悬索主共振响应的多尺度解与同伦分析解

利用哈密顿变分原理,引入拟静态假设,建立了悬索面内非线性运动方程,并采用Galerkin方法对其进行离散。接着运用多尺度法和同伦分析法得到了悬索前两阶模态主共振响应的近似解。为验证这两种分析方法的适用性,同时采用龙格-库塔法对方程直接进行了数值积分。数值计算结果表明,随着悬索垂跨比以及振幅的增加,由多尺度法与同伦分析法得到的幅频响应曲线存在明显的定性与定量的差别,而同伦分析法结果与数值法的结果更加接近。最后比较了两种分析方法得到的位移场与索力时程响应曲线。     

基于参数展开的同伦分析技术及其应用

提出了一种基于参数展开的新的同伦分析技术(PE-HAM)：结合参数展 开技术和同伦理论将一非线性动力系统(不要求系统内含有小参数)的求解问题转化为一组 线性微分方程的求解问题，并将之运用到强非线性振动领域. 用该方法研究了强非线性 Duffing系统的响应问题，得到了一阶近似解. 作为特例讨论了保守Duffing系统和受谐和 激励的耗散Duffing系统的稳态响应问题. 数值模拟的结果，说明了新方法的有效性.     

非线性振子极限环的实用分析法

本文提出一种研究极限环特性的实用分析法,适用于含有小参数的二阶常微分方程所表示的自治非线性振子.我们给出极限环存在性和稳定性的判别法以及极限环相图的实用求法.     

超越摄动:同伦分析方法基本思想及其应用

介绍一种新的、求解强非线性问题解析近似的一般方法------同伦分析方法.该方法从根本上克服了摄动理论对小参数的过分依赖, 其有效性与所研究的非线性问题是否含有小参数无关, 因此, 适用范围广.此外, 不同于所有其他解析近似方法,同伦分析方法提供了一个简单的途径, 确保所得到的级数解收敛, 从而获得 足够精确的解析近似.而且, 不同于所有其他解析近似方法, 同伦分析方法(HAM)提供了选取基函数之自由, 从而可以选择较好的基函数, 更有效地逼近问题的解. 同伦分析方法为非线性问题的解析近似求解提供了一个全新的思路, 为非线性问题(特别是不含小参数的强非线性问题)的 求解开辟了一个全新的途径.简要描述同伦分析方法的基本思想, 其在非线性力学、物理、化学、生物、金融、工程和 计算数学等领域的应用举例, 以及与摄动方法、Lyapunov 人工小参数法、$\delta$展开法、Adomian 分解法、同伦摄动方法之区别和联系.     

资金项目:国家自然科学基金资助项目

采用能计及非线性结构刚度的颤振方程为控制方程,和非定常N-S方程耦合求解,运用龙格-库塔方法在时域内求解结构响应的时间历程,从而确定颤振临界条件.计算了带结构刚度非线性的跨音速颤振特性.计算结果表明,结构刚度非线性对颤振特性有明显的影响.由于同时具有结构和气动力非线性,导致了具有复杂振荡极限环的特性.     

用同伦方法反演非饱和土中溶质迁移参数

非饱和土中溶质迁移参数反演问题可以归结为非线性算子方程的求解问题. 将同伦方法 引入该问题的求解，通过构造线性同伦将原问题转化为求解同伦函数最小值的无约束优化问 题. 同时在分析了同伦参数正则化效应的基础上，提出一种两段同伦参数修正方法. 即在求 解的初始阶段，根据拟Sigmoid函数调整同伦参数，以追踪同伦路径，保证计算稳定地进行； 在迭代的后期，采用与残差相关的同伦参数修正方法，以抵抗观测噪声对求解的影响. 数值 算例为求解带有平衡及非平衡吸附效应的一维非饱和土中溶质迁移模型参数反演问题，计算 结果表明了该方法的大范围收敛性及较强的抵抗观测噪声的能力.     

同伦分析方法进展综述

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.      

A brief review of the homotopy analysis method

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数, 适用范围更广,而且提供了一种简单的途径确保级数解收敛, 适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、 从未见报道的解. 这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.     

同伦分析方法进展综述

本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数,适用范围更广,而且提供了一种简单的途径确保级数解收敛,适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、从未见报道的解.这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.     

多体系统动力学方程的无违约数值计算方法

多体系统动力学方程为3阶微分代数方程,已有的约束违约稳定法存在位移违约问题,数值仿真准确性和稳定性不足。本文将求解高阶微分代数方程的降阶理论、ε嵌入处理方式与隐式龙格库塔法相结合,提出了直接满足位移约束条件的多体系统动力学方程的无违约算法,避免了约束违约问题。该方法先将多体动力学方程转化为2阶微分代数方程,并与位移约束方程联立;再应用ε嵌入隐式龙格库塔法进行数值求解。应用两种方法分别对单摆机构进行数值仿真,结果表明本文的方法不仅能适应较大步长,且准确性和稳定性均优于约束违约稳定法。     

Averaging Oscillations with Small Fractional Damping and Delayed Terms

We demonstrate the method of averaging for conservative oscillators which may be strongly nonlinear, under small perturbations including delayed and/or fractional derivative terms. The unperturbed systems studied here include a harmonic oscillator, a strongly nonlinear oscillator with a cubic nonlinearity, as well as one with a nonanalytic nonlinearity. For the latter two cases, we use an approximate realization of the asymptotic method of averaging, based on harmonic balance. The averaged dynamics closely match the full numerical solutions in all cases, verifying the validity of the averaging procedure as well as the harmonic balance approximations therein. Moreover, interesting dynamics is uncovered in the strongly nonlinear case with small delayed terms, where arbitrarily many stable and unstable limit cycles can coexist, and infinitely many simultaneous saddle-node bifurcations can occur.     

