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

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. .e infinite-dimensional modal system is reduced by using the Moiseev asymptotic relations. .e 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.


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 liquidfilled 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].e problems are usually investigated by the experimental and numerical methods [13][14][15].Carra et al. [16] investigated the linear and geometrically nonlinear dynamical response of a thin plate in contact with liquid.
ey 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 twodimensional 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.e 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 twodimensional nonlinear sloshing in both rectangular and cylindrical containers equipped with rigid baffles.
e 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.e 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 infinitedimensional modal system can be reduced to the finitedimensional modal system.e method is applicable to arbitrary excitations and a broad class of the container shapes [27][28][29][30].Lukovsky [31] developed the fivedimensional 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. is 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.
is can be a problem because if the excitation frequency is close to the natural frequency of the liquid, the resonantamplified impact force may cause damage to the container.
us, 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.

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 Figure 1, where Ox ′ y ′ z ′ is the absolute coordinate system and Orθz is the relative cylindrical coordinate system fixed at the container.e 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.
e inner radius of the container is r 2 , and the inner radius of the baffle is r 1 .z 1 and z 2 denote, respectively, the positions of the baffle and the unperturbed (hydrostatic) free surface.e thickness of the baffle is very small compared with the inner radius of the container.e density of the liquid is ρ. e time-dependent liquid domain Ω is bounded by the free surface , the wet container wall S w , the container bottom S b , and the baffle surface S p .e potential formulation of the nonlinear sloshing problem is given as follows [25]: where n is the outer normal of the boundary of the liquid domain Ω, z � f(r, θ, t) + z 2 is the time-dependent shape 2 Shock and Vibration equation of the free surface , v 0 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 ′ , r 0 ′ 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 e 1 and e 3 , respectively.e unit vector e 2 is given by e 2 � e 1 × e 3 .en, the vectors in equations ( 1)-( 6) can be written as When no overturning wave occurs, the continuum problem is reduced to a discrete conservative mechanical system with infinite degrees of freedom. is implies that the free surface and velocity potential can be expressed as where the vector function Θ denotes the Stokes-Zhukovsky potential, which was defined in [25], β p (t) and R q (t) are the generalized coordinates, the set η p (r, θ)   is a Fourier basis on the unperturbed free surface, and the set Φ q (r, θ, z)   should be complete in the unperturbed liquid domain.e 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: where I 1κ , I 2κ , I 3κ , A q , and A qq′ are the integrals over the time-dependent domain Ω(t) and only depend on β p and _ β p .eir 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 e mode shapes of the linear sloshing solutions are Φ q (r, θ, z)  .According to the Moiseev asymptotic relations [25], we have where Φ mn (m � 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 C 1 and could be calculated with continuity boundary conditions, as shown in Figure 2. e analytical solution for the velocity potential in each subdomain can be obtained using separation of variables.e 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].
In the following study, the perturbation parameter ε is introduced, which denotes the ratio of the excitation amplitude and the inner radius of the container.e 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 O(ε 1/3 ).e generalized coordinates for Φ 3 , Φ 4 , and Φ 5 are O(ε 2/3 ) and the other higher modes are O(ε) or o(ε).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.e results for the surface wave height and hydrodynamic forces are in good agreement with the experimental results.e higherorder terms than O(ε) 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 O(ε), one has the following asymptotic modal system: where the explicit expressions of K i (i � 1, . .., 9) have been given by [18] and σ mn (m � 0, 1, 2; n � 1) denotes the natural sloshing frequencies.

Resultant Hydrodynamic Force and Moment
e liquid velocity potential in each subdomain has the continuous boundary conditions of class C 1 .
e liquid domain with a single baffle can be divided into 4 subdomains (Ω 1 , Ω 2 , Ω 3 , Ω 4 ) with 3 artificial interfaces (Γ 1 , Γ 2 , Γ 3 ) as shown in Figure 4, namely, Obviously, Ω i (i � 1, 2) and Γ k (k � 1, 2) are timeindependent; Ω i (i � 3, 4) and Γ k (k � 3) are timedependent (it should be noted that Ω 3 and Ω 4 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, where Ω 3 and Ω 4 have the unperturbed free surfaces  1 and  2 , respectively.e resultant hydrodynamic force of the liquid pressure acting on the container can be obtained by the formula [31]: where M l is the liquid mass and r C is the position vector with respect to the mass center of the liquid in the relative cylindrical coordinate system Orθz.e container is subjected to the lateral ground movement, and the resultant hydrodynamic force can be rewritten as e hydrodynamic moment about the origin O due to the liquid pressure acting on the container can be obtained as follows: In the cylindrical coordinate system Orθz, the vectors in equations ( 21) and ( 22) can be represented as

