Hybrid model for bar and transducer

 

A mechanical-electrical 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

Y0    full distance of the condenser plates

Q0   total electric charge stored in the capacitor

mi    equivalent masses of the oscillators

ki    their equivalent elastic constants

bi    their equivalent damping factors

C1   capacitance of the decoupling pick-up capacitor

L1    inductance of the superconducting impedance-matching transformer

RL   resistance of the dissipative elements of the superconducting transformer

fG    force acting on the bar due to the passage of a gravitational wave

Electro-mechanical hybrid model of the sub-system made of the bar, the resonant transducer and the resonant impedance matching transformer.

NOTES:

A)   x, y and q are very small compared to Y0 and Q0 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 Y0 and Q0.

C)    the absolute value of the charge Q0 (or of the electric field E0) 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)=C0(1-y/Y0) the capacitance of the transducer ( ),

       Fel the mechanical reaction of the field E(t) on m2:

.

The dynamics equations of the hybrid model in the variables x, y e q are:


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