Analysis Of Flexible Multibody Systems Research Articles

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.

Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems

Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

Analysis of flexible multibody systems with spatial beams using mixed variational principles

A general finite element formulation is presented for dynamic analysis of spatial elastic beams, for small strains, in a multi-body configuration. The tangent maps associated to the finite rotation vector are used to compute the tangent matrices used to integrate implicitly the equations of motion in descriptor form. A corotational method and a mixed variational method are used to compute the tangent stiffness matrix. The tangent constraint matrices are obtained using consistent linearization of the constraint equations. The tangent inertia matrices, including the gyroscopic and centrifugal terms, are also obtained by using the tangent maps of rotation. The numerical examples analyzed in this paper include: dynamic analysis of flexible beam structures and multi-flexible body systems with open and closed kinematic loops. A comparison with the previous results in the literature shows a very good performance in terms of time integration step and number of elements used. © 1998 John Wiley & Sons, Ltd.

Analysis of flexible multibody systems with spatial beams using mixed variational principles

A general finite element formulation is presented for dynamic analysis of spatial elastic beams, for small strains, in a multi-body configuration. The tangent maps associated to the finite rotation vector are used to compute the tangent matrices used to integrate implicitly the equations of motion in descriptor form. A corotational method and a mixed variational method are used to compute the tangent stiffness matrix. The tangent constraint matrices are obtained using consistent linearization of the constraint equations. The tangent inertia matrices, including the gyroscopic and centrifugal terms, are also obtained by using the tangent maps of rotation. The numerical examples analyzed in this paper include: dynamic analysis of flexible beam structures and multi-flexible body systems with open and closed kinematic loops. A comparison with the previous results in the literature shows a very good performance in terms of time integration step and number of elements used. © 1998 John Wiley & Sons, Ltd.

APPLICATION OF THE ABSOLUTE NODAL CO-ORDINATE FORMULATION TO MULTIBODY SYSTEM DYNAMICS

The floating frame of reference formulation is currently the most widely used approach in flexible multibody simulations. The use of this approach, however, has been limited to small deformation problems. In this investigation, the computer implementation of the newabsolute nodal co-ordinate formulationand its use in the small and large deformation analysis of flexible multibody systems that consist of interconnected bodies are discussed. While in the floating frame of reference formulation a mixed set of absolute reference and local elastic co-ordinates are used, in the absolute nodal co-ordinate formulation only absolute co-ordinates are used. In the absolute nodal co-ordinate formulation, new interpretation of the nodal co-ordinates of the finite elements is used. No infinitesimal or finite rotations are used as nodal co-ordinates from beams and plates, instead, global slopes are used to define the element nodal co-ordinates. Using this interpretation of the element co-ordinates, beams and plates can be considered as isoparametric elements, and as a result, exact modelling of the rigid body dynamics can be obtained using the element shape function and the absolute nodal co-ordinates. Unlike the floating frame of reference approach, no co-ordinate transformation is required in order to determine the element inertia. The mass matrix of the finite elements is a constant matrix, and therefore, the centrifugal and Coriolis forces are equal to zero when the absolute nodal co-ordinate formulation is used. Another advantage of using the absolute nodal co-ordinate formulation in the dynamic simulation of multibody systems is its simplicity in imposing some of the joint constraints and also its simplicity in formulating the generalized forces due to spring-damper elements. The results obtained in this investigation show an excellent agreement with the results obtained using the floating frame of reference formulation when large rotation–small deformation problems are considered.

Selection of efficient coordinate partitioning methods in flexible multibody systems

In multibody dynamics, differential and algebraic equations which can satisfy both equation of motion and kinematic constraint equation should be solved. To solve these equations, coordinate partitioning method and constraint stabilization method are commonly used. In the coordinate partitioning method, the coordinates are divided into independent and dependent and coordinates. The most typical coordinate partitioning method are LU decomposition, QR decomposition, and SVD (singular value decomposition). The objective of this research is to find an efficient coordinate partitioning method in the dynamic analysis of flexible multibody systems. Comparing two coordinate partitioning methods, i.e. LU and QR decomposition in the flexible multibody systems, a new hybrid coordinate partitioning method is suggested for the flexible multibody analysis.

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.

Inverse Dynamic Analysis of Flexible Multibody System in the Joint Coordinate Space

Inverse Dynamic Analysis of Flexible Multibody System in the Joint Coordinate Space

Mixed-interface substructures for dynamic analysis of flexible multibody systems

This paper presents an efficient and accurate mixed-interface sub-structure formulation for dynamic analysis of multibody systems. The equation of motion is derived using the mixed-interface sub-structure approach, which is used to reduce the resulting number of elastic coordinates of multibody systems. The mixed-interface substructure procedure incorporates both the free- and fixed-interface boundary conditions that apply to different flexible and rigid bodies. The entire procedure requires a two-step transformation; first from one substructure to another substructure in the same body, and second from one body (or substructure) to another body (or substructure). This process is continued until it has been carried out for the entire multibody system. The applicability and accuracy of this formulation are illustrated on a two-link system with comparison made against the finite element method. It was found that the mixed-interface substructure procedure, although it uses fewer elements, provides numerical solutions comparable to the finite element method.

