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.     

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.     

车速和路面附着系数的滚动时域估计

基于Dugoff轮胎模型建立了车辆的非线性动力学方程,并给出了路面附着系数的约束条件.针对车速和路面附着系数约束的非线性估计,提出了一种基于滚动优化原理的滚动时域估计法(MHE),并给出了MHE法的具体步骤.在不同路面上对MHE法进行了多种工况的实验验证,并在同样条件下与扩展Kalman滤波法进行了比较.实验结果表明,MHE法的估计性能优于扩展Kalman滤波法.     

Accessible pre-design calculation tool to support the definition of EV components

The new freedoms in design that electric powertrains offer lead to a wide variety of configurations to consider when developing an electric vehicle (EV) from scratch. Furthermore, the strong relation of the battery size with vehicle weight, range and performances leads to a set of interrelated dependencies that can result in many design loops to fulfil the vehicle targets, market constraints and regulations simultaneously. The paper presents a pre-design tool to assist the electric vehicle development process by representing the different constraints and the possible feasible solutions in a single plot with the need of a small amount of inputs accesible to assess at pre-design phase. As a result, the tool depicts a set of feasible vehicle configurations that could fulfil the targets easing the interaction and loops among different expertise areas. To better assist selection, it also provides a sensitivity analysis of the performances to selected inputs and the user can introduce a cost function depending on vehicle weight and battery size. The tool is based on the vehicle longitudinal dynamics equations and equations that model the market and regulations constraints. It is aimed at providing an overview of the main specifications for component selection avoiding detailed vehicle modelling in the early pre-design phase at which the vehicle characteristics and even powertrain architecture are unknown. Finally, the tool results quality is evaluated by further developing one of its solutions for passenger car in four different vehicle configurations with the simulation software vemSim and AVL Cruise. The results of the simulations are compared to the solution of the pre-design tool to evaluate the level of fidelity and the deviations in the final result that can appear depending on the final architecture, components characteristics and control strategy.     

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.     

Direct method for optimal power management in hybrid electric vehicles

Hybrid electric vehicles are powered by an electric system and an internal combustion engine. The components of a hybrid electric vehicle need to be coordinated in an optimal manner to deliver the desired performance. This paper presents an approach based on direct method for optimal power management in hybrid electric vehicles with inequality constraints. The approach consists of reducing the optimal control problem to a set of algebraic equations by approximating the state variable which is the energy of electric storage, and the control variable which is the power of fuel consumption. This approximation uses orthogonal functions with unknown coefficients. In addition, the inequality constraints are converted to equal constraints. The advantage of the developed method is that its computational complexity is less than that of dynamic and non-linear programming approaches. Also, to use dynamic or non-linear programming, the problem should be discretized resulting in the loss of optimization accuracy. The propsed method, on the other hand, does not require the discretization of the problem producing more accurate results. An example is solved to demonstrate the accuracy of the proposed approach. The results of Haar wavelets, and Chebyshev and Legendre polynomials are presented and discussed.     

On Inverse Equations for Vehicle Dynamics

Instead of writing equations which when solved yield the response of a vehicle to an input such as the front wheel steer angle, one can often invert the equations so that a response quantity is specified as an input and a new set of equations is solved yielding the steer angle required as an output. Using these equations one can discover the input steer angle a driver would need to impose in order to accomplish a specific maneuver for various vehicles.

It is shown that there are many possible inverse equation sets and that the eigenvalues of the inverse equations are hard to interpret since they may have little to do with the vehicle parameters. The linear single-input single-output case is studied first to fix ideas using a simple example. For the bicycle model vehicle, it is shown that any vehicle may have unstable inverse equations depending upon the response quantity used. Extensions to nonlinear and multiple-input multiple output systems are discussed.     

