On Approximations of First Integrals for Strongly Nonlinear Oscillators

In this paper strongly nonlinear oscillator equations will be studied.It will be shown that the recently developed perturbation method based onintegrating factors can be used to approximate first integrals. Not onlyapproximations of first integrals will be given, butit will also be shown how in a rather efficient way the existence and stability oftime-periodic solutions can be obtained from these approximations. In particularthe generalized Rayleigh oscillator equation will be studied in detail, and it will beshown that at least five limit cycles can occur.     

On Approximations of First Integrals for a System of Weakly Nonlinear,Coupled Harmonic Oscillators

In this paper a system of weakly nonlinear, coupled harmonic oscillatorswill be studied. It will be shown that the recently developedperturbation method based on integrating vectors can be used toapproximate first integrals and periodic solutions. To show how thisperturbation method works the method is applied to a system of weaklynonlinear, coupled harmonic oscillators with 1:3 and 3:1 internalresonances. Not only approximations of first integrals will be given,but it will also be shown how, in a rather efficient way, the existenceand stability of time-periodic solutions can be obtained from theseapproximations. In addition some bifurcation diagrams for a set ofvalues of the parameters will be presented.     

Energy Method for Computing Periodic Solutions of Strongly Nonlinear Autonomous Systems with Multi-Degree-of-Freedom

In this paper with the use of conservation of average energy, a newmethod for computing the periodic solutions of strongly nonlinearautonomous systems with multi-degree-of-freedom is suggested. Thismethod cannot only decide the existence, but also give the approximateexpressions of the periodic solutions.     

The Existence Stability and Approximate Expressions of Periodic Solutions of Strongly Nonlinear Nonautonomous Systems with Multi-Degree-of-Freedom

Based upon the conservation of average energy a theorem about the periodic solutions of strongly nonlinear nonautomous systems with multi-degree-of freedom is proved. This theorem can not only decide the existence and stability of the periodic solutions, but at the same time can also give their first-order and second-order approximate expressions.     

强非线性振动系统周期解的能量迭代法

对于完全强非线性系统：x^.. g(x) f(x,x^.)x^.=0,提出求周期近似解析解以及这些解的稳定性的新方法。式中，g(x)、f(x,x^.)x^.分别是x,x、x^.的非线性函数。方法是基于能量原理，求出其一次近似解析解，然后引进牛顿迭代思想，得到周期系统数微分方程，最后根据谐波平衡原理及最小二乘法求其高次近似解，高次近似解的表达式由计算机辅助推导。计算参考文献[2]和[3]中的例题，令其中ε=1，研究该完全强非线性系统的周期解及其稳定性，本文方法与龙格-库塔数值法算得的结果对照如图1-3所示，它们表明本文方法不仅有效而且精度较高。     

Energy Method for Approximate Periodic Solution of Strongly Nonlinear Nonautonomous Systems

In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.     

含x^5项强非线性系统的共振解和亚谐解

本文应用强非线性系统频闪法,对公式进行适当变换,求出了方程x+ax+βx~3+γx~5=ε(-μx+δcosΩt)的共振解及亚谐解存在的条件及其解析表达式,并与数值计算结果作了比较.     

强非线性动力系统周期解分析

给出一类强非线性动力系统周期解存在性，唯一性和稳定性的简易差别法以及周期解的摄动法。本差别法把问题归结为干扰力在相应的未扰系统振动周期上的功函数及其导数的讨论，其限制条件比现有结果弱。本摄动法可以认为是经典Ｌｉｎｄｓｔｅｄｔ－Ｐｏｉｎｃａｒｅ（Ｌ－Ｐ）法在强非线性振动系统的推广。它与Ｌ－Ｐ法的主要区别在于假设系统的振动频率为相角的非线性函数。     

A Perturbation-Incremental Method for Strongly Nonlinear Autonomous Oscillators with Many Degrees of Freedom

The perturbation-incremental method is extended to determine thebifurcations and limit cycles of strongly nonlinear autonomousoscillators with many degrees of freedom. Coupled van der Poloscillators and coupled Rayleigh oscillators are taken as numericalexamples. Limit cycles of the oscillators can be calculated to anydesired degree of accuracy. The stabilities of limit cycles are alsodiscussed.     

