Fractional Birkhoffian method for equilibrium stability of dynamical systems

In this paper, we present a new method, i.e. fractional Birkhoffian method, for stability of equilibrium positions of dynamical systems, in terms of Riesz derivatives, and study its applications. For an actual dynamical system, the fractional Birkhoffian method of constructing a fractional dynamical model is given, and then the seven criterions for fractional Birkhoffian method of equilibrium stability are established. As applications, by using the fractional Birkhoffian method, we construct four kinds of actual fractional dynamical models, which include a fractional Duffing oscillator model, a fractional Whittaker model, a fractional Emden model and a fractional Hojman–Urrutia model, and we explore the equilibrium stability of these models respectively. This work provides a general method for studying the equilibrium stability of an actual fractional dynamical system that is related to science and engineering.     

Form invariance and conserved quantity for weakly nonholonomic system

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.     

Form invariance and noether symmetrical conserved quantity of relativistic Birkhoffian systems

IntroductionIn 1 92 7,theAmericanmathematicianG .D .BirkhoffmadeprimaryresearchesonBirkhoffiandynamics[1].In 1 983,theAmericanphysicistR .M .SantillistudiedthetransformationtheoryofBirkhoffequationsandgeneralizationofGalileirelativity ,andsummarizedcomprehensivelytheoriginofBirkhoffequationsandthelaterstudiesonthem[2 ].Since 1 992 ,theChinesemechanicianMeiFeng_xianganditsco_workershaveconstructedthedynamicsofBirkhoffiansystemonthebasisofRefs.[1 ,2 ] ,andgavethebasictheoreticalframe[3 - …     

Conformal invariance for nonholonomic system of Chetaev’s type with variable mass

Conformal invariance and conserved quantities for a nonholonomic system of Chetaev’s type with variable mass are studied. The conformal factor expressions are derived. The necessary and sufficient conditions are obtained to make the system’s conformal invariance Lie symmetrical. The conformal invariance of the weak and strong Lie symmetries for the system is given. The corresponding conserved quantities of the system are derived. Finally, an application of the result is shown with an example.     

Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems

Based on Riemann-Liouville fractional derivatives, conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems are investigated. Firstly, fractional generalized Birkhoff equations are obtained by studying fractional generalized Pfaff-Birkhoff principle. Secondly, the definition of fractional generalized quasi-symmetry is given, the criteria of fractional generalized quasi-symmetry and the corresponding conserved quantity are achieved for fractional generalized Birkhoffian systems. Thirdly, perturbation to symmetry and adiabatic invariants for disturbed fractional generalized Birkhoffian systems are presented. Finally, an example is given to illustrate the results.     

Lie transformation method for dynamical systems having chaotic behavior

The paper persents recent developments in a singular perturbation method, known as the Lie transformation method for the analysis of nonlinear dynamical systems having chaotic behavior. A general approximate solution for a system of first-order differential equations having algebraic nonlinearities is introduced. Past applications to simple dynamical nonlinear models have shown that this method yields highly accurate solutions of the systems. In the present paper the capability of this method is extended to the analysis of dynamical systems having chaotic behavior: indeed, the presence of small divisors in the general expression of the solution suggests a modification of the method that is necessary in order to analyze nonlinear systems having chaotic behavior (indeed, even non-simple-harmonic behavior). For the case of Hamiltonian systems this is consistent with the KAM (Kolmogorov-Arnold-Moser) theory, which gives the limits of integrability for such systems; in contrast to the KAM theory, the present formulation is not limited to conservative systems. Applications to a classic aeroelastic problem (panel flutter) are also included.     

