一种新的求解非线性系统周期解方法

本文利用切比雪夫多项式的若干良好性质,对非自治强非线性动力系统进行分析。将状态矢量在主周期上先展开谐波级数的形式,再将各谐波展开为切比雪夫多项式的形式,从而将求周期解的问题转变为非线性代数方程组的求解问题,得出一种可以方便、迅速地获得近似周期解的解析方法。此方法不依赖于小参数假设,可以用于分析强非线性问题和高维问题,而且对参数激励系统同样有效。以Duffing系统周期解的计算为例,通过与标准谐波平衡方法和四阶Runge-Kutta数值积分结果比较,说明此方法的有效性。     

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。     

带功能梯度材料的压电底层中周期裂纹对SH波的散射

本文研究了压电材料底层中周期裂纹对SH波的散射,通过渗透边界条件和界面上连续边界条件,将问题转化为一组带Hilbter核的奇异积分方程。利用利用切比雪夫多项式逼近方法求解Hilbter核的奇异积分方程,给出了标准动应力强度因子和电位移强度因子的表达式。最后通过数值算例说明了几何参数、物性参数,入射波频率和振幅等对强度因子的影响.     

基于信号分解和切比雪夫多项式的多体系统区间不确定性分析方法

参数的不确定性会对多体系统动力学响应产生显著影响, 区间分析方法只需根据不确定性参数的边界信息, 便可实现在多体系统动力学分析中考虑参数的不确定性. 考虑区间参数不确定性, 采用切比雪夫区间方法(Chebyshev interval method, CIM)在分析多体系统动力学响应时, 随着时间的增大, 响应边界精度会越来越低. 为解决CIM的这一问题, 本文将信号分解技术与切比雪夫多项式结合, 采用切比雪夫多项式分别对HHT变换(Hilbert-Huang transform, HHT)和局域均值分解(local mean decomposition, LMD)得到的瞬时幅值和瞬时相位近似, 提出CIM-HHT方法和CIM-LMD方法, 以获得含区间参数的长周期动力学响应边界. HHT和LMD分解能够将多体系统的多分量响应分解为多个单分量和一个趋势分量(残余分量)之和, CIM-HHT和CIM-LMD对每个分量的瞬时幅值和瞬时相位、和趋势分量采用切比雪夫多项式近似, 进而建立系统响应的耦合模型, 可以得到系统的动力学响应边界. 最后, 考虑单摆和曲柄滑块机构中的参数不确定性, 验证了CIM-HHT和CIM-LMD方法的有效性. 结果表明, 相比CIM, 在长周期区间动力学响应分析中CIM-HHT和CIM-LMD能够获得较准确的结果. 此外, 相比CIM-HHT, CIM-LMD具有更弱的末端效应, 计算精度更高.      

用切比雪夫多项式研究短梁的弯曲计算

考虑剪切效应,利用切比雪夫多项式构造严格满足表面切应力边界条件的轴向位移表达式,建立了短梁弯曲问题的新理论．利用奇异函数把作用在短梁上的复杂外载荷表示为分布载荷,推导出了短梁弯曲时的截面正应力公式及挠曲线表达式．把采用切比雪夫多项式推导出短梁的弯曲计算公式计算结果与弹性理论计算结果进行比较,可知该方法的计算精度较高．研究结果表明：在复杂外载荷作用下,当长高比小于等于6时,剪切变形对梁的弯曲挠度影响较大,而当长高比小于3时,剪切变形对梁的弯曲应力影响较大;因此建议采用切比雪夫多项式方法给出的挠度表达式、弯曲应力进行计算,因为切比雪夫多项式方法不但给出了复杂外载荷作用下梁截面挠度、弯曲应力的计算通式,而且该方法具有计算过程简便、精度高的优点．     

岩土力学参数概率分布的切比雪夫多项式推断

提出了较大样本岩土力学参数概率分布的切比雪夫多项式逼近法。基于数值逼近原理，直接根据试验样本矩，运用切比雪夫多项式推断岩土力学参数的概率密度函数，并用精度较高的K—S检验法，从理论上证明所求密度函数的正确性和实用性。该方法直接根据试验样本信息和统计方法推断，而不是事先假定成经典的理论概率分布，因此数学和物理意义更加充分。通过对各种经典分布曲线（正态分布、指数分布、对数正态分布等）数值检验，结果表明所得到的逼近表达式有很好的拟合性能。根据样本数据得出的某岩石抗压强度概率密度函数，与实际统计所得分布频率非常接近，可以满足岩土工程可靠性分析的要求。     

