In racecar vehicle dynamics, each known parameter is good or bad depending on execution. Engineers understand what is ideal or undesired, with full knowledge that everything is a compromise. Weight transfer is one parameter that is minimized – to aim for even loading on all four tires; resulting in maximum grip during cornering. In this paper, that issue is discussed with a focus on ride rates, roll rates and simple tire data analysis for a Formula SAE race car. Certain aspects covered will be from the suspension design experience of the 2011 University of California: Merced Formula SAE car. Included are calculations for the suspension design which was modified for this paper.
Formula SAE is a university level student competition setup by the Society of Automotive Engineers (SAE). Its goal is to promote top-notch engineering and careers in the automotive field. The competition requires students to handle all aspects of building a small marketable formula style racecar. This consists of design/analysis, building, testing, and all necessary documentations (cost, parts list, build log, etc).
Suspension systems in cars are setup to absorb shock from the road. In an extreme case, a stiff suspension would act similar to a rigid body. The wheels would skip and hop over imperfections in the road, creating an unpleasant and unstable ride characteristic. To fix this, ride rate of the un-sprung mass with respect to the ground would have to be zero. This is ideal but impossible to accomplish. Thus we want to run as soft of a spring rate as possible. Issue with an overly soft suspension is body roll, which becomes a major problem as we look into roll rates and its effect on overall tire grip.
Any book educating on vehicle dynamics or automotive design will start with tire discussions. Tires are the biggest factor in vehicle design as all forces acting on a car will in some way translate to or from the tires. Also, each application demands optimization of certain parameters (tire rate, peak grip, compound, etc). The key to optimum grip is for the tire’s contact patch to be perpendicular to the ground (0º camber) at all times. This insures that most of the tire is contacting the ground.
Figure 1 depicts forces on the tires; though there are many forces we will only discuss lateral and normal forces in this section.
Normally, the coefficient of friction (Cf) increases with greater vertical force but tires are the exact opposite. In the figures below we can see how a general tire behaves under load. For simplicity sake, a 3600lb vehicle with 50/50 weight distribution puts 900lbs of downward force on each tire. That puts us at a lateral grip handling of 1100lbs from figure 1; dividing the two values gives us 1.22g’s of cornering force.
Paralleling to figure 2, we see that Cf is 1.2. Now looking at a light sports car with a curb weight of under 2000lbs, each tire is loaded with 500lbs of force. This case we have lateral grip of 700 lbs and a coefficient of 1.4. Again, dividing it gives us cornering g-forces, this time we have 1.4g’s. As you can see, cornering force is due to Cf. These two graphs are linked using the basic frictional force equation. F_f = uF_n
Comparing the two, 1.4g is a much higher value and can definitely be felt when pushing a car to the tire’s limits.
Next up is tire rates – ratio comparing downward force to vertical displacement of the tire (K_T = F_z/deltaZ). Tire rate is solely affected by the side wall stiffness where a softer sidewall will have more vertical deflection. The significance of tire rate will be discussed in the next section.
Weight transfer occurs because lateral forces act through the CG point while the car rolls about its roll center (RC). This distance from CG to RC point creates a moment arm. You can imagine that the longer the moment arm, the greater this rolling force is.
Suspension systems attempt to provide constant contact between the ground and tire. Because of this, there is a rolling effect on the sprung mass; we deem this roll stiffness. The value is given by the roll rate formula: K(phi)_F/R = 1/2K_s(t_F/R)^2 where K_s = K_Spr(IR)^2.
The following are the principal weight transfer equations. With that, or (lateral weight transfer) and (longitudinal weight transfer) of a car can be easily found.
As stated, a softer suspension is desired to allow complete tire/ground contact during rough conditions. This practice is somewhat accepted in larger or pure luxury vehicles but become problematic elsewhere. Hitting a bump at high speeds can cause the car to oscillate erratically until the dampers can control and stop all the motion (dampers in this scenario are usually under-damped). When cornering even at low speeds, the sprung mass will roll about its roll axis (imaginary line connecting front and rear roll centers). Even during accelerating and braking, the car will shift to the front or rear respectively. Traveling or cornering too fast can cause the vehicle to tip over; an overly soft suspension will lower this threshold considerably. Additionally, the car will feel sluggish and unresponsive to operator’s inputs. Imagine having to accelerate quickly out of a dangerous situation, much of the power will be wasted in shifting weight towards the outside of your turn and to the rear before the car is able to drive away. This delay also causes a sluggish feeling. Thus a well-engineered suspension system is not only key in racing applications (where we want power directed down to the tires as quickly as possible), but also in everyday vehicles so drivers can feel confident when having to maneuver from a sticky situation.
Tires also add to the problem because they act as a non-linear spring (tire rate - K_T). A sample Hoosier tire designed for Formula SAE application has a tire rate of 600 lbs/in; hence adding to the weight transfer issue.
Scenario: A vehicle is cornering left while accelerating, the car’s sprung mass will roll about its roll axis towards the outside while also shifting weight to the rear axle pair. At this point, the outside rear tire is loaded much more than the other three. We know that higher loading translates to less cornering force and the inverse for less loading. We will look at a specific case in the next section.
The following calculations is modified for the simplicity of this paper from the suspension design for the Formula SAE race car from University of California: Merced
Refer to Appendix A for symbol definitions.
Front outside: = 282 - 74.53/2 + 91.69 = 336.425 lb
Front inside: = 282 - 74.53/2 – 91.69 = 153.045 lb
Rear outside: = 318 + 74.53/2 + 128.79 = 484.055 lb
Rear inside: = 318 + 74.53/2 – 128.79 = 226.475 lb
From the example scenario, our 600lb car has gone from a Cf of 1.74 in the front and 1.7 in the rear to the following:
FO – 1.55 FI – 1.73
RO – 1.40 RI – 1.58
The difference in Cf is 0.33 at the largest. This change due to weight transfer takes away from the overall grip of the car during turns. We can see here that the average rear grip is less than the average front grip; which gives the car an over-steering effect (easier to lose complete traction in the rear and spin out).
There are many ways to keep weight transfer from becoming an issue; this paper will overview several possible solutions. The obvious is reducing weight but that is usually not an easy thing to do. Easiest but usually overlooked is a performance alignment. One of the main reasons for an alignment is camber – most grip will come out of the tires when camber is set so the tires are perpendicular to the ground during a turn.
Rear toe is also a big factor because rear toe out will induce over-steering while toe in will help align the rear end and steady the car.
An easy fix is to increase the spring rates. This prevents excess lateral transfer but contradicts the beginning statement – the desire for soft springs. This is where optimizing comes into play.
From a design perspective, track width is the biggest factor. Increasing track width will increase roll stiffness and decrease lateral load transfer. Of course width must still be within reason (constrained by the width of the road or rules of the racing league). The roll center could also be raised to reduce the moment arm that was mentioned earlier. This will reduce roll but puts more lateral force on the car; which uses up some of the total possible lateral grip. This is where optimization brings it all together.
Heavier loads decrease a tire’s lateral grip potential. Lateral load transfers (from roll rates and tire rates) during cornering and accelerating/braking causes one tire to bear an even higher load; thus lowering cornering grip. In order to maintain maximum grip in a design perspective, we must do the following:
- Even loading of all 4 tires (reduce weight transfer)
- Reduce vertical load on the tires
Vehicle dynamics is still a mystery, even to those who have been designing suspension systems for their entire career. All aspects are accounted for and optimized depending on the application.
- W.F. Milliken and D.L. Milliken, ‘Race Car Vehicle Dynamics’, SAE International 1995
Appendix – A