Perturbation methods and proper orthogonal decomposition analysis for nonlinear aeroelastic systems

The modern engineering deals with applications of high complexity. From a mathematical point of view such a complexity means a large number of degrees of freedom and nonlinearities in the equations describing the process. To approach this difficult problem there are two levels of simplification. The first level is a physical reduction: the real problem is represented by mathematical models that are treated in order to be studied and their solution computed. At this level we can find all the discretization techniques like Galerkin projection or Finite Element Methods. The second level is a simplification of the original problem in order to study it in an easier way: a reduced order model is advocated. Simplification means to determine a dominant dynamics which drives the whole problem: not all the unknowns are considered independent being some of them functions of the remaining others. Two methodologies are considered in this a Thesis. The first is the Lie Transform Method based on the results of Normal Form Theory and the Center Manifold Theorem. For some conditions, called resonance or zero divisors, depending on combinations of the eigenvalues of the linearized system, the nonlinearity of the problem is reduced and a driving dynamics determined. The second is the Proper Orthogonal Decomposition (POD), which from the analysis of representative time responses of the original problem determines a subspace of state variables energetically significant spanned by the Proper Orthogonal Modes (POMs). The main issues related to the Lie Transform Method, as for all the Normal Form based Method, is the presence of small divisors for which there is no general rules to be determined when considering nonconservative systems. In the present work, this problem is considered and some physical parameters are related to such conditions determining qualitatively what small means for a divisor relatively to a perturbation parameter. Moreover, starting from the analytical results obtained the POD behavior in the neighborhood of a bifurcation point has been studied. In particular, POMs has been related to the linearized modes of the studied systems and it has been demonstrated their equivalence for systems experiencing a Hopf bifurcation. Moreover, some conditions of equivalence are addressed also in presence of static bifurcations with forcing loads. Finally, the relation i between modal activation and energy distribution has been studied and the possibility to relate POD behavior and nonlinearity (small divisors) of the response has been addressed.