Lie symmetries and conserved quantities of holonomic variable mass systems

In this paper, the Lie symmetries and the conserved quantities of the holonomic variable mass systems are studied. By using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the conserved quantities are given. And an example is given to illustrate the application of the result. Foundation item: the National Natural Science Foundation of China (19572038)     

Lie symmetries and conserved quantities of rotational relativistic systems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ1979,Ｒ.ＢｅｎｇｔｓｓｏｎａｎｄＳ.Ｆｒａｎｅｎｄｏｒｆａｃｃｕｒａｔｌｙｍｅａｓｕｒｅｄｔｈｅｍａｘｉｍｕｍｖａｌｕｅｓｏｆｔｈｅｓｐｉｎｖｅｌｏｃｉｔｙｏｆ14ｋｉｎｄｓｏｆｎｕｃｌｅｏｎｓ,ａｎｄｔｈｅｒｅｓｕｌｔｓｓｈｏｗｅｄｔｈａｔｔｈｅｍａｘｉｍｕｍｖａｌｕｅｏｆｔｈｅｓｐｉｎｖｅｌｏｃｉｔｙｏｆｏｎｅｎｕｃｌｅｏｎｗａｓｄｉｆｆｅｒｅｎｔｔｏｔｈｏｓｅｏｆｔｈｅｏｔｈｅｒｓ[1].Ｗｉｔｈｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ,…     

Adaptive explicit Magnus numerical method for nonlinear dynamical systems

Based on the new explicit Magnus expansion developed for nonlinear equations defined on a matrix Lie group, an efficient numerical method is proposed for nonlinear dynamical systems. To improve computational efficiency, the integration step size can be adaptively controlled. Validity and effectiveness of the method are shown by application to several nonlinear dynamical systems including the Duffing system, the van der Pol system with strong stiffness, and the nonlinear Hamiltonian pendulum system.     

A new Lie symmetrical method of finding conserved quantity for Birkhoffian systems

For a Birkhoffian system, a new Lie symmetrical method to find a conserved quantity is given. Based on the invariance of the equations of motion for the system under a general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations are obtained. Then, several important relationships which reveal the interior properties of the Birkhoffian system are given. By using these relationships, a new Lie symmetrical conservation law for the Birkhoffian system is presented. The new conserved quantity is constructed in terms of infinitesimal generators of the Lie symmetry and the system itself without solving the structural equation which may be very difficult to solve. Furthermore, several deductions are given in the special infinitesimal transformations and the results are reduced to a Hamiltonian system. Finally, one example is given to illustrate the method and results of the application.     

Simulation of the two-fluid model on incompressible flow with Fractional Step method for both resolved and unresolved scale interfaces

In the present paper, the Fractional Step method usually used in single fluid flow is here extended and applied for the two-fluid model resolution using the finite volume discretization. The use of a projection method resolution instead of the usual pressure-correction method for multi-fluid flow, successfully avoids iteration processes. On the other hand, the main weakness of the two fluid model used for simulations of free surface flows, which is the numerical diffusion of the interface, is also solved by means of the conservative Level Set method (interface sharpening) (Strubelj et al., 2009). Moreover, the use of the algorithm proposed has allowed presenting different free-surface cases with or without Level Set implementation even under coarse meshes under a wide range of density ratios. Thus, the numerical results presented, numerically verified, experimentally validated and converged under high density ratios, shows the capability and reliability of this resolution method for both mixed and unmixed flows.     

A method for finding the invariants and exact solutions of coupled non-linear differential equations with applications to dynamical systems

The polynomial invariants of (a set) non-linear differential equations are found by using a direct approach. The integrability of these invariants deserves the integrability of the given set of coupled differential equations. As applications, the Lorenz and Rikitake sets, among others, are studied. New invariants are obtained.     

Parametric open-plus-closed-loop control of chaos in continuous dynamical systems

This paper presents a parametric open-plus-closed-loop control approach to controlling chaos in continuous dynamical systems. As an example, chaos in the Lorenz model is controlled to demonstrate its application. Finally, the relations between the parametric open-plus-closed-loop control and the former control methods, such as the open-plus-closed-loop control and the parametric entrainment control, are discussed.Supported by the Science Foundation of the State Education Commission for Doctorate Program, and the Applied Science Foundation of the State Ministry of Metallurgical Industry.     

