# Study of Systems of Active Vibration Protection of Navigation Instrument Equipment
## 5 Numerical Analysis
\]  
\[\omega_{2 cf}/\omega_{o}\approx\frac{1}{\eta}\sqrt{1-2\Delta\eta(1+k_{m})^{-1} }=0.413\ \ (\mathrm{check}\ =0.46)\end{array}\]\[\left(\omega_{2 nf}^{d}\left/\omega_{o}\right)^{2}\approx\sqrt{\frac{\Delta}{\chi^{2 }\eta(1-\mathrm{k_{c}})+4\zeta_{\mathrm{a}}\chi\eta(\zeta_{\mathrm{o}}+\zeta_{ \mathrm{v}})+2\zeta_{\mathrm{c}}\chi^{2}}}=87.78\qquad\omega_{2 nf}^{d}/\omega_{o }=9.37\ \ (\mathrm{check}\ =10)\]  
\[\left(\omega_{3 nf}\left/\omega_{o}\right)^{2}\approx\frac{1+k_{m}}{\chi^{2}}= 1500\right.\]  
\[\omega_{3 nf}/\omega_{o}=38.73\ \ (\mathrm{check}\ =39.6)\]  
\[\omega_{3 cf}^{d}/\omega_{o}\approx\frac{\chi^{2}\eta(1-\mathrm{k_{c}})+4 \zeta_{\mathrm{a}}\chi\eta(\zeta_{\mathrm{o}}+\zeta_{\mathrm{v}})+2\zeta_{ \mathrm{c}}\chi^{2}}{2\chi^{2}\eta(\zeta_{\mathrm{o}}+\zeta_{\mathrm{v}})}=26.93\ (\mathrm{check}\ =23.74)\]  
\[\omega_{4 cf}/\omega_{o}\approx\frac{1}{\xi}=500\ (\mathrm{check}\ =479.5)\]  
\[\omega_{4 cf}^{d}/\omega_{o}\approx\frac{(\zeta_{\mathrm{o}}+\zeta_{\mathrm{v}} )}{\zeta_{\mathrm{o}}\xi_{\mathrm{v}}}=750\ (\mathrm{check}\ =740.7)\]  
Let us construct the logarithmic amplitude-frequency response of the vibration isolator. Using the developed model, let us summarize the data for building AFR (Table 2). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**M** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline Re3 & 0.3467768 & 0.256749317 & 0.117519317 & \(-0.11  
Figure 5 shows the AFR of the controlled and passive vibration isolator (dots show the asymptotic AFR), constructed for the given example as a function of the relative frequency of disturbance. From numerical calculations, for cases \(T_{v}\gg T_{a}(\xi>>\chi)\) and \(T_{a}\gg T_{v}\) (\(\chi>>\delta\)), and from the presented characteristics it can be seen that, even with the closeness of the low-frequency root modulus values, the use of simplified formulas makes it possible to use them to construct the asymptotic AFR of the controlled vibration isolator and, thus, determine the qualitative influence of this or that feedback parameter and executive mechanism on the effectiveness of vibration isolation. The application of the obtained results makes it possible to outline the main requirements for the characteristics of sensitive elements and the executive mechanism of the controlled vibration isolator, already at the first stage of design. At the same time, two cases are of practical interest:  
1. vibration protection against micro-vibration in the low-frequency range;
2. vibration protection against intense vibration in a wide frequency range. In the first case, we recommend using a low-frequency accelerometer and a high-frequency vibrator. This is explained by the fact that, in the vast majority of cases, the sensitivity threshold of low-frequency accelerometers is higher than the sensitivity threshold of high-frequency ones. From the point of view of technical implementation, it is easier to achieve a small time constant of the vibrator (or correct it). It should be noted that increasing the damping level of the accelerometer significantly reduces the resonant oscillations of the device in the frequency range:  
\[\omega/\omega_{o}\approx\sqrt{1+k_{m}}/\chi \tag{18}\]  
Figure 5: ARF of an active vibration isolator (\(\omega\)—frequency of vibration of a limited moving base). In the second case, on the contrary, the choice of a high-frequency accelerometer and a low-frequency vibrator is more appropriate. Figure 5 shows a graph of vibration level (dB) versus frequency (Hz). As can be seen, in the area from 0-6 of the \(T\)-axis, the vibration level is increased and ranges from 1.2-1.4 in this area. Moreover, the highest value corresponds to 1.47 dB. For some time, there has been a gradual decrease in the values of vibration indicators to the level of 0.001 dB. At the same time, at sections 21-26 of the \(T\)-axis, there is a jump in the values to the level of 0.09 dB. After that, the curve again monotonously demonstrates a decrease in vibration level.