Direct method of solving problems on the combined spatial motions of a body-fluid system

Limarchenko, O. S.

doi:10.1007/bf00884062

“…Let us consider a direct method for solving the nonlinear boundary-value problem (1.16)-(1. 19), assuming that the translational velocity r υ 0 ( ) t and instantaneous angular velocity r ω ( ) t of the body are known functions of time. Let the cavity be cylindrical near the free surface.…”

Section: Approximate Methods For Describing the Motion Of The Fluid Inmentioning

confidence: 99%

Variational methods of solving dynamic problems for fluid-containing bodies

Lukovskii

¹

2004

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A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid-body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.Introduction. The theory of motion of bodies fully or partially filled with a fluid, which is an important division of body mechanics, is widely used to solve a large variety of applied problems. Primarily, these are problems arising in designing modern vehicles for transportation of large amounts of fluid. Such problems include the determination of the mechanical interaction of a tank car and the fluid partially filling it, strength analysis of vessels with environmentally dangerous fluids in seismic areas, stability analysis and efficient control of air/space/sea craft, etc.This paper deals only with dynamic systems consisting of a perfectly rigid body and a perfect incompressible fluid. Formulations of problems for a body interacting with an internal or external fluid are detailed in [7, 13, 15, 52, 71, 79, 93-95, etc.].Mathematical models of such mechanical systems are based on the nonlinear dynamic equations of a body and the nonlinear equations of waves on the free surface of a finite fluid. Scientists working in this area of mechanics aimed their efforts at overcoming the chief difficulty of the theory-the necessity to describe the joint motion of two essentially different objects: a body and a fluid.The motion of a body is known to be described by nonlinear ordinary differential equations, whereas the description of the wave motions of the free liquid surface involves the formulation and solution of nonlinear initial-boundary-value problems for partial differential equations.

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“…19), assuming that the translational velocity r υ 0 ( ) t and instantaneous angular velocity r ω ( ) t of the body are known functions of time. Let us consider a direct method for solving the nonlinear boundary-value problem (1.16)-(1.…”

Section: Approximate Methods For Describing the Motion Of The Fluid Inmentioning

confidence: 99%

Variational methods of solving dynamic problems for fluid-containing bodies

Lukovskii

¹

2004

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A variational approach to solving linear and nonlinear problems for a body with cavities partially filled with a perfect incompressible fluid is enunciated. The approach applies a nonclassical variational principle to describe the spatial motion of a finite fluid with a free surface and the classical variational principle, which is widely used in rigid-body dynamics. These principles are used to formulate variational problems that are the basis of direct methods of solving nonlinear and linear dynamic problems for body-fluid systems. The approach allows us to derive an infinite system of nonlinear ordinary differential equations describing the joint motion of the rigid body and fluid and to develop an algorithm for determining the hydrodynamic coefficients. Linearized differential equations of motion of the mechanical system are presented and approximate methods are given to solve linear boundary-value problems and to determine the hydrodynamic coefficients.Introduction. The theory of motion of bodies fully or partially filled with a fluid, which is an important division of body mechanics, is widely used to solve a large variety of applied problems. Primarily, these are problems arising in designing modern vehicles for transportation of large amounts of fluid. Such problems include the determination of the mechanical interaction of a tank car and the fluid partially filling it, strength analysis of vessels with environmentally dangerous fluids in seismic areas, stability analysis and efficient control of air/space/sea craft, etc.This paper deals only with dynamic systems consisting of a perfectly rigid body and a perfect incompressible fluid. Formulations of problems for a body interacting with an internal or external fluid are detailed in [7, 13, 15, 52, 71, 79, 93-95, etc.].Mathematical models of such mechanical systems are based on the nonlinear dynamic equations of a body and the nonlinear equations of waves on the free surface of a finite fluid. Scientists working in this area of mechanics aimed their efforts at overcoming the chief difficulty of the theory-the necessity to describe the joint motion of two essentially different objects: a body and a fluid.The motion of a body is known to be described by nonlinear ordinary differential equations, whereas the description of the wave motions of the free liquid surface involves the formulation and solution of nonlinear initial-boundary-value problems for partial differential equations. 1 2 l l J J J J t t 12 1 / , (3.32) J J J J J J 1 0. (3.34)The generalized forces will appear on the right-hand sides of Eqs. (3.32) and (3.33), where

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“…That is the reason for surface wave antisymmetry [6][7][8]. During the translational motion of the vessel symmetrical coordinate functions that studied on the base of the linear approach does not perturb, but functions studied by the nonlinear theory are excited by virtue of nonlinear mechanical constraints.…”

Section: Additional Assumptions For Construction Of Mathematical Modelmentioning

confidence: 99%

“…Approximate satisfaction of the kinematic boundary condition on the free surface of the liquid is provided by the Galerkin method and methods of nonlinear mechanics [11] With the application of a smallness hypothesis on parameter a* [10,12], according to [6][7][8] we can determine with the third order smallness precision the dependencies bi= äi + Σ 0 n am7nmi + n,m + Σ ä n a m ai5^m li + Σ h n a m a,[a k h™ mlkl , n,m,l n,m,l,k Qi = Σ + Σ + Σ a j a k^klv j k (6) (the values of coefficients y™ mi , 6% mli , h™ mlki , ßjp, 7j* P , Sjkli are quadratures from Vi, Ω 0 and some their derivatives). Determining the form of interrelation between a{,bi and q i allows us to consider that for the third order smallness precision the boundary condition (3) will be fulfilled for any a* values.…”

mentioning

confidence: 99%

Nonlinear Properties for Dynamic Behavior of Liquid with a Free Surface in a Rigid Moving Tank

Limarchenko¹

2000

International Journal of Nonlinear Sciences and Numerical Simulation

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Mechanical and mathematical aspects of nonlinear properties for dynamical behavior of liquid with a free surface in tanks are described. Basic principles for dynamical simulation of the mentioned systems in the range of manifestation of nonlinear properties are proposed. Criteria for construction of lowdimensional models are developed. Some properties of such models for certain examples demonstrate the potential of the suggested approach.

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Direct method of solving problems on the combined spatial motions of a body-fluid system

Cited by 23 publications

References 1 publication

Variational methods of solving dynamic problems for fluid-containing bodies

Variational methods of solving dynamic problems for fluid-containing bodies

Variational methods of solving dynamic problems for fluid-containing bodies

Nonlinear Properties for Dynamic Behavior of Liquid with a Free Surface in a Rigid Moving Tank

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