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The response function of a network of springs and masses, an elastodynamicnetwork, is the matrix valued function $W(\omega)$, depending on the frequency$\omega$, mapping the displacements of some accessible or terminal nodes to thenet forces at the terminals. We give necessary and sufficient conditions for agiven function $W(\omega)$ to be the response function of an elastodynamicnetwork, assuming there is no damping. In particular we construct anelastodynamic network that can mimic a suitable response in the frequency ortime domain. Our characterization is valid for networks in three dimensions andalso for planar networks, which are networks where all the elements,displacements and forces are in a plane. The network we design can fit withinan arbitrarily small neighborhood of the convex hull of the terminal nodes,provided the springs and masses occupy an arbitrarily small volume.Additionally, we prove stability of the network response to small changes inthe spring constants and/or addition of springs with small spring constants.

Complete characterization and synthesis of the response function of elastodynamic networks