## 1 Introduction  
Due to the characteristics of interchanging the electric field with the mechanical field, the piezoelectric materials are often used as parts of the sensors or actuators. As the micro-fluid techniques develop with the design evolution of the micro-system, the piezoelectric materials are also used for micro-fluid element design. One of the common applications is the pumps being used in micro-fluidic systems in which the vibrating piezoelectric plate is used as the driving element. The vibration characteristics of piezoelectric plates have been investigated in many research studies. [PERSON] [1] proposed a complete liner piezoelectric model, and then derived the constitutive equations of the piezoelectric plates' vibration by using the electroelastic equations and the calculus of variations. [PERSON] [2] analyzed the vibration of piezoelectric shells, and discussed the static and dynamic characteristics of the smart materials made by bonding the piezoelectric layer on other structures. [PERSON] et al. [3] investigated the vibration modes of PZT-5H ceramics disks in different diameter-to-thickness ratios. [PERSON] et al. [4] determined the PZT-5A piezoelectric disks' vibration with 20 and 10 diameter-to-thickness ratios. The symmetric modes of vibration for circular piezoelectric plates was studied by [PERSON] [5] in order to optimize the dynamic electromechanical coupling coefficient. Most of the above-mentioned research studies that have analyzed the vibration of piezoelectric plates contain analytical and numerical results. The experimental results are needed to verify the theoretical computation. Numerous researchers have investigated the vibrations in the piezoelectric elements used in micropumps. [PERSON] and [PERSON] [6] used a mathematical algorithm and FEM analysis to derive the vibration characteristics of a circular Si thin film used in a piezoelectric micropump affected by solid-fluid coupling. They determined that the natural frequency in a coupled system is far lower than that in a non-coupled system. With a focus on flow rates, [PERSON] et al. [7] implemented a val valveless micropump in acrylic in which the vibration of a piezoelectric plate mimicked the caudal fins of a fish. [PERSON] [8] employed mathematical computation and finite element analysis to describe the vibrations in a micropump model with a focus on the means by which geometrical dimensions and fluid properties affect the resonant frequency. [PERSON] et al. in 2008 [9] examined the vibration characteristics of a micropump with a diaphragm bonded to a piezoelectric element. [PERSON] et al. [10; 11] developed a novel micropump that drives fluid via the vibration of a diaphragm. When implemented in the cooling system of a laptop computer, this device proved to be highly stable, thanks to its high flow rate. [PERSON] et al. [12] also developed an innovative one-sided actuating piezoelectric valveless micropump (OAPMP-valveless) that uses a secondary chamber to pump the fluid at higher flow rates. Then, experiments were conducted on the valveless micropump to verify the efficacy of the underlying three-dimensional (3D) numerical model. The fish-tailing type of valveless piezoelectric pump was also proposed in the study of [PERSON] el. al [13]. The dynamic analysis was performed theoretically; then, the flow rates under different driving frequencies were also studied through the experiment by [PERSON] et al. [14]. Then, [PERSON] et al. [15] presented the analytical and simulating model of a piezoelectric plate (PZT) actuated planar valveless PDMS micropump. The pump included a taper at the outlet to improve the flow rate, and the significant improvement of 28% was indicated in the research. The effect of the frequency and voltage on the flow rate and pressure-flow characteristics were also studied through the experiments.