Equivalent mechanical model of liquid sloshing in multi-baffled containers

Engineering Analysis with Boundary Elements 47 (2014) 82–95

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Equivalent mechanical model of liquid sloshing in multi-baffled containers M. Ebrahimian, M.A. Noorian, H. Haddadpour n Department of Aerospace Engineering, Sharif University of Technology, Tehran, P.O. Box 11155-8639, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 25 April 2013 Received in revised form 28 May 2014 Accepted 9 June 2014 Available online 17 July 2014

This study presents a method to determine an equivalent mechanical model (EMM) for multi-baffled containers with arbitrary geometries. The method is implemented for 2D and axisymmetric containers. The Laplace equation and Green's theorem are used to develop the fluid model and the boundary element method (BEM) is used to solve the fluid field governing equation. Moreover, a zoning method is utilized to model arbitrary arrangements of baffles in multi-baffled containers and a reduced order model is developed to model the free-surface sloshing. The exerted hydrodynamic pressure distribution, forces and moments on the walls of the container are determined based on the Bernoulli equation and a set of recursive formulation is presented to develop the model for multi-baffled containers. The results are validated in comparison with the literature and very good agreement is achieved. Furthermore, the effects of baffle attributes on the EMM parameters are also investigated and some conclusions are outlined. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Sloshing Equivalent mechanical model Multi-baffled container Boundary element method

1. Introduction Periodic motion of fluid with free-surface in a liquid container, which is called sloshing, and its effects on the dynamic of container's supporting system is a field of interest for many researchers. The liquid sloshing leads to exert the hydrodynamic forces on the walls of the container and can cause divergence or even damage of it or failure in the system's function. So, the investigation of the hydrodynamic forces and their effects on the dynamic characteristics of the system are important concerns. According to the importance of liquid sloshing, many valuable studies have been done in this field. Abramson [1] and Ibrahim [2] gathered lots of studies related to the sloshing in two distinct comprehensive literature. Moreover, there are some other individual research studies in this field. For example, Popov et al. [3] investigated the dynamic of liquid sloshing in compartmented and baffled rectangular road containers for some maneuvers and presented analytical steady-state and numerical transient solutions. Faltinsen and Timokha [4] developed a method to approximate the natural sloshing frequencies and modes for a 2D circular container. Wang et al. [5] determined the natural frequencies and vibration modes in a rigid cylindrical container with annular baffle.

n

Corresponding author. E-mail address: [email protected] (H. Haddadpour).

http://dx.doi.org/10.1016/j.enganabound.2014.06.001 0955-7997/& 2014 Elsevier Ltd. All rights reserved.

Numerical methods are widely used in the field of fluid dynamics to analyze the liquid sloshing in the containers with complex geometries. Since the interaction of the fluid and structure occurs at their interface, it is sufficient to consider the boundary of the fluid and container for evaluating the effect of liquid sloshing on the container. That is why the BEM, which concentrates on the boundary of the fluid and structure, is an appropriate method for analyzing this kind of problems. Many researchers such as Gedikli and Erguven [6], Firouz-Abadi et al. [7], Noorian et al. [8] and Ebrahimian et al. [9] used BEM for investigation of the linear and nonlinear sloshing, sloshing frequencies and the effect of baffle on them for different container geometries. The existence of fluid inside a system may alter its dynamic behavior. To approximate the fluid dynamics inside the containers, some EMMs have been developed. In these models, the linear planar liquid motion in a container is approximated by a series of mass–spring–dashpot systems or a set of simple pendulums. Graham and Rodriguez [10] introduced a mechanical model for liquid sloshing in a rectangular container based on linear potential theory. Roberts et al. [11] gathered design information and investigated the effects of propellant sloshing on the structural and control problems in a NASA report. Housner [12], Pinson [13], Bauer [14] and Li and Wang [15] are some other researchers who worked on the EMMs for different types of liquid containers. Although all of the listed researchers have had significant effects on development of the EMMs for simple geometries; however, when the geometry is getting complicated, these models

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

83

obtained [9]: Z n cp ϕp þ ðϕqn  ϕ qÞ dS ¼ 0

ð4Þ

S

where ϕ and qn are the fundamental solution of the Laplace equation in the flow region and its derivation, respectively and cp depends on the internal spatial angle at the source point p. Eq. (4) can be solved using a BEM model for 2D and axisymmetric flow fields [7–9]. By discretizing the boundary of the fluid into small elements, the following equation will be achieved [9]: n

Aϕ ¼ Bq

ð5Þ

where A and B are called the influence matrices of the fluid and ϕ and q are the vectors of nodal potential and flux density of the boundary element model, respectively. 2.2. Developing the BEM formulation for multi-baffled containers Fig. 1. Schematic view of a moving container in inertia coordinate system ðO X Y Z Þ.

lose their effectiveness. Moreover, the existence of baffle influences the dynamic behavior of the fluid and leads to change the EMM parameters which it has to be considered in the analysis. In this study, an efficient method is presented to determine the EMM parameters of liquid sloshing in 2D and axisymmetric multibaffled containers based on BEM formulation.

2. Governing equations For the inviscid and irrotational flow, by assuming the existence of a function as the potential of velocity, the governing equation of the fluid can be explained by the Laplace equation [8] ∇2 ϕ ¼ 0

ð1Þ

where ϕ is the velocity potential function. Consider a moving coordinate system (oxyz), which is named the slosh coordinate system, with small lateral acceleration a and small angular velocity ω in the inertia coordinate system ðOX Y Z Þ, as shown in Fig. 1. The slosh coordinate system is so defined that its z-axis is perpendicular to the free-surface of the liquid. The non-slipping condition on the walls of a rigid container is described as follows: ∂ϕ

T w ¼ rn ω ∂n rn ¼ ðrw  nÞ ¼ ½r nx r ny r nz T

ð2aÞ ð2bÞ

where n is defined as the normal outside vector of the flow field and rw ¼ ½x y zT denotes the position vector of wall point in the slosh coordinate system. Based on the definition of the velocity potential function, the kinematic condition of the free-surface and the unsteady Bernoulli equation, the following boundary condition is obtained for the free-surface of the fluid [7]:
 ∂ϕ

1 € ð3Þ ¼  rTf a_ þ ϕ f g ∂n
f where rf is the position vector of free-surface point in the slosh coordinate system and ϕf is the velocity potential function of the free-surface. 2.1. Boundary element model Using Green's second identity and divergence theorem and assuming ϕ and q ¼ ∂ϕ=∂n as well-behaved functions for a flow region with a boundary S, the following integral equation is

