# Coupled Responses in a Partially Liquid-Filled Cylindrical Tank with the Single Flexible Baffle under Pitching Excitations

Authors: Jiadong Wang; Jinxing Liu; Dan Wang
Journal: Shock and Vibration
 (2019)

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## Abstract

Antisloshing baffles are widely used in many engineering fields, including aerospace, ocean engineering, and nuclear engineering. The semianalytical scheme is proposed to explore the effect of the single flexible baffle on the coupled responses in the rigid cylindrical tank partially filled with ideal liquid undergoing the pitching excitation. The function series for the velocity potential, the dynamic deflection of the flexible baffle, and the surface wave height are given by introducing the time-dependent generalized coordinates. The Stokes–Joukowski potentials which are contained in the liquid velocity potential can be solved analytically. According to the dynamic and kinematic equations for the free surface and the coupled vibration equation for the flexible baffle, the coupled dynamic response equations are obtained. The additional damping terms are introduced to account for the sloshing damping. The semianalytical method is validated by the comparison study with the numerical results. Parameter studies are conducted to investigate the effects of the parameters of the flexible baffle on the coupled responses of the system.


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## Body

## 1. Introduction

The liquid sloshing in moving tanks partially filled with liquid is of great practical importance in many engineering applications, such as LNG (liquefied natural gas) tanks undergoing seismic excitation and fuel tanks in moving spacecrafts. Baffles are usually used to control the liquid sloshing.The experimental investigations for the effectiveness of the antisloshing baffle in various tanks were reported in several literatures. Goudarzi et al. [1] carried out experiments to investigate the sloshing damping which was provided by horizontal and vertical baffles installed in the rectangular tank, in which the liquid has the free surface. Xue et al. [2] designed the liquid sloshing experimental rig, which is driven by the wavemaker, to investigate the free surface fluctuation and pressure distribution in the rectangular tank with the perforated baffle. Hosseinzadeh et al. [3] conducted shaking table experiments to examine the effect of the antisloshing baffle on the surface wave height of the liquid in the cylindrical tank. Xue et al. [4] conducted experimental studies on the effects of different baffles on the hydrodynamic pressure in the rectangular tank partially filled with liquid undergoing the lateral excitations. Yu et al. [5] studied experimentally the effectiveness of two floating plates in reducing the liquid sloshing in the membrane-type LNG tank under pitching excitations. Chu et al. [6] performed experimental investigation on the sloshing phenomenon in the rectangular liquid-filled tank with multiple bottom-mounted baffles undergoing lateral excitation. Experimental results reveal that the antisloshing baffle can help in effectively reducing the surface wave height and change hydrodynamic pressure distribution of the tank wall.The motion of the free surface of the liquid in various tanks which are equipped with the antisloshing baffle has been widely simulated using the numerical methods. Xue and Lin [7] developed the three-dimensional numerical model NEWTANK to investigate the viscous liquid sloshing in the rectangular tank with internal baffles of different shapes and arrangements. Akyildiz [8] studied the effect of the vertical baffle on the nonlinear liquid sloshing in the two-dimensional rectangular tank by the numerical algorithm based on the volume of the fluid technique. The results showed that, as the baffle height increases, the intensity of the vortex generated at the sharp edge of the baffle became weaker. Using the modal analysis and the BEM (boundary element method), Noorian et al. [9] developed the reduced order model to explore the interactions of the liquid and the flexible structure in the baffled tanks. It was concluded that the baffles with a large flexibility has a significant effect on the coupled dynamics characteristics. Koh et al. [10] examined the effectiveness of the constrained floating baffle (CFB) in reducing the surface wave height in the tank. Based on the noninertial reference system, Lu et al. [11] investigated the sloshing of the viscous liquid in baffled rectangular tanks. The numerical results indicated that the dissipation due to the viscosity has a significant effect on the sloshing responses. Goudarzi and Danesh [12] proposed the finite volume-type model to study the sloshing damping of the vertical baffle in the liquid-filled tank. Based on the Open-FOAM code, Jin et al. [13] presented the numerical algorithm to explore the resonant sloshing in the swaying tank equipped with a horizontal perforated plate and provided a better explanation of the energy dissipation caused by the horizontal perforated plate. Wang et al. [14] presented the numerical technique to examine the effects of several different