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.     

A parametric study on the dynamic response of planar multibody systems with multiple clearance joints

A general methodology for dynamic modeling and analysis of multibody systems with multiple clearance joints is presented and discussed in this paper. The joint components that constitute a real joint are modeled as colliding bodies, being their behavior influenced by geometric and physical properties of the contacting surfaces. A continuous contact force model, based on the elastic Hertz theory together with a dissipative term, is used to evaluate the intrajoint contact forces. Furthermore, the incorporation of the friction phenomenon, based on the classical Coulomb’s friction law, is also discussed. The suitable contact-impact force models are embedded into the dynamics of multibody systems methodologies. An elementary mechanical system is used to demonstrate the accuracy and efficiency of the presented approach, and to discuss the main assumptions and procedures adopted. Different test scenarios are considered with the purpose of performing a parametric study for quantifying the influence of the clearance size, input crank speed, and number of clearance joints on the dynamic response of multibody systems with multiple clearance joints. Additionally, the total computation time consumed in each simulation is evaluated in order to test the computational accuracy and efficiency of the presented approach. From the main results obtained in this study, it can be drawn that clearance size and the operating conditions play a crucial role in predicting accurately the dynamic responses of multibody systems.     

A linear complementarity model for multibody systems with frictional unilateral and bilateral constraints

The Lagrange-I equations and measure differential equations for multibody systems with unilateral and bilateral constraints are constructed.For bilateral constraints,frictional forces and their impulses contain the products of the filled-in relay function induced by Coulomb friction and the absolute values of normal constraint reactions.With the time-stepping impulse-velocity scheme,the measure differential equations are discretized.The equations of horizontal linear complementarity problems(HLCPs),which are used to compute the impulses,are constructed by decomposing the absolute function and the filled-in relay function.These HLCP equations degenerate into equations of LCPs for frictional unilateral constraints,or HLCPs for frictional bilateral constraints.Finally,a numerical simulation for multibody systems with both unilateral and bilateral constraints is presented.     

Deregularization of a smooth system – example hydraulics

Many technical systems include steep characteristics for force laws, which as a rule lead to stiff differential equations and large computing times. For the dynamical performance such steep characteristics are very near to laws with set-valued properties and might therefore be replaced by set-valued force laws. This is true for multibody dynamics including unilateral contacts, and it is in an approximate way true for fluid mechanical systems like hydraulics. In the following we present a new modeling scheme for hydraulic systems, which establishes the hydraulic equations of motion in the form of multibody system eqations with bilateral and unilateral constraints, and which is able to reduce computing times by three to four order of magnitudes. A large industrial example illustrates the excellent performance of the new theory.     

The effect of the lubricated revolute joint parameters and hydrodynamic force models on the dynamic response of planar multibody systems

In this work a comprehensive methodology for dynamic modeling and analysis of planar multibody systems with lubricated revolute joints is presented. In general, this type of mechanical systems includes journal-bearings in which the load varies in both magnitude and direction. The fundamental issues associated with the theory of lubrication for dynamically loaded journal-bearings are revisited that allow for the evaluation of the Reynolds equation for dynamic regime. This approach permits the derivation of the suitable hydrodynamic force laws that are embedded into the dynamics of multibody systems formulation. In this work, three different hydrodynamic force models are considered, namely the Pinkus and Sternlicht approach for long journal-bearings and the Frêne et al. models for both long and short journal-bearings. Results for a planar slider?Ccrank mechanism with a lubricated revolute joint between the connecting-rod and slider are presented and utilized to discuss the assumptions and procedures adopted throughout the present study. Different test scenarios are taken into account with the purpose of performing a comparative study for quantifying the effect of the clearance size, lubricant viscosity, input crank speed and hydrodynamic force model on the dynamic response of multibody systems with lubricated revolute joints. From the global results obtained from computational simulations, it can be concluded that the clearance size, the lubricant viscosity and the operating conditions play a key role in predicting the dynamic behavior of multibody systems.     

Modeling intermittent contact for flexible multibody systems

This paper consists of two parts. The first part presents a complementarity based recursive scheme to model intermittent contact for flexible multibody systems. A recursive divide-and-conquer framework is used to explicitly impose the bilateral constraints in the entire system. The presented approach is an extension of the hybrid scheme for rigid multibody systems to allow for small deformations in form of local mode shapes. The normal contact and frictional complementarity conditions are formulated at position and velocity level, respectively, for each body in the system. The recursive scheme preserves the essential characteristics of the contact model and formulates a minimal size linear complementarity problem at logarithmic cost for parallel implementation. For a certain class of contact problems in flexible multibody systems, the complementarity based time-stepping scheme requires prohibitively small time-steps to retain accuracy. Modeling intermittent contact for this class of contact problems motivated the development of an iterative scheme. The second part of the paper describes this iterative scheme to model unilateral constraints for a multibody system with relatively fewer contacts. The iterative scheme does not require a traditional complementarity formulation and allows the use of any higher order integration methods. A comparison is then made between the traditional complementarity formulation and the presented iterative scheme via numerical examples.     

Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints

An approach is proposed for modeling and anal- yses of rigid multibody systems with frictional translation joints and driving constraints. The geometric constraints of translational joints with small clearance are treated as bilat- eral constraints by neglecting the impact between sliders and guides. Firstly, the normal forces acting on sliders, the driv- ing constraint forces （or moments） and the constraint forces of smooth revolute joints are all described by complementary conditions. The frictional contacts are characterized by a set- valued force law of Coulomb＇s dry friction. Combined with the theory of the horizontal linear complementarity problem （HLCP）, an event-driven scheme is used to detect the transi- tions of the contact situation between sliders and guides, and the stick-slip transitions of sliders, respectively. And then, all constraint forces in the system can be computed easily. Secondly, the dynamic equations of multibody systems are written at the acceleration-force level by the Lagrange multiplier technique, and the Baumgarte stabilization method is used to reduce the constraint drift. Finally, a numerical example is given to show some non-smooth dynamical behaviors of the studied system. The obtained results validate the feasibility of algorithm and the effect of constraint stabilization.     

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.      

Modeling of unilateral contact conditions with application to aerospace systems involving backlash, freeplay and friction

This paper describes an analysis procedure for the modeling of backlash, freeplay and friction in flexible multibody systems. The first two effects are formulated in a general manner as unilateral contact conditions in multibody dynamics. The incorporation of the effects of friction in joint elements is also discussed, together with an effective computational strategy. These non-standard effects are formulated within the framework of finite element based multibody dynamics that allows the analysis of complex, flexible systems of arbitrary topology. The versatility and generality of the approach are demonstrated by presenting applications to aerospace systems: the flutter analysis of a wing-aileron system with freeplay, the impact of an articulated rotor blade on its doop stop during engagement operation in high wind conditions, and the dynamic response of a space antenna featuring joints with friction.     

带约束非线性多体系统动力学方程数值分析方法

Lagrange方法是建立带约束多体系统动力学方程的普遍方法之一 ,其方程的形式为微分 代数方程组 ,数值计算与数值分析是研究多体系统动力学特性的重要方法。本文利用缩并法给出了带约束多体系统动力学方程的隐式数值计算方法和Lyapunov指数的计算方法。将数值仿真、Lya punov指数计算和Poincare映射有机结合 ,分析非线性多体系统动力学行为。通过一个算例 ,说明该方法的有效性     

Dynamics of spatial structure-varying rigid multibody systems

Summary Couplings in machines and mechanisms exhibiting backlash and friction phenomena can be modeled as multibody systems with unilateral constraints and Coulomb friction. The structure of the differential-algebraic equations describing the system depends on the state of the constraints. The contact forces occurring at active constraints are taken into account in the equations of motion as Lagrange multipliers. Additionally, the kinematic conditions of all active constraints are formulated on the acceleration level. Contact and friction laws are sufficient conditions for state transitions of active constraints, and are represented by nonsmooth characteristics. Several formulations, like the linear complementarity problem, and two different nonlinear systems of equations are presented together with their solution method. The theory is applied to a mechanical system containing three-dimensional and coupled unilateral constraints with friction. Received 14 May 1998; accepted for publication 5 January 1999     

A new model for dry and lubricated cylindrical joints with?clearance in spatial flexible multibody systems

A new approach to model and analyze flexible spatial multibody systems with clearance of cylindrical joints is presented in this work. The flexible parts are modeled by using absolute nodal coordinate formulation (ANCF)-based elements, while the rigid parts are described by employing the natural coordinate formulation (NCF), which can lead to a constant system mass matrix for the derived system equations of motion. In a simple way, a cylindrical joint with clearance is composed of two main elements, that is, a journal inside a bearing. Additionally, a lubricant fluid can exist between these two mechanical elements to reduce the friction and wear and increase the system??s life. For the case in which the joint is modeled as a dry contact pair, a technique using a continuous approach for the evaluation of the contact force is applied, where the energy dissipation in the form of hysteresis damping is considered. Furthermore, the frictional forces developed in those contacts are evaluated by using a modified Coulomb??s friction law. For the lubricated case, the hydrodynamic theory for dynamically loaded journal bearings is used to compute the forces generated by lubrication actions. The lubricated model is based on the Reynolds equation developed for the case of journal bearings with length-to-diameter ratios up to 1. Using this approach, the misalignment of the journal inside the bearing can be studied. Finally, two demonstrative examples of application are used to provide results that support the discussion and show the validity of the proposed methodologies.     

