两类弱非线性振动微分方程的插值摄动解法

本文分别用插值摄动法的两种不同方法（第一,第二解法）求解了两类弱非线性振动问题,用第二解法得到的Ｄｕｆｆｉｎｇ方程的解,精度很高,当小参数不是很小时,甚至比Ｌ－Ｐ法的结构更加精确,用第一解法求解有阻尼的自由振动问题时,由于可以公式化,故求解过程十分简便,本文选取的初始零级近似解,具有新的特色。     

保守系统弱非线性振动问题的插值摄动解法

用插值摄动法求得了一类保守系统奇次弱非线性振动问题的一级近似解，精度好，甚至比多尺度法还更好，计算过程简单。     

有正阻尼时强非线性振动问题的插值摄动解法

有正阻尼的非线性振动问题，无论是弱非线性或强非线性的情形，均不能用L－P法或改进的L－P法求解，本文用插值摄动法求解了有正阻尼的强非线性自由振动问题，运算简单，精度好。     

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。     

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。     

轴对称弹性圆锥扁壳非线性自由振动直接摄动解法

本文采用弹性圆锥扁壳中心无量纲振幅和壳体母线的倾角为参数，将挠度、应力函数的导数以及自由振动频率展开为双参量的幂级数形式．用直接摄动法获得各级递推线性偏微分方程．应用变分法求得各级递推方程的近似解答．从而给出弹性圆锥扁壳非线性自由振动频率的基本公式。     

轴对称弹性圆锥扁壳非线性自由振动直接摄动解法

本文采用弹性圆锥扁壳中心无量纲振幅和壳体母线的倾角为参数，将挠度、应力函数的导数以及自由振动频率展开为双参量的幂级数形式，用直接摄动法获得各级递推线性偏微分方程，应用变分法求得各级递推方程的近似解答，从而给出弹性圆锥扁壳非线性自由振动频率的基本公式。     

线化摄动法结合权残法求解有阻尼的强非线性振动问题

本文用线化摄动法和权残法相结合的方法求解了一类有强阻尼的强非线性振动问题，有较好的精度。     

弱粘弹性材料结构振动特性分析的摄动解法

本文提出了“弱粘弹性”概念,用以描述弱时变材料的动态性能,并构造了弱粘弹性模型进行结构振动分析。根据“弱”的特点,选择了小参数,给出弱粘弹性材料动态性能的描述形式。在此基础上,构造了两种摄动解法,第一种基于弹性解,第二种基于单粘弹性模量解。文中还特别讨论了松驰函数为指数形式时的情形。     

插值摄动法及其在力学中的应用

本文研究了最高阶导数乘以小参数，或出现奇点的微分方程的定解问题，用插值摄动法求得了一级近似解，它和通常的奇异摄动法（匹配法、多尺度法）的一级近似解的精度相同。     

On the Vibrations of a Simply Supported Square Plate on a Weakly Nonlinear Elastic Foundation

In this paper an initial-boundary value problem for a weakly nonlinear plate equation with a quadratic nonlinearity will be studied. This initial-boundary value problem can be regarded as a simple model describing free oscillations of a simply supported square plate on an elastic foundation. It is assumed that the foundation has a different behavior for compression and for expansion. An approximation for the solution of the initial-boundary value problem will be constructed using a two-timescales perturbation method. The existence and uniqueness of the solution of the problem will be proved. Also the asymptotic validity of the constructed approximations will be shown on long timescales. For specific parameter values, it turns out that complicated internal resonances occur.     

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string(or wave) equation with non-classical boundary conditions is considered. Oneend of the string is assumed to be fixed and the other end of the string isattached to a spring-mass-dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillationsof flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be usedto construct formal asymptotic approximations of the solution. It will also beshown that all solutions tend to zero for a sufficiently large value of thedamping parameter. For smaller values of the damping parameter it will be shownhow the string-system eventually will oscillate.     

Interpolation perturbation method for solving nonlinear problems

I.IntroductionInthispaper,onthebasisofRefll],usingtheinterPotationPerturbationmethod,theauthorseekstosolveseveralnon-Iinearproblems.Itsmainpointsare:Introducinganinterpolationfunction,wedeterminethisfunctionbyusingtheperturbationmethodandthenproceedtoseek…     

有正阻尼时强非线性自由振动问题的一种解法

在改进的L－P法的基础上,补充考虑振幅的变化,得到了一类有正阻尼的强非线性自由振动问题的一级近似解,近似解有较好的精度。     

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。     

A Perturbation Method for Nonlinear Two-Dimensional Maps

A new perturbation method for a weakly nonlinear two-dimensional discrete-time dynamical system is presented. The proposed technique generalizes the asymptotic perturbation method that is valid for continuous-time systems and detects periodic or almost-periodic orbits and their stability. Two equations for the amplitude and the phase of solutions are derived and their fixed points correspond to limit cycles for the starting nonlinear map. The method is applied to various nonlinear (autonomous or not) two-dimensional maps. For the autonomous maps we derive the conditions for the appearance of a supercritical Hopf bifurcation and predict the characteristics of the corresponding limit cycle. For the nonautonomous maps we show amplitude-response and frequency-response curves. Under appropriate conditions, we demonstrate the occurrence of saddle-node bifurcations of cycles and of jumps and hysteresis effects in the system response (cusp catastrophe). Modulated motion can be observed for very low values of the external excitation and an infinite-period bifurcation occurs if the external excitation increases. Analytic approximate solutions are in good agreement with numerically obtained solutions.     

The Asymptotic Perturbation Method for Nonlinear Continuous Systems

We apply the asymptotic perturbation (AP) method to the study of the vibrations of Euler--Bernoulli beam resting on a nonlinear elastic foundation. An external periodic excitation is in primary resonance or in subharmonic resonance in the order of one-half with an nth mode frequency. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing the harmonic terms with a simple iteration. We obtain amplitude and phase modulation equations and determine external force-response and frequency-response curves. The validity of the method is highlighted by comparing the approximate solutions with the results of the numerical integration and multiple-scale methods.     

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.     

线性哈密顿系统的本征摄动法

利用哈密顿算子辛自共轭的特点讨论了保守哈密顿体系的摄动问题,给出了哈密顿矩阵的本征值与本征向量的二阶摄动分析方法。即当系统在哈密顿框架下进行较小修改时,不重复求解大型哈密顿矩阵的本征问题,只需在原系统的模态参数基础上进行模态分析即可,这种矩阵摄动法给出了修改后矩阵的二阶本征值和本征向量,为一般线性保守体系的本征摄动求解提出了一个新方法。