Random Factor Method Research Articles

The dynamic analysis of stochastic thin-walled structures under thermal–structural–acoustic coupling

Random dynamic analysis of the thin-walled structure subjected to coupling loads from three fields is addressed in the frame of the finite element method. Based on the proposed dynamic finite element equation of the deterministic structure under thermal–structural–acoustic coupling, when the randomness of structural physical parameters, temperature load and fatigue test data is fully considered, the dynamic responses of random structure subjected to coupling loads from three fields are dealt with by random factor method. The numerical characteristic values of the random temperature field and that of random dynamic responses are then derived through the moment method of random variables. Subsequently, the dynamic reliability of the stochastic structure under three-field coupling is evaluated using residual strength model, dynamic stress-intensity interference theory and the cumulative damage equal principle. Finally, the results from the methodology proposed are compared with that from Monte-carlo method for a case of numerical example, along with an inspection of the impacts of random variables on dynamic analysis results.

Homogenized thermal properties of 3D composites with full uncertainty in the microstructure

In this work, random homogenization analysis for the effective thermal properties of a three-dimensional composite material with unidirectional fibers is presented by combining the equivalent inclusion method with Random Factor Method (RFM). The randomness of the micro-structural morphology and constituent material properties as well as the correlation among these random parameters are completely accounted for, and stochastic effective thermal properties as thermal expansion coefficients as well as their correlation are then sought. Results from the RFM and the Monte-Carlo Method (MCM) are compared. The impact of randomness and correlation of the micro-structural parameters on the random homogenized results is revealed by two methods simultaneously, and some important conclusions are obtained.

Integrated Structural Electromagnetic Analysis of Mesh Reflectors with Structural Random Dimensional Errors

A new procedure for the integrated structural electromagnetic analysis of cable mesh reflectors with structural random dimensional errors is presented. The procedure combines the integrated structural electromagnetic analysis with reliability analysis and aims to investigate the effects of structural random dimensional errors on the antenna far-field directivity pattern. Both the antenna far-field directivity pattern and structural member dimensions are taken as random variables in this procedure. The computational expressions of the mean value and variance of the far-field directivity are developed by means of the random factor method and the random variable’s functional moment method. The influences of structural random dimensional errors on the randomness of radiation directivity pattern are inspected via an offset cable mesh reflector.

Stochastic dynamic characteristics of FGM beams with random material properties

The random factor method is developed for the stochastic dynamic characteristics analysis of beams that are made of functionally graded materials (FGM) with random constituent material properties in this paper. The effective material properties of FGM beams are assumed to vary continuously through different directions according to the power law distribution. In the proposed method, the randomness of the dynamic characteristics is explicitly expressed by random factors of constituent material parameters, and the statistics (means and variances) of modal parameters can be analytically computed by the statistics of random inputs with little computational efforts. Compared with the first-order perturbation technique and the Monte-Carlo simulation method, the proposed method is illustrated and verified by a FGM cantilever beam with material variation through the thickness or axial direction, and the contributions of random material properties to the dispersion of dynamic characteristics for the structure are systematically investigated. The numerical results also show the dominant effect of constituent volume fraction index on the dispersion of dynamic characteristics.

Probability-based damage detection using model updating with efficient uncertainty propagation

Model updating method has received increasing attention in damage detection of structures based on measured modal parameters. In this article, a probability-based damage detection procedure is presented, in which the random factor method for non-homogeneous random field is developed and used as the forward propagation to analytically evaluate covariance matrices in each iteration step of stochastic model updating. An improved optimization algorithm is introduced to guarantee the convergence and reduce the computational effort, in which the design variables are restricted in search region by region truncation of each iteration step. The developed algorithm is illustrated by a simulated 25-bar planar truss structure and the results have been compared and verified with those obtained from Monte Carlo simulation. In order to assess the influences of uncertainty sources on the results of model updating and damage detection of structures, a comparative study is also given under different cases of uncertainties, that is, structural uncertainty only, measurement uncertainty only and combination of the two. The simulation results show the proposed method can perform well in stochastic model updating and probability-based damage detection of structures with less computational effort.

