Wishart Random Matrices in Probabilistic Structural Mechanics

Uncertainties need to be taken into account for credible predictions of the dynamic response of complex structural systems in the high and medium frequency ranges of vibration. Such uncertainties should include uncertainties in the system parameters and those arising due to the modeling of a complex system. For most practical systems, the detailed and complete information regarding these two types of uncertainties is not available. In this paper, the Wishart random matrix model is proposed to quantify the total uncertainty in the mass, stiffness, and damping matrices when such detailed information regarding uncertainty is unavailable. Using two approaches, namely, (a) the maximum entropy approach; and (b) a matrix factorization approach, it is shown that the Wishart random matrix model is the simplest possible random matrix model for uncertainty quantification in discrete linear dynamical systems. Four possible approaches for identifying the parameters of the Wishart distribution are proposed and compared. It is shown that out of the four parameter choices, the best approach is when the mean of the inverse of the random matrices is same as the inverse of the mean of the corresponding matrix. A simple simulation algorithm is developed to implement the Wishart random matrix model in conjunction with the conventional finite-element method. The method is applied vibration of a cantilever plate with two different types of uncertainties across the frequency range. Statistics of dynamic responses obtained using the suggested Wishart random matrix model agree well with the results obtained from the direct Monte Carlo simulation.     

Probabilistic Foundation Settlement on Spatially Random Soil

By modeling soils as spatially random media, estimates of the reliability of foundations against serviceability limit state failure, in the form of excessive differential settlements, can be made. The soil’s property of primary interest is its elastic modulus, E, which is represented here using a lognormal distribution and an isotropic correlation structure. Prediction of settlement below a foundation is then obtained using the finite element method. By generating and analyzing multiple realizations, the statistics and density functions of total and differential settlements are estimated. In this paper probabilistic measures of total settlement under a single spread footing and of differential settlement under a pair of spread footings using a two-dimensional model combined with Monte Carlo simulations are presented. For the cases considered, total settlement is found to be represented well by a lognormal distribution. Probabilities associated with differential settlement are conservatively estimated through the use of a normal distribution with parameters derived from the statistics of local averages of the elastic modulus field under each footing.     

中板轧机辊缝异常分析及解决办法

1　前言南钢中板厂 2 5 0 0 mm四辊可逆式中厚板轧机采用电动压下和液压压下的混合压下系统。压下系统的总速比 i=1 6.7,电动压下速度为0 .2 0 m /s。液压压下系统由“一重”设计和制造 ,其液压缸行程为 2 5 mm ,在压下过程中只对两边辊缝做微量的精确调整 ,轧制过程中辊缝主要由电动压下来实现。在 1 996年前该压下系统未出现过异常。此后一段时间内该系统在轧钢过程中时常会出现辊缝值发生变化这种不正常现象 ,使得压下工不能很好地控制厚度 ,造成“计划外”,甚至废品。一般情况下 ,在“咬钢”开始一段时间 (0～ 1 s内 ) ,两侧辊缝会增大 …     

Probabilistic Settlement Analysis by Stochastic and Random Finite-Element Methods

The paper discusses finite element models for predicting the elastic settlement of a strip footing on a variable soil. The paper then goes on to compare results obtained in a probabilistic settlement analysis using a stochastic finite element method based on first order second moment approximations, with the random finite element method based on generation of random fields combined with Monte Carlo simulations. The paper highlights the deficiencies of probabilistic methods that are unable to properly account for spatial correlation.     

环件轧制中导向辊运动的研究

以环件轧制过程中环件外径瞬时变化规律为依据,根据环轧过程中体积不变条件及芯辊、导向辊和驱动辊三者圆心的位置关系,推导出导向辊运动的速度和角速度的计算公式;并在Deform-3D软件中模拟导向辊的定位,对比分析了有无导向辊作用下环件的变形过程,同时在模拟中以一种简单有效的方法实现导向辊的定心、归圆作用。模拟结果表明,利用此方法对导向辊进行定位控制,可以使轧制过程稳定。     

