## 1 Introduction  
As a new type of artificial material, phononic crystal (PnC) possesses many properties that natural materials do not have, such as negative refraction property [1; 2], defect state property [3; 4; 5; 6], acoustic focusing property [7; 8; 9], and bandgap property [10; 11; 12; 13]. The passband is the frequency range in which elastic waves can propagate, while the frequency range of elastic waves that cannot propagate through the phononic crystal is termed \"stopband\". In addition, the passband and the stopband also represent the frequency range that allows and suppresses the vibration of the structure and the propagation of acoustic energy. Thus, this characteristic is known as the bandgap property [14; 15]. As one of the unique properties of the phononic crystals, there are two main mechanisms for the formation of the bandgaps, which are the Bragg scattering mechanism [16; 17] and the local resonance mechanism [18]. The periodicity of the structure is the major reason for the generation of the Bragg bandgaps, while the formation of the local resonance bandgaps is due to the resonance properties of a single scatter and the interaction of long-wave traveling waves in the substrate. For Bragg-type phononic crystals, to suppress elastic waves with longer wavelength, larger phononic crystals are required. This drawback not only limits the further development of phononic crystals but also produces considerable uncertainty in their practical application. The development of locally resonant phononic crystals resolves the limitation in Bragg phononic crystals due to the difference in the formation mechanism of the bandgaps, and the local resonance phononic crystals have a peculiar property of \"the small size controlling the large wavelength\" [18], which has attracted extensive attention from scientists. Due to the importance of controlling wave transmission in phononic crystals, various methods are developed to satisfy different requirements in adjusting wave bandgaps. On the one hand, the tuning of the band structure of phononic crystals can be achieved based on their tunable geometry and material properties [19; 20; 21]. On the other hand, adding piezoelectric material to traditional phononic crystals is also an efficient way to obtain desirable bandgaps [22; 23; 24]. Piezoelectric materials have been widely used in piezoelectric transducers, microphones, and pressure sensors. The largest difference between piezoelectric materials and other materials is their positive piezoelectric effect and inverse piezoelectric effect, which can achieve the mutual conversion of electrical energy and mechanical energy. Applying piezoelectric materials to phononic crystals allows the bandgaps to be adjusted without changing the intrinsic structure of the crystals [25; 26; 27]. A novel strategy was proposed by [PERSON] et al. [28] to investigate the vibration bandgap and active tuning characteristics of laminated composite metamaterial beams. [PERSON] et al. [29] studied the damping of resonators and derived the vibration transmissibility method (TM) for a finite electromechanical system based on the Timoshenko beam theory. To actively control the band structure in a system, [PERSON] et al. [30] periodically arranged piezoelectric shunt arrays on designed active beam resonators. [PERSON] et al. [31] applied piezoelectric materials (PZT-5H) and an epoxy-resin periodic array to study the effects of different parameters on the bandgap widths and initial frequencies of a nano-beam. Considering the synchronous switch damping technology, [PERSON] et al. [32] developed structural vibration control by using piezoelectric materials. [PERSON] et al. [33] proposed a tunable period metamaterial coupled with local resonators and changed the values of the resistance and inductance to adjust the constitutive properties of piezoelectric materials.