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 引用本文: 富立, 胡鸿奎, 富腾. 多体系统接触碰撞问题的牛顿-欧拉线性互补方法[J]. 力学学报, 2017, 49(5): 1115-1125.
Fu Li, Hu Hongkui, Fu Teng. CONTACT-IMPACT ANALYSIS IN MULTI-BODY SYSTEMS BASED ON NEWTON-EULER LCP APPROACH[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(5): 1115-1125.
 Citation: Fu Li, Hu Hongkui, Fu Teng. CONTACT-IMPACT ANALYSIS IN MULTI-BODY SYSTEMS BASED ON NEWTON-EULER LCP APPROACH[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(5): 1115-1125.

## CONTACT-IMPACT ANALYSIS IN MULTI-BODY SYSTEMS BASED ON NEWTON-EULER LCP APPROACH

• 摘要: 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.

Abstract: The contact-impact analysis in multibody systems based on the nonsmooth dynamics approach is a hot topic in the research of multibody system dynamics. Newton-Euler approach is adopted to develop dynamics model of contactimpact analysis in non-smooth multi-body systems, and a new LCP formula is presented in this work. Different from Lagrange methods, Newton-Euler modeling method incorporate equality constraints into dynamic models with noninterpenetration constraints and frictional constraints together. In Newton-Euler modeling method, the basic system is derived by removing the non-interpenetration constraints and frictional constraints from the original multi-body system. Newton-Euler eqution of basic system is established by using the maximum coordinates method. Because the coordinates of the basic system are not independent of each other, equality constraints are involved in modeling, the basic system dynamic equations is a set of DAE (differential algebra equation). With the aid of constraint Jacobian matrix, Lagrangian multipliers corresponding to the non-interpenetration constraint forces and Coulomb friction forces are added to the basic system DAE to obtain the dynamic equations of global motion of the multi-body system with characteristics of variable topological structure. The complete dynamic model is composed of basic system DAE, equality and inequality constraints. In order to simplify the derivation process of LCP, a decomposed matrix form is built. The LCP -based Time-stepping method is adopted for numerical simulation. Time-stepping algorithm is a popular non-smooth numerical algorithm, Its prominent feature is that it can avoid the tedious event-detection process in numerical integration. In the process of numerical integration, the contact-detachment state of the system can be determined by solving the LCP. Our method is carried out in slider-crank mechanism with a translational clearance joint, the simulation results indicate that this method is effective.

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