Nonlinear Longitudinal Stability of a Symmetric Aircraft

The longitudinal stability of a symmetric aircraft is examined beyond the well-known linearization about a steady mean state, leading to the phugoid and short-period modes. The exact equations of longitudinal motion of a symmetric aircraft are considered, i.e., balance of longitudinal and transverse force (without side force ) and balance of pitching moment; the drag terms included are friction and lift-induced drag, plus nonsymmetric lift ‐ drag polar; the mass density is taken as a constant, as well as thrust along the e ight path. Elimination would lead to a fourth-order nonlinear differential equation for the angle of attack relative to the angle of zero pitching moment, if the acceleration of the e ight-path angle is neglected, and the e ight-path angle is moderate. In the case of a small e ight-path angle, it simplie es to a thirdorder differential equation, containing a set of nonlinear corrections, to a second-order linear equation, specifying sinusoidal oscillation of the relative angle of attack. It is shown, by a small perturbation method, that forced oscillations occur at its harmonics (viz. double or triple frequency ), and free oscillations can have decaying or growing amplitude. In the case of statically stable aircraft, the oscillations have a short period and grow or decay slowly. In the case of statically unstable aircraft the growth is rapid, as in a pilot-induced oscillation.