Real-Time Simulation of Elastic Multibody Systems with Application in Vehicle Dynamics

Real-time simulation models are widely used for vehicle development, usually built up as rigid multibody systems. However, since lightweight structures are commonly used, body deformation is no longer negligible and rigid multibody simulations may be inaccurate. This work presents a real-time capable full vehicle model with a flexible car body, derived from a finite element model, whose performance has been improved by model order reduction. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Blast Damage Simulation With the Discontinuous Galerkin Finite Element Method of Bond⁃Based Peridynamics北大核心CSCD

近场动力学是一种积分型非局部的连续介质力学理论,已广泛应用于固体材料和结构的非连续变形与破坏分析中,其数值求解方法主要采用无网格粒子类的显式动力学方法.近年来,弱形式近场动力学方程的非连续Galerkin有限元法得到发展,该方法不仅可以描述考察体的非局部作用效应和非连续变形特性,还可以充分利用有限单元法高效求解的特点,并继承了有限元法能直接施加局部边界条件的优点,可有效避免近场动力学的表面效应问题.该文阐述了键型近场动力学的非连续Galerkin有限元法的基本原理,导出了计算列式,给出了具体算法流程和细节,计算模拟了脆性玻璃板动态开裂分叉问题,并对爆炸冲击荷载作用下混凝土板的毁伤过程进行了计算分析.研究结果表明,该方法能够再现爆炸冲击荷载作用下结构的复杂破裂模式和毁伤破坏过程,且具有较高的计算效率,是模拟结构爆炸冲击毁伤效应的一种有效方法.     

Transient Finite Element Simulation of the Temperature Field during Cryogenic Turning of Metastable Austenitic Steel AISI 347

The process chain in manufacturing often consists of many steps. As part of current researches the possibility of combining two process steps, turning and hardening, is investigated to optimize the manufacturing time and to decrease the energy consumption of the process. For metastable austenitic steels, deformation induced hardening during turning can be used to achieve surface hardening [1] and thus to increase the wear resistance [2] as well as the fatigue strength [3], by applying high passive forces onto the workpiece. This enables an austenite-martensite phase transformation, for which it is necessary to maintain low process temperatures, typically below room temperature. Thus, cryogenic coolants are applied [4]. For a better understanding of the influence of cutting parameters on the process temperatures and thus martensite formation, knowledge of the exact temperature distribution in the workpiece and in the contact zone between workpiece and tool is essential. Since the experimental determination of the temperature field is hardly possible, an inverse determination of the process temperatures via transient finite element simulation is performed. The present finite element approach only takes thermal loads into account. The simulations are performed in the finite element program FEAP (Finite Element Analysis Program) with an Eulerian mesh, which requires special consideration of the rigid body rotation of the workpiece. In order to prevent unphysical oscillations in the solution, introduced by the convective time derivative, a streamline upwind / Petrov–Galerkin stabilization scheme is utilized. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical and experimental investigation on stochastic dynamic load of a heavy duty vehicle

Numerical simulation and field test are used to investigate tire dynamic load. Based on multi-body dynamics theory, a nonlinear virtual prototype model of heavy duty vehicle (DFL1250A9) is modeled. The geometric structural parameters of the vehicle system, the nonlinear characteristics of shock absorber and leaf springs are precisely described. The dynamic model is validated by testing the data, including vertical acceleration of driver seat, front wheel, intermediate wheel and rear wheel axle head. The agreement between the response of the virtual vehicle model and the measurements on the test vehicle is satisfactory. Using the reliable model, the effects of vehicle speed, load, road surface roughness and tire stiffness on tire dynamic load and dynamic load coefficient (DLC) are discussed. The results demonstrate that the proposed model can offer efficient and realistic simulation for stochastic dynamic loads, so as to investigate vehicle road-friendliness.     

