# Adaptive Control of the Aerodynamic Flaps of the Savonius
## 3 Dynamic Analysis of the System
The aerodynamic force is defined by the following equation [35]:  
\[F_{w}=\frac{1}{2}C_{d}\rho A\left(v_{2}^{2}-v_{1}^{2}\right), \tag{28}\]  
where \(C_{d}=1.2\) is the drag coefficient; \(\rho=1.225\) kg/m\({}^{3}\); \(A=0.3\) m\({}^{2}\). According to (28), when the wind speed jumps from 15 m/s to 25 m/s, \(F_{w}=22.05\) N.  
This impulse is transferred to the flap over a time interval of \(t=0.1\) s, creating an impact torque:  
\[M_{w}=F_{w}R_{s}=22.05\cdot 0.08=1.7\) N \(\cdot\) m  
The flap receives an impulsive acceleration, defined by the equation of rotational motion [36]:  
\[I_{s}\ddot{\theta}_{s}=M_{w} \tag{30}\]  
Substituting \(I_{s}=0.0096\) kg\(\cdot\)m\({}^{2}\), we can obtain \(\ddot{\theta}_{s}=\frac{1.76}{0.0096}=183.3\) rad/s\({}^{2}\). This indicates the possibility of a sudden spike in flap speed, which can lead to overloads. The gearbox transmits this torque through the gear mechanism, taking into account the gear ratio \(i_{rad}=14\):  
\[M_{r}=\frac{M_{w}}{i_{r}}=\frac{1.76}{14}=0.126\;\mathrm{N}\cdot\mathrm{m} \tag{31}\]  
The reaction of the gearbox generates reverse impacts, which lead to oscillations [37]:  
\[I_{r}\bar{\theta}_{r}+b\bar{\theta}_{r}+k\theta_{r}=M_{r} \tag{32}\]  
where \(b\) is the viscous damping coefficient; \(k\) is gear transmission stiffness. The following parameters were used for numerical modeling: \(I_{r}=0.001\;\mathrm{kg}\cdot\mathrm{m}^{2}\), \(b=0.01\;\mathrm{N}\cdot\mathrm{m}\cdot\mathrm{s}/\mathrm{rad}\), \(k=5\;\mathrm{N}\cdot\mathrm{m}/\mathrm{rad}\). Figure 7 shows the result of the computed angular velocity of the gearbox. At the moment when the impulse torque is applied, a sharp increase in angular velocity and angle is observed, after which the system enters a regime of damped oscillations. The amplitude gradually decreases due to the effect of the damping torque \(\theta r\), but the nature of the oscillations confirms the system's resonance sensitivity to external disturbances. The solution of this equation demonstrates that the system is susceptible to resonant oscillations during sudden wind gusts. During sudden changes in wind load or in transitional modes (flap opening/closing), short-term interactions may occur between the mechanism's links, accompanied by impact loads. These impacts cause instantaneous acceleration spikes and generate inertial peaks, especially in the presence of clearances or backlash. The impact component is modeled as an impulse torque [38]:  
\[\delta_{r}=\frac{F_{s}L_{r}^{3}}{3 EI} \tag{22}\]0  
where \(J_{u}\) is the impact impulse (\(\mathrm{N}\cdot\mathrm{m}\cdot\mathrm{s}\)), which depends on the mass of the flap and the approach velocity; \(\delta(t-t_{0})\) is the delta function, describing the instantaneous nature of the impact at time \(t_{0}\). In numerical modeling, such loads can be approximated as the short-term application of a large torque over a small time interval. The moment of inertia of the flap \(I_{s}\) is a key  
\[\delta_{r}=\frac{F_{s}L_{r}^{3}}{3 EI} \tag{22}\]1 Gearbox angular velocity under impulse torque. parameter that defines the rotational inertia. For a semicircular plate rigidly mounted on a shaft with radius \(R_{s}\) and mass \(m\), the approximate equation is  
\[\delta_{r}=\frac{F_{s}L_{r}^{3}}{3 EI} \tag{22}\]2  
Substituting the values \(m\) = 0.4 kg, \(R_{s}\) = 0.08 m, we obtained \(I_{s}\approx\) 0.00128 kg\(\cdot\)m\({}^{2}\). This moment of inertia is used on the left-hand side of the dynamic equation:  
\[\delta_{r}=\frac{F_{s}L_{r}^{3}}{3 EI} \tag{22}\]3  
Thus, all physical effects acting on the flap are now quantitatively accounted for. This enables the construction of a complete controllable model of the system. Proper mass distribution helps reduce the system's inertia and minimize oscillations. The center of mass of the system is determined by  
\[\delta_{r}=\frac{F_{s}L_{r}^{3}}{3 EI} \tag{22}\]4  
where \(m_{i}\) is the mass of the link; \(x_{i}\), \(y_{i}\) are its coordinates.