Conditional Reliability Function Research Articles

Statistical Reliability Assessment with Generalized Intuitionistic Fuzzy Burr XII Distribution

Intuitionistic fuzzy sets provide a viable framework for modelling lifetime distribution characteristics, particularly in scenarios with measurement imprecision. This is accomplished by utilizing membership and non-membership degrees to accurately express the complexities of data uncertainty. Nonetheless, the complexities of some cases necessitate a more advanced approach of imprecise data, motivating the use of generalized intuitionistic fuzzy sets (GenIFSs). The use of GenIFSs represents a flexible modeling strategy that is characterized by the careful incorporation of an extra level of hesitancy, which effectively clarifies the underlying ambiguity and uncertainty present in reliability evaluations. The study employs a methodology based on generalized intuitionistic fuzzy distributions to thoroughly examine the uncertainty related to the parameters and reliability characteristics present in the Burr XII distribution. The goal is to provide a more accurate evaluation of reliability measurements by addressing the inherent ambiguity in the distribution’s shape parameter. Various reliability measurements, such as reliability, hazard rate, and conditional reliability functions, are derived for the Burr XII distribution. This extensive analysis is carried out within the context of the generalized intuitionistic fuzzy sets paradigm, improving the understanding of the Burr XII distribution’s reliability measurements and providing important insights into its performance for the study of various types of systems. To facilitate understanding and point to practical application, the findings are shown graphically and contrasted across various cut-set values using a valuable numerical example.

Nonlinear random vibration of the slender deep-water pier under seismic excitation

The present study investigates the nonlinear random vibration of the deep-water pier exposed to horizontal seismic excitation. First of all, the stochastic dynamic model of the pier is formulated. During the process, the pier is simplified as a cantilever beam fixed on the rock foundation, the seismic excitation is treated as Gaussian white noise, the hydrodynamic pressure is described with the radiation wave theory, and the equation for the nonlinear kinematic of the pier is deduced by the means of Kane’s method. Then, with the application of the stochastic averaging (SA) technique, the Fokker–Plank–Kolmogorov (FPK) equation governing the transient probability density function (PDF) of the amplitude envelope and the backward Kolmogorov (BK) equation for the conditional reliability function (CRF) are derived, respectively. The closed-form stationary PDF can be yielded directly from the reduced FPK equation, while the CRF and the conditional PDF of first-passage time are given after solving the BK equation numerically. Numerical discussions are performed to illustrate the trend of excitation intensity, mass ratio, immersion ratio, and inner and outer hydrodynamic effect on the stationary response and first-passage failure are examined, respectively. It has been shown that increases in the excitation intensity, mass ratio, and immersion ratio can amplify the response and reduce the reliability of the deep-water pier system. The hydrodynamic effect also leads to an amplification of the system response and a reduction in the reliability of the system. Similarly, the presence of inner water can also exacerbate these effects, and this phenomenon becomes more pronounced with increasing immersed ratios. Additionally, the analytical solution is validated by the result obtained by pertained Monte Carlo simulations (MCS). It is noted that this work will be helpful for the optimal seismic design of deep-water piers.

Reliability of the bi-stable piezoelectric energy harvester under random excitation

The bi-stable piezoelectric energy harvester can produce higher voltage and efficiency than traditional energy harvester. In this manuscript, the reliability of the bi-stable piezoelectric energy harvester excited by Gaussian white noise is investigated. The equivalent uncoupled equation of system is deduced on the basis of the generalized harmonic transformation for the bi-stable piezoelectric energy harvester original coupling system. The averaged stochastic differential equation for system energy of the equivalent uncoupled system is established using the stochastic averaging method. The conditional reliability function and the mean first-passage time are derived through the backward Kolmogorov equation and the generalized Pontryagin equation associated with the averaged stochastic differential equation. The influences of noise intensity, electromechanical coupling coefficient, and time constant ratio on the conditional reliability function and mean first-passage time are discussed. The analytical results are highly consistent with the numerical results that are obtained by the Monte Carlo numerical simulation.

Residual lifetime estimation for the mining truck tires

Reliability analysis has a very important role in supporting the operation of a system in harsh working conditions such as mines. Haul trucks are considering as a main asset of an open pit mine. Dump trucks are comprised of different sub-systems and among them, tires have a significant effect on the haulage economy. This study focused on the remaining life estimation of the mining truck tires considering the environmental operating conditions at an open pit iron ore mine in Iran. To achieve this, the proportional hazard model was used to estimate the conditional reliability functions and then the mean residual life of tires at the various initial survival times. The results of this study show that the ambient temperature has a significant effect on the reliability performance. By increasing the initial life of tires from 500 to 1000 h, the mean residual life will be decreased by 21% and 30% in low (below 15°C) and high (above 15°C) ambient temperature levels. These results are helpful to keep the truck tire performance at a desirable level by considering the appropriate preventive maintenance intervals.

