## 3 Results and Discussion
### Work Analysis of Unsteady Aerodynamic Force  
For the spring-suspended dynamic system, the cumulative work from \(t_{0}\) to \(t_{1}\) done by the aerodynamic self-excited forces when flutter occurred can be calculated as  
\[\left\{ \begin{array}{l}W_{F_{w}}=\int_{0_{0}}^{t_{1}}F_{\mathrm{e}}(t)\dot{h}(t )dt\\ W_{T_{w}}=\int_{0_{0}}^{t_{1}}T(t)\dot{\alpha}(t)dt\end{array} \right. \tag{6}\]  
The cumulative work histories on the two degrees of freedom done by aerodynamic self-excited forces for the two kinds of flutter were displayed in Figures 11 and 12. The self-excited forces and displacements were processed by a low-pass filter with different cut-off frequencies. Then, the first-order and higher-order harmonic components of the cumulative work were obtained. The work done by the first-order component of the self-excited force was the largest. Yet the higher-order components of the self-excited force provided little energy input to the dynamic system. For the classical flutter, the cumulative work done by vertical and torsional self-excited forces both increased exponentially over time. The cumulative work done by vertical self-excited forces was much  
Figure 11: The cumulative work done by self-excited force for classical flutter (Case: L60, \(U^{*}\) = 10.27): (a) vertical and (b) torsional. Figure 12: The cumulative work done by self-excited force for postflutter LCO (Case: L180, \(U^{*}\) = 6.12): (a) vertical and (b) torsional. bigger than that of the torsional self-excited force, which indicated that the vertical degree of freedom had the main effect on the classical flutter. By contrast, for the postflutter LCO, the cumulative work on vertical and torsional degrees of freedom increased almost linearly. The cumulative work done by the vertical self-excited force was much smaller than the torsional self-excited force. It can be inferred that the torsional degree of freedom was the dominant mode in the postflutter LCO. The aerodynamic hysteresis phenomenon during flutter resulted in a phase difference between vibration displacements and aerodynamic forces. This reflected the work characteristics of aerodynamic self-excited forces. The relationship of the work done by aerodynamic self-excited forces and the phase difference between vibration displacements and aerodynamic forces would be deduced further by the example of a vertical degree of freedom. When the flutter vibration was reducible to a harmonic vibration with a slow amplitude variation, the vertical displacement in a single period can be expressed as follows:  
\[h(t)=h_{Amp}\sin(\omega_{c}t) \tag{7}\]  
where \(h_{Amp}\) represents the slowly varied amplitude of harmonic vibration and can be regarded as a constant in a specific period. \(\omega_{c}\) is the coupled circular frequency of flutter vibration. Then, the vertical velocity is  
\[\dot{h}(t)=h_{Amp}\omega_{c}\cos(\omega_{c}t) \tag{8}\]  
Suppose the phase difference between vertical displacements and self-excited forces is  
\[\Delta\phi_{Flu}=\phi_{F}-\phi_{h} \tag{9}\]  
where \(\phi_{F}\) and \(\phi_{h}\) are the phases of vertical self-excited force and displacement, respectively. The vertical self-excited force can be written as  
\[F_{se}(t)=F_{st}+F_{Amp}\sin(\omega_{c}t-\Delta\phi_{Flu}) \tag{10}\]  
where \(F_{st}\) is the steady component of vertical self-excited force and \(F_{Amp}\) is the slowly varied amplitude of the unsteady component of vertical self-excited force, which is a constant in a specific period. Then the work done by the vertical self-excited force in a period can be calculated by  
\[W_{F_{se}}(t)=\int_{t_{0}}^{t_{0}+T_{c}}F_{se}(t)\Delta\dot{h}(t)dt=\left[F_{ Amp}h_{Amp}\omega_{c}\int_{t_{0}}^{t_{0}+T_{c}}\cos^{2}(\omega_{c}t)dt\right] (-\sin\Delta\phi_{Flu}) \tag{11}\]  
where \(F_{Amp}\), \(h_{Amp}\), \(\omega_{c}\) and \(\int_{t_{0}}^{t_{0}+T_{c}}\cos^{2}(\omega_{c}t)dt\) are all positive, so the sign of the work done by the vertical self-excited force is decided by the sign of \(\sin\Delta\phi_{Flu}\).