Inertia and Gyroscopic Loads: Method

Inertia and Gyroscopic Loads

What the tool computes

A structural loads report resolves every design condition into the forces that each mass item on the airplane applies to its attachment: the engines on their mounts, the batteries and baggage on their trays, the empennage on its fittings. In a steady straight pull-up these are simply the weights multiplied by the load factor. As soon as the airplane rotates, two more terms appear: the tangential acceleration of the angular acceleration and the centripetal acceleration of the angular velocity, both proportional to the item's distance from the centre of gravity. A spinning rotor adds a third effect that has nothing to do with the rotor's position, the gyroscopic couple that the mount must apply to turn the rotor's angular momentum with the airplane. The tool evaluates all of these for a table of mass items under one load-factor and angular-rate condition, writes the first item out term by term, tabulates the rest with their moments about the c.g., draws the items and their loads in side and plan view, and for the items marked as engines lists the mount load cases as 14 CFR Part 23 combines them. The equations are those Philip Donely collected for engine mounts in NACA TN 587 (1936), carried here in full vector form for any item.

Axes and sign conventions

Positions are entered in the drawing frame of a weight-and-balance report: fuselage station FS positive aft, buttline BL positive to starboard (the pilot's right), waterline WL positive up, each from the airplane's reference datum. This frame is right-handed (aft × starboard = up). The kinematics are worked in the standard body axes with the origin at the c.g.: x forward, y to starboard, z down; an item at (FS, BL, WL) sits at x = −(FS − FS_cg), y = BL − BL_cg, z = −(WL − WL_cg). Angular rates p, q, r and angular accelerations ṗ, q̇, ṙ are right-hand rotations about x, y, z: p positive right wing down, q positive nose up, r positive nose right. Loads and couples are reported in the same body axes: a positive F_z acts downward, a positive M_y pitches the nose up, a positive M_z yaws the nose right; the results page adds the sense of each component in words.

Load factors are entered in the structural sense of 14 CFR 25.321(a), which defines a positive load factor as one in which the aerodynamic force acts upward: n_z is +1 in level flight and +3.8 in a normal-category pull-up, n_x is positive when the airplane accelerates forward, n_y when it accelerates to starboard. Formally the load-factor vector is n = (A − g)/g with A the acceleration of the c.g. and g the gravity vector; in body axes with z down its components are (n_x, n_y, −n_z). Every item then feels an apparent gravity of −n g, so a positive n_z loads the items downward, as expected.

The inertia load of a mass item

A point fixed in a rigid body at r from the c.g. moves with the acceleration a = A + ω̇ × r + ω × (ω × r), where ω = (p, q, r) is the angular velocity and ω̇ = (ṗ, q̇, ṙ) the angular acceleration. The item of mass m applies to the structure the force F = m (g − a) (its weight together with its inertia reaction; the attachment applies −F to the item), which the tool writes as

F = −W (n_x, n_y, −n_z) − m (ω̇ × r) − m ω × (ω × r).

The first term is the load-factor part. The second is the tangential term: an angular acceleration about an axis through the c.g. accelerates the item perpendicular to both the axis and r, so a nose-up pitching acceleration q̇ pushes an item forward of the c.g. upward and the item loads its attachment downward by m q̇ x. The third is the centripetal term: ω × (ω × r) = ω (ω · r) − r |ω|² points from the item toward the rotation axis, so the item loads its attachment outward with m ω² d, d its distance from the axis. In a 2.5 rad/s yaw an engine 7 ft from the yaw axis feels 6.25 × 7 = 44 ft/s², 1.4 g, directed outboard and forward: a load of the same order as the normal load factor, and the reason the rule puts the yaw velocity beside the load factor. TN 587 wrote these terms for an engine on the x axis as the n force, the α force (W λ x/g for a pitching acceleration λ) and the centrifugal force (W x/g)(q² + r²); the full vector form here keeps the cross terms that appear when the item is off the axis of rotation, and gives TN 587's expressions exactly when it is on it.

The moment of each item's load about the c.g. is r × F, and when the item's c.g. is offset from its attachment point by Δ (entered as Δ = item c.g. minus attachment point in the FS/BL/WL frame) the attachment also carries Δ × F. The totals printed are the sums over the items entered; they equal the airplane's inertia reaction only if every mass item is entered, since the rotational terms of a complete airplane cancel about its own c.g. (Σ m r = 0) while those of a subset do not. The page prints the centroid offset of the items entered so that this can be judged.