Large Scale Simulation with Scaled Boundary Finite Element Method

Nowadays scientific and engineering applications often require wave propagation in infinite or unbounded domains. In order to model such applications we separate our model into near-field and far-field. The near-field is represented by the well-known finite element method (FEM), whereas the far-field is mapped by a scaled boundary finite element (SBFE) approach. This latter approach allows wave propagation in infinite domains and suppresses the reflection of waves at the boundary, thus being a suitable method to model wave propagation to infinity. It is non-local in time and space. From a computational point of view, those characteristics are a drawback because they lead to storage consuming calculations with high computational time-effort. The non-locality in space causes fully populated unit-impulse acceleration influence matrices for each time step, leading to immense storage consumption for problems with a large number of degrees of freedom. Additionally, a different influence matrix has to be assembled for each time step which yields unacceptable storage requirements for long simulation times. For long slender domains, where many nodes are rather far from each other and where the influence of the degrees of freedom of those distant nodes is neglectable, substructuring represents an efficient method to reduce storage requirements and computational effort. The presented simulation with substructuring still yields satisfactory results. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comportamiento dinámico de viaductos cortos considerando la interacción vehículo-vía-estructura-suelo

The dynamic vehicle-track-bridge-soil interaction is studied on high speed lines. The analysis is carried out using a general and fully three dimensional multi-body-finite element-boundary element model, formulated in the time domain to predict vibrations due to the train passage over the bridge. The vehicle is modelled as a multi-body system, the track and the bridge are modelled using finite elements and the soil is considered as a homogeneous half-space by the boundary element method. Usually, moving force model and moving mass model are employed to study the dynamic response of bridges. In this work, the multi-body system allows one to consider the quasi-static and dynamic excitation mechanisms. Soil-structure interaction is taken into account on the dynamic structure behaviour on simply-supported short span bridges. The influence of soil-structure interaction is analysed in both resonant and non-resonant regimes.     

Optimisation of road vehicle passive suspension systems. Part 1. Optimisation algorithm and vehicle model

Numerous problems have in the past been experienced during the development of military vehicle suspension systems. In order to solve some of these problems a two-dimensional multi-body vehicle dynamics simulation model has been developed for computer implementation. This model is linked to a mathematical optimisation algorithm in order to enable the optimisation of vehicle design parameters through the minimisation of a well defined objective function. In part 1 of this paper the concept of multi-disciplinary design optimisation is discussed. This is followed by the presentation of the up to six degrees of freedom vehicle model developed for this study, and a discussion of the specific gradient-based optimisation algorithm selected for the optimisation. In particular the derivation of the set of second-order differential equations, describing the acceleration of the different solid bodies of the vehicle model, is presented. In order to perform the optimisation of the non-linear suspension component characteristics, a six piece-wise continuous and linear approximation is used which is also described in this paper. Part 2 of this study will outline the simulation programme and the qualification of the programme. It will also present a typical case study where the proposed optimisation methodology is applied to improve the damper characteristics of a specific vehicle.     

频域有限元计算的扩展面向目标误差估计

研究了针对频域有限元直接动态分析的面向目标误差估计以及误差范围估计计算方法．面向目标的误差估计方法就是专门针对如何准确和经济地估算特定值误差的一种方法,利用原问题的共轭偶问题进行计算．频域有限元的直接动态分析是模拟频域扫描实验的一种计算方法,专门针对谐振激励的线性动态响应问题,利用将原自由度分解为实部和虚部描述频率的变化,从而计算变形体的动态响应．利用扩展针对有限元的面向目标误差估计的自由度,将该方法应用到直接动态分析中进行误差估计．通过建立同时包含实部和虚部自由度的能量弱形式及偶问题,并将其数值实现,估算频域直接动态分析有限元解的误差及误差范围,并通过悬臂梁的激振算例进行了验证．     

求解Brinkman方程的修正弱有限元方法

本文针对Brinkman方程引入了一种修正弱Galerkin(MWG)有限元方法.我们通过具有两个离散弱梯度算子的变分形式来逼近模型. 在MWG方法中, 分别用次数为$k$和$k-1$的不连续分段多项式来近似速度函数$u$和压力函数$p$. MWG方法的主要思想是用内部函数的平均值代替边界函数. 因此, 与WG方法相比, MWG方法在不降低准确性的同时, 具有更少的自由度, 对于任意次数不超过$k-1$ 的多项式,MWG方法均可以满足稳定性条件. MWG 方法具有高度的灵活性, 它允许在具有一定形状正则性的任意多边形或多面体上使用不连续函数. 针对$H^1$和$L^22$范数下的速度和压力近似解, 建立了最优阶误差估计. 数值算例表明了该方法的准确性, 收敛性和稳定性.     