baffles on the liquid oscillations in partially liquid-filled toroidal tanks. Sanapala et al. [15] conducted the numerical simulations to explore the sloshing dynamics of the liquid with free surface in the baffled rectangular tank undergoing vertical harmonic and seismic excitations. An advantage of the numerical method is that liquid sloshing in the complex-shaped tank can be simulated under generic excitation. In addition, the damping caused by vortex shedding at the sharp baffle edge can be considered in the numerical method. The disadvantage of the numerical method is that it is time-consuming, which makes the systematic parameter analysis very difficult.The analytical model, which has the high precision, can be used to solve the FSI (fluid-structure interaction) problem considering liquid sloshing efficiently, especially in the optimization design of baffles. Amabili [16] developed the analytical method to investigate the coupled frequencies and mode shapes of the horizontal circular cylindrical shells, which is partially filled with liquid. Amabili et al. [17] studied the coupled vibration of the vertical partially liquid-filled cylindrical container with the flexible bottom, in which the effect of the free surface wave was considered. Maleki and Ziyaeifar [18] presented the analytical method to investigate the sloshing damping in a tank equipped with the antisloshing baffle undergoing lateral excitations. They pointed out that the annular baffle has more effectiveness in reducing the liquid sloshing. Askari et al. [19] proposed an analytical method to investigate the effects of a rigid internal body on bulging and sloshing frequencies and modes of the flexible vertical cylindrical tank partially filled with liquid. Based on the liquid-domain decomposition, Faltinsen and Timokha [20] developed the analytical technology to explore the natural mode of the liquid sloshing in the rectangular tank with the slat-type screen installed at the tank middle. Based on the linear potential theory, Hasheminejad et al. [21] developed the rigorous analytical models for the two-dimensional transient sloshing of the liquid in the horizontal cylindrical tank with baffles under lateral excitations by using the conformal mapping technique. Kolaei et al. [22] investigated lateral sloshing force and overturning moment in the partially liquid-filled moving horizontal cylindrical tank with different longitudinal baffles. They concluded that the baffle mounted on the top of the tank has more effectiveness in reducing the resultant force caused by liquid sloshing. Wang et al. [23] provided the analytical solution of the coupled responses in a partially filled cylindrical tank with the flexible baffle undergoing lateral excitations. Sakai et al. [24] modeled the free surface of the liquid in a pitching horizontal cylindrical tank equipped with perforated plates. In the context of linear sloshing theory, Cho et al. [25] presented the semianalytical scheme to explore the effectiveness of the porous baffle in dissipating the sloshing energy and reducing the sloshing pressures acting on tank walls in the two-dimensional tank undergoing lateral excitations. The results suggested that the porous baffle positioned on the wall of the tank has more effectiveness than the porous baffle installed at the bottom. Analytical methods offer high computational efficiency and accuracy. They make it possible to derive the analytical expression for dynamic response, which can be used for revealing the physical mechanism of the liquid sloshing in baffled tanks. Analytical researches on the effectiveness of the flexible baffle in reducing the liquid sloshing in the cylindrical tank under pitching excitations in the literatures are relatively rare.Wang et al. [26] presented the analytical method to investigate the sloshing response of the liquid in a cylindrical tank equipped with the rigid annular baffles undergoing pitching excitations, and the dynamic deflections of the baffles and the sloshing damping were ignored. However, the fluid-structure interaction and the sloshing damping play important roles in the baffled tank. For this purpose, the semianalytical method is presented to study the coupled responses for the partially liquid-filled tank equipped with the single flexible annular baffle subjected to the pitching excitations. Using the analytical method [26], the Stokes–Joukowski potential can be solved. Substituting Stokes–Joukowski potential into the dynamic and kinematic equations for the free surface and the coupled vibration equation for the flexible baffle, the coupled response equations can be obtained. Additional damping terms are introduced to consider the energy dissipation of the liquid sloshing in the baffled tank. The semianalytical method is verified by comparing with the numerical results which were simulated by ADINA. The effect of the flexible baffle parameters on the coupled response is discussed in detail. In the subsequent study, we will quantitatively study the coupled responses in the partially liquid-filled cylindrical tank equipped with multiple flexible baffles with a different inner radius under lateral and pitching excitations.


