## 3 Results and Discussion
### Influence of Bending-Torsional Coupled Effect on Nonlinear Aerodynamics  
When the heaving and pitching are combined, one degree of freedom is bound to affect the other, and the interaction between the two motion modes leads to the existence of bending-torsional coupled term in aerodynamic forces. In order to observe the characteristic law of the interaction between the two degrees of freedom in heaving-pitching coupled motion, another motion form with different amplitudes is added to a specific basic motion, and the aerodynamic results are compared with the single degree of freedom (abbreviated as 1-DOF) motion. The variation in fundamental frequency, the second and third harmonic of aerodynamic lift and moment with heaving motion amplitude is reported in Figures 14 and 15. It can be seen from Figures 14a and 15a that in the state of 1-DOF pitching motion, with the increase in heaving amplitude, the fundamental frequency amplitude of aerodynamic forces has a roughly positive linear relationship with heaving amplitude. The linear relationship is destroyed when the foundational pitching motion exists, and the larger the foundational pitching motion amplitude is, the worse the linear relationship is, and the  
Figure 12: Aerodynamic lift spectrum under different coupled amplitude. (**a**) Heaving amplitude \(h_{0}=0.02B\) (**b**) Heaving amplitude \(h_{0}=0.1B\) (**c**) Heaving amplitude \(h_{0}=0.2B\). Figure 13: Aerodynamic moment spectrum under different coupled amplitude. (**a**) Heaving amplitude \(h_{0}=0.02B\) (**b**) Heaving amplitude \(h_{0}=0.1B\) (**c**) Heaving amplitude \(h_{0}=0.2B\). slower the growth rate of the fundamental frequency amplitude with heaving amplitude is. As shown in Figure 14b, for the aerodynamic lift, the curve shape of the second and third harmonic with the increase in heaving amplitude can hardly be damaged due to the change in the foundational pitching amplitude. The growth rate of the third harmonic is significantly higher than that of the second harmonic when the heaving amplitude is large (\(0.1B\)). Therefore, the phenomenon that the main components of high-order harmonic are transferred from the second harmonic to the third harmonic will appear with the increase in the heaving amplitude. As shown in Figure 15b, the curve shape of the second harmonic is basically not affected by the change in the foundational pitching amplitude, but the curve shape of the third harmonic is changed by it. In the heaving-pitching coupled motion, the pitching motion dominates the nonlinear characteristics of aerodynamic lift and moment by affecting the third harmonic, and the promotion of the nonlinear characteristics by the heaving motion is mainly reflected in the aerodynamic lift. As shown in Figures 16a and 17a, the linear relationship between fundamental frequency amplitude of aerodynamic forces and pitching amplitude involved is weak regardless of 1-DOF pitching motion or 2-DOF coupled motion, and the larger the amplitude, the weaker the linear characteristics. Therefore, pitching motion is considered to be the primary source form of nonlinear components. Figures 16b and 17b show that the curve shape of the second harmonic and third harmonic component with pitching amplitude is not significantly changed under different foundational heaving amplitudes, and both of  
Figure 14: Variation in Fourier amplitude of aerodynamic lift with heaving amplitude. (**a**) The fundamental frequency (**b**) Second harmonic and third harmonic. Figure 15: Variation in Fourier amplitude of aerodynamic moment with heaving amplitude. (**a**) The fundamental frequency; (**b**) Second harmonic and third harmonic. them show an increasing trend. With the increase in pitching amplitude, the growth rate of the third harmonic is higher than that of the second harmonic, and the growth range is significant under large amplitude. At this time, the importance of the third harmonic is prominent and should be paid attention to.