View Full Version : Torque arm anti-squat calculation

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.

Norm Peterson
02-06-2006, 04:11 PM
That's pretty close, and your approach of working with the tangents of angles is the way this whole business is presented in at least one reference (for any rear suspension, actually). BTW, that's an interesting description of "stolen" (lost?) antisquat.

The SVIC for a typical bushing-supported TA can be geometrically constructed from just four known points (five, if the front of the TA attaches to the chassis through a rod-ended link instead). It's perhaps best visualized by working with the concept that an instant center represents an intersection of force lines only.

In both TA cases, the trailing link pivots define one of the lines that intersect the SVIC.

A TA construction line runs between the axle center and its point of (mostly) vertical restraint at the chassis end, but this line is only indirectly involved with the IC. It's the force line of action from the chassis attachment that gives you the intersection that's your SVIC. For a bushed TA, that line is perpendicular to the TA construction line (which I'll note is not necessarily horizontal but is usually close). And as a somewhat picky note - all of this assumes that there is negligible friction between the TA and its chassis bushing.

For a rod-ended link TA, the link provides the second force line.

Something you can do to get roll understeer without giving up all your anti-squat or ending up with a too-short SVSA is to angle the trailing links inward toward the chassis. Then you can move the PHB vertically to alter roll steer independently of anti-squat, as the rollsteer line is no longer parallel to the side view inclination of the links.

The only real correction I have to offer is to note that a torque arm is not a special case of 3-link design, regardless of the longitudinally oriented "component count" being coincidentally the same. A TA is fixed to the axle and works through an entirely different mechanism, being loaded primarily in bending. All of the links in either a 3-link or 4-link are pivoted at both ends and are intended to work in tension/compression only. Any bending picked up in the links of a link suspension is due to bushing stiffness or insufficient rod end angular freedom, aka "bind".


02-06-2006, 06:52 PM
Thanks for the description of graphical construction for IC. Makes perfect sense when I think about it, as do many such things :)

I agree completely that the T/A is not a special case of 3-link design. One way to think about the difference is that, in a 3-link, the rotation of the SVSA and the rotation of the axle housing are coincident. In a T/A, they are not in general the same. This is made even more obvious by considering the graphical construction of IC - the center of axle housing rotation (the torque arm contact point) and the actual IC (lying on a line passing through the TA contact point or TA mounting link) are not the same, in general.

I think this analysis of T/A geometry helps to make the point that it is, purely from a geometric perspective, even more configurable than a 3-link. Whether that configurability is physically available in a real world implementation is another story.

Good point about angling the trailing arms inward, the roll axis now passes through two virtual points and is therefore more subject to manipulation.

Norm Peterson
02-07-2006, 06:43 AM
One further aspect that separates 3-link from TA arrangements is what happens to the "anti's" as the suspension moves. Typically, a TA will lose a few anti-squat percentage points as rear ride height drops, while a 3-link will normally gain a few (although its doing so with less linearity is a distinct possibility). Either can be used to advantage, though there might be some application-specificness about it all.


06-26-2006, 08:11 PM
Man I wish I had payed attn to math in school

I want to understand. heh can I get a tutor?:crying:

08-03-2006, 10:44 AM
Reading this I just had a Charlie Brown moment.I kept hearing this 'WHA-WHA-WHA-WHA-WHA-WHA' in my head

04-10-2007, 03:27 PM
Would the "bind" go away if the TA was decoupled?

Norm Peterson
04-11-2007, 08:38 AM
"Bind" in a torque arm is somewhat different from "bind" in a link. Simply decoupling it should make any amount of bind under trail braking differ from that present while leading the throttle down on corner exit, but won't make it go away. I think that would require that the chassis end(s) of the TA be unrestrained in the lateral direction. That way, chassis roll won't try to bend the TA (think plan view, with the TA attachment point vertically NOT on the line about which the car is rolling), which would then try to steer the axle in a direction that probably wouldn't agree with what the LCAs are trying to dictate.


05-02-2007, 01:46 PM
Another way of visualizing the position of the IC is to think of it as a 3link, but with a very short "odd" link that is positioned vertically. The normal construction procedure for a 3link will then give you the proper position.

The spreadsheet on Page 6 of my blog:


allows the user to input the locations of the rear pivot points for an asymmetric 3link, so I played around until I got an arrangement...with a vertical odd link...which would work. The torque arm ended up on the bottom, but it is possible to achieve dynamic cancellation of the driveshaft torque with a torque arm with reasonable anti-squat (the spreadsheet goes for 100% anti-squat).

05-03-2007, 05:58 PM
I realized, after that last post, that my comment about the torque arm being on the bottom made no sense at all, so I decided that, rather than try to make my 3link spreadsheet "work," I needed to start from scratch and do the derivations for a torque arm. So, to cut to the chase, the last page at my blog (Page 26) has a spreadsheet for torque arm setup:


05-04-2007, 12:00 PM
Nice work!

Now if you can only create some software which will automatically enlarge the driveline tunnel in a 1st gen, torque arms will be all the rage soon.

05-04-2007, 12:59 PM
Admittedly, the torque arm spreadsheet was largely a mental exercise. I was very curious whether I could get a torque arm setup that would give me what I wanted and still be halfway practical. And, as far as modifying a production torque arm car, I realize it isn't even halfway practical. But, if someone is building a tube frame car and is, for some reason, particularly enamored with the torque arm, it does become practical.

Mean 69
05-04-2007, 03:18 PM
Neat stuff guys, always love the tech goodies to make you think/challenge ideas that you've had.

Billy, nice to see you here, didn't realize you posted on this site. I looked at the blog for about two minutes and plopped in some typical values, or "packaged friendly" values onto the sheet. I am guessing that the offset of the T/A as you have it means laterally (plan view) with positive numbers meaning towards passenger? I assumed this and used 4" for a go. The output shows where the forward link location should be for torque cancellation, and in my example, there was a very large assymetry between left and right as a result, which doesn't surprise me given what the intent of the sheet is aimed at (specifically for torque cancellation). Not digging that at first gut check for a car that has to turn left and right, with (hopefully) similar characteristics for both directions (i.e. street/road course car).

I'd be interested to see where the T/A offset would need to be to do the same net effect, but utilizing symmetrical left and right links. I'm sure you could back into the results using your sheet through a bunch of iterations (i.e. offsetting the T/A until the forward locations of the links get really close).

Honestly though Billy, thanks a million for sharing your knowledge and ideas with all of us, it's extremely generous. For those of you that don't know, Billy is one of the original Ramcharger drag folks from before many of us were even born, and also contributed to perhaps the best text on performance suspension (Race Car Vehicle Dynamics). **This isn't a leg-hump, anyone that has ever met me knows for certain that I kiss no man's rear, but I DO respect my elders and teachers. (by the way, Norm, same goes for you, anyone that can convince Mark Ortiz to correct a monthly newsletter by using his exact text gets a head nod from me, nice work, sir).**

05-04-2007, 08:15 PM
Mean 69, the numbers that pop up in the spreadsheet yield a symmetrical setup, but the offset (which you interpreted correctly) is almost 11 inches. With 4 inches, you'd have to play with such numbers as the height of the rear attachment points and see if you could come up with something symmetrical. Won't guarantee anything practical.

I realized, after I put up the new page, that I should have added some words about the offset. It's the same as the offset on Page 6 (3link), of course. I'll probably go back and add something.

And, I contributed to the student workbook for RCVD. I hadn't yet met the Millikens when RCVD was first published. My work is mentioned in their second book (Chassis Design). I used RCVD as a text when I taught at the U. of Central Florida.