## 2. Physical Model

As plotted in Figure1, the rigid cylindrical tank equipped with the single flexible baffle is partially filled with ideal liquid. O-XYZ denotes the inertial coordinate, while o‐xyz and o‐rθz are, respectively, the Cartesian and cylindrical coordinates which are fixed to the moving tank. The outer boundary of the annular baffle is clamped to the container, and the inner boundary of the baffle is free. The surface wave height relative to the static liquid level can be expressed as z=fr,θ,z. The tank is undergoing the pitching excitation denoted as Θ2t. The rotation axis of the pitching tank is y-axis. The linearized boundary-value problem for the velocity potential Φ is(1)1r∂∂rr∂Φ∂r+1r2∂2Φ∂θ2+∂2Φ∂z2=0inΩ,(2)∂Φ∂r=zΘ˙2cosθon the wall of the tank,(3)∂Φ∂z=−rΘ˙2cosθon the bottom of the tank,in which Ω denotes the liquid domain. The velocity of the liquid particle on the free surface should be equal to the velocity of the liquid-free surface. One can obtain the kinematic boundary condition on the free surface [27]:(4)∂Φ∂z=−rΘ˙2cosθ+∂f∂ton the free surface.Figure 1
Cylindrical container with single elastic baffle under pitching excitations.The liquid pressure is equal to the gas pressure above the liquid-free surface. This gives the dynamic boundary condition on the free surface [27]:(5)∂Φ∂t=grΘ2cosθ−fgon the free surface,where g is the gravitational acceleration. There is no liquid outflow or inflow in this problem. Thus, the volume of the liquid remains constant, namely,(6)∫Σ0frdrdθ=0.As plotted in Figure2, the single flexible baffle is positioned at z1 and the liquid domain Ω can be divided into 4 subdomains by 3 artificial interfaces. The notation Φi (i = 1, 2, 3, and 4) represents the velocity potential within the subdomain Ωi. The compatibility condition requires the liquid velocity in contact with the flexible baffle to be the same as the velocity of the baffle, that is,(7)∂Φ∂zz=z1=−rΘ˙2cosθ+∂w∂t,where w is the dynamic deflection of the baffle. The baffle is modeled by a thin elastic plate and is assumed to be made of linearly elastic, homogeneous, and isotropic material. Moreover, the effect of shear deformation and rotary inertia is neglected. According to [28, 29], the dynamic deflection satisfies the coupled dynamic equation:(8)D∇4w+ρτ∂2w∂t2=ρl∂Φ3∂tz=z1−ρl∂Φ1∂tz=z1,D=Eτ3121−ν2,where τ, ρ, ν, and E denote the thickness, density, Poisson’s ratio, and Young’s modulus for the single flexible baffle, respectively; ρl denotes the density of the liquid; R2 is the baffle’s outer radius which is equal to the inner radius of the tank; h denotes the static liquid level; and R1 represents the baffle’s inner radius. Five dimensionless coordinates, i.e., ξ=r/R2, ξ1=R1/R2, ς=z/R2, ς1=z1/R2, and ς0=h/R2, are introduced. For purpose of solving the boundary-value problem (Equations (1)–(8)), the velocity potential can be decomposed into two parts: the potential ΦR representing the rigid body motion of the liquid which is concerned with the container motion and the potential ΦS representing the liquid motion relative to the container which is concerned with the liquid sloshing. In order to describe the pitching motion of the container, the Stokes–Joukowski potential is introduced. According to [27], the potential ΦR can be expressed as follows:(9)ΦR=Θ˙2Ψ2ξ,θ,ς,where Ψ2 denotes the component of the Stokes–Joukowski potential along y-axis, which is associated with the pitching motion about y-axis. The potential ΦS, the surface wave height f, and the dynamic deflection w can be expressed as a summation of the coupled modes. Introducing the time-dependent generalized coordinates qmnt, qmn′t, and Rmnt, one can obtain the following function series:(10)f=∑qmntfmnξcosmθ,(11)w=∑qmn′tWmnξcosmθ,(12)Φ=ΦR+ΦS=Θ˙2Ψ2ξ,θ,ς+∑Rmntφmnξ,ςcosmθ,in which φmn is the coupled mode of the liquid, fmnξ satisfies fmnξ=φmnξ,ς0, and Wmn is the coupled mode of the flexible baffle. The detailed definitions of coupled modes Wmn and φmn are referred to the coupled eigenvalue problem [30]. In the literature, the liquid in the baffled tank was partitioned into several subdomains. By the method of separation of variables, the trial solution for the velocity potential corresponding to each subdomain was solved. The eigenfrequency equation can be established by expanding the free surface condition, the artificial interface conditions and coupled vibration equations into the Fourier series in the liquid height direction, and the Bessel series in the radial direction. The natural modes and frequencies ωmn can be obtained by solving the eigenfrequency equation. According to the literature, the coupled modes satisfy the following equations:(13)∂φmn1∂ςς=ς1=∂φmn3∂ςς=ς1=R2Wmn,D∇4Wmncosmθ=gΛmn2R2ρτWmn+ρlφmn1−φmn3ς=ς1cosmθ,where φmni (i = 1, 2, 3, and 4) denotes the coupled mode corresponding to the subdomain Ωi and the dimensionless frequency Λmn satisfies Λmn2=ωmn2R2/g. Substituting Equations (10)–(12) into the kinematic conditions, Equations (4) and (7) on the free surface and the flexible baffle, respectively, gives(14)∑RmntΛmn2φmn=∑q˙mntφmnon the free surface,∑RmntgΛmn2R2ρτWmn+ρlφmn1−φmn3ς=ς1=∑q˙mn′tgΛmn2R2ρτWmn+ρlφmn1−φmn3ς=ς1on the baffle.Figure 2
Subdomains and interfaces.The orthogonality condition of the coupled modes [30] leads to the following relations among the generalized coordinates:(15)Rmnt=R2q˙mntΛmn2,qmn′t=R2qmntΛmn2.


## 3. Stokes–Joukowski Potential

The notationΨ⇀ is introduced as the Stokes–Joukowski potential. According to Wang et al. [26], Ψ⇀ satisfies the following equations:(16)ΔΨ⇀=0inΩ,(17)∂Ψ⇀∂n=r⇀×n⇀onS0∪Σ0,where Σ0 represents the mean liquid-free surface, S0 denotes the wetted surface of the tank and the baffle, r⇀ denotes the radius vector with respect to the origin of o‐xyz, and n⇀ is the outer normal vector of Σ0 and S0. According to Equation (12), only the component Ψ2 along y-axis needs to be considered. According to Equations (16) and (17), Ψ2 satisfy(18)ΔΨ2=0inΩ,(19)∂Ψ2∂ς=−R22ξcosθon the bottom of the tank,the free surface,and the baffle,(20)∂Ψ2∂ξ=R22ςcosθon the wall of the tank.Obviously, the dimensionless componentψ2=Ψ2/R22 can be introduced. According to Equations (19) and (20), we have(21)ψ2=ςξ−ψcosθ,where ψ satisfies the following equations:(22)∂2ψ∂ξ2+1ξ∂ψ∂ξ−ψξ2+∂2ψ∂ς2=0inΩ,∂ψ∂ξ=0on the wall of the tank,∂ψ∂ς=2ξon the bottom of the tank,the free surface, and the baffle.The liquid domain in the baffled tank is divided into 4 subdomains, and the functionψ within the subdomains can be expressed as(23)ψξ,ς=ψiξ,ςinΩii=1,2,3,and4,where ψi satisfies the Laplace equation Δψi=0 and Γk denotes the interface between two neighboring subdomains (Ωi and Ωi′). Apparently, the neighboring subdomains satisfy the following continuity conditions:(24)ψi=ψi′,∂ψi∂nk=∂ψi′∂nkonΓk,where n⇀k denotes the vector normal to the interface and ψi (i = 1, 2, 3, and 4) can be solved using the analytical method, which was proposed by Wang et al. [26].


