多体系统指标2运动方程HHT方法违约校正

马秀腾; 翟彦博; 谢守勇

doi:10.6052/0459-1879-16-275

多体系统指标2运动方程HHT方法违约校正

doi: 10.6052/0459-1879-16-275

马秀腾^{*, †, ,},
翟彦博^*,
谢守勇^*

*.
西南大学工程技术学院, 重庆 400715
†.
现代汽车零部件技术湖北省重点实验室, 武汉 430070

基金项目:

国家自然科学基金 51605391

和现代汽车零部件技术湖北省重点实验室开放基金 2012-04

详细信息

通讯作者:
马秀腾,副教授,主要研究方向:多体系统动力学建模与仿真算法.E-mail:maxt@swu.edu.cn

中图分类号: O313.7
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- 被引次数: 0
出版历程
- 收稿日期: 2016-09-29
- 修回日期: 2016-11-14
- 网络出版日期: 2016-11-30
- 刊出日期: 2017-01-18

HHT METHOD WITH CONSTRAINTS VIOLATION CORRECTION IN THE INDEX 2 EQUATIONS OF MOTION FOR MULTIBODY SYSTEMS

Ma Xiuteng^{*, †
, ,},
Zhai Yanbo^*,
Xie Shouyong^*

*.
College of Engineering & Technology, Southwest University, Chongqing 400715, China
†.
Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan 430070, China

摘要

摘要: 采用Cartesian绝对坐标建模方法，完整约束多体系统运动方程是指标3的微分——代数方程（differentialalgebraic equations，DAEs），数值求解指标3的DAEs属于高指标问题，通过对位置约束方程求导，可使运动方程的指标降为2.位置约束方程求导得到的是速度约束方程.直接求解指标3的运动方程，速度约束方程得不到满足，而且高指标DAEs的数值求解存在一些问题.论文首先采用HHT（Hilber——Hughes——Taylor）直接积分方法求解降指标得到的指标2运动方程，此时速度约束方程参与离散计算，从机器精度上讲速度约束自然得到满足，而位置约束方程没有参与计算，存在“违约”.针对违约问题，采用基于Moore——Penrose广义逆理论的违约校正方法，消除位置约束方程的违约.指标2运动方程HHT方法违约校正，将HHT方法和违约校正方法很好地结合，在数值求解指标2运动方程的过程中，位置约束方程和速度约束方程都不存在违约问题，而且新方法没有引入新的未知数向量，离散得到的非线性方程组的方程数量与原指标2运动方程的方程数量相同，求解规模没有扩大.新方法的实用和有效性通过算例的数值实验得到验证，数值实验也说明新方法保持了HHT方法本身具有的数值阻尼可以控制和二阶精度的特性.最后从非线性方程组的求解规模和计算速度上与其他方法进行了比较分析，说明新方法的优势所在.
- 运动方程 /
- 指标 /
- HHT方法 /
- Moore——Penrose广义逆 /
- 违约校正
Abstract: Equations of motion for multibody system with holonomic constraints in Cartesian absolute coordinates modeling method are index 3 differential-algebraic equations (DAEs).It is high index problem for numerical integration of index 3 DAEs.The index can be reduced to 2 by taking the derivative of position constraint equations, and velocity constraint equations can be obtained.During the integration of index 3 equations of motion, the velocity constraint equations are violated, and there are some problems in the integration of high index DAEs.Firstly, HHT (Hilber-Hughes-Taylor) direct integration method is used to the numerical integration of index 2 equations of motion.The velocity constraint equations involved in the integration, and they are satisfied in the view of computer precision.However, the position constraint equations are violated.Secondly, in order to eliminate the violation, the correction method based on MoorePenrose generalized inverse theory is adopted.HHT method with constraints violation correction for index 2 equations of motion is the combination of HHT and correction method.There are no position and velocity constraints violation during the integration in the view of computer precision.No new unknown variables are introduced, and the quantity of equations in nonlinear equations from discretization is the same as index 2 equations of motion.The new integration method is validated by numerical experiments.In addition, some characteristics of HHT method, such as controlled numerical damping and second-order accuracy, are persisted by the new integration method.Finally, the quantity of nonlinear equations from discretization and computational efficiency are compared with some other methods.The advantages of the new method are illustrated.
- equations of motion /
- index /
- HHT method /
- Moore-Penrose generalized inverse /
- violation correction

HTML全文

图 1 曲柄滑块机构简图

Figure 1. Sketch of slider crank mechanism

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图 2 机构$\theta $角时间历程

Figure 2. Time evolution of $\theta $ for mechanism

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图 3 速度约束方程违约曲线

Figure 3. Evolution of velocity constraint equations violation

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图 4 位置约束方程违约曲线

Figure 4. Evolution of position constraint equations violation

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图 5 速度约束方程违约曲线

Figure 5. Evolution of velocity constraint equations violation

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图 6 位置约束方程违约曲线

Figure 6. Evolution of position constraint equations violation

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图 7 求解过程中系统能量的变化曲线

Figure 7. Energy of the system during integration

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图 8 精度

Figure 8. Order of accuracy

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