Generalized homogeneous functions and the two-body problem

IntroductionInthispaper,wegeneralizetheclassicconceptofhomogeneousfunctionofdegreeαandwestudytherelationbetweenthehomogeneousfunctionconceptandthemovementofabodythatsatisfiestheKepler’ssecondlaw .Asanapplicationoftheinvolvedtechniquesthatwereused ,wepresentasolutionofthetwo_bodyproblem ,givingawaytoobtainatimeequationforthebodythatrotatesaroundtheother,usingtheconceptofhomogeneousfunction .WeobtainedaserieslikeasthatwaspresentedinRef.[1 ] .1　GeneralizedHomogeneousFunctionsLetUbeanopensub…     

Integrals of motion for the classical two-body problem with drag

Integrals of motion for the two-body problem with drag are obtained by operating on the second-order vector differential equation describing the motion. The force field consists of an inverse-square gravitational attraction and a drag force proportional to the velocity vector and inversely proportional to the square of the distance to the attracting center. The developed integrals are the analogs of the Keplerian scalar energy, the vector angular momentum, and the Laplace vector.     

Second-order theory for the two-body problem with varying mass including periastron effect

The model of varying mass function, including periastron effect, in terms of Delaunay variables will be expanded. The Hamiltonian of the problem is developed in the extended phase space by introducing a new canonical pair of variable (\(q_4, Q_4\)). The first “\(q_4 \)” is defined as explicit function of time and the initial mass of the system. The conjugate momenta “\(Q_4\)” is assigned as the momenta raises from the varying mass. The short-period analytical solution through a second-order canonical transformation using “Hori’s” method developed by “Kamel” is obtained. The variation equations for the orbital elements are obtained too. The results of the effect of the varying mass and the periastron effect in the case \(n = 2\) are analyzed.     

Geometric analysis of a two-body problem with quick loss of mass

Nonlinear Dynamics - We consider a two-body problem with quick loss of mass which was formulated by Verhulst (Verhulst in J Inst Math Appl 18: 87–98, 1976). The corresponding dynamical system...     

New Statement of the elasticity problem for a layer

The problem on the equilibrium of an inhomogeneous anisotropic elastic layer is considered. The classical statement of the problem in displacements consists of three partial differential equations with variable coefficients for the three displacements and of three boundary conditions posed at each point of the boundary surface. Sometimes, instead of the statement in displacements, it is convenient to use the classical statement of the problem in stresses [1] or the new statement of the problem in stresses proposed by B. E. Pobedrya [2]. In the case of the problem in stresses, it is necessary to find six components of the stress tensor, which are functions of three coordinates. The choice of the statement of the problem depends on the researcher and, of course, on the specific problem. The fact that there are several statements of the problem makes for a wider choice of the method for solving the problem. In the present paper, for a layer with plane boundary surfaces, we propose a new statement of the problem, which, in contrast to the other two statements indicated above, can be called a mixed statement. The problem for a layer in the new statement consists of a system of three partial differential equations for the three components of the displacement vector of the midplane points. The system is coupled with three integro-differential equations for the three longitudinal components of the stress tensor. Thus, in the new statement, just as in the other statements in stresses, one should find six functions. In the new statement, three of these functions (the displacements of the midplane points) are functions of two coordinates, and the other three functions (the longitudinal components of the stress tensor) are functions of three coordinates. It is shown that all equations in the new statement are the Euler equations for the Reissner functional with additional constraints. After the problem is solved in the new statement, three components of the displacement vector and three transverse components of the stress tensor are determined at each point of the layer. The new statement of the problem can be used to construct various engineering theories of plates made of composite materials.     

Bifurcations and equilibria in the extended N-body ring problem

We consider the motion of an infinitesimal particle under the gravitational field of (n+1) bodies in ring configuration, that consist of n primaries of equal mass m placed at the vertices of a regular polygon, plus another primary of mass m₀=βm located at the geometric center of the polygon. We analyze the phase flow, determine the equilibria of the system, their linear stability and the bifurcations depending on the mass of the central primary (parameter β).This study is extended to the case when the central body is an ellipsoid or a radiation source. In this case, the topology of the problem is modified.     

Statement of the problem of intraocular pressure measurement modeling by a pneumotonometric method

The process of intraocular pressure measurement by an optical analyzer is numerically modeled. The cornea and sclera are treated as axially symmetric deformable shells of revolution rigidly fixed along the edges; the space between the shells is filled with an incompressible liquid. The stress-strain state of the cornea and sclera is described by using a nonlinear theory of shells. The optical system is calculated on the basis of concepts of geometrical optics. Two types of boundary conditions are compared; for each of them, the relation between the pressure in the air jet and the area of the surface from which the reflected light is recorded by the photodetector is analyzed.     

