Erratum to: New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

New efficient method for dynamic modeling and simulation of flexible multibody systems moving in plane

Efficient, precise dynamic analysis for general flexible multibody systems has become a research focus in the field of flexible multibody dynamics. In this paper, the finite element method and component mode synthesis are introduced to describe the deformations of the flexible components, and the dynamic equations of flexible bodies moving in plane are deduced. By combining the discrete time transfer matrix method of multibody system with these dynamic equations of flexible component, the transfer equations and transfer matrices of flexible bodies moving in plane are developed. Finally, a high-efficient dynamic modeling method and its algorithm are presented for high-speed computation of general flexible multibody dynamics. Compared with the ordinary dynamics methods, the proposed method combines the strengths of the transfer matrix method and finite element method. It does not need the global dynamic equations of system and has the low order of system matrix and high computational efficiency. This method can be applied to solve the dynamics problems of flexible multibody systems containing irregularly shaped flexible components. It has advantages for dynamic design of complex flexible multibody systems. Formulations as well as a numerical example of a multi-rigid-flexible-body system containing irregularly shaped flexible components are given to validate the method.     

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.     

An Energy-Conserving and Filtering Method for Stiff Nonlinear Multibody Dynamics

The conservation of energy is necessary for accuracy oflong-term simulations, and also guarantees energystability, which is sought to alleviate any stabilityrestrictions on the time integration step size. We discusssome issues regarding the integration of stiff nonlineardynamics with traditional dissipative and energy andmomentum conserving methods and we introduce a new 2-stagemethod which is an adaptation of the time integrationscheme presented in [1] for rigid multibody dynamics. Bycombining an energy conserving scheme with a substageacceleration filter, we devise an implicit method thatpreserves the energy map and is capable of integratingstiff flexible multibody systems that require numericaldissipation. In order to avoid the artificial stiffnessand the resulting instabilities caused by the enforcementof kinematic constraints, a joint coordinate formulationis adopted to model the rigid components of the system,while a total Lagrangian approach is adopted to model theflexible elements in the system. As the resulting model istypically characterized by a large number of degrees offreedom, we also demonstrate how the method may beextended to incorporate a form of domain decomposition.     

Modal analysis of active flexible multibody systems

When performing modal analyses of active flexible multibody systems, both controller effects and flexible body dynamics should be included in a multidisciplinary system model. This paper deals with the theory of solving the closed-loop eigenvalue problem for active flexible multibody systems with multiple-input multiple-output proportional-integral-derivative (PID) type feedback controllers and multiple degrees of freedom finite element models. Modal analyses are performed on both a simple and complex active flexible multibody system in order to illustrate the difference between current modal analysis method for such systems and the proposed theory derived in this paper.     

Simulation process of flexible multibody systems with non-modal model order reduction techniques

One important issue for the simulation of flexible multibody systems is the reduction of the flexible body’s degrees of freedom. In this work, nonmodal model reduction techniques for flexible multibody systems within the floating frame of reference framework are considered. While traditionally in the multibody community modal techniques in many different forms are used, here other methods from system dynamics and mathematics are in the focus. For the reduction process, finite element data and user inputs are necessary. Prior to the reduction process, the user first needs to choose boundary conditions fitting the chosen reference frame before defining the appropriate in- and outputs. In this work, four different possibilities of modeling appropriate interface points to reduce the number of inputs and outputs are presented.     

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.     

On the influence of model reduction techniques in topology optimization of flexible multibody systems using the floating frame of reference approach

Employing the floating frame of reference formulation in the topology optimization of dynamically loaded components of flexible multibody systems seems to be a natural choice. In this formulation the deformation of flexible bodies is approximated by global shape functions, which are commonly obtained from finite element models using model reduction techniques. For topology optimization these finite element models can be parameterized using the solid isotropic material with penalization (SIMP) approach. However, little is known about the interplay of model reduction and SIMP parameterization. Also securing the model reduction quality despite major changes of the design during the optimization has not been addressed yet. Thus, using the examples of a flexible frame and a slider-crank mechanism this work discusses the proper choice of the model reduction technique in the topology optimization of flexible multibody systems.     

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.     

Object-Oriented Modelling of Flexible Beams

In this paper the problem of modelling flexible thin beams in multibody systems is tackled. The proposed model, implemented with the object-oriented modelling language Modelica, is completely modular, allowing the realization of complex systems by simple aggregation of basic components. The finite element method is employed as the basic scheme to spatially discretize the model equations. Exploiting the modular features of the language, the beam substructuring discretisation scheme (mixed finite element-finite volume) is derived as well. Selected simulation results are presented in order to validate the model with respect to both theoretical predictions and literature reference results.     

Geometric nonlinear effects on the planar dynamics of a pivoted flexible beam encountering a point-surface impact

Flexible-body modeling with geometric nonlinearities remains a hot topic of research by applications in multibody system dynamics undergoing large overall motions. However, the geometric nonlinear effects on the impact dynamics of flexible multibody systems have attracted significantly less attention. In this paper, a point-surface impact problem between a rigid ball and a pivoted flexible beam is investigated. The Hertzian contact law is used to describe the impact process, and the dynamic equations are formulated in the floating frame of reference using the assumed mode method. The two important geometric nonlinear effects of the flexible beam are taken into account, i.e., the longitudinal foreshortening effect due to the transverse deformation, and the stress stiffness effect due to the axial force. The simulation results show that good consistency can be obtained with the nonlinear finite element program ABAQUS/Explicit if proper geometric nonlinearities are included in the floating frame formulation. Specifically, only the foreshortening effect should be considered in a pure transverse impact for efficiency, while the stress stiffness effect should be further considered in an oblique case with much more computational effort. It also implies that the geometric nonlinear effects should be considered properly in the impact dynamic analysis of more general flexible multibody systems.     