Averaging Oscillations with Small Fractional Damping and Delayed Terms

We demonstrate the method of averaging for conservative oscillators which may be strongly nonlinear, under small perturbations including delayed and/or fractional derivative terms. The unperturbed systems studied here include a harmonic oscillator, a strongly nonlinear oscillator with a cubic nonlinearity, as well as one with a nonanalytic nonlinearity. For the latter two cases, we use an approximate realization of the asymptotic method of averaging, based on harmonic balance. The averaged dynamics closely match the full numerical solutions in all cases, verifying the validity of the averaging procedure as well as the harmonic balance approximations therein. Moreover, interesting dynamics is uncovered in the strongly nonlinear case with small delayed terms, where arbitrarily many stable and unstable limit cycles can coexist, and infinitely many simultaneous saddle-node bifurcations can occur.     

Nonlinear oscillations with parametric excitation solved by homotopy analysis method

An analytical technique, namely the homotopy analysis method （HAM）, is used to solve problems of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM is not dependent on any small physical parameters at all, and thus valid for both weakly and strongly nonlinear problems. In addition, HAM is different from all other analytic techniques in providing a simple way to adjust and control convergence region of the series solution by means of an auxiliary parameter h. In the present paper, a periodic analytic approximations for nonlinear oscillations with parametric excitation are obtained by using HAM, and the results are validated by numerical simulations.     

Existence and nonexistence of periodic solutions for the Liénard and related equations

The existence of periodic solutions of Liénard type equations is converted into the existence of special fixed point problems with auxiliary conditions. A general method for exact calculation of the limit cycles is given, and the corresponding numerical iterative procedure is carried out in significant cases, with comparison to standard Runge-Kutta numerical integration. On the basis of the general theory, criteria for the existence of limit cycles are given and tested in particular cases.     

HOMOTOPY ANALYSIS METHOD: A NEW ANALYTICMETHOD FOR NONLINEAR PROBLEMS

I.IntroductionAlthoughtherapiddevelopmentofdigitalcomputersmakesiteasierandeasiertonumericallysolvenolllinearproblems,itisstillratherditliculttogivethed'analyticapproximations.Currently,mostofour11onlinearanalytictechlliquesill'cunsatislllctory.Forinstance,althoughpel.turbatiolltechlliquesarewidelyappliedtoalZalyzcnolllillcarproblcllisillscienceandengineerillg,theyarehoweversostronglydependentonsmall13arall,etersappearedinequatiollsunderconsiderationthattheyarerestrictedonlytoweLlklynolllinea…     

Random perturbations of nonlinear oscillators: Homogenization and large deviations

This paper considers two prototypical strongly nonlinear oscillators with weak dissipation and noise: a pendulum which is acted on by a small noisy torque and opposed by a friction; and the noisy Duffing-van der Pol oscillator. But over a long time the weak dissipative and noise effects can be significant. This paper develops asymptotic techniques based on averaging and large deviations to study the effects of weak noise on the escape from the domain of attraction of stable equilibrium points and limit cycles in phase-space.     

Harmonic Balance Based Averaging: Approximate Realizations of an Asymptotic Technique

Averaging is a classical asymptotic technique commonly used to studyweakly nonlinear oscillations via small perturbations of the harmonicoscillator. If the unperturbed oscillator is autonomous and stronglynonlinear, but with a two-parameter family of periodic solutions, thenaveraging is allowed in principle but typically not considered feasibleunless (a) the required family of unperturbed periodic solutions can befound in closed form, and (b) the averaging integrals can be found inclosed form. Often, the foregoing requirements cannot be met. Here, itis shown how both these difficulties can be bypassed using the classicalbut heuristic approximation method of harmonic balance, to obtain approximate realizations of the asymptotic analytical technique. Theadvantages of the present approach are that (a) closed form solutions tothe unperturbed problem are not needed, and (b) the heuristic andasymptotic parts of the calculation are kept conceptually distinct, withscope for refining the former, while preserving the asymptotic nature ofthe latter. Several examples are provided, including oscillators with astrong cubic nonlinearity, velocity dependent nonlinear terms (includinga strongly nonconservative system), a nondifferentiable characteristic,and a strongly nonlinear but homogeneous function of order 1; dynamicphenomena investigated include damped oscillations, limit cycles, forcedoscillations near resonance, and subharmonic entrainment. Goodapproximations are obtained in each case.