# Nonlinear Sloshing of Liquid in a Rigid Cylindrical Container with a Rigid Annular Baffle under Lateral Excitation

Authors: Jiadong Wang; Sai Huen Lo; Ding Zhou; Yun Dong
Journal: Shock and Vibration
 (2019)

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## Abstract

Nonlinear response of liquid partially filled in a rigid cylindrical container with a rigid annular baffle subjected to lateral excitation is studied. A semianalytical approach is presented to determine the natural frequencies and modes of the liquid sloshing. Introducing the generalized time-dependent coordinates, the surface wave height and the velocity potential are expressed in terms of the natural modes of liquid sloshing. Based on the Bateman–Luke variational principle, the infinite-dimensional modal system is given by the variational procedure. The infinite-dimensional modal system is reduced by using the Moiseev asymptotic relations. The resultant hydrodynamic force and moment of the liquid pressure acting on the container mainly depend on the position vector of the mass center of the liquid. Expanding the integral about the weighted position coordinates into the Taylor series about the surface wave height at the unperturbed free surface gives the formula of the position vector of the mass center, which depends only on the generalized time-dependent coordinates. Excellent agreements have been achieved by comparing the present results with those obtained from Gavrilyuk’s solution and SPH solution. Finally, the surface wave height, resultant hydrodynamic force, and hydrodynamic moment for a container subjected to harmonic lateral excitation are discussed in detail.


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## Body

## 1. Introduction

Liquid sloshing can cause a serious problem in liquid containers subjected to lateral excitation. For instance, the failure of the floating roof and the fire of the oil storage containers due to liquid sloshing were observed in some earthquake disasters [1]. As is well known, the baffle mounted in a partially liquid-filled container could provide the energy dissipation of the liquid [2], which reduces the sloshing amplitude. As a fringe benefit, the baffles could enhance the stiffness of the container.Subjected to excitations, the linear dynamic response of liquid in a container equipped with baffles was investigated. Abramson [3] conducted some experiments to study the response of the liquid in a baffled fuel container, which is subjected to pitching and lateral excitations, respectively. Using the boundary element method, Gedikli and Ergüven [4] simulated the seismic response of the rigid liquid-filled container with an annular baffle. Isaacson and Premasiri [5] studied the dynamic damping due to baffles in a liquid-filled rectangular container or reservoir undergoing horizontal excitation. Using the finite element method, Biswal et al. [6] studied the effect of an annular baffle on the dynamic response of a partially liquid-filled cylindrical container. Based on the velocity potential formulation and linearization theory, Goudarzi et al. [7] developed an analytical model to estimate the dynamic damping ratio of liquid sloshing due to baffles in a liquid-filled rectangular container subjected to lateral excitation. Wang et al. [8, 9] developed a semianalytical method to solve the natural frequencies and modes of the liquid sloshing in the cylindrical container with single or multiple baffles. Based on the natural frequencies and modes obtained, the linear dynamic response of the liquid in a baffled container subjected to lateral excitation was analyzed [10, 11].Generally, the nonlinear sloshing problems of liquid are described by the Navier–Stokes equations and the dynamic and kinematic conditions on the free surface [12]. The problems are usually investigated by the experimental and numerical methods [13–15]. Carra et al. [16] investigated the linear and geometrically nonlinear dynamical response of a thin plate in contact with liquid. They proposed the experimental method to study the behavior of the liquid’s free surface during the nonlinear vibration of the thin plate undergoing harmonic excitations. Wu et al. [17] developed a time-independent finite difference scheme with fictitious cell technique to investigate viscous liquid sloshing in two-dimensional containers with baffles. However, this description creates some mathematical and computational difficulties. Due to these difficulties, a lot of researchers are interested in using the potential principle as described by Zhou et al. [18].Based on the potential formulations, there are two different approaches to study the nonlinear problems of liquid sloshing. The first is the numerical method [19, 20]. Hugo et al. [21] proposed the numerical method to investigate the sloshing response of cylindrical containers subjected to earthquake ground motion. Cho and Lee [22] and Cho et al. [23] simulated the large amplitude liquid sloshing in a two-dimensional baffled rectangular container under lateral excitation. Using the finite element method, Biswal et al. [24] studied the two-dimensional nonlinear sloshing in both rectangular and cylindrical containers equipped with rigid baffles.The second is the multimodal method, namely, the free surface and velocity potential are expressed in terms of the generalized Fourier expansion. Substituting the expansions into the original free-boundary value equations yields the modal system. The linear and nonlinear multimodal methods, which can be used to conduct the analytical investigations of resonant sloshing regimes, secondary resonance, and chaos, are presented in [25]. It also includes the nonlinear multimodal method developed by Faltinsen et al. [26]. Using the Bateman–Luke variational statement, they derived the infinite-dimensional modal system for the nonlinear sloshing of liquid in a container. Based on the Narimanov–Moiseev asymptotic relations, the infinite-dimensional modal system can be reduced to the finite-dimensional modal system. The method is applicable to arbitrary excitations and a broad class of the container shapes [27–30]. Lukovsky [31] developed the five-dimensional modal system for the nonlinear sloshing in the upright circular cylindrical container, which was validated by the experiments. Using the asymptotic modal method, Gavrilyuk et al. [32, 33] investigated the nonlinear resonant sloshing of liquid in a rigid circular cylindrical container with a rigid annular baffle. Raynovskyy and Timokha [34] studied the damped steady-state resonant sloshing in a circular base container by the nonlinear multimodal system, which is equipped with linear damping terms associated with the logarithmic decrements of the natural sloshing modes. It should be mentioned that the research on the multimodal system of containers equipped with baffles is insufficient.Using the variational multimodal method, Zhou et al. [18] investigated the nonlinearity of the free sloshing of the liquid in a baffled container caused by some initial displacement of the free surface from the viewpoint of no external excitation. This calculation focused on the surface wave height as an indicator of the liquid sloshing. However, the hydrodynamic forces due to the liquid sloshing were not at all investigated. This can be a problem because if the excitation frequency is close to the natural frequency of the liquid, the resonant-amplified impact force may cause damage to the container. Thus, the proposed study focuses on the nonlinear response of the forced sloshing of the liquid in a baffled container subjected to lateral harmonic or seismic excitation. By applying Taylor series expansion to the integral of the weighted position coordinates of the surface wave height, we derive formulae for the resultant hydrodynamic force and the moment of liquid pressure acting on the baffled container. Parametric analysis is applied to the resultant hydrodynamic force and moment, and the distribution of liquid pressure on the container wall is discussed. We also conduct steady-state analysis in the frequency domain.