Reliability analysis of a class of stochastically excited nonlinear Markovian jump systems

It is inevitable for the nonlinear systems to suffer from external stochastic disturbance. Meanwhile, the components failure will bring abrupt changes in its substructures, which can be considered as the internal stochastic disturbance. It is demonstrated that the components failure performs random jumpy factors switching between a finite number of modes. This salient feature allows us to identify this type of dynamic behaviors as response of nonlinear hybrid systems undergoing Markovian jumps. In this paper a novel method is presented to analyze the dynamical reliability of multi-degree-of-freedom (DOF) nonlinear stochastic systems undergoing Markovian jumps. Firstly, the Markovian jump process is introduced to formulate the aforementioned systems as continuous-discrete hybrid systems. Secondly, a two-step approximate method is applied to convert the hybrid systems to one governed by a set of averaged Itô stochastic differential equations (SDEs) of mechanical energies. Then the associated averaged backward Kolmogorov equation and the Pontryagin equation are constructed and solved to yield the conditional reliability function and the mean first-passage time of the original system respectively. Finally, an example of two coupled Duffing oscillators is illustrated to compare the analytical results and those from Monte Carlo simulation to verify the effectiveness of the proposed method.

Remaining useful Life Improvement for the Mining Railcars under the Operational Conditions

ABSTRACT The large tonnage mines use railway transportation as a system for mineral transportation. The rolling stock health condition study and improvement not only reduce the lifecycle cost of the assets but also ensures safe, reliable, punctual, and efficient transportation. The remaining useful life estimation is an efficient approach to condition-based maintenance, prognostics, and health management. This paper aims to predict the remaining life of a mining rolling stocks in the presence of operational environmental effects. In this study, the operation and failure data were obtained from Malmbana, LKAB, Sweden, a Swedish Railway Company, considering the effective operational factors. The failure behaviour of the railcar was evaluated and then the proportional hazard model was used to estimate the conditional reliability functions and accordingly the remaining useful life at the various initial survival time. Finally, the reliability-based time interval is applied to plan the maintenance operations. Results of this study show that the operator’s skill level and the maintenance quality had a significant influence on the reliability performance. By considering the proposed PM plan according to the desired reliability level, the remaining lifetime of the rolling stock will be improved by 78.10%, on average.

Multistate Diagnosis and Prognosis of Lubricating Oil Degradation Using Sticky Hierarchical Dirichlet Process–Hidden Markov Model Framework

In this study, we present a state-based diagnostic and prognostic methodology for lubricating oil degradation based on a nonparametric Bayesian approach, i.e., sticky hierarchical Dirichlet process–hidden Markov model (HDP-HMM). An accurate health state-space assessment for diagnostics and prognostics has always been unobservable and hypothetical in the past. The lubrication condition monitoring (LCM) data is generally segregated as “healthy or unhealthy”, representing a binary state-based perspective to the problem. This two-state performance-based formulation poses limitations to the precision and accuracy of the diagnosis and prognosis for real data wherein there may be multiple states of discrete performance that are characteristic of the system functionality. In particular, the reversible and nonlinear time-series trends of degradation data increase the complexity of state-based modeling. We propose a multistate diagnostic and prognostic framework for LCM data in the wear-out phase (i.e., the unhealthy portion of degradation data), accounting for irregular oil replenishment and oil change effects (i.e., nonlinearity in the degradation signal). The LCM data is simulated for an elementary mechanical system with four components. The sticky HDP sets the prior for the HMM parameters. The unsupervised learning over infinite observations and emission reveals four discrete health states and helps estimate the associated state transition probabilities. The inferred state sequence provides information relating to the state dynamics, which provides further guidance to maintenance decision making. The decision making is further backed by prognostics based on the conditional reliability function and mean residual life estimation.

