运动滑道含摩擦多体系统的建模和数值方法

研究了运动约束面含摩擦多体系统动力学方程的建立和算法问题．首先利用第一类Lagrange方程给出了系统的动力学方程,并以矩阵形式给出了这类系统摩擦力的广义力的一般表达式．为便于摩擦力和铰链约束力的分析与计算,采用笛卡尔坐标和约束方程的局部方法,使得系统的约束力与Lagrange乘子一一对应．应用增广法将微分一代数方程组转化为常微分方程组并用分块矩阵的形式给出,以便于方程的编程与计算,提高计算效率．最后用一个算例验证了该方法的有效性．     

摄动离散矩阵Lyapunov方程解的估计

研究摄动离散矩阵Lyapunov方程解的估计问题,利用矩阵运算性质及Lyapunov稳定性理论,给出在结构不确定性假设下方程解的存在条件及解的上下界估计,估计结果由一个线性矩阵不等式(LMI)和两个矩阵代数Riccati方程确定．针对几种不确定性假设,进一步给出矩阵代数Riccati方程的具体形式．最后通过一个算例说明了所得结果的有效性．     

线性离散周期系统输出反馈参数化极点配置

采用周期输出反馈的途径, 考虑了线性离散时间周期系统的极点配置问题. 通过将闭环单值性矩阵进行一系列转化, 周期输出反馈律的求解问题可以转化为一类Sylvester矩阵方程的求解问题. 利用Sylvester矩阵方程的最新结果, 可以将输出反馈增益表示为参数化形式, 从而得到了能够实现极点配置的所有输出反馈控制器. 数值算例验证了方法的有效性.     

线性最优控制系统的频域综合方法(多输入-多输出系统)

本文用频域方法讨论了二次性能指标下的多输入-多输出线性系统最优控制的综合问题. 利用多项式矩阵的谱分解方法,把求解最优综合函数的问题归结为求解两个多项式矩阵的 Diophantine方程,从而给出了该问题解的频域形式.     

一类广义Riccati矩阵方程对称解的双迭代算法

研究了一类广义系统控制理论导出的Riccati矩阵方程对称解的数值计算方法．运用牛顿算法将Riccati矩阵方程的对称解问题转化为线性矩阵方程的对称解或者对称最小二乘解问题,采用修正共轭梯度法解决导出的线性矩阵方程的对称解问题,可建立求Riccati矩阵方程对称解的双迭代算法．数值算例表明,双迭代算法是有效的．     

弹性力学极坐标公式利用矩阵形式的对比教学

利用弹性力学平面问题中有很强规律性且容易记忆的基本方程的直角坐标矩阵形式,按极坐标系与直角坐标系之间物理量、微分算子的对应关系,考虑特殊情况,可列出基本方程及物理量的极坐标矩阵表达形式.两种坐标系下的公式具有一定的对应规律,得到的极坐标矩阵形式体现出很强的规律性和记忆特点,矩阵形式易化成一般表达式.教学中可利用这些规律进行对比讲授,从而使学生对较为复杂且采用通常方法不易记忆的基本方程或物理量的极坐标形式变得有规可循,容易记忆和掌握.     

自洽场方法中广义本征值方程求解及其C++程序设计

本文研究自洽场方法中广义本征值方程求解的算法,并设计相应的C 程序来实现该算法。首先对重叠矩阵进行分解,并将广义本征值方程化为标准的本征值方程,再利用Householder变换将上一步变换所得的矩阵化为对称三对角矩阵,进而用QL方法求解这个三对角矩阵的本征值和本征矢量,从而得到自洽场方法中广义本征值方程的本征值和本征矢量。     

多元非线性数据拟合模型的数学推论及其回归方程的计算机拟合

本文推论了多元非线性数据拟合的通用数学模型,利用最小二乘法和极值原理,导出求解多元非线性回归方程的规范方程组。并用矩阵形式对规范方程组进行表述,在所表述的诸矩阵中,结构矩阵是其基础。用它可方便地转化出其他矩阵,这将大大简化程序的编制和规范方程组的解算。计算机根据输入数据自变量的个数和实验所作次数的多少,求解出相应的多元非线性回归方程及其评估方程质量的数据。     

