When thin cylindrical shells having freely supported end at the downstream side such as heat-shielding shells of afterburners, labyrinth air seals, annular structures in large diameter pipings and valves are subjected to axial leakage flows, an unstable vibration and a fatigue failure are apt to be occurred. In this paper, the unstable vibration of thin cylindrical shells is analytically investigated considering the fluid structure interaction between shells and fluids flowing through a narrow passage. The coupled equation of motion between shells and fluids is derived using the Flu¨gge’s shell theory and the Navier-Stokes equation. Especially, focusing on the higher circumferential vibrations, the unstable phenomenon of thin cylindrical shells is clarified by using root locus based on the complex eigenvalue analysis by using the mode functions obtained by the exact solution based on the Flu¨gge’s shell theory. The influence of shell-dimensions and so forth on the threshold of the instability of the coupled vibration of shells and flowing fluids are investigated and discussed.

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