This thesis analyses a lumped element circuit proposed for an analogue quantum simulation of opto-mechanics. The circuit consists of two resonators, a simple LC-resonator, and a similar resonator in which the inductor is replaced by a SQUID as a flux tuneable inductance. The interaction between the two resonators is established by mutual inductance between the inductor in the LC-resonator and the SQUID loop part of the other resonator. This makes the resonance frequency of the SQUID-resonator a function of the flux in the LC-resonator. It is shown that mutual and self inductances in the SQUID loop give rise to two transcendental constraints, for the magnetic flux in the SQUID loop, and the generalised flux across the SQUID. By an approximate solution for the constraints and under the assumption that the loop inductance is small compared to the SQUID inductance, we derive an approximate description for the circuit dynamics. We demonstrate that the circuit Hamiltonian contains the asymmetric opto-mechanical interaction, but in addition also a self-Kerr non-linearity in the analogue optical cavity, as well as a weak cross-Kerr interaction between the two resonators.