随机激励的耗散的Hamilton系统理论的研究进展

近几年中,利用Ｈａｍｉｌｔｏｎ系统的可积性与共振性概念及Ｐｏｉｓｓｏｎ括号性质等,提出了高斯白噪声激励下多自由度非线性随机系统的精确平稳解的泛函构造与求解方法,并在此基础上提出了等效非线性系统法,提出了拟Ｈａｍｉｌｔｏｎ系统的随机平均法,并在该法基础上研究了拟Ｈａｍｉｌｔｏｎ系统随机稳定性、随机分岔、可靠性及最优非线性随机控制,从而基本上形成了一个非线性随机动力学与控制的Ｈａｍｉｌｔｏｎ理论框架．本文简要介绍了这方面的进展．     

新型Poisson括号意义下的无穷维Lie代数

本文首先针对ＫｄＶ方程的Ｈａｍｉｌｔｏｎ形式，建立一种比较容易验证的新型Ｐｏｉｓｓｏｎ括号和无穷维Ｌｉｅ代数．其次，研究ＫｄＶ方程的Ｈａｍｉｌｔｏｎ形式的第一积分与新型Ｐｏｉｓｏｎ括号的关系，得到判定第一积分的充分必要条件．最后，构造ＫｄＶ方程的第一积分．     

流固耦合系统湿模态正交性的统一证明

本文基于Ｈａｍｉｌｔｏｎ原理和线性势流理论，严格证明了三维流固耦合系统湿模态的广义正交关系，并给出了运动方程的解耦条件。本文所用的推导方法由于不涉及具体复杂的微分算子，较之前入所用的方法具有明显的优越性。     

各向异性弹性力学问题Hamilton正则方程的一般形式

本文从修正后的Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ变分原理出发，导出了由２１个弹性常数组成的各向异性材料的混合方程，井证明它们即是Ｈａｍｉｌｔｏｎ正则方程。由该统一形式还给出了角铺设材料和正交各向异性材料的Ｈａｎｉｌｔｏｎ正则形式。     

一类偏微分方程的Hamilton正则表示

主要给出一系列关于力学中的偏微分方程的无穷维Ｈａｍｉｌｔｏｎ正则表示．其中包括变系数线性偏微分方程,ＫｄＶ方程,ＭＫｄＶ方程,ＫＰ方程,Ｂｏｕｓｉｎｅｓｑ方程等的无穷维Ｈａｍｉｌｔｏｎ正则表示．     

各向异性弹性力学问题Hamilton正则方程的一般形式

本文从修正后的Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ变分原理出发，导出了由２１个弹性常数组成的各向异性材料的混合方程，并证明它们即是Ｈａｍｉｌｔｏｎ正则方程。由该统一形式还给出了角铺设材料和正交各向异性材料的Ｈａｎｉｌｔｏｎ正则形式。     

热粘塑性介质中的温度率相关理论

本文提出了一种热粘塑性介质中热波的温度率相关理论，利用Ｃｌａｕｓｉｕｓ－Ｄｕｈｅｍ不等式和广义的Ｈｅｌｍｈｏｌｔｚ自由能，在一些合理的假设下，严格推导了温度率相关介质中的本构模型，即用Ｍｕｌｌｅｒ的冷度函数Ｔ代替经典热力学的绝对温度后，经典热力学中的本构理论可以方便地推广到温度率相关介质中。     

变分原理与非线性水波的Hamilton描述

本文用全变分方法导出水波动力学问题的基本方程组，再用平均势函数Ｆ（Ｘ，ｔ）渐近表示速度势函数（Ｘ，ｙ，ｔ）和Ｌａｇｒａｎｇｅ函数Ｌ（Ｘ，ｙ，ｔ），导出具有Ｂｏｕｓｉｎｅｓｑ形式的方程．研究了Ｈａｍｉｌｔｏｎ正则方程的简单推导问题．     

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...     

相空间中非保守系统的Herglotz广义变分原理及其Noether定理

与经典变分原理相比,基于由微分方程定义的作用量的Herglotz广义变分原理给出了非保守动力学系统的一个变分描述,它不仅能够描述所有采用经典变分原理能够描述的动力学过程,而且能够应用于经典变分原理不能适用的非保守或耗散系统.将Herglotz广义变分原理拓展到相空间,研究相空间中非保守力学系统的Herglotz广义变分原理与Noether定理及其逆定理.首先,提出相空间中Herglotz广义变分原理,给出相空间中非保守系统的变分描述,导出相应的Hamilton正则方程;其次,基于非等时变分与等时变分之间的关系,导出相空间中Hamilton-Herglotz作用量变分的两个基本公式;再次,给出Noether对称变换的定义和判据,提出并证明相空间中非保守系统基于Herglotz变分问题的Noether定理及其逆定理,揭示了相空间中力学系统的Noether对称性与守恒量之间的内在联系.在经典条件下,Herglotz广义变分原理退化为经典变分原理,与之相应的相空间中的Noether定理退化为经典Hamilton系统的Noether定理.文末以著名的Emden方程和平方阻尼振子为例说明上述方法和结果的有效性.     