非正交基空间理论及其在结构分析中的应用

本文采用Dirac符号表示基矢量,并引入伴基矢的定义,从而建立了单位算符的两种表示形式。利用单位算符即可求出矢量(位移及应力等)及算符(质量及刚度等)在非正交基空间中的矩阵表示,将结构分析方程转化为矩阵方程,并使方程的求解转化为一系列基矢的交换过程。附录中给出子程序SYMSOL,它具有多种功能:对称矩阵三角化、求侧向刚度矩阵、对称线性方程组求解、将广义特征值问题化为标准型等。     

一类离散时间代数Riccati矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在线性二次优化问题中遇到的含未知矩阵之逆的离散时间代数Riccati矩阵方程(DTARME)转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求DTARME的对称解的双迭代算法。双迭代算法仅要求DTARME有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定。数值算例表明双迭代算法是有效的。     

Singularity-free simulation of closed loop multibody systems by using null space of Jacobian matrix

Numerical simulation of closed loop multibody systems is associated with numerical solution of equations of motion which are, in general, in the form of DAE’s index-3 systems. For assuring continuous simulation, one should overcome some difficulties such as stabilization of the constraint equations, singular configuration of the system. In this paper, the system equations of motion with the Lagrange multipliers is rewritten by introducing generalized reaction forces. The combination with the condition of ideality of constraints leads to the system of equations which can be solved by numerical techniques smoothly, even over singular positions. Based on the new criterion of ideality of constraints, which relates generalized reaction forces and the null space matrix of Jacobian matrix, it is possible also to remove reaction forces and use only the reduced system of equations with null space matrix for passing singular positions. In order to prevent the constraint equations from the accumulated errors of integral time, the method of position and velocity projection has been exploited. Some numerical experiments are carried out to verify the proposed approach.     

Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces

This paper presents the topology optimization method for the steady and unsteady incompressible Navier–Stokes flows driven by body forces, which typically include the constant force (e.g. the gravity) and the centrifugal and Coriolis forces. In the topology optimization problem, the artificial friction force with design variable interpolated porosity is added into the Navier–Stokes equations as the conventional method, and the physical body forces in the Navier–Stokes equations are penalized using the power-law approach. The topology optimization problem is analyzed by the continuous adjoint method, and solved by the finite element method in conjunction with the gradient based approach. In the numerical examples, the topology optimization of the fluidic channel, mass distribution of the flow and local velocity control are presented for the flows driven by body forces. The numerical results demonstrate that the presented method achieves the topology optimization of the flows driven by body forces robustly.     

Two-dimensional team lifting prediction with floating-base box dynamics and grasping force coupling

The computational procedures of higher-order implicit integrators for mechanical systems with friction are provided in this paper. The dynamic equations are established using the augmented Lagrangian formulation, and set-valued friction forces are described by projection functions. To reduce the accuracy loss caused by event transitions and to eliminate spurious oscillations in the acceleration, a new robust event-driven scheme, which accurately detects event transitions and corrects the friction forces and accelerations at switching points, is proposed. The numerical performance of the proposed scheme is demonstrated by solving several benchmark problems. Numerical results show that the newly developed scheme can approximately achieve second-order accuracy, and they are more accurate than the classical Moreau time-stepping scheme under close computational efforts. Finally, a slider-crank system is simulated to prove the validity of the developed method for nonlinear mechanical systems with friction.

     

Modeling and simulation of longitudinal dynamics coupled with clutch engagement dynamics for ground vehicles

During the engagement of the dry clutch in automotive transmissions, clutch judder may occur. Vehicle suspension and engine mounts couple the torsional and longitudinal models, leading to oscillations of the vehicle body that are perceived by the driver as poor driving quality. This paper presents an effective formulation for the modeling and simulation of longitudinal dynamics and powertrain torsional dynamics of the vehicle based on non-smooth dynamics of multibody systems. In doing so friction forces between wheels and the road surface are modeled along with friction torque in the clutch using Coulomb’s friction law. First, bilateral constraint equations of the system are derived in Cartesian coordinates and the dynamical equations of the system are developed using the Lagrange multiplier technique. Complementary formulations are proposed to determine the state transitions from stick to slip between wheels and road surface and from the clutch. An event-driven scheme is used to represent state transition problem, which is solved as a linear complementarity problem (LCP), with Baumgarte’s stabilization method applied to reduce constraint drift. Finally, the numerical results demonstrate that the modeling technique is effective in simulating the vehicle dynamics. Using this method stick-slip transitions between driving wheel and the road surface and from the clutch, as a form of clutch judder, are demonstrated to occur periodically for certain values of the parameters of input torque from engine, and static and dynamic friction characteristics of tire/ground contact patch and clutch discs.     

