Using the symbolic method of homogeneous solutions, we study problems of steady vibrations of isotropic plates. On the planar faces of the plate we state various kinds of homogeneous boundary conditions. We obtain the homogeneous solutions of the equations of motion and construct the dispersion equations. We carry out numerical analyses of the dispersion equations for a plate with clamped and planar faces. Four figures. Bibliography: 8 titles.We consider an isotropic plate bounded by two parallel planes F + and F-at a distance 2h from each other and a series of cylindrical surfaces at (1 = 0, N) with generators perpendicular to the planes F + and F-. The plate is deformed by external forces that vary harmonically in time and act on the surfaces at. The stresses and displacements, or some combination of them are prescribed on the planar faces F + and F-.Solving this problem reduces to integrating the equations of motion in displacements under prescribed boundary conditions. In the dimensionless coordinate system xi (i = 1, 2, 3) we have [1; 2]: A2U + UlA2graddiv U = 0;(1) (Ti3(X1,X2,-4-1) =0 (i~---1,2,3);(2) u~(~l,~, +1) =0;(3) U3(xl,x2,4-1) =0, O'j3(Xl,X2,'+'I) =0 (j = 1,2), Uj(xl,x2,q_l) =0, ff33(Xl,X2,_l) =0; (4) O'i3(X1,X2,-1) ----O, Vi(Xl,X2,-1) = O, Here a~3(xl, z2, +1) = O, Ui(Xl, x2, +1) :--O, aiz(zl, x2, +1) = O, Ui(xl, z2, +1) = 0, U3(zl, z2, -1) = O, U3(Xl, x2, -1) = O, Uj(Xl, x2, -1) = O, Uj(xl,x2,-1) =0, aja(zl, z2, -1) = O, o'j3(Xl, x2, -1) : O, 0"33(Xl, x2, -1) = O, 033(Xl, x2, -1) : O, (5) U3(Zl, z2, +1) = O, ,rj3(Zl, x2, +1) = o, 0"33(Xl, X2,--1) = O, Uj(xl,x2,-1 ) -: O; anln=O = fl(s, x3); ~nSln=O = f2(s, x3); an3in=O = f3(s,x3). (6) 2 = h2w2c~-2; n and s are the natural coordinates associated with the directrix of the cylindrical surface az; a) is the frequency of vibrations; cl and c2 are the velocities of transverse and longitudinal waves, respectively; and fi(s, x3) are given functions. The rest of the notation is as in [2]. Using A. I. Lur'e's symbolic method [3] and applying the Laplace transform on the variable x3, we obtain the general integral of the system (1): Vj(

Using superposition of solutions and Fourier series we study the stressed state in a transversally isotropic layer with a cavity. We describe the results of numerical studies as functions of the elastic properties of the material, the geometric characteristics of the body, and the nature of the load variation. Four figures. Bibliography: 6 titles Various authors [1-6] have used Lur'e's method of superposition of solutions to solve problems involving the stressed state of thick plates under different boundary conditions on the plane faces. In the present paper we use the results of [1] to study the stressed state of a transtropic layer weakened by a cylindrical cavity.Consider an axisymmetric strain of a transtropic layer (1 <_ r < co, Ix3t _< 1) weakened by a cylindrical cavity of unit radius. Suppose that the stresses are zero at infinity (r ~ (x~), while on the surface of the cavity and the faces of the layer they are as follows:

[2] have used the Fourier transform and superposition of solutions to solve the problem of the vibrations of a strip and have also carried out a qualitative analysis and numerical studies of the dispersion equation when the upper boundary is load-free and the lower is rigidly restrained. The authors [3] have used the method of homogeneous solutions to study three-dimensional problems of steady vibrations of isotropic plates taking account of various forms of homogeneous boundary conditions on the planar faces and have obtained the solutions of the equations of motion, constructed the dispersion equations, and studied them for a plate with rigidly restrained ends. In the present paper the results of [3] are used to solve the problem of the vibrations of an isotropic plate when one of its planar faces is load-free and the other lies on a rigid base.

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