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

EFFECTS OF UNEQUAL-LENGTH HALFSHAFTS (INDEPENDENT SUSPENSION DRIVESHAFTS)

I have been building cars for Solo racing and hill climbs for a long time. Recently a question came up about unequal axle length in IRS on a rear wheel drive car. This also results in unequal angles in the (CV) axle joints.

We all know that unequal axle length in FWD cars results in Torque Steer. The question is what effect it has on a RWD vehicle where the hub cannot change position from the same forces. Specifically the car has a De Dion rear axle located by a watts link and parallel 4 bar trailing links. The car also has a limited slip differential of the clutch type.

I remember first reading about this in the 1970’s when the VW Rabbit (sold as the Golf in many markets) was introduced. The car had front-wheel drive. It had an off-center differential, which resulted in the halfshafts being of unequal length. VW made the longer shaft larger in diameter, to get the torsional rigidity of the two shafts to be equal. Press releases at the time said they had done that to eliminate torque steer.

I owned one of these cars, a 1983 GTI. I also drove a slightly earlier standard one, which had narrower tires, and a Scirocco, which used the same powertrain. I never noticed any effect from the unequal shaft lengths in any of these cars.

Note that making the shafts equal in torsional stiffness does not do anything about any effect from unequal joint angles. That car does not have large amounts of joint angularity at static condition, at least at stock ride height, but the steering feel was not noticeably different in right and left turns.

I never was convinced that the torsional stiffness of the shafts should make any difference, as long as the car had an open differential. Even if the two shafts torsionally deflect by different amounts when we get on the power, would that reasonably be expected to create a yaw input or a force through the

steering? Wouldn’t it merely cause a slight displacement of one differential side gear with respect to the other, while torque to both shafts remained identical – ergo no torque steer?

Now if there’s a limited-slip diff, or a locker, then it might be possible to feel the effect of the shafts twisting different amounts, even with rear-wheel drive. The car might turn toward the wheel with the longer shaft on sudden power application, and turn away from that side on sudden decel. However, this effect would be momentary. As soon as the shafts assumed a steady-state deflection, the wheel speeds would again be equal.

If I were building a car with a DeDion rear end and an offset diff, and a limited-slip, my approach would be to try building it the easy way first – probably meaning using equal-diameter shafts, simply ordered from the vendor – and then drive it and see if it veers detectably on abrupt changes of throttle setting. If there is a momentary effect as described above, then I would consider stiffening the longer shaft.

We generally want to keep joint angles small in any case, purely for reduced power loss and better joint life. This would be particularly true with unequal shaft lengths, but probably the biggest concern would be to make sure we don’t get really large joint angles on the shorter shaft in any condition.

BEAM AXLE ROLL CENTERS

I've just finish my master's and I've just been hired by a NASCAR cup team. I have several questions regarding how to model the force based roll centers of a beam axle rear suspension. A friend at work suggested that you would be able to help me find some good information regarding this.

In university, I was heavily involved with Formula SAE. While doing FSAE, I was able to derive, model, and validate a 14 degree of freedom model of a Formula SAE car. While creating this model, I learned a lot about the dynamics of FSAE cars. Working now in NASCAR, I would like to create the same type of model, except for a car with a beam rear axle. The sole purpose of this model is to help me better understand the dynamics of a NASCAR stock car.

The approach I took with my previous model involved calculating the car's front and side view instantaneous centers, and then calculating the n-line slope of the front and side view of each corner. Thus, knowing the front and side view n-line slopes, the front and side view geometric forces are calculated. Do you know of any good papers or books that go into more detail about this with beam rear axles? Any help would be greatly appreciated.

Roll centers could be said to be originally a beam axle concept, which we also find convenient as a simplification when dealing with independent systems. We imagine the front or rear wheel pair as a single system, laterally constrained with respect to the sprung structure at the roll center, even

though we actually have two separate systems. With a beam axle, it really is a single rigid unit, with a single set of links, or whatever, connecting it to the sprung mass.

