[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/series-and-parallel-springs-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/series-and-parallel-springs-wikipedia\/","headline":"Series and parallel springs – Wikipedia","name":"Series and parallel springs – Wikipedia","description":"From Wikipedia, the free encyclopedia Ways of coupling springs in mechanics In mechanics, two or more springs are said to","datePublished":"2021-06-21","dateModified":"2021-06-21","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1f\/SpringsInSeries.svg\/300px-SpringsInSeries.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1f\/SpringsInSeries.svg\/300px-SpringsInSeries.svg.png","height":"128","width":"300"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/series-and-parallel-springs-wikipedia\/","wordCount":3928,"articleBody":"From Wikipedia, the free encyclopediaWays of coupling springs in mechanicsIn 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:SeriesParallelMore 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.Table of ContentsFormulas[edit]Equivalent spring[edit]Partition formulas[edit]Derivations of spring formula (equivalent spring constant)[edit]See also[edit]References[edit]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 k1{displaystyle k_{1}} and k2{displaystyle k_{2}}.[1]  (The compliance c{displaystyle c} of a spring is the reciprocal 1\/k{displaystyle 1\/k} of its spring constant.)Partition formulas[edit]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:Fb=\u2212keq(x1+x2).{displaystyle 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 Fb. This means thatF1=\u2212k1x1=F2=\u2212k2x2=Fb.{displaystyle 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 x1{displaystyle x_{1},} and x2{displaystyle x_{2},}:x1\u00a0=\u00a0F1k1,x2\u00a0=\u00a0F2k2{displaystyle x_{1}~=~{frac {F_{1}}{k_{1}}},,qquad x_{2}~=~{frac {F_{2}}{k_{2}}}},and similarly,x1\u00a0+\u00a0x2\u00a0=\u00a0Fbkeq{displaystyle x_{1}~+~x_{2}~=~{frac {F_{b}}{k_{mathrm {eq} }}}}.Substituting x1{displaystyle x_{1},} and x2{displaystyle x_{2},} into the latter equation, we findF1k1\u00a0+\u00a0F2k2\u00a0=\u00a0Fbkeq{displaystyle {frac {F_{1}}{k_{1}}}~+~{frac {F_{2}}{k_{2}}}~=~{frac {F_{b}}{k_{mathrm {eq} }}}}.Now remembering that F1\u00a0=\u00a0F2\u00a0=\u00a0Fb{displaystyle F_{1}~=~F_{2}~=~F_{b}}, we arrive at1keq=1k1+1k2.{displaystyle {frac {1}{k_{mathrm {eq} }}}={frac {1}{k_{1}}}+{frac {1}{k_{2}}}.,}Energy StoredFor the series case, the ratio of energy stored in springs is:E1E2=12k1x1212k2x22,{displaystyle {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:E1E2=k1k2(k2k1)2=k2k1.{displaystyle {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,E1E2=12k1x212k2x2{displaystyle {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 toE1E2=k1k2.{displaystyle {frac {E_{1}}{E_{2}}}={frac {k_{1}}{k_{2}}}.,}See also[edit]References[edit]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/series-and-parallel-springs-wikipedia\/#breadcrumbitem","name":"Series and parallel springs – Wikipedia"}}]}]