Stochastic averaging of quasi-generalized Hamiltonian systems

A stochastic averaging method for generalized Hamiltonian systems (GHS) subject to light dampings and weak stochastic excitations is proposed. First, the GHS are briefly reviewed and classified into five classes, i.e., non-integrable GHS, completely integrable and non-resonant GHS, completely integrable and resonant GHS, partially integrable and non-resonant GHS and partially integrable and resonant GHS. Then, the averaged and FPK equations and the drift and diffusion coefficients for the five classes of quasi-GHS are derived. Finally, the stochastic averaging for a nine-dimensional quasi-partially integrable GHS is given to illustrate the application of the proposed procedure, and the results are confirmed by using those from Monte Carlo simulation.     

An investigation into natural vibrations of fluid–structure interaction systems subject to Sommerfeld radiation condition

A fluid-structure interaction system subject to Sommerfeld＇s condition is defined as a Sommerfeld system which is divided into three categories： Fluid Sommerfeld （FS） System, Solid Sommerfeld （SS） System and Fluid Solid Sommerfeld （FSS） System of which Sommerfeld conditions are imposed on a fluid boundary only, a solid boundary only and both fluid and solid boundaries, respectively. This paper follows the previous initial results claimed by simple examples to further mathematically investigate the natural vibrations of generalized Sommerfeld systems. A new parameter representing the speed of radiation wave for generalized 3-D problems with more complicated boundary conditions is introduced into the Sommerfeld condition which allows investigation of the natural vibrations of a Sommerfeld system involving both free surface and compressible waves. The mathematical demonstrations and selected examples confirm and reveal the natural behaviour of generalized Sommerfeld systems defined above. These generalized conclusions can be used in theoretical or engineering analysis of the vibrations of various Sommerfeld systems in engineering.     

全局分析的广义胞映射图论方法

应用广义胞映射理论的离散连续状态空间为胞状态空间的基本概念,依循Ｈｓｕ的将偏序集和图论理论引入广义胞映射的思想,以集论和图论理论为基础,提出了进行非线性动力系统全局分析的广义胞映射图论方法．在胞状态空间上,定义二元关系,建立了广义胞映射动力系统与图的对应关系,给出了自循环胞集和永久自循环胞集存在判别定理的证明,这样可借助国论的理论和算法来确定动力系统的全局性质．应用图的压缩方法,对所有的自循环胞集压缩后,在全局瞬态分析计算中瞬态胞的总数目得到有效地减少,并能借助于图的算法有效地实现全局瞬态的拓扑排序．在整个定性性质的分析计算中,仅采用布尔运算．     

Interpretation of internal tracer experiments and local sojourn time distributions

Tracer experiments are a valuable tool in the study of transport phenomena. Most of the theoretical work about tracers has been concerned either with systems having clearly defined inlet and outlet or with recirculating flows. In this article the methods are generalized to more complex situations and an attempt is made to provide a uniform theoretical treatment for local tracer experiments performed at several sites inside the flow systems. Local purging rate and mixing rate are defined and their measurements and applications are discussed. Several sojourn time distribution are discussed, especially that of local remaining life which might be useful in the study of local processes in large flow systems. The results should be of interest in a wide variety of areas in which tracer experiments are used, such as chemical reactor design, physiology, hydrology and the study of dispersion processes in the atmosphere and in oceans.     

On the existence and uniqueness of solutions generated by equivalent linearization

The existence and uniqueness of approximate solutions generated by the generalized method of equivalent linearization is considered. For the stationary analysis of systems with harmonic or Gaussian random excitation, it is shown that even though the equivalent linear system may not be unique, a simple element-by-element substitute system exists. Furthermore, this system is at least as good as any other similarly defined substitute system.     

铁摩辛柯基础梁的解析解和广义克雷洛夫函数

河南省自然科学基金项目（９８４０７０７００）资助摘要 定义并讨论了一类广义克雷洛夫函数，给出了用该函数表示的弹性地基上铁摩辛柯梁的解析解     

Nonholonomic motions of gyroscopic and wheeled systems

An approach to modeling the motion of mechanical systems with small generalized velocities is discussed. The motion of a gyroscope in a gimbal suspension and the lateral movements of a railway wheelset are discussed as examples for the situations when the solutions to the corresponding problems are evolved near the manifold defined by the primary Dirac constraints.     

An Augmented Formulation for Mechanical Systems with Non-Generalized Coordinates: Application to Rigid Body Contact Problems

In computational multibody algorithms, the kinematic constraintequations that describe mechanical joints and specified motiontrajectories must be satisfied at the position, velocity andacceleration levels. For most commonly used constraint equations, onlyfirst and second partial derivatives of position vectors with respect tothe generalized coordinates are required in order to define theconstraint Jacobian matrix and the first and second derivatives of theconstraints with respect to time. When the kinematic and dynamicequations of the multibody systems are formulated in terms of a mixedset of generalized and non-generalized coordinates, higher partialderivatives with respect to these non-generalized coordinates arerequired, and the neglect of these derivatives can lead to significanterrors. In this paper, the implementation of a contact model in generalmultibody algorithms is presented as an example of mechanical systemswith non-generalized coordinates. The kinematic equations that describethe contact between two surfaces of two bodies in the multibody systemare formulated in terms of the system generalized coordinates and thesurface parameters. Each contact surface is defined using twoindependent parameters that completely define the tangent and normalvectors at an arbitrary point on the body surface. In the contact modeldeveloped in this study, the points of contact are searched for on lineduring the dynamic simulation by solving the nonlinear differential andalgebraic equations of the constrained multibody system. It isdemonstrated in this paper that in the case of a point contact andregular surfaces, there is only one independent generalized contactconstraint force despite the fact that five constraint equations areused to enforce the contact conditions.