复杂非线性电路系统的动力学方程模型

Lagrange和Hamilton运动方程是分析力学的基本原理之一和方法论。应用Lagrange和Hamilton原理建立复杂非线性电路保守动力学方程模型是一种形式化可行的方法。对非保守的动力学系统,定义描述电路系统的荷控支路和链控支路的微观结构概念,应用Hamilton结构的方法,可以得到与La-grange结构等价的方程组;考虑大规模电路系统的复杂性,依据电路系统荷控支路和链控支路微观结构的概念,给出具有控制参量的Lagrange和Hamilton函数,以及具有相应关联矩阵和联接矩阵形式的Lagrange和Hamilton的动态方程;分析了保守和非保守复杂系统拓扑结构关系的描述和其动力学系统的建模,其建模过程具有规范性和方程具有对称性。虽然数学推导过程繁琐,但适合于计算机辅助形式化分析;基于Hamilton方法建立的电路模型为一阶微分动态方程组,特别适合进行理论分析和数值仿真计算。     

考虑刚-柔-热耦合的板结构多体系统的动力学建模

对热载荷作用下中心刚体与大变形薄板多体系统的动力学建模问题进行研究.基于Kirchhoff假设,从格林应变和曲率与绝对位移的非线性关系式出发,推导了非线性广义弹性力阵,用绝对节点坐标法建立了大变形矩形薄板的有限元离散的动力学变分方程.为了考虑刚体姿态运动、弹性变形和温度变化的相互耦合作用,推导了热流密度与绝对节点坐标之间的关系式.引入系统的运动学约束方程,建立了中心刚体-矩形板多体系统的考虑刚-柔-热耦合的热传导方程和带拉格朗日乘子的第一类拉格朗日动力学方程.为了有效地提高计算效率,将改进的中心差分法和广义-α法相结合,求解热传导方程和动力学方程,差分后的方程通过牛顿迭代法耦合求解.对刚-柔耦合和刚-柔-热三者耦合两种模型的仿真结果进行比较表明,刚体运动对温度梯度和热变形的影响显著.此外,本文建模方法考虑了几何非线性项,因此也考虑了热膨胀引起的轴向变形对横向变形的影响.     

基于LuGre模型非光滑柱铰链平面多体系统动力学的建模和数值方法

以含非光滑柱铰链平面多刚体系统为研究对象,将间隙充分小的柱铰链视为双边约束,用LuGre摩擦模型描述柱铰链内的摩擦;由第一类Lagrange方程导出该系统的动力学方程(微分 代数方程).铰链处的摩擦使得其动力学方程是关于Lagrange乘子的非线性代数方程组,由于LuGre摩擦模型具有很好的连续性,可将非线性代数方程组与常微分方程组的数值算法(如拟牛顿法和龙格 库塔法)相结合求解其动力学方程.最后,通过数值仿真算例说明了该算法的可行性和有效性,既能很好地反映柱铰链摩擦对系统动力学特性的影响,又能避免Coulomb干摩擦给方程求解带来的困难.     

位形空间中约束力学系统的Lagrange对称性与守恒量

研究位形空间中约束力学系统的Lagrange对称性,给出位形空间中约束力学系统的统一动力学方程,给出位形空间中约束力学系统统一方程的Lagrange对称性的判据,得到位形空间中约束力学系统统一方程的Lagrange对称性导致的守恒量及其存在的条件,并举例说明结果的应用.     

运动滑道含摩擦多体系统的建模和数值方法

研究了运动约束面含摩擦多体系统动力学方程的建立和算法问题．首先利用第一类Lagrange方程给出了系统的动力学方程,并以矩阵形式给出了这类系统摩擦力的广义力的一般表达式．为便于摩擦力和铰链约束力的分析与计算,采用笛卡尔坐标和约束方程的局部方法,使得系统的约束力与Lagrange乘子一一对应．应用增广法将微分一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,提高计算效率．最后用一个算例验证了该方法的有效性．     

考虑几何非线性和热效应的刚柔耦合系统的频域特性研究

研究温度场中旋转刚体-梁系统的刚-柔耦合动力学特性.考虑几何非线性和热效应,从精确的应变-位移关系式出发,用虚功原理和有限单元法建立了旋转刚体-梁系统的刚-柔耦合动力学方程.由于非线性刚度阵与变形的高次项有关,将非线性刚度阵的各元素表示为广义坐标阵和常值阵的乘积.数值计算表明,该方法可避免重复积分,提高计算效率.在此基础上研究了在温度递增的情况下几何非线性对系统的刚-柔耦合动力学特性的影响,用频谱分析方法研究了系统的固有频率随中心刚体转动惯量和温度的变化.     

