Periodic motion of microscale cantilevered fluid-conveying pipes with symmetric breaking on the cross-section

Abstract

A new theoretical model is developed for the three-dimensional (3D) nonlinear vibration analysis of cantilevered fluid-conveying micropipes with asymmetric cross-section. Fluid-conveying microscale pipes with ring-shaped cross-sections of O(2) symmetry are widely used in engineering. However, their cross-sections usually have a small symmetric breaking owing to manufacturing errors, which can be described by a small parameter. Hence, the dynamic behaviors of such geometrically imperfect micropipes are extremely required to be explored. This study mainly investigates the effects of the symmetric breaking parameter, flow velocity and dimensionless material length scale parameter on the dynamic behaviors of microscale cantilevered fluid-conveying micropipes. Based on the modified couple stress theory (MCST) and Hamilton's principle, the governing equation is derived, where the effects of the asymmetry of cross-section on the equation are reflected. Using the center manifold theory and normal form method, the original governing equation is rigorously reduced to a two-degree-of-freedom (2DOF) dynamical system, and the averaging equation of the reduced system is obtained by combining the averaging method. The existence and stability of the equilibrium point of the averaging equation are studied to obtain the results of the existence and stability of periodic motions of the original system. The results demonstrate that the types of periodic motions and stability when the pipe loses stability from different places on the curve of the critical flow velocity versus mass ratio (critical flow velocity curve) may differ. Meanwhile, the effects of various parameters on these periodic motions are presented. The increase in the dimensionless material length scale parameter increases the section corresponding to stable plane periodic motion and reduces the section corresponding to stable spatial periodic motion on the critical flow velocity curve. When dimensionless material length scale parameter is given, the existence and stability of various periodic motions of the system depend on the ratio of the increment of flow velocity around the critical value to the symmetric breaking parameter. In addition, some new phenomena are found in this paper. Stable 0 planar periodic motion of the macropipe can occur when pipe loses its stability from the hysteresis part of the critical flow velocity curve. Stable 0 planar periodic motion of microscale pipes can occur when pipe loses its stability from the non-hysteresis part of the critical flow velocity curve, and then stable 0 planar periodic motion and stable π planar periodic motion can coexist. Furthermore, stable planar and stable spatial periodic motions of pipes with symmetric breaking can coexist, and stable torus motion has also been observed; this is different from the dynamics of pipes with O(2) symmetry.