The straight pipe–curve fluid theory is further developed regarding in-plane nonlinear vibrations of curved pipes conveying fluid in this paper. Based on this theory, the finite element model of a U-shaped pipe conveying fluid with complex constraints is constructed by using the finite element method, and the chaotic motion characteristics of the system are examined. The results show that the chaotic characteristics of the U-shaped pipes conveying fluid with a free-end outlet deviate from traditional understandings under certain constraints, in that their chaotic motion can be triggered by inherent nonlinearity, notably the cubic nonlinearity associated with axial forces. Following a Hopf bifurcation, static deformation’s impact becomes significant, precipitating an earlier onset of Pitchfork bifurcation and chaos, and significantly altering the bifurcation diagram’s topological structure. The introduction of motion constraints and increasing collision stiffness leads to notable changes in the bifurcation diagram’s topology, with the chaotic region’s width initially widening before narrowing.
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