Plate vibration under irregular internal supports

In this paper, the symmetries and the conserved quantities for systems of generalized classical mechanics are studied. First, the generalized Noether’s theorem and the generalized Noether’s inverse theorem of the systems are given, which are based upon the invariant properties of the canonical action with respect to the action of the infinitesimal transformation of r-parameter finite group of transformation; second, the Lie symmetries and conserved quantities of the systems are studied in accordance with the Lie’s theory of the invariance of differential equations under the transformation of infinitesimal groups; and finally, the inner connection between the two kinds of symmetries of systems is discussed. PACC: 0320; 1110; 0220 I. INTRODUCTION The conservation laws (or first integrals) of the mechanical systems are always of mathematical importance and at the same time, they are regarded as the manifestation of some profound physical principles. The fact that the existence of a conserved quantity means that an inner dynamical symmetry in the mechanical system plays an irreplaceable role in the physical interpretation of motion. The modern methods in finding conservation laws are mainly two ways [1] with each having a concept of its own: the first one is based upon the invariant properties of Hamiltonian action with respect to the action of the infinitesimal transformation of the finite transformation group, i.e., Noether symmetries and conserved quantities; the second one is based upon the Lie’s theory of the invariance of differential equations under the action of infinitesimal groups. The differential equations of many physical problems appear to be the variational problems. The theory of the generalized classical mechanics, initiated by Ostrogradsky and Jacobi in the years from 1848 to 1858, has been gaining much development during the recent fifty years, and many important results have been made. [2 6] The Lagrange equations with high

http://iopscience.iop.org/article/10.1088/1009-1963/9/6/001/pdf

Symmetries and conserved quantities for systems of generalized classical mechanics

This paper addresses the free transverse vibrations of thin simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness with elastically restrained edges against rotation. The Gram-Schmidt process has been used to generate two-dimensional boundary characteristic orthogonal polynomials, which have been used in the Rayleigh-Ritz method to study the problem. The lowest three frequencies have been computed for various values of nonhomogeneous parameters, thickness parameters, aspect ratio and flexibility parameters. A comparison of the results with those available in the literature has been made. Three-dimensional mode shapes for the specified plate have been presented.

Free vibrations of simply supported nonhomogeneous isotropic rectangular plates of bilinearly varying thickness and elastically restrained edges against rotation using Rayleigh-Ritz method

Many types of engineering structures can be effectively modelled as orthotropic plates with opposite free edges such as bridge decks. The other two edges, however, are usually treated as simply supported or fully clamped in current design practice, although the practical boundary conditions are intermediate between these two limiting cases. Frequent applications of orthotropic plates in structures have generated the need for a better understanding of the dynamic behaviour of orthotropic plates with non-classical boundary conditions. In the present study, the transverse vibration of rectangular orthotropic plates with two opposite edges rotationally restrained with the remaining others free was studied by applying the method of finite integral transforms. A new alternative formulation was developed for vibration analysis, which provides much easier solutions. Exact series solutions were derived, and the excellent accuracy and efficiency of the method are demonstrated through considerable numerical studies and comparisons with existing results. Some new results have been presented. In addition, the effect of different degrees of rotational restraints on the mode shapes was also demonstrated. The present analytical method is straightforward and systematic, and the derived characteristic equation for eigenvalues can be easily adapted for broad applications.

Free Transverse Vibration of Rectangular Orthotropic Plates with Two Opposite Edges Rotationally Restrained and Remaining Others Free

We deduce a frequency equation for the natural coupled axially symmetric vibrations of elastic bases (in the form of annular plates) and a heavy two-layer incompressible perfect liquid in a rigid annular cylindrical vessel. We consider different limiting cases: the case of degeneration of annular plates into membranes, the case of absolutely rigid or circular plates, and the case of absence of the upper plate (liquid with free surface). For a broad range of parameters of the analyzed mechanical system, we investigate the frequency spectra and obtain a series of mechanical effects in the problem of hydroelasticity.

Axially Symmetric Vibrations of Elastic Annular Bases and a Perfect Two-Layer Liquid in a Rigid Annular Cylindrical Vessel

This paper studies the problem of free bending vibration of annular cylindrical tank partially filled with liquid in the consideration of surface wave. The exact formulae of the mode shape functions and frequencies are deduced. Results can be obtained by means of computer. The analysis shows that the effect of liquid on vibration of annular cylindrical tank is equivalent to different generalized distributive masses attached to inner and outer cylinders respectively.

Free bending vibration of annular cylindricaltank partially filled with liquid in theconsideration of surface wave

This paper presents an approximate solution for calculating eigen-frequencies of transverse vibration of rectangular plates elastically restrained against rotation along edges. The formulae are not only very simple and easily programmed but also have high accuracy. Finally, some numerical results are given and compared with other results obtained.

An approximate solution of eigen-frequencies of transverse vibration of rectangular plates with elastical restraints

This paper presents a new analytical solution of transverse vibration of rectangular plates simply supported at two opposite edges with arbitrary number of elastic line supports in one way. The reaction forces of the elastic line supports are regarded as the unknown external forces acted on the plate. The analytical solution of the differential equation of motion of the rectangular plate, which includes the unknown reaction forces, is gained. The frequency equation is derived by using the linear relutionships between the reaction forces of the elastic line supports and the transverse displacements of the plate along the elastic line supports. The representations of the frequency equation and the mode shape functions are different from those obtained by other methods.

An analytical solution of transverse vibration of rectangular plates simply supported at two opposite edges with arbitrary number of elastic line supports in one way

Zhou Ding

Papers

Free bending vibration of annular cylindricaltank partially filled with liquid in theconsideration of surface wave

An approximate solution of eigen-frequencies of transverse vibration of rectangular plates with elastical restraints

An analytical solution of transverse vibration of rectangular plates simply supported at two opposite edges with arbitrary number of elastic line supports in one way