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．     

结构可靠度响应面法的混沌动力学分析及其改进方法研究

引入混沌动力学理论讨论了结构可靠度响应面法收敛失败的非线性动力学根源.给出了几个典型非线性极限状态函数在参数区间上的可靠指标分岔图,展示了极限状态函数经过响应面法迭代成为非线性映射后计算结果的周期振荡、分岔和混沌等复杂动力学现象,说明了响应面法的收敛行为取决于极限状态函数的动力学性质和响应面法的迭代步长.在此基础上提出了改进响应面法用以改善经典响应面法收敛失败和计算误差大的缺点,算例结果证实了所提方法的可行性与精度.     

优化迭代步长的两种改进增量谐波平衡法

增量谐波平衡法(IHB法)是一个半解析半数值的方法, 其最大优点是适合于强非线性系统振动的高精度求解. 然而, IHB法与其他数值方法一样, 也存在如何选择初值的问题, 如初值选择不当, 会存在不收敛的情况. 针对这一问题, 本文提出了两种基于优化算法的IHB法: 一是结合回溯线搜索优化算法(BLS)的改进IHB法(GIHB1), 用来调节IHB法的迭代步长, 使得步长逐渐减小满足收敛条件; 二是引入狗腿算法的思想并结合BLS算法的改进IHB法(GIHB2), 在牛顿-拉弗森(Newton-Raphson)迭代中引入负梯度方向, 并在狗腿算法中引入2个参数来调节BSL搜索方式用于调节迭代的方式, 使迭代方向沿着较快的下降方向, 从而减少迭代的步数, 提升收敛的速度. 最后, 给出的两个算例表明两种改进IHB法在解决初值问题上的有效性.      

Z2—对称破缺音叉分歧点的分裂迭代算法

Ｗｒｒｎｅｒ和Ｓｐｅｎｃｅ在（５）中提出了一个正则的扩张系统，用以计算Ｚ２－对称破缺音叉分歧点。这种方法即是直接法。本文将用另一种方法－－分列迭代算法来计算Ｚ２－对称破缺音叉分歧点，分裂迭代算法超线性收敛，并明显地节约计算的工作量和计算所占用的内存。数值例子的计算成功地说明分裂迭代算法的有效笥。     

含外激励van der Pol-Mathieu方程的非线性动力学特性分析

本文针对含有自激励, 参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析, 发现其准周期运动的频谱中含有均匀边频带这一新的特性. 首先, 采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mathieu方程的周期响应, 得到了其非线性频率响应曲线; 再利用Floquet理论对周期解进行稳定性分析, 得到了两种类型的分岔及它们的位置. 然后, 基于van der Pol-Mathieu方程准周期运动的频谱中边频带相邻频率之间是等距的且含有两个不可约的基频的特性(其中一个基频是已知的, 另一个基频事先是未知的), 推导了相应的两时间尺度IHB法, 精确计算出van der Pol-Mathieu方程的准周期运动的另一个未知基频和所有的频率成份及其对应的幅值, 尤其在临界点附近处的准周期运动响应. 得到的准周期运动结果和利用四阶龙格-库塔(RK)数值法得到的结果高度吻合. 最后, 研究发现了含外激励van der Pol-Mathieu方程在不同激励频率时的一些丰富而有趣的非线性动力学现象.      

动态油膜-柔性转子的分岔与混沌

研究非稳态动载短轴承支撑的Jeffcott柔性转于系统的动力特性,将转速比、不平衡量、阻尼比、黏度作为控制参数,利用Floquet乘子预测周期解的局部稳定性,通过Lagrange插值精细积分法给出系统运动的数值结果并预测系统的长期性态,显示系统在4个参数组合的某些范围内还存在多形式次谐波解,以及由倍周期分岔、二次Hopf分岔通往混沌的现象.将动态油膜力模型和稳态油膜力模型的数值结果进行比较,表明动态非线性油膜力模型的合理性.     

