Nonlinear aeroelastic response of a flat panel. A normal form approach
The model of a nonlinear plate in a supersonic unsteady flow forced by a dynamic excitation and subject to a biaxial compressive load is considered. Indeed, the aeroelastic model can represent a variety of mechanical systems and it gives the opportunity to analyze the physical meaning of the concept of small divisors representing a relevant issue in the qualitative analysis of the response of a nonlinear system via perturbation approaches. Moreover, this aeroelastic system shows a rich dynamical behavior presenting both static and dynamic bifurcations. In this work the Von Karman model for a at panel is considered in order to represent the behavior of a nonlinear elastic plate and the aerodynamic loads are modeled using the Piston Theory, taking into account nonlinearities up to the third order. The nonlinear partial differential equations model is discretized by a Galerkin projection on a suitable functional basis obtaining a system of nonlinear ordinary differential equations. The discretized problem is studied via a perturbation approach based on the Lie Transform method in order to reduce the system to a simpler form defined by the resonances conditions, i.e., the Normal Form. By analyzing how different parameters work in the resonance conditions and in their definitions, a physical interpretation of small divisors is given. In particular, it is possible to study how the physical parameters influence the qualitative behavior of the aeroelastic system response. Moreover, using the same approach, the forced behavior of the system is studied in the neighborhood of both static and dynamic bifurcation points.
