The Study Of Limit Cycle Flutter For Airfoil With Cubic Stiffness

In airplane design, structural nonlinearities cannot be avoided．The structural nonlinearities arise from worn hinges of control surfaces, loose control linkages, material behavior and various other sources. In this paper, limit cycle oscillation of a two-degree-of-freedom airfoil with cubic nonlinearity stiffness flutter system in incompressible flow is investigated. The monition equations of airfoil are written as the four dimensional first order differential equations. Taking airspeed and the linear part of pitching stiffness as the bifurcation parameters. The supercritical or subcritical Hopf bifurcations are founded in the airfoil flutter system, i.e. the system has stable or unstable limit cycle oscillations motion. The system has the abundant dynamics characteristic when it subject to perturbation. In this paper, further and detailed investigations are given from theory, method and application. Some qualitative and quantitative rules are obtained, which benefit airfoil design and dynamics analysis.By means of harmonic balance method, taking different values, the figures of the amplitudes of limit cycle oscillations versus the airspeed are obtained.Taking linear pitching stiffness , the flutter system is investigated further. A two-degree-of-freedom dynamical system is reduced to a two dimensional first order differential equations based upon the Center Manifold Theorem and Mathematica program. The behaviors of bifurcation points is determined by using the method of successor function , which can be used to determine the stability of the equilibrium points.The Normal Form method and Mathematica program are applied to simplify the two-dimensional equations. Furthermore, the effects of the system parameters on flutter boundaries of the linear approximate system and amplitudes of the stable limit cycle oscillations are investigated.By means of constructing the relations of topology bifurcation solution of this type system with cubic stiffness and the physical parameters of system, universal unfolding bifurcation figures are obtained, which exhibit the motion models are plentiful. It is convenient to choice reasonable system parameters according to require motion models.