Spatial Dynamics of Multibody Tracked Vehicles Part I: Spatial Equations of Motion

In this paper, the nonlinear dynamic equations of motion of the three dimensional multibody tracked vehicle systems are developed, taking into consideration the degrees of freedom of the track chains. To avoid the solution of a system of differential and algebraic equations, the recursive kinematic equations of the vehicle are expressed in terms of the independent joint coordinates. In order to take advantage of sparse matrix algorithms, the independent differential equations of the three dimensional tracked vehicles are obtained using the velocity transformation method. The Newton-Euler equations of the vehicle components are defined and used to obtain a sparse matrix structure for the system dynamic equations which are represented in terms of a set of redundant coordinates and the joint forces. The acceleration solution obtained by solving this system of equations is used to define the independent joint accelerations. The use of the recursive equations eliminates the need of using the iterative Newton-Raphson algorithm currently used in the augmented multibody formulations. The numerical difficulties that result from the use of such augmented formulations in the dynamic simulations of complex tracked vehicles are demonstrated. In this investigation, the tracked vehicle system is assumed to consist of three kinematically decoupled subsystems. The first subsystem consists of the chassis, the rollers, the sprockets, and the idlers, while the second and third subsystems consist of the tracks which are modeled as closed kinematic chains that consist of rigid links connected by revolute joints. The singular configurations of the closed kinematic chains of the tracks are also avoided by using a penalty function approach that defines the constraint forces at selected secondary joints of the tracks. The kinematic relationships of the rollers, idlers, and sprockets are expressed in terms of the coordinates of the chassis and the independent joint degrees of freedom, while the kinematic equations of the track links of a track chain are expressed in terms of the coordinates of a selected base link on the chain as well as the independent joint degrees of freedom. Singularities of the transformations of the base bodies are avoided by using Euler parameters. The nonlinear three dimensional contact forces that describe the interaction between the vehicle components as well as the results of the numerical simulations are presented in the second part of this paper.     

Modelling,validation and analysis of a three-dimensional railway vehicle–track system model with linear and nonlinear track properties in the presence of wheel flats

This paper presents dynamic contact loads at wheel–rail contact point in a three-dimensional railway vehicle–track model as well as dynamic response at vehicle–track component levels in the presence of wheel flats. The 17-degrees of freedom lumped mass vehicle is modelled as a full car body, two bogies and four wheelsets, whereas the railway track is modelled as two parallel Timoshenko beams periodically supported by lumped masses representing the sleepers. The rail beam is also supported by nonlinear spring and damper elements representing the railpad and ballast. In order to ensure the interactions between the railpads, a shear parameter beneath the rail beams has also been considered into the model. The wheel–rail contact is modelled using nonlinear Hertzian contact theory. In order to solve the coupled partial and ordinary differential equations of the vehicle–track system, modal analysis method is employed. Idealised Haversine wheel flats with the rounded corner are included in the wheel–rail contact model. The developed model is validated with the existing measured and analytical data available in the literature. The nonlinear model is then employed to investigate the wheel–rail impact forces that arise in the wheel–rail interface due to the presence of wheel flats. The validated model is further employed to investigate the dynamic responses of vehicle and track components in terms of displacement, velocity, and acceleration in the presence of single wheel flat.     

Effect of curved track support failure on vehicle derailment

In order to investigate the effect of curved track support failure on railway vehicle derailment, a coupled vehicle–track dynamic model is put forward. In the model, the vehicle and the structure under rails are, respectively, modelled as a multi-body system, and the rail is modelled with a Timoshenko beam rested on the discrete sleepers. The lateral, vertical, and torsional deformations of the beam are taken into account. The model also considers the effect of the discrete support by sleepers on the coupling dynamics of the vehicle and track. The sleepers are assumed to move backward at a constant speed to simulate the vehicle running along the track at the same speed. In the calculation of the coupled vehicle and track dynamics, the normal forces of the wheels/rails are calculated using the Hertzian contact theory and their creep forces are determined with the nonlinear creep theory by Shen et al [Z.Y. Shen, J.K. Hedrick, and J.A. Elkins, A comparison of alternative creep-force models for rail vehicle dynamic analysis, Proceedings of the 8th IAVSD Symposium, Cambridge, MA, 1984, pp. 591–605]. The motion equations of the vehicle/track are solved by means of an explicit integration method. The failure of the components of the curved track is simulated by changing the track stiffness and damping along the track. The cases where zero to six supports of the curved rails fail are considered. The transient derailment coefficients are calculated. They are, respectively, the ratio of the wheel/rail lateral force to the vertical force and the wheel load reduction. The contact points of the wheels/rails are in detail analysed and used to evaluate the risk of the vehicle derailment. Also, the present work investigates the effect of friction coefficient, axle load and vehicle speed on the derailments under the condition of track failure. The numerical results obtained indicate that the failure of track supports has a great influence on the whole vehicle running safety.     

