A successive boundary element model for investigation of sloshing frequencies in axisymmetric multi baffled containers

This study presents a developed successive Boundary Element Method to determine the symmetric and antisymmetric sloshing natural frequencies and mode shapes for multi baffled axisymmetric containers with arbitrary geometries. The developed fluid model is based on the Laplace equation and Green's theorem. The governing equations of fluid dynamic and free surface boundary condition are also applied to proposed model. A zoning method is presented to model arbitrary arrangement of baffles in multi baffled axisymmetric tanks. The influence of each zone on neighboring zones is applied by introducing interface influence matrix which correlates the velocity potential of interfaces to their flux. By discretizing the flow boundaries, the integral equation governed on the boundary is formulated into a general matrix eigenvalue problem. The proposed method has a considerable effect on decreasing computational cost and a good accuracy in determining the sloshing natural frequencies. The obtained results for different types of container based on the application of the presented study are validated in comparison with the literature and very good agreement is achieved. Finally, the effect of baffle parameters on the sloshing natural frequencies was investigated and some conclusions are outlined.     

Reduced order modeling of liquid sloshing in 3D tanks using boundary element method

This paper presents the application of reduced order modeling technique for investigation of liquid sloshing in three-dimensional tanks. The governing equations of sloshing are written using a boundary element formulation for incompressible potential flow. Then, the governing equations are reduced to a more efficient form that is represented only in terms of the velocity potential on the liquid free surface. This particular form is employed for eigen-analysis of fluid motion and the sloshing frequencies and mode shapes are determined. Then, the sloshing frequencies and the corresponding right- and left-eigenvectors are used along the modal analysis technique to find a reduced order model (ROM) for fluid motion. Using a rectangular and a cylindrical tank, the results of the ROM are verified in comparison with the analytical results. Then some example tanks are used to examine the ROM in comparison with the full boundary element model for determination of the free surface motion under the angular and lateral motion of the tank. The obtained results demonstrate the capability and accuracy of the reduced order modeling technique for analysis of sloshing using a few modes.     

A reduced order model for liquid sloshing in tanks with flexible baffles using boundary element method

In order to study the interaction of sloshing and structural vibrations of baffled tanks, a reduced order model based on modal analysis of structure model and boundary element method for fluids motion is developed. For this purpose, the governing equations of elastic structure and incompressible flow are used to derive simple models to simulate both fields. Using the modal analysis technique, the structural motions are applied to the fluid model and on the other hand by using boundary element method, the fluid loads are applied to the structural model. Based on this formulation, a code is developed which is applicable to an arbitrary elastic tank with arbitrary arrangement of baffles. The obtained results are validated with literature data and then the effects of baffle flexibility on the sloshing frequencies and structural vibration frequencies are investigated. Copyright © 2011 John Wiley & Sons, Ltd.     

Boundary element analysis of liquid sloshing in baffled tanks

The paper concerns the natural frequencies and mode shapes of liquid sloshing in three dimensional baffled tanks with arbitrary geometries. The liquid is considered to be inviscid and incompressible and amplitudes of oscillations are assumed to be small. The tank bottom and the baffle are treated as rigid. The boundary element method is used to solve the considered problem. Triangular curvilinear 6-node boundary elements are applied. In the present formulation is not necessary to introduce the zoning method, because the baffles are treated as double layers immersed in liquid. After discretization the problem is formulated as the standard eigenvalue problem, which is limited to free surface degrees of freedom only. Several examples are given to verify the proposed method.     

The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems

This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.     

截面任意形状的柱形和环形容器内液体晃动特性的精确解法

本文研究截面任意形状的柱形和环形容器内液体的晃动特性，利用Ｆｏｕｒｉｅｒ级数匹配法将液体速度势沿容器壁面展开得频率方程，由行列式搜根法数值计算各阶固有振动频率，结果是精确的，方法具有通用性。     

二维晃动自然频率与阻尼比系数的试验识别

液体的晃动模态（自然频率、振型与阻尼比系数）是贮液结构设计以及振动控制的重要参数。在液体晃动的模态试验中,需要激发液面的模态运动,但液面的对称模态运动一般比较难以激发出来,使得对称模态参数（特别是阻尼比系数）难以精确识别。本文采用参数激振的方法对矩形、U形和圆形截面容器进行竖向激振,可容易激发出液体表面的前四阶模态(包括对称模态)运动,撤除激励后液体表面按某一特定的振型作自由衰减振动,通过激光测量液体表面波高的自由衰减曲线,从而精确得到液体晃动的自然频率与对应的阻尼比系数,测得晃动频率与理论频率结果吻合良好,表明本文试验识别方法有效。     

