The Science of …Differentials and Rear-End Gears

Michael S. Adams asks: “I hear that having the wrong stagger can ruin the handling in the corners, so I would think you must want to have a very limited slip differential. Is that correct or am I way off? How does that affect the car on straightaways? What differences are here in the drive train (and tires and suspension for that matter) between a car set up for an oval track and a car set up for a road course?”

Thanks for the question, Michael. Some background is probably going to be helpful for folks not as familiar with tire setups as you are. I also enlisted the help of Josh Browne, Chief Race Engineer for Red Bull Racing, to provide some expert information.

Let’s start by defining stagger. If you look at the path the inside tires take and the path the outside tires take when a car turns, you can see that the outside tires have to travel further than the inside tires, as I’ve shown in the picture below.

The outer wheels thus have to rotate faster than the inner wheels. Anyone who has been in a matching band is familiar with this problem: If you are on the outside of a row, you have to walk faster around a turn to keep up with the rest of your row. One way around this problem for race cars is for the outside tires to have a slightly larger circumference than the inside tires. At Bristol, the left-side circumference was 87.4 inches and the right-side circumference was 88.6 in. At Daytona, the circumferences were 87.9 inches on the left-hand side and 88.6 inches on the right-hand side. The tires on both sides had an 88.6 inch circumference at the road courses last year. Since all the tire sets have essentially the same amount of stagger (as determined by Goodyear), it really doesn’t allow the teams to use tire stagger as a variable. Josh notes:

“Regarding stagger – for the most part, stagger is a constant established by Goodyear. At the start of the race event, the tire specialist measures the circumference of each of the tires assigned to the team by Goodyear. There are very small (millimeters) differences between individual tires, so small changes in stagger can be achieved, but these are very minor effects.”

These small differences in the tires are because the tire-making process relies on molds, which prodcues small tire-to-tire variations; however, as Josh points out, the variations are really very small.

As Michael noted in his question, there are significant problems with the tires having to rotate at different speeds in terms of handling, especially when it comes to the rear tires. The front and rear suspensions are very different, because the two sets of wheels serve different functions. The front wheels steer and the rear wheels propel the car.

A NASCAR car has an independent front suspension–the two wheels operate independently of each other. The rear suspension, however, is a “solid axle”. In principle, the rear axle links the rear wheels so that they have to rotate at exactly the same rate. The problem is that this causes a certain amount of tension to build up because, although the right-side wheel is larger in circumference, the circumferential difference isn’t enough to compensate entirely for the greater distance it must travel. The only way to relieve the tension is for the wheel to skip or for something to break. Neither are very good alternatives. The solution is to make the ‘solid axle’ not quite solid. You get around this by using a differential, which is a complex arrangement of gears that allows the inside and outside rear wheels to rotate at slightly different speeds.

The rear-end gear is a little complicated because it serves multiple functions. Up until the point where the drivetrain meets the rear-end gear, the axis of rotational motion has been along the length of the car. The rotation has to be switched so that it is parallel to the axle and that’s the job of the rear-end gear. The rear-end gear is a ring-and-pinion gear. The pinion gear is the smaller of the two and sits at a right angle to the ring gear, which is the larger gear that has teeth on its face. The teeth are cut helically, to allow the gears to mesh smoothly together. (There were a number of rear gears broken at the Martinsville race.)

The differential connects to either side of the axle, as shown below, which is adapted from the How Stuff Works website. Look at their animation, as it is a great help in understanding how the differential works. In this figure, the pinion gear comes in from the right front of the drawing. As the pinion gear rotates, the ring gear rotates and the differential rotates with it. As long as the two wheels rotate at the same rate (for example, when the car is going straight), the differential doesn’t do anything.

The two outer gears (shown in red in the illustration above) can rotate in opposite directions and are connected the axles for the two rear wheels by the beige gears. When the car is going straight, the two gears rotate at the same speed; however, when the car turns, the gears for the outer wheel can rotate faster than the ring gear and the inside wheel. This allows the outer wheel to rotate slightly faster than the inner wheel. Problem solved.

Or not. The problem is that a standard differential transmits only as much torque to both wheels as the wheel with the least amount of traction can handle. If one wheel starts to slip, both wheels get less torque. In racing, you want some slip in your wheels, and if you hit a bump or one tire starts to slip, you don’t want the other tire to lose traction.  

A locking differential allows the two wheels to be locked together under certain conditions, but to unlock under other conditions. Advanced locking differentials may have viscous couplings, electrical or electromechanical mechanisms that control how well coupled the two axles are. Despite being really neat ways of allowing a little slip (but not too much), none of these types of locking differentials are allowed by NASCAR. NASCAR restricts teams to a particular type of locking differential that uses a mechanical means of keeping the two wheels. The Detroit Locker has a mechanical sensor that locks both wheels together as if they were on a solid shaft when the car is going straight, but allows them to turn at different speeds when wheel slip occurs (as happens during cornering).

The locker locks both wheels together when they are rotating at the same speed. When the car turns left and the right wheel is forced to rotate faster than the center of the differential, the locker ‘unlocks’ the right wheel as long as it continues to rotate faster than the center. The locking differential is actually applying drive torque through the inner wheel as the car goes around the corners. An unlocked differential provides the same torque to each wheel, but allows them to rotate at different rates. A locked differential forces each wheel to rotate at the same rate.

This locking process, which is controlled entirely by mechanical means, is crude and often very loud. The car basically switches between the drive coming from one and two wheels as the car corners, which can make the car handling unpredictable. Playing with the throttle (often done to help the car turn) can cause an unpredictable switch. They also make a loud clicking noise when engaging or disengaging. Again, here’s Josh’s take on the differential.

Regarding the “diff”, the only style that can be run per the rules is a “detroit locker” type.  The locking & unlocking of this “diff” primarily depends on the path curvature and stagger.  Small differences in rear slip ratio can be achieved with tire pressure adjustments, but you can assume the rear slip ratios to be constant in this case.  For most of the race tracks, the rear is “locked” on the straights, and unlocks either the left rear or right rear corner during deceleration.  Upon throttle application, the rear eventually locks back up.  The driver can, to a small degree, control the locker unlock/lock event with adjustments in his/her chosen path curvature, and the rate of throttle application.

During testing, we can run torque sensors on the axles, or, better yet, full-blown wheel-force-transducers.  This instrumentation accurately captures the locker behavior.  There are a few tracks at which the locker doesn’t unlock at all – and then every few laps, the rear will “pop”, as the rear winds up and skips a tooth.  If you watch for it, sometimes you can see it happen exiting the corners in qualifying (at a few tracks).

Josh’s answer helped me understand something I observed at Martinsville last year. When Elliott Sadler qualified (Josh was his crew chief at the time), we heard a very loud pop as Elliott was coming out of turn 2 on his first lap. Now I know that this was the differential unlocking. No doubt it also helped explain why the No. 19 qualified somewhere in the mid-twenties at Martinsville last year.

UPDATE 4/12/08: Josh tells me that the noise was the locker skipping a tooth, not unlocking.