Symmetric Formulations for Modal Analysis of Internal Fluid Structure Systems

This paper deals with appropriate computational methods for modal analysis of elastic structures containing an inviscid fluid (gas or liquid). These methods, based on Ritz-Galerkin projection using appropriate functional basis, allow us to construct reduced models expressed in terms of physical displacement vector field u in the structure, and generalized coordinates vector r describing the behavior of the fluid. Those reduced models lead to symmetric generalized eigenvalue matrix system involving a reduced number of degrees of freedom for the fluid. More precisely, we construct symmetric matrix models of the fluid considered as a subsystem, by considering the response of the fluid to a prescribed normal displacement of the fluid-structure interface. Two distinct situations are analyzed, namely linear vibrations of an elastic structure completely filled with a compressible gas or liquid and linear vibrations of an elastic structure containing an incompressible liquid with free surface effects due to gravity. The first case is a structural acoustic problem with modal interaction between structural modes with acoustic modes in rigid motionless cavity. Wall impedance can also be easily introduced in order to take into account fluid-structure interface dissipation, for further forced response studies. The second case is a hydroelastic-sloshing problem with modal interaction between incompressible hydroelastic structural modes with incompressible liquid sloshing modes in rigid motionless cavity, involving an elastogravity operator related to the wall normal displacement of the fluid-structure interface. For the construction of reduced models, the static behavior at zero frequency play an important role. This is why we start from “well-posed” variational formulations of the problem, in the sense that zero-frequency behavior must be well retrieved in the equations. It should be noted that the so-called “quasi-static correction” term plays a fundamental role in the Ritz-Galerkin procedure (error truncation). The general methodology corresponds to dynamic substructuring procedures adapted to fluid-structure modal analysis. For general presentations of computational methods using appropriate finite element and dynamic substructuring procedures applied to modal analysis of elastic structures containing inviscid fluids (sloshing, hydroelasticity and structural-acoustics), we refer the reader to Morand and Ohayon (1995).