# 10.3 The physics of springs

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This report summarizes work done as part of the Physics of Strings PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. This module describes experiments done on a spring system.

## A network of springs

Our research is centered on a network of springs, built by Jeff Hokansen and Dr. Mark Embree for use in the CAAM 335 Lab. Over the table, we set up a webcam on a beam and connected it to a computer running MATLAB. Springs are connected to pennies (nodes), two of which are fixed to the table. Along the outside pennies, strings run over pullies set along the edge of the table and are attached to hooks, upon which we hang masses. These masses cause the nodes to move. We use the webcam to capture an image of the network, then use a MATLAB script to find the center of each node; the pennies have been painted red to make it easier for MATLAB to detect them. This gives us the displacement of each node, from which we can compute the elongation of each spring. We also know the force applied to each node ( $9.8*mass$ in units of Newtons) and can calculate the spring constant k for each spring using Hooke's Law, ${f}_{\mathrm{restoring}}=-\left(\mathrm{elongation}\right)*k$

## A forward problem

In the forward problem, we seek to compare results from our physical model to the results predicted by solving a linear system of equations. Specifically, we wish to predict our displacements, given we know the load forces and spring constants in our system of springs.

Let us begin with an easier system of just two springs, three nodes, and two forces. Since only two of the nodes are moving, we will have two horizontal displacements denoted in the vector x . There are two elongations, one for each spring, denoted in the vector e .

$x=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right],\phantom{\rule{1.em}{0ex}}e=\left[\begin{array}{c}{e}_{1}\\ {e}_{2}\end{array}\right],$

Each spring elongation is a linear combination of node displacements. The equations can be written in the following manner.

$e=\left[\begin{array}{c}{e}_{1}\\ {e}_{2}\end{array}\right]=\left[\begin{array}{c}{x}_{1}\\ {x}_{2}-{x}_{1}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ -1& 1\end{array}\right]x=Ax$

Now we have our adjacency matrix, A . This translates us from node displacement to spring elongation. It will have one more property which will we shall see shortly. Now let us consider finding the restoring force, y , which will have one component for each spring.

$y=\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\end{array}\right],$

We assume that each spring follows Hooke's Law, $y=ke$ , where restoring force is directly proportional to elongation. Each spring has a corresponding stiffness, k i which comprise the the diagonal elements of matrix, K .

$y=\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\end{array}\right]=\left[\begin{array}{c}{k}_{1}{e}_{1}\\ {k}_{2}{e}_{2}\end{array}\right]=\left[\begin{array}{cc}{k}_{1}& 0\\ 0& {k}_{2}\end{array}\right]e=Ke=KAx$

The final step is to translate these restoring forces into the load forces acting on each node, denoted by vector f .

$f=\left[\begin{array}{c}{f}_{1}\\ {f}_{2}\end{array}\right]=\left[\begin{array}{c}{y}_{1}-{y}_{2}\\ {y}_{2}\end{array}\right]=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]={A}^{T}y$

Now we can see the second feature of the adjacency martrix. The transpose of A performs the reverse translation from edges to nodes. The final product of this example is the equation just shown: $f={A}^{T}KAx$ . Now we can expand the problem to any system of springs for which we can create an adjacency matrix A. For this project we focused on the spring network shown below.

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