# Study of Systems of Active Vibration Protection of Navigation Instrument Equipment
###### Abstract
First, it is necessary to determine the stiffness matrix and the mass matrix of the system. Next, the root method consists in finding the eigenvalues (frequencies) and eigenvectors of the stiffness matrix. Eigenvectors are vectors that determine the form of the system oscillation at the corresponding eigenfrequencies. The root method is one of the ways to determine the natural frequencies of a controlled vibration isolator. This method consists in measuring the amplitude of system oscillations at different frequencies of the external disturbance and constructing a graph of the amplitude  
Figure 1: Controlled vibration isolator, (1—object of vibration isolation; 2—moving platform; 3—block of measuring devices; 4—flat springs; 5—conditionally immobile platform; 6—base; 7—feedback unit; 8—electrodynamic vibrator; \(x\) and \(\mathbf{x_{c}}\)—displacement of the moving and conditionally fixed platform relative to the base, respectively). dependence on the frequency. The natural system frequency is defined as the frequency at which the amplitude of oscillations reaches its maximum value. The application of the root method requires the use of special devices, such as a vibrometer, which allows the measurement of the system oscillations' amplitude, and a signal generator to create an external disturbance [10]. When using the root method, it is also necessary to take into account the influence of factors that can affect the accuracy of measurements, for example, vibrations from other sources, electromagnetic interference, etc. [11; 12; 13]. In general, determining the natural frequencies of the controlled vibration isolator by the root method is a significant stage in the development and operation of precision systems [14; 15; 16; 17]. This method allows for maximum accuracy and stability of system operation and helps to avoid possible errors and unforeseen situations when working with such equipment. Internal resonances inside vibration isolators increase force transmission and, as a result, radiated noise from supporting structures [18]. This article proposes a new concept, i.e., a uniaxial vibration isolator, which includes transmission absorbers to suppress internal resonance. For vibration isolators with a high degree of damping, the bandwidth is already quite low at higher frequencies. However, internal resonances can have a significant effect on the sound power emitted by the supporting structure [19]. The article shows that this effect increases to 22 dB in the frequency range of 200-3000 Hz compared to an ideal vibration isolator. The system consisted of a primary mass with three degrees of freedom (rebound, pitch, and roll), supported by three vibration isolators on a rectangular plate. To reduce the effect of internal resonance, [PERSON] et al. [20] developed a composite vibration isolator that included two active dynamic vibration absorbers (DVAs) attached to intermediate masses. The power transfer in the two frequency bands was minimized by optimizing the passive properties of the DVA. Experiments demonstrated a decrease in power transmission and acoustic power of sound up to 20 dB and 4.3 dB, respectively. In [21], a similar system with vibration isolators consisting of periodic structures was studied to suppress the effect of internal resonance on the transmitted force and radiated noise. The importance of resonances in active vibration isolation systems was recognized in [22], which demonstrated velocity feedback and the failure to suppress this. In addition, internal resonances have been shown to cause stability problems. The behavior of an active isolation system with a biased connection is investigated in [23] subject to the limitations of the output signal of the actuator. Distributed parameter models are developed to analyze system response and generate a transfer matrix for designing an integrated passive-active isolation system.