位移法浅水波方程的解及其特性

不同于传统流体力学,在Lagrange坐标下推导浅水波方程.若将水平位移作为基本变量,则推导出的浅水波数学模型可描述为固体力学的非线性大位移问题.运用不可压缩条件,通过变分原理推导出位移法浅水波方程,给出椭圆函数形式的行波解,并分析孤波解产生的条件.该基础研究建立了在分析结构力学中分析浅水波问题的理论基础,有利于进一步开展水动力学的研究.     

利用变量变换构造耗散系统Lagrange函数

提出变系数耗散系统变分法逆问题的一种间接解法.首先利用坐标和时间变量变换将系统运动微分方程化为自伴随方程,计算得到Lagrange函数后,再变换为原来变量以得到给定方程的Lagrange函数.给出两个例子以说明所得结果的应用.     

刚弹耦合质量摆减振结构的动力学建模

利用Hamilton变分原理建立高层建筑质量摆减振结构在考虑刚弹耦合作用下的动力学模型．此模型中把高层建筑当作竖直悬臂梁即无线自由度结构处理,模型具有2个独立变量,为梁横向位移和质量摆的转角,同时考虑了几何非线性的约束条件,形成了一个非线性刚弹耦合方程．最后采用Galerkin方法对其离散化,导出有限维离散化动力学模型,并给出仿真算例,验证了该模型的正确性．     

变质量力学系统的Herglotz型Lagrange方程与Noether对称性和守恒量

Herglotz变分原理提供了非保守耗散问题的变分描述，同时变质量力学在自然界和工程领域有大量的应用，因此将Herglotz变分原理应用于变质量力学系统的Lagrange方程与守恒律研究，为研究变质量力学提供了一个新的途径.文中建立了变质量力学系统的Herglotz型广义变分原理，导出了变质量系统的Herglotz型Lagrange方程.定义了变质量力学系统的Herglotz型Noether对称性，建立并证明了Herglotz型Noether定理及其逆定理.文末给出两个变质量非保守系统的具体例子以说明结果的应用.     

基于Lagrange法的刚体与柔体动力学对比研究

对于机器人、航天器和车辆悬架部件等一些多体系统来说,其中一些部件的大尺寸、轻型化趋势使得传统的刚性体建模已经难以准确地模拟实际的工况,将部件所具有的柔性特性加入到多体系统模型中进行柔体动力学仿真,由于考虑了部件弹性变形与大范围刚性运动之间的耦合,故可以得到部件更为真实的动力学行为.文中通过与基于Lagrange法的刚体动力学基本方程进行对比研究,详细说明了柔体动力学方程中上述的耦合作用.以某大型雷达可展开天线为例,分别在刚体和柔体的假设下对其展开运动进行动力学仿真,结果表明采用柔性体仿真更能真实反映其动力学特性.     

弹性杆的Greenhill公式与Kirchhoff动力学比拟

Kirchhoff动力学比拟思想建立了弹性杆静力学与刚体定点转动动力学之间在概念和方法上的对应关系.受拉扭弹性直杆的平衡比拟于Lagrange重陀螺绕铅锤轴的永久转动.根据一次近似理论,考察了两者稳定判据的建立过程,表明其在稳定性上的比拟是Lyapunov意义上的.在此基础上进一步讨论了两端铰支时拉扭弹性直杆的Eule...     

Coupling characteristics of rigid body motion and elastic deformation of a 3-PRR parallel manipulator with flexible links

Modeling of multibody dynamics with flexible links is a challenging task, which not only involves the effect of rigid body motion on elastic deformations, but also includes the influence of elastic deformations on rigid body motion. This paper presents coupling characteristics of rigid body motions and elastic motions of a 3-PRR parallel manipulator with three flexible intermediate links. The intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions based on the assumed mode method (AMM). Using Lagrange multipliers, the fully coupled equations of motions of the flexible parallel manipulator are developed by incorporating the rigid body motions with elastic motions. The mutual dependence of elastic deformations and rigid body motions are investigated from the analysis of the derived equations of motion. Open-loop simulation without joint motion controls and closed-loop simulation with joint motion controls are performed to illustrate the effect of elastic motion on rigid body motions and the coupling effect amongst flexible links. These analyses and results provide valuable insight to the design and control of the parallel manipulator with flexible intermediate links.     

火炮振动分析与多体系统模型研究

研究火炮振动对于提高射击精度有重要意义. 冲击激励的特殊性和机械系统的复杂性,使火炮振动成为长期关注的经典问题,建立逼近火炮真实运动状态的动力学模型、分析掌握各种冲击作用下的振动特性,是火炮振动的核心研究分支. 本文在发展多刚体与多柔体系统动力学理论的基础上,研究了火炮动力学的 Lagrange 方程及 Guass 变分建模方法,并由其建立了火炮振动的刚性、刚柔耦合多体系统模型; 针对案例分别研究了火炮系统的冲击响应规律及固有振动特性,获得了火炮振动与其结构参数的内在联系. 研究结果对于改善火炮的振动性能、提高射击精度,具有理论指导意义和工程应用价值.     