## 2. Multimodal Method

A rigid cylindrical container with a thin rigid annular baffle, which is partially filled with inviscid, incompressible, and irrotational liquid, is depicted in Figure1, where Ox′y′z′ is the absolute coordinate system and Orθz is the relative cylindrical coordinate system fixed at the container. The origin of Orθz is located at the center of the bottom of the container, and the z-axis is orthogonal to the bottom of the container. The inner radius of the container is r2, and the inner radius of the baffle is r1. z1 and z2 denote, respectively, the positions of the baffle and the unperturbed (hydrostatic) free surface. The thickness of the baffle is very small compared with the inner radius of the container. The density of the liquid is ρ. The time-dependent liquid domain Ω is bounded by the free surface ∑, the wet container wall Sw, the container bottom S¯b, and the baffle surface S¯p. The potential formulation of the nonlinear sloshing problem is given as follows [25]:(1)Δϕ=0inΩt,(2)∂ϕ∂n=v0⋅n+ω⋅r×nonS¯b∪S¯p∪Sw,(3)∂ϕ∂n=v0⋅n+ω⋅r×n+f˙1+∇f2on∑,(4)∂ϕ∂t+∇ϕ⋅∇ϕ2−∇ϕ⋅v0+ω×r−g⋅r0′+r=0on∑,(5)∫ΩtdΩ=const,(6)z=fr,θ,tt=0=f0r,θ,ϕr,θ,z,tt=0=ϕ0r,θ,z,where n is the outer normal of the boundary of the liquid domain Ω, z=fr,θ,t+z2 is the time-dependent shape equation of the free surface ∑, v0 is the translation velocity vector of the origin O related to the coordinate system Ox′y′z′, ω is the angular velocity vector of the container, r′ is the position vector with respect to O′, r0′ is the position vector of the point O with respect to O′, r is the position vector with respect to O, and g is the gravity acceleration vector, and equation (6) denotes the initial conditions. In the cylindrical coordinate system Orθz, the unit vectors codirectional with the longitudinal axis (θ=0) and the z-axis (Oz) are e1 and e3, respectively. The unit vector e2 is given by e2=e1×e3. Then, the vectors in equations (1)–(6) can be written as(7)r=r1e1+r2e2+r3e3,ω=ω1e1+ω2e2+ω3e3,v0=v01e1+v02e2+v03e3,v˙0=v˙01e1+v˙02e2+v˙03e3.Figure 1
The partially liquid-filled rigid cylindrical container with an annular rigid baffle subjected to the lateral excitation.When no overturning wave occurs, the continuum problem is reduced to a discrete conservative mechanical system with infinite degrees of freedom. This implies that the free surface and velocity potential can be expressed as(8)fr,θ,t=∑p=1∞βptη¯pr,θ,ϕ=v0⋅r+ω⋅Θ+∑q=1∞RqtΦ¯qr,θ,z,where the vector function Θ denotes the Stokes–Zhukovsky potential, which was defined in [25], βpt and Rqt are the generalized coordinates, the set η¯pr,θ is a Fourier basis on the unperturbed free surface, and the set Φ¯qr,θ,z should be complete in the unperturbed liquid domain. The variational procedure gives the generalized form of the infinite-dimensional modal system [18], which is the complete analog of the original free-boundary problem given by equations (1)–(6) as follows:(9)dAqdt−∑q′=1∞Rq′Aqq′=0,(10)∑q=1∞∂Aq∂βpR˙q+∑q=1∞∑q′=1∞∂Aqq′∂βpRqRq′+ω˙1∂I21∂βp+ω˙2∂I22∂βp+ω˙3∂I23∂βp+ω1∂I31∂βp+ω2∂I32∂βp+ω3∂I33∂βp−ddtω1∂I31∂β˙p+ω2∂I32∂β˙p+ω3∂I33∂β˙p+v˙01+ωv203−ωv302−g1∂I11∂βp+v˙02+ωv301−ωv103−g2∂I12∂βp+v˙03+ωv102−ωv201−g3∂I13∂βp−ω122∂J11∂βp−ω222∂J22∂βp−ω322∂J33∂βp−ω1ω2∂J12∂βp−ω1ω3∂J13∂βp−ω2ω3∂J23∂βp=0,where I1κ, I2κ, I3κ, Aq, and Aqq′ are the integrals over the time-dependent domain Ωt and only depend on βp and β˙p. Their explicit formulations are given by Zhou et al. [18]. For the forced sloshing of liquid along the longitudinal axis (θ=0), equation (10) can be rewritten as(11)∑q=1∞∂Aq∂βpR˙q+∑q=1∞∑q′=1∞∂Aqq′∂βpRqRq′+∂I11∂βpx¨t+g∂I13∂βp=0.The mode shapes of the linear sloshing solutions areΦ¯qr,θ,z. According to the Moiseev asymptotic relations [25], we have(12)Φ¯1=Φ11cosθ,Φ¯2=Φ11sinθ,Φ¯3=Φ01,Φ¯4=Φ21cos2θ,Φ¯5=Φ21sin2θ,η¯1=Φ11z=z2cosθ,η¯2=Φ11z=z2sinθ,η¯3=Φ01z=z2,η¯4=Φ21z=z2cos2θ,η¯5=Φ21z=z2sin2θ,where Φmnm=0,1,2;n=1 are the mode shapes of liquid sloshing and can be obtained by the semianalytical approach, which was developed by Wang et al. [9]. Wang et al. [9] divided the liquid domain within a baffled container into several subdomains to ensure that the liquid velocity potential for each subdomain is of class C1 and could be calculated with continuity boundary conditions, as shown in Figure 2. The analytical solution for the velocity potential in each subdomain can be obtained using separation of variables. The eigenfrequency equation is obtained by expanding the free surface and artificial interface conditions into the Fourier series in the direction of the liquid height and the Bessel series in the radial direction. Obviously, the partition of the liquid domain is different from that applied by Gavrilyuk et al. [32], who determined the natural frequencies and modes using the Galerkin method, which is based on Green’s function. Wang et al. [9] compared the natural frequencies obtained from these two different methods and found that the difference between the results obtained by two methods is less than 0.2%. Figure 3 shows that the natural modes calculated by Wang et al. [9] are in good agreement with those calculated by Gavrilyuk et al. [32].Figure 2
Partition of the liquid domain developed by Wang et al. [9].Figure 3
Comparison of the natural modes calculated by Wang et al. [9] and Gavrilyuk et al. [32].
(a)
(b)
(c)In the following study, the perturbation parameterε is introduced, which denotes the ratio of the excitation amplitude and the inner radius of the container. The nonlinear modal system assumes that the forcing amplitude is sufficiently small. Miles [35, 36] demonstrated that generalized coordinates for Φ¯1 and Φ¯2 are dominating and are Oε1/3. The generalized coordinates for Φ¯3, Φ¯4, and Φ¯5 are Oε2/3 and the other higher modes are Oε or oε. In addition, Miles [35, 36] showed that the other higher modes contribute <1% to the response. Lukovsky [31] derived the five-dimensional nonlinear modal system. The results for the surface wave height and hydrodynamic forces are in good agreement with the experimental results. The higher-order terms than Oε will be neglected in the perturbation procedure, which means the perturbation parameter ε should be very small, i.e., the nonlinear analysis is considered. Substituting equation (8) in equations (9) and (11) and keeping terms up to Oε, one has the following asymptotic modal system:(13)β¨1+σ112β1+K1β1β˙12+β¨1β12+β1β˙22+β1β2β¨2+K2β¨1β22+2β˙1β˙2β2−β1β˙22−β1β2β¨2+K3β¨1β4+β˙1β˙4+β¨2β5+β˙2β˙5−K4β1β¨4+β2β¨5+K5β¨1β3+β˙1β˙3+K6β1β¨3−x¨tr23π3=0,(14)β¨2+σ112β2+K1β2β˙22+β¨2β22+β2β˙12+β1β2β¨1+K2β¨2β12+2β˙1β˙2β2−2β2β˙12−β1β2β¨1+K3β¨1β5+β˙1β˙5−β¨2β4−β˙2β˙4−K4β2β¨4−β1β¨5+K5β¨2β3+β˙2β˙3+K6β2β¨3=0,(15)β¨3+σ012β3+K8β˙12+β˙22+K10β1β¨1+β2β¨2=0(16)β¨4+σ212β4+K7β˙22−β˙12+K9β2β¨2+β1β¨1=0(17)β¨5+σ212β5−2K7β˙1β˙2−K9β2β¨1+β1β¨2=0where the explicit expressions of Ki (i = 1, …, 9) have been given by [18] and σmn (m = 0, 1, 2; n = 1) denotes the natural sloshing frequencies.


