一类非自治滞后-自激系统的主共振与锁模现象

研究一类受外激励作用的滞后－自激系统（van der Pol系统）的振动。对于主共振情况，用平均法求出了稳态响应方程，揭示了响应与系统参数的关系。结果表明，外激励的影响占主导地位，滞后因素对非线性共振频率有影响，而自激因素使得周期解在偏离非线性共振频率后发生Hopf分岔，转变为概周期运动。通过用映射法求得旋转数，揭示了主共振前、后的概周期运动中存在的锁模现象，其阶数分布符合Farey树规律。研究表明，锁模振动因滞后非线性因素而产生。随着滞后参数值（代表滞后程度）增大，各阶共振特别是亚谐振动的存在能力增强，对系统结构会带来较大影响。     

多尺度法二义性的一种解释

多尺度法是为解决含小参数系统发展起来的应用最广泛的摄动法之一. 在求解高阶近 似方程时,多尺度法一般只求特解. 用多尺度法求解van der Pol 方程的三阶解时 将出现矛盾. 以van der Pol方程为例,证明了忽略一阶修正量中的一阶谐波 项使得混合偏导数不能交换顺序,从而导致了多尺度法的二义性和另一个数学矛盾. 在求解一阶修正量时采用含有一阶谐波项的全解,消除了二义性和该矛盾. 该 方法所求得的近似解与数值解进行了比较,结果非常吻合,验证了其合理性.     

准周期响应对称破缺分岔点的一种快速计算方法

稳态响应如周期及准周期解的分岔计算, 是非线性动力学研究的难点问题之一. 与计算方法及分析理论相对完善的周期响应相比, 准周期响应的求解只是在近些年才得到较大进展, 而且其分岔分析更加棘手, 仍需要更有效的理论和方法. 目前, 稳态响应尤其是准周期响应的分岔计算, 一般需采用数值方法, 通过调节参数反复试算得到. 为此, 本文基于增量谐波平衡IHB法提出一种快速方法, 可以高效地确定准周期响应的对称破缺分岔点. 方法的理论基础是在准周期解的广义谐波级数表达基础上, 当响应发生对称破缺分岔时, 其偶次(含零次)谐波系数将逐渐由0变为小量. 基于此性质, 将零次谐波系数预先设定为小量, 同时将分岔控制参数视为可变的迭代变量, 进而通过IHB法构造迭代格式. 作为算例, 研究不可约频率作用下的双频激励Duffing系统以及Duffing-van der Pol耦合系统. 结果表明, 只要迭代格式收敛, 随着预设小量减小, 控制参数将逐渐接近分岔近似值; 同时, 通过提高谐波截断数可显著提高近似分岔值的计算精度. 所提方法无需反复试算, 只要迭代过程收敛、便可实现分岔点直接快速计算.      

两自由度耦合van der Pol振子的拟主振动解

本文运用非线性系统的模态方法研究了两自由度耦合van der Pol振子。从退化系统稳定的主振动解出发,得到了原系统的拟主振动解,并给出了系统周期运动的条件,讨论了系统周期解、概周期解的分叉。     

压电复合材料层合梁的分岔、混沌动力学与控制

研究了简支压电复合材料层合梁在轴向、横向载荷共同作用下的非线性动力学、分岔和混沌动力学响应. 基于vonKarman理论和Reddy高阶剪切变形理论,推导出了压电复合层合梁的动力学方程. 利用Galerkin法离散偏微分方程,得到两个自由度非线性控制方程,并且利用多尺度法得到了平均方程. 基于平均方程,研究了压电层合梁系统的动态分岔,分析了系统各种参数对倍周期分岔的影响及变化规律. 结果表明,压电复合材料层合梁周期运动的稳定性和混沌运动对外激励的变化非常敏感,通过控制压电激励,可以控制压电复合材料层合梁的振动,保持系统的稳定性,即控制系统产生倍周期分岔解,从而阻止系统通过倍周期分岔进入混沌运动,并给出了控制分岔图.      

