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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.

CALCULATING UPPER LINK GEOMETRY TO CANCEL TORQUE ROLL

I always enjoy seeing your newsletter. It usually makes me think outside my box and sometimes it really hurts my head.

As to the low speed understeer vs. high speed oversteer: my thinking has been that the degrees of rotation required against distance covered (yaw rate?) to negotiate the tighter radius turn induces the U/S. If the car is adjusted to work well on tight radius (fairly unstable or highly responsive to input), then on large radius turns that car becomes too loose for comfort at the high speeds and low rotation/distance required.

I wonder if I am missing something.

I have been following your articles on torque wedge relative to beam axle cars with interest.

Not knowing any better I have used 3 links with a centered third link with Watts or Panhard lateral location. I have made big improvements on some TransAm type cars’ exit traction by replacing the upper 2 links in the 4 link. We have to add a fair amount of ARB to the rear, of course.

Now I see that I really should be mounting the third link offset to try to balance torque equally L&R (Road Race assumed).

Obviously the torque applied depends on engine output and trans torque multiplication so it is a pretty dynamic number.

I am trying to figure a way to calculate the effects, and the more I think about the more it hurts. Torque is pretty easy once I decide which trans gear I am going to use and an average of the engine torque.

The part that balls me up is the anti-squat. I have learned that I can have the same anti-squat % with two totally different side view swingarms. And with the rear axle torque reacting thru the links, a short swing arm will make the car super reactive to throttle position, drop throttle loose and on throttle tight. I find that an IC somewhere near front axle seems pretty good with high power cars.

Now I am trying to figure out how to relate the third link offset and SVSA to the torque wedge.

I have addressed in past newsletters the reasons why cars tend to understeer more in low-speed turns than high-speed ones.

The questioner describes yaw velocity, and suggests that this adds understeer. It is true that the car will have greater yaw velocity (in degrees or radians per second) in a small-radius turn than in a large-radius one, for a given lateral acceleration.

However, I don’t think that yaw velocity adds understeer, in and of itself. Yaw acceleration (rate and direction of change of yaw velocity, in degrees or radians per second squared) – or more precisely, the inertial reaction to that acceleration – does add understeer if its direction is into the turn, as during entry. If its direction is out of the turn, as during exit, it adds oversteer instead. In other words, due to yaw inertia, the car doesn’t want to start rotating when it’s going straight, and once it is rotating it wants to keep rotating. This effect tends to add understeer on entry and oversteer on exit. The effect is more pronounced in cars that have larger polar moments of inertia in yaw (masses toward the ends) and less pronounced in cars with small polar moments of inertia in yaw (masses toward the middle, as with mid-engine cars). It is more pronounced in tight turns than in sweepers.

We could perhaps say that higher yaw velocity indirectly adds understeer, because the concomitant tighter turn radius implies that the front wheels will track further outside the rears, or not as far inside the rears, for a given set of slip angles at the tires. That is, on a wet road, in a tight turn the front tires will generally make tracks outside of the rear tire tracks, whereas in a sweeper the rear tire tracks will be outside the front tire tracks. With rear drive, the front tires are creating a drag force as they corner, and the rears are creating a thrust force. If the thrust force acts at a smaller radius than the drag force, as in a tight turn, a yaw couple results that tends to rotate the car out of the turn and therefore adds understeer. If the thrust force acts at a larger radius than the drag force, as in a sweeper, the effect acts the other way and adds oversteer.

For a given forward acceleration, including zero (constant speed), higher speed requires more thrust from the rear tires, to overcome the increased aerodynamic drag. That reduces the rear tires’ available lateral force capability, and adds oversteer.

If the car has a spool, or if the diff generates any locking torque in the conditions we are examining, that creates a yaw moment that adds understeer in any car that has to turn both ways and cannot use tire stagger. Up to a point at least, this effect is greater in tight turns than in sweepers.

Does more anti-squat/anti-lift produce more throttle push and/or more drop-throttle oversteer? I doubt that it necessarily does, although there might be a little very short-lived effect of that nature, due to the rear tires loading and unloading a little more abruptly. I would be more inclined to attribute any observation like that to changes in rear steer effects when the geometry is changed.

Now, to the question of what to do with a 3-link to cancel torque roll. Assuming that the lower links are symmetrical, the side view instant center isn’t what we need to pay attention to, nor is the overall amount of anti-squat.

What we need to do is create a roll moment with the longitudinal links that is equal and opposite to the roll moment created by the driveshaft torque. If the lower links are symmetrical, any lift or squat force they generate is the same on both sides of the car, so these forces do not create any roll moment. What matters is the lift force created by the upper link, and how far to the right of center this force acts.

