scholarly journals Sloshing of Liquid in Rigid Cylindrical Container with a Rigid Annular Baffle. Part II: Lateral Excitation

2012 ◽  
Vol 19 (6) ◽  
pp. 1205-1222 ◽  
Author(s):  
J.D. Wang ◽  
D. Zhou ◽  
W.Q. Liu
Keyword(s):  
Potential Function ◽  
Separation Of Variables ◽  
Superposition Principle ◽  
Surface Displacement ◽  
Hydrodynamic Pressure ◽  
Cylindrical Container ◽  
Generalized Coordinates ◽  
Total Potential ◽  
Free Surface Wave ◽  
Lateral Excitation

Sloshing response of liquid in a rigid cylindrical container with a rigid annual baffle subjected to lateral excitation has been studied. The complicated liquid domain is separated into several simple sub-domains by introducing the artificial interfaces. The analytical solutions of potential function corresponding to every sub-domain are obtained by using the method of separation of variables and the superposition principle. The total potential function under lateral excitation is taken as the sum of the container potential function and the liquid perturbed function. The expression of the liquid perturbed function is obtained by introducing the generalized coordinates. On the base of the natural frequencies and modes having been obtained by the sub-domain method, the orthogonality among the sloshing modes has been demonstrated. Substituting the potential functions into the free surface wave equation establishes the dynamic response equation of liquid. Then, the generalized coordinates are solved. The sloshing surface displacement, the hydrodynamic pressure distribution, the resultant hydrodynamic force and moment are discussed for the containers subjected to harmonic and seismic lateral excitation, respectively.

Response of Liquid in Cylindrical Tank with Rigid Annual Baffle Considering Damping Effect

2011 ◽  
Vol 255-260 ◽  
pp. 3687-3691 ◽  
Author(s):  
Jia Dong Wang ◽  
Ding Zhou ◽  
Wei Qing Liu
Keyword(s):  
Free Surface ◽  
Dynamic Response ◽  
Potential Function ◽  
Natural Frequencies ◽  
Damping Ratio ◽  
Cylindrical Tank ◽  
Generalized Coordinates ◽  
Damping Effect ◽  
Total Potential ◽  
Free Surface Wave

Sloshing response of liquid in a rigid cylindrical tank with a rigid annual baffle under horizontal sinusoidal loads was studied. The effect of the damping was considered in the analysis. Natural frequencies and modes of the system have been calculated by using the Sub-domain method. The total potential function under horizontal loads is assumed to be the sum of the tank potential function and the liquid perturbed function. The expression of the liquid perturbed function is obtained by introducing the generalized coordinates. Substituting potential functions into the free surface wave conditions, the dynamic response equations including the damping effect are established. The damping ratio is calculated by Maleki method. The liquid potential are obtained by solving the dynamic response equations of the system.

scholarly journals Sloshing of Liquid in Rigid Cylindrical Container with a Rigid Annular Baffle. Part I: Free Vibration

2012 ◽  
Vol 19 (6) ◽  
pp. 1185-1203 ◽  
Author(s):  
J.D. Wang ◽  
D. Zhou ◽  
W.Q. Liu
Keyword(s):  
Free Surface ◽  
Natural Frequencies ◽  
Separation Of Variables ◽  
Superposition Principle ◽  
Ideal Liquid ◽  
Frequency Equation ◽  
Cylindrical Container ◽  
Natural Frequencies And Modes ◽  
Free Surface Waves ◽  
Convergence Study

An analytical approach is presented to obtain the sloshing natural frequencies and modes of ideal liquid in a rigid cylindrical container with a rigid annular baffle. The free surface waves of the liquid are considered in the analysis. The artificial interfaces are introduced to divide the complicated liquid domain into several simple sub-domains. The exact analytical solutions of velocity potential of liquid corresponding to every sub-domain are obtained by using the method of separation of variables and the superposition principle. The Eigen-frequency equation is precisely derived by using the Fourier-Bessel expansion on the free surface and the artificial interfaces of the liquid. The convergence study shows high accuracy and fast convergence of the present approach. The comparative studies with those available from literature are made, excellent agreements have been achieved. Numerical results showing the variations of natural frequencies and modes versus position and inner diameter of the annular baffle are provided.

