Flutter prediction of its occurrence, amplitude and nonlinear behaviour

Abstract

Flutter prediction of its occurrence, limit cycle amplitude and chaotic behaviour is investigated using a general binary (two degree of freedom) coupled modal model to provide new closed form analytical predictions and insight into the aeroelastic phenomenon. The generalised unsteady flutter model, based on flutter derivatives (for any cross-section) and an approximate continuous bilinear lift curve, is used in a mass decoupled form to formulate and develop analytical criteria for the onset and amplitude of flutter under damping and offset angle of attack. This is based on modal coupling through stiffness and damping terms, to clearly identify and clarify two forms of flutter based on stiffness and viscous mode coupling. Closed form analytical criteria for flutter onset are developed including a new necessary dynamic divergence criterion (analogous to spragging in brake squeal) based on flutter derivatives for any structure. The model is first validated with the well-known classical numerical and experimental airfoil results of Theodorsen and analytical results of Pines, which are also shown to correspond to stiffness mode coupling. In addition, the new modelling enables the detection and corresponding conditions of viscous mode coupling flutter which does not require closely spaced modes. The occurrence and conditions of both stiffness and viscous mode coupling flutter is verified numerically with results highlighting important structural parameters i.e. damping factors which could cause flutter outside classical conditions. In particular, the conditions for viscous mode coupling are newly identified and found to occur at flow velocities substantially lower (up to approximately 10%) than classical stiffness mode coupling values. Dynamic divergence is shown to be the lower bound for the critical flutter speed under a range of conditons. The viscous mode coupling is shown to occur under high levels of torsional damping (20 times the nominal 1% of critical), for the airfoil case, compared to vertical damping, but does not require the modal frequencies to be close. The efficient analytical prediction for flutter amplitude is then numerically verified for Theodorsen's airfoil. The results show that flutter amplitude is primarily dependent upon flow velocity and stall conditions without structural nonlinearity. Finally, a further nonlinear investigation of flutter identifies useful analytical criteria for the occurrence of chaotic flutter which is confirmed numerically. These results, based on a bilinear lift coefficient model, highlight that the occurrence of chaotic flutter can be due to simplified aerodynamics alone i.e. without nonlinear structural stiffness. The closed form analyses provide critical insight into; the occurrence and underlying mechanisms of flutter based on stiffness and viscous mode coupling, dynamic divergence as well as the nonlinear amplitudes and behaviour under an offset angle of attack and stall conditions.