An investigation of stability of a control surface with structural nonlinearities in supersonic flow using Zubov's method

It is well known that the presence of nonlinearities may significantly affect the aeroelastic response of an aerospace vehicle structure. In this paper, the aeroelastic behaviour at high Mach numbers of an all-moving control surface with a nonlinearity in the root support is investigated. Very often, under certain flight conditions, a stable equilibrium point, corresponding to zero displacement of the structure, together with an unstable limit cycle arising from a sub-critical Hopf bifurcation results from the presence of the nonlinearity. The dynamic aeroelastic response to external excitation is also of interest, and when sinusoidal forcing is applied, the stable equilibrium point may then be replaced by a periodic attractor, and the limit cycle by an unstable multi-periodic solution. With or without this forcing, there is an attractor which will possess a domain of attraction. In this paper, the problem of estimating these domains of attraction is tackled using Zubov's method. In the absence of forcing, the method is applied directly to the aeroelastic equations, while for the forced system, the method of averaging is applied to approximate the aeroelastic equations by an autonomous system. The behaviour of the system with forcing is also investigated for flight speeds below a threshold which may occur where the unstable limit cycle of the unforced system disappears. In this regime, the nonlinear system may nevertheless still possess multiple attractors, and their domains of attraction are investigated, again using an averaged form of the aeroelastic equations. In this study, the nonlinearity in the root support was assumed to be due to a cubic hardening restoring moment. The Zubov approach, which always yields conservative estimates, was shown to be capable of rapidly giving a good indication of stability domain boundaries under many conditions. Although this investigation focuses on an aeroelastic system, the general form of equations considered arises in many other settings, so that the approach would be relevant to a whole range of engineering applications.