Variational Integrators for Dissipative Systems With One Degree of Freedom

2011 ◽  
Author(s):  
Masashi Iura
Keyword(s):  
Mixed Method ◽  
Degree Of Freedom ◽  
Dissipative Systems ◽  
Variational Integrators ◽  
Stationary Condition ◽  
Coupling Term ◽  
D’Alembert Principle ◽  
Discrete Algorithms ◽  
Numerical Integrators ◽  
Integration Techniques

Variational integrators are developed for dissipative systems with one degree of freedom. The dissipation considered herein is of simple Rayleigh dissipation type. The present formulation is based not on the Lagrange-d’Alembert principle, but on Hamilton’s principle. A benefit for using variational integration techniques is stressed in this paper. The discrete algorithms are obtained by a stationary condition of action integral, in which the Lagrangian is directly discretized. Unlike the existing algorithms, a coupling term between mass and dissipation exists in the present algorithms. A mixed method, in which a velocity is independent on a position coordinate, is presented for dissipative systems. In order to investigate an accuracy of numerical integrators, we introduce a new parameter in addition to the energy decay. Numerical examples show that the present variational, integrators are available for not only highly but also weakly dissipative systems.

Dynamics of Dissipative Systems with Hamiltonian Structures

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Xiaoming Zhang ◽  
Zhenbang Cao ◽  
Jianhua Xie ◽  
Denghui Li ◽  
Celso Grebogi
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Structure ◽  
Degree Of Freedom ◽  
Dissipative Systems ◽  
Integral Invariant ◽  
Hamiltonian Structures ◽  
Cartan Type ◽  
Two Degree Of Freedom

In this work, we study a class of dissipative, nonsmooth [Formula: see text] degree-of-freedom dynamical systems. As the dissipation is assumed to be proportional to the momentum, the dynamics in such systems is conformally symplectic, allowing us to use some of the Hamiltonian structure. We initially show that there exists an integral invariant of the Poincaré–Cartan type in such systems. Then, we prove the existence of a generalized Liouville Formula for conformally symplectic systems with rigid constraints using the integral invariant. A two degree-of-freedom system is analyzed to support the relevance of our results.

Numerical Analysis of the Galloping Character of Fan-Shaped Ice Cover Conductor

2014 ◽  
Vol 607 ◽  
pp. 176-180
Author(s):  
Zhi Hui Cheng ◽  
You Hui Chen ◽  
Ru Yu Zhang
Keyword(s):  
Numerical Simulation ◽  
Numerical Analysis ◽  
Air Force ◽  
Torsional Vibration ◽  
Sine Wave ◽  
Degree Of Freedom ◽  
Overhead Line ◽  
Wind Speeds ◽  
D’Alembert Principle ◽  
Three Degree Of Freedom

The hypothesis of half sine wave of cable galloping in span is adopted. For the problem of flabelliform iced cover conductor galloping, this paper presents a flabelliform cable galloping model of three degree-of-freedom based on D’Alembert Principle which considers the velocity coupling between air force and the cable torsional vibration. For the typical parameters of overhead line, based on the flabelliform cable galloping model of three degree-of-freedom, used the software Matlab for numerical simulation. Explore the flabelliform cable galloping characteristic of the wire on the condition of the different wind speeds, different angle of attack and span. According to the horizontal and vertical amplitude changes, determined the critical condition of transmission line dancing, and compared with the measured results are closer, according with the galloping mechanism of Nigol. It’s proved this theory is feasible, which is helpful to further study and the character of galloping.