Shock and Vibration
where F κ , M Oκ , g κ , and r κ C (κ � 1, 2, 3) are the projections of the vectors on the unit vectors e 1 , e 2 , e 3 , respectively.According to equations ( 21) and ( 22), we have e 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 vector r C of the mass center satisfies From equation ( 24), one obtains 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 over Ω i can be expanded into the Taylor series about the surface wave height at the unperturbed (hydrostatic) free surface (z � z 2 ).e Taylor expansions for r κ C (κ � 1, 2, 3) can be written as Substituting equations ( 8) and ( 12) into ( 28)- (30) gives Taking the second derivative of r κ C , we can obtain Substituting equations ( 31)-( 36) into ( 24) and ( 25) gives the resultant hydrodynamic force and moment, respectively.e container is subjected to the lateral ground movement x(t) along the longitudinal axis (θ � 0), namely, r 1 � x(t).Owing to the symmetry of the system, the resultant hydrodynamic force along θ � 0 axis (F 1 ) and the resultant hydrodynamic moment about θ � π/2 axis (M O2 ) are more significant than the other components.In the present analysis, we concentrate on the analysis of F 1 and M O2 .

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.e radius of the container is fixed at r 2 � 1 m. e position of the unperturbed free surface is taken as z 2 � 1 m. e container is subjected to the harmonic excitation, i.e., x(t) � X 0 sin ωt, where the amplitude of the container movement is X 0 � 0.02 m.
e liquid started movement from rest.According to equation (8), the initial condition for the modal system is taken as Shock and Vibration In the first case, the baffle is positioned at z 1 � 0.75 m and the inner radius of the baffle is taken as r 1 � 0.4 m. ree different excitation frequencies are considered: ω � 4 rad/s, 5 rad/s, and 6 rad/s.e 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.
In the second case, the baffle is positioned at z 1 � 0.5 m and the inner radius of the baffle is r 1 � 0.5 m. e excitation frequency is ω � 6.28 rad/s.e SPH analysis is carried out by using the commercial software ABAQUS.
e 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.
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.
e diameter of the container was 1.2 m. e liquid filling height was 0.6 m. e baffle was positioned at z 1 � 0.5 m, and the inner radius of the baffle was 0.55 m. e 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.e typical nonlinear phenomenon can be observed from the results: the peak value of the free liquid surface is always larger than the trough value.

Parametric Study
e 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.e container is subjected to a lateral harmonic excitation x(t) in the form of sinusoidal wave having the amplitude X 0 and the frequency ω. e radius of the container is fixed at r 2 � 1 m, and the liquid height is taken as z 2 � 1 m.

Effect of Position of the Baffle.
e inner radius of the baffle is fixed at r 1 � 0.5 m. e amplitude of the container movement is X 0 � 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.f max denotes the maximum amplitude of the surface wave height.F max and M max denote the maximum amplitudes of the resultant hydrodynamic force and moment, respectively.It is shown that f max decreases as the baffle is positioned towards the free surface.In contrast, F max increases with the increase of baffle position z 1 .As the baffle moves up from the bottom, M max decreases to the lowest point at z 1 � 0.6 m.However, when the baffle further moves up to the free surface from z 1 � 0.6 m, M max tends to increase.For different baffle positions z 1 � 0.1 m, 0.3 m, 0.5 m, and 0.7 m, the wallpressure 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 z 1 .e change   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: z 1 � 0.1 m, 0.3 m, 0.5 m, and 0.7 m.
e time histories of the surface wave height f wall on the wall at θ � 0, the resultant hydrodynamic force F 1 in the θ � 0 direction, and the resultant hydrodynamic moment M O2 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.erefore, the nonlinearity decreases as the maximum surface wave height decreases.

Effect of Inner Radius of the Baffle.
e baffle is fixed at z 1 � 0.5 m.
e amplitude of the container movement is 8 Shock and Vibration X 0 � 0.03 m, and the exciting frequency is ω � 6 rad/s.e 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 f max and M max increase with the increase of the inner radius of the baffle.However, F max decreases with the increase of baffle inner radius r 1 .For r 1 � 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 r 1 and the wall pressure below the baffle decreases with the increase of r 1 .e variations of F u , F d , M u , and M d with respect to the inner radius of the baffle are given in Table 4.It is seen that F u and M d increase with the increase of r 1 while F d decreases with the increase of r 1 .M u increases with the increase of r 1 when r 1 < 0.6 m.However, M u decreases with the increase of r 1 when r 1 > 0.6 m. e variation rates of F d and M d are significantly faster than those of F u and M u , respectively.
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: r 1 � 0.2 m, 0.4 m, 0.6 m, and 0.8 m, respectively.e time histories of the surface wave height f wall on the wall at θ � 0, the resultant hydrodynamic force F 1 in the θ � 0 direction, and the resultant hydrodynamic moment M O2 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 r 1 .

