Milliken Race Car Vehicle Dynamics

Understanding Steady-State Handling Analysis for Race Cars

Getting Started With Milliken Race Car Vehicle Dynamics

The approach most people refer to when they say "Milliken" comes from the methodology in Race Car Vehicle Dynamics by Ray and Bill Milliken. It is a steady-state handling analysis technique that linearizes tire behavior around a given operating point and uses that to predict understeer, load transfer, and grip potential. The method does not require expensive simulation software. A spreadsheet and some basic tire test data get you most of the way there. The core idea is simpler than it sounds. You take a car, determine how much vertical load each tire carries during cornering, look up the corresponding lateral force from tire data, and calculate what slip angle the tire needs to produce that force. Repeat across a range of lateral accelerations and you have a handle on the car's understeer gradient, roll distribution, and how your suspension choices affect balance. This is steady-state analysis, so it tells you what the car does at constant radius and constant speed. It does not tell you what happens during transition. I spent probably five years working through these calculations by hand before most of it got packaged into software. The reason I still go back to the spreadsheet method occasionally is that it forces you to understand what each parameter actually means. Software gives you a number. The spreadsheet makes you earn it.

What You Need Before You Start

You need a few inputs that most teams already have sitting around somewhere. The car's weight, wheelbase, track widths front and rear, static weight distribution, center of gravity height, roll center heights front and rear, and the roll stiffness distribution. You also need tire data. Ideally this comes from a tire tester at a place like Mooneyham or R&D Tire in the US. At minimum you need cornering stiffness values and peak lateral force data at several vertical loads. The cornering stiffness is the slope of the lateral force versus slip angle curve near zero slip angle. It is usually reported in N/rad or lb/deg. The Milliken method relies heavily on this value because the whole linearization is built on it. If your cornering stiffness numbers are off by even 10 percent, your predicted understeer gradient will be wrong. I learned that the hard way on a Formula SAE car where we used manufacturer-provided stiffness values instead of testing the actual tires we were running. The predicted balance was completely different from what the car actually did on track.

The Step-by-Step Analysis

Start with the static load on each tire. Divide the car weight by four if the distribution is even, but it rarely is. Multiply each axle's static load by the fraction of total weight on that axle to get the front and rear total vertical loads. During cornering, weight transfers laterally across each axle. The amount of lateral load transfer equals the lateral acceleration times the center of gravity height divided by the track width. Multiply that by the gravity constant to convert to g's if needed. The inside tire loses that amount and the outside tire gains it. Do this separately for the front and rear axles using their respective track widths. Next you need the roll-related load transfer, which is the portion of lateral transfer carried by the roll springs rather than the geometric path through the suspension. This is calculated as the lateral acceleration times the roll center height divided by the track width. The difference between total lateral transfer and roll transfer is the direct load transfer through the suspension geometry. This distinction matters because it determines how much load transfer is affected by anti-roll bar changes versus changes in suspension kinematics.

Here is where people commonly make mistakes. The front and rear load transfer do not simply divide the total car load transfer in proportion to axle distance. The roll center heights and roll stiffness distribution control how much of the load transfer each axle handles through roll. A car with a high front roll center and soft front anti-roll bar will behave very differently from one with the same total roll stiffness but distributed differently between front and rear. After you have the individual tire loads, go to your tire data and find the cornering stiffness at each of those loads. Cornering stiffness decreases as vertical load increases, roughly following a square root relationship. If you do not have data at your exact loads, interpolate. Then calculate the slip angle required at each tire to produce the lateral force at that load level. Lateral force equals cornering stiffness times slip angle in the linear region. The understeer gradient is simply the front slip angle minus the rear slip angle at a given lateral acceleration, divided by the lateral acceleration in g's. A positive value means the car understeers. Negative means it oversteers. Zero is neutral. The gradient changes with lateral acceleration because tire stiffness is not perfectly linear, so you should calculate it at multiple points, not just at low acceleration.

The Milliken method also gives you the slip stiffness ratio, which is the front slip angle divided by the rear slip angle at the same lateral acceleration. This is useful for understanding whether the car is front or rear heavy in terms of slip angle demand. A ratio greater than one means the front tires are working harder relative to the rear in terms of slip angle, which generally indicates understeer bias.

A Real Problem I Encountered

On a GT car we were setting up for a tracks that had significant camber changes through the main corners, I ran into a situation where the steady-state Milliken predictions and the actual car behavior diverged noticeably at higher lateral accelerations. The car was tracking the predicted understeer gradient at 0.8 g but became significantly more understeery at 1.2 g. The issue was tire temperature gradients across the contact patch. The outer shoulder of the outside tire was running 30 degrees hotter than the inner shoulder, which effectively reduced the usable contact patch and shifted the cornering stiffness down faster than the linear model predicted. The workaround was not to change the analysis method but to add a correction factor. I took the measured tire pressures and temperatures from the tires after a fast lap, compared the live data to the cold test data, and applied a stiffness reduction factor based on the observed temperature differential. It was approximately a 15 percent reduction in effective cornering stiffness at the high-load condition. That brought the prediction within about 5 percent of the actual behavior. Not perfect, but good enough for setup decisions. If you are doing this analysis for a car that runs very hot tires or very uneven tire temperatures, the standard Milliken approach will undershoot the understeer at high lateral acceleration. You need to account for that somehow, either through temperature-corrected tire data or by running transient simulation to complement the steady-state work.

