Coupled free vibration analysis of a fluid-filled rectangular container with a sagged bottom membrane

doi:10.1016/j.jfluidstructs.2009.11.001

Journal of Fluids and Structures

Volume 26, Issue 2, February 2010, Pages 236-252

https://doi.org/10.1016/j.jfluidstructs.2009.11.001 Get rights and content

Abstract

In the present paper, two-dimensional coupled free vibrations of a fluid-filled rectangular container with a sagged bottom membrane are investigated. This system consists of two rigid walls and a membrane anchored along two rigid vertical walls. It is filled with incompressible and inviscid fluid. The membrane material is assumed to act like an inextensible material with no bending resistance. First, the nonlinear equilibrium equation is solved and the equilibrium shape of the membrane is obtained using an analytical formulation neglecting the membrane weight. The small vibrations about the equilibrium configuration are then investigated. Along the contact surface between the bottom membrane and the fluid, the compatibility requirement is applied for the fluid–structure interactions and the finite element method is used to calculate the natural frequencies and mode shapes of the fluid–membrane system. The vibration analysis of the coupled system is accomplished by using the displacement finite element for the membrane and the pressure fluid-finite element for the fluid domain. The variations of natural frequencies with the pressure head, the membrane length, the membrane weight and the distance between two rigid walls are examined. Moreover, the mode shapes of system are investigated.

Introduction

Various types of fluid–structure interaction problems can be found in engineering and applied sciences. The most important ones are vibration analysis of the hydraulic gates (Daneshmand et al., 2004), sloshing motion of fluid in a tank (Yuanjun et al., 2007), earthquake response of water-retaining structures, pipes conveying fluid and containers (Paїdoussis et al., 2008), wind or fluid-induced vibration of bridge girders and vibration of offshore structures (Liang and Tai, 2006; Tao et al., 2007). Predicting the coupled motions of the fluid and the structures is generally a difficult task, and in most practical problems it is not possible to obtain closed form analytical solutions for coupled systems. The analysis of the fluid-filled storage containers and rigid or flexible boxes has been of great interest to many structural engineers in recent years (Jeong, 2006; Karagiozis et al., 2005; Biswal et al., 2004; Bermudez et al., 1997). Several analytical, semi-analytical and numerical approaches have been proposed to obtain the natural frequencies of fluid-filled storage containers and the other rigid or flexible rectangular boxes.

Bermudez and Rodriguez (1994) calculated the natural frequencies of the fluid-filled squared cavity with flexible boundaries. They solved the interior elastoacoustic problem by a finite element method which does not present spurious or circulation modes for nonzero frequencies. Chiba (1994) studied the axisymmetric free vibration of a flexible bottom plate in a cylindrical tank supported on an elastic foundation. Wang and Bathe (1997) calculated the natural frequencies and mode shapes of the fluid-filled rigid box. A new theory has been investigated for the dynamics of cylindrical shell-tanks with a flexible bottom and ring stiffeners by Amabili et al. (1998). The natural frequencies of an elastic thin-plate placed into a rectangular hole and connected to the rigid bottom slab of a rectangular container filled with fluid having a free surface have been studied by Cheung and Zhou (2000). Linear hydroelastic free vibration analysis of a cylindrical container with rigid side-wall and membrane bottom has been studied by Chiba et al. (2002). They considered small amplitude coupled free vibrations of a liquid and a membrane; in their study, the static deformation of the bottom membrane is neglected because of small membrane deformation.

In spite of the extensive field experiments and research in the area of the fluid–structure interaction problems, there has been no attempt to tackle the problem described in the present paper. Most researches have concentrated too much on the calculation of the natural frequencies of the rigid or flexible storage container including the small deformation of flexible bottom. Also, the effect of static deformation of the flexible bottom has been ignored in many previous studies.

The aim of this study is to present a framework for investigating the dynamic behavior of a fluid-filled rectangular container with a sagged bottom membrane. It should be noted that the problem considered in this paper is actually different because the static deformation of the bottom membrane is really large, and so more attention should be paid on it. Some physical applications, such as membrane roofs and floating roofs in petroleum container may be mentioned for the present study. Moreover, the proposed approach for the static and dynamic analysis can be used for static and dynamic study of the geomembrane tubes and foundations of the emergency bridges (Ghavanloo and Daneshmand, 2009) and breakwaters (Phadke and Cheung, 2003).

In the present paper, the natural frequencies and mode shapes of the coupled system are obtained. This system can be modeled as a sagged membrane which is anchored along two rigid vertical walls. The fluid inside the container is assumed to be incompressible and inviscid. The length of the membrane is longer than the distance between two walls. An inextensible membrane with no bending resistance is considered here. Because of inextensibility assumption, the tangential strain is equal to zero. A two-dimensional (plane-strain) problem is treated in which the membrane is an infinite cylinder with generators parallel to the horizontal plane. In other words, the membrane itself is assumed to be long and straight and the changes in cross-sectional area along the membrane length are neglected. These two assumptions facilitate the use of a two-dimensional model. To determine the equilibrium configuration of the membrane, the specific weight of the membrane is assumed to be negligible with respect to the specific weight of the fluid.

In this analysis, the nonlinear differential equations are solved analytically and the equilibrium shape of the membrane is obtained. Then, the linear, two-dimensional coupled vibration of the fluid–membrane system about the equilibrium shape is investigated. The vibration analysis of the coupled system is accomplished by using the displacement finite element for the membrane and the pressure finite element for the fluid domain. The first six frequencies and mode shapes of the coupled system are determined for different conditions. The effects of the pressure head, the membrane length, the membrane weight and the distance between two rigid walls on the natural frequencies are elucidated. Moreover, we will propose an approximate relationship between the natural frequencies of the fluid–membrane system and the pressure head when the other parameters are fixed.

