Stability analysis of a vertical cantilever pipe with lumped masses conveying two-phase flow

This study investigates the vibration behavior of a vertical cantilever pipe conveying gas–liquid two-phase flow, focusing on the influence of lumped masses attached to the vertical cantilevered pipe. The governing motion equation based on small deflection Euler–Bernoulli beam theory is solved by using the generalized integral transforms technique. The proposed solution approach was first validated against available numerical and experimental results in the literature. The effects of the mass ratios, number and position of lumped masses on the stability of the pipe are investigated. Numerical results show that the parameters of the lumped masses affect significantly the stability of the pipe conveying two-phase flow, by altering the fluid–structure interaction dynamics and impacting natural frequencies and vibration modes of the pipe. Specifically, as the position of a single lumped mass moves downward from the fixed end to the free end, the critical flow velocity initially increases and subsequently decreases, thereby reducing the stability of pipe. Moreover, increasing the number of lumped masses significantly impacts the critical flow velocity due to the mass ratios and locations. Notably, modal “jumping” phenomena are observed, which demonstrate continuous shifts between equilibrium and non-equilibrium states in the cantilever pipes. These findings are crucial for ensuring the safe operation of pipes with discrete masses across various engineering applications.