# Study on the Mechanism of the Variable-Speed Rotor Affecting Rotor Aerodynamic Performance
## 2 Modeling
From the above, the normal force can be expressed as  
\[C_{N}^{T}=C_{N}^{f}+C_{N}^{v} \tag{25}\]  
and the chord-wise force can be expressed as  
\[\left\{ \begin{array}{ll}C_{C}^{f}\left(k\right)=\eta C_{N}\left(Ma\right) \alpha_{E}\sin\alpha_{E}\left(k\right)\sqrt{f^{\prime\prime}\left(k\right)}&C _{N}^{\prime}\left(k\right)\leq C_{NI}\\ C_{C}^{f}\left(k\right)=\eta C_{N}\left(Ma\right)\alpha_{E}\sin\alpha_{E}\left( k\right)\sqrt{f^{\prime\prime}\left(k\right)}\Psi&C_{N}^{\prime}\left(k\right)>C_{NI} \end{array} \right. \tag{26}\]  
where \(\eta\) is a recovery factor and is often set to 0.95. The term \(\Psi=f^{Df\left(C_{N}^{\prime}-C_{NI}\right)}\) ensures a smooth chord-wise force with respect to the variation in AoA:  
\[\left\{ \begin{array}{ll}C_{L}=C_{N}^{T}\cos\alpha+C_{C}^{f}\sin\alpha\\ C_{D}=C_{d0}+C_{N}^{T}\sin\alpha-C_{C}^{f}\cos\alpha.\end{array} \right. \tag{27}\]  
In a series of ramp-up dynamic experiments by the University of Glasgow, [PERSON], [PERSON] and [PERSON] found that the original Leishman-Beddoes algorithm predicts the dynamic stall vortex formation ahead of the actual situation when the Mach number is 0.12 [24; 25]. Then, [PERSON] et al. proposed a new dynamic stall criterion in the original Leishman-Beddoes framework in a low Mach number situation \(\left(Ma<0.3\right)\). In the following study, the new criterion was found to be applicable in an oscillation freestream [26]. In the Sheng correction, there is a linear relationship between the reduced pitch rate, \(r\left(r=\frac{\delta c}{2V}\right)\), and the dynamic stall in AoA, \(\alpha_{ds}\):  
\[\alpha_{ds}=\alpha_{ds0}+T_{\alpha}r\qquad(r\geq r_{0}). \tag{28}\]  
The critical stall onset angle is defined as  
\[\left\{ \begin{array}{ll}\alpha_{cr}=\alpha_{ds0}&r\geq r_{0}\\ \alpha_{cr}=\alpha_{ss}+\left(\alpha_{ds0}-\alpha_{ss}\right)\frac{r}{r_{0}}&r< r_{0}\end{array} \right. \tag{29}\]  
where the steady stall angle, \(\alpha_{ss}\), and variables, \(\alpha_{ds0}\), \(T_{\alpha}\) and \(r_{0}\), are obtained via an airfoil experiment. The dynamic stall angle is derived by a one-step delay:  
\[\Delta\alpha^{\prime}=\left(\alpha\left(s\right)-\alpha\left(s-\Delta s\right) \right)\left(1-e^{-\frac{s}{\delta_{\alpha}}}\right) \tag{30}\]  
where the variable \(s\) expresses the non-dimensional time constant. In a low Mach number situation, the critical condition, \(\alpha^{\prime}>\alpha_{cr}\), replaces \(C_{N}^{\prime}>C_{NI}\) to be a new dynamic stall onset criterion. Accordingly, the Sheng correction implements an equivalent angle shift in the fitting function of the trailing edge separation point, Equation (21):  
\[\left\{ \begin{array}{ll}f^{\prime}=1-0.3e^{\left(\alpha^{\prime}-\alpha_{1} -\Delta\alpha_{1}\right)s_{1}^{-1}}&\alpha^{\prime}\geq\alpha_{1}\\ f^{\prime}=0.04+0.66e^{\left(\alpha_{1}+\Delta\alpha_{s}-\alpha^{\prime} \right)s_{2}^{-1}}&\alpha^{\prime}<\alpha_{1}\end{array} \right. \tag{31}\]  
where the shift angle matches  
\[\left\{ \begin{array}{ll}\Delta\alpha_{1}=\alpha_{ds0}-\alpha_{ss}&r\geq r_{0 }\\ \Delta\alpha_{1}=\left(\alpha_{ds0}-\alpha_{ss}\right)\frac{r}{r_{0}}&r<r_{0}. \end{array} \right. \tag{32}\]  
In the current work, the rigid flap assumption was applied, and a Jacobian method for trimming was implemented. It is known that the Jacobian matrix is used to determine the direction of updating feathering controls. We applied the C81 data and the linear inflow model to accelerate the matrix computation, and then used the dynamic stall algorithm and wake method to judge whether it was trimmed, maintaining trimming computation accuracy.