(Solved): Vertical tail is one of the most influential components in directional stability. Its contribution ...

Vertical tail is one of the most influential components in directional stability. Its contribution to the aircraft's directional stability can be calculated from the following equation \[ C_{n_{\beta_{v}}}=V_{v} \eta_{v} C_{L_{\alpha_{v}}}\left(1+\frac{d \sigma}{d \beta}\right) \] where \[ \eta_{v}\left(1+\frac{d \sigma}{d \beta}\right)=0.724+3.06 \frac{S_{v} / S}{1+\cos \Lambda_{c / 4 w}}+0.4 \frac{z_{w}}{d}+0.009 A R_{w} \] Let say a model of an airplane is tested in a wind tunnel with vertical tail off. Contributions of various components give \( C_{n_{\beta}}=-0.001 \mathrm{deg}^{-1} \). If the vertical tail is to be positioned at a point on the aft end of the fuselage giving a tail length of \( 4.8 \mathrm{~m} \), estimate the required vertical tail area to give an overall \( C_{n_{\beta}}=0.001 \mathrm{deg}^{-1} \) ? Given \( C_{L_{\alpha_{\nu}}}=0.0454, \Lambda_{c / 4 w}=0, S=18 \mathrm{~m}^{2}, \quad b= \) \( 10.6 \mathrm{~m} \) and the wing is set at the middle of the fuselage. \( (50 \) marks)