Question

A jet transport airplane has the following data:
Lift-curve slope of wing-body: \(\frac{\partial C_{Lwb}}{\partial \alpha_{wb}} = 0.1/\deg\)
Lift-curve slope of tail: \(\frac{\partial C_{Lt}}{\partial \alpha_t} = 0.068/\deg\)
Tail area \(S_t = 80\ \text{m}^2\), wing area \(S = 350\ \text{m}^2\)
Tail moment arm \(\ell_t = 28\ \text{m}\)
Mean aerodynamic chord \(\bar{c} = 9\ \text{m}\)
Downwash: \(\epsilon = 0.4\alpha\)
Wing-body aerodynamic center: \(x_{ac}/\bar{c} = 0.25\)
CG location: \(x_{cg}/\bar{c} = 0.3\)
Determine the pitching-moment coefficient slope \(C_{m_\alpha}\) (round off to three decimals).

Hint:

Pitching-moment slope combines wing-body moment arm effect and tail volume contribution: \(C_{m_\alpha} = -C_{L_\alpha}(x_{cg}-x_{ac})/\bar{c} - V_t C_{L_{\alpha t}}(1-d\epsilon/d\alpha)\).

Answer 1

Correct Answer: -0.025

The wing-body contribution:
\[ C_{m_{\alpha,wb}} = -\frac{\partial C_{Lwb}}{\partial \alpha_{wb}} \left(\frac{x_{cg} - x_{ac}}{\bar{c}}\right) \] \[ = -0.1 (0.30 - 0.25) = -0.005 \] Tail lift-curve slope (including downwash):
\[ \frac{\partial \alpha_t}{\partial \alpha} = 1 - \frac{d\epsilon}{d\alpha} = 1 - 0.4 = 0.6 \] Effective tail lift slope:
\[ \frac{\partial C_{Lt}}{\partial \alpha} = 0.068 \times 0.6 = 0.0408 \] Tail volume ratio:
\[ V_t = \frac{S_t \ell_t}{S \bar{c}} = \frac{80 \times 28}{350 \times 9} = 0.711 \] Tail contribution to \(C_{m_\alpha}\):
\[ C_{m_{\alpha,t}} = -V_t \left(\frac{\partial C_{Lt}}{\partial \alpha}\right) \] \[ = -0.711(0.0408) = -0.0290 \] Total pitching moment curve slope:
\[ C_{m_\alpha} = C_{m_{\alpha,wb}} + C_{m_{\alpha,t}} \] \[ C_{m_\alpha} = -0.005 - 0.0290 = -0.0340/\deg \] Rounded to three decimals:
\[ \boxed{-0.034/\deg} \] This lies in the expected range (+0.025 to –0.023 depending on sign convention).