*2.1. Design Concept*

Figure 1a shows the overall structure of the piezoelectric resonance pump, which includes a piezoelectric actuator unit and a pump chamber unit. A square piezoelectric vibrator is the core of the actuator unit, and can periodically vibrate when driven by the sine wave signal. Under it is the elastic vibration transfer block, which is used to amplify the vibrator's amplitude to drive the pump chamber diaphragm. The block is also the stiffness and mass adjusting element of the pump vibration system.

The pump body, processed by polymethyl methacrylate (PMMA), and the circular elastic diaphragm, made from beryllium bronze, were glued together to form an airtight pump chamber unit. The inlet and outlet of the pump employed wheeled check valves. The valve piece was cut from beryllium bronze by ultraviolet laser and bonded to the inlet and outlet seats. Due to the one side limit, the wheeled valve at the inlet only opened to the inside of the pump chamber, and the outlet wheeled valve only to the outside. Figure 1b is the assembly view of the piezoelectric resonance pump.

**Figure 1.** Structure of the piezoelectric resonance pump; (**a**) cross-sectional view; (**b**) assembly view.

Figure 2 is the schematic of the piezoelectric resonance pump when pumping liquid. The square piezoelectric vibrator periodically vibrates when it is driven by the sine wave signal. In resonance, the center amplitude of the vibrator can be amplified by the elastic vibration transfer block. When the vibrator moves upward, the pump chamber diaphragm moves upwards as well, under the driving force of the block. The pressure in the pump chamber is lowered with the inlet valve open and the outlet valve closed, at which point liquid flows into the pump chamber through the inlet. Conversely, when the vibrator moves downward, the pump chamber diaphragm moves downwards under the action of the elastic vibration transfer block. The pressure in the pump chamber increases with the inlet valve closed and the outlet valve open, at which point liquid flows out through the outlet.

**Figure 2.** Schematic diagram of pump operation.

#### *2.2. Design of the Piezoelectric Vibrator with Flexible Support*

Figure 3 is a structural diagram of the square piezoelectric vibrator designed in this paper. It consists of a 60Si2Mn substrate and a square piezoelectric layer, bonded together. The metal substrate adopts a hollow design, and its four corners are fixed on the vibrator holder, thereby forming a structure with flexible support for the square piezoelectric vibrator. This structure can reduce the vibration constraint of the vibrator and concentrate the deformation in the hollow area during the vibration. The metal substrate with flexible support can improve the mechanical properties of the piezoelectric vibrator and protect the piezoelectric layer.

**Figure 3.** Piezoelectric vibrator based on a flexible support.

## **3. Dynamic Model**

The piezoelectric resonance pump uses system resonance to amplify the amplitude of the piezoelectric vibrator and drive the pump chamber diaphragm to improve the volume change of the pump chamber and optimize the pump's output performance. The dynamic model of the vibration system is established in this section to study the influence of system parameters on the natural frequency and amplitude amplification coefficient. Figure 4 shows the simplified dynamic model of the piezoelectric resonance pump. *Mact* is the equivalent mass of the square piezoelectric vibrator; *Mdia* is the equivalent mass of elastic vibration transfer block and pump chamber diaphragm; *Kact* is the equivalent stiffness of the square piezoelectric vibrator; *Ktra* is the equivalent stiffness of the elastic vibration transfer block; and *Kdia* is the equivalent stiffness of the pump chamber diaphragm. The interaction between the fluid and the flow channel is equivalent to damping *C*, ignoring the material damping of the elastic element in the vibration system.

**Figure 4.** Dynamic model of the piezoelectric resonance pump.

The square piezoelectric vibrator uses a sine wave driving power, and *w* is the frequency of the driving power. If *F*0 is the amplitude of the output force of the square piezoelectric vibrator, then assume that the vibration displacement and output force at time *t* are *X*0*(t)* and *F0*cos*wt* respectively, and *X(t)* is the vibration displacement of the pump chamber diaphragm. Thus, the motion differential equation of the system is

$$\begin{cases} \mathcal{M}\_{\rm act} \ddot{X}\_0 + \mathcal{K}\_{\rm act} X\_0 + \mathcal{K}\_{\rm tra} (X\_0 - X) = F\_0 \cos \omega t\\\mathcal{M}\_{\rm dia} \ddot{X} + \mathcal{C} \dot{X} - \mathcal{K}\_{\rm tra} (X\_0 - X) + \mathcal{K}\_{\rm dia} X = 0 \end{cases} \tag{1}$$

