Steady-state equilibrium analysis of a multibody system driven by constant generalized speeds

A formulation which seeks steady-state equilibrium positions of constrained multibody systems driven by constant generalized speeds is presented in this paper. Since the relative coordinates are employed, constraint equations at cut joints are incorporated into the formulation. To obtain the steady-state equilibrium position of a multibody system, nonlinear equations are derived and solved iteratively. The nonlinear equations consist of the force equilibrium equations and the kinematic constraint equations. To verify the effectiveness of the proposed formulation, two numerical examples are solved and the results are compared with those of a commercial program.     

多体动力学的休斯敦方法及其发展

介绍多体动力学休斯敦方法的核心内容（含低序体阵列、变换矩阵、广义坐标及其导数、运动参数计算和动力学方程等）及其发展,即柔性体的有限段方法及综合模态分析方法。将变形表示为二阶小量形式,基于小变形原理,适时进行线性化,以获取动力刚化项和一致线性化动力学方程。     

Numerical integration scheme to reduce the errors in the satisfaction of constrained dynamic equation

Assuming that the existence of the constraints yields the change of the inertia force, this study derives the time-varying mass matrix for describing the constrained dynamic equation. It is displayed that the results corresponds with the ones by Udwadia and Kalaba. The numerical results obtained by integrating the constrained dynamic equation of second-order differential equations yield the errors in the satisfaction of the constraints. Modifying the derived dynamic equation this study presents a numerical algorithm to reduce the errors and to compute more precise motion. It is illustrated that the proposed method can be more precisely utilized in constrained mechanical systems through two applications of constrained nonlinear robotic systems.     

Effect of manufacturing tolerances on dynamic equilibria of multibody systems undergoing prescribed rotational motion

A general formulation is developed for the tolerance analysis of dynamic equilibria in a multibody system undergoing prescribed rotational motion, with applications including robots, spacecraft, propulsion and power generation systems, and sensors and actuators. In a state of dynamic equilibrium, a subset of the generalized coordinates assumes constant values while the remaining coordinates vary and respond in time. Manufacturing tolerances can be mathematically represented by probabilistic distributions or statistical variables through either an analytical approach or a Monte Carlo simulation. In the present tolerance work, the tolerances of design parameters including lengths, stiffnesses, inertias, and attachment positions are examined. In order to analytically calculate the statistical response of the dynamic equilibrium positions to such tolerances, the first-order sensitivities of the equilibria with respect to parameters are calculated. To illustrate the method’s accuracy and computational efficiency, two numerical examples are considered, and the statistical results obtained analytically for the equilibria are compared with those calculated through Monte Carlo simulation. In some cases, an equilibrium configuration can have an operating condition for which the response has zero standard deviation to perturbations of a design parameter. That condition can be a useful design point to the extent that typical manufacturing tolerances or other sources of variation would have no effect on the dynamic equilibrium configuration.     

闭环柔性多体系统的多点撞击问题

用于结构方法和单向递推组集方法研究闭环的柔性多体系统的多点撞击问题。在撞击过程中,在切割铰中增加撞击铰,导出了以铰坐标和柔性体模态坐标为系统广义坐标的闭环柔性多体系统在撞击阶段的动力学方程。利用相对坐标下约束方程的Ｌａｇｒａｎｇｅ乘子和利用绝对坐标下约束方程的Ｌａ－ｇｒａｎｇｅ乘子相等的关系求出切割铰的约束力,即撞击力。提出了撞击铰的切换方法。计算了卫星太阳电池卫帆板展开锁定时帆板之间多个接触处撞     

计及动力刚化的柔体动力学

提出两种计及动力刚化影响的动力学建模方法；有限段方法和一致线性化动力学方法。分析了这两种动力不主动力刚化的机理，有限段方法将柔体动力学问题转化为带有柔性的多刚体体系统动力学问题，计及了几何非线性的影响，适合于解决梁式结构的动力学问题；一致线性化动力学方法将变形场描述成为变形广义坐标的非线性形式，在适当的阶段线性化，可得到一致线性化动力学方程，自然计及了动力刚化项，适合于柔体的小变形问题。     

多体系统动力学方程的基于离散零空间理论的Newmark积分算法

考虑到多体系统动力学方程的数值计算方法是系统动力学分析的基础,提出一种基于离散零空间理论的Newmark积分算法。应用Newmark积分框架对多体系统动力学方程在时域上进行离散,通过离散零空间矩阵消去动力学方程中的拉格朗日乘子项,建立离散的达朗贝尔动力学方程,采用局部结点参数化进一步获得降维的达朗贝尔动力学方程。以空间双摆为算例,数值结果表明:该算法能够在实现系统降维、提高计算效率的同时,进行有效的数值分析,约束违约很小。     

A study on numerical solution method for efficient dynamic analysis of constrained multibody systems

A new and efficient computational method for constrained multibody systems is proposed. In the proposed method, local parametrization method is employed to apply the same solution method for position, velocity, and acceleration analyses since the coefficient matrices for each analysis have an identical matrix pattern. The skyline solution method is used to overcome numerical inefficiency when solving large scaled equations. Also, subsystem mpartitioning method is derived systematically to perform parallel processing for real time simulation. To show the numerical accuracy and efficiency of the proposed method, three numerical problems are solved.     

