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Skew-gradient representations of constrained mechanical systems

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Applied Mathematics and Mechanics(English Edition) 2015年第7期36卷 873-882页

作者： Fengxiang MEI Jinchao CUI School of Aerospace Engineering Beijing Institute of Technology School of Science Jiangnan University

The characteristics of stationary and non-stationary skew-gradient systems are studied. The skew-gradient representations of holonomic systems, Birkhoffian systems, generalized Birkhoffian systems, and generalized Hamiltonian systems are given. The characteristics of skew-gradient systems are used to study integration and stability of the solution of constrained mechanical systems. Examples are given to illustrate applications of the result.

关键词： constrained mechanical system skew-gradient system integration stability

Bifurcation for the generalized Birkhoffian system

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Chinese Physics B 2015年第5期24卷 419-420页

作者：梅凤翔吴惠彬 School of Mathematics Beijing Institute of Technology

The system described by the generalized Birkhoff equations is called a generalized Birkhoffian system. In this paper, the condition under which the generalized Birkhoffian system can be a gradient system is given. The stability of equilibrium of the generalized Birkhoffian system is discussed by using the properties of the gradient system. When there is a parameter in the equations, its influences on the stability and the bifurcation problem of the system are considered.

关键词： generalized Birkhoffian system gradient system stability bifurcation

Two kinds of generalized gradient representations for holonomic mechanical systems

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Chinese Physics B 2016年第1期25卷 653-656页

作者：梅凤翔吴惠彬 School of Aerospace Engineering Beijing Institute of Technology School of Mathematics Beijing Institute of Technology

Two kinds of generalized gradient systems are proposed and the characteristics of the two systems are studied. The conditions under which a holonomic mechanical system can be considered as one of the two generalized gradient systems are obtained. The characteristics of the generalized gradient systems can be used to study the stability of the holonomic system. Some examples are given to illustrate the application of the results.

关键词： holonomic mechanical system generalized gradient system Lyapunov function stability

Form invariance and conserved quantity for weakly nonholonomic system

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Applied Mathematics and Mechanics(English Edition) 2014年第10期35卷 1293-1300页

作者：吴惠彬梅凤翔 School of Mathematics Beijing Institute of Technology

The form invariance and the conserved quantity for a weakly nonholonomic system （WNS） are studied. The WNS is a nonholonomic system （NS） whose constraint equations contain a small parameter. The differential equations of motion of the system are established. The definition and the criterion of form invariance of the system are given. The conserved quantity deduced from the form invariance is obtained. Finally, an illustrative example is shown.

关键词： weakly nonholonomic system (WNS) form invariance conserved quantity

An application of a combined gradient system to stabilize a mechanical system

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Chinese Physics B 2016年第10期25卷 10-12页

作者：陈向炜张晔梅凤翔 Department of Physics and Information Engineering Shangqiu Normal University School of Mathematics and Physics Suzhou University of Science and Technology School of Aerospace Beijing Institute of Technology

A gradient system and a skew-gradient system can be merged into a combined gradient system. The differential equations of the combined gradient system are established and its property is studied. If a mechanical system can be represented as a combined gradient system, the stability of the mechanical system can be studied by using the property of the combined gradient system. Some examples are given to illustrate the applications of the results.

关键词： gradient system combined gradient system mechanical system stability

Skew-gradient representation of generalized Birkhoffian system

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Chinese Physics B 2015年第10期24卷 312-314页

作者：梅凤翔吴惠彬 School of Aerospace Engineering Beijing Institute of Technology School of Mathematics Beijing Institute of Technology

The skew-gradient representation of a generalized Birkhoffian system is studied. A condition under which the generalized Birkhoffian system can be considered as a skew-gradient system is obtained. The properties of the skew-gradient system are used to study the properties, especially the stability, of the generalized Birkhoffian system. Some examples are given to illustrate the application of the result.

关键词： generalized Birkhoffian system skew-gradient system Lyapunov function stability

One kind motion of controllable constrained Birkhoffian system:the absence of constraints

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Acta Mechanica Sinica 2020年第3期36卷 735-741页

作者： J.Chen F.X.Mei S.X.Liu Y.X.Guo School of Aerospace Engineering Beijing Institute of TechnologyBeijing 100081China School of Physics Liaoning UniversityShenyang 110000China

This paper is devoted to discuss the motion of controllable constrained Birkhoffian system along with its absence of *** first step is to establish the autonomous and non-autonomous differential equations of motion of the system,based on Pfaff-Birkhoff ***,the existence of constraint multipliers are exhaustively ***,the definition of one kind motion of the system,called free motion,is given,which is described and analyzed by the absence of constraints that are determined by constraint *** 2 illustrates that one system can realize its free motion by selecting proper control *** particular,theorem 2 provides that one system can naturally realize free motion when we consider the integral of the unconstrained Birkhoffian system as the constraints of constrained Birkhoffian ***,the results obtained are illustrated by several examples.

关键词： Constrained Birkhoffian system Kinematic control Absence of constraint Free motion

Birkhoff symmetry and Lagrange symmetry

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Journal of Beijing Institute of Technology 2015年第1期24卷 1-3页

作者：梅凤翔吴惠彬 School of Aerospace Engineering Beijing Institute of Technology School of Mathematics Beijing Institute of Technology

The purpose of this paper is to study the symmetry of first order Lagrangians and the cor- responding conserved quantity. The relation between the Lagrangians and the Birkhoff＇ s functions and the Birkhoff symmetry of Birkhoffian systems are used to obtain the symmetry of first order La- grangians and the corresponding conserved quantity. Two examples are given to illustrate the appli- cation of the result.

关键词： Birkhoffian system first order Lagrangian symmetry conserved quantity

Birkhoffian symplectic algorithms derived from Hamiltonian symplectic algorithms

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Chinese Physics B 2016年第1期25卷 407-411页

作者：孔新雷吴惠彬梅凤翔 College of Science North China University of TechnologyBeijing 100144China School of Mathematics and Statistics Beijing Institute of TechnologyBeijing 100081China School of Aerospace Engineering Beijing Institute of TechnologyBeijing 100081China

In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation, applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoftian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.

关键词： Birkhoffian equations Hamiltonian equations symplectic algorithm