Finite element formulations via the theorem of expended power in the Lagrangian, Hamiltonian and total energy frameworks

In this work we analyze the case of a vibrating beam, simply supported or clamped at both ends, under the effect of a high supersonic airflow along its axial direction. A complete aerodynamic model of the piston theory, which also takes into account the nonlinear components of the distributed aerodynamic transversal force, is used. The postcritical flutter behavior and its influence on the vibration state solutions of a fluttering beam without aerodynamic damping have been studied. This paper focuses particularly on the effects of these nonlinear aerodynamic forces on three frequencies, which are useful in characterizing the postcritical flutter solution set of the undamped beam in the whole frequency range: the minimum frequency, the frequency where the change of the modal shape with lower amplitude occurs, and the frequency corresponding to the solution with minimum amplitude of the vibration mode. Special attention has been given to the influence on the solution of the vibrating undamped beam with minimum modal amplitude, whose frequency is the most important among the three mentioned above; in fact, in the neighborhood of this particular solution, there exists the flutter state of the vibrating damped beam in limit cycle conditions. Three different schemes, two of them semianalytical (based on the classical and well known Rayleigh-Ritz and Galerkin methods) and one of them numerical (based on the finite element method), have been herein exploited, as in the author's previous papers, where beam flutter models with linear aerodynamic analysis were used. The good agreement between the results obtained by the three methods corroborates their effectiveness. More sophisticated models have been herein set up, considering that a more accurate analysis is necessary than in previous cases, where the aerodynamic numerical model was limited to within the framework of the quasisteady linearized piston theory, both for the coupling component between odd and even order vibrating modes, and for the aerodynamic damping component. The results obtained enable us to assess quantitatively the influence of these nonlinear aerodynamic forces on the postcritical beam flutter behavior, and particularly on the undamped beam solution with minimum amplitude of the vibration mode.