The relative height of the lower links and the upper, at the axle centerline plane, determine the magnitude of the tension force on the upper link. That plus the slope of the upper link determine the lift force from the link. The magnitude of the lift force, plus how far off center it acts, determine the roll moment countering the driveshaft torque.

The only gear ratio that matters is the ring and pinion ratio. This determines the ratio between the axle torque and the driveshaft torque. The equation for the inclination angle of the upper link, when the rest of the geometry is known, is:

tan ϴ_u = (R_t (H_u – H_l)) / (N_rp * H_l * L_yu)

where:

ϴ_u = upper link angle from horizontal (nose down positive)

R_t = effective radius of tire

H_u = height of upper link, at axle centerline plane

H_l = height of lower links, at axle centerline plane

N_rp = ring and pinion ratio

L_yu = lateral offset of top link from center

This equation does not allow for friction in the ring and pinion. This friction will reduce the axle torque slightly and therefore call for a slightly steeper upper link angle. However, since we are really only trying to get close to the correct value, this effect can be ignored.

A three-link with the top link offset is a very simple way to cancel driveshaft torque roll, but it has a drawback: it causes roll and wedge change when we brake. Unless we use a transmission brake, as some early cars actually did, the brakes don’t act through the driveshaft, so there is no driveshaft torque from them. Yet the asymmetrical three-link is still reacting axle housing torque asymmetrically. The car rolls to the right, and the right rear and left front unload.

We can make the brake torque react through different linkages than drive torque, and get even rear wheel loading under both power and braking. The simplest way to do this is to use a birdcage or brake floater just on the left. Alternatively, we can have birdcages or brake floaters on both ends of the axle. Obviously, this complicates the system. If we want to retain the simplicity of the basic three-link, we can compromise and accept having only partial cancellation of driveshaft torque, and a little wedge change and roll in braking.

EFFECT OF STAGGER ON REAR ANTI-LIFT/ANTI-SQUAT

I was reworking several of my Excel spreadsheets, specifically my calculations of anti-squat and longitudinal load transfer as defined by pitch centers and axis, and I started to wonder what the implications would be for an over staggered car on the forward longitudinal forces seen at the tire contact patch and then by definition in the longitudinal locating links on a solid rear axle car (torque arm and trailing links as per usual in my case).

At some point and time would it not be true that the outside (larger) tire would be driving with a positive longitudinal slip and a forward accelerating force and the inside (smaller) tire would be dragging with a negative longitudinal slip and a rearward longitudinal force?

And if the above statement is true, then is not the small tire side longitudinal linkage seeing what is in essence a force similar to braking?

And if all of the above is true, then there is really only one radius of curvature at which the car is stagger neutral with both rear tires providing forward force in proportion to their vertical loading and at that time and that time only we are getting the total anti-squat forces we calculate from our left and right longitudinal link geometry?

It definitely is true that anti-squat/anti-lift forces depend on the actual ground plane forces at the contact patches, and these can be dramatically influenced by tire stagger, with a locked axle or a limited-slip.

It is difficult to predict just what the ground plane forces will be. Their total magnitude, their distribution, and even their direction change depending on how close the tires are to the limit of adhesion. As these things change, not only do the resulting yaw moments change, but any roll moments created in the rear suspension by anti-squat/anti-lift effects also change.

With independent suspension, it is fairly simple to predict the jacking force at the wheel for a known or assumed ground plane force. With a live axle, it’s more complex. The longitudinal locating linkages are inboard of the wheels, so a portion of the longitudinal force from each tire reacts through the opposite side’s links. Also, the linkages act on the sprung structure inboard of the tires.

That’s just the thrust forces. On top of that, we have the torque. That acts on the axle housing as a unit, and reacts through whatever mechanism controls housing torque. The roll moment from the torque does not depend on the distribution of the individual wheel torques; it only depends on their sum.

If we have the brake calipers on floaters or birdcages, and a locked axle, the torques of the two brakes react through their individual birdcages or floaters. If we have dissimilar brakes, that changes the distribution of the torques. However, with a locked axle dissimilar brakes don’t change the distribution of the thrust forces. Both rotors act on the axle as a whole. Longitudinal forces (which, in sum at least, will be negative thrust, or positive retardation forces) are highly sensitive to stagger effects. On the other hand, if we have a locker or diff that unlocks on decel, dissimilar brakes will give us dissimilar ground plane forces. Stagger will affect the car, but in the same way that it does at the front in braking: a smaller tire gives us more retardation force for a given brake torque.