Linear Time-Invariant Approximations of Nonlinear Time-Periodic Systems

This paper discusses the development of a numerical method for the approximation of the nonlinear time-periodic rotorcraft flight dynamics with higher order linear time-invariant (LTI) models. The method relies on a per-rotor revolution perturbation scheme, which is of particular importance for the linearization of simulation models that do not allow for per-time-step perturbations, and for those output measures that necessitate the solution of partial differential equations and thus require several time steps to be computed. The paper demonstrates the application of the proposed methodology to obtain high-order LTI models capable of predicting vibrations for a generic utility helicopter. Simulations are used to validate the response of the linearized models against those from nonlinear simulations and from competing approaches in the literature. The proposed method is shown to predict accurately the nonlinear response for the case shown and for small amplitude maneuvers. Frequency-domain validation is also performed to compare the linear models derived with the proposed method with those obtained with harmonic decomposition, a competing approach based on a per-time-step perturbation scheme. Interestingly, the proposed algorithm yields nearly identical numerical results compared to harmonic decomposition, suggesting that the two methods are in fact equivalent but rely on different formulations.