# Adaptive Control of the Aerodynamic Flaps of the Savonius
## 2 Materials and Methods
Taking into account two reduction stages--the gearbox and the aerodynamic flap--we get [23] \[\omega_{m}(t)=\frac{\theta_{m}(t)}{t} \tag{2}\]1 (11) where \(\omega_{m}\)--angular velocity of the motor; \(\omega_{r}\)--angular velocity of the lever; \(\omega_{sh}\)--angular velocity of the flap; \(i_{r}\)--gear ratio of the gearbox; and \(i_{sh}\)--gear ratio between the lever and the flap. 2. Linear velocities of characteristic points: End of the lever (point \(P\)) [24]: \[\omega_{m}(t)=\frac{\theta_{m}(t)}{t} \tag{2}\]2 (12) Edge of the flap (point \(A\)): \[\omega_{m}(t)=\frac{\theta_{m}(t)}{t} \tag{2}\]3 (13) where: \(L_{r}\) is lever length and \(R_{s}\) is flap radius. 3. Backlash (dead zone) modeling. To model the phase shift caused by backlash \(\theta_{\mathit{lift}}\), the following condition was used [25]: \[\theta_{5h}^{\mathit{real}}(t)= \begin{cases}0,&\text{if }\theta_{sh(t)}< \theta_{\mathit{lift}}\\ \theta_{sh(t)}-\theta_{\mathit{lift}},&\text{otherwise}\end{cases} \] (14) where \(\theta_{\mathit{lift}}\approx 3^{\circ}=0.052\) rad. Figure 2 shows the dependence of the linear velocities of two key points of the mechanism--the end of the lever (\(V_{p}\)) and the point at the edge of the flap (\(V_{a}\))--on the motor's angular velocity \(\omega_{m}\). The modeling takes into account the gear ratios of both the gearbox and the flap mechanism. It is evident that both graphs increase linearly with \(\omega_{m}\); however, the velocity of the lever tip significantly exceeds that of the flap edge. This is explained by the fact that the lever is directly connected to the motor through a gearbox with a gear ratio of 1:14, while the flap receives motion through an additional stage (1:2.5), which further reduces its angular and linear speeds. These dependencies are critical for force calculations and material selection, as the high linear velocity of the lever tip can lead to significant inertial and impact loads. Figure 3 illustrates how changes in the lever length \(L_{r}\) affect the linear velocity of the lever tip \(V_{p}\) under fixed gearbox parameters. Curves are plotted for three lever lengths (0.1 m, 0.15 m, and 0.2 m). It is clear that as \(L_{r}\) increases, the linear velocity of the lever tip increases linearly. This behavior is consistent with the equation \(V_{p}=\omega_{r}\times L_{r}\), where the angular velocity \(\omega_{r}\) depends solely on \(\omega_{m}\) and the gear ratio. Thus, changing the lever length directly affects the kinematics and, consequently, the system dynamics. A longer lever increases the oscillation amplitude and the moment of inertia, which must be considered when selecting the servo motor and designing the damping system. The graph in Figure 4 illustrates the phase shifts between the links of the kinematic chain, the motor (\(\theta_{m}\)), the lever (\(\theta_{r}\)), and the flap (\(\theta_{sh}\)), both under ideal conditions and in the presence of backlash. In the absence of backlash, all angles are strictly proportional to each other, corresponding to the gear ratios. However, when mechanical clearance is present in the flap mechanism, a delay in response is observed: until the control signal exceeds the \"dead zone\" (in this example, \(3^{\circ}\), or 0.052 rad), the flap remains stationary. This creates a phase error that must be compensated for in software, especially in cases requiring micro-adjustments of flap position under light wind conditions. The delay in movement can lead to underactuation or oscillatory behavior. The conducted kinematic analysis of the flap control mechanism demonstrated its high structural consistency and predictable behavior under various input conditions. Clear dependencies were established between the angular and linear parameters of all links--from the servo motor to the flap--considering the two-stage transmission and gear ratios.