The derivation of the exact solution of the problem on natural vibrations of rectangular plates in closed form is possible only for plates of constant thickness and some form of boundary conditions. It should be noted that sufficient quantity of publications is known on this point where various numerical approaches are proposed to investigation of natural vibrations of thin plates under combined boundary conditions. In this case a single opinion about the most efficient approach is absent. Therefore, the necessity arises to develop different approximate techniques for solving the mentioned class of problems. The accuracy of such approximate techniques can be estimated by comparing an approximate solution with the exact one, which is possible only for a limited class of more simple problems. Therefore, in some cases, it is necessary to estimate approximate solutions comparing them with experimental data.

In the present study it is proposed to analyse the natural vibrations of rectangular plates of variable thickness with various boundary conditions based on a numerical-analytical solution using the spline-collocation method with the discrete-orthogonalization method [1], and the experimental method of holographic interferometry [2].

As an initial model it was considered the Kirchhoff-Love theory. The paper is dealt with the complicate boundary conditions such that the separation-of-variables procedure couldn't be used. In particular, it concerns the case when all plate edges fixed. The initial boundary problem in partial derivative was reduced to the eigenvalue problem for the systems of ordinary differential equations by means of representation of solution in the form of splain-functions of the fifth degree. One of the advantages of the spline-collocation method is a possibility to satisfy accurately the certain class of complex boundary conditions. The system was solved by the stable numerical method of discrete orthogonalization in combination with the step-by-step searching. The experimental investigations were performed by the holography interference method.

Holographic interferometry is based on the interference of a light wave that is reconstructed from a hologram and creates the image of some object being examined on the interference of two waves stored in the same hologram at different instants. Measurements of the dynamic fields of displacements, based on the method of holographic interferometry, have a number of advantages in comparison with traditional methods. These advantages include possibility to study complex objects made of different materials, and the absence of strict requirements to the treatment of diffusionly reflecting surfaces.

The method of holographic interferometry allows us to examine both models and real engineering devices. This feature is essential, because actual objects under studying, the fixing conditions, physical, geometrical, and rigidity characteristics are taken into account automatically.

The natural vibrations of the rectangular plates of constant and variable thickness, convex and concave under complex boundary conditions were studied. The dependence of dynamic characteristics of the natural vibrations on the thickness and plate profile, as well as the types of boundary conditions were analysed

1

A.M. Bercin, "Free vibration solution for clamped orthotropic plates using Kantorovich method", Int. J. Sound and Vibr., 196(2), 243-247, 1996. doi:10.1006/jsvi.1996.0479

2

R.B. Bhat, P.A.A. Laura, R.G.Gutierre, V.H.Cortinez, H.C. Sanzi, "Numerical expirements on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness", Int. J. Sound and Vibr., 138(2), 205-219, 1990. doi:10.1016/0022-460X(90)90538-B

3

A. Ya. Grigorenko, T.V. Tregubenko, "Numerical and experimental analysis of natural vibrations or rectangular plates with variable thickness", Int. Applied Mechanics, 36(2), 268-270, 2000. doi:10.1007/BF02682003

4

A.V. Leissa, "Vibrations of plates", NASA, S.P.,160, 1969.

5

S.F. Ng, Y. Araar, "Free vibration and bucling analysis of clamped rectangular plates of variable thickness by the Galerkin method", Int. J. Sound and Vibr., 152(2), 263-274, 1989. doi:10.1016/0022-460X(89)90725-6

6

I. West, "Holographic Interferometry", [Russian], Mir, Moscow, 474, 1982.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)