## 2 Methodology  
### Rotor Loads and Power Prediction Methods  
In the present work, a numerical model is deployed to predict isolated rotor loads and power in steady level flights. A fully articulated rotor is modelled with a hinge offset. It is assumed that the bending deformations are small when compared to the flap and lag motions and the elastic torsional deformations are involved in the optimal active twist schedules. A rigid blade model is adopted with flap(\(\beta\)) and lag(\(\xi\)) degrees of freedom, as shown in Figure 2. Four reference frames are used. The non-rotate frame named A is used for solving the trim equations. Rotation frame 5 is used for the blade load integral. Blade frame F is used in order to calculate the loads of blade. Blade section frame B is used to solve the local velocity and angle of attack.  
The aerodynamic model was established based on the blade element method. A look-up table for the airfoils is used to calculate the aerodynamic coefficients (\(C_{L}\), \(C_{D}\), and \(C_{M}\)). The quasi-steady correction of the blade element motion is considered. The Pitt-Peters inflow model obtained the induced velocity of the rotor. The motion response of the blades is calculated in the time domain with Euler method.  
\[\left\{\begin{array}{c}dF_{z}=dLcos(\delta)-dDsin(\delta)\\ dF_{X}=dF_{X-Induce}+F_{X-profile}=dLsin(\delta)+dDcs(\delta)\\ dM_{Z}=dM_{Z-Induce}+dM_{Z-profile}=\Big{(}F_{X-profile}+F_{X-profile}\Big{)} rR\\ C_{Z}=C_{L}cos(\delta)-C_{D}sin(\delta)\\ C_{X}=C_{Xi}+C_{Xp}=C_{L}sin(\delta)+C_{D}cos(\delta)\end{array}\right. \tag{1}\]  
where \(F_{Z}\) and \(F_{X}\) are the lift force and drag force of the blade section in frame \(F\), \(F_{X\text{-}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{} \text{}\text{}\text{}\text{}  
and profile components of the shaft moment of the blade section in frame \(S,L\) and \(D\) are the lift force and drag force of the blade section in frame \(B\), \(C_{()}\) is the coefficient, \(r\) is the dimensionless spanwise location of the blade section, \(R\) is the radius of the rotor, and the \(\delta\) is the induced angle of the blade section.  
The load integral of all blades is performed in order to obtain the forces and moments of the hub transferred by the hinges. The hub load is averaged with the rotation period to obtain the resultant forces and moments of the rotor, which is used for the trim and performance prediction. A propulsive trim is used for the helicopter steady-level flight solution. Three controls (collective and two cyclic controls) and two shaft attitudes (longitudinal and lateral attitudes) are obtained by solving the equilibrium equations of forces and moments with zero sideslip angle in frame \(A\). The tail rotor is assumed to automatically balance the torque from the main rotor. The fuselage is modelled as a rigid body. The center of gravity (CG) location of fuselage and the aerodynamic forces and moments that are derived from the attitudes and dynamic pressure are considered for the trim solution, as shown in Figure 3. The Newton-Raphson procedure is used to solve the weak nonlinear trim equations. The simplifying modeling assumptions mentioned above have proved to be sufficient and efficient in rotor power prediction in optimization problems of active control rotors [9; 13; 20].