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.     

Spatial Dynamics of Multibody Tracked Vehicles Part II: Contact Forces and Simulation Results

In this part of the paper, three dimensional computational capabilities, that includes significant details, are developed for the nonlinear dynamic analysis of large scale spatial tracked vehicles. Three dimensional nonlinear contact force models that describe the interaction between the track links and the vehicle components such as the rollers, sprockets, and idlers as well as the interaction between the track links and the ground are developed and used to define the generalized contact forces associated with the vehicle generalized coordinates. Tangential friction and contact forces are developed in order to maintain the stability of the track motion and avoid the slippage of the track or its rotation as a rigid body. Body and surface coordinate systems are introduced in order to define the spatial contact conditions. The nonlinear equations of motion of the tracked vehicle are solved using the velocity transformation procedure developed in the first part of this paper. This procedure is used in order to obtain a minimum set of differential equations, and avoid the use of the iterative Newton-Raphson algorithm. A computer simulation of a tracked vehicle that consists of one hundred and six bodies and has one hundred and sixteen degrees of freedom is presented in order to demonstrate the use of the formulations presented in this study.     

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 Motorcycle Multi-Body Model for Real Time Simulations Based on the Natural Coordinates Approach

This paper presents an eleven degrees of freedom, non-linear, multi-body dynamics model of a motorcycle. Front and rear chassis, steering system, suspensions and tires are the main features of the model. An original tire model was developed, which takes into account the geometric shape of tires and the elastic deformation of tire carcasses. This model also describes the dynamic behavior of tires in a way similar to relaxation models. Equations of motion stem from the natural coordinates approach. First, each rigid body is described with a set of fully cartesian coordinates. Then, links between the bodies are obtained by means of algebraic equations. This makes it possible to obtain simple equations of motion, even though the coordinates are redundant. The model was implemented in a Fortran code, named FastBike . In order to test the code, both simulated and real slalom and lane change maneuvers were carried out. A very good agreement between the numerical simulations and experimental test was found. The comparison of FastBike 's performance with those of some commercial software shows that first is much faster than others. In particular, real time simulations can be carried out using FastBike and it can be employed on a motorcycle simulator.     

Robust design optimization of the dynamic responses of a tracked vehicle system

This paper presents the robust design optimization of the dynamic responses of a heavy military tracked vehicle system. The tracked vehicle model addressed in this study has 954 degrees of freedom and consists of 189 bodies in total: 37 bodies for the chassis, such as sprockets, road wheels, road arms, etc.; 76 track link bodies for each track subsystem; 36 revolute joints; and 152 bushing elements. The design objectives were to minimize the maximum vertical acceleration of the hull and its variance while satisfying the wheel travel constraints for torsion bars and the hydro-pneumatic suspension units within ±1σ ranges. To avoid the difficulty of the design sensitivity analysis and to overcome the numerical noise, a progressive meta-model technique was employed in the optimization process. First, space-filling methods were used to determine the minimum number of sample points. Second, the simultaneous kriging method was used to construct the initial meta-models, and the augmented Lagrange multiplier (ALM) method was then used to solve the robust design problems of the meta-models. Third, the new design results were added to the analysis results for the initial sample points, and the meta-models were updated automatically. Next, the optimizer resolved the robust design problems of the updated meta-models. These processes were repeated until the convergence tolerances were satisfied. The robust design optimization of the tracked vehicle system, with 11 random design variables, was solved in only 26 analyses, including 12 analyses for the initial meta-models and 14 analyses added during the iterative optimization process.     

底盘测功机机械惯量电模拟方法的研究和实现

为开发汽车排气污染物简易瞬态工况法测试系统，研究了底盘测功机机械惯量电模拟的相关理论和方法。通过汽车在底盘测功机上运行状态的动力学分析，基于汽车驱动轮转动动态特性相同的原理，建立了汽车底盘测功机机械惯量的电模拟模型。分析了简易瞬态测试工况控制要求和风冷电涡流测功器的性能特点，构建了VMAS测试工况控制系统，应用预测控制和解耦控制理论和技术，设计了底盘测功机机械惯量电模拟控制方案，开发了基于DMC，具有模型增益自校正和解耦功能的VMAS测试扭矩控制器和简易瞬态工况控制试验系统。运行试验结果表明，该系统可以较好地模拟汽车加速运动惯量。     

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.     

