非对称强非线性振动特征分析

提出了构造一类非线性振子解析逼近周期解的的初值变换法.用Ritz-Galerkin法,将描述动力系统的二阶常微分方程,化为以振幅、角频率和偏心距为独立变量的不完备非线性代数方程组;关键是考虑初值变换,增加补充方程,构成了以角频率、振幅和偏心距为变量的完备非线性代数方程组.作为例子利用初值变换法求解了相对论修正轨道方程的六种分岔周期解.给出了非对称振动的幅频曲线和偏频(偏心距与角频率的关系)曲线.发现了固有角频率漂移现象.     

轮-带驱动系统稳态周期响应谐波平衡分析

研究含有单向离合器、两滑轮及附件的轮-带驱动系统稳定稳态周期响应.通过单向离合器连接从动轮与附属系统,并计入传送带的横向振动的影响,导出了由偏微分-积分方程与分段常微分方程组成的连续-离散型非线性耦合方程组.利用Galerkin方法将连续非线性方程组截断为一组非线性常微分方程组,再运用谐波平衡法得到轮-带驱动系统耦合非线性振动的稳态响应.通过比较有无单向离合器装置的系统稳定稳态幅频响应曲线,研究了单向离合器对驱动系统以及轮-带系统非线性动态特性的影响.并首次研究了高频激励下轮-带系统的稳态响应.最后,运用Runge-Kutta方法对比验证了基于谐波平衡法得到的稳态响应.     

纵横向耦合轴向运动梁的自由振动响应研究

研究轴向运动梁在纵向与横向振动耦合下的自由振动响应,尤其是在横向第1,2固有频率之比ω1/ω2接近1:3内共振条件下的系统响应.利用哈密顿原理建立非惯性参考系下轴向运动梁的振动微分方程,采用分离变量法分离时间变量和空间变量并利用Galerkin方法离散,得到了运动梁含有2次和3次非线性项的运动微分方程.利用增量谐波平衡法(IHB法)分析纵向与横向振动耦合时非线性振动复杂的频幅响应曲线,探讨了相互耦合下系统在横向前2阶固有频率附近没有横向外激励作用下的自由振动响应,揭示了很多复杂而有趣的非线性现象.     

轴向运动梁横向非线性振动研究

应用增量谐波平衡法(IHB法)研究轴向运动梁横向非线性振动的内部共振．根据哈密顿原理建立非惯性参考系下轴向运动梁的横向振动微分方程,采用分离变量法分离时间变量和空间变量并利用Galerkin方法离散运动方程,再应用IHB法进行非线性振动分析,研究了在固有频率之比ω20／ω10接近于3：1情况下,外激励频率ω在ω10,ω20附近的具有内部共振的基谐波和次谐波响应．数值结果表明了IHB法是一个求解轴向运动体系非线性振动的非常有效的半解析、半数值的方法。     

基于场方法的非线性系统求解

用场方法联立多尺度法求单自由度的非线性系统的近似解．将两个状态方程的一个状态变量看作是另一个状态变量和时间的一个场函数,把原系统化为求解具有初始条件的基本方程,通过多尺度展开,逐个摄动方程求解,获得了振幅和相位的一阶近似微分方程,作为例子,求得了非线性振动系统的一阶近似解,并和数值解进行比较,两者吻合较好。     

粘弹性板的非线性动力稳定性分析方法

将微分-积分型参数振动方程组转化成微分型,且基于增量谐波平衡法的一般应用途径,分析了受面内周期激励的粘弹性板的非线性动力稳定特性,揭示了主要动力不稳定区域的整体下移以及缩小和标准线性固体材料的粘性参数、板的振动频率之间的关系．同时给出了增量谐波平衡法直接应用于非线性微分-积分型参数振动方程的简化途径,并通过两种应用途径所得结果的对比,检验了这种简化途径的有效性．     

简支梁移动质量响应分析

随着行车速度与交通量不断增加,荷载不断加重,桥梁的移动荷载响应越来越得到人们的重视.考虑移动车辆的惯性效应与桥梁的阻尼效应时,需要把车辆荷载简化为移动质量进行研究,这时得到的控制方程是变系数偏微分方程,在数学上通常难以精确求解.经分离变量与模态叠加后,化为变系数常微分方程组.本文利用WKB法,得到了近似的动力学响应,并与数值解、移动常力、Inglis解进行了比较.     

