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.     

非完整约束系统几何动力学研究进展:Lagrange理论及其它

近10年来, 非完整力学的发展主要集中在两个相互关联的方向上, 一个是非完整运动规划, 另一个则是非完整约束系统的几何动力学, 这两个研究方向都充分地利用了现代几何学, 如纤维丛理论、辛流形和Poisson流形结构等等.本文主要综述非完整约束系统几何动力学的外附型和内禀型Lagrange理论, 包括非定常力学系统所需要的射丛几何学的基本概念、射丛按约束的直和分解、约束流形上的水平分布、D'Alembert-Lagrange方程与Chaplygin方程的整体描述、以及Riemann-Cartan流形上的非完整力学, 文中对Chetaev条件和d-δ交换关系的几何意义作了深入讨论.除此之外, 简要评述非完整力学的Hamilton理论与赝Poisson结构、Noether对称性和Lie对称性、动量映射与对称约化、Vakonomic动力学等几个非常重要专题的研究进展.      

弹性力学的复变量数值流形方法

数值流形方法通过引入数学和物理双重网格,将插值域和积分域分别定义在两个不同的覆盖上来完成系统能量泛函积分运算. 当采用高阶函数构造位移函数时,广义节点自由度将大大增加. 在求解系统的平衡方程中,运算量是与自由度的三次方成正比的,因此数值流形方法的计算量是较大的. 为此,在复变量理论的基础上,采用一维基函数建立二维问题的逼近试函数,然后将其应用于弹性力学的数值流形方法,提出了复变量数值流形方法,推导了弹性力学的复变量数值流形方法的公式. 与传统的数值流形方法相比,复变量数值流形方法具有计算量小、精度高的优点.      

完整约束只限制质点系的位置吗?

本文主要讨论完整约束与非完整约束对力学系统的限制条件.在分析力学中首先遇到的重要概念是约束.约束应定义为:“在非自由系统中,那些预先加定的,与初始条件及受力条件无关的,限制系统位置或速度的运动学条件”.约束可分为完整与非完整约束.在一般理论力学与分析力学教材上,通常以约束方程可能表达的函数形式来定义这两类约束.     

完整约束只限制质点系的位置吗?

本文主要讨论完整约束与非完整约束对力学系统的限制条件.在分析力学中首先遇到的重要概念是约束.约束应定义为:"在非自由系统中,那些预先加定的,与初始条件及受力条件无关的,限制系统位置或速度的运动学条件".约束可分为完整与非完整约束.在一般理论力学与分析力学教材上,通常以约束方程可能表达的函数形式来定义这两类约束.     

有多余坐标完整系统的自由运动

对于完整力学系统,若选取的参数不是完全独立的,则称为有多余坐标的完整系统. 由于完整力学系统的第二类Lagrange 方程中没有约束力,故为研究完整力学系统的约束力,需采用有多余坐标的带乘子的Lagrange方程或第一类Lagrange 方程. 一些动力学问题要求约束力不能为零,而另一些问题要求约束力很小. 如果约束力为零,则称为系统的自由运动问题. 本文提出并研究了有多余坐标完整系统的自由运动问题. 为研究系统的自由运动,首先,由d'Alembert-Lagrange 原理, 利用Lagrange 乘子法建立有多余坐标完整系统的运动微分方程;其次,由多余坐标完整系统的运动方程和约束方程建立乘子满足的代数方程并得到约束力的表达式;最后,由约束系统自由运动的定义,令所有乘子为零,得到系统实现自由运动的条件. 这些条件的个数等于约束方程的个数,它们依赖于系统的动能、广义力和约束方程,给出其中任意两个条件,均可以得到实现自由运动时对另一个条件的限制. 即当给定动能和约束方程,这些条件会给出实现自由运动时广义力之间的关系. 当给定动能和广义力,这些条件会给出实现自由运动时对约束方程的限制. 当给定广义力和约束方程,这些条件会给出实现自由运动时对动能的限制. 文末,举例并说明方法和结果的应用.      

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)。     

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.     

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.     

Dynamic buckling of a beam with transverse constraints

A nonlinear dynamic system with continuously distributed mass is studied using several approaches: experimentally, numerically as well as analytically. The nonlinearity of the system consists of geometrical constraints imposed on the motion. It is harmonically loaded and it is demonstrated that for certain choices of the loading parameters, periodic, quasi-periodic or chaotic behaviour may occur depending on the initial conditions. An important issue is to investigate the number of degrees of freedom needed in order to analytically model the system accurately enough that the important characteristics of the motion are retained in the solution. It is found that the impact conditions at the constraints are of crucial importance and a new approach is proposed for modelling of the impacts. The method is based on the fact that the free motion can be approximated with quite a few degrees of freedom, while at impact all the infinite number of degrees of freedom are considered.     

