非对称非线性随机系统的等价线性化方法及其应用

本文提出了非对称非线性系统在随机干扰下的等价线性化方法以及系统随机反应统计矩的迭代求解方法；在此基础上，利用文献（３）中建立的有损伤结构非对称滞变恢复力模型，研究了结构在主余震作用下的随机反应，最后通过算例比较，验证了本文所提方法的合理性。     

机床非线性颤振的分段线性化及其数学模型

通过引入非线性调制环节N(S),本文实现了非线性颤振的分段线性化,建立了统一的切削动力学系统模型。这一模型使线性颤振理论、非线性颤振理论趋于统一,并使超声振动切削抑制颤振的机理从理论上得到解释。     

基于等效线性化法的弱非线性体系下的地震响应分析

对弱非线性单自由度非自由振动系统结构在白噪声地震激励下的随机响应问题进行了研究。首先,建立了结构的弱非线性响应方程;然后根据能量平衡原理对运动方程进行了等效线性化;最后获得了金井清谱(Kanai-Tajimi谱)下等效线性化方程的位移响应及速度响应的方差,将弱非线性随机响应问题简化为求解线性随机响应问题,并给出了算例验证。结果表明,该方法可以准确得到弱非线性随机响应问题的解。同时,建立了单自由度结构弱非线性随机响应频域分析的统一解析解法。     

一般强非线性振动微分方程式的线性化和逐步积分法

介绍了对一般强非线性振动微分方程进行线性化并采用相应的Wilson-法进行逐步积分计算的数值计算方法。大量应用实践表明该方法计算结果正确、可靠、计算速度快,是一种非常有效的计算分析的手段。可以应用于一般非线性动力学方程的求解。     

非线性泊松问题的拟线性化边界积分方法研究

非线性泊松问题在热传导和多孔催化粒子的扩散反应等问题中是非常常见的,为此,利用广义拟线性化迭代理论,提出了一种非线性泊松问题的新的数值迭代方法.该方法将非线性方程转化成一序列线性方程的迭代,其优点是初始值的选取具有一定的理论基础,并且在一定的初始值条件下,迭代结果将单调地收敛于非线性问题的解.将此迭代方法与边界元和双互易杂交边界点方法结合,并用于非线性泊松问题的求解,比较了两种方法的结果精度,收敛速度及不同初始值下的稳定性.结果显示,基于拟线性化的双互易杂交边界点法具有较高的稳定性和计算效率,并且收敛速度为平方阶.     

三阶非线性发展方程的线性化有理逼近方法

研究了三阶非线性发展方程的初边值问题的解。采用基于Sinc函数的微分求积法发展了线性化有理逼近方法。通常的配点法不适用于上述三阶问题的求解。本文把提出的方法用于求解KdV方程,取得了良好的效果。     

多自由度柔性结构非线性随机振动响应分析方法

随机等效线性化方法是多自由度结构非线性随机振动分析的有效方法,然而柔性结构隐式的系统方程限制了该方法的应用.本文首先提出了把多自由度柔性结构隐式系统方程显式化的等效非线性系统建立方法,该方法结合模态分析,将系统几何非线性作用等效为模态坐标的高次多项式组合,把多自由度物理系统转化为容易求解的模态系统;在此基础上运用随机等效线性化技术建立柔性结构非线性随机振动响应分析方法.该方法通过引入虚拟激励原理,大幅提高了计算效率,使对多自由柔性结构的非线性随机振动响应分析成为可能.通过算例分析,本文分析结果和精确解与数值模拟解的比较结果表明,本文方法具有较好的计算精度和较高的计算效率,能够应用于实际柔性工程结构的随机振动响应分析.     

一种改进的等效线性化方法

提出了一种改进的等效线性化方法.将现有方法中忽略的高阶谐波项作为等效线性方程的外激励,得到非齐次的等效线性化方程,利用谐波平衡法将该方程分解为一系列常微分方程组,用摄动法求解.算例表明,本文方法不仅提高了等效线性化方法的精度,而且对现有方法不能处理的含偶次非线性系统的分析同样有效.     

