On Multimode Nonlinear Vibrations of Filled Cylindrical Shells

Kubenko, V. D.; Koval’chuk, P. S.; Kruk, L. A.

doi:10.1023/a:1023672201760

Cited by 33 publications

(21 citation statements)

References 7 publications

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“…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]: …”

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confidence: 99%

“…(2.6) depend on the material and geometrical parameters of the shell, the wave numbers, and the added masses of the fluid [11].…”

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confidence: 99%

“…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].…”

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confidence: 99%

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

Koval’chuk

Kruk

2006

Int Appl Mech

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The natural frequencies of cylindrical shells filled with a fluid and having the ends either simply supported or clamped are determined. Conditions are studied under which the natural frequencies of the shell are close or multiple Keywords: cylindrical shell, perfect viscous fluid, natural frequency, internal resonance Introduction. In designing fluid-filled elastic cylindrical shells against nonlinear forced and parametric vibration, one needs preliminary information on the spectrum of their natural frequencies that would account for the presence of the fluid. The possible closeness or multiplicity of these frequencies (internal resonances [1, 5]) creates prerequisites for strong energy coupling and interaction of different modes of the shells during vibration [4,5]. Because of this, the uncoupled (single-mode) vibration of these shells becomes unstable, and, simultaneously, complex, coupled (multimode) vibration may occur.Occurrence of internal resonances, including combinational ones, in the shell-fluid system is usually the starting point for approximating the expected dynamic deflection of shells.In the present paper, we study the frequency spectrum of cylindrical shells of finite length completely filled with a fluid. We will examine the influence of the geometry of the shell and fluid on the feasibility of internal resonances that most often occur in real shell-fluid systems vibrating with large deflections.1. To describe the dynamic behavior of a shell filled with a fluid, we use the well-known medium-deflection equations in mixed form [2, 3]:

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“…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]: …”

mentioning

confidence: 99%

“…(2.6) depend on the material and geometrical parameters of the shell, the wave numbers, and the added masses of the fluid [11].…”

mentioning

confidence: 99%

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

Koval’chuk

Kruk

2006

Int Appl Mech

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“…To construct approximate periodic solutions of the system (1.14), we will apply the asymptotic averaging method developed by Bogolyubov and Mitropolsky [3]. We will also use the results we obtained and published in [5,6,[8][9][10][11]. The solutions of Eqs.…”

Section: Periodic Solutions In the Case Of Internal Resonancesmentioning

confidence: 99%

“…(3.1) are not coupled because δ 1 = δ 2 = 0. To study the interaction of the modes at ω 1 ≠ ω 2 , it is necessary to take into account the second and higher approximations [10]. Table 1 contains the natural frequencies ω i = ω i /2π of this shell for m = 1 and different values of n. For comparison, the table includes (in the first row) the frequencies ω 0i = ω 0i /2π of the same, yet "dry" shell.…”

Section: Periodic Solutions In the Case Of Internal Resonancesmentioning

confidence: 99%

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

2005

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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]:

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Variational methods of solving dynamic problems for fluid-containing bodies

Lukovskii

2004

Int Appl Mech

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A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid-body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.Introduction. The theory of motion of bodies fully or partially filled with a fluid, which is an important division of body mechanics, is widely used to solve a large variety of applied problems. Primarily, these are problems arising in designing modern vehicles for transportation of large amounts of fluid. Such problems include the determination of the mechanical interaction of a tank car and the fluid partially filling it, strength analysis of vessels with environmentally dangerous fluids in seismic areas, stability analysis and efficient control of air/space/sea craft, etc.This paper deals only with dynamic systems consisting of a perfectly rigid body and a perfect incompressible fluid. Formulations of problems for a body interacting with an internal or external fluid are detailed in [7, 13, 15, 52, 71, 79, 93-95, etc.].Mathematical models of such mechanical systems are based on the nonlinear dynamic equations of a body and the nonlinear equations of waves on the free surface of a finite fluid. Scientists working in this area of mechanics aimed their efforts at overcoming the chief difficulty of the theory-the necessity to describe the joint motion of two essentially different objects: a body and a fluid.The motion of a body is known to be described by nonlinear ordinary differential equations, whereas the description of the wave motions of the free liquid surface involves the formulation and solution of nonlinear initial-boundary-value problems for partial differential equations. 1 2 l l J J J J t t 12 1 / , (3.32) J J J J J J 1 0. (3.34)The generalized forces will appear on the right-hand sides of Eqs. (3.32) and (3.33), where

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On Multimode Nonlinear Vibrations of Filled Cylindrical Shells

Cited by 33 publications

References 7 publications

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

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

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

Variational methods of solving dynamic problems for fluid-containing bodies

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