Eigenvalue problems for free vibrations of rectangular elastic plates with random inhomogeneities

1976 ◽  
pp. 533-539
Author(s):  
A. D. Wood ◽  
F. D. Zaman
Keyword(s):  
Eigenvalue Problems ◽  
Free Vibrations ◽  
Elastic Plates

The Loading-Frequency Relationship in Multiple Eigenvalue Problems

1971 ◽  
Vol 38 (4) ◽  
pp. 1007-1011 ◽  
Author(s):  
K. Huseyin ◽  
J. Roorda
Keyword(s):  
Eigenvalue Problems ◽  
Stability Boundary ◽  
Conservative System ◽  
Free Vibrations ◽  
Degree Of Freedom ◽  
External Loading ◽  
Loading Frequency ◽  
Multiple Loading ◽  
The Stability ◽  
Frequency Relationship

The free vibrations of a linear conservative system with multiple loading parameters are studied, attention being restricted to pure eigenvalue problems. It is shown that the smallest frequency and external loading parameters of such a system constitute a strictly convex (synclastic) surface which cannot have convexity toward the origin of the “parameter space.” It is further proved that in the case of systems with one degree of freedom only, the surface takes the form of a plane. The practical implications of these results regarding the estimation of the frequencies and/or the stability boundary of the system are discussed. Thus it is observed that, on the basis of the established theorems, lower bounds to the frequencies at any stage of external loading and/or upper bounds to the stability boundary are readily obtainable. A two-degree-of-freedom illustrative example is discussed.

Nonlinear Vibration of Thin Elastic Plates, Part 1: Generalized Incremental Hamilton’s Principle and Element Formulation

1984 ◽  
Vol 51 (4) ◽  
pp. 837-844 ◽  
Author(s):  
S. L. Lau ◽  
Y. K. Cheung ◽  
S. Y. Wu
Keyword(s):  
Nonlinear Problems ◽  
Free Vibrations ◽  
Element Formulation ◽  
Hamilton’S Principle ◽  
Elastic Plates ◽  
Hamilton's Principle ◽  
Nonlinear Bending ◽  
Nodal Displacements ◽  
Kirchhoff Theory ◽  
Thin Elastic Plates

The finite element method has been widely used for analyzing nonlinear problems, but it is surprising that so far only a few papers have been devoted to nonlinear periodic structural vibrations. In Part 1 of this paper, a generalized incremental Hamilton’s principle for nonlinear periodic vibrations of thin elastic plates is presented. This principle is particularly suitable for the formulation of finite elements and finite strips in geometrically nonlinear plate problems due to the fact that the nonlinear parts of inplane stress resultants are functions subject to variation and that the Kirchhoff assumption is included as part of its Euler equations. Following a general formulation method given in this paper, a simple triangular incremental modified Discrete Kirchhoff Theory (DKT) plate element with 15 stretching and bending nodal displacements is derived. The accuracy of this element is demonstrated via some typical examples of nonlinear bending and frequency response of free vibrations. Comparisons with previous results are also made. In Part 2 of this paper, this incremental element is applied to the computation of complicated frequency responses of plates with existence of internal resonance and very interesting seminumerical results are obtained.

Inverse Problems in Vibration

1986 ◽  
Vol 39 (7) ◽  
pp. 1013-1018 ◽  
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.

On the Equivalence of Dispersion Relations Resulting from Rayleigh-Lamb Frequency Equation and the Operator Plate Model

2001 ◽  
Vol 123 (4) ◽  
pp. 417-420 ◽  
Author(s):  
Nikolay A. Losin
Keyword(s):  
Velocity Dispersion ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Dispersion Relations ◽  
Plate Model ◽  
Elastic Plates ◽  
Flexural Waves ◽  
Frequency Spectra ◽  
Infinite Power ◽  
Antisymmetric Vibrations

The Rayleigh-Lamb frequency equations for the free vibrations of an infinite isotropic elastic plate are expanded into the infinite power series and reduced to the polynomial frequency and velocity dispersion relations. The latter are compared to those of the operator plate model developed in [Losin, N. A., 1997, “Asymptotics of Flexural Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 64, No. 2, pp. 336–342; Losin, N. A., 1998, “Asymptotics of Extensional Waves in Isotropic Elastic Plates,” ASME J. Appl. Mech., 65, No. 4, pp. 1042–1047] for both symmetric and antisymmetric vibrations. As a result of comparative analysis, the equivalence of the corresponding dispersion polynomials is established. The frequency spectra, generated by Rayleigh-Lamb equations, are illustrated graphically and briefly discussed with reference to those published in [Potter, D. S., and Leedham, C. D., 1967, “Normalized Numerical Solution for Rayleigh’s Frequency Equation,” J. Acoust. Soc. Am., 41, No. 1, pp. 148–153].

Structural-Acoustic Vibration Problems in the Presence of Strong Coupling

2012 ◽  
Vol 135 (1) ◽  
Author(s):  
Heinrich Voss ◽  
Markus Stammberger
Keyword(s):  
Eigenvalue Problems ◽  
Acoustic Vibration ◽  
Projection Methods ◽  
Free Vibrations ◽  
Coupled Systems ◽  
Approximation Properties ◽  
Strongly Coupled ◽  
Weakly Coupled ◽  
Weakly Coupled Systems

Free vibrations of fluid–solid structures are governed by unsymmetric eigenvalue problems. A common approach which works fine for weakly coupled systems is to project the problem to a space spanned by modes of the uncoupled system. For strongly coupled systems, however, the approximation properties are not satisfactory. This paper reports on a framework for taking advantage of the structure of the unsymmetric eigenvalue problem allowing for a variational characterization of its eigenvalues and structure preserving iterative projection methods. We further cover an adjusted automated multilevel substructuring (AMLS) method for huge fluid–solid structures. The reliability and efficiency of the method are demonstrated by the free vibrations of a structure completely filled with water.

scholarly journals Buckling and Free Vibrations of Nanoplates – Comparison of Nonlocal Strain and Stress Approaches

2019 ◽  
Vol 9 (7) ◽  
pp. 1409 ◽  
Author(s):  
Małgorzata Chwał ◽  
Aleksander Muc
Keyword(s):  
Eigenvalue Problems ◽  
Scale Effects ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Small Scale ◽  
Governing Equations ◽  
First Order ◽  
Fundamental Equations

The buckling and free vibrations of rectangular nanoplates are considered in this paper. The refined continuum transverse shear deformation theory (third and first order) is introduced to formulate the fundamental equations of the nanoplate. In addition, the analysis involves the nonlocal strain and stress theories of elasticity to take into account the small-scale effects encountered in nanostructures/nanocomposites. Hamilton’s principle is used to establish the governing equations of the nanoplate. The Rayleigh-Ritz method is proposed to solve eigenvalue problems dealing with the buckling and free vibration analysis of the nanoplates considered. Some examples are presented to investigate and illustrate the effects of various formulations.

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