# Adaptive Control of the Aerodynamic Flaps of the Savonius
## 2 Materials and Methods  
The Savonius wind turbine considered in this study is a vertical-axis machine with a rotor composed of two tiers, each featuring three blades. Each blade is equipped with a semicircular aerodynamic flap designed to control airflow based on the current wind speed. Each flap is made of 0.5 mm thick sheet metal, shaped as a semicircle with a 25 cm arc length and 30 cm height. The flap rotates around its own axis, and the angle of rotation determines the extent to which it obstructs the airflow and, consequently, influences the torque generated by the corresponding blade. Flap actuation is performed using an MG996R servo motor manufactured by Tower Pro, based in Taichung City, Taiwan. The motor is located at the rotor base, near the central axis. Mechanical transmission to the flap is accomplished through a lever mechanism that includes [19] the following:  
* The servo motor's output shaft;
* A steel rod lever (6 mm in diameter);
* A connecting joint (axle) linking the lever to the flap and enabling its rotational movement. The servo motors are automatically controlled based on signals from a controller that monitors the current wind speed. When a defined threshold (25 m/s) [20] is exceeded, the forced flap-closing mechanism is activated to reduce rotor load and prevent mechanical damage. Every component of the mechanism must be incorporated into the system's kinematic and dynamic models, as the interaction between the links affects the system's response to control actions and external disturbances. The kinematic scheme of the aerodynamic flap control mechanism is a closed system of lever and rotational links, ensuring the transmission of control motion from the servo to the flap. It is designed for analyzing displacements, velocities, and accelerations of all links under predefined input motion laws (i.e., rotation angle of the servo motor's output shaft). The system consists of the following main kinematic links:  
1. Servo motor (Link 1)--provides rotational motion. A shaft rigidly connected to the lever is mounted on the motor's output. The rotation angle is denoted as \(\theta_{m}(t)\). 2. Gearbox (Link 2)--built into the servo housing, it reduces speed and increases output torque. Its gear ratio is denoted as \(i_{r}\), and the output angle as \(\theta_{r}(t)\). * Lever (Link 3)--a straight link of length \(L_{r}\), rotating around the gearbox axis. It moves the flap connection point along an arc of a certain radius. * Connecting joint and flap (Link 4)--the flap receives rotation from the lever. The flap rotation is denoted by the angle \(\theta_{s}(t)\), which is linked to the input angle via the complete kinematic transmission (Figure 1). Let \(O\) be the center of the drive shaft, and \(P\) be the end of the lever. Then the coordinates of point P over time are expressed as [21]  
\[x_{P}(t)=L_{r}cos\theta_{r}(t)\text{and }y_{P}(t)=L_{r}sin\theta_{r}(t) \tag{1}\]  
where \(\theta_{r}(t)=\frac{\theta_{m}(t)}{t_{r}}\) is angle of lever rotation. The diagram shows the mechanical linkage between the servo motor and the aerodynamic flap. Point O represents the center of the motor shaft, which generates the input angular motion \(\theta_{m}(t)\). A rigid lever of length \(L_{r}\) is attached to the shaft and rotates around point \(O\). The end of the lever is denoted as point \(P\), which follows a circular trajectory around \(O\). Point \(A\) is located on the outer edge of the semicircular flap, at a distance \(R_{s}\) from its axis of rotation. The motion is transmitted from the motor to the flap via the lever and connecting joint at point \(P\), enabling the flap to rotate with an angle \(\theta_{s}(t)\). The diagram also shows the angular relationships and the direction of motion for each link, which are critical for analyzing velocities, accelerations, and control dynamics. The dependencies in (1) describe a circular trajectory, which is important for further analysis of the flap's movement, as it is connected to this point through the connecting joint.