一类非线性奇异振动方程的渐近解

本文讨论了一类型典的非一性奇异振动方程－ＷＣＭ方程的斩近解法、分析了该方程解的稳定性问题，并将结果与摄动法求解的结果以及数值积分的结果相比较以说明所用方法能用于此类非线性奇异方程的分析。     

非线性振动的一个新的渐近解法

本文在渐近法的基础上，引进谐波平衡思想，得到了一个新的渐近解法。与传统方法相比，应用本文方法求渐近解，不必解微分方程和依靠消除永年项建立补充工程，而是将求解过程转化为一系列的代数运算。因此，本文解法便于手算，更有利于用计算机计算高阶近似。     

描述非线性振动的一个数学模型

本文运用保守振动系统的能量守恒定律，论述了一种确定振动系统运动微分方程阻尼函数的方法，以求得与保守系统运动微分方程的解相一致的相平面边界曲线。     

一类非线性离散人口模型的稳定性和振动性

本文讨论了非线性差分方程ｘｎ＋１＝ｘｎ／１－ａ－ｂｘｎ－ｋ＋ｃｘ＾２ｎ－ｋ平衡点的稳定性，并获得了此方程在条件ａ∈（１，∞），ｂ∈（－∞，＋∞）＝Ｒ，ｃ∈（０，∞）下其正解的渐近性质及其关于正平衡点振动的充分条件，且当Ｋ＝０时建立了正平衡点渐近稳定的充分必要性准则。’     

有阻尼的强非线性Duffing方程强迫振动问题的插值摄动解法

用插值摄动法和改进的L－P法相结合的方法求解了一类有正阻尼的强非线性Duffing方程强迫振动的初值问题，计算过程中仅需代数和求导运算，不需求积或求解微分方程，并能反映振幅的变化，因而优于一般的改进的L－P法。     

弦的非线性振动摄动法及其K—dV方程的孤波解

本文用摄动法导出了弦的非线性振动的Ｋ－ｄＶ方程，讨论了只有耗散和色散的特殊情况，然后详细讨论了Ｋ－ｄＶ方程的孤波解的性态，并给出了弦的非线性振动孤波的主要特征。     

载流导线的横向非线性振动

研究二长直电流间载流导线的横向非线性振动，采用广义加权残值法偏偏微运动方程简化。导线载直流时得到Ｄｕｆｆｉｎｇ方程，求出解和频率的精确形式和近似形式。导线载交流时得到非线性Ｍａｔｈｉｅｕ方程，共振解摄动法得到。交直流载的频幅流形都反应出二长直电流间载流导线横向振动典型的非线性。     

谐振式加速度计非线性振动的建模与优化

实验发现硅微谐振式加速度计非线性效应的弱化可以降低测量噪声,对非线性振动进行了推导与建模,利用扫频数据得到模型中的未知参数,并根据模型计算出使非线性效应消失所需的激励电压,给闭环工作点的设计提供了理论指导。实验结果表明:模型与实际系统在各激励电压幅值下均保持一致,非线性效应消失点和实验吻合。根据闭环工作点的设置进行了方案优化,最终在0.25 mV激励电压下的闭环实验中,实现了5.5×10~(-5) Hz的频率输出噪声,对应加速度噪声为1.7μg,与之前30μg的噪声水平相比改善了数十倍。     

Duffing方程强迫振动问题的插值摄动解法

本文用插值摄动法求解了一类弱非线性非自治系统有正阻尼的Duffing方程的初值问题。此法把一个求解非线性二阶常微分方程问题转化成为求解二次代数方程的问题,计算过程十分简单,并能反映振幅的变化,因而优于L－P解法。     

齿轮系统非线性振动研究进展

围绕圆柱齿轮系统的参数振动和间隙非线性振动问题, 较为详细地评述了20年来国 际上的研究进展情况. 文中首先说明了齿轮系统啮合过程非线性振动的基本概念, 包括基本 的力学模型、数学模型、不同类型的分析系统和求解方法; 然后分别评述了时变轮齿啮合刚 度参数振动问题和齿侧间隙非线性振动问题的研究进展. 此后讨论了同时 包含齿侧间隙和时变啮合刚度时齿轮非线性振动问题方面的研究. 最后,建议了齿轮系统 非线性振动方面今后的研究重点.     

Dynamic stabilization of an asymmetric nonlinear bubble oscillator

The principal objective of this study is to present a new numerical scheme based on a combination of q-homotopy analysis approach and Laplace transform approach to examine the Fitzhugh–Nagumo (F–N) equation of fractional order. The F–N equation describes the transmission of nerve impulses. In order to handle the nonlinear terms, the homotopy polynomials are employed. To validate the results derived by employing the used scheme, we study the F–N equation of arbitrary order by using the fractional reduced differential transform scheme. The error analysis of the proposed approach is also discussed. The outcomes are shown through the graphs and tables that elucidate that the used schemes are very fantastic and accurate.     

