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Spring system
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<h2 id="known-spring-lengths">Known spring lengths</h2> <p>Consider the simple case of three nodes, in one dimension x = [ x 1 x 2 x 3 ] {\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}} , connected by two springs. If the nominal lengths, <i>L</i>, of the springs are known to be 1 and 2 units respectively, i.e. L = [ 1 2 ] {\displaystyle \mathbf {L} ={\begin{bmatrix}1\\2\end{bmatrix}}} , then the system can be solved as follows: </p><p> The stretching of the two springs is given as a function of the positions of the nodes by </p> Δ L = B ⊤ x − L = [ − 1 1 0 0 − 1 1 ] x − L {\displaystyle \Delta \mathbf {L} =B^{\top }\mathbf {x} -\mathbf {L} ={\begin{bmatrix}-1&1&0\\0&-1&1\end{bmatrix}}\mathbf {x} -\mathbf {L} } <p>where B ⊤ {\displaystyle B^{\top }} is the <a href="/facts/Matrix_transpose/8wmsagGS">matrix transpose</a> of the oriented <a href="/facts/Incidence_matrix/eMXGWQBw">incidence matrix</a> </p> B = [ − 1 0 1 − 1 0 1 ] , {\displaystyle B={\begin{bmatrix}-1&0\\1&-1\\0&1\end{bmatrix}},} <p>relating each degree of freedom to the direction each spring pulls on it. The forces on the springs are </p> F springs = − W Δ L = − W ( B ⊤ x − L ) = − W B ⊤ x + W L {\displaystyle F_{\text{springs}}=-W\Delta \mathbf {L} =-W(B^{\top }\mathbf {x} -\mathbf {L} )=-WB^{\top }\mathbf {x} +W\mathbf {L} } <p>where <i>W</i> is a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> giving the stiffness of every spring. Then the force on the nodes is given by left multiplying by B {\displaystyle B} , which we set to zero to find equilibrium: </p> F nodes = − B W B ⊤ x + B W L = 0 {\displaystyle F_{\text{nodes}}=-BWB^{\top }\mathbf {x} +BW\mathbf {L} =0} <p>which gives the linear equation: </p> B W B ⊤ x = B W L {\displaystyle BWB^{\top }\mathbf {x} =BW\mathbf {L} } . <p>Now, the matrix B W B ⊤ {\displaystyle BWB^{\top }} is singular, because all solutions are equivalent up to rigid-body translation. Let us prescribe a <a href="/facts/Dirichlet_boundary_condition/m9wsgG89">Dirichlet boundary condition</a>, e.g., x 1 = 2 {\displaystyle x_{1}=2} . </p><p>As an example, let <i>W</i> be the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> then </p> B W B ⊤ = [ 1 − 1 0 − 1 2 − 1 0 − 1 1 ] {\displaystyle BWB^{\top }={\begin{bmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{bmatrix}}} <p>is the <a href="/facts/Laplacian_matrix/Iwea2mm9">Laplacian matrix</a>. Plugging in x 1 = 2 {\displaystyle x_{1}=2} we have </p> B W B ⊤ x = [ 1 − 1 0 − 1 2 − 1 0 − 1 1 ] [ 2 x 2 x 3 ] = B W L = [ − 1 − 1 2 ] {\displaystyle BWB^{\top }\mathbf {x} ={\begin{bmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{bmatrix}}{\begin{bmatrix}2\\x_{2}\\x_{3}\end{bmatrix}}=BW\mathbf {L} ={\begin{bmatrix}-1\\-1\\2\end{bmatrix}}} . <p>Incorporating the 2 to the left-hand side gives </p> [ 2 − 2 0 ] + [ − 1 0 2 − 1 − 1 1 ] [ x 2 x 3 ] = [ − 1 − 1 2 ] {\displaystyle {\begin{bmatrix}2\\-2\\0\end{bmatrix}}+{\begin{bmatrix}-1&0\\2&-1\\-1&1\end{bmatrix}}{\begin{bmatrix}x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}-1\\-1\\2\end{bmatrix}}} . <p>and removing rows of the system that we already know, and simplifying, leaves us with </p> [ − 2 0 ] + [ 2 − 1 − 1 1 ] [ x 2 x 3 ] = [ − 1 2 ] {\displaystyle {\begin{bmatrix}-2\\0\end{bmatrix}}+{\begin{bmatrix}2&-1\\-1&1\end{bmatrix}}{\begin{bmatrix}x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}-1\\2\end{bmatrix}}} . [ 2 − 1 − 1 1 ] [ x 2 x 3 ] = [ 1 2 ] {\displaystyle {\begin{bmatrix}2&-1\\-1&1\end{bmatrix}}{\begin{bmatrix}x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}1\\2\end{bmatrix}}} . <p>so we can then solve </p> [ x 2 x 3 ] = [ 2 − 1 − 1 1 ] − 1 [ 1 2 ] = [ 3 5 ] {\displaystyle {\begin{bmatrix}x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}2&-1\\-1&1\end{bmatrix}}^{-1}{\begin{bmatrix}1\\2\end{bmatrix}}={\begin{bmatrix}3\\5\end{bmatrix}}} . <p>That is, x 1 = 2 {\displaystyle x_{1}=2} , as prescribed, and x 2 = 3 {\displaystyle x_{2}=3} , leaving the first spring slack, and x 3 = 5 {\displaystyle x_{3}=5} , leaving the second spring slack. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Gaussian_network_model/Kg220Imr">Gaussian network model</a></li> <li><a href="/facts/Anisotropic_Network_Model/6lm1YEmS">Anisotropic Network Model</a></li> <li><a href="/facts/Stiffness_matrix/kjsEp3rj">Stiffness matrix</a></li> <li><a href="/facts/Spring-mass_system/7IQNToGt">Spring-mass system</a></li> <li><a href="/facts/Laplacian_matrix/Iwea2mm9">Laplacian matrix</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://cnx.org/content/m32657/latest/">The Physics of Springs</a></li></ul>