Bounds on the phase speed and growth rate of the extended Taylor-Goldstein problem

In this paper we consider the extended Taylor-Goldstein problem of hydrodynamic stability dealing with the stability of stratified shear flows in sea straits of arbitrary cross section. For this problem we have obtained the upper and lower bounds on the phase velocity of neutral modes and estimates for the growth rate of unstable modes. Furthermore, we have found a semielliptical instability region depending on the minimum Richardson number and parabolic instability regions which intersect the semielliptical instability region for a class of basic flows.     

Complete group classification and exact solutions to the extended short pulse equation

In this paper, the complete group classification is performed on the extended short pulse equation (ESPE), which including many important non-linear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary parameters of the equation. Furthermore, the symmetry reductions and exact solutions to the short pulse types of equations are investigated, and the physical significance of the solutions are considered from the transformation group point of view.     

基于广义卡尔曼滤波的桥梁结构物理参数识别

基于广义卡尔曼滤波提出了随机荷载作用下桥梁结构物理参数的识别方法。首先,以荷载为观测对象,推导出基于有限元模型的桥梁结构系统的观测方程,以结构待识别的物理参数为状态向量,建立系统状态方程;然后,对该状态方程和观测方程构成的非线性参数系统应用广义卡尔曼滤波,从而识别出结构的物理参数。对一座简支梁桥和一座三跨连续梁桥在不同工况下的物理参数识别进行了数值仿真,结果表明本文方法能够准确地识别桥梁结构全部刚度参数、质量参数和阻尼参数,且具有很强的抗噪性能,从而验证了本文方法的有效性和鲁棒性,可应用于识别大型桥梁结构的物理参数。     

基于修正的Rodrigues参数的EKF算法在姿态估计中的应用

以低成本航姿测量系统为研究对象,针对四元数描述姿态时存在冗余的不足,提出使用三维独立矢量的修正的Rodrigues参数进行姿态解算,利用与其影子参数相互切换的方法解决了修正的Rodrigues参数在描述姿态时存在奇异的问题,根据系统中器件的特点,以陀螺零偏和修正的Rodrigues参数为状态向量,以加速度计、磁强计输出为观测向量建立了基于扩展Kalman滤波的姿态估计算法,避免了利用四元数进行姿态估计时状态误差方差阵产生奇异的问题,并在一定程度上减小了估计算法的计算量。最后通过全姿态以及摇摆基座数据仿真验证了算法的有效性。     

A fictitious domain approach for the Stokes problem based on the extended finite element method

In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by extended finite element method and studied for the Poisson problem in a paper of Renard and Haslinger of 2009. The method allows computations in domains whose boundaries do not match. A mixed FEM is used for the fluid flow. The interface between the fluid and the structure is localized by a level‐set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf‐sup condition between the spaces for the velocity and the Lagrange multiplier. Convergence analysis is given, and several numerical tests are performed to illustrate the capabilities of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

A boundary value problem in extended thermodynamics - one-dimensional steady flows with heat conduction

Abstract. One-dimensional steady flows with heat conduction, treated by the 13-moment theory of extended thermodynamics are considered. The usual well-posed boundary conditions for the corresponding problems in the Navier-Stokes-Fourier (NSF) theory are insufficient to give unique solutions. In order to have unique physically sensible solutions, minimization of the deviation of iterative approximations from the exact solutions, proposed in [1], is used as a criterion. Moreover, the solutions are shown to be invariant with respect to change of consistent boundary conditions - a requirement abide by our physical intuition. In the problems of plane shearing flow and Couette flow, the minimization with respect to two uncontrollable parameters is involved. The examples are carried out numerically, and the results are compared with the classical results of NSF theory.Received: 16 August 2002, Accepted: 20 April 2003, Published online: 5 December 2003PACS: 83.20.Lr, 83.50.Ax Correspondence to: I-S. Liu Dedicated to Professor Ingo Müller on the occasion of his 65th birthday     

An extended thermodynamic approach to non-Newtonian fluids and related results in Marangoni instability problem

It is shown that extended irreversible thermodynamics provides a simple and coherent modelling of non-Newtonian fluids. The basic hypothesis underlyig the present formalism is to raise the stress tensor to the status of independent variable. Restrictions are placed on the constitutive equations and the material coefficients by the second law of thermodynamics, stability of equilibrium and the objectivity principle. The general procedure is applied to derive the Reiner-Rivlin and Rivlin-Ericksen second-order fluids. As an illustration, Marangoni convection in a thin horizontal layer of a non-Newtonian fluid submitted to a temperature gardient and placed in a microgravity environment is treated.     

