A unified formulation for mechanical joints with and without clearances/bushings and/or stops in the framework of multibody systems

Virtually all machines and mechanisms use mechanical joints that are not perfect from the kinematic point of view and for which tolerances, in the fitting of their components, are specified. Together with such controlled clearances, mechanical joints may require the use of bushing elements, such as those used in vehicle suspensions. Furthermore, in many situations the joints exhibit limits (stops) in their translational or rotational motion that have to be taken into account when modeling them. The dynamic response of the mechanical systems that use such realistic mechanical joints is largely dependent on their characteristic dimensions and material properties of the compliant elements, implying that correct models of these systems must include realistic models of the bushing/clearance joints and of the joint stops. Several works addressed the modeling of imperfect joints to account for the existence of clearances and bushings, generally independently of the formulation of the perfect kinematic joints. This work proposes a formulation in which both perfect and clearance/bushing joints share the same kinematic information making their modeling data similar and enabling their easy permutation in the context of multibody systems modeling. The proposed methodology is suitable for the most common mechanical joints and easily extended to many other joint types benefiting the exploration of a wide number of modeling applications, including the representation of cut-joints required for some formulations in multibody dynamics. The formulation presented in this work is applied to several demonstrative examples of spatial mechanisms to show the need to consider the type of imperfect joints and/or joints with stops modeling in practical applications.     

Development of generic multibody road vehicle models for crashworthiness

The crashworthiness analysis of road vehicles requires detailed data of the vehicles that the automotive manufacturers are, generally, unable to release due to commercial or legal restrictions. In the development of passive safety subsystems or substructures, the overall crash response of a vehicle model used to support it, must mimic that of the real vehicle; if this exists, regardless of any particular constructive detail of its structure provided that it is not located in the vicinity of such subsystem. This work proposes a methodology for the development of multibody models of road vehicles, for passive safety analysis, which include all general structural and mechanical features of real vehicles and start by exhibiting impact dynamic responses similar to the top of line vehicles. These vehicle models, designated as generic, do not require the knowledge of most of the particular details of the design of the real vehicle, which the manufacturers are unable to release, but can be adjusted to have crash responses similar to those of the real vehicle. Based on an existing finite element model of a car, which has all constructive features of vehicles of the chosen class, a multibody model is built applying the plastic hinge approach. By using a selected number of crash scenarios, defined in international standards such as the EuroNCAP, selected parameters of the vehicle multibody model are adjusted to ensure a good correlation between its impact responses and those of the finite element model. The crash responses are measured in terms of structural deformations, velocities and accelerations, occupant injury measures and structural energy absorption capabilities. Assuming that the plastic hinge constitutive equations of the multibody model are not exactly known, their parameters are used here as the multibody vehicle model that are adjusted. The methodology proposed is demonstrated by its application to the identification of the vehicle multibody model of a large family car for which the reference vehicle is available as a detailed finite element model.     

履带动态张紧力的动力学仿真

履带车辆的履带动态张紧力在很大程度上决定行动部分各元件的载荷、履带寿命、功率损失和履带脱落的概率.该文用多体动力学软件RecurDyn,以某履带车辆为原形建立履带车辆的多刚体模型,并建立有效的不同等级的路面模型.模型中履带子系统有六个自由度,能够真实的模拟履带运动的实际情况.通过对车体上几个固定位置处履带张紧力的变化和履带上参考履带板的受力变化,来检测履带动态张紧力的变化.分别改变路面、速度、预张紧力等相关影响因素做多组计算,对比分析了履带动态张紧力的变化情况.     

Identification of multibody vehicle models for crash analysis using an optimization methodology

This work proposes an optimization methodology for the identification of realistic multibody vehicle models, based on the plastic hinge approach, for crash analysis. The identification of the design variables and the objective function and constraints are of extreme importance for the success of the optimization. The characteristics of the plastic hinges are used as design variables while the objective functions are formulated with measures of the difference between the dynamic response of the model and a reference response. The sequential application of genetic and gradient-based optimization methods is used to solve the optimization problem constituting a systematic approach to the automatic identification of vehicle multibody models. The methodology is demonstrated with the identification of the multibody model of a large family car for side and front crash. The vehicle model is developed in the MADYMO multibody code which is linked with the optimization algorithms implemented in the Matlab Optimization Toolbox.     

