# Study on the Propulsion Characteristics of a Flapping Flat-Plate Pumping Device
## 2 Physical Model
### Motion Model  
The flapping flat-plate hydrofoil pumping device is inspired by fish locomotion patterns, where coordinated movements of the body and tail generate jet-like vortices that produce thrust, maintaining high propulsion efficiency during swimming and flapping. Fish locomotion can be considered as a coupled heaving and pitching motion of the tail, based on which the motion trajectory model of the flat-plate hydrofoil is established as shown in Figure 3. In the figure, \(A_{max}\) represents the heaving amplitude of the flat-plate hydrofoil; \(\theta_{max}\) denotes the pitching amplitude; and T indicates the motion period.  
The synchronous belt drive mechanism is controlled by a synchronous belt, and its motion follows simple harmonic motion. The motion equation of the flapping flat-plate hydrofoil is given as follows:  
\[\left\{\begin{aligned} y(t)=A_{max}\sin(\omega t)\\ \theta(t)=\theta_{max}\sin(\omega t+\varphi)\end{aligned}\right. \tag{1}\]  
where \(y(t)\) represents the heaving displacement of the flapping flat-plate hydrofoil; \(\theta(t)\) represents the pitching displacement; \(\omega\) is the angular frequency of flapping; \(\varphi\) denotes the phase difference between heaving and pitching motions.  
The motion equation of the crank-slider drive mechanism follows quasi-harmonic motion, expressed as follows:  
\[\left\{\begin{aligned} y(t)=\frac{\overline{r_{\text{cross}}(wt)(l _{\text{offset}}-rsin(wt))}}{\theta(t)=-\frac{(rnsin(\theta_{20}+wt)-rnsin( \theta_{20}+wt))}{b(\frac{(rnsin(\theta_{20}+wt)-rnsin(\theta_{20}+wt))^{2}}{ b^{2}}+l)}}\end{aligned}\right. \tag{2}\]  
where \(r\) represents the crank radius; \(w\) is the crank angular frequency; \(l_{\text{offset}}\) denotes the offset distance; \(\theta_{20}\) is the initial phase of the heaving crank; \(\theta_{30}\) is the initial phase of the pitching crank; \(b\) is the distance between sliders; and \(l\) is the length of the connecting rod.  
Figure 3: (**a**) Schematic diagram of simple harmonic motion; (**b**) Schematic diagram of quasi-harmonic motion.  
In this study, the actual heaving and pitching trajectories for both the crank-slider mechanism and synchronous belt mechanism were obtained using ADAMS-2020 software. These actual motion trajectories were then compared with the theoretical trajectories derived from the kinematic models, and the results are presented in Figure 4. The comparison reveals that the actual motion trajectories align well with the theoretical predictions, validating the reliability of the modeled kinematics for both drive mechanisms.  
To investigate the effect of the flow channel outlet angle on the hydraulic performance of the device, comparative analyses were conducted to select the optimal outlet design. Figure 5 illustrates the flapping flat-plate hydrofoil device structures under various outlet design schemes. In this study, to standardize the notation of the outlet angle, the following convention is adopted: when the outlet opens outward relative to the baseline reference line (horizontal line), the angle is considered positive; conversely, when the outlet contracts inward, the angle is negative.  
Figure 4: (**a**) Actual motion trajectories of simple harmonic motion; (**b**) Actual motion trajectories of quasi-harmonic motion.