含空洞非线性材料的本构势和空洞扩展率

本文基于体胞模型的解析分析,分析了含空洞非线性材料的宏观本构势,得到了各种幂硬化指数下宏观应力和基体平均流变应力之间的相关曲线.当基体是遵循经典塑性全量理论时,这些曲线方程就是一簇依赖于空洞体积比和硬化指数的屈服面方程.当基体是粘性体时,这些方程就是粘性约束方程.通过曲线拟合的方法,本文发展了修正的Gurson方程,使之适合于不同硬化指数的情况.最后本文计算了粘性体中空洞的相对扩展率,结果与已有体胞模型的数值模拟计算结果相当一致.     

一类强非线性系统周期解的能量迭代法

应用能量原理，引入牛顿线性化思想进行迭代，求强非线性Ｌｉｅｎａｒｄ方程的周期解。例子说明本方法行之有效且精度高。近似解析解表达式可借助计算机推出     

Stationary and non-stationary probability density function for non-linear oscillators

A method for the evaluation of the stationary and non-stationary probability density function of non-linear oscillators subjected to random input is presented. The method requires the approximation of the probability density function of the response in terms of C-type Gram-Charlier series expansion. By applying the weighted residual method, the Fokker-Planck equation is reduced to a system of non-linear first order ordinary differential equations, where the unknowns are the coefficients of the series expansion. Furthermore, the relationships between the A-type and C-type Gram-Charlier series coefficient are derived.     

激光陀螺惯性测量单元系统级标定方法

传统的分立标定方法必须依靠高精度的转台提供姿态基准,不满足带减振器的惯性测量单元(IMU)和现场标定需求.首先建立了附加约束条件的陀螺和加速度计安装坐标系数学模型,根据陀螺和加速度计的输出误差方程,从惯性导航基本误差方程出发推导了惯性测量单元的系统级误差参数标定Kalman滤波模型,该模型包含了陀螺和加速度计零偏、比例因子、安装误差在内共21维标定误差状态变量,且仅以速度解算误差为观测量.依据所建立的模型和设计的标定路径对此系统级标定方法进行了仿真,仿真结果表明,陀螺和加速度计零偏估计精度分别优于0.005°/h和0.005 mg,安装误差估计精度优于1″,比例因子误差优于1ppm,满足高精度惯导系统的标定需求.     

激光陀螺捷联惯导系统多位置标定方法

在建立惯性仪表简化误差模型的基础上,提出了一种多位置标定方法.该方法充分考虑标定条件、设备以及时间等因素,设计了一种多位置连续转动标定方案,充分激励惯性仪表各项误差参数,从而建立起所有误差参数与系统导航误差之间的关系,通过测量每个位置静态导航状态下的速度误差,采用最小二乘估计,全面辨识出所有21个误差参数.理论分析和实验结果表明,与传统标定方法相比,该方法对标定设备要求低,无需北向基准,实现简单方便,在较短的时间内就可以一次标定出惯性仪表所有21个误差参数,标定精度与基于精密转台的标定精度相当,具有较强的工程实用性.     

基于匀速率26位置法的iIMU-FSAS光纤陀螺仪标定

针对iIMU-FSAS光纤陀螺仪静态26位置法标定结果无法收敛的问题,提出了由外框轴提供稳定转速的匀速率26位置法标定方案,从理论上与静态26位置法进行了对比分析;采用该方法对i IMU-FSAS陀螺仪进行了多天的标定实验,结合静态26位置法有限次迭代的结果,对匀速率26位置法标定参数(零偏、标度因数和安装误差)的合理性和重复性等进行了分析。结果表明:与静态26位置法相比,匀速率26位置法的标定结果不仅能够收敛,且标定结果合理、性能稳定。     

Geometrically non-linear generalized hybrid element and refined element method of non-conforming modes