Modelling of Torsion Beam Rear Suspension by Using Multibody Method

The multibody systems analysis has become one of the main simulation techniques to calculate the elasto-kinematics characteristics of a car suspension under wheel loads or to realize complex full vehicle models in order to predict the handling performances or the NVH quality. The modelling of torsion beam rear suspensions—widely adopted in cars belonging to B or C class—presents some problems arising from the structural behaviour of this component. A linear method based on component mode synthesis was used to represent the flexible torsion beam within the multibody model. This kind of approach was compared with a non-linear FE analysis. The elasto-kinematics analysis of the suspension was performed by using SIMPACK multibody code. The main suspension parameters (toe angle, camber angle, wheelbase and track variation) were calculated by changing wheel travel and loads. Static analyses, involving great displacements, were performed and a different number of modes were considered in the modal condensation of the torsion beam. The results of multibody simulations were compared with those obtained from a non-linear FE model. Different stiffness values of the bushings that connect the torsion beam to the vehicle chassis were taken into account.     

Dynamic Analysis for Planar Multibody Mechanical Systems with Lubricated Joints

The dynamic analysis of planar multibody systems with revolute clearance joints, including dry contact and lubrication effects is presented here. The clearances are always present in the kinematic joints. They are known to be the sources for impact forces, which ultimately result in wear and tear of the joints. A joint with clearance is included in the multibody system much like a revolute joint. If there is no lubricant in the joint, impacts occur in the system and the corresponding impulsive forces are transmitted throughout the multibody system. These impacts and the eventual continuous contact are described here by a force model that accounts for the geometric and material characteristics of the journal and bearing. In most of the machines and mechanisms, the joints are designed to operate with some lubricant fluid. The high pressures generated in the lubricant fluid act to keep the journal and the bearing surfaces apart. Moreover, the lubricant provides protection against wear and tear. The equations governing the dynamical behavior of the general mechanical systems incorporate the impact force due to the joint clearance without lubricant, as well as the hydrodynamic forces owing to the lubrication effect. A continuous contact model provides the intra-joint impact forces. The friction effects due to the contact in the joints are also represented. In addition, a general methodology for modeling lubricated revolute joints in multibody mechanical systems is also presented. Results for a slider-crank mechanism with a revolute clearance joint between the connecting rod and the slider are presented and used to discuss the assumptions and procedures adopted.     

A compact smoothing-differentiation and projection approach for the kinematic data consistency of biomechanical systems

The estimation of the skeletal motion obtained from marker-based motion capture systems affects the results of the kinematic and dynamic analysis of biomechanical systems. The main source of error is the inaccuracy of velocities and accelerations derived from experimentally measured displacements of markers placed on the skin of joints. This error is mainly due to the amplification of high-frequency low-amplitude noise introduced by the motion capture system when the raw displacement signals are differentiated. Another source of error is the skin motion artifact that produces violations of the kinematic constraint equations of the multibody system. An integrated smoothing-differentiation-projection approach to ensure the kinematic data consistency in the context of the analysis of biomechanical systems is presented. The raw data differentiation problem is solved by applying a single-step smoothing-differentiation technique based on the Newmark integration scheme. A systematic multibody procedure is proposed based on the projection of the positions and its smoothed derivatives into their corresponding constraint manifolds to ensure the kinematic data consistency. Several benchmark kinematic signals that include an acquired nonstationary mono-dimensional motion of biomechanical origin and computer generated data of a four-bar mechanism were processed using the proposed method to study its performance.     

Complex Flexible Multibody Systems with Application to Vehicle Dynamics

A formulation to describe the linear elastodynamics offlexible multibody systems is presented in this paper. By using a lumpedmass formulation the flexible body mass is represented by a collectionof point masses with rotational inertia. Furthermore, the bodydeformations are described with respect to a body-fixed coordinateframe. The coupling between the flexible body deformation and its rigidbody motion is completely preserved independently of the methods used todescribe the body flexibility. In particular, if the finite elementmethod is chosen for this purpose only the standard finite elementparameters obtained from any commercial finite element code are used inthe methodology. In this manner, not only the analyst can use any typeof finite elements in the multibody model but the same finite elementmodel can be used to evaluate the structural integrity of any systemcomponent also. To deal with complex-shaped structural models offlexible bodies it is necessary to reduce the number of generalizedcoordinates to a reasonable dimension. This is achieved with thecomponent mode synthesis at the cost of specializing the formulation toflexible multibody models experiencing linear elastic deformations only.Structural damping is introduced to achieve better numerical performancewithout compromising the quality of the results. The motions of therigid body and flexible body reference frames are described usingCartesian coordinates. The kinematic constraints between the differentsystem components are evaluated in terms of this set of generalizedcoordinates. The equations of motion of the flexible multibody systemare solved by using the augmented Lagrangean method and a sparse matrixsolver. Finally, the methodology is applied to model a vehicle with acomplex flexible chassis, simulated in typical handling scenarios. Theresults of the simulations are discussed in terms of their numericalprecision and efficiency.     

