A general method for impact dynamic analysis of a planar multi-body system with a rolling ball bearing joint

Impact affects the dynamic characteristics of mechanical multi-body systems and damages those rotating parts, such as the joint rolling element bearings, which are high-precision, defect intolerant components. Based on multi-body dynamic theory, Hertzian contact theory, and a continuous contact model, this study proposed a modelling method that can describe the dynamic behaviour of planar mechanical multi-body systems containing a rolling ball bearing joint under impact. In this method, the rigid bodies and bearing joint were connected according to their joint force constraints; the impact constraint between the multi-body system and the target rigid body was constructed using a continuous contact force model. Based on this method, the reflection relationship between the external impacts of the mechanical multi-body system and the variation law governing the dynamic load on the rolling bearing joint were revealed. Subsequently, an impact multi-body system, which was composed of a sliding–crank mechanism containing a rolling ball bearing joint and the target rigid body with an elastic support, was analysed to explore the dynamic response of such a complex discontinuous dynamic system andthe relevant relationship governing the dynamic load on the rolling bearing joint. In addition, a multi-body dynamic simulation software was used to build a virtual prototype of the impact slider–crank system. Compared with the theoretical model, the prototype had an additional deep groove ball bearing. That is to say, the prototype model took account of the specific geometric structural characteristics and the complex contact relationship of the inner and outer races, rolling balls, and bearing cage. Finally, the effectiveness of the theoretical method proposed in this study was verified by comparative analysis of the results. The results suggested that the external impact of a mechanical multi-body system was prone to induce sudden changes in the equivalent reaction force on its bearing joint and the dynamic load carried on its rolling balls. This study provided an effective method for exploring the distribution characteristics of dynamic loads on rolling ball bearing joints under working impact load conditions. Moreover, it offered support for the parameter optimisation of geometric structure, performance evaluation, and dynamic design of the rolling ball bearings.     

Modal test and analysis of cantilever beam with tip mass

The phenomenon of dynamic stiffening is a research field of general interest for flexible multi-body systems. In fact, there are not only dynamic stiffening but also dynamic softening phenomenon in the flexible multi-body systems. In this paper, a non-linear dynamic model and its linearization characteristic equations of a cantilever beam with tip mass in the centrifugal field are established by adopting the general Hamilton Variational Principle. Then, the problems of the dynamic stiffening and the dynamic softening are studied by using numerical simulations. Meanwhile, the modal test is carried out on our centrifuge. The numerical results show that the system stiffness will be strengthened when the centrifugal tension force acts on the beam (i.e. the dynamic stiffening). However, the system stiffness will be weakened when the centrifugal compression force acts on the beam (i.e. the dynamic softening). Furthermore, the equilibrium position of the system will lose its stability when the inertial force reaches a critical value. Through theoretical analysis, we find that this phenomenon comes from the effect of dynamic softening resulting from the centrifugal compression force. Our test results verify the above conclusions and confirm that both dynamic stiffening and softening phenomena exist in flexible multi-body systems. The project supported by the National Natural Science Foundation of China (19972002) and the Doctoral Programme from The State Education Commission China (20010001011)     

Parameterisation,validation and implementation of an all-terrain SUV FTire tyre model

This paper focuses on the parameterisation, validation and implementation of an FTire model of a Michelin LTX A/T² 235/85R16 tyre. This tyre is designed for both on- and off-road use and is commonly used on all wheel drive SUVs. Quasi-static laboratory and dynamic field tests were conducted to acquire parameterisation and validation test data for the FTire model. Quasi-static parameterisation tests include acquiring vertical tyre stiffness over a flat plate and cleats, tyre footprint sizes and shapes, longitudinal, lateral and torsional tyre stiffness for various tyre normal loads, as well as vibrational tyre responses. Dynamic parameterisation tests include dynamic cleat test data. An Adams model of the tyre testing equipment is implemented to simulate the FTire model and validate it against dynamic validation test results. Finally, the model is implemented on a fully nonlinear multi-body dynamics model of a Land Rover Defender. It is found that the FTire model is able to predict the lateral tyre behaviour well on a smooth road surface. The vertical and longitudinal tyre behaviour on a smooth road surface and on a rough surface are predicted accurately.     

