Gyroscopically Coupled In-Plane and Out-of-Plane Vibrations of Rotating Hollow Circular Plate: Case of In-Plane Axis of Rotation

In this work, we construct and study a model of the coupled plane-transverse vibrations of a circular thin plate with a concentric hole under the action of Coriolis and centrifugal inertial forces caused by rotation of the system around an axis located in the plane of the plate. The partial differential equations of oscillations are obtained using the Hamilton-Ostrogradsky variational principle. Under the assumption that the angular velocity of rotation is small relative to the frequency of the operational skew-symmetric bending mode of plate vibrations, an approximate analytical solution is obtained for the radial, circumferential, and transverse components of the displacement field in the free vibration mode. Using the Galerkin projection, the problem was reduced to a system of two second-order linear differential equations for modal coordinates of mutually orthogonal basic skew-symmetric vibration modes of the plate. It is discovered that the regime of initially excited harmonic oscillations in the presence of rotation is transformed into a regime of amplitude-modulated beats. Analytical expressions are derived both for the frequency of the slow beat envelope and for the relative amplitude-modulation factor. We show that it is fundamentally possible to determine the modulus of the projection of the angular-velocity vector onto the plane of the plate from the measured value of the envelope frequency. We consider the problem of choosing the optimal geometric shape of the resonator for maximizing the sensitivity of the system to changes in the angular velocity of rotation. We also address the question of determining the direction of the projection of the angular velocity vector onto the plane of the plate from the measured depth of amplitude modulation of the beat regime.