THE AEROELASTIC RESPONSE OF A 2-DIMENSIONAL AIRFOIL WITH BILINEAR ANDCUBIC STRUCTURAL NONLINEARITIES

A two-dimensional airfoil with either a bilinear or cubic structural n onlinearity in pitch, and subject to incompressible flow has been anal ysed; 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 airfoi l motion, or solved in a semi-analytical manner using a dual-input des cribing function technique. For both types of nonlinearity regions of limit cycle oscillation (LCO) are detected for velocities well below t he divergent flutter boundary. Using the finite difference method it i s shown that the existence of the LCOs is strongly dependent on the in itial conditions of the airfoil. Although the describing function meth od cannot predict the effect of initial conditions, it does give reaso nable predictions of the velocity at which LCOs commence, and good pre dictions of the magnitude of the LCOs-at least for those cases where t he LCO motion is predominantly period-one. The existence of the LCOs i s 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-p lane plots and Poincare sections of the airfoil time histories; the ex istence of chaos was confirmed for the cubic nonlinearity via calculat ion of the Lyapunov exponents, one of which is positive. The fact that chaotic motion is obtained with both bilinear and cubic nonlinearitie s suggests that it is not the discontinuous nature of the stiffness, a ssociated with the bilinear nonlinearity, which is responsible for pro ducing this chaotic motion.