## 3 Results
### Loads and Transfer Matrices for Propeller Hub Motion  
The previous subsection assessed the transfer behavior for zero frequency about the propeller hub. In this subsection, this is extended by showing transfer functions which describe the response to harmonic perturbations about the propeller hub. This subsection mainly focuses on perturbation response with respect to disc pitch \(\theta\). Due to axial symmetry, the transfer functions for disc yaw (\(\psi\)) are equivalent (with some signs being changed). The transfer functions with respect to harmonic disc in-plane (y and z) translation can be derived, as a steady velocity \(\dot{z}\) is almost equivalent to a steady disc pitch \(\theta\). For the whirl flutter studies shown later, the translation terms are still computed separately, but they are not shown in this subsection for the sake of brevity. Figure 9 shows these transfer functions for a selection of aerodynamic methods for operating point \(C_{T}=0,\ V=V_{D}\) from Table 3. The top row shows the force components, and the bottom row, the hub moments. Each plot contains the real (solid line) and imaginary (dashed line) parts of the complex transfer functions plotted over perturbation frequency. The markers indicate the samples identified with harmonic (for the mid-fidelity methods) or pulse (for the low-fidelity methods) perturbation. Only UPM transfer functions are shown for the mid-fidelity codes for better visibility, representing the other mid-fidelity methods which give similar results. All transfer functions are real-valued for zero frequency and equivalent to the 1P hub loads shown in Figure 8. A constant term in the real part (e.g., in the bottom-left plot for \(M_{2\theta}\)) over frequency indicates aerodynamic stiffness. A linear slope in the imaginary part (e.g., bottom-left plot for \(M_{y\theta}\)) indicates aerodynamic damping in this component. All other contributions (e.g., the nonlinear decrease in the real part of \(M_{2\theta}\)) have their origin in an unsteady aerodynamic behavior which is nonlinear with frequency. From Figure 9, some differences from the Wagner method, which predicts higher amplitudes in the transfer functions, can already be observed. This is consistent with the observations for the 1P hub loads in Figure 8. The other methods match reasonably well. To obtain a more quantitative comparison of all methods, the transfer functions are linearized with respect to frequency. This allows for a split into the derivative form comprising aerodynamic stiffness and damping, similar to the description of the Houbolt/Reed method in Equations (5) and (6). The linearization is conducted by using the zero-frequency samples (to obtain correct transfer functions at zero frequency) and a sample at a chosen frequency (in this work, at 10 Hz perturbation frequency). The real part of the zero-frequency transfer sample is used as aerodynamic stiffness, and the linear slope in the imaginary part, as aerodynamic damping. The resulting values are non-dimensionalized according to Equation (4) and can then be compared to the derivatives from the Houbolt/Reed method. The resulting linearized transfer functions would look similar to the Houbolt/Reed sample in Figure 9, which are already in linear form. Appendix A gives an example for the linearization procedure. Figure 10 shows a comparison of the eight unique derivatives obtained from the frequency-dependent transfer functions with the derivatives from the Houbolt/Reed method (marked with horizontal dashed lines in each plot). In general, good agreement is found among most methods for derivatives \(C_{y\theta}\), \(C_{z\theta}\), \(C_{m\theta}\) and \(C_{m\theta}\). These (similar to the 1P hub loads) are the in-phase components, which are mainly related to kinematics [11]. The [PERSON] method has the highest deviations from the Houbolt/Reed method and the mid-fidelity codes. The BEM+Wagner method matches again very well with the mid-fidelity methods.