Series and Parallel Springs

Multiple spring elements are often used; in such cases, we must obtain the equivalent spring constant for the combined elements.

Parallel Springs

Parallel springs are connected side by side, as shown in the left side below:

More in general, parallel springs have the same deflection, but are subject to different forces.

Assuming that the force $f$ is applied so that both springs have the same deflection $x$ but different forces $f_1$ and $f_2$, we have:

$$x=\frac{f_1}{k_1}=\frac{f_2}{k_2}$$

If the system is in static equilibrium, then we have:

$$f=f_1+f_2=k_{1}x+k_{2}x=(k_1+k_2)x$$

Thus, for the equivalent system, where $f=k_{e}x$, we have:

$$k_e=k_1+k_2$$

In general:

$$k_e=\sum _{i=1}^n{k}_i$$

Series Springs

Series springs are connected to end to end.

More in general, series springs have the same torque or force, but different deflections.

Assuming both strings are in static equilibrium and are subject to the same force $f$, their deflections $f/k_1$ and $f/k_2$ will not be the same unless $k_1=k_2$. The total deflection $x$ of the system is obtained from:

$$x=\frac{f}{k_1}+\frac{f}{k_2}=\left(\frac{1}{k_1}+\frac{1}{k_2}\right)f$$

For the equivalent system $f=k_{e}x$, we then have:

$$\frac{1}{k_e}=\frac{1}{k_1}+\frac{1}{k_2}$$

In general:

$$\frac{1}{k_e}=\sum _{i=1}^n\frac{1}{k_i}$$