Dynamics of a bubble rising in gravitational field

AbstractThe rising motion in free space of a pulsating spherical bubble of gas and vapour driven by the gravitational force, in an isochoric, inviscid liquid is investigated. The liquid is at rest at the initial time, so that the subsequent flow is irrotational. For this reason, the velocity field due to the bubble motion is described by means of a potential, which is represented through an expansion based on Legendre polynomials. A system of two coupled, ordinary and nonlinear differential equations is derived for the vertical position of the bubble center of mass and for its radius. This latter equation is a modified form of the Rayleigh-Plesset equation, including a term proportional to the kinetic energy associated to the translational motion of the bubble.

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Dynamics of a spherical bubble rising in gravity, subject to traveling pressure disturbance

Communications in Applied and Industrial Mathematics ◽

10.2478/caim-2018-0020 ◽

2018 ◽

Vol 9 (1) ◽

pp. 149-158

Author(s):

Giorgio Riccardi ◽

Enrico De Bernardis

Keyword(s):

Gravitational Field ◽

Wave Front ◽

Crucial Role ◽

Equations Of Motion ◽

Center Of Mass ◽

Bubble Radius ◽

Spherical Bubble ◽

Pressure Step ◽

Bubble Rising ◽

Traveling Speed

Abstract The motion of a spherical bubble rising in a gravitational field in presence of a traveling pressure step wave is investigated. Equations of motion for the bubble radius and center of mass are deduced and several sample cases are analysed by means of their numerical integration. The crucial role played by the traveling speed of the wave front and by the intensity of the pressure step are discussed. A first comparison with the axisymmetric dynamics is discussed.

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The motion of rotating cylinders sliding on pebbled ice

Canadian Journal of Physics ◽

10.1139/p99-074 ◽

2000 ◽

Vol 77 (11) ◽

pp. 847-862 ◽

Cited By ~ 1

Author(s):

MRA Shegelski ◽

M Reid ◽

R Niebergall

Keyword(s):

Thin Film ◽

Liquid Film ◽

Contact Area ◽

Relative Velocity ◽

Thin Liquid Film ◽

Center Of Mass ◽

Translational Motion ◽

Rotating Cylinder ◽

Outer Edge ◽

Slow Rotation

We consider the motion of a cylinder with the same mass and sizeas a curling rock, but with a very different contact geometry.Whereas the contact area of a curling rock is a thin annulus havinga radius of 6.25 cm and width of about 4 mm, the contact area of the cylinderinvestigated takes the form of several linear segments regularly spacedaround the outer edge of the cylinder, directed radially outward from the center,with length 2 cm and width 4 mm. We consider the motion of this cylinderas it rotates and slides over ice having the nature of the ice surfaceused in the sport of curling. We have previously presented a physicalmodel that accounts for the motion of curling rocks; we extend this modelto explain the motion of the cylinder under investigation. In particular,we focus on slow rotation, i.e., the rotational speed of the contact areasof the cylinder about the center of mass is small compared to thetranslational speed of the center of mass.The principal features of the model are (i) that the kineticfriction induces melting of the ice, with the consequence that thereexists a thin film of liquid water lying between the contact areasof the cylinder and the ice; (ii) that the radial segmentsdrag some of the thin liquid film around the cylinder as it rotates,with the consequence that the relative velocity between the cylinderand the thin liquid film is significantly different than the relativevelocity between the cylinder and the underlying solid ice surface.Since it is the former relative velocity that dictates the nature of themotion of the cylinder, our model predicts, and observations confirm, thatsuch a slowly rotating cylinder stops rotating well before translationalmotion ceases. This is in sharp contrast to the usual case of most slowlyrotating cylinders, where both rotational and translational motion ceaseat the same instant. We have verified this prediction of our model bycareful comparison to the actual motion of a cylinder having a contactarea as described.PACS Nos.: 46.00, 01.80+b

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Turning Gait Planning Method for Humanoid Robots

Applied Sciences ◽

10.3390/app8081257 ◽

2018 ◽

Vol 8 (8) ◽

pp. 1257 ◽

Cited By ~ 3

Author(s):

Tianqi Yang ◽

Weimin Zhang ◽

Xuechao Chen ◽

Zhangguo Yu ◽

Libo Meng ◽

...

Keyword(s):

Inverted Pendulum ◽

Center Of Mass ◽

Translational Motion ◽

Humanoid Robots ◽

Inverted Pendulum Model ◽

Straight Line ◽

Pendulum Model ◽

The Stability ◽

And Control ◽

Present Simulation

The most important feature of this paper is to transform the complex motion of robot turning into a simple translational motion, thus simplifying the dynamic model. Compared with the method that generates a center of mass (COM) trajectory directly by the inverted pendulum model, this method is more precise. The non-inertial reference is introduced in the turning walk. This method can translate the turning walk into a straight-line walk when the inertial forces act on the robot. The dynamics of the robot model, called linear inverted pendulum (LIP), are changed and improved dynamics are derived to make them apply to the turning walk model. Then, we expend the new LIP model and control the zero moment point (ZMP) to guarantee the stability of the unstable parts of this model in order to generate a stable COM trajectory. We present simulation results for the improved LIP dynamics and verify the stability of the robot turning.

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Force Acting on a Spherical Bubble Rising through a Quiescent Liquid.