Explicit motion of dynamic systems with position constraints

Although many methodologies exist for determining the constrained equations of motion, most of these methods depend on numerical approaches such as the Lagrange multiplier’s method expressed in differential/algebraic systems. In 1992, Udwadia and Kalaba proposed explicit equations of motion for constrained systems based on Gauss’s principle and elementary linear algebra without any multipliers or complicated intermediate processes. The generalized inverse method was the first work to present explicit equations of motion for constrained systems. However, numerical integration results of the equation of motion gradually veer away from the constraint equations with time. Thus, an objective of this study is to provide a numerical integration scheme, which modifies the generalized inverse method to reduce the errors. The modified equations of motion for constrained systems include the position constraints of index 3 systems and their first derivatives with respect to time in addition to their second derivatives with respect to time. The effectiveness of the proposed method is illustrated by numerical examples.     

Measurement and correlation of high frequency behaviors of a very flexible beam undergoing large deformation

A correlation method of high frequency behaviors of a very flexible beam undergoing large displacement is presented. The suggested method based on the experimental modal analysis leads to more accurate correlation results because it directly uses the modal parameters of each mode achieved from experiment. First, the modal testing and the parameter identification method are suggested for flexible multibody dynamics. Due to the flexibility of a very thin beam, traditional testing methods such as impact hammer or contact type accelerometer are not working well. The suggested measurement with high speed camera, even though the test beam is very flexible, is working well. Using measurements with a high speed camera, modal properties until the 5th mode are measured. And After measuring each damping ratio until the 5th mode, a generic damping model is constructed using inverse modal transformation technique. It’s very interesting that the modal transformation technique can be also applied even in the ANCF simulation which uses the global displacement and finite slope as the nodal coordinates. The results of experiment and simulation are compared until the 5th mode frequency, respectively, by using ANCF forced vibration analysis. Through comparison between numerical simulation and experiment, this study showed that the proposed generic damping matrix, modal testing and parameter identification method is very proper in flexible multibody dynamic problems undergoing large deformation.     

Estimation of muscle and joint forces in the human lower extremity during rising motion from a seated position

Biomechanical models are often employed to predict in vivo muscle or joint forces in the human body because measuring these forces is difficult. Even though the rising motion from a seated position frequently occurs in daily life and the force acting on the knee joints during the motion is important for aged or infirmed people, limited studies related to the motion have been conducted. This study aims to propose a numerical procedure to estimate the muscle and joint forces in the human lower extremity during rising motion from a seated position. The human lower extremity is idealized as a multibody system in which the Hill-type muscle force model is employed. The multibody system consists of four bodies (shank, thigh, pelvis, and upper body), three revolute joints, and ten forces. The motion of the multibody system is assumed constrained to the sagittal plane, and the muscles in the human lower extremity are idealized by nine action/reaction forces. The nine forces are determined by minimizing the metabolic energy, which is consumed during the rising motion. Metabolic energy consists of the energy consumed by heat generation of muscles and the mechanical work done by muscles. For the accuracy validation of the proposed estimation method, numerical results obtained with the proposed method are compared with existing experimental results.     

一种线性化圆平板挤压膜力模型

提出一种形式简单的线性化圆平板挤压膜力模型,用最小均方差方法得到等效的线性化挤压膜力的系数,用数值法研究了系统的响应。通过计算不同间隙时线性化系统和非线性系统的响应及其误差,对线性化模型在描述挤压膜系统时的可行性及近似程度做了分析研究。     

Sensitivity analysis approach to multibody systems described by natural coordinates

The classical natural coordinate modeling method which removes the Euler angles and Euler parameters from the governing equations is particularly suitable for the sensitivity analysis and optimization of multibody systems. However, the formulation has so many principles in choosing the generalized coordinates that it hinders the implementation of modeling automation, A first order direct sensitivity analysis approach to multibody systems formulated with novel natural coordinates is presented. Firstly, a new selection method for natural coordinate is developed. The method introduces 12 coordinates to describe the position and orientation of a spatial object. On the basis of the proposed natural coordinates, rigid constraint conditions, the basic constraint elements as well as the initial conditions for the governing equations are derived. Considering the characteristics of the governing equations, the newly proposed generalized-ct integration method is used and the corresponding algorithm flowchart is discussed. The objective function, the detailed analysis process of first order direct sensitivity analysis and related solving strategy are provided based on the previous modeling system Finally, in order to verify the validity and accuracy of the method presented, the sensitivity analysis of a planar spinner-slider mechanism and a spatial crank-slider mechanism are conducted. The test results agree well with that of the finite difference method, and the maximum absolute deviation of the results is less than 3%. The proposed approach is not only convenient for automatic modeling, but also helpful for the reduction of the complexity of sensitivity analysis, which provides a practical and effective way to obtain sensitivity for the optimization problems of multibody systems.     

