STUDYING THE FOCAL VALUE OF ORDINARY DIFFERENTIAL EQUATIONS BY NORMAL FORM THEORY

ＳＴＵＤＹＩＮＧＴＨＥＦＯＣＡＬＶＡＬＵＥＯＦＯＲＤＩＮＡＲＹＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＢＹＮＯＲＭＡＬＦＯＲＭＴＨＥＯＲＹＺｈａｎｇＱｉｃｈａｎｇ（张琪昌）（ＴｉａｎｊｉｎＵｎｉｖｅｒｓｉｔｙ．Ｔｉａｎｊｉｎ３０００７２．Ｐ．Ｒ．Ｃ...     

范式理论与平均法的等价性分析

分析了范式理论与平均法的等价性。得到的结论是：对含有一对纯虚根的二维非线性系统，使用两种方法得到的结果是等价的，并提供了两个算例来证实其结论的正确性。虽然分析是针对一类二维非线性系统，但其结论同样适合于高维非线性系统。     

Normal form of the nonsemi-simple bifurcation problem

I.IntroductionBynow,agreatamountoftheoreticalresultshavebeenachievedinnonlinearscience.Tofindthenewmotivationforthefurtherdevelopmentinnonlinearscience,moreattentionarepayedtoapplicationsoftheseachievementandnewbreakthroughhasbeenexpectedsince1990s.Thesit…     

Convergence of Birkhoff Normal form for Essentially Isochronous Systems

We reconsider the problem of the convergence of Birkhoff's normal form for a system of perturbed harmonic oscillators, under the condition that the system is essentially isochronous. In contrast with previous proofs based on the so called quadratically convergent method, the present proof uses only classical expansions in a parameter. This allows us to bring into light some mechanisms of accumulation of small divisors, which can be useful in more complicated and interesting cases. These same mechanisms allow us to prove the theorem with the Bruno condition on the frequencies in a very natural way.     

随机平均规范形方法

计算随机规范形系数是应用随机规范形方法的关键．提出一种应用随机平均计算随机规范形系数的方法．为了说明方法的有效性,对白噪声激励的Duffing系统,经过变换,对于相应的平均方程,比较了精确解、规范形方法解和能量包络方法解的稳态概率密度,结果表明,当非线性项系数较小时,三者完全一致．当非线性项系数较大时,规范形方法所得解与精确解相差不大．     

Complex Inner Product Averaging Method for Calculating Normal Form of Ode

ＩｎｔｒｏｄｕｃｔｉｏｎＭｏｓｔｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓｙｓｔｅｍｓａｒｅｍｕｌｔｉ＿ｄｅｇｒｅｅ＿ｏｆ＿ｆｒｅｅｄｏｍｓｙｓｔｅｍｓｉｎｔｈｅｆｉｅｌｄｏｆｅｎｇｉｎｅｅｒｉｎｇ[1,2 ],ｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓｙｓｔｅｍｓｄｏｎｏｔｈａｖｅｅｘａｃｔｌｙｔｈｅｏｒｅｔｉｃｓｏｌｕｔｉｏｎｓ.Ｔｈｅｒｅｆｏｒｅ,ｉｔｉｓａｆｏｃａｌｐｏｉｎｔｃｏｎｃｅｒｎｅｄｂｙｓｃｉｅｎｔｉｓｔｓｔｏｓｅａｒｃｈｆｏｒａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｓｆｏｒａｌｏｎｇｔｉｍｅ ,ａｎｄｒ…     

The new geometric criterion of polynomial stability

In this paper, we apply the coningacy and boundedness of the zeros for a polynomial f_n(z) with real coefficienta (i=0,1,2,…,n). A new simple geometric criterion for stability of f_n(z) is given which is very convenient for application.     

A note on uniqueness of the normal form for quasi-integrable systems

We provide a simple argument that at non-resonant actions the normal form for a quasi-integrable Hamiltonian system, as defined by von Zeipel-Poicaré and Lie perturbation algorithms, is unique.

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Sommario Si fornisce una semplice dimostrazione dell'unicità della forma normale di un sistema hamiltoniano quasi-integrabile, come definito dagli algoritmi perturbativi di VonZeipel-Poincarè e Lie.

     

On the sensitivity of non-generic bifurcation of non-linear normal modes

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r=1,3,5,… . Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears, and generic bifurcation appears in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.     