Consider a multi-baffled 2D or an axisymmetric container as shown in Fig. 2. Based on the discretizing method which is presented in Ref. [9], one can divide a baffled container into a number of zones so that the boundary of each zone can be divided into two or three parts for interface and wall nodes, similar to Fig. 2. For a zone with one interface, such as the first zone of the containers in Fig. 2, Eq. (5) can be written in the form of the following set of equations: " 1 #8 1 9 " 1 #( 1 ) qi A11 A112 < ϕi = B11 B112 ¼ ð6Þ q1w B121 B122 A121 A122 : ϕ1w ; where i and w indices refer to nodes on the interface and walls, respectively. A1kj and B1kj are the associated blocks of the influence 1 1 matrices of a zone with one interface. ϕi , q1i , ϕw and q1w denote the nodal potential and flux density of the first interface and the walls of the first zone, respectively. Substituting Eq. (2a) into Eq. (6) gives the following equation for the nodal potential of the walls of the first zone:

ϕ1w ¼ D1i ϕ1i þ D1w ω

ð7aÞ

D1i ¼ ðB121 ðB111 Þ  1 A112  A122 Þ  1 ðA121  B121 ðB111 Þ  1 A111 Þ

ð7bÞ

D1w ¼ ðB121 ðB111 Þ  1 A112  A122 Þ  1 ðB121 ðB111 Þ  1 B112  B122 ÞR 1n

ð7cÞ

R 1n

where is a matrix that contains the of the first zone's wall points. Using these definitions, one can write the following equation for the flux density of the first interface: rTn

q1i ¼ Z1i ϕi þ Z1w ω

ð8aÞ

Z1i ¼ ðB111 Þ  1 ðA111 þ A112 D1i Þ

ð8bÞ

Z1w ¼ ðB111 Þ  1 ðA112 D1w  B112 R1n Þ

ð8cÞ

1

Z1i

Z1w

where and are the interface influence matrices of the first zone. For the mth zone of the container which has more interfaces, Eq. (5) can be written as the following set of equations [9]: 9 2 9 2 m 38 38 > > ϕm qm Bm Bm Bm A11 Am Am > > i i 11 12 13 > 12 13 > = = < < m m 7 6 A m A m Am 7 6 m ϕm qm ¼ 4 B21 B22 B23 5 ð9Þ 4 21 w w 22 23 5 > > > m m m ; > :  qm  1 > ; : ϕm  1 > B B B Am Am Am i 31 32 33 31 32 33 i By substituting Eq. (2a) into Eq. (9), one can achieve the nodal flux density of the mth interface as follows: m m qm i ¼ Zi ϕi þ Zw ω m

ð10aÞ

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Fig. 2. Schematic view of multi-baffled containers: (A) 2D container and (B) axisymmetric container. m where Zm i and Zw are the interface influence matrices of the mth zone and defined as m m m 1 m Zm A21 þC5 C1 Þ i ¼ Z0 ðA11  A12 ðA 22 Þ

ð10bÞ

m m 1 m 1 C5 ¼ Am A23 þC6 Zm 13  A12 ðA 22 Þ i

m m 1 m C6 ¼ Bm B23 13  A 12 ðA 22 Þ

ð10jÞ

ð10kÞ

The coefficient matrices in these equations are defined as follows:

Based on these equations and with some mathematical simplifications, one can write the following recursive equation between the nodal potential of the interfaces:

m 1 m 1 Z0 ¼  ðAm B21  Bm 11 þC5 C2 Þ 12 ðA 22 Þ

1 m m ϕm ¼ Hm i i ϕi þHw ω

m m 1 m m m1 Zm B22  Bm þ C5 C3 Þ w ¼ Z0 ððA 12 ðA22 Þ 12 ÞR n þ C6 Zw

ð10cÞ

ð10dÞ

Hm i

ð11aÞ

Hm w

m m 1 m 1 1 A23 þ C4 Zm Þ C0 ¼  ðAm 33  A 32 ðA22 Þ i

ð10eÞ

and can be named as adaptive interface matrices of where the mth zone and defined as

m m 1 m C1 ¼ C0 ðAm A21 Þ 31  A 32 ðA 22 Þ

ð10f Þ

m Hm i ¼ ðC1 þ C2 Zi Þ

ð11bÞ

m 1 m C2 ¼ C0 ðAm B21  Bm 32 ðA 22 Þ 31 Þ

ð10gÞ

m Hm w ¼ ðC3 þ C2 Zw Þ

ð11cÞ

m 1 m m m1 B22 Bm Þ C3 ¼ C0 ððAm 32 ÞR n þC4 Zw 32 ðA 22 Þ

ð10hÞ

m m 1 m C 4 ¼ Bm B23 33  A 32 ðA22 Þ

ð10iÞ

Using Eq. (11a), one can write the nodal potential of an interface in terms of the nodal potential of the other interfaces and free-surface. Moreover, using the presented equations, the nodal potential of the walls of the mth zone can be calculated

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

3.1. 2D Containers

as follows:

ϕ

¼ Dm i

Dm i

m m m m 1 ¼ ðAm ðBm 22 Þ 21 Zi D0 Hi  A 21 Þ

m w

ϕ

m m i þDw

85

ω

ð12aÞ ð12bÞ

m 1 m m m1 m m m ðBm þ Bm Dm w ¼ ðA22 Þ 22 R n  B23 Zw 21 Zw  D0 Hw Þ

ð12cÞ

m m1 þAm Dm 23 0 ¼ B23 Zi

ð12dÞ

2.3. Free-surface governing equation

Ns

ð16Þ

where ρ is the density of the liquid, is a matrix which contains the position vectors of the wall points of mth zone in the slosh m _ is the time derivation of the nodal potential coordinate and ϕ w vector of the walls of this zone. Using Eq. (12a), one can rewrite Eq. (16) as follows: Rm w

m

ð13Þ

where R f is a matrix that contains the transpose vectors of the free-surface node positions in the slosh coordinate system. Using the reduced order modeling technique, the solution of Eq. (13) can be written in the form of a summation of the natural mode shapes as follows [7]:

