A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloidal and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components ur, uθ, and uz in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

The vibrations problem of thin orthotropic skew plates of linearly varying thickness is analyzed using the small deflection theory of plates. Using dimensionless oblique coordinates, the deflection surface can be expressed as a polyonmial series satisfying the boundary conditions. For orthotropic plates which is clamped on all the four edges, numerical results for the first two natural frequencies are presented for various combinations of aspect ratio, skew angle and taper parameter. The properties of material used are one directional glass fibre reinforced plastic GFRP. The results obtained may be summarised as follows: 1. In case of the first mode vibration of plates with increase in the skew angle, the natural frequencies of plates decrease. 2. As the aspect ratio decrease, the natural frequencies of plates decrease. 3. For the identical skew angle, natural frequencies of plates increase with the taper parameter of thickness.