## 1 Introduction  
Piezoelectric materials are smart materials which serve as crucial components in sensors or actuators. When the need arises for vibration control in lightweight flexible structures with complex shapes, piezoelectric sensors and actuators can be used to prevent adding to the overall mass of the structure and to achieve precision control. Piezoelectric materials have superior electromechanical properties, are lightweight and easy and inexpensive to produce, and have low power consumption [1; 2; 3; 4]. For instance, the puncture systems used in cellular biology research have precise and fine vibration control [5]. In addition to applying the vibration characteristics of inverse piezoelectricity, the puncture systems also apply energy extraction systems with direct piezoelectric effects. Recently, vibration energy harvesting devices using piezoelectric materials have garnered the attention of researchers. These devices convert the vibration energy of mechanical structures into usable electrical energy, which can detect the inhibiting effects of vibration signals [6]. Energy harvesting systems based on lead zirconate titanate (PZT) can be embedded into vibrating objects to generate tens to hundreds of microwatts of energy for low-power integrated circuits (ICs). Thus, piezoelectric materials can be combined with low-power ICs to create continuously-working miniature wireless sensors, such as implantable biomedical sensors and wearable devices [7], wireless sensor networks [8], and smart buildings and special structures [9; 10; 11]. The vibration characteristics of piezoelectric materials can be analyzed in advance using finite element software for greater convenience in the optimization of piezoelectric effects. To obtain accurate analysis results, the boundary conditions of the model must be clearly defined, and the right piezoelectric material parameters must be used. The parameters provided with commercially-available piezoelectric materials are often inadequate for numerical calculations. Complete dynamic characteristics cannot be derived from the mechanical parameters of anisotropic materials, and thus, measuring effective and complete material parameters is key to analysis accuracy. In this study, the material constants of the piezoelectric specimens were measured using the resonance method (RM). This method has been widely applied. In 1954, [PERSON] et al. [12] used the resonance characteristics of an equivalent circuit simulating a piezoelectric crystal and employed static methods, quasi-static methods, dynamic methods, and hydrostatic methods to measure the elastic, piezoelectric, and dielectric constants of 45\({}^{\circ}\) Rochelle salt. Based on these measurement methods, they then developed RM to measure the material properties of piezoelectric crystals. In this method, the elastic and piezoelectric constants of the piezoelectric crystal are measured and the dielectric constants are inversely calculated from measured capacitance. Although the compliance \(S_{11}^{E}\) and \(S_{11}^{D}\), electromechanical coupling coefficients \(k_{33}\) and \(k\), and permittivity \(\varepsilon_{33}^{T}\) were obtained, the measurement was lack of guideline for prepared the standard specimens. In 1961, the IRE Standards on Piezoelectric Crystals [13] presented five specimens with different geometric shapes, aspect ratios, polarization orientations, and electrode surface orientations. The mode shapes corresponding to these five specimens were the thickness extension (TE) mode, length extension (LE) mode, thickness shear extension (TS) mode, length thickness extension (LTE) mode, and radial extension (RAD) mode. As polarized PZT has a hexagonal lattice (6 mm), it has 10 material constants, including five elastic constants, three piezoelectric constants, and two dielectric constants. With the TE, LE, TS, LTE, and RAD modes, the resonant and anti-resonant frequencies can be measured, and then the material constants of its various forms can be inversely calculated.