Bond graph modeling of a railway truck on curved track

Rail vehicles become unstable beyond a critical speed because of speed dependant creep forces at the rail-wheel contact surface. There is a constant demand for improved rail vehicle model for design of high-speed vehicles, and track. Works available in the literature are based on models of trucks with various simplifying assumptions such as reduced degree of freedom, small displacements and without kinematic nonlinearities and inclination of the contact surface. In this paper an integrated bond graph model of a truck is developed for the first time without the aforementioned assumptions. It is found that truck dynamical behavior is significantly different from those found in the literature in some cases.     

Hard real-time multibody simulations using ARM-based embedded systems

The real-time simulation of multibody models on embedded systems is of particular interest for controllers and observers such as model predictive controllers and state observers, which rely on a dynamic model of the process and are customarily executed in electronic control units. This work first identifies the software techniques and tools required to easily write efficient code for multibody models to be simulated on ARM-based embedded systems. Automatic Programming and Source Code Translation are the two techniques that were chosen to generate source code for multibody models in different programming languages. Automatic Programming is used to generate procedural code in an intermediate representation from an object-oriented library and Source Code Translation is used to translate the intermediate representation automatically to an interpreted language or to a compiled language for efficiency purposes. An implementation of these techniques is proposed. It is based on a Python template engine and AST tree walkers for Source Code Generation and on a model-driven translator for the Source Code Translation. The code is translated from a metalanguage to any of the following four programming languages: Python-Numpy, Matlab, C++-Armadillo, C++-Eigen. Two examples of multibody models were simulated: a four-bar linkage with multiple loops and a 3D vehicle steering system. The code for these examples has been generated and executed on two ARM-based single-board computers. Using compiled languages, both models could be simulated faster than real-time despite the low resources and performance of these embedded systems. Finally, the real-time performance of both models was evaluated when executed in hard real-time on Xenomai for both embedded systems. This work shows through measurements that Automatic Programming and Source Code Translation are valuable techniques to develop real-time multibody models to be used in embedded observers and controllers.     

On the modeling of the intervertebral joint in multibody models for the spine

The need to develop feasible computational musculoskeletal models of the spine has led to the development of several multibody models. Central features in these works are models for the ligaments, muscles, and intervertebral joint. The purpose of the present paper is to show how experimental measurements of joint stiffnesses can be properly incorporated using a bushing element. The required refinements to existing bushing force functions in musculoskeletal software platforms are discussed and further implemented using a SpineBushing element specific to the intervertebral joint. Four simple lumbar spine models are then used to illustrate the accompanying improvements. Electronic supplemental material for this article includes a complementary review of formulations of stiffness matrices for the intervertebral joint.     

Design of characteristics of air-pressure-controlled hydraulic shock absorbers in an intercity bus

The air-pressure-controlled shock absorber is capable of changing its damping force depending on the air pressure in the air springs. Due to the possibility of improving dynamic properties of all vehicles that use the axles’ air suspensions, BRANO Inc. (the Czech producer of shock absorbers) started to develop semi-active air-pressure-controlled hydraulic telescopic shock absorbers. The SOR C 12 intercity bus is the reference vehicle for which the research and development of controlled shock absorbers is done and on which the shock absorbers are verified. Force–velocity characteristics of the controlled shock absorbers of the axles’ air suspension were designed on the basis of results of computer simulations with the bus multibody models created in the alaska simulation tool. Multibody models of an empty vehicle, a fully loaded vehicle and three variants of a partly loaded vehicle were created. For each weight of the bus two multibody models of various levels of complexity were created. Since the bus multibody models should be used especially for designing force–velocity characteristics of air-pressure-controlled shock absorbers, great attention (in the framework of the possibilities of multibody dynamics) was paid to the correct interpretation of the real behaviour of hydraulic shock absorbers and air springs of the axles’ suspension. As a criterion for the design of the optimum force–velocity characteristics of the controlled shock absorbers, the maximum similarity of the dynamic responses of multibody models of the bus of all the considered weights to dynamic response of the reference multibody model of the bus with the same load as during the experimental measurements on the real vehicle (approx. 71.5% of the maximum loaded vehicle weight) was chosen. In the course of the measurements the non-controlled shock absorbers’ characteristics were optimally tuned for that vehicle weight. Time histories of relative deflections of the axles’ air springs determined during the simulations of the vehicle running over the vertical artificial obstacle were compared. The approach based on the evaluation of the correlation coefficient of two time series was used for the evaluation of the dynamic responses accordance.     

