Synchronous behavior and energy transfer mechanism of pipe-in-pipe system with external and internal resonances

The fluid-conveying pipe-in-pipe system is a typical coupled pipeline system, which exhibits rich dynamic phenomena under the coupling effect of two pipes and the gyroscopic effect caused by the internal flow. Complex vibration responses mean complex energy transfer between the inner and outer pipes or between different modes of the system.This study focuses on the influence of the 1:3 internal resonance of the PIP system on the synchronization patterns and the triggering conditions of complex motions. A Galerkin method based on coupled mode shapes is proposed to study nonlinear vibrations of the PIP system, and its accuracy and computational efficiency are verified. The frequency-response curves, time histories, phase portraits, and synchronization diagrams are obtained using the Runge-Kutta method and pseudo-arclength continuation technique. The results show that imperfect and failed synchronization can emerge in the system under internal resonance conditions due to energy transfer between distinct vibrational modes. The presence of internal resonance generates either double-jump resonance peaks or frequency response curves with complex interlaced patterns, leading to multi-valued regions. Furthermore, quasiperiodic and chaotic motions are observed, confirming a torus-doubling bifurcation route to chaos. This work elucidates the mechanisms underlying complex motions in PIP systems and provides critical insights for their structural optimization and design.