside wheel, and therefore requires a smaller steering angle. If a car has 100% Ackermann, then for any turn the difference in steer angle from the inside and outside will allow both wheels to navigate the corner without any sliding. In a race car, however, things get more complicated.
Source: “Ackermann Geometry.” Wikipedia, en.wikipedia.org/wiki/Ackermann_steering_geometry.
The slip angle at which peak lateral force is generated on a tire can change with the normal load on said tire. As the vehicle goes through a corner, you want to be able to maximize the force generated by both front tires, but each one may have a different slip angle requirement. This can be analyzed by tracing the "Line of Peaks" as you increase load on the tire. The amount of Ackermann you use can be tuned to achieve the peak slip angle difference between the inside and outside tire for a range of turn radii.
This plot compares the steering angle difference (inside to outside) for 100% Ackermann, an idealized line of peaks optimization, and the final configuration of 60% Ackermann. Note how the relative steer angles are optimized for high speed corners, while the relatively higher Ackermann at low speed will help the vehicle navigate tight corners where geometric effects are more important.
Instant Center Locations
The kinematic centers of the suspension linkages control the force coupling characteristics of the suspension. Any force acting on the tire can be broken into a component perpendicular to the virtual swing arm, which actuates the linkage and is fed into the spring/damper assembly, and a component parallel to the swing arm which is reacted through the suspension links into the chassis. High roll and pitch center placement can reduce the pitch/roll moment arm, improving body control. It will also generate a higher amount of geometric load transfer, which is faster than elastic load transfer from the springs. This, however, comes at the expense of increased chassis jacking forces, tire scrub due to track/wheelbase variation, and overall suspension ride quality. Force coupling and instant centers truly represent a 3-D problem, but to simplify the analysis I treated the side view and front view geometry as two separate cases and joined them together at the end.
For roll centers, I wanted to explore the effects of absolute height on the lateral response time to enter a corner, comparing the trade-off to the increased jacking forces. After that, I wanted to explore the effect of tuning the front/rear heights relative to each other to minimize the lag/phasing between the front and rear response in corner entry. To explore these effects I generated step steer plots using a model described here:
Below are a few example results:
Some things worth paying attention to, beyond raw lateral acceleration, is the relative build of of slip angle and roll angle between the front and rear
Another interesting relationship to explore is the effect on the loading of the tires, and subsequent effect on transient grip capability.
As expected, raising the roll center height did improve the response times for lateral load transfer, and subsequently, lateral acceleration. This was the case for both rise and settling time. However, the magnitude of effects were small - raising the roll centers by 6 inches only yielded a 2.5% improvement in response time.
For such a drastic change, the effect on jacking forces to the chassis and suspension were comparatively more severe, in some cases increasing link loading up to 12%! This was not considered a worthy trade-off, so the decision was made to keep the roll centers low.
Moving on to explore relative RC heights, I kept the front height constant at 0.5 inches, and repeated the same analyses while varying only the rear height.
Typically, since steering inputs are made at the front, there is a slight lag in the rear axle response as the vehicle begins to yaw in the corner. As can be seen in this figure, raising the rear roll center only (therefore increasing the geometric, instantaneous weight transfer) can help load up the rear axle faster. For a high enough roll center, the rear (orange curve) may even surpass the front (blue).
Ultimately, this did not actually translate to faster rear end response. This is because the rear can't generate a slip angle until the front end induces initial vehicle yaw. This plot displays the rise time for slip angle buildup with increasing roll center - you can see that raising rear roll center actually only hurts the rear end response.
Subsequently, total lateral acceleration response was significantly compromised by raising the rear center only. With this in mind, the final decision was to keep both front and rear roll center heights close to each other, and at a low height (< 1")
After roll centers were decided, the next step was to select the side view instant center locations, which control the suspension force-coupling relationship in pitch motions. In roll, a dedicated spring element meant that I had more freedom in selecting a roll center height. However, the springs that control the pitch behavior of the vehicle are also responsible for the heave control, meaning pitch center placement has more downstream effects. Ultimately, this portion of the design was actually carried out in the suspension platform control development, so I will direct you to that page to learn more about that part of the process.
Pickup Point Optimization
With target values selected for camber rates, kingpin axis, and instant center locations, the final step was to design a set of 3D points that could achieve those characteristics, which is a surprisingly flexible endeavor. There is an infinite amount of configurations that exist that can achieve a given combination of targets, and not all are created equal. Further constraints and criteria were identified to guide the process:
1) Outboard points must fit within specified bounding boxes for wheel, brakes, packaging and upright clearance
2) Suspension configurations should minimize lateral roll center migration in roll
3) Suspension configurations should minimize loading through the suspension links
4) Suspension configurations should integrate with the frame and other nearby components to create lightweight and stiff packaging solutions
Packaging can get pretty tight inside a 10" wheel!
To start things off, I needed a tool that can enable me to efficiently generate feasible suspension configurations. To do this I created a SOLIDWORKS model that constructs a complete geometry defined purely off of equations. When I open up the equations table, I can update any field I like, such as the front view swing arm length, and the geometry automatically updates.
In order to evaluate roll center migration, I used a MATLAB tool developed by my friend Derek Moore that calculates and plots vertical and lateral roll center migration for a variety of scenarios. It is a great tool! To learn more about it, and the rest of his great work, be sure to visit his website here.
To evaluate suspension loads, a series of worst-case load cases were created using data from the lap time simulation:
To estimate link loads, suspension links were approximated as truss members and solved for steady state forces and moments. While not perfect, this enables an easy side to side comparison to evaluate the effect of moving pickup points on link loading. A few examples are summarized here:
With these tools, I was able to collect a portfolio of various geometry solutions that achieved all of my kinematic targets, minimized link loading, and minimized roll center migration.
This catalog was submitted to the frame designers, who would rate and rank them, returning the catalog to me with comments. An example summary is pictured below. This kicked off an iterative process that helped us converge on a final control arm geometry that achieves all of my targets while integrating with the frame effectively.
Final Design
The final suspension characteristics are summarized in the table below, followed by renderings of the final assembly:
Appendix
Along the process of pickup point selection, many compromises were made in order to improve packaging, manufacturability, or load paths. Pictured below are some snippets from my design presentation outlining some of the major trade-off decisions made.
