Spring system
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In engineering and physics, a spring system or spring network is a model of physics described as a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
with a position at each vertex and a spring of given stiffness and length along each edge. This generalizes
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
to higher dimensions. This simple model can be used to solve the pose of static systems from
crystal lattice In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystal, crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that ...
to springs. A spring system can be thought of as the simplest case of the
finite element method Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...
for solving problems in
statics Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in mechanical equilibrium, equilibrium with its environment ...
. 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 In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of th ...
or equivalently as an
energy minimization In the field of computational chemistry, energy minimization (also called energy optimization, geometry minimization, or geometry optimization) is the process of finding an arrangement in space of a collection of atoms where, according to some com ...
problem.


Known spring lengths

Consider the simple case of three nodes, in one dimension \mathbf = \begin x_1 \\ x_2 \\ x_3 \end, connected by two springs. If the nominal lengths, ''L'', of the springs are known to be 1 and 2 units respectively, i.e. \mathbf = \begin 1\\ 2 \end, then the system can be solved as follows:  The stretching of the two springs is given as a function of the positions of the nodes by : \Delta \mathbf = B^\top \mathbf - \mathbf = \begin-1 & 1 & 0\\ 0 & -1 & 1\end \mathbf - \mathbf where B^\top is the
matrix transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
of the oriented
incidence matrix In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element o ...
: B = \begin-1 & 0\\ 1 & -1\\ 0 & 1\end, relating each degree of freedom to the direction each spring pulls on it. The forces on the springs are :F_\text = -W\Delta \mathbf = -W (B^\top \mathbf - \mathbf) = -W B^\top \mathbf + W \mathbf where ''W'' is a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagon ...
giving the stiffness of every spring. Then the force on the nodes is given by left multiplying by B, which we set to zero to find equilibrium: : F_\text = -B W B^\top \mathbf + B W \mathbf = 0 which gives the linear equation: : B W B^\top \mathbf = B W \mathbf. Now, the matrix B W B^\top is singular, because all solutions are equivalent up to rigid-body translation. Let us prescribe a
Dirichlet boundary condition In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the boundary of the domain are fixed. The question of finding solutions to such equat ...
, e.g., x_1 = 2. As an example, let ''W'' be the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
then :B W B^\top = \begin1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end is the
Laplacian matrix In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Lap ...
. Plugging in x_1 = 2 we have :B W B^\top \mathbf = \begin1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end\begin2 \\ x_2 \\ x_3\end = B W \mathbf = \begin-1\\-1\\2\end. Incorporating the 2 to the left-hand side gives :\begin2\\-2\\0\end + \begin -1 & 0 \\ 2 & -1 \\ -1 & 1\end\beginx_2 \\ x_3\end = \begin-1\\-1\\2\end. and removing rows of the system that we already know, and simplifying, leaves us with :\begin-2\\0\end + \begin2 & -1 \\ -1 & 1\end\beginx_2 \\ x_3\end = \begin-1\\2\end. : \begin2 & -1 \\ -1 & 1\end\beginx_2 \\ x_3\end = \begin1\\2\end. so we can then solve :\beginx_2 \\ x_3\end = \begin2 & -1 \\ -1 & 1\end^ \begin1\\2\end = \begin3\\5\end. That is, x_1=2, as prescribed, and x_2=3, leaving the first spring slack, and x_3=5, leaving the second spring slack.


See also

*
Gaussian network model The Gaussian network model (GNM) is a representation of a biological macromolecule as an elastic mass-and-spring (device), spring network to study, understand, and characterize the mechanical aspects of its long-time large-scale dynamics (mechanic ...
* Anisotropic Network Model * Stiffness matrix * Spring-mass system *
Laplacian matrix In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Lap ...


External links


The Physics of Springs
{{DEFAULTSORT:Spring System Springs (mechanical) Elasticity (physics) Solid mechanics