Coupled hydroelastic vibrations of an elliptical cylindrical tank with an elastic bottom*

HASHEMINEJAD Seyyed M., TAFANI Mostafa

Acoustics Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran, E-mail: hashemi@iust.ac.ir

Coupled hydroelastic vibrations of an elliptical cylindrical tank with an elastic bottom*

HASHEMINEJAD Seyyed M., TAFANI Mostafa

Acoustics Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran, E-mail: hashemi@iust.ac.ir

(Received November 17, 2012, Revised January 8, 2014)

An exact three-dimensional analysis based on the linear potential theory and the elaborated method of eigenfunction expansion in elliptic coordinates are presented to study the free coupled elasto-hyrodynamic characteristics of an upright non-deformable cylindrical container of elliptical planform with a flexible bottom plate, filled to an arbitrary depth with an inviscid incompressible liquid. Extensive numerical data are presented in an orderly fashion for the first few symmetric/anti-symmetric coupled hydroelastic natural frequencies as a function of fluid depth parameter for two plate aspect ratios. Also, selected hydrodynamic and structural deformation modes shapes are presented in graphical form. The effects of liquid level, bottom plate elasticity, and cross sectional aspect ratio on the sloshing frequencies and hydrodynamic pressure modes are examined. The validity of the results is examined through computations using a commercial finite element package as well as by comparison with the data available in literature.

exact solution, fluid-structure interaction, free surface oscillations, vertical elliptical vessel, Mathieu functions

Introduction

The dynamics of the free liquid surface or “liquid sloshing” is a fascinating physical phenomenon that has been an important area of research in regard to stability, safeguarding, and structural integrity of partially filled externally excited liquid storage tanks as well as moving cargo vessels with many applications in a wide range of technologies and engineering disciplines (e.g., pressure vessels, storage tanks, dams, reactors, nuclear vessels, trucks, ships, rockets, aircrafts, and spacecrafts[1]). Also, the coupled vibration analysis of flexible tank-liquid systems has turned into an interesting field of study in regard to many complex engineering problems. In particular, the coupled elastohyrodynamic analyses of partially-filled cylindrical containers with flexible components (walls, bottoms, covers, or internal baffles) have been the subject of several studies over the past few decades. For example, Chiba[2]carried out theoretical and experimental studies on the coupled hydroelastic free axisymmetric vibrations of a liquid-filled rigid upright circular cylindrical container with an elastic bottom, taking into account the in-plane forces in the plate due to the static liquid pressure. Amabili and Dalpiaz[3]used the Rayleigh-Ritz method to study the free vibrations of elastic base plates in partially-filled vertically standing circular and annular cylindrical tanks with ring stiffeners. Bauer and Chiba[4]investigated the coupled motion of a viscous liquid in an upright rigid circular cylindrical tank with an elastic bottom (a flexible membrane or an elastic plate), including the free surface tension effect. Cheng and Zhou[5]used the separation of variables and Galerkin methods to investigate the hydroelastic vibration of a flexible thin plate placed into a circular hole and elastically connected to the rigid bottom slab of a partially filled rigid circular cylindrical container. Chiba et al.[6]carried out a linear free hydroelastic vibration analysis of a frictionless liquid with a free surface contained in a rigid cylindrical tank with a flexible (membrane) bottom. Biswal et al.[7]used a coupled liquid-structure finite element approach to study effects of the tank wall and annularbaffle flexibility on the natural sloshing frequencies and dynamic response of a partially filled upright cylindrical tank. Askari and Daneshmand[8]proposed a semi-analytic method based on Love’s thin shell theory and eigenfunction expansions to investigate the free vibration of an elastic bottom plate of a partially fluidfilled rigid cylindrical container with an internal body in form of a rigid cylindrical block that is concentrically and partially submerged inside the container. Biswal and Bhattacharyya[9]employed a finite element approach along with Newmark’s time integration scheme to study the coupled response of a partially filled upright elastic circular cylindrical tank-baffle system fabricated from lightweight laminated composite materials under translational base excitation. Shekari et al.[10]used finite shell elements for the tank structure and boundary elements for the liquid region to investigate the seismic response of upright elastic cylindrical liquid storage tanks under long-period ground motions and isolated by bilinear hysteretic bearing elements. Bohun and Trotsenko[11]developed an analytic modal solution for coupled free hydroelastic oscillations in an upright circular cylindrical container with the unperturbed free surface of the fluid being covered by an elastic membrane or plate. Amiri and Sabbagh-Yazdi[12]both experimentally and numerically (FEM-Ansys) studied the dynamic characteristics (natural frequencies and mode shapes) of fixed roof, ground supported, tall cylindrical liquid storage tanks. Moslemi and Kianoush[13]proposed a rigorous three-dimensional finite element procedure to study the dynamic behavior of open-top ground-supported partially filled shallow and tall flexible cylindrical tanks. The effects of a wide range of parameters such as tank wall elasticity, vertical component of earthquake, sloshing of liquid free surface, tank aspect ratio, and base fixity condition on dynamic behavior were addressed. Matsui and Nagaya[14]proposed a hybrid analytical and finite element method to investigate the nonlinear sloshing in a rigid-walled cylindrical oil storage tank with a single-deck-type elastic floating roof subjected to long-period seismic ground motion. Noorian et al.[15]utilized a reduced order BEM/FEM model to study the interaction of sloshing and structural vibrations of cylindrical tanks with arbitrarily arranged flexible baffles. Wang et al.[16]employed the mode superposition method to study liquid sloshing response in a rigid cylindrical container with multiple annual rigid baffles under lateral harmonic and seismic excitations. Nicolici and Bilegan[17]investigated the wall elasticity effect on the free surface motion in a partially filled vertical cylindrical storage tank subjected to ground acceleration based on fully coupled 3D CFD/FEM models developed in the Ansys workbench environment.

