非自治二阶Hamilton系统的周期解

本文推广了Willem的一个结果,Willem研究的受迫周期振动要求受迫势能关于空间变量是周期的,而本文只要求受迫势能对时间变量积分后关于空间变量是周期的,该结果包括了单摆的受迫振动.本文将用直接变分最小方法和Rabinowtz的鞍点定理来研究当势函数对时间变量积分后是周期时受迫摆方程的周期解.     

海洋立管双模态动力学分岔分析

为研究剪切流作用下顶张力立管的涡激振动响应规律,将立管简化为Euler-Bernoulli梁模型,用van der Pol尾流振子描述流体的作用,建立了立管涡激振动的非线性动力学模型.基于二阶Galerkin模态离散所得常微分方程组,采用谐波平衡法、Poincaré映射方法和Lyapunov指数法分析系统响应特点.研究结果表明:随着流速的增加,系统响应在周期运动和概周期运动间多次转换,其中周期解区域对应系统的涡激共振区;谐波平衡法结果能够较准确地预测涡激共振区周期解的振幅和频率,以及非涡激共振区概周期解的主要频率成分.     

轴向运动三参数黏弹性梁弱受迫振动的渐近分析

研究了轴向运动三参数黏弹性梁的弱受迫振动．建立了轴向运动三参数黏弹性梁受迫振动的控制方程．使用多尺度法渐近分析了运动梁的稳态响应,导出了解稳定性边界方程、稳态振幅的表达式以及稳态响应非零解的存在条件．依据Routh-Hurwitz定律决定了非线性稳态响应非零解的稳定性．     

弹性厚矩形板受迫振动的功的互等定理法

本文将功的互等定理法（ＲＴＭ）推广应用于求解基于Ｒｅｉｓｓｎｅｒ理论的厚矩形板受迫振动问题·本文导出了厚矩形板动力基本解；给出了三边固定一边自由厚矩形板在均布简谐干挠力作用下稳态响应的精确解析解·这是计算厚矩形板受振动稳态响应的一个简便通用的方法·     

剪切变形对直线型正交异性层合圆板大幅度受迫振动的影响*

本文研究了计及横向剪切变形的直线型正交异性层合圆板在简谐载荷q₀cosωt作用下的非线性受迫振动问题.采用伽辽金方法得到强振频率与振幅关系的解析解.最后,分析了横向剪切对板振动的影响,并给出了板的非线性自由振动的非线性周期对线性周期的比值.     

有阻尼受迫sine-Gordon方程的全局周期吸引子

本文证明了当阻尼与扩散系数在一定的参数范围内时,有阻尼的受迫sineGordon方程的狄氏问题对于任意非自治时间周期受迫力均具有唯一的指数吸引有界集的周期解．并且,如果受迫力是自治的,则全局吸引子恰是系统唯一的指数吸引有界集的平衡解．     

受迫广义KdV-Burgers方程行波解的存在唯一性

本文研究具有受迫性的广义二维KdV-Burgers方程的周期行波解,为了获得周期行波解的存在唯一性定理,使唤用特定系数法和Schauder不动点定理获得了受迫广义KdV-Burgers方程周期行波解存在唯一性的条件.并获得了周期行波解的一些先验估计式.     

流动致振引出的非线性四阶微分方程周期解的存在性研究

流动引起的振动问题是力学上比较著名的问题。本文应用一些不动点的基本原理研究了这一问题，并给出了周期解存在性条件和尾流振动方程周期解存在的参数范围；另外，在周期解的稳定性及渐近表达方面也做了一些工作，获得一些结果．     

非线性振动分析中的正交函数法

本文根据谐波平衡法假设周期解的基本思想，提出了一种分析非线性振动特性的正交函数法。将位移展开为谐波的级数形式，根据线性模态和三角级数的正交性导出了一组形式简单的特征方程。有效地解决了平方非线性系统存在漂移项的困难，算例表明：本文方法精度高，收敛快，工作量小。     

两自由度非线性振动系统主参数激励下的分岔分析

本文给出了参数激励作用下两自由度非线性振动系统,在1:2内共振条件下主参数激励低阶模态的非线性响应.采用多尺度法得到其振幅和相位的调制方程,分析发现平凡解通过树枝分岔产生耦合模态解,采用Melnikov方法研究全局分岔行为,确定了产生Smale马蹄型混沌的参数值.     

Frequency dependent iteration method for forced nonlinear oscillators

A new iteration method for nonlinear vibrations has been developed by decomposing the periodic solution in two parts corresponding to low and high harmonics. For a nonlinear forced oscillator, the iteration schema is proposed with different formulations for these two parts. Then, the schema is deduced by using the harmonic balance technique. This method has proven to converge to the periodic solutions provided that a convergence condition is satisfied. The convergence is also demonstrated analytically for linear oscillators. Moreover, the new method has been applied to Duffing oscillators as an example. The numerical results show that each iteration schema converges in a domain of the excitation frequency and it can converge to different solutions of the nonlinear oscillator.     

