Nonlinear modeling of langevin transducers using the rayleigh law in the piezoelectric ceramics

Ultrasonic Langevin transducers are used in several power ultrasound applications. However, nonlinear effects can influence their performance, especially in a high vibration amplitude level. These nonlinear effects produce variations in the resonant frequency, harmonics of the excitation frequency, loss of symmetry in the frequency response around the resonances and a “frequency domain hysteresis”, also known as jumping and dropping phenomena. This work presents a simplified model to take into account the most relevant nonlinear effects using only two parameters, one to reproduce the changes in the resonance frequency and the other to introduce the dependence with the history in the frequency sweep. The piezoelectric constitutive equations are extended using a linear dependence of the elastic constant with the amplitude of the mechanical displacement. In order to introduce the frequency hysteresis, the elastic constant is computed using a combination of the current value of the amplitude with the previous state amplitude. The results were tested using different Langevin type transducers. In this work a 26 kHz transducer, originally developed by Mectron Medical Technology (Italy) for bone surgery, is evaluated in a wide range of input voltages and sweeping the frequency up and down in every case.