Viscous Effect on the Roll Motion of a Rectangular Structure

The viscous effect on the roll motion of a rectangular structure was investigated in a two-dimensional wave tank. The structure was used to simulate a simplified barge in the beam sea condition. The structure with a draft one-half of its height was hinged at the center of gravity and free to roll (1 degree of freedom) by waves. The dynamic characteristics of the structure were first identified, including its roll natural period. The dynamic response of the barge-like structure under wave actions was then tested with regular waves with a range of wave periods that are shorter, equal to, and longer than its roll natural period. Particle image velocimetry was used to obtain the velocity field in the vicinity of the structure. The coupled interactions between the incident waves and the structure were demonstrated by examining the vortical flow fields to elucidate the effect of viscous damping (also called the eddy making damping) to the roll motion of the structure over wave periods. For incoming waves with a wave period same as the roll natural period, the structure roll motion was, as expected, greatly reduced by the viscous-damping effect. At wave periods shorter than the roll natural period, the structure roll motion was slightly reduced by the viscous effect. However, at wave periods longer than the roll natural period, the viscous effect due to flow separation at structure corners indeed amplified the roll motion. This indicates that not only can the viscous effect damp out the roll motion, it can also amplify the roll motion.     

Random Uncertainties Model in Dynamic Substructuring Using a Nonparametric Probabilistic Model

This paper presents a new approach, called a nonparametric approach, for constructing a model of random uncertainties in dynamic substructuring in order to predict the matrix-valued frequency response functions of complex structures. Such an approach allows nonhomogeneous uncertainties to be modeled with the nonparametric approach. The Craig–Bampton dynamic substructuring method is used. For each substructure, a nonparametric model of random uncertainties is introduced. This nonparametric model does not require identifying uncertain parameters in the reduced matrix model of each substructure as is usually done for the parametric approach. This nonparametric model of random uncertainties is based on the use of a probability model for symmetric positive-definite real random matrices using the entropy optimization principle. The theory and a numerical example are presented in the context of the finite-element method. The numerical results obtained show the efficiency of the model proposed.     

Consistent Method for PDF Solutions of Random Oscillators

The probability density function (PDF) of the responses of random oscillators with both external and parametric excitations is formulated as an exponential function of polynomials in state variables. The PDF is assumed to be governed by the Fokker-Planck-Kolmogorov equation. Special measure is taken to satisfy the Fokker-Planck-Kolmogorov equation in the average sense of integration with the assumed function, and quadratic algebraic equations are obtained for determining the unknown PDF. Examples are given to show the application of the method to the random oscillators with both external and parametric excitations. The PDFs obtained with the proposed method are compared with the obtainable exact ones. Numerical results showed that the PDF solutions obtained with the proposed method approach the exact solutions.     

Response Spectrum Analysis of Suspension Bridges for Random Ground Motion

The response spectrum method of analysis for suspension bridges subjected to multicomponent, partially correlated stationary ground motion is presented. The analysis is based on the relationship between the power spectral density function and the response spectrum of the input ground motion and fundamentals of the frequency domain spectral analysis. The analysis duly takes into account the spatial correlation of ground motions between the supports, the quasi-static component of the response, and the modal correlation between different modes of vibration. A suspension bridge is analyzed under a set of important parametric variations in order to (1) compare between the responses obtained by the response spectrum method of analysis and the frequency domain spectral analysis; and (2) investigate the behavior of suspension bridges under seismic excitation. The parameters include the spatial correlation of ground motion, the angle of incidence of the earthquake, the ratio between the three components of ground motion, the number and nature of modes considered in the analysis, and the nature of the power spectral density function of ground motion. It is shown that the response spectrum method of analysis provides a fair estimate of responses under parametric variations considered in the study.     

双边剪夹送辊在实际生产中的运动分析

文章从理论上分析了夹送辊夹持钢板时的运动情况,结合夹送辊运输钢板的状态解释了几种常见钢板跑偏现象,为夹送辊的跑偏调整提供了一些理论依据。     

Solutions to Boundary Value Problem of Nonlinear Impulsive Differential Equation of Fractional Order

This paper is devoted to study the existence and uniqueness of solutions to a boundary value problem of nonlinear fractional differential equation with impulsive effects. The arguments are based upon Schauder and Banach fixed-point theorems.We improve and generalize the results presented in [B. Ahmad, S. Sivasundaram,Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Analysis: Hybrid Systems, 3(2009), 251-258].     

