有多余坐标的完整系统形式不变性导致的新守恒量

研究有多余坐标的完整力学系统由形式不变性直接导出的新型守恒量。用有多余坐标的双面理想完整约束力学系统的运动微分方程和约束方程在无限小变换下的形式不变性，给出系统形式不变性的定义和判据。得到形式不变性导致守恒量的条件以及守恒量的形式，并给出三种特殊情形下的推论。举例说明结果的应用。     

长条形基坑带顶部支撑排桩支护体系空间变形研究

提出了一种建立具有固定双面约束多点摩擦的多体系统动力学方程的方法. 用笛卡尔坐标阵 描述系统的位形,根据局部方法的递推关系建立系统的约束方程,应用第一类Lagrange方程 建立该系统的动力学方程,使得具有摩擦的约束面的法向力与Lagrange乘子一一对应,便于 摩擦力的分析与计算,并用矩阵形式给出了摩擦力的广义力的一般表达式. 应用增广法将微 分-代数方程组转化为常微分方程组,并用分块矩阵的形式给出,以便于方程的编程与计算. 给出了一种改进的试算法,可提高计算效率. 最后给出了一个算例,应用试算法和RK法对算 例进行了数值仿真.     

双面约束多点摩擦多体系统的建模和数值方法

提出了一种建立具有固定双面约束多点摩擦的多体系统动力学方程的方法. 用笛卡尔坐标阵描述系统的位形,根据局部方法的递推关系建立系统的约束方程,应用第一类Lagrange方程建立该系统的动力学方程,使得具有摩擦的约束面的法向力与Lagrange乘子一一对应,便于摩擦力的分析与计算,并用矩阵形式给出了摩擦力的广义力的一般表达式. 应用增广法将微分-代数方程组转化为常微分方程组,并用分块矩阵的形式给出,以便于方程的编程与计算.给出了一种改进的试算法,可提高计算效率. 最后给出了一个算例,应用试算法和RK法对算例进行了数值仿真.      

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.     

非稳定约束反力作的功

<正> 在Lagrange函数不显含时间t的条件下,对于完整、非稳定约束的保守力学系统,Lagrange方程给出了广义能量积分(Jacobi 积分),即T_(?)-T_0+V=const (1)(1)式表明:系统的机械能并不守恒,这是由于Lagrange 力学将力分为主动力和约束力,而约束力在任何虚位移中不作功,因而 Lagrange 方程中并不出     

约束力学系统引入待定乘子的两种方式

讨论约束力学系统中引入待定乘子两种不同的途径及其对应的物理意义.一种方式引入待定乘子是处理非独立坐标变更,乘子与约束反力相关;另一种方式是变分原理条件极值要求,乘子与系统的拉格朗日函数相关.对完整系统由两种途径导出的带乘子运动方程是相同的,但是对非完整系统则分别导出两种不同的运动方程:罗斯方程和维科(Vacco)方程.     

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)。     

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.     

非线性非完整系统的运动方程及其广义能量积分

本文研究了一阶非线性非完整系统的运动规律,由 Routh 方程出发,用了矩阵的右乘零因子的有关理论,给出了该系统的不含待定乘子的一般方程,本文还讨论了一阶非线性非完整系统的运动方程存在广义能量积分的条件,并给出了相应的广义能量积分.     

On application of Newton’s law to mechanical systems with motion constraints

This work is devoted to deriving and investigating conditions for the correct application of Newton’s law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton’s law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space and not on the dual space of a manifold, which is the natural geometrical space for this. Finally, the Euler–Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated and clarified further.     

A new approach to the vakonomic mechanics

The aim of this paper was to show that the Lagrange–d’Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them we consider the generalization of the Hamiltonian principle for nonholonomic systems with non-zero transpositional relations. We apply this variational principle, which takes into the account transpositional relations different from the classical ones, and we deduce the equations of motion for the nonholonomic systems with constraints that in general are nonlinear in the velocity. These equations of motion coincide, except perhaps in a zero Lebesgue measure set, with the classical differential equations deduced with the d’Alembert–Lagrange principle. We provide a new point of view on the transpositional relations for the constrained mechanical systems: the virtual variations can produce zero or non-zero transpositional relations. In particular, the independent virtual variations can produce non-zero transpositional relations. For the unconstrained mechanical systems, the virtual variations always produce zero transpositional relations. We conjecture that the existence of the nonlinear constraints in the velocity must be sought outside of the Newtonian mechanics. We illustrate our results with examples.     

RELATIVISTIC VARIATION PRINCIPLES AND EQUATION OF MOTION FOR VARIABLE MASS CONTROLLABLE MECHANICAL SYSTEM

ＲＥＬＡＴＩＶＩＳＴＩＣＶＡＲＩＡＴＩＯＮＰＲＩＮＣＩＰＬＥＳＡＮＤＥＱＵＡＴＩＯＮＯＦＭＯＴＩＯＮＦＯＲＶＡＲＩＡＢＬＥＭＡＳＳＣＯＮＴＲＯＬＬＡＢＬＥＭＥＣＨＡＮＩＣＡＬＳＹＳＴＥＭ￥(罗绍凯)ＬｕｏＳｈａｏｋａｉ（ＳｈａｎｇｑｉｕＴｅａｃｈｅｒｓ...     

