Nonlinear Galerkin Methods Applied in the Dimension Reduction of Vibrating Fluid Conveying Tubes

Abstract

In this paper we use the nonlinear Galerkin method to reduce the infinite dimensional system describing the oscillations of a fluid conveying tube to a finite low dimensional system. In fact we are able to replace a system of nonlinear partial differential equations by a set of nonlinear ordinary differential equations which are the amplitude equations of the critical modes. Three important problems are addressed in this respect. First, the choice of the form of the critical modes (ansatz functions), second the choice of the number m of the critical modes and finally the construction of the reduced system. For the latter point the so-called approximate inertial manifold (AIM) theory is used. Its explanation is one of the central goals of this paper. By means of numerical simulations for large amplitude oscillations of a fluid conveying tube we compare the results of various choices of ansatz functions and various numbers m. Further we compare the inertial manifold approximation with the flat Galerkin method which usually is used by engineers.