Continuous contact force models for impact analysis in multibody systems

One method for predicting the impact response of a multibody system is based on the assumption that the impacting bodies undergo local deformations and the contact forces are continuous. In a continuous analysis, the integration of the system equations of motion is carried out during the period of contact; therefore, a model for evaluating the contact forces is required. In this paper, two such contact force models are presented, both Hertzian in nature and based upon the direct-central impact of two solid particles.At low impact velocities, the energy dissipation during impact can be represented by material damping. A model is constructed based on the general trend of the Hertz contact law in conjuction with a hysteresis damping function. The unknown parameters are determined in terms of a given coefficient of restitution and the impact velocity. When local plasticity effects are the dominant factor accounting for the dissipation of energy at high impact velocities, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are again evaluated. The two particle models are generalized to an impact analysis between two bodies of a multibody system.     

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.     

含分数阻尼特性元件的多体系统动力学研究

在绝对节点坐标体系下研究了具有分数导数阻尼特性元件的多体系统动力学建模、求解问题. 采用基于绝对节点坐标的无闭锁效应剪变梁单元离散柔性构件,建立了含常数质量矩阵的系统动力学方程, 并采用数值耗散可控的广义a方法求解. 通过数值算例计算,对比研究了算法参数与阻尼项的分数指数对系统动力学响应的影响规律.该方法可以进一步扩展到众多工程实际问题研究中.      

Lubricated revolute joints in rigid multibody systems

The main purpose of this work is to present a general methodology for modeling lubricated revolute joints in constrained rigid multibody systems. In the dynamic analysis of journal-bearings, the hydrodynamic forces, which include both squeeze and wedge effects, generated by the lubricant fluid, oppose the journal motion. The hydrodynamic forces are obtained by integrating the pressure distribution evaluated with the aid of Reynolds’ equation, written for the dynamic regime. The hydrodynamic forces built up by the lubricant fluid are evaluated from the system state variables and included into the equations of motion of the multibody system. Numerical examples are presented in order to demonstrate the use of the methodologies and procedures described in this work.     

Stability for manifolds of equilibrium states of fractional generalized Hamiltonian systems

For a fractional generalized Hamiltonian system, in terms of Riesz derivatives, stability theory for the manifolds of equilibrium states is presented. The gradient representation and second order gradient representation of a fractional generalized Hamiltonian system are studied, and the conditions under which the system can be considered as a gradient system and a second order gradient system are given, respectively. Then, equilibrium equations, disturbance equations, and first approximate equations of a fractional generalized Hamiltonian system are obtained. A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to a fractional generalized Hamiltonian system, and three propositions on the stability of the manifolds of equilibrium states of the system are investigated. As the special cases of this article, the conditions which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system, a fractional Hamiltonian system and a Hamiltonian system are given, respectively, and the stability theory for the manifolds of equilibrium states of these systems are obtained. Further, a fractional dynamical system and a fractional Volterra model of the three species groups are given to illustrate the method and results of the application. Finally, by using the method in this paper, we construct a new kind of fractional dynamical model, i.e. the fractional Hénon–Heiles model, and we study its stability of the manifolds of equilibrium states.     

Three-Dimensional Large Deformation Analysis of the Multibody Pantograph/Catenary Systems

To accurately model the nonlinear behavior of the pantograph/catenary systems, it is necessary to take into consideration the effect of the large deformation of the catenary and its interaction with the nonlinear pantograph system dynamics. The large deformation of the catenary is modeled in this investigation using the three-dimensional finite element absolute nodal coordinate formulation. To model the interaction between the pantograph and the catenary, a sliding joint that allows for the motion of the pan-head on the catenary cable is formulated. To this end, a non-generalized arc-length parameter is introduced in order to be able to accurately predict the location of the point of contact between the pan-head and the catenary. The resulting system of differential and algebraic equations formulated in terms of reference coordinates, finite element absolute nodal coordinates, and non-generalized arc-length and contact surface parameters are solved using computational multibody system algorithms. A detailed three-dimensional multibody railroad vehicle model is developed to demonstrate the use of the formulation presented in this paper. In this model, the interaction between the wheel and the rail is considered. For future research, a method is proposed to deal with the problem of the loss of contact between the pan-head and the catenary cable.     

