Conserving Properties in Constrained Dynamics of Flexible Multibody Systems

The context of this work is the non-linear dynamics ofmultibody systems (MBS). The approach followed for parametrisation ofrigid bodies is the use of inertial coordinates, forming a dependent setof parameters. This approach mixes naturally with nodal coordinates in adisplacement-based finite element discretisation of flexible bodies,allowing an efficient simulation for MBS dynamics. An energy-momentumtime integration algorithm is developed within the context of MBSconstraints enforced through penalty methods. The approach follows theconcept of a discrete derivative for Hamiltonian systems proposed byGonzalez, achieving exact preservation of energy and momentum. Thealgorithm displays considerable stability, overcoming the traditionaldrawback of the penalty method, namely numerical ill-conditioning thatleads to stiff equation systems. Additionally, excellent performance isachieved in long-term simulations with rather large time-steps.     

Superelements Modelling in Flexible Multibody Dynamics

We present a new implementation of substructuring methods forflexible multibody analysis. In previous developed formulations, wefixed the local axes of the superelement to one node. In thisformulation, the reference frame is floating and close, in some sense,to the body center. The local frame is selected based on the positionsof the interface nodes of the superelement, and completely independentof the order in which the nodes of the superelement are given.Therefore, the superelement itself depends only on the nodes positions,and on the mass and stiffness properties, thus allowing a very easyinterfacing between the finite element program which computed thesuperelement and the mechanism analysis program.     

Dynamics of Flexible Multibody Systems with Non-Holonomic Constraints: A Finite Element Approach

In this article it is shown how non-holonomic constraints can beincluded in the formulation of the dynamic equations of flexiblemultibody systems. The equations are given in state space formwith the degrees of freedom, their derivatives and the kinematiccoordinates as state variables, which circumvents the use ofLagrangian multipliers. With these independent state variables forthe system the derivation of the linearized equations of motion isstraightforward. The incorporation of the method in a finiteelement based program for flexible multibody systems is discussed.The method is illustrated by three examples, which show, amongother things, how the linearized equations can be used to analysethe stability of a nominal steady motion.     

Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints

The use of multibody formulations based on Cartesian or naturalcoordinates lead to sets of differential-algebraic equations that haveto be solved. The difficulty in providing compatible initial positionsand velocities for a general spatial multibody model and the finiteprecision of such data result in initial errors that must be correctedduring the forward dynamic solution of the system equations of motion.As the position and velocity constraint equations are not explicitlyinvolved in the solution procedure, any integration error leads to theviolation of these equations in the long run. Another problem that isvery often impossible to avoid is the presence of redundant constraints.Even with no initial redundancy it is possible for some systems toachieve singular configurations in which kinematic constraints becometemporarily redundant. In this work several procedures to stabilize thesolution of the equations of motion and to handle redundant constraintsare revisited. The Baumgarte stabilization, augmented Lagrangian andcoordinate partitioning methods are discussed in terms of theirefficiency and computational costs. The LU factorization with fullpivoting of the Jacobian matrix directs the choice of the set ofindependent coordinates, required by the coordinate partitioning method.Even when no particular stabilization method is used, a Newton–Raphsoniterative procedure is still required in the initial time step tocorrect the initial positions and velocities, thus requiring theselection of the independent coordinates. However, this initialselection does not guarantee that during the motion of the system otherconstraints do not become redundant. Two procedures based on the singlevalue decomposition and Gram–Schmidt orthogonalization are revisited forthe purpose. The advantages and drawbacks of the different procedures,used separately or in conjunction with each other and theircomputational costs are finally discussed.     

