Hopf Bifurcation of a Standing Cantilever Pipe Conveying Fluid with Time Delay

In this article, the dynamical behavior of a standing cantilever pipe conveying fluid is investigated with time 
 
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 delay. By applying piezoelectric materials and considering the time delay of voltage, the motion equation is built with motion-limiting constraints and elastic support. The motion equation is discretized into ordinary differential equations by the Galerkin method. A stability analysis of the equilibrium point with three parameters is obtained. The system will lose stability at the equilibrium point and generate Hopf branches. The central manifold theorem and the canonical type theory are used to study the stability of Hopf branching directions and branching period solutions. Finally, numerical results validate the theoretical analysis.