Navier-Stokes方程的奇异性对大气运动方程组的影响

综述了大气运动基本方程组在光滑函数类中的稳定性和Navier-Stokes方程的不稳定性的若干结论．在此基础上,以大气运动方程组的Boussinesq近似为例,阐述了Navier-Stokes方程的不稳定性导致的大气运动基本方程组的某些简化模式的不稳定性,从而得到在简化基本方程过程中应该遵守的一个原则,以保证简化方程的稳定性．     

大气运动基本方程组的稳定性分析

以分层理论提供的基本方法分析大气运动基本方程组的拓扑学特征;证明局地直角坐标系中的大气运动基本方程组在无穷可微函数类中是稳定方程;给出局部解意义下使方程组典型定解问题适定的充要条件;讨论大气动力学中有关“以过去推测未来”以及当涉及应用问题时如何修改定解条件和下垫面的选择等问题;指出在通常假设下,基本方程组中的3个运动方程和连续方程完全决定了这个方程组的性质．     

Model-based control strategies for systems with constraints of the program type

The paper presents a model-based tracking control strategy for constrained mechanical systems. Constraints we consider can be material and non-material ones referred to as program constraints. The program constraint equations represent tasks put upon system motions and they can be differential equations of orders higher than one or two, and be non-integrable. The tracking control strategy relies upon two dynamic models: a reference model, which is a dynamic model of a system with arbitrary order differential constraints and a dynamic control model. The reference model serves as a motion planner, which generates inputs to the dynamic control model. It is based upon a generalized program motion equations (GPME) method. The method enables to combine material and program constraints and merge them both into the motion equations. Lagrange’s equations with multipliers are the peculiar case of the GPME, since they can be applied to systems with constraints of first orders. Our tracking strategy referred to as a model reference program motion tracking control strategy enables tracking of any program motion predefined by the program constraints. It extends the “trajectory tracking” to the “program motion tracking”. We also demonstrate that our tracking strategy can be extended to a hybrid program motion/force tracking.     

Hamiltonian formulation of generalized quantum dynamics

The Hamiltonian formulation of the usual complex quantum mechanics in the theory of generalized quantum dynamics is discussed. After the total trace Lagrangian, total trace Hamiltonian and two kinds of Poisson brackets are introduced, both the equations of motion of some total trace functionals which are expressed by total trace Poisson brackets and the equations of motion of some operators which are expressed by the without-total-trace Poisson brackets are obtained. Then a set of basic equations of motion of the usual complex quantum mechanics are obtained, which are also expressed by the Poisson brackets and total trace Hamiltonian in the generalized quantum dynamics. The set of equations of motion are consistent with the corresponding Heisenberg equations. Project supported by Prof. T.D. Lee’s NNSC Grant, the National Natural Science Foundation of China, the Foundation of Ph. D. Directing Programme of Chinese University, and the Chinese Academy of Sciences.     

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).     

On effects of discretization and the selection of constitutive equations on stability of continua

In classical continuum mechanics the set of basic equations consists of the equation of motion, the kinematic equation and the constitutive equations. The study concentrates on constitutive modeling and the effects of discretization on stability problems. The method of investigation is analytic, the spectra of linear mapping operators of continuous and discrete dynamical systems are studied. As results cases are found, when a hidden incursive nature of a material model leads to unstable behavior of the continuum. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Visualization in mechanics: the dynamics of an unbalanced roller

It is well known that mechanical engineering students often find mechanics a difficult area to grasp. This article describes a system of equations describing the motion of a balanced and an unbalanced roller constrained by a pivot arm. A wide range of dynamics can be simulated with the model. The equations of motion are embedded in a graphical user interface for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the system dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration the simulation tool has been applied to a range of roller–pivot arm configurations. In addition, approximations to the equations of motion are explored and a second-order model is shown to be accurate for a limited range of parameters.     

Modelling of the controlled motion ofa point on a plane

A solution of the problem of feedback control of the motion of a point on a plane is presented. The equations of the controlprogramme (the objective) are set up as a system of differential equations with a given set of singular trajectories in the domain of admissible positions of the controlled point, as well as a given topological structure of the partition into trajectories. These equations define the vector field of velocities of the programmed motions of the point and are used to find the corresponding control forces.     

Well-posedness of a model of point dynamics for a limit of the Keller-Segel system

The purpose of this paper is to prove well-posedness for a problem that describes the dynamics of a set of points by means of a system of parabolic equations. It has been seen in Velázquez (Point dynamics in a singular limit of the Keller-Segel model. (1) motion of the concentration regions, SIAM J. Appl. Math., to appear) that the considered model is the limit of a singular perturbation problem for a system of the Keller-Segel type.     

Rigid–flexible coupling dynamic analysis on a mass attached to a rotating flexible rod

A rigid–flexible coupling dynamic analysis is presented where a mass is attached to a massless flexible rod which rotates about an axis. The rod is limited to small deformation so that the mass is constrained to move in the plane of rotation. A strongly nonlinear model of the system is established based on the couplings between the elastic deflections of the mass and rigid rotation, in which the mass deflection and rigid rotation are both treated as unknown variables. The additional inertia forces on the mass and coupling mechanism are elucidated in the system model. In the case of varied but prescribed rigid rotation, a set of time-varying differential equations governing mass motion is obtained. The trajectories of mass motion are examined for the spin-up and spin-down rotation. Under constant rigid rotation, a set of ordinary differential equations is further attained, and the issues with dynamic frequencies and critical angular velocity of the system are analyzed. The effects of the centrifugal, Coriolis and tangential inertia forces on the dynamic responses are discussed.     