The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

This paper considers the computation of the simplest parameterized normal forms (SPNF) of Hopf and generalized Hopf bifurcations. Although the notion of the simplest normal form has been studied for more than two decades, most of the efforts have been spent on the systems that do not involve perturbation parameters due to the restriction of the computational complexity. Very recently, two singularities – single zero and Hopf bifurcation – have been investigated, and the SPNFs for these two cases have been obtained. This paper extends a recently developed method for Hopf bifurcation to compute the SPNF of generalized Hopf bifurcations. The attention is focused on a codimension-2 generalized Hopf bifurcation. It is shown that the SPNF cannot be obtained by using only a near-identity transformation. Additional transformations such as time and parameter rescaling are further introduced. Moreover, an efficient recursive formula is presented for computing the SPNF. Examples are given to demonstrate the applicability of the new method.     

On general form of Navier-Stokes equations and implicit factored scheme

A general weak conservative form of Navier-Stokes equations expressed with respect to non-orthogonal Curvilinear coordinates and with primitive variables was obtained by using tensor analysis technique, where the contravariant and covariant velocity components were employed. Compared with the current coordinate transformation method, the established equations are concise and forthright, and they are more convenient to be used for solving problems in body-fitted curvilinear coordinate system. An implicit factored scheme for solving the equations is presented with detailed discussions in this paper. For n-dimensional flow the algorithm requires n-steps and for each step only a block tridiagonal matrix equation needs to be solved. It avoids inverting the matrix for large systems of equations and enhances the speed of arithmetic. In this study, the Beam- Warming’s implicit factored schceme is extended and developed in non-orthogonal curvilinear coordinate system.     

Resonant non-linear normal modes. Part I: analytical treatment for structural one-dimensional systems

Approximations of the resonant non-linear normal modes of a general class of weakly non-linear one-dimensional continuous systems with quadratic and cubic geometric non-linearities are constructed for the cases of two-to-one, one-to-one, and three-to-one internal resonances. Two analytical approaches are employed: the full-basis Galerkin discretization approach and the direct treatment, both based on use of the method of multiple scales as reduction technique. The procedures yield the uniform expansions of the displacement field and the normal forms governing the slow modulations of the amplitudes and phases of the modes. The non-linear interaction coefficients appearing in the normal forms are obtained in the form of infinite series with the discretization approach or as modal projections of second-order spatial functions with the direct approach. A systematic discussion on the existence and stability of coupled/uncoupled non-linear normal modes is presented. Closed-form conditions for non-linear orthogonality of the modes, in a global and local sense, are discussed. A mechanical interpretation of these conditions in terms of virtual works is also provided.     

Numerical computation of three‐dimensional incompressible Navier–Stokes equations in primitive variable form by DQ method

In this paper, the global method of differential quadrature (DQ) is applied to solve three‐dimensional Navier–Stokes equations in primitive variable form on a non‐staggered grid. Two numerical approaches were proposed in this work, which are based on the pressure correction process with DQ discretization. The essence in these approaches is the requirement that the continuity equation must be satisfied on the boundary. Meanwhile, suitable boundary condition for pressure correction equation was recommended. Through a test problem of three‐dimensional driven cavity flow, the performance of two approaches was comparatively studied in terms of the accuracy. The numerical results were obtained for Reynolds numbers of 100, 200, 400 and 1000. The present results were compared well with available data in the literature. In this work, the grid‐dependence study was done, and the benchmark solutions for the velocity profiles along the vertical and horizontal centrelines were given. Copyright © 2003 John Wiley & Sons, Ltd.     

A form function method for form finding of tension structure

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｅｎｇｉｎｅｅｒｉｎｇ ,ｉｔｉｓｏｆｔｅｎｒｅｑｕｉｒｅｄｔｏｅｘｔｅｎｄａｔｅｎｓｉｏｎｓｔｒｕｃｔｕｒｅｏｎａｇｉｖｅｎｂｏｕｎｄａｒｙ ,ｓｕｃｈａｓｃａｂｌｅｎｅｔｏｒｍｅｍｂｒａｎｅ.Ｔｏａｎａｌｙｚｅｔｈｅｓｔｒｕｃｔｕｒｅ ,ｉｔｉｓｎｅｃｅｓｓａｒｙｔｏｋｎｏｗｔｈｅｉｎｉｔｉａｌｆｏｒｍｏｆｉｔ.Ｂｕｔｉｔｉｓｎｏｔｅａｓｙｔｏｇｅｔａｎａｃｃｕｒａｔｅｏｎｅｏｎａｎａｖｅｒａｇｅｂｏｕｎｄａｒｙ .Ｔｈｅｆｏｒｍｏｆｔｅｎｓｉｏｎｓｔｒｕｃｔｕｒｅｕｎｄｅ…     