Shape sensitivity analysis and design studies for CAD flume sections

This paper presents shape design sensitivity analysis (DSA) and design studies for recreational waterslides represented in computer-aided design (CAD) environment. The mathematical representations of a number of commonly used flume sections that serve as the building blocks for waterslide configurations are created in CAD tools. Geometric dimensions of the individual sections that affect not only their geometric shape but also the overall configurations are identified as design variables. These design variables can be varied to search for better design alternatives, for example, safer waterslides. A set of coupled differential equations based on Lagrange’s equations of motion that describe the motion of the riding object are derived. The equations of motion incorporate friction forces between the riding object and the surface of the flume sections. These second-order differential equations are then solved using Mathematica. Based on the equations of motion and design variables identified, a set of differential equations are derived for calculating shape DSA coefficients. These equations are solved numerically again using Mathematica. The major contribution of the paper are (1) extending waterslide design parameterization and shape DSA computation to true CAD-based flume sections, which greatly alleviates the design for manufacturing issues previously encountered, (2) incorporating friction forces into shape DSA computation, and (3) developing a design scenario that includes sensitivity display and what-if studies for a compromised design that is safer and with a larger acceleration, therefore, higher excitement levels. Incorporating friction forces into the computation supports design for rider’s excitement levels, which are related to accelerations. Waterslide design will not be realistic without including friction forces.     

On the mass and momentum transport in the Navier–Stokes slip layer

The Navier–Stokes slip boundary conditions are considered as conditions following from the mass and momentum balances within a thin, shell-like moving boundary layer. A problem of consistency between different models, that describes the internal and external friction in a viscous fluid, is stated within the framework of a proper form of the layer momentum balance. Appropriate constitutive equations for friction forces are formulated. The common features of the Navier, Stokes, Reynolds, and Maxwell concepts of a boundary slip layer are revalorized and discussed. Different mobility mechanisms connected with the transpiration phenomena, important for flows in micro- and nanochannels, are classified as a part of equations for the external friction.     

A constrained motion perspective of railway vehicles collision

Impacts, friction, and normal contact forces occur in the railway vehicles couplers. This paper presents a novel nonsmooth model of the train collision; until now, only penalized models have been approached. The train dynamics is described by an equality of measures formulated at the velocity level. The equations of motion are integrated using the Moreau time-stepping algorithm. Impulsive and normal contact forces are described by a set-valued law of Signorini type, while friction forces are described by a set-valued law of Coulomb type. The constrained forces are computed deducing a particular, simplified formulation of the Udwadia–Kalaba equations. The resulting algorithm is simple and straightforward. Both impulsive and nonimpulsive dynamics are casted in the same framework. Any feature or situation regarding train collisions may be modeled. A demonstrative application is presented. Simulations reveal nonsmooth phenomena like simultaneous multiple collisions, stick-slip, captures, and offset in the final equilibrium position.     

基于Karnopp摩擦的柔性滑移铰的建模与仿真

对含Karnopp摩擦的柔性滑移铰系统进行动力学建模和仿真.将滑移铰中的滑块视为柔性体,滑道视为刚性接触面,考虑滑道与滑块之间的间隙.由于柔性滑块与滑道的接触状态和摩擦情况比较复杂,采用有限元方法建立了柔性滑块的力学模型,基于罚函数方法建立含Karnopp摩擦柔性滑移铰接触力模型,通过试算迭代法判断柔性滑块各节点的接触状态,基于KED方法和Newmark方法给出了含该滑移铰机械系统动力学方程的数值算法.最后,以含Karnopp摩擦的柔性滑移铰和驱动摆杆构成的机械系统为例进行动力学仿真,分析了其动力学特性,验证了本文给出的方法的有效性.     

Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints

A computational methodology for analysis of spatial flexible multibody systems, considering the effects of the clearances and lubrication in the system spherical joints, is presented. The dry contact forces are evaluated through a Hertzian-based contact law, which includes a damping term representing the energy dissipation. The frictional forces are evaluated using a modified Coulomb’s friction law. In the case of lubricated joints, the resulting lubricant forces are derived from the corresponding Reynolds’ equation. An absolute nodal formulation is utilized in flexible body formulation. The generalized-α method is used to solve the resulting equations of motion. The effectiveness of the methodology is demonstrated by two numerical examples.