Reliability of elastic impact system with Coulomb friction excited by Gaussian white noise

Abstract The reliability of nonlinear Elastic impact oscillator with Coulomb friction subjected to Gaussian white noise excitation is investigated. Firstly, the stochastic averaging method for the nonlinear Elastic impact system with Coulomb friction is proposed and the averaged Ito equation for the system energy is obtained. Secondly, the Backward Kolmogorov(BK) equation and Generalized Pontryagin(GP) equation are established to achieve the conditional reliability function and the mean first-passage time of the Elastic impact system. Finally, two examples are worked out in detail to illustrate the feasibility of the proposed procedures. The accuracy and efficiency of the proposed procedures are validated by Monte Carlo simulations. We can conclude that the increase of the Coulomb friction, noise intensity and damping constants of the elastic impact force can improve the reliability of the Elastic impact system.

Remaining useful life prediction of lubricating oil with dynamic principal component analysis and proportional hazards model

Lubricating oil contains a lot of tribological information of the machine and plays an important role in machine health. Oil degrades with serving time and causes severe wear afterwards, which is a complex dynamic process, and difficult to be accurately described by a single property. Therefore, the main purpose of deterioration prediction is to estimate the remaining useful life that the oil can still fulfill its functions by analyzing oil condition monitoring data. With a large amount of oil condition monitoring data collected, a vector autoregressive model is applied to the original oil data to describe the dynamic deterioration process. Then dynamic principal component analysis, an effective dimensionality reduction method, is employed to obtain the principal components capturing the most information of the oil data. The proportional hazards model is then built to calculate the failure risk of the lubricating oil based on the condition monitoring information, where its baseline function represents the aging process assuming to follow the Weibull distribution and its positive link function represents the influence of covariates (the principal components) on the failure risk. Finally, the remaining useful life prediction of lubricating oil can be obtained by explicit formulas of the characteristics such as the conditional reliability function and the mean residual life function. This work provides an approach to assess the health of lubricating oil, and a guidance for oil maintenance strategy.

Model parameter estimation and residual life span prediction of pneumatic diaphragm pump based on hidden Markov model in intelligent spraying

Pneumatic diaphragm pump is an important part in intelligent spraying. When pneumatic diaphragm pump does not work normally, the entire intelligent spraying product line will be malfunctioned. To maintain and manage pneumatic diaphragm pump effectively, the grade analysis of the health status of pneumatic diaphragm pump is generally used according to its working state. Due to the effects of condition monitoring and random faults, some observable health predictions are often inaccurate. There are very few papers dealing with the health monitoring of pneumatic diaphragm pump and their estimation of residual life span. In this article, a method with vector autoregressive model and continuous-time hidden Markov model was proposed to analyze and evaluate the life span of pneumatic diaphragm pump based on the estimation error of the health condition and the cumulative deterioration of pneumatic diaphragm pump. It is modeled through a continuous-time Markov chain with three states, which includes unobservable healthy state 0, unobservable warning state 1, and observable fault state 2. The expectation–maximization algorithm is used to estimate the model parameters of the fitted hidden Markov. Through the posterior probability of pneumatic diaphragm pump in warning state 1, the derived conditional reliability function and mean residual life span formula can be calculated to evaluate the residual life span of pneumatic diaphragm pump. The results showed that the method can effectively predict the residual life span of pneumatic diaphragm pump, illustrate the effectiveness of the model, and improve the accuracy of the health status rating.

Reliability assessment for multi-state systems with state transition dependency

As multi-state system reliability models are capable of characterizing the multi-state nature of engineered systems in deteriorating process, they have received considerable concerns in the past few decades. Most reported works on multi-state system reliability modeling are, however, based on the premise that transitions among states of a system/component are stochastically independent. Sometimes, a system may experience the same environmental/working conditions when deteriorating from a better state to worse ones, and thus, state transitions of a system/component could be stochastically dependent. In this paper, a new reliability model for multi-state systems/components with state transition dependency is put forth. The dependency among state transitions is implicitly characterized by copula functions which offer a great flexibility of linking arbitrary marginal distributions together to construct a multivariate distribution. The reliability function of a population of multi-state systems/components and the conditional reliability function of each individual multi-state system/component given a set of observations are derived. The likelihood functions for model parameter estimation are formulated for two cases, i.e., continuous and discontinuous inspection strategies, and model selection criteria are customized to identify the most preferable model among candidates. Two illustrative examples, together with comparative studies, are presented to demonstrate the effectiveness of the proposed method.