On the normal forms of Hamiltonian systems

We propose a novel method to analyze the dynamics of Hamiltonian systems with a periodically modulated Hamiltonian. The method is based on a special parametric form of the canonical transformation , using Poincaré generating function Ψ (t,x,y). As a result, stability problem of a periodic solution is reduced to finding a minimum of the Poincaré function. The proposed method can be used to find normal forms of Hamiltonians. It should be emphasized that we apply the modified concept of Zhuravlev [Introduction to Theoretical Mechanics. Nauka Fizmatlit, Moscow (1997); Prikladnaya Matematika i Mekhanika 66(3), (2002) in Russian] to define an invariant normal form, which does not require any partition to either autonomous – non-autonomous, or resonance – non-resonance cases, but it is treated in the frame of one approach. In order to find the corresponding normal form asymptotics, a system of equations is derived analogous to Zhuravlev's chain of equations. Instead of the generator method and guiding Hamiltonian, a parametrized guiding function is used. It enables a direct (without the transformation to an autonomous system as in Zhuravlev's method) computation of the chain of equations for non-autonomous Hamiltonians. For autonomous systems, the methods of computation of normal forms coincide in the first and second approximations. Using this method we will present solutions of the following problems: nonlinear Duffing oscillator; oscillation of a swinging spring; dynamics of solid particles in the acoustic wave of viscous liquid, and other problems.     

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

Two methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Weak formulation and first order form of the equations of motion for a class of constrained mechanical systems

Some new theoretical results are presented on modeling the dynamic response of a class of discrete mechanical systems subject to equality motion constraints. Both the development and presentation are facilitated by employing some fundamental concepts of differential geometry. At the beginning, the equations of motion of the corresponding unconstrained system are presented on a configuration manifold with general properties, first in strong and then in a primal weak form, using Newton׳s law of motion as a foundation. Next, the final weak form is obtained by performing a crucial integration by parts step, involving a covariant derivative. This step required the clarification and enhancement of some concepts related to the variations employed in generating the weak form. The second part of this work is devoted to systems involving holonomic and non-holonomic scleronomic constraints. The equations of motion derived in a recent study of the authors are utilized as a basis. The novel characteristic of these equations is that they form a set of second order ordinary differential equations (ODEs) in both the coordinates and the Lagrange multipliers associated to the constraint action. Based on these equations, the corresponding weak form is first obtained, leading eventually to a consistent first order ODE form of the equations of motion. These equations are found to appear in a form resembling the form obtained after application of the classical Hamilton׳s canonical equations. Finally, the new theoretical findings are illustrated by three representative examples.     

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对于连续截面直梁平面横力弯曲情形, 分析了剪切变形引起横截面翘曲的影响. 基于位移模式假定, 用材料力学方法得到相应的正应力与切应力表达式, 并以均布力作用的矩形截面简支梁为例说明截面翘曲对于应力的影响. 该分析方法及其结果适用于材料力学教学.     

On the absolute stability of a class of indirect control systems

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｉｎｄｉｒｅｃｔｃｏｎｔｒｏｌｓｙｓｔｅｍ ｘ =Ａｘ Ｂφ(σ) , σ =ＣＴｘ ρφ(σ) ,(1 )ｗｈｅｒｅｘ∈Ｒｎ,σ ∈Ｒ ,Ａｉｓｎ×ｎｓｔａｂｌｅｍａｔｒｉｘ ,ＢａｎｄＣａｒｅｎｄｉｍｅｎｓｉｏｎａｌｖｅｃｔｏｒｓ,ｔｈｅｓｙｍｂｏｌＴｓｔａｎｄｓｆｏｒｔｒａｎｓｐｏｓｉｔｉｏｎ ,ρｉｓａｃｏｎｓｔａｎｔ,ｆｕｎｃｔｉｏｎ φ(σ) :Ｒ →Ｒｉｓｃｏｎｔｉｎｕｏｕｓａｎｄｓａｔｉｓｆｉｅｓｃｏｎｄｉｔｉｏｎσφ(σ) >0　　 ( σ∈Ｒ ,σ≠ 0 ) . (2 )Ｍａｎｙｓｐｅ…