Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

Marsden, Jerrold E.; Wendlandt, Jeffrey M.

doi:10.1007/978-1-4612-2012-1_18

Cited by 24 publications

(15 citation statements)

References 64 publications

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“…The Moser-Veselov equations (4.1)-(4.3) can be obtained by a discrete variational principle, as was done in Moser and Veselov (1991). This variational principle has the general form of that in discrete mechanics described in, for example, Marsden and Wendlandt (1997), Bobenko and Suris (1999) and Marsden and West (2001). See also the following sections on optimal control.…”

Section: Mv-algorithm 2 Define the Mapmentioning

confidence: 99%

The symmetric representation of the rigid body equations and their discretization

et al. 2002

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This paper analyses continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO(n) × SO(n) and study its associated symplectic structure. We describe the relationship of these ideas with the Moser-Veselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in this paper may be found in Bloch et al (Bloch A M, Crouch P, Marsden J E and Ratiu T S 1998 Proc. IEEE Conf. on Decision and Control 37 2249-54).

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Section: Mv-algorithm 2 Define the Mapmentioning

confidence: 99%

The symmetric representation of the rigid body equations and their discretization

et al. 2002

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“…On one hand, the kinetic energy of the rod model is calculated as 9 For a theoretical treatment see Refs. [156][157][158].…”

Section: Equilibrium Equationsmentioning

confidence: 98%

“…For conservative systems, usual variational principles of mechanics are used while for dissipative or forced systems, the Lagranged'Alembert principle is preferred. 13 A complete survey about variational integrators can be reviewed in [133,157,158]. Additionally, Kane et al [116] discus about variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems.…”

Section: Time-stepping Schemementioning

confidence: 98%

Constitutive and Geometric Nonlinear Models for the Seismic Analysis of RC Structures with Energy Dissipators

Mata

Barbat

Oller

et al. 2008

Arch Computat Methods Eng

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Nowadays, the use of energy dissipating devices to improve the seismic response of RC structures constitutes a mature branch of the innovative procedures in earthquake engineering. However, even though the benefits derived from this technique are well known and widely accepted, the numerical methods for the simulation of the nonlinear seismic response of RC structures with passive control devices is a field in which new developments are continuously preformed both in computational mechanics and earthquake engineering. In this work, a state of the art of the advanced models for the numerical simulation of the nonlinear dynamic response of RC structures with passive energy dissipating devices subjected to seismic loading is made. The most commonly used passive energy dissipating devices are described, together with their dissipative mechanisms as well as with the numerical procedures used in modeling RC structures provided with such devices. The most important approaches for the formulation of beam models for RC structures are reviewed, with emphasis on the theory and numerics of formulations that consider both geometric and constitutive sources on nonlinearity. In the same manner, a more complete treatment is given to the constitutive nonlinearity in the context of fiber-like approaches including the corresponding cross sectional analysis. Special attention is paid to the use of damage indices able of estimating the remaining load carrying capacity of structures after a seismic action. Finally, nonlinear constitutive and geometric formulations for RC beam elements are examined, together with energy dissipating devices formulated as simpler beams with adequate constitutive laws. Numerical examples allow to illustrate the capacities of the presented formulations.

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“…The essential aspects of VI's can be reviewed in Marsden and Wendlandt 1997; and in (Leok and Shingel 2012a;Leok 2005) general techniques for constructing variational integrators are provided. Spectral variational integrators are described in (Hall and Leok 2014a) and prolongation-collocation methods in (Leok and Shingel 2012b).…”

Section: Variational Integrationmentioning

confidence: 99%

A Brief Introduction to Variational Integrators

Lew

2016

Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics

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In this chapter, a brief introduction to the formulation of variational methods for finite-dimensional Lagrangian systems is presented. To this end, the first two sections focus on describing the Lagrangian and Hamiltonian points of view of mechanics for systems evolving on manifolds. Special attention is paid to the construction of the Lagrangian function and to the role of Hamilton's variational principle in the deduction of the balance equations. The relation between the symmetries of the Lagrangian function and the existence of invariants of the dynamics along with the symplectic nature of the flow are also addressed. In the third section, the discussion turns towards the formulation of a time-discrete analogue of the theory. The cornerstone of such a construction is given by a discrete analogue of Hamilton's variational principle which provides a systematic procedure to construct discrete approximations to the exact trajectory of a mechanical system on both the configuration space and the phase space. The approximation properties and the geometric characteristics of the resulting discrete trajectories are explained. Finally, we apply the variational methodology to construct symplectic and momentum-conserving time integrators for two problems of practical interest in engineering and science.

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Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

Cited by 24 publications

References 64 publications

The symmetric representation of the rigid body equations and their discretization

The symmetric representation of the rigid body equations and their discretization

Constitutive and Geometric Nonlinear Models for the Seismic Analysis of RC Structures with Energy Dissipators

A Brief Introduction to Variational Integrators

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