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Research Papers

# Analysis of Nonlinear Free Vibration of Circular Plates With Cut-Outs Using $R$-Function Method

[+] Author and Article Information
K. V. Avramov, O. Tyshkovets

Department of Nonstationary Vibrations, National Academy of Sciences of Ukraine, 2/10 Dm. Pozharskoho Street, 61046 Kharkiv, Ukraine

K. V. Maksymenko-Sheyko

Department of Applied Mathematics and Numerical Methods, A.N. Podgorny Institute for Mechanical Engineering Problems, National Academy of Sciences of Ukraine, 2/10 Dm. Pozharskoho Street, 61046 Kharkiv, Ukrainekvavr@kharkov.ua

J. Vib. Acoust 132(5), 051001 (Aug 18, 2010) (11 pages) doi:10.1115/1.4001496 History: Received August 05, 2008; Revised February 23, 2010; Published August 18, 2010; Online August 18, 2010

## Abstract

Geometrically nonlinear vibrations of circular plate with two cut-outs are simulated by the von Karman equations with respect to displacements. The combination of the Rayleigh–Ritz method and the $R$-function method, which allows satisfying all boundary conditions, is applied to obtain the vibration modes of the plate. The nonlinear vibrations are expanded using these vibrations modes. The dynamical system with three degrees of freedom is derived by Galerkin method. The influence of cut-outs size on linear and nonlinear vibrations of the plate is analyzed. For different parameters of cut-outs, different internal resonances occur in the plate. The nonlinear vibrations of the system for different internal resonances are analyzed by multiple scale method.

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## Figures

Figure 1

Circular plate with two cut-outs

Figure 2

The steps of R-function derivation

Figure 3

Function ω2

Figure 4

The eigenmodes of bending vibrations

Figure 5

Backbone curves for the case of internal resonance 2ν1≈ν2

Figure 6

Backbone curves. (a) Cases A and C of Eq. 38. (b) Case Е of Eq. 38.

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