多刚体系统的Kane方程

本文从理论上探讨了多刚体系统在受有任意阶非线性、非完整约束下Kane方程的具体形式。为建立复杂的多体系统的动力学方程,提供了理论依据和方法。     

关于Kane方程

1.引言 在分析动力学的应用研究中,T.R.Kane提出了一种方法,可以不通过寻求系统的动力学函数而直接建立系统的不带乘子的动力学方程。但是,Kane在建立他的方法时,其论证有如下缺点: 1) Kane在讨论中,一下子引进很多被记为ν_(q_r)~(p_i),ω_(q_r)~R的量,并称之为所谓“偏速度”,“偏角速度”。而这些量一般说来并不是速度。实际上,它们只是一些变换系数。 2) Kane在建立他的方程时,是从质点系每一个单个质点的D'Alembert方程出发,乘以他引进的“偏速度”,然后累加而得到的。这种方法很容易使人们认为,这种动力学方     

弹性杆-刚性体系统动力学分析的广义逆矩阵方法

将多刚体系统的广义逆矩阵方法推广到含弹性杆与刚性体的混合系统的动力学分析中,建立了以节点坐标表示的基于全局惯性坐标系的刚体-柔性体混合系统动力学方程.首先以两端节点坐标为变量推导了弹性杆的动力学方程,以刚性体节点坐标为变量推导了刚性体节点速度约束方程和刚性体动力学方程,最后得到弹性杆与刚性体混合系统的动力学方程和速度约束方程.本方法在同一个惯性坐标系对刚柔多体系统进行描述,具有方法简洁、便于计算建模的特点.论文最后给出两个数值算例,检验了方法的有效性.     

多体系统接触碰撞问题的牛顿-欧拉线性互补方法

基于非光滑动力学方法的多体系统接触碰撞分析是目前多体系统动力学的研究热点.本文采用牛顿-欧拉方法建立多体系统接触、碰撞问题的动力学模型,给出一种牛顿-欧拉型线性互补公式.该建模方法与目前一般采用的拉格朗日建模方法的不同之处是约束条件中除了库仑摩擦、单边约束之外还含有光滑等式约束.在建立系统动力学模型时,首先解除摩擦约束和单边约束得到原系统对应的基本系统.牛顿-欧拉方法采用最大数目坐标建立基本系统的动力学方程,由于坐标不相互独立,因此基本系统中带有等式约束,其数学模型为一组微分代数方程.借助约束雅可比矩阵,在基本系统微分代数方程中添加摩擦接触和单边约束对应的拉氏乘子,就可以得到系统全局运动的具有变拓扑结构特征的动力学方程,再结合非光滑约束互补条件便可构成完备的系统动力学模型.完备的动力学模型由动力学微分方程以及等式约束和不等式约束组成.线性互补公式采用分块矩阵形式进行推导,简化了推导过程.数值计算采用基于线性互补的时间步进算法.时间步进算法是目前流行的非光滑数值算法,其突出特点是可以免去数值积分中繁琐的事件检测过程,而数值积分过程中通过对线性互补问题的求解可以确定系统的触-离状态.通过对典型的曲柄滑块间隙机构进行数值分析,验证本文方法的有效性.     

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.      

航天器太阳阵多体展开的动力学分析

本文研究航天器太阳阵多体系统展开过程的动力学问题.文中引入 Denavit-Hartenberg 座标系及齐次座标变换理论对系统进行描述,应用 Kane 动力学方程建立了太阳阵展开和撞击动力学模型,编制了仿真软件 K-DIDAS,并对带大型太阳阵的航天器进行了多体系统展开动力学分析计算,给出了供工程应用的有关数据和参数.     

