## 2 Materials and Methods
### [PERSON]/[PERSON] Method  
[PERSON] and [PERSON]'s theory [5] represents a pioneering effort in the field of whirl flutter prediction methods in conjunction with an analytical description of unsteady propeller aerodynamics. According to [PERSON] and [PERSON], load vector \(\underline{f}\) in Equation (2) can be defined as  
\[\underline{f}_{HR}=\frac{\rho}{2}V^{2}2\pi R^{3}\bigg{[}\underline{K}_{Aero}(V, \Omega)+s\underline{D}_{Aero}(V,\Omega)\bigg{]}\Delta\underline{x}_{Hub} \tag{4}\]  
where \(\rho,V,\Omega\) and \(R\) are the air density and velocity, the rotational speed and the radius of the propeller, respectively. Central to their approach is the formulation of unsteady propeller derivatives. These define the aerodynamic stiffness and damping matrices (\(\underline{K}_{Aero}\) and \(\underline{D}_{Aero}\)) (see Equations (5) and (6)). \[\underline{K}_{Aero}(V,\Omega)\ =\ \begin{cases}0&0&\frac{C_{\mu\theta}}{2R}&\frac{C_{ \mu\theta}}{2R}\\ 0&0&\frac{C_{\mu\theta}}{2R}&\frac{C_{\mu\theta}}{2R}\\ 0&0&\frac{C_{\mu\theta}}{C_{\mu\theta}}&\frac{C_{\mu\theta}}{2R}\\ 0&0&C_{\mu\theta}&C_{\mu\theta}\end{cases}  \tag{5}\]  
\[\underline{D}_{Aero}(V,\Omega)\ =\ \begin{cases}-\frac{C_{\mu\theta}}{2R}&\frac{C_{ \mu\theta}}{2R}&\frac{C_{\mu\theta}}{2R}&\frac{C_{\mu\theta}}{2R}\\ -\frac{C_{\mu\theta}}{2R}&\frac{C_{\mu\theta}}{2R}&\frac{C_{\mu\theta}}{2R}\\ -\frac{C_{\mu\theta}}{2R}&\frac{C_{\mu\theta}}{2R}&\frac{C_{\mu\theta}}{2R}\\ -\frac{C_{\mu\theta}}{2R}&\frac{C_{\mu\theta}}{2R}&\frac{C_{\mu\theta}R}{2R} \end{cases}  \tag{6}\]  
The derivatives describe the forces and moments on the propeller plane due to the corresponding displacements and velocities of the propeller hub (\(\Delta\underline{x}_{Hub}\)) (see Equation (1)). Each of the derivatives corresponds to the change in a load (first index) in response to a perturbation (second index). As an example, \(C_{m\theta}\) denotes the change in moment \(M_{y}\) at the propeller hub due to a displacement in pitch \(\theta\). Note that according to [PERSON] and [PERSON]'s notation for the derivatives, \(y\) and \(z\) stand for the forces in \(F_{y}\) and \(F_{z}\), while \(m\) and \(n\) represent the moments in \(M_{y}\) and \(M_{z}\). \(q\) and \(r\) denote rotational velocities \(\theta\) and \(\dot{\psi}\). Although stiffness properties are a function of translations and rotations, Equation (5) only contains derivatives with respect to rotations \(\theta\) and \(\dot{\psi}\). Translatory displacements do not add a perturbation to the propeller disc, so that the first two columns in Equation (5) are zero. In contrast, Equation (6) contains derivatives with respect to rotations (first two columns) and rotational velocities (last two columns). The derivatives in response to \(\dot{y}\) and \(\dot{z}\) in the first two columns of Equation (6) are described in terms of \(\theta\) and \(\dot{\psi}\) rather than \(\dot{y}\) and \(\dot{z}\) because they linearly depend on the generalized coordinates (refer to Equation (1)). Generally speaking, due to the quasi-steady assumption, a velocity in \(y\) or z can be interpreted as a steady induced flow angle. In summary, a total of 16 derivatives are defined, although the inherent symmetry of the model reduces the number of unique derivatives to 8 (e.g., \(C_{z\theta}=-C_{y\phi}\)) [9]. These derivatives are calculated using blade integrals. Exemplary for \(C_{m\theta}\), the equation reads [9]  
\[C_{m\theta}\ =\ -\ \bigg{(}\frac{N_{b}}{4}\bigg{)}\bigg{(}\frac{1}{\pi R} \bigg{)}\int_{\eta_{0}}^{1}\frac{\eta^{2}c(\eta)C_{la}(\eta)G}{\sqrt{(\frac{l }{\pi})^{2}+\eta^{2}}}\frac{A_{R}}{\bigg{[}2+A_{R}\sqrt{1-Ma^{2}(1+\eta^{2}( \frac{\pi}{l})^{2})}\bigg{]}}d\eta \tag{7}\]  
with  
\[J\ =\ \frac{V\pi}{\Omega R} \tag{8}\]  
and  
\[k\ =\ \frac{c}{2R\sqrt{\bigg{(}\frac{l}{\pi}\bigg{)}^{2}+\eta^{2}}} \tag{9}\]  
As seen in Equation (7), the derivative is determined based on an integration along the blade. Here, \(\eta\) describes the normalized radius, while \(\eta_{0}\) stands for the hub radius.