Flextensional Transducer Modeling Using Variational Principles

Variational principles are an approximation method that allows one to obtain accurate estimates for a quantity of interest using relatively crude representations, known as trail functions, for the physical behavior of the system. This method is applied to flextensional transducer analysis by coupling a variational principle developed for the resonance frequency of the piezoelectric driving element to one for the resonance frequency of the shell, carefully ensuring that the boundary conditions at the driver-shell interface are satisfied. The in-vacuo mode shapes and resonance frequencies for a Class V ring-shell transducer calculated in this manner are compared with finite element modeling and experimental data for the first two modes of the transducer. There is excellent agreement between the methods in the calculation of the resonance frequencies, even though the mode shapes calculated variationally do not agree exactly with the finite element predictions. Fluid-loading effects on the transducer are introduced by coupling the in-vacuo variational transducer model to a variational principle for the surface pressure of a radiating body based on the Helmholtz integral equation. The surface pressures determined using the variational formulation and the finite element method for a single Class V ring-shell projector are compared for the first two resonant modes.