A combined Multiple Time Scales and Harmonic Balance approach for the transient and steady-state response of nonlinear aeroelastic systems

The majority of methods for calculating the dynamic response of nonlinear aeroelastic systems considers only steady-state periodic behaviour. An exception is the Multiple Time Scales method, which can estimate both the transient and steady-state solutions of such systems; nevertheless, this approach is only accurate close to the Hopf bifurcation. This paper proposes a novel combined approach whereby the transient response is obtained from the Multiple Time Scales method and the asymptotic periodic behaviour is corrected using the Harmonic Balance method. This consistent and efficient framework mutually empowers both techniques and accounts for large parameter variations around the critical condition. The effect of cubic aero-structural nonlinearities on the dynamic response of a generic aeroelastic system is then investigated. Both the Multiple Time Scales and Harmonic Balance methods are adopted and perfect agreement of the explicit results is demonstrated, albeit near the system instability. In contrast, the proposed combined solution is valid for a wider range of perturbations, is analytical and has negligible computational cost while retaining accuracy. The role of key parameters and terms on the core mechanism of the dynamic behaviour is rigorously identified and discussed, from both physical and mathematical points of view. Galloping is finally considered as the simplest but complete application to a fundamental yet practical problem, featuring full conceptual complexity while exploiting the solid synthesis capability of the newly proposed analytical approach. Excellent agreement was found in all cases with results from the numerical integration of the nonlinear equations of motion in the time domain.