## 3. Resultant Hydrodynamic Force and Moment

The liquid velocity potential in each subdomain has the continuous boundary conditions of classC1. The liquid domain with a single baffle can be divided into 4 subdomains (Ω¯1,Ω¯2,Ω3,Ω4) with 3 artificial interfaces (Γ¯1,Γ¯2,Γ3) as shown in Figure 4, namely,(18)ϕr,θ,z,t=ϕir,θ,z,t,r,θ,z∈Ω¯ii=1,2,ϕir,θ,z,t,r,θ,z∈Ωii=3,4.Figure 4
Cross section, subdomains, and artificial interfaces of the time-dependent liquid domainΩ.
(a)
(b)Obviously,Ω¯ii=1,2 and Γ¯kk=1,2 are time-independent; Ωii=3,4 and Γkk=3 are time-dependent (it should be noted that Ω3 and Ω4 have the free surfaces ∑1 and ∑2, respectively.). As shown in Figure 5, the unperturbed (hydrostatic) liquid domain Ω¯ can also be divided into 4 subdomains (Ω¯i,i=1,2,3,4) with three artificial interfaces (Γ¯k,k=1,2,3), namely,(19)Φmnr,z=Φmnir,z,r,θ,z∈Ω¯i,where Ω¯3 and Ω¯4 have the unperturbed free surfaces ∑¯1 and ∑¯2, respectively. The resultant hydrodynamic force of the liquid pressure acting on the container can be obtained by the formula [31]:(20)F=Mlg−Mlv˙0+ω×v0+ω×ω×rC+ω˙×rC+2ω×rC+r¨C,where Ml is the liquid mass and rC is the position vector with respect to the mass center of the liquid in the relative cylindrical coordinate system Orθz. The container is subjected to the lateral ground movement, and the resultant hydrodynamic force can be rewritten as(21)F=Mlg−Mlv˙0+r¨C.Figure 5
Cross section, subdomains, and artificial interfaces of the unperturbed liquid domainΩ¯.
(a)
(b)The hydrodynamic moment about the originO due to the liquid pressure acting on the container can be obtained as follows:(22)MO=MlrC×g−v˙0.In the cylindrical coordinate systemOrθz, the vectors in equations (21) and (22) can be represented as(23)F=F1e1+F2e2+F3e3,M=MO1e1+MO2e2+MO3e3,g=g1e1+g2e2+g3e3,rC=rC1e1+rC2e2+rC3e3,where Fκ, MOκ, gκ, and rCκκ=1,2,3 are the projections of the vectors on the unit vectors e1, e2, e3, respectively. According to equations (21) and (22), we have(24)Fκ=Mlgκ−Mlv˙0κ+r¨Cκ,(25)MO1=MlrC2g3−v˙03−rC3g2−v˙02,MO2=MlrC3g1−v˙01−rC1g3−v˙03,MO3=MlrC1g2−v˙02−rC2g1−v˙01.The mass center of the liquid is the unique point in the liquid domain with the property that the summation of the weighted position vectors relative to this point is zero. Hence, the position vectorrC of the mass center satisfies(26)rC=ρl∫ΩrdvMl.From equation (24), one obtains(27)rCκ=∑i=12∫Ω¯irκdv+∑i=34∫ΩirκdvV,where V denotes the volume of the liquid.If the surface wave height is small compared to the inner radius of the container, the integrals overΩi can be expanded into the Taylor series about the surface wave height at the unperturbed (hydrostatic) free surface (z=z2). The Taylor expansions for rCκ (κ=1,2,3) can be written as(28)rC1=∑i=14∫Ω¯ir1dv+∑i=34∫Σ¯i−2ηrcosθdsV,(29)rC2=∑i=14∫Ω¯ir2dv+∑i=34∫Σ¯i−2ηrsinθdsV,(30)rC3=∑i=14∫Ω¯ir3dv+12∑i=34∫Σ¯i−2η2dsV.Substituting equations (8) and (12) into (28)–(30) gives(31)rC1=β1∫0r1Φ113r,z2r2dr+∫r1r2Φ114r,z2r2drz2r22,(32)rC2=β2∫0r1Φ113r,z2r2dr+∫r1r2Φ114r,z2r2drz2r22,(33)rC3=β12+β22∫0r1Φ113r,z22rdr+∫r1r2Φ114r,z22rdr2z2r22+β32∫0r1Φ013r,z22rdr+∫r1r2Φ014r,z22rdrz2r22+β42+β52∫0r1Φ213r,z22rdr+∫r1r2Φ214r,z22rdr2z2r22+z22.Taking the second derivative ofrCκ, we can obtain(34)r¨C1=β¨1∫0r1Φ113r,z2r2dr+∫r1r2Φ114r,z2r2drz2r22,(35)r¨C2=β¨2∫0r1Φ113r,z2r2dr+∫r1r2Φ114r,z2r2drz2r22,(36)r¨C3=β˙12+β1β¨1+β˙22+β2β¨2∫0r1Φ113r,z22rdr+∫r1r2Φ114r,z22rdrz2r22+2β˙32+β3β¨3∫0r1Φ013r,z22rdr+∫r1r2Φ014r,z22rdrz2r22+β˙42+β4β¨4+β˙52+β5β¨5∫0r1Φ213r,z22rdr+∫r1r2Φ214r,z22rdrz2r22+z22.Substituting equations (31)–(36) into (24) and (25) gives the resultant hydrodynamic force and moment, respectively. The container is subjected to the lateral ground movement xt along the longitudinal axis (θ=0), namely, r1=xt. Owing to the symmetry of the system, the resultant hydrodynamic force along θ=0 axis (F1) and the resultant hydrodynamic moment about θ=π/2 axis (MO2) are more significant than the other components. In the present analysis, we concentrate on the analysis of F1 and MO2.


