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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: mortiz49@earthlink.net. Readers are invited to subscribe to this newsletter by e-mail. Just e-mail me and request to be added to the list.

ANOTHER TRIANGULATED FOUR-LINK AXLE

I am using a very light (aluminum) rear beam axle on a front wheel drive autocross car. I have read your discusssion of live rear axle location by Watts, Mumford and others but I am using a four link location system without a Panhard or Watts type system. I have the two lower links wide based at the front and meeting at the middle of the axle. They are about 3" off the ground and basically horizontal, thus lateral location of the axle is about 3 inches off the ground. My question is where is the roll center? I once thought it had to be at the level of the lower links, but perhaps not.

The upper links go straight back about 9 inches above the lower links and have provision to move the forward attachment points up and down to induce a bit of rear wheel steer in cornering (not currently doing this).

In this case, the rear axle roll axis, or rear axle axis of rotation in roll, is a line passing through the lower link centerline intersection, and parallel to the upper links. The roll center is the axle plane intercept of this line, as with any rear axle roll axis. If the lower links met at the exact center of the axle, the upper link angle would not change the roll center at all. If their attachment points to the axle are very close, so that their intersection is a short distance behind the axle, then upper link angle affects the roll center, but only a little. As the lower link centerline intersection gets further behind the axle, the upper link angle has more influence on the roll center height.

In all cases, if all the links are horizontal, the roll center is indeed at the height of the lower links, and the rear axle axis of rotation in roll is horizontal. True to this model, there is no roll steer. If we raise the front of the upper links, the rear axle axis of rotation slopes upward toward the front. Again true to this model, we have roll oversteer.

Anti-squat (in forward acceleration) is zero in all cases, because the rear wheels are not driven and can make no forward force. Anti-lift in braking is affected by the side-view link angles, however. If the lower links are horizontal and the upper links slope upward at the front, the rear suspension has pro-lift in braking. This is not necessarily a serious problem, but we can have some anti-lift, and still

have any desired roll steer and geometric anti-roll (roll center height), if we make the lower links slope upward toward the front.

LARGE VS. SMALL TIRE AND WHEEL DIAMETER

I thoroughly enjoyed your article "Tall vs Short sidewalls" in the April 2007 issue of Racecar Engineering magazine. In that article you discussed about the benefits of short sidewall tyres fitted on larger diameter rims. So, you explained how beneficial (or not) would be a different configuration of two wheels with exactly the same overall diameter. I would like here to enter another parameter to the discussion. What about two wheels with exactly the same tire sidewall characteristics but with different overall diameter? For example, let's say we have a 15" overall diameter (not rim diameter) wheel (rim + tire) and a 17" overall diameter wheel. And now let's assume that the sidewall characteristics of the two tires are exactly the same. Also the widths of the two tires are the same.

Which one do you think will be more beneficial to overall handling? It is quite sure that we will have two footprints of different shape (I am not sure if they will have a different area though).

I think that with the low wheel we will have a more round footprint shape, while with the tall wheel the print will be more long and narrow. Do you think this will have any effect to the centroid of the lateral force with consequences to pneumatic trail and aligning torque? What do you think?

15" and 17" are pretty small for tire outside diameters, but the question is really the same regardless of what hypothetical sizes we might posit.

As with tire width, we need to define our basis for comparison, and this is not as straightforward as one might suppose. Do we hold inflation pressure constant? Static deflection or tire vertical spring rate? Do we assume identical tread compound? Do we compare fully optimized cases, which probably implies different inflation pressures, static deflections, and tread compounds? One can make a rational case for any of these approaches. In an actual design or modification situation, availability of tires and compounds may constrain us. Since we are examining the principles here, we will look at the question from a variety of angles.

What happens if we hold tire pressure constant? In theory, the contact patch should stay the same area at a given load as the tire diameter increases, just as it should with increasing tire width. However, with a wider tire, the contact patch gets wider and shorter. With a larger diameter, the contact patch stays the same length and width. In both cases, the tire's static deflection decreases, and its vertical spring rate increases, as we add size. In both cases, tensile loads in the carcass increase. A tire is approximately round in section, so when its sectional circumference increases, the hoop stresses acting transversely and radially increase. That increases cord tension when tire width increases. A tire is also round in side view, so it also has hoop stresses acting circumferentially in side view. As we increase pressure, we can measure the increase in diameter and circumference.

