Mar 08, 2018 Navigating the Tension Between Agility and Structure. We need to be able to navigate the tension between being flexible and being rigid—to recognize what is possible to change and adapt, and just as importantly, to understand what must not change. We need to be willing to interrogate the structures around us, and to tear them down if they.
Based on the spring–mass model, a novel mechanical model of sloshing with fluid–structure interaction under a horizontal excitation is proposed, and the coupled dynamic equation of sloshing system is established. Considering the flexibility of bulkhead, the effects of certain factors, such as bulkhead bending stiffness and filling ratios, on the mode of coupled sloshing system are investigated. It is found that these factors have significant influence on the mode. By comparing the present results with the results of ADINA based on the linear potential flow theory and published literatures, the proposed coupled sloshing model is verified. The results show that the simplified rigid mass, , dominates the contributions to bending moment near the bottom of a bulkhead and the spring–mass, , , to bending moment near the liquid-free surface of a bulkhead. Furthermore, the computational cost is greatly reduced by using the proposed mechanical model with fluid–structure interaction for a rectangular tank sloshing.
Keywords Sloshing, fluid–structure interaction, spring–mass model, water wave mode, coupling effect
Sloshing is the phenomenon of two or more kinds of fluid restrictedly moving in limited space. The prominent feature of sloshing is the existence of free surface and highly nonlinear characteristics. The sloshing phenomenon is usually encountered in many engineering fields, such as very large crude carrier (VLCC), liquefied petroleum gas (LPG), liquefied natural gas (LNG) liquid cargo ships in non-fully loaded navigation, liquid storage tanks in the earthquake, moving oil tank cars, and rocket bunker tanks. When sloshing occurs, the tank wall is subjected to long-term sloshing pressure, which may cause structural fatigue or even ultimate failure. Sloshing can also affect motion stability and cause disastrous accidents for liquid cargo ships, rocket, and oil tankers. These accidents will not only cause economic loss but also result in the environmental pollution because of oil spill. So, it is of great importance to investigate the sloshing characteristics in engineering problems.
Many researchers have studied the fluid–structure interaction (FSI) in sloshing. The main research methods are as follows.
In the aspect of analytic solutions, the study of nonlinear analytic solution of sloshing and analytic solution of coupled sloshing system was limited, so researchers developed equivalent mechanical models and linear analytic solutions based on potential flow theory for rigid tanks of different shapes. Kim1 studied eigenvalue of sloshing and adopted reduction integral scheme of rotational penalty method to exclude pseudo mode. Ibrahim2 obtained the analytic solution of sloshing for rigid and flexible tank based on linear potential flow theory. He also used the pendulum model and spring–mass model to establish the simplified sloshing model for a rigid tank and the equivalent force and moment acting on the bulkhead were given. By means of the spring–mass mechanical model based on linear potential theory, Livaoglu3 carried out the coupled dynamic response calculation for partially filled tank fluid–structure–soil/foundation and analyzed the effect of foundation stiffness and wall-foundation connection forms on sloshing behavior. On the basis of linear potential theory, Tabri et al.4 adopted a simplified spring–mass model to study the coupling effect during liquid carrier collision, and the results showed that simplified model is reasonable by comparing with experimental data. Tsukamoto et al.5 studied the effect of flexible connection on sloshing between moving link and wall in partially filled tank based on linear potential flow theory, and the calculation results showed that the analytic solution is reasonable through comparison with meshless moving-particle semi-implicit (MPS) method. They also showed that the moving link can weaken sloshing in inner fluid. Based on linear potential flow theory, Li and Wang6 deduced the analytic solution of sloshing pressure and velocity in a two-dimensional rectangular tank, and they provided an accurate simplified mechanical model for sloshing fluid in a rigid rectangular tank.