Rigid rotor dynamic stability using Floquet theory

The dynamic behaviour of a rigid rotor elastically connected by a constant speed joint is investigated. The effects of inertial inequalities and stiffness inequalities are evaluated and the stability characteristics are worked out. The Floquet theory is extensively applied to this system in order to obtain the stability limit curves. Comparisons with experimental results obtained by other authors, available in the literature, are reported in order to verify the procedure. Results concerning the unsymmetrical rotors are widely shown through parameters values restricted to realistic cases.     

A novel method for slant crack detection in rotors based on turning in two directions

This paper presents a novel way to detect fatigue slant cracks in rotors based on theoretical discussion. Hence, the dynamic behaviour of a Jeffcott rotor system with a mid-span slant crack under arbitrary crack orientations is studied. First, using concepts of fracture mechanics, the flexibility matrix and subsequently the system’s stiffness matrix are calculated. A symmetric relation for a global stiffness matrix is presented and proved. Next, the motion equations of the system that are obtained in four directions, two transverse, one torsional and one longitudinal, are solved using the Runge–Kutta numerical method. The characteristics of crack orientations for angles greater than 90° (transverse crack) are investigated in detail and their influence on the elements of the crack compliance matrix is presented. Also, slant crack characteristics with complementary angles are compared to each other. It is shown that the difference between cracked systems with complementary angles is only in 3rd row (3rd column) of the crack compliance matrix, and also it is shown that due to the presence of a slant crack, the system responses in forward and backward motion are different. Using the frequency responses of the shaft obtained, a technique to detect the existence of slant cracks on the shaft was proposed. This novel method is a simple way that can be used for slant crack detection in rotors.     

海洋核动力平台定位系统多体动力学建模与分析

海洋核动力平台可以为海上油气开发、偏远岛屿和海水淡化等提供稳定的能源供给, 是具有潜力的重要海洋装备. 定位系统作为核动力平台的核心部位, 主要由单点转塔、YOKE刚臂、系泊腿和系泊支架组成, 属于典型的多刚体动力学系统. 对定位系统进行多体动力学分析可以提高核动力平台长期作业的可靠性. 基于多刚体动力学理论, 结合定位系统多铰连接的拓扑结构, 建立了定位系统的多体动力仿真模型; 进而考虑核动力平台作业海洋环境, 通过谱分析与线性叠加原理得到核动力平台在多年重现期下的六自由度运动时程. 以我国首座海洋核动力平台为例, 利用多体动力学模型分别计算了一年一遇、十年一遇和百年一遇海况下定位系统的系泊回复力与连接结构受力行为. 计算结果与准静力学模型、Kane动力学模型进行对比, 给出了定位系统的系泊回复刚度曲线, 并提出了系泊回复力动力放大系数. 研究可为定位系统系泊能力评估和各连接结构受力分析提供参考和借鉴.      

Dynamic characteristic and stability analysis of a beam mounted on a moving rigid body

The phenomenon of dynamic stiffening has drawn general interest in flexible multi-body systems. In fact, approximately analytical, numerical and experimental research have proved that both dynamic stiffening and dynamic softening can occur in flexible multi-body systems. In this paper, the nonlinear dynamic model of a beam mounted on both the exterior and the interior of a rigid ring is established by adopting the general Hamiltons variational principle. The dynamic characteristics of the system are studied using a theoretical method when the rigid ring translates with constant acceleration or rotates steadily. The research proves theoretically that both dynamic stiffening and dynamic softening can occur in both the translation as well as the rotation state of multi-body systems. Furthermore, the approximate vibration frequency, critical value and post-buckling equilibria of the translational beam with constant acceleration are obtained by employing the assumed modes method, which validates the theoretical results. The L² norm stability of the trivial equilibrium of the system with the external beam and the L norm stability of the bending of the external beam are proved by employing the energy–momentum method.This research was supported by the National Natural Science Foundation of China (10272002) and the Doctoral Program from the Ministry of Education of China (20020001032).     