Stability and vibration analysis of a complex flexible rotor bearing system

This paper presents the non-linear dynamic analysis of a flexible rotor having unbalanced and supported by ball bearings. The rolling element bearings are modeled as two degree of freedom elements where the kinematics of the rolling elements are taken into account, as well as the internal clearance and the Hertz contact non-linearity. In order to calculate the periodic response of this non-linear system, the harmonic balance method is used. This method is implemented with an exact condensation strategy to reduce the computational time. Moreover, the stability of the non-linear system is analyzed in the frequency-domain by a method based on a perturbation applied to the known harmonic solution in the time domain.     

夹层椭圆形板的1/3亚谐解

研究了夹层椭圆形板的非线性强迫振动问题。在以5个位移分量表示的夹层椭圆板的运动方程的基础上,导出了相应的非线性动力方程。提出一类强非线性动力系统的叠加-叠代谐波平衡法。将描述动力系统的二阶常微分方程,化为基本解为未知函数的基本微分方程和派生解为未知函数的增量微分方程。通过叠加-叠代谐波平衡法得出了椭圆板的1/3亚谐解。同时,对叠加-叠代谐波平衡法和数值积分法的精度进行了比较。并且讨论了1/3亚谐解的渐近稳定性。     

Triple scale analysis of periodic solutions and resonance of some asymmetric non linear vibrating systems

We consider small solutions of a vibrating mechanical system with smooth non-linearities for which we provide an approximate solution by using a triple scale analysis; a rigorous proof of convergence of the triple scale method is included; for the forced response, a stability result is needed in order to prove convergence in a neighbourhood of a primary resonance. The amplitude of the response with respect to the frequency forcing is described and it is related to the frequency of a free periodic vibration.     

具有holling Ⅲ类功能反应的捕食者食系统差分方程周期解的存在性[英文]

本文利用重合度理论研究了一类具有holling Ⅲ类功能反应捕食者食差分方程的正周期解的存在性问题。     

二端面转轴相对转动非线性动力学系统的主共振与次共振解

本文研究在简谐激励力作用下二端面弹性转轴相对转动的主共振、超谐波共振和亚谐波共振.用平均法研究了系统的主共振,得到了系统的渐进稳态周期解,采用多尺度法求得了系统的3次超谐波共振解和1/3次亚谐波共振解.     

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

Analytical approximations to primary resonance response of harmonically forced oscillators with strongly general nonlinearity

This paper presents an innovative analytical approximate method for constructing the primary resonance response of harmonically forced oscillators with strongly general nonlinearity. A linearization process is introduced prior to harmonic balancing (HB) of the nonlinear system and a linear system is derived by which the accurate analytical approximation procedure is easily and innovatively implemented. The main cutting edge of the proposed method is that complicated and coupled nonlinear algebraic equations obtained by the classical HB method is avoided. With only one iteration, very accurate analytical approximate primary resonance response can be determined, even for significantly nonlinear systems. Another advantage is the direct determination of the maximum oscillation amplitude. This is due to the appropriate form chosen for the approximation with no extra processing required. It is concluded that the result of an initial approximate solution from the two-term (constant plus the first harmonic term) harmonic balance is not reliable especially for strongly nonlinear systems and a correction to the initial approximation is necessary. The proposed method can be applied to general oscillators with mixed nonlinearities, such as the Helmholtz-Duffing oscillator. Two examples are presented to illustrate the applicability and effectiveness of the proposed technique.     

Hybrid numerical methods for periodic initial value problems involving second-order differential equations

Computer simulation of problems in celestial mechanics often leads to the numerical solution of the system of second-order initial value problems with periodic solutions. When conventional methods are applied to obtain the solution, the time increment must be limited to a value of the order of the reciprocal of the frequency of the periodic solution.In this paper hybrid methods of orders four and six which are P-stable are developed. Further, the adaptive hybrid methods of polynomial order four and trigonometric order one have also been discussed. The numerical results for the undamped Duffing equation with a forced harmonic function are listed.     

Bifurcating periodic solution for a class of first-order nonlinear delay differential equation after Hopf bifurcation

In this paper, we predict the accurate bifurcating periodic solution for a general class of first-order nonlinear delay differential equation with reflectional symmetry by constructing an approximate technique, named residue harmonic balance. This technique combines the features of the homotopy concept with harmonic balance which leads to easy computation and gives accurate prediction on the periodic solution to the desired accuracy. The zeroth-order solution using just one Fourier term is applied by solving a set of nonlinear algebraic equations containing the delay term. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. It is shown that the solutions are valid for a wide range of variation of the parameters by two examples. The second-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration. Moreover, the residue harmonic balance method works not only in determining the amplitude but also the frequency of the bifurcating periodic solution. The method can be easily extended to other delay differential equations.