Rheological representation of fractional order viscoelastic material models

We develop rheological representations, i.e., discrete spectrum models, for the fractional derivative viscoelastic element (fractional dashpot or springpot). Our representations are generalized Maxwell models or series of Kelvin-Voigt units, which, however, maintain the number of parameters of the corresponding fractional order model. Accordingly, the number of parameters of the rheological representation is independent of the number of rheological units. We prove that the representations converge to the corresponding fractional model in the limit as the number of units tends to infinity. The representations extend to compound fractional derivative models such as the fractional Maxwell model, fractional Kelvin-Voigt model, and fractional standard linear solid. Computational experiments show that the rheological representations are accurate approximations of the fractional order models even for a small number of units.     

含摩擦滑移铰平面多刚体系统动力学的数值算法

基于LuGre摩擦模型和线性互补问题(LCP)的数值算法,给出了具有双边约束含摩擦滑移铰平面多体系统动力学的数值算法.首先,根据滑移铰的特点,当间隙充分小时,将其视为双边约束,给出了滑移铰中滑道作用于滑块上的法向接触力的互补关系;LuGre摩擦模型能有效地描述机械系统中的黏滞与滑移运动,将该模型用于描述滑块与滑道间的摩擦力.其次,结合Baumgarte约束稳定化方法,应用第一类Lagrange方程,建立了该多体系统的动力学方程,给出了Lagrange乘子与滑移铰中作用于滑块上的法向接触力的关系式.然后,将滑块与滑道间多种接触状态的判断以及作用于滑块上的法向接触力的计算转换为线性互补问题的求解,并用常微分方程的数值算法求解该多体系统的动力学方程.最后,通过数值仿真算例揭示了滑移铰中滑块的黏滞与滑移现象,以及滑块在滑道内的多种接触状态;另外,在文中分别采用Coulomb干摩擦模型和LuGre摩擦模型,对算例中的某些工况进行了数值仿真,并且分别用本文方法得到的数值仿真结果与已有方法得到的数值仿真结果对比,表明了本文给出的方法的有效性.      

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.     

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.     

考虑弹簧阻尼作动器解析雅可比矩阵的多刚体动力学分析

弹簧-阻尼-作动器(spring-damper-actuator,SDA)是多体系统中常见的力元,在工程领域中有着广泛的应用.采用绝对坐标方法建立的多体系统动力学控制方程通常是复杂的非线性微分-代数方程组.为了保证数值解的精度和稳定性,通常需要采用隐式算法求解动力学方程,而雅可比矩阵的计算在隐式数值求解过程中至关重要.对于含有SDA的多体系统,SDA造成的附加雅可比矩阵是与广义坐标和广义速度相关的高度非线性函数.目前的很多研究工作专注于广义力向量的计算,然而对附加雅克比矩阵的计算则少有关注.针对含SDA的多刚体系统进行动力学分析,首先基于Newmark算法研究其在动力学方程求解中的雅可比矩阵的构成形式;然后推导SDA的广义力向量对应的附加雅可比矩阵,其中包括广义力向量对广义坐标和对广义速度的偏导数矩阵.最后通过两个数值算例研究附加雅可比矩阵对动力学分析收敛性的影响;数值分析表明:当SDA的刚度、阻尼和作动力数值较大时,SDA导致的附加雅可比矩阵对数值解的收敛性有重要影响;当考虑SDA对应的附加雅可比矩阵时,动力学分析可以以较少的迭代步实现收敛,从而减少分析时间.     

Decay of vortex velocity and diffusion of temperature in a generalized second grade fluid

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid was introduced. The velocity and temperature fields of the vortex flow of a generalized second fluid with fractional derivative model were described by fractional partial differential equations. Exact analytical solutions of these differential equations were obtained by using the discrete Laplace transform of the sequential fractional derivatives and generalized Mittag-Leffier function. The influence of fractional coefficient on the decay of vortex velocity and diffusion of temperature was also analyzed.     