## 4. Comparison Study

For checking the validity of the present method, the forced nonlinear sloshing of liquid in a rigid circular cylindrical container with a rigid annular baffle is studied, respectively, by using the Gavrilyuk’s modal system [33], the smoothed particle hydrodynamics (SPH), and the present method. The radius of the container is fixed at r2 = 1 m. The position of the unperturbed free surface is taken as z2 = 1 m. The container is subjected to the harmonic excitation, i.e., xt=X0sinωt, where the amplitude of the container movement is X0 = 0.02 m. The liquid started movement from rest. According to equation (8), the initial condition for the modal system is taken as(37)β1=0,β2=0,β3=0,β˙1=0,β˙2=0,β˙3=0.In the first case, the baffle is positioned atz1 = 0.75 m and the inner radius of the baffle is taken as r1 = 0.4 m. Three different excitation frequencies are considered: ω = 4 rad/s, 5 rad/s, and 6 rad/s. The surface wave heights on the wall at θ = 0 obtained by the present modal system are compared with those obtained by Gavrilyuk’s modal system, as shown in Figures 6–8. It is seen from Figures 6–8 that the present results are in good agreement with those from the Gavrilyuk’s modal system.Figure 6
Time history of the surface wave height on the wall atθ=0 for ω¯=4rad/s.Figure 7
Time history of the surface wave height on the wall atθ=0 for ω¯=5rad/s.Figure 8
Time history of the surface wave height on the wall atθ=0 for ω¯=6rad/s.In the second case, the baffle is positioned atz1 = 0.5 m and the inner radius of the baffle is r1 = 0.5 m. The excitation frequency is ω = 6.28 rad/s. The SPH analysis is carried out by using the commercial software ABAQUS. The surface wave profiles across the container (at θ = 0) at four different times (t = 0.5 s, 1.5 s, 2.5 s, and 3.5 s) are compared with those obtained by the SPH method, as shown in Figure 9. It is seen that good agreement is achieved between the SPH solution and the present solution.Figure 9
Surface wave profiles across the container at different times forω¯ = 6.28 rad/s. (a) t=0.5s. (b) t=1.5s. (c) t=2.5s. (d) t=3.5s.
(a)
(b)
(c)
(d)To verify the correctness of the present multimodal system under seismic excitation, the present results of free surface wave height are compared with the experiment results reported by Hosseinzadeh et al. [37]. A scaled steel container with 0.002 m wall thickness was used in the experiment. The diameter of the container was 1.2 m. The liquid filling height was 0.6 m. The baffle was positioned at z1 = 0.5 m, and the inner radius of the baffle was 0.55 m. The Tabas earthquake record (Iran, 1978), scaled to PGA = 0.4 g, was considered as the base excitation. In Figure 10, the free surface wave heights at the wall are compared with the experiment ones. As seen from Figure 8, the present results are in good agreement with the experiment ones. The typical nonlinear phenomenon can be observed from the results: the peak value of the free liquid surface is always larger than the trough value.Figure 10
Time history of the surface wave height on the wall atθ=0 subjected to Tabas earthquake (Iran, 1978).


## 5. Parametric Study

The effects of baffle parameters (position and inner radius) and excitation parameters (amplitude and frequency) on nonlinear response of liquid in a container are discussed in detail. The container is subjected to a lateral harmonic excitationxt in the form of sinusoidal wave having the amplitude X0 and the frequency ω. The radius of the container is fixed at r2 = 1 m, and the liquid height is taken as z2 = 1 m.


### 5.1. Effect of Position of the Baffle

The inner radius of the baffle is fixed atr1 = 0.5 m. The amplitude of the container movement is X0 = 0.03 m, and the exciting frequency is ω = 6 rad/s. Under the given lateral excitation, the maximum amplitudes of nonlinear responses are presented in Table 1. fmax denotes the maximum amplitude of the surface wave height. Fmax and Mmax denote the maximum amplitudes of the resultant hydrodynamic force and moment, respectively. It is shown that fmax decreases as the baffle is positioned towards the free surface. In contrast, Fmax increases with the increase of baffle position z1. As the baffle moves up from the bottom, Mmax decreases to the lowest point at z1 = 0.6 m. However, when the baffle further moves up to the free surface from z1 = 0.6 m, Mmax tends to increase. For different baffle positions z1 = 0.1 m, 0.3 m, 0.5 m, and 0.7 m, the wall-pressure distribution (at θ = 0) resulting in the maximum resultant hydrodynamic force and moment is shown in Figure 11. It is observed from Figure 11 that the wall pressure above the baffle decreases with the increase of z1. The change of the wall pressure below the baffle versus the position of the baffle is relatively small. Fu and Mu denote the resultant hydrodynamic force and the resultant moment of the liquid above the baffle. Fd and Md denote the resultant hydrodynamic force and the resultant moment of the liquid below the baffle. The variations of Fu, Fd, Mu, and Md resulting in the maximum resultant hydrodynamic force and moment with respect to the position of the baffle are given in Table 2. It is seen from Table 2 that Fu and Mu decrease with the increase of z1 while Fd and Md increase with the increase of z1. It is seen from Table 2 that the increase rate of Fd exceeds the decrease rate of Fu. This results in the increase of the maximum resultant hydrodynamic force (Fmax = Fu + Fd) with the increase of z1. The increase rate of Md is slower than the decrease rate of Mu when z1 < 0.6 m. However, the increase rate of Md exceeds the decrease rate of Mu when z1 > 0.6 m. This results in the nonmonotonic variation of the maximum resultant moment (Mmax = Mu + Md).Table 1
The maximum amplitudes of the nonlinear response versus the position of the baffle forr1=0.5m.
Maximum amplitudes of the nonlinear response
The position of the bafflez1 (s)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
f
max (m)
0.230
0.224
0.214
0.205
0.188
0.171
0.147
0.099
F
max (N)
2532.4
2563.2
2596.3
2615.9
2651.2
2662.7
2817.8
2923.1
M
max (N·m)
1843.1
1832.4
1808.8
1762.7
1725.6
1616.4
1718.8
1723Figure 11
Wall-pressure distributions (atθ=0) resulting in the maximum amplitudes Fmax and Mmax for r1 = 0.5 m and z1=0.1m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.Table 2
The resultant hydrodynamic force and moment from liquid below/above the baffle versus the position of the baffle forr1=0.5m.
Resultant force and moment from liquid below/above the baffle
The position of the bafflez1 (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
F
u (N)
2215.5
1924.5
1632.9
1326.8
1038.5
752.3
557.9
336
ΔFu (N)
/
−291
−291.6
−306.1
−288.3
−286.2
−194.4
−221.9
F
d (N)
316.9
638.7
963.4
1289.1
1612.7
1910.4
2259.9
2587.1
ΔFd (N)
/
321.8
324.7
325.7
323.6
297.7
349.5
327.2
M
u (N·m)
1783.9
1722.6
1615.1
1451.7
1264.3
979
857.9
612.8
ΔMu (N·m)
/
−61.3
−107.5
−163.4
−187.4
−285.3
−121.1
−245.1
M
d (N·m)
59.2
109.8
193.7
311
461.3
637.4
860.9
1110.2
ΔMd (N·m)
/
50.6
83.9
117.3
150.3
176.1
223.5
249.3
ΔFu, ΔFd, ΔMu, and ΔMd denote the increments of the resultant force and moment from liquid below/above the baffle for the increment of the inner radius of the baffle Δz1 = 0.1 m.To investigate the effect of the baffle’s position on nonlinearity, the present nonlinear response is compared with the linear response [10, 11] within the first 30 seconds. Four different positions of the baffle are, respectively, considered: z1 = 0.1 m, 0.3 m, 0.5 m, and 0.7 m. The time histories of the surface wave height fwall on the wall at θ = 0, the resultant hydrodynamic force F1 in the θ = 0 direction, and the resultant hydrodynamic moment MO2 about the θ = π/2 axis are depicted in Figures 12–14, respectively. It is seen from Figures 12–14 that the nonlinearity increases with the time and the nonlinearity decreases as the baffle moves close to the free surface. According to equations (1)–(6), the nonlinearity only appears on the free surface. Therefore, the nonlinearity decreases as the maximum surface wave height decreases.Figure 12
Time history of the surface wave height at the container wall under the lateral harmonic excitation forr1 = 0.5 m and z1=0.1m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.
(a)
(b)
(c)
(d)Figure 13
Time history of the resultant hydrodynamic force in theθ=0 direction under the lateral harmonic excitation for r1 = 0.5 m and z1=0.1m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.
(a)
(b)
(c)
(d)Figure 14
Time history of the resultant hydrodynamic moment about theθ=π/2 axis under the lateral harmonic excitation for r1 = 0.5 m and z1=0.1m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.
(a)
(b)
(c)
(d)


