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.     

A new approximation for the three-point probability function

Statistical continuum theory based approaches are commonly used for the computation of the effective properties of heterogeneous materials. Statistical distribution and morphology of the microstructure are represented by n-point probability function. One-point probability function statistical representation of the microstructure leads to volume fraction dependent homogenization. However, second and higher order probability functions include the information of phase distribution and morphology. Most statistical based homogenization methods are limited to two-point probability function due to the lack of simple approximation of higher order probability functions that can be easily exploited. In this paper, a new approximation of the three-point probability function is proposed and discussed. The new approximation results are compared to existing approximations from the literature and to the real probability functions calculated from a computer generated two-phase micrographs.     

Nonuniform reinjection probability density function in type V intermittency

Nonlinear Dynamics - In this paper, type V intermittency is studied using the M function methodology developed in the last years. This methodology is applied on two different maps to evaluate the...     

Variations in steepness of the probability density function of beam random vibration

Dynamic behaviour of a beam, subjected to stationary random excitation, has been investigated for the situation in which the response is different from the model of a Gaussian random process. The study was restricted to the case of symmetric non-Gaussian probability density functions of beam vibrations. There are two possible causes of deviations of the system response from the Gaussian model: the first, nonlinear behaviour, concerns the system itself and the second is external when the excitation is not Gaussian. Both cases have been considered in the paper. To clarity the conclusions for each case and to avoid interference of these different types of system behaviour, two beam structures, clamped-clamped and cantilevered, have been studied. A numerical procedure for prediction of the nonlinear random response of a clamped-clamped beam under the Gaussian excitations was based on a linear modal expansion. Monte Carlo simulation was undertaken using Runge–Kutta integration of the generalised coordinate equations. Probability density functions of the beam response were analysed and approximated making use of different theoretical models. An experimental study has been carried out for a linear system of a cantilevered beam with a point mass at the free end. A pseudo-random driving signal was generated digitally in the form of a Fourier expansion and fed to a shaker input. To generate a non-Gaussian excitation a special procedure of harmonic phase adjustment was implemented instead of the random choice. In so doing, the non-Gaussian kurtosis parameter of the beam response was controlled.     

非线性随机结构动力可靠度的密度演化方法

建议了一类新的非线性随机结构动力可靠度分析方法。基于非线性随机结构反应分析的概率密度演化方法,根据首次超越破坏准则对概率密度演化方程施加相应的边界条件,求解带有初、边值条件的概率密度演化方程,可以给出非线性随机结构的动力可靠度。研究了数值计算技术,建议了具有自适应功能的TVD差分格式。以具有双线型恢复力性质的8层框架结构为例进行了地震作用下的动力可靠度分析,与随机模拟结果的比较表明,所建议的方法具有较高的精度和效率。     

随机结构非线性动力响应的概率密度演化分析

提出了随机结构非线性动力响应分析的概率密度演化方法．根据结构动力响应的随机状态方程，利用概率守恒原理，建立了随机结构非线性动力响应的概率密度演化方程．结合Newmark-Beta时程积分方法与Lax-Wendroff差分格式，提出了概率密度演化方程的数值分析方法．通过与Monte Carlo分析方法对比，表明所给出的概率密度演化方法具有良好的计算精度和较小的计算工作量．研究表明：随机结构非线性动力响应概率密度具有典型的演化特征，随着时间增长，概率密度曲线分布趋于复杂．     

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.     

基于概率密度演化方法的随机结构可靠度分析

随机结构反应的概率密度演化方法能够给出随机荷载作用下随机结构反应的概率密度函数。在此基础上,根据给定的正常使用位移限值要求,直接进行积分给出了随机结构的正常使用可靠度及其失效概率。在实例分析中,与一类情况下的精确解答及基于反应正态分布假定的二阶矩方法分析结果进行了比较。研究表明：基于密度演化方法的随机结构可靠度分析具有很高的精度,而二阶矩方法的可靠度分析结果则往往具有一定的偏差,在失效概率较低时可能给出虚假的失效概率。     

结构动力非线性随机反应的联合概率分布

在密度演化理论基本思想的框架下,对广义密度 演化方程进行推广,导出了结构不同反应量的联合概率密度函数演化方程. 结合确定性结构 非线性动力反应分析与二维偏微分方程求解的有限差分方法,可以获取结构不同反应量的联 合概率密度函数的数值解答. 分析实例表明：结构反应的联合概率密度函数呈丘陵状不规则 分布,而不同反应量之间的相关系数是时变的.     

