“…Recently, special application, such as the ultrasonic cleaning with the requirement for multi-frequencies (Lin and Xu, 2008), put forward the new demand for the performance of piezoelectric stack transducers. In the previous reports about the longitudinally sandwiched and radially cylindrical piezoelectric transducers, some methods have been tried to obtain the multi-frequency and wideband characteristics, including the resistance tuning method (Lin et al, 2018; Tian et al, 2018; Wang et al, 2015a, 2017, 2018), the approach of adjusting the ratio of piezoelectric layer numbers between the actuator and sensor parts (Wang et al, 2015a), the elastic foundation stiffness variable approach (Wang and Luo, 2016), the inductive matching technique (Lin, 2017; Lin and Xu, 2017), and the prestress control strategy (Arnold and Mühlen, 2001a, 2001b). Among these methods, the resistance tuning method and the ratio adjusting method are easy to implement and effective to obtain the multi-frequency characteristics.…”