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Mark Ortiz Automotive is a chassis consulting service primarily serving oval track and road racers. This newsletter is a free service intended to benefit racers and enthusiasts by offering useful insights into chassis engineering and answers to questions. Readers may mail questions to: 155 Wankel Dr., Kannapolis, NC 28083-8200; submit questions by phone at 704-933-8876; or submit questions by e-mail to: firstname.lastname@example.org. Readers are invited to subscribe to this newsletter by e-mail. Just e-mail me and request to be added to the list.
FRONT TO REAR TRACK RATIO
I've been building a wide-body Porsche 914 for a few years now, and had always assumed that the wider rear track would stabilize the rear of the car better (especially since I now have a somewhat heavy sprint car engine in the back). But lately I've been questioning whether that's correct. I found that F3 chassis currently in use run a wider front track. And on a trip to Laguna Seca Raceway I noticed a Ferrari F430 which appeared to have a wider front track as well. I even went to speak with some local FSAE students to get some answers. But the only thing they said is, "A narrower rear track gets around the cones faster." That was not the answer I was looking for.
Published lateral acceleration numbers from many of the current 3-wheeled chassis are impressive. But I cannot find any material explaining why.
I am in desperate need of someone to explain the chassis dynamics involved.
First, letís make sure we define what we mean by track width. This is the transverse distance between the center planes of the wheels. I sometimes encounter people who think it means the overall width at the outside of the tires instead. Engineers generally use the term to mean the center-to-center dimension.
Exactly what is the center can have some nuances, if we are trying to really be accurate. When the wheels have some camber, we may mean the distance between the contact patch centers, or more properly, load centroids. This is probably the most realistic and accurate, but those points are a bit hard to ascertain in the design phase. The most convenient approximation to this is the intersection of the wheel center plane and the ground, which is not quite the same point as the load centroid, but is easier to define in the design phase. Or, we can say itís the distance between the wheel midpoints, at hub height. This is roughly equivalent to taking the other two values at zero camber, and treating that as design track width. If we are just after general information, these subtleties arenít necessarily important.
There is no perfect ratio of front to rear track width, but there are practical considerations that may lead us one way or another. In the case of your V8-powered Porsche 914, you are probably keeping the suspension as nearly stock as practicable. The car will be tail-heavy, so it needs more tire in the rear. The only way you can add a lot of tire and rim width without major chassis fabrication and suspension redesign is to add it on the outboard side of the wheels, and flare the fenders. At the front, you donít need to add as much tire, and you want to keep the scrub radius from getting too big, so the front track wonít end up growing as much.
The car will be okay this way, although I think youíll need to make a conscious effort to apex wide on tight turns to avoid clouting curbs with your inside rear. I have a street car thatís wider at the back myself, and I have to constantly remind myself that the rear is wider than the part in my field of view. The car handles well, but this aspect requires some driver adaptation.
As to whether the car gets tighter or looser if you narrow or widen one end relative to the other, in most cases it will change, for various reasons. However, the factors involved vary from one case to the next, and we canít reduce the whole matter to a blanket generalization, even regarding the general direction of the behavior change.
If we are designing a race car from scratch, we arenít constrained by a need to preserve existing structure. For road course or oval track applications, we generally will be constrained by limits on tire and/or rim width, perhaps even a spec tire, and a limit on overall width. For wide, high-speed tracks, usually the widest car will be fastest, because there will be less lateral load transfer and we will be working the four tires more nearly evenly. In most rear-engined cars, we will have wider tires at the rear than at the front. If we have the same overall width limit at both ends, and the rear tires are wider, we need to have the track narrower at the rear than at the front, simply to take full advantage of the width limit at both ends.
In Formula SAE, and for SCCA solo events and other autocross events, the constraints are somewhat different. The turns are tight, the track is narrow, and some of the turns are defined only by the need to clear cones at the inside Ė slaloms are an obvious example. There is no maximum width limit in FSAE. The event itself prevents us from wanting a wide car. In tight turns, the rear tires track inside the front ones, even when there are significant slip angles. Not surprisingly, the car has a tendency to collect cones with the inside rear. It is consequently common to make the car about 4 inches narrower across the rear than across the front. The rules actually impose a limit on the track ratio: the narrower track canít be less than 75% of the wider. Most teams donít get anywhere near violating that rule.
There are certain cases in which we can readily predict whether adding rear track width will add understeer or oversteer. If we have a live axle, and we leave the springs alone and add track width by moving the tires out, the car should get tighter (more understeer). The car will have the same amount of roll, the same angular roll resistance at both ends, the same amount of load transfer at the front tires, and the same amount of load transfer at the rear springs, but less load transfer at the rear
tires, because the rear roll-resisting moment will be taken out over a wider base. The total load transfer will be less, and the reduction will all come at the rear. That should make the rear stick better compared to the front. Of course, the car can always be re-balanced using other changes.
