In this paper, three displacement functions are introduced to simplify the basic equations of a radially polarized, spherically isotropic, piezoelectric medium with radial inhomogeneity. For the general non-axisymmetric free vibration problem, it is shown that the controlling equations are "nally reduced to an uncoupled second order ordinary di!erential equation and a coupled system of three second order ordinary di!erential equations. Solutions to these di!erential equations are given for the case that material constants are of power functions of the radial co-ordinate. For free vibrations of multilayered piezoelastic spherical shells, it is shown that there are two separated classes of vibrations. The "rst class is independent of the electric e!ect and is identically the same as that for pure elasticity, while the second is a!ected by the electric "eld. Numerical results are given for the non-axisymmetric free vibration of a single-layered, inhomogeneous piezoelastic spherical shell and e!ects of some involved parameters are discussed.
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It is further assumed that the body force components F G (i"r, , ) can also be decomposed in the same way, i.e.,
The most common case is that the body force vector is potential, for which one has
By employing equations ( 5) and ( 6), through some lengthy manipulations, we can transfer the basic equations to the following equations: * * [A#( c )w!( c )( G!G)#( e ) !r;#r G G ] (8) ! 1 sin * * [B#( c ) ( ! )#r<!r G ]"0, * * # *G * "0, * * !