## 4. Coupled Response Equations

Substituting Equations (11) and (12) into the dynamics conditions on the flexible baffle, Equation (8) yields(25)∑qmn′tΛmn2gR2ρτWmn+ρlφmn1−φmn3ς=ςjcosmθ+∑q¨mn′tρτWmncosmθ=R22ρlΘ¨2ψ1−ψ3ς=ς1cosθ+∑R˙mntρlφmn3−φmn1ς=ς1cosmθ.According to Equation (15), Equation (25) can be rewritten as(26)∑qmntgρτWmn+ρlφmn1−φmn3ς=ς1cosmθ+∑R2q¨mntΛmn2ρτWmncosmθ=R22ρlΘ¨2∑ψ1−ψ3ς=ς1cosθ+∑R2q¨mntΛmn2ρlφmn3−φmn1ς=ς1cosmθ.Substituting Equations (10) and (12) into the dynamics condition on the free surface, Equation (5) gives(27)∑R2q¨mntΛmn2fmnξcosmθ+g∑qmntfmnξcosmθ−gΘ2ξ+Θ¨2ψΣ0−ς0ξR2R2cosθ=0.According to the orthogonality of the cosine functions, Equations (26) and (27) can be rewritten as(28)∑q¨1ntΛ1n2τW1nρρl+φ1n1−φ1n3ς=ς1+gR2∑q1ntτW1nρρl+φ1n1−φ1n3ς=ς1+R2∑Θ¨2ψ3−ψ1ς=ς1=0.∑q¨1ntΛ1n2f1nξ+gR2∑q1ntf1nξ−gΘ2ξ+Θ¨2ψΣ0−ς0ξR2=0,According to the orthogonality condition of the coupled modes [30], the dynamics equation corresponding to the generalized coordinate qnt can be obtained:(29)M1nq¨1nt+q1ntK1n=Ft,in which M1n, K1n, and Ft can be written as(30)M1n=∫01f1nf1nξdξ+∫ξ11τW1nρ/ρl+φ1n1−φ1n3ς=ς1W1nξdξΛ1n2,(31)K1n=Λ1n2gR2M1n,(32)Ft=gΘ2∫01f1nξ2dξ+Θ¨2∫01f1nψΣ0ξdξ−ς0∫01f1nξ2dξR2+R2Θ¨2∫ξ11ψ1−ψ3ς=ςjW1nξdξ.The liquid sloshing in the baffled container will be damped. The convenient way to account for the sloshing damping phenomenon generally happening in the real world is to introduce additional damping terms. So, Equation (33) takes the form:(33)M1nq¨1nt+2ζq˙1nM1nK1n+q1ntK1n=Ft,in which ζ denotes the damping ratio. According to the Bernoulli principle, the hydrodynamic pressure is solved using the following formula:(34)Pξ,θ,ς,t=−ρ∂Φ∂t=−ρR2cosθΘ¨2ςξ−ψR2+∑q¨1ntφ1nΛ1n2.Integrating the pressure functionPξ,θ,ς,t on the tank wall, the resultant force for the hydrodynamic pressure which acts on the tank wall can be calculated. The component of the resultant force along the θ=0 direction can be written as follows:(35)Fx=−ρR23π∑q¨1ntΛ1n2∫0ς0φ1nξ=1dς+ρR24πΘ¨2∫0ς0ψξ=1dς−ς022.The moment caused by liquid pressure which acts on the tank wall about theθ=π/2 axis can be written as follows:(36)Mwally=ρR24π∑q¨1ntΛ1n2∫0ς0φ1nξ=1ςdς−ρR25πΘ¨2∫0ς0ψξ=1ςdς−ς033.The moment caused by liquid pressure which acts on the bottom of the tank about theθ=π/2 axis can be written as follows:(37)Mbottomy=ρR24π∑q¨1ntΛ1n2∫01φ1nς=0ξ2dξ−ρR25πΘ¨2∫01ψς=0ξ2dξ.The moment caused by liquid pressure which acts on the flexible baffle about theθ=π/2 axis can be written as follows:(38)Mbaffley=ρR24π∑q¨1ntΛ1n2∫ξ11φ1n3−φ1n1ς=ς1ξ2dξ−R2Θ¨2∫ξ11ψ3−ψ1ς=ς1ξ2dξ.According to Equations (36)–(38), the resultant moment can be written as(39)My=Mwally+Mbottomy+Mbaffley.