Another case we might consider would be to increase the track width at one end, while keeping the wheel rates in roll unchanged.
I probably should spell out exactly what I mean by the wheel rate in roll, as I have encountered some confusion on this from various quarters. Wheel rate in roll, as I use the expression, is the rate of elastic change in wheel load with respect to linear suspension displacement, when the two wheels of a front or rear pair each move the same amount in opposite directions Ė in English units, the pounds of load change per wheel when one side compresses one inch and the other side extends one inch. This relates to the angular roll resistance as follows:
Kφ = Ĺ * Kroll * t2 * π/180
Kroll = (2 * Kφ) / (t2 * π/180) = 360Kφ / πt2
Or, approximating 180/π to three significant figures:
Kφ = Ĺ * Kroll * t2 / 57.3
Kroll = 2 * 57.3 * Kφ / t2
Kφ = angular roll resistance, lb-in/deg
Kroll = linear wheel rate in the roll mode, lb/in
t = track width, inches
In the case mentioned above, where we leave the springs alone on a beam axle and move the wheels out, we are increasing t, leaving Kφ unchanged and decreasing Kroll. Or at least Kφ remains unchanged if we are ignoring effects of tire compliance.
Now, let's take a simple case where all roll resistance is elastic, the c.g. is in the middle of the car, and Kφ, Kroll, and t are all equal at both ends. Then suppose we increase t at the rear by a factor of two, leaving the front end unchanged, and leaving Kroll unchanged at all wheels. This is a ridiculously exaggerated example of course, but the numbers are easy to work with in your head.
At the rear, Kφ increases by a factor of four. The total of Kφ at the front and Kφ at the rear increases by a factor of 5/2, or 2.5. Roll at a given lateral acceleration decreases by a factor of 2.5. Total load transfer at a given lateral acceleration is 2/3 as great, because the mean track width is 3/2 as great.
The rear suspension is now providing 80% of the roll resistance instead of 50%, but doing it with tires spaced twice as far apart. The load transfer at the front is (2/3)*(20/50) = 26.67% of what it was before we widened the rear track, and the load transfer at the rear is (2/3)*(80/50) = 106.67% of what it was.
As a check, (106.67% * 50%) + (26.67% * 50%) = 66.67%. That is, the sum of the new rear load transfer and the new front load transfer is 2/3 of the old total load transfer, as it should be with a 3/2 wider mean track.
Note that in this case we actually increased load transfer at the rear a bit when we widened the track, and dramatically decreased load transfer at the front. That should add oversteer rather than understeer.
This example is particularly relevant to go-karts, where the suspension mainly consists of the tires, and consequently wheel rates in all modes tend to remain constant as we change other things, provided we use the same tires and inflation pressures. Kart racers tend to find that the kart gets freer (understeers less/oversteers more) when the front track is narrower compared to the rear, although it depends to some extent on the turn in question and on how we widen or narrow the front track. If we widen the front track by spacing the wheels out on the spindles, we increase caster jacking with steer. This can mean that the vehicle is tighter in sweepers yet freer in tight turns.
So far, we have been analyzing effects of track width, and track width relationships, purely in terms of effects on load transfer. There are some other effects as well. If we have a locked rear, and we are road racing, meaning we can't use rear tire stagger, we generally have a greater tendency toward locked-axle push when we widen the rear track. With a limited-slip diff, there is a similar effect, only more subdued. However, if rear load transfer increases, that can make it easier to unload the inside rear tire in hard cornering, reducing locked-axle push instead.
It should now be apparent why I say the whole business can't be reduced to a blanket generalization. However, with sufficient thought it is possible to understand the various factors at play.
Regarding 3-wheeled vehicles, these can produce similar lateral acceleration numbers to 4-wheelers, if they don't tip over. They can be made very light, so with suitable tires they can generate plenty of lateral g's.
The key to giving a trike good overturning resistance is to put the preponderance of the weight at the 2-wheeled end. Do that, and keep the c.g. low and the track wide, and you can compete with cars on the skidpad. Put the heavy bits at the one-wheeled end, and you are creating job security in the legal profession.
In pure lateral acceleration, all lateral load transfer occurs at the 2-wheeled end. This will generally lead us to want larger tires at the 2-wheeled end, even though there are twice as many of them. The outside one will be doing most of the work.
The July-August 2005 issue of the Newsletter is entirely about trikes.