JAPANESE JOURNAL OF MULTIPHASE FLOW ◽

10.3811/jjmf.10.264 ◽

1996 ◽

Vol 10 (3) ◽

pp. 264-273 ◽

Cited By ~ 8

Author(s):

Shu TAKAGI ◽

Yoichiro MATSUMOTO

Keyword(s):

Spherical Bubble ◽

Bubble Rising ◽

Quiescent Liquid

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SUB-keV ATOM INDUCED SPUTTERING OF CONDENSED N2: A MOLECULAR DYNAMICS STUDY

Modern Physics Letters B ◽

10.1142/s0217984993000850 ◽

1993 ◽

Vol 07 (13n14) ◽

pp. 857-863 ◽

Cited By ~ 5

Author(s):

HEINO KAFEMANN ◽

HERBERT M. URBASSEK

Keyword(s):

Molecular Dynamics ◽

Energy Distribution ◽

Time Dependence ◽

Degrees Of Freedom ◽

Center Of Mass ◽

Translational Motion ◽

Original Position ◽

Time And Space

By molecular dynamics, the sputtering of a condensed N 2 sample due to 100 eV N atom bombardment is studied. The features observed in general parallel those of previous studies of Ar sputtering. Time- and space-resolved measurements give novel information on the original position of sputtered molecules and the time dependence of their energy distribution. Rotational and vibrational degrees of freedom are underpopulated with respect to center-of-mass translational motion.

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Translational motion of a spherical bubble in an acoustic standing wave of high intensity

Physics of Fluids ◽

10.1063/1.1458597 ◽

2002 ◽

Vol 14 (4) ◽

pp. 1420-1425 ◽

Cited By ~ 61

Author(s):

Alexander A. Doinikov

Keyword(s):

Standing Wave ◽

High Intensity ◽

Translational Motion ◽

Spherical Bubble ◽

Acoustic Standing Wave

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SYNTHESIS OF AN ADAPTIVE AUTOMATIC SYSTEM OF AIRCRAFT CONTROLS WITH MULTIDIMENSIONAL PI-REGULATOR

Journal of Rocket-Space Technology ◽

10.15421/51913 ◽

2019 ◽

Vol 27 (4) ◽

pp. 79-85

Author(s):

Aramis Viktorovich Tishchenko ◽

Anatoly Mikhailovich Kulabukhov ◽

Victor Alexandrovich Masalskiy

Keyword(s):

Differential Equations ◽

Nonlinear Differential Equations ◽

Minimum Mean Square Error ◽

Center Of Mass ◽

Control Object ◽

The State ◽

Quadratic Functional ◽

State Variables ◽

Control Actions ◽

Conditional Minimization

The article presents the synthesis of a functional diagram of an adaptive automatic control system (ACS) for controlling an aircraft with an automatically reconfigurable multidimensional PI controller, which provides the minimum static and minimum mean square error of control with minimal energy consumption for the formation of the control exposure. The synthesis of ACS algorithms is performed as a result of solving the problem of conditionally minimizing the quadratic functional of the generalized work (taking into account restrictions on state variables and control actions given by differential equations of the control object (CO) and inequalities). The mathematical description of the multidimensional CO is carried out using the CO model in the state space, which automatically takes into account the mutual influence of individual control loops on each other. As the state variables of the aircraft, linear displacements, speeds and accelerations of the center of mass of the aircraft, and angular displacements, speeds and accelerations of the rotational movement of the aircraft relative to the center of mass are used. The matrix equation of dynamics of the aircraft is formed by a system of nonlinear differential equations of the first order of forces and moments of forces acting on the aircraft. To ensure the minimum static control error, integrators are included in the ACS (for each control action). The algorithm for the formation of control actions of the extended CO, providing the declared properties of the ACS, is obtained as a result of solving the problem of conditional minimization of the generalized work functional. The task of conditional minimization of a functional with constraints is performed by the maximum principle. The resulting two-point boundary value problem is transformed by the invariant immersion method into a Cauchy problem for optimal values of state variables. The evaluation of the characteristics of a specific adaptive ACS for the spacecraft is expected to be obtained as a result of further research by mathematical modeling.

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Algorithm for solving optimal open-loop terminal control problem for spacecraft rendezvous with account of constraints on the state

VESTNIK of the Samara State Aerospace University ◽

10.18287/2541-7533-2021-20-1-46-64 ◽

2021 ◽

Vol 20 (1) ◽

pp. 46-64

Author(s):

A. F. Shorikov ◽

A. Yu. Goranov

Keyword(s):

Control Problem ◽

Nonlinear Differential Equations ◽

Center Of Mass ◽

Algebraic Method ◽

Open Loop ◽

Geometric Constraints ◽

Linear Recurrence ◽

Terminal Control ◽

Finite Dimensional Vector Space ◽

Maneuvering Spacecraft

The paper proposes an algorithm for solving the optimal open-loop terminal control problem of two spacecraft rendezvous with constraints on their states. A system of nonlinear differential equations that describes the dynamics of the active (maneuvering) spacecraft relative to the passive spacecraft (station) in the central gravitational field of the Earth in the orbital coordinate system of coordinates related to the passive spacecraft center-of-mass is considered as an initial model. The obtained nonlinear model of the active spacecraft dynamics is linearized relative to the specified reference state trajectory of the passive spacecraft, and then it is discretized and reduced to linear recurrence relations. Mathematical formalization of the spacecraft rendezvous problem under consideration is carried out at a specified final moment of time for the obtained discrete-time controlled dynamical system. The quality of solving the problem is estimated by a convex functional taking into account the geometric constraints on the active spacecraft states and the associated control actions in the form of convex polyhedral-compacts in the appropriate finite dimensional vector space. We propose a solution of the problem of optimal terminal control over the approach of the active spacecraft relative to the passive spacecraft in the form of a constructive algorithm on the basis of the general recursive algebraic method for constructing the availability domains of linear discrete controlled dynamic systems, taking into account specified conditions and constraints, as well as using the methods of direct and inverse constructions. In the final part of the paper, the computer modeling results are presented and conclusions about the effectiveness of the proposed algorithm are made.

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