Nonlinear vibrations of dissipative systems with one degree of freedom

1984 ◽  
Vol 16 (5) ◽  
pp. 676-682
Author(s):  
V. A. Bazhenov ◽  
A. A. Grom ◽  
V. I. Gulyaev ◽  
P. P. Lizunov
Keyword(s):  
Nonlinear Vibrations ◽  
Degree Of Freedom ◽  
Dissipative Systems

Computational Techniques for Finite Element Method Based Kineto-Elastodynamic Analysis of Mechanisms

1994 ◽  
Author(s):  
Vinod K. Mahna ◽  
Nanik T. Asnani
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
Solution Procedure ◽  
Computational Techniques ◽  
Coupling Term ◽  
Geometric Stiffness ◽  
Integration Techniques ◽  
Computational Aspects ◽  
Element Method

Abstract The aim of computational techniques employed for the kineto-elastodynamic analysis of mechanisms is to evaluate the strain induced at any prescribed location in an elastic constituent link of the mechanism. The problem to be grappled is one of multi-degree-of-freedom coupled differential equation and is amenable to its solution by the modal analysis and numerical integration techniques. This formidable problem calls for imposition of boundary conditions and inter-link compatibility requirements, assembly of elemental matrices, incorporation of geometric stiffness matrix, evaluation of non-symmetric coupling term matrices, elimination of rigid-body degrees of freedom, transformation of local constraints pressed upon by the kinematic joints to the global frame before the stage gets propitious to embark upon the solution procedure. This paper enlarges upon the computational aspects involved in the kineto-elastodynamic analysis of mechanisms using finite-element method.

scholarly journals Structure-preserving integrators for dissipative systems based on reversible– irreversible splitting

2020 ◽  
Vol 476 (2234) ◽  
pp. 20190446 ◽  
Author(s):  
Xiaocheng Shang ◽  
Hans Christian Öttinger
Keyword(s):  
Dissipative Systems ◽  
Symplectic Method ◽  
Accuracy Control ◽  
Structure Preserving ◽  
Thermodynamic Structure ◽  
Degeneracy Condition ◽  
Numerical Integrators ◽  
Frame Work ◽  
Irreversible Dynamics ◽  
Damped Systems

We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.

scholarly journals Variational Integrators for Dissipative Systems

2020 ◽  
Vol 65 (4) ◽  
pp. 1381-1396 ◽  
Author(s):  
David J. N. Limebeer ◽  
Sina Ober-Blobaum ◽  
Farhang Haddad Farshi
Keyword(s):  
Dissipative Systems ◽  
Variational Integrators

scholarly journals Variational integrators, the Newmark scheme, and dissipative systems

Equadiff 99 ◽  
2000 ◽  
pp. 1009-1011 ◽  
Author(s):  
M. West ◽  
C. Kane ◽  
J.E. Marsden ◽  
M. Ortiz
Keyword(s):  
Dissipative Systems ◽  
Variational Integrators ◽  
Newmark Scheme

scholarly journals Variational integrators for stochastic dissipative Hamiltonian systems

2020 ◽  
Author(s):  
Michael Kraus ◽  
Tomasz M Tyranowski
Keyword(s):  
Hamiltonian Systems ◽  
Generating Function ◽  
Numerical Test ◽  
Planck Equation ◽  
Geometric Approach ◽  
Variational Integrators ◽  
Discrete Version ◽  
Structure Preserving ◽  
D’Alembert Principle ◽  
Long Time

Abstract Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian, which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange–d’Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange–d’Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether’s theorem. Furthermore, mean-square and weak Lagrange–d’Alembert Runge–Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behaviour compared to nongeometric methods. The Vlasov–Fokker–Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.

scholarly journals A General Numerical Scheme for the Optimal Control of Fractional Birkhoffian Systems

2021 ◽  
Author(s):  
Lin He ◽  
Chunqiu Wei ◽  
Jiang Sha ◽  
Delong Mao ◽  
Kangshuo Wang
Keyword(s):  
Optimal Control ◽  
Sequential Quadratic Programming ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problem ◽  
State Constraints ◽  
Variational Integrators ◽  
Final State ◽  
D’Alembert Principle ◽  
Algebraic Constraints

Abstract This paper deals with the optimal control of fractional Birkhof-fian systems based on the numerical method of variational integrators. Firstly, the fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d'Alembert principle. Secondly, by directly discretizing the fractional Pfaff–Birkhoff–d'Alembert principle, we develop the equivalent discrete fractional forced Birkhoff equations, which are served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by the existing methods such as sequential quadratic programming. Finally, an example is given to show the efficiency and simplicity of the proposed method.

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