Flutter of an airfoil with a cubic nonlinear restoring force

Abstract

In this paper, the effect of a cubic structural restoring force on the flutter characteristics of a twodimensional airfoil placed in an incompressible flow is investigated. The aeroelastic equations of motion are written as a system of eight first-order ordinary differential equations. Given the initial values of plunge and pitch displacements and their velocities, the system of equations is integrated numerically using a 4 order Runge-Kutta scheme. Results for softand hard-springs are presented for a pitch degree-of-freedom nonlinearity. The study shows the dependence of the divergence flutter boundary on initial conditions for a soft spring. For a hard spring, the nonlinear flutter boundary is independent of initial conditions for the spring constants considered. The flutter speed is identical to that for a linear spring. Divergent flutter is not encountered, but instead limit cycle oscillation occurs for velocities greater than the flutter speed. The behaviour of the airfoil is also analyzed using analytical techniques developed for nonlinear dynamical systems. The Hopf-bifurcation point is determined analytically and the amplitude of the limit cycle oscillation in postHopf-bifurcation for a hard spring is predicted using an asymptotic theory. The frequency of the limit cycle *Principal Officer and Head, Experimental Aerodynamics and Aeroelasticity Group. Also adjunct professor, Depl. of Mathematical Sciences, University of Alberta. Associate Fellow AIAA. Research Associate, Experimental Aerodynamics and Aeroelasticity Group. 'Professor, Dept. of Mathematical Sciences. Copyright © 1998 by B.H.K. Lee. L.Y. Jiang and Y.S. Wong, Published by the American Inst i tute of Aeronautics and Astronautics Inc. with permission oscillation is estimated from an approximate method. Comparisons with numerical simulations are carried out and the accuracy of the approximate method is discussed. The analysis can readily be extended to study limit cycle oscillation of airfoils with nonlinear polynomial spring forces in both plunge and pitch degrees of freedom. NOMENCLATURE ah non-dimensional distance from airfoil midchord to elastic axis b airfoil semi-chord CL aerodynamic lift coefficient CM pitching moment coefficient h plunge displacement m airfoil mass R response amplitude of pitch motion r response amplitude of plunge motion ra radius of gyration about elastic axis t time U free stream velocity U non-dimensional velocity, U/bcoa UL non-dimensional linear flutter speed xa non-dimensional distance from elastic axis to centre of mass X system variable vector XE system equilibrium point y variable vector a pitch angle of airfoil OCA pitch angle amplitude of l imit cycle oscillation EI, £2 constants in Wagner's function Pa, p= coefficients of cubic spring in pitch and plunge C,a, viscous damping ratios in pitch and plunge H airfoil/air mass ratio, m/Kpb