Fundamentals of vehicle–track coupled dynamics

This paper presents a framework to investigate the dynamics of overall vehicle–track systems with emphasis on theoretical modelling, numerical simulation and experimental validation. A three-dimensional vehicle–track coupled dynamics model is developed in which a typical railway passenger vehicle is modelled as a 35-degree-of-freedom multi-body system. A traditional ballasted track is modelled as two parallel continuous beams supported by a discrete-elastic foundation of three layers with sleepers and ballasts included. The non-ballasted slab track is modelled as two parallel continuous beams supported by a series of elastic rectangle plates on a viscoelastic foundation. The vehicle subsystem and the track subsystem are coupled through a wheel–rail spatial coupling model that considers rail vibrations in vertical, lateral and torsional directions. Random track irregularities expressed by track spectra are considered as system excitations by means of a time–frequency transformation technique. A fast explicit integration method is applied to solve the large nonlinear equations of motion of the system in the time domain. A computer program named TTISIM is developed to predict the vertical and lateral dynamic responses of the vehicle–track coupled system. The theoretical model is validated by full-scale field experiments, including the speed-up test on the Beijing–Qinhuangdao line and the high-speed running test on the Qinhuangdao–Shenyang line. Differences in the dynamic responses analysed by the vehicle–track coupled dynamics and by the classical vehicle dynamics are ascertained in the case of vehicles passing through curved tracks.     

A model for vehicle–track random interactions on effects of crosswinds and track irregularities

A stochastic mathematical model is developed to evaluate the dynamic behaviours and statistical responses of vehicle–track systems when random system excitations including crosswinds and track irregularities are imposed. In this model, the railway vehicle is regarded as a multi-rigid-body system, the track system is modelled by finite element theory. These two systems are spatially coupled by the nonlinear wheel–rail contact forces and unsteady aerodynamic forces. The high efficiency and accuracy of this stochastic model are validated by comparing to the robust Monte-Carlo method. Numerical studies show that crosswinds have a great influence on the dynamic performance of vehicle–track systems, especially on transverse vibrations. When the railway vehicle initially runs into the wind field, it will experience a severe vibration stage, and then stepping into a relatively steady state where the fluctuating winds and track irregularities will play deterministic roles in the deviations of system responses. Moreover, it is found that track irregularities should be properly considered in the safety assessment of the vehicle even in strong crosswinds.     

Coupling Model of Vertical and Lateral Vehicle/Track Interactions

A new dynamic model of vehicle/track interaction is presented. The model considers the vehicle and the track as a whole system and couples the vertical interaction with the lateral interaction. The vehicle subsystem is modeled as a multi-body system with 37 degrees of freedom, which runs on the track with a constant velocity. The track substructure is modeled as a discretely supported system of elastic beams representing the rails, sleepers and ballasts. The normal contact forces between wheels and rails are described by Hertzian nonlinear elastic contact theory and the tangential wheel/ rail forces are decided by the creep theory. Numerical results are compared with those of conventional dynamic models of railway vehicles. Applications of the coupling model to the investigation of safety limits against derailment due to the track twist and the combined alignment and cross-level irregularities are reported at the end of the paper.     

