An Implementation Method for Constrained Flexible Multibody Dynamics Using a Virtual Body and Joint

A convenient implementation method for constrained flexiblemultibody dynamics is presented by introducing a virtual rigid body andjoint. The general purpose program for rigid and flexible multibodydynamics consists of three major parts of a set of inertia modules, aset of force modules, and a set of joint modules. Whenever a new forceor joint module is added to the general purpose program, the modules forthe rigid body dynamics are not reusable for the flexible body dynamics.Consequently, the corresponding modules for the flexible body dynamicsmust be formulated and programmed again. Since the flexible bodydynamics handles more degrees of freedom than the rigid body dynamicsdoes, implementation of the module is generally complicated and prone tocoding mistakes. In order to overcome these difficulties, a virtualrigid body is introduced at every joint and force reference frames. Newkinematic admissibility conditions are imposed on two-body referenceframes of virtual and original bodies by introducing a virtual flexiblebody joint. There are some computational overheads due to the additionalbodies and joints. However, since computation time is mainly dependenton the frequency of flexible body dynamics, the computational overheadof the presented method is not a critical problem, while implementationconvenience is dramatically improved.     

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.     

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.     

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.     

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.     

Dynamic Modelling of Electromechanical Multibody Systems

A unified methodology for modelling electromechanical multibody systemsis presented. The systems are comprised of rigid or flexible multibodysub-systems and electrical networks of analog components. The sub-systemsare coupled by transducers such as DC motors, moving-plate capacitors,and moving-coil inductors. The electromechanical system is represented bya single graph representation; linear graph theory is then used to generatea relatively small number of system equations. The graph-theoretic formulationis efficient, unifying, and systematic, andwas readily implemented in a computer algorithm using symbolic programming.The formulation and computer implementation are demonstrated using twoexamples of electromechanical systems: a simple condensator microphone anda robot manipulator actuated by DC motors.     

Flexible Multibody Systems Models Using Composite Materials Components

The use of a multibody methodology to describe the large motion of complex systems that experience structural deformations enables to represent the complete system motion, the relative kinematics between the components involved, the deformation of the structural members and the inertia coupling between the large rigid body motion and the system elastodynamics. In this work, the flexible multibody dynamics formulations of complex models are extended to include elastic components made of composite materials, which may be laminated and anisotropic. The deformation of any structural member must be elastic and linear, when described in a coordinate frame fixed to one or more material points of its domain, regardless of the complexity of its geometry. To achieve the proposed flexible multibody formulation, a finite element model for each flexible body is used. For the beam composite material elements, the sections properties are found using an asymptotic procedure that involves a two-dimensional finite element analysis of their cross-section. The equations of motion of the flexible multibody system are solved using an augmented Lagrangian formulation and the accelerations and velocities are integrated in time using a multi-step multi-order integration algorithm based on the Gear method.     

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.     

Analysis of Flexible Multibody Systems with Intermittent Contacts

This paper is concerned with the dynamic analysis of flexible,nonlinear multibody systems undergoing intermittent contacts. Contact isassumed to be of finite duration, and the forces acting between thecontacting bodies which can be either rigid or deformable are explicitlycomputed during simulation. The modeling of contact consists of threeparts: a number of holonomic constraints that define the candidatecontact points on the bodies, a unilateral contact condition which istransformed into a holonomic constraint by the addition of a slackvariable, and a contact model which describes the relationship betweenthe contact force and the local deformation of the contacting bodies.This work is developed within the framework of energy preserving anddecaying time integration schemes that provide unconditional stabilityfor nonlinear, flexible multibody systems undergoing intermittentcontacts.     

