On Free Nonlinear Vibrations of Fluid-Filled Cylindrical Shells with Multiple Natural Frequencies

Abstract

Nonlinear free vibrations of a cylindrical shell fully filled with a perfect incompressible fluid are studied. The case is examined where two natural frequencies of the shell are close Keywords: cylindrical shell, perfect incompressible fluid, nonlinear vibrations, first integrals, traveling wave Introduction. The nonlinear vibrations of fluid-filled thin shells represented by simple (usually one-degree-of-freedom) models have been studied in sufficient detail (see [1, 2, 7, 9, 12, etc.] for review of such studies). However, such models fail to describe many complex nonlinear phenomena associated with the interaction of two or more different modes of shells filled with a fluid. The coupling of such modes may be a cause of intensive energy exchange among them, in addition to specific ("resonant") conditions. This may result in qualitatively new deformation modes differed from traditional standing waves. These new modes are, in particular, waves traveling in the circumferential direction. We also investigate a superposition of standing and traveling waves, chaotic modes, and other phenomena.The present paper sets forth a method for analysis of the free multimode vibrations with internal resonances of finite-length cylindrical shells filled with a fluid.1. Initial Dynamic Equations. For dynamic equations of a fluid-filled cylindrical shell we use mixed geometrically nonlinear equations of the theory of flexible shallow shells [14]:

On the spectrum of natural frequencies of circular cylindrical shells completely filled with a fluid

“…Here, s 1 = n 1 /R; s 2 = n 2 /R; and λ m = mπ/l. When q ≡ 0 (free vibration), the governing equations (1.5) for both cases take the following form [6,[9][10][11][12]: …”

“…Of prime practical interest are the first three resonances, since the frequency ω 3 in shells of medium length [2] is usually much greater than the frequencies ω 1 and ω 2 [12]. The frequency ω 3 can become comparable to ω 1 and ω 2 only in modes with a sufficiently large wave number n.…”

“…nonlinear with respect to the generalized displacements {f i } = { f i 01 , f i 02 , …, f i 11 , f i 12 , …} (i = 1, 2) (up to the third power inclusively[5,[7][8][9][10][11][12]); and Q nm1 2 , are constants dependent on the form of the function q 0 (x, y). It follows from (1.5) that there are always internal resonances[5].…”

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

“…Expansion (1.3) is valid in the case where the shell with fluid has no close and multiple frequencies [10,14,15]. Otherwise (if there are internal resonances [1,7,15,20]), the deflection function w should include more modes. The last term in (1.3) accounts for the effect of preferential inward buckling [3,7], which was discovered experimentally [5] and is characteristic of nonlinear vibrations.…”

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.…”