Continuation of Higher Order Harmonic Balance Solutions for Nonlinear Aeroelastic Systems

The Harmonic Balance method is a very useful tool for characterizing and predicting the response of nonlinear dynamic systems undergoing periodic oscillations, either self-excited or due to harmonic excitation. The method and several of its variants have been applied to nonlinear aeroelastic systems over the last two decades.

This paper presents a detailed description of several Harmonic Balance methods and a continuation framework allowing the methods to follow the response of dynamic systems from the bifurcation point to any desired parameter value, while successfully negotiating further fold bifurcations. The continuation framework is described for systems undergoing sub-critical and super-critical Hopf bifurcations as well as a particular type of explosive bifurcation.

The methods investigated in this work are applied to a nonlinear aeroelastic model of a Generic Transport Aircraft featuring polynomial or freeplay stiffness nonlinearity in the control surface. It is shown that high order Harmonic Balance solutions will capture accurately the complete bifurcation behavior of this system for both types of nonlinearity. Low order solutions can become inaccurate in the presence of numerous folds in the Limit Cycle Oscillation branch but can still yield practical engineering information at a fraction of the cost of higher order solutions. Time domain Harmonic Balance schemes are shown to be more computationally expensive than the standard Harmonic Balance approach.