Free and forced vibration analysis of a nonlinear system with cyclic symmetry: Application to a simplified model

This work program is devoted to studying the nonlinear dynamics of a structure with cyclic symmetry under conditions of geometric nonlinearity, through the use of the harmonic balance method (HBM). In order to study the influence of nonlinearity due to the large deflection of blades, a simplified model has been developed. This approach leads to a system of linearly coupled, second-order nonlinear differential equations, in which nonlinearity appears via cubic terms. Periodic solutions, in both the free and forced cases, are sought by applying HBM coupled with an arc-length continuation method. Solution stability has been investigated using Floquet's theorem. In addition to featuring similar and nonsimilar nonlinear modes, the unforced system is known to contain localized nonlinear modes that arise from branching point bifurcation at certain vibration amplitudes. In the forced case, these nonlinear modes give rise to a complex dynamic behavior. Many bifurcations can take place, thus leading to strong or weak localization that may or may not be stable. In this study, special attention has been paid to the influence of excitation on dynamic responses. Several cases of excitation have been analyzed herein: localized excitation, and low-engine-order excitation. In the case of low-engine-order excitation, sensitivity of the response to a perturbation of this excitation type has been investigated, and it has been shown that for a localized, or sufficiently detuned excitation, several solutions can coexist, some of which are represented by closed curves in the Frequency-Amplitude domain. These various solutions overlap when increasing the force amplitude, leading to forced nonlinear localization. Because closed curves are not tied up with the basic nonlinear solution, they can easily be overlooked. In this study, they have been calculated using a sequential continuation with the force amplitude as a parameter.     

Bifurcation and chaos analysis of a nonlinear electromechanical coupling relative rotation system

Hopf bifurcation and chaos of a nonlinear electromechanical coupling relative rotation system are studied in this paper. Considering the energy in air-gap field of AC motor, the dynamical equation of nonlinear electromechanical coupling relative rotation system is deduced by using the dissipation Lagrange equation. Choosing the electromagnetic stiffness as a bifurcation parameter, the necessary and sufficient conditions of Hopf bifurcation are given, and the bifurcation characteristics are studied. The mechanism and conditions of system parameters for chaotic motions are investigated rigorously based on the Silnikov method, and the homoclinic orbit is found by using the undetermined coefficient method. Therefore, Smale horseshoe chaos occurs when electromagnetic stiffness changes. Numerical simulations are also given, which confirm the analytical results.     

Forced nonlinear resonance in a system of coupled oscillators

We consider a resonantly perturbed system of coupled nonlinear oscillators with small dissipation and outer periodic perturbation. We show that for the large time t～?(-2) one component of the system is described for the most part by the inhomogeneous Mathieu equation while the other component represents pulsation of large amplitude. A Hamiltonian system is obtained which describes for the most part the behavior of the envelope in a special case. The analytic results agree with numerical simulations.     

Similarity solution and lie symmetry for a coupled nonlinear system

The Lie point symmetries of a set of coupled nonlinear partial differential equations are considered. The system is an extended version of the usual nonlinear Schrödinger equation. In the similarity variable deduced from the symmetry analysis, the system is equivalent to the Painlevé III in Ince's classification. By starting from a solution of the Painlevé equation, one can reproduce various classes of solutions of the original PDEs. Such solutions include both rational and progressive types or a combination of the two.     

双法布里—珀罗干涉仪传感模型的理论分析

本文提出了一种用发光二极管作为光源、用自聚焦透镜构成法布里-珀罗腔的双法布里-珀罗干涉仪光纤位移传感模型.根据部分相干光的干涉理论,得到了这个传感模型输出光强与两个法布里一珀罗干涉仪腔长之差的函数曲线.     

Modulation instabilities in a system of four coupled, nonlinear Schrödinger equations

The modulation instability of continuous waves for a system of four coupled nonlinear Schrödinger equations, two of which are in the unstable regime, is studied. In earlier studies, plane or continuous waves for a system of two coupled, nonlinear Schrödinger equations is shown to exhibit modulation instability (MI), even if both modes are in the normal dispersion regime, provided that the coefficient of cross phase modulation (XPM) is larger than that of self phase modulation (SPM). Requirements for MI in this system of four coupled, nonlinear Schrödinger equations can be relaxed. MI can occur even if the magnitude of XPM is less than that of SPM, and the magnitude of instability is generally larger than that of each mode alone. The implications for parametric process and wavelength exchange in optical physics with two pump waves are discussed.     