Nonholonomic Multibody Dynamics

The paper deals with the nonholonomic multibody system dynamics from apoint of view which is caused by some actual applications in high-tecareas like high-speed train technology or biomechanics of somedisciplines in high-performance sports. Obviously, looking at suchproblems, there are very close connections between classical analyticaldynamics, differential geometry and modern control theory. But theseconnections cannot be used to get new composed results in solvingcomplicated problems of multibody system dynamics because correspondingsoftware tools are not enough in tune with each other. This paper willgive some ideas for developing a unified basis for modeling, simulationand control of nonholonomic multibody systems.First, a derivative-free approach for generating Lagrangian motionequations of multibody systems with kinematical tree structure as wellas for constrained multibody systems is given. This has been done byusing differential-geometric concepts in a Riemannian space. Secondly,the well-known theorem of Frobenius is considered with respect to itsclassical interpretation by the so-called object of nonholonomy as wellas by its modern interpretation in the nonlinear control theory usingLie-brackets. The ideas are illustrated by the classical rollingcondition and edge condition on double-curved surfaces. Specialnumerical problems in simulation of multibody systems subject toadditional kinematic constraints are discussed. Finally threeapplications are given.     

Quadratic and Higher-Order Constraints in Energy-Conserving Formulations of Flexible Multibody Systems

The treatment of constraints is considered here within the framework ofenergy-momentum conserving formulations for flexible multibody systems.Constraint equations of various types are an inherent component of multibodysystems, their treatment being one of the key performance features ofmathematical formulations and numerical solution schemes.Here we employ rotation-free inertial Cartesian coordinates of points tocharacterise such systems, producing a formulation which easily couples rigidbody dynamics with nonlinear finite element techniques for the flexiblebodies. This gives rise to additional internal constraints in rigid bodies topreserve distances. Constraints are enforced via a penalty method, which givesrise to a simple yet powerful formulation. Energy-momentum time integrationschemes enable robust long term simulations for highly nonlinear dynamicproblems.The main contribution of this paper focuses on the integration of constraintequations within energy-momentum conserving numerical schemes. It is shownthat the solution for constraints which may be expressed directly in terms ofquadratic invariants is fairly straightforward. Higher-order constraints mayalso be solved, however in this case for exact conservation an iterativeprocedure is needed in the integration scheme. This approach, together withsome simplified alternatives, is discussed.Representative numerical simulations are presented, comparing the performanceof various integration procedures in long-term simulations of practicalmultibody systems.     

A Modal Multifield Approach for an Extended Flexible Body Description in Multibody Dynamics

This paper presents a new methodology to simulate the behaviour of flexible bodies influenced by multiple physical field quantities in addition to the classical mechanical terms. The theoretical framework is based on the extended Hamilton Principle and an adapted modal multifield approach. Furthermore, the use of finite element analysis for the necessary data preprocessing is explained. Numerical solution strategies for the coupled system of differential equations with different time scale properties are mentioned. The method is applied to simulate a structure with distributed piezo-ceramic devices inducing an additional electrostatic field. Two thermoelastic problems, which have to consider the influence of spatial temperature distribution, also demonstrate the benefits of the presented approach.     

Flexible Multibody Dynamics: Review of Past and Recent Developments

In this paper, a review of past and recent developments in the dynamics of flexible multibody systems is presented. The objective is to review some of the basic approaches used in the computer aided kinematic and dynamic analysis of flexible mechanical systems, and to identify future directions in this research area. Among the formulations reviewed in this paper are the floating frame of reference formulation, the finite element incremental methods, large rotation vector formulations, the finite segment method, and the linear theory of elastodynamics. Linearization of the flexible multibody equations that results from the use of the incremental finite element formulations is discussed. Because of space limitations, it is impossible to list all the contributions made in this important area. The reader, however, can find more references by consulting the list of articles and books cited at the end of the paper. Furthermore, the numerical procedures used for solving the differential and algebraic equations of flexible multibody systems are not discussed in this paper since these procedures are similar to the techniques used in rigid body dynamics. More details about these numerical procedures as well as the roots and perspectives of multibody system dynamics are discussed in a companion review by Schiehlen [79]. Future research areas in flexible multibody dynamics are identified as establishing the relationship between different formulations, contact and impact dynamics, control-structure interaction, use of modal identification and experimental methods in flexible multibody simulations, application of flexible multibody techniques to computer graphics, numerical issues, and large deformation problem. Establishing the relationship between different flexible multibody formulations is an important issue since there is a need to clearly define the assumptions and approximations underlying each formulation. This will allow us to establish guidelines and criteria that define the limitations of each approach used in flexible multibody dynamics. This task can now be accomplished by using the absolute nodal coordinate formulation which was recently introduced for the large deformation analysis of flexible multibody systems.     

