Autoparametric resonance in a structure containing a liquid,Part II: Three mode interaction

Autoparametric coupling of the first antisymmetric liquid sloshing mode with two orthogonal structure freedoms in a simple structure containing a liquid is investigated theoretically and experimentally. Asymptotic approximation up to the first order shows four possible conditions of internal resonance. The response of the system is obtained analytically and numerically in the neighbourhood of the internal resonance conditions. Under the principal internal resonance (i.e., when one of the normal mode frequencies is twice one of the other mode frequencies) the system possesses a steady-state response. Under the summed or differenced internal resonance (i.e., when one of the normal mode frequencies equals the sum or difference of two other natural mode frequencies) the system does not achieve a constant amplitude steady-state response.Experimental investigations confirm the possible existence of most of the internal resonance conditions considered in the analytical study; however, theoretical amplitude-frequency response curves are rather higher than the experimental results. Experimental observations showed that other kinds of instabilities occur when the liquid free surface exhibits rotational flow at a forcing frequency just above twice the liquid sloshing frequency.     

Autoparametric resonances in a structure/fluid interaction system carrying a cylindrical liquid tank

The nonlinear-coupled vibrations of an elastic structure and liquid sloshing in a cylindrical container are investigated. The behavior of the liquid surface is governed by a kind of the Mathieu equation because the structure is subjected to a vertical and sinusoidal excitation. Modal equations for liquid sloshing governing the coupled motions are derived when the natural frequency of the structure is equal to twice the natural frequency of an anti-symmetric mode of sloshing. The theoretical resonance curves are determined by using van der Pol's method. The influences of a liquid level and a detuning parameter on the theoretical resonance curves are investigated when only the excitation frequency is selected as a control parameter. The inclination of a frequency response curve depends on the liquid level. Furthermore, a small deviation of the tuning condition may cause amplitude- and phase-modulated motions and chaotic vibrations. This deviation also leads to separate the occurrence region of the coupled vibration into two regions of the excitation frequency. The theoretical resonance curves are quantitatively in agreement with the experimental data. Lastly, the amplitude- and phase-modulated motions and chaotic vibrations were observed in experiments.     

Frequency response of sloshing in an annular cylindrical tank subjected to pitching excitation

Frequency responses of stable planar and rotary motions in a partially filled annular cylindrical tank, subjected to a pitching excitation at a frequency in the neighborhood of the lowest resonant frequency, are investigated. The nonlinearity of the liquid surface oscillation and the nonlinear coupling between the dominant modes and other modes (e.g., an axisymmetric mode) are considered in the response analysis of the sloshing motion. The basic equations of the liquid motion are derived by using the variational principle and the nonlinear equations of motion of the liquid surface displacement are formulated. The characteristics of the liquid motion in an annular cylindrical tank are discussed. The equations governing the amplitude of the stable planar and rotary liquid motions are derived and the stability of each motion is analyzed. An experiment was carried out using a model tank. It is shown that the nonlinear characteristic of the liquid motion in an annular cylindrical tank is more complicated than that in a circular cylindrical tank. Furthermore, it is shown that the nonlinear analysis is important for estimating the sloshing responses.     

Non-linear coupled slosh dynamics of liquid-filled laminated composite containers: a two dimensional finite element approach

This paper brings into focus some of the interesting effects arising from the non-linear motion of the liquid free surface, due to sloshing, in a partially filled laminated composite container along with the associated coupling due to fluid-structure interaction effects. The finite element method based on two-dimensional fluid and structural elements is used for the numerical simulation of the problem. A numerical scheme is developed on the basis of a mixed Eulerian-Lagrangian approach, with velocity potential as the unknown nodal variable in the fluid domain and displacements as the unknowns in the structure domain. The FE formulation based on Galerkin weighted residual method along with an iterative solution procedure are explained in detail followed by a few numerical examples. Numerical results obtained by the present investigation for the rigid containers are first compared with the existing solutions to validate the code for non-linear sloshing without fluid-structure coupling. Thereafter the computational procedures are advanced to obtain the coupled interaction effect of non-linear sloshing in laminated composite containers.     