Research on Shock Mechanism for Flying Gangue in Steeply Dipping Seam Mining

Aim at the random problems of flying gangue movement in steeply dipping seam mining, based on the spherical flying gangue movement model, computational formula of collision restitution coefficient was established when flying gangue shocks bottom plate of working face according to the theory of Hertz classical elastic collision; Considering the influences of the randomness such as the density, radius size, impact velocity of flying gangue, the elastic modulus, Poisson’s ratio of contact bottom plate of working face, the motion model of flying gangue with stochastic parameter was constructed by applying the random factor method; and the mean value and the variance of the collision restitution coefficient were derived by using the algebra synthesis method and moments method. The numerical examples show that the randomness of parameters has a rather large influence on the collision restitution coefficient of flying gangue in steeply dipping seam mining.

Frequency Reliability Sensitivity Analysis for Rotor System with Random Parameters

The random factor method (RFM), an effective numerical simulation method, is employed to analyze the probabilistic characteristics and reliability sensitivity of uncertain structure system. The natural frequencies of the system can be expressed as the function of the random factors. By using the random variable’s algebra synthesis method, the expressions for the mean value and standard deviation of natural frequencies are derived from Rayleigh quotient. The reliability mode and the safety probability of random systems are defined, and then the frequency reliability sensitivity analysis method is presented. The formulas of failure probability sensitivity are derived when random variables are normally distributed. The reliability index is evaluated, and its sensitivity to mean values and variances are mathematically expressed, then the efficiency and accuracy of the proposed method are demonstrated by a numerical example.

Stochastic homogenized effective properties of three-dimensional composite material with full randomness and correlation in the microstructure

In this work, random homogenization analysis for the effective properties of composite materials with unidirectional fibers is addressed by combining the equivalent inclusion method with the Random Factor Method (RFM). The randomness of the micro-structural morphology and constituent material properties as well as the correlation among these random parameters are fully considered, and stochastic effective properties including effective elastic tensor and effective elastic properties together with their correlation are sought. Results from the RFM and the Monte-Carlo Method (MCM) are compared, and the impact of randomness and correlation of the micro-structural parameters on the random homogenized results are investigated by the two methods. Finally, the correlation coefficients of the effective properties are obtained by the MCM. The RFM is found to deliver rapid results with comparable accuracy to the MCM approach.

DYNAMIC ANALYSIS OF MULTIBODY SYSTEMS WITH PROBABILISTIC PARAMETERS

Dynamic analysis of multibody systems with probabilistic parameters was presented. Dynamic modeling of multibody systems was obtained by Lagrange's method. The probabilistic differential algebraic equations were transformed into pure probabilistic differential equations by generalized coordinate partitioning method. The Newmark step-by-step integration method was used to calculate the results. Using the method of random factor method, the numerical characteristics of the system response were derived, and the results were expressed in statistic view. As an illustrating example, dynamic modeling of a rotating bar and sliding block system considering the probabilistic of load, geometric and physical parameters was presented. Compared with the result of Monte-Carlo numerical simulation method, the accuracy and efficiency of the method are verified. The results illustrate that the probabilistic parameters affect the dynamic response of the multibody system and the dynamic modeling with probabilistic parameters can objectively reflect the dynamic behavior of the objective systems.

Dynamic Characteristics Analysis of Supercavitating Vehicle Structure with Stochastic Parameters

Stiffness and mass matrixes were given for frequency calculation of supercavitating vehicle structure (SVS) based on semi-analytical finite element method. Considering the randomicity of physical parameters and their relativity, the function of stochastic frequency and probability density function were obtained for SVS by random factor method. The randomicity of frequency for SVS with stochastic physical parameters was analyzed and the characteristics of probability distribution were given for front five order modes. The calculation results show that SVS is the most prone to bending mode, it’s extremely disadvantage for SVS stability sliding motion in supercavitation. In addition, the frequency have strong randomicity, the external incent frequency in a larger range may lead to load characteristics more bad for SVS.

