Vibrations of Pipes with Internal Flow and Rigid Body Degrees of Freedom

In the present paper the nonlinear dynamics of elastic pipes conveying fluid at arbitrary flow rates are investigated. The nonlinear equations of motion are derived using a unified form of the Lagrange Equations for non-material volumes formulated by Irschik and Holl 1], see also Chapter 3 of 2]. In a first step cantilevered pipes are considered using elastic degrees of freedom combined with a Ritz-Galerkin Ansatz of arbitrary order for modelling the deformations of the pipes. The Lagrange Equations for non-material volumes include a nonzero surface integral of the kinetic energy due to the moving outlet surface at the end of the pipe. The linear equations of motion obtained from this model are then analytically investigated utilizing the corresponding Eigenvalue problem. The results are visualized in an Argand representation of the corresponding Eigenvalues of the system matrix and compared to existing results obtained by using different formulations, such as the Hamilton Principle for Open-Systems, formulated by Benjamin 4], as demonstrated by Païdoussis 5], see also chapter 3.5 of 6]. In a next step an elastic pipe with a rigid body degree of freedom combined with a Ritz-Galerkin Ansatz is modelled with one supported and one free end. The derivation of the equations of motion is performed by using a floating-frame of reference formulation which leads to a system of nonlinear second order differential equations describing the motion of the pipe. Finally, the stability of the solutions of the equations of motion for varying flow rate is studied numerically. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)