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Front and rear roll angles

Front and rear roll angles

I have done a fair bit of mathematical and 3D modelling of independent automotive suspension systems, and have come to the conclusion that the in pure steady state cornering no matter what the combination of springs, ARB/sway bars, motion ratios or linkage geometry, the roll angle (ignoring the tyre flex component) is always determined by the deflection of the spring and that the load they see on each axle will directly correspond to the rate of the spring, meaning that they will deflect such that the wheel movement and thus roll angle, is the same at each end of the vehicle.

It has been suggested by an experienced engineer, that this is not the case and that the front and rear axles DO roll different amounts, however the mechanics of this were not explained. Surely if such a case was true, this means the chassis must be twisting to accommodate the difference in roll angle? In a typical modern chassis with say 15,000Nm/° torsional rigidity, surely the amount of differential roll, if any, must be negligible to the point of not being worth considering?

RE: Front and rear roll angles

He may be thinking about the tire deflection contribution. He may be thinking about the torsional stiffness. He may be thinking about something subtle. Or he may be wrong.

However 'the load they see on each axle will directly correspond to the rate of the spring' is a bit worrying as it ignores the effect of RCH, at first glance.


Greg Locock

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RE: Front and rear roll angles

By that I mean the roll angle is dependent on the spring deflection, which is dependent on the spring load, which is dependant on the spring rate AND the other contributory factors, RCH included. Assuming all else held equal, then the load is proportional to the spring rate.

RE: Front and rear roll angles

For modern unibodies, the amount of torsion in the unibody is not very much. Body-on-frame trucks might have a bit more twist. Older models with less frame rigidity, might have a lot more.

If you neglect the torsion in the unibody and you are on flat ground, then there is no choice but for the roll angle to be the same front and rear (neglecting tire compliance). The geometry dictates it ... up to the point where you lift a wheel off the ground in hard cornering. There are some front-drive vehicles notorious for lifting up the inside rear, and some rear-drive vehicles will lift up the inside front.

RE: Front and rear roll angles

I guess my next question is, should we, or can we neglect it?

At what torsional stiffness does it become enough of an issue to warrant attention? The car in question is relatively weak, only 5000Nm/degree.

RE: Front and rear roll angles

What's the total roll moment at, say, 1 g of lateral acceleration?

If the vehicle weighs (say) 1500 kg and the center of gravity is (say) 0.667 m above ground then the total roll moment is 1500 kg x 9.807 m/s2 x 0.667 m = 9807 N.m. Call it 10,000 N.m.

So if the total roll moment is applied to the chassis - implying that all the weight of the car is at one end with no roll stiffness on that end, and none of the weight of the car is on the other end which is somehow being constrained to have no roll - it'll twist by two degrees.

Back that assumption off to something realistic, and the chassis twist is probably not going to be very much compared to the suspension deflections, but just enough to drive owners crazy with squeaks and rattles - or make it so that the doors won't open if someone uses a scissor jack under one corner.

RE: Front and rear roll angles

I see. So lets assume a 1000kg car, 1G acceleration, 55% front weight distribution and CGh of .5m

The total roll moment is 9810N x .5m = 4905Nm.

The front roll moment is then 4905Nm x .55 = 2698Nm

The rear roll moment is then 4605Nm x .45 = 2207Nm

The difference is 490Nm, which at 5000Nm/degree is .098° or 0°5'53"

Correct thinking on that one? I would consider that level of twist fairly irrelevant if it correct.

RE: Front and rear roll angles

Should the roll stiffness distribution not be counted in that calculation somewhere also?

RE: Front and rear roll angles

Yes, because the chassis torsion distributes in accordance with it as well as in accordance with the longitudinal CG position.

I'm still not sure what your basic question is.

Are you perhaps referring to Bob Bolles?


RE: Front and rear roll angles

"I'm still not sure what your basic question is."

Don't worry, neither am I. I guess I am just trying to expand my understanding of vehicle dynamics, it would be nice to have some way of checking the amount of twist imparted on a chassis for a given set of parameters, even if it is only a rough estimate.

I have no idea who Bob Bolles is.

