PTAddict

02-04-2006, 07:08 PM

The recent post by Schmoov69 about torque arm 3-links caused me to wonder just how one does calculate the anti-squat of a T/A setup. I couldn't find formulas in any of my texts, so I tried to retrieve my basic static analysis skills from 25 years ago, and I think I've got it. Everyone here can check me, or more likely tell me exactly where the formulas are already documented :)

The key for me is that you can't analyze a T/A the same way as a ladder bar, or any multi-link suspension with a known SVSA. If you think of the forces in the suspension not as "jacking" effects, but as torques which must be reacted by the suspension members, it makes this easier.

Example: car with 100" wheelbase, 20" C.G. Call the ratio of these two, 20/100 = .20, the weight transfer ratio (WTR). If we have a ladder bar with forward mount point 10" off the ground and 50" forward of the axle center, we can construct another ratio, call it the anti-squat force ratio (AFR). In this case, the AFR is 10/50 = .20, and the percentage anti-squat = AFR / WTR = .2 / .2 = 100 percent. Easy.

Torque arm is more problematic, because the effective lengths we need for the above ratios are no longer determined strictly by a physical (ladder bar) or virtual (multi-link intersection) center point. Here's the torque arm analysis, starting with trailing links that are level with the ground. Think of the force on the contact patch generating a torque by acting on a lever arm which is the height of the level trailing links. This torque is reacted by an essentially horizontal arm, the torque arm. Thus, the anti-squat force ratio (AFR) becomes the height of the level trailing link, divided by the length of the torque arm. Back to our above example, if the level trailing link is 10" high and the torque arm is 50" long, the AFR is .2 and our anti-squat is 100 percent again. Note that for nearly level torque arms - which is always the case - the height of the front torque arm mounting point is essentially irrelevant.

If the rear trailing links are angled up or down in the side view, the analysis is complicated somewhat because the trailing links will add their own "jacking effects". If the links are angled down toward the front, they will reduce anti-squat, if up, they increase it. This effect is also simple to calculate: divide the vertical gain in distance from rear to front pivot by the horizontal distance from rear to front pivot. For instance, front trailing arm pivot is 10" high, rear pivot is 11", horizontal distance between pivots 20", the anti-squat force ratio contributed by the trailing links is (10 - 11) / 20 = -.05. Angling the trailing links upward to the rear "stole" a fourth of our anti-squat in this case. So the formula for anti-squat is: (AFRtorquearm + AFRtrailinglinks) / WTR. AFRtrailinglinks is a positive number for links angled down to the rear, negative number for trailing links angled up to the rear.

What if we do want to make our trailing links angle upward to the rear, to induce a bit of roll understeer? Well, we can shorten the torque arm proportionally to recover the desired anti-squat - in the above example, we need to increase AFRtorquearm to .25, which we can do by shortening the T.A. to 40". Won't that shorter torque arm lead to more brake hop? Curiously, maybe not in this specific instance. Remember that the front of the torque arm is constrained only in the vertical direction, it is not a pivot point. The actual side view arc of motion (the SVSA) will be larger in radius than the torque arm, because the links angled up to the rear will "push" the axle backwards somewhat as the rear of the car lifts. Of course, once the trailing arms move past level, the opposite will occur, and the SVSA will be shorter than the T/A length.

Anyway, that's my quick 'n dirty analysis. You can see, based on the ability to play around with geometry parameters more independently of each other, why the T/A is fairly popular.

The key for me is that you can't analyze a T/A the same way as a ladder bar, or any multi-link suspension with a known SVSA. If you think of the forces in the suspension not as "jacking" effects, but as torques which must be reacted by the suspension members, it makes this easier.

Example: car with 100" wheelbase, 20" C.G. Call the ratio of these two, 20/100 = .20, the weight transfer ratio (WTR). If we have a ladder bar with forward mount point 10" off the ground and 50" forward of the axle center, we can construct another ratio, call it the anti-squat force ratio (AFR). In this case, the AFR is 10/50 = .20, and the percentage anti-squat = AFR / WTR = .2 / .2 = 100 percent. Easy.

Torque arm is more problematic, because the effective lengths we need for the above ratios are no longer determined strictly by a physical (ladder bar) or virtual (multi-link intersection) center point. Here's the torque arm analysis, starting with trailing links that are level with the ground. Think of the force on the contact patch generating a torque by acting on a lever arm which is the height of the level trailing links. This torque is reacted by an essentially horizontal arm, the torque arm. Thus, the anti-squat force ratio (AFR) becomes the height of the level trailing link, divided by the length of the torque arm. Back to our above example, if the level trailing link is 10" high and the torque arm is 50" long, the AFR is .2 and our anti-squat is 100 percent again. Note that for nearly level torque arms - which is always the case - the height of the front torque arm mounting point is essentially irrelevant.

If the rear trailing links are angled up or down in the side view, the analysis is complicated somewhat because the trailing links will add their own "jacking effects". If the links are angled down toward the front, they will reduce anti-squat, if up, they increase it. This effect is also simple to calculate: divide the vertical gain in distance from rear to front pivot by the horizontal distance from rear to front pivot. For instance, front trailing arm pivot is 10" high, rear pivot is 11", horizontal distance between pivots 20", the anti-squat force ratio contributed by the trailing links is (10 - 11) / 20 = -.05. Angling the trailing links upward to the rear "stole" a fourth of our anti-squat in this case. So the formula for anti-squat is: (AFRtorquearm + AFRtrailinglinks) / WTR. AFRtrailinglinks is a positive number for links angled down to the rear, negative number for trailing links angled up to the rear.

What if we do want to make our trailing links angle upward to the rear, to induce a bit of roll understeer? Well, we can shorten the torque arm proportionally to recover the desired anti-squat - in the above example, we need to increase AFRtorquearm to .25, which we can do by shortening the T.A. to 40". Won't that shorter torque arm lead to more brake hop? Curiously, maybe not in this specific instance. Remember that the front of the torque arm is constrained only in the vertical direction, it is not a pivot point. The actual side view arc of motion (the SVSA) will be larger in radius than the torque arm, because the links angled up to the rear will "push" the axle backwards somewhat as the rear of the car lifts. Of course, once the trailing arms move past level, the opposite will occur, and the SVSA will be shorter than the T/A length.

Anyway, that's my quick 'n dirty analysis. You can see, based on the ability to play around with geometry parameters more independently of each other, why the T/A is fairly popular.