# Adaptive Control of the Aerodynamic Flaps of the Savonius
## 2 Materials and Methods
Tracking the motion of this point allows one to determine linear velocity and acceleration, as well as to understand the mechanical linkage to the flap. To analyze the motion of the mechanism, it is important to determine the angular velocities and accelerations of each of the rotating links:  
(1) The servo motor generates rotational motion with a rotation angle \(\theta_{m}(t)\), then [22]  
\[\omega_{m}(t)=\frac{\theta_{m}(t)}{t} \tag{2}\]  
(2) Angular velocity of the lever:  
Figure 1: Kinematic diagram of the aerodynamic flap control mechanism, illustrating the key points and geometric relationships. Since the lever is rigidly connected to the gearbox output, its angle of rotation is fully determined by the output motion:  
\[\theta_{r}(t)=\frac{\theta_{m}(t)}{i_{r}}\Rightarrow\omega_{m}(t)=\frac{\theta_{ r}(t)}{t}=\frac{1}{i_{r}}\cdot\omega_{m}(t) \tag{3}\]  
Angular velocity of the aerodynamic flap:  
The rotation of the flap is caused by the movement of the lever. If there is a mechanical transmission between the flap and its axis with a transmission ratio \(i_{mex}\), then  
\[\theta_{s}(t)=\frac{\theta_{r}(t)}{i_{mex}}\Rightarrow\omega_{s}(t)=\frac{1} {i_{mex}}\cdot\omega_{r}(t)=\frac{\omega_{m}(t)}{i_{r}i_{mex}} \tag{4}\]  
Angular accelerations:  
The derivatives of the angular velocities are determined similarly:  
\[\alpha_{m}(t)=\frac{\omega_{m}(t)}{t} \tag{5}\]  
\[\alpha_{r}(t)=\frac{\alpha_{m}(t)}{i_{r}}\;\alpha_{s}(t)=\frac{\alpha_{m}(t)}{ i_{r}i_{mex}} \tag{6}\]  
Thus, knowing the motion law of the motor \(\theta_{m}(t)\) allows for the complete determination of the dynamic characteristics of all links in the kinematic chain. The linear characteristics of specific points in the mechanism are important for determining joint reactions, calculating loads, and analyzing inertial forces. Let us consider the linear velocities and accelerations of two key points: the end of the lever (point \(P\)) and a characteristic point on the flap (point \(A\)). Since point \(P\) rotates along an arc of radius \(L_{r}\) around the center of the gearbox, its linear velocity is defined as [22]  
\[v_{P}(t)=L_{r}\cdot\omega_{r}(t)=\frac{L_{r}}{i_{r}}\cdot\omega_{m}(t) \tag{7}\]  
Similarly, the tangential acceleration at the lever tip (point \(P\)):  
\[\alpha_{P}(t)=L_{r}\cdot\alpha_{r}(t)=\frac{L_{r}}{i_{r}}\cdot\alpha_{m}(t) \tag{8}\]  
Let point \(A\) be located at the outer edge of the flap at a distance \(R_{s}\) from the flap's axis of rotation. Then its linear velocity and acceleration are expressed similarly:  
\[v_{A}(t)=R_{s}\cdot\omega_{s}(t)=\frac{R_{s}}{i_{r}i_{mex}}\cdot\omega_{m}(t) \tag{9}\]  
Similarly, the tangential acceleration at the flap edge (point \(A\)):  
\[\omega_{m}(t)=\frac{\theta_{m}(t)}{t} \tag{2}\]0  
These dependencies are critically important for evaluating the inertial moments arising in the links and for subsequent dynamic analysis. They also make it possible to relate control parameters (via \(\omega_{m}\)) to the kinematics of the point of application of aerodynamic force. To verify the correctness of the constructed kinematic scheme and to obtain quantitative estimates of the motion of the flap control mechanism's links, numerical modeling was performed based on analytical expressions derived from the equations of rotational and translational motion kinematics. The goal of the modeling was to determine the behavior of angular and linear parameters (velocities, accelerations, and rotation angles) of the system's links under varying control signals (motor angular velocity), as well as to conduct a sensitivity analysis regarding the influence of design parameters such as lever length or gear ratio. In addition, special attention was paid to studying phase shifts between components in the presence of backlash, as well as to spatial visualization of the trajectories of key points of the mechanism. The following key formulas were used in the numerical modeling:  
1. Transformation of angular velocity from the motor to the flap.