Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.     

考虑谐波互作用的行波管欧拉非线性理论模型

基于拉格朗日体系的考虑谐波互作用的理论模型,将离散的粒子近似处理为流体,得到电子相位的连续分布函数.对电子相位连续分布函数进行傅里叶一阶展开,并结合贝塞尔母函数关系式,建立了考虑谐波互作用的欧拉非线性理论模型.应用考虑谐波互作用的欧拉非线性理论模型对一支L波段空间行波管和一支C波段空间行波管进行大信号分析,并与拉格朗日理论模型进行对比.结果表明:在增益1dB压缩点之前,考虑谐波互作用的欧拉非线性理论模型与拉格朗日理论模型十分符合,增益最大误差不超过4%.考虑谐波互作用的欧拉非线性理论模型能够有效的对增益1 dB压缩点之前的谐波进行分析.仿真结果验证了考虑谐波互作用的欧拉非线性理论模型的正确性和有效性.考虑谐波互作用的欧拉非线性理论不但提供了一个谐波快速计算模型,而且为后续研究行波管谐波的产生机理与抑制方法奠定了基础.     

Basin boundaries in asymmetric vibrations of a circular plate

In order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation, we reexamine a primary resonance, studied by Yeo and Lee [Corrected solvability conditions for non-linear asymmetric vibrations of a circular plate, Journal of Sound and Vibration 257 (2002) 653-665] in which at most three stable steady-state responses (one standing wave and two traveling waves) are observed to exist. Further examination, however, tells that there exist at most five stable steady-state responses: one standing wave and four traveling waves. Two of the traveling waves lose their stability by Hopf bifurcation and have a sequence of period-doubling bifurcations leading to chaos. When the system has five attractors: three equilibrium solutions (one standing wave and two traveling waves) and two chaotic attractors (two modulated traveling waves), the basin boundaries of the attractors on the principal plane are obtained. Also examined is how basin boundaries of the modulated motions (quasi-periodic and chaotic motions) evolve as a system parameter varies. The basin boundaries of the modulated motions turn out to have the fractal nature.     

Experimental verification of the mechanisms for nonlinear harmonic growth and suppression by harmonic injection in a traveling wave tube

Understanding the generation and growth of nonlinear harmonic (and intermodulation) distortion in microwave amplifiers such as traveling wave tubes (TWTs), free electron lasers (FELs), and klystrons is of current research interest. Similar to FELs, the nonlinear harmonic growth rate scales with the harmonic number in TWTs. In klystrons, the wave number scaling applies to the nonlinear harmonic bunching and associated nonlinear space-charge waves. Using a custom-modified TWT that has sensors along the helix, we provide the first experimental confirmation of the scaling of nonlinear harmonic growth rate and wave number in TWTs. These scalings of a nonlinearly generated harmonic mode versus an injected linear harmonic mode imply that suppression by harmonic injection occurs at a single axial position that can be located as desired by changing the injected amplitude and phase.     

Non-linear resonances in the forced responses of plates,part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.     

Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method

Mathematical modeling of many autonomous physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear evolution equations plays a significant role in the study of nonlinear physical phenomena. In this article, the enhanced (G′/G)-expansion method has been applied for finding the exact traveling wave solutions of longitudinal wave motion equation in a nonlinear magneto-electro-elastic circular rod. Each of the obtained solutions contains an explicit function of the variables in the considered equations. It has been shown that the applied method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering fields.     

圆管结构中周向导波非线性效应的模式展开分析

在二阶微扰近似条件下, 采用导波模式展开分析方法研究了圆管结构中周向导波的非线性效应. 伴随基频周向导波传播所发生的二次谐波, 可视为由一系列二倍频周向导波模式叠加而成. 从动量定理出发, 结合柱坐标系下非线性应力张量及其散度的数学表达式, 针对圆管中某一基频周向导波模式, 推导出相应的二倍频应力张量及二倍频彻体驱动力的数学表达式, 建立了确定二倍频周向导波模式展开系数的控制方程, 得到了伴随基频周向导波传播所发生的二次谐波声场的形式解. 理论分析和数值计算表明, 当构成二次谐波声场的某一二倍频周向导波模式与基频周向导波的相速度匹配时, 该二倍频周向导波模式的位移振幅表现出随传播周向角积累增长的性质; 当两者的相速度失配时, 二倍频周向导波的振幅随传播周向角表现出“拍”效应.     