ϕf ¼ ∑ ϕ n ξn ðtÞ ¼ Φξ

m _m pm w ¼  ρðR w a þ ϕ w Þ

m m _ m _ pm w ¼  ρðR w a þ Di ϕ i þ Dw ω Þ

Using Eqs. (3) and (10a), the following equation can be written for a container with N zones: 1€ 1 ϕ þ ZNi ϕf ¼  Rf a_ ZNw ω g f g

The nodal values of the hydrodynamic pressure exerted on the walls of the mth zone of a 2D container are evaluated as

ð14aÞ

n¼1

ð17Þ

_ m is the time derivation of the nodal potential of the mth where ϕ i interface which can be calculated using the recursive equation (11a). By calculating the nodal potential of all interfaces in terms of the nodal potential of free-surface and using Eq. (14a), one can determine the pressure distribution on the walls of all zones. By discretizing the walls of the container to small elements and adding the applied forces on all elements, one can determine the total hydrodynamic force exerts on the walls of the container. The hydrodynamic force exerted on the jth element of a zone can be calculated using the following equation: !8 j 9 Z lj Z 1

where

ξ ¼ fξ1 ⋯ ξNs gT

ð14bÞ

in which N s is the number of sloshing modes, ϕ n is the right eigenvector corresponding to the nth sloshing mode and ξn ðtÞ is the corresponding generalized modal coordinate. Also Φ is the modal matrix which contains the right eigenvectors and ξ is the vector of generalized modal coordinates. The modal matrix satisfies the following equation: 1 T Φ Φ¼I g L

ð14cÞ

where ΦL is the left eigenvectors matrix. By substituting Eq. (14a) into Eq. (13) and multiplying by the left eigenvectors from the left hand side, one can write the following undamped modal form of the sloshing equation [7]: 1 g

ξ€ þ Kξ ¼  Ea a_  Eω ω

ð15aÞ

The coefficient matrices of Eq. (15a) are calculated as follows: K ¼ Φ L ZN i Φ

ð15bÞ

Kkk ¼ ω2k

ð15cÞ

Ea ¼ ΦL R f

ð15dÞ

Eω ¼ Φ L ZN w

ð15eÞ

T

T

T

where lj is the length of the jth element of the wall, pj ðsÞ is the hydrodynamic pressure distribution along this element, nj is its normal outside vector, jJ j j is the determinant of the Jacobian matrix of the transformation from the global coordinate system to the local coordinate system, pj1 and pj2 are the nodal hydrodynamic pressures of the jth element and N 1 and N2 are the element's shape functions. By adding the hydrodynamic forces of all elements of a zone, one can achieve the net hydrodynamic force exerts on the zone as !8 j 9 Z 1

8 m9 > I > > = < xm > m IF ¼ Iy > > > ; : Im >

ð20aÞ

ð20bÞ

z

in which ωk is the natural frequency corresponding to the kth mode of sloshing.

where Im F is a matrix which is formed by assembling the matrices which are calculated from the integral terms of Eq. (19) for the mth zone. For a container with one zone, using Eq. (17) and the time derivation of Eq. (14a), one can write the following equation to calculate the vector of hydrodynamic force as _Þ F 1 ¼  ρðE1F a þS1F Φξ_ þ W1F ω

ð21aÞ

where 3. Hydrodynamic pressure and resultant forces and moments Based on the Bernoulli equation, the hydrodynamic pressure distribution on the walls of the container can be calculated. In the following two sections, the required equations for determining the hydrodynamic pressure and resultant forces and moments of 2D and axisymmetric multi-baffled containers are presented.

E1F ¼ I1F R 1w

ð21bÞ

S1F ¼ I1F D1i

ð21cÞ

W1F ¼ I1F D1w

ð21dÞ

But for a multi-baffled container with more than one zone, one can determine the net hydrodynamic forces by adding the

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M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

hydrodynamic forces of all zones. In general, for a multi-baffled container with N zones, using Eqs. (11a), (12a) and (14a), the vector of net hydrodynamic force is calculated as follows: F ¼ ρ

N ðEN F aþ SF

Φξ_

þW N F

ω_ Þ

ð22aÞ

where þ1 mþ1 mþ1 ¼ Em Rw Em F F þ IF

ð22bÞ

þ1 Sm F

mþ1 þ1 mþ1 Sm þ Im Di F Hi F

ð22cÞ

m mþ1 þ1 þ1 mþ1 Wm ¼ Wm þ Im Dw F F þSF Hw F

ð22dÞ

¼

Moreover, the hydrodynamic moment exerted on a zone can be expressed as Z lj Ne M¼ ∑ Uj pj ðsÞnj ds ð23aÞ j¼1

0

2

zj

0 6 U ¼ 4 zj  yj j

0 x

j

3

yj

x 7 5 0 j

ð23bÞ

j

j

m M m ¼ Im M pw

where 2

3

Im M¼

7 7¼ 5

Ref. [9] has used the Fourier expansion to develop a boundary element model for axisymmetric containers. The hydrodynamic pressure which is exerted on the walls of the mth zone of an axisymmetric container can also be expressed using this expansion as follows: Nc

m _m pm w ¼ ∑  ρðR w aþ ϕ w Þ cos ðnc θ Þ

ð24aÞ

where nc is the circumferential mode number and N c is the total number of active circumferential modes. By substituting the time derivation of Eq. (12a), Eq. (27) can be rewritten as N

c m m _m m _ pm w ¼ ∑  ρðR w aþ Di ϕ i þ Dw ω Þ cos ðnc θ Þ

ð28Þ

nc ¼ 1

The applied hydrodynamic force on a ring corresponding to the jth element of the wall of a zone in an axisymmetric container can be determined as follows: Z π Z lj Fj ¼ pj ðs; θÞGz nj r j ds dθ ð29aÞ 0

j

where r is the distance of the center of the jth element from the axis of symmetry and Gz is the rotation matrix about the z-axis 2 3 cos ðθÞ  sin ðθÞ 0 6 7 ð29bÞ Gz ¼ 4 sin ðθÞ cos ðθÞ 0 5 0

0

1

j

Im yz 6 m 6 Izx 4 Im xy

2

m~ ~ Im z Y w  Iy Z w 6 m~ m 6 Ix Z w  Iz X~ w 4 m~ ~ Im y X w  Ix Y w

3 7 7 5

ð24bÞ

Also n is the normal vector of jth element in the xz plane and its component along the y-axis is zero. Thereupon, based on Eq. (27), the net hydrodynamic force exerted on a zone is calculated as follows: Ne

