## Bifurcation theory applied to aircraft motionsBifurcation theory is used to analyze the nonlinear dynamic stability characteristics of single-degree-of-freedom motions of an aircraft or a flap about a trim position. The bifurcation theory analysis reveals that when the bifurcation parameter, e.g., the angle of attack, is increased beyond a critical value at which the aerodynamic damping vanishes, a new solution representing finite-amplitude periodic motion bifurcates from the previously stable steady motion. The sign of a simple criterion, cast in terms of aerodynamic properties, determines whether the bifurcating solution is stable (supercritical) or unstable (subcritical). For the pitching motion of a flat-plate airfoil flying at supersonic/hypersonic speed, and for oscillation of a flap at transonic speed, the bifurcation is subcritical, implying either that exchanges of stability between steady and periodic motion are accompanied by hysteresis phenomena, or that potentially large aperiodic departures from steady motion may develop. On the other hand, for the rolling oscillation of a slender delta wing in subsonic flight (wing rock), the bifurcation is found to be supercritical. This and the predicted amplitude of the bifurcation periodic motion are in good agreement with experiments. |

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accompanied by hysteresis aerodynamic damping vanishes AIAA aircraft or flap airfoil in supersonic/hypersonic Ames Research Center amplitude angle of attack APPLIED TO AIRCRAFT bifurcation diagram bifurcation is subcritical bifurcation parameter bifurcation periodic motion bifurcation periodic solution bifurcation solution BIFURCATION THEORY APPLIED critical value Damping-in-pitch derivative delta wing versus eigenvalues eigenvector experimental results finite finite-amplitude periodic motion Flap Oscillation Flight Dynam Hopf Bifurcation hysteresis phenomena incompressible flow large aperiodic departures linear theory loses its stability mean deflection angle Moffett Field Murray Tobak NASA nonlinear dynamic stability oscillations in transonic periodic roll oscillation pitching motion potentially large aperiodic Rock of Slender single-degree-of-freedom motions slender delta wing stability characteristics stability theory stable periodic solutions stable steady motion stable supercritical steady-state subsonic flow supersonic/hypersonic airfoil supersonic/hypersonic flow Taylor series theory predicts transonic flow trim angle trim condition trim position University of Waterloo Waterloo wing in subsonic wing versus angle