rear twist beam motion ratios

[sambeeb]

Hi,

In trying to work out my F:R wheel rates in my FWD er, I'm stuck on how to calculate the motion ratio of a twist beam rear end.
The things that I find difficult to get my head around are the fact that the left and right wheels are always coupled to a certain extent, and difficulty in figuring where to measure trailing arm pivots from.
Regarding the permanently partially coupled left and right wheels (through the twist beam) I'm guessing that I can never think in terms of one wheels motion ratio - one wheels is always dependant on the other yeah? I'm thinking I'd need to calculate the whole axle at once with the piggy back RARB disconnected, but then also reconnnceted at the soft, medium and hard settings to get the full picture. And I'd have to repeat all that again if I ever changed rear springs.

Regarding finding the pivot to measure from, you have the pivot mounted to the chassis, but the trailing arms of the axle don't actually pivot from there, they pivot from the common twist/torsion beam itself don't they? I'm sure there'd be some theoretical coupling point between the chassis pivot and the twist beam that should be a reference point but I just can't find any info.

any help greatly appreciated
sam

[snailpace85]

The twist beam won't have a huge effect on where the trailing arms pivot because the bending stiffness of the trailing arms is pretty high. Just think of the twist beam as an anti-roll bar that can't be disconnected and think in terms of both rear wheels moving together. Measuring the wheel hub centre, spring mounting point and the trailing arm pivot point leads to the usual relationship between spring rate and wheel rate.

I don't know if you're going to the extent of calculating anti-roll bar rates but the twist beam itself is a little tricky to get right. It has an open section and you can look up the torsional stiffness for that in standard text books. The problem is that the end effects where the twist beam joins the trailing arms mean that the formulas will underestimate the real anti-roll stiffness.

Hope that's some help,

Scott

[sambeeb]

it just seems way more complex the more I think about it. For example a one wheel bump would be a combination of the spring stiffness but also the torsional force that goes through the twist beam. But a two wheel bump will be the pair of springs together, but no additional stiffness coming from the beam. You get the feeling that when cornering or when hitting a one wheel bump that the wheel rate comes from some degree of both springs + beam + RARB with the trailing arm pivoting from the trailing arms attachment to the twist beam. But a two wheel bump/launch compression will come from both springs only and pivot more from the whole rear ends pivot at the chassis mounting point because the compression loading is the same for both wheels. See what I mean - it seems as though its pivot point would vary depending upon what the car is doing, like a variable motion ratio, and its going to be inherently way stiffer in ride than in roll.
And my head really starts to hurt when you think about what might be happening when it lifts an inside rear.
When you guys work out wheel rates for normal Mac strut or double wishbone rear corners, do you do so with the bar disconnected or connected?

[snailpace85]

Engineers and physicists are fond of saying ‘As a first approximation….’ before launching into a simplified calculation that’ll probably get things good enough (within 10 to 20% with a bit of luck). You’re quite right though. The real motion of the axle won’t quite go through the pivots on roll.

But using the dreaded ‘as a first approximation’ we can assume that the pivots act as perfect pivots which means that they are completely free in rotation but perfectly stiff against normal movement. This has the effect of forcing the pivoting to occur at those points. The twist beam is relatively flexible in torsion so the torsion moments about an axis parallel to the beam will put a bending moment into the trailing arms that won’t bend them very much simply because they are so much stiffer. There will be a bending moment in the twist beam about an axis along the length of the car. The twist beam is quite stiff in this direction. That could deform the trailing arms a little but the twist beam’s length does mean it is relatively flexible in bending just not as flexible as it is in torsion. Therefore you can reasonably assume, up to a point, that the twist beam simply has an anti-roll effect with the suspension otherwise acting as a pair of independent trailing arms pivoting about the bushed pivots.

There is an issue that the behaviour of open sections in torsion doesn’t perfectly comply with standard stress analysis formulas due to the stiffening effect at the ends where the twist beam is attached. That means that any calculations are likely to underestimate the roll stiffness of the suspension. You might be able to search the internet to find academic papers on the subject which might give you an estimate of this effect. If you’ve got experience and access to Finite Element Analysis (FEA) facilities that would be the easiest way to calculate more accurately. That contrasts with traditional circular section anti-roll bars which are reasonably easy to calculate by hand. I don’t know how familiar you are with beam bending and torsion hand calculations but a popular book with engineers here in the UK is Roark’s ‘Formulas for Stress and Strain’ and contains the relevant information.