水平激励下任意截面柱形储液罐内液体的晃动响应

采用半解析法研究水平激励下任意截面柱形储液罐内液体的晃动响应。基于叠加原理,将液体运动速度势分为两部分:刚体速度势和摄动速度势。满足非齐次边界条件的刚体速度势可解析得到;以自由晃动模态为广义坐标、时间为变量的摄动速度势可由模态叠加法设出。将整个速度势函数代入自由表面波方程,利用模态的正交性得到动力响应方程。结果与有限元非常吻合。     

液体非线性晃动的同步Hopf分叉现象

针对受俯仰激励作用的圆柱形贮箱中液体非线性晃动,用变分原理建立了液体晃动的压力体积分形式的Lagrange函数;并将速度势函数在自由液面处作波高函数的级数展开,从而导出自由液面运动学和动力学边界条件非线性方程组;最后用四阶Runge-Kutta法求解非线性方程组。计算发现,在一定的激励频率内面外主模态和次生模态发生同步Hopf分叉,并给出了发生分叉的频率区域。     

内置有水平挡板的矩形储液器非线性晃动分析

该文以内置有水平挡板的矩形储液器为研究对象,对储液器的非线性晃动问题展开研究。利用势流理论和虚功原理,推导了由水平挡板引起的储液器非线性阻尼比(非线性体现于波高有关)的计算公式,同时考虑水平挡板对储液器晃动频率的影响而对非线性阻尼比计算公式进行了修正。结合液体晃动的非线性分析理论,研究了水平挡板处于不同位置、挡板长度不同时储液器的液面波高与晃动力变化情况。利用Fluent软件进行了数值模拟,并与理论模型分析的结果进行对比。结果表明：当水平挡板靠近储液器底部或长度较小时,储液器内液体的非线性晃动现象较明显,利用非线性三阶模态方程推导得到的波高、晃动力与数值模拟结果较接近,而仅考虑一阶线性响应会明显低估储液器液面波高,但其求解的晃动力却与考虑非线性值的状况基本一致;随着挡板到自由液面距离的变小或挡板长度的增大,储液器液面波高、晃动力幅值减小,液体晃动呈线性变化,说明水平挡板靠近自由液面或增大其长度时能够提高储液器的阻尼比,进而也更能抑制液体的非线性晃动。同时水平挡板逐渐靠近自由液面或长度逐渐增大时,储液器晃动频率逐渐减小,减小的幅度分别可达到5.7%～28%。     

温度影响下Winkler-Pasternak弹性地基上多孔FGM矩形板的自由振动分析

基于经典薄板理论和Hamilton原理研究温度影响下Winkler-Pasternak弹性地基上多孔功能梯度材料(FGM)矩形板的自由振动特性。采用Voigt混合幂率模型和孔隙任意分布模型来表征多孔FGM矩形板的材料属性,并考虑多孔FGM矩形板内部均匀温升和材料具有温度依赖特性;应用物理中面推导弹性地基上多孔FGM矩形板自由振动的控制微分方程并进行无量纲化;采用微分变换法(DTM)对无量纲控制微分方程及其边界条件进行变换,引入典型的六种边界在MATLAB统一编程且保证计算精度一致,经过迭代收敛,求解出无量纲固有频率;通过算例研究了边界条件、梯度指数、升温、孔隙率、长宽比、边厚比、无量纲弹性刚度系数和无量纲剪切刚度系数对多孔FGM矩形板振动特性的影响。     

Free Vibration Analysis of Generally Layered Composite Beams with Arbitrary Boundary Conditions

A hyperbolic shear deformation theory is used for the free vibration analysis of generally layered composite beams with arbitrary boundary conditions. The variationally consistent governing differential equations and boundary conditions are derived by employing Hamilton's principle. The dynamic stiffness method is applied to calculate the vibration frequencies of the laminated beams with the help of Wittrick–Williams algorithm. Examples of application of the hyperbolic shear deformation theory for the free vibration analysis of the laminated beams with a couple of different boundary conditions are presented. The present results are compared to the numerical solutions and experimental results available in the literature.     

Analysis of the small amplitude sloshing of a liquid in a rigid container of arbitrary shape using a low‐order boundary element method

A boundary element method for estimating the natural sloshing frequencies and forces developed due to an inviscid, incompressible liquid oscillating in a rigid‐walled tank is presented here. The amplitude of oscillation is assumed to be small enough to allow linearization of the boundary condition at the free surface. Both free and forced oscillation cases are studied. Comparison with results available in the literature shows good agreement. The present method offers significant economy of computing resources compared to the finite element method. Copyright © 2000 John Wiley & Sons, Ltd.     

