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 discre ...
with a position at each vertex and a
spring Spring(s) may refer to: Common uses * Spring (season) Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
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 (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
to higher dimensions. This simple model can be used to solve the pose of static systems from
crystal lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
to springs. A spring system can be thought of as the simplest case of the
finite element method The 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 ...
for solving problems in
statics Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with ...
. 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 one 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 three ...
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

If the nominal lengths, ''L'', of the springs are known to be 1 and 2 units respectively, then the system can be solved as follows: Consider the simple case of three nodes connected by two springs. Then the stretching of the two springs is given as a function of the positions of the nodes by : \Delta \mathbf = \begin1 & -1 & 0\\ 0 & 1 & -1\end \mathbf - \mathbf = B^\top \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
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 ...
: B = \begin1 & 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 diagonal ma ...
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 the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
, 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. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
then :B W B^\top = \begin1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1\end is the Laplacian matrix. 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 * Anisotropic Network Model *
Stiffness matrix In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution ...
*
Spring-mass system In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
* Laplacian matrix


External links


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