Nonlinear dynamics and gust response of a two-dimensional wing

In this paper, aeroelastic equations for a two-dimensional wing encountering a wind gust are presented and solved numerically. The indicial aerodynamic theory based on Wagner's function is adopted to obtain aerodynamic forces and moments acting on the wing in the time domain. The dynamics of the wing is approximated via two degrees-of-freedom, i.e. pitch and plunge. The structural stiffness is modeled by a linear translational and a nonlinear torsional spring. Two different types of stiffness nonlinearities are examined: (i) flat spot or dead zone to model free-play, and (ii) hysteresis. For given system parameters, bifurcation diagrams are presented, and time-history, power-spectral density, phase-plane and Poincare diagrams are shown at different flow velocities. We show that the dynamics of the system with the free-play nonlinearity may be very complex with the possibility of the occurrence of chaos through period-doubling bifurcations. In contrast, the dynamics with the hysteresis nonlinearity is found to be rather simple, where a Hopf bifurcation is the only bifurcation observed. The response of the nonlinear system to sharp-edged and 1-cosine gust profiles are also obtained at different flow velocities and compared to the time response of the system with no gust input. It was found that the gust input may cause the nonlinear system to get attracted to a periodic orbit in the subcritical flow regime. In addition, basin of attraction is obtained for various amplitudes of the sharp-edged gust. It is discussed that as the gust becomes stronger, the likelihood of the occurrence of limit-cycle oscillation increases while the stable points become less dispersed inside the stability map and form a finite region confined to large values of initial conditions.