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Series and parallel springs

Ways of coupling springs in mechanics

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 (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 an ensemble (or system) of two springs, in series or in parallel, whose spring constants are k_1 and k_2. (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}
Equivalent compliance	c_\mathrm{eq} = c_1 + c_2
Deflection (elongation)	x_\mathrm{eq} = x_1 + x_2
Force	F_\mathrm{eq} = F_1 = F_2
Stored energy	E_\mathrm{eq} = E_1 + E_2

Partition formulas

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 formulas (equivalent spring constant)

:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !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:

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 ::F_1 = -k_1 x_1= F_2=-k_2 x_2 = F_b.\, Working in terms of the absolute values, we can solve for x_1\, and x_2\,: ::x_1 ~=~ \frac{F_1}{k_1}\,,\qquad x_2 ~=~ \frac{F_2}{k_2}, and similarly, ::x_1 ~+~ x_2 ~=~ \frac{F_b}{k_\mathrm{eq}}. Substituting x_1\, and x_2\, into the latter equation, we find ::\frac{F_1}{k_1} ~+~ \frac{F_2}{k_2} ~=~ \frac{F_b}{k_\mathrm{eq}}. Now remembering that F_1 ~=~ F_2 ~=~ F_b, we arrive at ::\frac{1}{k_\mathrm{eq}} = \frac{1}{k_1} + \frac{1}{k_2}. \, |} :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Equivalent Spring Constant (Parallel) |- |Both springs are touching the block in this case, and whatever distance spring 1 is compressed has to be the same amount spring 2 is compressed. The force on the block is then: ::{| |F_b \, |= F_1 + F_2 \, |- | |= -k_1 x - k_2 x \, |} So the force on the block is ::F_b = - (k_1 + k_2) x. \, Which is how we can define the equivalent spring constant as ::k_\mathrm{eq} = k_1 + k_2 . \, |} :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Compressed Distance |- |In the case where two springs are in parallel it is immediate that: ::{| |- |} |In the case where two springs are in series, the force of the springs on each other are equal: ::{| |- |-k_1 x_1 = -k_2 x_2. \, |} From this we get a relationship between the compressed distances for the **in series** case: ::\frac{x_1}{x_2} = \frac{k_2}{k_1}. \, |} :{| class="toccolours collapsible collapsed" width="60%" style="text-align:left" !Energy Stored |- |For the **series** case, the ratio of energy stored in springs is: ::\frac{E_1}{E_2} = \frac{\frac{1}{2} k_1 x_1^2}{\frac{1}{2}k_2 x_2^2}, \, but there is a relationship between x1 and x2 derived earlier, so we can plug that in: ::\frac{E_1}{E_2} = \frac{k_1}{k_2} \left(\frac{k_2}{k_1}\right)^2 = \frac{k_2}{k_1} . \, For the **parallel** case, ::\frac{E_1}{E_2} = \frac{\frac{1}{2} k_1 x^2}{\frac{1}{2}k_2 x^2} \, because the compressed distance of the springs is the same, this simplifies to ::\frac{E_1}{E_2} = \frac{k_1}{k_2}. \, GATO Ignace notes about spring associations |} ## References ## References 1. Keith Symon (1971), ''Mechanics.'' Addison-Wesley. {{ISBN. 0-201-07392-7 ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Series_and_parallel_springs) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Series_and_parallel_springs?action=history). ::

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