Low-gravity liquid nonlinear sloshing analysis in a tank under pitching excitation

Under pitch excitation, the sloshing of liquid in circular cylindrical tank includes planar motion, rotary motion and rotary motion inside planar motion. The boundaries between stable motion and unstable motion depend on the radius of the tank, the liquid height, the gravitational intension, the surface tensor and the sloshing damping. In this article, the differential equations of nonlinear sloshing are built first. And by variational principle, the Lagrange function of liquid pressure is constructed in volume integration form. Then the velocity potential function is expanded in series by wave height function at the free surface. The nonlinear equations with kinematics and dynamics free surface boundary conditions through variation are derived. At last, these equations are solved by multiple-scales method. The influence of Bond number on the global stable response of nonlinear liquid sloshing in circular cylinder tank is analyzed in detail. The result indicates that the system's amplitude–frequency response changes from a ‘soft-spring’ to a ‘hard-spring’ in the planar motion with the decreasing of the Bond number, while it changes from a ‘hard-spring’ to a ‘soft-spring’ in the rotary motion. At the same time, jump, lag and other nonlinear phenomena of liquid sloshing are discovered.     

The sloshing of stratified liquid in a two-dimensional rectangular tank

The sloshing of inviscid liquid of stratified density in a rectangular tank is analyzed.As the flow is no longer irrotional,the governing equation is found to be quite different from the Laplace equation used for the liquid of constant density.In particular it contains terms of mixed temporal and spatial derivatives.The problem is solved based on the variable separation method and Laplace transform for the constant Vaisala-Brunt frequency.It is found that the stratification of density may have small effects...     

A numerical study of three-dimensional liquid sloshing in tanks

A numerical model NEWTANK (Numerical Wave TANK) has been developed to study three-dimensional (3-D) non-linear liquid sloshing with broken free surfaces. The numerical model solves the spatially averaged Navier–Stokes equations, which are constructed on a non-inertial reference frame having arbitrary six degree-of-freedom (DOF) of motions, for two-phase flows. The large-eddy-simulation (LES) approach is adopted to model the turbulence effect by using the Smagorinsky sub-grid scale (SGS) closure model. The two-step projection method is employed in the numerical solutions, aided by the Bi-CGSTAB technique to solve the pressure Poisson equation for the filtered pressure field. The second-order accurate volume-of-fluid (VOF) method is used to track the distorted and broken free surface. Laboratory experiments are conducted for both 2-D and 3-D non-linear liquid sloshing in a rectangular tank. A linear analytical solution of 3-D liquid sloshing under the coupled surge and sway excitation is also developed in this study. The numerical model is first validated against the available analytical solution and experimental data for 2-D liquid sloshing of both inviscid and viscous fluids. The validation is further extended to 3-D liquid sloshing. The numerical results match with the analytical solution when the excitation amplitude is small. When the excitation amplitude is large where sloshing becomes highly non-linear, large discrepancies are developed between the numerical results and the analytical solutions, the former of which, however, agree well with the experimental data. Finally, as a demonstration, a violent liquid sloshing with broken free surfaces under six DOF excitations is simulated and discussed.     

An investigation and modelling of energy dissipation through sloshing in an egg-shaped shell

Sloshing absorbers work on a similar principle to that of tuned vibration absorbers. A sloshing absorber consists of a tank, partially filled with liquid. The absorber is attached to the structure to be controlled, and relies on the structure's motion to excite the liquid. Consequently, a sloshing wave is produced at the liquid free surface possessing energy dissipative qualities to suppress excessive vibrations of the structure.The hen's egg has evolved to dissipate vibration energy rapidly to protect its contents. An uncooked hen's egg's capability to rapidly dissipate potentially harmful energy, is due to sloshing of its contents. Hence, there may be lessons to learn from the natural design of an egg which could be employed in the engineered (artificial) design of a sloshing absorber.The primary objective of this work is to identify the physical events responsible for effective energy dissipation in an eggshell, at different fill levels. A secondary objective is to demonstrate the suitability of the Smoothed Particle Hydrodynamics (SPH) method for numerical predictions in such an unusually shaped shell. Through numerical predictions, the possibility of modifying the egg's design to further encourage dissipation patterns is explored briefly. Simple experiments are also presented to check the validity of the numerical predictions.     

