M.A. Basarab − Dr.Sc. (Phys.-Math.), Associate Professor, 

Bauman Moscow State Technical University

E-mail: bmic@mail.ru

B.S. Lunin − Dr.Sc. (Eng.), Leading Research Scientist,

Department of Chemistry, Moscow State University named after M.V. Lomonosov

A.V. Kolesnikov − Ph.D. (Eng.), Associate Professor, 

Bauman Moscow State Technical University

The functioning of Coriolis vibratory gyroscopes (CVGs) is based on the effect of the precession of elastic waves excited in axisymmetric bodies (the Bryan effect). In practice, these bodies (resonators), are usually thin shells of revolution made of elastic materials with high quality factor. The standing wave excited in them moves along a circumferential angle in a plane perpendicular to the axis of symmetry, similarly to a solid body, thus demonstrating its inertial properties. However, it rotates in space by an angle smaller than the angle of rotation of the basement, which allows one to measure the angle of rotation of the latter. The full mathematical description of the SVG dynamics contains systems of partial differential equations of the theory of thin shells. Significant computational power is required for the numerical integration of such systems with sufficient accuracy. In connection with this, given the fact that the main part of the resonator vibrational energy (as well as the maximum amplitude of a standing wave) corresponds to a region adjacent to its edge, a ring model is often considered instead of a shell. The aim of this work is to study the dynamics of free oscillations of a ring resonator mounted on a base rotating at an arbitrary angular velocity. Earlier, for some types of laws of angular velocities, analytical solutions of the equations of dynamics were obtained. However, firstly, the class of possible analytical solutions is rather narrow and does not allow one to study many practically important non-stationary operating modes of the device, including the dynamics of transients, etc. Secondly, analytical solutions are usually presented in the form of infinite series for special functions and require the use of certain asymptotic approximations, depending on the dynamic range of angular velocity. The latter fact practically negates all the advantages of the analytical approach. The proposed numerical-analytical method of solution is based on the combined use of the method of integral transforms in the angle and the method of lines (Rothe) in the time variable. It allows one to take advantage of the exact (analytical) approach, as well as the flexibility of computational finite-difference schemes. Issues related to error estimation and computational complexity of the implemented algorithm are considered. The results are presented that make it possible to evaluate the proposed approach for cases of constant and variable angular velocities when the resonator is excited by the second form of oscillation. As an example of the variable angular velocity, the quadratic law of rotation of the base is chosen, which requires the use of the apparatus of the special Airy functions, as well as their asymptotic approximations through Bessel functions. The method allows one to effectively explore the complex dynamic regimes of operation of a CVG, both of ring type and of common practical designs with resonators in the form of shells of revolution (hemispherical − HRG, cylindrical − CRG, etc.).

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