Phase filtering in nonlinear-imaging technique with a phase object

Using the nonlinear-imaging technique with a phase object (NIT-PO), we have studied third-order nonlinearities of various samples. In this work, we develop, for pure nonlinear refractive materials, an approximate method to calculate the nonlinear refractive coefficient analytically. By decomposing the object field passing through the phase object into two top-hat beams of different phases and beam radius, we acquire the approximate phase contrast, from which we extract the nonlinear refractive coefficient. This approximation is valid when the on-axis nonlinear phase shift by the sample is less than π. In addition, this approximation serves to estimate the sensitivity and monotonic interval for nonlinearity measurements more easily and thus helps us to maximize both the sensitivity and monotonic interval of measurements. We test this method with CS(2), a well-characterized third-order nonlinear refractive material using 21 ps laser pulses at 532 nm. We expect this method can be applied to high-order nonlinear refraction cases.     

解非线性方程的一类可微优化方法

本文提出了一种求解非线性方程的新方法,通过把问题等价为一个带不等式约束的优化问题,利用极大熵函数和罚函数建立了求解非线性方程的一个可微优化方法,证明了方法的收敛性,给出了数值算例,理论和算例均表明该方法是可靠和有效的。     

柔性转子─支承系统瞬态响应分析的模态叠加─Riccati传递矩阵法

利用模态叠加概念和Ｎｅｗｍａｒｋβ差分公式，采用加速度型状态变量，导出了非线性柔性转子一支承系统中分布质量轴段的场传递矩阵，然后用Ｒｉｃｃａｔｉ传递矩阵法稳定空间边界值问题，提出了分析柔性转子一支承系统瞬态响应的模态叠加一Ｒｉｃｃａｔｉ传递矩阵法。此法未对非线性元件作线性化近似处理，所以对弱、强非线性系统均适用，此法的传递矩阵维数低、稳定性好、计算精度和效率较高、易程序化，尤其适合于分析含强非线性元件大型柔性转于一支承系统的瞬态响应。     

一种求解信号包络曲线端点值的新方法

近年来出现了一种能有效分析非线性、非稳定振动信号的新方法,即希尔伯特-黄变换法。该方法通过三次样条插值分解信号时会在其边界处出现端点效应问题。在深入研究端点效应问题基础上提出一种能有效解决该问题的新方法,即最近一次极值镜像延拓法。将该方法应用到非线性信号分析中,结果表明该方法不仅能有效抑制端点效应,而且还能提高非线性信号分解的准确性,同时具有简单易行、适应性强、通用性好等特点,对水泵机组运行时泵站厂房结构的振动监测具有一定的实用价值。     

Detection of nonlinear distortions with multisine excitations in the case of nonideal behavior of the input signal

A method for the detection, qualification, and quantification of nonlinear distortions on a frequency response function (FRF) measurement was described in previous papers by Schoukens et al.. The kernel idea of that method is to use well-chosen periodic excitations where only some of the considered frequency components are excited. The nonexcited frequency lines (detection lines) are used to detect, qualify, and quantify the nonlinear distortions. Due to the presence of a low-quality generator, a nonlinear actuator, or due to the interaction of the generator with the nonlinear system, unwanted excitation power can be present at the detection lines at the input of the system. In this case, the basic conditions necessary to apply the detection method are no longer valid. In this paper, a first-order compensation method is described that allows the detection, qualification, and quantification of nonlinear distortions on the FRF measurement with the original technique of previous papers by Schoukens et al., in the case that the desired input cannot be applied at the device under test. This first-order compensation method is illustrated on simulations and a real measurement example.     

A fast implementation of explicit time-stepping algorithms with the finite element method for a class of nonlinear evolution problems

In this paper a class of nonlinear evolution problems is considered. It is shown that, under special conditions, the application of the product approximation method for nonlinear problems in the finite element method results in constant (i.e. time-independent) matrices. In those cases the amount of computing required to solve these equations with an explicit time-stepping algorithm is decreased considerably compared to the standard Galerkin formulation in which the matrices are time-dependent. The method is applied to two practical two-dimensional problems: the shallow water equations and a nonlinear heat conduction problem.     

