## 4 Conclusions  
The amplitude dependence of nonlinear self-excited aerodynamic forces for the semi-closed box deck section under heaving-pitching coupled motion is investigated in this work. The combination form of forced amplitude is changed by dynamic grid technology, and the self-excited aerodynamic forces are obtained by forced vibration numerical simulation for discussion and analysis. Key conclusions deriving from the study include:  
With the increasing pitching amplitude, the main components of the high-order harmonic of aerodynamic forces are transferred from the second harmonic to third in the heaving-pitching coupled motion. With the increasing heaving amplitude, the energy transfer phenomenon of aerodynamic lift is promoted, while that of aerodynamic moment is inhibited. Therefore, when the pitching amplitude is large (\(\alpha_{0}>10^{\circ}\)), the importance of the third harmonic in the aerodynamic lift should be prioritized in nonlinear flutter calculation, while when the pitching amplitude is small (\(\alpha_{0}<10^{\circ}\)), the importance of the second harmonic in the aerodynamic moment should be considered first.  
Compared with the 1-DOF pitching motion, the proportion of nonlinear aerodynamic nonlinear component is reduced when the heaving motion of small amplitude is added. At this time, the small amplitude heaving motion plays a positive damping role in the coupled  
Figure 23: Normalized statistical diagram of aerodynamic nonlinear components in different pitching amplitude. (**a**) foundational heaving amplitude \(h_{0}=0.02B\); (**b**) foundational heaving amplitude \(h_{0}=0.1B\); (**c**) foundational heaving amplitude \(h_{0}=0.2B\).  
Figure 22: Normalized statistical diagram of aerodynamic nonlinear components in different heaving amplitude. (**a**) foundational pitching amplitude \(\alpha_{0}=2^{\circ}\); (**b**) foundational pitching amplitude \(\alpha_{0}=8^{\circ}\); (**c**) foundational pitching amplitude \(\alpha_{0}=16^{\circ}\).  
motion. The heaving motion has little promoting effect on the aerodynamic nonlinear components, which mainly contributes to the aerodynamic lift, and has a complex influence on the nonlinear characteristics.  
In the heaving-pitching coupled motion, the proportion of nonlinear components generally increases with the pitching amplitude, and the proportion of high-order harmonic components can reach more than 20% under large pitching amplitude (\(\alpha_{0}=20^{\circ}\)). The high harmonic components of aerodynamic forces are mainly generated by pitching motion, and the nonlinear characteristics of aerodynamic forces, especially aerodynamic moment, are mainly affected by pitching motion.  
In general, the proportion of nonlinear components of aerodynamic lift is higher than that of aerodynamic moment, so more attention should be paid to the nonlinear research of aerodynamic lift. It is worth noting that the extreme value heaving amplitude of positive damping action is observed in the nonlinear aerodynamic lift, and the nonlinear component of aerodynamic moment has a higher growth rate with the increase in coupled amplitude. Therefore, the nonlinear research of aerodynamic moment should also be emphasized in the case of large amplitude and extreme value heaving amplitude.  
Conceptualization, H.L. and J.J.; methodology, H.L.; software, H.L.; validation, H.L., J.J. and L.Z.; formal analysis, H.L. and J.J.; investigation, H.L. and L.D.; resources, J.J.; data curation, H.L.; writing--original draft preparation, H.L.; writing--review and editing, H.L.; visualization, H.L.; supervision, J.J.; project administration, J.J.; funding acquisition, J.J. All authors have read and agreed to the published version of the manuscript.  
This study was financially supported by NSFC under grant No. 51908108.  
The authors declare no conflict of interest.