求解非线性动力系统周期解推广的打靶法

提出一种确定非线性系统周期轨道及周期的改进打靶算法。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入打靶法的迭代过程，从而能迅速确定出系统的周期轨道及其周期。该方法对初始迭代参数没有苛刻要求，可以用于分析强非线性系统，而且对参数激励系统同样有效，对高维系统也能迅速、准确地求得周期解。文中应用该方法对三维Rǒssler系统和八维非线性柔性转子-轴承系统的周期轨道和周期进行了求解，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性。     

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。     

轴向功能梯度变截面梁的自由振动研究

摘 要：本文引入一种新的、简单易行的近似方法,求解轴向非均匀变截面梁的自由振动固有频率。将位移展开成切比雪夫多项式,从而变系数控制微分方程转化为含未知系数的齐次线性方程组。利用非零解的存在条件,进而得到含固有频率的特征方程。通过和特定梯度下已有的精确解进行比较,验证了该方法的精度和有效性,并分析了梯度参数、支承条件等对固有频率的影响。     

基于吸收边界条件的有限元方法和离散切比雪夫展开的二维缺陷识别

结合考虑了吸收边界条件的有限元正演方法,通过离散切比雪夫多项式在特定离散点的展开逼近待识别函数,实现了无限区域内二维缺陷的识别。正问题计算中引入的吸收边界条件虽然不能严格地模拟无限域对近场波动的影响,但能够以满足实际需要的精度模拟这一影响,且具有解耦特性,极大地简化了数值计算,在反演计算中,离散点取在切比雪夫多项式的零点处,使逼近时最大偏差最小,数值算例表明此方法的有效性。     

A new numerical technique for the analysis of parametrically excited nonlinear systems

A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem to the solution of a set of linear algebraic equations in each iteration. The method is equally applicable for nonlinear systems which are represented in state-space form or by a set of second-order differential equations. The proposed technique is found to duplicate the periodic, multi-periodic and chaotic solutions of a parametrically excited system obtained previously using the conventional numerical integration schemes with comparable CPU times. The technique does not require the inversion of the mass matrix in the case of multi degree-of-freedom systems. The present method is also shown to offer significant computational conveniences over the conventional numerical integration routines when used in a scheme for the direct determination of periodic solutions. Of course, the technique is also applicable to non-parametrically excited nonlinear systems as well.     

树形多体系统非线性动力学的数值分析方法

研究了树形多体系统大线性动力学分析的数值方法,利用多体系统的正则方程及其线性化程,给出了多体系统Ｌｙａｐｕｎｏｖ指数和Ｐｏｉｎｃａｒｅ映射的计算方法,该算法具有较好的计算精度和通用性,既适用于说明该算法的有效性,并对该系统的动力学行为进行分析,最后用算例说明该算法的有效性,并对该系统的动力学特征（周期解、准周期解、分岔、混沌以及通往混沌的道路等）进行了分析。     

一种确定非线性裂纹转子解的形式的新方法

将小波变换与Ｐｏｉｎｃａｒｅ映射相结合，即用Ｐｏｉｎｃａｒｅ映射确定周期解，用谐波小波变换区分拟周期响应和混沌运动，提出了一种分析非线性裂纹转子系统解的形式随参数变化的新方法．结果表明这种方法是非常有效的，它比以前所用的计算Ｌｉａｐｕｎｏｖ指数的方法节约了计算时间，并且较易实施．     

