Stability Boundaries

Until the advent of electronic analog and digital computers, numerical solutions of the equations of airplane motion were essentially limited to finding stability boundaries, the combinations of airplane stability derivatives and other parameters that divide stability from instability. Stability boundaries are found by Routh’s Criterion, developed by the Briton E. J. Routh in the early 1900s.

Airplane stability boundaries were first calculated in Britain (Bryant, Jones, and Pawsey, 1932). This was in a study of dynamic stability beyond the stall. Bryant and his co-authors found stability derivatives for a number of airplanes up to an angle of attack of 40 degrees. With these data, they produced stability boundaries as functions of static directional and lateral stability derivatives, both nondimensionalized by Glauert’s airplane relative density parameter /г.

There was an earlier British paper by S. B. Gates that presented contours of constant damping ratio and natural frequency for the longitudinal phugoid, as functions of tail volume and center-of-gravity position (Gates, 1927). While not strictly a stability boundary analysis, the Gates work certainly laid the groundwork for Bryant’s boundaries.

Two NACA reports by Charles H. Zimmerman (1935 and 1937) carried on Gates’ and Bryant’s pioneering stability boundary work. Zimmerman’s ambitious goal was to produce charts for the rapid estimation of the dynamics of any airplane. The Zimmerman reports have charts for both longitudinal and lateral motions, 40 of the former and 22 of the latter (Figure 18.6). As in Bryant’s work, the results are normalized using Glauert’s airplane density parameter г. The Zimmerman charts include period and damping estimates for the phugoid and Dutch roll motions.