Integrating finite rotations

Bottasso, Carlo L.; Borri, M.

doi:10.1016/s0045-7825(98)00031-0

Cited by 93 publications

(106 citation statements)

References 15 publications

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“…This must, in particular, include flexible bodies undergoing large deformations. In this regard, Lie group formulations were already (at least implicitly) used in [18][19][20][21][22][23]63] . …”

Section: Discussionmentioning

confidence: 99%

“…The matrix (7) has appeared under different names, such as 'spatial cross product' in [61,62,82], or the 'north-east cross product' [22]. The closed form of the exp mapping on SE (3) is found evaluating the matrix exponential…”

Section: Instantaneous Screws and Canonical Coordinatesmentioning

confidence: 99%

“…A closed form of the right-trivialized differential of the exp mapping on SE (3) was reported in [18,22,128] dexp X = dexp ξ 0 P dexp ξ (21) for X = (ξ , η) ∈ se (3), where…”

Section: Rigid Body Velocities: Twistsmentioning

confidence: 99%

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Geometric methods and formulations in computational multibody system dynamics

Müller

Terze

2016

Acta Mech

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Multibody systems are dynamical systems characterized by intrinsic symmetries and invariants. Geometric mechanics deals with the mathematical modeling of such systems and has proven to be a valuable tool providing insights into the dynamics of mechanical systems, from a theoretical as well as from a computational point of view. Modeling multibody systems, comprising rigid and flexible members, as dynamical systems on manifolds, and Lie groups in particular, leads to frame-invariant and computationally advantageous formulations. In the last decade, such formulations and corresponding algorithms are becoming increasingly used in various areas of computational dynamics providing the conceptual and computational framework for multibody, coupled, and multiphysics systems, and their nonlinear control. The geometric setting, furthermore, gives rise to geometric numerical integration schemes that are designed to preserve the intrinsic structure and invariants of dynamical systems. These naturally avoid the long-standing problem of parameterization singularities and also deliver the necessary accuracy as well as a long-term stability of numerical solutions. The current intensive research in these areas documents the relevance and potential for geometric methods in general and in particular for multibody system dynamics. This paper provides an exhaustive summary of the development in the last decade, and a panoramic overview of the current state of knowledge in the field.

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Section: Discussionmentioning

confidence: 99%

Section: Instantaneous Screws and Canonical Coordinatesmentioning

confidence: 99%

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Geometric methods and formulations in computational multibody system dynamics

Müller

Terze

2016

Acta Mech

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“…Specifically, we need to set up P(q n , q n+1 ) ∈ R n×(n−m) such that, range(P(q n , q n+1 )) = null(G(q n , q n+1 )) (11) Note that due to the properties of the discrete constraint Jacobian G(q n , q n+1 ) ∈ R m×n (cf. Equation (18) in Part I), the mid-point velocities may be expressed as v n+1/2 = P(q n , q n+1 )w (12) with w ∈ R n−m . The last equation can be interpreted as discrete version of (6).…”

Section: Discrete Size-reductionmentioning

confidence: 99%

The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: multibody dynamics

Betsch

Leyendecker

2006

Numerical Meth Engineering

117

101

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SUMMARYIn the present work, rigid bodies and multibody systems are regarded as constrained mechanical systems at the outset. The constraints may be divided into two classes: (i) internal constraints which are intimately connected with the assumption of rigidity of the bodies, and (ii) external constraints related to the presence of joints in a multibody framework. Concerning external constraints lower kinematic pairs such as revolute and prismatic pairs are treated in detail. Both internal and external constraints are dealt with on an equal footing. The present approach thus circumvents the use of rotational variables throughout the whole time discretization. After the discretization has been completed a size-reduction of the discrete system is performed by eliminating the constraint forces. In the wake of the size-reduction potential conditioning problems are eliminated. The newly proposed methodology facilitates the design of energy-momentum methods for multibody dynamics. The numerical examples deal with a gyro top, cylindrical and planar pairs and a six-body linkage.

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“…Energy decaying schemes can be generally divided into two approaches. In the first approach, the time-discontinuous Galerkin formulation was utilized to predict the behavior of nonlinear beams [8, 12,13] and multi-body systems [9,10,11,14]. These algorithms involve a multistage computation per time step and require solving a larger system than those solved with classical schemes, due to the coupling between the displacement and velocity.…”

Section: Introductionmentioning

confidence: 99%

A robust composite time integration scheme for snap-through problems

et al. 2015

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A robust time integration scheme for snapthrough buckling of shallow arches is proposed. The algorithm is a composite method that consists of three sub-steps. Numerical damping is introduced to the system by employing an algorithm similar to the backward differentiation formulas (BDF) method in the last substep. Optimal algorithmic parameters are established based on stability criteria and minimization of numerical damping. The proposed method is accurate, numerically stable, and efficient as demonstrated through several examples involving loss of stability, large deformation, large displacements and large rotations.

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Integrating finite rotations

Cited by 93 publications

References 15 publications

Geometric methods and formulations in computational multibody system dynamics

Geometric methods and formulations in computational multibody system dynamics

The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: multibody dynamics

A robust composite time integration scheme for snap-through problems

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