This comes from the circumferential strain that goes with the circumferential stress. Tire unloaded circumference is a measurement we commonly take, especially in oval track racing, to control tire stagger. Road racers often ignore this measurement, but they shouldn't. You don't necessarily want stagger on a road course, but you don't want to have it unintentionally. It is a bit harder to compare tires of different dimensions at the same pressure, but we know that a larger-diameter tire has more surface area for its pressure to act upon, and it therefore has to have greater cord tension at a given pressure.

This means that the larger-diameter tire acts stiffer, at a given pressure, just like a wider tire does. Therefore, we will probably end up running lower pressure in the larger-diameter tire. Then the contact patch will get longer.

What happens if we set the tire pressure such that the larger tire and the smaller one both have the same static deflection – say, ½"? Taking the questioner's 17" and 15" unloaded diameters as our basis, for a 17" diameter a ½" static deflection theoretically gives us a contact patch 5.74" long. For a 15" diameter, ½" static deflection theoretically equates to a 5.38" contact patch length. That's a contact patch length ratio of 1.067, compared to a diameter ratio of 1.133. The contact patch length does not quite go up proportionally to the diameter; it goes up by roughly half the percentage. ½" is a fairly large deflection for a tire that small. The contact patch length in both cases is more than 1/3 of the tire diameter.

It turns out that for all reasonable diameters and static deflections, the amount of non-linearity does not change much. If we doubled both the diameters, and calculated theoretical contact patch length at ½" deflection, we still get a contact patch length ratio of 1.065 – not much change from 1.067. Cut the deflection to ¼", and the length ratio becomes 1.064 – still about the same.

My conclusion is that within the range of diameters and static deflections we are likely to consider for a particular car, contact patch length for a given static deflection has a surprisingly linear relationship to outside tire diameter: the contact patch length ratio stays at about .94 times the tire diameter ratio. The percentile increment of contact patch length gain remains roughly half of the percentile increase in diameter.

Using the simplifying assumption that, for constant contact patch width, contact patch length is simply inversely proportional to inflation pressure, the 17" tire should need only 1/1.067 times as much pressure and should have about 1.067 times the contact patch area, for the same static deflection and vertical spring rate. At this somewhat lower pressure, it will have around 1.062 times the side-view circumferential hoop stress and around 1/1.067 times the sectional circumferential hoop stress – meaning that overall cord tension and tire rigidity should be similar despite the reduced pressure.

More contact patch, similar rigidity; that looks good. Of course, there is a weight penalty, just as there is if we widen the tire. But overall, the bargain looks surprisingly similar.

One might think there would be a disproportionate penalty in rotational inertia, since the mass of a larger-diameter tire not only is greater, but also acts on a larger radius of gyration. However, a larger-diameter tire also turns at a lower rotational speed for a given road speed, and rotationally

accelerates at a lower rate for a given longitudinal acceleration. The linear speed at the tread of any tire has to be approximately equal to the road speed of the car. Correspondingly, the linear acceleration of the tread has to be approximately equal to the linear acceleration of the car. So the rotational inertia penalty is roughly proportional to the weight gain, but not significantly greater.

There is a further weight penalty in the wheel. This is somewhat variable depending on wheel design, but in most cases it's safe to say that we add more weight to the wheel center and rim by increasing the wheel diameter than we add to the rim by increasing the width.

As a broad generalization, it is probably true that we can increase the contact patch area more weight-efficiently by adding width than by adding diameter. However, the difference is not dramatic, and there are advantages to using diameter. One big advantage is that when we add contact patch area by adding diameter, the tire does not become more camber-sensitive, as it does when we add width. There is generally an an advantage in resistance to aquaplaning when we go tall instead of wide, and better performance in snow.

Depending on how we play other tradeoffs, with a taller tire we may very well be able to run a softer and/or hotter-running tread compound.

As for self-aligning torque and pneumatic trail, if the contact patch is longer, those will increase. The centroid of lateral force should move rearward in the contact patch, at least for moderate slip angles. It should also move forward more as the tire approaches the limit of adhesion, increasing self-aligning torque falloff as the limit of adhesion draws near. Whether that improves steering feel or not is a matter of driver preference and the design of the rest of the car.