Multibody Dynamics of Very Flexible Damped Systems

An efficient multibody dynamics formulation is presented for simulating the forward dynamics of open and closed loop mechanical systems comprised of rigid and flexible bodies interconnected by revolute, prismatic, free, and fixed joints. Geometrically nonlinear deformation of flexible bodies is included and the formulation does not impose restrictions on the representation of material damping within flexible bodies.The approach is based on Kane's equation without multipliers and the resulting formulation generates 2ndof+m first order ordinary differential equations directly where ndof is the smallest number of system degrees of freedom that can completely describe the system configuration and m is the number of loop closure velocity constraint equations. The equations are integrated numerically in the time domain to propagate the solution.Flexible bodies are discretized using a finite element approach. The mass and stiffness matrices for a six-degree-of-freedom planar beam element are developed including mass coupling terms, rotary inertia, centripetal and Coriolis forces, and geometric stiffening terms.The formulation is implemented in the general purpose multibody dynamics computer program flxdyn. Extensive validation of the formulation and corresponding computer program is accomplished by comparing results with analytically derived equations, alternative approximate solutions, and benchmark problems selected from the literature. The formulation is found to perform well in terms of accuracy and solution efficiency.This article develops the formulation and presents a set of validation problems including a sliding pendulum, seven link mechanism, flexible beam spin-up problem, and flexible slider crank mechanism.     

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.     

An Efficient Implementation of the Recursive Approach to Flexible Multibody Dynamics

This paper deals with the efficient extension of the recursive formalism (articulated body inertia) for flexible multibody systems. Present recursive formalisms for flexible multibody systems require the inversion of the mass matrix with the dimension equal to the number of flexible degrees of freedom of particular bodies. This is completely removed. The paper describes the derivation of equations of motion expressed in the local coordinate system attached to the body, then the discretization of these equations of motion based on component mode synthesis and FEM shape functions and, finally, two versions of the new recursive formalism.     

Teaching Multibody Dynamics from Modeling to Animation

This paper presents a student project which takes place just after the lecture in classical mechanics for undergraduate students in engineering. The pedagogical objectives of this learning layer cover various aspects, namely: give the student the opportunity to exploit and analyze the equations of motion for a real application, make them able to formulate consistent hypotheses for such applications and promote an actual multi-disciplinarity activity (mechanics, numerical methods, computer science and CAD). The project is performed by groups of 6–8 students and is organized in the frame of a global active pedagogical process which characterizes our undergraduate engineering.     

多体系统动力学数值解法

多体系统动力学研究的主要内容动力学建模与数值解法是多体系统动力学研究的主要内容之一。对多体系统动力学方程及其动力学数值解法的研究成果进行了较为全面的阐述。多体系统动力学及动力学方程进行了简单的归纳和总结,多体系统动力学数值求解,特别是刚柔耦合多体系统微分／代数方程的数值解法等研究热点进行了详细的阐述,并简要展望了多体系统动力学数值解法今后的发展趋势,为多体系统动力学计算机仿真奠定了基础。     

Mooring Systems – A Multibody Dynamic Approach

A method for simulating the motion behaviour of moored floatingoffshore structures in the time domain is presented. Theinteraction between the fluid and the floating structure isconsidered using linear potential theory. The hydrodynamic forcesacting on the mooring lines are computed using a modified Morisonequation. A complete three-dimensional model of the structure andthe mooring lines is generated using a multibody system approachincluding a subsystem technique. The model results in a largenumber of degrees of freedom. In order to illustrate a practicalapplication of this method, an analysis of a moored ponton in arandom sea is presented. Different configurations of the systemare examined in order to evaluate the motion behaviour and therestoring forces. Comparisons are made with the naturalfrequencies of the damped system.     