轴向运动薄板非线性振动及其稳定性研究

应用增量谐波平衡法(IHB法)研究轴向运动薄板横向非线性振动特性及其稳定性.通过Hamilton原理推导出了非惯性参考系下四边简支轴向运动薄板的横向振动微分方程,然后利用Galerkin方法离散运动方程.对离散后的非线性方程组应用IHB法进行非线性振动分析,研究了在固有频率之比ω20/ω10接近于3:1情况下,外激励频率ω在ω10附近的具有内部共振的基谐波响应.最后用多元Floquet理论分析了系统周期解的稳定性,其中采用Hsu方法来计算转移矩阵.通过对具体例子的数值计算,分别得到了自由振动和不同外激励下的频幅相应曲线,通过对比运动梁模型和运动薄板模型的计算结果,分析了各种模型的适用范围.     

一类高振荡微分方程组的一个对称数值解法

讨论形如（x）＋Ω^2x=g（x）（g（x）=-（（δ）U）/（（δ）x））的一类高振荡微分方程组数值解法构造问题.我们给出了计算该类方程组的一个对称数值解法.并以FPU问题为例进行数值实验,与脉冲法相比较,数值实验结果显示该解法具有较好的能量保守性.     

电站并联运行的不对称非线性振荡

本文给出一个结构不完全对称并联电网的等价定理,它把双输入双输出非线性耦合的微 分方程组等价为单输入单输出的非线性微分方程,然后用渐近方法和谐波线性化方法求其一 次近似解,得到一些新的物理性质,有助于合理选择电网结构,以提高其结构稳定性.     

Practical frequency response analysis of non-linear time-delayed differential or difference equation models

A systematic algorithm for generating the polyharmonic balance equations for any system within a broad class of time-delayed differential, or non-linear difference equation models, is presented. The method, which is readily automated, enables the balance equations to be written down directly in terms of the coefficients of the governing equation, and the complex amplitudes of a general harmonic waveform. The system frequency response or amplitude dependent describing function is then readily computed. The method is illustrated by means of examples including both a time-delayed differential system example and a discrete time NARX model application. The results are validated against detailed numeric simulation which confirms the accuracy and efficacy of the approach.     

Harmonic Balance Nonlinear Identification of a Capacitive Dual-Backplate MEMS Microphone

This paper describes the application of a nonlinear identification method to extract model parameters from the steady-state response of a capacitive dual-backplate microelectromechanical systems microphone. The microphone is modeled as a single-degree-of-freedom second-order system with both electrostatic and mechanical nonlinearities. A harmonic balance approach is applied to the nonlinear governing equation to obtain a set of algebraic equations that relate the unknown system parameters to the steady-state response of the microphone. Numerical simulations of the governing equation are also performed, using theoretical system parameters, to validate the accuracy of the harmonic balance solution for a weakly nonlinear microphone system with low damping. Finally, the microphone is experimentally characterized by extracting the system parameters from the response amplitude and phase relationships of the experimental data.     

多频激励磁悬浮能量采集

研究多频激励下磁力悬浮非线性磁电能量器采集系统的动力学特性.结合谐波平衡法、牛顿迭代法和弧长延伸法近似分析非线性电力耦合的常微分方程组,研究多简谐频激励下系统的非线性稳态幅频响应特征.通过改变激励的频率,研究磁力悬浮非线性振动能量采集器的幅频特性.研究结果表明,多频激励的稳态幅频响应随非线性系数的增大而位移幅频响应的共振峰变小但带宽变宽.另外,还通过对比电学参数对共振响应幅度以及区域的影响,确定了电阻、电感和耦合系数对增强两个共振强度、扩大两个共振区域,也就是提高能量采集的强度和带宽的影响.数值模拟验证了近似解析分析结果.     

Automatic computation of polyharmonic balance equations for non-linear differential systems

An automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross-product non-linear differential system to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of a general harmonic waveform. The system frequency response, in the form of a multi-input, amplitude dependent describing function, is therefore readily computed. The method is illustrated by means of an example, and the results validated against detailed numeric simulation.     

