Computational Schemes for Flexible,Nonlinear Multi-Body Systems

This paper deals with the development of computational schemes for the dynamic analysis of flexible, nonlinear multi-body systems. The focus of the investigation is on the derivation of unconditionally stable time integration schemes for these types of problem. At first, schemes based on Galerkin and time discontinuous Galerkin approximations applied to the equations of motion written in the symmetric hyperbolic form are proposed. Though useful, these schemes require casting the equations of motion in the symmetric hyperbolic form, which is not always possible for multi-body applications. Next, unconditionally stable schemes are proposed that do not rely on the symmetric hyperbolic form. Both energy preserving and energy decaying schemes are derived that both provide unconditionally stable schemes for nonlinear multi-body systems. The formulation of beam and flexible joint elements, as well as of the kinematic constraints associated with universal and revolute joints. An automated time step selection procedure is also developed based on an energy related error measure that provides both local and global error levels. Several examples of simulation of realistic multi-body systems are presented which illustrate the efficiency and accuracy of the proposed schemes, and demonstrate the need for unconditional stability and high frequency numerical dissipation.     

On the Modeling of Friction and Rolling in Flexible Multi-Body Systems

This paper is concerned with the dynamic analysis of flexible, nonlinear multi-body systems undergoing contact involving friction and rolling. A continuous friction law is used to model the friction forces between contacting bodies. This avoids the numerical problems associated with the discontinuity inherent to Coulomb's friction law and eliminates the need for different sets of equations modeling sliding and rolling as distinct phenomena. On the other hand, continuous friction laws eliminate specific physical phenomena implied by Coulomb's friction law. The condition of vanishing relative velocity between two contacting bodies is not possible: sticking or rolling are replaced by creeping with a small relative velocity. Discrete events such as transition from slipping to rolling or rolling to slipping are eliminated, together with the high frequency phenomena they are likely to cause. The computational issues associated with the continuous friction law and with the enforcement of the non-holonomic rolling constraint are addressed in this paper. This work is developed within the framework of energy preserving and decaying time integration schemes that provide unconditional stability for nonlinear, flexible multi-body systems undergoing contact involving friction and rolling.     

Robust Optimal Design of Multi-Body Systems

Techniques in robust optimal design of multi-body systems are presented.In order to define the concept of robust optimal design, a performancevariance minimization method is applied on variations of designvariables. The regression model is estimated through a response surfacemethodology to evaluate the performance variance of multi-body systems.The experiments necessary for the response surface methodology are notconducted through physical tests but through nonlinear multi-bodydynamic simulation. The techniques presented are applied to arotary-type compressor in order to apply them to the robust optimaldesign of a weight balancer for use in vibration control and to show theeffectiveness of the techniques.     

多体系统仿真分析平台子系统建模技术

针对多体系统建模效率低、模型重用困难等问题,研究了复杂多体系统模型的子系统建模技术.提出支持系统分解和模型重用的子系统结构,采用虚部件定义子系统外部拓扑连接关系,通过主控参数封装模型设计知识和内部参数信息,从而建立自顶向下的层次化子系统结构模型;然后依据子系统主控参数关联关系及层次化结构特点,提出复杂多体系统参数化求解与装配算法.最后以卡车整车建模实例,介绍复杂多体系统的子系统建模过程.     

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.     

Multi-body dynamics including the effects of flexibility and compliance

New and recently developed concepts and ideas useful in obtaining efficient computer algorithms for solving the equations of motion of multi-body mechanical systems with flexible links are presented and discussed. These ideas include the use of Euler parameters, Lagrange's form of d'Alembert's principle, generalized speeds, quasi-coordinates, relative coordinates, and structural analysis techniques. The mechanical systems considered are linked bodies forming a tree structure, but with no “closed loops” permitted. An explicit formulation of the governing equations is presented.     

A theoretical and numerical comparison of elastic nonlinear finite element methods

A theoretical and numerical comparison is made of five finite element formulations which have been presented for the nonlinear elastic analysis of planar structures. The nonlinear force-displacement equations associated with these formulations are derived from a common starting point. In this manner the assumptions and, therefore, the limits of applicability inherent in each method is readily seen. Three solution procedures (iteration, load incrementation and displacement incrementation) are used to apply these formulations to a group of problems that covers a wide range of types of nonlinearity. Among the types of behavior examined are unlimited hardening behavior, unlimited softening behavior, softening-hardening behavior with no instability, and softening-hardening behavior with instability.     