含非理想约束多柔体系统递推建模方法

基于多体系统中邻接物体运动学递推关系,可以证明树状多体系统中末端物体的作用体现为传递给其内接物体的惯性和外力. 由于闭环系统切断铰约束反力和非理想约束反力可看作为系统外力,任何复杂系统都可以转化为等效的树系统,并且系统约束方程中所涉及的广义加速度可以系统化地用描述约束反力的拉氏乘子替换. 基于以上结果,提出了针对含非理想约束多柔体系统递推建模方法. 利用该方法可以将复杂多体系统动态减缩为单个物体,从而在求解系统加速度时不需对整个系统的质量矩阵进行求逆运算,同时大幅度地降低了非理想约束反力方程的维数. 通过一个算例具体说明了所提方法的求解过程,算例结果与现有商业软件所得结果一致.      

车辆纵向非光滑多体动力学建模与数值算法研究

在建立车辆纵向多体系统的动力学模型中, 将车身与车轮视为刚体, 两者通过减振器链接; 将传动系统视为一个圆盘通过扭簧和阻尼器与驱动轮连接; 将车轮与路面间的接触力简化为法向约束力、摩擦力和滚阻力偶,其中摩擦力的模型采用库仑干摩擦模型. 采用笛卡尔坐标作为该系统的广义坐标用于描述该系统的位形, 给出系统单双边的约束方程, 应用第一类拉格朗日方法建立了系统的动力学方程. 由于摩擦与滚阻的非光滑性, 使得该系统动力学方程不连续. 为便于计算, 建立了车轮与路面接触点的相对切向加速度与摩擦力余量的互补条件、车轮角加速度与滚阻力偶余量的互补条件, 以及车轮轮心法向加速度与路面法向约束力的互补条件. 将接触—分离、黏滞—滑移的判断问题转化成线性互补问题的求解, 并给出了具有约束稳定化的基于事件驱动法的数值计算方法. 最后, 应用该方法对车辆纵向多体系统进行了仿真, 分析了输出扭矩、摩擦及滚阻系数对其动力学行为的影响.      

Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics

Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation' for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.     

多体系统动力学动态最优化设计与灵敏度分析

基于多体系统的动态最优化设计过程包括传统的多体系统仿真分析、系统设计灵敏度分析、 系统最优化设计等过程, 针对多体系统运动学、用二阶常微分方程和微分代数方程描述 的动力学,基于含设计参数的通用数学模型及通用的积分型目标函数,采用高效的系统灵 敏度分析伴随变量方法及易于实施的惩罚函数最优设计方法,建立了多体系统最优设计数学 模型和算法. 通过双摆系统、曲柄-滑块系统、弹簧/阻尼器-滑块系统3个算例对上述 算法的有效性进行了验证.     

A velocity transformation method for the nonlinear dynamic simulation of railroad vehicle systems

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.     

含摩擦与碰撞平面多刚体系统动力学线性互补算法

基于接触力学理论和线性互补问题的算法, 给出了一种含接触、碰撞以及库伦干摩擦, 同时具有理想定常约束(铰链约束) 和非定常约束(驱动约束) 的平面多刚体系统动力学的建模与数值计算方法. 将系统中的每个物体视为刚体, 但考虑物体接触点的局部变形, 将物体间的法向接触力表示成嵌入量与嵌入速度的非线性函数,其切向摩擦力采用库伦干摩擦模型. 利用摩擦余量和接触点的切向加速度等概念, 给出了摩擦定律的互补关系式; 并利用事件驱动法, 将接触点的黏滞-滑移状态切换的判断及黏滞状态下摩擦力的计算问题转化成线性互补问题的求解. 利用第一类拉格朗日方程和鲍姆加藤约束稳定化方法建立了系统的动力学方程, 由此可降低约束的漂移, 并可求解该系统的运动、法向接触力和切向摩擦力, 还可以求解理想铰链约束力和驱动约束力. 最后以一个类似夯机的平面多刚体系统为例, 分析了其动力学特性, 并说明了相关算法的有效性.