### 5.2. Effect of Inner Radius of the Baffle

The baffle is fixed atz1 = 0.5 m. The amplitude of the container movement is X0 = 0.03 m, and the exciting frequency is ω = 6 rad/s. The maximum amplitudes of the nonlinear response versus the inner radius of the baffle are given in Table 3. It is seen from Table 3 that fmax and Mmax increase with the increase of the inner radius of the baffle. However, Fmax decreases with the increase of baffle inner radius r1. For r1 = 0.2 m, 0.4 m, 0.6 m, and 0.8 m, the wall-pressure distribution (at θ = 0) resulting in the maximum resultant hydrodynamic force and moment is shown in Figure 15. It is observed that the wall pressure above the baffle increases with the increase of r1 and the wall pressure below the baffle decreases with the increase of r1. The variations of Fu, Fd, Mu, and Md with respect to the inner radius of the baffle are given in Table 4. It is seen that Fu and Md increase with the increase of r1 while Fd decreases with the increase of r1. Mu increases with the increase of r1 when r1 < 0.6 m. However, Mu decreases with the increase of r1 when r1 > 0.6 m. The variation rates of Fd and Md are significantly faster than those of Fu and Mu, respectively.Table 3
The maximum amplitudes of the nonlinear response versus the inner radius of the baffle forz1=0.5m.
Maximum amplitudes of the nonlinear response
The inner radius of the baffler1 (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
f
max (m)
0.168
0.170
0.172
0.178
0.188
0.199
0.215
0.227
F
max (N)
2707.8
2706.9
2704.7
2680.6
2651.2
2640.2
2598.1
2568.2
M
max (N·m)
1667.3
1678.6
1690.4
1701.5
1725.6
1778.4
1806.5
1852.4Figure 15
Wall-pressure distributions (θ=0) corresponding to the maximum amplitudes Fmax and Mmax for z1 = 0.5 m and r1= 0.2 m, 0.4 m, 0.6 m, and 0.8 m, respectively.Table 4
The resultant hydrodynamic force and moment from liquid below/above the baffle versus the inner radius of the baffle forz1=0.5m.
Resultant force and moment from liquid below/above the baffle
The inner radius of the baffler1 (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
F
u (N)
1012.7
1016.5
1030.4
1031.6
1038.5
1073.7
1085.9
1130.1
ΔFu (N)
/
3.8
13.9
1.2
6.9
35.2
12.2
44.2
F
d (N)
1695.1
1690.4
1674.3
1649
1612.7
1566.5
1512.2
1438.1
ΔFd (N)
/
−4.7
−16.1
−25.3
−36.3
−46.2
−54.3
−74.1
M
u (N·m)
1251.3
1256.5
1262
1262.5
1264.3
1277
1239.8
1194.5
ΔMu (N·m)
/
5.2
5.5
0.5
1.8
12.7
−37.2
−45.3
M
d (N·m)
416
422.1
428.4
439
461.3
501.4
566.7
657.9
ΔMd (N·m)
/
6.1
6.3
10.6
22.3
40.1
65.3
91.2
ΔFu, ΔFd, ΔMu, and ΔMd denote the increments of the resultant force and moment from liquid below/above the baffle for the increment of the inner radius of the baffle Δr1 = 0.1 m.To study the effect of the baffle’s inner radius on the nonlinearity, the nonlinear response is compared with the linear response within the first 30 seconds. Four different inner radii of the baffle are considered:r1 = 0.2 m, 0.4 m, 0.6 m, and 0.8 m, respectively. The time histories of the surface wave height fwall on the wall at θ = 0, the resultant hydrodynamic force F1 in the θ = 0 direction, and the resultant hydrodynamic moment MO2 about the θ = π/2 axis are depicted in Figures 16–18. It is observed that the nonlinearity increases with the increase of the inner radius r1.Figure 16
Time history of the surface wave height at the container wall under the lateral harmonic excitation forz1 = 0.5 m and r1=0.2m (a), 0.4 m (b), 0.6 m (c), and 0.8 m (d), respectively.
(a)
(b)
(c)
(d)Figure 17
Time history of the resultant hydrodynamic force in theθ=0 direction under the lateral harmonic excitation for z1 = 0.5 m and r1=0.2m (a), 0.4 m (b), 0.6 m (c), and 0.8 m (d), respectively.
(a)
(b)
(c)
(d)Figure 18
Time history of the resultant hydrodynamic moment about theθ=π/2 axis under the lateral harmonic excitation for z1 = 0.5 m and r1=0.2m (a), 0.4 m (b), 0.6 m (c), and 0.8 m (d), respectively.
(a)
(b)
(c)
(d)


