Series and parallel springs


In mechanics, two or more springs are said to be in series when they are connected end-to-end or point to point, and they are said to be in parallel when they are connected side-by-side; in both cases, so as to act as a single spring:


More generally, two or more springs are in series when any external stress applied to the ensemble gets applied to each spring without change of magnitude, and the amount of strain of the ensemble is the sum of the strains of the individual springs. Conversely, they are said to be in parallel if the strain of the ensemble is their common strain, and the stress of the ensemble is the sum of their stresses.
Any combination of Hookean springs in series or parallel behaves like a single Hookean spring. The formulas for combining their physical attributes are analogous to those that apply to capacitors connected in series or parallel in an electrical circuit.

Formulas

Equivalent spring

The following table gives formulas for the spring that is equivalent to an ensemble of two springs, in series or in parallel, whose spring constants are and.

Partition formulas

Derivations of spring formulas (equivalent spring constant)

The force that each spring experiences will have to be same since otherwise, the springs would buckle. Moreover, this force will be the same as Fb. This means that
Working in terms of the absolute values, we can solve for and :
and similarly,
Substituting and into the latter equation, we find
Now remembering that, we arrive at
The force on the block is then:
but there is a relationship between x1 and x2 derived earlier, so we can plug that in:
For the parallel case,
because the compressed distance of the springs is the same, this simplifies to
GATO Ignace notes about spring associations