Theoretical analysis of hot-image effect in a high-power laser system with cascaded nonlinear medium

We present a universally applicable and quick method to forecast the intensity and location of the hot-image effects in a high-power laser system structured by cascaded Kerr medium plates. The analytical expressions for the locations and the peak intensity of the hot-images are deduced by using propagation matrix method. The results are useful for laser designers to estimate and minimize the threat of optical damage. Theoretical analysis demonstrates that a maximum of N hot-images may appear in a laser system structured by N cascaded Kerr medium plates and the distance between two adjacent hot-images is two times the interval between two adjacent Kerr medium plates. The number and locations of the hot-images are related with the number of the Kerr medium plates, the distance from the scatterer to the front of the first Kerr medium plate, and the interval between two adjacent Kerr medium plates. The peak intensity of the hot-images depends on the number of the Kerr medium plates, the B-integral of each Kerr medium plate, the amplitude and phase modulation coefficients of the scatterer and the peak intensity of the input beam. The hot-image effects in a laser system with cascaded Kerr medium plates from two to eight are discussed in detail and numerically analyzed. Numerical simulation results are in agreement with the theoretical results.     

Natural frequencies of a shaft with periodically placed rotors and bearings

Flexural and torsional natural frequencies of a shaft with periodically placed rotors and bearings are investigated by using a “wave approach”. By a judicious choice of parameters, it is possible to obtain ranges of operating speed that are free from both flexural and torsional resonances. Suggestions for widening such speed ranges are included.     

耦合Schrödinger系统的周期振荡折叠孤子

引入对称延拓和非线性变换, 将(G'/G)展开法扩展到研究(1+1)维非线性耦合Schrödinger系统, 构造出该系统的一些分离变量形式的精确解. 通过对解中的任意函数进行适当的设置, 获得了两类周期振荡折叠孤子. 关键词： 耦合Schrö dinger系统 G'/G)展开法')" href="#">(G'/G)展开法 精确解 周期振荡折叠孤子     

Multi-scale energy exchanges between a nonlinear oscillator of Bouc-Wen type and another coupled nonlinear system

The concept of energy exchange between coupled oscillators can be endowed for wide variety of applications such as control and energy harvesting. It has been proved that by coupling an essential nonlinear oscillator (cubic nonlinearity) to a main system (mostly linear), the latter system can be controlled in a one way and almost irreversible manner. The phenomenon is called energy pumping and the coupled nonlinear system is named as nonlinear energy sink (NES). The process of energy transfer from the main system to the nonlinear smooth or non-smooth attachment at different scales of time can present several scenarios: It can be attracted to periodic behaviors which present low or high energy levels for the main system and/or to quasi-periodic responses of two oscillators by persistent bifurcations between their stable zones. In this paper we analyze multi-scale dynamics of two attached oscillators: a Bouc-Wen type in general (in particular: a Dahl type and a modified hysteresis system) and a NES (nonsmooth and cubic). The system behavior at fast and first slow times scales by detecting its invariant manifold, its fixed points and singularities will be analyzed. Analytical developments will be accompanied by some numerical examples for systems that present quasi-periodic responses. The endowed Bouc-Wen models correspond to the hysteretic behavior of materials or structures. This paper is clearly connected with the dynamics of systems with hysteresis and nonlinear dynamics based energy harvesting.     

Geometry and transport in a model of two coupled quadratic nonlinear waveguides

This paper applies geometric methods developed to understand chaos and transport in Hamiltonian systems to the study of power distribution in nonlinear waveguide arrays. The specific case of two linearly coupled chi((2)) waveguides is modeled and analyzed in terms of transport and geometry in the phase space. This gives us a transport problem in the phase space resulting from the coupling of the two Hamiltonian systems for each waveguide. In particular, the effect of the presence of partial and complete barriers in the phase space on the transfer of intensity between the waveguides is studied, given a specific input and range of material properties. We show how these barriers break down as the coupling between the waveguides is increased and what the role of resonances in the phase space has in this. We also show how an increase in the coupling can lead to chaos and global transport and what effect this has on the intensity.     

Theoretical and nonlinear behavior analysis of a flexible rotor supported by a relative short herringbone-grooved gas journal-bearing system

This paper considers the bifurcation and nonlinear behavior of a flexible rotor supported by a relative short herringbone-grooved gas journal bearing system. A numerical method is employed to a time-dependent mathematical model. A finite difference method with successive over relation method is employed to solve the Reynolds’ equation. The system state trajectory, Poincaré maps, power spectra, and bifurcation diagrams are used to analyze the dynamic behavior of the rotor and journal centers in the horizontal and vertical directions under different operating conditions. The analysis reveals a complex dynamic behavior comprising periodic and quasi-periodic response of the rotor and journal centers. It further shown the dynamic behavior of this type of system varies with changes in bearing number and rotor mass. The results of this study contribute to a better understanding of the nonlinear dynamics of herringbone-grooved gas journal bearing systems.     

一类非线性耦合系统的复合型双孤子新解

首先给出一种函数变换,把一类非线性耦合系统化为两个第一种椭圆方程组.然后利用第一种椭圆方程的新解与B?cklund变换,构造了一类非线性耦合系统的无穷序列复合型双孤子新解.     

