Sloshing Frequency Research Articles

Analytically approximate natural sloshing modes and frequencies in two-dimensional tanks

Analytical approaches to nonlinear and linear sloshing problems need to know approximate natural sloshing modes expressed by continuously-differentiable harmonic functions. A new method for constructing those approximate modes as well as for a fast computing of the corresponding natural sloshing frequencies is proposed in the two-dimensional case. The method facilities a parametric study of the natural sloshing frequencies in a prismatic tank associated with LNG (Liquefied Natural Gas) containers. The results are extensively compared with other approximate analytical solutions.

Investigation on natural frequency of an optimized elliptical container using real-coded genetic algorithm

This study introduces a method based on real-coded genetic algorithm to design an elliptical shaped fuel tank. This method enhances the advantage of the system such as roll stability, and reduces disadvantages like fluid c.g. height and overturning moment. These parameters corresponding to the elliptical tanks with different filling levels are properly optimized. Moreover the effects of these optimized shapes on natural sloshing frequency are investigated. Comparing presented results with experimental ones indicate the reliability and accuracy of the present work. In addition, a new method based on genetic algorithm, which enhances tank rollover threshold, is presented. This optimization enhances roll stability, although reducing the natural sloshing frequency in comparison to cylindrical tanks. In contrast, the sloshing frequency of the optimized elliptical tank is enhanced in compare with conventional elliptical tanks, which is considered as an advantageof the presented work.

Estimation of Natural Frequency and Suppression Control of Sloshing during Ladle Transfer and Tilt Motion by Automatic Pouring Machine

Estimation of Natural Frequency and Suppression Control of Sloshing during Ladle Transfer and Tilt Motion by Automatic Pouring Machine

Weakly nonlinear sloshing in a truncated circular conical tank

Sloshing of an ideal incompressible liquid in a rigid truncated (tapered) conical tank is considered when the tank performs small-magnitude oscillatory motions with the forcing frequency close to the lowest natural sloshing frequency. The multimodal method, the non-conformal mapping technique and the Moiseev type asymptotics are employed to derive a finite-dimensional system of weakly nonlinear ordinary differential (modal) equations. This modal system is a generalization of that by Gavrilyuk et al 2005 Fluid Dyn. Res. 37 399–429. Using the derived modal equations, we classify the resonant steady-state wave regimes occurring due to horizontal harmonic tank excitations. The frequency ranges are detected where the ‘planar’ and/or ‘swirling’ steady-state sloshing are stable as well as a range in which all steady-state wave regimes are not stable and irregular (chaotic) liquid motions occur is established. The results on the frequency ranges are qualitatively supported by experiments by Matta E 2002 PhD Thesis Politecnico di Torino, Torino.

Sloshing characteristics of an annular cylindrical tuned liquid damper for spar-type floating offshore wind turbine

The natural sloshing frequencies of annular cylindrical TLD are parametrically investigated by experiment, aiming at the exploration of its successful use for suppressing the structural vibration of spar-type floating wind turbine subject to multidirectional wind, wave and current excitations. Five prototypes of annular cylindrical TLD are defined according to the inner and outer radii of acryl container, and eight different liquid fill heights are experimented for each TLD prototype. The apparent masses near the first and second natural sloshing frequencies are parametrically investigated by measuring the apparent mass of interior liquid sloshing to the acceleration excitation. It is observed from the parametric experiments that the first natural sloshing frequency shows the remarkable change with respect to the liquid fill height for each TLD model with different container dimensions. On the other hand, the second natural sloshing frequency is not sensitive to the liquid fill height but to the gap size, for all the TLD models, convincing that the annular cylindrical sloshing damper can effectively suppress the wave- and wind-induced tilting motion of the spar-type floating wind turbine.

내부 슬로싱 현상에 따른 사각상자 형태의 부유체 서지 거동에 대한 수치적 고찰

In this paper, possibility of controlling motion of a floating structure using a tuned liquid damper (TLD) is numerically investigated. A TLD is a tank partially filled with liquid. Sloshing motion of liquid inside a tank is known to suppress movement of the tank subject to external excitations at specific frequency. The effects of sloshing phenomena inside a rectangular floating body on its surge motion are investigated by varying external excitation frequency. First, a grid-refinement study is carried out to ensure validity of grid independent numerical solutions using present numerical techniques. Then, surge motion of the floating body subjected to external wave is simulated for five different excitation frequencies of which the center frequency equals to the natural frequency of internal liquid sloshing. The normalized amplitudes of surge motion of the target floating body are compared according to the excitation frequency, for the cases with and without water inside the floating body. It is shown that the motion of the floating body can be minimized by matching the sloshing natural frequency to the excitation frequency.