RE: Front and rear roll angles

You could do "springs in series" calculations combining the chassis torsional stiffness forward of the CG with the front suspension roll stiffness (you'd scale that 5000 overall torsional stiffness upward), doing a similar calc for the rear, and dividing the total roll moment accordingly. It may be better to determine the roll moment based on the CG longitudinal position and height and a line drawn between the front and rear geometric roll centers if there either roll center height is at a significant height (as it tends to be with stick axles). Picky and probably unwarranted for this purpose would be to use the sprung mass and its CG rather than the overall numbers.

Bob is a US circle track chassis guy (author, magazine editor, consultant) who frequently writes in terms of what the front and rear of a car "want to do" in terms of roll.


RE: Front and rear roll angles

I guess what I am not understanding first and foremost, is that my calculations for roll angle on each axle will give me exactly the same amount of wheel displacement each end, no matter what the combination of springs, bars, track width, weight dsitribution, motion ratio or roll centers. Now this to me just makes sense, but then if it is true, it does seem to suggest that in pure roll, the chassis is not twisting?

Obviously, add in braking or acceleration and it all goes a lot more complicated...

RE: Front and rear roll angles

The only way the chassis is not twisting (at all) are if the torsional stiffness is infinite, or if the distribution of roll stiffness taking into account the geometric roll centers and springs and antiroll bars at each end is exactly the same as the weight distribution.

Calculate the amount of total body roll based on total roll stiffness front and rear of the complete vehicle, then knowing that number, calculate how much resisting moment there is from the front end and from the rear end. I betcha they won't match the weight distribution. They shouldn't except for a minor miracle, or unless the suspension was specifically designed for them to match (which is not a good way of doing it).

RE: Front and rear roll angles

The cars in question are around 62% front weight, but have around 65% of the roll resistance taken by the rear axle.

If the load transfer calcs tell me I get wheel displacements of .63" front and rear, then chassis has one roll angle of 1.3° over a 58" track width. I am having difficulty seeing how this then relates back to the chassis.

Are you suggesting that the numbers I should see for front and rear wheel displacements should be different? That would obviously cause twist.

RE: Front and rear roll angles

Crudely, you can think in terms of the rear axle dragging something like 27% of the roll moment away from the front axle (where it would go if the chassis was infinitely rigid torsionally and the front and rear suspension roll resistances were equal). That 27% of roll moment being dragged off to where it wouldn't go otherwise will certainly cause chassis torsional deflection once you let the chassis become realistically flexible.

If a glorified order-of-magnitude guess helps you any, a single mass model I put together some time ago for a car of similar torsional stiffness with 54% front weight and roll resistance taken 59% up front resulted in a little under a quarter degree of chassis twist at 1g lateral acceleration. I wouldn't attempt to extrapolate from that if I were you.


RE: Front and rear roll angles

If it makes it easier to visualize/understand, you can try an calculate it for two "half" cars.
Imagine the car is split at the CG, and that the front part and the rear part would need to resist roll independently.
If there is a different roll angle for the front half and the rear half of the car, then you "twist" your chassis/structure,
which can be seen as a torsional spring. If the chassis would be perfectly rigid, the difference in roll angle will be taken
up by the tyre as deflection. Ignoring chassis twist and other structural deflections for a moment, the tyre deflection is
inverse proportional to the axle deflection (if we assume same vertical stiffnesses of the tyres).
Add the tyres as additional springs in series to the suspension springs into your model, and then you can see, that while
roll angle in relation the ground plane is the equal front and rear for a rigid body, the displacement can be split between the
two springs.
I think (but could be wrong) that this is what your mate referred too, as this is quite commonly done in the racing fraternity.
They will measure the suspension deflection and then calculate a "roll angle" for the front and rear, which most of the time is
not equal. The difference is taken up by chassis deflection (which for a half decent modern race car, shouldn't be much), and the
tyre deflection, thereby this gives and indication of what the tyre loadings are at the different axles.
If the chassis structure is sufficient stiff (> 10x tyre stiffness), the difference in suspension roll angle gives and indication
about Lateral Roll Moment Distribution or Lateral Load Transfer, which can be used as a handling tuning parameter.
If the structure which connects front and rear is not stiff enough, you twist a additional spring, which is also largely undamped.
If this is the case, the amount of LRMD you can achieve is limited, you can stiffen one end, but all it does is twist the chassis more.
If you have a stiffness issue (maybe with some historic race cars)you are better off, trying to aim for close to equal roll suspension
roll angles front to rear, to minimize the load put into the chassis (torsional spring).