Contact-impact formulation for multi-body systems using component mode synthesis

The efficiency and accuracy are two most concerned issues in the modeling and simulation of multi-body systems involving contact and impact. This paper proposed a formulation based on the component mode synthesis method for planar contact problems of flexible multi-body systems. A flexible body is divided into two parts： a contact zone and an un-contact zone. For the un-contact zone, by using the fixed-interface substructure method as reference, a few low-order modal coordinates are used to replace the nodal coordinates of the nodes, and meanwhile, the nodal coordinates of the local impact region are kept unchanged, therefore the total degrees of freedom （DOFs） are greatly cut down and the computational cost of the simulation is significantly reduced. By using additional constraint method, the impact constraint equations and kinematic constraint equations are derived, and the Lagrange equations of the first kind of flexible multi-body system are obtained. The impact of an elastic beam with a fixed half disk is simulated to verify the efficiency and accuracy of this method.     

Gauge principle and variational formulation for flows of an ideal fluid

A gauge principle is applied to mass flows of an ideal compressible fluid subject to Galilei transformation. A free-field Lagrangian defined at the outset is invariant with respeet to global SO(3) gauge transformations as well as Galilei transformations. The action principle leads to the equation of potential flows under constraint of a continuity equation. However, the irrotational flow is not invariant with respect to local SO(3) gauge transformations. According to the gauge principle, a gauge-covariant derivative is defined by introducing a new gauge field. Galilei invariance of the derivative requires the gauge field to coincide with the vorticity, i.e. the curl of the velocity field. A full gauge-covariant variational formulation is proposed on the basis of the Hamilton‘‘s principle and an assoicated Lagrangian. By means of an isentropic material variation taking into account individual particle motion, the Euler‘‘s equation of motion is derived for isentropic flows by using the covariant derivative. Noether‘‘s law associated with global SO(3) gauge invariance leads to the conservation of total angular momentum. In addition, the Lagrangian has a local symmetry of particle permutation which results in local conservation law equivalent to the vorticity equation.     

多体系统动力学方程违约修正的数值计算方法

多体系统动力学方程为微分代数方程,一般将其转化成常微分方程组进行数值计算,在数值积分的过程中约束方程的违约会逐渐增大.本文对具有完整、定常约束的多体系统,在修改的带乘子Lagrange正则形式的方程的基础上,根据Baumgarte提出的违约修正的方法,给出了一种多体系统微分代数方程违约修正法和系统的动力学方程的矩阵表达式.通过对曲柄-滑块机构的数值仿真,计算结果表明本文给出的方法在计算精度和计算效率上好于Baumgarte提出的两种违约修正的方法.     

整车多体动力学模型验证

迫切需要建立一个通用的汽车仿真开发平台为多种电子控制系统提供统一开发环境.采用Roberson和Wittenberg (R-W)多体动力学理论,建立了自由度(DOF)包含底盘单项和集成控制所需运动参数的整车模型,分别进行了动力学仿真和试验对比验证,结果表明模型的纵向、侧向和垂向动力学性能的正确性.     

木卫停泊轨道间低耗能小推力转移轨道设计方法研究

对木卫停泊轨道间的低耗能小推力转移轨道设计方法进行了研究,提出基于“类halo轨道截面”法的低耗能转移轨道参数化方法和基于配点法的多体Lambert问题求解算法,并利用全局优化算法得出了燃耗最少的初步优化结果;利用多体同伦法和固定近心点高度的多圈转移控制律得到了各段小推力转移轨道的有效设计结果.所提方法同样适用于其他天体间的转移轨道设计,为多体环境下低耗能小推力转移轨道提出了新的设计思路和方法.     

