Kinematics and Roll Centres

At its basest level, the role of the suspension is to hang the wheel off the chassis and stop it flailing around. Any object can have six ways of moving - three displacements (up and down, inwards and outwards and forwards and backwards) and three rotations. The suspension is meant to restrict five of those movements relative to the chassis so that, in theory, the only movement is up and down. If you have ball joints at each end of the suspension links, you need five separate links to control five out of the six possible ways of moving - hence the term multi-link suspension.

I've already decided to use double wishbone suspension for the car and each of those arms is equivalent to 2 separate links. This means we need an extra link - the toe link, joining the upright to the chassis, otherwise the upright will be free to rotate about an axis running between the top and bottom ball joints on the upright. On the rear this link can be fixed, but on the front it is joined to the ends of the steering rack so that the racer can actually negotiate corners. I'm actually considering having identical suspension front and rear so that it cuts down on the component rate, i.e. four identical uprights with either a fixed toe link or a steering arm rather than having four separate uprights per car. As anyone who has worked in a manufacturing environment will tell you it's far easier and cheaper to build four the same than four different.

Anyway, I digress... Analysis of five separate links is not a trivial matter but a bit of careful inspection of a suspension will show that for a given position of wheel and chassis, you can replace the entire suspension with a single arm and its kinematic and force behaviour will be the same. The point where this single arm pivots is called the instant centre of the suspension. The further away from the wheel this instant centre is, the less change there is in camber as the wheel moves in bump and rebound. The classic case is a solid axle suspension where the instant centre is effectively positioned off at infinity so that there is no camber change as the suspension rolls. A similar case is where the wishbones are parallel. Movement in bump causes no change in camber, but roll causes significant camber change.

Once you know the location of the instant centre, you can calculate the position of the roll centre. This is defined as a point where a lateral force can be applied to the car which would cause no roll of the body. Effectively it's a line of action where the side force from the tyres is transmitted into the body. If the line of action happens to pass through the centre of gravity, there is no roll and if the line of action is above the centre of gravity then the car will lean into the curve. Most of the time, the roll centre is below the centre of gravity and the car leans outwards when you steer. The roll centre is found by joining a line from the centre of the contact patch to the instant centre. Where these two lines (one for each side of the car) intersect is the kinematic roll centre.

Roll centres are important as they control how much weight is transferred to which wheel when you corner. The side force generated by a tyre is a function of three things: the weight supported by the tyre, the camber angle of the tyre and the slip angle - the angle between the direction of travel of the tyre and its heading. Controlling how much weight is on each tyre is a way of controlling its handling and it's something we'll come back to later when we get to sorting out the spring and damper rates.

We can't control the total amount of weight transferred - that's a function of how much 'g' we're pulling in a corner, the height of the centre of gravity and the track of the car. F1 engineers have been trying to put the c of g below ground level for many years with little success, and I'm sure I'm going to have similar success on that front. Having a roll centre close to the c of g is achievable and will limit the amount of roll of the car, making it more predictable. The only problem we have to consider is jacking. Anyone of a certain age might remember the joy of driving a VW Beetle close to the limit. The Beetle has swing arm suspension at the rear - instead of wishbones it used the half shaft to support the wheel. When you cornered the force was transmitted up the half shaft and with a lot of roll this would jack the rear of the car upwards and you could get the rear wheel to tuck under the car and lead to a Beetle-shaped hole in the nearest hedge.

Anyway, enough waffle, back to design. In the last post I stated that I wanted no camber change in roll and minimal migration of the roll centre. The only way to achieve this is by having the instant centres of both sides of the suspension meeting each other on the centreline of the car. This will result in shorter equivalent swing arms (i.e. the length of the imaginary bar joining the centre of the upright to the instant centre) than could otherwise be achieved. In other words, using this method there will be more camber change in bump using this system than there perhaps could be. The only way to get a reduction in camber change is to have as wide a track as possible. Until I can get out and measure someone's bodywork (no part of the tyre is allowed outside the envelop of the bodywork under the rules, which sort of rules out a Formula Ford as a design concept), I don't know how large a track I can get away with. So for now, while I work with the concept, I'll make a few assumptions and assume the maximum width will be around 1400 mm.

BTW the picture above (click it to enlarge) is of a quick and dirty concept, which has a reasonably long equivalent swing arm length (where the three thin red and green lines converge is the instant centre), but around 40mm of lateral roll centre migration (the diagram is for when the car has rolled through two degrees). It's an OK concept, except the top wishbones are too close together to match with a sane set of rollbar backstays - in an attempt to keep a handle on weight, I'm trying to use as little spaceframe as possible, so integrating the suspension pickups with the roll bar seems a fairly good idea. I'll be back once I've done some tweaking to get the instant centres aligned.