Health Assessment Method for Electronic Components Subject to Condition Monitoring and Hard Failure

In this paper, a health assessment method for electronic components subject to condition monitoring (CM) is presented, which can be used to estimate the conditional reliability characteristics given the current operational age and the corresponding degradation state. The degradation process is characterized by a continuous-time Markov chain, which is incorporated into a Cox’s proportional hazard model to describe the hazard rate of the time to failure. The two main challenges encountered in the health assessment of electronic components subject to CM and hard failure, i.e., the large number of degradation states and the general deteriorating transition mechanism, can be properly addressed by the proposed method. To illustrate the effectiveness of the proposed method, a case study of power metal-oxide-semiconductor field-effect transistors is performed as a representative example of a general electronic application. The method for the model parameter estimation is developed and applied using the degradation and failure data obtained from an accelerated testing experiment, based on which the conditional reliability function, the mean residual life, and other health characteristics can be explicitly calculated using the proposed discretization technique. The excellent accuracy of the health assessment demonstrates the effectiveness and advantages of the proposed method applied to a real electronic component subject to CM and hard failure.

Dynamical reliability of internally resonant or non-resonant strongly nonlinear system under random excitations

Abstract The dynamical reliability of multi-degrees-of-freedom (MDOF) strongly nonlinear system under Gaussian white noise excitations is studied, including resonance and non-resonance. Firstly, the equations of motion of the original system with or without internal resonance are reduced to a set of Ito stochastic differential equations after stochastic averaging. Then, the backward Kolmogorov equation and the Pontryagin equation associated with the resonantly or non-resonantly averaged Ito stochastic differential equations, which determine the conditional reliability function and the mean first-passage time of the original random system, are constructed under appropriate boundary and (or) initial conditions, respectively. In particular, if the non-resonantly averaged system is completely decoupled, the conditional reliability function and the mean first-passage time of the original non-resonant system can be obtained by solving a set of simplified backward Kolmogorov equations. A system comprising two weakly coupled and strongly nonlinear mechanical oscillators is given as a concrete example to show the application of the proposed method. The 1:1 internal resonance or non-resonance is discussed. The corresponding high-dimensional backward Kolmogorov equation and Pontryagin equation are established and solved numerically. All theoretical results are validated by a Monte Carlo digital simulation.

Reliability Estimation of Stochastic Dynamical Systems with Fractional Order PID Controller

In this paper, the reliability of stochastic dynamical systems under Gaussian white noise excitations with fractional order proportional–inegral–derivative (FOPID) controller is estimated. First, the FOPID controller is approximated by a set of combination of displacement and velocity based on the generalized van der Pol transformation. Then, the stochastic averaging method of energy envelope is applied to obtain a diffusive differential equation, from which the Backward Kolmogorov equation, governing the conditional reliability function, and the Generalized Pontryagin equation, governing the statistical moments of first-passage time, are derived from the averaged equation and solved numerically. Finally, in the two examples, the critical parameters in the FOPID controller are shown to be capable of improving the reliability of the stochastic dynamical system apparently, and all numerical results are verified to be efficient and correct by the Monte Carlo simulation.

Context-driven mean residual life estimation of mining machinery

ABSTRACTMaintenance is crucial to ensure production/output and customer satisfaction in the mining sector. The cost of maintenance of mechanised and automated mining systems is very high, necessitating efforts to enhance the effectiveness of maintenance systems and organisation. For effective maintenance planning, it is important to have a good understanding of the reliability and availability characteristics of the systems. Determining the Mean Residual Life (MRL) of systems allows organisations to more effectively plan maintenance tasks. In this paper, we use a statistical approach to estimate MRL and consider a Weibull proportional hazard model (PHM) with time-independent covariates to model the hazard function so that the operating environment could be integrated into the reliability analysis. The paper explains our methods for calculating the conditional reliability function and computing the MRL as a function of the current conditions. The model is verified and validated using data from the hydraulic system of LHD equipment in a Swedish mine. The results are useful to estimate the remaining useful life of such systems; the method can be used for maintenance planning, helping to control unplanned stoppages of highly mechanised and automated systems.

Parameter estimation and remaining useful life prediction of lubricating oil with HMM

Lubricating oil plays a vital role in the full life-span performance of the machine. Lubricating oil deterioration, which leads to the attenuation of oil performance and severe wear afterwards, is a slow degrading process, which can be observed by condition monitoring, but the actual degree of the oil degradation is often very difficult to examine. The main purpose of lubricating oil degradation prediction is to estimate the failure time when the oil no longer fulfills its functions. We suppose that the state process evolution of lubricating oil degradation can be modeled using a hidden Markov model (HMM) with three states: healthy state, unhealthy state, and failure state. Only the failure state is observable. While the lubricating oil is in service, vector data that are stochastically related to the deterioration state are obtained through on-line condition monitoring by an OLVF (On-line Visual Ferrograph) sensor at regular sampling epochs. A method of Time Series Analysis (TSA) is applied to the healthy portions of the oil data histories to get the residuals as the observable process containing partial information to fit the hidden Markov model. The unknown parameters of the fitted hidden Markov model are estimated by the Expectation-Maximization (EM) algorithm. The remaining useful life (RUL) of lubricating oil can be evaluated through explicit formulas of the characteristics such as the conditional reliability function (CRF) and mean residual life (MRL) function in terms of the posterior probability.