Dynamic modelling of the double wishbone motor-vehicle suspension system

In this paper the dynamic analysis of the double wishbone motor-vehicle suspension system using the point-joint coordinates formulation is presented. The mechanical system is replaced by an equivalent constrained system of particles and then the laws of particle dynamics are used to derive the equations of motion. Due to the presence of large number of geometric and kinematic constraints the velocity transformation approach is used to eliminate some constraints. The equations of motion in terms of the Cartesian coordinates of the particles are transformed to a reduced set in terms of relative joint variables by defining differential-algebraic equations in terms of the joint variables are equal to the number of degrees of freedom of the whole system plus the number of cut-joint constraints corresponding to cut of kinematical closed loops. Use of both the Cartesian and relative joint variables produces an efficient set of equations without loss of generality. The chosen suspension includes open and closed loops with quarter-car model.     

On the elastic stability of static non-holonomic systems

This paper presents a new formulation for the elastic stability of static non-holonomic structural systems. The theory is developed within the tradition of discrete (or discretized) systems written in terms of a set of generalized coordinates and control parameters. The non-holonomic conditions are written as constraint functions. The formulation employs a Lagrangian functional in terms of the total potential energy, the constraint functions and multipliers. Critical states are identified and the solution is next expanded by regular perturbations. This allows to establish a classification of critical states and identify the initial postcritical behavior. This solution is valid provided that there is no change in the active constraints of the system. The paper presents a mathematical analysis of the critical condition, and concludes with simple examples of two degree-of-freedom systems previously investigated by other authors.     

THE DETERMINATION OF THE COMPUTATIONAL DEGREES OF FREEDOM OF BAR SYSTEMS1)

为了简化和统一结构力学教材中杆件体系计算自由度$W$的算法,本文提出了“直接法”。该方法以相邻节点间的杆件和结点为分析对象,对于含$m$根杆件、$g$ 个单刚结点和$h$个单铰结点的平面杆件体系,直接法给出计算自由度公式$W=3m-(3g+2h$)。相比于已有方法,直接法研究对象清晰,数学运算统一,适合于编程计算,便于初学者准确而快速计算复杂体系的$W$;也为结构超静定次数的确定提供了一种简单有效的数值算法。     

Discrete bright–dark soliton solutions and parameters controlling for the coupled Ablowitz–Ladik equation

Analysis of piecewise-linear nonlinear dynamical systems is critical for a variety of civil, mechanical, and aerospace structures that contain gaps or prestress that are caused by cracks, delamination, joints or interfaces among components. Recently, a technique referred to as bilinear amplitude approximation (BAA) was developed to estimate the response of bilinear systems that have no gap or prestress. The method is based on an idea that the dynamics of a bilinear system can be treated as a combination of linear responses in two time intervals both of which the system behaves as a distinct linear system: (1) the open state and (2) the closed or sliding state. Both geometric and momentum constraints are then applied as compatibility conditions between the states to couple the linear vibrational response for each time interval. In order to estimate the response for more general cases where there are either gaps or prestress in the system, a generalized BAA method is proposed in this paper. The new method requires inclusion of contact stiffness and damping to model contact behavior in the sliding state, and new equilibrium positions for each state to establish proper coordinates. The new method also finds the bilinear frequency of the system, which cannot be computed using the bilinear frequency approximation method previously developed since that method is only accurate for the zero gap and no prestress case. The generalized BAA method is demonstrated on a single degree of freedom system, a three degree of freedom system, and a cracked cantilever beam model for various gap sizes and prestress levels.     

大学物理《力学》中的自由度分析

对《力学》中的物体自由度进行多方面分析,以深化教学、提高学生正 确分析物理问题的能力.使用实际教学分析的研究方法,在《力学》范围内讨论自由度与坐标、 自由与约束的关系并得以下结论: (1) 同一物体的自由度随其所在的``空间'不同而不同, 不因坐标系的选取不同而 异, 在同类参考系中不因参考系的动静而有别;(2)自由度遵循叠加原理. 讨论了质点系的总自由度及相关计算问题,并指出研究《力学》中自由度的意义.     

On new forms of motive differential equations of a mechanical system relative to non-inertial frame

Based on D'Alembert's principle of a mechanical system relative to non-inertial frame and by introducing the concept of the generalized inertial potential, new forms of differential equations of motion of a mechanical system with holonomic and the non-holonomic constraints relative to the non-inertial frame are obtained. The merits and demerits between our method and the Newtonian dynamic method as well as the analytic dynamic method are discussed comparatively. Finally, two examples are given to illustrate the application of the motive differential equations in the new forms.