Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

/pdf/liquid-sloshing-dynamics-4apevf1yp2.pdf

Liquid Sloshing Dynamics

Recent Advances in Liquid Sloshing Dynamics

The nonlinear coupled vibrations of an elastic structure and liquidsloshing in a rectangular tank partially filled with liquid, are investigated.The structure on which the liquid tank is attached is vertically subjected to a sinusoidal excitation when the natural frequency of the structure is equal to twicethe natural frequency of one of the sloshing modes. In the theoretical analysis, the modal equations are derivedby taking nonlinear fluid force into account. Responses of the structure and the liquid surface are presented asresonance curves using the harmonic balance method. From this theoreticalanalysis the following predictions are obtained: (a) due to the nonlinearity of the fluid force, harmonic oscillations appear in the structure, while subharmonic oscillations occur on the liquid surface; (b) the shapes of the resonance curves markedly change depending on the liquid level; and (c) when the tuning condition is slightly deviated, amplitudemodulated motions and chaotic oscillations appear during a certain range of the excitation frequency. These were qualitatively in agreement with the experimental results.

Nonlinear Parametric Vibrations of an Elastic Structure with a Rectangular Liquid Tank

Recent Advances in Physics of Fluid Parametric Sloshing and Related Problems

An analytical model for describing the effectiveness of deepwater-type tuned liquid damper (TLD) with submerged nets for suppressing horizontal vibration of structures is proposed. TLD is a damping device for suppressing the vibration of long-period structures such as high-rise buildings, tall towers, the pylons of cable-stayed bridges, and so on. The damping force is created by the hydrodynamic force caused by the sloshing of water contained in rectangular tanks located on top of such structures. In this study, we proposed the dynamical model for analyzing deepwater-type TLD (DTLD) where the liquid depth is deep compared with the length of the rectangular tank. In particular, the effect of hydraulic resistance produced by submerged nets installed in the tank is examined intensively. In the analysis of DTLD, employing finite amplitude wave theory, we obtained the hydrodynamic force and the dissipation energy by using Galerkin method, taking the effect of submerged nets into account. The calculated results thus obtained are compared with experimental results, by which the validity of the modeling methodology is confirmed. Finally, the case in which DTLD with nets is installed on an actual structure is investigated both theoretically and experimentally and the the performance of DTLD is illustrated.

Modeling of Deepwater-Type Rectangular Tuned Liquid Damper With Submerged Nets

Nonlinear liquid motion in a partially filled annular cylindrical tank, in response to a pitching excitation, is investigated. The nonlinearity of the liquid surface oscillation is considered in the response analysis of the sloshing motion. Basic equations are derived by employing the variational principle. The nonlinear ordinary differential equations governing the liquid surface oscillation are then derived by applying Galerkin's method to the basic equations. The effects of the ratio of inner radius to outer one on liquid surface oscillation are discussed. The time histories of the liquid surface displacement are calculated to the harmonic pitching excitations. An experiment was conducted using a model tank. A good agreement was found between the theoretical and experimental results. It is shown that the nonlinear analysis is important for estimating the sloshing responses.

Nonlinear Liquid Oscillation in an Elastic Circular Cylindrical Tank Subjected to Pitching Excitation.

Three-dimensional nonlinear liquid motion in a rigid rectangular tank partially filled with liquid, in response to pitching excitation, is investigated. The nonlinear ordinary differential equations governing the liquid surface oscillation are derived by applying Galerkin's method to basic equations. The fundamental modes in two horizontal directions of the tank are considered dominant in the admissible functions that are assumed to be represented by combining the modal functions obtained by the linearization analysis. The nonlinear coupling characteristics of two dominant modes are discussed. Calculations are presented for the time histories of the liquid surface displacement to the harmonic pitching excitation. It is shown that the three-dimensional nonlinear analysis is important for estimating the sloshing responses in a rectangular tank when the aspect ratio of the tank is an integer.

Three-Dimensional Sloshing Analysis in a Rectangular Tank Subjected to Pitching Excitation.

Frequency responses of stable planar and rotary motions in a partially tilled circular cylindirical 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 equations governing the amplitude of those two types of liquid surface motions are derived and the stability analysis of each motion is conducted. An experiment was carried out using a model tank. Good agreement was found between the theoretical and experimental results. It is shown that the nonlinear analysis is important for estimating the sloshing responses.

Frequency Response of Sloshing in a Circular Cylinirical Tank Subjected to Pitching Excitation.

Free vibration analysis is presented for liquid surface motion in a rigid circular cylindrical tank which is rotating around its symmetrical axis at constant angular velocity. It is considered that the equilibrium surface forms a paraboloid of revolution. Basic equations of liquid motion are derived by using the acceleration potential under the assumptions that the gravity. effect is dominant in comparison with the centrifugal force and that the liquid is inviscid and incompressible. It is confirmed that both elliptic and parabolic vibration modes of free surface oscillation appear due to tank rotation. The characteristic equations for the elliptic modes are derived by applying Galerkin's method. It is noted that the natural frequencies of the elliptic mode are separated by the Coriolis force. An experiment was conducted using a model tank. A good agreement was found between the theoretical and experimental results.

Free Vibration Analysis of Liquid Motion in a Rotating Circular Cylindrical Tank (Elliptic Mode).

Parametric vibration of the liquid surface in a partially filled circular cylindrical tank is investigated. The tank is subjected to horizontal, vertical and pitching excitations. Nonlinearity of the liquid surface oscillation is considered in the response analysis of the sloshing motion. The nonlinear and parametric ordinary differential equations, governing the axisymmetric liquid surface oscillation and including the higher order radial modes, are derived by applying Galerkin's method. It is confirmed that the vertical excitation causes the parametric excitation. In addition, it is noted that the pitching excitation also causes the parametric excitation when the pitching axis does not intersect the symmetrical axis of the circular cylindrical tank. The time histories of the liquid surface displacement to the harmonic vertical excitations are calculated in cases where the dominant mode is axisymmetric and where it has nodal radiuses. An experiment was conducted using a model tank. Good agreement was found between the theoretical and experimental results.

Hiroki Takahara

Papers

Nonlinear Liquid Oscillation in an Elastic Circular Cylindrical Tank Subjected to Pitching Excitation.

Three-Dimensional Sloshing Analysis in a Rectangular Tank Subjected to Pitching Excitation.

Frequency Response of Sloshing in a Circular Cylinirical Tank Subjected to Pitching Excitation.

Free Vibration Analysis of Liquid Motion in a Rotating Circular Cylindrical Tank (Elliptic Mode).

Effects of Higher Order Radial Modes upon Nonlinear Sloshing in a Circular Cylindrical Tank Subjected to Pitching Excitation.