Influence of compliance for an elastokinematic model of a proposed rear suspension

The research is carried out to improve passenger’s comfort to increase the vehicles stability in dynamic conditions. The literature available in the automotive engineering considers different topics for studying suspensions. An example represents mechanisms structure and analysis (synthesis, kinematics, and dynamics) under various operating conditions. These aspects have been approached before analytically, numerical. The current paper studies the influence of the lateral force on the contact patch of the wheel and the corresponding variations of vehicle stability parameters, such as camber angle and wheel rear track. The study is performed for a newer innovative rear suspensions mechanism which does not have a wheel track and camber angle variation, relative to the chassis, when the suspension components was considered rigid. A numerical solution is obtained through a virtual model on several commercial codes: MSC Adams, Patran, Nastran. Concerning the analysed parametes, their variation increases as the applied force is increased. Moreover, the largest variation corresponds to the case were elastic bushings and deformable links are considered.     

考虑驾驶人特性的一主多从混合博弈容错控制

为保证线控底盘电动汽车在遭遇执行器失效时的稳定性，并考虑人-车交互行为，提出了以驾驶人为领导者的一主多从（Single-leader-multiple-follower，SLMF）混合博弈容错控制框架。为实现驾驶人-车辆的交互控制，首先建立了两者的耦合模型。其次，将驾驶人及5个底盘子系统即主动前轮转向（Active Front Steering，AFS）系统和4个轮毂电机建模为博弈中的6个参与者，基于Stackelberg主从博弈与多人合作博弈设计了SLMF混合博弈控制框架。考虑驾驶人具有优先控制权限及执行器对驾驶人行为的补偿作用，基于Stackelberg博弈理论建立了驾驶人与底盘子系统的主从博弈模型，其中驾驶人作为领导者通过感知跟随者的行为做出转向决策，而5个底盘子系统被建模为跟随者。由于跟随者追求共同的横向稳定控制目标，因此基于合作博弈理论建立了合作模型，并对领导者的转向策略做出最优响应。最后，为研究跟随者之间追求不同目标导致不合作时的控制效果，设计了非合作Nash博弈与Stackelberg博弈相结合的混合博弈为对比方法，通过实时硬件在环测试验证并对比了2种方法。结果表明：针对不同风格的驾驶人，所设计的方法可以保证遭遇执行器卡死失效车辆的稳定性。与不合作的情况相比，2种不同风格的驾驶人驾驶的车辆在底盘子系统合作时，车辆稳定性分别提升了54.62%和53.78%，驾驶人工作负荷分别降低了31.79%和36.07%。     

Kinematic and dynamic analysis of the McPherson suspension with a planar quarter-car model

McPherson suspension modelling poses a challenging problem due to its nonlinear asymmetric behaviour. The paper proposes a planar quarter-car analytical model that not only considers vertical motion of the sprung mass (chassis) but also: (i) rotation and translation for the unsprung mass (wheel assembly), (ii) wheel mass and its inertia moment about the longitudinal axis, and (iii) tyre damping and lateral deflection. This kinematic–dynamic model offers a solution to two important shortcomings of the conventional quarter-car model: it accounts for geometry and for tyre modelling. The paper offers a systematic development of the planar model as well as the complete set of mathematical equations. This analytical model can be suitable for fast computation in hardware-in-the-loop applications. Furthermore, a reproducible Simulink implementation is given. The model has been compared with a realistic Adams/View simulation to analyse dynamic behaviour for the jounce and rebound motion of the wheel and two relevant kinematic parameters: camber angle and track width variation.     

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.     

Intelligent Electronic Systems in Commercial Vehicles for Enhanced Traffic Safety

Looking at the future trends of the road traffic, one will recognize that the commercial vehicle participation will not decrease, although it is required from the environmental and social viewpoints. The reason is that the other means of freight transport (water, railway, air) do not provide the same flexibility as the road transport, and direct business interest of those companies, who are using this transport form is larger than the eventual loss caused by the penalties to be paid (taxes, compensation of higher axle load). This conflict is hard to solve, but the effect can be minimized. The commercial vehicle industry attempts to introduce systems to the vehicles, which are targeting on reduction of the environmental impacts caused by heavy vehicles. These systems, which are named generally as “intelligent chassis systems”, electronically control the operation of the chassis subsystems (engine, transmission, brake, suspension) and co-ordinate their operation on a higher level (vehicle controller, intelligent control systems, such as adaptive cruise control, video camera based lane change recognition system, etc.). This paper reviews the state-of-the-art of the commercial vehicle chassis systems, and tries to project their future development.     