一维非线性声波传播特性

针对一维非线性声波的传播问题进行了有限元仿真和实验研究. 首先推导了一维非线性声波方程的有限元形式, 含有高阶矩阵的非线性项导致声波具有波形畸变、谐波滋生、基频信号能量向高次谐波传递等非线性特性. 编制有限元程序对一维非线性声波进行了计算并对仿真得到的畸变非线性声波信号进行处理, 分析其传播性质和物理意义. 为验证有限元计算结果, 开展了水中的非线性声波传播的实验研究, 得到了不同输入信号幅度激励下和不同传播距离的畸变非线性声波信号. 然后对基波和二次谐波的传播性质进行详细讨论, 分析了二次谐波幅度与传播距离和输入信号幅度的变化关系及其意义, 拟合出二次谐波幅度随传播距离变化的方程并阐述了拟合方程的物理意义. 结果表明, 数值仿真信号及其频谱均与实验结果有较好的一致性, 证实计算方法和结果的正确性, 并提出了具有一定物理意义的二次谐波随传播距离变化的简单数学关系. 最后还对固体中的非线性声波传播性质进行了初步探讨. 本研究工作可为流体介质中的非线性声传播问题提供理论和实验依据.     

圆波导梳状慢波结构的耦合阻抗

 对目前慢波结构的模拟计算中存在的问题进行了深入分析，并提出了新的计算耦合阻抗的方法。研究了圆波导梳状慢波结构基波的耦合阻抗,其值可以在较宽频带内大于1 W。该结构内孔径为波长的1/4~1/3，是同一频段螺旋线慢波结构的5倍左右，能容纳比螺旋线大25倍的电流，能够承受比螺旋线大25倍的平均功率。互作用计算表明采用这种慢波结构的行波管在Ka波段可获得的最高功率达170 kW。     

Enhancement of the SHG efficiency in a doubly resonant 2D-photonic crystal microcavity

In this paper we discuss the conditions to obtain the enhancement of second harmonic generation in a two-dimensional circular photonic crystal AlGaAs cavity. The photonic crystal circular cavity offers the possibility of having high-Q resonance modes with respect to those obtained with other types of photonic crystal lattices. The crystallographic cut of the AlGaAs provides a strong nonlinear coupling between a transverse-magnetic (TM) polarized resonant mode at the fundamental wavelength and a transverse-electric (TE) polarized resonant mode at second harmonic wavelength. The double resonance condition leads to a strong improvement of the second harmonic generation process. A preliminary linear analysis has been performed by using the finite-difference time-domain method, which includes the dispersive response of the material, modeled using the well-known one-pole pair Lorentzian function.     

Observation of traveling thermoacoustic shock waves (L)

Shock waves were explored in the thermoacoustic spontaneous gas oscillations occurring in a gas column with a steep temperature gradient. The results show that a periodic shock occurs in the traveling wave mode in a looped tube but not in the standing wave mode in a resonator. Measurements of the harmonic components of the acoustic intensity reveal a clear difference between them. The temperature gradient acts as an acoustic energy source for the harmonic components of the shock wave in the traveling wave mode but as an acoustic energy sink of the second harmonic in the standing wave mode.     

Parametric excitation of traveling waves in a circular non-dispersive medium

This paper discusses a special type of propagating waves created by parametric excitation in a circular taught string. The string, being a non-dispersive medium propagates deformations in a similar manner to electromagnetic waves in vacuum, both have simple wavelength–frequency relationship that play an important role here. Nonlinear equations are derived under the assumption of finite deformations, whose solution produces a square-wave like, limited-amplitude, traveling wave. Closed-form expressions are obtained for the parametric excitation characteristics of the nonlinear system and the steady-state traveling waves are described by a generalized eigenvalue problem. The latter relates the nonlinear elongation of the neutral axis to the participating wavelengths forming the propagating wave. Detailed numerical simulations are provided to validate the solution and to illustrate graphically the waveforms. It is shown that propagating sinusoidal parametric excitation gives rise to various square-wave like deformation shapes which a unique phenomenon is arising in non-dispersive media.     

