摘要
当多体系统的约束全部是摩擦接触时,其动力学问题可归结为一个常微分方程(ordinary differentialequation,ODE)与线性互补问题(linear complementarity problem,LCP)的混合动力学问题.如果除了摩擦接触之外还增加了光滑的双边约束,则需要将ODE-LCP混合动力学模型推广为微分代数方程(differentialalgebra equation,DAE)与LCP的混合动力学模型.该文采用DAE与LCP混合动力学方法求解不考虑碰撞但同时含有持续摩擦接触及光滑等式约束的多体系统动力学问题.在建立系统动力学模型时,首先将含摩擦的约束从系统中移去得到基本动力学系统.由于基本系统中带有等式约束,所以基本系统的动力学方程为一组DAE.结合基本系统的DAE与约束的互补条件便可以得到DAE-LCP混合动力学模型.数值计算采用基于DAE与LCF的步进(time-stepping)算法,将系统动力学方程及其约束离散化并转化为一个混合LCP进行求解.该算法无需进行滞-滑状态检测,避免了事件检测导致的繁复计算.利用所提方法对典型机构的非光滑非线性特征进行了数值分析,验证了该文方法的正确有效性.
Presently,dynamics of nonsmooth multibody systems is a hot research topic.The usual approach in treating such systems is to derive basic system from the original system by removing the nonsmooth constraints firstly.The Lagrange equations of the second kind of basic system combine with the complementarity condition of the nonsmooth constraints to set up at each discrete moment in time a Linear Complementarity Problem （LCP）. This article focuses on the problem of dynamic modeling and numerical simulating of multibody systems with friction contacts.By neglecting the clearance and the effect of impact between rigid bodies and constraints, the state variables in the differential equations are continuous.Due to the set-value mapping characteristic of dry friction forces,the differential equations of motion have discontinuous right-hand vector fields,therefore to allow our system to be classified as a Filippov system.In addition to the friction constraints,our model also incorporates frictionless bilateral ones.If the simulation has only friction constraints,then the problem is an ODE-LCP model.Combining friction contacts with frictionless bilateral constaints,the ODE-LCP model has to be extended to DAE-LCP（DAE,Differential Algebra Equation） mixed model.In order to obtain the model, the basic system is derived from the original system by removing the friction constraints firstly.Because the equality constraints are retained in the basic system,the dynamic model of basic system is a set of DAE.With the aid of constraint Jacobian matrix,the normal contact forces and tangential friction forces of nonsmooth constraints,which obey the complementarity contact laws,are added to the DAE of the basic system to obtain the DAE-LCP mixed model. Approaches used in the past for simulating rigid multibody dynamics with friction contacts include piece-wise DAE approaches,acceleration-force LCP approaches,and velocity-impulse LCP-based time-stepping methods. Recognizing that the nature of the frictional constraint can induce sick-slip motion,the last approach is used in this work,which has the advantage that it does not suffer from the detection for stick-slip transition that could appear in the first two approaches.This framework is based on a LCP,but it is different from acceleration-force LCP approaches that attempt to find the accelerations of the bodies.Our approach considers impulses and velocities as the fundamental unknowns.Acceleration-force LCP approaches solve for accelerations from the dynamics equations and then use the accelerations in an integration procedure.Because the complementarity law between acceleration and friction surplus is valid only when the relative tangential velocity is zero,zero crossing detection for velocity is required.However the complementarity law between velocity and friction surplus remains valid through a full-range of motion,so in contrast to acceleration-force schemes,the velocity-impulse methods need no event-detection.In the new framework,the integration and dynamical resolution steps are combined.The main achievement of this approach is that it has solutions for any configuration. As the time-step tends to zero,a subsequence of the numerical solutions approaches the solution of a differential inclusion. Our method is carried out in a numerical example,and the simulation results indicate that this method is effective.
出处
《力学学报》
EI
CSCD
北大核心
2011年第2期400-407,共8页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家自然科学基金资助项目(10672007)~~