基于可观测状态的轴承-转子系统周期解计算及稳定性分析

分析了轴承-转子系统的稳定性和分岔,基于系统可观测状态信息给出1种求解系统周期解及识别周期解稳定性的方法,同时将该方法与Floquet理论相结合分析系统周期解的稳定性及失稳分岔形式,将转速作为分岔参数分析系统响应的周期、拟周期、多解共存和跳跃现象.结果表明,采用该方法计算系统周期解及稳定性时,利用系统可观测稳态和瞬态信息,即可求解出系统Jacobian矩阵而无需实时求解轴承非线性油膜力的Jacobian矩阵.与传统PNF方法相比,该方法不仅具有很高的精度而且可以节约计算量,同时可以预测追踪随控制参数变化的系统周期解及其稳定性,可用于指导轴承-转子系统的非线性动力学设计.     

概率约束评估的功能度量法的混沌控制

功能度量法是基于可靠度的结构优化设计中评估概率约束的一种方法,其改进均值(AMV)迭代格式具有简洁、高效的优点,但对一些非线性功能函数搜索最小功能目标点时可能陷入周期振荡或混沌解,本文利用混沌反馈控制的稳定转换法对功能度量法的AMV迭代格式实施收敛控制.首先展示一些功能函数应用功能度量法AMV格式迭代计算产生了周期解和混沌解现象,并对迭代算法进行了混沌动力学分析.然后利用稳定转换法对功能度量法迭代失败的参数区间进行混沌控制,使嵌入周期和混沌轨道的不稳定不动点稳定化,获得了稳定收敛解,实现了迭代解的周期振荡、分岔和混沌控制.     

Duffing系统解的转迁集的解析表达式

通过对非线性Ｄｕｆｉｎｇ方程解的稳定性进行研究，得到了其周期一解失稳的转迁集的解析表达式，同时应用广义牛顿法，得到了Ｄｕｆｉｎｇ方程对称破缺分岔转迁集的解析表达式，与Ｕｅｄａ用模拟计算机的方法和Ａ．Ｙ．Ｔ．Ｌｅｕｎｇ用增量谐波平衡数值方法的结果吻合良好，克服了用模拟计算机或数字计算机确定物理参数平面上的转迁集计算工作量十分大的困难．     

一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学

考虑一类具有对称性的三自由度碰撞振动系统.系统的庞加莱映射在一定条件下存在对称不动点,对应于系统的对称周期运动.根据对称性导出庞加莱映射P是另外一个隐式虚拟映射Q的二次迭代.推导了庞加莱映射对称不动点的解析表达式.根据映射不动点的稳定性及分岔理论,映射P的对称不动点发生内伊马克沙克-音叉(Neimark--Saker-pitchfork)分岔对应于映射Q发生内伊马克沙克-倍化(Neimark--Sakerflip)分岔.利用隐式虚拟映射Q,通过对范式作两参数开折分析,研究了映射P的对称不动点在内伊马克沙克-音叉分岔点附近的局部动力学行为.碰撞振动系统在这个余维二分岔点附近的局部动力学行为可能表现为投影后的庞加莱截面上的单一对称不动点、一对共轭不动点、单一对称拟周期吸引子以及一对共轭拟周期吸引子.数值模拟得到了内伊马克沙克-音叉分岔点附近的各种可能情况.内伊马克沙克-分岔和音叉分岔互相作用可能产生新的结果:对称不动点虽然首先分岔为两个共轭不动点,但是这两个共轭不动点是不稳定的,最终收敛到同一个对称拟周期吸引子.     

Improving convergence of incremental harmonic balance method using homotopy analysis method

We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.     

Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

Bifurcations and route to chaos of the Mathieu–Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.     

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.     

A modified incremental harmonic balance method for rotary periodic motions

The determination of periodic solutions is an essential step in the study of dynamic systems. If some of the generalized coordinates describing the configuration of a system are angular positions relative to certain reference axes, the associated periodic motions divide into two types: oscillatory and rotary periodic motions. For an oscillatory periodic motion, all the generalized coordinates are periodic in time. On the other hand, for a rotary periodic motion, some angular coordinates may have unbounded magnitude due to the persistent circulation about their pivots. In this case, although the behaviour of the system is periodic physically, those angular coordinates are not periodic in time. Although various effective methods have been developed for the determination of oscillatory periodic motion, the rotary periodic motion can only be determined by brute force integration. In this paper, the incremental harmonic balance (IHB) method is modified so that rotary periodic motions can be determined as well as oscillatory periodic motions in a unified formulation. This modified IHB method is applied to a practical device, a rotating disk equipped with a ball-type balancer, to show its effectiveness.     

