An analytical model is presented of axisymmetric circular hollow piezoelectric ceramic cylinders with arbitrary dimensions and boundary conditions. Forced vibrations of the cylinders with specified potentials on the electroded surfaces and displacement or stress on the boundaries are considered. The exact, linearized, axisymmetric governing equations are used in the analysis. 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 series expressions for components of displacement, stress, electric potential, and electrical displacement are products of Bessel and sinusoidal functions and are orthogonal to other terms. 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 on all surfaces of the hollow piezoelectric cylinder. Numerical results are presented for hollow piezoelectric cylinders of various dimensions. Input electrical admittance and displacements are computed for three special cases in bands that include several resonance frequencies, and they are in excellent agreement with those computed using atila-a finite element package.
An exact series method is presented to analyze classical Langevin transducers with arbitrary boundary conditions. The transducers consist of an axially polarized piezoelectric solid cylinder sandwiched between two elastic solid cylinders. All three cylinders are of the same diameter. The length to diameter ratio is arbitrary. Complex piezoelectric and elastic coefficients are used to model internal losses. Solutions to the exact linearized governing equations for each cylinder include four series. Each term in each series is an exact solution to the governing equations. Bessel and trigonometric functions that form complete and orthogonal sets in the radial and axial directions, respectively, are used in the series. Asymmetric transducers and boundary conditions are modeled by using axially symmetric and anti-symmetric sets of functions. All interface and boundary conditions are satisfied in a weighted-average sense. The computed input electrical admittance, displacement, and stress in transducers are presented in tables and figures, and are in very good agreement with those obtained using atila-a finite element package for the analysis of sonar transducers. For all the transducers considered in the analysis, the maximum difference between the first three resonance frequencies calculated using the present method and atila is less than 0.03%.
An exact method is presented to analyze classical Langevin transducers with internal losses. The transducers consist of an axially polarized piezoelectric cylinder sandwiched between two elastic cylinders. All three cylinders are of the same diameter. Exact solutions to the exact equations of motion of the piezoelectric and elastic cylinders and the Gauss electrostatic condition are used. Complex piezoelectric and elastic coefficients are used to model internal losses. For each cylinder, the first set of solutions contains Bessel functions that form a complete set in the radial direction. The second and third sets contain trigonometric functions that form complete sets in the axial direction. They are used to represent fields that are symmetric and anti-symmetric with respect to the plane midway between the ends of the cylinder, respectively. The interface and boundary conditions are satisfied by using the orthogonal properties of the functions. Transducers with identical elastic cylinders at the ends as well as those with a light head mass and a heavy tail mass are analyzed. Numerical results are presented to illustrate the input electrical admittances of transducers. They are compared with those obtained using ATILA - a finite element package for the analysis of sonar transducers.
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