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

N. F. Morozov, A. V. Lukin, I. A. Popov
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Abstract

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.

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旋转中空圆板的陀螺耦合平面内和平面外振动:旋转轴在平面内的情况
摘要 在这项工作中,我们构建并研究了一个带有同心孔的圆形薄板在科里奥利力和离心惯性力作用下的平面-横向耦合振动模型。振荡偏微分方程是利用 Hamilton-Ostrogradsky 变分原理求得的。假设旋转角速度相对于板振动的偏斜对称弯曲工作模式的频率较小,则可得到自由振动模式下位移场的径向、圆周和横向分量的近似解析解。利用伽勒金投影法,将问题简化为两个二阶线性微分方程系,用于计算板的相互正交基本偏斜对称振动模式的模态坐标。研究发现,在存在旋转的情况下,最初激发的谐波振荡机制转变为振幅调制节拍机制。我们得出了慢节拍包络频率和相对振幅调制因子的分析表达式。我们证明,从根本上可以根据测量到的包络频率值确定角速度矢量在平板平面上的投影模量。我们考虑的问题是选择谐振器的最佳几何形状,使系统对旋转角速度变化的灵敏度最大化。我们还讨论了根据测量到的节拍振幅调制深度确定角速度矢量在板平面上的投影方向的问题。
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来源期刊
CiteScore
0.70
自引率
50.00%
发文量
44
期刊介绍: Vestnik St. Petersburg University, Mathematics  is a journal that publishes original contributions in all areas of fundamental and applied mathematics. It is the prime outlet for the findings of scientists from the Faculty of Mathematics and Mechanics of St. Petersburg State University. Articles of the journal cover the major areas of fundamental and applied mathematics. The following are the main subject headings: Mathematical Analysis; Higher Algebra and Numbers Theory; Higher Geometry; Differential Equations; Mathematical Physics; Computational Mathematics and Numerical Analysis; Statistical Simulation; Theoretical Cybernetics; Game Theory; Operations Research; Theory of Probability and Mathematical Statistics, and Mathematical Problems of Mechanics and Astronomy.
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