多刚体系统运动方程的推导

本文简要论述多刚体系统运动方程各种不同推导方法的利弊,并提出一种新的方法。此法依据Kane伪坐标方程,以刻划系统整体结构的两个矩阵为基础,通过对矩阵求导的方法和一系列矩阵推演,建立系统的动量矩方程组。文末导出了多刚体卫星的姿态方程。     

非完整树形多刚体系统的动力学方程

本文用Jourdain-Bertrand原理及图论的一些基本概念,导出非完整树形多刚体系统的动力学方程.当约束是完整的,且广义坐标相互独立时,则得到Wittenburg 的动力学方程,最后也由该原理得到Kane 方程. ...     

树形带球铰的多刚体系统动力学方程的建立方法

本文提出了一种建立树形带球铰多刚体系统动力学方程的新方法,运用约束系统动力学研究成果,提出了广义刚体的概念以代替多刚体系统的子系统,并借助于广义刚体的不断扩充,求得了多刚体系统动力学方程建立的递推方法,该方法简单直观,几何概念清楚,并允许多刚体系统的扩充,且便于进行计算机符号处理。     

A matrix formulation for the dynamic analysis of planar mechanisms using point coordinates and velocity transformation

This paper presents a matrix formulation for the dynamic analysis of planar mechanisms consisting of interconnected rigid bodies. The formulation initially uses the rectangular Cartesian coordinates of an equivalent constrained system of particles to define the configuration of the mechanical system. This results in a simple and straightforward procedure for generating the equations of motion. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open-loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed-loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. Two examples are used to demonstrate the generality and efficiency of the proposed method.     

Forces associated with non-linear non-holonomic constraint equations

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by non-holonomic equations that are inherently non-linear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known non-holonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as non-holonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor first to form the former, larger set and subsequently perform matrix multiplications.     

Nonlinear dynamic analysis on rigid-flexible coupling system of an elastic beam

Previous work examined the effect of the attached stiffness matrix terms on stability of an elastic beam undergoing prescribed large overall motion. The aim of the present work is to extend the nonlinear formulations to an elastic beam with free large overall motion. Based on initial stress method, the nonlinear coupling equations of elastic beams are obtained with free large overall motion and the attached stiffness matrix is derived by solving sub-static formulation. The angular velocity and the tip deformation of the elastic pendulum are calculated. The analytical results show that the simulation results of the present model are tabled and coincide with the one-order approximate model. It is shown that the simulation results accord with energy conservation principle.     

Unsteady compressible flow due to relative rotation and translation of a viscous fluid on a disk

Summary The modification of the axisymmetric viscous flow due to relative rotation of the disk or fluid by a translation of the boundary is studied. The fluid is taken to be compressible, and the relative rotation and translation velocity of the disk or fluid are time-dependent. The nonlinear partial differential equations governing the motion are solved numerically using an implicit finite difference scheme and Newton's linearisation technique. Numerical solutions are obtained at various non-dimensional times and disk temperatures. The non-symmetric part of the flow (secondary flow) describing the translation effect generates a velocity field at each plane parallel to the disk. The cartesian components of velocity due to secondary flow exhibit oscillations when the motion is due to rotation of the fluid on a translating disk. Increase in translation velocity produces an increment in the radial skin friction but reduces the tangential skin friction.     

Hyperbolicity of Velocity-Stress Equations for Waves in Anisotropic Elastic Solids

This paper reports mathematical properties of the three-dimensional, first-order, velocity-stress equations for propagating waves in anisotropic, linear elastic solids. The velocity-stress equations are useful for numerical solution. The original equations include the equation of motion and the elasticity relation differentiated by time. The result is a set of nine, first-order partial differential equations (PDEs) of which the velocity and stress components are the unknowns. Cast into a vector-matrix form, the equations can be characterized by three Jacobian matrices. Hyperbolicity of the equations is formally proved by analyzing (i) the spectrum of a linear combination of the three Jacobian matrices, and (ii) the eigenvector matrix for diagonalizing the linearly combined Jacobian matrices. In the three-dimensional space, linearly combined Jacobian matrices are shown to be connected to the classic Christoffel matrix, leading to a simpler derivation for the eigenvalues and eigenvectors. The results in the present paper provide critical information for applying modern numerical methods, originally developed for solving conservation laws, to elastodynamics.     