½X~ w ii ¼ xiw ;

½Y~ w ii ¼ yiw ;

½Z~ w ii ¼ ziw

ð24cÞ

F ¼ ∑ ρ j¼1

Z π π

Nc

j n~ ∑ cos ðnc θÞ dθ

Z

nc ¼ 1

and are the position components of the ith node on the walls of the mth zone. For a container with one zone, similar to the process of calculation of the hydrodynamic force, one can write the following equation to calculate the hydrodynamic moment: _Þ M 1 ¼  ρðE1M aþ S1M Φξ_ þ W1M ω

ð25aÞ

where E1M ¼ I1M R 1w

ð25bÞ

S1M ¼ I1M D1i

ð25cÞ

W1M ¼ I1M D1w

ð25dÞ

For a multi-baffled container with more than one zone, the net hydrodynamic moment can be determined by adding the hydrodynamic moments of all zones. In general, for a multi-baffled container with N zones, using Eq. (17) and the time derivation of Eq. (14a), the net hydrodynamic moment can be calculated as follows: N ðEN M a þ SM

Φξ_ þ WN ω_ Þ M

ð26aÞ

where þ1 mþ1 mþ1 ¼ Em Rw Em M M þIM

ð26bÞ

þ1 mþ1 þ1 mþ1 ¼ Sm þ Im Di Sm M M Hi M

ð26cÞ

m mþ1 þ1 þ1 mþ1 Wm ¼ Wm þ Im Dw M M þ SM Hw M

ð26dÞ

1 1

9 !8 _ Þ =j < r 1 ðrT a þ ϕ 1 1 _ Þ; : r ðrT a þ ϕ

½N 1 N 2 jJ j j dξ

2

2

2

w

ð30aÞ

xiw ; yiw ; ziw

M ¼ ρ

ð27Þ

nc ¼ 1

π

where x ; y ; z are the position components of the center of jth element with respect to the coordinate axes which the components of the moment vector are calculated about them. Eq. (23a) can be rewritten in the matrix form for the mth zone as j

3.2. Axisymmetric containers

where 8 9j > < nx cos ðθÞ > = j n~ ¼ nx sin ðθÞ > > : ; n

ð30bÞ

z

in which nx and nz are the components of the normal outside vector n. Using the orthogonality of the trigonometric functions, the parenthesized term of Eq. (30a) can be written as follows: ( R π ½njx 0 njz T 1 1 ½N1 N 2 jJ j j dξ; nc ¼ 1 ð30cÞ 0; nc a 1 Eq. (30a) can be rewritten in the matrix form for the mth zone as follows: m Fm ¼ Im F p0w

ð31aÞ

where m m _m m _ pm 0w ¼  ρR w ðR w a þ Di ϕ i þ Dw ω Þ

ð31bÞ

½R w kk ¼ r k

ð31cÞ

in which r k is the vertical distance of the kth node of the wall from the axis of symmetry. IF is a matrix formed by assembling the matrices calculated from the integral terms of Eq. (30a) based on Eq. (30c). For a container with one zone, one can use Eq. (21a) to calculate the vector of hydrodynamic force; but the coefficient matrices of this equation for axisymmetric geometry are defined

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

87

as follows: 1

ð32aÞ

1

ð32bÞ

E1F ¼ I1F R w R 1w S1F ¼ I1F R w D1i 1

W1F ¼ I1F R w D1w

ð32cÞ

It is also possible to use Eq. (22a) to determine the net hydrodynamic force for a multi-baffled axisymmetric container with N zones, using the following coefficient matrices: mþ1

þ1 mþ1 Em ¼ Em Rw F F þ IF

þ1 Rm w

mþ1

þ1 mþ1 þ1 Sm ¼ Sm þIm Rw F F Hi F

ð33aÞ

þ1 Dm i

ð33bÞ

mþ1

m mþ1 þ1 þ1 ¼ Wm þ Im Rw Wm F F þSF Hw F

þ1 Dm w

ð33cÞ

The hydrodynamic moment exerted on a zone can be expressed as Ne

M ¼ ∑ ρ

Z π

j¼1

π

Nc

j U n~ ∑ cos ðnc θ Þ dθ j

nc ¼ 1

Z

9 !8 _ Þ =j < r 1 ðrT a þ ϕ 1 1 ½N 1 N 2 jJ j dξ _ Þ; : r ðrT a þ ϕ 1 1

j

2

2

2

w

Fig. 3. Mass–spring–dashpot EMM.

ð34Þ Eq. (34) can be rewritten in the matrix form for the mth zone as m M m ¼ Im M p0w

ð35Þ

Im M

is defined in Eq. (24b). where For an axisymmetric container with one zone, the hydrodynamic moment is calculated using Eq. (25a) with the following coefficient matrices: 1

E1M ¼ I1M R w R1w

ð36aÞ

dashpot, cn , located at the vertical position of hn with respect to the CG of the solidified fluid. The following conditions should be satisfied with the presented model [2]:

 Total fluid mass: 1

mF ¼ mq þ ∑ mn S1M

¼

1 I1M R w D1i

ð36bÞ

1

W1M ¼ I1M R w D1w

ð36cÞ

For a multi-baffled axisymmetric container with N zones, the net hydrodynamic moment can be determined using Eq. (26a), but with the following coefficient matrices: mþ1

þ1 mþ1 ¼ Em Rw Em M M þIM

þ1 Rm w mþ1

þ1 mþ1 þ1 ¼ Sm þ Im Rw Sm M M Hi M þ1 Wm M

¼

þ1 Dm i

m mþ1 þ1 mþ1 mþ1 Wm þ Im R w Dw M þ SM Hw M

ð37aÞ ð37bÞ

ð38aÞ

n¼1

 Equilibrium about CG of the solidified fluid: 1

mq hq ¼ ∑ mn hn

ð38bÞ

n¼1

where mF is the total mass of the fluid. Based on the definition of the spring constant of a mass–spring system, kn is defined as kn ¼ mn ω2n ð38cÞ where ωn is the natural frequency of the nth sloshing mode. The effective moment of inertia about the y-axis which passes through the CG of the solidified fluid is also defined as follows: 2