Total Roll Angle (Body to ground) = suspension roll angle + tyre roll angle
I think your mate referred to suspension roll angle not total roll angle, and the suspension roll angle can be different front to rear
and most often is.

RE: Front and rear roll angles

My model is done as you say, as two different, separate axles each with their own roll stiffness.

When calculating each axle, the total elastic load transfer/total wheel roll rate will always end up equal.

Equal roll rates in the suspension would mean no chassis twist.

Here's some of my workings.

Vehicle mass = 2575lbs.
Track front = 58"
Track Rear = 58"
Front weight distribution 62%
CGh = 22"
RChF = 2"
RChR = 5"
Axle height = 11.75" (F+R)

Spring rate
Front - 250lbs/in
Rear - 250lbs/in
Motion Ratio
Front - .7
Rear -.7

Gives a wheel rate of 123.5lbs/in front and rear.

Unsprung mass
Front - 66lbs
Rear - 62lbs

ARB (simple model)
Diameter - 26mm
Length - 32"
Arm - 12"
Motion ratio - .55

Gives a wheel rate of 161lbs/in

Diameter - 22mm
Length - 32"
Arm - 6"
Motion ratio - .59

Gives a wheel rate of 379lbs/in

In total, we have a front wheel roll rate of 161+123.5 = 284.5lbs/in and a rear wheel roll rate of 379+123.5 = 503.5lbs/in.

Now, based on those parameters, at 0.8g lateral acceleration I get

Total load transfer - 744lbs
Front LT - 280lbs
Rear - 464lbs

Front unsprung LT - 22.17lbs
Front geometric LT - 40.40lbs
Front elastic LT = 290lbs - 22.17lbs - 40.40lbs = 218.7lbs.

Rear unsprung LT - 19.76lbs
Rear geometric LT - 58.99lbs
Rear elastic LT = 464lbs - 19.76lbs - 58.99lbs = 385.39lbs.

218.7lbs into 284.5lbs/in = .77" wheel deflection at the front.

385.4lbs into 503.5lbs/in = .77" wheel deflection at the rear.

RE: Front and rear roll angles

I think your "problem" is that you assume infinite stiff tyres in your model.
Calculate what the force are the contact point would be, for your 0.77" wheel deflection or better any given roll angle, in your model
You can assume tyre vertical stiffness with ~300-330 N/mm --> 1700-1900# for a common touring car racing slick or whatever values you may have from your tyres.
If we assume a flat road and equal tyres front and rear, the tyres will deflect differently, if I assume 300N/mm stiffness I get a difference of ~ 0.288° for your values. With total values of ~1.65° and ~1.36° roll rear and front, with the suspension providing ~1.05° roll and the difference comes from the tyre deflection.

RE: Front and rear roll angles

More basic than that.

In order calculate chassis twist, the model must assume that the chassis might actually twist, and where the necessary torsional input is located. "Springs in series" should have been a big enough hint.


RE: Front and rear roll angles

TC3000, that makes a lot of sense, thank you.

RE: Front and rear roll angles

What is the wheelbase of this car?


RE: Front and rear roll angles

Sorry Greg there was an error somewhere.

Total load transfer is 781 lbs.

Geometric LT
Front 40.4lbs
Rear 58.99lbs

Unsprung LT
Front 21.17 lbs
Rear 19.76 lbs

Elastic load transfer is therefore 781 lbs minus the sum of the above (140.32 lbs) = 641.06 lbs.

The elastic load transfer is split according to the RSD, 36.14% front in this case

So front load transfer is 231.7lbs, rear is 409.4lbs.

Wheel deflection is then 231.7/284.37 = .815 at the front, and 409.4/502.55 = .815" at the rear.

RE: Front and rear roll angles

Norm, the wheel base is 2560mm, however there is no input for it on the model.

