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 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
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