Nonlinear thermal–mechanical vibration of flow-conveying double-walled carbon nanotubes subjected to random material property

The present study deals with the dynamic response variability of nonlinear thermal–mechanical vibration of the fluid-conveying double-walled carbon nanotubes (DWCNTs) by considering the effects of the temperature change, geometric nonlinearity and nonlinearity of van der Waals (vdW) force. The nonlinear governing equations of the fluid-conveying DWCNTs are derived based on the Hamilton’s principle and theory of thermal elasticity. The Young’s modulus of elasticity of the DWCNTs is assumed as stochastic with respect to position to actually describe the random material property of the DWCNTs. By utilizing the Monte Carlo simulation, the nonlinear coupled governing equations of the fluid-conveying DWCNTs become deterministic. Then we adopt the harmonic balance method in conjunction with Galerkin’s method to solve the nonlinear coupled deterministic differential equations for many different sample functions. Some statistical dynamic response of the DWCNTs such as the mean values and standard deviations (SDs) of the amplitude of the displacement are calculated, meanwhile the effects of the temperature change and flow velocity on the statistical dynamic response of the DWCNTs are investigated. It is concluded that the mean value and SD of the amplitude of the displacement increase nonlinearly with the increase of the frequencies in both low and high temperatures. Furthermore, the mean value of the amplitude of the displacement for any fixed frequency decreases due to the temperature change in low temperature; on the contrary, it increases under the temperature change in high temperature.     

归一化频率自适应梳状滤波器的稳定性分析

采用多个归一化频率估计器并联形成梳状滤波器, 以跟踪和检测平稳概周期信号各正弦成分的未知频率和未知幅值. 滤波器包括相互耦合的状态估计和频率估计两个非线性微分方程. 运用慢积分流形实现两个微分方程之间的解耦, 获得关于多个频率估计值的概周期非线性动力系统, 再应用平均方法导出估计频率的非线性自治方程. 分析了自治系统的三种局部稳定性: 孤立平衡点的指数稳定性, 中心流形存在性与半稳定性以及结构扰动下的有界性. 说明幅值估计与信号跟随的收敛性和有界性. 给出滤波器参数对频率跟踪和幅值估计的暂态和稳态性能的影响. 算法实现了在给定频率区间而不是给定数值条件下的正弦分量及其幅值的准确跟随, 并且响应速度不受正弦分量幅值大小的影响. 通过仿真验证了算法的有效性.     

正交各向异性叠层板的非线性主共振分析

研究了在四边简支的边界条件下,正交各向异性矩形叠层板在横向简谐激励作用下的非线性主共振及其稳定性问题．在给出了正交各向异性叠层板的振动微分方程的基础上,利用伽辽金法导出了相应的达芬型非线性强迫振动方程．应用平均法对主共振问题进行求解,得到了系统在稳态运动下的幅频响应方程．基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件．作为算例,分别给出了不同条件下,系统运动的幅频响应曲线图、振幅-激励幅值响应曲线图和动相平面图,并对解的稳定性进行了分析,讨论了各参数对系统非线性振动特性的影响．     

Theoretical analysis and experimental study for nonlinear hybrid piezoelectric and electromagnetic energy harvester

A nonlinear hybrid piezoelectric (PE) and electromagnetic (EM) energy harvester is proposed, and its working model is established. Then the vibration response, output power, voltage and current of nonlinear hybrid energy harvester subjected to harmonic excitation are derived by the method of harmonic balance, and their normalized forms are obtained by the defined dimensionless parameters. Through numerical simulation and experimental test, the effects of nonlinear factor, load resistance, excitation frequency and the excitation acceleration on amplitude and electrical performances of hybrid energy harvester are studied, which shows that the numerical results are in agreement with that of experimental tests. Furthermore, it can be concluded that the bigger nonlinear factor, the lower resonant frequency; moreover, there is an optimal nonlinear factor that make the harvester output the maximum power. In addition, the output power of nonlinear hybrid energy harvester reaches the maximum at the optimal loads of PE and EM elements, which can be altered by the excitation acceleration. Meanwhile, the resonant frequency corresponding to the maximum power rises firstly and then falls with PE load enhancing, while it rises with EM load decreasing; furthermore, the frequency lowers with the acceleration increasing. Besides, the larger acceleration is, the bigger power output and the wider 3 dB bandwidth are. Compared with performances of linear hybrid energy harvester, the designed nonlinear energy harvester not only can reduce the resonant frequency and enlarger the bandwidth but also improve the output power.