Common Pitfalls

The biggest mistake I see is treating the understeer gradient as a constant. It is not. It changes with lateral acceleration, with tire temperature, with wear, and with compound. Reporting a single number like "the car has a 3 degree/g understeer gradient" is meaningless without specifying the condition. Always report it at a specific lateral acceleration level. Another mistake is ignoring the difference between kinematic roll center and instantaneous roll center. The Milliken method uses the kinematic roll center for load transfer calculations, but the actual load path through the suspension depends on the instantaneous center geometry. For most road course cars the difference is small. For cars with high scrub radius or unusual suspension layouts, it can be significant. I have seen teams miss a full degree of understeer this way on a car with a very high front roll center. A third issue is the assumption that the tires are operating in their linear region. The Milliken method works best between 0.3 and 0.8 g of lateral acceleration. Below 0.3 g the tire behavior is so linear that the analysis adds little value. Above 0.8 g the tires are entering the nonlinear region and the linearization breaks down. If you need to predict behavior at the limit, you need a nonlinear tire model or actual tire test data across the full slip angle range.

What the Method Cannot Do

The steady-state Milliken approach has clear limitations. It cannot predict transient response. It does not account for aerodynamic downforce, which for most race cars becomes the dominant vertical load at speed. It assumes the tire operates on a single plane and does not consider camber thrust effects unless you add them separately. It also cannot model self-aligning torque or the relationship between aligning torque and slip angle, which matters for steering feel and feedback. For aerodynamic cars, you need to add the aero downforce to each tire's vertical load before doing the analysis. The downforce distribution is rarely the same as the weight distribution, so you cannot simply use the static numbers. You need aero balance data from wind tunnel testing or CFD, or you can estimate it from drag and downforce measurements on track.

Software Options

There are several tools that implement the Milliken methodology. The original VeDYNA software from VeSyMA does a complete steady-state and transient analysis based on the same principles. It is commercial and fairly expensive. For teams that want something lighter, there is the free TUNE software from Milliken Research, which is a direct implementation of the steady-state handling analysis. It is not the most polished interface but it gets the calculations right and the source code is available for review. If you are working in an academic environment, SimuTools has a vehicle dynamics module that includes the Milliken steady-state analysis. It is less comprehensive than VeDYNA but sufficient for learning and preliminary setup work. For Formula SAE and similar student competitions, these tools are commonly used and the results are reliable enough for competition-level decisions. Some teams build their own spreadsheet implementations. This is actually the best way to learn the method because you have to enter every equation yourself. I would recommend starting with a spreadsheet before moving to any packaged software. Once you understand the equations, the software becomes a calculator instead of a black box.

Practical Tips

Measure your roll centers properly. Many teams use CAD models or approximate formulas, but the actual roll center height can vary significantly through the suspension travel, especially on cars with double wishbone suspensions. A change of half an inch in roll center height can shift the understeer gradient by a degree or more. Use actual measured positions, not theoretical ones. Test your tires at the loads they will see in the car. If your car puts 1200 lbs on each front tire during cornering, do not use tire data from a 800 lb load point and extrapolate. The cornering stiffness at 1200 lbs could be 20 percent lower than at 800 lbs, and that difference compounds through the entire analysis. Track your weight transfer with load cells if possible. Having actual measured load transfer data lets you validate your calculations and spot issues like suspension binding or unwanted compliance that the theory does not account for. A discrepancy between calculated and measured load transfer is usually a sign that something in the suspension is not behaving as modeled.

The analysis takes about 30 minutes to set up once you have all the data, and another 15 minutes per setup change you want to evaluate. That is the main advantage over full simulation, which can take hours per iteration. For quick comparison of anti-roll bar changes, spring rate changes, or camber adjustments, the Milliken steady-state method is fast and accurate enough to guide decisions.

When to Move Beyond This Method

If your car generates significant aero load, if you need to predict transient behavior like throttle lift-off oversteer, or if you are working with a tire that has a very nonlinear cornering stiffness curve, you will outgrow the basic Milliken approach. In those cases, transition to a transient simulation tool that uses a Pacejka tire model or direct tire test data. The steady-state analysis still has value as a sanity check even in those situations, but it is not sufficient on its own. The fundamental principles do not change between the steady-state and transient methods. Understanding what the Milliken analysis is telling you about your car's balance will make you a better user of any simulation tool. The spreadsheet will not replace advanced simulation, but it will make you understand what the simulation is actually doing instead of treating it as an opaque calculation engine.