I have a question about deriving kinematic roll centres and instant centres that I hope someone here can help me with.

When calculating kinematic roll centres, you first derive the instant centre. The instant centre is the intersection of a line extending through the suspension's upper control arm (UCA) and another line extending through its lower control arm (LCA). This is shown in the attached diagrams. In the first two cases, calculating the instant centre is straightforward. However, in the third case, the UCA and LCA are parallel to one another and the two lines never intersect. How is the instant centre - and by extension the roll centre - derived in this situation?

Thanks.

I know nothing of this topic but would venture to guess that the center is distributed evenly between the upper and lower control arms? Therefore an average amount could be seen at the midpoint between both arms?

im not exactly sure but my hunch is you should get the correct result if you start from a suspension with both arms tilted in the same direction and from there let the instant centre move by slowly decreasing the angle
so in the end it sould end up right in the middle at infinite distance

Thanks, guys.

The kinematic approach seems a bit questionable, particularly in the way it decouples the lateral and the logitudinal instant centres, so I may explore a force-based approach to deriving roll centres instead.

BuddhaBing,

I used to read about what happens in these cases, but I can't recall it right now (I don't have that book on me either), so I probably won't answer your question directly.

However, what I can offer (I think) some insight on, is the kinematics in general. So here goes...

When you calculate the instant/instantaneous centre of rotation (ICR), what you do is the following. Suppose you have a rigid body (here, wheel/rim) with two links attached to it (UCA & LCA, say at points U and L, respectively), both having some kind of angular velocity. The IRC would then be the intersection of the lines, perpendicular to the linear velocity vectors of the rigid body at points U and L. Which is what you have in your first two cases, essentially.

In the third case, since such lines would be parallel to one another, as you've promptly noticed, no IRC can be defined. What this means, is that the wheel can't rotate about the axis coming out of the page. In essence, its camber will stay fixed. Another way to think of it is a rectangular sign suspended from above by two equal length, parallel cables. It can swing sideways, but its centreline will remain horizontal...i.e. it won't tilt.

Same thing happens here - the wheel (in theory) can move up and down, but its upper and lower ends can't go left and right (as viewed from the front of the car).

Of course, this doesn't mean that the car's body won't roll, as it's attached to the wheels by means of compressible springs.

As a guess, I would say that the RC is in the middle of the track, halfway up between the LCA and UCA. At least, common sense would suggest so.

At the same time, if the above hypothesis is correct, it is another reason why equal length, parallel control arms are not very popular - high RC.

Hope the above was halfway cohesive.

Cheers,

SpeedyPro is mostly correct. Instantaneous centres can only be defined for bodies having angular velocity. If there is no angular velocity, the instantaneous centre is at infinity.

If you want to work out the roll centre of the vehicle, it seems like you're going about it the wrong way. The instantaneous centres of the wheels probably won't tell you much about how the chassis is moving. Try drawing your rigid body diagrams to include the chassis.

Well, that approach is taken straight from the book 'Race Car Vehicle Dynamics' by Milliken and Milliken. It's considered to be one of the canonical books on vehicle dynamic theory so I'm reluctant to say that the approach they've taken is wrong.

The attached figure is taken from the book and demonstrates how a roll centre is derived using this approach. The instant centre for each wheel is calculated and a line is then projected from the middle of each wheel's tyre-ground contact patch through its instant centre. The intersection of these two projected lines is the roll centre.

As the chassis moves, the suspension geometry will change and the instant centres and roll center will move.

Quote from BuddhaBing :

Well, that approach is taken straight from the book 'Race Car Vehicle Dynamics' by Milliken and Milliken. It's considered to be one of the canonical books on vehicle dynamic theory so I'm reluctant to say that the approach they've taken is wrong.

The attached figure is taken from the book and demonstrates how a roll centre is derived using this approach. The instant centre for each wheel is calculated and a line is then projected from the middle of each wheel's tyre-ground contact patch through its instant centre. The intersection of these two projected lines is the roll centre.

As the chassis moves, the suspension geometry will change and the instant centres and roll center will move.

Actually, looking at your scans, and seeing that they use lines through tire contact patches (and I've seen it done before, so no arguement here), the roll centre for the body of a car using parallel control arms would be at the ground surface, in the centre of the car's track.

The reason for this is that ICRs for both wheels are now essentially at infinity. So if you try to draw a line from the tire contact patch to a point at infinity, this line will, practically, have zero slope, which means it's horizontal and located at H=0.

But of course, to be absolutely precise, the line's slope will be just above (and just below) zero, so the two of them will intersect in the middle of the vehicle's track.

This sort of places a cap on where your RC will be positioned. With the right angling of the wishbones, you can get the RC below, or above the ground, as desired.

Thanks Speedy Pro, that makes a lot of sense. As the control arms tend near parallel, the instant centre tends towards infinity and the line projected from the tyre contact patch to the instant centre tends to flatten along the ground. This builds on what Shotglass said above. In practice, I think I'll treat parallel control arms as a special case and either perturb one slightly so that they're no longer parallel and process as usual or just set the roll centre automatically in those cases.

Glad to help. If it's no secret, what are you planning to do with these design principles? Just curious.

Nothing secret, just pottering around with a little hobby project to keep my C++ and OpenGL skills current. I'm taking the opportunity to learn something new at the same time and, given my interest in motorsports, introductory vehicle dynamics seemed a logical choice.

Quote from BuddhaBing :

Well, that approach is taken straight from the book 'Race Car Vehicle Dynamics' by Milliken and Milliken. It's considered to be one of the canonical books on vehicle dynamic theory so I'm reluctant to say that the approach they've taken is wrong.

The attached figure is taken from the book and demonstrates how a roll centre is derived using this approach. The instant centre for each wheel is calculated and a line is then projected from the middle of each wheel's tyre-ground contact patch through its instant centre. The intersection of these two projected lines is the roll centre.

As the chassis moves, the suspension geometry will change and the instant centres and roll center will move.

Interesting stuff! I'd not seen those diagrams before. I was confused because I thought you were trying to calculate the roll centre of the vehicle using only the instantaneous centre of one wheel, which seemed unlikely. Now that I've seen the diagrams it makes sense!

Is it absolutely necessary to have a roll centre? In the 3rd picture, there is just no roll centre, because the wheel doesn't roll in the given plane.

Quote from detail :

Is it absolutely necessary to have a roll centre? In the 3rd picture, there is just no roll centre, because the wheel doesn't roll in the given plane.

Roll centre is the point about which the car rotates. Its nothing to do with wheel roll/rotation.

I've got some good info on this. Give me a couple of hours and I'll post some jpegs for ya.

"However, in the third case, the UCA and LCA are parallel to one another and the two lines never intersect. How is the instant centre - and by extension the roll centre - derived in this situation?"

The short answer is it is at ground level, slap bang in the middle. Just waiting for my camera to charge and I'll post some images.

These photos come from Allan Staniforth's Race and Rally Car Source Book. A fantastic read and a must for any self respecting petrol head.

First one shows the situation with parrallel equal wishbones as asked. I added the others cos it's all interesting stuff.

Ofcourse the effects of roll centres are a million miles more complicated then working them out

Excellent, thanks for those pics, Gentlefoot. I'll have to see if I can get my hands on that book too.