Stability Analysis of a Reach Truck

2009 ◽  
Author(s):  
Jong-Su Bae ◽  
Taewung Kim ◽  
Hyun-Yong Jeong
Keyword(s):  
Stability Analysis ◽  
Variational Method ◽  
Potential Function ◽  
Fe Simulation ◽  
Safety Concern ◽  
Mathematical Tool ◽  
Total Potential ◽  
Pitch Direction ◽  
Simulation Results ◽  
The Stability

There is a need for a higher mast of a reach truck in the market, but a higher mast brings a safety concern. Usually, it is more plausible to fall in the roll direction than in the pitch direction. Since a reach truck with a high mast is a heavy and its center of gravity is high, it is not easy to conduct tests to evaluate its stability. If there is a mathematical tool to evaluate the stability of a reach truck, it is easy to evaluate a design in terms of stability and to modify the design in order to increase its stability. In this study, a variational method using a total potential function was used to make a mathematical means to evaluate the stability of a reach truck. By using the mathematical means the stability of a reach truck was evaluated and compared with FE simulation results.

Anti-Sloshing Effects of Longitudinal Partial Baffles in a Partly-Filled Container Under Lateral Excitation

2014 ◽  
Author(s):  
Amir Kolaei ◽  
Subhash Rakheja ◽  
Marc J. Richard
Keyword(s):  
Free Surface ◽  
Boundary Element Method ◽  
Eigenvalue Problem ◽  
Boundary Element ◽  
Lateral Force ◽  
Computational Time ◽  
Generalized Coordinates ◽  
Liquid Slosh ◽  
Lateral Excitation ◽  
Element Method

This study is aimed at analysis of transient lateral slosh in a partially-filled cylindrical tank with different designs of longitudinal partial baffles using a coupled multimodal and boundary-element method. A boundary element method is initially formulated to solve the eigenvalue problem of free liquid slosh, assuming inviscid, incompressible and irrotational flows. Significant improvement in computational time is achieved by reducing the generalized eigenvalue problem to a standard one involving only the velocity potentials on the half free-surface length using the zoning method. The generalized coordinates of the free-surface oscillations under a lateral excitation are then obtained from superposition of the natural slosh modes. The lateral slosh force is also formulated in terms of the generalized coordinates and hydrodynamic coefficients. The validity of the model is illustrated through comparisons with available analytical solutions. Two different designs of longitudinal baffles are considered: bottom- and top-mounted baffles. The effect of different baffle designs on the normalized slosh frequencies, modes and lateral force are investigated. It is shown that the multimodal method yields computationally efficient solutions of liquid slosh within moving baffled containers. The results suggest that the effectiveness of baffles in suppressing the liquid oscillations is strongly affected by the baffle length relative to the free-surface height. The top-mounted baffle yields the greatest effectiveness, when it pierces the free-surface. The bottom-mounted baffle, however, may not be considered as an efficient mean for controlling the liquid slosh in tank vehicles where the liquid fill height is above 50%.

