The paper is concerned with the forced nonlinear multimode vibrations of thin cylindrical shells fully filled with a perfect incompressible fluid. The frequency response characteristics of shells undergoing steady-state vibration as simple (standing wave) and compound (traveling wave) deformation modes are plotted and examined Introduction. The nonlinear vibrations of thin cylindrical shells filled with a fluid are addressed in many scientific studies reviewed in [6,10,11]. These vibrations were mainly analyzed using one-or two-parameter models. These models fail to describe many nonlinear phenomena with involving the interaction of several energy-equivalent flexural modes. The nonlinear interrelation of these modes and internal resonances favor intensive energy exchange between different flexural modes, thus giving rise to qualitatively new deformation modes different from ordinary standing waves. Of greatest practical interest are periodic circumferential traveling waves, standing waves with a complex spatial relief, irregular (chaotic) waves, etc.The free multimode nonlinear vibrations of fluid-filled cylindrical shells are studied in [11][12][13][14][15][16][17]. In the present paper, we use asymptotic methods of nonlinear mechanics to analyze the multimode vibrations of fluid-filled shells subject to external periodic transverse loads. Special emphasis will be given to the case where natural frequencies are in a certain (resonant) relationship. The dynamic deflection will be considered to include conjugate modes (having the same wave numbers but shifted in phase along the circumference [4,7,8]) and general modes (with different wave numbers).1. Problem Formulation. Governing Equations. Consider a circular cylindrical shell (of radius R, length l, and thickness h) fully filled with a fluid and subject to a pressure nonuniformly distributed over the lateral surface and periodically varying with time as q x y t q x y t ( , , ) ( , )cos = 0 Ω , where q x y 0 ( , )is a function of spatial coordinates x and y. The x-axis originates at one of the ends, and the y-axis is directed along the circumference. To describe the dynamic multimode deformation of the shell, we use mixed-form equations [2, 3]:
The motion and localization of bubbles in a liquid subject to two-frequency excitation are studied. Plane and spherical waves are considered. Stationary solutions are obtained and the conditions of their stability are analyzed
The motion of gas bubbles in an inhomogeneous standing wave is examined. The nonlinear system of equations is solved by the averaging method. Stationary solutions (bubble clusters) are found, and the conditions for their stability are established
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...
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