Reliability of dynamic systems under limited information

A method is developed for reliability analysis of dynamic systems under limited information. The available information includes one or more samples of the system output; any known information on features of the output can be used if available. The method is based on the theory of non-Gaussian translation processes and is shown to be particularly suitable for problems of practical interest. For illustration, we apply the proposed method to a series of relevant examples and compare with results given by traditional statistical estimators. It is demonstrated that the method delivers accurate results for the case of linear and nonlinear dynamic systems, and can be applied to analyze experimental data and/or mathematical model outputs.     

Estimation of component failure probability from masked binomial system testing data

The component failure probability estimates from analysis of binomial system testing data are very useful because they reflect the operational failure probability of components in the field which is similar to the test environment. In practice, this type of analysis is often confounded by the problem of data masking: the status of tested components is unknown. Methods in considering this type of uncertainty are usually computationally intensive and not practical to solve the problem for complex systems. In this paper, we consider masked binomial system testing data and develop a probabilistic model to efficiently estimate component failure probabilities. In the model, all system tests are classified into test categories based on component coverage. Component coverage of test categories is modeled by a bipartite graph. Test category failure probabilities conditional on the status of covered components are defined. An EM algorithm to estimate component failure probabilities is developed based on a simple but powerful concept: equivalent failures and tests. By simulation we not only demonstrate the convergence and accuracy of the algorithm but also show that the probabilistic model is capable of analyzing systems in series, parallel and any other user defined structures. A case study illustrates an application in test case prioritization.     

Validation of reliability computational models using Bayes networks

This paper proposes a methodology based on Bayesian statistics to assess the validity of reliability computational models when full-scale testing is not possible. Sub-module validation results are used to derive a validation measure for the overall reliability estimate. Bayes networks are used for the propagation and updating of validation information from the sub-modules to the overall model prediction. The methodology includes uncertainty in the experimental measurement, and the posterior and prior distributions of the model output are used to compute a validation metric based on Bayesian hypothesis testing. Validation of a reliability prediction model for an engine blade under high-cycle fatigue is illustrated using the proposed methodology.     

Numerical solutions of Fokker-Planck equation of nonlinear systems subjected to random and harmonic excitations

A new approach for an efficient numerical implementation of the path integral (PI) method based on non-Gaussian transition probability density function (PDF) and the Gauss-Legendre integration scheme is developed. This modified PI method is used to solve the Fokker-Planck (FP) equation and to study the nature of the stochastic and chaotic response of the nonlinear systems. The steady state PDF, periodicity, jump phenomenon, noise induced changes in joint PDF of the states are studied by the modified PI method. A computationally efficient higher order, finite difference (FD) technique is derived for the solution of higher-dimensional FP equation. A two degree of freedom nonlinear system having Coulomb damping with a variable friction coefficient subjected to Gaussian white noise excitation is considered as an example which can represent a bladed disk assembly of turbo-machinery blades. Effects of normal force and viscous damping on the mean square response are investigated.     

Estimation of exponential component reliability from uncertain life data in series and parallel systems

Estimating reliability of components in series and parallel systems from masking system testing data has been studied. In this paper we take into account a second type of uncertainty: censored lifetime, when system components have constant failure rates. To efficiently estimate failure rates of system components in presence of combined uncertainty, we propose a useful concept for components: equivalent failure and equivalent lifetime. For a component in a system with known status and lifetime, its equivalent failure is defined as its conditional failure probability and its equivalent lifetime is its expectation of lifetime. For various uncertainty scenarios, we derive equivalent failures and test times for individual components in both series and parallel systems. An efficient EM algorithm is formulated to estimate component failure rates. Two numerical examples are presented to illustrate the application of the algorithm.     

