Nonlinear Theory for a Thin-Walled Piezoelectric Ceramic Ring Excited in the Radial Breathing Mode.
NAVAL UNDERWATER SYSTEMS CENTER NEW LONDON CONN NEW LONDON LAB
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The nonlinear differential equation governing a thin-walled piezoelectric ceramic ring excited in the radial breathing mode is derived from the field equation of motion for an elastic continuum, the expanded nonlinear constitutive equations, and the appropriate boundary conditions on the stress. A somewhat pedagogic development of fundamental analytic concepts, such as tensors, tensor manipulations, and coordinate systems, is first presented to fix these necessary concepts clearly in mind. The elastic field equation is derived from basic concepts, and the general definition of strain is presented. The appropriate coordinate transformations, from spatially fixed Cartesian to materially embedded cylindrical, are demonstrated. Then the explicit nonlinear differential equation is derived to the next higher order from linear. It was assumed that the elastic nonlinearities were separable because of the typically high Qs approximately 1000 for the suspended rings dealt with. A perturbation solution is presented displaying analytically the dependence of the second harmonic on the nonlinear elastic parameters. In the concluding comments suggestions are made as to how the elastic and electrical nonlinear parameters might be measured.
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