Inertial Coupling in Rapid Maneuvers
We saw in the last section how to include nonlinear gravity effects in control response and how such effects manifest themselves in the response of a relatively sedate vehicle. Of the other two categories of nonlinearity—aerodynamic and inertial— little in a general way can be said about the first. Aerodynamic characteristics, especially for flexible vehicles at high subsonic Mach numbers, are too varied and complex to admit of useful generalizations. A very elaborate (and very costly!) aerodynamic model is required for full and accurate simulation or computation. Not so, however, for the second category of nonlinearity. There is a class of problems, all generically connected, known by names such as roll resonance, spin-yaw coupling, inertia coupling, and so on (Heppe and Celinker, 1957; Phillips, 1948; Pinsker, 1958) that pertain to large-angle motions, or even violent instabilities, that can occur on missiles, launch vehicles, and slender aircraft performing rapid rolling maneuvers. These have their source in the pq and pr terms that occur in the pitching and yawing moment equations. A detailed analysis of these motions would take us beyond the scope of this text. Some is given in Etkin (1972), and much more is given in the cited references. One very important conclusion, due to Phillips (1948), is that there is a band of roll rates for airplanes within which the airplane is unstable. At lower roll rates, the usual stability criteria apply. At rates above the band the airplane is gyrosta – bilized in the way a spinning shell or top is. The lower of the critical roll rates for a normally stable airplane is given approximately by the lesser of
If the roll rate in a maneuver approaches or exceeds this value the possibility of a dangerous instability exists.