In the aspect of numerical methods, researchers mainly used finite element method (FEM), meshless method and arbitrary Lagrangian–Eulerian (ALE) method to handle the sloshing problem with FSI. Bucchignani et al.7 studied the motion of an incompressible inviscid flow in a deformable tank. Idelsohn et al.8 used particle FEM to study a tanker sinking. They also found that the method can provide a very advantageous and efficient way for solving contact and free surface problems. By means of ALE method, Zhang and Suzuki9 established the FE model of sloshing with fluid–structure coupling effect under ship collision situations. By comparing Zhang’s model with linear sloshing model and Lagrangian FE model from the view of energy and computational costs, Zhang et al. drew a conclusion that coupling effect has a significant influence on dynamic response of the system and ALE method has certain advantages for sloshing problems. Lee et al.10 used MPS approach and FEM to simulate sloshing in the two-dimensional rectangular tank and dam break successfully. Kim et al.11 adopted impulse response function approach to solve linear motion of liquid ship while finite differential method (FDM) is employed to discrete flow field, and they also studied coupling effects between liquid sloshing and ship motion. Mitra and Sinhamahapatra12 established the coupled dynamic FE equation of sloshing for a two-dimensional elastic tank by using Galerkin method, and the conclusion was that the sloshing pressure in flexible bulkhead condition was larger than that in rigid bulkhead condition. Taking flexible tanks into consideration, Degroote et al.13 numerically calculated tank sloshing related to FSI and found that the impact pressure at the bottom of the rigid cylindrical tank was twice as much as the flexible tank. By means of ALE approach, Kassiotis et al.14 calculated impact pressure that acted on a nonlinear structure. Cao et al.15 numerically calculated the sloshing response with coupling effect between fluid inside the tank and flexible wall under the wind loads by using ALE method. The results showed that reinforcement hoop can reduce tank stress as well as improve the tank stability in wind loads. In addition, Pan et al.16 employed MPS method to simulate large deforming free surface of sloshing.
In the aspect of experimental researches, large-scale model tests and sloshing experiments of a flexible tank hardly have been done although lots of sloshing experiments have been carried out for rigid tanks currently. For a rigid tank, the effects of excitation parameters, such as frequency and amplitude, on sloshing loads was studied by Akyildiz and Unal17 and they also found that baffles can significantly reduce fluid motion. Nevertheless, the effect of fluid viscosity on impact pressures was not investigated in sloshing tests. Specific drop tests of a tank were carried out by Anghileri et al.,18 and the effect of sloshing on tank structural damages was studied. Compared with various numerical results obtained from FE, Eulerian, ALE, and smoothed particles hydrodynamics (SPH) model, they also concluded that the SPH method was the most feasible for the analysis of the sloshing. Tabri et al.4 experimentally investigated the effect of FSI on sloshing load and structural deformation energy during ship collisions, and they found that the deformation energy in the wet test was only about 80% of that in the dry tests. For a standing beam and hanging beam arranged in a rolling tank, a coupled experiment between an elastic beam and a sloshing liquid was carried out by Degroote et al.,13 and the numerical method was validated by the experimental data and the effect of grid scale on calculation results was also discussed. Carra et al.19 experimentally studied the linear and geometrically nonlinear dynamical response of a thin plate in contact with water on one or both sides. They found that the plate deflection due to hydrostatic pressure played an important role in changing the plate nonlinearity, but tests with liquid on both sides can eliminate this effect.
By using numerical methods to study sloshing coupled with structure, the computational scale is very large, especially in dealing with some complex engineering problems. On the basis of above, we propose an alternative mechanical model to investigate liquid sloshing inside a flexible rectangular tank coupled to structure. For a rectangular tank subjected to a horizontal excitation, a novel mechanical model of sloshing involved in coupling effect is proposed. Although the proposed model cannot focus on the details of the flow field distribution, it can be easily used to handle the coupling effect between equivalent sloshing liquid and bulkhead. In this article, first, a simplified fluid–structure coupling model for sloshing fluid in a flexible rectangular tank is proposed, and the dynamic equation of sloshing system is established. Second, the effects of bulkhead bending stiffness and filling ratio on coupled modal frequencies are analyzed for this coupled mechanical model. Then, dynamic response analysis of the model is carried out, and the effects of rigid mass and convection mass on bending moment of bulkhead are investigated. The results are compared with the numerical results of coupled FE model of sloshing based on classical linear potential flow theory and other results from the literatures of Bucchignani et al.,7 Lee et al.,10 and Löhner et al.20 The results demonstrate the proposed model is reasonable. Furthermore, the proposed model has the following advantages:
- It does not need to resolve large-scale data transfer between the flow field and structural field.
- The number of degrees of freedom in nonlinear equation is small for the proposed mechanical sloshing system during numerical calculation of fluid–structure coupling, and the solving efficiency can be increased greatly.