Topology optimization of geometrically nonlinear structures using an evolutionary optimization method

The topology optimization using isolines/isosurfaces and extended finite element method (Iso-XFEM) is an evolutionary optimization method developed in previous studies to enable the generation of high-resolution topology optimized designs suitable for additive manufacture. Conventional approaches for topology optimization require additional post-processing after optimization to generate a manufacturable topology with clearly defined smooth boundaries. Iso-XFEM aims to eliminate this time-consuming post-processing stage by defining the boundaries using isovalues of a structural performance criterion and an extended finite element method (XFEM) scheme. In this article, the Iso-XFEM method is further developed to enable the topology optimization of geometrically nonlinear structures undergoing large deformations. This is achieved by implementing a total Lagrangian finite element formulation and defining a structural performance criterion appropriate for the objective function of the optimization problem. The Iso-XFEM solutions for geometrically nonlinear test cases implementing linear and nonlinear modelling are compared, and the suitability of nonlinear modelling for the topology optimization of geometrically nonlinear structures is investigated.     

基于Bessel和Meijer-G函数的楔形和锥形悬臂梁振动分析

基于欧拉-伯努利梁理论,利用Lagrange法建立了楔形和锥形截面梁在外激作用下的非线性微分方程.提出了一种基于Bessel函数和Meijer-G函数线性组合的无需迭代及近似截断的振型函数,且该振型函数不依赖于楔形和锥形变截面梁的弯曲振动的运动方程是否为标准的Bessel形式,该方法能快速求解线性基频和模态函数.随后将...     

参数激励下碳纳米谐振器的非线性振动研究

该文研究了两端固支情况下的碳纳米管谐振器在直交流电激励作用下的非线性振动响应。考虑外激励频率接近其固有频率的主共振情况下以及几何非线性,忽略静电力的非线性、范德瓦耳斯力和初始静变形的影响,运用欧拉伯努利连续梁理论,通过哈密尔顿原理得到非线性控制方程。然后利用伽略金方法离散得到降阶方程,最后应用摄动分析方法获得其振动响应的幅频曲线。主要分析了阻尼、直交流电荷载、几何三次非线性项对响应的影响。结果表明几何三次非线性项直接影响碳纳米管谐振器在外激励作用的振动响应以及碳纳米管在交流电荷载作用下的振动响应相对于直流电荷载更加敏感,并在一定程度上发生软化行为等。     

A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

Gaussian closure method is commonly used in the analysis of nonlinear stochastic systems. However, Gaussian closure may lead to unacceptable errors when system response is very much different from being Gaussian, and accuracy of the method decreases as the nonlinearity of the system increases. The need for better accuracy in strongly non-linear problems has caused the development of non-Gaussian closure schemes. In this paper, we develop a new copula-based Gaussian mixture closure method for randomly excited nonlinear systems. Our method relies on the assumption of marginal PDF of response in terms of finite Gaussian mixture model, and the derivation of joint PDF with aid of dependence modeling of Gaussian copula. By substituting the non-Gaussian PDF representation into moment equations of nonlinear system, we further develop an optimization-based closure scheme for the solution of the unknown parameters in joint PDF. In this way, PDF and thus, moments of response of highly nonlinear system can be described in a more flexible and robust way. Effectiveness of the new closure method is demonstrated by a nonlinear and a Duffing oscillator that are subjected to Gaussian white noise. The results are compared with the Gaussian closure and exact solution. It has been shown that Gaussian closure is a special case of the new closure method, and accuracy of Gaussian closure is the lower bound of that of the new closure method.     

单级柔顺正交位移放大机构非线性建模与优化

同时实现操作的灵活性与稳定性是精密工程领域的研究热点。单级柔顺正交位移放大机构可通过正交位移转换实现平行夹持,并通过位移放大提高操作的灵活性,但该机构的传统建模方法主要基于小变形假定并忽略剪切作用,导致模型精度较低。为此,对典型的单力输入单级柔顺正交位移放大机构进行精确的非线性建模与优化。考虑到剪切作用与几何非线性因素,对单级柔顺正交位移放大机构输出位移进行两步法半解析建模,以实现非线性结果的快速预测。第1步,基于能量法与欧拉-伯努利梁理论,建立该机构输出位移的线弹性解析模型,并结合小变形静力学有限元分析,拟合剪切非线性修正系数;第2步,结合几何非线性静力学有限元分析与数值拟合,建立该机构输出位移的几何非线性修正系数模型。为最大化输出位移并抑制几何非线性作用,提出机构平面尺寸和厚度综合优化策略,并利用ANSYS Workbench有限元仿真验证了机构输出位移非线性模型与优化结果的有效性。仿真结果显示,机构输出位移非线性模型的误差小于5%,且可依据不同优化策略显著增大输出位移或将几何非线性程度约束于指定范围内。研究表明利用所提出的方法对单级柔顺正交位移放大机构进行非线性建模与优化,可有效提高压电驱动柔顺微夹钳的位移输出性能与开环控制的精度和实时性,有利于实现稳定灵活的微操作。     

