Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies

Krysl, Petr; Endres, L.

doi:10.1002/nme.1272

Cited by 57 publications

(53 citation statements)

References 43 publications

(58 reference statements)

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“…Figure 4 illustrates the convergence behaviour in the norm R − R converged 2 and the norm − converged 2 , where the orientation matrix R converged = R (t = 100) and the angular momentum in the body frame converged = (t = 100) have been obtained with an extremely small step size of 0.001. On this problem the LIEMID algorithm is on par with the currently best explicit second-order algorithm, the Newmark algorithm of Krysl and Endres [4]. In fact, the performance of the best second-order implicit algorithm, the canonical midpoint rule of Austin, Krishnaprasad and Wang [5] is marginally more accurate for the body-frame momentum, but lags behind significantly in the accuracy of the attitude matrix.…”

Section: Freely Spinning Body [3]mentioning

confidence: 94%

Explicit momentum-conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

Krysl

2005

Int. J. Numer. Meth. Engng

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SUMMARYWe reformulate the midpoint Lie algorithm, which is implicit in the torque calculation, to achieve explicitness in the torque evaluation. This is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step. Thus, we obtain two related variants, both of which are explicit in the torque calculation, but only first order in the time step. Both variants are momentum conserving and both are symplectic. Consequently, drawing on the properties of the composition of maps, we introduce another algorithm that combines the two variants in a single time step. The resulting algorithm is explicit, momentum conserving, symplectic, and second order. Its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators.

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Section: Freely Spinning Body [3]mentioning

confidence: 94%

Explicit momentum-conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

Krysl

2005

Int. J. Numer. Meth. Engng

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“…Krysl [86] proposed a version of a generalized α scheme for the solution of ODE on a Lie group. This was one of the first formlations that initiated the developments for DAE.…”

Section: Geometric Integration Of Mbs Models In Absolute Coordinatesmentioning

confidence: 99%

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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“…Consider a rigid body with body-fixed BFR F b = {Ω; e b,1 , e b,2 , e b,3 } located at an arbitrary point Ω. Denote the inertia matrix with respect to this BFR by M b ; see (41). The configu-ration of the BFR F b is described by C = (R, r).…”

Section: Spatial Representationmentioning

confidence: 99%

“…The ODEs (90) can be solved with any vector space integration scheme (originally the Munthe-Kaas scheme uses a Runge-Kutta method) with initial value X(t k−1 ) = 0. Recently, the geometric integration concepts were incorporated in the generalized α method [17,41] for MBS described in absolute coordinates. In this case the representation of proper rigid body motions is crucial, as discussed in [57,58], which is frequently incorrectly represented by SO(3) × R 3 .…”

Section: Geometric Integrationmentioning

confidence: 99%

Screw and Lie group theory in multibody dynamics

Müller

2017

Multibody Syst Dyn

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Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS), and at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows to use arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit-Hartenberg convention) and to avoid introduction of joint frames. The computational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formulations. In this paper, recursive O(n) Newton-Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These formulations are related to the corresponding algorithms that were presented in the literature. Two forms of MBS motion equations are derived in closed form using the Lie group formulation: the so-called Euler-Jourdain or "projection" equations, of which Kane's equations are a special case, and the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives. The geometric modeling allows for direct application of Lie group integration methods, which is briefly discussed.

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Explicit Newmark/Verlet algorithm for time integration of the rotational dynamics of rigid bodies

Cited by 57 publications

References 43 publications

Explicit momentum-conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

Explicit momentum-conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

Geometric methods and formulations in computational multibody system dynamics

Screw and Lie group theory in multibody dynamics

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