### 5.3. Effect of Excitation Amplitude

The inner radius of the baffle is fixed atr1 = 0.8 m, and the baffle is positioned at z1 = 0.7 m. Four different excitation amplitudes are considered: X0 = 0.01 m, 0.02 m, 0.03 m, and 0.04 m, respectively. The circular frequency is taken as ω = 6 rad/s. Under the lateral excitation, the maximum amplitudes of liquid response are given in Table 5. It is seen that the maximum amplitudes of liquid response increase with the increase of the excitation amplitude. To investigate the effect of the excitation amplitude on the nonlinearity of response, the present results are compared with the linear ones within the first 30 seconds. The time histories of fwall, F1, and MO2 are plotted in Figures 19–21, respectively. It is seen that the nonlinearity increases with the increase of the excitation amplitude.Table 5
The maximum amplitudes of the nonlinear response versus the excitation amplitude forr1=0.8m and z1=0.7m.
Maximum amplitudes of the nonlinear response
Excitation amplitudeX0 (m)
0.01
0.02
0.03
0.04
f
max (m)
0.065
0.137
0.214
0.293
F
max (N)
860.2
1722.8
2593.1
3490.3
M
max (N·m)
600.7
1208.0
1820.8
2457.8Figure 19
Time history of the surface wave height at the container wall under the lateral harmonic excitation forω = 6 rad/s and X0 = 0.01 m (a), 0.02 m (b), 0.03 m (c), and 0.04 m (d), respectively.
(a)
(b)
(c)
(d)Figure 20
Time history of the resultant hydrodynamic force in theθ=0 direction under the lateral harmonic excitation for ω = 6 rad/s and X0 = 0.01 m (a), 0.02 m (b), 0.03 m (c), and 0.04 m (d), respectively.
(a)
(b)
(c)
(d)Figure 21
Time history of the resultant hydrodynamic moment about theθ=π/2 axis under the lateral harmonic excitation for ω = 6 rad/s and X0 = 0.01 m (a), 0.02 m (b), 0.03 m (c), and 0.04 m (d), respectively.
(a)
(b)
(c)
(d)


### 5.4. Effect of Excitation Frequency

The inner radius of the baffle is fixed atr1 = 0.7 m, and the baffle is positioned at z1 = 0.7 m. The fundamental frequency of the liquid is 3.77 rad/s. The container is undergoing the lateral harmonic excitation, and the amplitude is fixed at X0 = 0.02 m. Using the method developed by Faltinsen and Timokha [25], the steady-state analysis in the frequency domain for the equations (13)–(17) is conducted. Considering the harmonic excitation, the generalized coordinates for the two dominant modes can be assumed as follows:(38)β1t=Asinωt+A¯cosωt+oε1/3,(39)β2t=B¯sinωt+Bcosωt+oε1/3.Substituting equations (38) and (39) into (13)–(17), we have(40)β3t=c11+c12cos2ωt+c13sin2ωt+oε2/3,β4t=c21+c22cos2ωt+c23sin2ωt+oε2/3,β5t=c31+c32cos2ωt+c33sin2ωt+oε2/3,where ciji=1,2,3;j=1,2,3 satisfies(41)c11=l0B2+B¯2+A¯2+A2,c12=h0B2−B¯2−A2+A¯2,c13=2h0BB¯+AA¯,c21=l2B2+B¯2−A2−A¯2,c22=h2B2−B¯2+A2−A¯2,c23=2h2BB¯−AA¯,c31=−2l2BA¯+AB¯,c32=2h2B¯A−BA¯,c33=−2h2AB+A¯B¯,where l0, h0, l2, and h2 satisfies(42)h0=K10+K82σ¯012−4,h0=K9+K72σ¯212−4,l0=K10−K82σ¯012,l2=K9−K72σ¯212,σ¯mn=σmnω.The initial displacements and velocities for the free surface are assumed to be zero (i.e., the liquid is in a static state at the initial time). According to Faltinsen and Timokha [25], one can conclude the existence of the only type of motion: planar waves (i.e., B=B¯=A¯=0). For this case, the generalized coordinate A satisfies(43)Aσ¯112−1+d1A2=X0r23π3,where d1 satisfies(44)d1=K5h02−l0−K3h22−l2−2K6h0−2K4h2−K12.The response curves in the frequency domain, which are illustrated in Figure22, can be obtained from equation (43). As shown in Figure 22, there are two branches in the ω,A-plane, which have soft-spring behavior. According to the stability conditions for the planar waves given by Faltinsen and Timokha [25], the unstable planer waves are associated with points on the branches that are located between the two auxiliary curves γd1 (σ¯112−1+d1A2=0) and γ3d1 (σ¯112−1+3d1A2=0).Figure 22
Steady-state planar wave amplitude response (ω, A) in the frequency domain for X0 = 0.02 m.