A simplified mechanical model of coupled sloshing system
In this section, based on spring–mass mechanical model of sloshing in a rigid tank, a simplified mechanical model considering the coupling effect between liquid and flexible bulkhead is proposed, and the coupled dynamic equation is established.
A two-dimensional rectangular tank is given as shown in Figure 1; it is of width B, height L, and liquid filling level h.
Figure 1. Coupled sloshing system model (L = 1 m, B = 1 m, , µ = 0.363, E = 3 × 109 Pa).
Assuming the tank wall is rigid, sloshing liquid inside the tank can be simplified by a series of spring–mass, as shown in Figure 1. The model is also shown in the literatures of Ibrahim2 and Livaoglu.3 It can be expressed by the following formulae (1)–(5)
(1) |
(2) |
(3) |
(4) |
(5) |
where denotes the rigid mass moving with the tank; and denote the equivalent mass and the equivalent spring stiffness constant of each sloshing mode, respectively; denotes the distance from the equivalent mass points to fluid center of gravity, respectively; and denote the total mass of fluid in the tank and the fluid filling height, respectively.
Based on the simplified spring–mass model for a rigid tank sloshing, a novel mechanical model involving in coupling to flexible bulkhead is proposed for a rectangular tank sloshing. The bulkhead is simplified as a plane strain beam, and its density, modulus of elasticity, section moment of inertia, external load, and beam bending displacement are denoted by , , , , and , respectively. The simplified fluid point mass, corresponding displacements, and spring stiffness are denoted by , , and , respectively.
Tank sloshing with fluid–structure coupling effect is investigated, as shown in Figure 1. Considering the flexible bulkhead, dynamic equation of fluid–structure coupling system is established by using Lagrange equation derived from Hamilton principle and variational method.
The total kinetic energy of coupled system is
(6) |
The total potential energy of coupled system is
(7) |
The non-conservative force virtual work of system is
(8) |
According to the superposition solution of vibration modes, the beam bending displacement can be expressed by
(9) |
(10) |
where , , , , , , and denote the sectional area of the beam, the external load, the bending displacement of the left and right bulkhead, the generalized coordinates of the left and right bulkhead, and mode of vibration of the beam, respectively, as shown in Figure 1.
Then
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
where
(17) |
where .
According to Lagrange equation, we have
(18) |
![Rigid element ansys Rigid element ansys](/uploads/1/2/3/7/123764530/347862112.jpg)
(19) |
(20) |
Then, the following equations can be yielded by equations (18)–(20)
(21) |
(22) |
(23) |
Definition
Then, equations (21)–(23) are expressed as
(24) |
where and are the generalized coordinates of the left and right bulkhead, respectively; is the spring–mass point generalized coordinates; and are the external excitations which act on the left and right bulkhead, respectively. In equation (24), external excitation, Q, includes force boundary and displacement boundary, such as external force, F, and forced displacement, x(t), and where m is the nodal mass.
Considering the flexibility of bulkhead for a two-dimensional rectangular tank sloshing, the coupled dynamic equation of the proposed mechanical model is established and shown in equation (24).
Liquid sloshing coupling to flexible bulkheads inside a two-dimensional rectangular tank is investigated in this article. The tank is made up of plexiglass; its width (B), height (L), density (ρ), Young’s modulus (E), and Poisson’s ratio (μ) are 1 m, 1 m, 1200 kg/m3, 3 × 109 Pa, and 0.363, respectively. The fluid density is 1000 kg/m3. Aiming to study sloshing problem related to coupling effect, a coupled mechanical model of sloshing is proposed based on the spring–mass model, and modal analysis and dynamic response calculations are carried out under different conditions.
In order to study the effect of FSI on the sloshing, uncoupled sloshing frequencies can be obtained as according to formulae (1)–(5) first. We select the first five order frequencies (n = 5) to analyze the sloshing characteristics. The filling ratio ranges from 30% to 80% with an interval of 10%, and then uncoupled sloshing frequencies are calculated under different filling ratios, as shown in Table 1.
|
Table 1. Liquid sloshing frequencies without coupling effect (Hz).
The bulkhead thickness is taken as follows: 0.01, 0.013, 0.016, 0.02, 0.024, 0.027, and 0.03 m, respectively, and natural frequencies of empty tanks for different thickness of the bulkhead are calculated, as shown in Table 2.
|
Table 2. Tank natural frequencies without loading liquid (Hz).