车辆避障驾驶控制方法研究

受机器人基于人工势场的路径控制方法的启发,在吴镜开等提出方法的基础上,提出了车辆避障控制机制。定义了基于停车视距的车辆行驶动态目标位置;依据人工势场建模原理构建道路和障碍物势场;为了平滑车辆行驶路线,采用了贝塞尔（Bezier）曲线车辆轨迹拟合的方法;针对车辆系统是典型的非完整控制系统,将车辆运动学模型转换为链式系统模型,实现相对简化系统控制设计目的。仿真控制实验说明这种方法对车辆避障控制具有较好的轨迹跟踪效果和全局稳定性。     

Calculating road input data for vehicle simulation

Multibody system simulation is an important tool in the development process in vehicle engineering. Without much effort, different vehicle variants and designs can be simulated, analyzed, and optimized. This is of particular relevance in an early stage of development, when no physical prototypes are available yet. In order to simulate the vehicle models under realistic conditions, suitable input data is needed for the simulation. We present an approach to derive a virtual road profile based on a tire-surrogate model, measured spindle forces, and a multibody system model of the measurement vehicle. In contrast to the measured spindle forces, the road profile together with the tire-surrogate model can be used to simulate other vehicle variants, for which no measurements are available. The road profile is derived by solving an inverse control problem. We formulate this inverse problem in the context of system simulation and provide a short mathematical analysis. Additionally, we discuss a solution approach, the method of control-constraints, which is also applied in a numerical simulation study to compute a virtual road profile.     

Modeling and simulation of longitudinal dynamics coupled with clutch engagement dynamics for ground vehicles

During the engagement of the dry clutch in automotive transmissions, clutch judder may occur. Vehicle suspension and engine mounts couple the torsional and longitudinal models, leading to oscillations of the vehicle body that are perceived by the driver as poor driving quality. This paper presents an effective formulation for the modeling and simulation of longitudinal dynamics and powertrain torsional dynamics of the vehicle based on non-smooth dynamics of multibody systems. In doing so friction forces between wheels and the road surface are modeled along with friction torque in the clutch using Coulomb’s friction law. First, bilateral constraint equations of the system are derived in Cartesian coordinates and the dynamical equations of the system are developed using the Lagrange multiplier technique. Complementary formulations are proposed to determine the state transitions from stick to slip between wheels and road surface and from the clutch. An event-driven scheme is used to represent state transition problem, which is solved as a linear complementarity problem (LCP), with Baumgarte’s stabilization method applied to reduce constraint drift. Finally, the numerical results demonstrate that the modeling technique is effective in simulating the vehicle dynamics. Using this method stick-slip transitions between driving wheel and the road surface and from the clutch, as a form of clutch judder, are demonstrated to occur periodically for certain values of the parameters of input torque from engine, and static and dynamic friction characteristics of tire/ground contact patch and clutch discs.     

3D road traffic situation simulation system

The 3D traffic situation simulation system combines the multibody based mathematical model of a vehicle, the multibody mathematical model of human body, the database of vehicle and human body data and the display subsystem. Together with the model of driving surface the system can be used to simulate and analyse vehicle and its occupant behaviour under different road conditions and different driving regimes. The result obtained this way can be used to investigate safety related parameters and optimise the driver–vehicle–road system regarding to arbitrary criteria (safety, comfort, speed, etc.). The results of simulations are available as numerical data as well as animations in virtual 3D environment.     

Multi-Level Planning for Semi-autonomous Vehicles in Traffic Scenarios Based on Separation Maximization

The planning of semi-autonomous vehicles in traffic scenarios is a relatively new problem that contributes towards the goal of making road travel by vehicles free of human drivers. An algorithm needs to ensure optimal real time planning of multiple vehicles (moving in either direction along a road), in the presence of a complex obstacle network. Unlike other approaches, here we assume that speed lanes are not present and that different lanes do not need to be maintained for inbound and outbound traffic. Our basic hypothesis is to carry forward the planning task to ensure that a sufficient distance is maintained by each vehicle from all other vehicles, obstacles and road boundaries. We present here a 4-layer planning algorithm that consists of road selection (for selecting the individual roads of traversal to reach the goal), pathway selection (a strategy to avoid and/or overtake obstacles, road diversions and other blockages), pathway distribution (to select the position of a vehicle at every instance of time in a pathway), and trajectory generation (for generating a curve, smooth enough, to allow for the maximum possible speed). Cooperation between vehicles is handled separately at the different levels, the aim being to maximize the separation between vehicles. Simulated results exhibit behaviours of smooth, efficient and safe driving of vehicles in multiple scenarios; along with typical vehicle behaviours including following and overtaking.