Dmitry Maslov

This article is concerned with investigating the nonlinear dynamics of the cylindrical resonator of a wave solid-state gyroscope. The nonlinearity of oscillations caused by the nonlinear properties of electrostatic control sensors is considered. This nonlinearity is derived by taking into account the finite ratio of resonator flexure to the small gap of electrostatic control sensors. The equations of the electromechanical system that in interconnected form describe the nonlinear mechanical oscillations of the gyroscope resonator and electrical oscillations in the control circuit are derived. The resulting differential equations belong to the class of Tikhonov systems, since the equation of electrical processes in the control circuit is singularly perturbed. By taking into account the low electrical resistance of the oscillation control circuit, which determines a small parameter at the derivative in the singularly perturbed equation of electrical processes, the nonlinear oscillations of the wave solid-state gyroscope resonator are studied. The small parameter method is used to obtain a mathematical model of the resonator dynamics, which jointly takes into account the nonlinearity of the resonator oscillations and the electrical resistance of the oscillation control circuit. A special method is proposed to reduce the nonlinear equations of the resonator dynamics to the standard form of the system of differential equations for averaging and the equations of the dynamics of the wave solid-state gyroscope resonator are averaged. It is shown that, in the case of nonlinear oscillations, consideration of the electrical resistance of the oscillation control circuit does not affect the angular velocity of the gyroscope drift, but causes slight dissipation of the oscillations, which also leads to an insignificant correction of the resonant frequency.

The dynamics of a vibrating ring microgyroscope operating in the forced oscillation mode is investigated. The elastic and viscous anisotropy of the resonator and the nonlinearity of oscillations are taken into consideration. Additional nonlinear terms are suggested for the mathematical model of resonator dynamics. In addition to cubic nonlinearity, nonlinearity of the fifth degree is considered. By using the Krylov – Bogolyubov averaging method, equations containing parameters characterizing damping, elastic and viscous anisotropy, as well as coefficients of oscillation nonlinearity are deduced. The parameter identification problem is reduced to solving an overdetermined system of algebraic equations that are linear in the parameters to be identified. The proposed identification method allows testing at large oscillation amplitudes corresponding to a sufficiently high signal-to-noise ratio. It is shown that taking nonlinearities into account significantly increases the accuracy of parameter identification in the case of large oscillation amplitudes.

A wave solid-state gyroscope with a cylindrical resonator and electrostatic control sensors is considered. The gyroscope dynamics mathematical model describing nonlinear oscillations of the resonator under voltage on the electrodes is used. The reference voltage causes a cubic nonlinearity and the alternating voltage causes a quadratic nonlinearity of the control forces.
Various regimes of supplying voltage to gyro sensors are investigated. For the linearization of oscillations the form of voltages on the electrodes is presented. These voltages compensate for both nonlinear oscillations of the resonator caused by electrostatic sensors and those caused by other physical and geometric factors. It is shown that the control forces have a nonlinearity that is eliminated by the voltage applied to the electrode system according to a special law.
The proposed method can be used to eliminate nonlinear oscillations and to linearize power characteristics of sensors for controlling wave solid-state gyroscopes with hemispherical, cylindrical and ring resonators.

The dynamics of a vibrating ring microgyroscope resonator with open-loop and closed feedback is investigated. We use a mathematical model of forced oscillations for thin elastic resonator, taking into account the nonlinearity coefficient, uneven stiffness, difference in Q-factors and control impact parameters. Using the Krylov–Bogolyubov averaging method, the resonator dynamics in slow variables measured by microgyroscope electronics has been investigated. Formulas with algorithmic compensation of the above defects for determining the angular velocity of the resonator under nonlinear oscillations and without feedback have been obtained. Control signals taking into account the defects are presented for feedback of the microgyroscope operating in the compensation mode of the angular velocity sensor. Numerical modeling of angular velocity determination in the operation modes considered has been carried out.

This paper develops the holomorphic regularization method of the Cauchy problem for a special type of Tikhonov system that arises in the mathematical modeling of wave solidstate gyroscope dynamics. The Tikhonov system is a system of differential equations a part of which is singularly perturbed. Unlike other asymptotic methods giving approximations in the form of asymptotically converging series, the holomorphic regularization method allows one to obtain solutions of nonlinear singularly perturbed problems in the form of series in powers of a small parameter converging in the usual sense. Also, as a result of applying the holomorphic regularization method, merged formulas for an approximate solution are deduced both in the boundary layer and outside it. These formulas allow a qualitative analysis of the approximate solution on the entire time interval including the boundary layer.
This paper consists of two sections. In Section 1, the holomorphic regularization method of the Cauchy problem for a special type of Tikhonov system is developed. The special type of Tikhonov system means the following: singularly perturbed equations are linear in the variables included in them with derivatives, the matrix of the singularly perturbed part of the system is diagonal, the remaining equations have separate linear and nonlinear parts. An algorithm for deriving an approximate solution to the Cauchy problem for the Tikhonov system of special type by using the holomorphic regularization method is presented. In Section 2, the mathematical model describing in interconnected form the mechanical oscillations of the gyroscope resonator and the electrical processes in the oscillation control circuit is considered. The algorithm for deriving an approximate solution proposed in Section 1 is used. Formulas for an approximate solution taking into account the structure of the Tikhonov system are deduced.