Flexible Bodies in Multibody System Codes

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.     

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.     

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

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.     

Exact Equations for Tractrix Curves Associated with Vehicle Offtracking

Abstract

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.     

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。     

Symbolic-numerical Solution of Multibody Systems with Closed Loops

SUMMARY

For multibody systems with closed kinematic Loops a set of ordinary differential equations and decoupled algebraic equations is formulated which can be solved with explicit multistep integration algorithms. This is achieved by introducing a minimal set of generalized coordinates being specified during numerical integration. For avoiding restart of the integration algorithm after changing these variables transformation relationships are given. Velocity and acceleration constraints are satisfied exactly, position constraints are fulfilled approximately by a dynamic invariant projection onto the constraint manifold. The method is demonstrated by an application to a five-point wheel suspension.     

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.     

A Modular Computer Model for the Design of Vehicle Dynamics Control Systems1

SUMMARY

This paper describes a multiport approach to computer-aided modeling of vehicle dynamics. The modeling approach produces models that are suitable for the interactive design and evaluation of complex control strategies. The vehicle model which can be used for ride and handling analysis, is built from modular components. The components are programmed using the syntax of the computer aided control system design (CACSD) program EASYS. Seven modeling components are used to create a three-dimensional vehicle dvnamics model. The model is flexible enoug-h to simulate any suspension design with revolute joints.

Each component of the model consists of a FORTRAN subroutine and a main calling module called a macro. To simplify the process of model building, the modeling components in the car model are designed to represent physical elements, such as the spring, damper, link or tire. To create a model, the components, which are represented by blocks, are interconnected through points, located on the blocks, called pons. These ports have been designed to simulate the location of the connection points between the physical elements, as observed in real systems. The construction of multibody models within a CACSD program offers the flexibility of simultaneous interactive simulation of the three-dimensional dvnamics and evaluation of the desien of the controls.

Although modeling of multibody systems using FORTRAN components has been pioneered by Chace, Haug and Orlandea; and bond graph modeling of multibody systems has been investigated by Bos, this approach is novel because:-

The model is included in the control system design program (EASYS). This arrangement allows the designer to exploit the advanced control design tools available in the program. Furthermore, this approach significantly reduces the computation time required for running the model after parameters modification.

The model is built from components that are interconnected by ports which represent the actual physical location of the connection points between the elements. The multiport approach simplifies the model building process for multibody systems. This simplification is achieved by reducing the model of a multibody system to a block diagram form.     

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.     

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.     

Stabilization of High-Speed Railway Vehicles

SUMMARY

The problems of critical speeds of railway vehicles with dry friction units determination are discussed. A new approach is used which extends the field of application of dynamic response linear analysis methods to vital nonlinear multibody systems. The special features concerning the influence of dry friction forces in the body supports on the trucks and parameters of horizontal constraints of wheelsets and truck frames on critical speed are indicated. It is shown that a significant rising of railway vehicle critical speeds can be reached by changing the structure of constraints between the body and the trucks.     

Effect of the combined centre of gravity on the running safety of freight wagons

ABSTRACT

This paper presents an analysis of loaded freight wagon dynamics in curve alignments. We investigate the effects of the combined centre of gravity (CCOG) on the running safety of freight wagons and examine proper position of the CCOG. A simple wagon-rail model is implemented using the multibody dynamics software ADAMS/Rail. The simulation model is operated on curve tracks with various radii and velocities and the curving performances are evaluated. The results indicate that the CCOG can be located within a flexible and accurate range. The longitudinal offset is good for the curving performance and the permissible lateral offset should be assessed based on the curve radius and cant deficiency.     

Effect of Frame Compliance on the Lateral Dynamics of Motorcycles

SUMMARY

The lateral dynamics of an uncontrolled motorcycle, running on a straight, level road surface, is investigated in this paper. The structural compliances in the front and the rear frames of the motorcycle are taken into account by introducing additional degrees of freedom in the analysis. The kinematics of the tires is represented by linear differential equations which are based on the taut-string model of pneumatic tires. The linear differential equations of motion are solved to yield the eigensolutions of the system. Numerical results, obtained for parameters corresponding to a Honda CB750 motorcycle, are presented and discussed.     

Self-Excited Vibrations of Railway Vehicle with Dry Friction Units

SUMMARY

Some results obtained during developing a new approach to analysis of conditions of self-excitation of lateral vibrations of railway vehicles as systems of variable structure are given in this article. Structure variability is conditioned by the state of constraints with dry friction. An idea of the replacement of a nonlinear multibody system by a number of subsystems, which are linearized in a certain way, is worked out. The efficiency of the method suggested for the determination of the characteristics of self-excited vibrations of active multibody systems with dry friction is illustrated by way of an example of the investigation of the dynamic behaviour of a freight car.