Modelling and simulation of a sphere rolling on the inside of a rough vertical cylinder

A classical problem of nonholonomic system dynamics—the motion of a sphere on the inside of a rough vertical cylinder—is extended to rolling friction. The case study is modelled in independent coordinates. Due to the nonholonomic constraints imposed on the sphere, the governing equations arise as a set of differential-algebraic equations. The results of numerical simulations show the transition of the sphere from a sinusoid path on the vertical cylinder surface to a fall with slip. The physics of the ‘paradoxical’ motion is explained in detail.     

Regular and singular components of periodic flows in the fluid interior

The structure of infinitesimal periodic motions in the interior of a rotating compressible fluid which has been stratified using salt is analyzed taking account of dissipation effects. In the general case, the system of fundamental equations of motion belongs to the class of singularly perturbed equations, the solutions of that consist of functions which are regular and singular with respect to the dissipative coefficients that describe both propagating hybrid waves as well as several types of accompanying singular components including boundary layers. The thicknesses of the singular components are determined by the kinematic viscosity, the diffusion coefficient of the salt and the characteristic frequencies of the problem. In the model of a barotropic or homogeneous fluid, the singular components of spatial periodic flows combine together, which is indicative of degeneracy of the system of equations. Taking account of the full set of components, which are regular and singular with respect to the dissipative characteristics, enables one to construct exact solutions of problems of the generation and non-linear interaction of waves.     

Existence of oscillatory solutions along the path of longitudinal flight equilibriums of an unmanned aircraft, when the automatic flight control system fails

The motion around the center of mass of a rigid unmanned aircraft, whose flight control system fails, in an “Aero Data Model In a Research Environment” is described, by a set of nine nonlinear ordinary differential equations. The longitudinal flight with constant forward velocity is described by a subset of three nonlinear differential equations, obtained from the general system. In this paper, the existence of oscillatory solutions of this system of three differential equations is proved by means of coincidence degree theory and Mawhin's continuation theorem.     

可修复复杂系统脆性故障的研究

提出了可修复复杂系统的脆性概念,主要针对可修复复杂系统的脆性及其特性进行研究,建立了无储备和有储备可修复复杂系统的结构方程模型,用范数指标函数作为衡量控制变量的标准研究了可修复复杂系统的最优控制问题.     

A nonlinear truck model and its treatment as a multibody system

A planar vertical truck model with nonlinear suspension and its multibody system formulation are presented. The equations of motion of the model form a system of differential-algebraic equations (DAEs). All equations are given explicitly, including a complete set of parameter values, consistent initial values, and a sample road excitation. Thus the truck model allows various investigations of the specific DAE effects and represents a test problem for algorithms in control theory, mechanics of multibody systems, and numerical analysis. Several numerical tests show the properties of the model.     

寿险品种优化理论模型

分析了寿险品种创新依附于寿险定价的实际情况.以个体公平的现代产品创新理念,创建独立的寿险品种创新方法体系,并利用遗传算法原理给出了寿险品种优化设计的理论模型,使复杂的寿险品种创新过程得以量化,从而变得简洁直观,并富有方便的可操作性,适合计算机并行处理.这一模型蕴涵了动态的思维方式,利用它可随时对寿险品种进行调整.     

Two-step generation of the equations of motion of closed-chains

This paper presents a two-step generation of the equations of motion of planar mechanisms using point and joint coordinates. First, the formulation replaces a rigid body by a dynamically equivalent constrained system of particles and uses Newton’s second law to study the motion of the particles without introducing any rotational coordinates. Then, the equations of motion are transformed to a reduced set in terms of selected relative joint variables using a velocity transformation matrix. For an open-chain, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For a closed-chain, suitable joints should be cut and few cut-joints constraint equations are included. An example of a closed-chain is used to demonstrate the generality and efficiency of the proposed method.     

Coloured noise for dispersion of contaminants in shallow waters

In this article, we explore the application of a set of stochastic differential equations called particle model in simulating the advection and diffusion of pollutants in shallow waters. The Fokker–Planck equation associated with this set of stochastic differential equations is interpreted as an advection–diffusion equation. This enables us to derive an underlying particle model that is exactly consistent with the advection–diffusion equation. Still, neither the advection–diffusion equation nor the related traditional particle model accurately takes into account the short-term spreading behaviour of particles. To improve the behaviour of the model shortly after the deployment of contaminants, a particle model forced by a coloured noise process is developed in this article. The use of coloured noise as a driving force unlike Brownian motion, enables to us to take into account the short-term correlated turbulent fluid flow velocity of the particles. Furthermore, it is shown that for long-term simulations of the dispersion of particles, both the particle due to Brownian motion and the particle model due to coloured noise are consistent with the advection–diffusion equation.     

Mathematical analysis of radionuclides displacement in porous media with nonlinear adsorption

This paper is devoted to the mathematical analysis of the miscible displacement of a set of radionuclides in a flow occurring in a heterogeneous porous medium. The flow is governed by Darcy's law, and the motion of the chemical species is given by a nonclassical advection-diffusion-reaction equations system. The novelty of the model lies in the adsorption phenomenon that leads to a time derivative of a nonlinear term in these equations. A semi-discretization method is used to establish the existence of weak solutions to this system. Uniform L^∞-estimates on the solutions are specified.