Free Vibrations of Elastic Systems in Elliptic Domains

Inverse Problems in Vibration

Applied Mechanics Reviews ◽

10.1115/1.3149517 ◽

1986 ◽

Vol 39 (7) ◽

pp. 1013-1018 ◽

Cited By ~ 48

Author(s):

Graham M. L. Gladwell

Keyword(s):

Inverse Problems ◽

Eigenvalue Problems ◽

Discrete Systems ◽

Free Vibrations ◽

Continuous Systems ◽

Thin Beam ◽

Inverse Eigenvalue Problems ◽

Elastic Systems ◽

Inverse Eigenvalue ◽

Sturm Liouville

This article concerns infinitesimal free vibrations of undamped elastic systems of finite extent. A review is made of the literature relating to the unique reconstruction of a vibrating system from natural frequency data. The literature is divided into two groups—those papers concerning discrete systems, for which the inverse problems are closely related to matrix inverse eigenvalue problems, and those concerning continuous systems governed either by one or the other of the Sturm–Liouville equations or by the Euler–Bernoulli equation for flexural vibrations of a thin beam.

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Application of Variational Equation of Motion to the Nonlinear Vibration Analysis of Homogeneous and Layered Plates and Shells

Journal of Applied Mechanics ◽

10.1115/1.3630109 ◽

1963 ◽

Vol 30 (1) ◽

pp. 79-86 ◽

Cited By ~ 18

Author(s):

Yi-Yuan Yu

Keyword(s):

Nonlinear Vibration ◽

Nonlinear Vibrations ◽

Variational Equation ◽

Equations Of Motion ◽

Equation Of Motion ◽

Free Vibrations ◽

Layered Plates ◽

Variational Approximations ◽

Elastic Systems ◽

Plates And Shells

An integrated procedure is presented for applying the variational equation of motion to the approximate analysis of nonlinear vibrations of homogeneous and layered plates and shells involving large deflections. The procedure consists of a sequence of variational approximations. The first of these involves an approximation in the thickness direction and yields a system of equations of motion and boundary conditions for the plate or shell. Subsequent variational approximations with respect to the remaining space coordinates and time, wherever needed, lead to a solution to the nonlinear vibration problem. The procedure is illustrated by a study of the nonlinear free vibrations of homogeneous and sandwich cylindrical shells, and it appears to be applicable to still many other homogeneous and composite elastic systems.

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Free Vibrations of Elastic Systems in Elliptic Domains

High-Precision Methods in Eigenvalue Problems and Their Applications ◽

10.1201/9780203401286.ch15 ◽

2004 ◽

pp. 215-232

Author(s):

Leonid Akulenko

Keyword(s):

Free Vibrations ◽

Elastic Systems

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Amplitude Incremental Variational Principle for Nonlinear Vibration of Elastic Systems

Journal of Applied Mechanics ◽

10.1115/1.3157762 ◽

1981 ◽

Vol 48 (4) ◽

pp. 959-964 ◽

Cited By ~ 169

Author(s):

S. L. Lau ◽

Y. K. Cheung

Keyword(s):

Variational Principle ◽

Nonlinear Vibrations ◽

Elliptic Integral ◽

Nonlinear Problems ◽

Free Vibrations ◽

Linear Solution ◽

Starting Point ◽

Highly Nonlinear ◽

Elastic Systems ◽

Incremental Step

The incremental method has been widely used in various types of nonlinear analysis, however, so far it has received little attention in the analysis of periodic nonlinear vibrations. In this paper, an amplitude incremental variational principle for nonlinear vibrations of elastic systems is derived. Based on this principle various approximate procedures can be adapted to the incremental formulation. The linear solution for the system is used as the starting point of the solution procedure and the amplitude is then increased incrementally. Within each incremental step, only a set of linear equations has to be solved to obtain the data for the next stage. To show the effectiveness of the present method, some typical examples of nonlinear free vibrations of plates and shallow shells are computed. Comparison with analytical results calculated by using elliptic integral confirms that excellent accuracy can be achieved. The technique is applicable to highly nonlinear problems as well as problems with only weak nonlinearity.

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