作大范围运动柔性梁的一种碰撞动力学求解方法

研究作大范围运动柔性梁的碰撞动力学问题。针对梁碰撞前、碰撞过程、碰撞后三个阶段,分别建立各阶段的动力学方程。基于柔性多体系统刚柔耦合动力学理论,建立梁无碰撞时的刚柔耦合动力学方程。结合冲量动量法和接触约束法提出一种碰撞动力学求解方法,先以冲量动量法实现运动转换得到协调的碰撞初始条件,再以接触约束法求解碰撞过程。导出系统的刚柔耦合碰撞动力学方程,给出了接触、分离判据,实现三个不同阶段的转换与动力学求解。编制仿真软件,对实例进行动力学仿真,得到系统不同阶段的动力学响应。研究表明,所提碰撞求解方法可以较方便地用于计算柔性多体系统的碰撞问题。梁的柔性、大范围运动及碰撞效应相互耦合,碰撞行为对于柔性多体系统碰撞过程和碰撞后的全局动力学性态均产生较大的影响。     

弹性支承体与旋转叶片耦合系统的固有频率分析

应用多体系统动力学理论建立弹性支承体与旋转叶片耦合系统的动力学方程 ,在系统稳定运动状态下 ,对耦合系统的动力学方程进行线性化处理 ,然后计算其系统固有频率 ;分析弹性支承体变形对系统固有频率的影响     

Dynamic modeling, input-shaped maneuvering and vibration suppression of flexible body using quasi-coordinates and euler parameters

In dealing with the dynamics of a flexible body, the rigid-body motions and elastic vibrations are analyzed separately. However, rigidbody motions cause vibrations, and elastic vibrations affect rigid-body motions, indicating the inherent coupling between rigid-body motions and elastic vibrations. The coupled equations of motion for a flexible body can be derived by means of Lagrange’s equations in terms of quasi-coordinates. The resulting equations of motion are hybrid and nonlinear. This paper proposes a unified approach for the equations of motion for a flexible body based on the perturbation method and the Lagrange’s equations of motion in terms of quasicoordinates and Euler parameters to analyze a more general case maneuvering. The resulting equations consist of zero-order nonlinear equations of motion which depict rigid-body motions and first-order time-varying linear equations of motion which depict perturbed rigid-body motions and elastic vibration. Hence, the input-shaped maneuvering can be applied to the zero-order equation considering the induced vibrations. Since the input-shaped maneuvering alone cannot achieve vibration suppression, the vibration suppression controller combined with the input-shaped maneuvering is proposed in this study. As a numerical example, a hub with elastic appendages is considered. Numerical results show that the unified modeling approach proposed in this paper is effective in numerical simulation and control design.     

Structural and mechanical systems subjected to constraints

The characteristics of dynamic systems subjected to multiple linear constraints are determined by considering the constrained effects. Although there have been many researches to investigate the dynamic characteristics of constrained systems, most of them depend on numerical analysis like Lagrange multipliers method. In 1992, Udwadia and Kalaba presented an explicit form to describe the motion for constrained discrete systems. Starting from the method, this study determines the dynamic characteristics of the systems to have positive semidefinite mass matrix and the continuous systems. And this study presents a closed form to calculate frequency response matrix for constrained systems subjected to harmonic forces. The proposed methods that do not depend on any numerical schemes take more generalized forms than other research results.     

Constraint embedding in kinematics and dynamics of hybrid manipulator systems

An approach to kinematics and dynamics based on kinematic influence coefficient matrices is proposed for applications to the analysis and control of motion controlled multibody systems, e.g., hybrid robotic manipulator systems. The scheme is unique in a sense that all the kinematic constraints are completely embedded into the formulations at the kinematics level and equation of motion of the system is obtained in a closed form with respect to the minimal set of independent joint coordinates. Furthermore, all kinematic and dynamic formulations are expressed compactly in the same format by using two special algebraic operators. This isomorphic formalism allows systematic transformations of kinematic and dynamic informations between different sets of coordinates in a purely algebraic way.     

Dynamic analysis of harmonically excited non-linear structure system using harmonic balance method

An analytical method is presented for evaluation of the steady state periodic behavior of nonlinear structural systems. This method is based on the substructure synthesis formulation and a harmonic balance procedure, which is applied to the analysis of nonlinear responses. A complex nonlinear system is divided into substructures, of which equations are approximately transformed to modal coordinates including nonlinear term under the reasonable procedure. Then, the equations are synthesized into the overall system and the nonlinear solution for the system is obtained. Based on the harmonic balance method, the proposed procedure reduces the size of large degrees-of-freedom problem in the solving nonlinear equations. Feasibility and advantages of the proposed method are illustrated using the study of the nonlinear rotating machine system as a large mechanical structure system. Results obtained are reported to be an efficient approach with respect to nonlinear response prediction when compared with other conventional methods.     

气膜端面密封角向摆动自振稳定性

提出造成气膜端面密封角向摆动自振的内在机理，既有半频摆动自振又有角向气锤自振。在其稳定性分析中给出了不同端面结构、浮环支撑弹簧和二次密封阻尼等的影响。分析按照小扰动线性化的分布参数法，联立气膜微扰雷诺方程和浮环微扰运动方程，对密封系统的角向摆动自振稳定性界限进行了数值迭代。