Three-dimensional vibration of cantilevered fluid-conveying micropipes—Types of periodic motions and small-scale effect

Abstract

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of fluid-conveying cantilevered micropipes. Particular attention is given on the derivation and analysis of the reduced equations, and the small-scale effect on the periodic motions. Based on the modified couple stress theory (MCST), the governing equations are derived by using Hamilton’s principle. The material length scale parameter and large-deflection-induced geometric nonlinearities given by the Lagrangian strain tensor are incorporated into the governing equations. Utilizing the center manifold theory, normal form method and O(2) symmetry, the original governing equations can be rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system. Then two possible types of periodic motions, i.e. planar periodic and spatial periodic motions, together with their stabilities are investigated by means of averaging methods and numerical simulations. Results show that the larger the dimensionless material length scale parameter is, the wider the region of mass ratio for stable planar periodic motion is. Particularly, the presence of small length scale parameter makes micropipes be more likely to oscillate in a plane. It is also shown that for mass ratio corresponding to the hysteresis of the curves of critical flow velocity versus mass ratio, the stabilities for bifurcating periodic motions at lower, moderate and higher critical flow velocities may be different.