The state-space representation is based on a state function \(x\) satisfying the first order ODE \(\tau_{1} \dot{x}=x_{o}\left(\alpha-\tau_{2} \dot{\alpha}\right)-x\), where argument \(\left(\alpha-\tau_{1} \dot{\alpha}\right)\) indicates shift for the angle of attack rate \(\dot{\alpha}\) of the static variation of \(0 \leq x_{o}(\alpha) \leq 1\). Here, \(\tau_{1}\) and \(\tau_{2}\) are the time constants expressed with the chord to free stream speed ratio \(c / U\). The output functions as the force and moment coefficients then become

\[ c_{l}(x, \alpha)=\frac{\pi}{2}(1+\sqrt{x})^{2} \sin \alpha, \quad c_{m}(x, \alpha)=c_{l}(x, \alpha) \frac{5(1-\sqrt{x})^{2}+4 \sqrt{x}}{16} \]

Obtain the lift and the moment coefficient variation wrt \(\alpha\) for an airfoil pitching with \(\alpha(t)=30^{\circ} \sin (\omega t)\) about its quarter chord point. Assume: \(x_{o}(\alpha)=\cos ^{2}(3 \alpha), 0 \leq \alpha \leq 30^{\circ}\) and take \(\tau_{1}=0.5 \mathrm{c} / U\) and \(\tau_{2}=4.0 \mathrm{c} / U\) with \(k=\omega c / U=0.05\).