Stability, Bifurcations And Chaos In Pipes Conveying Fluid

In this paper, the stability and nonlinear dynamics of pipes conveying fluid, both for straight and curved pipes, are investigated. Particular attention is concentrated on the complex bifurcation routes and chaotic motions in various system parameter regions of several pipe models. Two effective numerical methods so-called DQM and GDQR are proposed to analyze these pipe models. Based on theory analysis and numerical calculations, a series of important phenomena were obtained and some valuable conclusions put forward. Especially, some important nonlinear dynamics shown in this paper have not been examined before. In contrast to the works by formerly investigators, the present paper represents several features as follows:1. The nonlinear dynamics of pined-pined straight pipes conveying pulsating fluid is further studied. Previous work has shown that the pined-pined straight pipes conveying pulsating fluid may be subjected to parameter resonances when the mean fluid velocity is low and the vibrations were shown to be periodic or quasi-periodic motions in the vicinity of stability boundaries. In this work, the pined-pined pipe model and a variant of this system with higher mean fluid velocity are considered and complex bifurcations are examined for the pipe system. Moreover, chaotic motions are also detected in several parameter spaces of the system. These results extend the understanding of the dynamical mechanism for the pined-pined straight pipes conveying pulsating fluid.2. It is successful to extend the Differential Quadrature Method (DQM) to the nonlinear analysis of straight pipes conveying fluid. For the straight pipe conveying fluid with motion constraints, the equation of motion is a high order partial differential one. Thus two adjacentΔ-points at two boundaries of the pipe are introduced to apply the boundary conditions for the DQM. Numerical simulations show that, the results calculated by DQM represent reasonable agreement with those by the Galerkin method. Hence, the application field of DQM is effectively extended. Further, this work supplies a theory base for the DQM applied in the nonlinear problems of curved pipes conveying fluid.3. The nonlinear dynamics of a curved pipe conveying fluid and subject to motion constraints are investigated. In terms of the balances of the forces acting on elements of the pipe and fluid, the nonlinear equation of motion for the curved pipe with motion constraints, which has a six order partial differential form, was derived. The nonlinearity in this equation is associated with the nonlinear impact force due to the motion constraints. After the discretazion by DQM, the nonlinear dynamical equations obtained for the system are further solved via numerical iteration technique. It is shown that there exist complex bifurcation branches and chaotic motions in several key parameter spaces. The route to chaos is examined to be via period-doubling bifurcations.4. The forced vibrations of a curved pipe conveying fluid and subject to motion constraints are studied further. For the applying of DQM, the harmonic excitation at the free end of the pipe need be considered and the dynamical equations formed. Numerical analysis shows that it has significant difference between the curved pipes with and without harmonic excitation. For the pipe with harmonic excitation, the routes to chaos are shown to be via both period-doubling bifurcations and quasi-periodic course.5. The nonlinear dynamics of a curved pipe conveying fluid and subject to nonlinear foundations are investigated. Based on the balances of the forces acting on elements of the pipe and fluid, considering the effect of nonlinear foundations, the nonlinear equation of motion for the system is derived. This equation is further discretized via DQM and transformed into a set of nonlinear dynamical equations for the system. Numerical calculations have shown that there exist chaotic transients in such a curved pipe system. Numerical calculations have further detected three final steady vibrations in this system. However, when the initial calculating conditions are varied even a bit, the final vibrations of the system may change from one to another one.6. The General Differential Quadrature Rule (GDQR) is extended to analyze the vibrations and stability of pipes conveying fluid. In contrast to DQM, GDQR can accurately apply the boundary conditions for the system without adjacentΔ-points at boundaries. Numerical calculations indicate that, GDQR is an effective and valid method to discrete the equation of motion for the pipes conveying fluid. It is easy to deal with the boundary conditions while the accuracy of the results obtained is perfect. Thus this method can meet the requirement in engineering application with less computation.From these contents listed above, it has bended itself to search for high effective numerical methods in such a fluid-solid interaction dynamical pipe model. Moreover, the stability and nonlinear dynamics of pipes conveying fluid are represented. Especially, the bifurcations and chaotic mechanism obtained in the pined-pined straight pipe and curved pipe conveying fluid might have not been examined before. The conclusions obtained in this study are significant for the design of pipes conveying fluid.