The above review clearly indicates that while there exists a notable body of literature that addresses the hydroelastic effects in partially-filled upright circular cylindrical containers, rigorous theoretical or numerical solutions for an upright cylindrical container of elliptical planform (with or without structural flexibility effects) seems to be nonexistent. The primary purpose of the current paper is to employ the classical linear potential theory and modal expansions in terms of radial and angular Mathieu functions to fill this gap. Particular attention is paid to the effects of tank cross sectional aspect ratio, liquid fill depth, and bottom plate elasticity on the natural sloshing characteristics. The proposed model is of noble interest essentially due to its inherent value as a canonical problem in hydroelasticity. It can be of practical value in seismic analysis of ground-supported upright elliptical tanks[18]. It can serve as an alternative to the numerical (experimental) methods, which may encounter a number of drawbacks such as time-consuming modeling process and numerical computations. Also, the presented analytic solution can provide a valuable benchmark for comparison to other solutions obtained by strictly numerical or asymptotic approaches.

1. Formulation

The elliptic coordinate system used is defined by the linear transformation x+iy=ccosh(ξ+iη) (i=-1) where ξ∈[0,∞) is the radial coordinate, η∈ [0,2π] is the angular coordinate, and the “ξ ,η= constant” curves form a family of ellipses and hyperboles, where the foci of ellipse is located at (x=±c, y=0) (see Fig.1(a)).

Fig.1(a) The elliptic coordinate system

The edge of elliptical plate is signified by ξ=ξ0, with the semi-major axis a=ccosh ξ0, and the semiminor axis b=csinh ξ0. Also, the liquid-tank system is composed of a rigid-walled vertically standing cylindrical elliptical tank with an elastic elliptical bottom plate with the mass density ρ, the thickness h,Young’s modulus E, and Poisson’s ratio ν, coupled with the incompressible and inviscid filling liquid (density0ρ) of depth L, as depicted in Fig.1(b).

Fig.1(b) Problem configuration

According to the classical small-deflection plate theory, the differential equation of motion governing the transverse displacement response of the thin elastic bottom plate, w, subjected to arbitrary distributed time-dependent transverse net external forces per unit area, P(ξ ,η,t), with clamped boundary conditions, is written as[19]

where t is the time, and D=Eh3/12(1-ν2) is the flexural rigidity of the plate. Following the standard free vibration analysis procedure and assuming timeharmonic motions of frequency Ω, one may expand the displacement of elastic plate in a series of admissible (in-vacu) eigen-functions, in the form[20]

Where the superscripts SS, SA, AS and AA, respectively stand for: symmetrical mode about both the major and minor axes, symmetrical mode about the major axis and antisymmetrical about minor axis, antisymmetrical mode about the major axis and symmetrical about the minor axis, and antisymmetrical mode about both the major and minor axes. Also, Xm

kare unknown modal coefficients, k signifies any of SS, SA, AS and AA modes, and the course of action for obtaining the (in-vacu) natural frequencies and the corresponding elastic vibrational modes of the elliptical plate is outlined in the Appendix (see also Ref.[20]).