The gyroscopic couple

The rotating parts of an engine (propeller, spinner, crankshaft and the rotating accessories referred to the propeller speed; TN 587 puts the crankshaft at about 10 percent of the propeller's inertia, times the gear ratio) carry the angular momentum H = I_p Ω along the shaft, forward for a propeller turning clockwise seen from behind (the right-hand rule about +x, the usual sense of American engines), aft for the other sense. When the airplane turns with angular velocity ω the shaft must turn with it, so H changes at the rate ω × H and the mount has to apply that moment to the rotor; the reaction on the mount, the gyroscopic couple, is M = −ω × H = H × ω. With H = s I_p Ω along x and s = ±1 for the sense,

M = s I_p Ω (0, −r, q):

a nose-right yaw rate r gives a nose-down pitching couple and a nose-up pitch rate q gives a nose-right yawing couple for a clockwise rotor, both reversed for a counter-clockwise one. This is the rule every pilot learns: raising the tail of a tailwheel airplane with a clockwise propeller (a nose-down pitch rate) yaws the nose left (Pilot's Handbook of Aeronautical Knowledge, Figure 5-50). The magnitude is I_p Ω √(q² + r²), independent of the roll rate and of where the engine sits. TN 587 writes the same couples as M = −I_p ω r and N = +I_p ω q. For a two-blade propeller the couple is not steady: it fluctuates between 0 and 2 I_p Ω ω at twice the propeller speed in the plane of the steady couple, with a couple of amplitude I_p Ω ω at right angles to it (TN 587; Conway 1943); the tool reports the steady value of a rotor with three or more blades and notes the two-blade amplitude. The torque reaction on the mount is opposite to the rotor's sense (a clockwise propeller rolls the airplane left, PHAK Figure 5-47), and the mean torque is the shaft power divided by the rotor angular velocity, Q = P/Ω.

The rules

14 CFR 23.371(a) (2016 edition, Amendment 23-48 text): each engine mount and its supporting structure must be designed for the gyroscopic, inertial and aerodynamic loads that result, with the engine and propeller at maximum continuous rpm, under either the yawing and horizontal-tail conditions of 23.351 and 23.423 or all possible combinations of a yaw velocity of 2.5 radians per second, a pitch velocity of 1.0 radian per second, a normal load factor of 2.5 and maximum continuous thrust. The tool lists the eight sign combinations of the two rates with n_z = 2.5 and the thrust for every item marked as an engine, taking the engine's inertia force at its position (load-factor part plus the centripetal part of the rates), the gyroscopic couple and the thrust together. The yaw velocity of 2.5 rad/s descends from the spin: Conway's 1943 survey of design practice records British requirements of 3.0 rad/s for single-engine trainers, 2.5 for twin-engine trainers and 2.0 for other aerobatic types, and German formulas giving about 3 rad/s. Part 25 of the 2016 edition (25.371, Amendment 25-141) no longer prescribes rates: it refers the gyroscopic loads of a transport airplane to the maneuver, gust and landing conditions of 25.331, 25.341, 25.349, 25.351, 25.473, 25.479 and 25.481 at the rotating speed appropriate to each, the pitch maneuver of 25.331(c)(1) being carried to point A2. The fill labelled 23.371 is therefore the Part 23 rule; a transport airplane's rates come from its own maneuver analysis.

14 CFR 23.361: the mount must carry a limit engine torque at takeoff power and propeller speed acting with 75 percent of the limit loads of flight condition A, and a limit torque at maximum continuous power and speed with the full condition A loads; the limit torque is the mean torque multiplied by 1.25 for turbopropellers, 1.33 for engines with five or more cylinders, and 2, 3 or 4 for engines with four, three or two cylinders (23.361(c)). A turbopropeller installation also takes the takeoff limit torque multiplied by 1.6 for a propeller-control malfunction with 1 g level-flight loads (23.361(a)(3)). The 2016 Part 25 (25.361) keeps the 1.25 for turbopropellers and asks for the maximum accelerating torque of other turbine engines, which may be entered through the torque-factor field. The tool applies the condition A loads as n_z = 0.75 n_A and n_A on the engine's weight with no rotation, and the torque reaction about x; thrust is not named in 23.361 and is not added there.

