The Modes of Airplane Motion

Small-perturbation airplane motions are characterized by modes, just as the dis­turbed motions of two spring-coupled masses are a composite of a high-frequency mode of motion in which the masses move toward and away from each other, and a low-frequency mode in which the masses move in the same direction. The five classical modes of airplane motion are found as factors of the airplane’s longitudinal and lateral characteristic equations (Jones, 1934).

The characteristic equations are of fourth or higher degree, so that factors must be found by successive approximations, rather than in closed form. The factors are either real or they occur in pairs, in conjugate complex form. The real factors are characterized by times to double or halve amplitude following a disturbance or by the inverse of the factor, the time constant. The complex factors are usually characterized by their periods or frequencies (damped or undamped) and by their dimensionless damping ratios. The five classical modes are

Phugoid, a low-frequency motion involving large pitch attitude and height changes at essentially constant angle of attack. Damping is low, especially for aerody­namically clean airplanes.

Longitudinal short period, a rapid, normally heavily damped motion at essentially constant airspeed. Damping is provided by wing lift in plunging, as well as horizontal tail lift in rotation. Rapid pitch maneuvers occur in this mode.

Dutch roll, a rolling, yawing, and sideslipping motion of generally low damping, especially at high altitudes.

Roll, essentially a pure rolling motion about the airplane’s longitudinal axis, heavily damped. The primary response to lateral controls is in this mode.

Spiral, a very slow divergence or convergence involving large heading changes, moderate bank angles, and near-zero sideslip.

Additional or combined modes appear in special circumstances, such as the supersonic height mode, discussed in Chapter 11. Notable combined modes are

Coupled roll-spiral or lateral phugoid, the conversion of two simple, aperiodic modes into one oscillatory mode. This mode occurs on airplanes with high effective dihedral and low roll damping (Ashkenas, 1958; Newell, 1965). It has been observed on some V/STOL and high-speed airplanes.

Lateral divergence, a degeneration of the Dutch roll mode into two aperiodic modes, one divergent.

Longitudinal divergence, a degeneration of the phugoid or short-period modes into two aperiodic modes, one divergent. For the phugoid case, the divergent mode is called speed instability or tuck; for the short-period case the divergent mode is called pitchup.

Kinematically constrained modes of motion are those in which some flight variable such as altitude or bank angle is suppressed entirely by theoretical control surface or thrust closed loops, representing pilot control actions. The object is to get approximate stability criteria for flight conditions where the pilot is actively controlling a variable. Two such modes are

Constrained airspeed mode, in which altitude is maintained by some control mo­ment, such as would be produced by the elevator. This produces a mathematical demonstration of speed stability (Neumark, 1957). The constraint results in a first-order differential equation in perturbation airspeed. There is an unstable real root for flight on the back side of the lift-drag polar, corresponding to lift coefficients above that for minimum drag. Section 2 of Chapter 12 discusses the implications of speed stability for naval aircraft.

Constrained yaw mode, in which zero bank angle is maintained by the ailerons (Pinsker, 1967). This constraint results in a first-order differential equation in perturbation yawing velocity. Pinsker demonstrated an aperiodic divergence at angles of attack greater than 18 degrees for an airplane with a low-aspect-ratio wing. This is similar to the nose slice experienced by some modern fighters. Stability of this aperiodic mode is governed by the LCDP parameter (Chapter 9, Sec. 15) Nv — (Nsa/LSa) Lv, where Nv and Lv are the yawing and rolling moments due to sideslip and NSa and LSa are the yawing and rolling moments due to aileron deflection.

The useful concept of airplane modes of motion has been extended to rotary-wing aircraft. In forward flight, their modes of motion are similar to those of fixed-wing aircraft. However, many of the usual stability derivatives disappear in hovering flight, giving quite different results for the modes of motion in hover.

By adding apparent mass effects to the stability derivatives, one can obtain modes of motion for lighter-than-air vehicles. Cook (2000) used earlier models by Lipscombe, Gomes, and Crawford and recent wind-tunnel data to derive modes of motion for a modern nonrigid airship.