基于概率密度演化方法的随机MSCSS构造研究

附加柱是巨子型有控结构体系（Mega-Sub Controlled Structure System,即MSCSS）中的关键因素,为了确定MSCSS的细部构造,本文对附加柱的连接方式进行了研究。针对四种附加柱的构造方案,利用切球选点方法,确定了91个不同材料参数及地震强度的随机工况,在此基础上进行了随机巨子型有控结构体系在随机激励下的动力特性分析;基于概率密度演化理论,对各种附加柱构造情况下MSCSS的位移、加速度、附加柱应力的均值和标准差进行对比,确定了最佳的附加柱构造方案。结果表明,释放MSCSS附加柱顶端的水平约束,可以有效地降低MSCSS的加速度、位移、附加柱应力响应,使MSCSS抗震性能的优越性得到充分的发挥。本文提出的MSCSS基本构造形式可为MSCSS的进一步深入研究提供依据。     

Improved probability hypothesis density filter for multitarget tracking

Multitarget tracking (MTT) is important in radar and/or sonar surveillance systems. However, as a popular implementation of MTT, the standard probability hypothesis density (PHD) filter has computerized intractability and imprecise estimation. To overcome these drawbacks, an improved PHD filter is proposed in this paper. First, we apply particle approximation to achieve the closed-form solution to the PHD propagations. Afterward, the weight formula is simplified by the method of kernel estimate. Meanwhile, the excess weights of newborn particles are increased to survival particles on average. As a result, the overestimated target number and the missed detection are balanced, and the accuracy of MMT is improved by our lemmas. Finally, simulations are presented to compare the performance of the proposed filter with that of the standard one. The results show that our filter can achieve MTT with better performance.     

Stationary response probability density of nonlinear random vibrating systems: a data-driven method

Nonlinear Dynamics - A data-driven method is established to derive the (approximately) analytical expression of the stationary response probability density of nonlinear random vibrating system,...     

一种求算结构可靠度指标的新方法

为解决结构功能函数在结构设计点附近非线性程度较高时,一次可靠度计算方法(如JC法)不收敛的问题,提出一种新的可靠度计算方法.该算法根据结构可靠指标的几何意义,先将迭代点靠近极限状态面,再通过在极限状态面上搜索下一迭代点,逐渐减小极限状态面上的点到标准正态空间中坐标原点的距离,从而达到收敛的目的.与其他方法相比,该方法不...     

基于概率密度演化理论的LRB隔震支座时变可靠度研究

铅芯橡胶隔震支座LRB （Lead Rubber Bearing）的等效水平刚度是隔震建筑物的重要力学指标,橡胶材料由于其复杂活跃的化学组分,在使用过程中极易发生老化,是LRB支座中随时间变化的活跃因素。考虑铅芯橡胶隔震支座中橡胶材料退化的时变性和随机性,引入时变变异系数。基于概率密度演化方法,得到LRB支座等效水平刚度的概率密度演化曲线和时变可靠度。结果表明,随着时间的增长,支座等效水平刚度产生增大的趋势,支座的可靠度发生非线性降低,老化对支座等效水平刚度的离散性产生了明显的影响。     

Numerical investigation of a response probability density function of stochastic vibroimpact systems with inelastic impacts

The paper is devoted to numerical calculations of a response probability density function of stochastic vibroimpact systems excited by additive Gaussian white noise. An impact in the considered systems is modeled as a classical, inelastic one against motionless barrier(s) with values of a restitution coefficient both very small and close to unity. Certain empirical formulas are derived based on the results of numerical simulation.     

A highly accurate GPU Lattice Boltzmann method with directional interpolation for the probability distribution functions

In this article, a highly accurate and graphics processing unit (GPU)-accelerated Lattice Boltzmann Method (LBM) is presented. The methodology is derived from a combination of conventional and recent LBM algorithms, mainly focusing on reducing the computational time, memory allocation, and complexity of existing algorithms. The general implementation focuses on accelerating the overall methodology using GPGPU technology based on Compute Shaders from OpenGL and avoids the storage of the distribution function components to reduce the memory allocation size. Furthermore, an efficient spatial interpolation of the probability distribution function components is described, based on a directional interpolation, without unnecessary control points for the reconstruction of virtual nodes data. The present methodology, tested for spatial accuracy via two- and three-dimensional Lid-Driven Cavity benchmark cases, shows excellent agreement with the results reported in the literature. Additionally, time efficiency is analyzed by comparing different configurations for the construction of virtual streaming points.