Hyperbolic chaos in a system of resonantly coupled weakly nonlinear oscillators

We show that a hyperbolic chaos can be observed in resonantly coupled oscillators near a Hopf bifurcation, described by normal-form-type equations for complex amplitudes. The simplest example consists of four oscillators, comprising two alternatively activated, due to an external periodic modulation, pairs. In terms of the stroboscopic Poincaré map, the phase differences change according to an expanding Bernoulli map that depends on the coupling type. Several examples of hyperbolic chaos for different types of coupling are illustrated numerically.     

耦合相对转动非线性动力系统的稳定性与近似解

研究了一类含三次非线性耦合项的相对转动非线性动力系统的动力学行为. 建立了具有非线性弹性力、广义摩阻力耦合项的系统动力学方程. 运用多尺度法求解谐波激励下耦合非自治系统的近似解,通过讨论系统的主共振和内共振特性,分析了耦合项对系统响应的影响. 应用奇异性理论研究了主振稳态响应分岔方程的稳定性,得到了系统的转迁集和分岔曲线的拓扑结构. 关键词： 相对转动 非线性耦合动力系统 奇异性理论 稳定性     

Propagation of high-order Bessel–Gaussian beam through a misaligned first-order optical system

The generalized diffraction integralis used to derive a generalized formula for high-order Bessel–Gaussian beams (HBGBs) through a misaligned first-order ABCD optical system. It is found that, when a HBGB propagates through a misaligned optical system, the beam shape of the output beam is unchanged. However, the center of the output beam is deviated from the optical axis, forming a decentered HBGB. The position of the output beam may be controlled by adjusting the misaligned parameters. Based on the derived formula, the diffraction patterns of HBGBs propagating through a simple misaligned lens system have been calculated numerically. These results may be useful in the application of laser beams for trapping and manipulating a wide variety of particles.     

Transition from quasi-periodicity to chaos in a system of coupled nonlinear oscillators

The global bifurcation structure for a model of coupled nonlinear oscillators has been analysed numerically. It is shown that destruction of the two-torus preceding chaos is usually observed in this system. The critical surface of the invariant two-torus and its collapse in the course of rotation are firstly observed in a realistic differential equation system. A scaling property for the fine structure of phase-locking regions has also been confirmed.     

光学微盘腔与三能级量子点系统中的模耦合研究

用一种全量子理论方法研究了波导、光学微盘腔与三能级量子点耦合系统的动力学过程,求出其耦合后的透射模和反射模的解析解. 由于微腔表面粗糙引起反向散射,在微腔内形成两简并回音壁耦合共振模,其耦合率为β;量子点的两激发态分别以耦合率g₁,g₂与回音壁耦合共振模产生耦合. 在实数空间里,得出透射光谱和反射光谱的数值解,这些三能级模型结果比二能级模型结果更接近真实光学微盘腔系统,能更好地显示耦合系统的动力学特性. 关键词： 模耦合 光学微盘腔 三能级量子点 全量子理论     

Full time-domain nonlinear coupled dynamic analysis of a truss spar and its mooring/riser system in irregular wave

A new full time-domain nonlinear coupled method has been established and then applied to predict the responses of a Truss Spar in irregular wave.For the coupled analysis,a second-order time-domain approach is developed to calculate the wave forces,and a finite element model based on rod theory is established in three dimensions in a global coordinate system.In numerical implementation,the higher-order boundary element method(HOBEM)is employed to solve the velocity potential,and the 4th-order Adams-Bashforth-Moultn scheme is used to update the second-order wave surface.In deriving convergent solutions,the hull displacements and mooring tensions are kept consistent at the fairlead and the motion equations of platform and mooring-lines/risers are solved simultaneously using Newmark-integration scheme including Newton-Raphson iteration.Both the coupled quasi-static analysis and the coupled dynamic analysis are performed.The numerical simulation results are also compared with the model test results,and they coincide very well as a whole.The slow-drift responses can be clearly observed in the time histories of displacements and mooring tensions.Some important characteristics of the coupled responses are concluded.     

一类动力学系统通过函数耦合实现混沌同步

非线性动力学系统的混沌同步, 一般采用单向线性耦合的控制方式, 对于函数耦合方式研究的比较少. 这就存在一个问题, 对于非线性动力学系统, 在线性耦合实现混沌同步后, 是否其他函数的耦合方式都可以实现混沌同步? 本文对于一类非线性动力学系统, 研究了其线性耦合同步与函数耦合同步的关系, 证明当线性耦合实现同步后, 函数在满足一定的条件下, 可以通过函数耦合实现系统的混沌同步. 最后对于Duffing系统采用两种函数耦合进行了仿真计算, 证明了结论的正确性.