1

2

I E ¼ I q þ mq h q þ ∑ mn h n

ð38dÞ

n¼1

ð37cÞ

4. Equivalent mechanical model

4.1. Lateral excitation

The dynamic behavior of a system which contains rigidly connected closed liquid containers can be formulated by approximating the liquid dynamics using EMMs. An EMM for a linear planar liquid motion can be developed using a series of mass– spring–dashpot systems (Fig. 3). In this figure mq is a rigid mass, called quiescent mass, moving in unison with the container and located at a vertical position hq which is measured with respect to the Center of Gravity (CG) of the solidified fluid, and I q is its moment of inertia about the axis that passes through its CG. The associated equivalent mass of nth sloshing mode is indicated by mn , called modal mass, and is restrained by a spring, kn , and a

Consider a container which is located under the pure translational excitation as x ¼ x0 eiωt

ð39Þ

where ω is the excitation frequency. Ignoring the damping effects of liquid sloshing and assuming the mass center of the solidified fluid as the center of the slosh coordinate system, one can rewrite Eq. (15a) as follows: 1 g

ξ€ þ Kξ ¼  Ea a_

ð40aÞ

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coordinate system, one can rewrite Eq. (15a) as follows:

where Ea ¼ ½Eax Eay Eaz 

ð40bÞ

The solution of Eq. (40a) can be expressed as ð41Þ

Substituting Eqs. (39) and (41) into Eq. (40a) yields the following result for the nth mode of sloshing: 1 g

ω4

ðω2n  ω2 Þ

x0 eiωt

ð42Þ

n

where ðEax Þ is the nth row of the vector Eax which corresponds to the nth sloshing mode with the natural frequency of ωn . By substituting Eq. (42) into the time derivation of Eq. (14b) and using Eqs. (22a) and (26a) with appropriate coefficient matrices based on the container geometry, one can calculate the net hydrodynamic force and moment caused by the lateral excitation for a 2D or an axisymmetric container. In planar motion, the x component of the net hydrodynamic force and the hydrodynamic moment about the y-axis for a rigid container in pure lateral excitation can be determined using the following equations:   ρ Ns N ω2 F x ¼ ρEN ∑ ðS Fx Þn ðEax Þn 2 ω2 x0 eiωt ð43aÞ Fx þ gn¼1 ðωn  ω2 Þ   ρ Ns N ω2 ∑ ðS My Þn ðEax Þn 2 ω2 x0 eiωt M y ¼ ρEN My þ 2 gn¼1 ðωn  ω Þ

ð43bÞ

¼ SN FΦ¼

N SM

¼ SN M

N ½S Fx

Φ¼

N S Fy

N ½S Mx

ð43cÞ

N S Mz 

ð43dÞ

N ðS Fx Þn

N and ðS My Þn and respectively, and ðEax Þn

N S Fx

N

denote the nth column of and S My , is the nth row of Eax . The hydrodynamic force and moment equations for pure lateral excitation of undamped models, which represented by Ref. [2], are   Ns ω2 F x ¼ mF þ ∑ mn 2 ω2 x0 eiωt ð44aÞ 2 ð ωn  ω Þ n¼1  My ¼

Ns

g

n¼1

ω2n

∑ mn

Ns

ð47bÞ

The solution of Eq. (47a) can also be expressed as Eq. (41). Substituting Eqs. (41) and (46) into Eq. (47a) yields the following result for the nth mode of sloshing:

ξ_ n ¼ ðEωy Þn

ω2

θ0 eiωt

ð48Þ

ðω2n  ω2 Þ

where ðEωy Þn is the nth row of the vector Eωy which corresponds to the nth mode of sloshing with ωn as the corresponding natural frequency. By substituting Eq. (48) into the time derivation of Eq. (14b) and using Eqs. (22a) and (26a) with appropriate coefficient matrices based on the container geometry, one can calculate the net hydrodynamic force and moment caused by pitching excitation for a 2D or an axisymmetric container. In planar motion, the x component of the net hydrodynamic force and the hydrodynamic moment about the y-axis for a rigid container in pure pitching excitation can be determined using the following equations:   Ns 1 N n F x ¼ ρWN ω2 θ0 eiωt ð49aÞ Fx  ρ ∑ ðS Fx Þn ðEωy Þ 2 2 ð ωn  ω Þ n¼1

n¼1

N S Fz 

N S My

Eω ¼ ½Eωx Eωy Eωz 

 Ns N n M y ¼ ρW N My  ρ ∑ ðS My Þn ðEωy Þ

where N SF

ð47aÞ

where

ξ ¼ ξ0 eiωt

ξ_ n ¼  ðEax Þn

ξ€ þ Kξ ¼  Eω ω



þ ∑ mn hn þ n¼1

g

ω2n





ω2 ω2 x0 eiωt 2 ðωn  ω2 Þ

ð44bÞ



1 ðω

2 n

ω

2Þ

ω2 θ0 eiωt

The force and moment equations for pure pitching excitation of the undamped models are represented by Ref. [2] as follows:   Ns Ns 1 F x ¼  ∑ mn hn þ ∑ mn ðhn ω2n þ gÞ 2 ω2 θ0 eiωt ð50aÞ 2 ðωn  ω Þ n¼1 n¼1 My ¼

2 I q þ mq h q þ

Ns



∑ mn hn ωn þ

n¼1

g

ω2n

2

! 1 ω2 θ0 eiωt ðω2n  ω2 Þ ð50bÞ

Comparing Eqs. (49a) and (49b) with Eqs. (50a) and (50b) yields the following beneficial relation for determining the EMM parameters: I q þ mq hq ¼ ρWN My 2

where g is the gravity acceleration. Comparing Eqs. (43) with Eqs. (44) leads to the following useful relations for determining the EMM parameters: mF ¼ ρEN Fx

ρ

ð45aÞ

N

mn ¼ ðS Fx Þn ðEax Þn g  mn hn þ

g

ω2n



ρ

ð45bÞ N

¼ ðS My Þn ðEax Þn g

ð45cÞ

and also a supplementary equation as Ns

g

n¼1

ω2n

∑ mn

¼ ρEN My

ð45dÞ

So the EMM parameters can be determined using Eqs. (38) and (45). 4.2. Pitching excitation Pure pitching excitation about the y-axis is expressed as

θ ¼ θ0 eiωt

ð46Þ

Ignoring the damping effects of liquid sloshing and assuming the mass center of the solidified fluid as the center of the slosh

