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Stability and Control

4 tools in the Aerodynamics area.

Longitudinal Static Stability

Stick-fixed longitudinal static stability and trim of a conventional airplane from its wing, fuselage and horizontal-tail data: the airplane lift-curve slope, the pitching-moment derivatives C_m0 and C_malpha about the centre of gravity, the stick-fixed neutral point solved exactly from the moment balance (with Nelson's approximate form beside it), the static margin, the angle of attack and elevator angle to trim at the flight condition, the tail load, the elevator-angle gradients dδ_e/dα (the stability criterion of NACA Report 711, with its equation (1) printed in the report's symbols) and dδ_e/dC_L, the aft c.g. limit for a required static margin and the forward limit from the up-elevator travel at C_Lmax. The fuselage moment slope may be entered, estimated by Multhopp's strip method from a width table (Munk's apparent-mass factor from Lamb's spheroid coefficients, the wing upwash by the thin-airfoil closed form) or by the fuselage-and-nacelle factor K_f of Report 711 Figure 8; the tail slope by the Report 688 aspect-ratio formula, the elevator effectiveness from Report 711 Figure 3 and the downwash gradient by the lifting-line relation 2 a_w/(pi AR) or the DATCOM fit. Charts of C_m against C_L at the entered c.g., the limits and the neutral point and of the elevator angle to trim against speed, and a side view with the a.c., c.g. and neutral point marked. The default is a light airplane (S 160 ft^2, mean chord 4.5 ft, a_w 4.5/rad, tail 32 ft^2 at 15 ft, a_t 3.5/rad, dε/dα 0.35, η_t 0.9, c.g. at 25 % of the chord, fuselage C_malpha 0.12/rad) at 120 knots: neutral point at 50.4 % of the chord, static margin 25.4 %, trim at 1.10 degrees and 2.07 degrees of down elevator.

1 worked example reproduced, listed on the tool page.

Weight and Balance

Aircraft loading by the weight, arm and moment method of the FAA Aircraft Weight and Balance Handbook: a station table (empty aircraft, crew, passengers, baggage, cargo, ballast) with the fuel entered by volume and density or by weight and an optional fuel-arm table, the moment and the moment index of every item, and the weight and centre of gravity at zero fuel, at the ramp, at take-off and at landing in length units aft of the datum and in percent of the mean aerodynamic chord. Each condition is tested against the maximum weights and against the centre-of-gravity envelope entered as a polygon of weight and c.g. points (normal and utility category), with the forward and aft limits at its weight and the distance to each, and the path of the c.g. as the fuel burns is drawn through the envelope. The page also works the weight-shift, weight-addition and ballast formulas, the lateral c.g., the most forward and most aft adverse loadings, the empty weight and its c.g. from a weighing with tare and from an equipment-change record, and the floor loading of a pallet. The default is the four-seat single-engine sample loading problem of FAA-H-8083-1B (Figures 5-1 to 5-5, pages 5-2 to 5-4): 3,027 lb at 43.54 in.

1 worked example reproduced, listed on the tool page.

Aircraft Stability Analysis

Longitudinal and lateral-directional stability of an aeroplane, static and dynamic, from its stability derivatives. The derivatives come from a pasted derivative set (the JSON text the Vortex Lattice Analysis tool writes, or one typed by hand), from typed nondimensional or dimensional derivatives, or from five public example aircraft: the Lockheed Jetstar and Boeing 747 of NASA CR-2144 and the Navion of NASA CR-96008. Static results: the sign of C_m alpha, static margin, neutral and manoeuvre points, elevator to trim, elevator per g, speed stability, c.g. limits; weathercock and dihedral stability, the spiral criterion, the steady sideslip of a crosswind landing, roll helix angle, rudder for a failed engine, adverse yaw. Dynamic results: the 4 by 4 state matrices in body-fixed stability axes with the alpha-dot terms and the product of inertia, eigenvalues by the QR algorithm and by the closed solution of the quartic, the short period, phugoid, Dutch roll, roll and spiral modes (frequency, damping ratio, period, time to half or double amplitude), the classical approximations beside them, eigenvector phasors, flying-quality Levels by MIL-F-8785C, time histories for control inputs, initial disturbances and 1 - cos gusts, root loci against c.g. position and airspeed, transfer-function zeros, and pitch and yaw dampers. The default is the Jetstar in the power approach, for which the report prints short period zeta 0.528 at 1.66 rad/s, Dutch roll zeta 0.0832 at 1.45 rad/s, roll 1/T 1.95 per second and a spiral that doubles in 62 s; the tool gives 0.527 at 1.664, 0.0830 at 1.453, 1.950 and 62.3 s. A linear small-disturbance study aid, not a certification analysis.

1 worked example reproduced, listed on the tool page.

Flutter and Divergence

Aeroelastic design checks for light aircraft, models and fins from the public NACA reports. The bending-torsion flutter of a typical section is solved with Theodorsen's unsteady aerodynamics (the circulation function C(k) from Bessel functions) by the U-g method and by the p-k method, with the damping and frequency curves against speed, the flutter speed, frequency and reduced frequency, the divergence speed, the aileron reversal speed and the quasi-steady and steady approximations beside them, and the flutter speed against altitude at constant mass. Wing estimates give the torsional divergence of a straight, swept or tapered wing by the Diederich and Budiansky formula of NACA TN 1680 and the aileron reversal of a uniform wing by strip theory. A separate case works the bending-torsion flutter criterion of NACA TN 4197 for missile and rocket fins. The critical speeds are compared with 1.15 times the design dive speed. The default section is the worked case of NACA Report 685, which flutters at 1.542 times the semichord times the torsion frequency.

1 worked example reproduced, listed on the tool page.

Other Aerodynamics sections

Airfoils and Wings · Rotors and Ducted Fans · Aircraft Performance · Atmosphere and Flight Conditions · Propulsion