## 5. Method Validation

The analytical model was validated by comparing with the numerical results, which were simulated using the finite element software ADINA (version 9.3.4). The potential-based fluid elements of ADINA are the efficient way to model the fluid. The elements have only one degree of freedom per node and can be coupled to ADINA structural elements and the pressure boundary condition by the interface elements. In many cases, ADINA can automatically generate the interface elements along the boundary of the liquid. The fluid-structure interface element connects the potential-based fluid element with an adjacent structural element. The free surface interface element is placed onto the boundary of a potential-based fluid element where the pressure has to be prescribed and the displacements of the fluid are required.The angular displacement of the pitching excitation follows a sinusoidal functionΘ2=Θ0sinω0t, where ω0 and Θ0 denote the frequency and amplitude of the excitation, respectively. The parameters for the liquid, the flexible baffle, and the tank in the comparison study have been listed in Table 1. The inner radius of the baffle was set as R1=0.5m, and the baffle was installed at z1=0.6m. The excitation frequencies were taken as ω0=3rad/s, 4 rad/s, and 5 rad/s. As depicted in Figures 3–5, the deflection of the flexible baffle at the inner edge for θ=0 and the surface wave heights on the wall fwall at θ=0 obtained from the present analytical method were compared with those which were obtained from the ADINA model. Figures 3–5 show that the analytical solutions are in good agreement with the numerical results simulated by the ADINA.Table 1
Parameters for the liquid and the baffled container in the comparison analysis.
Description
Parameter
Value
Inner radius of the container (m)
R
2
1
Height of the liquid (m)
h
1
Density of the liquid (kg/m3)
ρ
l
1000
Thickness of the baffle (mm)
τ
2
Density of the baffle (kg/m3)
ρ
2870
Poisson ratio of the baffle
ν
0.3
Elastic modulus of the baffle (Pa)
E
7.2e10
Amplitude of the excitation (rad)
Θ
0
0.01Figure 3
The time histories of the surface wave height on the wall (θ=0) and the dynamic deflection of the baffle at the inner edge (θ=0) for ω0=3rad/s. (a) surface wave height on the wall, (b) dynamic deflection of the baffle at the inner edge.
(a)
(b)Figure 4
The time histories of the surface wave height on the wall (θ=0) and the dynamic deflection of the baffle at the inner edge (θ=0) for ω0=4rad/s. (a) surface wave height on the wall, (b) dynamic deflection of the baffle at the inner edge.
(a)
(b)Figure 5
The time histories of the surface wave height on the wall (θ=0) and the dynamic deflection of the baffle at the inner edge (θ=0) for ω0=5rad/s. (a) surface wave height on the wall, (b) dynamic deflection of the baffle at the inner edge.
(a)
(b)When the inner radius of the baffle approaches that of the tank, the radial width of the annular baffle is very small. Currently, the baffle has very small effects on the liquid sloshing. To further verify the correctness of this method, the solution from the present analytical method was compared with the exact solution without the baffle developed by Ibrahim [31]. In this case, the inner radius of the baffle was set as R1=0.95m and the baffle was installed at z1=0.5m. The excitation frequency was taken as ω0=5rad/s. It can be observed from Figure 6 that the difference between the solution obtained by the present analytical and the exact solution is very small. To verify the correctness of the ADINA model, a liquid storage model without the baffle was established to compare the numerical solution with the exact solution. From Figure 6, the numerical solution is in good agreement with the exact solution.Figure 6
Comparisons of the time histories of the surface wave height on the wall (θ=0) among the exact solution, ADINA solution and present analytical result for ω0=5rad/s.


## 6. Steady-State Response

The steady-state solution of the coupled response undergoing the pitching sinusoidal excitations is conducted to explore the effectiveness of the flexible baffle in reducing the liquid sloshing. Substituting the pitching sinusoidal excitations into Equation (32), one can obtain the following formula:(40)Ft=F1nsinω0t,in which F1n satisfies the following equation:(41)F1n=Θ0g∫01f1nξ2dξ−ω02R2∫01ψ−ςξΣ0f1nξdξ+∫ξ11ψ1−ψ3ς=ς1W1nξdξ.According to Equation (33), the steady-state response can be obtained in the following form:(42)q1nt=A1nsinω0t−B1ncosω0t,in which A1n and B1n satisfy(43)A1n=F1nK1n1−β1n21−β1n22+2ζβ1n2,B1n=F1nK1n2ζβ1n1−β1n22+2ζβ1n2,β1n=ω0/ω1n.Substituting Equation (42) into Equation (10) gives the steady-state response for the surface wave height on the wall of the tank at θ=0:(44)fwall=sinω0t∑A1nf1nξ=1−cosω0t∑B1nf1nξ=1.Substituting Equation (42) into Equation (35) gives the steady-state response for the resultant force along the x-axis:(45)Fx=Fx1sinω0t+Fx2cosω0t,in which Fx1 and Fx2 satisfy the following equation:(46)Fx1=ρR23πω02∑A1nΛ1n2∫0ς0φ1nξ=1dς−R2Θ0∫0ς0ψξ=1dς−ς022,Fx2=−ρR23πω02∑B1nΛ1n2∫0ς0φ1nξ=1dς.Substituting Equation (42) into Equation (39) gives the steady-state response for the resultant moment about the y-axis:(47)My=My1sinω0t+My2cosω0t,in which My1 and My2 satisfy the following equation:(48)My1=ρR24πω02R2Θ0∫0ς0ψξ=1ςdς+∫01ψς=0ξ2dξ+∫ξ11ψ3−ψ1ς=ς1ξ2dξ−ς033−∑A1nΛ1n2∫01φ1nς=0ξ2dξ+∫0ς0φ1nξ=1ςdς+∫ξ11φ1n3−φ1n1ς=ς1ξ2dξ,My2=ρR24πω02∑B1nΛ1n2∫01φ1nς=0ξ2dξ+∫0ς0φ1nξ=1ςdς+∫ξ11φ1n3−φ1n1ς=ς1ξ2dξ.According to Equations (44), (45), and (47), fwall, Fx, and My can be written as(49)fwall=fmaxsinω0t−ϕ1,Fx=Fmaxsinω0t−ϕ2,My=Mmaxsinω0t−ϕ3,in which fmax, Fmax, and Mmax are the maximum amplitudes of the steady-state responses. ϕ1, ϕ2, and ϕ3 are the phases of fwall, Fx, and My. The maximum amplitudes of the steady-state responses satisfy the following equations:(50)fmax=∑A1nf1nξ=12+∑B1nf1nξ=12,Fmax=Fx12+Fx22,Mmax=My12+My22.The coupled steady-state responses versus the parameters of the elastic baffle (the inner radiusR1, the position z1, and the thickness τ) are systematically investigated. Five different artificial damping ratios are considered: ζ=0.01, 0.02, 0.03, 0.04, and 0.05. The excitation frequency is taken as ω0=4.14rad/s, which is close to the first natural frequency of the liquid sloshing in the container with no baffle. To study the influence of the baffle’s inner radius R1, the baffle is installed at z1=0.7m and the thickness of the baffle is fixed at τ=2mm. To study the effect of the baffle’s position z1, the inner radius of the baffle is taken as R1=0.4m and the thickness of the baffle is fixed at τ=2mm. The effect of the thickness of the baffle is discussed when the baffle is located at z1=0.7m, and the baffle’s inner radius is fixed at R1=0.5m. Other parameters for the steady-state analysis have been listed in Table 2.Table 2
The parameters for the baffled container in the steady-state analysis.
Description
Parameter
Value
Inner radius of the container (m)
R
2
1
Height of the liquid (m)
h
1
Density of the liquid (kg/m3)
ρ
l
1000
Density of the baffle (kg/m3)
ρ
2870
Poisson ratio of the baffle
ν
0.3
Elastic modulus of the baffle (Pa)
E
7.2e9
Amplitude of the excitation (rad)
Θ
0
0.01