ð49bÞ

ð51aÞ

and also the following supplementary equations: Ns

∑ mn hn ¼  ρWN Fx

ð51bÞ

n¼0

N

mn ðhn ω2n þ gÞ ¼  ρðS Fx Þn ðEωy Þn 

mn hn ωn þ

g

ωn

2

N

¼  ρðS My Þn ðEωy Þn

ð51cÞ ð51dÞ

Using the combination of Eqs. (38), (45) and (51), all of the EMM parameters for a liquid container with specified dimensions are calculable. Based on the presented method, one can calculate these parameters for any arbitrary container geometry and arrangement of baffles. It should be noted that the presented mass–spring model was developed based on linear planar liquid motion and simulates only the linear motion of the fluid. Very close to the free surface, the nonlinear effects of sloshing will be considerable and the accuracy of the presented model will be decreased. So, to model the nonlinear sloshing phenomena, such as rotational and chaotic sloshing, some other EMMs like spherical or compound pendulum should be used. However, most of the time, because of the low

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

volume of the sloshing liquid near the free-surface, its effect on the dynamic of the container is not considerable except for small containers.

1 tanhðð2n  1Þπ h=2aÞ IE 4 768 ∑ ¼ 1 þ 2 2 5 Is ð2n  1Þ5 1 þðh=aÞ π ðh=aÞð1 þ ðh=aÞ Þ n ¼ 1

ð52gÞ Iq

5. Numerical results and discussion

mF h

For investigation of the ability of the presented method in determining the EMM parameters, some 2D and axisymmetric containers are provided in two distinct sections. In both sections, first the EMM parameters of a simple container are calculated and the results are compared with the literature. Then, the effects of the baffle parameters on the equivalent models are investigated and finally, the performance of the presented method for arbitrary container geometry and arrangement of baffles are survived using multi-baffled containers with complex geometries. It should be noticed that for calculation of the EMM parameters in the following sections, only the first three sloshing modes are included.

¼ 2

"

Is mF h



2

1

4 1 þ ðh=aÞ

þ 2

1

768

∑

π 5 ðh=aÞð1 þ ðh=aÞ2 Þ n ¼ 1

tanhðð2n  1Þπ h=2aÞ

#

ð2n  1Þ5

 2  2 1 m m q hq n hn  ∑ mF h h n ¼ 1 mF

ð52hÞ

In these equations n is the sloshing mode number, mF and h are the total mass and depth of the fluid, a is the width of the container, W F is the weight of the fluid and I s and I E are the mass moment of inertia of the solidified fluid and effective mass moment of inertia of the fluid about the y-axis. Figs. 5–8 illustrate the variations of the EMM parameters against the non-dimensional fluid depth for a rectangular container and give a comparison between the present method and the 1

5.1. 2D containers In this section, three cases are provided to validate the results of presented method and investigate the effects of baffles on EMM parameters of 2D containers. For all cases in this section, b, which is the length of the container along the y-axis, is considered as a unit.

0.8

m1 /mF

0.6

m2 /mF

m q , mn ______ mF

5.1.1. Case 1: simple rectangular container Fig. 4 shows a rectangular cross-section container. The EMM parameters of this container are presented in Ref. [2] as follows: mn 8 tanhðð2n  1Þπ h=aÞ ¼ mF π 3 ð2n  1Þ3 h=a

89

m3 /mF mq /mF

0.4

ð52aÞ 0.2

hn 1 tanh ðð2n  1Þπ h=2aÞ ¼  ð2n  1Þπ h=2a h 2

ð52bÞ

hkn 8 tanh ðð2n  1Þπ h=aÞ ¼ W F π2 ð2n  1Þ2

ð52cÞ

1 m mq n ¼ 1 ∑ mF n ¼ 1 mF

ð52dÞ

hq mF 1 mn hn ¼ ∑ h mq n ¼ 1 mF h

ð52eÞ

  Is 1  a 2 ¼ þ1 2 12 h mF h

ð52f Þ

0

2

0

0.5

1

1.5 h/a

2

2.5

3

Fig. 5. Ratios of the equivalent masses to the fluid mass vs. non-dimensional fluid depth: solid lines, analytical; symbols, present method.

1 h1 /h h2 /h

0.8

h3 /h 0.6

hq /h

hq ,hn _____ h

0.4 0.2 0 -0.2 -0.4 -0.6

Fig. 4. Schematic view of a rectangular cross-section container.

0

0.5

1

1.5 h/a

2

2.5

3

Fig. 6. Ratios of the vertical positions of equivalent masses to the fluid depth vs. non-dimensional fluid depth: solid lines, analytical; symbols, present method.

90

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

analytical results. For calculation of the EMM parameters in this section, only the first three sloshing modes are included. 5.1.2. Case 2: baffled rectangular container Fig. 9 shows a schematic view of a rectangular cross-section container with a row of baffle at height d from the bottom of the container. The effect of baffle position on the EMM parameters is studied in this case. Figs. 10–13 show the variations of the EMM parameters against the non-dimensional baffle position. As Fig. 10 shows, the presence of baffle on the equivalent masses is significant when it is near the free-surface; but far away from it, variation of the baffle position has no great influence on the equivalent masses, their vertical positions from CG and also spring constants. Nevertheless, even far away from the free-surface, changing the position of the baffle is still effective on the ratios of mass moments of inertia. As Fig. 13

Fig. 9. Schematic view of a baffled rectangular cross-section container (e=a ¼ 0:15; h=a ¼ 1).

1 0.9 0.8

0.8

0.7

0.6 hk n ____ WF

mq ,mn ______ mF

hk2 /WF hk3 /WF

0.4

m1 /mF

0.6

hk1 /WF

m2 /mF

0.5

m3 /mF 0.4

mq /mF

0.3

0.2 0.2 0.1

0 0

0.5

1

1.5 h/a

2

2.5

3

0 0

Fig. 7. Non-dimensional equivalent spring constants vs. non-dimensional fluid depth: solid lines, analytical; symbols, present method.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/h Fig. 10. Ratios of the equivalent masses to the fluid mass vs. non-dimensional baffle position.

1

1 I E /I s

h1 /h

I q /I s

0.8

h2 /h

0.8

h3 /h hq /h

0.6 hq ,hn _____ h

I E , Iq _____ Is

0.6

0.4

0.4

0.2 0.2

0 0 0

0.5

1

1.5 h/a

2

2.5

3

Fig. 8. Ratios of the effective and fixed mass moments of inertia to the moment of inertia of solidified fluid vs. non-dimensional fluid depth: solid lines, analytical; symbols, present method.

