Series and parallel springs

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

Parallel Series

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 strain (deformation) 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 (linear-response) 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 a system of two springs, in series or in parallel, whose spring constants are k_1 and k_2.[1] (The compliance c of a spring is the reciprocal 1/k of its spring constant.)

Quantity In Parallel In Series
Equivalent spring constant k_{eq} = k_1 + k_2 \frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2}
Equivalent compliance \frac{1}{c_{eq}} = \frac{1}{c_1} + \frac{1}{c_2} c_{eq} = c_1 + c_2
Strain (elongation) x_{eq} = x_1 = x_2 x_{eq} = x_1 + x_2
Stress (applied force) F_{eq} = F_1 + F_2 F_{eq} = F_1 = F_2
Stored energy E_{eq} = E_1 + E_2 E_{eq} = E_1 + E_2

Partition formulas

Quantity In Parallel In Series
Strain (elongation) x_1 = x_2 \, \frac{x_1}{x_2} = \frac{k_2}{k_1} = \frac{c_1}{c_2}
Stress (applied force) \frac{F_1}{F_2} = \frac{k_1}{k_2} = \frac{c_2}{c_1} F_1 = F_2  \,
Stored energy \frac{E_1}{E_2} = \frac{k_1}{k_2} = \frac{c_2}{c_1} \frac{E_1}{E_2} = \frac{k_2}{k_1} = \frac{c_1}{c_2}

Derivations

See also

References

  1. Keith Symon (1971), Mechanics. Addison-Wesley. ISBN 0-201-07392-7
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