### 6.1. Coupled Responses versus the Inner Radius of the Baffle

The amplitudes of the surface wave height, the resultant force, and the moment versus the inner radius of the baffle are plotted in Figures7–9, respectively. The corresponding phases are also illustrated. According to Figures 7–9, when the inner radius of the flexible baffle is smaller than 0.3 m, the coupled dynamic responses do not change significantly as the inner radius increases. In fact, the change in the area of the baffle is the main factor leading to the variation of the coupled response. When the inner radius of the baffle is small, the rate of change of its area is small. Therefore, the variation of the coupled response is negligible. When the inner radius is bigger than 0.3 m, the coupled dynamic responses increase rapidly with the increase of the inner radius. When the inner radius is 0.45 m, there are peaks in the curves. The excitation frequency is close to the natural frequency of the coupled system, and the corresponding phases are 90°. Therefore, the resonance will occur, which increases the amplitude of the coupled response rapidly. However, such an increase can be effectively reduced by the damping ratio. Figure 10 gives the resonant time histories of the surface wave height for different damping ratios. As the inner radius increases continuously, fmax, Fmax, and Mmax decrease rapidly, reaching the minimum values when the inner radius is 0.628 m, 0.652 m, and 0.626 m, respectively. At the minimum of the coupled response, the corresponding phase will undergo the abrupt change from close to 0° to almost 180°. When Fmax reaches the lowest point (R1=0.652m and z1=0.7m), the pressure distributions on the tank wall and baffle are plotted in Figure 11. The effect of sloshing is more profound in the liquid above the baffle, and the pressure due to the sloshing portion at θ=0 is out of phase with the excitation. The pressure due to the liquid below the baffle at θ=0 is in phase with the excitation. The effect of the damping ratio on the pressure due to the sloshing portion is more significant than on the pressure due to the liquid below the baffle. The coupled dynamic responses increase rapidly with the continuously increasing inner radius. When the baffle is narrow, the effects of the baffle on liquid sloshing are sharply reduced with the increase of the inner radius of the baffle, and the natural frequency is close to the case without the baffle, which causes the resonance.Figure 7
The amplitudes and the phases of the surface wave height versus the inner radius of the flexible baffle forz1=0.7m and τ=2mm. (a) the amplitude of the surface wave height, (b) the phase of the surface wave height.
(a)
(b)Figure 8
The amplitudes and the phases of the resultant force versus the inner radius of the flexible baffle forz1=0.7m and τ=2mm. (a) the amplitude of the resultant force, (b) the phase of the resultant force.
(a)
(b)Figure 9
The amplitudes and the phases of resultant moment versus the inner radius of the flexible baffle forz1=0.7m and τ=2mm. (a) the amplitude of the resultant moment, (b) the phase of the resultant moment.
(a)
(b)Figure 10
The resonant time histories of the surface wave height on the wall (θ=0) for z1=0.7m, R1=0.45m and τ=2mm.Figure 11
The pressure distributions on the tank wall and baffle atθ=0 when ω0t=π/2 for z1=0.7m, R1=0.652m and τ=2mm. (a) The pressure distribution of the tank wall above the baffle, (b) The pressure distribution of the tank wall below the baffle, (c) The pressure distribution acting on the upper surface of the baffle, (d) The pressure distribution acting on the lower surface of the baffle.
(a)
(b)
(c)
(d)