Section snippets

Problem statement

We consider the problem of determining the small amplitude motions of an inviscid and incompressible fluid contained in a rectangular container with a sagged bottom membrane. Let Ω_f and Γ_s be the domains occupied by the fluid and the membrane, respectively, as shown in Fig. 1. Let us denote by Γ₀ the free boundary of the fluid and by Γ_s the interface between the membrane and the fluid.

Our objective is to obtain the governing equations of the membrane under the influence of the fluid pressure.

Static analysis

Considering the time-independent terms in Eqs. (2a), (2b) and neglecting the membrane weight with respect to the fluid weight, the following nondimensional governing equations for the static analysis can be obtained: $\frac{d x_{e}}{d s} - \cos (ψ_{s}) = 0, \frac{d y_{e}}{d s} - \sin (ψ_{s}) = 0,$ $\frac{d n}{d s} = 0, n \frac{d ψ_{s}}{d s} - (h_{f} + y_{e}) = 0,$ where x_e and y_e are the coordinates of the membrane in equilibrium position. ψ_s is the tangential angle of the membrane with the horizontal direction in the equilibrium configuration. For determining the equilibrium shape of

Solid domain

In order to analyze the vibrations of the membrane about its equilibrium configuration, the equations of motion in the tangential and normal directions can be written in nondimensional form as $η \frac{\partial^{2} v_{t}^{d}}{\partial t^{2}} = \frac{\partial (n_{d} + n_{0})}{\partial s} = \frac{\partial n_{d}}{\partial s},$ $η \frac{\partial^{2} v_{n}^{d}}{\partial t^{2}} = (n_{d} + n_{0}) \frac{\partial ψ}{\partial s} - \hat{p} = (n_{d} + n_{0}) \frac{\partial ψ}{\partial s} - (p_{s} + p_{d}),$ where $v_{t}^{d}$ , $v_{n}^{d}$ and n_d are tangential displacement, normal displacement and additional tension in the membrane, respectively. To eliminate n_d, one can solve Eq. (22b) for n_d, differentiate the result with respect to s, substitute it

Weak formulation

For developing the weak form of Eqs. (29), (32), we multiply the equations (29) and (32) by the test functions ξ and ζ such that ξ=0 and ∂ξ/∂s=0 at s=0,l, ζ=0 on Γ₀ and integrate over the element domains. Using integration by parts on some of the terms and the boundary conditions we obtain $\begin{matrix} \int_{s_{1}^{e}}^{s_{2}^{e}} (\frac{η}{n_{0}} ({(A^{e})}^{2} ξ \frac{\partial^{2} v_{t}^{d}}{\partial t^{2}} - 2 {(A^{e})}^{- 1} (B^{e}) \frac{\partial ξ}{\partial s} \frac{\partial^{2} v_{t}^{d}}{\partial t^{2}} + \frac{\partial ξ}{\partial s} \frac{\partial^{3} v_{t}^{d}}{\partial t^{2} \partial s}) + \frac{\partial^{2} ξ}{\partial s^{2}} \frac{\partial^{2} v_{t}^{d}}{\partial s^{2}} - a_{1}^{e} \frac{\partial ξ}{\partial s} \frac{\partial^{2} v_{t}^{d}}{\partial s^{2}} \\ - a_{2}^{e} \frac{\partial ξ}{\partial s} \frac{\partial v_{t}^{d}}{\partial s} + a_{3}^{e} ξ \frac{\partial v_{t}^{d}}{\partial s} + a_{4}^{e} ξ v_{t}^{d}) d s \\ + ξ (\frac{η}{n_{0}} (2 {(A^{e})}^{- 1} (B^{e}) \frac{\partial^{2} v_{t}^{d}}{\partial t^{2}} - \frac{\partial^{3} v_{t}^{d}}{\partial t^{2} \partial s}) + \frac{\partial^{3} v_{t}^{d}}{\partial s^{3}} + a_{1}^{e} \frac{\partial^{2} v_{t}^{d}}{\partial s^{2}} + a_{2}^{e} \frac{\partial^{2} v_{t}^{d}}{\partial s}) \\ - A^{e} p_{d}) \end{matrix}$

Numerical results and discussion

A computer program based on finite element method is written according to the mathematical description given in the previous sections. In this section, the first six natural frequencies and mode shapes of a fluid–membrane system are obtained. The effects of different parameters including the membrane weight, the membrane length and the fluid level are also studied.

Concluding remarks

Two-dimensional coupled free vibrations of a fluid-filled rectangular container with a sagged bottom membrane were analyzed in this paper. The membrane material was assumed to be inextensible, and its weight was neglected in the computation of the equilibrium state. The container was filled with an incompressible and inviscid fluid. To facilitate the use of a two-dimensional analysis, the system was assumed to be extremely long and straight.

The governing equations for small vibrations about the

Acknowledgments

The authors would like to sincerely thank Professor Païdoussis for his comments and suggestions. We also thank the anonymous reviewers for their valuable comments on the manuscript.

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¹: Currently, Visiting Professor, Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street W., Montreal, Québec, Canada H3A 2K6.

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Coupled free vibration analysis of a fluid-filled rectangular container with a sagged bottom membrane

Abstract

Introduction

Section snippets

Problem statement

Static analysis

Solid domain

Weak formulation

Numerical results and discussion

Concluding remarks

Acknowledgments

Journal of Sound and Vibration

Computer Methods in Applied Mechanics and Engineering

Journal of Sound and Vibration

Journal of Fluids and Structures

Journal of Sound and Vibration

Journal of Sound and Vibration

European Journal of Mechanics A/Solids

Journal of Fluids and Structures

Journal of Fluids and Structures

Ocean Engineering