Revisited Harmonic Balance Trim Solution Method for Periodically-Forced Aerospace Vehicles

In this paper, a numerical method is proposed for determining the periodic state and control solutions of nonlinear time-periodic systems. Starting from an initial guess at the solution, the algorithm uses a harmonic balance technique to refine the solution through a gradient-based optimization approach. The algorithm introduces three major innovations when compared to previous techniques: it does not rely on state transition matrices, it simultaneously solves for the approximate higher-order linear time-invariant dynamics about the periodic solution, and it can be used to compute the harmonic control inputs that attenuate arbitrary state harmonics. Following a description of the algorithm, it is applied to three different aerospace vehicles with nonlinear time-periodic dynamics: a flapping-wing micro aerial vehicle, a helicopter, and a flapping-tail airplane. In all cases, the algorithm derives periodic solutions for the states and controls even when starting from very poor initial guesses. The algorithm has clear application in the development of advanced flight control laws that attenuate certain state harmonics and in the prediction of loads and vibrations.