View source for Physics:Spring system

{{Short description|Physics model}}
[[Image:elastic network model.png|class=skin-invert-image|thumb|A 2-dimensional spring system.]]

In engineering and physics, a '''spring system''' or '''spring network''' is a model of physics described as a [[Graph (discrete mathematics)|graph]] with a position at each vertex and a [[Engineering:Spring (device)|spring]] of given stiffness and length along each edge. This generalizes [[Physics:Hooke's law|Hooke's law]] to higher dimensions. This simple model can be used to solve the pose of static systems from [[Physics:Crystal lattice|crystal lattice]] to springs. A spring system can be thought of as the simplest case of the [[Finite element method|finite element method]] for solving problems in [[Physics:Statics|statics]]. Assuming linear springs and small deformation (or restricting to one-dimensional motion) a spring system can be cast as a (possibly overdetermined) [[System of linear equations|system of linear equations]] or equivalently as an [[Chemistry:Energy minimization|energy minimization]] problem.

== Known spring lengths ==
Consider the simple case of three nodes, in one dimension <math>\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}</math>, connected by two springs. If the nominal lengths, ''L'', of the springs are known to be 1 and 2 units respectively, i.e. <math>\mathbf{L} = \begin{bmatrix} 1\\ 2 \end{bmatrix}</math>, then the system can be solved as follows:

&nbsp;The stretching of the two springs is given as a function of the positions of the nodes by
: <math>\Delta \mathbf{L} = B^\top \mathbf{x} - \mathbf{L} = \begin{bmatrix}-1 & 1 & 0\\ 0 & -1 & 1\end{bmatrix} \mathbf{x} - \mathbf{L}</math>
where <math>B^\top</math> is the matrix transpose of the oriented [[Incidence matrix|incidence matrix]]
: <math>B = \begin{bmatrix}-1 & 0\\ 1 & -1\\ 0 & 1\end{bmatrix},</math>
relating each degree of freedom to the direction each spring pulls on it.
The forces on the springs are
:<math>F_\text{springs} = -W\Delta \mathbf{L} = -W (B^\top \mathbf{x} - \mathbf{L}) = -W B^\top \mathbf{x} + W \mathbf{L}</math>
where ''W'' is a [[Diagonal matrix|diagonal matrix]] giving the stiffness of every spring. Then the force on the nodes is given by left multiplying by <math>B</math>, which we set to zero to find equilibrium:
: <math>F_\text{nodes} = -B W B^\top \mathbf{x} + B W \mathbf{L} = 0</math>
which gives the linear equation:
: <math>B W B^\top \mathbf{x} = B W \mathbf{L}</math>.
Now, the matrix <math>B W B^\top</math> is singular, because all solutions are equivalent up to rigid-body translation. Let us prescribe a [[Dirichlet boundary condition]], e.g., <math>x_1 = 2</math>. 

As an example, let ''W'' be the [[Identity matrix|identity matrix]] then 
:<math>B W B^\top = \begin{bmatrix}1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end{bmatrix}</math>
is the [[Laplacian matrix]]. Plugging in <math>x_1 = 2</math> we have
:<math>B W B^\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} = B W \mathbf{L} = \begin{bmatrix}-1\\-1\\2\end{bmatrix}</math>.
Incorporating the 2 to the left-hand side gives
:<math>\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}</math>.
and removing rows of the system that we already know, and simplifying, leaves us with
:<math>\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}</math>.
:<math> \begin{bmatrix}2 & -1 \\ -1 & 1\end{bmatrix}\begin{bmatrix}x_2 \\ x_3\end{bmatrix} = \begin{bmatrix}1\\2\end{bmatrix}</math>.
so we can then solve
:<math>\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}</math>.
That is, <math>x_1=2</math>, as prescribed, and <math>x_2=3</math>, leaving the first spring slack, and <math>x_3=5</math>, leaving the second spring slack.

== See also ==
* [[Physics:Gaussian network model|Gaussian network model]]
* [[Physics:Anisotropic Network Model|Anisotropic Network Model]]
* [[Stiffness matrix]]
* Spring-mass system
* [[Laplacian matrix]]

== External links ==
* [http://cnx.org/content/m32657/latest/ The Physics of Springs]

{{DEFAULTSORT:Spring System}}
[[Category:Springs (mechanical)]]
[[Category:Elasticity (physics)]]
[[Category:Solid mechanics]]

{{Sourceattribution|Spring system}}