Substructure synthesis methods for dynamic analysis of multi-body systems

The formulation for the dynamic analysis of flexible multi-body systems that undergo large rigid body motion, leads to geometrically non-linear inertia properties due to large rotations. These inertia non-linearities that represent the coupling between gross rigid body motion and small elastic deformation, are dependent on the assumed displacement field. As alternatives to the finite element methods, deformable body shape functions and shape vectors are commonly employed to describe elastic deformation of linear structures. In this paper, substructure shape functions and shape vectors are used to describe elastic deformation of non-linear inertia-variant multi-body systems. This leads to two different representations of inertia nonlinearities; one is based on a consistent mass formulation, while the other is a lumped mass technique. The multi-body systems considered are collections of interconnected rigid and flexible bodies. Open and closed loop systems are permitted.     

Comparison of different formulations of 2D beam elements based on Bond Graph technique

Bond Graphs are well suited for modelling multibody systems. In this paper modelling of planar flexible beams undergoing large overall motions are studied based on finite element (FE) technique. Two well-known approaches are used – the co-rotational (CR) and absolute nodal coordinate (ANC) formulation. Two ANC formulations are analyzed – one in which elastic forces is described using classical beam theory in a local coordinate frame, and another based on a global continuum mechanics approach. Starting from these classical formulations velocity formulations are developed and used to develop Bond Graph FE components. The effect of gravity has been considered as well. These components can be put in libraries and used for systematic Bond Graph flexible body model development. It is shown that Bond Graph technique is capable of dealing with different flexible body formulations and can be used as a general approach in parallel to other modelling approaches. Models are developed and simulations are performed using the object oriented environment of BondSim. Owing to the object oriented approach, transformation from one to the other model is relatively simply. The results are illustrated by suitable examples and they confirm accuracy of the developed models. It was shown that the CR approach offers much better performance than the both ANC formulations.     

Experimental Validation of a Flexible MBS Dynamic Formulation through Comparison between Measured and Calculated Stresses on a Prototype Car

Numerical methods for the dynamics of flexible multi-body systems havebeen developed by the related scientific and technical community duringthe last two or three decades. However, most examples which can be foundin the bibliography are either very simple and/or have not beenconfronted with the results of real tests. The solution of realistic andcomplex problems along with their corresponding experimental validation,will be useful to evaluate the different formulations available, andwill make them more valuable for the industrial world.On the other hand, stress calculation is essential for themechanical design of machines and mechanisms. An adequate way tonumerically solve this problem consists of modeling the flexibility ofthose links whose stresses are of interest for the designer. In theexperimental field, the well-established technique of extensometryenables the measurement of the stresses suffered by the actualcomponents.Therefore, the dynamic simulation of realistic and complexmulti-body systems giving as output the stresses undergone by componentsduring the motion, and the comparison with the correspondingexperimental results, should be attempted in order to credit thevalidity of a certain numerical method. In this work, the abovementionedobjective has been addressed. A prototype car has been taken as anexample of complex and realistic multi-body system. The actual vehiclehas been built and its virtual counterpart has been implemented on acomputer.     

Mimetic discretization of two-dimensional Darcy convection

We consider discretization of the planar convection of the incompressible fluid in a porous medium filling rectangular enclosure. This problem belongs to the class of cosymmetric systems and admits an existence of a continuous family of steady states in the phase space. Mimetic finite-difference schemes for the primitive variables equation are developed. The connection of a derived staggered discretization with a finite-difference approach based on the stream function and temperature equations is established. Computations of continuous cosymmetric families of steady states are presented for the case of uniform and nonuniform grids.     

Analysis of a class of non-linear and time-varying parameter feedback systems†

The paper presents some simple methods of analysis for a class of feedback systems. A method of feedback iteration is presented, directly applicable to non-linear and time-varying parameter feedback systems, without involving any intermediate formulations to obtain the response to a given input. Then, a numerical method closely following that of Naumov is described. Next, methods of predetermining the time-varying parameters for specified output are presented. One is a direct method of backward signal flow and the other is a numerical method based on that of Naumov (1957). These methods have the advantage of lending themselves to digital computation. The method of successive iteration round the feedback loop was first used in the classic paper by Nyquist (1932).     

Sensitivity analysis of steel moment frames accounting for geometric and material nonlinearity

A procedure for sensitivity analysis of planar steel moment frameworks accounting for second-order displacement effects and inelastic material behavior under combined bending moment and axial force is presented in the context of performance-based design for seismic loading. Analytical formulations defining the sensitivity of nodal displacements to modifications in member cross-section sizes are derived. A nine-story moment frame example illustrates the applicability and accuracy of the developed formulations.     