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.     

Experimental modeling of controlled suspension vehicles from onboard sensors

In this article, identification of vertical dynamics of vehicles with controlled suspensions is considered. Identification is performed from experimental data measured on a four-poster bench test of a segment C car, equipped with a CDC-Skyhook dampers control system. The measurements are obtained from the onboard accelerometers needed by the control system. A nonlinear model in regression form is identified, having the road profile and damper control currents as inputs and chassis accelerations as outputs. The model is identified by means of a set membership structured identification method, which takes advantage of physical information on the structure of the system, decomposing the system into three subsystems: one represents the chassis and engine and the other two represent the overall behavior of front and rear suspensions, wheels and tires. This decomposition allows us to avoid the complexity accuracy problems derived from the high dimension of required regression space. Indeed, the overall high-dimensional identification problem is reduced to the identification of lower dimensional subsystems and to the estimation of their interactions. An iterative scheme is used for solving the decomposed identification problem. As the chassis pitch is small for the usual road profiles, the chassis-engine block is considered linear and standard linear methods are used for its identification. The other two subsystems are the main sources of nonlinearities in the system, mainly due to the significant nonlinearities of controlled dampers and of tires. Owing to the complexity/accuracy problems of a physical modeling of these subsystems, an input–output approach is taken. In particular, a nonlinear set membership method that does not require the search of the functional form of involved nonlinearities is used for the identification of these subsystems. The iterative algorithm converged in two iterations to a model providing a quite satisfactory simulation accuracy for all the considered road profiles and CDC-Skyhook settings.     

Experimental modeling of controlled suspension vehicles from onboard sensors

In this article, identification of vertical dynamics of vehicles with controlled suspensions is considered. Identification is performed from experimental data measured on a four-poster bench test of a segment C car, equipped with a CDC-Skyhook dampers control system. The measurements are obtained from the onboard accelerometers needed by the control system. A nonlinear model in regression form is identified, having the road profile and damper control currents as inputs and chassis accelerations as outputs. The model is identified by means of a set membership structured identification method, which takes advantage of physical information on the structure of the system, decomposing the system into three subsystems: one represents the chassis and engine and the other two represent the overall behavior of front and rear suspensions, wheels and tires. This decomposition allows us to avoid the complexity accuracy problems derived from the high dimension of required regression space. Indeed, the overall high-dimensional identification problem is reduced to the identification of lower dimensional subsystems and to the estimation of their interactions. An iterative scheme is used for solving the decomposed identification problem. As the chassis pitch is small for the usual road profiles, the chassis-engine block is considered linear and standard linear methods are used for its identification. The other two subsystems are the main sources of nonlinearities in the system, mainly due to the significant nonlinearities of controlled dampers and of tires. Owing to the complexity/accuracy problems of a physical modeling of these subsystems, an input-output approach is taken. In particular, a nonlinear set membership method that does not require the search of the functional form of involved nonlinearities is used for the identification of these subsystems. The iterative algorithm converged in two iterations to a model providing a quite satisfactory simulation accuracy for all the considered road profiles and CDC-Skyhook settings.     

Performance of spade-less wheeled military vehicles with passive and semi-active suspensions during mortar firing

Many armies are replacing heavy slow tracked vehicles with their lighter wheeled counterparts for their high mobility and better shoot and scoot capabilities. These features make the vehicle hard to track and target in counter-battery fire. However, when firing high calibre guns, spades are needed to connect the vehicle chassis to the ground, so as to transmit parts of the large firing force directly to the ground. Use of spades hinders the vehicle mobility, while elimination of them paves the way for having quicker and more mobile wheeled vehicles. In this article, vibration response of a spade-less High Mobility Multi-purpose Wheeled Vehicle with a mounted mortar is studied and controlled using stock passive, optimised passive, and optimised semi-active dampers as primary suspensions. The spade-less vehicle with optimised passive and semi-active dampers has a better response in heave, pitch, and fore-aft motions and can fire with better accuracy compared to a spade-less vehicle with stock passive dampers. Simulation results indicate that the spades can be removed from wheeled military vehicles if the precautions are taken for the tyres.