## 5.1. Effect of Position of the Baffle

The inner radius of the baffle is fixed atr1 = 0.5 m. The amplitude of the container movement is X0 = 0.03 m, and the exciting frequency is ω = 6 rad/s. Under the given lateral excitation, the maximum amplitudes of nonlinear responses are presented in Table 1. fmax denotes the maximum amplitude of the surface wave height. Fmax and Mmax denote the maximum amplitudes of the resultant hydrodynamic force and moment, respectively. It is shown that fmax decreases as the baffle is positioned towards the free surface. In contrast, Fmax increases with the increase of baffle position z1. As the baffle moves up from the bottom, Mmax decreases to the lowest point at z1 = 0.6 m. However, when the baffle further moves up to the free surface from z1 = 0.6 m, Mmax tends to increase. For different baffle positions z1 = 0.1 m, 0.3 m, 0.5 m, and 0.7 m, the wall-pressure distribution (at θ = 0) resulting in the maximum resultant hydrodynamic force and moment is shown in Figure 11. It is observed from Figure 11 that the wall pressure above the baffle decreases with the increase of z1. The change of the wall pressure below the baffle versus the position of the baffle is relatively small. Fu and Mu denote the resultant hydrodynamic force and the resultant moment of the liquid above the baffle. Fd and Md denote the resultant hydrodynamic force and the resultant moment of the liquid below the baffle. The variations of Fu, Fd, Mu, and Md resulting in the maximum resultant hydrodynamic force and moment with respect to the position of the baffle are given in Table 2. It is seen from Table 2 that Fu and Mu decrease with the increase of z1 while Fd and Md increase with the increase of z1. It is seen from Table 2 that the increase rate of Fd exceeds the decrease rate of Fu. This results in the increase of the maximum resultant hydrodynamic force (Fmax = Fu + Fd) with the increase of z1. The increase rate of Md is slower than the decrease rate of Mu when z1 < 0.6 m. However, the increase rate of Md exceeds the decrease rate of Mu when z1 > 0.6 m. This results in the nonmonotonic variation of the maximum resultant moment (Mmax = Mu + Md).Table 1
The maximum amplitudes of the nonlinear response versus the position of the baffle forr1=0.5m.
Maximum amplitudes of the nonlinear response
The position of the bafflez1 (s)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
f
max (m)
0.230
0.224
0.214
0.205
0.188
0.171
0.147
0.099
F
max (N)
2532.4
2563.2
2596.3
2615.9
2651.2
2662.7
2817.8
2923.1
M
max (N·m)
1843.1
1832.4
1808.8
1762.7
1725.6
1616.4
1718.8
1723Figure 11
Wall-pressure distributions (atθ=0) resulting in the maximum amplitudes Fmax and Mmax for r1 = 0.5 m and z1=0.1m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.Table 2
The resultant hydrodynamic force and moment from liquid below/above the baffle versus the position of the baffle forr1=0.5m.
Resultant force and moment from liquid below/above the baffle
The position of the bafflez1 (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
F
u (N)
2215.5
1924.5
1632.9
1326.8
1038.5
752.3
557.9
336
ΔFu (N)
/
−291
−291.6
−306.1
−288.3
−286.2
−194.4
−221.9
F
d (N)
316.9
638.7
963.4
1289.1
1612.7
1910.4
2259.9
2587.1
ΔFd (N)
/
321.8
324.7
325.7
323.6
297.7
349.5
327.2
M
u (N·m)
1783.9
1722.6
1615.1
1451.7
1264.3
979
857.9
612.8
ΔMu (N·m)
/
−61.3
−107.5
−163.4
−187.4
−285.3
−121.1
−245.1
M
d (N·m)
59.2
109.8
193.7
311
461.3
637.4
860.9
1110.2
ΔMd (N·m)
/
50.6
83.9
117.3
150.3
176.1
223.5
249.3
ΔFu, ΔFd, ΔMu, and ΔMd denote the increments of the resultant force and moment from liquid below/above the baffle for the increment of the inner radius of the baffle Δz1 = 0.1 m.To investigate the effect of the baffle’s position on nonlinearity, the present nonlinear response is compared with the linear response [10, 11] within the first 30 seconds. Four different positions of the baffle are, respectively, considered: z1 = 0.1 m, 0.3 m, 0.5 m, and 0.7 m. The time histories of the surface wave height fwall on the wall at θ = 0, the resultant hydrodynamic force F1 in the θ = 0 direction, and the resultant hydrodynamic moment MO2 about the θ = π/2 axis are depicted in Figures 12–14, respectively. It is seen from Figures 12–14 that the nonlinearity increases with the time and the nonlinearity decreases as the baffle moves close to the free surface. According to equations (1)–(6), the nonlinearity only appears on the free surface. Therefore, the nonlinearity decreases as the maximum surface wave height decreases.Figure 12
Time history of the surface wave height at the container wall under the lateral harmonic excitation forr1 = 0.5 m and z1=0.1m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.
(a)
(b)
(c)
(d)Figure 13
Time history of the resultant hydrodynamic force in theθ=0 direction under the lateral harmonic excitation for r1 = 0.5 m and z1=0.1m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.
(a)
(b)
(c)
(d)Figure 14
Time history of the resultant hydrodynamic moment about theθ=π/2 axis under the lateral harmonic excitation for r1 = 0.5 m and z1=0.1m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.
(a)
(b)
(c)
(d)


## 5.2. Effect of Inner Radius of the Baffle

The baffle is fixed atz1 = 0.5 m. The amplitude of the container movement is X0 = 0.03 m, and the exciting frequency is ω = 6 rad/s. The maximum amplitudes of the nonlinear response versus the inner radius of the baffle are given in Table 3. It is seen from Table 3 that fmax and Mmax increase with the increase of the inner radius of the baffle. However, Fmax decreases with the increase of baffle inner radius r1. For r1 = 0.2 m, 0.4 m, 0.6 m, and 0.8 m, the wall-pressure distribution (at θ = 0) resulting in the maximum resultant hydrodynamic force and moment is shown in Figure 15. It is observed that the wall pressure above the baffle increases with the increase of r1 and the wall pressure below the baffle decreases with the increase of r1. The variations of Fu, Fd, Mu, and Md with respect to the inner radius of the baffle are given in Table 4. It is seen that Fu and Md increase with the increase of r1 while Fd decreases with the increase of r1. Mu increases with the increase of r1 when r1 < 0.6 m. However, Mu decreases with the increase of r1 when r1 > 0.6 m. The variation rates of Fd and Md are significantly faster than those of Fu and Mu, respectively.Table 3
The maximum amplitudes of the nonlinear response versus the inner radius of the baffle forz1=0.5m.
Maximum amplitudes of the nonlinear response
The inner radius of the baffler1 (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
f
max (m)
0.168
0.170
0.172
0.178
0.188
0.199
0.215
0.227
F
max (N)
2707.8
2706.9
2704.7
2680.6
2651.2
2640.2
2598.1
2568.2
M
max (N·m)
1667.3
1678.6
1690.4
1701.5
1725.6
1778.4
1806.5
1852.4Figure 15
Wall-pressure distributions (θ=0) corresponding to the maximum amplitudes Fmax and Mmax for z1 = 0.5 m and r1= 0.2 m, 0.4 m, 0.6 m, and 0.8 m, respectively.Table 4
The resultant hydrodynamic force and moment from liquid below/above the baffle versus the inner radius of the baffle forz1=0.5m.
Resultant force and moment from liquid below/above the baffle
The inner radius of the baffler1 (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
F
u (N)
1012.7
1016.5
1030.4
1031.6
1038.5
1073.7
1085.9
1130.1
ΔFu (N)
/
3.8
13.9
1.2
6.9
35.2
12.2
44.2
F
d (N)
1695.1
1690.4
1674.3
1649
1612.7
1566.5
1512.2
1438.1
ΔFd (N)
/
−4.7
−16.1
−25.3
−36.3
−46.2
−54.3
−74.1
M
u (N·m)
1251.3
1256.5
1262
1262.5
1264.3
1277
1239.8
1194.5
ΔMu (N·m)
/
5.2
5.5
0.5
1.8
12.7
−37.2
−45.3
M
d (N·m)
416
422.1
428.4
439
461.3
501.4
566.7
657.9
ΔMd (N·m)
/
6.1
6.3
10.6
22.3
40.1
65.3
91.2
ΔFu, ΔFd, ΔMu, and ΔMd denote the increments of the resultant force and moment from liquid below/above the baffle for the increment of the inner radius of the baffle Δr1 = 0.1 m.To study the effect of the baffle’s inner radius on the nonlinearity, the nonlinear response is compared with the linear response within the first 30 seconds. Four different inner radii of the baffle are considered:r1 = 0.2 m, 0.4 m, 0.6 m, and 0.8 m, respectively. The time histories of the surface wave height fwall on the wall at θ = 0, the resultant hydrodynamic force F1 in the θ = 0 direction, and the resultant hydrodynamic moment MO2 about the θ = π/2 axis are depicted in Figures 16–18. It is observed that the nonlinearity increases with the increase of the inner radius r1.Figure 16
Time history of the surface wave height at the container wall under the lateral harmonic excitation forz1 = 0.5 m and r1=0.2m (a), 0.4 m (b), 0.6 m (c), and 0.8 m (d), respectively.
(a)
(b)
(c)
(d)Figure 17
Time history of the resultant hydrodynamic force in theθ=0 direction under the lateral harmonic excitation for z1 = 0.5 m and r1=0.2m (a), 0.4 m (b), 0.6 m (c), and 0.8 m (d), respectively.
(a)
(b)
(c)
(d)Figure 18
Time history of the resultant hydrodynamic moment about theθ=π/2 axis under the lateral harmonic excitation for z1 = 0.5 m and r1=0.2m (a), 0.4 m (b), 0.6 m (c), and 0.8 m (d), respectively.
(a)
(b)
(c)
(d)


