  The response function of a network of springs and masses, an elastodynamic
network, is the matrix valued function $W(\omega)$, depending on the frequency
$\omega$, mapping the displacements of some accessible or terminal nodes to the
net forces at the terminals. We give necessary and sufficient conditions for a
given function $W(\omega)$ to be the response function of an elastodynamic
network, assuming there is no damping. In particular we construct an
elastodynamic network that can mimic a suitable response in the frequency or
time domain. Our characterization is valid for networks in three dimensions and
also for planar networks, which are networks where all the elements,
displacements and forces are in a plane. The network we design can fit within
an 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 in
the spring constants and/or addition of springs with small spring constants.