INVESTIGATION ON KANE DYNAMIC EQUATIONS BASED ON SCREW THEORY FOR OPEN-CHAIN MANIPULATORS

IntroductionDynamicsofroboticmanipulatorsisthebasisofdynamiccontrolandmechanicaldesignofroboticmanipulators.Dynamicanalysisofrobotscanemploytheapproachesoftheoreticalmechanicsanddynamicequationsofanalyticalmechanics,suchasNewton_Eulerdynamicequations,Lagrange_Eulerdynamicequations ,Routhequation ,Appellequation ,andKaneequations.Theultimateresultsarethesamewhenweinvestigateononekindofmotionsaboutoneobjectivebydifferentdynamicequations,butthespeedandquantitiesofcomputationoftheseequationsare…     

Stratified magnetohydrodynamic flow of tangent hyperbolic nanofluid induced by inclined sheet

This paper studies stratified magnetohydrodynamic(MHD) flow of tangent hyperbolic nanofluid past an inclined exponentially stretching surface. The flow is subjected to velocity, thermal, and solutal boundary conditions. The partial differential systems are reduced to ordinary differential systems using appropriate transformations.The reduced systems are solved for convergent series solutions. The velocity, temperature,and concentration fields are discussed for different physical parameters. The results indicate that the temperature and the thermal boundary layer thickness increase noticeably for large values of Brownian motion and thermophoresis effects. It is also observed that the buoyancy parameter strengthens the velocity field, showing a decreasing behavior of temperature and nanoparticle volume fraction profiles.     

Laminar boundary layer flow of a nanofluid along a wedge in the presence of suction/injection

The behavior of an incompressible laminar boundary layer flow over a wedge in a nanofluid with suction or injection has been investigated. The model used for the nanofluid integrates the effects of the Brownian motion and thermophoresis parameters. The governing partial differential equations of this problem, subjected to their boundary conditions, are solved by the Runge-Kutta-Gill technique with the shooting method for finding the skin friction and the rate of heat and mass transfer. The result are presented in the form of velocity, temperature, and volume fraction profiles for different values of the suction/injection parameter, Brownian motion parameter, thermophoresis parameter, pressure gradient parameter, Prandtl number, and Lewis number. The conclusion is drawn that these parameters significantly affect the temperature and volume fraction profiles, but their influence on the velocity profile is comparatively smaller.     

Use of Cholesky Coordinates and the Absolute Nodal Coordinate Formulation in the Computer Simulation of Flexible Multibody Systems

In a previous publication, procedures that can be used with the absolute nodal coordinate formulation to solve the dynamic problems of flexible multibody systems were proposed. One of these procedures is based on the Cholesky decomposition. By utilizing the fact that the absolute nodal coordinate formulation leads to a constant mass matrix, a Cholesky decomposition is used to obtain a constant velocity transformation matrix. This velocity transformation is used to express the absolute nodal coordinates in terms of the generalized Cholesky coordinates. The inertia matrix associated with the Cholesky coordinates is the identity matrix, and therefore, an optimum sparse matrix structure can be obtained for the augmented multibody equations of motion. The implementation of a computer procedure based on the absolute nodal coordinate formulation and Cholesky coordinates is discussed in this paper. Numerical examples are presented in order to demonstrate the use of Cholesky coordinates in the simulation of the large deformations in flexible multibody applications.     

Motion and Pinning of Discrete Interfaces

We describe the motion of interfaces in a two-dimensional discrete environment by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuous analysis. We show that below a critical ratio of the time and space scalings we have no motion of interfaces (pinning), while above that ratio the discrete motion is approximately described by the crystalline motion by curvature on the continuum described by Almgren and Taylor. The critical regime is much richer, exhibiting a pinning threshold (small setsmove, large sets are pinned), partial pinning (portions of interfaces may not move), pinning after an initial motion (possibly to a non-convex limit set), “quantization” of the interface velocity, and non-uniqueness effects.