Flexible Bodies in Multibody System Codes

SUMMARY

Multibody codes are'efficient tools to simulate nonlinear dynamic behaviour of rigid and flexible multibody systems undergoing large overall motions overlaid by small elastic deformation. This paper gives an overview of common approaches for the equations of motion of flexible body models and presents a general way to prepare the required data including geometric stiffening terms. In particularly, for the nodal approach the data are derived using standard results of a finite element analysis of the body. The computation of coefficient matrices describing the equations of motion is done outside the finite element code by matrix manipulations only. The data are stored in a standardised object-oriented structure. Consequently, the data set is independent of the formulation of the multibody system code.     

Dynamic and Vibration Analysis of a Vehicle Rear Axle System

A detailed finite element model for the rear axle system of a sport utility vehicle is developed in this investigation. The axle system is treated as a multibody system that consists of nine bodies that include the input shaft, two output shafts, the carrier and tube system, four control arms and a track bar. The rotating input and output shafts are mounted on the carrier and tube system using six bearings. The four control arms and the track bar are connected to the carrier system and the frame of the vehicle using rubber bushings. In the model developed in this investigation, three dimensional beam elements are used to develop the finite element model for the input and output axle shafts, the control arms, and the track bar. A non-conventional finite element formulation is used to develop the equations of motion of the rotating input and output shafts in order to account for the effect of their angular velocities. These equations are expressed in terms of inertia shape integrals that depend on the assumed displacement field. The inertia shape integrals are first evaluated for each finite element. The inertia shape integrals of the rotating shafts are obtained by assembling the inertia shape integrals of its finite elements using a standard finite element assembly procedure. A conventional finite element formulation is used for the control arms and the track bar. The model developed in this investigation includes the effect of the bearing stiffness, the effect of the stiffness of the helical springs of the suspension system, and the effect of the stiffness of the tires. Using the Lagrangian dynamics and the finite element method, the equations of motion of the axle system are developed and expressed in terms of the nodal coordinates of the shafts, the control arms and the track bar as well as the degrees of freedom of the carrier. This finite dimensional model is used to determine the mode shapes and the natural frequencies of the axle system. The discrepancies between several of the natural frequencies predicted using the dynamic model developed in this investigation and natural frequencies determined experimentally are found to be less than 2%. A parametric study is performed in order to investigate the effect of the axle system parameters on the natural frequencies and mode shapes. Using the modal transformation, a set of differential equations of motion of the axle system is developed and used to examine the system dynamics under given loading conditions. The solutions of the resulting equations of motion are obtained using numerical methods.     

Dynamic analysis of a pantograph-catenary system using absolute nodal coordinates

The dynamic interaction between the catenary and the pantographs of high-speed trains is a very important factor that affects the stable electric power supply. In order to design a reliable current collection system, a multibody simulation model can provide an efficient and economical method to analyze the dynamic behavior of the catenary and pantograph. In this article, a dynamic analysis method for a pantograph-catenary system for a high-speed train is presented, employing absolute nodal coordinates and rigid body reference coordinates. The highly flexible catenary is modeled using a nonlinear continuous beam element, which is based on an absolute nodal coordinate formulation. The pantograph is modeled as a rigid multibody system. The analysis results are compared with experimental data obtained from a running high-speed train. In addition, using a derived system equation of motion, the calculation method for the dynamic stress in the catenary conductor is presented. This study may have significance in providing an example that a structural and multibody dynamics model can be unified into one numerical system.     

