# Study of Systems of Active Vibration Protection of Navigation Instrument Equipment
## 2 Materials and Methods
The use of these methods to determine the qualitative influence picture of the parameters of individual feedback links on the vibration isolator characteristics is complicated if the number of such parameters exceeds 3,..., 4, and the order of equations exceeds 4,..., 5. In this work, a simplified method of visual assessment of the influence of feedback parameters on the effectiveness of the controlled vibration isolator has been proposed by using expressions for the approximate determination of its zeros and poles. A modal regulator is considered as a state regulator. Modal control refers to the root methods of synthesis of linear automatic control systems. With small deviations of the pendulum, the linear state regulator ensures the stability of the equilibrium position. Dynamic vibration isolators are used to achieve a local effect: reducing the vibration activity of the object in the places of their attachment. Often, this can even be associated with the deterioration of the vibrational state of the object in other, less suitable locations. Dynamic vibration isolators can be constructively implemented on the basis of passive elements (masses, springs, dampers) and active elements that have their own energy sources. On the basis of active elements, automatic regulation systems are used, which use electrically, hydraulically and pneumatically controlled elements. The use of active elements expands the possibilities of dynamic vibration damping, as it allows continuous adjustment of the parameters of the dynamic vibration isolator as a function of excitations under conditions of variable vibration loads. A similar result can sometimes be achieved with the help of passive devices with non-linear characteristics. Dynamic vibration damping is used for all types of vibrations: longitudinal, bending, torsional, etc. Such research methods as the use of linear and nonlinear differential equations, methods of automatic control theory, the theory of electromechanical systems and the theory of elasticity, numerical methods, and calculation methods using MatLAB programs, will be used in the research. The structure of the suspension usually consists of mass-produced vibration isolators. There are vibration isolators with elastic damping characteristics, different combinations of vibration-isolating and shock-proofing properties, variations in durability and the ability to function in certain climatic conditions, as well as purely constructive features. All the listed properties are preserved to a certain extent for vibration isolators of the same type for all standard sizes. Let us consider the controlled vibration isolator (Figure 1) [56,57] with feedback, formed according to the principle of introducing \"fictitious\" parameters into the system (Table 1) [50]. Accordingly, the control system contains three parallel control channels with FB: a control channel for the absolute acceleration of the moving platform \(\ddot{x}\), which is measured by a linear accelerometer with a time constant T\({}_{a}\) and a relative damping coefficient \(\zeta_{a}\), a control channel for the speed of the moving platform relative to a stationary one with a low-pass filter (LPF) with the time constant T\({}_{f}\) and the relative displacement control channel with the transmission gain coefficient k\({}_{c}\). The executive mechanism of the controlled vibration isolator is an electrodynamic vibrator with a time constant T\({}_{v}\). Let us use the parameters: T\({}_{0}\) and \(\zeta_{0}\)--time constant and the relative attenuation coefficient of the passive vibration isolator; \(\chi\), \(\xi\), and \(\eta\)--the ratio of the time constants of the accelerometer, vibrator, and LPF to the time constant of the passive vibration isolator; \(\zeta_{\rm c}\) and \(\zeta_{\rm c}\)--the equivalent relative attenuation coefficients introduced by the vibrator and the velocity control channel.