When coupling effect between liquid and tank is considered, modal frequencies of the coupled system are listed in Tables 3–8. The first five modes mainly represent water wave modes, while the last five modes mainly represent lower modes of tank.
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Table 3. Modal frequencies of coupled system for 30% filling ratio (Hz).
|
Table 4. Modal frequencies of coupled system for 40% filling ratio (Hz).
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Table 5. Modal frequencies of coupled system for 50% filling ratio (Hz).
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Table 6. Modal frequencies of coupled system for 60% filling ratio (Hz).
|
Table 7. Modal frequencies of coupled system for 70% filling ratio (Hz).
|
Table 8. Modal frequencies of coupled system for 80% filling ratio (Hz).
The parameters including liquid filling ratios and structural stiffness influence the modal frequency of the coupled sloshing system. Through comparison of the results obtained from Tables 1–8, some characteristics for coupled sloshing system are found as follows.
Liquid filling ratio has effects on the modal frequency of the coupled sloshing system. It is observed from Tables 1–8 that the first-order frequency of the coupled sloshing system is nearly equal to the one of the uncoupled system at 30% and 40% fill depth. But the sixth-order frequency represented by tank in the coupled sloshing system is smaller than that in the uncoupled system. At 50%, 60%, and 70% fill depth, the first-order frequencies of the coupled sloshing system are also smaller than those in the uncoupled system, while the sixth-order frequencies in the coupled sloshing system are greater than those in the uncoupled system for more flexible bulkhead, such as the thickness of the bulkhead of 0.01 m. It is also found that the first-order frequencies and the sixth-order frequencies in the coupled sloshing system are also smaller than those in the uncoupled system at 80% fill depth.
Structural stiffness has effects on the modal frequency of the coupled sloshing system. If thickness of the bulkhead is smaller, namely, smaller bending stiffness of the bulkhead, the coupling effect becomes stronger, especially on the first two order frequencies of fluid. When the bulkhead is thickened, such as 0.03 m, the bending stiffness becomes larger, and the fluid modal frequencies of coupled sloshing system are closer to the theoretical ones of uncoupled system, namely, coupling effect is weaker. It is also found that the water wave modal frequencies decrease as bending stiffness of the bulkhead decreases. Meanwhile, the water wave modal frequencies of the coupled sloshing system are smaller than those of the uncoupled system for a given liquid filling ratio.
When bending stiffness of the bulkhead becomes larger (i.e. bulkhead thickness is greater than 0.02 m), the sixth-order frequency represented by tank in the coupled sloshing system first increases and then decreases as filling ratio increases, and the first-order frequency of non-loading liquid is the greatest. When the bending stiffness becomes smaller (i.e. bulkhead thickness is smaller than 0.02 m), the sixth-order frequency represented by tank in the coupled sloshing system first increases from below the sixth-order frequency under non-loading liquid to the above of it and then decreases the other way around as filling ratio increases. This phenomenon illustrates that the coupling effect has a significant influence on the mode of coupled system. The fluid–structure coupling strength is represented by both structural stiffness and filling ratio, and the coupled strength between structural stiffness and a series of springs is exhibited in the non-diagonal element of the stiffness matrix in equation (24).
Figure 2 shows the effect of bulkhead thickness and filling ratio on the modal shapes. When the bulkhead thickness is 0.01 m, the sixth and seventh modal shapes represented by tank are asymmetric, such as the sixth and seventh modal frequencies corresponding to the fluid–structure coupling system shown in Tables 3–8. When the bulkhead thickness is large enough, such as 0.03 m, the sixth and seventh modal shapes are symmetric. It illustrates that the asymmetrical characteristic of modal shapes is more obvious when bending stiffness of bulkhead is smaller, and coupling effect should be adequately considered in this case.
Figure 2. Modal shape of coupled sloshing system for bulkhead: (a) filling ratio: 30%, bulkhead thickness: 0.01 m; (b) filling ratio: 30%, bulkhead thickness: 0.03 m; (c) filling ratio: 80%, bulkhead thickness: 0.01 m; and (d) filling ratio: 80%, bulkhead thickness: 0.03 m.
