Dynamic stability of a spinning tube conveying a fluid through a symmetrical noncircular cross-section

When a fluid flows inside a tube, the deformations of the tube can interact with the fluid flowing within it and these dynamic interactions can result in significant lateral motions of the tube and the flowing fluid. The purpose of this report is to examine the dynamic stability of a spinning tube through which an incompressible frictionless fluid is flowing. The tube can be considered as either a hollow beam or a hollow cable. The analytical results can be applied to spinning or stationary tubes through which fluids are transferred; e.g., liquid coolants, fuels and lubricants, slurry solutions, and high explosives in paste form. The coupled partial differential equations are determined for the lateral motion of a spinning Bernoulli-Euler beam or a spinning cable carrying an incompressible flowing fluid. The beam, which spins about an axis parallel to its longitudinal axis and which can also be loaded by a constant axial force, is straight, uniform, simply supported, and rests on a massless, uniform elastic foundation that spins with the beam. Damping for the beam and foundation is considered by using a combined uniform viscous damping coefficient. The fluid, in addition to being incompressible, is frictionless, has a constant density, and flows at a constant speed relative to the longitudinal beam axis. The Galerkin method is used to reduce the coupled partial differential equations for the lateral motion of the spinning beam to a coupled set of 2N, second order, ordinary differential equations for the generalized beam coordinates. By simplifying these equations and examining the roots of the characteristic equation, an analytical solution is obtained for the lateral dynamic instability of the beam (or cable). The analytical solutions determined the minimum critical fluid speed and the critical spin speeds, for a specified fluid speed, in terms of the physical parameters of the system.