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Free in-plane and out-of-plane vibrations of rotating thin ring based on the toroidal shell theory

Senjanović I., Ćatipović I., Alujević N., Čakmak D., Vladimir N.

Archives of Mechanics

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2018

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Vol. 70, nr 5

429--455

EN

In this paper rigorous formulae for natural frequencies of in-plane and out-of-plane free vibrations of a rotating ring are derived. An in-plane vibration mode of the ring is characterised by coupled flexural and extensional deformations, whereas an out-of-plane mode is distinguished by coupled flexural and torsional deformations. The expressions for natural frequencies are derived from a generalised toroidal shell theory. For the in-plane vibrations, the ring is considered to be a short top segment of a toroidal shell. For the out-of-plane vibrations, the ring is considered to be a side segment of the shell. Natural vibrations are analysed by the energy approach. The expressions for the ring strain and kinetic energies are deduced from the corresponding expressions for the torus. It is shown that the ring rotation causes bifurcation of natural frequencies of the in-plane vibrations only. Bifurcation of natural frequencies of the out-of-plane vibrations does not occur. Otherwise, for non-rotating rings, the derived formulae for the natural frequencies of the in-plane and the out-of-plane flexural vibrations are very similar. The derived analytical results are validated by a comparison with FEM and FSM (Finite Strip Method) results, as well as with experimental results available in the literature.

2

Simplified procedure for the free vibration analysis of rectangular plate structures with holes and stiffeners

Cho D. S., Vladimir N., Choi T. M.

Polish Maritime Research

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2015

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nr 2

71--78

EN

Thin and thick plates, plates with holes, stiffened panels and stiffened panels with holes are primary structural members in almost all fields of engineering: civil, mechanical, aerospace, naval, ocean etc. In this paper, a simple and efficient procedure for the free vibration analysis of such elements is presented. It is based on the assumed mode method and can handle different plate thickness, various shapes and sizes of holes, different framing sizes and types as well as different combinations of boundary conditions. Natural frequencies and modes are determined by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equations. Mindlin theory is applied for a plate and Timoshenko beam theory for stiffeners. The applicability of the method in the design procedure is illustrated with several numerical examples obtained by the in-house developed code VAPS. Very good agreement with standard commercial finite element software is achieved.

3

Natural vibrations of thick circular plate based on the modified Mindlin theory

Senjanovic I., Hadzic N., Vladimir N., Cho D.-S.

Archives of Mechanics

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2014

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Vol. 66, nr 6

389--409

EN

Outline of the modified Mindlin theory is presented in which the Mindlin mathematical model with three differential equations of motion for total deflection and rotations is decomposed into a single equation for pure bending vibrations with transverse shear and rotary inertia effects and two differential equations for in-plane shear vibrations. The governing equations are transformed from orthogonal to polar coordinate system for the purpose of circular plate vibration analysis. The fourth order differential equation of flexural vibrations is split further into two second order equations of Bessel type. Also, the in-plane shear differential equations are transformed to Bessel equation by introducing displacement potential functions. The exact values of natural frequencies are listed and compared with FEM results.