RE: Front and rear roll angles


let's look at this another way round:
You have calculated a combined wheel rate (springs + ARB) of 284.5 lb/in front and 503.5 lb/in rear, if we combine the two, you get a total rate of 788 lb/in to resist roll of the suspended body/mass.

In other words you resist ~63.9% of the total roll moment at the rear of the car.
If you roll your body any given amount, what will be the torque experience by the body at the front and the rear?
The difference will try to "twist" your body.
If you assume a rigid body, it can't be twisted, but the reaction forces at the ground will be different --> your tyre "springs" will see a different load.
What is the twist angle of the body needed to make the reaction forces to the ground equal --> think about "cutting" your body in two parts.

On a different note, if you carry ~63% of the mass on the front, why is your front LT/OM lower then your rear?
What would be the overturning moment for each individual axle?
How is this overturning moment resisted/counter acted --> what is the resulting deflection/ roll angle of each axle?

RE: Front and rear roll angles

So I make that 381.5lbs load transfer per degree of roll.

Of that 381.5lbs, for every degree of roll, the rear would take 247.7lbs and the front would take 133.4lbs.

The roll couple is 20" front and 17" rear. So 247.7 x 20 / 12 = 412lbft front and 133.4 x 17 / 12 = 188lbft at the rear.

The difference is 224lbft. Given the torsional rigidity of 5000Nm/° or 6779lbft/° the chassis twist is therefore .03°.

Is that on the right lines?

As for the wonky roll stiffness, it's a FWD car.

RE: Front and rear roll angles

Got that the wrong way round.

133.4 x 20 / 12 = 222lbft front
247.7 x 17 / 12 = 350lbft rear

Difference is 128lbft and 0.018° twist.

RE: Front and rear roll angles

What would be the difference, if you assume a chassis stiffness of "0", as in the front and the rear of the car are linked via a revolute / hinge, or if the two systems would roll independently (as two systems)?

RE: Front and rear roll angles

I don't really understand the question. That is what I just did was it not?

RE: Front and rear roll angles

Quote (Komodo)

Norm, the wheel base is 2560mm, however there is no input for it on the model.
Knowing that little piece of information better fits something I might use for sanity checking.

What the wonky roll stiffness distribution sounds like is FWD being adapted for autocrossing or track day use. In which case if you're running unusually high rear tire inflation pressures (relative to street specs), that will have an effect on all this.


RE: Front and rear roll angles


Yes, it is a FWD car but not one set up for autocross. This car leaves the factory with that setup and is renowned for it's excellent handling. It's not the first time I have discussed the setup here and apparently it baffles the old school guys.


You could well be right, I'll admit I am a little skeptical of such low numbers, however the maths all seems to check out. I'm open to suggestions on what you think might be more reasonable values for the given parameters?

RE: Front and rear roll angles

Total load transfer - 744lbs
Front LT - 280lbs
Rear - 464lbs

How did you arrive at the proportions of front and rear load transfer?

If that was arrived at by proportioning the amount of wheel loadings based on their spring rates with the assumption of a rigid bodyshell between them (which appears to be the case) then of course you are going to get the same calculated wheel deflections, because it was a constraint that was put into your calculations - implied by assuming the load transfers are in proportion to total spring rates.

Do the calculations again by making the front and rear load transfers in proportion to the front and rear weight distribution.

I'm only doing rough calculations in my head but it's enough to indicate that the numbers are going to be different.

RE: Front and rear roll angles

But that would be even more incorrect. The load transfer is only proportional to the weight distribution at 'full load transfer', that is when both inside wheels have fully unloaded. Up until this point, you have a partial load transfer, and the rates of transfer at each end are going to be dictated by the roll stiffness.

The stiffer axle transfers load faster than the softer one. This is a known fact.

The weight distribution at full load transfer (the car on two outside wheels) is the same as the static load distribution. Unless someone shows me test results that say otherwise, I consider this fact also.