Mesoscale simulations of particulate flows with parallel distributed Lagrange multiplier technique

Fluid particulate flows are common phenomena in nature and industry. Modeling of such flows at micro and macro levels as well establishing relationships between these approaches are needed to understand properties of the particulate matter. We propose a computational technique based on the direct numerical simulation of the particulate flows. The numerical method is based on the distributed Lagrange multiplier technique following the ideas of Glowinski et al. [16] and Patankar [30]. Each particle is explicitly resolved on an Eulerian grid as a separate domain, using solid volume fractions. The fluid equations are solved through the entire computational domain, however, Lagrange multiplier constrains are applied inside the particle domain such that the fluid within any volume associated with a solid particle moves as an incompressible rigid body. Mutual forces for the fluid-particle interactions are internal to the system. Particles interact with the fluid via fluid dynamic equations, resulting in implicit fluid-rigid body coupling relations that produce realistic fluid flow around the particles (i.e., no-slip boundary conditions). The particle-particle interactions are implemented using explicit force-displacement interactions for frictional inelastic particles similar to the DEM method of Cundall et al. [10] with some modifications using a volume of an overlapping region as an input to the contact forces. The method is flexible enough to handle arbitrary particle shapes and size distributions. A parallel implementation of the method is based on the SAMRAI (Structured Adaptive Mesh Refinement Application Infrastructure) library, which allows handling of large amounts of rigid particles and enables local grid refinement. Accuracy and convergence of the presented method has been tested against known solutions for a falling particle as well as by examining fluid flows through stationary particle beds (periodic and cubic packing). To evaluate code performance and validate particle contact physics algorithm, we performed simulations of a representative experiment conducted at the U.C. Berkeley Thermal Hydraulic Lab for pebble flow through a narrow opening.     

A Lagrange duality characterisation for stability under arbitrary switching in switched positive linear systems

The present communication is concerned with uniform exponential stability, under arbitrary switching, in discrete-time switched positive linear systems. Lagrange duality is used in order to obtain a new characterisation for uniform exponential stability which is in terms of sets of inequalities involving each of the matrices that represent the modes of the system. These sets of inequalities are shown to generalise the classical linear Lyapunov inequality that characterises, in positive matrices, the property of being Schur. Each solution to these sets of inequalities is shown to provide a representation, in terms of a number of linear functionals, for a common Lyapunov function for the switched positive linear system. A result is further presented which conveys to, a conservative upper bound on the minimum required number of linear functionals (in the above mentioned representation), and also to a method for computing them. Our proof for the aforementioned characterisation is based on another (equivalent) characterisation, in terms of the solvability of a dynamic programming equation associated to the switched positive linear system, which is also reported in the paper. In particular, it is shown that the associated dynamic programming equation has at most one solution. And this solution is shown to be convex, monotonic, positively homogeneous, and it yields a common Lyapunov function for the switched positive linear system.     

Dynamics analysis of linear elastic planar mechanisms

This paper presents a formulation for the dynamics analysis of an elastic mechanism and integrating a stiff system using efficient numerical methods. Because all the elastic degrees of freedom are included in the vector of generalized variables, the size of the equations is much larger than that obtained using either the assumed mode or the distributed parameter finite element approach. However, the resulting system matrix is sparse and the elastic coordinates are absent from the system matrix, and these are useful properties for subsequent numerical analysis. Techniques for solving a system of linear time-variant equations are applied to the dynamics equations, assuming that the system matrix is slow-changing, and thus, may be approximated by a series of piecewise constant matrices. It is argued that the problem of determining the integration time step is transformed into the problem of computing the exponential of the system matrix with automatic time scaling. A numerical example is given to show that the behavior of the rigid coordinates converges to that of an all-rigid-body model by artificially increasing the Young’s modulus of the elastic components, despite the very-high-frequency vibrations of the elastic coordinates induced by the increment of the stiffness.     

弹性支撑下风电机组传动系统结构动力分析

考虑水平轴风力发电机组齿轮箱弹性支撑的柔性连接特性,基于集中质量思想和拉格朗日方法,建立风力发电机传动系统多体动力学模型,研究了齿轮箱弹性支撑对传动系统结构动力学特性的影响.利用动力学模型和模态分析方法,得到了由弹性支撑耦合到系统后的模态频率,并获取了在该模态激励下的模态动能分布.采用变参数方法进行传动系统模态对齿轮箱弹性支撑刚度变化的敏感性分析,利用模态叠加法进行齿轮箱体的动响应分析.数值求解结果和分析表明,考虑齿轮箱弹性支撑的传动系统某阶固有频率即为弹性支撑下齿轮箱体振动主模态;弹性支撑线刚度对传动系统低频率固有模态存在一定影响;齿轮箱体振动分析时应考虑1阶和2阶的低频模态较为合理.本研究工作对传动链系统方案可靠性设计和抑制传动链振动的加阻控制提供了一定理论基础.