## 4 Results and Discussion
### Aerodynamic Damping
#### 4.3.3 Aerodynamic Damping Calculation  
As previously presented, the non-aerodynamic and total damping values are so small that the influence of the damping on the vibration frequency can be ignored. When \(f(t)\) is the undamped response of displacement, then the damped response in the vacuum is \(e^{-\xi_{m}\omega_{k}t}f(t)\), where \(\omega_{k}\) is the natural circle frequency associated with the natural frequency. The vibration damping in flow is composed of two parts, i.e., non-aerodynamic damping and aerodynamic damping. Thus, the damped response in flow can be described by \(e^{-(\xi_{a}+\xi_{m})\omega_{k}t}f(t)\). The total damping in flow can be expressed as \(\xi_{tot}=\xi_{a}+\xi_{m}\), where \(\xi_{tot}\) and \(\xi_{m}\) are predicted quantitatively, as discussed above. Therefore, aerodynamic damping is calculated by the following formula:  
\[\xi_{a}=\xi_{tot}-\xi_{m}. \tag{14}\]  
The total damping only includes the aerodynamic and numerical damping when the Rayleigh damping constants \(\alpha\) and \(\beta\) are both zero. Generally, the numerical damping is so small that it can be neglected, and the total damping with the Rayleigh damping constants of \(\alpha=0,\ \beta=0\) are considered to be mainly due to aerodynamics. Figure 17 shows the aerodynamic damping values which are calculated according to Equation (14) with in accordance with the Rayleigh damping constants \(\alpha=0,\ \beta=6\times 10^{-6}\) and \(\alpha=0,\ \beta=0\) for the first bending modal excitation, respectively. As shown in Figure 17, these results basically conform to the predictions. The aerodynamic damping is reduced when the non-dimensional mass flow rate decreases. In particular, the aerodynamic damping value reaches 0 when the non-dimensional mass flow rate is about 0.825, which represents the flutter boundary. This is basically consistent with the experimental results, where the non-dimensional mass flow rate of the flutter boundary is 0.795 [18]. Figure 16: Tip leading-edge displacement in flow for the second modal excitation after filtering. Figure 17: Comparison of aerodynamic damping. Figure 18 shows the blade tip leading-edge displacement response for zero Rayleigh damping (\(\alpha=0,\ \beta=0\)) at different mass flow rates corresponding to A, B, and C operating conditions, as presented in Figure 17. From A to C, the convergent blade displacement response gradually becomes divergent, indicating that the blade tends to suffer from instability. Clearly, the \(\xi_{m}\) and \(\xi_{tot}\) are affected by mechanical damping. Furthermore, the influence of mechanical damping (\(\alpha,\beta\)) on the aerodynamic damping using the fluid-structure interaction simulation is studied. The Rayleigh damping constants are calculated using Equation (10) and Equation (11) from a considered frequency range of \(f_{lo}\) to \(f_{up}\). The aerodynamic damping results with different Rayleigh damping constants, which are calculated using different frequency ranges, are shown in Table 4.  
The first three natural frequencies in the dynamic response are \(f_{1}\), \(f_{1}\), and \(f_{3}\). Table 4 shows the significant difference in total damping \(\xi_{tot}\) and non-aerodynamic damping \(\xi_{m}\) with various Rayleigh damping, whereas the values of aerodynamic damping \(\xi_{a}\) are almost the same except for those defined by the frequency range of 0 to \(f_{1}\), which correspond to the largest \(\beta\). This indicates that the damping constant \(\beta\) is not too large for the accurate prediction of aeroelastic stability. However, the computational cost generally increases as \(\beta\) decreases. The results in Table 4 suggest that the Rayleigh damping constants defined by the frequency range of 0 to the second natural mode are appropriate when considering both computational accuracy and cost for the coupled fluid-structure simulation. Furthermore, the effect of the amplitude of the excitation force on predicted aerodynamic damping is also studied.