14 CFR 23.363 and 25.363: the mount must carry a limit lateral load factor of not less than 1.33 or one-third of the condition A load factor, and this side load may be assumed independent of the other flight conditions. The tool lists it as a case on its own. Because the rules keep these cases apart, the tool does not add the gyroscopic, torque and side loads into one envelope; it marks the case with the largest force and the one with the largest couple so the governing combination for a given mount member can be picked by inspection.

Pitching accelerations. Neither Part 23 nor the 2016 Part 25 prescribes a numerical angular acceleration. The text of 25.331(c)(2) from 1978 to 2014 (Amendment 25-46) required the checked maneuver to produce a nose-up pitching acceleration of at least 39 n (n − 1.5)/V rad/s² at 1 g and a nose-down acceleration of at least −26 n (n − 1.5)/V at the limit load factor, V the equivalent airspeed in knots; the tool prints these for the entered n_A and speed as a guide to what may be typed into the acceleration fields. TN 587's own suggestion for the abrupt pull-up, λ = (1875/I_Y)(l_t/15) n_Z with the airplane's pitching inertia in slug·ft² and the tail length in feet, is reproduced in the test script for its F6C-4 example (8.05 rad/s²).

Reading the results

The first section restates the frame, the condition and the vectors n, ω and ω̇. The second writes the first mass item's load out: ω × r, the centripetal term ω × (ω × r), the tangential term ω̇ × r, and each component of F with the substituted numbers, followed by the resultant, the effective load factors the item feels and the moment about the c.g. The table that follows gives the same quantities for every item, with the totals and a sketch in side and plan view whose red arrows are the items' loads on one scale. With engine data entered, the gyroscopic section gives H, the couple for the entered rates with the sense of each component, and the 23.371 combinations; the mount section gives the torque values and, for each engine item, a table of the rule cases with the force at the engine c.g., the couple (gyroscopic plus torque reaction) and, when a mount offset was entered, the moment about the attachment point Δ × F + C. All loads are limit loads; the factor of safety of 23.303 / 25.303 is applied afterwards.

Default example

The default is a twin-engine light airplane with four mass items: 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 and a 140 lb empennage at FS 330, WL 130, about a c.g. at FS 120, WL 95; each engine turns a 3 slug·ft² rotor clockwise at 2,575 rpm maximum continuous (2,700 at takeoff), develops 200 hp at takeoff and 180 hp maximum continuous with 600 lb of thrust, and has four cylinders (torque factor 2). Under the 23.371 yaw condition (r = 2.5 rad/s, n_z = 2.5) the left engine sits at (58, −84, −5) in, its centripetal acceleration is (−30.2, 43.8, 0) ft/s², and it loads its mount with 422 lb forward, 612 lb outboard and 1,125 lb down (1,349 lb resultant, 3.0 g effective) plus a nose-down gyroscopic couple of I_p Ω r = 3 × 269.7 × 2.5 = 2,022 lb·ft = 24,270 lb·in; the pitch condition gives a 9,708 lb·in yawing couple, and the combined case 26,140 lb·in. The takeoff mean torque is 389 lb·ft (4,669 lb·in), the limit torque twice that, and the 23.363 side load 1.33 × 450 = 598 lb. Every value is checked in the tool's test script against the rule text, closed forms, TN 587's F6C-4 example and an independent numpy computation.

Limits

The tool treats the airplane as rigid and the items as point masses: no elastic or dynamic amplification, no distributed inertia of the items themselves (an engine's own moments of inertia resist the airplane's angular acceleration with couples I_e ω̇, which TN 587 includes and which are small; they are not computed here). The aerodynamic loads on the cowling and nacelle that 23.371 also names, the slipstream and P-factor, the engine seizure and blade-loss cases of 25.361(a)(3) and 25.362, and the propeller's periodic two-blade couples are not included. The totals are those of the items entered, not the airplane's balance. The landing-gear reactions of the ground conditions belong to the Landing Gear Loads tool, whose load factors may be entered here.