# Multiple Spring Systems

The spring rate/stiffness calculation has been covered in the previous blog article. Springs are not designed to be stretched or compressed beyond their elastic limits, i.e. springs obey Hooke’s law,

F = k x d.

F = force applied on the springk = spring rate d = displacement

Two or more springs can be arranged in series or parallel. The representation of spring arrangements is very useful in model analysis such as finite element analysis. For example, a crankshaft with varied diameters in axial direction can be presented by a series of springs with different stiffnesses.

For series arrangements, equivalent spring rate, keq can be calculated as follows:

1/keq = 1/k1 + 1/k2

If, there are more than two springs, the equation becomes:

1/keq = 1/k1 + 1/k2 + 1/k3 …+ 1/kn , where n is the number of springs.

The displacement, d is given by:

d = d1 + d2 …+ dn

For parallel arrangements, equivalent spring rate, keq can be calculated as follows:

keq = k1 + k2

If, there are more than two springs, the equation becomes:

keq = k1 + k2 + k3 …+ kn , where n is the number of springs.The equivalent displacement, d is given by:

d = d1 = d2 …= dn