Dynamics of a system of articulated pipes conveying fluid - I.Theory

Abstract

A general theory is derived to account for the free motions of a chain of articulated pipes through which there is a constant flow of incompressible fluid. It is assumed that the position of the joint at the inlet end of the chain is fixed, and that the pipes are subject to conservative forces, which might include their weights and the stresses in resilient joints between them. The motion is observed to be approximately independent of the effects of fluid friction, even when they are of considerable magnitude, and accordingly the fluid is taken to be frictionless in the theoretical model. The Lagrangian method of analysis is used, and a slightly unusual aspect of the method arises in that the complete physical system has unbounded energy; this approach to the problem incidentally provides a clear physical interpretation of the collective effects of the fluid upon the pipes. Lagrangian equations are established in a form where the main dependent variables are the potential and kinetic energies of the `finite part' of the system, i.e. the pipes and the space enclosed by them; hence an appropriate statement of Hamilton's principle is deduced. An elementary model for centrifugal pumps and turbines is noted to be a system of the present class though having only a single degree of feedom, and this is briefly considered as an illustration of energy transfer to or from the fluid. Infinitesimal motions about a state of equilibrium in which the pipes are alined are next investigated, and the stability of the equilibrium is discussed. In this connexion there appear some remarkable properties owing essentially to the fact that the hydrodynamical forces on the pipes are conservative when the outlet end of the chain is simply supported, whereas they are in general non-conservative when the end is free. Two different forms of instability are recognized; one is termed `buckling', being similar to the failure of structures under static loading, and the other consists of self-excited oscillations. It is pointed out that a continuously flexible elastic tube comprises an extreme example of the present type of system in which the number of degrees of freedom is infinite, and Hamilton's principle is shown in this case to imply that a certain partial differential equation is satisfied by the lateral displacement of a tube during bending motions; this derivation of the equation is a fairly delicate matter, and the only previous attempt at it is believed to be seriously in error. In $\S$3 the conditions of stability for a particular system with two degrees of freedom are examined in detail.