The Problem of Forced Nonlinear Vibrations of Cylindrical Shells Completely Filled with Liquid

Abstract

The forced nonlinear vibrations of a thin cylindrical shell completely filled with a liquid are studied. A refined mathematical model is used. The model takes into account the nonlinear terms up to the fifth power of the generalized displacement of the shell. The Bogolyubov-Mitropolsky averaging method is used to plot amplitude-frequency response curves for steady-state vibrations. The steady-state vibrations at the frequency of principal harmonic resonance are analyzed for stability Keywords: cylindrical shell, perfect incompressible liquid, amplitude-frequency response, stability, single-mode deformationThe nonlinear dynamic deformation of cylindrical shells partially or completely filled with a liquid is of considerable interest for the strength and stability analyses of elements of aviation and space systems, main pipeline systems, etc. [11]. The publications [4-10, 12, etc.] report on some results obtained on the assumption that the shell-fluid system either freely vibrates or experiences the action of an external periodic load. In both cases, the vibration analysis was based on simplified models that include nonlinear terms of the third order in generalized displacements. Such models are known to allow a reliable analysis of the amplitude-frequency responses (AFRs) of a shell-fluid system, no matter whether the type of nonlinearity is soft or hard. When the skeletal curves of shells posses a combined nonlinearity (soft-hard), it is necessary to use more accurate models that would account for quartic terms.Note that such compound AFRs are very frequently observed in experiments on "dry" (without liquid) shells [2]. Naturally, a similar pattern is likely for liquid-filled shells.In the present paper, we will use refined models to analyze the forced nonlinear vibrations of a liquid-filled cylindrical shell under the action of external harmonic pressure nonuniformly distributed over the lateral surface. Our main tasks are to plot and analyze the AFRs of the shell in the vicinity of the principal resonance using analytical methods (averaging method) and to study the influence of the liquid on the AFRs and the stability regions of steady-state single-mode vibrations (standing waves).1. Problem Formulation. Original Equations. Consider a liquid-filled circular cylindrical shell of length l, thickness h, and radius R. Its external lateral surface is subjected to transverse pressure described by the formula q(x, y, t) = q 0 (x, y)cosΩt, where q 0 (x, y) is a given function of spatial coordinates x and y (x is the longitudinal coordinate originating at one of the shell's ends and running along its axis, and y is the circumferential coordinate). The shell ends are assumed to be simply supported, i.e.,where w is the radial deflection of the shell (positive in the direction of the center of curvature); v is the circumferential displacement of the median surface of the shell; and M x is the bending moment.To describe the deformation of the shell, we will use mixed equations [2, 3]:

Non-linear vibrations of shells: A literature review from 2003 to 2013

“…Karagiozis et al [110] developed two numer ical models based on Donnell's non linear theory with and with out in plane inertia to study non linear vibrations of clamped fluid filled shells. In a series of papers Koval'chuk and Kruk [111,112], Koval'chuk et al [113] and Kubenko et al [114] discussed the problem of non linear forced vibrations of completely filled simply supported circular cylindrical shells using Donnell's non linear shallow shell theory and the Krylov Bogolyubov Mitro pol'skii averaging technique. Particularly, in Refs.…”

“…Particularly, in Refs. [111,112,114] a three degree of freedom model was used that included the driven and companion modes plus a quartic sine term as a function of longitudinal coordinate to describe the axisymmetric contraction of the shell, whereas in Ref. [113] a five degree of freedom model that contained four asymmetric modes was considered to study the non linear interaction of different flexural modes.…”

High order asymptotic dynamics of a nonlinearly coupled electromechanical system

“…From the analytical point of view it is worth mentioning the similarity between the nonlinearly coupled behavior in liquid-shell systems and the electromechanical system at hand. The analogy in the governing differential equations is obtained by adopting a simplified liquid-shell system in which only one liquid mode is considered [7,10].…”