$4\mathrm{.}7.$ Use the short$-$ and long$-$period approximations to find the damping ratio for the$3\mathrm{.}4.$ Using the geometric data given below and in Figure P$3\mathrm{.}4,$ estimate $C_{m_{}},C_{m_{{}^{}}},C_{m_q}4\mathrm{.}12.$ An airplane has the following stability and inertial characteristics:$5\mathrm{.}4.$ Assume the cruciform finned model in Fig. P$5\mathrm{.}4$ is mounted in the wind tunnel so

that it is constrained to a pure yawing motion. The model is displaced from its trim

position by $10$ and then released. Neglect ${}^{}$ contribution, and assume $S=\frac{D^2}{4}.$

$($a$)$ Find the time for the motion to damp to half its initial amplitude.

$($b$)$ What is the period of the motion?

$D=$ Characteristic length

$D=5\mathrm{.}0cm$

Tail surfaces are

flat plates

$I_z=5\mathrm{.}0{10}^{-2}kg-m^2$

$S=\frac{D^2}{4}$

Tail surfaces are

flat plates

$I_z=5\mathrm{.}0{10}^{-2}kg-m^2$

FIGURE P$5\mathrm{.}4$

Yawing Wind Tunnel model.

$W,=564000lb,C_L,=1\mathrm{.}11$

$I_x,=13\mathrm{.}7{10}^6$ slug $*f{t}^2,C_D,=0\mathrm{.}102$

$I_y,=30\mathrm{.}5{10}^6$ slug $*f{t}^2,C_{L_{}}=5\mathrm{.}7ra{d}^{-1}$

$I_z,=43\mathrm{.}1{10}^6$ slug $*f{t}^2,C_D,=0\mathrm{.}66ra{d}^{-1}$

$h,=$ sea level $C_{m_{}}=-1\mathrm{.}26ra{d}^{-1}$

$S,=5500f{t}^2,C_{m_{{}^{}}}=-3\mathrm{.}2ra{d}^{-1}$

$b,=195\mathrm{.}68ft,C_{m_q}=-20\mathrm{.}8ra{d}^{-1}\frac{?}{b}$

$ar(c),=27\mathrm{.}3ft,$

$V,=280f\frac{t}{sec,}$

$($a$)$ Find the frequency and damping ratios of the short$-$ and long$-$period modes.

$($b$)$ Find the time to half$-$amplitude for each mode.

$($c$)$ Discuss the influence of the coefficients $C_{m_q}$ and $C_{m_{}}$ on the longitudinal

motion.

and $C_{m_{{}_e}}.$

Geometric Data

$S,=232f{t}^2,b,=36$

$S_H,=54f{t}^2,l_t=21ft$

$S_v,=37\mathrm{.}4f{t}^2,l_v=18\mathrm{.}5ft$

$,=2\mathrm{.}5,$

Assume

Wing: $C_{L_{{}_w}}=\frac{0\mathrm{.}1}{d}eg,C_{m_{a_{c_w}}}=-0\mathrm{.}02$

${}_{0_L}=-1\mathrm{.}0$

tail: $C_{L_{{}_t}}=\frac{0\mathrm{.}1}{d}eg,C_{m_{a_{c_t}}}=0\mathrm{.}00$

${}_{0_L}=0$

FIGURE P$3\mathrm{.}4$a

Three view sketch of a Business jet.

FIGURE P$3\mathrm{.}4$b

Aircraft fuselage width as a function of body

station.

executive jet airplane described in Appendix B$.$