## 5.3. Effect of Excitation Amplitude

The inner radius of the baffle is fixed atr1 = 0.8 m, and the baffle is positioned at z1 = 0.7 m. Four different excitation amplitudes are considered: X0 = 0.01 m, 0.02 m, 0.03 m, and 0.04 m, respectively. The circular frequency is taken as ω = 6 rad/s. Under the lateral excitation, the maximum amplitudes of liquid response are given in Table 5. It is seen that the maximum amplitudes of liquid response increase with the increase of the excitation amplitude. To investigate the effect of the excitation amplitude on the nonlinearity of response, the present results are compared with the linear ones within the first 30 seconds. The time histories of fwall, F1, and MO2 are plotted in Figures 19–21, respectively. It is seen that the nonlinearity increases with the increase of the excitation amplitude.Table 5
The maximum amplitudes of the nonlinear response versus the excitation amplitude forr1=0.8m and z1=0.7m.
Maximum amplitudes of the nonlinear response
Excitation amplitudeX0 (m)
0.01
0.02
0.03
0.04
f
max (m)
0.065
0.137
0.214
0.293
F
max (N)
860.2
1722.8
2593.1
3490.3
M
max (N·m)
600.7
1208.0
1820.8
2457.8Figure 19
Time history of the surface wave height at the container wall under the lateral harmonic excitation forω = 6 rad/s and X0 = 0.01 m (a), 0.02 m (b), 0.03 m (c), and 0.04 m (d), respectively.
(a)
(b)
(c)
(d)Figure 20
Time history of the resultant hydrodynamic force in theθ=0 direction under the lateral harmonic excitation for ω = 6 rad/s and X0 = 0.01 m (a), 0.02 m (b), 0.03 m (c), and 0.04 m (d), respectively.
(a)
(b)
(c)
(d)Figure 21
Time history of the resultant hydrodynamic moment about theθ=π/2 axis under the lateral harmonic excitation for ω = 6 rad/s and X0 = 0.01 m (a), 0.02 m (b), 0.03 m (c), and 0.04 m (d), respectively.
(a)
(b)
(c)
(d)


## 5.4. Effect of Excitation Frequency

The inner radius of the baffle is fixed atr1 = 0.7 m, and the baffle is positioned at z1 = 0.7 m. The fundamental frequency of the liquid is 3.77 rad/s. The container is undergoing the lateral harmonic excitation, and the amplitude is fixed at X0 = 0.02 m. Using the method developed by Faltinsen and Timokha [25], the steady-state analysis in the frequency domain for the equations (13)–(17) is conducted. Considering the harmonic excitation, the generalized coordinates for the two dominant modes can be assumed as follows:(38)β1t=Asinωt+A¯cosωt+oε1/3,(39)β2t=B¯sinωt+Bcosωt+oε1/3.Substituting equations (38) and (39) into (13)–(17), we have(40)β3t=c11+c12cos2ωt+c13sin2ωt+oε2/3,β4t=c21+c22cos2ωt+c23sin2ωt+oε2/3,β5t=c31+c32cos2ωt+c33sin2ωt+oε2/3,where ciji=1,2,3;j=1,2,3 satisfies(41)c11=l0B2+B¯2+A¯2+A2,c12=h0B2−B¯2−A2+A¯2,c13=2h0BB¯+AA¯,c21=l2B2+B¯2−A2−A¯2,c22=h2B2−B¯2+A2−A¯2,c23=2h2BB¯−AA¯,c31=−2l2BA¯+AB¯,c32=2h2B¯A−BA¯,c33=−2h2AB+A¯B¯,where l0, h0, l2, and h2 satisfies(42)h0=K10+K82σ¯012−4,h0=K9+K72σ¯212−4,l0=K10−K82σ¯012,l2=K9−K72σ¯212,σ¯mn=σmnω.The initial displacements and velocities for the free surface are assumed to be zero (i.e., the liquid is in a static state at the initial time). According to Faltinsen and Timokha [25], one can conclude the existence of the only type of motion: planar waves (i.e., B=B¯=A¯=0). For this case, the generalized coordinate A satisfies(43)Aσ¯112−1+d1A2=X0r23π3,where d1 satisfies(44)d1=K5h02−l0−K3h22−l2−2K6h0−2K4h2−K12.The response curves in the frequency domain, which are illustrated in Figure22, can be obtained from equation (43). As shown in Figure 22, there are two branches in the ω,A-plane, which have soft-spring behavior. According to the stability conditions for the planar waves given by Faltinsen and Timokha [25], the unstable planer waves are associated with points on the branches that are located between the two auxiliary curves γd1 (σ¯112−1+d1A2=0) and γ3d1 (σ¯112−1+3d1A2=0).Figure 22
Steady-state planar wave amplitude response (ω, A) in the frequency domain for X0 = 0.02 m.


## 6. Conclusions

Based on the multimodal method carried out in a recent paper [18], the nonlinear response of the liquid partially filled in a rigid cylindrical container with a rigid annular baffle and subjected to lateral excitation is investigated. According to Raynovskyy and Timokha [34], the significant damping can be caused by the vortex shedding at the sharp edges of the baffles. In the subsequent study, we will quantitatively study this damping based on experimental and numerical methods and then introduce it into the multimodal system. In the present study, the surface wave height, resultant hydrodynamic force, and moment with respect to the position and inner radius of the baffle and the parameters (amplitude and frequency) of the excitation are discussed in detail. The following observations are highlighted:(1)
The resultant hydrodynamic force increases as the baffle moves towards the free surface and increases with the decrease of inner radius of the baffle(2)
The surface wave height decreases as the baffle is positioned towards the free surface and decreases with the decrease of inner radius of the baffle(3)
The nonlinearity increases with the increase of the exciting amplitude and the accumulation of the sloshing time



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*Source: 5398038-2019-01-23.xml*