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  • 1
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; finite elementmethod ; QR decomposition ; Cholesky decomposition ; absolute nodal coordinate formulation ; floating frameof reference formulation ; incremental methods ; large deformation ; large rotation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation' for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.
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  • 2
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; nonlinear dynamics ; dynamic simulation ; penalty methods ; constraint stabilization ; constraint violation ; constraint convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents stability and convergence results on a novel approach for imposing holonomic constraints for a class of multibody system dynamics. As opposed to some recent techniques that employ a penalty functional to approximate the Lagrange multipliers, the method herein defines a penalized dynamical system using penalty-augmented kinetic and potential energies, as well as a penalty dependent constraint violation dissipation function. In as much as the governing equations are not typically cocreive, the usual convergence criteria for linear variational boundary value problems are not directly applicable. Still numerical simulations by various researchers suggest that the method is convergent and stable. Despite the fact that the governing equations are nonlinear, the theoretical convergence of the formulation is guaranteed if the multibody system is natural and conservative. Likewise, stability and asymptotic stability results for the penalty formulation are derived from well-known stability results available from classical mechanics. Unfortunately, the convergence theorem is not directly applicable to dissipative multibody systems, such as those encountered in control applications. However, it is shown that the approximate solutions of a typical dissipative system converge to a nearby collection of trajectories that can be characterized precisely using a Lyapunov/Invariance Principle analysis. In short, the approach has many advantages as an alternative to other computational techniques: (1) Explicit constraint violation bounds can be derived for a large class of nonlinear multibody dynamics problems (2) Sufficient conditions for the Lyapunov stability, and asymptotic stability, of the penalty formulation are derived for a large class of multibody systems (3) The method can be shown to be relatively insensitive to singular configurations by selecting the penalty parameters to dissipate ‘constraint violation energy’ (4) The Invariance Principle can be employed in the method, in certain cases, to derive the asymptotic behavior of the constraint violation for dissipative multibody systems by identifying ‘constraint violation limit cycles’ Just as importantly, these results for nonlinear systems can be ‘sharpened’ considerably for linear systems: (5) Explicit spectral error estimates can be obtained for substructure synthesis (6) The penalty equations can be shown to be optimal in the sense that the terms represent feedback that minimizes a measure of the constraint violation
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  • 3
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; contact-impact ; Hertz models ; flexibility ; structural damping ; crashworthiness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A computer based formulation for the analysis of mechanical systems is investigated as a feasible method to predict the impact response of complex structural systems. A general methodology for the dynamic analysis of rigid-flexible multibody systems using a number of redundant Cartesian coordinates and the method of the Lagrange multipliers is presented. The component mode synthesis is then used to reduce the number of flexible degrees of freedom. In many impact situations, the individual structural members are overloaded giving rise to plastic deformations in highly localized regions, called plastic hinges. This concept is used by associating revolute nonlinear actuators with constitutive relations corresponding to the collapse behavior of the structural components. The contact of the system components is described using a continuous force model based on the Hertz contact law with hysteresis damping. The effect and importance of structural damping schemes in flexible bodies are also addressed here. Finally, the validity of this methodology is assessed by comparing the results of the proposed models with those obtained in different experimental tests where: a beam collides transversally with a rigid block; a torque box impacts a rigid barrier.
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  • 4
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; flexible plate ; symbolic implementation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The modelling of flexible elements in mechanical systems has been investigated via several methods issuing from both the field of multi-body dynamics and the area of structural mechanics and vibration theory. As regards the multibody approach, recursive formulations in relative coordinates are quite suitable and efficient for a large variety of applications. Such a formalism is developed here for a general multibody system containing flexible plates and in such a way that its full symbolic generation is possible within the ROBOTRAN program [1].
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  • 5
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; nonlinear vibration ; internal resonance ; energy balance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents the ground-work of implementing the multibody dynamics codes to analyzing nonlinear coupled oscillators. The recent developments of the multibody dynamics have resulted in several computer codes that can handle large systems of differential and algebraic equations (DAE). However, these codes cannot be used in their current format without appropriate modifications. According to multibody dynamics theory, the differential equations of motion are linear in the acceleration, and the constraints are appended into the equations of motion through Lagrange's multipliers. This formulation should be able to predict the nonlinear phenomena established by the nonlinear vibration theory. This can be achieved only if the constraint algebraic equations are modified to include all the system kinematic nonlinearities. This modification is accomplished by considering secondary nonlinear displacements which are ignored in all current codes. The resulting set of DAE are solved by the Gear stiff integrator. The study also introduced the concept of constrained flexibility and uses an instantaneous energy checking function to improve integration accuracy in the numerical scheme. The general energy balance is a single scalar equation containing all the energy component contributions. The DAE solution is then compared with the solution predicted by the nonlinear vibration theory. It also establishes new foundation for the use of multibody dynamics codes in nonlinear vibration problems. It is found that the simulation CPU time is much longer than the simulation of the original equations of the system.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 9 (1996), S. 73-90 
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; linear graph theory ; absolute and joint coordinates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Multibody dynamics involves the generation and solution of the equations of motion for a system of connected material bodies. The subject of this paper is the use of graph-theoretical methods to represent multibody system topologies and to formulate the desired set of motion equations; a discussion of the methods available for solving these differential-algebraic equations is beyond the scope of this work. After a brief introduction to the topic, a review of linear graphs and their associated topological arrays is presented, followed in turn by the use of these matrices in generating various graph-theoretic equations. The appearance of linear graph theory in a number of existing multibody formulations is then discussed, distinguishing between approaches that use absolute (Cartesian) coordinates and those that employ relative (joint) coordinates. These formulations are then contrasted with formal graph-theoretic approaches, in which both the kinematic and dynamic equations are automatically generated from a single linear graph representation of the system. The paper concludes with a summary of results and suggestions for further research on the graph-theoretical modelling of mechanical systems.
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  • 7
    ISSN: 1573-269X
    Keywords: Multibody dynamics ; object-oriented programming ; differential geometry ; sparse matrices ; Jacobians
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Discussed in this paper is a novel method for the generation of Jacobian matrices which is particularly suitable for object-oriented implementations of multibody dynamics programs. The method starts from a description of multibody kinematics as a series of general mappings between manifolds, from which the overall Jacobian results — via the chain rule — as a sequence of matrix products. For these matrices, a new sparse-matrix scheme is suggested. Their “elements” are, besides zeroes, the well-known spatial transformation matrices and the local Jacobians of the individual transmission elements. It is shown how the main approaches for calculation of Jacobians in robotics can be viewed as particular decompositions and multiplication schemes of the sparse-matrices discussed above. Furthermore, two new schemes are derived which may be advantageous for dynamics calculations. The exposition is complemented by a comparison of Jacobian-based methods with composite rigid body and recursive methods for the generation of dynamical equations together with some comments on our current C++-implementation.
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