The case of simultaneous in-plane and out-of-plane lateral vibrations of small amplitude of a horizontally rotating cantilever tube conveying fluid is examined. The rotation is with respect to the fixed end of the tube at a constant angular velocity. The diameter of the tube is constant and much smaller than its length. There is no nozzle attached at the free end. The tube is inextensible. Two inter-dependent partial differential equations in the two directions of deflection of the system are derived by means of the equilibrium of forces on a fluid-tube element. The same system of equations is derived by means of Hamilton’s principle. An approximate solution is sought in the case of linearized out-of-plane lateral vibrations in the form of a series of normalized eigenfunctions from the linear cantilever beam theory using Galerkin’s method. The threshold of instability, namely the critical dimensionless circular frequency and speed of flow at the onset of instability of the fluid-tube cantilever are investigated. The numerical results show that the externally applied constant rotational speed increases the critical circular frequency and speed of flow compared to the non-rotating fluid-tube cantilever, thus, increasing the stiffness of the system.

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