## 4 Evaluation of UAV Flight Performance Data  
Our theory above applies to the five parameters \(M,\ R,\ f,\ m_{u}\), and \(C=\frac{M^{\prime}}{M}\). The radius \(R\) and masses \(M\) and \(M^{\prime}\) are assumed to be fixed parameters during flight, which cannot be changed. The flight performance can be controlled by precisely adjusting the active unbalance mass \(m_{u}\) (or \(\frac{m_{u}}{M}\)) and/or by adjusting the initial rotation rate \(f_{0}\) at the start (or  
Figure 4: Decrease in rotational energy \(E_{rot}\) and increase in translational energy \(E_{trans}\) in dependence of flight path length \(x(t)\). \(E_{0}\), \(E_{1/2}\): max and half-max of rotation energy, \(A^{*}\) Point of energetic equivalence. the actual rotation rate \(f\) during flight). In most cases, a high initial rotation frequency \(f_{0}\) is desired to store a large amount of energy on board, making the key steering parameter \(\frac{m_{u}}{M}\). The following formulas, derived above, are applied here for flight performance calculations:  
\[\#\text{Energy mechanically stored}\;\;E_{rot}=MR^{2}\;\frac{\omega_{0}^{2}}{4} \tag{31}\]  
This formula calculates the amount of pure mechanical energy stored in the system. Energy storable on board depends on the square of radius \(R\) and square of rotation frequency \(f_{0}\) and is proportional to mass \(M\) of the rotor. Any electrical power supply on board is ignored here. There is no influence of construction factor \(C\) on the magnitude of storable rotation energy. \[\#\text{Initial trust}\;\;F_{0}^{\prime}=M^{\prime}a_{0}^{\prime} \tag{32}\]  
This formula calculates the initial thrust generated by the product of the mass of the vehicle (\(M^{\prime}\)) and its initial acceleration (\(a_{0}^{\prime}\))  
\[\#\text{Initial acceleration}\;\;a_{0}^{\prime}=\Big{(}\frac{m_{u}}{M^{ \prime}}\Big{)}R\;\omega_{0}^{2} \tag{33}\]  
determined by the magnitude of \(m_{u}\) and/or rotation frequency \(f_{0}\). Typical values of \(m_{u}=M^{\prime}*10^{-4}\), respectively \(M^{\prime}*10^{-6}\), are assumed in calculation here, while \(f_{0}\) is variable and \(R\) is fixed by design. \[\#\text{Height of vertex}\;\;h_{max}=R^{2}\frac{\omega_{0}^{2}}{g_{E}}. \tag{34}\]  
Horizontal spin orientation of the rotor disk has to be assumed here. The height of the vertex is independent of rotor mass \(M\) and \(m_{u}\). Acceleration due to Earth gravity: \(g_{E}=9.81\frac{m}{g^{2}}\). \[\#\text{Max. velocity}\;\;v_{max}=R\;\frac{\omega_{0}}{\sqrt{2}} \tag{35}\]  
Vertical spin orientation of the rotor disk is assumed here. Maximum velocity is independent of rotor mass \(M\) and \(m_{u}\). For preliminary evaluation of potential in-flight performance, we take here three hypothetical UAVs of different sizes and mass and calculate the height of vertex, initial thrust, initial acceleration, and maximum velocity in dependence on initial rotation rate (Tables 1-4). All data calculated here are obtained under the simplifying assumption \(M=M^{\prime}\), thus \(C=1\). For the evaluation of a specific vehicle, we have to set its specific \(C>1\). Thus, initial acceleration \(a^{\prime}\) and initial trust \(F_{0}^{\prime}\) of the complete vehicle will be reduced because of bigger vehicle mass \(M^{\prime}\) compared to rotor mass \(M\). This is according to Equation _a_\(M=a^{\prime}M^{\prime}=m_{u}R\omega^{2}\). For instance, assuming \(C=2\), the height of the vertex as well as the initial thrust of the vehicle, are smaller, namely half the value of the corresponding rotor data. Additionally, the way back to the Earth's surface and managing a safe landing needs