The irrotational flow induced within the tank fluid due to the plate motion may be described in terms of a velocity potential that satisfies the Laplace equation ∇2φ( ξ ,η,z, t)=0, which after assuming time-harmonic dependence of the form φ( ξ ,η,z, t)= φ( ξ ,η,z )iΩeiΩt[8], is written in the elliptical coordinate system of Fig.1(a) in the form[21]

Next, by going through the usual separation of variable procedure, the axial, radial, and angular components of the potential solution, (,,z) φ ξ η, may respectively be written in terms of transcendental, and radial/ angular Mathieu functions as[21]

Consequently, using the spatial solutions (4), while keeping the boundary condition (5) in mind, the fluid velocity potential may be expanded as

Subsequent application of the dynamic and kinematic boundary conditions at the free surface of the liquid implies[22]

where Γ( ξ ,η,t) denotes the free liquid surface displacement, and g is the gravitational acceleration. Directly substituting the field expansions (2) and (6) into the compatibility conditions (7) and (8), and making use of the general orthogonality relation,)dA =0 (n, m≠i, j) between the distinct characteristic modes, where A represents the cross sectional surface area of the elliptical tank, after some manipulations, one obtains the key relationships between the unknown coefficients () andin the form

Table 1(a) Calculated dimensionless system natural frequencies for a partially water-filled, circular tank with a thin elastic bottom plate

Now, in the absence of the external loads, the net pressure acting on the elastic plate, P(ξ ,η,t), which stems from the sloshing liquid, is written as[22]

Directly substituting the expansions (2) and (6) into Eq.(10), subsequently substituting the result into the equation of motion of plate (1), and making use of the orthogonality properties of the eigenfunctions(ξ, η), while keeping the linear relation (9) in mind, along with the fact that the normal (in-vacu) modesmust satisfy the classical relation=

k SA, AS, and AA modes, and the (Μ×Μ) coefficient matrix Rk(Ω) is given asThus, the coupled system natural frequencies,(where l is the root number), and the unknown modal coefficients,, along with {} can directly obtained from (12), after finding the roots of characteristic equation=0. This completes the nece-ssary background required for complete analysis of the problem. Next we consider some numerical examples.

Table 1(b) Calculated system natural frequencies for a partially water-filled circular tank with a thin elastic bottom plate

Fig.2(a) The first few liquid free surface displacement and bottom plate structural mode shapes of the circular tank system for selected fluid column depths

Fig.2(b) The first few liquid free surface displacement and bottom plate structural mode shapes of the elliptical tank system for selected fluid column depths

2. Numerical examples

In order to examine the general character of thesolution, a number of numerical examples are considered in this section. Realizing the wide range of input geometric and material parameters, while keeping in mind the intense computations involved here, we shall not attempt an exhaustive presentation of numerical results and will thus confine our attention to a particular model. From the collection of data presented here, certain trends are noted and general conclusions are reached about the relative importance of certain parameters. Accordingly, thin clamped elliptical steel plate (ν=0.3, ρ=7 860kg/m3, E=2×1011N/m2, b=5 m, h=0.01 m) of selected aspect ratios (a/ b=1,2), coupled to water columns (ρ=1000 kg m3)of sele-

0 cted depths (L/2b=0.1,1,2), are considered. Two distinct general MAPLE codes were constructed for calculating the natural frequencies of coupled system and the associated mode shapes of circular (a/ b= 1.001), and elliptical (a/ b=2), tanks with rigid or elastic bottom plates. The root findings were accomplished by implementing the standard bisection technique, while the numerical integrations were performed by employing the two dimensional Gaussian quadrature method. The convergence of numerical calculations was ensured by increasing the truncation size, M, while looking for steadiness in the numerical value of the computed natural frequencies. The value of M=10 was found to yield satisfactory results (i.e., four decimal points convergence) in all cases considered.

Table 2(a) The first few calculated circular tank coupled system natural frequencies for selected fluid column depths

Before presenting the main results, we shall first demonstrate the overall validity of the formulation. To do this, we first set (a/ b=1.01, b=2 m, L=2 m) in our general MAPLE code to compute the dimensionless system natural frequenciesρh/ D (k=SS, SA, AS, AA; l=1,2,…,20) for a partially water-filled, nearly circular tank with a thin elastic bottom plate (h=0.005 m, ρ=7 860kg/m3, E= 188 GPa, ν=0.3). The outcome, as presented in Table 1(a), shows good agreements with the data presented in Tables 3 and 4 of Ref.[5].

