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Twin-engine light airplane in the 14 CFR 23.371 yaw condition: item inertia loads and the engine gyroscopic couple

Tool	Inertia and Gyroscopic Loads
Source	Federal Aviation Administration (2016). Code of Federal Regulations, Title 14, Part 23, Sections 23.361, 23.363 and 23.371 (1 January 2016 edition); Donely, P. (1936). The Forces and Moments on Airplane Engine Mounts. NACA Technical Note 587 (equations p. 6, F6C-4 example pp. 11–15).
Level	Undergraduate

Problem

A twin-engine light airplane (c.g. at FS 120, WL 95 in) carries two 450 lb engines at FS 62, BL ±84, WL 100 with their c.g. 8 in forward of the mount, 150 lb of nose baggage at FS 30, WL 92 and a 140 lb empennage at FS 330, WL 130. Each engine turns a 3 slug·ft² rotor clockwise at 2,575 rpm maximum continuous (2,700 rpm at 200 hp takeoff; 180 hp maximum continuous; 600 lb thrust; four cylinders). Find the load each item applies to the structure in the 23.371(a)(2) yaw condition (yaw velocity 2.5 rad/s, normal load factor 2.5), the totals about the c.g., the gyroscopic couple of each engine, and the engine-mount cases of 23.361, 23.363 and 23.371.

Inputs to Enter

c.g. FS 120, BL 0, WL 95 in; n_A 3.8; V 150 KEAS (information only)
Items: Left engine 450 lb (62, −84, 100), ΔFS −8; Right engine 450 lb (62, 84, 100), ΔFS −8; Nose baggage 150 lb (30, 0, 92); Empennage 140 lb (330, 0, 130); engines with the role Engine
Condition § 23.371(a)(2) yaw: n_z 2.5, r 2.5 rad/s, all else zero
Engine: I_p 3.0 slug·ft², 2,575 rpm maximum continuous, 2,700 rpm takeoff, clockwise from behind, 600 lb thrust, 200 / 180 hp, four cylinders (factor 2)

Open the Inertia and Gyroscopic Loads tool (loaded with this example)

Answers

Left engine at r = (58, −84, −5) in: ω × (ω × r) = (−30.21, 43.75, 0) ft/s²; F = −W(0, 0, −2.5) − m ω×(ω×r) = (422.5, −611.9, 1,125) lb, |F| = 1,349 lb (3.0 g effective); M about the c.g. (−97,560, −67,360, 0) lb·in; Δ × F = (0, −9,000, −4,895) lb·in about the mount
Totals of the four items: F = (587.6, 0, 2,975) lb, |F| = 3,032 lb; M_y = −77,660 lb·in (nose down); the load-factor part is 2.5 × 1,190 = 2,975 lb exactly, the rest is the yaw rate's centripetal term
Gyroscopic couple: H = 3 × 269.7 = 809.0 slug·ft²/s; yaw 2.5 rad/s gives M_y = −24,270 lb·in (2,022 lb·ft, nose down); pitch 1.0 rad/s gives M_z = +9,708 lb·in (nose right); both 26,140 lb·in
Torque: Q_to = 200 hp / (2π 2,700/60) = 389.0 lb·ft = 4,669 lb·in, limit 2 × Q = 9,337 lb·in rolling left; Q_mc = 4,406 lb·in; 23.363 side load 1.33 × 450 = 598.5 lb
Left engine mount, yaw +2.5 with thrust: F = (1,023, −612, 1,125) lb, couple −24,270 lb·in pitch, moment about the mount (0, −33,270, −4,895) lb·in; largest force and couple in the pitch +1.0 / yaw +2.5 combination (1,845 lb, 26,140 lb·in)

Notes

The tool's default inputs are this airplane; press Submit. The case is the tool's own example checked against closed forms (m ω² r, m α r, the H × ω rule with the Pilot's Handbook sign check), the rule text and an independent numpy computation (References/Inertia-Loads/checks/inertia_check.py); TN 587's F6C-4 terms are reproduced in the same test script (test_inertia_gyroscopic_loads.php, 179 checks). The 2.5 rad/s / 1.0 rad/s / 2.5 g combination is the Part 23 rule; the 2016 § 25.371 refers transport airplanes to their maneuver, gust and landing conditions instead.