As seen from Table 2, the first two modal frequencies of empty tank are identical, and the modal shapes are symmetrical. But the deviation between the sixth and seventh modal frequencies represented by tank is presented at certain filling ratio in coupled sloshing system, as seen from Tables 3–8. The deviations lead to asymmetry of the sixth and seventh modal shapes. As shown in Figure 2(a) and (c), the sixth and seventh modal shapes present the asymmetrical characteristic at 30% and 80% filling ratio, and the asymmetrical characteristic is more obvious at 80% filling ratio. Although sloshing geometric model with fluid–structure coupling effect is symmetric, the structural modal shapes are asymmetric when strong coupling effect is taken into consideration.
As seen from Tables 1–8, some conclusions can be drawn:
- Under a certain filling ratio, coupling effect leads to a decrease in water wave modal frequency, especially when bending stiffness of the bulkhead becomes smaller and filling ratio becomes higher.
- The parameters of filling ratio and bending stiffness of the bulkhead have an important effect on the first-order frequency of sloshing water wave involving coupling effect. When the bending stiffness becomes larger, the first-order water wave frequency increases as filling ratio increases, and the first water wave frequency of the coupled system is slightly smaller than that of the uncoupled system.
- When the bending stiffness becomes smaller, the first-order water wave frequency first increases (filling ratio ranges from 30% to 40%) and then decreases, and the first-order frequency of water wave in the coupled system is smaller than that of the uncoupled system.
- As filling ratio increases under smaller thickness of the bulkhead, the relative deviation of frequencies between coupled and uncoupled systems becomes larger.
- Considering stronger coupling effect, the first two modal shapes represented by structure are asymmetric although sloshing geometric model is symmetric.
Analysis of dynamic response for the proposed coupled sloshing model
In this section, in order to validate the proposed mechanical model, first, some test cases in the literatures (Bucchignani et al.,7 Lee et al.,10 and Löhner et al.20) are numerically calculated by using the proposed mechanical model. Then, comparative analysis between the results of this model and the results of the literatures are carried out. Finally, this mechanical sloshing model and FE model based on linear potential flow theory are numerically simulated by commercial FSI code ADINA. Also, this model is verified by the two comparative results and the effects of simplified mass on bulkhead bending moment are discussed.
A square tank (L = H = 1 m) derived from the literature of Bucchignani et al.7 is filled by water. The vertical walls are made up of steel (E = 198 × 109 Pa, μ = 0.3) and are characterized by a square section with size s = 5 cm. Free surface is subjected to an initial shape, η(x, 0), under tank bottom motionless condition and initial potential energy, , is obtained; then, we assume that the energy totally converted into initial kinetic energy of spring–mass point, , meanwhile, initial velocity (v0 = 0.022 m/s) of mass, , is obtained. Finally, the proposed model is employed to simulate this case. For the point located on the left wall at the tip, the horizontal displacement obtained from the present results and the other from Bucchignani’s et al.7 is shown in Figure 3. It can be observed that both the results are essentially consistent despite a slight difference between the two.
Figure 3. Transient history of the horizontal displacement of the point located on the left wall at y = 1 m, for λ = L/2.
For an elastic tank in the literature of Lee et al.10 with 50% filling ratio, its width and height are 5 and 8 m, respectively. Young’s modulus, Poisson’s ratio, thickness, and density of the wall are 5 GPa, 0.3, 0.11034 m, and 7860 kg/m3, respectively. The tank is excited with a horizontal acceleration, a, given by a = 0.05·g·sin(0.4t) where g is the gravitational acceleration. Using MPS method, numerical computation for the tank sloshing with coupling effect is carried out by Lee et al.10 By using the proposed method, numerical computation of the tank sloshing is also carried out. The bending deflections of elastic wall are shown in Figure 4. It is found that the deflections from the proposed method are consistent with those from Lee et al.10 except the right wall in the second tank of Figure 4(a). It is also found that highly nonlinearity is presented due to the splashing and breaking of free surface, as shown in the second tank of Figure 4(b). Because the spring–mass model is obtained from linear potential theory in this study, this mechanical model could not deal with the nonlinearity problem including swirling, splashing, and breaking of free surface. As a result, the difference of the bending deflection of the right wall between the second tanks of Figure 4(a) and (b) occurs when free surface presents highly nonlinearity.