Optimal CBM policy with two sampling intervals

PurposeThe purpose of this paper is to propose a novel condition-based maintenance (CBM) policy with two sampling intervals for a system subject to stochastic deterioration described by the Cox’s proportional hazards model (PHM).Design/methodology/approachIn this paper, the new or renewed system is monitored using a longer sampling interval. When the estimated hazard function of the system exceeds a warning limit, the observations are taken more frequently, i.e., the sampling interval changes to a shorter one. Preventive maintenance is performed when either the hazard function exceeds a maintenance threshold or the system age exceeds a pre-determined age. A more expensive corrective maintenance is performed upon system failure. The proposed model is formulated in the semi-Markov decision process (SMDP) framework.FindingsThe optimal maintenance policy is found and a computational algorithm based on policy iteration for SMDP is developed to obtain the control thresholds as well as the sampling intervals minimizing the long-run expected average cost per unit time.Research limitations/implicationsA numerical example is presented to illustrate the whole procedure. The newly proposed maintenance policy with two sampling intervals outperforms previously developed maintenance policies using PHM. The paper compares the proposed model with a single sampling interval CBM model and well-known age-based model. Formulas for the conditional reliability function and the mean residual life are also derived for the proposed model. Sensitivity analysis has been performed to study the effect of the changes in the Weibull parameters on the average cost.Practical implicationsThe results show that considerable cost savings can be obtained by implementing the maintenance policy developed in this paper.Originality/valueUnlike the previous CBM policies widely discussed in the literature which use sequential or periodic monitoring, the authors propose a new sampling strategy based on two sampling intervals. From the economic point of view, when the sampling is costly, it is advantageous to monitor the system less frequently when it is in a healthy state and more frequently when it deteriorates and enters the unhealthy state.

First-exit problem of MDOF strongly nonlinear oscillators under wide-band random excitations without internal resonances

The first-exit problem of strongly nonlinear oscillators with multi-degrees-of-freedom (MDOF) is studied theoretically in this paper. The excitations are modelled as wide-band random noises. When there is no internal resonance, the equations of motion of the original system are reduced to a set of averaged Ito stochastic differential equations (SDEs) after stochastic averaging. Two partial differential equations (PDEs), namely the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-exit time, are established under appropriate boundary and (or) initial conditions. The theoretical method is applied to study a 3-DOF strongly nonlinear system. Long expressions of drift and diffusion coefficients of the averaged Ito SDEs are obtained by using software Mathematica. The qualitative boundary conditions of the associated backward Kolmogorov equation and the Pontryagin equation are quantified by examining the behaviour of drift coefficients at the lower boundary. The method of finite difference is used to solve the corresponding high-dimensional PDEs. Monte Carlo simulation is performed to verify the validity of the proposed theoretical method.

Reliability estimation of a system subject to condition monitoring with two dependent failure modes

ABSTRACTA new competing risk model is proposed to calculate the Conditional Mean Residual Life (CMRL) and Conditional Reliability Function (CRF) of a system subject to two dependent failure modes, namely, degradation failure and catastrophic failure. The degradation process can be represented by a three-state continuous-time stochastic process having a healthy state, a warning state, and a failure state. The system is subject to condition monitoring at regular sampling times that provides partial information about the system is working state and only the failure state is observable. To model the dependency between two failure modes, it is assumed that the joint distribution of the time to catastrophic failure and sojourn time in the healthy state follow Marshal–Olkin bivariate exponential distributions. The Expectation–Maximization algorithm is developed to estimate the model's parameters and the explicit formulas for the CRF and CMRL are derived in terms of the posterior probability that the system is in the warning state. A comparison with a previously published model is provided to illustrate the effectiveness of the proposed model using real data.

First-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations

In this paper, first-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations is studied theoretically. By using stochastic averaging method, the equations of motion of the original internally resonant Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations. The backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are established under appropriate boundary and (or) initial conditions. An example is given to show the accuracy of the theoretical method. Numerical solutions of high-dimensional backward Kolmogorov and Pontryagin equation are obtained by finite difference. All theoretical results are verified by Monte Carlo simulation.