Elastic Plates in Multibody Systems: A Treatment Using Existing Rigid Body Codes

In the present paper, it is shown how one can employ existing rigid body codes to handle systems containing elastic plates by using a Rayleigh–Ritz discretization procedure. The equations of motion are formulated for a rectangular plate undergoing large rigid body motions but small elastic deformations. Geometric nonlinearities in the elastic coordinates are taken into account to include the effect of dynamic stiffening. As an example, a spin-up maneuver for a simply-supported plate is treated.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

An Efficient Multibody Dynamics Model for Internal Combustion Engine Systems

The equations of motion for the major components in an internalcombustion engine are developed herein using a recursive formulation.These components include the (rigid) engine block, pistons, connectingrods, (flexible) crankshaft, balance shafts, main bearings, and enginemounts. Relative coordinates are employed that automatically satisfy allconstraints and therefore lead to the minimum set of ordinarydifferential equations of motion. The derivation of the equations ofmotion is automated through the use of computer algebra as the precursorto automatically generating the computational (C or Fortran) subroutinesfor numerical integration. The entire automated procedure forms thebasis for an engine modeling template that may be used to supportthe up-front design of engines for noise and vibration targets.This procedure is demonstrated on an example engine under free(idealized) and firing conditions and the predicted engine responses arecompared with results from an ADAMS model. Results obtained by usingdifferent bearing models, including linear, nonlinear, and hydrodynamicbearing models, are discussed in detail.     

Multibody Dynamics Modeling of Variable Length Cable Systems

This paper presents a procedure for studying the dynamics ofvariable length cable systems. Such systems commonly occur in deploymentand retrieval (pay-out and reel-in) in cable towing systems such as inship and marine applications.The cable is modeled as a chainand treated as a multibody system. The chain links in turn are modeledas lumped masses. The pay-out/reel-in process is modeled with variablelength links near the towing point.Application in marine systems are presented and discussed.     

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.     

Finite Element Method in Dynamics of Flexible Multibody Systems: Modeling of Holonomic Constraints and Energy Conserving Integration Schemes

In this work we discuss an application of the finite elementmethod to modeling of flexible multibody systems employing geometricallyexact structural elements. Two different approaches to handleconstraints, one based on the Lagrange multiplier procedure and anotherbased on the use of release degrees of freedom, are examined in detail.The energy conserving time stepping scheme, which is proved to be wellsuited for integrating stiff differential equations, gouverning themotion of a single flexible link is appropriately modified and extendedto nonlinear dynamics of multibody systems.     

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.     

A DAE Approach to Flexible Multibody Dynamics

The present work deals with the dynamics of multibody systems consisting ofrigid bodies and beams. Nonlinear finite element methods are used to devise a frame-indifferent spacediscretization of the underlying geometrically exact beam theory. Both rigid bodies and semi-discrete beams are viewed as finite-dimensional dynamical systems with holonomic constraints. The equations of motion pertaining to the constrained mechanical systems under considerationtake the form of Differential Algebraic Equations (DAEs).The DAEs are discretized directly by applying a Galerkin-based method.It is shown that the proposed DAE approach provides a unified framework for the integration of flexible multibody dynamics.     

Active Vibration Control of Flexible Robots Using Virtual Spring-damper Systems

When using robots for heavy loads and huge operating ranges, elastic deformations of the links have to be taken into account during modeling and controller design. Whereas for conventional rigid multilink industrial robots modeling can schematically be done by standard techniques, it is a massive problem to obtain an accurate analytic model for multilink flexible robots. But an accurate analytic model is essential for most modern controller design techniques, and modeling errors can lead to instability of the controlled system due to spillover since the eigenvalues of the system are only slightly damped. A new approach to active damping control for flexible robots is presented in this paper where the actuators act like virtual spring-damper-systems. As the spring-damper-element is a passive energy dissipative device, it will never destabilize the system and thus the control concept will be very insensitive to modeling errors. Basically, the two parameters, spring stiffness and damping constant of this system, are arbitrary and model independent. To satisfy performance requirements they are adjusted using knowledge of the system model. The more it is known about the system model, the better these parameters may be adjusted. The new input of the controlled system is a virtual variation of the spring base. The paper illustrates this technique with the help of a simple and easy to model one link flexible robot which is also available as a real laboratory testbed.