-0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/h Fig. 11. Ratios of the vertical positions of equivalent masses to the fluid depth vs. non-dimensional baffle position.

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

shows, the maximum ratios of mass moments of inertia occur when the baffle is at the middle of the fluid depth. 5.1.3. Case 3: multi-baffled oval cross-section container An oval cross-section container with two rows of baffles is shown in Fig. 14. In this case, the ability of the proposed method in determining the EMM parameters for multi-baffled 2D containers with complex geometries is surveyed. The results are summarized in Table 1. This table shows the variations of the EMM parameters against the non-dimensional fluid depth for two different nondimensional baffle sizes. When the free-surface is close to the upper baffles, almost all of the fluid above the upper baffles moves under the shallow water condition and the ratio of the equivalent mass of the first sloshing mode to the total fluid mass is increased by increasing the non-dimensional fluid depth, initially. But with a further increment of the depth of the fluid, the free-surface is getting far away from the upper baffles and

0.9 0.8 0.7

also its size is decreased considerably and the liquid oscillation is reduced. Therefore, the boundaries of the quiescent fluid region are expanded gradually and the quiescent mass becomes closer to the CG. The ratios of the quiescent mass to the total fluid mass and its moment of inertia to the moment of inertia of solidified fluid are increased in this situation. 5.2. Axisymmetric containers The performance of the presented method for axisymmetric containers is surveyed in this section. Three cases are provided to validate the results of presented method and investigate the effects of baffles on EMM parameters of axisymmetric containers. 5.2.1. Case 1: simple circular cylindrical container A circular cylindrical container is illustrated in Fig. 15. The EMM parameters for this container are presented in Ref. [2] as follows: mn 2a ¼ tanhðλ1n h=aÞ mF λ1n hðλ2 1Þ 1n

ð53aÞ

  hn 1 4a tanhðλ1n h=2aÞ ¼ 1 h 2 λ1n h

ð53bÞ

hkn 2 tanh ðλ1n h=aÞ ¼ 2 WF ðλ1n  1Þ

ð53cÞ

2

0.6

hk n ____ WF

91

2

mF h

hk 2/WF

0.4



Is

hk 1/WF

0.5

 1 1 a 2 þ 12 4 h

¼ "

hk 3/WF

I E ¼ mF

0.3

ð53dÞ

1 1  ð2a=λ hÞ tanhðλ h=2aÞ h a2 1n 1n þ  8a2 ∑ 12 4 n¼1 λ2 ðλ2  1Þ 2

1n

0.2

Iq 2

mF h

0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/h Fig. 12. Non-dimensional equivalent spring constants vs. non-dimensional baffle position.

¼

IE 2

mF h



# ð53eÞ

1n

 2  2 1 m mq hq n hn  ∑ mF h h n ¼ 1 mF

ð53fÞ

In these equations, a is the radius of the container and λ1n are the roots of the derivative of the Bessel function of the first kind. The other parameters definition and also the ratios of mq =mF and hq =hF are the same as those defined in Section 5.1.1. Figs. 16–19 illustrate the variations of the EMM parameters against the non-dimensional fluid depth for this case and present a comparison between the results of the proposed method and the

0.3 0.28 0.26 0.24

IE , Iq _____ Is

0.22 0.2 0.18 0.16 IE/I s

0.14

I q /I s

0.12 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/h Fig. 13. Ratios of the effective and fixed mass moments of inertia to the moment of inertia of solidified fluid vs. non-dimensional baffle position.

Fig. 14. Schematic view of a multi-baffled oval cross-section container.

92

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

Table 1 Variations of the EMM parameters against the non-dimensional fluid depth for the container shown in Fig. 14. e=a ¼ 0:1

h=a

0.51 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 0.99

e=a ¼ 0:3

m1 mF

mq mF

h1 hF

hq hF

hk1 WF

IE Is

Iq Is

m1 mF

mq mF

h1 hF

hq hF

hk1 WF

IE Is

Iq Is

0.684 0.740 0.705 0.659 0.603 0.537 0.460 0.371 0.267 0.145 0.031

0.242 0.250 0.271 0.306 0.355 0.417 0.494 0.586 0.697 0.832 0.963

 0.389  0.402  0.405  0.406  0.408  0.406  0.405  0.398  0.389  0.369  0.332

1.264 1.167 1.029 0.863 0.696 0.536 0.395 0.270 0.163 0.072 0.013

0.297 0.407 0.472 0.532 0.591 0.644 0.686 0.707 0.688 0.583 0.305

0.780 0.755 0.721 0.685 0.644 0.599 0.551 0.500 0.450 0.405 0.380

0.280 0.250 0.221 0.209 0.208 0.220 0.237 0.262 0.292 0.329 0.367

0.134 0.449 0.551 0.565 0.545 0.502 0.440 0.361 0.263 0.144 0.031

0.459 0.445 0.435 0.433 0.445 0.476 0.528 0.603 0.704 0.834 0.963

 0.207  0.265  0.288  0.303  0.316  0.325  0.334  0.337  0.337  0.326  0.297

0.465 0.436 0.408 0.391 0.369 0.331 0.277 0.208 0.135 0.063 0.011

0.015 0.157 0.288 0.388 0.477 0.556 0.623 0.666 0.667 0.576 0.305

0.811 0.794 0.771 0.745 0.715 0.681 0.642 0.600 0.556 0.514 0.491

0.625 0.609 0.586 0.557 0.521 0.488 0.460 0.445 0.443 0.457 0.480

0.5 0.4 0.3 0.2

hq ,hn _____ h

0.1 0 -0.1 h1 /h

-0.2

h2 /h -0.3

h3 /h

-0.4

hq /h

-0.5 Fig. 15. Schematic view of a circular cylindrical container.

0

1

2

3

4 h/a

5

6

7

8

Fig. 17. Ratios of the vertical positions of equivalent masses to the fluid depth vs. non-dimensional fluid depth: solid lines, analytical; symbols, present method.

1

1

0.9

0.9

0.8

0.8

0.7

0.7

m 1/mF 0.6

m 2/mF

0.5

m 3/mF

0.4

mq /mF

hk n ____ WF

mq ,mn ______ mF

0.6

hk 1/WF 0.5

hk 2/WF

0.4

hk 3/WF

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0

1

2

3

4 h/a

5

6

7

8

Fig. 16. Ratios of the equivalent masses to the fluid mass vs. non-dimensional fluid depth: solid lines, analytical; symbols, present method.

