Dynamics of Multibody Systems

Roberson, Robert E.; Schwertassek, R.; Huston, Ronald L.

doi:10.1115/1.3176059

Cited by 59 publications

(31 citation statements)

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“…In this paper, the HHT formulas are used as the integration formulas in the TLISMNI algorithm. In general, the differential equations of motion and the algebraic constraint equations of a multibody system can be written, respectively, in the following forms [8,9,12]:…”

Section: Tlismni Methodsmentioning

confidence: 99%

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Sparse matrix implicit numerical integration of the Stiff differential/algebraic equations: Implementation

Hussein

Shabana

2010

Nonlinear Dyn

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The finite element absolute nodal coordinate formulation (ANCF) is often used in modeling very flexible bodies in multibody system (MBS) applications. This formulation leads to a constant mass matrix, allowing for an efficient sparse matrix implementation. Nonetheless, the use of the ANCF finite elements to model stiff structures can lead to high frequencies associated with ANCF coupled deformation modes, as discussed in the literature. Implicit numerical integration methods can be effectively used to develop efficient procedures for the solution of MBS differential/algebraic equations. Most existing implicit integration algorithms, however, require numerical differentiation of the equations of motion, and some of these integration methods do not ensure that the kinematic algebraic constraint equations are satisfied at all levels (position, velocity, and acceleration). Because of these limitations, existing implicit integration methods can be less accurate and less efficient when used to solve large scale MBS applications. In order to circumvent this problem, the two-loop implicit sparse matrix numerical integration (TLISMNI) method was proposed for the solution of MBS differential/algebraic equations. The TLISMNI method does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation. This paper discusses TLISMNI implementation issues including the step size selection, the error control, and the effect of the numerical damping. The relation between the step size selection and the structure stiffness is also discussed. The use of the computer implementation described in this paper is demonstrated by solving very stiff structure problems using the Hilber-Hughes-Taylor (HHT) method, which includes numerical damping. An eigenvalue analysis and Fast Fourier Transform (FFT) are performed in order to identify the fundamental modes of deformation and demonstrate that the contributions of these fundamental modes can be erroneously damped out when some other implicit integration methods are used. The TLISMNI method, on the other hand, captures the contributions of these fundamental modes. The results, obtained using the TLISMNI method, are compared with the results obtained using other methods including the implicit HHT-I3 and the explicit Adams integration methods. The results obtained show that the TLISMNI method can be five times faster than the other two methods when no numerical damping is considered.

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Section: Tlismni Methodsmentioning

confidence: 99%

“…By using the embedding technique, the equations of motion can be written as function of the independent coordinates only as follows [8,9,12]:…”

Section: Explicit Adams Methodsmentioning

confidence: 99%

Sparse matrix implicit numerical integration of the Stiff differential/algebraic equations: Implementation

Hussein

Shabana

2010

Nonlinear Dyn

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“…In the following the motion during this time interval is considered. The motion equations for a cluster comprising N disks, when each disk moves as a free body, are given in the form is the incidence matrix of order P 2 N, it characterises the connectivity between the disks as defined in (Roberson and Schwertassek, 1988). The meaning of the incidence matrix is best explained by an example of a graph shown in Fig.…”

Section: Governing Equationsmentioning

confidence: 99%

A multibody approach in granular dynamics simulations

Vinogradov

Sun

1997

Computational Mechanics

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A plane model of a granular system made out of interconnected disks is treated as a multibody system with variable topology and one-sided constraints between the disks. The motion of such a system is governed by a set of nonlinear algebraic and differential equations. In the paper two formalisms (Lagrangian and Newton-Euler) and two solvers (Runge-Kutta and iterative) are discussed. It is shown numerically that a combination of the Newton-Euler formalism and an iterative method allows to maintain the accuracy of the fourth order Runge-Kutta solver while reducing substantially the CPU time. The accuracy and efficiency are achieved by integrating the error control into the iterative process. Two levels of error control are introduced: one, based on satisfying the position, velocity and acceleration constraints, and another, on satisfying the energy conservation requirement. An adaptive time step based on the rate of convergence at the previous time step is introduced which also allows to reduce the simulation time. The efficiency and accuracy is investigated on a physically unstable vertical stack of disks and on multibody pendulums with 50, 100, 150 and 240 masses. An application to the problem of jamming in a two-phase flow is presented. IntroductionThe granular models are used to describe many physical systems, such as soils, sand, grain, rock, pills, broken ice, etc., to name a few. Since granular systems are discrete, they may have some very distinct features, such as the variable spatial density (initial or developed during the dynamic process); the existence of internal degrees of freedom which may lead to microforms in a macrosystem; the formation of clusters which behave (at least during some time) as quasi-rigid bodies; and the variable interconnectivity between the particles.From the mathematical point of view a granular system of interconnected particles is a multibody system with one-sided constraints and is described by a system of nonlinear differential and algebraic equations. A model of a granular system based on a multibody approach allows to take all of the above features of the system into account. The difficulty of this approach is computational, i.e. it is associated with the numerical stability of the solution for an initial value problem and the efficiency of computations for large systems. The two requirements, efficiency and accuracy, are, of course, interconnected.The existing approach to simulating discrete granular systems is based on a model developed by P.A. Cundall, and it is widely known as a distinct element method (DEM). According to this method ''the interaction of particles is viewed as a transient problem with states of equilibrium developing whenever the internal forces balance'' (Cundall and Strack, 1979). It is reminiscent of the molecular dynamics models in a sense that the interactions are localized by choosing such a time step that the propagation of disturbances is limited only to the neighbours. In other words, a particle moves under the actions of constant external and...

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“…See Roberson and Schwertassek (1988), Schielen (1990), and Shabana (1989Shabana ( , 1997 for details and for further references. The previous work addresses more general joints than the single hinges treated here; some of it applies to¯exible as well as rigid bodies; systems that are not physical trees are treated by introducing inter-branch constraints.…”

Section: Introductionmentioning

confidence: 97%

“…Existing multibody work is couched in matrix form, sometimes under the misimpression that this is a requirement for computer implementation (Roberson and Schwertassek (1988) state on p. 177:``The equations must be converted from vector/dyadic form to matrix form for implementation on a digital computer. '').…”

Section: Introductionmentioning

confidence: 99%

Global recursive dynamics of articulated tree structures

Katz

Waits

1999

Computational Mechanics

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A rigorous global recursive formulation is developed for the dynamics of a classical articulated tree. The system in question is composed of rigid parts interconnected by spring loaded single hinges to form a tree. The global formulation addresses the dynamics of complete branches (subtrees). The external moments applied to the branch are brought to bear on the instantaneous moment of inertia of the branch. Corrections for deviations from rigidity are accumulated recursively from sub-branches. The translational motion is likewise treated by addressing the structure as a whole and accumulating effects of internal motion recursively. Global recursive dynamics is a rigorous reexpression of the equations of motion of the system in terms that are, at the same time, intuitive and directly computable. Applied sequentially, global recursive dynamics gives rise to a family of approximations, progressing from the rigid to the fully articulated, which converge to the solution as the time step tends to zero.

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Dynamics of Multibody Systems

Cited by 59 publications

References 0 publications

Sparse matrix implicit numerical integration of the Stiff differential/algebraic equations: Implementation

Sparse matrix implicit numerical integration of the Stiff differential/algebraic equations: Implementation

A multibody approach in granular dynamics simulations

Global recursive dynamics of articulated tree structures

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