Natural sloshing frequencies and modes in a rectangular tank with a slat-type screen

Being installed in tanks, screens play the role of slosh-suppressing devices which may strongly change resonant sloshing frequencies and yield an extra nonlinear damping due to cross-flow resulting in either flow separation or jet flow. Employing the linear sloshing theory and domain decomposition method, we construct an accurate analytical approximation of the natural sloshing modes in a rectangular tank with a slat-type screen at the tank middle. Two-dimensional irrotational flow of an ideal incompressible liquid is assumed. Because the considered flow model does not account for flow separation and jet flow at the screen, the velocity field is locally singular at the sharp edges. The constructed solution captures this singularity. Analyzing this solution establishes a complex dependence of the natural sloshing frequencies on the solidity ratio, the number of submerged screen gaps, the liquid depth, and the position of perforated openings relative to the mean free surface. Results are compared with experimental data. Natural surface wave profiles are discussed in the context of a jump of the velocity potential at the screen and the local inflow component to the screen.     

Analysis of liquid sloshing of a tuned magnetic fluid damper for single and co-axial cylindrical containers

A tuned magnetic fluid damper (TMFD) is a dynamic absorber using a magnetic fluid. A characteristic of the TMFD is changing natural frequency of a magnetic fluid sloshing under a magnetic field. The magnetic fluid sloshing in a coaxial cylindrical container was analyzed theoretically under the axisymmetrical magnetic field. The theoretical results showed that a radial component of the magnetic field also changed the natural sloshing frequency. A policy of the suitable design of the TMFD was presented and the effect of the radial component of the magnetic field was verified experimentally.     

基于液体晃动干扰观测器的航天器混沌姿态H_∞控制

针对航天器受液体燃料晃动及内外周期性微小激励耦合影响产生混沌运动的问题,提出了基于神经网络干扰观测器的自适应H∞鲁棒控制方案,以实现充液航天器混沌姿态运动的消除与液体燃料晃动的抑制.基于神经网络的非线性逼近能力设计干扰观测器,自适应跟踪补偿液体晃动、参数不确定及外扰引起的耦合扰动,解决液体燃料晃动角速度及外扰不易直接测量的问题,提高控制器对系统不确定的自适应能力及液体晃动的抑制能力.同时考虑观测误差与模型不精确问题,利用H∞控制策略提高控制器的鲁棒性.通过与现有常用控制策略的对比仿真研究,验证了控制方案的有效性及优势.     

VIBRATIONS OF PARTIALLY FILLED CYLINDRICAL TANKS WITH RING-STIFFENERS AND FLEXIBLE BOTTOM

The dynamics of a tank partially filled with a liquid having a free surface is investigated. In the analysis, the effect of free surface waves is taken into account, so that both bulging and sloshing modes are studied. The structure is completely flexible, and is composed of a vertically standing circular cylindrical shell, with ring-stiffeners, and a flexible bottom that consists of a circular plate generally resting on an elastic Winkler foundation. The plate edge is joined to the shell by coupling rotational springs distributed around the edge. The interaction between the plate and the shell via the fluid is included. Application to the particular cases of (i) a flexible shell with a rigid bottom and (ii) a flexible bottom plate with a rigid cylinder are also presented, as well as a comparison with approximate theories. Ring-stiffeners on the shell are also considered in order to investigate their influence on the modal behaviour of the tank. Solution of this problem is achieved via the method of artificial springs within the framework of the Rayleigh–Ritz theory, while considering strong fluid–structure interaction.     

Sloshing in partially filled liquid containers—Numerical and experimental study for 2-D problems

This paper deals with the numerical and experimental studies of sloshing of liquid in partially filled prismatic containers subjected to external excitation. Meshless local Petrov-Galerkin (MLPG) method is used for computing the nonlinear sloshing response of liquid in a two-dimensional rigid prismatic tank. At every instant of time, velocity potential is computed at each node and the nodal positions are updated. A local symmetric weak form (LSWF) for nonlinear sloshing of liquid 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. An experimental set-up is also designed to study the behavior of liquid sloshing in partially filled prismatic tank. The resulting slosh heights for various excitation frequencies and amplitudes are compared with the data obtained from the numerical studies. The numerical results are close to that obtained experimentally and little variations in the data are due to ineptness of the experimental set-up and the input parameters.     