### 6.2. Coupled Responses versus the Position of the Baffle

The amplitudes of the surface wave height, the resultant force, and the moment versus the position of the baffle are plotted in Figures12–14, respectively. The corresponding phases are also illustrated. According to Figures 12–14, the coupled dynamic responses become stronger when the baffle approaches the bottom of the container. When the flexible baffle is far from the free surface of the liquid, the baffle has relatively little influence on the liquid sloshing, and the excitation frequency is close to the first natural frequency of the coupled system. Therefore, the system generates resonance under the external excitation, leading to large amplitude of the coupled response. Currently, the amplitude of the coupled response decreases with the increase of the damping ratio. In addition, the phases for ζ=0.01 approximate 90°. When the baffle is located at z1=0.1m, the resonant time histories of the surface wave height for different damping ratios are plotted in Figure 15. As the flexible baffle ascends, the coupled dynamic responses corresponding to different damping ratios gradually reduce to the same minimum value. Meanwhile, the natural frequency of the system is less than the excitation frequency, and the amplitude of the coupled response is very small. This is mainly because the direction of external excitation changes so fast that the liquid sloshing cannot change accordingly as a result of the inertial effects. At this moment, the phases for ζ=0.01 approximate 180°. Thus, the effectiveness of the sloshing reduction can be improved when the flexible baffle is installed near the free surface (Figures 16–18).Figure 12
The amplitudes and the phases of the surface wave height versus the position of the flexible baffle forR1=0.4m and τ=2mm. (a) the amplitude of the surface wave height, (b) the phase of the surface wave height.
(a)
(b)Figure 13
The amplitudes and the phases of the resultant force versus the position of the flexible baffle forR1=0.4m and τ=2mm. (a) the amplitude of the resultant force, (b) the phase of the resultant force.
(a)
(b)Figure 14
The amplitudes and the phases of resultant moment versus the position of the flexible baffle forR1=0.4m and τ=2mm. (a) the amplitude of the resultant moment, (b) the phase of the resultant moment.
(a)
(b)Figure 15
The resonant time histories of the surface wave height on the wall (θ=0) for z1=0.1m, R1=0.4m and τ=2mm.Figure 16
The amplitudes and the phases of the surface wave height versus the thickness of the flexible baffle forR1=0.5m and z1=0.7m. (a) the amplitude of the surface wave height, (b) the phase of the surface wave height.
(a)
(b)Figure 17
The amplitudes and the phases of the resultant force versus the thickness of the flexible baffle forR1=0.5m and z1=0.7m. (a) the amplitude of the resultant force, (b) the phase of the resultant force.
(a)
(b)Figure 18
The amplitudes and the phases of resultant moment versus the thickness of the flexible baffle forR1=0.5m and z1=0.7m. (a) the amplitude of the resultant moment, (b) the phase of the resultant moment.
(a)
(b)


### 6.3. Coupled Responses versus the Thickness of the Baffle

In this part, the thickness of the baffle ranges between 0.001 m and 0.005 m and the inner radius of the baffle is taken as 0.5 m. Obviously, the thickness of the baffle is much less than the dimensions of the baffle. As a result, the assumption of the thin plate can be applied here. The amplitudes of the surface wave height, the resultant force, and the moment versus the thickness of the flexible baffle are plotted in Figures12–14, respectively. The corresponding phases are also illustrated. The coupled responses increase rapidly the increase of the thickness of the baffle. When the thickness of the baffle increases to 1.83 mm, there are peaks in the curves. This is because the excitation frequency is close to the third natural frequency of the coupled system, and the peak decreases with the increasing damping ratio. At this point, the phases are 90°. As the thickness increases continuously, fmax, Fmax, and Mmax decrease rapidly, reaching the minimum values when the thickness of the baffle is 2.88 mm, 3.43 mm, and 2.82 mm, respectively, and the phases approximate 90°. The influence of the damping ratio on the amplitudes of the coupled dynamic response can be ignored when the thickness of the baffle is bigger than 2.55 mm. This is mainly because the baffle has a greater influence on the natural frequency of the liquid sloshing as the stiffness increases, making the natural frequency of the liquid sloshing far away from the excitation frequency.


## 6.1. Coupled Responses versus the Inner Radius of the Baffle

The amplitudes of the surface wave height, the resultant force, and the moment versus the inner radius of the baffle are plotted in Figures7–9, respectively. The corresponding phases are also illustrated. According to Figures 7–9, when the inner radius of the flexible baffle is smaller than 0.3 m, the coupled dynamic responses do not change significantly as the inner radius increases. In fact, the change in the area of the baffle is the main factor leading to the variation of the coupled response. When the inner radius of the baffle is small, the rate of change of its area is small. Therefore, the variation of the coupled response is negligible. When the inner radius is bigger than 0.3 m, the coupled dynamic responses increase rapidly with the increase of the inner radius. When the inner radius is 0.45 m, there are peaks in the curves. The excitation frequency is close to the natural frequency of the coupled system, and the corresponding phases are 90°. Therefore, the resonance will occur, which increases the amplitude of the coupled response rapidly. However, such an increase can be effectively reduced by the damping ratio. Figure 10 gives the resonant time histories of the surface wave height for different damping ratios. As the inner radius increases continuously, fmax, Fmax, and Mmax decrease rapidly, reaching the minimum values when the inner radius is 0.628 m, 0.652 m, and 0.626 m, respectively. At the minimum of the coupled response, the corresponding phase will undergo the abrupt change from close to 0° to almost 180°. When Fmax reaches the lowest point (R1=0.652m and z1=0.7m), the pressure distributions on the tank wall and baffle are plotted in Figure 11. The effect of sloshing is more profound in the liquid above the baffle, and the pressure due to the sloshing portion at θ=0 is out of phase with the excitation. The pressure due to the liquid below the baffle at θ=0 is in phase with the excitation. The effect of the damping ratio on the pressure due to the sloshing portion is more significant than on the pressure due to the liquid below the baffle. The coupled dynamic responses increase rapidly with the continuously increasing inner radius. When the baffle is narrow, the effects of the baffle on liquid sloshing are sharply reduced with the increase of the inner radius of the baffle, and the natural frequency is close to the case without the baffle, which causes the resonance.Figure 7
The amplitudes and the phases of the surface wave height versus the inner radius of the flexible baffle forz1=0.7m and τ=2mm. (a) the amplitude of the surface wave height, (b) the phase of the surface wave height.
(a)
(b)Figure 8
The amplitudes and the phases of the resultant force versus the inner radius of the flexible baffle forz1=0.7m and τ=2mm. (a) the amplitude of the resultant force, (b) the phase of the resultant force.
(a)
(b)Figure 9
The amplitudes and the phases of resultant moment versus the inner radius of the flexible baffle forz1=0.7m and τ=2mm. (a) the amplitude of the resultant moment, (b) the phase of the resultant moment.
(a)
(b)Figure 10
The resonant time histories of the surface wave height on the wall (θ=0) for z1=0.7m, R1=0.45m and τ=2mm.Figure 11
The pressure distributions on the tank wall and baffle atθ=0 when ω0t=π/2 for z1=0.7m, R1=0.652m and τ=2mm. (a) The pressure distribution of the tank wall above the baffle, (b) The pressure distribution of the tank wall below the baffle, (c) The pressure distribution acting on the upper surface of the baffle, (d) The pressure distribution acting on the lower surface of the baffle.
(a)
(b)
(c)
(d)


