Articulated Models of Cantilevers Conveying Fluid: The Study of a Paradox

Abstract

A general theory is presented for the dynamics of nth-degree-of-freedom articulated (lumped flexibility) models of cantilevers conveying fluid, of which the two-degree-of-freedom model of a column subjected to follower forces (first investigated by Ziegler) is a particular case. The ability of the articulated system to predict the dynamical behaviour of the continuous system modelled is investigated, and in particular the paradox that, whereas the continuous system is subject to only oscillatory instability (at sufficiently high flow), the model is generally subject to both oscillatory and buckling instabilities, and sometimes only to the latter. Complex frequency calculations show that buckling is associated with the higher modes of the articulated system, which, irrespective of the number of degrees of freedom, do not model well the corresponding modes of the continuous system. The critical flow velocities for buckling and oscillatory instabilities are calculated extensively, the latter showing good convergence to the corresponding values of the continuous system. The theory is supported by a set of experiments. Agreement between theory and experiment is satisfactorily good.

Flutter instability of freely hanging articulated pipes conveying fluid

“…With these assumptions, the calculation of the lagrangian does not present particular difficulties, details can be found in [1] or [3]. For convenience we introduce the constant angle φ 0 = 0 (thenφ 0 =φ 0 = 0) in such a way that the different terms write…”

“…The latter shows that the works devoted to articulated pipes have mainly focused on two-segment pipes and that the experimental results are fairly rare and incomplete. It should be pointed out that while the seminal works of Benjamin [1,2] and the study of PaÏdoussis and Deksnis [3] were undertaken to deduce general statements about the stability of pipes conveying fluid, subsequent studies, including the present one, were mainly motivated by the fact the articulated pipes are examples of dynamical systems governed by ordinary differential equation system which can then be easily integrated and, moreover, that are amenable to experimental investigations. In the present study, we perfom experiments with articulated pipes made of 2 to 5 segments, our experimental setup is presented in §II.…”

“…Benjamin noted that the occurence of flutter, resulting from growth of small perturbations to attain limit cycle oscillations, requires a net transfer of energy from the fluid to the pipe during this phase. Later, Païdoussis et Deksnis [3] generalized the linear stability analysis of Benjamin to an arbitrary number of segments to study the transition, as n is increased, between articulated pipe and continuous flexible pipe. They found that the critical velocities and frequencies for the flutter of articulated pipes converge to the values of the continuous system as n is increased.…”

Flow-Induced Vibration of Circular Cylindrical Structures

“…However, the characteristics of an articulated pipe are not necessarily the same as those of a continuous pipe (Paidoussis and Deksnis 1970). The static deformation increases with flow velocity; it is caused by fluid centrifugal force.…”

Three routes to chaos for a three-degree-of-freedom articulated cylinder system subjected to annular flow and impacting on the outer pipe

“…This derivation draws upon previous work on articulated systems by Paidoussis and Deksnis [10] and Paidoussis [11]. The forces associated with the structure itself, i.e.…”