Variational Problem on Vibrations of Unequal-Thickness Rings and Its Application for Calculating Ultrasonic Vibration Concentrators

The paper considers a method for calculating the natural frequencies of vibrations of unequal-thickness rings, based on application of Hamilton’s variational principle and theories of vibrations of curved beams of the Euler-Bernoulli and Timoshenko type. Solutions of the problem are represented as Fourier series providing possibility of its reduction to the system of linear algebraic equations. The problem of determining natural frequencies is reduced to a generalized problem for the eigenvalues of matrices. Based on a comparison of the numerical results obtained for an eccentric ring with the results of calculations by the finite element method, the advantages of using the Timoshenko theory are shown, including increased calculation accuracy and the possibility to identify radial and radial-flexural eigenmodes. The possibility of reducing computational costs when using the Timoshenko theory is explored by representing the determinant of the block matrix describing the problem as a product of lower-order determinants. It is shown that the relations obtained on the basis of the Euler-Bernoulli theory, in the particular case of equal-thickness ring, lead to the well-known analytical formulas for the natural frequencies of the ring oscillations. The obtained results can be used to calculate ring concentrators of ultrasonic vibrations. The advantage of the proposed method in comparison with other known approaches, for example, the harmonic balance me-thod, consists in no need for the work with differential or integral-differential equations of vibrations, which are a rather complex structure for the case of unequal-thickness rings and require the use of computationally expensive operations, for example, discrete convolution, for their solution.

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