Random homogenization analysis in linear elasticity based on analytical bounds and estimates

In this work, random homogenization analysis of heterogeneous materials is addressed in the context of elasticity, where the randomness and correlation of components’ properties are fully considered and random effective properties together with their correlation for the two-phase heterogeneous material are then sought. Based on the analytical results of homogenization in linear elasticity, when the randomness of bulk and shear moduli, the volume fraction of each constituent material and correlation among random variables are considered simultaneously, formulas of random mean values and mean square deviations of analytical bounds and estimates are derived from Random Factor Method. Results from the Random Factor Method and the Monte-Carlo Method are compared with each other through numerical examples, and impacts of randomness and correlation of random variables on the random homogenization results are inspected by two methods. Moreover, the correlation coefficients of random effective properties are obtained by the Monte-Carlo Method. The Random Factor Method is found to deliver rapid results with comparable accuracy to the Monte-Carlo approach.

A new method for random vibration analysis of stochastic truss structures

A new method called the random factor method (RFM) for the natural frequency, mode shape and random vibration analysis of stochastic truss structures is presented in this paper. Using the RFM, the structural physical parameters and geometry can be considered as random variables. The structural stiffness and mass matrices can then, respectively, be described by the product of two parts corresponding to the random factors and the deterministic matrix. The structural natural frequencies, mode shapes and random response can be expressed as the function of the random factors. By means of the random variable's functional moment method and the algebra synthesis method, computational expressions for the mean value, standard deviation and variation coefficient of the dynamic characteristics, mean square value of the stationary random displacement and stress response are developed. The influences of the randomness of the structural parameters on the dynamic characteristics, structural displacement and stress responses are demonstrated using a truss structure and Monte-Carlo simulations illustrate the efficiency of the proposed method.

Probabilistic and Interval Static Response Analysis of Truss Structures with Uncertain Parameters

This paper presents the random factor method (RFM) and interval factor method (IFM) for the static analysis of truss structures with uncertainty. From the static governing equations of finite element method, the computational expressions for the mean value and standard deviation of structural displacement and stress are developed by using the RFM. The computational expressions for the lower and upper bounds of structural static response are also derived by means of the interval operations when system parameters are interval variables. The effect of uncertainty of individual structural parameters and loads on structural static response is demonstrated by truss structures.

Optimal active random vibration control for smart truss structures based on reliability

The current paper presents the reliability-based optimization of the location and feedback gains of active bar in a closed-loop control system for random smart truss structures under stationary random excitation. The mathematical model with reliability constraints on the root mean square value of the structural random dynamic displacement and stress response is developed based on maximization of dissipation energy due to the control action. The randomness of the structural physical materials, geometry, and damping are included in the analysis, and the applied forces are considered as stationary random excitation. The numerical characteristic corresponding to mean values and standard deviations of the stationary random responses of random smart structures is developed by the random factor method. Numerical examples of truss structures are presented to demonstrate the rationality and validity of the active control model.

Dynamic response analysis of stochastic truss structures under non-stationary random excitation using the random factor method

The dynamic characteristics and responses of stochastic truss structures under non-stationary random excitation are investigated using a new method called the random factor method (RFM). Using the RFM, the structural physical parameters and geometry can be considered as random variables. Based on the assumption that the randomness on each element is the same for all elements, the structural stiffness and mass matrices can then respectively be described by the product of two parts corresponding to the random factor and the deterministic matrix. From the expressions of structural random response in the frequency domain, computational expressions for the mean value, standard deviation and variation coefficient of the mean square value of the non-stationary random displacement and stress response are developed by means of the random variable’s functional moment method and the algebra synthesis method. The influences of the randomness of the structural parameters on the dynamic characteristics, structural displacement and stress responses are demonstrated using a truss structure.