One consequence of this is that we can have a lot of geometric anti-roll without having net upward jacking. We may have net upward or downward jacking, and it may be asymmetrical, but the portion of it that comes from lateral force does not depend on the right/left tire ground plane force distribution. It does depend on that with an independent system.

As with independent suspension, with a beam axle geometric anti-roll does directly relate to contact patch lateral movement as the suspension moves. With a beam axle, this is different for roll and ride. It is the behavior in roll that mainly concerns us when assigning roll center height.

If we have either a good computer kinematics program, or the ability to measure lateral contact patch movement during roll displacement, we can infer geometric anti-roll from that, very much as we might with independent suspension. The rate of y movement with respect to z movement (dy/dz) will be the same for both wheels, and so will the rate of change of that quantity (d²y/dz²). With most independent systems, if the geometry is the same on both sides at static, dy/dz is instantaneously the same on both sides at static condition, but it is not a constant as the suspension moves, so it becomes unequal on the two sides as soon as there is any roll.

A beam axle with the roll center at hub height has the same amount of geometric anti-roll as a swing axle independent system with both swing axles pivoted exactly on the car centerline and at hub height – or, more precisely, it has the amount of geometric anti-roll that such an independent system would have if it didn’t start jacking as soon as the lateral ground plane forces become unequal.

In a swing axle system, the suspension on the outside of the turn tries to jack the car up, and the suspension on the inside of the turn tries to jack the car down. The result is a roll resisting moment. If the ground plane forces are equal, the jacking forces will be equal and opposite, and their sum – the net jacking force – will be zero. However, as load transfers laterally, the ground plane force on the outside tire becomes greater than the ground plane force at the inside tire. The result of this is a net upward jacking force that increases non-linearly with cornering force. That is why swing axle suspensions do not jack noticeably in gentle driving, but exhibit sudden and dramatic jacking in hard cornering.

The same forces that try to jack the car up in a swing axle system are present in a beam axle, but they are reacted within the axle assembly and not through the locating linkage. Equal ground plane forces at the two tires try to bend the axle into an S shape. As the ground plane forces become more unequal, a second effect is superimposed on the first, tending to bend the axle into an inverted U shape. The axle will actually deflect in response to these forces, but if it is adequately stiff the deflections will not be noticeable.

Cup car rear suspensions use truck arms and a Panhard bar. For readers unfamiliar with truck arms, those are I-section beams, bolted to the axle near its ends, extending forward and inward to anchor points some inches apart, near the forward end of the driveshaft. They are the sole longitudinal location of the axle, and react axle torque for both propulsion and braking. Since the front bushings on the truck arms are a few inches apart, the system actually relies on small deflections of its components – mainly the truck arms – to be able to move in roll at all.

For modeling or calculation purposes, I suggest approximating the system by supposing that the axle assembly moves in roll about an axis defined by two points. The rear point is where the Panhard bar centerline intercepts the car’s centerplane (xz origin plane). The front point is an average of the centerplane intercepts of two lines, each passing through a rear tire contact patch center and the corresponding truck arm pivot bushing center. The roll center then is the point where that axle axis of rotation in roll intercepts the axle (yz) plane. This will be very close to the midpoint height of the Panhard bar.

When the Panhard bar is inclined, it will jack the rear of the car up or down. When the bar is approximately centered in the car, this will not in itself create wheel load changes. However, in the presence of rear spring split, it will create wheel load changes. Also, there will be aerodynamic effects.

Geometric roll resistance is the lion’s share of rear roll resistance on a Cup car. The front end, in contrast, has very little and may even have net pro-roll. Along with that, the front has huge elastic roll resistance, from a big anti-roll bar, often with asymmetrical arm geometry, and from the snubbers it usually rides on when cornering. This means that the car responds relatively greatly to rear roll center height changes, and comparatively little to rear spring and anti-roll bar changes.