The Floquet Analysis of Liquid Interface Wave in a Double-Frequency Excited Cylindrical Container

2014 ◽  
Vol 541-542 ◽  
pp. 505-513
Author(s):  
Long Chang ◽  
Yong Jun Jian ◽  
Yan Jun Sun ◽  
Jie Su ◽  
Quan Sheng Liu
Keyword(s):  
Surface Tension ◽  
Separation Of Variables ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Amplitude Equation ◽  
Single Frequency ◽  
Cylindrical Container ◽  
Interface Waves ◽  
Double Frequency ◽  
Frequency Excitation

In generally cylindrical container, by the method of separation of variables solve the Laplace equation and linearized boundary conditions and obtain amplitude equation of double-frequency excitation fluid interface waves considering surface tension effects. We use double-layer single-frequency viscous fluid damping coefficient to modify the amplitude equation, and then analyze the instability of the new amplitude equation using Floquet theory. By numerical calculation, the instability region determined by different depths, excitation frequency, excitation amplitude, relatively amplitude factor, the phase difference and the surface tension is given. Moreover, the double-frequency excitation spectrum of double critical phenomena is also discussed.

scholarly journals Effect of Double-Side Curved Baffle on Reducing Sloshing in Tanks under Surge and Pitch Excitations

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Aimeng Zhu ◽  
Mi-An Xue ◽  
Xiaoli Yuan ◽  
Feng Zhang ◽  
Wei Zhang
Keyword(s):  
Effective Stress ◽  
Laboratory Experiments ◽  
Hydrodynamic Pressure ◽  
Structural Safety ◽  
Tank Wall ◽  
Motion Platform ◽  
Liquid Storage ◽  
Wave Elevation ◽  
Wave Heights ◽  
Free Surface Wave

Sloshing is associated with the structural safety of liquid storage vessel. Installing the baffles inside the containers would be beneficial for the mitigating the damage due to the severe sloshing. In this study, an innovative type of double-side curved baffle was proposed to evaluate its effect on reducing sloshing in a rectangular tank under surge and pitch excitation. For comparison with conventional baffles, effects of the vertical baffle and the T-type baffle on mitigating sloshing were also studied experimentally and numerically by analyzing the free surface wave elevation as well as the hydrodynamic pressure on the tank wall. The effective stress at the double-side curved baffle along the height direction of the baffle is much smaller than that at the T-type baffle although they have the same mitigation effect on sloshing wave heights. The sloshing-induced effective stress on the double-side curved baffles was analyzed by varying their radian. Findings show that the effective stress on the baffle tends to decrease with the increase in the radian. The velocity field was presented to observe effect of the baffles on sloshing with the aid of ADINA and laboratory experiments conducted on a hexapod motion platform.

scholarly journals Artificial Potential Function Safety and Obstacle Avoidance Guidance for Autonomous Rendezvous and Docking with Noncooperative Target

2019 ◽  
Vol 2019 ◽  
pp. 1-17
Author(s):  
Jianghui Liu ◽  
Haiyang Li
Keyword(s):  
Potential Function ◽  
Obstacle Avoidance ◽  
Motion Equation ◽  
Reference State ◽  
Attractive Potential ◽  
Total Potential ◽  
Autonomous Rendezvous ◽  
Artificial Potential Function ◽  
Rendezvous And Docking ◽  
Target Spacecraft

The problem of artificial potential function (APF) safety and obstacle avoidance guidance for autonomous rendezvous and docking of chaser spacecraft with noncooperative spacecraft is studied. The relative motion equation of the chaser and the target is established based on the line-of-sight coordinate system, the reference state is designed, and the corresponding state error is deduced. The attitude motion equation of the noncooperative target spacecraft in space is established. The safety and obstacle avoidance guidance problem of autonomous rendezvous and docking with noncooperative target is transformed into a path planning problem in a dynamic environment. The attractive potential function is designed according to the state error. In order to ensure that the chaser can safely approach the noncooperative target spacecraft, a safe corridor with ellipse cissoid is designed in the final approaching stage of autonomous rendezvous and docking. The obstacle is assumed to be a sphere with a certain radius to avoid its influence in the approach, and the obstacle potential function is designed based on the Gaussian function method. The total potential function of the system is designed according to the attractive potential function, the safe potential function, and the obstacle potential function. The total potential function of the system is modified to ensure that the reference state is the minimum of the total potential function of the system. The stability of the system is proven according to the Lyapunov stability principle, and the conditions for satisfying the monotonic decrease in the total potential function of the system are deduced. Finally, the effectiveness of the proposed method is verified by three sets of numerical simulations.

