# On the Lock-In Phenomena near the Transonic Buffet Onset of a Prescribed Pitching Airfoil
## 2 Energy Map and Model for Amplitude Growth  
For a forced spring-mass-damper oscillator illustrated in Figure 2 that uses a linear torsional spring with the spring constant \(k\) and the damping coefficient \(b\), mounted on a rigid airfoil with a chord length of \(c\) at the elastic axis \(X_{e}\), the governing equation is written as:  
\[I\ddot{\theta}+b\dot{\theta}+k(\theta-\theta_{0})=M \tag{2}\]  
where \(I\), \(M\), and \(\theta_{0}\) are the moments of inertia with respect to \(X_{e}\), the aerodynamic pitch moment with respect to \(X_{e}\) and the equilibrium angular position of the spring, respectively.  
[PERSON] et al. (2020) established a model for the amplitude growth and the equilibrium for the flow-induced flutter of the airfoil, given that the pitch motion \(\theta\) and \(\dot{\theta}\) are sinusoidal, i.e., \(\theta=A_{\theta}sin\omega t\) and \(\omega=2\pi/T\). The multiplication of \(\dot{\theta}\) to Equation (2) results in  
\[I\ddot{\theta}\dot{\theta}+k(\theta-\theta_{0})\dot{\theta}=M\dot{\theta}-b \dot{\theta}\dot{\theta} \tag{3}\]  
which can be described by the growth of the energy in the system as follows:  
\[\frac{d}{dt}\left(\frac{1}{2}I\dot{\theta}^{2}\right)+\frac{d}{dt}\left(\frac {1}{2}k\theta^{2}\right)=\frac{de_{f}^{*}}{dt}-\frac{de_{d}^{*}}{dt}=\frac{de ^{*}}{dt}. \tag{4}\]  
In Equation (4) \(e_{f}^{*}\) and \(e_{d}^{*}\) denote the energy absorbed from the flow by the airfoil and the energy lost due to the damping, respectively. By integrating Equation (4) over an oscillation cycle period \(T\) with the cycle beginning at \(t_{n}\) we can further achieve  
\[I\Big{[}\dot{\theta}^{2}(t_{n}+T)-\dot{\theta}^{2}(t_{n})\Big{]}+k\Big{[} \theta^{2}(t_{n}+T)-\theta^{2}(t_{n})\Big{]}=2[e^{*}(t_{n}+T)-e^{*}(t_{n})]=2 E^{*} \tag{5}\]  
where \(E^{*}\) represents the net energy extraction of the airfoil within a cycle. Under the assumption of sinusoidal pitch motion, the amplitude shift over a cycle is derived from Equation (5) as follows:  
\[A_{\theta}^{2}(t_{n}+T)-A_{\theta}^{2}(t_{n})=\frac{2E^{*}}{I\omega^{2}cos^{2 }\omega t+ksin^{2}\omega t}. \tag{6}\]  
where \(A_{\theta}(t_{n})=\frac{\theta(t_{n})}{sin(\omega t)}=\frac{\dot{\theta}(t_{ n})}{\omega cos(\omega t)}\). If the pitch oscillation occurs at the natural structural frequency \(\omega=\sqrt{k/T}\) the amplitude growth can be simplified as follows:  
\[A_{\theta}^{2}(t_{n}+T)-A_{\theta}^{2}(t_{n})=\frac{2E^{*}}{k} \tag{7}\]  
or  
\[A_{\theta}(t_{n}+T)-A_{\theta}(t_{n})=\frac{2E^{*}}{k\overline{A_{\theta}}} \tag{8}\]  
where \(\overline{A_{\theta}}=[A_{\theta}(t_{n}+T)-A_{\theta}(t_{n})]/2\).  
The equilibrium is obtained when \(A_{\theta}(t_{n}+T)=A_{\theta}(t_{n})\), i.e.,  
\[E^{*}=0 \tag{9}\]  
Figure 2: The aeroelastic structural system for a one degree-of-freedom (1-DOF) pitch.  
according to the Poincare map. Only when  
\[\frac{d}{dA_{\theta}}\bigg{(}A_{\theta}+\frac{E^{*}}{kA_{\theta}}\bigg{)}<1 \tag{10}\]  
or  
\[\frac{dE^{*}}{dA_{\theta}}<0 \tag{11}\]  
can the system aforementioned be stable. The fact that Equations (9) and (11) are both satisfied determines whether the system will equilibrate if the airfoil is allowed to flutter freely.