Dynamic and Vibration Analysis of a Vehicle Rear Axle System

A detailed finite element model for the rear axle system of a sport utility vehicle is developed in this investigation. The axle system is treated as a multibody system that consists of nine bodies that include the input shaft, two output shafts, the carrier and tube system, four control arms and a track bar. The rotating input and output shafts are mounted on the carrier and tube system using six bearings. The four control arms and the track bar are connected to the carrier system and the frame of the vehicle using rubber bushings. In the model developed in this investigation, three dimensional beam elements are used to develop the finite element model for the input and output axle shafts, the control arms, and the track bar. A non-conventional finite element formulation is used to develop the equations of motion of the rotating input and output shafts in order to account for the effect of their angular velocities. These equations are expressed in terms of inertia shape integrals that depend on the assumed displacement field. The inertia shape integrals are first evaluated for each finite element. The inertia shape integrals of the rotating shafts are obtained by assembling the inertia shape integrals of its finite elements using a standard finite element assembly procedure. A conventional finite element formulation is used for the control arms and the track bar. The model developed in this investigation includes the effect of the bearing stiffness, the effect of the stiffness of the helical springs of the suspension system, and the effect of the stiffness of the tires. Using the Lagrangian dynamics and the finite element method, the equations of motion of the axle system are developed and expressed in terms of the nodal coordinates of the shafts, the control arms and the track bar as well as the degrees of freedom of the carrier. This finite dimensional model is used to determine the mode shapes and the natural frequencies of the axle system. The discrepancies between several of the natural frequencies predicted using the dynamic model developed in this investigation and natural frequencies determined experimentally are found to be less than 2%. A parametric study is performed in order to investigate the effect of the axle system parameters on the natural frequencies and mode shapes. Using the modal transformation, a set of differential equations of motion of the axle system is developed and used to examine the system dynamics under given loading conditions. The solutions of the resulting equations of motion are obtained using numerical methods.     

基于多体动力学的柴油机曲轴疲劳寿命分析

为分析4100QBZL柴油机曲轴的疲劳寿命,建立该曲柄连杆机构的刚柔耦合多体动力学模型,将多组试验测量的缸内压力作为驱动力,进行耦合仿真得到曲轴在柔性体模型下的主轴颈、连杆轴颈负荷仿真结果,并根据载荷结果对曲轴进行静强度校核。最后结合由多体动力学软件得到的载荷谱与有限元分析所得的曲轴在各个工况下的应力应变分析结果,以及通过材料的各项属性拟合出的S-N曲线,对曲轴进行了疲劳寿命预测。结果表明:曲轴的静强度及疲劳寿命均达到了工程设计要求,曲轴最危险部位的寿命次数也达到了1013以上,认为曲轴不会发生疲劳破坏。     

Vertical Vibration Analysis of Vehicle/Imperfect Track Systems

Summary This paper studies the vertical vibration of a vehicle traveling on an imperfect track system. The car body and sleepers are modeled as Timoshenko beams with finite length, and the rail is assumed as an infinite Timoshenko beam with discrete supports. Imperfection of the track system comes from a sleeper lost partial support by the ballast. Since deflection of the rail is limited within a certain interval where the vehicle is passing over, the infinite domain problem can be transformed into a finite domain problem with moving boundary. In this work, the equations of motion of the car body, rail and sleepers are discretized first by the finite element method. The discretized equations of motion for the vehicle and track systems are then assembled, respectively. Finally, the Newmark method is applied to obtain the response of the vehicle and track systems at each time step. The effect of the vehicle speed on the response of the vehicle and track systems is investigated.     

Influence of the aerodynamic forces on the pantograph–catenary system for high-speed trains

Most of the high-speed trains in operation today have the electrical power supply delivered through the pantograph–catenary system. The understanding of the dynamics of this system is fundamental since it contributes to decrease the number of incidents related to these components, to reduce the maintenance and to improve interoperability. From the mechanical point of view, the most important feature of the pantograph–catenary system consists in the quality of the contact between the contact wire of the catenary and the contact strips of the pantograph. The catenary is represented by a finite element model, whereas the pantograph is described by a detailed multibody model, analysed through two independent codes in a co-simulation environment. A computational procedure ensuring the efficient communication between the multibody and finite element codes, through shared computer memory, and suitable contact force models were developed. The models presented here are contributions for the identification of the dynamic behaviour of the pantograph and of the interaction phenomena in the pantograph–catenary system of high-speed trains due to the action of aerodynamics forces. The wind forces are applied on the catenary by distributing them on the finite element mesh. Since the multibody formulation does not include explicitly the geometric information of the bodies, the wind field forces are applied to each body of the pantograph as time-dependent nonlinear external forces. These wind forces can be characterised either by using computational fluid dynamics or experimental testing in a wind tunnel. The proposed methodologies are demonstrated by the application to real operation scenarios for high-speed trains, with the purpose of defining service limitations based on train and wind speed combination.     