Resource-aware integration of AUTOSAR-compliant ECUs with an empirical wcet prediction model

This paper presents an integration method of AUTOSAR-compliant ECUs which can evaluate resource constraints in an early-stage of development. There are three types of resources for an ECU (timing, memory, and interface) which should be carefully managed for successful ECU integration. The proposed method consists of three steps: measurement, prediction, and evaluation. In the first step, a method to measure resource factors for AUTOSAR-compliant software architecture is introduced. Based on the method, a worst-case execution cycle of a runnable, memory section usages of a software component, and interface of legacy ECUs can be obtained. In the second step, the obtained factors are quantitatively predicted according to the architectural designs of the integration ECU. In the case of the timing resource, the worst-case execution time of the integration ECU can be precisely predicted by a proposed empirical model. In the last step, the resource constraints such as CPU, memory, network utilizations can be evaluated with predicted resource factors before implementation. The proposed method was applied to the integration of an in-house engine management system composed of two ECUs. The method successfully provided quantitative measures to evaluate architectural designs of three different integration scenarios.     

Wheel-rail contact models for vehicle system dynamics including multi-point contact

Advanced modelling of rail vehicle dynamics requires realistic solutions of contact problems for wheels and rails that are able to describe contact singularities, encountered for wheels and rails. The basic singularities demonstrate themselves as double and multiple contact patches. The solutions of the contact problems have to be known practically in each step of the numerical integration of the differential equations of the model. The existing fast, approximate methods of solution to achieve this goal have been outlined. One way to do this is to replace a multi-point contact by a set of ellipses. The other methods are based on so-called virtual penetration. They allow calculating the non-elliptical, multiple contact patches and creep forces online, during integration of the model. This allows nearly real-time simulations. The methods are valid and applicable for so-called quasi-Hertzian cases, when the contact conditions do not deviate much from the assumptions of the Hertz theory. It is believed that it is worthwhile to use them in other cases too.     

On Inverse Equations for Vehicle Dynamics

Abstract

Instead of writing equations which when solved yield the response of a vehicle to an input such as the front wheel steer angle, one can often invert the equations so that a response quantity is specified as an input and a new set of equations is solved yielding the steer angle required as an output. Using these equations one can discover the input steer angle a driver would need to impose in order to accomplish a specific maneuver for various vehicles.

It is shown that there are many possible inverse equation sets and that the eigenvalues of the inverse equations are hard to interpret since they may have little to do with the vehicle parameters. The linear single-input single-output case is studied first to fix ideas using a simple example. For the bicycle model vehicle, it is shown that any vehicle may have unstable inverse equations depending upon the response quantity used. Extensions to nonlinear and multiple-input multiple output systems are discussed.     

Schedule construction under precedence constraints in flexray in-vehicle networks

As embedded time-triggered applications have widely replaced mechanical systems in modern automobiles, holistic scheduling of tasks and messages of such applications on in-vehicle networks has become a critical issue. For offering QoS (Quality of Service) guarantees, the holistic schedule must satisfy numerous constraints such as protocol specifications, delay constraints and precedence constraints between tasks schedules and messages transmissions. Existing approaches to this problem search through a vast design space of all possible joint task and message schedules. This leads to a high complexity and limits the scalability of such approaches for scheduling the large scale systems. To cope with this problem, we propose an approach that divides the holistic scheduling problem to two sub-problems: the sub-problem of message scheduling and the sub-problem of task scheduling, while precedence relations and end-to-end information passing between task instances and messages are preserved and the end-to-end deadlines are guaranteed. This helps to reduce the workload on the problem solvers and improves efficiency and scalability. In the first sub-problem, our approach optimizes scheduling the set of messages and allocates time windows for scheduling each task with respect to precedence constraints, end-to-end deadlines and FlexRay protocol specifications. The length of each time window helps to preserve the respective tasks schedulability and to provide flexibility for both task and message scheduling. The objective is defined with respect to extensibility issues. In the second sub-problem, our approach optimizes schedule of the set of tasks with respect to their allocated time windows and timing constraints. The objective is defined with respect to latency issues. We optimize the solution to each sub-problem using Mixed Integer Linear Programming optimization framework. Performance evaluations show that, compared with existing holistic scheduling approaches, our approach is more scalable and obtains better solutions in a reasonable amount of time.     