Effect of Excitation Amplitude.
e inner radius of the baffle is fixed at r 1 � 0.8 m, and the baffle is positioned at z 1 � 0.7 m.Four different excitation amplitudes are considered: X 0 � 0.01 m, 0.02 m, 0.03 m, and 0.04 m, respectively.e 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.e time histories of f wall , F 1 , and M O2 are plotted in Figures 19-21, respectively.It is seen that the nonlinearity increases with the increase of the excitation amplitude.

Effect of Excitation Frequency.
e inner radius of the baffle is fixed at r 1 � 0.7 m, and the baffle is positioned at z 1 � 0.7 m.
e fundamental frequency of the liquid is 3.77 rad/s.e container is undergoing the lateral harmonic excitation, and the amplitude is fixed at X 0 � 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: Substituting equations ( 38) and (39) into ( 13)-( 17), we have , where c ij (i � 1, 2, 3; j � 1, 2, 3) satisfies where l 0 , h 0 , l 2 , and h 2 satisfies

Shock and Vibration
e 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 where d 1 satisfies

Shock and Vibration
e response curves in the frequency domain, which are illustrated in Figure 22, 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 c d 1 (σ 2  11    Shock and Vibration

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

Figure 1 :
Figure 1: e partially liquid-filled rigid cylindrical container with an annular rigid baffle subjected to the lateral excitation.

Figure 2 :
Figure 2: Partition of the liquid domain developed by Wang et al. [9].

Figure 4 :
Figure 4: Cross section, subdomains, and artificial interfaces of the time-dependent liquid domain Ω.

Figure 5 :
Figure 5: Cross section, subdomains, and artificial interfaces of the unperturbed liquid domain Ω.

Figure 6 :Figure 7 :
Figure 6: Time history of the surface wave height on the wall at θ � 0 for ω � 4 rad/s.

Figure 8 :
Figure 8: Time history of the surface wave height on the wall at θ � 0 for ω � 6 rad/s.

Figure 12 :Figure 13 :
Figure 12: Time history of the surface wave height at the container wall under the lateral harmonic excitation for r 1 � 0.5 m and z 1 � 0.1 m (a), 0.3 m (b), 0.5 m (c), and 0.7 m (d), respectively.

Figure 16 :
Figure 16: Time history of the surface wave height at the container wall under the lateral harmonic excitation for z 1 � 0.5 m and r 1 � 0.2 m (a), 0.4 m (b), 0.6 m (c), and 0.8 m (d), respectively.

Table 2
. It is seen from Table2that F u and M u decrease with the increase of z 1 while F d and M d increase with the increase of z 1 .It is seen from Table2that the increase rate of F d exceeds the decrease rate of F u . is results in the increase of the maximum resultant hydrodynamic force (F max � F u + F d ) with the increase of z 1 .e increase rate of M d is slower than the decrease rate of M u when z 1 < 0.6 m.However, the increase rate of M d exceeds the decrease rate of M u when z 1 > 0.6 m. is results in the nonmonotonic variation of the maximum resultant moment (M max � M u + M d ).

Table 1 :
e maximum amplitudes of the nonlinear response versus the position of the baffle for r 1 � 0.5 m.

Table 2 :
e resultant hydrodynamic force and moment from liquid below/above the baffle versus the position of the baffle for r 1 � 0.5 m.ΔF u , ΔF d , ΔM u , and ΔM d 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 Δz 1 � 0.1 m.

Table 3 :
e maximum amplitudes of the nonlinear response versus the inner radius of the baffle for z 1 � 0.5 m.

Table 4 :
e resultant hydrodynamic force and moment from liquid below/above the baffle versus the inner radius of the baffle for z 1 � 0.5 m.
12 u , ΔF d , ΔM u , and ΔM d 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 Δr 1 � 0.1 m.12Shock and Vibration

Table 5 :
e maximum amplitudes of the nonlinear response versus the excitation amplitude for r 1 � 0.8 m and z 1 � 0.7 m.