A linear complementarity method for the solution of vertical vehicle–track interaction

A new method is proposed for the solution of the vertical vehicle–track interaction including a separation between wheel and rail. The vehicle is modelled as a multi-body system using rigid bodies, and the track is treated as a three-layer beam model in which the rail is considered as an Euler-Bernoulli beam and both the sleepers and the ballast are represented by lumped masses. A linear complementarity formulation is directly established using a combination of the wheel–rail normal contact condition and the generalised-α method. This linear complementarity problem is solved using the Lemke algorithm, and the wheel–rail contact force can be obtained. Then the dynamic responses of the vehicle and the track are solved without iteration based on the generalised-α method. The same equations of motion for the vehicle and track are adopted at the different wheel–rail contact situations. This method can remove some restrictions, that is, time-dependent mass, damping and stiffness matrices of the coupled system, multiple equations of motion for the different contact situations and the effect of the contact stiffness. Numerical results demonstrate that the proposed method is effective for simulating the vehicle–track interaction including a separation between wheel and rail.     

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 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.     

A Model Reduction Procedure for the Dynamic Analysis of Rail Vehicles Subjected to Linear Creep Forces

The set of differential equations governing the motion of an unrestrained coned wheelset travelling on a tangent section of track and acted upon by creep forces arising from the contact between wheel and rail are, in the terminology of numerical analysis, extremely "stiff". This stiffness can be attributed to the existence of two negative real eigenvalues in the solution of the eigenproblem associated with the linearized equations of motion. Compared with the two complex conjugate eigenvalues that complete this solution, the real eigenvalues have large magnitudes and necessitate that relatively. small timesteps be used in order to obtain an accurate numerical integration of the full set of equations of motion. However, by truncating the set of left and right eigenvectors to eliminate these real eigenvalues in a modal analysis of the wheelset, it was found that their contribution to the overall dynamic response is negligible. This same modal truncation approach was then applied to the sub-structured equations of motion for a simple rail vehicle system consisting of two wheelsets connected to a main body by linear springs and dampers. Essentially, the physical degrees of freedom for each wheelset substructure were replaced by a single complex coordinate obtained from the previous normal modes analysis. Using this model reduction procedure, accurate numerical results for the motion of the rail vehicle were generated several times faster than the results obtained by numerically integrating the full set of differential equations directly.     

Influence of uneven rail irregularities on the dynamic response of the railway track using a three-dimensional model of the vehicle–track system

A mathematical model of the vehicle–track interaction is developed to investigate the coupled behaviour of vehicle–track system, in the presence of uneven irregularities at left/right rails. The railway vehicle is simplified as a 3D multi-rigid-body model, and the track is treated as the two parallel beams on a layered discrete support system. Besides the car-body, the bogies and the wheel sets, the sleepers are assumed to have roll degree of freedom, in order to simulate the in-plane rotation of the components. The wheel–rail interface is treated using a nonlinear Hertzian contact model, coupling the mathematical equations of the vehicle–track systems. The dynamic interaction of the entire system is numerically studied in time domain, employing Newmark's integration method. The track irregularity spectra of both the left/right rails are taken into account, as the inputs of dynamic excitations. The dynamic responses of the track system induced by such irregularities are obtained, particularly in terms of the vertical (bounce) and roll displacements. The numerical model of the present research is validated using several benchmark models reported in the literature, for both the smooth and unsmooth track conditions. Four sample profiles of the measured rail irregularities are considered as the case studies of excitation sources, examining their influences on the dynamic behaviour of the coupled system. The results of numerical simulations demonstrate that the motion of track system is significantly influenced by the presence of uneven irregularities in left/right rails. Dynamic response of the sleepers in the roll direction becomes more sensitive to the rail irregularities, as the unevenness severity of the parallel profiles (quantitative difference between left and right rail spectra) is increased. The severe geometric deformation of the track in the bounce–pitch–roll directions is mainly related to such profile unevenness (cross-level) in left/right rails.     

