Bifurcations Analysis Of Nonlinear Aeroelastic Systems

The flutter phenomenon of aeroelastic systems is one of the most important problems in many areas, such as aviation, wind power generation, civil engineering etc. The bifurcation and complex responses of the nonlinear aeroelastic systems are the research hotspot in this area. Based on the modern nonlinear dynamics theory, the bifurcations of two-dimensional airfoils are investigated by analytical method in this dissertation, and the main contributions are as follows:The two-dimensional airfoil with cubic pitching stiffness is studied, and the second order differential equations of motion for the airfoil are rewritten as a four dimensional differential equations in state space. Applying the eigenvalue theory, the analytic solutions of critical boundaries of pitchfork bifurcations and Hopf bifurcations are obtained in 2 dimensional parameter plane. The number and the stabilities of the equilibrium points and the limit cycles in different regions of 2 dimensional parameter plane are analyzed.According to the symmetry of the system, the equilibrium points and the limit cycles of the system are symmetrical about the origin, so the equilibrium points and the limit cycles are conjugated. The other bifurcations maybe occur in some parameter zones, such as pitchfork bifurcations of closed orbits, the period doubling bifurcation of closed orbits, chaos etc. These bifurcation phenomena are found with numerical simulation, and the period-5 windons are found in chaos zones.Taking the nondimensional flight speed and the nondimensional linear part of pitching stiffness as parameters, the analytic solutions of critical boundaries of pitchfork bifurcation and Hopf bifurcation are obtained in the 2 dimensional parameter planes. The results show that when the nondimensional linear part of pitching stiffness is less than a certain value, the limit cycle will not occur because of the existence of Hopf bifurcations. To find out the reason of limit cycle appearance, with the help of harmonic balance method, the approximate critical boundaries of 2-multiple semi-stable limit cycle bifurcation are obtained, and the bifurcation points of supercritical or subcritical Hopf bifurcations are also found.The co-dimension two bifurcations of the two-dimensional airfoil with cubic pitching stiffness are studied, the parameter plane is divided into eight regions by each bifurcation boundary, in each region the stabilities and the numbers of equilibrium points and limit cycles are analyzed. Some numerical simulation results are given for illustration. The complex dynamical behaviors around the equilibrium points are found out by the numerical simulation of the nonlinear systems.For the two-degree-of-freedom nonlinear aeroelastic systems with piecewise linear nonlinearity in pitching direction, the bifurcations conditions of the equilibrium points are derived with the analytical method. The necessary conditions for the existence of the limit cycles are obtained. The existence of the limit cycle is studied by the method of coupling chart and the numerical simulation method. The result indicates that the coupling chart method is only applicable when the inflexion does not locate at the coordinate origin.