Stability of cylindrical shells with added mass in fluid flow

Abstract

The paper proposes a technique for the stability analysis of an elastic cylindrical shell with added mass interacting with a flowing fluid. Three cases of mass-shell contact are considered: the added mass (i) is attached to the shell at one point, (ii) has the form of a circular ring, and (iii) is distributed uniformly over the length. For each case, equations to determine the critical speeds of the fluid corresponding to the quasistatic (divergent) and dynamic (flutter) buckling of the shell are derived Keywords: cylindrical shell, ideal incompressible fluid, added mass, critical speed, stability, divergence, flutterIntroduction. Some sections of pipelines quite often have local "inclusions" such as rigidly attached concentrated masses. A mass may be attached to a structure at a point or linearly distributed over a shell surface. Such "inclusions" may significantly affect both quasistatic (divergence) and dynamic (flutter) buckling modes of pipelines interacting with the fluid flowing inside them [4,5].The present paper outlines a technique for the stability analysis of a finite pipeline section modeled by an elastic cylindrical shell loaded by a small added mass and filled with a moving fluid. Three cases of mass-shell contact are considered: (a) the mass is concentrated at one point of the free surface; (b) the mass is uniformly distributed over a circular ring contacting with the shell; (c) the mass is distributed along the length. Problem Formulation. Starting Equations of Motion of a Shell with a Concentrated Mass.Consider an elastic closed cylindrical shell containing a fluid moving with a constant speedU. The geometry of the shell is shown in Fig. 1. A point mass M is rigidly attached to the shell at a point with coordinates x 0 and y 0 . We will introduce the following assumptions and restrictions, which are usually used to solve most dynamic problems for shells with concentrated masses [1, 5-8, etc.]. Since the tangential rigidity of the shell is much higher than its radial rigidity (especially at lower vibration frequencies), the tangential inertial forces can be neglected. Assume that the added concentrated mass transmits only the radial force to the shell. The rotational inertia of the mass showing up during the vibrations of the shell is also neglected. The fluid filling the shell is ideal and incompressible, and its motion is potential.If there is no fluid, the problem of the concurrent vibrations of the shell and added mass is usually solved by determining the natural frequencies and modes by the well-known method of separation of motions [1, 6-8, etc.]. This method "separates" the mass from the shell and models its effect by some concentrated reaction force. The displacements of the shell caused by this force are determined from the equations of its motion. Finally, the frequency equation is derived from the displacement compatibility conditions to find the frequency spectrum of the discrete-continuous system under consideration. The frequencies can be used to identify the natural modes of the sh...