Three-dimensional nonlinear dynamics of a cantilevered pipe conveying fluid subjected to loose constraints

Abstract

The dynamics of pipes conveying fluid has become a hot topic in the research field of fluid-structure interactions. Perhaps one of the main reasons why the dynamics of pipes conveying fluid has remained of intense interest to dynamicists is the fact that it displays interesting and sometimes unexpected nonlinear dynamical behavior and it has become a handy tool in developing or testing modern dynamics theory. In physical terms, the dynamical system of a loosely supported pipe conveying fluid is an example of large class of problems involving self-excited oscillations and interactions with loose constraints. Hence, understanding and modeling the dynamics of such systems is of both fundamental and practical interests. The two-dimensional (2-D) vibration of a cantilevered pipe conveying fluid has been studied for a long time, but for the moment, only few studies discussed the three-dimensional (3-D) planar and non-planar dynamics of cantilevered pipes subjected to nonlinear loose constraints. This paper establishes the 3-D governing equations and explores the 3-D dynamics of a cantilevered pipe with loose constraints somewhere along its length, with consideration of the friction effect during impacting between the pipe and the loose constraints. The loose constraints consist of two parallel bars (TPBs) on both sides of the pipe in one fixed lateral direction, with free gaps between the pipe and the restraining bars. The impacting force between the pipe and loose constraints is depicted by a smoothened-trilinear spring. In the theoretical analysis, the two governing equations were discretized via Galerkin′s approach and solved using a fourth-order Runge-Kutta method. As the flow velocity becomes sufficiently high, flutter instability of the pipe occurs and limit cycle motions would be generated. When the lateral displacement of the pipe exceeds the free gap, effective impacting occurs. During impacting between the pipe and loose constraints, either static or dynamic friction forces would occur along the other lateral direction. The nonlinear behavior in two perpendicular planes and the influence of friction coefficient were analyzed with special attention. The dynamic responses of the pipe system for various internal flow velocities are exhibited in the form of bifurcation diagrams, time traces and phase plots. Comparisons of the planar and non-planar motions of the pipe with or without loose constraints are conducted for various internal flow velocity ranges. Results show that the constrained pipe is capable of displaying interesting dynamics in the presence of nonlinear impacting force induced by the loose constraints. Both 3-D periodic and quasi-periodic oscillations are observed in a wide range of internal flow velocities. It is found that the introducing of nonlinear impacting constraints slightly enlarges the internal flow velocity range for non-planar motions. It is also shown that the orientation of 2-D planar vibrations of the pipe may be changed with the increase of the friction coefficient. The results obtained in this work may be useful for further understanding other problems in fluid-structure interactions involving slender structures and axial flows, such as the dynamics of slender cylinders in axial flow and deep water risers concurrently subjected to axial and cross flows.