# Study of Systems of Active Vibration Protection of Navigation Instrument Equipment
## 3 The Mathematical Model  
Figure 2 shows a diagram of the controlled vibration isolator of a gyro-theodolite (GT) built on vertically mounted flat springs, with an electrodynamic actuator (EA) and feedback proportional to the absolute acceleration of the GT body, and its displacement and speed relative to the moving base. In accordance with the adopted control algorithm, the measuring device 3 (MD) of the vibration isolator includes an accelerometer MXS3334 UL and capacitive sensors measuring the relative speed and relative displacement of the moving cylindrical coil 8 of the GT. From the physical point of view, the accelerometer and relative displacement meter signals are used to increase the inertial mass of the GT and reduce the stiffness of its elastic connection with the base, and the relative velocity meter signal is used to increase the damping of low-frequency vibrations of the GT body. In order to reduce the effect of controlled damping on the effectiveness of vibration isolation in the high-frequency region, a low-pass filter (LPF) with a time constant T\({}_{f}\) is installed in the relative velocity meter circuit. Taking into account the dynamics of the accelerometer and the actuator, the system of equations of the controlled vibration isolator takes the form  
\[ \begin{array}{l}m\ddot{x}+\ddot{b}\dot{x}+cx-U\dot{i}=b\ \dot{x}_{e}+cx_{e},\\ v\dot{x}-k_{n}x+L\frac{d\dot{i}}{d\dot{i}}+Ri+U_{a}+U_{f}=v\dot{x}_{e}-k_{f}, \\ -k_{a}\ddot{x}+T_{a}^{2}\ddot{U}_{a}+2\zeta T_{a}U_{a}+U=0,\\ -k_{f}\dot{x}+T_{f}U_{f}+U_{f}=-k_{f}x_{e},\end{array}  \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline
**Name** & **Values, Units of Measurement** \\ \hline Gyroscope kinetic moment & H = 0. 4 Nms \\ \hline Mass of the sensing element & m = 1 kg \\ \hline Distance from the suspension point to the center of mass & 1 = 0.1 m \\ \hline The moment of the SE inertia relative to the \(X\)-axis & J\({}_{\rm x}=1.3\times 10^{-2}\)  kg/m\({}^{2}\) \\ \hline The moment of the SE inertia relative to the \(Y\)-axis & J\({}_{\rm y}=4\times 10^{-4}\)  kg/m\({}^{2}\) \\ \hline The moment of the SE inertia relative to the \(Z\)-axis & J\({}_{\rm z}=1.13\times 10^{-2}\)  kg/m\({}^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the gyro-theodolite. where \(x\) and \(x_{e}\)--displacement of the GT body and the movable base relative to the OXY coordinate system; \(U_{a}\) and \(U_{f}\)--output voltages of the accelerometer and the LPF; \(L\) and \(R\)--inductance and active resistance of the moving coil winding of the EA; \(v\)--electromagnetic coupling coefficient; \(m\)--mass of the device; \(b\) and \(c\)--coefficients of resistance and stiffness of the elastic connection of the GT with the base; T\({}_{a}\) and \(\zeta_{a}\)--time constant and the relative attenuation coefficient of the accelerometer; k\({}_{a}\), k\({}_{f}\) and k\({}_{n}\)--static transmission coefficients of the accelerometer, the LPF, and the relative displacement meter, respectively; T\({}_{f}\)--LPF time constant. In order to analyze the influence of feedback parameters on the stability of the controlled vibration isolator, let us present its characteristic equation in a dimensionless form:  
\[