Studying the forced vibrations of fluid-filled cylindrical shells with regard to the nonlinear interaction of different flexural modes

Abstract

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]:

Nonlinear vibrations of cylindrical shells filled with a fluid and subjected to longitudinal and transverse periodic excitation

“…The superposition principle fails here because of the nonlinearity of such a problem formulation: the response of the dynamic shell-fluid system to a combination of longitudinal and transverse periodic loads is not the sum of the responses of this system to the individual loads. Here we may expect qualitatively new nonlinear effects that were not observed in the partial problems of forced [7,[10][11][12][13][14]18] or parametric [2,4,11,16] vibrations of shell-fluid objects.The present paper sets out to develop a method and to apply it to study the nonlinear deformation of elastic cylindrical shells filled with a fluid and subjected to longitudinal-and-transverse periodic excitation. We will primarily analyze the dynamic behavior of filled shells in the worst (with respect to dynamic stress) case where the natural frequencies of the shell-fluid system are in resonance relations with the frequencies of both external periodic forces.…”

“…The superposition principle fails here because of the nonlinearity of such a problem formulation: the response of the dynamic shell-fluid system to a combination of longitudinal and transverse periodic loads is not the sum of the responses of this system to the individual loads. Here we may expect qualitatively new nonlinear effects that were not observed in the partial problems of forced [7,[10][11][12][13][14]18] or parametric [2,4,11,16] vibrations of shell-fluid objects.…”

Stability of elastic cylindrical shells interacting with flowing fluid

“…The problems of stability and vibrations of thin cylindrical shells interacting with a fluid moving inside them are of substantial interest for the dynamic strength and operational reliability of various pipeline systems. The complexity of the formulations and solution of such problems is determined by a number of factors, which were partially discussed in [2,4,6,[9][10][11][13][14][15]. In the general case, such problems should be given a nonlinear formulation taking into account the geometrical nonlinearity of the shells and nonlinear damping, which would allow a more adequate description of the dynamic deformation both during and after buckling.…”

Numerical study of the dependence of the critical flutter speed and time of a plate on rheological parameters

“…The second-order equations of the theory of thin shallow shells [2,4,11,12] will be used. In the modern literature, these equations go by the name of Marguerre [5] who proposed them to analyze thin-walled structures such as aircraft wings for strength, stiffness, and stability.…”