Everything you needed to know about roll centres
Most people who have an active interest in cars have probably heard of roll centres and some may even think that they understand the concept. The subject is however, not all that it seems and has confused a lot of people who are supposed to know better. The misunderstanding probably comes from the dual roll (sorry) that the roll centre has. The first and most obvious is the geometric effect it has on the wheel movement and the second and not so obvious is the roll it plays in transferring forces between the wheel and chassis.
When a car goes round a corner, the laws of physics and the layout of the car cause it to roll by some greater or lesser extent. It should be pretty obvious that there must be some axis that this rolling motion takes place about. No prizes for guessing that this is called the roll axis. It is an imaginary line that joins the front and rear roll centres. It is possible to find out where the roll centres are located by photographing the car in question as it negotiates a bend. Unfortunately this technique doesn't work too well if the car does not exist. The good news is though, that it can be determined geometrically as shown in the diagram below. Similar methods can be used for other types of suspension.
If we redraw the diagram but this time introduce the effect of body roll we get a situation shown in the next diagram where the roll centre has moved.
Having stated that the roll centre is the point about which a car rolls it is reasonable to assume that this new location is now the point about which the car rotates. Think about it for a while and don't feel too bad if you agree. A lot of so-called experts think so as well. If you can read the otherwise excellent book Race and Rally Car Source Book by Alan Staniforth, pay particular attention to the chapter on suspensions where he speculates increasing the down force on the car by arranging the roll centre to move outside the track of the car. Similarly in his book Tune to Win Carroll Smith shows a slightly moved roll centre for a wishbone set up with equal length parallel arms even though, using the construction line method shown previously, there would be no intersection point.
The diagram above is scanned in from the book (yes I do have a copy).
If non of the above had been mentioned, the chances are that most of you would have assumed that, given the symmetrical nature of a car, it would roll somewhere along its centre line. Isn't ignorance bliss! In most instances it is practical to assume that the car rolls along its centre axis. So what is all the fuss about roll centres?
Along side the geometric considerations mentioned above, the roll centre has an important roll to play with regards to the forces transmitted between the wheel and chassis. Before getting too involved you need to know that for theoretical purposes it is possible to replace two forces with a single one or vice versa in order to simplify calculations. Take a look at the next two diagrams.
The single force in the second diagram can be used to replace the two forces in the first. Note that it must act through the point (usually imaginary) where the other two forces would meet. If you have trouble visualising the concept imagine that the lines are pieces of string that are being pulled. The single sting in the second diagram would move the object in the same direction as the two strings in the first. It is also worth noting that the lengths of the string make no difference to how the object moves. You will also need to know that a rod, which is pivoted at either end, can only transmit a force along its length. Any off axis force will produce a turning motion. Imagine reversing a trailer and you should get the idea.
I can now draw the second diagram again.
This time imagine that the lines represent forces. The force at the tyre can only act on the chassis through the suspension arms, which are pivoted at each end. If you understood the above paragraph you should realise that theoretically we can replace all the forces involved with the right-hand suspension assembly with a single force acting at the point of intersection of the construction lines that have been drawn. Similarly we can do the same thing with the left-hand suspension, only in this example the intersection point is off the side of the page. Finally we can replace these two forces with a single one acting at the roll centre.
It is important to remember that this single force acting at the roll centre is just a concept to make calculations and predictions easier. In reality the force at the tyre exerts a force through the wheel onto the hub carrier, which in turn pushes and pulls the suspension arms, which finally transmit the forces to the chassis. Also worth a mention is the fact that because these forces act directly on chassis, any vertical component (which will be upwards if the roll centre is above ground level) will lift the chassis. This is the cause of the infamous jacking effect on early swing axle cars. The problem afflicts all independent suspension systems but is less pronounced the lower the roll centre.
OK. If you haven't already fallen asleep, you may remember that when a car rolls its roll centre moves but it wasn't that big a deal. Now we are dealing in forces however the single substitute force will always act through the roll centre wherever it is. The question is now, what effect will this movement have on the behaviour of the car? Unfortunately, without knowing the forces involved at each tyre, we can't say exactly which way the resultant force will act. It will, however, be between the two extremes of no weight transfer and total weight transfer as the inner wheel just starts to lift. Any point between these two is valid and can be used to analyse the effect of the movement.
Initially the sideways shift of the roll centre looks alarming but going back to lines of force and bits of string, it is possible to replace the force at the roll centre with an equivalent one on the centre line of the car. The problem has now been reduced to one where the force through the roll centre only moves vertically. We can now start to look at the practical implications, and why designers harp on so much about roll centres in the first place.
For a given car cornering at a fixed lateral acceleration the weight transfer between the inner and outer wheel will always be the same. Changing the suspension linkage or spring rates will do nothing to alter it. If the roll centre height coincides with that of the centre of gravity the entire weight transfer will be due to the forces through the roll centre and because there will be no turning moment between the two the chassis will not roll. As the roll centre moves below the centre of gravity some of the weight transfer will be caused by body roll that acts through the springs. The total transfer will still be the same. Eventually none of the weight will be transferred through the roll centre but will be a result of the body roll alone. The total amount will still be the same however. Now comes the interesting part. A car usually consists of two axles, one at the front the other at the back. Each will have its own roll centre. Imagine that we make the one at the front coincide with the centre of gravity while the one at the back is at ground level. When the car corners the front end doesn't want to roll but the back end does. Hopefully the car is stiff enough not to twist (which also answers the question of why designers are interested in the torsional stiffness of the chassis) so it will roll through some intermediary angle. This in turn means that the front axle will transfer more weight than it otherwise would and the rear axle less causing the car to under steer more than it otherwise would.
Somewhere way back towards the start of this article we were discussing the fact that the roll centre moves as the car rolls. We can now analyse what the effects of this movement is. If we start with the same car as the one above we know that initially the roll centre layout will increase the understeering characteristics of the car. If, however, the roll centre height moves upwards as the car starts to roll the understeering effect will get less. Depending on the rest of the suspension set up, this may make the car unstable and in an extreme case the car after seeming reluctant to turn into a corner would suddenly spin. Although the above might be an extreme example, it is important if you are designing or modifying suspension systems to be aware of how the roll centre is likely to move and the effect this will have on the handling of the car.