0

1

2

3

4 h/a

5

6

7

8

Fig. 18. Non-dimensional equivalent spring constants vs. non-dimensional fluid depth: solid lines, analytical; symbols, present method.

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

93

analytical results. For calculation of the EMM parameters in this section, only the first three sloshing modes are included.

1 0.9

5.2.2. Case 2: baffled axisymmetric container Fig. 20 shows a schematic view of a circular cylindrical container with a row of baffles at height d from the bottom. The effect of baffle position on the EMM parameters of an axisymmetric container is studied in this section. Figs. 21–24 show the variations of the EMM parameters against the non-dimensional baffle position. The effects of baffle position on the EMM parameters for axisymmetric containers are the same as 2D containers described in Section 5.1.2.

0.8 0.7

IE , I q _____ Is

0.6 0.5 0.4 0.3

IE /I s

0.2

I q /I s

0.1 0

0

1

2

3

4 h/a

5

6

7

8

5.2.3. Case 3: multi-baffled spherical container A spherical container with two rows of baffles is shown in Fig. 25. In this case, the ability of the proposed method in determining the EMM parameters for multi-baffled axisymmetric containers with complex geometries is surveyed. The results are 0.5

Fig. 19. Ratios of the effective and fixed mass moments of inertia to the moment of inertia of solidified fluid vs. non-dimensional fluid depth: solid lines, analytical; symbols, present method.

h 1/h h2 /h

0.4

hq /h

hq ,hn _____ h

0.3

0.2

0.1

0

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/h

Fig. 20. Schematic view of a baffled circular cylindrical container (e=a ¼ 0:3; h=a ¼ 2).

Fig. 22. Ratios of the vertical positions of equivalent masses to the fluid depth vs. non-dimensional baffle position.

1

0.9

hk1 /WF 0.8

hk2 /WF 0.8

0.7 m1 /m F

0.6

mq /mF

0.6 hk n ____ WF

mq ,mn ______ mF

m 2/m F 0.5 0.4

0.4 0.3 0.2

0.2

0.1

0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/h Fig. 21. Ratios of the equivalent masses to the fluid mass vs. non-dimensional baffle position.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/h Fig. 23. Non-dimensional equivalent spring constants vs. non-dimensional baffle position.

94

M. Ebrahimian et al. / Engineering Analysis with Boundary Elements 47 (2014) 82–95

Table 2 Variations of the EMM parameters against the non-dimensional fluid depth for the container shown in Fig. 25. h=a

1.02 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.98

e=a ¼ 0:1

e=a ¼ 0:3

m1 mF

mq mF

h1 hF

hq hF

hk1 WF

IE Is

Iq Is

m1 mF

mq mF

h1 hF

hq hF

hk1 WF

IE Is

Iq Is

0.488 0.505 0.459 0.405 0.347 0.285 0.222 0.157 0.094 0.037 0.004

0.472 0.488 0.522 0.570 0.626 0.687 0.752 0.821 0.890 0.955 0.995

 0.238  0.235  0.225  0.214  0.201  0.186  0.169  0.147  0.119  0.082  0.032

 0.282  0.235  0.190  0.147  0.109  0.077  0.050  0.029  0.013  0.003  1.2E  4

0.681 0.882 0.957 1.003 1.020 1.005 0.950 0.843 0.668 0.402 0.096

0.385 0.342 0.290 0.239 0.191 0.147 0.109 0.078 0.056 0.044 0.041

0.090 0.092 0.082 0.071 0.063 0.056 0.051 0.047 0.044 0.042 0.041

0.115 0.316 0.361 0.351 0.317 0.270 0.215 0.154 0.093 0.037 0.004

0.681 0.658 0.635 0.638 0.664 0.707 0.762 0.824 0.891 0.955 0.995

 0.069  0.092  0.098  0.102  0.104  0.103  0.098  0.087  0.070  0.042 0.001

 0.098  0.066  0.053  0.049  0.044  0.036  0.026  0.016  0.007  0.001 3.9E  5

0.045 0.348 0.589 0.745 0.847 0.896 0.888 0.814 0.658 0.400 0.096

0.521 0.504 0.484 0.459 0.430 0.397 0.364 0.334 0.311 0.296 0.291

0.424 0.432 0.434 0.420 0.397 0.371 0.345 0.324 0.306 0.295 0.291

The effects of baffle position on the EMM parameters for this case are the same as those presented in Section 5.1.3.

0.35

6. Conclusion

I____ E , Iq Is

0.3

0.25

0.2 IE /I s I q /I s 0.15 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

d/h Fig. 24. Ratios of the effective and fixed mass moments of inertia to the moment of inertia of solidified fluid vs. non-dimensional baffle position.

In this study a method for determining the EMM parameters of multi-baffled containers was presented. The fluid model was developed based on the Laplace equation, Green's theorem and potential flow. A zoning method was used to model arbitrary arrangements of baffles and a reduced order model was utilized to solve the free-surface governing equation. The exerted hydrodynamic pressure, force and moment were determined for 2D and axisymmetric containers and the method was developed for multi-baffled containers using a set of recursive equations. The accuracy of the results of the proposed method was validated with the EMM parameters of simple rectangular crosssection and circular cylindrical containers and good agreements were achieved. The ability of this method for investigation of the EMM parameters of multi-baffled 2D and axisymmetric containers was also illustrated. The effect of size and position of baffle and liquid depth on the EMM parameters was also surveyed and it was showed that the increment of the baffle size and decreasing its distance from free-surface affect the EMM parameters significantly especially for the ratios of moments of inertia. It is also inferable from the results that the equivalent masses corresponding to the second or higher order sloshing modes are very less than the first mode. However, the influence of each sloshing mode on the dynamic of the fluid depends on the proximity of the excitation frequency to the corresponding sloshing natural frequency of that mode. For example, if the excitation frequency is more close to the second sloshing natural frequency than the first one, the second mode would be more active than the first mode. So, choosing the number of required equivalent masses in the EMM depends on the excitation frequency of the liquid container. References

Fig. 25. Schematic view of a spherical baffled container.

summarized in Table 2. This table presents the variations of the EMM parameters against the non-dimensional fluid depth for two different non-dimensional baffle sizes.

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