# Study on the Propulsion Characteristics of a Flapping Flat-Plate Pumping Device
## 4 Results
### Impact of Flapping Mode on Water Body  
#### 4.1.1 Effect of Flapping Mode on Thrust and Lift  
To study the effect of different flapping modes of the two devices' flapping flat-plate hydrofoils on thrust and lift, calculations were performed for the conditions corresponding to a flapping frequency of \(f=1\) Hz for both modes. Figure 9 shows the instantaneous thrust coefficient curves for the two flapping modes over two complete motion cycles. For both modes, the thrust coefficient exhibits a double-peak pattern within each cycle, corresponding to the reciprocal oscillatory motion of the hydrofoil. It can be observed that the variation trends of the instantaneous thrust coefficient in the two flapping modes shown in the figure exhibit similar periodic evolution characteristics. When the hydrofoil begins its periodic flapping, the instantaneous thrust coefficient gradually decreases with the increase in flapping amplitude, reaching the minimum value of the cycle at the maximum flapping amplitude. As the hydrofoil moves toward its equilibrium position, the thrust coefficient begins to increase and reaches the maximum value of the cycle just before equilibrium. During the reverse flapping phase, the thrust coefficient again undergoes an asymmetric fluctuation process, first decreasing and then increasing, ultimately forming a bimodal periodic variation. It is also noteworthy that the fluctuations in the instantaneous thrust coefficient differ between the two flapping modes. The simple harmonic motion, driven by the synchronous belt, exhibits a smooth sinusoidal waveform with typical periodic fluctuations, while the quasi-harmonic motion, driven by the crank-slider mechanism, generates two distinct peaks within two periods. Since the synchronous belt drive provides uniform periodic motion, the amplitude and frequency of the hydrofoil's motion remain consistent, resulting in a smooth and symmetric variation in the thrust coefficient. The thrust coefficient follows a typical sinusoidal waveform with a fluctuation frequency corresponding to the flapping frequency of the hydrofoil, exhibiting a symmetric and smooth trend over one period. Moreover, the thrust coefficient remains entirely positive throughout the cycle, indicating continuous forward thrust generation and stable propulsion. However, the instantaneous thrust coefficient of quasi-harmonic motion exhibits more complex variations, particularly with two distinct peaks occurring over two periods. This phenomenon arises from the nonlinear motion characteristics of the crank-slider  
Figure 9: Instantaneous thrust coefficient curves for different flapping modes. mechanism. Unlike the smooth motion of the synchronous belt drive, the crank-slider mechanism causes asymmetric and nonlinear oscillations of the hydrofoil. During the hydrofoil's oscillation, the crank-slider mechanism can induce significant acceleration at certain moments, especially during the acceleration phase, where the speed variation of the hydrofoil is more pronounced, resulting in sharp fluctuations in the thrust coefficient. As a result, in quasi-harmonic motion, the instantaneous thrust coefficient exhibits multiple peaks, which are closely related to the acceleration changes, asymmetric motion frequency, and amplitude of the hydrofoil. Notably, the thrust coefficient becomes negative during parts of the cycle, indicating transient drag generation. This behavior results from the nonlinear acceleration profile introduced by the crank-slider mechanism, particularly near the turning points of the flapping cycle. The two unequal thrust peaks per cycle result from the non-uniform motion characteristics of the quasi-harmonic flapping, where the hydrofoil experiences asymmetric velocity and acceleration profiles across different phases of the cycle.