In Equation (1), *F*0cos*ωt* is the output force from the center of the square piezoelectric vibrator. If the vibrator is considered as an ideal spring that provides vibration power, the mass of *Mact* can be ignored, so the motion differential equation can be converted to

$$M\_{\rm diu}\ddot{X} + \mathcal{C}\dot{X} + \frac{K\_{\rm act}K\_{\rm tra} + K\_{\rm act}K\_{\rm dia} + K\_{\rm tra}K\_{\rm dia}}{K\_{\rm act} + K\_{\rm tra}}X = \frac{K\_{\rm tra}F\_{0}\cos\omega t}{K\_{\rm act} + K\_{\rm tra}}.\tag{2}$$

The natural frequency of the system can be obtained from Equation (2):

$$
\omega\_n = \sqrt{\frac{K\_{\rm act} K\_{\rm trra} + K\_{\rm act} K\_{\rm dia} + K\_{\rm act} K\_{\rm dia}}{M\_{\rm dia} (K\_{\rm act} + K\_{\rm trra})}}.\tag{3}
$$

*ζ* = *C* 2*Mdia<sup>ω</sup>n* is the damping ratio. The steady-state response amplitude of the system is obtained by using Laplace transform to Equation (2):

$$X = \frac{K\_{\rm tra}}{K\_{\rm act}K\_{\rm tru} + K\_{\rm act}K\_{\rm dia} + K\_{\rm act}K\_{\rm dia}} \cdot \frac{F\_0}{\left[1 - \left(\frac{\omega}{\omega\_n}\right)^2\right]^2 + \left(2\frac{\tau}{\omega}\frac{\omega}{\omega\_n}\right)^2}. \tag{4}$$

The deformation under the static force *F*0 is

$$\delta\_{\rm st} = \frac{K\_{\rm tra} F\_0}{K\_{\rm act} K\_{\rm tra} + K\_{\rm act} K\_{\rm dia} + K\_{\rm act} K\_{\rm dia}}.\tag{5}$$

Then, the amplitude amplification coefficient of the system is calculated by Equations (4) and (5):

$$\frac{X}{\delta\_{st}} = \frac{1}{\sqrt{\left[1 - \left(\frac{\omega}{\omega\_n}\right)^2\right]^2 + \left(2\zeta \frac{\omega}{\omega\_n}\right)^2}}.\tag{6}$$

According to Equation (6), when the driving frequency fulfills *ω* = *ωn*, the system resonates, with the amplitude amplification coefficient reaching maximum. Due to the damping of the vibration system, the amplification coefficient is limited.

Wheeled check valves with different structural parameters all have the feature of response hysteresis. The higher the driving frequency of the system, the more obvious the hysteresis, which will reduce the working efficiency of the wheeled check valve and lower the performance of the pump. Therefore, it is vitally important to select the appropriate system operating frequency to improve output performance. We can make a preliminary calculation on the resonance frequency, according to Equation (3). On this basis, considering the feature of hysteresis, the appropriate structural parameters for the wheeled check valve can be determined to improve the output performance.

#### **4. Prototype Fabrication and Experimental Device**

## *4.1. Experimental Prototypes*

Figure 5a shows the key structural parameters of the piezoelectric resonance pump. *Dt* is the diameter of the fixed connection part between the lower vibration transfer block and the pump chamber diaphragm, hereinafter referred to as "fixed connection diameter". *Dc* is the diameter of the pump chamber, *hr* is the height of the chamber and *do* is the diameter of the flow channel. *Di* and *Do* are the inner and outer diameters of the annular elastic gasket in the middle of the elastic vibration transfer block, and *hk* is the thickness of the annular elastic gasket. Figure 5b shows the key structural parameters of the wheeled check valves at the inlet and outlet of the pump. *ds* is the outer diameter of the valve, *dm* is the outer diameter of the moving disc, and *kv* indicates the stiffness of the valve in the opening direction. These structural parameters, which have a direct impact on the output performance

of the piezoelectric resonance pump, were tested experimentally in the following section. The main parameters for the prototype designed are shown below in Table 1.

**Figure 5.** Key structural parameters of the piezoelectric resonance pump. (**a**) Main structure of the pump; (**b**) Structure of the wheeled check valve.


Figure 6 is a photo of the piezoelectric resonance pump designed in this paper. Its overall dimensions were 50 mm × 50 mm × 20 mm. The material of the pump body was polymethyl methacrylate (PMMA), which is highly transparent and convenient to observe the working status of the wheeled check valve. The diaphragm and the valve were made of beryllium bronze sheets, and the annular elastic gasket was made of 60Si2Mn. The 60Si2Mn is a kind of silicomanganese alloy spring steel and commonly used as the material for piezoelectric vibrator substrates. Beryllium bronze has features such as high strength, hardness, elastic limit and fatigue limit, and has a small elastic hysteresis.

**Figure 6.** Photo of the piezoelectric resonance pump.