Figure 4. The bending deflection of elastic wall: (a) the results of the proposed method and (b) the result of Lee et al.10
Based on volume of fluid (VOF) method, Löhner et al.20 performed numerical computations for a rigid tank sloshing under a given filling ratio (35%) and a sway motion condition (x(t) = A sin(2πt/T), A = 0.025 m, T = 1.27 s). Its length (L), width (b), and height (H) are all 1 m. Corresponding rigid and flexible tank are also studied by this method. For the flexible wall, its density, elastic modulus, Poisson’s ratio, and thickness are 1200 kg/m3, 2 × 109 Pa, 0.363, and 0.02 m, respectively.
The comparison of the horizontal force, Fx, calculated by this method and by Löhner et al. is shown in Figure 5. It is found that the transient response of the horizontal force obtained from this method and VOF method lasted for 40 cycles for the rigid wall. When sloshing response turns into a steady-state regime, both the results for the rigid wall are basically consistent although the amplitude of this result is slightly smaller than that of Löhner’s et al. The reason of the slight distinction is that the large-amplitude motion of free surface is not taken into consideration in the proposed model. Considering the coupling effect, the transient response of the horizontal force lasted for 25 cycles for the flexible wall, which is less than that for the rigid wall. This is because part of the kinetic energy of sloshing liquid is converted into strain energy of the flexible wall, and the steady state of the horizontal force is easily built up. Through comparison of the amplitude of the horizontal force of steady state between the rigid wall and the flexible wall, the former is larger than the latter. The conclusion from the above analysis is that the proposed mechanical model is reasonable for sloshing with coupling effect.
Figure 5. Time history of force Fx for a tank at A/L = 0.025, T/T1 = 1.
Aiming to investigate characteristics of dynamic response of tank sloshing with coupling effect, the tank bottom is subjected to a horizontal harmonic excitation of displacement, x(t) = A sin(2πft), as shown in Figure 1, where excitation frequency is f = 0.5 Hz, displacement amplitude is A = 0.01 m, and bulkhead thickness is 0.013 m. Filling ratio ranges from 30% to 80% with an interval of 10%. The proposed sloshing model with coupling effect in this article and classical coupled FE model of sloshing based on linear potential flow theory are numerically calculated by commercial software ADINA.
In this section, bending moments at p1 and p2 are investigated. The positions of p1 and p2 are shown in Figure 1. p1 is located at the bottom of bulkhead and p2 at an intermediate position from the free surface to bilges.
The coupled FE sloshing model based on linear potential flow theory is modeled, as shown in Figure 6. The model is calculated by the commercial FSI code ADINA. Time histories of bulkhead bending moment for various filling ratios are shown in Figures 7–18. The relationship between filling ratios and amplitude of bending moment is shown in Figures 19 and 20. The solid line represents the results of the proposed mechanical model and the dashed line represents the calculation results of ADINA.
Figure 6. Coupled FE sloshing model based on linear potential flow theory is modeled by ADINA.
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Figure 7. Time histories of moment at p2 (30%).
Figure 8. Time histories of moment at p1 (30%).
Figure 9. Time histories of moment at p2 (40%).
Figure 10. Time histories of moment at p1 (40%).
Figure 11. Time histories of moment at p2 (50%).
Figure 12. Time histories of moment at p1 (50%).
Figure 13. Time histories of moment at p2 (60%).
Figure 14. Time histories of moment at p1 (60%).
Figure 15. Time histories of moment at p2 (70%).
Figure 16. Time histories of moment at p1 (70%).
Figure 17. Time histories of moment at p2 (80%).
Figure 18. Time histories of moment at p1 (80%).
Figure 19. Amplitude of bending moment at p1 under various filling ratios.
Figure 20. Amplitude of bending moment at p2 under various filling ratios.
Through analysis of time histories of bending moment (shown in Figures 7–18) under different filling conditions, it can be observed that the bending moment caused by sloshing is almost the same in the two kinds of calculation models. It proves once again that the proposed mechanical model is reasonable. The time history curves are periodical. The period of bending moment at p1 is equal to the excitation period. Also, it is found that the time histories of bending moment at p2 include some harmonic components though the curves are periodical. As a result, it is concluded that the bending moment near the liquid-free surface contains different components of frequencies.
