Geometrically exact equation of motion for large-amplitude oscillation of cantilevered pipe conveying fluid

Abstract

Theoretical modeling and dynamic analysis of cantilevered pipes conveying fluid are presented with particular attention to geometric nonlinearities in the case of large-amplitude oscillations. To derive a new version of nonlinear equation of motion, the rotation angle of the centerline of the pipe is utilized as the generalized coordinate to describe the motion of the pipe. By using variational operations on energies of the pipe system with respect to either lateral displacement or rotation angle of the centerline, two kinds of new equations of motion of the cantilever are derived first based on Hamilton’s principle. It is interesting that these two governing equations are geometrically exact, different-looking but essentially equivalent. With the aid of Taylor expansion, one of the newly developed equations of motion can be degenerated into previous Taylor-expansion-based governing equation expressed in the form of lateral displacement. Then, the proposed new equation of motion is linearized to determine the stability of the cantilevered pipe system. Finally, nonlinear analyses are conducted based on the current geometrically exact model. It is shown that the cantilevered pipe would undergo limit-cycle oscillation after flutter instability is induced by the internal fluid flow. As expected, quantitative agreement between geometrically exact model and Taylor-expansion-based model can be achieved when the oscillation amplitude of the pipe is relatively small. However, remarkable difference between the results of oscillation amplitudes predicted using these two models would occur for large-amplitude oscillations. The main reason is that in the Taylor-expansion-based model, high-order geometric nonlinearities have been neglected when applying the Taylor expansion, thus yielding some deviation when large-amplitude oscillations are generated. Consequently, the proposed new geometrically exact equation of motion is more reliable for large-amplitude oscillations of cantilevered pipes conveying fluid.