柔性多体系统动力学实验研究综述

介绍了国内外柔性多体系统动力学实验研究现状,分为三个方面,即理论模型验证实验、动力学特性的实验研究和其它实验.柔性多体系统动力学建模理论的发展经历了3个阶段:运动-弹性动力学(KED)方法、传统混合坐标方法和计及了动力刚化效应的各种非线性理论.关于这些理论的模型验证实验均在本文中作了重点介绍.文中还对柔性多体系统动力学性态的研究实验也作了介绍,包括系统模态特性和共振等非线性力学行为.关于机械臂控制和碰撞研究实验虽有提及,但不作为重点.随后,着重介绍了柔性体弹性振动位移的测量和阻尼因素的处理这两个在实验不可避免但又难以解决的问题,尤其是结构阻尼和大范围运动引起的空气阻力.最后指出了今后的研究方向.文中对一些较为重要的实验装置也着重予以介绍,并给出了部分实验图片及数据曲线,以给读者一个更好的理解和参考.      

Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

In this paper, we will give conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle’s invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapunov function and an elegant derivation of the proof. Moreover, an instability theorem for assessing the instability of equilibrium sets of non-linear mechanical systems with frictional bilateral constraints is formulated. These results are illustrated by means of examples with both unilateral and bilateral frictional constraints.     

Unilateral problems of dynamics

Summary Contact processes may be described by local discretizations, by rigid representation or by mixed methods incorporating both ideas. A rigid body approach is proposed for the dynamics of mechanical systems, achieving good results also for multiple-contact problems. Contacts in multibody systems are mainly considered, with the corresponding contact constraints varying with time, thus generating structure-variant systems. The equations of motion for dynamical systems with such unilateral behavior are discussed, solution methods and applications are presented. Received 3 March 1999; accepted for publication 5 May 1999     

Simulation of planar flexible multibody systems with clearance and lubricated revolute joints

Modeling of clearance joints plays an important role in the analysis and design of multibody mechanical systems. Based on the absolute nodal coordinate formulation (ANCF), a new computational methodology for modeling and analysis of planar flexible multibody systems with clearance and lubricated revolute joints is presented. A planar absolute nodal coordinate formulation based on the locking-free shear deformable beam element is implemented to discretize the flexible bodies. A continuous contact-impact model is used to evaluate the contact force, in which energy dissipation in the form of hysteresis damping is considered. A force transition model from hydrodynamic lubrication forces to dry contact forces is introduced to ensure continuity in the joint reaction force. A comprehensive study with different lubrication force models has also been carried out. The generalized-α method is used to solve the equations of motion and several efficient methods are incorporated in the proposed model. Finally, the methodology is validated by two numerical examples.     

约束多体系统的基于离散零空间的隐式龙格库塔法

将离散零空间理论应用于多体系统动力学方程的数值计算,可降低多体系统动力学方程的维数。通过给出离散零空间理论与IRK法相结合的一般数学框架,提出了多体系统动力学的基于离散零空间理论的IRK法。数值算例表明：该算法可获得较满意的数值结果,约束违约程度很小,三种积分算法算例的范数均在10^-16之内。     

非光滑多体系统动力学数值算法的研究进展

非光滑多体系统动力学数值计算方法是多体系统动力学研究的重要内容之一. 本文介绍了近年来含摩擦与碰撞的非光滑多体系统动力学数值算法方面的研究进展. 首先, 讨论了库仑摩擦模型和修正的库仑摩擦模型, 以及具有单边和双边约束的多体系统中法向约束力的特点. 其次, 回顾了基于连续模型和非连续模型的多体系统动力学方程的数值计算方法, 详细介绍了基于互补概念的非光滑多体系统动力学的事件驱动法和时间步进法, 分析比较了相关的数值算法. 最后, 指出了一些需要进一步研究的问题.     

结构碰撞的动力响应分析

本文研究结构在动力作用下发生碰撞时的动力响应的分析方法。其中碰撞包括结构与刚（弹）性支座的碰撞和两个结构物之间的碰撞；碰撞的速度假定为中低速度，分析时不考虑局部的破坏问题。本文方法的关键点是提出了碰撞过程中的碰撞反力的模拟表达式。它是通过碰撞时结构与碰撞物体的碰撞力和碰撞变形关系的假定，并利用能量守恒原理，动量守恒原理和冲量定理建立的，既可描述完全弹性碰撞过程也可描述非完全弹性碰撞过程，当然也可考虑结构阻尼的影响。文末给出了几个算例，其中与有解析解的做了比较，符合得很好；至于没有解析解可比较的，在理论上也是合理的，算法上采用了中央差分的逐步积分法，在碰撞过程中采用非常小的积分步长，获得了预想的理想结果。