POD analysis for free response of linear and nonlinear marginally stable aeroelastic dynamical systems

The aim of this work is to study the relationship between Proper Orthogonal Modes (POMs) and Values (POVs) – as obtained by the free response analysis of linear and nonlinear aeroelastic (namely, marginally stable) systems – and the intrinsic invariants of such systems like modes (linear case) or manifolds (nonlinear case). The present investigation is first conducted for linear systems, by extending the results obtained in previous works, and then for nonlinear systems in the neighborhood of and beyond a Hopf bifurcation, where interesting features like limit cycle oscillations and chaotic response appear. The key point of the present theoretical analysis consists of avoiding the use of complex state-space variables to identify the subspaces where the distinguishing system dynamics – each associated to a couple of complex conjugated eigenvalues – live. Thus, a relation between the invariants obtained by the asymptotic expansion provided by Multiple Time Scale (MTS) method and POMs has been established. The typical section and the panel flutter problems do not only provide significant numerical test-cases for validation, but also suggest some insight about an alternate way of performing the Karhunen–Loève decomposition by combining POD analysis with Galerkin representation of the solution.