## 4 Energy Consumption Modeling
### Subsystem Model
unit theory, which calculates the propeller's thrust and torque, and another based on the momentum theory, which calculates the aerodynamics of the propeller inflow. Figure 4 illustrates the force and velocity analyses on the cross-section of the blade. The angle between the blade cross-section's zero-lift line and the airflow direction is the aerodynamic angle of attack. The lift coefficient is directly proportional to this angle and can be obtained from both the propeller blade's torsion angle and the incoming flow angle. Following the blade unit theory, Equation (11) is derived to express the lift and drag in a blade section located extremely close to the rotor motor's axis of rotation. \[\dot{J}\dot{\mathbf{\Omega}}=M-\mathbf{\Omega}\times J\mathbf{\Omega} \tag{2}\]0  
where \(dL\) is the blade unit lift, \(u\) is the air inflow velocity, \(c\) is the blade chord length, \(k\) is the linearity coefficient, \(\theta\) is the twist angle of the blade, \(\varphi\) is the airflow incidence angle (\(\theta\) and \(\varphi\) vary with \(r\)), \(dD\) is the drag, \(c_{d}\) is the drag coefficient (normally the coefficient of drag is considered constant), and \(r\) is the radial distance of the blade unit from the rotor motor shaft. The horizontal and vertical components of the FPTLIR's airspeed can be determined by coordinate transformations, as shown in Equation (12), when the FPTLIR is flying at a given speed. \[\dot{J}\dot{\mathbf{\Omega}}=M-\mathbf{\Omega}\times J\mathbf{\Omega} \tag{2}\]1  
where \(v_{x}\) is the horizontal component of the airspeed relative to the fuselage rigid body, \(v_{z}\) is the vertical component of the airspeed relative to the fuselage rigid body, \(V_{w}\) is the wind speed in the inertial reference frame, \(R^{-1}\) is the rotation matrix from the inertial reference system to the rigid-body coordinate system, and \(V\) is the flight velocity of the FPTLIR in the inertial coordinate system. As depicted in Figure 4, the airflow incidence angle and the airspeed can be derived from the horizontal and vertical components of the airspeed, which are calculated according to Equation (13). \[ \begin{array}{l}u_{pl}(r,\psi)=\omega r+v_{x}\sin(\psi)\\ u_{pr}(r)=v_{in}+v_{z}\\ u(r,\psi)=\sqrt{{u_{pl}}^{2}+{u_{pr}}^{2}}\\ \varphi(r,\psi)=\tan^{-1}(u_{pr}/u_{pl})\end{array}  \tag{13}\]  
where \(\omega\) is the angular velocity of the propeller, \(v_{in}\) is the air induced velocity, \(u_{pl}\) is the horizontal component of the air inflow velocity \(u\), \(u_{pr}\) is the vertical component of the air inflow velocity \(u\), and \(\psi\) is the angular position of the propeller blade in the direction of rotation. According to the force analysis shown in Figure 4, the thrust and torque acting on the blade unit cross-section can be calculated using Equation (14). \[ \begin{array}{l}dF_{T}(r,\psi)=N(dL\cos\varphi-dD\sin\varphi)\\ dF_{Q}(r,\psi)=Nr(dL\cos\varphi+dD\sin\varphi)\end{array}  \tag{14}\]  
where \(N\) is the number of blades of the propeller. Considering that the horizontal component of the airflow incidence velocity typically greatly exceeds its vertical counterpart, and given that the drag force is generally perpendicular to the thrust direction and considerably less than the lift force, it is common practice to ignore the reverse drag component in the calculated torque equation. The determination of the propeller thrust and torque involves integrating the solution and averaging over one rotation cycle of the propeller in Equation (15). \[ \begin{array}{l}F_{T}=\int_{R_{1}}^{0.97R_{2}}\int_{0}^{2\pi}0.5N\rho u_{pl} ^{2}ck(\theta-u_{pr}/u_{pl})d\psi dr/(2\pi)\\ F_{Q}=\int_{R_{1}}^{R_{2}}\int_{0}^{2\pi}0.5 Nr\rho u_{pl}^{2}c(\varphi k( \theta-u_{pr}/u_{pl})+c_{d})d\psi dr/(2\pi)\end{array}  \tag{15}\]where \(R_{1}\) is the blade bottom length and \(R_{2}\) is the blade length. It should be noted that the thrust of the paddle element is calculated by integration only up to 97% of the paddle blade length, and the tip losses can be approximated to be negligible in this calculation [40].