Random seismic response analysis of truss structures with uncertain parameters

This paper presents an approach called random factor method (RFM) for the random vibration analysis of seismic response of random truss structures. An antenna and a truss bridge with random material parameters and geometric dimensions are considered to illustrate the application of the RFM in the seismic response analysis of random structures under stationary or non-stationary random earthquake excitation. Using the RFM, the structural stiffness and mass matrices can respectively be described by the product of two parts corresponding to the random factors of structural parameters and the deterministic matrix. From the expressions of structural seismic random response in the frequency domain, computational expressions for the mean value and standard deviation of the mean square value of the seismic random displacement and stress response are developed by means of the random variable’s functional moment method and the algebra synthesis method. The influences of the randomness of the structural parameters on the structural seismic responses are investigated.

Natural frequency and mode shape analysis of structures with uncertainty

Two methods called random factor method (RFM) and interval factor method (IFM) for the natural frequency and mode shape analysis of truss structures with uncertain parameters are presented in this paper. Using the RFM, the structural physical parameters and geometry can be considered as random variables. The structural stiffness and mass matrices can then, respectively, be described by the product of two parts corresponding to the random factors and the deterministic matrix. The structural natural frequencies, mode shapes and random response can be expressed as the function of the random factors. By means of the random variable's algebra synthesis method, the computational expressions for the mean value and standard deviation of natural frequencies and mode shapes are derived from the Rayleigh quotient. Using the IFM, the structural parameters can be considered as interval variables and the computational expressions for the lower and upper bounds of the natural frequency and mode shape are derived by means of the interval operations. The effect of uncertainty of individual structural parameters on structural dynamic characteristics, and the comparison of structural natural frequency and mode shape using the RFM and IFM are demonstrated by truss structures.

Dynamic characteristic analysis of fuzzy-stochastic truss structures based on fuzzy factor method and random factor method

A new fuzzy stochastic finite element method based on the fuzzy factor method and random factor method is given and the analysis of structural dynamic characteristic for fuzzy stochastic truss structures is presented. Considering the fuzzy randomness of the structural physical parameters and geometric dimensions simultaneously, the structural stiffness and mass matrices are constructed based on the fuzzy factor method and random factor method; from the Rayleigh’s quotient of structural vibration, the structural fuzzy random dynamic characteristic is obtained by means of the interval arithmetic; the fuzzy numeric characteristics of dynamic characteristic are then derived by using the random variable’s moment function method and algebra synthesis method. Two examples are used to illustrate the validity and rationality of the method given. The advantage of this method is that the effect of the fuzzy randomness of one of the structural parameters on the fuzzy randomness of the dynamic characteristic can be reflected expediently and objectively.

Seismic random vibration analysis of stochastic structures using random factor method

地震随机的颤动分析随机捆绑结构被介绍。调用随机的因素方法的一个新方法与不明确的参数被用于结构的动态分析，由于在他们的材料性质和几何学的可变性。用随机的因素方法，自然频率和随机的结构的模式形状能被二部分的产品分别地描述，相应于有不确定性的结构的参数的随机的因素、确定由常规有限元素分析的自然频率和 modeshapesobtained 的值。随机捆绑结构受到静止或非静止的随机的地震刺激。为平均数的平均数和标准差的计算表情摆平排水量，吝啬的方形的应力借助于随机的变量的功能的时刻方法和代数学合成方法被开发。一条天线和一座 truss 桥被用作实际工程例子在静止 ornon 静止的随机的地震刺激下面在随机的结构的地震反应分析说明随机的因素方法的应用程序。

Reliability-Based Optimization of Active Nonstationary Random Vibration Control

The reliability-based optimization of active bars' placement and feedback gains of a closed-loop control system for stochastic intelligent truss structural active nonstationary random vibration control is presented. The optimal mathematical model with the reliability constraints on the mean square value of structural dynamic displacement and stress response are built based on the maximization of dissipation energy due to control action, where not only the structural physics parameters, geometric dimensions, and damping are considered as random variables, but also the applied forces are considered as nonstationary random excitation. The numerical characteristics of the nonstationary random responses of stochastic intelligent structure are developed by the random factor method. The rationality and validity of the presented model are demonstrated via an engineering example, and the effect of randomness of the structural parameters on the control performance is also examined.