圆柱形容器内液体横向晃动的有限元法求解

提出了用变分有限元方法计算容器内液体晃动谐振频率的方法和步骤 ,这种方法可适用于各种形状复杂的容器内液体晃动问题 ,其正确性已经得到验证。按照这一方法计算了卧式圆柱形容器内液体的横向晃动问题 ,并对相关结果进行了总结 ,得出了该类型容器内液体晃动的一般规律。     

不同截面水槽流体参数晃动的SPH模拟

用光滑粒子流体动力学方法(SPH)对几种不同截面水槽内的流体参数晃动问题进行数值模拟.以二维矩形截面水槽为例,对其一阶正对称参数晃动过程进行分析,验证这一非线性晃动问题特征:包括水体波动特性,频率关系.对U形和圆形截面的前二阶反对称与正对称参数晃动过程进行数值模拟.结果表明:SPH方法可以有效地实现槽内流体在竖向谐振作用下的参数晃动过程.     

一种基于双目视觉的箭体晃动在线监测方法

为监测液体运载火箭箭体长期储存状态下的晃动情况,保证产品的安全稳定,提出一种基于双目视觉测量的箭体晃动在线监测方法.利用全站仪等设备对像机内、外参数进行高精度标定;测量在箭体坐标系下同一标识点不同时刻的精确坐标;通过不同时刻标识点的坐标变化及约束条件构建方程组并求解位姿变换参数,进而得到实时箭体晃动量.实验验证结果表明...     

Analysis of the modal energy distribution of an excited vibrating panel coupled with a heavy fluid cavity by a dual modal formulation

This paper describes the modal interaction between a panel and a heavy fluid cavity when the panel is excited by a broad band force in a given frequency band. The dual modal formulation (DMF) allows describing the fluid–structure coupling using the modes of each uncoupled subsystem. After having studied the convergence of the modal series on a test case, we estimate the modal energies and the total energy of each subsystem. An analysis of modal energy distribution is performed. It allows us to study the validity of SEA assumptions for this case. Added mass and added stiffness effects of the fluid are observed. These effects are related to the non-resonant acoustic modes below and above the frequency band of excitation. Moreover, the role of the spatial coupling of the resonant cavity modes with the non-resonant structure modes is also highlighted. As a result, the energy transmitted between the structure and the heavy fluid cavity generally cannot be deduced from the SEA relation established for a light fluid cavity.     

Self-induced rotary sloshing caused by an upward round jet in a cylindrical container

Jet-induced rotary sloshing of a Newtonian fluid in a partially filled cylindrical container is numerically and experimentally visualized to investigate a flow pattern inside the rotary sloshing wave. Visualization techniques employed in this study include Computational Fluid Dynamics (CFD) and Particle Image Velocimetry (PIV). Result for the sloshing period is compared with the theoretical result of a small time-harmonic irrotational flow of an inviscid and incompressible fluid, and several results for 3D view of the rotary sloshing and the velocity vector in the vertical and horizontal cross-sections are given.     

用紧束缚方法描述光子晶体缺陷耦合的共振频率移动

通常采用含两个耦合参数的紧束缚近似，就能很好地描述光子晶体缺陷因耦合而导致的共振频率分裂.然而，缺陷耦合造成的共振频率移动，即包含奇数个缺陷的耦合系统的中央共振频率位置与原来单缺陷时的共振频率位置存在差异，则只有采用含三个耦合参数的严格紧束缚方法才能正确描述.根据耦合参数与共振频率的关系，利用三缺陷耦合系统的模拟计算结果确定了三个耦合参数的具体值，从而在理论上能够预言由任意个缺陷构成的耦合系统的共振频率的移动和分裂.理论预言与基于有限时域差分法的数值计算完全相符.     

Bistability of a compliant cavity induced by acoustic radiation pressure

We report on the first observation of multiple-order bistability due to acoustic radiation pressure in a compliant acoustic cavity formed between a spherical ultrasonic transducer immersed in water and the free liquid surface located at its focus. The hysteretic behavior of the cavity length, observed both with amplitude ramps and frequency sweeps, is accurately described using a one-dimensional model of a compliant Fabry-Pérot resonator assuming the acoustic radiation pressure to be the only coupling between the cavity and the acoustic field.