Nonlinear bendings of unsymmetrically layered anisotropic rectangular plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

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Calculation of Non-Homogeneous Anisotropic Rectangular Plates with Arbitrary Fixation on the Contur

Materials Science Forum ◽

10.4028/www.scientific.net/msf.968.444 ◽

2019 ◽

Vol 968 ◽

pp. 444-449

Author(s):

Nikolay Zavrak

Keyword(s):

Main Idea ◽

Isotropic Case ◽

Rectangular Plates ◽

Orthotropic Plates ◽

Anisotropic Plates ◽

Practical Applications ◽

Internal Forces ◽

The Relationship ◽

Difficult Cases ◽

Anisotropic Rectangular Plates

The developing of the effective methodic of elastic orthotropic plates’ calculation and the research on the base of their state under different boundary conditions are of great importance nowadays. The representation of the received results in the form, convenient for practical use, is also important. For practical applications in engineering are important tables for determining deflections and internal forces of structures. Such tables for the isotropic case under various conditions of plate support on the contour are given in many works. As for the anisotropic plates, there are no such tables, with the exception of one Huber table compiled for a freely supported rectangular orthotropic plate, depending on the relationship between the stiffness values. Here is a method of calculating the non-homogeneous anisotropic rectangular plates with arbitrary fixation on the contour is set forth, which is reduced to a boundary value problem. The main idea of a calculated general methodic of linear marginal differential tasks calculation is based on underlying of the main part of a solution. Such approach is proved by means of development and some generalization of common positions of a variational method of marginal tasks of mathematical physics of self-conjugated tasks solution. To solve a system of equations in terms of displacements using finite difference method (FDM) in combination with different variations of analytical solutions. It is advisable to construct a numerical solution of the problem so that in difficult cases the support fixing and uploading solution sought, not directly, but in the form of amendments to the known solution for simple cases of reference to consolidate and uploading at finding the solutions which the analytical methods or the FDM with sparse mesh may be used. Given as examples are the results of calculation for a series of square orthotropic plates with a fixed boundary under the action of uniformly distributed load.

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TRANSVERSE VIBRATIONS OF SIMPLY SUPPORTED ANISOTROPIC RECTANGULAR PLATES CARRYING AN ELASTICALLY MOUNTED CONCENTRATED MASS

Journal of Sound and Vibration ◽

10.1006/jsvi.1998.1653 ◽

1998 ◽

Vol 215 (5) ◽

pp. 1195-1202 ◽

Cited By ~ 6

Author(s):

H.A. Larrondo ◽

D.R. Avalos ◽

P.A.A. Laura

Keyword(s):

Rectangular Plates ◽

Concentrated Mass ◽

Simply Supported ◽

Transverse Vibrations ◽

Anisotropic Rectangular Plates

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The transmission of sound through anisotropic rectangular plates

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1988.0034 ◽

1988 ◽

Vol 416 (1850) ◽

pp. 199-215 ◽

Cited By ~ 2

Keyword(s):

Rectangular Plate ◽

Sound Wave ◽

Compressible Fluid ◽

Sound Power ◽

Rectangular Plates ◽

Small Frequency ◽

Plate Equations ◽

Plane Sound ◽

Rigid Baffle ◽

Anisotropic Rectangular Plates

A plane sound wave is incident upon an anisotropic rectangular plate set in an otherwise rigid baffle surrounded by a light compressible fluid. In the limit of small wavelengths, compared with plate dimensions, the transmitted sound power is estimated, averaged over a small frequency band and over all possible angles of incidence. Explicit results are presented for a special type of anisotropy, and they have different forms according as the operating frequency ω is above coincidence, below coincidence or with in a range of frequencies (ω 2 < ω < ω 1 ) that corresponds to coincidence. Transition formulae are given for frequencies near the boundaries of the three régimes. The results are extended to allow for more general plate equations.

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