Equivalent higher order linear and non-linear systems

The object of this study is to put forth the concept of equivalence of classes of linear and non-linear systems of higher order. In particular, the equivalence of classes of non-linear (non-autonomous/autonomous) systems of order n described by partial/ordinary differential equations with corresponding classes of linear systems of order (n + 1) is established through a differential transformation of the dependent variable. In view of the fact that the resulting linear systems are amenable to existing state-space techniques, this approach can be expected to be of value in the study of many non-linear problems arising in a variety of disciplines. The possible applications of the technique are illustrated with an example.     

Equivalence conditions for classes of linear and non-linear distributed parameter systems

Equivalence of certain classes of second-order non-linear distributed parameter systems and corresponding linear third-order systems is established through a differential transformation technique. As linear systems are amenable to analysis through existing techniques, this study is expected to offer a method of tackling certain classes of non-linear problems which may otherwise prove to be formidable in nature.     

Supersymmetric origin of equivalent quantum systems

The previously established equivalence of certain multidimensional quantum hamiltonians is shown to be a consequence of the supersymmetry in quantum mechanics. Thereby the supersymmetric quantum mechanics can serve as a regular source of equivalent quantum systems in arbitrary space dimensions.     

Equilibrium point damage prognosis models for structural health monitoring

Most structural dynamic systems are of high order; however, they often exhibit phenomena that can be dealt with effectively using low order models. This paper presents a method for describing certain kinds of damage evolution in mechanical systems. The method relies on a simple principle that as damage evolves in a structural dynamic system, the damage indicator (i.e., diagnostic feature) behaves like a stable quasi-stationary equilibrium point in a subsidiary non-linear bifurcating system within the so-called damage center manifold. It is shown that just as linear normal modes govern the behavior of linear structures with idealized damping, so too do non-linear normal forms govern the evolution of damage within structures in many instances. The method is justified with citations from the literature on certain types of mechanical failure and then applied in an experimental case involving reversible damage in a bolted fastener. Off-line experiments on a rotorcraft fuselage show that the evolution of damage is sensitive to both temporal and spatial bifurcation parameters. A diagnostic sensing strategy whereby output-only transmissibility features are used to decrease the order of high order structural dynamic measurements is also described.     

Analytical theory for the description of powder systems under compression

In this work, a new theoretical approach to modelling some properties of powder systems under compression is presented. This new theoretical route consists of modelling an actual powder system (with particles of unequal size and irregular form) by means of a system of deforming spheres in a simple cubic arrangement and with a certain global porosity that, in some way, makes it equivalent to the actual one. The study of the evolution of the effective contact area between particles and the effective path of the electric or thermal flow through the powder aggregate is the starting point for establishing the equivalence relationship between the actual system and the simple cubic one. In order to exemplify the utility of this new theoretical tool, two classic problems of practical interest have been studied: the electrical conduction in sintered powders and the law governing the powders’ cold die compaction. The proposed solutions to these problems, as well as the equations allowing one to obtain the equivalence relationship, are validated by experiments carried out in actual powder systems.     

Effective Hamiltonian approach to doubly degenerate electronic states

Effective vibronic Hamiltonian models are built for E ⊗e Jahn-Teller systems and analytical solutions are obtained through Lie algebraic methods. Although approximate, we show that these models allow in particular to recover the possible ground state crossover when quadratic couplings are present. The equivalence of E ⊗e and G' ⊗e vibronic systems in cubic symmetry is precisely established through a particular realization of the electronic operators for an orbital quadruplet. We show how this equivalence is broken by a rovibronic interaction which, for a G' ⊗e system, still gives an exactly solvable model.     

一类分数阶混沌系统的研究

推广了一类分数阶混沌系统并证明了这类分数阶混沌系统的拓扑等价性,指出分数阶系统产生混沌吸引子的必要条件是系统平衡点的稳定性不变.通过数值模拟表明,此分数阶系统与整数阶系统一样仍然产生一对互不相交的2-涡卷混沌吸引子. 关键词： 分数阶 混沌 稳定性     

Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.     

Non-linear Liouville and Shrödinger equations in phase space

Unitary representations of the Galilei group are studied in phase space, in order to describe classical and quantum systems. Conditions to write in general form the generator of time translation and Lagrangians in phase space are then established. In the classical case, Galilean invariance provides conditions for writing the Liouville operator and Lagrangian for non-linear systems. We analyze, as an example, a generalized kinetic equation where the collision term is local and non-linear. The quantum counter-part of such unitary representations are developed by using the Moyal (or star) product. Then a non-linear Schrödinger equation in phase space is derived and analyzed. In this case, an association with the Wigner formalism is established, which provides a physical interpretation for the formalism.     

Vibration of cracked shafts in bending

Cracked rotating shafts exhibit a certain particular dynamic response due to the local flexibility of the cracked section. In this response, most of the features of the response of a shaft with dissimilar moments of inertia can be identified. Moreover, the non-linear behavior of the closing crack introduces the characteristics of non-linear systems. For many practical applications, the system can be considered bi-linear and analytical methods can be applied. A de Laval rotor with an open crack is investigated by way of application of the theory of shafts with dissimilar moments of inertia. Furthermore, analytical solutions are obtained for the closing crack under the assumption of large static deflections, a situation common in turbomachinery. Finally, a solution is developed for the case in which the local flexibility function is found experimentally.     