一类非线性系统随机振动的等效线性化

本文根据机械振动系统中弹性力、阻尼力只决定于弹性元件的变形和阻尼元件的相对速度的特点，建立了一种处理一类多自由度非线性系统随机振动的新方法。此方法物理意义明确，计算量小，是处理这类非线性系统随机振动行之有效的方法。     

Efficient uncertainty quantification of dynamical systems with local nonlinearities and uncertainties

Idealized modeling of most engineering structures yields linear mathematical models, i.e., linear ordinary or partial differential equations. However, features like nonlinear dampers and/or springs can render nonlinear an otherwise linear model. Often, the connectivity of these nonlinear elements is confined to only a few degrees-of-freedom (DOFs) of the structure. In such cases, treating the entire structure as nonlinear results in very computationally expensive solutions. Moreover, if system parameters are uncertain, their stochastic nature can render the analysis even more computationally costly. This paper presents an approach for computing the response of such systems in a very efficient manner. The proposed solution procedure first segregates the DOFs appearing in the nonlinear and/or stochastic terms from those DOFs that involve only linear deterministic operations. Second, the responses of nonlinear/stochastic terms are determined using a non-standard form of a nonlinear Volterra integral equation (NVIE). Finally, the responses of the remaining DOFs are computed through a convolution approach using the fast Fourier transform to further increase the computational efficiency. Three examples are presented to demonstrate the efficacy and accuracy of the proposed method. It is shown that, even for moderately sized systems (∼1000 DOFs), the proposed method is about three orders of magnitude faster than a conventional Monte Carlo sampling method (i.e., solving the system of ODEs repeatedly).     

The reliability engineer looks at stress screening

Stress screening (as a final step in the manufacture of electronics) adds brief intervals of severe broadband vibration to temperature shock. The goal is to precipitate as hard defects any latent manufacturing defects. Applying such stressing to subassemblies and assemblies is much faster and much more effective than the more familiar ‘burn-in’ or temperature shock alone.     

Some thoughts on averaging techniques in stochastic dynamics

Stochastic averaging and quasi-conservative averaging are used essentially to obtain a one-dimensional Markov approximation describing the response of a randomly excited one-degree-of-freedom system. Difficulties arise when the original physical system is more than one degree of freedom with both low and high damping modes, or/and it has a strongly nonlinear stiffness and the stochastic excitation is non-white. The ways in which such difficulties can be overcome are discussed.     

非线性结构振动及其控制系统的多维ARMA模型

本文研究了结构非线性振动及其控制系统的ARMA模型的建立问题。对结构施行等效线性化处理：对状态变量的观测,仅测量结构的前几个位移或速度。将随机激励和观测噪声等维化处理后,建立了等效结构振动及其控制系统的多维ARMA模型。基于多维ARMA模型,给出了结构受风与地震作用下的ARMA模型和模态分析的ARMA模型。     

Estimation under the Burr type XII failure model based on censored data: a comparative study

Summary Based on type-2 censored samples, the maximum likelihood, uniformly minimum variance unbiased, Bayes and empirical Bayes estimators of one of the two shape parameters (k) and reliability functionR(t) of the Burr type XII failure model are computed and compared. Computations show that when the censoring sizer=10, the EBE’s of κ andR(t),t=0.9, are better than the corresponding UMVUE’s for as few asm ^*=7 past samples fork andm ^*=11 past samples forR(0.9), in the sense of having smaller estimated risks, when the gamma conjugate prior is used.     

Harmonic wavelets based statistical linearization for response evolutionary power spectrum determination

A novel harmonic wavelets based statistical linearization approach is proposed for determining the evolutionary power spectrum (EPS) of the response of nonlinear oscillators subject to stochastic excitation. Specifically, first a mathematically rigorous wavelet-based representation of non-stationary stochastic processes is presented. Next, a representation of the process corresponding to a specific scale and translation level is derived. This procedure leads to an EPS estimation approach which is applicable for estimating not only separable but non-separable in time and frequency EPS as well. Several numerical results are presented in this context. Next, focusing on the case of the stochastic response of a linear system and relying on the orthogonality properties of the developed representation an excitation-response EPS relationship is derived. It is further shown that the excitation-response EPS relationship is valid even for linear time-variant (LTV) systems since the approach possesses inherently the element of time-dependence. Further, an extension via statistical linearization of the input-output EPS relationship for the case of a nonlinear system is developed. The approach involves the concept of assigning optimal and response dependent equivalent stiffness and damping elements corresponding to the specific frequency and time bands. This leads to an iterative determination of the EPS of the system response. Pertinent Monte Carlo simulations demonstrate the reliability and versatility of the approach.