Analysis of time-periodic nonlinear dynamical systems undergoing bifurcations

In this study a new procedure for analysis of nonlinear dynamical systems with periodically varying parameters under critical conditions is presented through an application of the Liapunov-Floquet (L-F) transformation. The L-F transformation is obtained by computing the state transition matrix associated with the linear part of the problem. The elements of the state transition matrix are expressed in terms of Chebyshev polynomials in timet which is suitable for algebraic manipulations. Application of Floquet theory and the eigen-analysis of the state transition matrix at the end of one principal period provides the L-F transformation matrix in terms of the Chebyshev polynomials. Since this is a periodic matrix, the L-F transformation matrix has a Fourier representation. It is well known that such a transformation converts a linear periodic system into a linear time-invariant one. When applied to quasi-linear equations with periodic coefficients, a dynamically similar system is obtained whose linear part is time-invariant and the nonlinear part consists of coefficients which are periodic. Due to this property of the L-F transformation, a periodic orbit in original coordinates will have a fixed point representation in the transformed coordinates. In this study, the bifurcation analysis of the transformed equations, obtained after the application of the L-F transformation, is conducted by employingtime-dependent center manifold reduction andtime-dependent normal form theory. The above procedures are analogous to existing methods that are employed in the study of bifurcations of autonomous systems. For the two physical examples considered, the three generic codimension one bifurcations namely, Hopf, flip and fold bifurcations are analyzed. In the first example, the primary bifurcations of a parametrically excited single degree of freedom pendulum is studied. As a second example, a double inverted pendulum subjected to a periodic loading which undergoes Hopf or flip bifurcation is analyzed. The methodology is semi-analytic in nature and provides quantitative measure of stability when compared to point mappings method. Furthermore, the technique is applicable also to those systems where the periodic term of the linear part does not contain a small parameter which is certainly not the case with perturbation or averaging methods. The conclusions of the study are substantiated by numerical simulations. It is believed that analysis of this nature has been reported for the first time for this class of systems.     

Dynamical behavior analysis and bifurcation mechanism of a new 3-D nonlinear periodic switching system

This paper presents a new periodic switching chaotic system, which is topologically non-equivalent to the original sole chaotic systems. Of particular interest is that the periodic switching chaotic system can generate stable solution in a very wide parameter domain and has rich dynamic phenomena. The existence of a stable limit cycle with a suitable choice of the parameters is investigated. The complex dynamical evolutions of the switching system composed of the Rössler system and the Chua’s circuit are discussed, which is switched by equal period. Then the possible bifurcation behaviors of the system at the switching boundary are obtained. The mechanism of the different behaviors of the system is investigated. It is pointed out that the trajectories of the system have obvious switching points, which are decided by the periodic signal. Meanwhile, the system may be led to chaos via a period-doubling bifurcation, resulting in the switching collisions between the trajectories and the non-smooth boundary points. The complicated dynamics are studied by virtue of theoretical analysis and numerical simulation. Furthermore, the control methods of this periodic switching system are discussed. The results we have obtained clearly show that the nonlinear switching system includes different waveforms and frequencies and it deserves more detailed research.     

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.     

Perturbation methods for nonlinear autonomous discrete-time dynamical systems

Two perturbation methods for nonlinear autonomous discrete-time dynamical systems are presented. They generalize the classical Lindstedt-Poincaré and multiple scale perturbation methods that are valid for continuous-time systems. The Lindstedt-Poincaré method allows determination of the periodic or almost-periodic orbits of the nonlinear system (limit cycles), while the multiple scale method also permits analysis of the transient state and the stability of the limit cycles. An application to the discrete Van der Pol equation is also presented, for which the asymptotic solution is shown to be in excellent agreement with the exact (numerical) solution. It is demonstrated that, when the sampling step tends to zero the asymptotic transient and steady-state discrete-time solutions correctly tend to the asymptotic continuous-time solutions.     

碰撞振动系统分岔与混沌的研究进展

针对工程实际中普遍存在的碰撞振动系统这种典型的非光滑动力系统, 其研究具有重要的理论意义和工程实用价值. 碰撞振动系统动力学的分析与研究方法主要有理论分析、数值模拟以及应用与实验研究. 为了研究碰撞振动系统的周期运动稳定性、分岔及混沌, 采用的手段有建立Poincar\'{e}映射、中心流形和范式方法, 映射的分岔与混沌理论是碰撞振动系统研究的理论基础. 首先简述了碰撞振动系统的分析与研究方法, 光滑非线性系统动力学的分析方法部分可以推广到碰撞振动系统, 碰撞振动的不连续性导致一些方法的适用性和有效性问题. 进一步综述了碰撞振动系统周期运动稳定性、分岔、混沌及奇异性的理论研究和工程应用现状. 最后着重结合相关离散型映射系统的动力学发展, 对碰撞振动系统的分岔与混沌研究及存在的主要问题进行了讨论, 并展望了其发展趋势.