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LONGITUDINAL ANTI IN INDEPENDENT SUSPENSION WITH DROP GEARS

In the May-June 2009 newsletter, I wrote:

With sprung diffs or inboard brakes, dx/dz is the value for a wheel that does not rotate with the upright. For outboard brakes, live axles, or odd cases such as Humvees and VW Transporters where the upright contains drop gears, dx/dz is the value for a wheel that does rotate with the upright or axle.

Upon further reflection, I have concluded that actually the case of an independent suspension with drop gears is not quite the same as the case of an outboard brake or live axle, or the case of a hydrostatic or electric motor on the upright. In most cases, torque does react through the linkage, but its magnitude is not the same as in those other cases. There is even one possible case where the vehicle has drop gears in the upright, yet no torque reacts through the linkage!

Drop-geared uprights are generally only found in off-road vehicles, or perhaps vehicles intended for very poor roads, and the whole question is largely irrelevant for most forms of racing. However, it does represent an interesting analytical puzzle.

The basic principles that apply are these:

1.The reaction torque applied to the vehicle as a whole has to be equal and opposite to the wheel torque.

2.The reaction torque applied through the diff or transaxle mounts has to be equal and opposite to the shaft torque.

3.The reaction torque acting through the suspension linkage has to be the difference between the wheel reaction torque and the shaft reaction torque.

When the wheel and the shaft turn opposite directions, the reaction torques are of opposite sign. Therefore, their difference has an absolute value greater than either the wheel torque or the shaft torque.

Taking the simplest case, suppose the upright contains a simple two-gear set, with a 1:1 ratio. This would be used when the drop gears are used only to gain ground clearance, and not torque multiplication. The shaft turns backward when the wheel turns forward, and with the same torque (ignoring friction). If we are driving forward, the reaction torque at the diff mounts is equal to wheel torque, and forward. The reaction torque through the suspension linkage is rearward, and twice the magnitude of wheel torque.

Does that make the anti-squat or anti-lift twice as great? Not exactly, in most cases. It makes the torque anti twice as great, but not the thrust anti. The total anti is twice as great, if the side view swing arm length is not infinite or undefined, and if the side view instant center is at wheel center height, i.e. if thrust anti is zero.

To find effective force line slope graphically for such a case, the correct method would be:

1.In side view, draw a line from wheel center to side view instant center.

2.Find the midpoint of that line.

3.Draw a line from the contact patch center to that midpoint. That is your effective force line.

The slope, or dz/dx, of that line will be equal to the instantaneous side-view inclination from vertical, or dx/dz, of the contact patch center, as measured on a K rig or K&C rig – provided of course that the wheel is rotationally locked in an appropriate manner, i.e. at the inboard end of the shaft. The jacking force coefficient, or dF_z/dF_x, as measured on a K&C rig, will also agree with the measured contact patch dx/dz.

The only thing that will be different in K&C testing is measurement of caster change. That can no longer be taken as equal to wheel rotation. It is now half of wheel rotation.

Now let's consider another case. Suppose we have a three-gear train in the upright: an idler between the driving gear and the driven gear, with the driving and driven gears still the same size, for the same 1:1 drive ratio. Now the shaft turns the same direction as the wheel, and the same speed. We now have a situation where the reaction torque acting through the diff mounts is exactly equal and opposite to wheel torque, just as it would be with no drop gears. Correspondingly, the difference between overall reaction torque and shaft reaction torque is zero, and there is no torque reacted through the suspension linkage.

What happens if we use the drop gears to get torque multiplication, not just ground clearance? If the shaft turns the opposite direction to the wheel, and twice as fast, what happens then? Reaction torque at the diff is now ½ as great as wheel torque – forward when driving forward – and reaction torque through the linkage is 1.5 times wheel torque, rearward. To graphically construct our effective force line, we use an effective SVIC 2/3, or 1/1.5, of the way out to the geometric SVIC, rather than halfway. Wheel rotation with respect to suspension displacement is now 1.5 times caster change.

Taking the same case, except with same-direction rotation, diff reaction torque is now ½ of wheel torque, rearward, and linkage reaction torque is also ½ of wheel torque, rearward. Effective SVIC is now twice as far away from the wheel center as geometric SVIC.

Taking the same two cases, with an overall drop gear ratio of 1.5:1, with opposite rotation diff reaction torque when driving forward is 1/1.5, or .67, of wheel torque, forward, and linkage reaction torque is 1.67 times wheel torque, rearward. Effective SVIC is 1/1.67, or .60, as far from the wheel as geometric SVIC.

Same case, except same-direction rotation: diff reaction torque is .67 times wheel torque, rearward. Linkage reaction torque is .33 times wheel torque, rearward. Effective SVIC is 3 times as far from the wheel as geometric SVIC.

Generalizing for any drop gear ratio n, with opposite rotation diff reaction torque when driving forward is wheel torque times 1/n, forward. Linkage reaction torque is wheel torque times (1 + 1/n), rearward. Effective SVIC is 1/(1 + 1/n) times as far from the wheel as the geometric SVIC. Caster change with respect to suspension displacement is 1/(1 + 1/n) times wheel rotation.

Generalizing similarly for same-direction rotation, diff reaction torque when driving forward is wheel torque times 1/n, rearward. Linkage reaction torque is wheel torque times (1 – 1/n), rearward. Effective SVIC is 1/(1 – 1/n) times as far from the wheel as geometric SVIC. Caster change with respect to suspension displacement is 1/(1 – 1/n) times wheel rotation. Note that when n = 1, the quantity 1/(1 – 1/n) is undefined. Physically, this means that effective side-view swing arm length is infinite, i.e. there is no torque anti, and that caster change cannot be computed from wheel rotation, as wheel rotation is zero regardless of caster change.