Formulation of the Aeroelastic Instability Problem of Rectangular Plates in Uniform Flow Based on the Hamiltonian Mechanics for the Constrained System

This paper addresses a formulation and an aeroelastic instability analysis of a plate in a uniform incompressible and irrotational flow based on the classical variational principle framework. Because of an intrinsic algebraic relation between the plate displacement and the velocity potential, this system has to be formulated as the constrained system. In this study, we tried to apply the Hamiltonian mechanics to the formulation of the fluid-structure interaction problem with mixed boundary condition. As a result, we obtain the canonical equations, that consist of the evolution equations for the plate displacement, the velocity potential, the Lagrange multiplier and canonically conjugate momenta for those physical quantities. In particular, it was found that the Lagrange multiplier was just the pressure. In other words, the equations of time evolution could be derived for not only the plate displacement and the velocity potential but also the pressure (the Lagrange multiplier). The stability of this system was analyzed by the eigenvalue analysis. Then, flutter modes, their frequencies and growth rates were discussed. The proposed technique has the advantage that it can reduce iteration procedures in the stability analysis. As a consequence, it can be expected that the stability of this system can be evaluated efficiently. This paper introduces a formulation of the only two dimensional problem, and the stability analysis of a clamped-free plate is implemented as an numerical example. Howerver, this formulation can be applied to three dimensional problems without intrinsic difficulties.