## 3 Results and Discussion
### Amplitude Dependence of High-Order Harmonic Components in Heating-Pitching Coupled Motion  
The aerodynamic time-domain data are identified through numerical simulation, and the aerodynamic time history curves of five stable periods are taken as shown in Figure 9. As can be seen from Figure 9a, slight waveform distortion can be observed in the aerodynamic time history curve under heaving-pitching coupled motion with small amplitude, and the harmonic characteristics of aerodynamic lift are worse than that of aerodynamic moment. This indicates that the high-order harmonic components have begun to emerge in the case of small amplitude. After increasing the combined amplitude, severe waveform distortion can be observed in both aerodynamic lift and moment, as reported in Figure 9b. That is to say, with the increase in amplitude, the content of high-order harmonic components increases, which makes the harmonic characteristics of time history curve destroyed obviously.  
In order to deeply explore the variation law of nonlinear aerodynamic components under different combined amplitudes, ten stable periods of time history curve are taken to  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Figure** & **Reduced Velocity** & **Heaving Amplitude** & **Pitching Amplitude** \\ \hline \multirow{4}{*}{4} & \multirow{4}{*}{6} & 0.02\(B\) & 2\({}^{\circ}\) \\  & & 0.04\(B\) & 4\({}^{\circ}\) \\  & & 0.08\(B\) & 8\({}^{\circ}\) \\  & & 0.1\(B\) & 10\({}^{\circ}\) \\  & & 0.16\(B\) & 16\({}^{\circ}\) \\  & & 0.2\(B\) & 20\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Operation setting.  
Figure 9: Time history curve of aerodynamic forces. (**a**) Small amplitude motion (\(\alpha_{0}=2^{\circ}\), \(h_{0}=0.02B\)). (**b**) Large amplitude motion (\(\alpha_{0}=20^{\circ}\), \(h_{0}=0.2B\)).  
obtain the frequency domain data by fast Fourier transform (FFT), and the nonlinear components of aerodynamic forces represented by the second to fifth harmonics are analyzed.  
Heaving motions with different amplitudes (\(h_{0}\) = 0.02\(B\), 0.04\(B\), 0.08\(B\), 0.1\(B\), 0.16\(B\), 0.2\(B\)) are added to three groups of foundation pitching motions (\(\alpha_{0}\) = 2\({}^{\circ}\), 8\({}^{\circ}\), 16\({}^{\circ}\)), and the spectrum diagram is shown in Figures 10 and 11. The high-order harmonic components of the aerodynamic forces are increased with the added heating amplitude, and the high-order harmonic main components of the aerodynamic lift are transferred from the second harmonic to the third. Moreover, when the foundation pitching amplitude is larger, the heating amplitude required for the third harmonic to dominate is smaller. The higher harmonic components of aerodynamic moment increase evenly with the amplitude, and the main component is always the second harmonic.  
Therefore, in the flutter calculation of the semi-closed box deck section, the importance of the third harmonic component should be given priority in the aerodynamic lift when the pitching amplitude is larger (\(\alpha_{0}>10^{\circ}\)). For the aerodynamic moment, heaving and pitching amplitudes should be considered comprehensively to determine the importance of the high-order harmonic component in large amplitude motion. When the pitching amplitude is small (\(\alpha_{0}<10^{\circ}\)), the importance of the second harmonic component should be given priority.