非保守力学系统的Lie对称性和守恒量

本文讨论非保守力学系统的Ｌｉｅ对称性，给出由Ｌｉｅ对称性得到力学系统守恒量的条件，并给出了说明性的例子，在文章最后，还说明对于非保守系统，它的Ｎｏｅｔｈｅｒ对称性并不一定是其自身的Ｌｉｅ对称性。     

理论力学中的概率问题

讨论了含有随机变量的力学系统. 通过4个例题给出了研究力学系统概率问题的具体方法.     

完整力学系统的共形不变性与守恒量

将Birkhoff方程的共形不变性和共形因子的概念拓展到完整力学系统,研究一般完整力学系统在无限小变换下的共形不变性与守恒量.给出了一般完整力学系统的共形不变性的定义和确定方程;研究了系统的Noether对称性与共形不变性之间的关系,研究表明,当Noether对称变换的生成元和非势广义力满足一定条件时,变换也是共形不变的,给出了相应的共形因子表达式,得到了一般完整力学系统的共形不变性直接导致的Noether守恒量;研究了系统的Lie对称性与共形不变性之间的关系,给出了与Lie对称性相应的无限小变换共形不变的充分必要条件,得到了一般完整力学系统的共形不变性直接导致的Lutzky守恒量.文中还举例说明结果的应用.     

Hamilton原理四种表达形式的异同

Ｈａｍｉｌｔｏｎ原理四种表达形式的异同陈滨（北京大学力学系北京１００８７１）１．引言Ｈａｍｉｌｔｏｎ原理是经典力学中的重要原理。它是以泛函（Ｈａｍｉｌｔｏｎ作用量）驻值的变分形式给出的力学系统动力学原理，并可以作为整个力学系统动力学研究的基础，Ｈａｍ...     

旁撑对一端支承一端自由压杆长度系数的影响

本文讨论了旁撑对一端支承一端自由的压杆长度系数的影响，并给出了简单的计算公式，供设计者参考。     

动力学与控制论力学系统的自由度

力学系统的自由度定义源自描述系统位形的独立坐标数.在分析力学发展过程中,人们通过对非完整约束的研究,将其拓展为独立的坐标 变分数.本文指出,对于含非完整约束的力学系统,该定义存在不妥之处,给出的自由度会过度限制系统的力学行为.文中研究力学系统在状态空间中的可达流形,指出可达流形维数与描述系统动力学的一阶常微分方程组的最少未知函数个数一致,例如Gibbs-Appell方程与广义速度方程联立的未知函数个数,进而将可达流形维数的一半定义为系统自由度.通过含黏弹性支承的振动系统、在倾斜平面上运动的冰橇等案例,讨论了单个非完整约束导致的半自由度概念,指出其力学意义和与相邻整数自由度的关系.此外,文中还给出两个非完整约束导致系统减少一个自由度的案例,讨论了系统的切丛和余切丛维数.      

论力学系统的自由度

力学系统的自由度定义源自描述系统位形的独立坐标数.在分析力学发展过程中,人们通过对非完整约束的研究,将其拓展为独立的坐标变分数.本文指出,对于含非完整约束的力学系统,该定义存在不妥之处,给出的自由度会过度限制系统的力学行为.文中研究力学系统在状态空间中的可达流形,指出可达流形维数与描述系统动力学的一阶常微分方程组的最少未知函数个数一致,例如Gibbs-Appell方程与广义速度方程联立的未知函数个数,进而将可达流形维数的一半定义为系统自由度.通过含黏弹性支承的振动系统、在倾斜平面上运动的冰橇等案例,讨论了单个非完整约束导致的半自由度概念,指出其力学意义和与相邻整数自由度的关系.此外,文中还给出两个非完整约束导致系统减少一个自由度的案例,讨论了系统的切丛和余切丛维数.     

变质量可控力学系统的Gauss原理和Appell方程

力学系统的运动依赖于力和约束,人们可以借助于力来控制运动(称为动力学控制),也可以借助于约束来控制运动(称为运动学控制)。我们研究一类力学系统,它的约束依赖于某些控制参数。得到可控力学系统的Lagrange方程、Hamilton方程和AppelI方程,用Gauss原理导出一阶非线性非完整系统广义坐标下的Appell 方程。本文考虑非线性非完整系统,导出了变质量可控力学系统在广义坐标和准坐标下的Gauss原理及Appell方程,最后举例说明其应用。本文主要结果是(1.5),(1.6),(1.20),(2.1),(3.13)及(3.14)。     

参激屈曲梁的倍周期分岔和混沌运动的实验研究

本文对一端固定一端滑动承受轴向简谐载荷的屈曲梁的非线性响应进行了实验。研究了其基本参数共振和主参数共振两种情况，揭示了系统的倍周期分岔和混沌运动等复杂动力学行为。在某些情况下，混沌吸引子和周期吸引子共存，另一些则存在间歇混沌。给出了响应的时间历程、相图、频率谱和Ｐｏｉｎｃａｒｅ映射     