复杂曲面汽车车身设计可视化网络开发环境的实现

本文讨论复杂曲面汽车车身可视化设计环境的具体实现。在使设计师充分掌握、便捷使用相关信息的原则下，以实现以相似类比的方法完成汽车车身的几何建模和动力学分析的紧密结合，设计了要求的开发环境的具体功能及有关模块的分配，并指出以基于Web的Internet/Intranet网络技术来实现可视化车身设计环境和CAD／CAE信息集成是可行的，且有兼容性好、易于加入全球虚拟企业网等优点。     

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.     

Simulation and Visualisation of the Dynamic Behaviour of an Overhead Power System with Contact Breaking

For high speed rail traffic it is necessary to design overhead power systems which minimize the contact loss between pantograph head and contact wire. To predict how different design solutions will behave it is favourable to model and simulate the dynamic behaviour. In this paper a model of an overhead power system is specified and used in simulation. The model is suitable for simulation with contact loss since it includes specifications of impact conditions between pantograph head and contact wire. Two sets of equations of motion are specified, one for the contact case and one for the non-contact case. The model also includes lateral movement of the wire due to the zigzag span and friction between the pantograph head and the contact wire. It is shown how to make animations of the system behaviour using a MCAE-system. The animations are made using a geometrical model of the system together with results from numerical simulations.

Through the examples provided, use of the mathematical model and the geometrical model is presented. The response is visualised as time histories and phase plane diagrams of different coordinates and as animations of the total system response. The different types of visualisations make an excellent combination when studying the system behaviour of different design solutions.

In one example, simulation using the linearised set of equations gives the same results as simulation using the set of fully nonlinear equations, due to periodic response and the simple alternation of contact conditions. It is shown that the situation when any of the parameters vary suddenly is possible to simulate using the fully nonlinear equations of motion.     

对呼和浩特市城乡公交一体化的设想

国务院机构改革将原建设部的城市公共交通职能划归交通运输部后,使呼和浩特市发展城乡公交一体化成为可能,也为建立真正的城乡一体化管理体制打下了坚实的基础。文章就如何发展呼和浩特市城乡公交一体化谈一些设想。     

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.     

Modelling and controlling of car-following behavior in real traffic flow using ARMAX identification and Model Predictive Control

Nowadays, car following models, as the most popular microscopic traffic flow modeling, are increasingly being used by transportation experts to evaluate new Intelligent Transportation System (ITS) and Advanced Driver Assistance Systems (ADAS) applications. The control of car following is essential due to its safety and its operational efficiency. For this purpose, this paper builds a model of car following behavior based on ARMAX structure from a real traffic data set and presents a Model Predictive Control (MPC) controller. An important advantage of this type of control is its ability to cope with constraints on controls. Since safety and operational efficiency are constraints for car following, therefore we have recruited this type of controller in this study to deal with these constraints. Based on the relative distance and relative acceleration of each instant, the MPC predicts the future behavior of the leader vehicle (LV) and according to this behavior, the acceleration of the follower vehicle (FV) is controlled. The MPC tries to control this acceleration in a way to keep the relative distance at a safe region. To investigate the performance of the designed controller, the result of the system is compared with the behavior of human drivers with similar initial conditions. Also, some other test performances were accomplished to investigate other features such as robustness and the stability of the designed MPC. The simulation results show that the MPC controller has a behavior much safer than that of real drivers and it can provide a pleasant trip for passengers.     

Real-time capable vehicle-trailer coupling by algorithms for differential-algebraic equations

The real-time simulation of vehicle trains requires an accurate and numerically feasible representation of the vehicle-trailer coupling. Although the equations of motion for the chassis instances can be reduced to systems of ordinary differential equations, additional constraints on the relative motion of vehicle and trailer are introduced when considering the hitch. In this article, we present a strategy for the simulation of vehicle-trailer combinations, where the algebraic constraints of the coupling are treated explicitly. Although this approach allows exact modeling of the respective joint geometry and realistic calculation of the coupling forces, a suitable numerical algorithm is required in order to solve the resulting differential-algebraic system of index 3 in real-time. The implementation in a commercial vehicle dynamics program is discussed and real-time simulation results are shown, which prove its feasibility for different coupling joints and demanding driving maneuvers.     

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。