Adjoint operator method and normal forms of higher order for nonlinear dynamical system

I.IntroductionInordertostudybifurcationsofnonlineardynamicalsystemsinthedegeneratecasesofhighercodimensionnumber(>3),wemustcomputenormalformsofhigherorderfornonlineardynamicalsystems.Inrecenttwentyyearsmanyscientistsmadeveryimportantcontributionstodevelop…     

On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems

Recently Yuan and Agrawal [L. Yuan and O.P. Agrawal, A numerical scheme for dynamic systems containing fractional derivatives, Journal of Vibration and Acoustics 124 (2002) 321–324] presented a new numerical scheme to calculate the dynamic response of mechanical systems, the damping forces of which are described by fractional derivatives. When solving the resulting equation of motion by time integration, it is necessary to store the entire displacement history of the system due to the non-local character of the fractional derivatives. The cited scheme appears to overcome this drawback by transforming the equation of motion with the fractional term into a set of ordinary differential equations. It can be shown that this scheme is equivalent to a classical spring-dashpot representation and thus does not imply the benefits derived from fractional-derivative models. In addition, it is less flexible and incorrectly predicts the asymptotic behavior.     

Screw-matrix method in dynamics of multibody systems

In the present paper the concept of screw in classical mechanics is expressed in matrix form, in order to formulate the dynamical equations of the multibody systems. The mentioned method can retain the advantages of the screw theory and avoid the shortcomings of the dual number notation. Combining the screw-matrix method with the tool of graph theory in Roberson/Wittenberg formalism. We can expand the application of the screw theory to the general case of multibody systems. For a tree system, the dynamical equations for eachj-th subsystem, composed of all the outboard bodies connected byj-th joint can be formulated without the constraint reaction forces in the joints. For a nontree system, the dynamical equations of subsystems and the kinematical consistency conditions of the joints can be derived using the loop matrix. The whole process of calculation is unified in matrix form. A three-segment manipulator is discussed as an example. This work is supported by the National Natural Science Fund.     

不确定动力系统平稳随机响应分析

由于设计、建造以及测量等诸多不确定因素的影响,通常的有限元力学分析模型只是原型结构的一种均值近似,采用随机结构模型是更为合理的.本文应用随机矩阵模拟不确定线性动力系统有限元模型中质量阵、阻尼阵和刚度阵的随机不确定性,并进一步建立此类非参数概率系统在平稳随机外载作用下动力响应的虚拟激励高效求解算法.数值结果表明,均值有限元模型和随机矩阵模型的动力响应具有很大的差异.对于精细制造,模型的随机性是不能忽略的,本文提出的算法为此类问题求解提供了一条有效途径.     

A tensor method for the derivation of the equations of rigid body dynamics

A tensor method for the derivation of the equations of rigid body dynamics,based onthe concepts of continuum mechanics,is presented.The formula of time derivative of theinertia tensor with zero corotational rate is used to prove the equivalences of five methods,namely,Lagrange’s equations,Nielsen’s equations,Gibbs-Appell’s equations,Kane’sequations and the generalized momentum type of Kane’s equations.Some differentialidentities on angular velocity and angular acceleration are given.     

一种非线性系统周期解的延拓算法

提出了一种非线性系统周期解的延拓算法。指出了非线性系统周期解在分岔点处由于雅可比矩阵奇异而导致一般延拓方法延拓失败问题;然后基于推广的打靶法的思想,将普通延拓算法推广,提出了一种用于周期解延拓的算法。对于非线性动力系统,该算法可以在已知某一参数下的周期解的基础上,求解出在一定参数范围内非线性动力系统的解随参数的连续变化情况。应用该方法对非线性柔性转子-轴承系统的周期解与参数的依赖关系进行了求解,验证了方法的有效性。     

Lattice with long-range interaction of power-law type for fractional non-local elasticity

Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the transform operation, we map the lattice equations into continuum equation with Riesz derivatives of non-integer orders. The continuum equations that are obtained from the lattice model describe fractional generalization of non-local elasticity models. Particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case.