Nabla导数下Hamilton系统的约化

提出并研究nabla导数下Hamilton系统的约化问题.依据nabla导数下力学系统的Hamilton原理,建立Hamilton系统的正则方程,给出系统的能量积分和循环积分;并利用这些积分,约化系统的Hamilton正则方程.结果表明:约化后的方程仍保持系统的Hamilton正则方程形式,Nabla导数下力学系统的约化理论是连续和离散力学系统的约化理论的统一和拓展.文中讨论了时间尺度等于实数集和整数集两种特殊情形下Hamilton系统的约化,并举例说明了结果的应用.     

THE MATHEMATICAL MODELS AND GENERALIZED VARIATIONAL PRINCIPLES OF NONLINEAR ANALYSIS FOR PERFORATED THIN PLATES

ＴＨＥＭＡＴＨＥＭＡＴＩＣＡＬＭＯＤＥＬＳＡＮＤＧＥＮＥＲＡＬＩＺＥＤＶＡＲＩＡＴＩＯＮＡＬＰＲＩＮＣＩＰＬＥＳＯＦＮＯＮＬＩＮＥＡＲＡＮＡＬＹＳＩＳＦＯＲＰＥＲＦＯＲＡＴＥＤＴＨＩＮＰＬＡＴＥＳＣｈｅｎｇＣｈａｎｇｉｕｎ（程昌钧）；ＹａｎｇＸｉａｏ...     

Numerical study on effects of check valve closure flow conditions on pressure surges in pumping station with air entrainment

This paper proposes a numerical procedure to better compute the characteristics of pressure surges when check valves close under different flow conditions in a pumping station. Studies have shown that the effects of check valve closure on the pressure transients are predominantly dependent on the magnitude and gradient of the flow velocities immediately downstream of the check valve at the instant of valve closure. Through the present study, it was noted that the transient flow velocities near the check valve of a fluid system are also dependent on the characteristics of the air entrained into the fluid system. An improved numerical computational procedure for the fluid system with air entrainment under different transient conditions downstream of the check valve is also proposed in this paper. With a fluid system operating within the critical range of air entrainment values, the present analysis showed that there is a possibility of ‘high pressure surges’ when the check valves were closed at flow rates other than the positive flow conditions. This phenomenon was confirmed through field observations. This study thus concludes that a detailed numerical transient analysis of the fluid system, with various assumed amounts of entrained air, is necessary whenever there is the possibility of air entrainment into the fluid system, and that the flow conditions at the instant of check valve closure need to be modelled. Copyright © 2001 John Wiley & Sons, Ltd.     

Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media

Traditional mathematical models of multiphase flow in porous media use a straightforward extension of Darcys equation. The key element of these models is the appropriate formulation of the relative permeability functions. It is well known that for one-dimensional flow of three immiscible incompressible fluids, when capillarity is neglected, most relative permeability models used today give rise to regions in the saturation space with elliptic behavior (the so-called elliptic regions). We believe that this behavior is not physical, but rather the result of an incomplete mathematical model. In this paper we identify necessary conditions that must be satisfied by the relative permeability functions, so that the system of equations describing three-phase flow is strictly hyperbolic everywhere in the saturation triangle. These conditions seem to be in good agreement with pore-scale physics and experimental data.     

捷联式惯性导航系统算法研究

对高动态环境下的高精度捷联惯性导航系统的算法进行了深研究。中提出了一种新的三回路捷联惯尼算法,该算法具有姿态圆锥补偿和速度的划船和转动效应补偿。中推导出三回路算法中的导航系,机体系和地球系更新的数学模型,给出了姿态圆锥补偿、速度划船效应和转动效应的数学模型。并对该算法进行了仿真研究,最后给出了计算仿真结果。     

A generalized mathematical model to analyze the nonlinear behavior of a controlled gyroscope in gimbals

In this paper, the effect of the parameter variation on the stability and dynamic behavior of a gyroscope in gimbals with a feedback control system, formed by a Proportional + Integral $+$ Derivative (PID) controller and a DC motor with an ideal train gear is researched. The generalized mathematical model of the gyro is obtained from the Euler-Lagrange equations by using the nutation theory of the gyroscope. The use of approximated models of the control system are deduced from the mathematical model of the gyro, taking into account that the integral action of the PID controller is constrained and that the inductance of the DC motor may be negligible. The analysis and choice of appropriate state variables to simulate the dynamic behavior of different models of the gyro are also considered. The paper shows that from the analysis of the equilibrium points, a Bogdanov Takens and a Poincaré-Andronov-Hopf bifurcation can appear. These bifurcations are analyzed from the calculation of a parameter which is known as the first Lyapunov value, showing that it is possible to deduce a procedure to find out when a complicated model can be substituted by a simpler one. In particular, the possibility of self-oscillating and chaotic behavior for different models of the system by using the first Lyapunov value as a function of the parameters of the PID controller is researched. Numerical simulations have been performed to evaluate the analytical results and the mathematical discussions.     