A Lagrange multiplier/fictitious domain method for the numerical simulation of incompressible viscous flow around moving rigid bodies: (I) case where the rigid body motions are known a priori

We describe in this Note a method for the numerical simulation of incompressible viscous flow around moving rigid bodies; we suppose the rigid body motions a priori known. The computational technique takes advantage of a time discretization by operator splitting à la Marchuk-Yanenko and of a finite element space discretization on a fixed mesh, to combine a Lagrange multiplier/fictitious domain treatment of the rigid body motions with an L²-projection technique, to force the incompressibility condition. The results of numerical experiments concerning flow around moving disks at Reynolds number of the order of 100 are presented.     

Dynamical modeling and attitude analysis for the spacecraft with lateral solar arrays

The dynamics and attitude motion of the three-axis stabilized spacecraft installed with lateral solar arrays is investigated in terms of the rigid-flexible coupled global modes of the system. The spacecraft consists of a rigid platform with small moment of inertia and two groups of flexible solar arrays with relatively large moment of inertia installed on the rigid rotation shafts. The rigid-flexible coupled dynamic model of the spacecraft is established by using the Hamiltonian Principle. The global mode method is employed to work out the natural frequency and global modal shapes of the rigid-flexible coupled dynamic model combined with corresponding boundary conditions. To validate the effectiveness of the analytical results obtained by global mode method, the natural frequencies and mode shapes obtained from finite element model using MSC.Patran software are used as a reference. A numerical example is given to show that the results obtained from both methods are matched very well (the relative errors of the corresponding frequencies are small enough) and the rigid motion of the platform is coupled with the vibration mode of the flexible solar arrays. This implies that the global analytical modes can be used to accurately describe the rigid-flexible coupled motion of the spacecraft. By comparing with the finite element model, the reduced dynamical model derived in terms of the global modes of the system has a lower dimension. Numerical simulations for the system with variations of parameters and dynamic responses analysis for different applied forces are performed to illustrate that, the characteristics of the model are affected by inner and external factors.     

Hybrid simulation strategy for multiple planar collisions with changing topologies and local deformation

The development of a hybrid strategy for the simulation of multiple plastic-elastic collisions is presented. The strategy attempts to bridge the gap between finite element methods (FEM), which typically require excessively long computation times for multiple impact simulations, and lumped parameter approaches that cannot provide accurate local deformation information. The proposed strategy employs a finite element routine solely to simulate the impact phase, thereby obtaining detailed local deformation information. The simulation of the flight phase between impacts, however, proceeds under rigid body dynamics, resulting in significant reduction in computation time. The transfer of control between FEM and rigid body dynamics is automatic and the points of contact need not be known a priori. The progressive object internal plastic strain, determined from FEM, is retained from one impact to the next, thereby ensuring a certain degree of continuity of the physical properties of the body. An example is presented to demonstrate the efficacy of this approach.     

大垂度柔索的动力学建模与仿真

用多刚体-球铰模型对大垂度柔索进行离散化建模,利用多刚体系统动力学理论建立了该模型的动力学方程.采用矩阵广义逆理论对其位移和速度进行修正,以消除该微分-代数方程在数值分析中的违约现象.数值仿真证明了该方法的正确性.     

Transfer matrix method for the free and forced vibration analyses of multi-step Timoshenko beams coupled with rigid bodies on springs

Multi-step Timoshenko beams coupled with rigid bodies on springs can be regarded as a generalized model to investigate the dynamic characteristics of many structures and mechanical systems in engineering. This paper presents a novel transfer matrix method for the free and forced vibration analyses of the hybrid system. It is modeled as a chain system, where each beam and each rigid body with its supporting spring are dealt with one element, respectively. The transfer equation of each element is deduced based on separation of variables method. The system overall transfer equation is obtained by substituting an element transfer equation into another. Then, the free vibration characteristics are acquired by solving exact homogeneous linear equations. To compute the forced vibration response with modal superposition method, the body dynamic equations and augmented eigenvectors are established, and the orthogonality of augmented eigenvectors is mathematically proved. Without high-order global dynamic equation or approximate spatial discretization, the free and forced vibration analyses of the hybrid system are achieved efficiently and accurately in this study. As an analytical approach, the present method is easy, highly stylized, robust, powerful and general for the complex hybrid systems containing any number of Timoshenko beams and rigid bodies. Four numerical examples are implemented, and the results show that this method is computationally efficient with high precision.     

