【作者】 张志刚；

【导师】 吴志刚； 齐朝晖；

【作者基本信息】 大连理工大学 ， 动力学与控制， 2015， 博士

【摘要】 细长梁是机械系统中应用最广泛的柔性部件,也是柔性多体系统动力学中最受关注的研究对象之一。随着轻质材料的广泛应用和现代机械系统运行速度的提高,柔性梁的动力学特性变得极其复杂,表现为大范围刚体运动与梁自身大变形的耦合效应越来越显著。基于小变形、小转动假设的传统柔性多体系统建模方法已经无法为这类系统的分析和设计提供可靠的数值仿真结果。本文对大变形梁的建模理论,以及刚柔耦合系统动力学方程的数值求解方法做了深入研究。几何精确梁理论在刚性截面假设基础上定义了具有客观性的应变度量,可用于分析大位移大转动情况下梁的变形。基于平面几何精确Euler-Bernoulli梁内力虚功率,提出了一种适用于多体系统中大变形平面梁建模的应变插值梁单元。由于与梁应变能直接相关的是应变而非位移,因而选取单元轴向应变和曲率作为基本变量进行离散,不受单元自身刚体位移大小影响,且能得到形式简洁的单元节点力和单刚矩阵。单元两端截面的形心位移和转角由几何方程积分得到,充分计及了“动力刚化项”。所构造应变插值梁单元不仅适用于小变形问题,同样能够用于大变形平面梁的刚柔耦合动力学分析。现有的空间几何精确梁单元大多采取了对轴线变形和截面转动独立插值的策略,在对细长梁建模时常常引起剪切闭锁等困难。本文充分考虑了细长梁变形耦合关系,构造了一种适用于多体系统中大变形空间梁建模的空间Euler-Bernoulli梁单元。以惯性坐标系下节点处的位移矢量和截面转动矢量为单元参数,通过对轴线变形和截面转动进行耦合插值,构造了梁截面始终与轴线切向保持垂直的单元变形场。以此为基础,利用几何精确梁理论推导了空间梁单元的节点力列阵、切线刚度矩阵和一致质量矩阵。所提出空间Euler-Bernoulli梁单元不仅适用于多体系统动力学中大变形空间梁的建模,也可用于细长结构的几何非线性分析。柔性多体系统的动力学方程往往是一组刚性方程,其数值求解具有相当难度。目前普遍应用的刚性方程数值解法,其基本思想是通过数值阻尼来滤除高频响应。虽然也成功地解决了许多问题,但其计算效率仍然不能令人满意。通过将系统内力表达式中瞬时应变修正为一小段时间间隔内的平均应变,提出了一种柔性多体系统动力学滤频建模的平均应变方法。该方法能够为系统方程引入应变阻尼项和应变惯性项,通常情况下所建模型可用非刚性方程求解器进行仿真求解,计算效率也明显提高。最后,将平均应变方法应用到大变形梁的动力学建模中,通过数值算例验证了所提方法的有效性。更多还原

【Abstract】 The slender beam is not only the most used flexible component in mechanical systems, but also one of the most popular research objects in flexible multibody system dynamics. With the wide application of light materials and the improvement of the operating speed in modern mechanical systems, the dynamic characteristics of the flexible beam become extremely complex, and the coupling between large rigid body motions and large deformations is becoming more and more significant. Previous researches have indicated that the traditional modeling method in flexible multibody system dynamics based on small deformations and rotations assumption has been unable to provide reliable numerical simulation results for this kind of mechanical system. This paper presents an in-depth study on the modeling theory of large deformation beams and the numerical methods for dynamic equations of rigid-flexible coupling systems.With the assumption of rigid cross-section, the strain measures defined in the geometrically exact beam theory are of objectivity, which can be used to analyze the deformation of the beam with large displacements and finite rotations. Based on the internal virtual power of the geometrically exact plane Euler-Bernoulli beam, a strain-interpolation beam element, which is suitable for molding of large deformation plane slender beam in flexible multibody systems, is proposed. It is the strain rather than the displacement directly related to the beam strain energy. So, when structuring the finite element, the axial strain and the curvature of the beam element are selected to disperse as the basic variables. This is not affected by the rigid-body displacement of the element, but also can get simple element nodal forces and stiffness matrix. The centroid displacement and angle of the cross-section at two ends are obtained by integrating the geometry equation, which automatically captures the dynamic stiffening terms. The proposed beam element with strain interpolation is not only suitable for the small deformation problems of plane beams, but also the large deformation rigid-flexible coupling dynamics.The independent interpolation of finite rotations is widely adopted in the existing geometrically exact spatial beam element, which causes the problem of shear locking when modeling the flexible slender beam. On considering the deformation coupling relationships of the slender beam, a spatial Euler-Bernoulli beam element is proposed, which is suitable for the modeling of large deformation spatial slender beam in flexible multibody systems. The global displacement and rotation vectors are selected as the nodal coordinates and an element deformation field is constructed, in which the beam cross-sections can keep perpendicular to the current neutral axes by employing a special coupled interpolation of the centroid position and the cross-section orientation. On this basis, the beam element nodal force, tangent stiffness matrix and consistent mass matrix are derived according to the geometrically exact beam theory. The proposed spatial Euler Bernoulli beam element is not only suitable for the large deformation flexible beam’s modeling in multibody system dynamics, but also can be used to the geometric nonlinear problem of large deformation slender structure.The dynamic equations of flexible multibody systems are often a set of stiff equations, which are very difficult to solve. At present, stiff equation solvers are generally adopted, of which the basic idea is to filter out the high frequency response by numerical damping. Although many problems have been solved successfully, the computational efficiency of the stiff equation solvers is still unsatisfactory. By replacing the original instantaneous strains in the system internal force expression with an average strains over a small time interval, an average strain method is proposed, which can filter out the high frequencies of the system equation. The strain damping and inertia terms are introduced to the system equations through this method, and generally the modified system dynamic equations can be solved by using nonstiff solvers. Finally, the average strain method is applied to the dynamic modeling of the large deformation beams, and the efficiency of the proposed method is verified by the numerical simulations.更多还原