圆管结构中的非线性周向导波*

采用将二阶微扰近似与模式展开分析相结合的方法,可从理论上有效求解圆管结构中的非线性周向导波问题。通过数值计算和有限元仿真发现,基频与二倍频周向导波模式的相速度匹配程度,可显著影响二倍频周向导波模式随传播周向角的积累增长程度。针对基频与二倍频周向导波模式的相速度和群速度均相匹配的情形,通过实验研究发现,周向导波确可存在强烈的非线性效应,且周向导波的二次谐波发生效应可对圆管早期损伤状态做出敏感的响应。文中给出的有关结果,可为进一步开展非线性周向导波的研究工作奠定理论和实验基础。     

A Note on Second Harmonic Resonance of a Magneto-Acoustic Wave in Planar Plasma-Filled Waveguide

The second harmonic generated by non-linear self interaction of a magneto-acoustic wave in a planar plasma-filled waveguide is a freely propagating mode. Coupled-mode equations for resonant nonlinear interaction of this second harmonic mode with the fundamental mode are derived. Using these equations the possibility of generation of the second harmonic is analysed.     

行波管中二次谐波注入对三阶互调的抑制

 利用Christine 1维多信号非线性互作用物理模型,对螺旋线行波管中注入二次谐波后三阶互调的输出进行了模拟。模拟结果表明：在螺旋线大功率行波管中,通过二次谐波注入能有效地抑制三阶互调输出,从而降低了行波管的非线性输出,并存在最佳的注入功率和相位条件,使三阶互调可被抑制到最低。介绍了二次谐波注入抑制三阶互调的实验研究,并与模拟结果进行了比较,两者基本一致。     

光子带隙谐振腔回旋管振荡器的自洽非线性理论

定义了光子带隙谐振腔(photonic-band-gap cavity, PBGC)的等效半径,论证了使用该半径将PBGC等效为具有模式选择性的金属圆柱谐振腔的有效性,揭示了其在PBGC设计过程中的指导性作用.基于等效半径的运用,建立起光子带隙谐振腔回旋管振荡器(PBG回旋管)的自洽非线性理论,并对工作于TE₃₂模的PBG回旋管作了理论分析和数值计算.目前的研究表明:高频电磁场沿角向呈行波或驻波的不同极化形式对PBG回旋管的注-波互作用过程具有较大的影响;较之电子回旋基波,该器件中二 关键词： 光子带隙谐振腔 等效半径 回旋管 自洽非线性理论     

弹性微管内气泡的非线性受迫振动

将弹性管壁视为膜弹性结构, 探索在外部声场作用下弹性微管内液柱-气泡-管壁构成耦合振动系统的非线性特征. 利用逐级近似法对系统非线性共振频率、基频和三倍频振动幅值响应、 分频激励共振机理等进行了理论分析. 基频和三倍频振动的幅-频响应数值结果表明: 气泡的轴向共振和管壁共振不能同时出现; 两垂直方向的振动均表现出幅值响应多值性, 进而可能引起系统的不稳定声响应; 三倍频振动在低频区的声响应强于高频区. 关键词： 弹性微管 受迫振动 非线性振动 气泡声响应     

Modulation of Weakly Nonlinear Traveling Waves in the Presence of a Damped Large-Scale Mode

Radiophysics and Quantum Electronics - We study the modulation of weakly nonlinear traveling waves in the presence of a damped large-scale mode. The traveling wave is generated by the Marangoni...     

Bifurcations and Dynamics of Traveling Wave Solutions to a Fujimoto-Watanabe Equation

In this paper, we study the bifurcations and dynamics of traveling wave solutions to a Fujimoto-Watanabe equation by using the method of dynamical systems. We obtain all possible bifurcations of phase portraits of the system in different regions of the parametric space. Then we show the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, compactions and kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions and periodic wave solutions are implicitly given,while the expressions of kink-like and antikink-like wave solutions are explicitly shown. The dynamics of these new traveling wave solutions will greatly enrich the previews results and further help us understand the physical structures and analyze the propagation of the nonlinear wave.     

Ka波段二次谐波损耗波导回旋行波管非线性分析

 建立了由损耗段和铜段组成的波导结构的Ka波段二次谐波回旋行波放大器理论模型,并进行了非线性理论研究。基于稳定性分析,确定35 GHz TE₀₂模二次谐波回旋行波放大器的基本工作参数:波导半径为1.02 cm,电子注工作电压为90 kV,工作电流为25 A,工作磁场为0.642 6 T,横纵速度比为1.2;然后通过模拟详细分析了工作电流、波导损耗和速度零散等因素对该放大器性能的影响。研究表明：在该结构中损耗段可以有效地抑制模式竞争从而提高输出功率和带宽;工作电流对输出功率的影响存在最大值;速度零散对输出功率有很大的影响。