具有可积微分约束的力学系统的Lie对称性

研究具有可积微分约束的力学系统的Ｌｉｅ对称性与守恒量。采用两种方法：一是用不可积微分约束系统的方法;另一是用积分后降阶系统的方法,研究两种方法之间的关系。     

用具有负定非对称矩阵的梯度系统构造稳定的广义Birkhoff系统1）

Birkhoff系统是一类比Hamilton系统更广泛的约束力学系统,可在原子与分子物理,强子物理中找到应用.非定常约束力学系统的稳定性研究是重要而又困难的课题,用构造Lyapunov函数的直接方法来研究稳定性问题有很大难度,其中如何构造Lyapunov函数是永远的开放问题.本文给出一种间接方法,即梯度系统方法.提出一类梯度系统,其矩阵是负定非对称的,这类梯度系统的解可以是稳定的或渐近稳定的.梯度系统特别适合用Lyapunov函数来研究,其中的函数V通常取为Lyapunov函数.列出广义Birkhoff系统的运动方程,广义Birkhoff系统是一类广泛约束力学系统.当其中的附加项取为零时,它成为Birkhoff系统,完整约束系统和非完整约束系统都可纳入该系统.给出广义Birkhoff系统的解可以是稳定的或渐近稳定的条件,进一步利用矩阵为负定非对称的梯度系统构造出一些解为稳定或渐近稳定的广义Birkhoff系统.该方法也适合其他约束力学系统.最后用算例说明结果的应用.     

RESEARCH ON MODELING AND NUMERICAL METHOD OF FREE STANDING BODY ON PLANAR RIGID MULTIBODY DYNAMICS

采用非光滑多体系统动力学的方法研究浮放物体与基础平台组成的多体系统,建立其非光滑接触的动力学方程与数值算法.浮放物体由主体部分和支撑腿组成,其间通过含黏弹性阻力偶的转动铰连接.支撑腿与基础平台间的接触力简化为接触点的法向接触力和摩擦力,采用扩展的赫兹接触力模型描述接触点的法向接触力,采用库伦干摩擦模型描述其摩擦力.采用笛卡尔坐标系下的位形坐标作为系统的广义坐标.首先,将基础平台运动看作非定常约束,用第一类拉格朗日方程建立系统的动力学方程,并采用鲍姆加藤约束稳定化的方法解决违约问题.随后给出基于事件驱动法和线性互补方法的数值算法.当相对切向速度为零时,构造静滑动摩擦力的正负余量和正、负向加速度的互补关系,从而将接触点黏滞——滑移切换的判断以及静滑动摩擦力的计算转化为线性互补问题进行求解,并采用Lemke算法求解线性互补问题.最后,通过数值仿真选择合适的步长;通过仿真结果说明浮放物体运动中存在的黏滞-滑移切换现象以及基础平台运动、质心位置对浮放物体运动的影响.     

Weak formulation and first order form of the equations of motion for a class of constrained mechanical systems

Some new theoretical results are presented on modeling the dynamic response of a class of discrete mechanical systems subject to equality motion constraints. Both the development and presentation are facilitated by employing some fundamental concepts of differential geometry. At the beginning, the equations of motion of the corresponding unconstrained system are presented on a configuration manifold with general properties, first in strong and then in a primal weak form, using Newton׳s law of motion as a foundation. Next, the final weak form is obtained by performing a crucial integration by parts step, involving a covariant derivative. This step required the clarification and enhancement of some concepts related to the variations employed in generating the weak form. The second part of this work is devoted to systems involving holonomic and non-holonomic scleronomic constraints. The equations of motion derived in a recent study of the authors are utilized as a basis. The novel characteristic of these equations is that they form a set of second order ordinary differential equations (ODEs) in both the coordinates and the Lagrange multipliers associated to the constraint action. Based on these equations, the corresponding weak form is first obtained, leading eventually to a consistent first order ODE form of the equations of motion. These equations are found to appear in a form resembling the form obtained after application of the classical Hamilton׳s canonical equations. Finally, the new theoretical findings are illustrated by three representative examples.     

Well posed formulations of holonomic mechanical system dynamics and design sensitivity analysis

Abstract

Manifold theoretic ordinary differential equations of motion for holonomic mechanical systems that depend on problem data, or design variables, are shown to be well posed; i.e., they have a unique solution that depends continuously on problem data. It is proved that these differential equations are equivalent to the d’Alembert variational formulation and the index 3 Lagrange multiplier formulation of differential-algebraic equations of motion, which are also shown to be well posed. These results provide a foundation for dynamic system design sensitivity analysis, which requires differentiability of solutions of the equations of motion with respect to design variables.