As seen from Figure 19, when filling ratio is no more than 50%, the amplitude of bending moment at p2 calculated by ADINA for the FE model based on classical linear potential flow theory is greater than that from this mechanical model in this article. When filling ratio is above 50%, the results are on the contrary. The maximum deviation of the amplitude of bending moment at p2 is 1.5 N m, and the relative deviation is 14.56% under 70% filling ratio condition. The bending moment at p2 nonlinearly increases as filling ratio increases. The reason is that sloshing loading at p2 near the liquid-free surface nonlinearly varies as filling ratio increases.
As can be seen in Figure 20, the amplitude of bending moment at p1 calculated by ADINA for the FE model based on classical linear potential flow theory is smaller than that of this mechanical model under all six loading conditions. The maximum deviation of the amplitude of bending moment is 11.5 N m, and the relative deviation is 6.18% under 70% filling ratio condition too. It is also found that the amplitude of bending moment at p1 basically linear increases as filling ratio increases.
By using fast Fourier transform (FFT) technology, time histories of the bending moment at p1 and p2 are analyzed in frequency domain. As shown in Figure 21, frequency component of the bending moment at p2 contains exciting frequency (0.5 Hz) and the first-order and the second-order water wave frequency. Under all six loading conditions, the amplitude of bending moment corresponding to the first-order water wave frequency is larger than that corresponding to the exciting frequency. The amplitude of bending moment corresponding to the second-order water wave frequency is very small and even negligible. The results reveal that the bending moment of the bulkhead near the free surface is dominant contributed by the dynamic load, and the load is induced by the movement of liquid-free surface, that is, the mass point, , and spring, , in this mechanical model with coupling effect.
Figure 21. The amplitude spectrum analysis of bending moment at p2.
As can be seen in Figure 22, frequency component of the bending moment at p1 contains excitation frequency (0.5 Hz) and the first-order water wave frequency. At this location, the amplitude of bending moment corresponding to the first-order frequency is almost negligible by comparing with that corresponding to the exciting frequency under all six loading conditions. The results demonstrate that the bending moment of the bulkhead near the bottom is dominant contributed by the inertia of rigid body mass, , while the dynamic loads caused by mass, , and spring, , have little effect on the bending moment of bulkhead near the bottom.
Figure 22. The amplitude spectrum analysis of bending moment at p1.
In this article, the simplified mechanical model with coupling effect for a rectangular tank sloshing subjected to a horizontal excitation is proposed and the coupled dynamic equation of sloshing is deduced. The effects of bulkhead bending stiffness and filling ratios on the mode of coupled system are investigated, and dynamic response of the coupled system is carried out. In this study, the main research results are summarized as follows:
- Coupling effect leads to a decrease in water wave frequencies.
- As filling ratio increases, structural modal frequencies first increase and then decrease.
- Higher filling ratio and smaller bulkhead thickness cause stronger coupling effect, which can lead to asymmetry of modal shapes for the symmetrical structure.
- The simplified mass, , dominates the contributions to bending moment near the bottom of the bulkhead, and the simplified spring–mass, and , dominate the contributions to bending moment near the liquid-free surface of the bulkhead.
- By comparing the results of this mechanical model with those of other literatures, the proposed mechanical model of sloshing with coupling effect is reasonable for a rectangular tank under a horizontal excitation.
In this study, the proposed mechanical model of sloshing is only effective for a rectangular tank subjected to a horizontal excitation, since the rotational inertia of spring–mass point and tank is not considered for the excitation of pitching. Further researches on the nonlinearity of sloshing under combined excitation including heaving and pitching excitation will be carried out based on the proposed mechanical model with coupling effect.
Academic Editor: Filippo Berto
Declaration of conflicting interests
The authors declare that there is no conflict of interests regarding the publication of this article.
The authors declare that there is no conflict of interests regarding the publication of this article.
Funding
The work presented in this article has been carried out under the co-support provided by the Ministry of Education and Ministry of Finance of China (Grant No. 201335), the Key Project of Natural Science Foundation of China (Grant No. 51239007) and the NSFC (51239007).
The work presented in this article has been carried out under the co-support provided by the Ministry of Education and Ministry of Finance of China (Grant No. 201335), the Key Project of Natural Science Foundation of China (Grant No. 51239007) and the NSFC (51239007).
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