Object-oriented modelling and simulation of a motorcycle

The design of a control oriented motorcycle model for the simulation of two-wheeled vehicles is widely recognized to be a very challenging task, as a complete analytical model is not directly available, due to its complexity and its high sensitivity to parameters' variations. Accordingly, a reliable model should be based on multibody modelling tools endowed with automated symbolic manipulation capabilities. This paper presents a simulation model for the dynamic behaviour of a motorcycle based on the object-oriented modelling paradigm developed in Modelica, within the Dymola environment. Specifically, we illustrate the modular approach to motorcycle modelling and discuss the tire-road interaction model, which is the crucial part of the proposed model. The validity of the proposed simulation model is assessed on real data, measured on an instrumented test vehicle. Further, to perform the verification phase a virtual driver model has been implemented, which allows to track both a roll angle and a target speed profile during different maneuvers. In particular, the behaviour of the driver is modelled as an inverse pendulum, with a rotational degree of freedom along the forward axis. This allows accounting for the driver lean angle, which is necessary to fully capture the real driving behaviour and its effects on the overall vehicle dynamics.     

On the modelling of finite growth considering the mechanics of cell division

Mitotic cells grow in volume and divide themselves into two identical cells producing at macroscopic scale a volume expansion in living bodies. Due to inhomogeneous distributions of the growth factors, growth occurs at different rates and directions. Focusing into the direction of growth, some living bodies alter their growing behaviour influenced by mechanical loads. If loads appear during the growth process, cell division is reorientated following the main direction of the elastic deformations. Therefore, new cells will be created in this direction while relaxing the stress state of the body at the same time. In this work, we present a modelling approach for growing bodies which change their growth direction depending on mechanical loads. The model is implemented into a finite element framework to be an useful tool for predicting morphological changes in growing bodies. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Correlation between the properties of eigenfrequencies and eigenmodes in a chain of rigid bodies with torque connections

A chain of N rigid bodies with elastic connections transmitting a torque interaction is considered. Similar individual inertial elements of the chain have one degree of freedom. An exact analytical solution of the problem of oscillation eigenfrequencies and eigenmodes is constructed for an arbitrary value of N. The properties of frequencies and modes of such a chain are compared with those of a canonical Newton chain. It is established that the correlation between eigenfrequencies and the properties of the corresponding eigenmodes fundamentally differs from that for a canonical Newton chain. Lower eigenfrequencies of a chain of rigid bodies with inertialess torque connections correspond to modes with a larger number of nodes, while higher eigenfrequencies correspond to the smoother modes. The nontypical correlation between the oscillation eigenfrequencies and eigenmodes discovered based on the exact analytical solution to the problem of free oscillations of a chain of N rotating rigid bodies contradicts the ideas underlying theoretical studies in the field of solid state physics devoted to simulation of mechanical and thermal dynamic processes in crystalline lattices.     

三维横向流体诱发直管振动的数值模拟

基于有限体积法和有限元法,结合动网格控制技术,建立了横向流体作用下三维弹性直管流致振动计算的数值模型,实现了计算结构动力学与计算流体力学之间的联合仿真．首先,通过对刚性管的静止绕流计算,研究了网格离散方式和不同湍流模型对圆柱类结构静止绕流流场特征的影响和预测能力,得到了适用于双向耦合分析的CFD模型;其次,利用基于双向流固耦合方法的流致振动模型,计算并分析了流体力与结构位移间的相位关系,指出流体力与位移间的相位差是由流体力引起的,同时对双向耦合和单向耦合进行了比较分析;最后通过对直管流致振动的数值计算,联合管表面压力、尾流区时均速度、分离角等时均量,分析了尾流区的流场特征.