# Study of Systems of Active Vibration Protection of Navigation Instrument Equipment
## 4 Building the Boundaries of Sustainability
### Vibration Isolation Efficiency  
The effectiveness of vibration isolation in case of kinematic disturbance is understood as the degree of realization of the goals of vibration protection by the anti-vibration device. In the case of force harmonic excitation, the goal of protection may be to reduce the amplitude of the force transmitted to a stationary object, or to reduce the amplitude of constant forced oscillations of the source. In the case of kinematic harmonic excitation, the protection consists in reducing the amplitude of absolute acceleration (overload) of the object, as well as in reducing the amplitude of its oscillations relative to the base of the object. Quantitatively, the degree of implementation of the goal of vibration protection can be characterized by the values of dimensionless efficiency coefficients. The conditions for the effectiveness of vibration protection are formulated in the form of inequalities according to the relevant criteria. Since the specified coefficients depend on frequency, we can talk about the effectiveness of vibration protection at a given frequency or in a given frequency range. The value of the coefficient \(k_{m}\), selected according to the condition of ensuring the stability of the controlled vibration isolator, depends on the ratio between the time constants of the accelerometer and the vibrator, and takes the largest possible value at \(T_{v}\gg T_{a}\)) (\(\xi>>\chi\)), or \(T_{v}\ll T_{a}\)) (\(\xi<<\chi\)) [44; 45]. With a power disturbance, the effectiveness of vibration isolation is determined by the poles of the frequency response, as in the case of the actual parameters of the controlled vibration isolator--\(\zeta_{o}<<1\), \(\xi_{v}<<1\), \(\xi<<\eta\) and \(\chi<<\eta\), and the choice of the maximum possible \(k_{m}\)that which does not violate the stability of the vibration isolator--are separated. Taking into account the above, the ratio of the lower natural and coupled frequencies of the controlled vibration isolator to the natural frequency of the passive vibration isolator, \(\
u_{1,2,3}\), can be estimated by the following formulas:  
\[\
u_{1,2}^{2}\approx\frac{1-k_{c}}{1+k_{m}}\cdot\frac{1}{\sqrt{1-2\eta\Delta(1+k _{m})^{-1}}};\ v_{3}\approx\frac{1}{\eta}\sqrt{1-2\Delta\eta(1+k_{m})^{-1}}, \tag{8}\]  
where \(\Delta=2\Big{(}\varsigma_{\rm c}+\varsigma_{\rm o}+\varsigma_{\rm v}\Big{)} +\eta(1-\varsigma_{\rm c})\). A similar estimate for higher frequencies \(\
u_{4,5,6}\) depends on the ratio between \(T_{v}\) and \(T_{a}\): where \(\xi>>\chi\)  
\[\
u_{4}\approx\frac{1+k_{m}}{\xi},\hskip 28.452756 pt\
u_{5,6}\approx\frac{1}{ \chi}, \tag{9}\]  
and when \(\chi>>\xi\)  
\[\
u_{4,5}^{2}\approx\frac{1+k_{m}}{\chi^{2}},\hskip 28.452756 pt\
u_{6}\approx \frac{1}{\xi}. \tag{10}\]  
The damping of low-frequency vibrations of the controlled vibration isolator is determined by the real part of  
\[\text{Re}\
u_{1,2}\approx-\frac{1}{2\eta}\bigg{(}1-\sqrt{1-2\Delta\eta(1+k_{m} )^{-1}}\bigg{)}. \tag{11}\]  
The highest of these frequencies approaches, in the case of \(T_{v}\gg T_{a}\), the natural frequency of the accelerometer and, in the case of \(T_{v}\gg T_{a}\), the coupled frequency of the vibrator. The relative attenuation coefficient of low-frequency vibrations of the controlled vibration isolator can be estimated by the formula  
\[\varsigma_{1}\approx\frac{1}{2\eta}\sqrt{\frac{1+k_{m}}{1-k_{c}}}\ (1-\sqrt{1-2\Delta\eta(1+k_{m})^{-1}} \bigg{)}\cdot\sqrt[4]{1-2\eta\Delta(1+k_{m})^{-1}}. \tag{12}\]  
Under a kinematic disturbance, the efficiency of vibration isolation systems is determined not only by the vibration isolator's own dynamics, but also by the mechanism of forming an external disturbance. Under real parameters, the zeros of the controlled vibration isolator are also separated and determined at \(\xi>>\chi\) by the roots of the equation  
\[