多柔体系统碰撞动力学研究综述

多柔体系统碰撞动力学研究具有重要的研究价值和工程实际意义,本文针对多柔体系统碰撞动力学研究中的几个基本问题进行了全面的分析和评述,其中包括多柔体系统动力学方程的描述、碰撞模型的建立、铰接间隙引起的碰撞问题、数值算法、实验研究、控制等几个方面,并根据目前的发展现状和研究中存在的问题,指出了今后多柔体系统动力学碰撞研究中的发展方向      

A multi-body optimization framework with a knee kinematic model including articular contacts and ligaments

Multi-body optimization is one of the methods proposed to reduce the errors due to soft-tissue artifact in gait analysis based on skin markers. This method uses a multi-body kinematic model driven by the marker trajectories. The kinematic models developed so far for the knee joint include a lower pair (such as a hinge or a spherical joint) or more anatomical and physiological representations including articular contacts and the main ligaments. This latter method allows a better representation of the joint constraints of a subject, potentially improving the kinematic and the subsequent static and dynamic analyses, but model definition and mathematical implementation can be more complicated. This study presents a mathematical framework to implement a kinematic model of the knee featuring articular contacts and ligaments in the multi-body optimization. Two penalty-based methods (minimized and prescribed ligament length variations) consider deformable ligaments and are compared to a further method (zero ligament length variation) featuring isometric ligaments. The multi-body optimization is performed on one gait cycle for five asymptomatic male subjects by means of a lower limb model including the foot, shank, thigh and pelvis. The mean knee kinematics, ligament lengthening and contact point positions are compared over the three methods. The results are also consistent with results from the literature obtained by bone pins or biplanar fluoroscopy. Finally, a sensitivity analysis is performed to evaluate how the joint kinematics is affected by the weights used in the penalty-based methods. The approach is purely kinematic, since the penalty-based framework does not require the solution of the joint static or dynamic analyses and makes it possible to consider ligament deformations without the definition of ligament stiffness that generally cannot be identified through in vivo measurements. Nevertheless, as far as a knee kinematic model is concerned, particularly in musculoskeletal modeling, this approach appears to be a good compromise between standard non-physiological kinematic models and complex deformable dynamic models.     

Dynamics and stability of turbocharger rotors

The paper discusses the bifurcation and stability behavior of (automotive) turbochargers with full-floating ring bearings. Turbocharger rotors exhibit a highly nonlinear behavior due to the nonlinearities introduced by the floating ring bearings. A flexible multibody model of the rotor/bearing system is presented. Numerical run-up simulations are compared with corresponding test rig measurements. The nonlinear oscillation effects are thoroughly investigated by means of simulated and measured rotor vibrations. The influence of various system parameters on the bifurcation behavior of the rotor/bearing system is analyzed. The article examines rotors supported in full-floating ring bearings with plain circular bearing geometry in the inner and outer oil gap. By recapitulating the well-known oil whirl and oil whip phenomena for single and double oil film bearings, the paper gives an overview on the fundamental dynamic effects occurring in turbocharger systems.     

多体系统接触碰撞问题的牛顿-欧拉线性互补方法

基于非光滑动力学方法的多体系统接触碰撞分析是目前多体系统动力学的研究热点.本文采用牛顿-欧拉方法建立多体系统接触、碰撞问题的动力学模型,给出一种牛顿-欧拉型线性互补公式.该建模方法与目前一般采用的拉格朗日建模方法的不同之处是约束条件中除了库仑摩擦、单边约束之外还含有光滑等式约束.在建立系统动力学模型时,首先解除摩擦约束和单边约束得到原系统对应的基本系统.牛顿-欧拉方法采用最大数目坐标建立基本系统的动力学方程,由于坐标不相互独立,因此基本系统中带有等式约束,其数学模型为一组微分代数方程.借助约束雅可比矩阵,在基本系统微分代数方程中添加摩擦接触和单边约束对应的拉氏乘子,就可以得到系统全局运动的具有变拓扑结构特征的动力学方程,再结合非光滑约束互补条件便可构成完备的系统动力学模型.完备的动力学模型由动力学微分方程以及等式约束和不等式约束组成.线性互补公式采用分块矩阵形式进行推导,简化了推导过程.数值计算采用基于线性互补的时间步进算法.时间步进算法是目前流行的非光滑数值算法,其突出特点是可以免去数值积分中繁琐的事件检测过程,而数值积分过程中通过对线性互补问题的求解可以确定系统的触-离状态.通过对典型的曲柄滑块间隙机构进行数值分析,验证本文方法的有效性.     