Optimization of curving performance of rail vehicles

The curving performance of a transit rail vehicle model with 21 degrees of freedom is optimized using a combination of multibody dynamics and a genetic algorithm (GA). The design optimization is to search for optimal design variables so that the noise or wear, arising from misalignment of the wheelsets with the track, is reduced to a minimum level during curve negotiations with flange contact forces guiding the rail vehicle. The objective function is a weighted combination of angle of attack on wheelsets and ratios of lateral to vertical forces on wheels. Using the combination of the GA and a multibody dynamics modelling program, A'GEM, the generation of governing equations of motion for complex nonlinear dynamic rail vehicle models and the search for global optimal design variables can be carried out automatically. To demonstrate the feasibility and efficacy of the proposed approach of using the combination of multibody dynamics and GAs, the numerical simulation results of the optimization are offered, the selected objective function is justified, and the sensitivity analysis of different design parameters and different design parameter sets on curving performance is performed. Numerical results show that compared with suspension and inertial parameter sets, the geometric parameter set has the most significant effect on curving performance.     

Optimization of curving performance of rail vehicles

The curving performance of a transit rail vehicle model with 21 degrees of freedom is optimized using a combination of multibody dynamics and a genetic algorithm (GA). The design optimization is to search for optimal design variables so that the noise or wear, arising from misalignment of the wheelsets with the track, is reduced to a minimum level during curve negotiations with flange contact forces guiding the rail vehicle. The objective function is a weighted combination of angle of attack on wheelsets and ratios of lateral to vertical forces on wheels. Using the combination of the GA and a multibody dynamics modelling program, A’GEM, the generation of governing equations of motion for complex nonlinear dynamic rail vehicle models and the search for global optimal design variables can be carried out automatically. To demonstrate the feasibility and efficacy of the proposed approach of using the combination of multibody dynamics and GAs, the numerical simulation results of the optimization are offered, the selected objective function is justified, and the sensitivity analysis of different design parameters and different design parameter sets on curving performance is performed. Numerical results show that compared with suspension and inertial parameter sets, the geometric parameter set has the most significant effect on curving performance.     

Vibrations Generated by Rail Vehicles: A Mathematical Model in the Frequency Domain

Movement of railway vehicles generates mechanical vibrations of a wide range of frequency. Depending on track materials, dissipation in form of viscous and hysteretic damping is present, and stiffness depends on strain-rate. In a previous paper (Castellani et al., 1998), a mathematical model to describe track materials has been developed in the frequency domain. The present paper applies this model, and attempts an analytical formulation of vehicle-track and soil interaction in the frequency domain. Rail vibrations during the passage of a vehicle are generated by three families of forces: a) the weight of the moving vehicle, b) the inertial reaction of the vehicle under the effect of corrugations over an undeformable rail, and, c) the vehicle inertial forces due to displacements of the rail. The first two groups of forces do not depend on the rail displacement, and the related mathematical formulation is a simple problem of forces at a mobile point of application. Formulation of the vehicle inertial forces, related to the rail vibration, requires reference to the acceleration of the rail, as seen by an observer in motion with the vehicle itself. Moreover, it is necessary to express the equilibrium equation of two dynamic systems, the vehicle and the track, at a the movable point of contact. There is no straight numerical procedure to solve this equation in the frequency domain. In the paper two theoretical propositions (Fryba, 1988; Grassie et al., 1982) are revisited with reference to the effect of the transit of a single wheel. Fryba infers that, in the absence of corrugations, the forces c) are null. Grassie et al. (1982) present a mathematical formulation of the interaction between wheel and rail, at mobile point of contact. At each position, the interaction force is of impulsive type. They presume that for a corrugation of harmonic type, of wavelength ?, the wheel is subject to a harmonic motion, of the frequency f = V/?, where V is the wheel velocity. All other frequency components, due to the impulse, are disregarded. Both these assumptions are shown to be inconsistent from a theoretical point of view, however they suggest suitable approaches to the solution.     

An Investigation into Stick-Slip Vibrations on Vehicle/Track Systems

A model for the numerical simulation of vehicle/track interaction and stick-slip vibration is presented. A finite element model is developed to calculate vertical contact forces. These forces are then coupled through the contact patch into a non-linear time-domain model by which the stick-slip vibration behaviour of a wheel-rail system is analysed. The investigation suggests that stick-slip vibration may occur if a vehicle which has a maligned or an initial 'wind-up' wheeiset meets a vertical irregularity or contaminants on the track.