A mathematical model of the ring-spinning process

This paper develops a mathematical model of the ring-spinning process that takes into account its non-stationary nature. A complex system of differential equations is obtained, which from a mathematical point of view constitutes a ‘free-boundary’ problem. Its solution involves definition of suitable boundary conditions related to the mechanical characteristics of the process and of the spinning machine itself. The boundary conditions which determine the solution are pointed out. A numerical solution of the system of differential equations can be obtained by the Finite-Segments method, as shown in an example.     

Study of a new macro-particle model for the low-temperature pyrolysis of dried wood chips

 This paper presents a transient one-dimensional mathematical model which simulates the pyrolysis of a single dried wood particle. The porous wood particle is considered as a two-phase system: the solid phase consisting of wood and char and the gas phase consisting of volatiles and tar. Conservation equations for mass, momentum and energy are developed for each phase. Chemical processes are described through an existing one-stage three-reactions scheme, leading separately to char, tar and volatiles evolution dynamics. The variation of transport and physical properties with temperature and with composition is presented by algebraic equations. The model presented in this paper is more advanced than the current models found in literature, since it contains physical effects not included in past models, such as cross diffusion, differing solid and gas phase temperatures and a transient gas phase momentum equation incorporating the wall friction experienced by a fluid flowing through a porous medium. Furthermore, an adequate reference system for enthalpy, based on temperature dependent reaction heats, is used. The mathematical equations with initial and boundary conditions are solved numerically by means of a commercial CFD code (PHOENICS). The validity of the pyrolysis kinetics scheme is examined through comparison with experimental data. Furthermore, the macro-particle model is validated by (1) examining the limitations and importance of the newly-modelled effects (different solid phase and gas phase temperature, cross diffusion and wall friction) and (2) making a comparison between predicted and experimental results for large particles. Received on 18 December 2000     

Modeling and dynamic analysis of a magnetically actuated butterfly valve

In this work, a magnetically actuated butterfly valve is considered and a novel and accurate mathematical model is derived. The equilibrium of the system is investigated and the effects of the inlet velocity and direct current voltage (DC) on the stable rotation angle of the valve are presented. Considering a time periodic perturbation arising from electric circuit, the effects of the operating angle, inlet velocity, and driving parameters on the periodic and chaotic dynamics of the system are investigated. It is observed that, for an opening angle less than the cut-off angle, there exists a unique DC voltage for a stable equilibrium. The stability of this equilibrium depends nonlinearly on the inlet velocity and the seating torque. An expression is derived for the threshold value for the stability of the valve. Under periodic voltage, the inlet velocity and stable angle induce a backward shift on the resonant frequency, and jump phenomena and subharmonics are observed for some values of the driving amplitude. The highest amplitudes of vibration are detected for a fully open valve, for an almost closed valve, and for a valve with large inlet velocity. Using bifurcation diagrams and Lyapunov exponents, it is shown that the system exhibits a route to chaos with windows of period doubling and unbounded motion. Some guidance for design of magnetically actuated butterfly valves is proposed as well as recommendations for future work.     

变结构多刚体系统的动力学问题

本文深入地研究了多刚体系统结构变化的动力学问题,建立下相应的数学模型,讨论了有关的数字仿真,并给出了一个算例,结果是满意的,对所论课题,本文工作同以往相比更准确完整,更便于计算机编程。     

Modeling of Axisymmetric Two-phase Dilute Flows

This paper deals with the numerical treatment of Eulerian approach for dilute two-phase compressible flows (gas-particles mixtures) in axisymmetric configurations. For dilute flows, two classes of models depending on the dispersed phase volumetric fraction can be found. The volume occupied by the particles may be considered, that yields a model in which the gas phase and the dispersed phase equations are coupled through the void fraction and the source terms (Delhaye model). The void fraction effects can be neglected, that means the gas phase is a carrier phase for the particles (Ishii model). The mathematical nature of the two models is demonstrated from analysis of characteristic directions. For the Delhaye's model, a centered scheme is used to solve the system of partial differential equations, while an upwind TVD scheme is used for the Ishii's one. Then, it is shown that the problem of symmetry boundary conditions does not depend on the physical approach, as long as the flow remains dilute. However, a classical treatment for symmetry boundary conditions at the geometrical axis leads to large errors. A particular treatment for this boundary is presented: a new class of particles, described by a supplementary system of equations, is required.