10.6 Networks of uniform strings

Networks of Uniform Strings

In this module we take an analytic approach to determining the eigenvalues and eigenmodes forvarious planar string networks. In future work we may ask under what conditions can one determinethe structure of the network from eigenvalue information. The networks we considered are uniform,all the strings have the same transverse and longitudinal stiffness. For each network we determined thecharactersitic equation and observed the behavior of the eigenvalues as we varied the transverse stiffness.

We wish to determine eigenvalues and eigenmodes of the planar network wave equation.The planar network wave equation has the following form.

The above says the vector displacement, u _i , of each string in our network, which we have numbered 1 through N , satisfies the wave equation. P _i is a matrix which encodes the stiffness of the string in the longitudinal(along the string)and the transvere(perpendicular to the string) directions. The P _i will have the form

where σ is the longitudinal stiffness, k is the transverse stiffness and ${\widehat{v}}_i$ is a unit vector in the direction of the undistrubed string assuming that the undisturbed string is straight. Without any loss in generality we shall always assume $\sigma =1$

The nodes or vertices in our network are where the strings are joined or “tied”. As a way of keepingtrack of how the strings our connected we number the nodes, 1 through n .

If one is dealing with just one network, to indicate orientation and connectivityof the strings it suffices to draw and label network. To emphasize that at some point we may wish to distinguish betweendifferent networks, we shall be more precise. The connectivity, how the strings are connected,and orientation, how s relate to a strings nodes, can be specified mathemaically using what are called adjacency $\left(\mathcal{E}\right)$ and incidence $\left(\mathcal{D}\right)$ matrices. The adjacency matrix is a square matrix whose dimensionis equal to the number of nodes, n . The incidence matrix is a rectangular matrix whose number of rows is equalto the number of nodes in the network, n , and whose number of columns is equal to the number of edges, N .

There are several boundary conditions which must be satisfied at the nodes.

1. Clamped: Zero Displacement on boundary of network.
2. Continuity: Displacement at a node must be the same for all the strings joined at the node
3. Force Balance: The net force acting on a node due to the strings joined at the node must be zero.