Simulation of Multibody Vehicles Moving over Elastic Guideways1

This paper describes the present state of a general purpose computer program for calculating the dynamic response of vehicles travelling over guideways which may be elastic.

The linearized state-equations of motion for general multibody vehicles are constructed automatically by the program, these equations are supplemented by the equations for the active subsystems. Finally, the vehicle system equations are combined with the modal equations for elastic guideways and the complete set of coupled equations is solved simultaneously by numerical integration.     

Simulation of Multibody Vehicles Moving over Elastic Guideways1

SUMMARY

This paper describes the present state of a general purpose computer program for calculating the dynamic response of vehicles travelling over guideways which may be elastic.

The linearized state-equations of motion for general multibody vehicles are constructed automatically by the program, these equations are supplemented by the equations for the active subsystems. Finally, the vehicle system equations are combined with the modal equations for elastic guideways and the complete set of coupled equations is solved simultaneously by numerical integration.     

Fatigue Life Prediction of a Railway Bogie under Dynamic Loads through Simulation

For dynamically loaded lightweight structures fatigue strength is an important design criteria. In this paper a new method to predict fatigue lifetime is shown. This is based on the combination of frequency domain and time domain calculations, which allows lifetime prediction with reduced computational effort. The method is implemented to work in a concurrent engineering software environment together with a computer aided design (CAD), a finite-element-method (FEM) and a multibody system (MBS) program. The benefits of the new approach are demonstrated by application to the bogie of a freight locomotive. The dynamic loads acting on the bogie are computed by multibody simulation. The bogie frame is considered as an elastic body of the MBS and the highly nonlinear wheel rail contact is modeled quasi-elastically. For the ride on a straight track the equations of motion can be linearized and the covariance matrix of the loads is calculated. The ride through a ramp is simulated by using the nonlinear differential equations. FEM yields the stresses in the most stressed locations of the bogie depending on the loads calculated by multibody simulation. Based on these stresses the fatigue life prediction is carried out in the MBS post-processing program FATIGUE.     

Flexible multibody simulation of automotive systems with non-modal model reduction techniques

The stiffness of the body structure of an automobile has a strong relationship with its noise, vibration, and harshness (NVH) characteristics. In this paper, the effect of the stiffness of the body structure upon ride quality is discussed with flexible multibody dynamics. In flexible multibody simulation, the local elastic deformation of the vehicle has been described traditionally with modal shape functions. Recently, linear model reduction techniques from system dynamics and mathematics came into the focus to find more sophisticated elastic shape functions. In this work, the NVH-relevant states of a racing kart are simulated, whereas the elastic shape functions are calculated with modern model reduction techniques like moment matching by projection on Krylov-subspaces, singular value decomposition-based reduction techniques, and combinations of those. The whole elastic multibody vehicle model consisting of tyres, steering, axle, etc. is considered, and an excitation with a vibration characteristics in a wide frequency range is evaluated in this paper. The accuracy and the calculation performance of those modern model reduction techniques is investigated including a comparison of the modal reduction approach.     

索结构六节点等参元非线性有限元法

针对索结构几何非线性的特点,提出了一种六节点等参数单元有限元模型,采用5次多项式作为位移插值函数及单元初始形状函数,假定索是理想柔性,且满足胡克定律,基于修正的Lagrangian坐标描述法,建立了非线性有限元基本方程和切线刚度矩阵,利用Newton-Raphson法进行求解.     