Flexible multibody dynamics — An object-oriented approach

An approach to the computer aided analysis of flexible multibody systems using object-oriented programming methods is presented. The aim is to support the rapid generation of specialized programs by providing an open, extensible C++ toolkit. This toolkit contains modules (C++ classes) which allow the declaration and manipulation of multibody components such as joints, bodies and actuators in an intuitive manner. New components (e.g., new finite elements) are easily introduced to extend the toolkit. The equations of motion for a multibody system consisting of these components are formulated by direct application of the principle of virtual work using symbolic techniques. It is possible to use absolute as well as relative coordinates in a problem-dependent manner.

Development of energy absorbing devices using a kinetostatic multibody dynamics methodology

Abstract A kinetostatic approach based on flexible multibody dynamics for crashworthiness and structural impact of vehicles is presented here. The equations of motion for a partially flexible body composed of a rigid part and a flexible part are derived accounting for the inertial coupling between each component's gross motion and its elastodynamics. For components having a mass for the flexible part much smaller than the mass of their rigid part the inertial coupling can be neglected. Under these assumptions the flexible part of the body behaves as a massless structure attached to the rigid body and it can be analysed as a separate non-linear structure. The deformations of this flexible part are added to the motion of the rigid part to obtain the complete motion of the flexible body. This computer-based procedure is a kinetostatic method for the analysis of flexible multibody systems subjected to non-linear deformations and impact conditions which is applied to the study of low speed impact of vehicles. The proposed methodology is applied to the study of a new design for energy absorbing devices. Several simulations of the vehicle impact against a rigid wall with different incoming speeds and impact angles are performed and the behaviour of the vehicle occupant during the impact is analysed.

Effects of Mode Selection, Scaling, and Orthogonalization on the Dynamic Analysis of Flexible Multibody Systems∗

ABSTRACT This paper analyzes the effects of mode selection, scaling, and orthogonalization on dynamic analysis of flexible mechanical systems. The effects of mode selection are investigated by combining attachment modes and constraint modes with vibration normal modes to represent elastic deformations of flexible components. Four sets of input data that depend on the options for mode scaling and orthogonalization are generated. The dynamic response and computer simulation times for each set are compared. Two examples are chosen: (1) a cantilever beam and (2) a vehicle with a flexible chassis. With vibration normal modes and attachment modes, it was found that mode scaling was helpful for efficient simulation, while the effect of mode orthogonalization was problem-dependent. With vibration modes and constraint modes, however, mode scaling had no effect or detracted from numerical efficiency.

An efficient computational method for dynamic stress analysis of flexible multibody systems

This paper presents an efficient computational method of dynamic stress history calculation for a general three-dimensional flexible body by combining flexible multibody dynamic simulation and quasi-static finite element analysis (FEA). In the dynamic simulation of flexible multibody systems, flexible components can undergo nonsteady gross motion and small elastic deformation that is described with respect to the body reference frame by using the assumed mode method. D'Alembert inertia loads from the gross body motion and the elastic deformation are expressed as a combination of space-dependent and time-dependent terms that are obtained from the dynamic simulation. D'Alembert inertia loads that are associated with each unit value of the time-dependent terms are then distributed to all finite element nodes in order to compute a corresponding stress influence coefficient through quasi-static structural analyses. Total dynamic stresses due to D'Alembert inertia loads are obtained by multiplying actual magnitude of time-dependent terms with the associated stress influence coefficients. By the proposed method, it is shown that, for a general three-dimensional component, the required number of FEAs can be significantly reduced.

Modelling of superelements in mechanism analysis

AbstractThe paper deals with substructuring for dynamic analysis of flexible multibody systems. Three different techniques based on component synthesis are discussed, corresponding respectively to fully consistent mass discretization, lumped mass discretization and corotational approximation of inertia forces. To simplify the computer implementation, only the lumped mass and corotational approximations have been considered in detail and programmed. Both approaches are validated on simple examples of rotating beams for which a full elastic model is available using a fully non‐linear beam element. The computational efficiency of the corotational inertia approach is also demonstrated on the deployment of a large flexible satellite antenna.

Direct determination of periodic solutions of the dynamical equations of flexible mechanisms and manipulators

AbstractA method based on finite elements, characterized by generalized strains as functions of the nodal co‐ordinates, is the basis for the computer program SPACAR for the kinematic and dynamic analysis of flexible multibody systems. This program allows also the evaluation of the linearized dynamical equations, which makes possible the implementation of a method for the direct determination of stationary and periodic solutions and of a continuation method for these solutions if some parameter is varied. As an illustration, these methods are applied to a Duffing equation, a double pendulum and a slider‐crank mechanism with an elastic connecting rod.