Van der Pol时滞弱耦合系统多尺度分析

对于非线性耦合项中带有时滞的van der Pol系统,采用多尺度法对该系统进行定性以及定量的分析.研究结果表明,对于van der Pol时滞耦合系统,时滞的存在影响了系统的稳定性,使系统的周期解发生了静态分岔和Hopf分岔.研究还发现,对于耦合强度较弱的情形,利用多尺度法对系统进行定嚣分析是合理可靠的.我们取不同的耦合强度作用了系统的时间历程图,相图和分岔图,分析了解析解与数值解之间产生误差的原因.本文所研究的系统来源于耦合的激光振荡器.     

垂直冲击消振系统简谐激励响应及稳定性分析

运用迭代映射及其稳定性分析原理,研究了垂直冲击消振系统的简谐激励响应及其周期响应的稳定性．首先建立了稳定周期响应的参数区域边界方程,分析了稳定周期运动向混沌转变的一般规律．然后以典型的二阶主振系为例,得到了几个对消振效果影响较大的稳态周期响应区域的详细数值结果,讨论了稳态周期响应区域及附近的消振效果．     

悬臂输流管道在基础激励与脉动内流联合作用下的参激振动

首先建立了悬臂输流管道在基础激励与脉动内流联合作用下的运动方程;然后基于Galerkin法研究了该系统的非线性动力学行为,分析了系统运动状态随激励频率和相位差的变化,以及混沌百分比随频率比(基础激励频率与脉动频率之比)和相位差的变化。结果表明,无论以激励频率还是以相位差为分岔参数,系统都具有多种形式的动态响应,包括周期运动、概周期运动和混沌运动,但进入和脱离混沌的途径不同。相位差和频率比对系统的混沌百分比有重要影响:相位差为π/2时系统混沌百分比最大;频率比为1时系统混沌百分比最小,频率比较小或较大时系统混沌百分比与只有基础激励时接近。     

谐激励作用下输流曲管的混沌振动研究

研究了谐激励作用下输流曲管在系统参数区域内的混沌振动.基于牛顿法导出了输流曲管模型的非线性控制方程,并利用微分求积法对此方程在空间域进行离散,导出了输流曲管的非线性动力学方程组.在此基础上,对输流管道的动力响应进行了数值模拟.采用分岔、相平面、时间历程和庞加莱映射图等手段分析发现,在流速和激励频率的参数区域内,系统将可能发生包括混沌振动在内的多种运动形式.系统可经由倍周期分岔或概周期运动通向混沌.分析结果为工程输流管道模型的合理设计提供了参考.     

横向荷载作用下弹性地基上梁的主共振分析

基于Winkler地基模型及Euler-Bernoulli梁理论,建立了弹性地基上有限长梁的非线性运动方程.运用Galerkin方法对运动方程进行一阶模态截断,并利用多尺度法求得该系统主共振的一阶近似解.分析了长细比、地基刚度、外激励幅值和阻尼系数等参数对系统主共振幅频响应的影响,然后通过与非共振硬激励情况对比分析主共振对其动力响应的影响.结果表明:主共振幅频响应存在跳跃和滞后现象;阻尼对主共振响应有抑制作用;主共振显著增大系统稳态动力响应位移.     

强迫Van der Pol振子的动力学特性

采用增量谐波平衡方法导出强迫Van der Pol振子稳态周期响应的IHB计算格式．以外激励频率为参数进行跟踪延续获得了系统主共振时的幅频响应特性，并作出了特定系统参数下的周期响应极限环．其结果与Runge—Kutta方法进行了对比，结果表明该算法精度可以灵活控制，且收敛速度快，结果可靠，是非线性电路系统等工程应用中强非线性问题动力学特性分析的有效方法．     

Effect of External Excitations on a Nonlinear System with Time Delay

The trivial equilibrium of a two-degree-of-freedom autonomous system may become unstable via a Hopf bifurcation of multiplicity two and give rise to oscillatory bifurcating solutions, due to presence of a time delay in the linear and nonlinear terms. The effect of external excitations on the dynamic behaviour of the corresponding non-autonomous system, after the Hopf bifurcation, is investigated based on the behaviour of solutions to the four-dimensional system of ordinary differential equations. The interaction between the Hopf bifurcating solutions and the high level excitations may induce a non-resonant or secondary resonance response, depending on the ratio of the frequency of bifurcating periodic motion to the frequency of external excitation. The first-order approximate periodic solutions for the non-resonant and super-harmonic resonance response are found to be in good agreement with those obtained by direct numerical integration of the delay differential equation. It is found that the non-resonant response may be either periodic or quasi-periodic. It is shown that the super-harmonic resonance response may exhibit periodic and quasi-periodic motions as well as a co-existence of two or three stable motions.     

