On the stationary vibrations of a rectangular plate subjected to stress prescribed partially at the circumference

M G El Sheikh,D K Callebaut,A H Khater

doi:10.1155/s0161171291000789

Abstract

The stationary periodical problem of a vibrating rectangular plate, stressed at a segment while fixed elsewhere at one of its edges, is considered. Using the finite Fourier transformation, the problem is converted to a singular integral equation that in turn can be reduced to an infinite system of algebraic equations. The truncation of the algebraic system is justified.

Highlights

Expressed in terms of the longitudinal and transversal potentials t and xp, respectively, the stresses and displacements satisfy (Nowacki 1]) the followng equations oy-2 v 1-2v
Carrying out procedures similar to those used in obtaining equations (1.20) and (1.21), the uniform conditions (1.3)-(1.5) lead to an additional three equations of the same type
Recalling that G(x) 0 for -c < x < c and integrating eq (2.1) with respect to 0, we obtain over the interval

Summary

Ixl undetermined

The application of the Fourier transform to the completed conditions (1.16) and (1.17) whose left-hand sides are given explicitly by (1.6) and (1.7), together with (1.15) yields. The Fourier components P,_ belong to the extension P*(x) of the constant P* by defining it to be zero 101 outside < c. Carrying out procedures similar to those used in obtaining equations (1.20) and (1.21), the uniform conditions (1.3)-(1.5) lead to an additional three equations of the same type. Solving the system which consists of these additional three equations together with eq (1.21) for the coefficients A., B., C. The discrete problem (1.22) can be written in the standard form where (1.23) (1.24)

Here we have used the expansion

Substituting in the formula