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Lockheed Jetstar in the power approach (NASA CR-2144): short period, phugoid, Dutch roll, roll and spiral modes, flying-quality Levels and static margins

Tool	Aircraft Stability Analysis
Source	Heffley, R. K., and Jewell, W. F. (1972). Aircraft Handling Qualities Data. NASA Contractor Report 2144, National Aeronautics and Space Administration, Washington, DC. Table VII-1 (p. 171), Table VII-2 (p. 183), Figure VII-2 (p. 169); printed transfer-function factors in Tables VII-4 (p. 185) and VII-8 (p. 189).
Level	Graduate

Problem

From the nondimensional stability derivatives the report gives for the Lockheed Jetstar in the power-approach configuration at sea level and 224 ft/s, find the longitudinal and lateral-directional modes, compare them with the transfer-function factors the report prints, rate them against MIL-F-8785C for a Class II-L airplane in Flight Phase Category C, and find the static and manoeuvre margins, the elevator per g and the controls for a 10° steady sideslip.

Inputs to Enter

S = 542.5 ft², b = 53.75 ft, c = 10.93 ft; V = 224 ft/s, q̄ = 59.4 lb/ft², α_0 = 6.5°; W = 23,905 lb; I_x = 42,275, I_y = 126,106, I_z = 160,113, I_xz = 5,470 slug·ft² (body axes)
C_L 0.737, C_D 0.095, C_Lα 5.0, C_Dα 0.75, C_mα −0.80, C_mα̇ −3.0, C_mq −8.0, C_Lδe 0.4, C_mδe −0.81 per rad
Body axes: C_yβ −0.72, C_nβ 0.137, C_lβ −0.103, C_lp −0.37, C_np −0.14, C_lr 0.11, C_nr −0.16, C_nδa −0.0075, C_lδa 0.054, C_yδr 0.175, C_nδr −0.063, C_lδr 0.029 per rad
Lateral derivatives and inertias rotated through 6.5° to stability axes (C_lβ −0.0868, C_nβ 0.1478, I_xz −7,924 slug·ft²); constant thrust; elevator doublet of 1° for 2 s

Open the Aircraft Stability Analysis tool (loaded with this example)

Answers

Short period ζ 0.527, ω_n 1.664 rad/s (report 0.528, 1.66); phugoid ζ 0.0516, ω_n 0.182 rad/s (report 0.0521, 0.188, which includes a thrust-line term the table does not carry)
Dutch roll ζ 0.0830, ω_n 1.453 rad/s (report 0.0832, 1.45); roll 1/T_R 1.950 1/s (report 1.95); spiral 1/T_s −0.0111 1/s, doubling in 62.3 s (report −0.0112); |φ/β| 1.218
Eigenvalues by the QR algorithm and by the closed quartic agree to 1e-14; θ/δ_e numerator −2.246 (s + 0.0360)(s + 0.917) against the printed −2.25, 0.0360, 0.919
n/α 6.74 g per rad, ω_sp²/(n/α) 0.411; every checked quantity is Level 1 (Dutch roll ζ 0.083 against the 0.08 minimum; unstable spiral doubling in 62 s against the 12 s minimum)
Static margin 16.0 % of c (neutral point 1.73 ft behind the c.g.), manoeuvre margin 19.8 %; elevator −11.3° per g; dδ_e/dV +0.81° per 10 ft/s
Spiral criterion C_lβ C_nr − C_nβ C_lr = −0.0060 (divergent, as the root shows); 10° sideslip (23 kt crosswind) needs 7.9° aileron, 20.8° rudder and 4.8° bank; 20° aileron gives p b/2V = 0.050

Notes

The tool's default inputs are this case; press Submit. The page prints the comparison with the report's factors both from the derivative set and directly from the report's dimensional derivatives (Tables VII-3 and VII-7). The method is checked by test_aircraft_stability.php (1,276 checks) against five flight conditions of NASA CR-2144 and NASA CR-96008 (the Navion of CR-96008 within 0.15 %; CR-2144, printed to three figures, within 0.7 % except the phugoid damping and the spiral root), the limits of MIL-F-8785C read from the specification, and an independent numpy / scipy implementation (References/Aircraft-Stability/checks/).