I’ll leave this post here as I’m sure there must be a word limit but I’ll do more posts in the next few days for you on the role and limitations of calculations in motorsport and the motor industry generally (my background) and say something about the limitations of my first approximation assumptions above which turn out to be quite large really and become larger as you stiffen the suspension for motorsport. At that point what you said in your posts becomes more relevant and it isn’t completely straightforward.

Hope that makes some sort of sense,

Scott

[sambeeb]

Well thanks for the reply. It both gives me a sense of relief in having some confirmation that these twist beam rear ends are in fact complex beasts, but also shows me that I'm way out of my depth when it comes to the proper engineering side of looking at it.
Perhaps I should give a bit of back ground as to why I'm looking at the rear end. The car I have is a vw polo 9n3 gti 1.8t. I use it in street registered class hillclimbs. Following mark Ortiz and dick shine advice on these rear ends I have lowered it as far as practical to get the COG and roll centre as low as possible. I have added a 20mm RARB set on hard to get the car to lift its inside rear - the reasoning being that this is just about the only way to get good traction out of the open diff front end when going up hills. Of course I still have problems getting the power down though. So I was wondering if my rear springs weren't too soft. Yes it will currently lift the inside rear but I think it is doing so because I have added the most 'retail' torsional stiffness to the twist beam that I can. But when it does this though, the whole rear is then supported on virtually a standard spring rate on the outside rear. I'm worried this will be inducing an outside rear 'squat' that will try to lift the inside front. So I want to run stiffer rear springs to try to help with the above scenario and also help with a bit of 'anti-squat' on launch and when powering up straight uphill sections - I can reduce the RARB stiffness to compensate on roll if I have too.
So I guess what I'm wondering is 1. is my reasoning in the ballpark/worth a shot 2. that's why I was asking about motion ratio's so that I can take a punt at a better rear spring rate.
MY front has 150lb springs and 22mm FARB on hardest of 2 settings, rears are 130lb springs with 20mm RARB on hardest of 3 settings. The car weighs 1100kg with a 65/35 f:r weight spilt in event trim.

[pipcarder]

This thread peaked my interest after recognising the question from one of Mark Ortiz's newsletters so I thought I should register and chime in.
First of do you have a link to the regulations? It would be interesting to see what's allowed in regards to chassis modifications. I would definitely try harder rear springs and a softer RARB as an ARB will increase lateral load transfer and lift on the inside rear, you should be able to achieve a higher degree of roll resistance before you start to lift the inside rear with stiffer springs over an ARB. From reading your discussion with Mark it would be beneficial to increase rear roll resistance on the rear so you can use a softer front ARB that should decrease front lateral load transfer, increasing front cornering force but also keep the front from rolling to much as it sounds like you are in some real need of more camber.

It may be worth getting some second-hand springs and cutting a coil or two off since they have a higher spring rate that will get higher when they are cut shorter, by no means the best solution but I get the impression you don't have a massive budget for racing. it will also allow you to try progressively harder springs with the more coils cut off. With regards to increasing roll resistance on the rear, increasing the track width at the rear with spacers or a lower offset wheel will increase the amount of roll resistance that you could achieve before lifting the inside rear which is the limiting factor to roll resistance.

With regards to your worry about the inside front unweighting due to outside rear squat, this is a dynamic situation and the main effect of dynamic load transfer in the CG height and the track width/wheelbase, do not confuse suspension displacement for weight transfer; suspension displacement is the effect of weight transfer and not the cause. If anything the squatting of the rear would decrease load transfer since the CG would be lower.

A thought on corner weights, using shims under a spring would allow the corner weights to be adjusted, I imagine the front will be sensitive to static weight distribution on launch with an open diff.

If anything doesn't make sense I apologise, I have just finished a very long weekend at Castle Combe.