A variational formulation of acousto-structural vibrations, with application to flutter problems

Abstract

Presented herein is a general variational formulation for acousto-structural vibrations and its application to elastic plates. First, a variational formulation for gas-structure media is obtained by means of Hamilton's principle under the usual constraint conditions [B. Tabarrok, Int. J. Num. Meth. Eng. 13, 197–201 (1978)]. Then, the constraints, frequently undesirable in computation, are relaxed through Friedrichs's transformation, as illustrated by the authors [Finite Elements in Water Resources, (Springer-Verlag, New York, 1982) pp. 2.101–110]. Thus a variational formulation with no constraints is established for acousto-structural vibrations. Next, the field equations governing the motion of elastic plate in a potential gas flow are derived by using the unconstrained variational formulation together with the kinematic hypotheses of R. D. Mindlin [J. Appl. Mech. 18, 31–38 (1951)]. The governing equations accommodate all the types of motions of the plate. Further, the sufficient conditions are enumerated to ensure the uniqueness in solutions of the governing equations by the logarithmic convexity arguments. [Sarigül's work was supported by the Amelia Earhart fellowship.]