Next, we set (a/ b=1.01, b=0.144 m, L= 0.144 m) in our general MAPLE code to compute the system natura l fre qu encies,(k=SS, SA, AS, AA, l =1,2,...,20),forapartially water-filled,nearlycircular tank with a thin elastic bottom plate (h= 0.002 m, ρ=7 860kg/m3, E=206 GPa, ν=0.25). The results, as presented in Table 1(b), exhibit fair agreements with the data presented in Refs.[2,3,8,23].

Table 2(b) The first few calculated elliptical tank coupled system natural frequencies for selected fluid column depths

Also shown in the last column of the Table 1(b) are the excellent agreements obtained with the results computed by using the finite element package ABAQUS[24].

It is noteworthy that in the latter validation, a maximum of 20 000 eight-noded quadrilateral axisymmetric “S8R” elements were used to model the elliptical plate, while about 4 000 000 eight-noded quadrilateral axisymmetric “AC3D8R” elements were used to model the liquid column.

Also, mesh size sensitivity analysis was carried out for numerical convergence checking.

Figures 2(a) and 2(b) respectively present the first few liquid free surface displacement and bottom plate structural mode shapes of the water-coupled circular and elliptical tank systems (a/ b=1,2), for selected fluid column depths (L/2b=0.1,1,2).

Also shown are the free surface mode shapes for the circular and elliptical tanks with rigid bottom plates. Furthermore, the corresponding coupled system natural frequencies along with the excellent agreements obtained with the calculated FEM values are shown in Tables 2(a) and 2(b), respectively. The most important observations are as follows. Increasing liquid elevation has nearly no effect on the liquid free surface mode shape of the rigid-bottom tanks as well as on the structural modes of the elastic plate. It particularly influences the liquid free surface mode shapes of the partially filled tanks with an elastic bottom, which may simply be linked to the increased hydroelastic coupling effects. Also, there appears to be a wider variety of liquid free surface mode shapes associated to the tanks with an elastic bottom (dotted lines) in comparison with the tanks with a rigid bottom (dashed lines), especially for tanks of circular cross sections (a/ b=1). Moreover, it is clear from natural frequency values listed in Tables 2(a) and 2(b) and also the mode shapes shown in Figs.2(a) and 2(b) that increasing the bottom plate aspect ratio (i.e., changing from a circular to an elliptical cross section), leads to a notable decrease in the system natural frequencies in addition to a remarkable decrease in the mode shape oscillations (number of nodal points).

Fig.3 Variation of the first few nondimensional system natural frequencies with the fluid column depth for circular and elliptical tanks

This notable drop in system natural frequencies could make the upright cylindrical tanks of elliptical cross sections more vulnerable to external resonant excitations in comparison to the circular tanks. It is also interesting to note that for low liquid fill depths (e.g., L/2b=0.1, the liquid free surface mode shapes (dotted lines) more or less follow the structural mode shapes (solid lines), and as the liquid fill depth increases, the mode shapes tend to deviate from each other as expected.

Lastly, a further examination of the hydroelastic mode shapes clearly demonstrates the absence of socalled “mode localization” effects in the fluid-coupled system (i.e., confinement of the vibrational energy to a region close to the elastic bottom plate or the liquid free surface)[25].

Figure 3 shows the variation of the first few nondimensional system natural frequencies,(n=0,1, 2,3;l =1,2,...,5)and(k=SS, SA, AS, AA; l=1,2, ...,5) with the fluid column depth, L/2 b for watercoupled circular and elliptical elastic and rigid bottom tanks. The key observations are as follows. Two families of modes are observed, as marked in the elastic bottom subplots, namely the sloshing- and the bulging-type modes[3-6]. Sloshing modes are predominantly caused by the oscillations of the free liquid surface, while bulging modes are linked to vibrations of the container structure. A simple look at the subplots clearly indicates that by increasing the liquid column height, the interaction (coupling) between the sloshing and bulging modes increases, as the corresponding natural frequency curves get closer to each other (i.e., they become of the same order), for both circular and elliptical bottom plates. Also, increasing the elastic plate aspect ratio leads to an overall slight decrease in the bulging natural frequencies, while thesloshing frequencies associated with the rigid bottom tanks notably decrease with increasing aspect ratio. This implies an overall increase in coupling between the sloshing and bulging modes as the plate ellipticity increases, which may be linked to the increased liquid added mass effects associated with the elliptic bottom plate. Finally, increasing (decreasing) the plate stiffness (flexibility) causes an overall decrease in sloshing natural frequencies, which in turn leads to a reduced interaction between the bulging and sloshing modes, nearly regardless of plate aspect ratio.