A novel approach for generating multi-direction multi-double-scroll attractors

Attention is focused in this work on quasiperiodic motion of nonlinear systems whose spectrum contains uniformly spaced sideband frequencies with a distance \(\omega _{d}\) apart, around a frequency \(\omega \) with \(\omega \gg \omega _{d}\) and its integer multiples, which are referred to as carrier frequencies. The ratio of the two frequencies \(\omega \) and \(\omega _{d}\) is an irrational number. A new method based on the traditional incremental harmonic balance (IHB) method with multiple timescales, referred to as Lau method, where two timescales, \(\tau _{1}=\omega t\) (a fast timescale) and \(\tau _{2}=\omega _{d}t\) (a slow timescale), are introduced, is presented to analyze quasiperiodic motion of nonlinear systems. An amplitude increment algorithm is adapted to deal with cases where the two frequencies \(\omega \) and \(\omega _{d}\) are    unknown a priori, in order to automatically trace frequency response of quasiperiodic motion of nonlinear systems and accurately calculate all frequency components and their corresponding amplitudes. Results of application of the present IHB method to quasiperiodic free vibration of a hinged–clamped beam with internal resonance between two transverse modes are shown and compared with previously published results with Lau method and those from numerical integration. While differences are noted between results predicted by the present IHB method and Lau method, excellent agreement is achieved between results from the present IHB method and numerical integration even in cases of strongly nonlinear vibration. The present IHB method is also used to analyze quasiperiodic free vibration of high-dimensional models of the hinged–clamped beam.     

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton's principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited. Bifurcations including the saddle-node, Hopf, perioddoubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.     

Energy harvesting in a Mathieu–van der Pol–Duffing MEMS device using time delay

This paper investigates quasi-periodic vibration-based energy harvesting in a delayed nonlinear MEMS device consisting of a delayed Mathieu–van der Pol–Duffing type oscillator coupled to a delayed piezoelectric coupling mechanism. We use the multiple scales method to approximate the quasi-periodic response and the related power output near the principal parametric resonance. The effect of time delay on the energy harvesting performance is studied. It is shown that for appropriate combination of time delay parameters, there exists an optimum range of excitation frequency beyond the resonance where quasi-periodic vibration-based energy harvesting is maximum. Numerical simulations are performed to confirm the analytical predictions.     

Simultaneous combination and 1:3:5 internal resonances in a parametrically excited beam-mass system

The nonlinear dynamics of a base-excited slender beam carrying a lumped mass subjected to simultaneous combination parametric resonance of sum and difference type along with 1:3:5 internal resonances is investigated. Method of normal form is applied to the governing nonlinear temporal differential equation of motion to obtain a set of first-order differential equations which are used to obtain the steady-state, periodic, quasi-periodic and chaotic responses for different control parameters viz., amplitude and frequency of external excitation and damping. Frequency response, phase portraits, time spectra and bifurcation diagram are plotted to visualize the system behaviour with variation in the control parameters. Here, two distinct zones of trivial instability, blue sky catastrophe phenomena, jump down phenomena, simultaneous occurrence of periodic and chaotic orbits, period doubling of the mixed-mode periodic orbits leading to chaos, attractor merging crisis, boundary crisis, type II and on-off intermittencies are observed. Bifurcation diagram is plotted to facilitate the designer to choose a safe operating zone.     

Dynamics of two strongly coupled van der pol oscillators

A perturbation method is used to study the steady state behavior of two Van der Pol oscillators with strong linear diffusive coupling. It is shown that a bifurcation occurs which results in a transition from phase-locked periodic motions to quasi-periodic motions as the coupling is decreased or the detuning is increased. The analytical results are compared with a numerically generated solution.