Symmetry Transformations and Exact Solutions of a Generalized Hyperelastic Rod Equation

In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as some special constants, the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients, which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials. Using the same method to solve the nonlinear wave equation, the validity and rationality of this method are verified.     

膜结构极小曲面找形的一种自适应有限元分析

找形分析是膜结构设计中的关键环节,但在数学上,膜结构的极小曲面找形分析是一个高度非线性问题,一般无法求得其解析解,因此数值方法成为重要工具。近年来,基于单元能量投影法（EEP法）的一维非线性有限元的自适应分析已经取得成功,基于EEP法的二维线性有限元自适应分析也被证实是有效、可靠的。在此基础上,该文提出一种基于EEP法的二维非线性有限元自适应方法,并成功将之应用于膜结构的找形分析。其主要思想是,通过将非线性问题用Newton法线性化,引入现有的二维线性问题的自适应求解技术,进而实现二维有限元自适应分析技术从线性到非线性的跨越,将非线性有限元的自适应分析求解从一维问题拓展到二维问题。该方法兼顾求解的精度和效率,对网格自适应地进行调整,最终得到优化的网格,其解答可按最大模度量逐点满足用户设定的误差限。该文综述介绍了这一进展,并给出数值算例用以表明该方法的可行性和可靠性。     

基于混合粒子群优化算法辨识Hammerstein模型

非线性系统辨识是现代辨识领域中的一个主要问题。在非线性系统辨识中,系统常被表示为一系列块连接。针对非线性系统中的Hammerstein模型,本文提出了利用混合粒子群优化算法对非线性系统模型进行辨识。该方法的基本思想是将非线性系统的辨识问题转化为参数空间上的优化问题,然后采用粒子群优化算法(PSO)获得该优化问题的解。为了进一步增强粒子群优化算法的辨识性能,提出利用一种混合粒子群优化算法。最后,给出仿真实验,其结果验证了本文给出的辨识方法是有效的。     

钢筋混凝土框剪结构非线性地震反应分析

本文采用壁式框架来分析框架─剪力墙结构的非线性地震反应,假定弯曲刚度沿杆轴按抛物线分布,导出了考虑剪切变形的单元刚度矩阵,将古典刚度法或高斯消元法用以形成该时间间隔内的瞬时抗侧移刚度矩阵,用适合非线性动力计算的数值方法求解动力方程。     

求解载流薄壳二维磁弹性的差分法

本文针对在电磁场和机械场耦合作用下的载流薄壳的非线性变形问题进行了研究。给出了载流薄壳在耦合场作用下的二维电动力学方程、磁弹性非线性运动方程和Lorentz力表达式,通过变量代换将描述载流薄壳的磁弹性状态方程整理成含有10个基本未知函数的标准Cauchy型。并通过差分法及准线性化方法,将标准Cauchy型非线性偏微分方程组,变换成为能够用正交离散法编程求解的准线性微分方程组,实现了载流薄壳的磁弹性应力与变形的数值解。     

一类非线性三阶边值问题的可解性

本文利用上下解方法和微分不等式技巧研究了一类非线性三阶边值问题的可解性,在非线性项满足某些局部性条件时得到了一些存在性结论。     

Stochastic linearization method with random parameters for SDOF nonlinear dynamical systems: prediction and identification procedures

This paper describes firstly, the calculation of the Power Spectral Density Function (PSDF) for the stationary response of SDOF nonlinear second-order dynamical systems excited by a white or a broad-band Gaussian noise, and secondly, the identification of a single-degree-of-freedom (SDOF) nonlinear dynamical second-order dynamical system driven by a broad-band or a colored Gaussian noise. The two aspects are based on the use of a stochastic linearization method with random parameters which is an efficient way of approximating the PSDF. The gain obtained by this method is shown on a SDOF nonlinear dynamical system. In addition, it is shown that the stochastic linearization method with random parameters is an efficient approach for identifying a SDOF nonlinear dynamical system.