scholarly journals Weak formulation of free-surface wave equations

2001 ◽  
Vol 5 (2) ◽  
pp. 75-85
Author(s):  
A. D. Sneyd
Keyword(s):  
Free Surface ◽  
Evolution Equation ◽  
Wave Equations ◽  
Evolution Equations ◽  
Surface Displacement ◽  
Wave Dispersion ◽  
The Body ◽  
Weak Formulation ◽  
Free Surface Displacement ◽  
Free Surface Wave

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion, η=∑n=1∞αn(t)En where the αn(t) are time-dependent coefficients. For a tank with vertical sides the En are eigenfunctions of the eigenvalue problem, ∇2+λ2E=0,  ∇E⋅n^=0 on the tank side walls. Evolution equations for the αn(t) can be obtained by taking inner products of the linearised equation of motion, ρ∂v∂t=−1ρ∇P+F with the normal irrotational wave modes. For unforced waves each evolution equation is a simple harmonic oscillator, but the method is most useful when the body force F is something more exotic than gravity. It can always be represented by a forcing term in the SHM evolution equation, and it is not necessary to assume F irrotational. Several applications are considered: the Faraday experiment, generation of surface waves by an unsteady magnetic field, and the metal-pad instability in aluminium reduction cells.

Design of Fixed Offshore Platforms to Dynamic Ship Impact Loads

1992 ◽  
Vol 114 (3) ◽  
pp. 146-153 ◽  
Author(s):  
M. J. Sterndorff ◽  
J. Waegter ◽  
C. Eilersen
Keyword(s):  
Dynamic Behavior ◽  
Superposition Principle ◽  
Hydrodynamic Pressure ◽  
Dynamic Loads ◽  
Mode Shapes ◽  
Offshore Platforms ◽  
Detailed Calculation ◽  
Pull Out ◽  
Realistic Description ◽  
The Impact

A ship impact is a dynamic phenomenon and the dynamic global load effects can be significant, especially for small platforms where dynamic loads from the ship impact can be larger than the extreme environmental loads and the ship impact can govern the design of the platform. This paper describes a detailed procedure for dynamic analysis of fixed offshore platforms exposed to ship impacts. The procedure includes: • a consistent description of the motion of the vessel and dynamic interaction with the platform during the impact; • a realistic description of the global dynamic behavior of the platform during the impact; • detailed calculation of the transient hydrodynamic pressure forces acting on the vessel during the impact; and • a realistic description of the local deformation zone at the point of impact. The equations of motion for the vessel and the platform are solved simultaneously in the time domain, and the overall dynamic loads acting on the platform during the impact are determined by means of the modal superposition principle. The procedure has been applied for the design and subsequent risk analysis of three small tripod tower-type platforms to impacts from drifting supply vessels. The effect of the number of mode shapes used for representation of the dynamic behavior of the platforms, and the influence of the transient hydrodynamic pressure forces have been investigated. Critical velocity tables for different impact situations have been developed. For nearly all the situations investigated, the critical collapse criterion was overturning of the platform due to pull-out of piles in tension.

The Application of MATLAB in Classical Electrostatics Boundary-Value Problems

2013 ◽  
Vol 378 ◽  
pp. 602-608
Author(s):  
Fu Jian Zong ◽  
Jin Ma
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value Problems ◽  
Potential Function ◽  
Laplace Equation ◽  
Electrostatic Field ◽  
Computer Image ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Partial Differential

In this paper we introduce the use of a computer image and the Partial Differential Equation (PDE) Toolbox in MATLAB, and discuss the electrostatic field, the potential function and the solution of the Laplace equation by separation of variables and the PDE toolbox. It is convenient to figure out the classical electrostatics problem with MATLAB.

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