Unstable oscillation of tubular cantilevers conveying fluid I. Theory

Abstract

When the velocity of fluid flow in a tube, fixed at the upstream end and free at the other, is increased beyond a certain critical value, the system becomes unstable and small random perturbations grow into lateral oscillations of large amplitude. This paper is concerned with establishing the conditions of stability in the case of a system which is constrained to move in a horizontal plane. Neglecting internal friction in the material of the tube and the effect of the surrounding fluid, a universal stability curve is constructed corresponding to conditions of neutral stability and hence separating stable and unstable régimes. This is done by finding solutions to the equations of motion both by exact methods and also approximately by expressing the motion as the sum of the first few eigenfunctions of a cantilever beam. The complex frequency of the four lowest modes of the system is calculated in two representative cases for successively increasing values of the flow velocity to demonstrate how transition from stability to instability takes place.