# Piezoelectric Composite Vibrator with a Bilaminated Structure for Bending Vibration
## 1 Introduction  
In the past 30 years, due to the rapid development of science and technology, low-frequency transducers have been more and more applied in sonar system and Marine science research. As a part of sonar systems, transducers have played a very important role in underwater sounding, locating, object recognition, communication, navigation, remote control, seabed remote sensing (bottom, stratum, landform), searching for oil and gas, and developing mineral resources. Piezoelectric vibrator is the core element of low frequency transducer. Reducing the resonant frequency and increasing the vibration displacement are the key technologies in the study of low frequency transducers. Bilaminated piezoelectric vibrators have several advantages, such as low resonance frequency, small size, light weight, and simple structure, and are an ideal low-frequency vibration sound source. Several studies on bilaminated vibrators made of piezoelectric ceramics and trilaminar vibrators made of piezoelectric ceramics and metal sheets have been reported. [PERSON] developed a theory of bimorph and trilaminar modifications of circular plate transducers for underwater applications. The approach used in his work was based on presenting deflection curves under flexure using a polynomial function, and [PERSON]'s principle for the cases of simply supported and clampedboundary conditions [1]. [PERSON] studied the effective electromechanical coupling coefficient and resonant frequency of bilaminated and trilaminated circular plate transducers by using the energy method [2,3]. [PERSON] et al. presented a general solution of displacement, resonance, and free-field voltage sensitivity for metal piezoceramic bimorph transducers with a fixed boundary [4]. [PERSON] et al. derived the displacement distribution function and resonance frequency equation of a piezoelectric ceramic bimorph, and studied the influence of the vibrator's radius on resonance frequency and vibration displacement [5]. [PERSON] et al. studied the circular composite piezoelectric vibrator and calculated and analyzed the relationship among resonance frequency, the effective electromechanical coupling coefficient, and the size of the vibrator [6]. [PERSON] studied three laminated plate transducers under three different boundary conditions, and derived the expressions of the resonance frequency and the effective electromechanical coupling coefficient by using the Raleigh method. In this study, numerical calculations and finite element simulations of the resonance frequency and the effective electromechanical coupling coefficient of the transducer varying with structural parameters were performed [7]. [PERSON] et al. studied the radiation impedance of a thin circular flat plate in flexural vibration with a free boundary, and the results showed that the radiation impedance was the function of the wave number \(\kappa\), the radius \(\alpha\), and the nodal line [8]. Studies on the performance of rectangular piezoelectric laminated vibrators are limited. [PERSON] studied the flexural vibration of a piezoelectric ceramic rectangular thin plate. Under the boundary conditions of free edges and simply supported edges, the resonance frequency equations were derived, and the relationship between the vibrational modes and geometrical dimensions was analyzed [9]. [PERSON] et al. studied the resonance frequencies of a rectangular trilaminar vibrator under five boundary conditions [10]. [PERSON] et al. determined the effects of modifying the transducer's physical size and other performance parameters on its frequency character and radiation property [11]. [PERSON] et al. found that the sensitivity and amplitude of a piezoelectric metal/ceramic composite transducer could be improved by changing the thickness of the piezoelectric material and the spatial distribution of the vibrator [12].