A generalized hybrid method of non-conforming modes based on a non-linear generalized variational principles with relaxed interelement continuity requirements, is developed, and the plane quadrilateral geometrically non-linear element is presented, furthermore, non-linear refined element method is devised by orthogonal approach. It is shown that the refined element can improve the computational accuracy for non-conforming modes. The project supported by the National Natural Science Foundation of China     

THE METHOD OF DERIVATION OF MAC－MILLAN＇S EQUATION FOR THE NON－LINEAR NON－HOLONOMIC SYSTEM

ＴＨＥＭＥＴＨＯＤＯＦＤＥＲＩＶＡＴＩＯＮＯＦＭＡＣ－ＭＩＬＬＡＮ＇ＳＥＱＵＡＴＩＯＮＦＯＲＴＨＥＮＯＮ－ＬＩＮＥＡＲＮＯＮ－ＨＯＬＯＮＯＭＩＣＳＹＳＴＥＭ￥ＱｉｕＲｏｎｇ（邱荣）（ＤｅｐａｒｔｍｅｎｔｏｆＰｈｙｓｉｃｓ，ＦｕｚｈｏｕＵｎｉｖｅｒｓｉ...     

A transport approach to the convolution method for numerical modelling of linearized 3D circulation in shallow seas

A new method for solving the linearized equations of motion is presented in this paper, which is the implementation of an outstanding idea suggested by Welander: a transport approach to the convolution method. The present work focuses on the case of constant eddy viscosity and constant density but can be easily extended to the case of arbitrary but time-invariant eddy viscosity or density structure. As two of the three equations of motion are solved analytically and the main numerical ‘do-loop’ only updates the sea level and the transport, the method features succinctness and fast convergence. The method is tested in Heaps' basin and the results are compared with Heaps' results for the transient state and with analytical solutions for the steady state. The comparison yields satisfactory agreement. The computational advantage of the method compared with Heaps' spectral method and Jelesnianski's bottom stress method is analysed and illustrated with examples. Attention is also paid to the recent efforts made in the spectral method to accelerate the convergence of the velocity profile. This study suggests that an efficient way to accelerate the convergence is to extract both the windinduced surface Ekman spiral and the pressure-induced bottom Ekman spiral as a prespecified part of the profile. The present work also provides a direct way to find the eigenfunctions for arbitrary eddy viscosity profile. In addition, mode-trucated errors are analysed and tabulated as functions of mode number and the ratio of the Ekman depth to the water depth, which allows a determination of a proper mode number given an error tolerance.     

Response probability density functions of strongly non-linear systems by the path integration method

The paper discusses challenges in numerical analysis and numerical/analytical results for strongly non-linear systems—systems with “signum”-type non-linearities. Such non-linearities are implemented for instantaneous variations of the systems’ parameters, to reduce their mean energy response when subjected to random excitations. Numerical results for displacement and velocity response probability density functions (PDFs), energy response PDFs and various order moments are obtained by the path integration technique. Attention is also given to evaluation of mean upcrossing rate, related to the system's half period, via Rice's formula informally applied to discontinuous response PDFs.     

计算圆板大振幅非线性振动频率的平均刚度法

本文用平均刚度法研究圆板大振幅非线性振动的频率问题，导出了相应的非线性广义特征值方程，构造了一种避免发散并能加速收敛的加权平均迭代法，计算结果与Ｋａｎｔｏｒｏｖｉｃｈ时间平均法的解十分吻合。     

EPC procedure for PDF solution of non-linear oscillators excited by Poisson white noise

The stationary probability density function (PDF) solution of the responses of non-linear stochastic oscillators subjected to Poisson pulses is analyzed. The PDF solutions are obtained by the exponential-polynomial closure (EPC) method. To assess the effectiveness of the solution procedure numerically, non-linear oscillators are analyzed with different impulse arrival rates, degree of oscillator non-linearity and excitation intensity. Numerical results show that the PDFs obtained with the EPC method yield good agreement with those obtained from Monte Carlo simulation when the polynomial order is 4 or 6. It is also observed that the EPC procedure is the same as the equivalent linearization procedure under Gaussian white noise in the case of the polynomial order being 2.