Nonlinear and chaotic vibrations of cantilevered micropipes conveying fluid based on modified couple stress theory

The aim of this paper is to develop a nonlinear theoretical model for cantilevered micropipes/microbeams conveying fluid and to explore the possible size-dependent nonlinear responses based on the modified couple stress theory. Compared to previous work, this newly developed nonlinear model can be utilized for predicting the post-instability nonlinear dynamics of fluid-conveying micro-cantilever more than its linear dynamics. By considering the geometric nonlinearities, the gravity, and the effect of loose supports at the downstream end, the nonlinear equation of motion is derived using the Hamilton's principle. The governing partial differential equation is further discretized with the aid of Galerkin's approach. Numerical results show that the fluid-conveying micro-cantilever is capable of displaying rich dynamical behaviors. For a ‘horizontal’ or ‘hanging’ micropipe conveying fluid, it is found that flutter instability occurs at a critical flow velocity, beyond which the micropipe would undergo a limit cycle motion; for a ‘standing’ micropipe with relatively long length, however, both buckling and flutter instabilities could occur. More interesting dynamical behavior has been obtained for a modified micropipe system with loose supports at its tip end. In the presence of nonlinear constraining force resulted from loose supports, construction of bifurcation diagrams with the flow velocity as variable parameter and some corresponding phase-plane portraits have shown that chaotic vibrations do indeed arise, following a sequence oof period-doubling bifurcations. It is also demonstrated that the presence of small length scale can enhance the stability of the micropipe. However, size dependence of the post-flutter responses for the system is not pronounced.