一种二阶混合有限体元格式的GAMG预条件子

针对一种含跳系数椭圆问题的二阶混合有限体元格式,讨论求解相应离散系统PGMRES法的预条件子构造问题.通过严格的理论分析,建立分层基下该二阶混合有限体元刚度矩阵和二次有限元刚度矩阵的谱等价关系,并利用关于二次有限元刚度矩阵的一种基于分层思想的GAMG预条件子,为二阶混合有限体元刚度矩阵设计一种高效GAMG预条件子.数值结果验证理论分析的正确性和新预条件子的高效性与稳定性.     

铅基反应堆自然循环与应急余热排出研究

自然循环特性是铅基反应堆一回路的关键运行特性,对反应堆的非能动应急余热排出具有重要的影响,自然循环特性与余热排出能力是反应堆热工水力研究的重要内容。采用多孔介质方法,建立了CiADS铅基堆1/4三维计算模型,使用FLUENT程序对额定工况与低功率工况进行稳态计算。为了研究全厂断电事故下的余热排出过程,从热工水力的等效原则出发,尝试建立二维等效模型以提高瞬态计算效率。结果表明,CiADS铅基堆具备低功率自然循环运行能力和一定的事故容错能力;二维等效模型与三维模型计算结果吻合较好,可用于瞬态下的简化分析;CiADS铅基堆的非能动余热排出系统能够较好地应对全厂断电事故,反应堆具有良好的固有安全性。     

平行双线BCI等效替代强场连续波电磁辐射实验研究

由于互联系统信号传输的基础是双线,以低频线缆耦合通道中比较典型的平行双线为研究对象,针对互联系统单端为非线性受试设备的电磁辐射敏感度进行实验研究。讨论了典型互联系统辐射与注入的等效以及在高场强下的外推条件,对所选非线性器件是否满足实验要求进行了实验验证。分析了大电流注入法等效替代强场连续波电磁辐射的实验结果,并对辐照时天线极化方式不同以及有无大电流注入探头对辐射与注入等效研究的影响进行了分析。     

Linearization of non-linear stochastically excited dynamic systems

A “second order” linearization method for non-linear stochastically excited dynamic systems is described. Second order probabilistic functions are used to evaluate the linearized stiffness or damping coefficients of the non-linear elements. The values of these coefficients are frequency dependent, in contrast to the nowadays generally used “first order” linearization methods. Principles of both “first order” and “second order” methods are discussed. So-called “linearization functions” of typical non-linearities greatly simplifying practical computations are introduced. The actual linearization procedure is discussed. A simple approximate criterion for the validity of the linearization is established.     

基于Hamilton模型的永磁同步风力发电系统中混沌运动的H∞控制

针对永磁同步风力发电系统的混沌运动现象, 提出了基于系统Hamilton模型的H_∞控制方案, 使得系统脱离混沌, 运行稳定. 首先将永磁同步风力发电系统模型经过一系列状态变换, 转化为类Lorenz经典数学模型, 并验证了系统在一定参数区域内存在混沌现象. 随后基于Hamilton系统充分利用系统物理结构和无需补偿“无功力”的优点, 建立了混沌系统的Hamilton模型, 并考虑了系统存在外扰情况下的H_∞控制方法. 本文所设计的控制器     

Wavelength-controlled variable-order optical fractional Fourier transform

The relationship between optical fractional Fourier transforms (OFRTs) obtained at different wavelengths is derived by use of the ABCD matrix formalism. It is shown that varying the wavelength while retaining the same optical system can be used to control the order of the OFRT. The advantage of this method of varying OFRT order is that no variation in the characteristics of the bulk optics is required. A general experimental verification of the theory is provided by showing the exact equivalence of two OFRT systems of different order when they are replayed using the same input function at different wavelengths.     

Vibration analysis of harmonically excited non-linear system using the method of multiple scales

An analytical method is presented for evaluation of the steady state periodic behavior of non-linear systems. This method is based on the substructure synthesis formulation and a multiple scales procedure, which is applied to the analysis of non-linear responses. A complex non-linear system is divided into substructures, of which equations are approximately transformed to modal co-ordinates including non-linear term under the reasonable procedure. Then, the equations are synthesized into the overall system and the solution of the non-linear system can be obtained. Based on the method of multiple scales, the proposed procedure reduces the size of large-degree-of-freedom problem in solving the non-linear equations. Feasibility and advantages of the proposed method are illustrated by the application of the analytic procedure to the non-linear rotating machine system as a large mechanical structure system. Results obtained are reported to be an efficient approach with respect to non-linear response prediction when compared with other conventional methods.     

Dynamical symmetry breaking due to vacuum misalignment

We show that under a certain assumption systems with second-class constraints can be regarded as gauge-fixed systems with first-class constraints. The massive Yang-Mills theory, a point particle on a sphere and theO(N) symmetric non-linear σ-model are considered as concrete examples.