The TLCD U-Tube Liquid Sloshing Finite Element Modal Analysis

The paper studies the dynamic characteristics of liquid sloshing in TLD rectangular container, and verifies the feasibility of finite element numerical simulation, by the theory and finite element numerical simulation of liquid sloshing. Similarly employing the verified numerical simulation method, we get the sloshing frequency and mode of liquid sloshing in TLCD u-tube container.

Slosh dynamics of liquid-filled composite containers—A two dimensional meshless local Petrov–Galerkin approach

This paper deals with the studies of sloshing of liquid in partially filled rectangular composite containers subjected to external excitation. The governing equation for inviscid fluid is written as pressure variable form. At each time step, the pressure is evaluated using the meshless local Petrov–Galerkin (MLPG) approach. A local symmetric weak form (LSWF) for linearized sloshing is developed, and a truly meshless method, based on LSWF and moving least squares (MLS) approximation, is presented for the solution of Laplace equation with the requisite boundary conditions. The effectiveness of the developed algorithm is demonstrated through few numerical examples. The comparison of results in terms of natural slosh frequencies, sloshing amplitudes and hydrodynamic pressures obtained in the present investigation are made with those available in the reported literature. To observe the change in the total liquid response due fluid–structure interaction effects, parametric studies are carried out for different cases by varying the fibre orientations and wall thicknesses in the laminated container wall. The present meshless method based on LSWF is found to be simple and attractive with a great potential in engineering applications.

Multimodal analysis of weakly nonlinear sloshing in a spherical tank

AbstractSloshing in a spherical tank due to horizontal excitation is studied by using the nonlinear multimodal method which involves the analytically approximate sloshing modes by Faltinsen & Timokha (J. Fluid Mech., vol. 703, 2012, pp. 391–401). General fully and weakly nonlinear modal equations are derived but an emphasis is on the Moiseev–Narimanov asymptotic modal system which implies that the forcing frequency is close to the lowest natural sloshing frequency and there are no secondary resonances in the forcing frequency range leading to a nonlinear resonant amplification of double and triple harmonics in higher modes. The Moiseev–Narimanov modal system is used to construct an asymptotic time-periodic solution and, thereby, classify the corresponding steady-state wave regimes appearing as stable and unstable planar waves and swirling. The results on the stability boundaries are compared with experiments by Sumner & Stofan (1963,Tech. Rep.TN D-1991, NASA Technical Note) and Sumner (1966,Tech. Rep.TN D-3210, NASA). A good agreement is established for$0. 2\leq h\lesssim 1$. Discrepancy for higher liquid depths$1\lesssim h\lt 2$are explained by secondary resonance.

Dynamic simulation of drum level sloshing of heat recovery steam generator

Drum level sloshing is the latest discovery in the application of heat recovery steam generator (HRSG) in combined cycle, and shows certain negative influence on drum level controlling. In order to improve drum level controlling, influence factors on the drum level sloshing were investigated. Firstly, drum sub-modules were developed using the method of modularization modeling, and then the model of drum level sloshing was set up as well. Experiments were carried out on the experimental rig, and the model was validated using the obtained experimental results. Dynamic simulation was made based on the model to get a 3-D graph of drum level sloshing, which shows a vivid procedure of drum level sloshing. The effect of feed-water flow rate, main-steam flow rate and heating quantity on the drum level sloshing was analyzed. The simulation results indicate that the signals with frequency higher than 0.05 Hz are that of drum level sloshing, the signals with frequency of 0.0-0.05 Hz are that of drum level trendy and "false water level", and variation of the feed-water flow rates, main-steam flow rates and heating quantities can change the frequency of drum level sloshing, i.e., the frequency of sloshing increases with the increase of feed-water flow rate, or the decrease of the main-steam flow rate and the heating quantity. This research work is fundamental to improve signal-to-noise ratio of drum level signal and precise controlling of drum level.

부유식 해상풍력발전기의 자세제어장치 개발을 위한 환형 실린더 탱크의 슬로싱 실험

The floating offshore wind turbines are usually exposed to the wave and wind excitations which are irregular and undirected. In this paper, the sloshing characteristics of annular cylindrical tank were experimentally investigated to reduce the structural dynamic motion of floating offshore wind turbine which is robust to the irregular change of excitation direction of wind and wave. The formula for the natural sloshing frequencies of this annular cylindrical tank was derived theoretically. In order to validate this formula, the shaking equipment was established and frequency response functions were measured. Two types of tank were considered. The first and second natural sloshing frequencies were investigated according to the depth of the water. It has been observed that between theoretical and experimental results shows a good agreement.