My method satisfies both of these. Distributing the load according the weight distribution, for use at partial load transfers, does not. It suggests that A. the front axle will transfer the load faster, even if the rear is stiffer and B, that there is longitudinal load transfer at some point before full load transfer, which either instantly transfers back to the original axle at full load transfer OR, that the weight distribution at full load transfer is somehow different to the static distribution.

This, to me, sounds ill thought out.

RE: Front and rear roll angles

"I don't really understand the question. That is what I just did was it not?"

I had in mind to model the front and the rear as two independent systems, and I think Brian is hinting at the same thing.

Something along the lines of this:
Front sprung mass * lateral acceleration * moment arm (CG height - RC height) = Overturning Moment
Overturning Moment / half track width is a force acting at the wheel plane
Force @ wheel divided by 2*wheel rate (which is your lumped spring & ARB rate) = deflection at the wheel
sum of deflection at the wheel(s) / track width = roll angle

If you run this calculation, you will see that if unconstrained your front and rear would like to roll a different amount to reach equilibrium of moments.
The difference between the two individual roll angles is largest if the roll stiffness distribution is 100% at one axle.

RE: Front and rear roll angles

OK, so lets run those numbers again. (Sorry, this is taking a while isn't it!)

SMf = 1465lbs
SMr = 855 lbs

MAf = 20"
MAr = 17"

OMf = (1465 * .8 * 20)/12 = 1953 lbft/°
OMr = (855 * .8 * 17)/12 = 969 lbft/°

Ff = OMf / (TRf/2) = 67lbs
Fr = OMr / (TRr/2) = 33lbs

Front roll = Ff / (2 * 284 lb/in) = 6.78°
Rear roll = Fr / (2 * 503 lb/in) = 1.9°.

OK, I see where you are coming from now.

RE: Front and rear roll angles

If you replace your rear springs with solid links, what does that mean to your model?
What happens in reality if the car can't roll around it's suspension, but does not have infinite stiff tyres?
If the rear rolls on it's tyres, but the front suspension is a lot softer then your tyre stiffness, will there be deflection in the front suspension or not?
If yes, but you still have no deflection in your rear suspension, what does this say about your suspension roll angle?

the front tyres and rear tyres have a individual plane, which can warp/twist against each other (front axle vs. rear axle), these plane(s) lay between the rigid body plane and the ground (which if flat is a plane too) plane. The rigid body can have only one overall angle in roll in relation to the ground plane (if the ground is flat), but this angle is made up of two angles: rigid body vs. wheel plane + wheel plane vs. ground plane.
The deflection of the tyre(s) (axle wise) in inverse proportional to the suspension deflection at the axle.
The stiffest axle will deflect it's tyres more/most.

RE: Front and rear roll angles

The purpose of getting you to model the front and rear as two completely independent systems was not to imply that doing so had any semblance to reality, only to convince you that the bodyshell is indeed being subject to torsional loads. I believe that point has now sunk in.

The next step is to introduce a spring rate between the front and rear subassemblies, and find out where the equilibrium lies.

RE: Front and rear roll angles

Yes I see how that works now, thankyou!

The spring rate being that 5000Nm/degree torsion stiffness?

Went and dug my copy of Millikens RCVD out of storage last night. I'm sure the answers are in there somewhere...

RE: Front and rear roll angles

Greg, with regards to the calculated wheel displacements, I think I know what is causing them to come out much lower than expected. Just yesterday a thread popped up on the owners club about sway bar stiffness and someone had measured the stiffness of the bar and found the results to be some 60% of the values given by the calculations floating about for the bar rates. This is owing to the bending of the arm and the flexibility between the bushing and the arm, both of which are not accounted for in the calculations.

Not only this, but it seems I had the length of the bars a fair bit shorter than actual.

Adjusting the bar rates for this would suggest that the car has a roll rate of around 5°/g which is right on the money according to RCVD.

RE: Front and rear roll angles

this may help you along
the two angles φf and φr are the two individual angles as discussed and calculated yesterday.
ctor is the torsional stiffness of your chassis.
I made a mistake in the formula yesterday, should have read (half track)^2, but I think you get the idea anyway

RE: Front and rear roll angles

if you calculate your spring rates as @ the wheel and lump your ARB rates into these wheel rates as well ssp,f&r becomes your track width.

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