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TECHNICAL PAPERS

# Nonlinear Vibrations of Viscoelastic Composite Cylindrical Panels

[+] Author and Article Information
Bakhtiyor Eshmatov

Institute of Mechanics and Seismic Stability of Structures, Akademgorodok, Tashkent, 700143, Uzbekistanebkh@mail.ru

Subrata Mukherjee

Department of Theoretical and Applied Mechanics, Kimball Hall, Cornell University, Ithaca, NY 14853sm85@cornell.edu

J. Vib. Acoust 129(3), 285-296 (Nov 04, 2006) (12 pages) doi:10.1115/1.2730532 History: Received March 02, 2006; Revised November 04, 2006

## Abstract

This paper is devoted to mathematical models of problems of nonlinear vibrations of viscoelastic, orthotropic, and isotropic cylindrical panels. The models are based on Kirchhoff-Love hypothesis and Timoshenko generalized theory (including shear deformation and rotatory inertia) in a geometrically nonlinear statement. A choice of the relaxation kernel with three rheological parameters is justified. A numerical method based on the use of quadrature formulas for solving problems in viscoelastic systems with weakly singular kernels of relaxation is proposed. With the help of the Bubnov-Galerkin method in combination with a numerical method, the problems in nonlinear vibrations of viscoelastic orthotropic and isotropic cylindrical panels are solved using the Kirchhoff-Love and Timoshenko hypothesis. Comparisons of the results obtained by these theories, with and without taking elastic waves propagation into account, are presented. In all problems, the convergence of Bubnov-Galerkin’s method has been investigated. The influences of the viscoelastic and anisotropic properties of a material, on the process of vibration, are discussed in this work.

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Topics: Vibration

## Figures

Figure 1

Model geometry

Figure 2

The numerical convergence of the Bubnov-Galerkin method: 1—N=1; 2—N=3; 3—N=7; 4—N=11

Figure 3

Dependence of the deflection on time for various values of A: 1—A=0.0; 2—A=0.05; 3—A=0.1

Figure 4

Dependence of the deflection on time for various values of α: 1—α=0.15; 2—α=0.25; 3—α=0.75

Figure 5

The influence of the material orthotropicity (qkl=1): 1—Δ=1; 2—Δ=2; 3—Δ=3

Figure 6

Dependence of the deflection on time for various kernels of relaxation: 1—exponential kernel; 2—weakly-singular kernel of Koltunov-Rzhanitsyn

Figure 7

Dependence of the deflection on time for various values of geometrical parameter θ: 1—θ=6; 2—θ=12; 3—θ=18

Figure 8

Dependence of the deflection on time for various theories: 1—q=3, λ=1; 2—q=1, λ=1.5, with account 3—q=3, λ=1; 4—q=1, λ=1.5, without account of elastic waves propagation

Figure 9

The influence of the initial imperfections: 1—w0=10−4; 2—w0=0.6∙10−4; 3—w0=0.2∙10−4

Figure 10

Dependence of the deflection on time for various values of λ: 1—λ=1; 2—λ=2; 3—λ=3

Figure 11

The influence of the static shearing edge load q: 1—q=0; 2—q=0.001; 3—q=0.002

Figure 12

The influence of the nonlinear properties of the plate material: 1—λ=1; q=1; w0=10−1; 2—λ=1; q=3; w0=10−4; 3—λ=3; q=1; w0=10−4, linear case; 4—λ=1; q=1; w0=10−1; 5—λ=1; q=3; w0=10−4; 6—λ=3; q=1; w0=10−4, nonlinear case

Figure 13

The influence different viscoelastic properties of the material (qkl=1): 1—A=A13=A23=0.05, Aij=0, i,j=1,2; 2—A=Aij=0.05, i=1,2, j=1,2,3; 3—A=0.05, A11=0.06, A12=0.07, A21=0.08, A22=0.09, A13=0.1, A23=0.11

Figure 14

Dependence of the deflection on time for various theories: 1—Timoshenko; 2—Kirchhoff-Love

Figure 15

Dependence of the deflection on time for various values of geometrical parameter δ: 1—δ=15; 2—δ=20; 3—δ=25

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