Aircraft Stability Analysis Including Unsteady Aerodynamic Effects

35possible cases forC φ, where the arrows indicate the direction in time. When the loopencloses an area in a clockwise sense, energy is being added to the system ( Figure 1-9a),whereas counterclockwise loops indicate energy dissipation from the system (Figure1-9b). If a limit cycle occurs, the net energy exchange is zero (Figure 1-9c). Figure 1-10shows the actual roll angle time-history of a slender delta wing at an angle of attackwhere the roll oscillations damp out. It was obtained from wind tunnel tests by Arena,and reported in [46] and [51]. The correspondingC × φ loop is shown in Figure 1-11.Figure 1-12 presents a time history of wing rock buildup at θ = 30°for tests reported in[51]. The dynamic rolling moment characteristics of the buildup cycle A are shown inFigure 1-13, where the clockwise loop in the plot indicates a dynamic instability. Noticethat the restoring moment is roughly linear with roll angle. Since energy is being fed tothe system, the roll angle amplitude is increasing. When the loop is very thin like the oneshown, it accounts for the fact that the buildup happens very slowly. The analogous plotof the rolling moment coefficient after the system has reached steady state (cycle B) canbe seen on Figure 1-14. There still is an unstable region between -20° and 20°, but twostable damping “lobes” have formed for the larger roll angles. The limit cycle oscillationis sustained because the area of the lobes equals the area of the unstable portion of theplot such that the net energy exchange is zero.l0