# Research on the Coupled Bending-Torsional Flutter Mechanism for Ideal Plate
## 4 Ideal Flat Plate Flutter Analysis
### Flutter Frequency  
Figure 3 shows the variation curves of the vertical and torsional modes of the ideal plate with wind speed obtained by the methods of \"complex mode analysis\", \"stiffness drive\" mechanism, and \"incentive feedback\" mechanism. From the diagram, it can be concluded that:  
1. The vertical and torsional mode frequencies identified by the two theories of \"complex mode analysis\" and \"incentive feedback\" mechanism are consistent with each other. The torsional mode frequencies identified by the \"stiffness-driven\" mechanism theory are consistent with the \"complex mode analysis\" and \"incentive feedback\" mechanisms, and the vertical vibration frequency is equal to the torsional vibration frequency in the flutter critical state.
2. The torsional vibration damping ratio is close to zero at all levels of wind speed, while the vertical vibration damping ratio gradually increases with the increase of wind speed, and is obviously greater than zero. Therefore, the variation curve of the torsional mode frequency with wind speed identified by the \"stiffness-driven\" mechanism is consistent with the torsional vibration frequency identified by the complex mode analysis and the incentive feedback mechanism, while the vertical mode frequency identified by the \"stiffness-driven\" mechanism is deviated from the modal frequency identified by the complex mode analysis and the \"incentive-feedback\" mechanism. However, in the flutter critical state, the damping ratios of vertical and torsional vibrations are zero, which is consistent with the assumption of the \"stiffness drive\" mechanism, where the vertical and torsional vibration frequencies are the same, and the flutter critical frequency is located on the torsional vibration frequency change curve, indicating that the ideal plate flutter is a bending-torsion-coupled vibration driven by its torsional mode. When the wind speed is greater than the flutter critical wind speed, the torsional mode damping is identified as negative damping and the vertical mode damping is positive by the \"complex mode analysis\" and \"incentive feedback\" mechanisms. Therefore, when the wind speed exceeds the flutter critical wind speed, the flutter morphology of the ideal plate is due to the negative damping of the torsional mode.  
Based on the above analysis, the results of the \"stiffness-driven\" mechanism analysis show that flutter is a bending-torsional-coupled vibration generated by torsional mode drive, while the results of \"complex mode analysis\" and \"incentive feedback\" mechanism analysis show that the reason why flutter is divergent is that the torsional mode damping  
Figure 3: Vibration frequency of an ideal plate.  
is negative. It can be seen that there is no contradiction between the two theories of the \"stiffness-driven\" mechanism and \"torsional negative damping driven\" mechanism in flutter analysis, and they expound the characteristics of flutter bending-torsional-coupled vibration and self-excitation divergence from the aspects of \"flutter frequency\" and \"flutter damping\", respectively.