THC Science

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

Series		Parallel

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[edit]

Equivalent spring[edit]

The following table gives formula 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 Series	In Parallel

Equivalent spring constant	${\frac {1}{k_{\mathrm {eq} }}}={\frac {1}{k_{1}}}+{\frac {1}{k_{2}}}$	$k_{\mathrm {eq} }=k_{1}+k_{2}$
Equivalent compliance	$c_{\mathrm {eq} }=c_{1}+c_{2}$	${\frac {1}{c_{\mathrm {eq} }}}={\frac {1}{c_{1}}}+{\frac {1}{c_{2}}}$
Deflection (elongation)	$x_{\mathrm {eq} }=x_{1}+x_{2}$	$x_{\mathrm {eq} }=x_{1}=x_{2}$
Force	$F_{\mathrm {eq} }=F_{1}=F_{2}$	$F_{\mathrm {eq} }=F_{1}+F_{2}$
Stored energy	$E_{\mathrm {eq} }=E_{1}+E_{2}$	$E_{\mathrm {eq} }=E_{1}+E_{2}$

Partition formulas[edit]

Quantity	In Series	In Parallel

Deflection (elongation)	${\frac {x_{1}}{x_{2}}}={\frac {k_{2}}{k_{1}}}={\frac {c_{1}}{c_{2}}}$	$x_{1}=x_{2}\,$
Force	$F_{1}=F_{2}\,$	${\frac {F_{1}}{F_{2}}}={\frac {k_{1}}{k_{2}}}={\frac {c_{2}}{c_{1}}}$
Stored energy	${\frac {E_{1}}{E_{2}}}={\frac {k_{2}}{k_{1}}}={\frac {c_{1}}{c_{2}}}$	${\frac {E_{1}}{E_{2}}}={\frac {k_{1}}{k_{2}}}={\frac {c_{2}}{c_{1}}}$

Derivations of spring formula (equivalent spring constant)[edit]

Equivalent Spring Constant (Series)

When putting two springs in their equilibrium positions in series attached at the end to a block and then displacing it from that equilibrium, each of the springs will experience corresponding displacements x₁ and x₂ for a total displacement of x₁ + x₂. We will be looking for an equation for the force on the block that looks like:

F_{b}=-k_{\mathrm {eq} }(x_{1}+x_{2}).\,

The force that each spring experiences will have to be same, otherwise the springs would buckle. Moreover, this force will be the same as F_b. This means that