## 2 Integrated Dynamics Model of the Helicopter Engine/Transmission System/Rotor
### Rotor Dynamic Load Modeling  
According to the simplified rotor performance calculation and analysis of [PERSON] [25; 26], and [PERSON] [26], the main rotor model is established by the blade element analysis theory, and the rotor is divided into several micro-elements along the wing spread direction. The aerodynamic force acting on the rotor blade is decomposed into a two-dimensional profile aerodynamic force and rotor-induced velocity aerodynamic force. Assuming that the angular velocity of the helicopter is zero, the force of each microelement segment is calculated, and the total tension and demand torque of the main rotor blade are obtained via radial and axial integrations.  
The aerodynamic diagram of the blade section is shown in Figure 6. Furthermore, the air density is denoted by \(\rho_{a}\), blade cross-section velocity is denoted by \(v\), blade equivalent chord length is denoted by \(c_{e}\), rotor pitch is denoted by \(\theta\), inflow angle is denoted by \(\alpha\), lift coefficient is denoted by \(C_{L}\), \(C_{L}=a_{0}\alpha\), and drag coefficient is denoted by \(C_{D}\). Then, the microelement lift \(\mathrm{d}L\) and resistance \(\mathrm{d}D\) at the radial distance \(r_{b}\) can be expressed as follows:  
\[\mathrm{d}L=\frac{1}{2}\rho_{a}\phi^{2}c_{e}C_{L}\mathrm{d}r_{b} \tag{7}\]  
\[\mathrm{d}D=\frac{1}{2}\rho_{a}\phi^{2}c_{e}C_{D}\mathrm{d}r_{b} \tag{8}\]  
The flow speed in the blade section \(\vartheta\) depends on the main rotor angular velocity \(\omega_{\mathrm{MR}}\), air inflow velocity \(v_{c}\), and rotor-induced velocity \(v_{ind}\).  
\[\vartheta^{2}=(\omega_{MR}r_{b})^{2}+(v_{c}+v_{ind})^{2} \tag{9}\]  
\[\tan\varphi=\frac{v_{c}+v_{ind}}{\omega_{MR}r_{b}} \tag{10}\]  
Given that \(\omega_{MR}r\gg v_{c}+v_{ind}\), \(\varphi\) denotes a small angle as follows:  
\[\varphi\approx(v_{c}+v_{ind})/\omega_{MR}r_{b} \tag{11}\]  
\[\alpha=\theta-\varphi\approx\theta-(v_{c}+v_{ind})/\omega_{MR}r_{b} \tag{12}\]  
\[\vartheta=\omega_{MR}r_{b} \tag{13}\]  
Figure 6: Rotor blade profile.  
Substitute Equations (11)-(13) into Equations (7) and (8), as follows:  
\[\mathrm{d}L=\frac{1}{2}\rho_{a}(\omega_{MR}r_{b})^{2}c_{e}a_{0}\bigg{(}\theta- \frac{v_{c}+v_{ind}}{\omega_{MR}r_{b}}\bigg{)}\mathrm{d}r_{b} \tag{14}\]  
\[\mathrm{d}D=\frac{1}{2}\rho_{a}(\omega_{MR}r_{b})^{2}c_{e}C_{\mathrm{D}} \mathrm{d}r_{b} \tag{15}\]  
For the rotor plane, the blade lift and drag are decomposed in the vertical and horizontal directions, respectively.  
\[\mathrm{d}F_{MR}=\mathrm{d}L\cos\varphi-\mathrm{d}D\sin\varphi\approx\mathrm{ d}L \tag{16}\]  
\[\mathrm{d}T_{MR}=(\mathrm{d}L\sin\varphi+\mathrm{d}D\cos\varphi)r_{b}\approx( \mathrm{d}L\varphi+\mathrm{d}D)r_{b} \tag{17}\]  
By integrating (17) in the span direction, the required torque generated in the rotor can be obtained as follows:  
\[T_{MR}\approx\int\limits_{0}^{r}(\mathrm{d}L\varphi+\mathrm{d}D)r_{b}=\frac{ 1}{2}\rho_{a}\int\limits_{0}^{r}(\omega_{MR}r_{b})^{2}c_{e}\bigg{[}a_{0}\bigg{(} \theta-\frac{v_{c}+v_{ind}}{\omega_{MR}r_{b}}\bigg{)}\frac{v_{c}+v_{ind}}{ \omega_{MR}r_{b}}+C_{D}\bigg{]}r_{b}\mathrm{d}r_{b} \tag{18}\]  
Here, only the effects of the variation in air inflow velocity \(v_{c}\) and main rotor speed \(\omega_{MR}\) on the rotor demand torque are considered, and Equation (18) is simplified as follows:  
\[T_{MR}\approx\int\limits_{0}^{r}(\mathrm{d}L\varphi+\mathrm{d}D)r_{b}=\frac{ 1}{2}\rho_{a}c_{e}\bigg{[}\frac{r^{3}}{3}a_{0}\theta(v_{c}+v_{ind})\omega_{MR} +\frac{r^{4}}{4}C_{D}\omega_{MR}{}^{2}\bigg{]}-T_{A} \tag{19}\]  
where \(T_{A}=\frac{1}{2}\rho_{a}c_{e}\int\limits_{0}^{r}(v_{c}+v_{ind})^{2}a_{0}r_{b} \mathrm{d}r_{b}\).