Hybrid model for bar and transducer

A mechanicalelectrical hybrid model of detector before the SQUID amplifier is drawn in the picture below. It is made of two mechanical oscillators and an electrical oscillator We use the following notation: x differential longitudinal strain of the bar y relative distance of the capacitor plates q the electric charge flowing through the output inductance Y_{0} full distance of the condenser plates Q_{0} total electric charge stored in the capacitor m_{i} equivalent masses of the oscillators k_{i} their equivalent elastic constants b_{i} their equivalent damping factors C_{1} capacitance of the decoupling pickup capacitor L_{1} inductance of the superconducting impedancematching transformer R_{L} resistance of the dissipative elements of the superconducting transformer f_{G} force acting on the bar due to the passage of a gravitational wave Electromechanical
hybrid model of the subsystem made of the bar, the resonant transducer
and the resonant impedance matching transformer. NOTES: A) x, y and q are very small compared to Y_{0} and Q_{0} so linear approximation is always possible. B) we shall not report the static components of y and q, we can always remove them from the equations redefining Y_{0} and Q_{0}. C) the absolute value of the charge Q_{0} (or of the electric field E_{0}) enter the dynamic equations as a sensitivity tuning coefficient. Let us call · the condenser charge at time t, · the effective electric field between the plates, · C(t)=C_{0}(1y/Y_{0}) the capacitance of the transducer ( ), · F_{el} the mechanical reaction of the field E(t) on m_{2}:
. The dynamics equations of the hybrid model in the variables x, y e q are:
