Abstract
Several mechanical models exist on elastic pipes containing fluid flow. In this paper those models are considered, where the fluid is incompressible, frictionless and its velocity relative to the pipe has the same but time-periodic magnitude along the pipe at a certain time instant. The pipe can be modelled either as a chain of articulated rigid pipes or as a continuum. The dynamic behaviour of the system strongly depends on the different kinds of boundary conditions and on the fact whether the pipe is considered to be inextensible, i.e. the cross-sectional area of the pipe is constant. The equations of motion are derived via Lagrangian equations and Hamilton’s principle. These systems are non-conservative, and the amount of energy carried in and out by the flow appears in the model.
It is well-known, that intricate stability problems arise when the flow pulsates and the corresponding mathematical model, a system of ordinary or partial differential equations, becomes time-periodic. There are several standard techniques, like perturbation method, harmonic balance, finite difference, etc., to analyze these models. The method which constructs the state transition matrix used in Floquet theory in terms of the shifted Chebyshev polynomials of the first kind is especially effective for stability analysis of large systems. The implementation of this method using computer algebra enables us to obtain more accurate results and to investigate more complex models. The stability charts are created with respect to three important parameters: the forcing frequency ω, the perturbation amplitude υ and the mean flow velocity U.
Models of random seas based on the Lagrangian equations of motion / by Willard J. Pierson, Jr.
