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A three-dimensional (3D) method of analysis is presented for determining the natural frequencies of shallow spherical domes with non-uniform thickness. Unlike conventional shell theories, which are mathematically two dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components $u_{\varphi }$, $u_{\theta }$, and $u_z$ in the meridional, circumferential, and normal directions, respectively, are taken to be periodic in $\theta$ and in time, and algebraic polynomials in the $\varphi$ and z directions. Potential (strain) and kinetic energies of the shallow spherical domes with non-uniform thickness 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. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from a 2D exact method, a 2D thick shell theory, and a 3D finite element method by previous researchers.

The dynamic stability of bidirectional woven fiber laminated glass/epoxy composite shallow shells subjected to harmonic in-plane loading in hygrothermal environment is considered. An eight-noded isoparametric shell element with five degrees of freedom is used in the analysis. In the present finite element formulation, a composite doubly curved shell model based on first-order shear deformation theory (FSDT) is used for the dynamic stability analysis of shell panels subjected to hygrothermal loading. A program is developed using MATLAB for the parametric study on the dynamic stability of shell panels under the hygrothermal field. The effects of various parameters like static load factor, curvature, shallowness, temperature, moisture, stacking sequence and boundary conditions on the dynamic instability regions of woven fiber glass/epoxy shell panels are investigated. The location of dynamic instability regions is shown to affect significantly due to presence of the hygrothermal field.

In this paper, the possibility of using piezoelectric elements to create a prestress state in a plate and a shallow shell leading to a change in the natural frequencies is demonstrated numerically and experimentally. Strains in the thin-walled structure are determined using the non-linear relations based on the Reissner–Mindlin theory, which are linearized with respect to the state with a small deviation from the initial equilibrium caused by the inverse piezoeffect. A mathematical formulation of the dynamics problem is based on the variational principle of virtual displacements, taking into account the prestress state. The solution was developed by the finite element method. The validity of the obtained results is confirmed by comparing the natural frequencies and mode shapes of a rectangular plate with a piezoelectric element obtained numerically and experimentally at different values of electrical voltage. A series of calculations has been carried out to analyze the influence of the elasticity modulus, length, thickness, curvature of the shallow shell and voltage on the spectrum of natural frequencies of vibration. It was found that the effectiveness of using this technique decreases with increasing stiffness of the structure as a whole.