The Aeroelastic Response of a Two-Dimensional Airfoil with Bilinear and Cubic Structural Nonlinearities

Abstract

A two-dimensional airfoil with either a bilinear or cubic structural nonlinearity in pitch, and subject to incompressible flow has been analysed; the aerodynamic forces on the airfoil are evaluated using Wagner's function. The resulting equations are either integrated numerically using a finite difference method to give time histories of the airfoil motion, or solved in a semi-analytical manner using a dual-input describing function technique. For both types of nonlinearity regions of limit cycle oscillation (LCO) are detected for velocities well below the divergent flutter boundary. Using the finite difference method it is shown that the existence of the LCOs is strongly dependent on the initial conditions of the airfoil. Although the describing function method cannot predict the effect of initial conditions, it does give reasonable predictions of the velocity at which LCOs commence, and good predictions of the magnitude of the LCOs—at least for those cases where the LCO motion is predominantly period-one. The existence of the LCOs is strongly dependent on the properties of the airfoil. In some cases, most notably those with small structural preloads, regions of chaotic motion are obtained, as suggested by power spectral densities, phase-plane plots and Poincaré sections of the airfoil time histories; the existence of chaos was confirmed for the cubic nonlinearity via calculation of the Lyapunov exponents, one of which is positive. The fact that chaotic motion is obtained with both bilinear and cubic nonlinearities suggests that it is not the discontinuous nature of the stiffness, associated with the bilinear nonlinearity, which is responsible for producing this chaotic motion.