## 6.2. Coupled Responses versus the Position of the Baffle

The amplitudes of the surface wave height, the resultant force, and the moment versus the position of the baffle are plotted in Figures12–14, respectively. The corresponding phases are also illustrated. According to Figures 12–14, the coupled dynamic responses become stronger when the baffle approaches the bottom of the container. When the flexible baffle is far from the free surface of the liquid, the baffle has relatively little influence on the liquid sloshing, and the excitation frequency is close to the first natural frequency of the coupled system. Therefore, the system generates resonance under the external excitation, leading to large amplitude of the coupled response. Currently, the amplitude of the coupled response decreases with the increase of the damping ratio. In addition, the phases for ζ=0.01 approximate 90°. When the baffle is located at z1=0.1m, the resonant time histories of the surface wave height for different damping ratios are plotted in Figure 15. As the flexible baffle ascends, the coupled dynamic responses corresponding to different damping ratios gradually reduce to the same minimum value. Meanwhile, the natural frequency of the system is less than the excitation frequency, and the amplitude of the coupled response is very small. This is mainly because the direction of external excitation changes so fast that the liquid sloshing cannot change accordingly as a result of the inertial effects. At this moment, the phases for ζ=0.01 approximate 180°. Thus, the effectiveness of the sloshing reduction can be improved when the flexible baffle is installed near the free surface (Figures 16–18).Figure 12
The amplitudes and the phases of the surface wave height versus the position of the flexible baffle forR1=0.4m and τ=2mm. (a) the amplitude of the surface wave height, (b) the phase of the surface wave height.
(a)
(b)Figure 13
The amplitudes and the phases of the resultant force versus the position of the flexible baffle forR1=0.4m and τ=2mm. (a) the amplitude of the resultant force, (b) the phase of the resultant force.
(a)
(b)Figure 14
The amplitudes and the phases of resultant moment versus the position of the flexible baffle forR1=0.4m and τ=2mm. (a) the amplitude of the resultant moment, (b) the phase of the resultant moment.
(a)
(b)Figure 15
The resonant time histories of the surface wave height on the wall (θ=0) for z1=0.1m, R1=0.4m and τ=2mm.Figure 16
The amplitudes and the phases of the surface wave height versus the thickness of the flexible baffle forR1=0.5m and z1=0.7m. (a) the amplitude of the surface wave height, (b) the phase of the surface wave height.
(a)
(b)Figure 17
The amplitudes and the phases of the resultant force versus the thickness of the flexible baffle forR1=0.5m and z1=0.7m. (a) the amplitude of the resultant force, (b) the phase of the resultant force.
(a)
(b)Figure 18
The amplitudes and the phases of resultant moment versus the thickness of the flexible baffle forR1=0.5m and z1=0.7m. (a) the amplitude of the resultant moment, (b) the phase of the resultant moment.
(a)
(b)


## 6.3. Coupled Responses versus the Thickness of the Baffle

In this part, the thickness of the baffle ranges between 0.001 m and 0.005 m and the inner radius of the baffle is taken as 0.5 m. Obviously, the thickness of the baffle is much less than the dimensions of the baffle. As a result, the assumption of the thin plate can be applied here. The amplitudes of the surface wave height, the resultant force, and the moment versus the thickness of the flexible baffle are plotted in Figures12–14, respectively. The corresponding phases are also illustrated. The coupled responses increase rapidly the increase of the thickness of the baffle. When the thickness of the baffle increases to 1.83 mm, there are peaks in the curves. This is because the excitation frequency is close to the third natural frequency of the coupled system, and the peak decreases with the increasing damping ratio. At this point, the phases are 90°. As the thickness increases continuously, fmax, Fmax, and Mmax decrease rapidly, reaching the minimum values when the thickness of the baffle is 2.88 mm, 3.43 mm, and 2.82 mm, respectively, and the phases approximate 90°. The influence of the damping ratio on the amplitudes of the coupled dynamic response can be ignored when the thickness of the baffle is bigger than 2.55 mm. This is mainly because the baffle has a greater influence on the natural frequency of the liquid sloshing as the stiffness increases, making the natural frequency of the liquid sloshing far away from the excitation frequency.


## 7. Conclusion

The coupled dynamic response of partially liquid-filled cylindrical tank equipped with single flexible baffle subjected to the pitching excitation has been analytically investigated. Introducing the generalized coordinates, the velocity potential containing the Stokes–Joukowski potential is formulated. The Stokes–Joukowski potential can be calculated using the analytical method [26]. According to coupled conditions on the flexible baffle and the boundary conditions on the free surface, the coupled dynamic response equation can be obtained. Compared with the numerical results which were simulated by ADINA, the analytical method is verified. The following observations are drawn:(1)
The introduction of the damping term can reflect the dissipation effect when the excitation frequency is close to the natural frequency of the coupled system.(2)
The effectiveness of the sloshing reduction can be improved when the flexible baffle is installed near the free surface.(3)
The baffle has a greater influence on the natural frequency of the liquid sloshing as the stiffness increases, making the natural frequency of the liquid sloshing far away from the excitation frequency.



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*Source: 9854187-2019-07-01.xml*