Modal Analysis of Liquids in Rectangular Container

The disturbance on the free surface of the liquid when the liquid-filled tanks are excited is called sloshing. It is important to know the fundamental natural frequencies of the system to analyze how a system vibrates under external excitations. The aim of the present paper is to perform find the natural sloshing frequencies of liquid filled containers experimentally and numerically. A swine sweep test is carried out on a shake table to find the slosh frequencies experimentally. Finite element technique is employed to solve slosh frequencies numerically. The governing equation of motion of the fluid is considered homogeneous, isotropic, inviscid and incompressible. The governing equations are expressed in terms of the pressure variable alone. The numerical natural frequencies of sloshing are calculated using 3D finite element model. The liquid domain is discretized using eight-node isoperimetric brick elements. Sloshing frequencies obtained from experiment and numerical model are in excellent agreement. The sloshing frequencies obtained are also compared with analytical solution available in literature and found to be in good agreement.

基于Bessel展开法的微重下圆柱贮箱内液体小幅晃动特性研究

以微重环境下部分充液圆柱贮箱受横向简谐激励作用时液面小幅晃问题为研究背景，利用一阶Bessel函数构造了满足该问题中存在的Laplace方程和边界条件的速度势和波高函数。由于微重下的自由静液面呈复杂的“弯月形”，这使得要求速度势和波高函数精确满足自由液面处的边界条件存在一定的困难。在前人研究的基础上，运用第二类边界条件下的Bessel函数展开法，对微重下的自由液面条件进行精确处理，得到晃动速度势和波高函数的一种半解析法。进一步推导了晃动力和晃动力矩的求解式，并基于等效准则建立了液体自由晃动的弹簧-振子等效力学模型。通过数值计算和对比，验证了文中方法的准确性，并研究了等效力学模型中的参数与Bond数之间的变化规律。 The present study is based on the problem of liquid sloshing with small amplitude in a partial filling cylindrical cavity under harmonic excitation and low-gravity environment. Liquid sloshing velocity potential and wave height function, which are the problems belongs to Laplace equation and boundary conditions, were formulated using Bessel expansion method in this paper. Since the phenomenon o “crescent” of free surface due to low-gravity, the solving process of velocity potential and wave height become more difficult. And the study of characters of liquid sloshing with small amplitude in a cylindrical cavity in low-gravity condition was presented. Based on the consideration of surface tension and the foundation of the predecessor studies, the analytical solution of Laplace equation was calculated after the liquid velocity potential and wave height function were formulated firstly. Then the liquid sloshing force and sloshing moment ware given, and frequency of liquid free sloshing and velocity potential were computed. The equivalent mechanical model of mass-spring was established by mechanical equivalent principle. The present method was certified by comparison between numerical numerical solution and analytical solution. The relationship between parameters of the equivalent mechanical models and Bond number were revealed.

Shaping Container Motion for Multimode and Robust Slosh Suppression

The majority of slosh-control techniques have required feedback control to suppress liquid oscillations induced by container motion. Input shaping is an alternative method for generating motion commands that reduce slosh without the need for sensors. This paper provides a rigorous theoretical process for designing multimode and robust input shapers for slosh suppression. Furthermore, it demonstrates that the input-shaping method is more effective than traditional command-smoothing methods. This paper describes the design of four different input shapers that reduce multiple slosh frequencies and have robustness to frequency changes that can occur in containers of sloshing liquids. The performance of the input shapers is then compared with typical unshaped and smooth commands. Experimental results show that input shaping can robustly reduce slosh amplitude over a range of liquid depths.

Uncertainty Quantification of Sloshing Motion in Partially Filled Rectangular Tanks

Increasing computing power has made it not only possible but also highly practical to apply the Monte Carlo technique to a large variety of engineering problems. In this paper, the Monte-Carlo technique is used for calculating the statistical properties of inviscid incompressible fluid partially filled in a rectangular tank. First, uncertainty in the slosh and acoustic modes is estimated considering the gravitational forces and bulk modulus of the fluid as random variables. Second, uncertainty in the vibration frequencies in the rectangular tank with a baffle fixed inside it is studied. Later, linear slosh wave response analysis is carried out considering uncertainty in the external forcing function. Numerical results of the dynamic analyses show that the local minima and maxima of the slosh wave heights shift in time due to uncertainties in the frequency, whereas the local minima and maxima of the slosh wave heights change in magnitude only at a given time due to uncertainties in the amplitude. It is also found that the baffle, which is typically used to create desirable mode shifts, can also lead to increase in dispersion of the slosh and acoustic frequencies due to uncertainty.

Asymptotic nonlinear multimodal modeling of liquid sloshing in an upright circular cylindrical tank. I. Modal equations

Combining the Lukovsky‐Miles variational method and the Narimanov‐Moiseev asymptotics, we deduce a nonlinear modal system describing the resonant liquid sloshing in an upright circular cylindrical tank. The sloshing occurs due to a small-amplitude periodic or an almost-periodic excitation with forcing frequency close to the lowest natural sloshing frequency. In contrast to the existing nonlinear modal systems based on the Narimanov‐Moiseev asymptotic intermodal relations, the derived modal equations (i) contain all necessary (infinitely many) generalized coordinates of the second and third orders and (ii) include exclusively nonzero hydrodynamic coefficients, for which (iii) fairly simple computational formulas are found. As a consequence, the modal equations can be used in analytical studies of nonlinear sloshing phenomena, which will be demonstrated in the forthcoming Part II.

