## 2 Materials and Methods
### Transfer Matrix Method  
The goal of this work is to compare different aerodynamic methods with special focus on the unsteady aerodynamics and their influence on whirl flutter predictions. Hence, a method is required to derive the relevant unsteady aerodynamic quantities and to conduct whirl flutter stability predictions based on them. This paper employs the transfer matrix (TM) method recently developed by one of the authors [26] to identify the relevant transfer functions based on time-domain simulations. The TM-method is centered around describing the unsteady perturbation response of the propeller hub loads, \(\underline{f}\), to hub motion \(\Delta\underline{x}_{hub}\) in the frequency domain according to a frequency-dependent transfer matrix, \(\underline{H}_{prop}\):  
\[\underline{f}_{TM}=\underline{H}_{prop}(i\omega,V,\Omega,\rho)\Delta \underline{x}_{hub} \tag{11}\]  
with  
\[\underline{H}_{prop}(i\omega,V,\Omega,\rho)\ =\ \left\{ \begin{array}{cccc}F_{yy}&F_{ yz}&F_{y\vartheta}&F_{y\psi}\\ F_{zy}&F_{zz}&F_{z\vartheta}&F_{z\psi}\\ M_{yy}&M_{yz}&M_{y\vartheta}&M_{y\psi}\\ M_{zy}&M_{xz}&M_{z\vartheta}&M_{z\psi}\end{array} \right\} \tag{12}\]  
The transfer matrix captures all features included in the time-domain method, such as unsteady aerodynamics and wake descriptions. It is used to extract aerodynamic transfer functions, similar to [PERSON] and [PERSON]'s aerodynamic derivatives but capturing the full frequency dependency of the methods studied. The workflow is described in the following. The steps in italics are not required for the original TM-method workflow but are employed in this work to gain further insights into the unsteady aerodynamic behavior:  
1. **Trimming** to a steady operating point in axial flow The propeller is trimmed to achieve a target steady thrust coefficient \(C_{T}\) and to bring the system to an equilibrium, around which the perturbations are employed. Trimming is achieved by iteratively changing the blade pitch angle until \(C_{T}\) matches the desired thrust. For each iteration, the system is integrated in time with a fixed blade pitch until a steady state in the hub loads is reached. The resulting equilibrium is saved for further restarts. 2. _Response of the system to steady disc pitch perturbation (1P hub loads)_ Instead of directly identifying the full, frequency-dependent transfer functions for hub motion perturbations, the quasi-steady response to disc pitch (or disc angle of attack) is identified first. Starting from the equilibrium point in \"1.\", the disc pitch angle is perturbed by a steady 1\({}^{\circ}\) inclination, and the system is integrated in time until a steady state in the hub loads is achieved over one period. 3. **Perturbation of the hub motion** and recording of the hub load response To identify the frequency-dependent transfer matrices, the propeller is perturbed in  
Figure 3: Flow chart for the blade lift and wake equilibrium. y-translation and disc pitch \(\theta\) using unsteady pulse-shaped excitation (for low-fidelity methods) or harmonic forced motion (for mid-fidelity codes). The response of the hub load components in the time domain is recorded. The system is integrated in time until the transient response has decayed. For harmonic forced motion, this process is repeated for several perturbation frequencies (5 Hz, 10 Hz, 20 Hz) to cover the desired range for the frequency-domain flutter analysis. 4. Identification of the **frequency-domain transfer function** from motion to loads By converting both motion and load time histories into the frequency domain using Fourier transformation, the frequency-domain transfer function from hub motion to hub loads can be identified through division. The scalar transfer functions (e.g., disc pitch to moment about the z-axis) are then arranged into the transfer matrices in Equation (12). Due to axial symmetry, the transfer functions for z-translation and disc yaw \(\psi\) can be derived from symmetry and do not have to be calculated separately. 5.