摘要
柔性多体系统是由多个刚体和柔性体相互连接而组成的系统,在人们生产生活中应用广泛。随着高速机械、机器人和航天结构的发展,对包含整体运动和弹性变形的柔性多体系统的研究越来越重要。
本硕士学位论文是参加与香港城市大学合作项目 ——柔性多体系统动力学问题研究的基础上完成的。中心内容是应用有限元和模态分析方法研究柔性多体系统的动力学响应。主要内容是
1、采用拉格朗日乘子法导出了柔性多体系统的控制方程,发展了一种基于有限元法和模态分析的方法来研究柔性体的动力响应。
2、在应用有限元方法时,通过连续梁的模态分析导出了形函数,该形函数称为精确模态形函数。进而采用精确模态形函数导出了欧拉-伯努利梁单元的质量矩阵和刚度矩阵。
3、在用应变能求刚度矩阵时,使用了非线性几何关系,将刚度矩阵分为常规的线性刚度矩阵和非线性的几何刚度矩阵。本文的工作考虑了整体位移和弹性变形耦合,同时考虑了梁的纵向刚化效应。
4、由于柔性多体系统快变的弹性变形和相对慢变的刚体运动相耦合,动力方程离散成一组非线性刚性微分代数方程组。为了求解这种方程组,本文基于欧拉积分法提出了一种的非线性积分方法,将柔性体的控制方程降为一阶。为了避免积分累积误差,引入违约修正来校正位移和速度。
5、对旋转梁、双连杆机构和曲臂连杆机构进行了仿真计算,仿真结果和已有的结果对比表明,本文的方法是有效的,具有优越性,能保证较高的计算精度。
Flexible Multibody Systems (FMBS) is systems that consist of rigid bodies and flexible bodies. Due to the development of high speed machinery, robots and aerospace structures, the research of FMBS undergoing both gross motion and elastic deformation is getting more and more important.
This thesis is based on the investigation on “Dynamics Response of Flexible Multibody Systems” which is a cooperative project with City University of Hong Kong. In this thesis, the Finite Element Method (FEM) and mode analysis are applied to FMBS. Major contents are as follows:
1. A new method based on mode analysis is developed to investigate dynamic response of FMBS in this thesis and the governing equation for FMBS is derived by using the method of Lagrange multipliers.
2. Comparing to the conventional FEM, the shape function, which is named Accurate Mode Shape Function, is derived from motion equations for continuous beams, and the mass and stiffness matrix for Euler-Bounerlli beam are derived using the Accurate Mode Shape Function.
3. By applying quadratic strain-displacement relationship, the Stiffness matrix is divided into two matrices. One is conventional stiffness matrix; the other is geometric stiffness matrix which is non-linear. Not only the coupling of gross motion and elastic deformation, but also the stiffness effect of axial stiffening effect is taken into account.
4. As motion equations for FMBS can be discretized into a set of non-linear Differential Algebraic Equations (DAEs), in this thesis, a non-linear numerical integration method based on Euler's method, which can reduce DAEs to first-order ones, is proposed to solve these DAEs. A position and velocity correction are introduced to avoid cumulate errors.
5. Simulations on a rotating beam, a two-bar mechanism and a slider crank are used to verify the proposed method. The computing results of numerical examples show that efficiency of the present method.
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