This paper presents a Fourier spectral element method (FSEM) to analyze the free vibration of conical-cylindrical-spherical shells with arbitrary boundary conditions. Cylindrical-conical and cylindrical-spherical shells as special cases are also considered. In this method, each fundamental shell component (i.e., cylindrical, conical, and spherical shells) is divided into appropriate elements. The variational principle in conjunction with first-order shear deformation shell theory is employed to model the shell elements. Since the displacement and rotation components of each element are expressed as a linear superposition of nodeless Fourier sine functions and nodal Lagrangian polynomials, the global equations of the coupled shell structure can be obtained by adopting the assembly procedure. The Fourier sine series in the displacement field is introduced to enhance the accuracy and convergence of the solution. Numerical results show that the FSEM can be effectively applied to vibration analysis of the coupled shell structures. Numerous results for coupled shell structures with general boundary conditions are presented. Furthermore, the effects of geometric parameters and boundary conditions on the frequencies are investigated.

Vibration analysis of coupled conical-cylindrical-spherical shells using a Fourier spectral element method.

This paper deals with 3-D modeling of piezoelectric materials. The model is based on an exact description of the potential and electric field inside a material. Moreover, coherent piezoelectric equations are used. Modeling has been applied to rectangular and cylindrical elements. In each case, the exact equations of the displacements along the three coordinate axes and the corresponding electric impedance are calculated. The general resonance conditions are stated for these two geometries. It is shown that, contrary to the 1-D models, a unique equation describes lateral and thickness vibrations, or radial and thickness vibrations. These properties enable us to analytically calculate the frequency spectrum of rectangular elements, thick disks, or cylinders and also thick rings or hollow cylinders versus the width to thickness ratio. It is then very easy to determine the corresponding dispersion diagram related to each geometry sample. These resonance conditions are similar to those deduced from the 1-D model described in the IEEE standard but are more general and necessitate no cancelling out assumptions. In addition, contrary to 1-D models, the wave velocities and the permittivity are independent of the element geometry (parallelepiped or cylindrical). The wave velocities are equal to those stated for the wave propagation in infinite medium and measured with pulse-echo techniques. It is the coupling inside the material which modifies the resonance conditions and not the geometrical dimensions of the vibrating element. 3-D modeling and 1-D radial mode of the admittance of a thick disk are calculated and compared with experimental measurements. Theoretical and measured admittances are compared and discussed.

Three-dimensional modeling of piezoelectric materials

Scattering and active acoustic control from a submerged piezoelectric-coupled orthotropic hollow cylinder

Three-dimensional vibrations of cylindrical elastic solids with V-notches and sharp radial cracks

An analytical model of axisymmetric vibrations of hollow elastic circular cylinders with arbitrary boundary conditions is presented. Free vibrations of cylinders with free or fixed boundaries and forced vibrations of cylinders with specified non-uniform displacement or stress on the boundaries are considered. Three series solutions are used and each term in each series is an exact solution to the exact governing equations of motion. The terms in the expressions for components of displacement and stress are products of Bessel and sinusoidal functions and are orthogonal to each other. Complete sets of functions in the radial and axial directions are formed by terms in the first series and the other two, respectively. It is therefore possible to satisfy arbitrary boundary conditions. It is shown that just two terms in each series are sufficient to determine several resonance frequencies of cylinders with certain specified boundary conditions. The error is less than 1%. Numerical results are also presented for forced vibration of hollow steel cylinders of length 10 mm and outer diameter 10 mm with specified normal displacement or stress. Excellent agreement with finite element results is obtained at all frequencies up to 1 MHz. Convergence of the series is also discussed.

Free and forced vibrations of hollow elastic cylinders of finite length.

A method is presented to determine the forced responses of piezoelectric cylinders using weighted sums of only certain exact solutions to the equations of motion and the Gauss electrostatic conditions. One infinite set of solutions is chosen such that each field variable is expressed in terms of Bessel functions that form a complete set in the radial direction. Another infinite set of solutions is chosen such that each field variable is expressed in terms of trigonometric functions that form a complete set in the axial direction. Another solution is used to account for the electric field that can exist even when there is no vibration. The weights are determined by using the orthogonal properties of the functions and are used to satisfy specified, arbitrary, axisymmetric boundary conditions on all the surfaces. Special cases including simultaneous mechanical and electrical excitation of cylinders are presented. All numerical results are in excellent agreement with those obtained using the finite element software ATILA. For example, the five lowest frequencies at which the conductance and susceptance of a stress-free cylinder, of length 10 mm and radius 5 mm, reach a local maximum or minimum differ by less than 0.01% from those computed using ATILA.

Forced responses of solid axially polarized piezoelectric ceramic finite cylinders with internal losses

Forced vibrations of solid elastic cylinders

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Free and forced vibrations of hollow elastic cylinders of finite length.

Forced responses of solid axially polarized piezoelectric ceramic finite cylinders with internal losses

Forced vibrations of solid elastic cylinders