The extended Graetz problem with arbitrary boundary conditions in an axially heat conducting tube

An analytical solution for the extended Graetz problem with prescribed wall temperature, heat flux or a linear combination of temperature and heat flux is presented. The present solution is derived from a superposition principle which is proposed here in order to reduce the problem to equivalent integral equations. The complex Fourier transform and the convolution theorem is used to obtain analytical expansions in terms of exponential functions for the temperature distribution and heat flux at the wall. The solutions obtained in this way are proved to be mathematically simple and easy to handle in small sized computers. A general solution for the particular geometry of a pipe is presented. This solution is extended to the case of an axially heat conducting pipe. The combined effect of both the Péclet number and the wall to the fluid conductivity ratio on the temperature and the heat flux distributions at the wall is analysed.Nomenclature e thickness of the tube wall - g complex function given by Eqs. (31) and (46) - h film coefficient - k thermal conductivity of the fluid - k _s thermal conductivity of the wall - q _i heat flux at the inner wall - q _o heat flux at the outer wall - r radial coordinate - v _x velocity component in the axial direction - x axial coordinate - B _i Biot number (=hR/k) - C _p fluid specific heat at constant pressure - K conductance ratio (=k _se/kR) - M(a, b, z) Kummer's function - O(x) function of order x - P _e Péclet number (=c _pVR/k) - R tube inner radius - T _b temperature of the environment outside the pipe - T _s wall temperature - T ₀ fluid temperature upstream at x=– - T _f fluid temperature downstream at x=+ - V characteristic velocity - _m ^± characteristic roots for >0 and <0 - dimensionless axial coordinate (=x/P _eR) - dimensionless radius (=r/R) - dimensionless temperature (=(T–T ₀)/(T _f–T ₀)) - _s dimensionless wall temperature (=(T _s-T₀)/(T _f-T₀)) - _n modified eigenvalues (= _n P _e) - dimensionless velocity (=v _x/V) - fluid density - dimensionless heat flux (=q _iR/k(T _f–T ₀)) - dimensionless heat flux (=q _o R/k(T _f–T ₀)) - complex number     

Matrix perturbation method for the vibration problem of structures with interval parameters

ＭＡＴＲＩＸＰＥＲＴＵＲＢＡＴＩＯＮＭＥＴＨＯＤＦＯＲＴＨＥＶＩＢＲＡＴＩＯＮＰＲＯＢＬＥＭＯＦＳＴＲＵＣＴＵＲＥＳＷＩＴＨＩＮＴＥＲＶＡＬＰＡＲＡＭＥＴＥＲＳＱｉｕＺｈｉ－ｐｉｎｇ（邱志平）ＣｈｅｎＳｕ－ｈｕａｎ（陈塑寰）Ｌｉｕｚｈｏｎｇ－ｓｈｅｎ...     

On the correspondence between the three- and four-dimensional parameters of the three-dimensional rotation group

A fundamental kinematic theorem due to Euler permits synthesizing a series of three- and four-dimensional orientation parameters that correspond to each other in spaces of the same dimension. We use the theorem about the homeomorphism of two topological spaces (the three-dimensional sphere S ³ ? R ⁴ with a single punctured (removed) point and the three-dimensional space R ³) to establish a one-to-one mutually continuous correspondence between the four- and three-dimensional kinematic parameters prescribed in these spaces. The latter can be proved using the stereographic projection of points of the sphere S ³ onto the hyperplane R ³. For the normalized (Hamiltonian) Rodrigues-Hamilton parameters, we present a method of stereographic projection of a point belonging to the three-dimensional sphere S ³ onto the oriented space R ³. We present a family of local kinematic parameters obtained by the method of mapping four symmetric kinematic parameters of the space R ⁴ onto the oriented real space R ³. In contrast to the well-known four symmetric global parameters of the Rodrigues-Hamilton orientation, the synthesized three-dimensional orientation parameters are local (have two singular points ±360°). The differential equations of rotation in the three-dimensional orientation parameters are obtained by the projection method. We present the three-dimensional parameters corresponding to the classical Hamiltonian quaternions defined in the four-dimensional vector space R ⁴.