Boundary integral equation solution of viscous flows with free surfaces

Summary solutions of the biharmonic equation governing steady two-dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.An iterative modification of the classical BBIE is presented which is able to solve a large class of (nonlinear) viscous free surface flows for a wide range of surface tensions. The method requires a knowledge of the asymptotic behaviour of the free surface profile in the limiting case of infinite surface tension but this can usually be obtained from a perturbation analysis. Unlike space discretisation techniques such as finite difference or finite element, the BBIE evaluates only boundary information on each iteration. Once the solution is evaluated on the boundary the solution at interior points can easily be obtained.     

Large-displacement nonlinear sloshing in 2-D circular rigid containers — prescribed motion of the container

The two-dimensional large-displacement non-linear sloshing analysis of liquids in circular rigid containers is revisited. The updating of the free surface position is carried out using an adaptive technique for repositioning the computational nodes on the free surface avoiding the use of remeshing algorithms. Smoothing and volume correction approaches using polar co-ordinates are also presented. The fluid is modelled with potential flow theory using modified Rayleigh damping. All non-linear terms in the boundary conditions are taken into account. The known prescribed motion of the container is arbitrary. Boundary elements are used to solve the potential equations and standard techniques are used for the time integration. The analysis can be applied to arbitrarily shaped containers and is limited to the case of non-breaking waves. © 1998 John Wiley & Sons, Ltd.     

Non‐linear sloshing in partially liquid filled containers with baffles

A finite element method is used for computing the non‐linear sloshing response of liquid in a two‐dimensional rigid rectangular tank with rigid baffles. The potential formulation is considered for the liquid domain and a mixed Eulerian–Langrangian scheme is adopted. The solution is obtained by the Galerkin method. The fourth‐order Runge–Kutta method is employed to advance the solution in the time domain. A regridding technique is applied to the free surface of the liquid, which effectively eliminates the numerical instabilities without the use of artificial smoothing. Through the comparison with the available results for the rectangular tank without baffle, the validity of the present formulation is checked and then extended to the solution of tanks with rigid baffles. The effects of baffle parameters such as position, dimension and numbers on the non‐linear sloshing response are examined. The present numerical solution procedure is also applied to the non‐linear sloshing problems in a circular cylindrical container with annular baffle. Copyright © 2006 John Wiley & Sons, Ltd.     

任意的拉-欧边界元法解大晃动问题

本文针对可动边界问题提出了一个任意的拉格朗日-欧拉边界元法.运用该方法不仅边界上的控制点可以精确地跟踪自由面,而且可以避免单元畸变,保持良好的数值稳定性,实例计算表明这对求解大振幅晃动问题特别有效.另外文中引入了临界振幅的概念,对晃动分析中使用小振幅假设的条件进行了论证,首次给出了它的量级界限.     

贮箱内液体小幅晃动的频率和阻尼计算

根据小幅晃动条件下利用速度势函数推导的计算液体晃动频率和内部Rayleigh阻尼的特征值方程,利用简化处理转化为一般的广义特征值问题,并针对这种非对称大型稀疏矩阵采用了Arnoldi迭代方法,求解得到晃动频率和内部阻尼。将利用Stokes边界层理论计算得到边界上的阻尼与内阻尼之和作为晃动的等效阻尼,通过两个算例比较计算结果接近实验结果。这种方法可以计算任意形状贮箱内液体的小幅晃动频率和晃动阻尼。     

平行四边形加肋板自由振动分析的无网格法

基于一阶剪切理论,提出一种求解平行四边形加肋板自由振动问题的无网格法,通过用一系列点来离散平板及肋条,得到加肋板的无网格模型。基于一阶剪切理论及移动最小二乘近似求出位移场,以梁模拟肋条,求出平行四边形加肋板总动能及总势能。再由Hamilton原理导出加肋板自由振动的控制方程,采用完全转换法引入边界条件,求解方程得出结构自振频率。以不同参数的加肋板为例,将该文解与ABAQUS有限元解进行比较分析。研究表明,该方法能有效地分析平行四边形加肋板自由振动问题,在肋条位置改变时,又避免了网格重构。     

Noise due to sloshing within automotive fuel tanks

The effect of liquid sloshing upon the acoustics of automotive vehicles is becoming more pronounced with the increasing size of fuel tanks and the recent progress in minimising tyre, engine and aero-acoustical noise generation. This paper deals with experiments and numerical simulations of sloshing in automotive fuel tanks. A correlation between recorded slosh noise and simulated pressure fluctuations within the sloshing liquid is derived. The presented numerical simulations are based on the volume-of-fluid method and are able to track the free-surface-flow within complex fuel tank shapes subjected to arbitrary outer acceleration profiles. Some typical applications of the numerical approach are presented, too.