Fig.4(a) The first few hydrodynamic pressure mode shapes of the circular tank system for selected fluid column depths

Figures 4(a) and 4(b) respectively present the first few hydrodynamic pressure mode shapes of the water-coupled circular and elliptical tank systems (a/ b=1,2) for selected fluid column depths (L/ 2b=0.1,1). Also shown are the pressure mode shapes for the circular and elliptical tanks with rigid bottom plates. The key observations are as follows. The effect of bottom plate elasticity on the hydrodynamic pressure mode shapes decreases when increasing the liquid fill level. In particular, the main effect of including the bottom plate elasticity is to shift the localization of hydrodynamic pressure mode shapes from the liquid free surface (in the rigid-bottom situation) towards the tank bottom (in the elastic bottom configuration), especially for higher mode numbers of the circular tank. Lastly, by increasing the plate aspect ratio to (a/ b= 2), the spatial variations in the hydrodynamic pressure mode shapes significantly decrease and a more uniform pressure pattern is observed, nearly regardless bottom plate elasticity and/or liquid fill depth.

Fig.4(b) The first few hydrodynamic pressure mode shapes of the elliptical tank system for selected fluid column depths

3. Conclusions

A general normal mode series solution in terms of radial/angular Mathieu functions, and based on the linearized theory of surface gravity waves, is presented for investigating the free hydroelastic coupled vibrations of a partially-filled vertically standing rigidwalled elliptic cylindrical container with an elastic bottom plate. Extensive numerical data are presented for the first few fluid-coupled system resonant frequencies as a function of fluid column depth for selected container cross sectional ellipticities. The presented results confirm that the dynamic characteristics of the fluid-coupled system are significantly influenced by the plate elasticity, aspect ratio and filling liquid depth. The most important observations are summarized as follows. Increasing liquid elevation particularly influences the liquid free surface mode shapes of the partially filled tanks with an elastic linked to the increased hydroelastic coupling effects. Increasing the bottom plate aspect ratio causes a notable decrease in the system natural frequencies and mode shape oscillations which could make tanks of elliptical cross sections more vulnerable to low frequency external resonant excitations in comparison to the circular tanks. For low liquid fill depths, the liquid free surface mode shapes roughly follow the structural mode shapes, and they tend to deviate from each other as the liquid fill depth increases. By increasing the liquid column height, the interaction between the sloshing and bulging modes increases for both circular and elliptical bottom plates. Increasing the elastic plate aspect ratio leads to an overall increase in the sloshing/bulging mode coupling, linked to the increased liquid added mass effects. Increasing the plate stiffness causes an overall decrease in sloshing natural frequencies, which in turn leads to a reduced interaction between the bulging and sloshing modes. The effect of bottom plate elasticity on the hydrodynamic pressure mode shapes decreases when increasing the liquid fill level. In particular, including the bottom plate flexibility leads to a notable shift in the localization of hydrodynamic pressure mode shapes from the liquid free surface towards the elastic tank bottom.

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Appendix

Assume that the harmonic displacement is w= w(ξ ,η,t)=W(ξ ,η)sin(ωt ) in the absence of the forcing function (P=0), then the equation of motion (1) can favorably be factored in the form

where ζ4=ρh ω2/D , and the solution of (A1) may be decomposed as W=W1+W2, which satisfy

where e1,2=±ζ2c2/4. The general solution of (A2) may favorably be expressed as a linear combination of even and odd modes in terms of the radial and angular Mathieu functions[20]:

The frequency equations for each type of oscillation modes (SS, SA, AS and AA) can be determined by implementation of solutions (A3) in the clamped plate edge conditions (see Eq.(1)). Accordingly, after some manipulations, it can be shown that the following linear systems of equations must hold respectively:

SS modes

The procedure for obtaining the unknown modal coefficientsthroughis as follows. First, one can multiply Eqs.(A4a)-(A4d) by each of the factors, cos(2m)η, cos[(2m+1)], sin[(2m+1)]η, and sin[(2+2)] mη respectively, and then integrating from 0 to 2π. These integrals may subsequently be analytically evaluated by taking advantage of the Fourier expansions of the angular Mathieu functions in terms of the transcendental functions, while taking advantage of the classical orthogonality of these functions, leading to four truncated sets oflinear eigenproblems, which can subsequently be solved for the natural frequencies of the elliptical plate,(m=1,2, 3,…), along with the associated vibrational modes, xκ(), as follows:

and the coefficient matrix Tk(ωi) is written as

10.1016/S1001-6058(14)60030-5

* Biography: HASHEMINEJAD Seyyed M. (1962-), Male, Ph. D.