A fully nonlinear oscillator with contact and friction

In order to understand the effects of the coupling between smooth and nonsmooth nonlinearities in dynamical systems, we introduce a simple mechanical model containing the exact expression of nonlinear strains together with non-regularized unilateral contact and Coulomb friction conditions. This model is suggested both by previous work dealing with unilateral contact conditions and by work closer to classical dynamical systems which has recently led to the introduction of a new attractor. The main part of the paper concerns the investigation of the set of equilibrium states which turns out to be much more intricate than when the strains were linearized. Then the paper ends by suggesting some possible paths towards the study of the stability of these equilibria and of the qualitative dynamics.     

A nonlinear oscillator concept for one-dimensional pulsating detonations

To interpret the results of direct numerical simulations for the one-dimensional pulsating detonation, a nonlinear oscillator model is proposed based on the integral conservation considerations. A gauging procedure is suggested in which the results from the direct numerical simulation are used to compute the coefficients of the oscillator equation. Various terms of the oscillator equation obtained are compared with those of mechanical nonlinear oscillators in order to illustrate their analogy. Among many important features captured by this nonlinear oscillator equation, the oscillatory behavior of the detonation wave front can be interpreted as resonant excitation of the chemical energy release. Received 24 March 1997 / Accepted 7 May 1998     

Monohaemonically driven nonlinear elastically hereditary oscillator

An analysis of the steady-state forced oscillatons of an elastically hereditary single-mass system with three terms included in the Volterra-Fréchet multiple-integral relations [1] makes it possible to determine the fundamental laws governing the frequency dependence of the amplitude and phase [2]. Also of interest is the analogous problem with all terms included in the multiple-integral relations. The authors realize this possibility in the general case. Separable weighting functions consisting of products of exponential relaxation kernels with a separated instantaneous part are considered as a concrete example.     

Chaotic behavior of a nonlinear oscillator

Behavior of bifurcation and chaos in a forced oscillator(?)containing a square nonlinear term is investigated by using Mel’nikov method and digital computer simulations.     

Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator

We present a theoretical study of the dynamics of the coupled system of Jiang, McFarland, Bergman, and Vakakis. It comprises a harmonically excited linear subsystem weakly coupled to an essentially nonlinear oscillator. We explored the rich dynamics exhibited by this coupled system by determining its periodic responses and their bifurcations. Not surprisingly, we found a lot of interesting dynamics over a broad frequency range: cyclic-fold, Hopf, symmetry-breaking, and period-doubling bifurcations; phase-locked motions; regions with multiple coexisting solutions; hysteresis; and chaos. We did not find any occurrence of energy transfer via modulation (also known as zero-to-one internal resonance); theoretically, the possibility of its occurrence was ruled out for systems with weak nonlinearity and damping. Finally, we investigated the ef fectiveness of the so-called nonlinear energy sink (NES) in vibration attenuation of forced linear structures. We found that the NES results in an increase in the vibration amplitude of the linear subsystem, especially when the damping is low, contrary to the claim made by Jiang et al. Also, we did not find any indication of nonlinear energy pumping or localization of energy in the NES, away from the directly forced linear subsystem, indicating that the NES is not ef fective for controlling the vibrations of forced linear structures.     

Forced nonlinear oscillator with nonsymmetric dry friction

In this paper we discuss an approximately steady motion of an oscillator as a single whole with a constant “on the average” velocity. For that purpose we analyze the position and stability of some special points of the phase portrait. In the presence of internal excitation and nonsymmetric Coulomb dry friction, a motion of the oscillator with a constant “on the average” velocity is possible. The algebraic equation for this constant velocity is found. For different parameters of the model there exist at most three regimes of motion with a constant velocity, but only one or two of them are stable. The theoretical results obtained can be used for the design of worm-like moving robots.     

Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator

An asymmetric nonlinear oscillator representative of the finite forced dynamics of a structural system with initial curvature is used as a model system to show how the combined use of numerical and geometrical analysis allows deep insight into bifurcation phenomena and chaotic behaviour in the light of the system global dynamics.Numerical techniques are used to calculate fixed points of the response and bifurcation diagrams, to identify chaotic attractors, and to obtain basins of attraction of coexisting solutions. Geometrical analysis in control-phase portraits of the invariant manifolds of the direct and inverse saddles corresponding to unstable periodic motions is performed systematically in order to understand the global attractor structure and the attractor and basin bifurcations.