Relative Finite Element Coordinates in Multibody System Simulation

In the paper a numerical approach for deriving the nonlinear explicitform dynamic equations of rigid and flexible multibody systems ispresented. The dynamic equations are obtained as Ordinary DifferentialEquations for generalized coordinates and without algebraic constraints.The Finite Element Theory is applied for discretization of flexiblebodies. The minimal set of the generalized coordinates includesindependent joint motions, as well as independent small flexibledeflections of finite element nodes. The node deflections and stiffnessmatrices are calculated with respect to the moving relative coordinatesystems of the flexible bodies. The positions and orientations ofelement and substructure coordinate systems are updated according to thenode deflections. A major step of the numerical process is the kinematicanalysis and calculation of matrices of partial derivatives of thequasi-coordinates (dependent joint motions and coordinates of points andnodes) with respect to the generalized coordinates. The inertia terms inthe dynamic equations are obtained multiplying the matrices of thepartial derivatives by the mass matrices of the rigid and flexiblebodies. Stiffness properties of flexible bodies are presented in thedynamic equations by stiff forces that depend on the generalizedrelative flexible deflections only. Several examples of large motion ofbeam structures show the effectiveness of the algorithm.     

Dynamics of Flexible Multibody Systems Using Virtual Work and Linear Graph Theory

By combining linear graph theory with the principle of virtualwork, a dynamic formulation is obtained that extends graph-theoreticmodelling methods to the analysis of flexible multibody systems. Thesystem is represented by a linear graph, in which nodes representreference frames on rigid and flexible bodies, and edges representcomponents that connect these frames. By selecting a spanning tree forthe graph, the analyst can choose the set of coordinates appearing inthe final system of equations. This set can include absolute, joint, orelastic coordinates, or some combination thereof. If desired, allnon-working constraint forces and torques can be automaticallyeliminated from the dynamic equations by exploiting the properties ofvirtual work. The formulation has been implemented in a computerprogram, DynaFlex, that generates the equations of motion in symbolicform. Three examples are presented to demonstrate the application of theformulation, and to validate the symbolic computer implementation.     

A Gluing Algorithm for Network-Distributed Multibody Dynamics Simulation

The improved performance and capacity of networks has made thecombined processing power of workstation clusters a potentiallypromising avenue for solving computationally intensive problems acrosssuch distributed environments. Moreover, networks provide an idealplatform to employ heterogeneous hardware and software to solvemultibody dynamics problems. One fundamental difficulty with distributedsimulation is the requirement to couple and synchronize the distributedsimulations. This paper focuses on the algorithms necessary to coupletogether separately developed multibody dynamics modules so that theycan perform integrated system simulation. To identify a useful couplingstrategy, candidate numerical algorithms in the literature are reviewedbriefly – namely, stiff time integration, local parameterization,waveform relaxation, stabilized constraint and perturbation. Anunobtrusive algorithm that may well serve this `gluing' role ispresented. Results from numerical experiments are presented and theperformance of the gluing algorithm is investigated.     

Dynamics of Multibody Systems Subjected to Impulsive Constraints

This paper presents a procedure for studying dynamics of multibodysystems subjected to impulsive constraints which may be either holonomicor nonholonomic. The procedure automatically incorporates the effects ofimpulsive constraints through its analysis. The governing equationsthemselves are developed from Kane's equations, using partial velocityvectors and partial angular velocity vectors. Explicit expressions forthe coefficients and terms of the governing equations are presented. Twosubcases are studied: (1) the constraints are instantaneously applied andcontinue to act; and (2) the constraints are instantaneously applied andlifted immediately after completion of impact. The internal impulses ateach joint and constraint impulses associated with the impulsiveconstraints are calculated. The procedure is checked with two examples,whose solutions are established.     

A Subsystem Synthesis Method for Efficient Vehicle Multibody Dynamics

A subsystem synthesis method has been proposed for dynamicanalysis of a vehicle multibody system that consists of severalsubsystems. In this method, each subsystem can be independently analyzedwith a virtual reference body. For overall vehicle system analysis,subsystems can be synthesized to the chassis with effective inertiamatrix and force vector from the virtual reference body of eachsubsystem. Using this matrix and vector, a fixed size (6 × 6) of equationsof motion for a chassis can be formed and independently solved. There after, equations of motion for each subsystem are solved, subsystem bysubsystem. Computational efficiency of the proposed method has been alsoinvestigated theoretically through the operational counting method. Inorder to show the effectiveness of the method, a Short Long Arm (SLA)vehicle suspension subsystem has been analyzed.