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Series and parallel springs
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<h2 id="formulas">Formulas</h2> <h3>Equivalent spring</h3> <p>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 <a href="/facts/Hooke%2527s_law/7hTUcKUm">spring constants</a> are k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} .<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> (The <a href="/facts/Stiffness/xlabLkfT">compliance</a> c {\displaystyle c} of a spring is the reciprocal 1 / k {\displaystyle 1/k} of its spring constant.) </p> <table><tbody><tr><th>Quantity</th><th>In Series</th><th></th><th>In Parallel</th></tr><tr><td colspan="4"></td></tr><tr><td>Equivalent spring constant</td><td> 1 k e q = 1 k 1 + 1 k 2 {\displaystyle {\frac {1}{k_{\mathrm {eq} }}}={\frac {1}{k_{1}}}+{\frac {1}{k_{2}}}} </td><td></td><td> k e q = k 1 + k 2 {\displaystyle k_{\mathrm {eq} }=k_{1}+k_{2}} </td></tr><tr><td>Equivalent compliance</td><td> c e q = c 1 + c 2 {\displaystyle c_{\mathrm {eq} }=c_{1}+c_{2}} </td><td></td><td> 1 c e q = 1 c 1 + 1 c 2 {\displaystyle {\frac {1}{c_{\mathrm {eq} }}}={\frac {1}{c_{1}}}+{\frac {1}{c_{2}}}} </td></tr><tr><td>Deflection (elongation)</td><td><i> x e q = x 1 + x 2 {\displaystyle x_{\mathrm {eq} }=x_{1}+x_{2}} </i></td><td></td><td><i> x e q = x 1 = x 2 {\displaystyle x_{\mathrm {eq} }=x_{1}=x_{2}} </i></td></tr><tr><td>Force</td><td> F e q = F 1 = F 2 {\displaystyle F_{\mathrm {eq} }=F_{1}=F_{2}} </td><td></td><td> F e q = F 1 + F 2 {\displaystyle F_{\mathrm {eq} }=F_{1}+F_{2}} </td></tr><tr><td>Stored energy</td><td> E e q = E 1 + E 2 {\displaystyle E_{\mathrm {eq} }=E_{1}+E_{2}} </td><td></td><td> E e q = E 1 + E 2 {\displaystyle E_{\mathrm {eq} }=E_{1}+E_{2}} </td></tr></tbody></table> <h2 id="partition-formulas">Partition formulas</h2> <table><tbody><tr><th>Quantity</th><th>In Series</th><th>In Parallel</th></tr><tr><td colspan="3"></td></tr><tr><td>Deflection (elongation)</td><td> x 1 x 2 = k 2 k 1 = c 1 c 2 {\displaystyle {\frac {x_{1}}{x_{2}}}={\frac {k_{2}}{k_{1}}}={\frac {c_{1}}{c_{2}}}} </td><td> x 1 = x 2 {\displaystyle x_{1}=x_{2}\,} </td></tr><tr><td>Force</td><td> F 1 = F 2 {\displaystyle F_{1}=F_{2}\,} </td><td> F 1 F 2 = k 1 k 2 = c 2 c 1 {\displaystyle {\frac {F_{1}}{F_{2}}}={\frac {k_{1}}{k_{2}}}={\frac {c_{2}}{c_{1}}}} </td></tr><tr><td>Stored energy</td><td> E 1 E 2 = k 2 k 1 = c 1 c 2 {\displaystyle {\frac {E_{1}}{E_{2}}}={\frac {k_{2}}{k_{1}}}={\frac {c_{1}}{c_{2}}}} </td><td> E 1 E 2 = k 1 k 2 = c 2 c 1 {\displaystyle {\frac {E_{1}}{E_{2}}}={\frac {k_{1}}{k_{2}}}={\frac {c_{2}}{c_{1}}}} </td></tr></tbody></table> <h3>Derivations of spring formulas (equivalent spring constant)</h3> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Truss/bJXJyUtz">Truss</a></li> <li><a href="/facts/Duality_(mechanical_engineering)/N7vNQvAY">Duality (mechanical engineering)</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Keith Symon (1971), Mechanics. Addison-Wesley. ISBN 0-201-07392-7 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>