Residue harmonic balance for two-degree-of-freedom airfoils with cubic structural nonlinearity

Abstract

A residue harmonic balance method is extended to determine the approximations and bifurcations for a coupled airfoil system with cubic structural nonlinearity analytically. The approximate solutions to any desired accuracy can be obtained easily by solving a set of linear algebraic equations in each step. The strongly cubic nonlinear pitching and plunging stiffness terms in the two-degree-of-freedom self-excited equations of airfoil motions are considered. The approximations to the angular frequency and limit cycle are analytically found. The results show that the second-order approximations obtained by the proposed technique are in excellent agreement with the numerical integration results. The existence of cubic nonlinearity in the stiffness term does not affect the linear flutter speed of airfoil motion a great deal. However, the responses are different not only after supercritical bifurcation flutter, but also before the linear flutter speed in the subcritical bifurcation. Furthermore, the supercritical and subcritical Hopf bifurcations of three nonlinear cases of airfoil motions are analyzed taking the flow speed as the bifurcation parameter.