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Acoust. Soc. 1596-1608. , 1987, "Vibration Isolation for Spacecraft Using the Piezoelectric Polymer PVF 2," Proc. of the 114th Meeting of the Acoustic Soc. , Nov. 1987. , 1983, Distributed Parameter System Theory, Part 1 & 2, Hutchinson Ross Pub. Co. Stroudsburg, P A. , 1969, Linear Piezoelectric Plate Vibrations, Plenum, New York. , 1959, "Piezoelectric Relation and the Rachial Deformation of Polarized Spherical Shell," J. Acoust. Soc. 315-318. 1201-1206. • 1. (10), December 1988, 51-58. , 1989a, "Development of a Light-weight Robot End-effector using Polymeric Piezoelectric Bimorph," Proceedings of 1989 IEEE Intl.
0:3 « R2; thus, the ratio of finite distance to radius of • Piezoelectric Shells. [ 40 ] 1 + a3 ~ 1 and lrl- 1+~~1. 6) = phA1A2ii3 . ) I IJ I h/2 I-h/2 , can be treated as boundary forces on the top and bottom surfaces; or set to zero if both surfaces are force free surfaces. (a33 - e33E3) Ih/2 -h/2 denotes the transverse normal stress effect. • 2. 9) where superscript "m" denotes the mechanically induced component and "e" the electrically induced component. The resultant forces and moments, both mechanical and electric, are defined next.
A final variational equation is derived, which leads to all electromechanical system equations and boundary conditions (Tzou and Zhong, 1991b). 1 Variation of Kinetic Energy The variation of kinetic energy Jr is Note that integration by parts was used in the kinetic energy variation. , a) the mechanical strains Sij and b) the electric fields E j which are presented separately. • 2. Piezoelectric Shell Vibration Theory [ 25 ] a) Variation of Mechanical Strain Energy There are three normal strains and three shear strains needed to be considered in the derivation.