# Adaptive Control of the Aerodynamic Flaps of the Savonius
## 3 Dynamic Analysis of the System  
Dynamic analysis is essential for determining the mechanism's response to control and external influences, such as wind load, friction, backlash, and impact torques. Unlike kinematics, which focuses on the geometric parameters of motion, dynamics takes into account the acting forces and torques, providing a complete and more accurate model of the system's behavior. The rotational motion of the flap around its axis is governed by [PERSON]'s second law for rotational motion [26]:  
\[I_{s}\cdot\ddot{\theta}_{s}(t)=M_{ctrl}(t)-M_{w}(t)-M_{f}(t)-M_{l}(t)-M_{u}(t), \tag{15}\]  
where \(I_{s}\) is the moment of inertia of the flap relative to its axis of rotation; \(M_{ctrl}(t)\) is control torque from the servo motor, generated by the PID controller; \(M_{w}(t)\) is torque due to wind pressure on the flap; \(M_{J}(t)\) is torque due to friction resistance (dry and viscous); \(M_{l}(t)\) is torque caused by mechanical clearances and backlash; and \(M_{u}(t)\) is impact and short-term dynamic disturbances arising from sudden wind changes. For a semicircular metal sheet, it can be approximately assumed as  
\[I_{s}=\frac{1}{2}mR_{s}^{2}, \tag{16}\]  
Taking into account the mass \(m\approx 0.4\) kg and radius \(R_{s}=0.08\) m, \(I_{s}\approx 0.00128\) kg\(\cdot\)m\({}^{2}\). The torque generated by the wind flow on the flap is one of the main external loads on the system. It depends on the current wind speed and the geometry of the flap. For an approximate calculation, we use the aerodynamic formula [27]:  
\[M_{w}(\theta)=\frac{1}{2}C_{d}\cdot\rho\cdot A\cdot v^{2}\cdot R_{s} \tag{17}\]  
where \(C_{d}\) is the drag coefficient. For a semicircular flap, it is typically assumed to be \(C_{d}\approx 1.2\); \(\rho=1.225\) kg/m\({}^{3}\) is air density under standard conditions; \(A=0.3\) m\({}^{2}\) is the effective area of the flap; \(v\) is instantaneous wind speed (in m/s); \(R_{s}=0.08\) m is the distance from the axis of rotation to the center of pressure on the flap (moment arm). Thus, the external load increases with the square of the wind speed, which is critically important for control system design and the selection of actuator parameters. This is precisely why adaptive correction of control coefficients must be implemented in the system. Table 1 shows the calculated torque values at different wind speeds. As seen in Table 1, the torque \(M_{w}\) depends on the square of wind speed \(v^{2}\), so when the speed increases from 10 to 20 m/s, the torque increases fourfold. This is important to consider in control system design--under strong wind gusts, the flap experiences significantly higher loads, which must be compensated by the control torque. In a real mechanical system, the motion of the links is accompanied by losses caused by internal friction, mechanical clearances (backlash), and possible impact interactions. These components significantly affect the system's behavior during control and must be included in the motion equation. The dry friction model assumes the presence of a constant resisting torque acting in the opposite direction of motion. It is described as [28]  
\[M_{dry}=\mu\cdot N\cdot R_{s}\cdot sign(\omega_{s}) \tag{18}\]  
where \(\mu\) is the friction coefficient (approximately 0.05 for metal surfaces); \(N\) is the normal contact force in the axial joint; \(R_{s}\) is the flap radius; \(sign(\omega_{s})\) is the sign of angular velocity (indicates the direction of friction). Viscous friction accounts for resistance proportional to the angular velocity [29]:  
\[M_{viscous}=B\cdot\omega s \tag{19}\]  
where \(B\) is viscous damping coefficient and \(B=0.01\). Backlash is the clearance between transmission links, resulting in a range where changes in the input signal do not produce changes in the output. It manifests as hysteresis or a \"dead zone\". The torque caused by backlash can be approximated as follows:  
\[M_{l}= \begin{cases}K_{l}\cdot\mathit{sign}(e)&|e|>\delta\\ 0,&|e|\leq\delta\end{cases}