A linearized input–output representation of flexible multibody systems for control synthesis

In this paper, a linearized input–output representation of flexible multibody systems is proposed in which an arbitrary combination of positions, velocities, accelerations, and forces can be taken as input variables and as output variables. The formulation is based on a nonlinear finite element approach in which a multibody system is modeled as an assembly of rigid body elements interconnected by joint elements such as flexible hinges and beams. The proposed formulation is general in nature and can be applied for prototype modeling and control system analysis of mechatronic systems. Application of the theory is illustrated through a detailed model development of an active vibration isolation system for a metrology frame of a lithography machine.     

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.     

Geometrically exact physics-based modeling and computer animation of highly flexible 1D mechanical systems

This paper presents a geometrically exact beam theory and a corresponding displacement-based finite-element model for modeling, analysis and natural-looking animation of highly flexible beam components of multibody systems undergoing huge static/dynamic rigid-elastic deformations. The beam theory fully accounts for geometric nonlinearities and initial curvatures by using Jaumann strains, concepts of local displacements and orthogonal virtual rotations, and three Euler angles to exactly describe the coordinate transformation between the undeformed and deformed configurations. To demonstrate the accuracy and capability of this nonlinear beam element, nonlinear static and dynamic analysis of two highly flexible beams are performed, including the twisting a circular ring into three small rings and the spinup of a flexible helicopter rotor blade (Graphical abstract). These numerical results reveal that the proposed nonlinear beam element is accurate and versatile for modeling, analysis and 3D rendering and animation of multibody systems with highly flexible beam components.     

A unified solution scheme for inverse dynamics

In this paper, a completely new solution scheme for inverse dynamics, which can be commonly applied in different types of link systems such as open- or closed-loop mechanisms, or ones constituting rigid or flexible link members, is presented. The scheme is developed using the finite element method (FEM), which evaluates the entire system as a continuum with the equation of motion in Cartesian coordinates and in dimension of force. The inverse dynamics is calculated by using a matrix-form relation to the nodal forces obtained by the FEM. The matrix-form equations are divided individually into terms of force, transformation between coordinates and length, which makes the scheme potentially better in terms of applicability and expansibility. The scheme cannot only deal with open- and closed-loop link systems independently, but it can also deal seamlessly with those that gradually change their forms and dynamics. There is also no need to revise the basic numerical algorithm of the scheme, regardless of the stiffness of the constituting link member, i. e. rigid or flexible. The main objective of this paper is to present the extensive ability of the scheme as a unified scheme, by carrying out calculations on several types of rigid and flexible manipulators, along with an application to feed-forward control of a link mechanism which continuously changes its form from an open- to a closed-loop.     

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.     

Friction Models and Stress Recovery Methods in Vehicle Dynamics Modelling

This paper presents two examples of calculations for vehicles with flexible bodies by using mixed multibody and finite element methods. The first example deals with dynamics computations for a bimodal train with a flexible cistern, whereas the second example concerns the dynamics calculations for the PW-6 glider. In the first example, the influence of the chosen friction model on the train dynamics calculations results was discussed. The second example presents several methods of stress calculations and a comparison of results. The achieved conclusions may be used as suggestions towards a modelling method choice for a given problem.Both issues being discussed are of great importance in dynamics of flexible multibody systems modelling practice and durability assessment. In both examples, the kinematics of the system was presented in absolute coordinates, the motion equations in the DAE form, and the reduction of the number of degrees of freedom was achieved by means of the Craig–Bampton (CB) method.     

Interface reduction of flexible bodies for efficient modeling of body flexibility in multibody dynamics

The floating frame of reference techniques is an established technique to incorporate flexibility in multibody models. The model dimension of the body flexibility models can be reduced by model reduction techniques such as Component Mode Synthesis (CMS) or Krylov subspace-based techniques, but the efficiency of these techniques is limited by the number of interface nodes in which the flexible body is or can be loaded. A common solution to this problem is condensing the different nodes of a given interface surface into a single node, which represents the net motion of the interface surface. Commercial finite element packages offer two modeling techniques to condense interface surfaces: rigid multipoint constraints and interpolation multipoint constraints. Rigid multipoint constraints will typically result in stiffness overestimation, whereas interpolation multipoint constraints will lead to an underestimation. Which approximation of both is most suitable depends on the application. However, the default definition of interpolation multipoint constraints does not allow generation of reduced body flexibility models for multibody models. This paper gives a theoretical background of the problem cause, and offers a practical solution. The two modeling techniques result in significantly different approximations of the body flexibility dynamics, as is shown in a numerical example.     

Flexible multibody modeling of reeving systems including transverse vibrations

This paper presents a discretization procedure for the flexible multibody modeling of reeving systems. Reeving systems are assumed to include a set of rigid bodies connected by wire ropes using a set of sheaves and reels. The method is capable to model the deformation of the varying-length wire-rope spans. Wire ropes are assumed to deform axially, transversally and in torsion. This paper shows the capability of the presented method to model transverse vibrations. The discretization procedure uses a combination of absolute position coordinates, relative-transverse deformation coordinates and longitudinal material coordinates. Each wire-rope span is modeled using a single two-noded element under an arbitrary Lagrangian–Eulerian approach. The discretization method is validated using analytical and numerical reference solutions found in the literature that describe the dynamics of varying-length strings. In addition, the dynamics of a three-dimensional tower crane is simulated.