Data structures for the distributed iterative solution of non-conventional finite element models

A class of specialised data structures designed for the distributed solution of non-conventional finite element formulations, which are equally effective when used in conjunction with conventional formulations, is presented. We begin by briefly discussing how the non-conventional finite element formulations being developed within the structural analysis group at IST [Freitas JAT, Almeida JPM, Pereira EMBR. Non-conventional formulations for the finite element method. Comput Mech 1999;23(5–6):488–501] lead to systems of equations that appear to be naturally suited for parallel processing, but we also recognise that to take full advantage of the characteristics of these systems – large dimension, non-overlapping block structure and sparsity – it is necessary to use appropriate data structures. The approach presented, which references the logical subdivisions of the system matrices, was designed to fulfil these objectives. Examples of parallel performance and efficiency on an homogeneous distributed platform are presented.     

基于微分/代数方程的多体系统动力学设计灵敏度分析的伴随变量方法

基于多体系统动力学微分/代数方程数学模型和通用积分形式的目标函数,建立了多体系统动力学设计灵敏度分析的伴随变量方法,避免了复杂的设计灵敏度计算,对于设计变量较多的多体系统灵敏度分析具有较高的计算效率.文中给出了通用公式以及具体的计算过程和验证方法,并将目标函数及其导数积分形式的计算转化为微分方程的初值问题,进一步提高了计算效率和精度.文末通过一曲柄-滑块机构算例对算法的有效性进行了验证.     

On relaxed LMI-based designs for fuzzy regulators and fuzzy observers

Relaxed conditions for stability of nonlinear, continuous and discrete-time systems given by fuzzy models are presented. A theoretical analysis shows that the proposed methods provide better or at least the same results of the methods presented in the literature. Numerical results exemplify this fact. These results are also used for fuzzy regulators and observers designs. The nonlinear systems are represented by fuzzy models proposed by Takagi and Sugeno (1985). The stability analysis and the design of controllers are described by linear matrix inequalities, that can be solved efficiently using convex programming techniques. The specification of the decay rate, constrains on control input and output are also discussed.     

考虑热效应的复合材料多体系统动力学研究

研究离心力和温度变化引起的附加弯曲变形对复合材料柔性多体系统振动特性的影响．从本构关系和非线性应变与位移关系式出发,用虚功原理和有限单元法建立了复合材料柔性梁的动力学变分方程,在此基础上建立了复合材料柔性多体系统的动力学方程．对曲柄-连杆-滑块机构的数值仿真表明,对于非对称的复合材料梁,各层弹性模量和热膨胀系数的差异会引起附加弯曲变形,从而影响系统的振动特性．     

Position and stiffness analysis of a new asymmetric 2PRR–PPR parallel CNC machine

In this paper, structural stiffness analysis of a new 3-axis asymmetric planar parallel manipulator, a 2 P RR–P P R structural kinematic chain, is investigated. The manipulator is proposed as a tool holder for a 5-axis hybrid computer numerical control (CNC) machine. First, the structure of the robot is introduced and inverse kinematics solution is presented. Secondly, stiffness matrix of the robot is determined using a continuous method based on Castigliano’s theorem and calculation of strain energy of the robot components. This method removes the need for commonly used simplifying assumptions and, therefore, results in good accuracy. For this purpose, force and strain energy for each segment of the robot are analyzed. Finally, to verify the analytical results, commercial FEM software is used to simulate the physical structure of the manipulator. A numerical example is presented which confirms the correctness of the analytical formulations.     

Finite time agreement protocol design of multi‐agent systems with communication delays

In this paper, the finite‐time agreement problem of continuous‐time multi‐agent systems with communication delays is considered. First, the multi‐agent system researched in the paper is described. Second, some notations and lemmas used in the paper are given. Then, agreement protocols for continuous (but non‐smooth) multi‐agent systems are proposed, which ensure that the states of agents reach agreement in finite time. The stability analysis and simulations are presented to show the effectiveness of the method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Nonsmooth output feedback stabilization of a class of genuinely nonlinear systems in the plane

In this note, we address the problem of output feedback stabilization for a class of planar systems that are inherently nonlinear in the sense that the linearized system at the origin is neither controllable nor observable. Moreover, the uncontrollable modes contain eigenvalues on the right-half plane. By the well-known necessary condition, such planar systems cannot be stabilized, even locally by any smooth output feedback, and hence must be dealt with by nonsmooth output feedback. The main contribution of this work is the development of a non-Lipschitz continuous output feedback design method that leads to a solution to the problem. The proposed output feedback control scheme is not based on the separation principle but rather, relies on the design of a reduced-order nonlinear observer from an earlier paper with an appropriate twist, and the tool of adding a power integrator. A non-Lipschitz continuous output feedback controller is explicitly constructed, achieving global stabilization of the planar systems without imposing the high-order growth conditions required in a previous paper.