# Simulation Analysis and Experiment of Piezoelectric Pump with Tapered Cross-Section Vibrator
## 4 Analysis of Pumping Principle  
The piezoelectric vibrator in the flow field plays a major role to drive the fluid and form a unidirectional flow. Pumping fluid flow is the result of complex interactive mechanical, electrical, and hydraulic multi-physics fields. It is difficult to obtain an accurate analytical expression of the pumping flow. To investigate the main factors of the pumping flow and derive an approximate analytical expression and simplify the complexity of the research problem, this study only considered the effect of the piezoelectric vibrator on the fluid flow. Therefore, the piezoelectric vibrator is regarded as a whole body and defined as a rigid body. When driven by AC voltage, the piezoelectric vibrator swings around the fixed end in a sinusoidal pattern [21]. Experimental results demonstrate that when the piezoelectric vibrator operates in the second-order bending mode, it achieves a superior driving effect. As shown in Figure 5, point O represents the first node of the second-order mode, N is the second node, and T denotes the end of the vibrator. The ON segment undergoes vertical oscillations, which only induce up-and-down oscillations of the fluid within the pump chamber and do not contribute to unidirectional fluid flow. The only contribution to the unidirectional flow comes from the pitching motion indicated by the shaded region; hence, the analysis can be limited to the pitching motion within this shaded portion. The fluid surrounding the vibrator undergoes arc swing motion under the influence of the vibrating vibrator, producing a motion away from point N along the vibrator direction due to inertial forces, as illustrated in Figure 6. Within the given coordinate system, assuming the vibrator begins oscillating at time \(t=0\), starting from an initial angular position of \(\theta=-\theta_{0}\), the angular displacement of the vibrator can be expressed as:  
\[\theta=-\theta_{0}cos2\pi ft \tag{1}\]  
where \(f\) represents the vibration frequency of the vibrator, which is identical to the frequency of the applied excitation. \(\theta_{0}\) is the angular amplitude of the vibration, and \(t\) is time. Figure 5: Simplified model of the pump vibrator. Figure 6: Centrifugal force on a fluid particle. The circumferential velocity of a fluid particle can be expressed as:  
\[v^{-}=r\dot{\theta}=r\theta_{0}2\pi fsin2\pi ft \tag{2}\]  
The centrifugal force acting on a fluid particle at a given position is:  
\[dF=\rho\cdot b\cdot\theta_{0}^{3}(2\pi f)^{3}sin^{3}(2\pi ft)r^{2}drdt \tag{3}\]  
where \(\rho\) is the density of water and \(b\) is the width of the vibrator. Starting from time \(t=0\), when the vibrator begins moving from its maximum negative angular displacement \(-\theta_{0}\), and reaching time \(s\): \(0\leq s\leq\frac{1}{4}\). The increase in fluid momentum due to the vibrator's action equals the momentum of the fluid leaving the pipeline:  
\[P_{add}=\int_{0}^{s}v(t)dm=\int_{0}^{s}v(t)A\rho v(t)dt=A\rho\int_{0}^{s}v^{2} (t)dt \tag{4}\]  
where \(v\) represents the velocity component of the fluid particle in the direction of the pump chamber and \(A\) is the cross-sectional area of the outlet pipe. During the time interval \(0\sim s\), the impulse experienced by the fluid along the direction of the pump chamber is given by:  
\[I=\iiint\ (cos\theta)dFdt \tag{5}\]  
Given that the value of \(\theta\) is relatively small, it can be approximated as:  
\[I=\iiint\ dFdt \tag{6}\]  
By combining Equations (4) and (6), we obtain:  
\[A\rho\int_{0}^{s}v^{2}(t)dt=\iiint\ dFdt \tag{7}\]  
\[\rho Av^{2}(s)=\iint dF=\rho\cdot b\cdot\theta_{0}^{3}(2\pi f)^{3}\iint \limits_{ \begin{subarray}{c}0<s<R\\ 0<t\leq s\end{subarray}