## 3 Results
### Loads for Steady Propeller Disc Pitch (1P Hub Loads)  
Before progressing to the unsteady results, the response of the propeller models to a steady perturbation of the propeller disc pitch (as induced by an aircraft angle of attack), also called 1P hub loads [16], is discussed. Only small perturbations and thus linearized transfer functions are shown. The name 1P hub loads originates from the once-per-revolution (1P) angle-of-attack variation (see Equation (13) and Figure 7a) that  
Figure 6: Comparison of steady blade loading for operating point \(C_{T}=0.1,\ V=V_{D}\). The top plot shows the out-of-plane force distribution (\(F_{x}\), in N/m) along the blade span. The bottom plot shows the corresponding tangential or in-plane blade loading (\(F_{tun}\)). each individual blade experiences during one revolution due to kinematics [11]. The first-harmonic nature of the resulting blade loads is filtered out for the hub loads due to the summation over all blades at the propeller hub, resulting in steady loads in the non-rotating system. \[\alpha_{dyn}=-\theta\cdot\sin(\Omega t) \tag{13}\]  
Figure 7b shows the resulting out-of-plane blade load distribution over one revolution (in Figure 7b, results from UPM are shown). The propeller disc in this example is tilted around the global y-axis, so the top is tilted forwards. Solid contour lines (on the right side of the disc) indicate an increase in local thrust in these sectors, and dashed contour lines (left side of the disc), a loss in thrust. As Figure 7b shows, the lift for a tilt around the global y-axis is mainly anti-symmetric with respect to the global z-axis, resulting in the coupling torque (\(M_{z\theta}\)), which is one of the main drivers of whirl flutter instability (see \(C_{n\theta}\) in Figure 2). Additionally, it can be observed that the maximum of the unsteady- lift lags in phase behind the angle of attack, which has its maximum and minimum exactly at 270\({}^{\circ}\) and 90\({}^{\circ}\), respectively (see Equation (13) and Figure 7a). This (and the corresponding lift deficiency) is caused by the unsteady aerodynamic liftlag effects [11]. This unsteadiness leads to a reduction in the amplitude of the 1P hub loads. A further reduction in the lift amplitude can be attributed to effects derived from induced velocity. Sectors with more lift generate more circulation shed in the wake and thus higher induced velocities affecting the disc, reducing the local angle of attack and the resulting lift. A second consequence of the phase lag between angle of attack and lift maximum is the development of a moment about the global y-axis. This can be visualized as a phase shift in the resulting global moment from the z-axis to higher azimuth angles, resulting in an \(M_{y}\) component. This (and the corresponding side force, \(F_{y}\)) are called off-axis terms, because they are perpendicular to the primary load response (also called in-phase terms in this paper) along the z-axis. Further explanations for the other hub loads can be found in [11]. Figure 8 compares the linear relation between disc pitch and steady hub in-plane loads for the different methods for operating point \(C_{T}=0,~{}V=V_{D}\) from Table 3. The top row shows the force transfer functions, and the bottom one, the corresponding moments. The plots on the right side correspond to the in-phase terms, mainly caused by the kinematic angle-of-attack variation in the disc. The plots on the left show the off-axis terms caused by the unsteady lift lag effects. Figure 7: Distribution of variation in out-of-plane blade forces and angle of attack during one revolution with steady disc pitch angle. (**a**) Local blade angle of attack perturbation due to disc pitch \(\theta\); (**b**) Variation in out-of-plane force in UPM simulation. As apparent from Figure 8, the z-force and -moment are predicted to be much higher than the off-axis terms by all methods.