中心有刚体质量的环形薄板的非线性强迫振动

本文研究中心带有刚体质量外部固定铰支或活动铰支的环形薄板的非线性强迫振动。考虑板的弯曲变形、面内位移和几何非线性，用哈米尔顿原理建立板的运动方程，用Ｋａｎｔｏｒｏｖｉｃｈ平均法消去时间变量，然后用数值积分求得非线性振动的振幅随激振力的大小及激振力的频率而变化的关系。求解过程中用打靶法逐步改进未知参量，以保证边界条件的满足。最后讨论薄膜力、激振力的分布、板的内外半径比等因素对响应的影响     

一类多自由度强非线性振动系统主共振的渐近解法

提出一种渐近方法用来处理一类多自由度强非线性自治振动系统，它是新渐近方法￣［１］的推广。本方法适用于主共振情形，我们建立了振幅和相位所满足的方程。文末用两个例子说明本方法的有效性。     

保守系统强非线性强迫振动问题的一种解法

本文用新的改进的L-P法求得了一类保守系统强非线性强迫振动问题的一级近似共振周期解,近似解比已有改进的L-P法的相应解更加准确。     

Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators

Redistribution of energy in a highly asymmetric system consisting ofcoupled linear and highly nonlinear damped oscillators isinvestigated. Special attention is paid to the excitation of a nonlinearnormal mode while the energy is initially stored in other modes of thesystem. The transition proceeds via the mechanism of subharmonicresonance which is possible because of the strong nonlinearity of thesystem. The conditions of the energy transition to NNM being effectiveare revealed and guidelines to design such a systems are formulatedin detail.     

On the Asymptotic Solutions of Boundary Value Problems for a Class of Systems of Nonlinear Differential Equations (I)

A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations . The asymptotic expansions of solutions are constructed, the remainders are estimated. The former works are improved and generalized.     

On Mode Interactions for a Weakly Nonlinear Beam Equation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a Rayleigh perturbation will be studied. It will be shown that the calculations to find internal resonances in this case are much more complicated than and differ substantially from the calculations for the weakly nonlinear wave equation with a Rayleigh perturbation as for instance presented in [3] or [7]. The initial-boundary value problem can be regarded as a simple model describing wind-induced oscillations of flexible structures like suspension bridges or iced overhead transmission lines. Using a two-timescales perturbation method approximations for solutions of this initial-boundary value problem will be constructed.     

被动隔振体非线性振动的能量迭代解法

研究了由基础振动激励、弹性材料隔离的被动隔振体的强非线性动力响应。用变形的三次多项式函数表征隔振材料的非线性刚度特性,建立了被动隔振体的非线性动力学方程,得到有阻尼受迫振动Duffing方程。将求解强非线性自治系统的能量迭代方法加以改进,推广应用到强非线性非自治系统,求出周期响应的近似解析解表达式,以及幅频关系、相频关系和隔振系数的近似表达式。算例中应用本方法与Runge-Kutta方法进行了对照,结果表明求解精度较高。本文利用计算机进行了辅助推导。     

RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

ＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦ　ＮＯＮＬＩＮＥＡＲＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＲＥＳＥＡＲＣＨＯＦＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＦＯＲＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＤＩＦＦＥ...     

求解非线性动力系统周期解推广的打靶法

提出一种确定非线性系统周期轨道及周期的改进打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参入打靶法的迭代过程，从而能迅速确定出系统的周期轨道及其周期。该方法对初始迭代参数没有苛刻要求，可以用于分析强非线性系统，而且对参数激励系统同样有效，对高维系统也能迅速、准确地求得周期解。文中应用该方法对三维Rǒssler系统和八维非线性柔性转子-轴承系统的周期轨道和周期进行了求解，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性。     

On the Weakly Nonlinear,Transversal Vibrations of a Conveyor Belt with a Low and Time-Varying Velocity

In this paper the weakly nonlinear, transversal vibrations of aconveyor belt will be considered. The belt is assumed to move witha low and time-varying speed. Using Kirchhoff's approach a singleequation of motion will be derived from a coupled system ofpartial differential equations describing the longitudinal andtransversal vibrations of the belt. A two time-scalesperturbation method is then applied to approximate the solutionsof the problem. It will turn out that the frequencies of the belt speed fluctuations play an important role in the dynamic behaviourof the belt. It is well-known in linear systems that instabilitiescan occur if the frequency of the belt speed fluctuations is thesum of two natural frequencies. However, in the weakly nonlinearcase as considered in this paper this is no longer true. It turns out that the weak nonlinearity stabilizes the system.     

Asymptotic Solution for a Kind of Boundary Layer Problem

This paper studies the partial differential equation with a small coefficient in the highest-order item. This kind of equation is also named as boundary layer problem. The Burgers equation and modified Burgers equation are analyzed in this approach. First, these equations are transferred into the strong nonlinear ones, and then the corresponding strong nonlinear equations are solved based on the perturbation method. The results from the asymptotic method are comparable with those obtained from numerical computation. An erratum to this article is available at .