客车车身骨架动应力研究的现状与未来

综述了多体系统动态模型的发展及动态应力历程求解的进展,给出了国际上的最新研究成果,并以CJ6121GCHK型客车车骨架为例,给出了计算其动应力历程的步骤。     

柔性曲梁多体系统的研究现状和展望

本文对近几年来柔性多体系统建模理论的研究进展进行了评述, 详细阐述了曲梁结构多体系统动力学研究的理论背景和工程意义. 结合连续介质力学和多体系统动力学理论, 介绍了平面曲梁、空间曲梁应变位移场的描述以及几何非线性问题研究现状. 通过国内外文献及已有的研究, 综述了现有曲梁的离散化方法, 以及这些方法对于定曲率、变曲率曲梁的适用性. 然后, 总结了曲梁刚柔耦合动力学建模和数值计算中的难点问题, 介绍了平面曲梁、空间曲梁实验研究的现状. 最后, 综合全文, 提出了目前研究存在的难点问题, 以及计划解决这些难点问题的方法.      

Numerical simulation of Gerotor pumps considering rotor micro-motions

Gerotor units are widely used in low-pressure (up to 30 bar) fluid power applications, injection as well as lubrication systems, due to their compact package and low cost. Their performance in terms of volumetric efficiency, flow pulsations, internal pressure peaks or localized cavitation depends on many parameters, such as the rotors’ profiles and the manufacturing tolerances. This paper proposes a multi-domain simulation approach for the numerical analysis of the performance of Gerotor units. Characterized by simulation swiftness, the model can be used for virtual prototyping of units considering the actual geometry of the rotors, their geometrical tolerances and the properties of the working fluid. The approach is based on the coupling of different models: a numerical geometric model evaluates the instantaneous volumes and flow areas inside the unit; a lumped parameter fluid dynamic model describes the displacing process of the tooth space volumes; finally, a mechanical model evaluates the internal micro-motions of the rotors axes according to their tolerances. In this way, the model determines the actual loading of the rotors, considering also the actual location of the points of contact. After presenting the approach, the paper describes the potentials of the proposed method with reference to a particular Gerotor pump design. Specific experiments were performed within this research to permit a detailed model validation, and comparisons in terms of significant steady-state as well as transient pressure and flow features are presented. The approach used in the current paper can be considered valuable when studying the impact of real-life technological clearances on the fluid-dynamic performance of the pump.     

Implementation of 3-D isoparametric finite elements on supercomputer for the formulation of recursive dynamical equations of multi-body systems

Kinematic formulation of the versatile three-dimensional isoparametric eight-noded brick element with six degrees of freedom at each node (three-translational and three-rotational), suitable for the discretization of flexible bodies with intricate geometric configurations, has been developed and implemented on the supercomputer IBM-3090 for the simulation of dynamical mechanical systems. The pipelining feature of the above vector-processor has been exploited for achieving a significant order of magnitude in computational efficiency. The concepts of indexed reference arrays have been utilised in the development of dynamical equations of motion, eliminating expensive Boolean matrix multiplication operations. The algorithm developed is an improvement and extension of [7], with the implementation of the brick element formulation. The recursive Kane's equations, modal analysis technique and strain energy principles are integrated into the procedure. The above technique is also applied to the constrained multi-body systems. An illustrative example of an spin-up maneuver of a space robot with three flexible links carrying a solar panel is presented. The prediction of dynamic behaviour of the system is carried out under a constrained environment and the effects of geometric stiffening and its subsequent restoring elastic forces are demonstrated.