步行车辆运动弹性动力学分析

步行车辆在快速运动中,既有大刚体运动又有小弹性位移,且还有不连续的边界约束条件,是一个很复杂的动力学问题。本文用有限元方法建立了该系统的动力学模型,采用分段的直接积分法求解其弹性动力响应。     

Modeling and Analysis of Nonlinear Multibody Systems

SUMMARY

The method of multibody systems is introduced for the modeling of nonlinear vehicle systems. The equations of motion and the equations of reaction are found. For the dynamical analysis a classification of nonlinear phenomena is presented using objective and subjective criteria. As an illustrative example the dynamical behaviour of a four-body pendulum is investigated.     

General Purpose Vehicle System Dynamics Software Based on Multibody Formalisms

This paper pursues two objectives: Firstly, to review the state-of-the-art of general purpose vehicle system dynamics software and secondly, to describe two representatives, the program MEDYNA and the program NEWEUL. The general modeling requirements for vehicle dynamics software, the multibody system approach and a comparative discussion of multibody software are given. The two programs NEWEUL and MEDYNA are described with respect to modeling options, computational methods, software engineering as well as their interfaces to other software. The applicability of these programs is demonstrated on two selected examples, one from road vehicle problems and the other from wheel/rail dynamics. It is concluded that general purpose software based on multibody formalisms will play the same role for mechanical systems, especially vehicle systems, as finite element methods play for elastic structures.     

General Purpose Vehicle System Dynamics Software Based on Multibody Formalisms

SUMMARY

This paper pursues two objectives: Firstly, to review the state-of-the-art of general purpose vehicle system dynamics software and secondly, to describe two representatives, the program MEDYNA and the program NEWEUL. The general modeling requirements for vehicle dynamics software, the multibody system approach and a comparative discussion of multibody software are given. The two programs NEWEUL and MEDYNA are described with respect to modeling options, computational methods, software engineering as well as their interfaces to other software. The applicability of these programs is demonstrated on two selected examples, one from road vehicle problems and the other from wheel/rail dynamics. It is concluded that general purpose software based on multibody formalisms will play the same role for mechanical systems, especially vehicle systems, as finite element methods play for elastic structures.     

PantoCat statement of method

The pantograph–catenary dynamic interaction analysis program (PantoCat) addresses the need for a dynamic analysis code able to analyse models of the complete overhead energy collecting systems that include all mechanical details of the pantographs and the complete topology and structural details of the catenary. PantoCat is a code based on the finite element method, for the catenary, and multibody dynamics methods, for the pantograph, integrated via a co-simulation procedure. A contact model based on a penalty formulation is selected to represent the pantograph–catenary interaction. PantoCat enables models of catenaries with multiple sections, including their overlap, the operation of multiple pantographs and the use of any complex loading of the catenary or pantograph mechanical elements including aerodynamic effects. The models of the pantograph and catenary are fully spatial being simulated in tangential or curved tracks, with or without irregularities and perturbations. User-friendly interfaces facilitate the construction of the models while the post-processing facilities provide all quantities of interest of the system response according to the norms and industrial requirements.     

Exact Equations for Tractrix Curves Associated with Vehicle Offtracking

Vehicle offtracking behavior at low speeds is closely approximated by a geometric entity called a tractrix. This paper presents differential equations for generalized coordinates of a planar multibody vehicle model based on tractrix behavior. The equations are exact, can be used with any type of input path, are valid for forward and backward movements, and are much simpler than previously published formulations used to compute transient offtracking. The differential equations can be integrated using conventional numerical integration algorithms to obtain plots of the low-speed tracking performance of articulated vehicles. The equations were formulated symbolically by a computer program used to analyze the kinematic and dynamic behavior of multibody systems. Example numerical results are plotted.