# Study of Systems of Active Vibration Protection of Navigation Instrument Equipment
## 5 Numerical Analysis  
To simulate the operation of the controlled vibration isolator, a software dynamic model was developed in the Simulink/MATLAB package. The parameters of the device are selected on the basis of known designs of accelerometers, the values of which are presented in Table 1.  
The case \(\mathrm{T}_{v}>>\mathrm{T}_{a}\) (\(\xi>>\chi\))  
\[k_{m}=8;\,\chi=\mathrm{T}_{a}/\mathrm{T}_{\mathrm{o}}=0.002;\,\xi=\mathrm{T}_{v}/ \mathrm{T}_{\mathrm{o}}=0.05;\,\zeta_{\mathrm{o}}=0.04;\,\zeta_{\mathrm{p}}=0.02; \,\zeta_{\mathrm{a}}=0.2;\]  
\[\beta=\omega_{v}/\omega_{\mathrm{o}}=3;\,k_{c}=0.5;\,\delta=\omega_{c}/\omega_{ \mathrm{o}}=0.24;\,\eta=\mathrm{T}_{f}/\mathrm{T}_{\mathrm{o}}=1.19;\,\zeta_{ \mathrm{c}}=1.13;\]  
\[\Delta=2(\zeta_{\mathrm{c}}+\zeta_{\mathrm{o}}+\zeta_{\mathrm{v}})+\eta(1- \mathrm{k}_{\mathrm{c}})=2.97.\]  
\[\sqrt{1-2\eta\Delta(1+k_{m})^{-1}}=0.4632\]  
\[\left(\omega_{1 nf}/\omega_{o}\right)^{2}\approx\frac{1-k_{c}}{1+k_{m}}\cdot \frac{1}{\sqrt{1-2\eta\Delta(1+k_{m})^{-1}}}\]  
\[=0.055555\cdot 2.1589=0.12\omega_{1 nf}/\omega_{o}=0.346\ (\mathrm{check}\ -0.346)\]  
\[\omega_{1 cf}^{d}/\omega_{o}\approx\frac{1-k_{c}}{\Delta}=0.168\ \ (\mathrm{check}\ =0.167)\]  
\[ \begin{array}{l}\xi_{1}\approx\frac{1}{2\eta}\sqrt{\frac{1+k_{m}}{1-k_{c}}} (1-\sqrt{1-2\Delta\eta(1+k_{m})^{-1}})\cdot\sqrt[4]{1-2\eta\Delta(1+k_{m})^{- 1}}\\ \phantom{\xi_{1}\approx}=0.42\cdot 4.24\cdot 0.537\cdot 0.68=0.65\ \ (\mathrm{check}\ =0.56)\end{array} \]  
\[\omega_{2 cf}/\omega_{o}\approx\frac{1}{\eta}\sqrt{1-2\Delta\eta(1+k_{m})^{-1} }=0.39\ \ (\mathrm{check}\ -0.47)\]  
\[\omega_{2 nf}^{d}/\omega_{o}\approx\sqrt{\frac{\Delta}{2\zeta_{\mathrm{o}}\eta \xi}}=24.98\ \ (\mathrm{check}\ =25.06)\]  
\[\omega_{3 cf}/\omega_{o}\approx\frac{1+k_{m}}{\xi}=180\ (\mathrm{check}\ =182)\]  
\[\omega_{3 nf}^{d}/\omega_{o}\approx\frac{1}{\chi}=500\ (\mathrm{check}\ =499.9)\]  
\[\omega_{4 nf}/\omega_{o}\approx\frac{1}{\chi}=500\ (\mathrm{check}\ 494)\]  
The case \(\mathrm{T}_{a}>>\mathrm{T}_{v}\) (\(\chi>>\xi\))  
\[ \begin{array}{l}k_{m}=14;\,\chi=\mathrm{T}_{a}/\mathrm{T}_{\mathrm{o}}=0.1; \,\zeta=\mathrm{T}_{v}/\mathrm{T}_{\mathrm{o}}=0.002;\,\zeta_{\mathrm{o}}=0.04; \,\zeta_{\mathrm{o}}=0.02;\,\zeta_{\mathrm{a}}=0.2;\\ \phantom{\xi_{1}\approx}\beta=\omega_{v}/\omega_{\mathrm{o}}=3;\,k_{c}=0.5;\, \delta=\omega_{c}/\omega_{\mathrm{o}}=0.18;\,\eta=\mathrm{T}_{f}/\mathrm{T}_{ \mathrm{o}}=1.35;\,\zeta_{\mathrm{c}}=1.52;\\ \Delta=2(\zeta_{\mathrm{c}}+\zeta_{\mathrm{o}}+\zeta_{\mathrm{v}})+\eta(1- \mathrm{k}_{\mathrm{c}})=3.83.\end{array} \]  
\[\sqrt{1-2\eta\Delta(1+k_{m})^{-1}}=0.5573\]  
\[\chi^{2}\eta(1-\mathrm{k}_{\mathrm{c}})+4\zeta_{\mathrm{a}}\chi\eta(\zeta_{ \mathrm{o}}+\zeta_{\mathrm{v}})+2\zeta_{\mathrm{c}}\chi^{2}=0.00675+0.00648+0. 0304=0.04363\]  
\[\left(\omega_{1 nf}/\omega_{o}\right)^{2}\approx\frac{1-k_{c}}{1+k_{m}}\cdot \frac{1}{\sqrt{1-2\eta\Delta(1+k_{m})^{-1}}}=0.03333\cdot 1.794=0.0598\]  
\[\omega_{1 nf}/\omega_{o}=0.244\ \ (\mathrm{check}\ =0.233)\]  
\[\omega_{1 cf}^{d}/\omega_{o}\approx\frac{1-k_{c}}{\Delta}=0.1305\ \ (\mathrm{check}\ =0.131)\]  
\[ \begin{array}{l}\xi_{1}\approx\frac{1}{2\eta}\sqrt{\frac{1+k_{m}}{1-k_{c}}} (1-\sqrt{1-2\Delta\eta(1+k_{m})^{-1}})\cdot\sqrt[4]{1-2\eta\Delta(1+k_{m})^{- 1}}\\ \phantom{\xi_{1}\approx}=0.37\cdot 5.477\cdot 0.4427\cdot 0.7465=0.67\ \ (\mathrm{check}\ =0.64)\end{array}