Constrained motion of deformable bodies

AbstractIn this paper, issues related to the dynamic modelling of constrained deformable bodies that undergo large rigid body displacements are discussed. Particular attention is focused on finite element formulations. It is shown that the use of nodal co‐ordinates and shape functions to describe the finite rotation of some of the commonly used finite elements leads to a linearization of the kinematics and dynamic relationships. The structure of the non‐linear dynamic equations that govern the motion of deformable bodies that undergo large displacements is examined. Comments on the finite element formulation of the invariants of motion, the definition of the generalized forces and moments in flexible body dynamics and the computational strategy used for the automatic generation of the equations of motion are made. The computer formulation of the joint constraints between deformable bodies as well as the numerical algorithms currently used in many of the general purpose computer programs that are based on the augmented formulation are discussed. A decoupled joint‐elastic acceleration recursive formulation is also presented. This formulation leads to a small system of acceleration equations whose dimensions are independent of the number of the elastic degrees of freedom of the system. In this paper, the coupling between the displacements of the deformable bodies is classified as kinematic, inertia and elastic. In view of this classification, comments on the validity of using the updated finite element Lagrangian formulation and the 4 × 4 transformation matrix in the dynamic analysis of flexible multibody systems are made. The coupling between the finite rotation and the wave motion in constrained deformable bodies is also discussed.

A decoupled flexible‐relative co‐ordinate recursive approach for flexible multibody dynamics

AbstractAn extended kinematic graph concept and a variational‐vector calculus approach are employed to develop a new recursive formulation for the dynamic analysis of flexible multibody systems. The extended graph concept introduced defines frames and transformations between frames as nodes and edges, respectively, rather than the more traditional body and joint convention. Kinematic relationships between adjacent flexible bodies are derived, using joint relative co‐ordinates and a state vector notation that represents both translational and rotational components of velocity. Deformation kinematics are formulated in terms of modal co‐ordinates, under small deformation assumptions. Joint relative co‐ordinates are decoupled from deformation modal co‐ordinates in both kinematic relations and in the recursive dynamics algorithm, leading to a significant reduction in the dimension of matrices that must be inverted. Dynamic analysis of a flexible closed‐loop spatial robot is performed to illustrate the efficiency of the algorithm.

Vibrations analysis of flexible multibody systems

This paper is the continuation of a previous work (Pascal, 1988a) in which we consider the dynamical analysis of flexible space vehicles modelled by a chain of rigid and elastic bodies with tree structure. The multibody system consists of n + 1 bodies (S$iiei:) (i = 0, 1,…n) interconnected by n hinges la (a = 1,…n). The only external forces and torques are exerted on the first body which is assumed to be rigid. On each individual flexible appendage (Si), the only external forces and torques are those introduced by the hinges. Assuming that the multibody system undergoes small vibrations around an equilibrium position, we define in the frequency domain the linear transformation giving the resultant forces and torques on the boundaries of each flexible appendage (Si) in terms of the displacements of these boundaries. The motion of each flexible appendage is represented by a set of component modes, which are the modes obtained when the appendage vibrates independently with respect to the other parts of the whole system. Two different sets of vibration modes are used to give an expansion of the transfer function between forces and torques exerted on the boundaries of the body (Si) and displacements of these boundaries

Dynamic analysis of flexible multi-body systems using mixed modal and tangent coordinates

This paper presents a mixed modal and tangent coordinate technique for computer aided analysis of flexible mechanical systems whose components undergo large translations and large rotations. In this model the configuration of a flexible component is identified by using two sets of generalized coordinates, namely rigid body and elastic coordinates. The rigid body coordinates define the location and orientation of a body axis, whereas the elastic coordinates define the displacement field of a component with respect to its body axis. The elastic coordinates are introduced by using finite element discretization to model flexible components with complex geometries. A modal analysis technique is used to identify the elastic mode shapes and to eliminate insignificant higher frequency modes. An orthonormalization of constraint Jacobian matrix associated with rigid body coordinates is used to identify the rigid body tangent coordinates. The resulting modal and tangent coordinates are used to develop an automated numerical integration scheme to solve the system differential and algebraic equations. Two numerical examples are considered to show the feasibility of dynamic analysis of flexible mechanical systems using this scheme.

Timoshenko beams and flexible multibody system dynamics

A method for the dynamic analysis of flexible multibody systems that accounts for rotary inertia and shear deformation effects is presented. Flexible components in the system are discretized by using the finite element method. Because of the large rotations of the system components, a set of reference co-ordinates are employed to describe the motion of a selected body reference. Kinetic and strain energies are derived for each element, thus identifying the element mass and stiffness matrices which account for the rotary inertia and shear deformation effects. A new set of time invariant element matrices that describe the coupling between the reference motion and elastic deformations is developed. These matrices need be evaluated only once in advance for the dynamic analysis. It has been, shown that the form of these matrices as well as the mass and stiffness matrices are significantly affected by the inclusion of rotary inertia and shear deformation. The formulation presented in this paper is exemplified by a two-dimensional flexible multibody aircraft. Numerical results indicate that rotary inertia and shear have a significant effect on the dynamic response of large scale flexible multibody systems with large angular rotations and in which the reference motion and elastic deformation are coupled.