A flextensional transducer with an Omega shape and its algorithmic method of the resonant frequency and the shape functions are suggested.The Omega transducer is separated into four parts treated respectively as a thin shell of revolution and the theories of thin shells of revolution and piezoelectricity are used to obtain the energy functional of each part so that the sum of the energy functionals of the four parts is the energy functional of the whole Omega transducer.By substituting the shape functions with undetermined coefficients and the geometrical boundary conditions into the energy functional of the Omega transducer,the resonant frequency of the Omega transducer is firstly determined with the Rayleigh-Ritz method.With the gotten resonant frequency,the constant coefficients of the shape functions are following solved through the Rayleigh-Ritz partial differential equations and the geometrical boundary condition equations.The solving method of the resonant frequency and the shape functions is also extended to the cymbal transducer.Such an analytical method is verified to be feasible by the results of the finite element analysis and experiments.The research indicates that(1) The radial vibration of the piezoelectric ceramic is in phase with the longitudinal vibration of the top of metal cap,and it cut down the reversed phase component in the sound field. The Omega transducer can be a low frequency transducer.(2) The determination method of the resonant frequency and the shape functions give a solution to the optimum designs of the Omega transducer and the cymbal transducer.(3) The determination method of the resonant frequency and the shape functions can also be used in other flextensional transducers or other structures which are composed of thin shells of revolution,so it is universal.