New Variational Solutions for the Rolling of Flanged Sections in Universal Roll Passes

Metallurgist - The variational principle of minimum total power and modern computational means are used to describe the deformation of metal in and to determine the power/force conditions for...     

Nonlinear Response of Double-Wall Cylindrical Shell Vibrations under Random Excitation

An analytical model is presented to predict the nonlinear response of a double-wall sandwich cylindrical shell system subjected to random excitation. Nonlinear spring-dashpot models are integrated into the system to characterize the behavior of the soft core. Donnell’s thin shell theory is used to develop the governing nonlinear equations of motion. A Monte Carlo simulation of stationary random processes, multimode Galerkin-type approach, and numerical integration procedures are used to develop linear and nonlinear response solutions of simply supported cylindrical shells. Numerical results include time domain response histories, root-mean-square values and response spectral densities. Parametric studies are performed to investigate the effects of nonlinearity, shell thickness, core stiffness, and thickness.     

关于一类非线性微分方程非振动解的渐近性

本文研究了二阶非线性微分方程(a(t)φ(y'(t)))'+f(t,y(t))=0,t≥t0的非振动解的渐近性.     

Linearized Stability and Accuracy for Step-by-Step Solutions of Certain Nonlinear Systems

In this work, stability and accuracy of the Newmark method for nonlinear systems are obtained from a linearized analysis. This analysis reveals that an unconditionally stable integration method for linear elastic systems is unconditionally stable for nonlinear systems and a conditionally stable integration method for linear elastic systems remains conditionally stable for nonlinear systems except that its upper stability limit might vary with the step degree of nonlinearity and step degree of convergence. A sufficient condition to have a stable computation for nonlinear systems in a whole step-by-step integration procedure is also developed in this study. Furthermore, it is also found that numerical accuracy in the solution of nonlinear systems is closely related to the step degree of nonlinearity and step degree of convergence although its characteristics are similar to those of the preceding works for linear elastic systems. Since these results are obtained from a linearized analysis, they can be applicable to the nonlinear systems that satisfied the simplifications for the analysis but may not be applicable to general nonlinear systems.     

Equivalent Linearization for the Nonstationary Response Analysis of Nonlinear Systems with Random Parameters

This paper presents an approach for analyzing nonlinear systems with parameter uncertainty subjected to stochastic excitation. The uncertain parameters are modeled as time-independent random variables. A general solution procedure based on equivalent linearization is presented. The set of orthogonal polynomials associated with the probability density function is used as the solution basis for the response moments. In addition, the instantaneous equivalent stiffness and damping matrices are approximated as quadratic random functions. The resulting Liapunov system with explicit random coefficients can then be converted into a deterministic system using the method of weighted residuals. Applications to single-degree-of-freedom uncertain systems are given and the accuracy of the results is validated.     

非线性转子动力学中的分叉与混沌运动研究概述

介绍了非线性动力学的分叉与混沌问题研究的基本方法 ,分析了非线性动力学中存在的主要问题 ,对非线性转子动力学中的分叉与混沌运动研究的国内外概况进行了较全面的综述     

Bayesian Probabilistic Approach to Structural Health Monitoring

A Bayesian probabilistic methodology for structural health monitoring is presented. The method uses a sequence of identified modal parameter data sets to compute the probability that continually updated model stiffness parameters are less than a specified fraction of the corresponding initial model stiffness parameters. In this approach, a high likelihood of reduction in model stiffness at a location is taken as a proxy for damage at the corresponding structural location. The concept extends the idea of using as indicators of damage the changes in structural model parameters that are identified from modal parameter data sets when the structure is initially in an undamaged state and then later in a possibly damaged state. The extension is needed, since effects such as variation in the identified modal parameters in the absence of damage, as well as unavoidable model error, lead to uncertainties in the updated model parameters that in practice obscure health assessment. The method is illustrated by simulating on-line monitoring, wherein specified modal parameters are identified on a regular basis and the probability of damage for each substructure is continually updated.