# Study on the Mechanism of the Variable-Speed Rotor Affecting Rotor Aerodynamic Performance
## 4 Results and Discussion  
It is clear that the blade dynamic pressure varies with the square of rotor speed, thus directly affecting the rotor required power. At the same time, the blade aerodynamic conditions, such as the airfoil AoA and Mach number, change with variation in the rotor speed which also affects the aerodynamic characteristics, such as the airfoil lift coefficient and the drag coefficient. The current work applied a dimensionless parameter analysis. This kind of analysis method could separate the variable-speed rotor performance from the dynamic pressure effects and focus on the aerodynamic characteristics. Based on a previous study, it is known that the rotor speed affects the induced power and profile power more significantly [30], which is proportional to the cubic rotor speed:  
\[P_{i}\sim\omega^{3}\int_{0}^{2\pi}\int_{\varepsilon}^{1}\lambda_{i}dC_{T} \tag{33}\]  
\[P_{o}\sim\omega^{3}\int_{0}^{2\pi}\int_{\varepsilon}^{1}\left(\mu_{h}dC_{Ho} +\frac{1}{R}dC_{Qo}\right) \tag{34}\]  
where \(\lambda_{i}\) and \(\mu_{h}\) are, respectively, the induced velocity ratio and the advance ratio at the HP (hub plane) coordinate. The increased backward thrust force derived from the airfoil drag and the shaft torque derived from the airfoil drag are defined as \(dC_{T}\), \(dC_{Ho}\) and \(dC_{Qo}\), respectively. Figure 4: Bo 105 trim plot. Therefore, we can use the remaining terms  
\[\eta_{i}=\int_{0}^{2\pi}\int_{\varepsilon}^{1}\lambda_{i}dC_{T} \tag{35}\]  
\[\eta_{o}=\int_{0}^{2\pi}\int_{\varepsilon}^{1}\left(\mu_{h}dC_{Ho}+\frac{1}{R}dC _{Qo}\right) \tag{36}\]  
to represent the variable-speed rotor characteristics. It is noteworthy that the values of variables \(\eta_{i}\) and \(\eta_{o}\) equal the corresponding power coefficients. The sum of the induced power and profile power is defined as the direct power loss (DPL), which represents the part of total required power directly affected by the rotor speed. Figure 5 depicts the relationship between the DPL and the rotor speed for different flight speeds under the condition of given take-off weights. The four classic flight speeds are selected, representing hover, slow flight, cruise, and fast flight, respectively. Furthermore, the cubic rotor speed curve is also given. The variable-speed rotor aerodynamic characteristics are indicated by comparing the DPL variation with the cubic curve. When the rotor speed changes from a high value to a low value, if the DPL variation trend is slower than the cubic curve, this indicates that the dynamic pressure reduction is the main reason why a lower rotor speed could decrease the required power. If the DPL variation trend is faster, this indicates that the rotor aerodynamic performance is improved. That is, \(\eta_{o}\) and \(\eta_{i}\) are reduced. In this situation, both the advanced aerodynamic performance and the reduced dynamic pressure contribute to the required rotor power reduction. It can be seen from Figure 5 that, in hover and slow flight, the DPL variation trend is slower than the cubic curve, and the DPL variation with rotor speed is slight, which means that, primarily, the blade dynamic pressure reduction improves the rotor aerodynamic performance. During cruise and fast flight modes, the DPL variation trend is faster than the cubic curve, implying that both the aerodynamic performance improvement and dynamic pressure reduction play roles. A dimensionless parameter analysis is applied to separate the variable-speed rotor performance from the dynamic pressure. The conditions of slow flight are listed in Table 2. It is noticed that the three cases have different true flight parameters, but the same advance ratio and blade loading coefficient. Figure 5: Variation in direct power loss (DPL) with rotor speed for different take-off weights. Figures 6-8 respectively depict the time history of airfoil AoA, lift coefficient, \(C_{L}\), and drag coefficient, \(C_{D}\), with respect to the azimuth, for the above three kinds of slow flight cases.