Estimation of the hopf bifurcation point for aeroelastic systems

The estimation of the Hopf bifurcation point is an important prerequisite for the non-linear analysis of non-linear instabilities in aircraft using the classical normal form theory. For unsteady transonic aerodynamics, the aeroelastic response is frequency-dependent and therefore a very costly trial-and-error and iterative scheme, frequency-matching, is used to determine flutter conditions. Furthermore, the standard algebraic methods have usually been used for systems not bigger than two degrees of freedom and do not appear to have been applied for frequency-dependent aerodynamics. In this study, a procedure is developed to produce and solve algebraic equations for any order aeroelastic systems, with and without frequency-dependent aerodynamics, to predict the Hopf bifurcation point. The approach performs the computation in a single step using symbolic programming and does not require trial and error and repeated calculations at various speeds required when using classical iterative methods. To investigate the validity of the approach, a Hancock two-degrees-of-freedom aeroelastic wing model and a multi-degree-of-freedom cantilever wind model were studied in depth. Hancock experimental data was used for curve fitting the unsteady aerodynamic damping term as a function of frequency. Fairly close agreement was obtained between the analytical and simulated aeroelastic solutions with and without frequency-dependent aerodynamics.