This is a problem very familiar to physical chemists, because a polyatomic molecule is a set of atoms connected by bonds, which can be approximated by springs. The molecular vibrations are quantized, but apart from that, it is the same problem (the classical version is easier), and it is one that undergrads routinely learn to solve.

The end points/joints on the bicycle frame are the "atoms", and the tubes that connect them are the "bonds."

In both cases, the bonds can be treated, to excellent approximation, as simple harmonic oscillators (springs). The motions are the normal vibrational modes.