Wave energy extraction by coupled resonant absorbers

In this article, a range of problems and theories will be introduced that will build towards a new wave energy converter (WEC) concept, with the acronym 'ROTA' standing for resonant over-topping absorber. First, classical results for wave power absorption for WECs constrained to operate in a single degree of freedom will be reviewed and the role of resonance in their operation highlighted. Emphasis will then be placed on how the introduction of further resonances can improve power take-off characteristics by extending the range of frequencies over which the efficiency is close to a theoretical maximum. Methods for doing this in different types of WECs will be demonstrated. Coupled resonant absorbers achieve this by connecting a WEC device equipped with its own resonance (determined from a hydrodynamic analysis) to a new system having separate mass/spring/damper characteristics. It is shown that a coupled resonant effect can be realized by inserting a water tank into a WEC, and this idea forms the basis of the ROTA device. In essence, the idea is to exploit the coupling between the natural sloshing frequencies of the water in the internal tank and the natural resonance of a submerged buoyant circular cylinder device that is tethered to the sea floor, allowing a rotary motion about its axis of attachment.

The Vibration of Partially Filled Cylindrical Tank Subjected to Variable Acceleration

In this study, the vibration of a cylindrical tank partially filled with liquid under motion modeled as mass lumped is investigated. A three-dimensional quasi-static model of a partially-filled tank of circular cross-section is developed and integrated into a comprehensive three-dimensional vehicle model to study its dynamic performance as a function of acceleration, and the fill volume. The liquid load movement occurring in the roll and pitch planes of the tank is derived as a function of the longitudinal acceleration, and then the corresponding shifted load is expressed in terms of center of mass coordinates and mass moments of inertia of the liquid bulk, assuming negligible influence of fundamental slosh frequency and viscous effects. The vibration characteristics of the partially filled tank vehicle are evaluated in terms of load shift, forces and moments induced by the cargo movement, and dynamic load transfer in the longitudinal direction. The semi analytical response is obtained by means of SimuLink? Matlab Software. The effects of longitudinal acceleration of the tank system on the liquid surface inclination and consequently shifting of centroids and moment of inertia are illustrated.

A Boundary Element Investigation of Liquid Sloshing in Coupled Horizontal and Vertical Excitation

Sloshing flows in a two‐dimensional rigid rectangular tank under specified excitations in the coupled horizontal and vertical modes are simulated by using a higher‐order boundary element method (BEM). The liquid sloshing is formulated as an initial‐boundary‐value problem based on the fully nonlinear potential flow theory. And a semi‐mixed Eulerian‐Lagrangian technique combined with the 4th‐order Runge‐Kutta scheme is employed to advance the solutions in the time marching process. A smoothing technique is applied to the free surface at every several time steps to avoid the possible numerical instabilities. Numerical results obtained are compared with the available solutions to validate the developed model. The parametric studies are carried out to show the liquid sloshing effects in terms of the slosh frequencies and excitation amplitudes in horizontal and vertical modes, the second‐order resonance frequency, a bottom‐mounted vertical rigid baffle, free surface displacement, and hydrodynamic forces acting on the tank.

Equivalent mechanical models of sloshing fluid in arbitrary‐section aqueducts

SUMMARYThis paper reports on a semi‐analytical/numerical method to model sloshing water in an arbitrarily shaped aqueduct. The water motion is assumed to be inviscid, compressible, and linear (small displacement). The transverse sloshing fluid in an aqueduct is equivalently simplified as a fixed rigid mass M0 and a mass–spring system (M1, K1). According to a rule that the actual fluid (computed with finite element model) and its equivalent mechanical model have the same first sloshing frequency and acting effects on the aqueduct, the analytical solutions of the fixed (impulsive) mass M0, sloshing (convective) massM1, spring stiffness K1, and their locations in the aqueduct body are acquired by the least squares (curve fitting) algorithm. Applying this equivalent principle, the equivalent mechanical models are respectively obtained for the sloshing water in rectangular, semicircular, U‐shaped, and trapezoid aqueducts. The equivalent principle and fluid models are validated through comparison investigations involving rectangular and U‐shaped aqueducts. The dynamic properties and seismic responses of the original and equivalent systems are simulated, compared, and discussed for a U‐shaped aqueduct bridge. The main purpose of this paper is to provide a simplified model of sloshing fluid for the seismic/wind‐resistant computation of the support structures of the aqueduct bridge. Copyright © 2011 John Wiley & Sons, Ltd.