非自治时滞反馈控制系统的周期解分岔和混沌

研究时滞反馈控制对具有周期外激励非线性系统复杂性的影响机理,研究对应的线性平衡态失稳的临界边界,将时滞非线性控制方程化为泛函微分方程,给出由Hopf分岔产生的周期解的解析形式．通过分析周期解的稳定性得到周期解的失稳区域,使用数值分析观察到时滞在该区域可以导致系统出现倍周期运动、锁相运动、概周期运动和混沌运动以及两条通向混沌的道路：倍周期分岔和环面破裂．其结果表明,时滞在控制系统中可以作为控制和产生系统的复杂运动的控制“开关”．     

非线性转子-轴承系统的周期解及近似解析表达式

通过对普通打靶方法进行改造提出一种确定非线性系统周期轨道及周期的新型打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参与打靶法的迭代过程，迭代过程包含对周期轨道和周期的求解，迭代过程中的增量通过优化方法选择，从而能迅速确定出系统的周期轨道及其周期。应用所求的结果结合谐波平衡方法求得了非线性系统的周期轨道的近似解析表达式，理论上通过增加谐波的阶数任何精度的周期解都可以得到。最后将该方法应用于非线性转子轴承系统，求出了在某些参数下转子的周期解及其近似解析表达式，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性，计算结果对于转子系统运动的定量控制有重要理论指导意义。     

Bifurcation analysis of milling process with tool wear and process damping: regenerative chatter with primary resonance

In this paper, the occurrence of various types of bifurcation including symmetry breaking, period-doubling (flip) and secondary Hopf (Neimark) bifurcations in milling process with tool-wear and process damping effects are investigated. An extended dynamic model of the milling process with tool flank wear, process damping and nonlinearities in regenerative chatter terms is presented. Closed form expressions for the nonlinear cutting forces are derived through their Fourier series components. Non-autonomous parametrically excited equations of the system with time delay terms are developed. The multiple-scale approach is used to construct analytical approximate solutions under primary resonance. Periodic, quasi-periodic and chaotic behavior of the limit cycles is predicted in the presence of regenerative chatter. Detuning parameter (deviation of the tooth passing frequency from the chatter frequency), damping ratio (affected by process damping) and tool-wear width are the bifurcation parameters. Multiple period-doubling and Hopf bifurcations occur when the detuning parameter is varied. As the damping ratio changes, symmetry breaking bifurcation is observed whereas the variation of tool wear width causes both symmetry breaking and Hopf bifurcations. Also, under special damping specifications, chaotic behavior is seen following the Hopf bifurcation.     

Large Uniform Expansions of Periodic Solutions to Strongly Non-Linear Evolution Equations with Odd Polynomial Non-Linearity

A new method of uniform expansions of periodic solutions to ordinary differential equations with arbitrary odd polynomial non-linearity is constructed to study quasi-harmonic processes in non-linear dynamical systems, in particular when a small parameter of non-linearity is absent. The main idea of the method consists in using the ratio of the amplitudes of higher harmonics to the amplitude of the first harmonic of a periodic solution as a small formal parameter. In the particular case of a single-periodic solution, this small parameter appears due to descending the amplitudes of harmonics monotonically with increasing their number. Due to uniform expansion the amplitudes of higher harmonics turn out to be rational and fractional functions in the amplitude of the first harmonic and the frequency of oscillations. We show that the method of uniform expansions is an effective tool for obtaining convergent expansions of periodic solutions in explicit form all over the domain, where periodic solutions exist, independently of the magnitude of non-linearity. In each subsequent approximation, one more higher harmonic is taken into account, with all the other harmonics being corrected. We demonstrate the effectiveness of the method on the examples of the harmonically forced Duffing oscillator; free vibrations of the oscillator with fifth-power non-linearity and mathematical pendulum.