Spring rates are
usually expressed as lbs/in. In other words, if we compress a given
spring by one inch, how much resistance does it offer? If a spring
that is compressed 1" pushes back with a force of 500 lbs than that spring
is rated as 500 lbs/in (assuming it is linear).
What we want to do here is determine a similar type of rate for the wheel.
How much force does it take to push the wheel up a distance of one
inch? The force required will be equal to the "effective wheel
rate". A torque balance on the swing arm will help us answer
this question.
If we push up on the wheel and displace it by 1 inch
(dw = 1" in Figure 1)  how far does this compress the spring
(ds in Figure 1)? From similar triangles we calculate ds as:
L1/(ds) = L2/(dw) which leads to: ds = (L1/L2)dw

We will call the spring rate k1 (lbs/in).
The force exerted on the wheel by the compressed spring is:
F1 = k1(ds) = k1(L1/L2)dw

Now use the moment balance on the swing arm to calculate F2:
F2 = (L1/L2)F1 = k1(L1/L2)²dw

Divide through by dw to yield the "wheel rate":
Wheel Rate = F2/dw = k1(L1/L2)²

Wheel Rate = Spring Rate x (Motion Ratio)²

NOTE the power of 2 on the motion ratio. The motion ratio must be squared when
calculating wheel rate. This never fails to mess people up, as it easy to overlook
unless you develop the equations as we did here.
Summary of Results
For the E46 Coupe chassis we get the following table. If you know your spring rates
then you can calculate your wheel rates using the yellow formulas. Note that the
front wheel rate is just about equal to the spring rate. The motion ratios are
obtained by calculating the slope of the curve of displacement of the center of the front and rear
brake rotors vs. change in distance between spring perches.
