An Efficient Method to Calculate the Failure Rate of Dynamic Systems with Random Parameters Using the Total Probability Theorem

2015 ◽  
Vol 8 (3) ◽  
pp. 623-631 ◽  
Author(s):  
Monica Majcher ◽  
Zissimos P. Mourelatos ◽  
Vasileios Geroulas ◽  
Igor Baseski ◽  
Amandeep Singh
Keyword(s):  
Failure Rate ◽  
Efficient Method ◽  
Dynamic Systems ◽  
Total Probability ◽  
Random Parameters ◽  
Total Probability Theorem

Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters

2016 ◽  
Vol 138 (3) ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vasileios Geroulas
Keyword(s):  
Large Scale ◽  
Probability Of Failure ◽  
Time Dependent ◽  
Total Probability ◽  
Conditional Probabilities ◽  
Random Parameters ◽  
Random Vibrations ◽  
Input Parameters ◽  
Total Probability Theorem ◽  
Dependent Probability

The field of random vibrations of large-scale systems with millions of degrees-of-freedom (DOF) is of significant importance in many engineering disciplines. In this paper, we propose a method to calculate the time-dependent reliability of linear vibratory systems with random parameters excited by nonstationary Gaussian processes. The approach combines principles of random vibrations, the total probability theorem, and recent advances in time-dependent reliability using an integral equation involving the upcrossing and joint upcrossing rates. A space-filling design, such as optimal symmetric Latin hypercube (OSLH) sampling, is first used to sample the input parameter space. For each design point, the corresponding conditional time-dependent probability of failure is calculated efficiently using random vibrations principles to obtain the statistics of the output process and an efficient numerical estimation of the upcrossing and joint upcrossing rates. A time-dependent metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure. The proposed method is demonstrated using a vibratory beam example.

Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters

2015 ◽  
Author(s):  
Zissimos P. Mourelatos ◽  
Monica Majcher ◽  
Vasileios Geroulas
Keyword(s):  
Large Scale ◽  
Probability Of Failure ◽  
Time Dependent ◽  
Total Probability ◽  
Conditional Probabilities ◽  
Random Parameters ◽  
Random Vibrations ◽  
Input Parameters ◽  
Total Probability Theorem ◽  
Dependent Probability

The field of random vibrations of large-scale systems with millions of degrees of freedom is of significant importance in many engineering disciplines. In this paper, we propose a method to calculate the time-dependent reliability of linear vibratory systems with random parameters excited by non-stationary Gaussian processes. The approach combines principles of random vibrations, the total probability theorem and recent advances in time-dependent reliability using an integral equation involving the up-crossing and joint up-crossing rates. A space-filling design, such as optimal symmetric Latin hypercube sampling, is first used to sample the input parameter space. For each design point, the corresponding conditional time-dependent probability of failure is calculated efficiently using random vibrations principles to obtain the statistics of the output process and an efficient numerical estimation of the up-crossing and joint up-crossing rates. A time-dependent metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure. The proposed method is demonstrated using a vibratory beam example.

A Random Process Metamodel for Time-Dependent Reliability of Dynamic Systems

2014 ◽  
Author(s):  
Dorin Drignei ◽  
Igor Baseski ◽  
Zissimos P. Mourelatos ◽  
Vijitashwa Pandey
Keyword(s):  
Random Process ◽  
Dynamic Systems ◽  
Random Variables ◽  
Kriging Model ◽  
Random Processes ◽  
Time Dependent ◽  
Total Probability ◽  
Input And Output ◽  
System Input ◽  
Total Probability Theorem

A new metamodeling approach is proposed to characterize the output (response) random process of a dynamic system with random variables, excited by input random processes. The metamodel is then used to efficiently estimate the time-dependent reliability. The input random processes are decomposed using principal components or wavelets and a few simulations are used to estimate the distributions of the decomposition coefficients. A similar decomposition is performed on the output random process. A Kriging model is then built between the input and output decomposition coefficients and is used subsequently to quantify the output random process corresponding to a realization of the input random variables and random processes. In our approach, the system input is not deterministic but random. We establish therefore, a surrogate model between the input and output random processes. The quantified output random process is finally used to estimate the time-dependent reliability or probability of failure using the total probability theorem. The proposed method is illustrated with a corroding beam example.

Reliability Model of Gear With Correlated Failure Modes Based on Joint Distribution

2014 ◽  
Author(s):  
Lai Yue-Hua ◽  
Dong Hai-Ping ◽  
Yi Xiao-Jian ◽  
Ding Juan ◽  
Lei Hua-Jin
Keyword(s):  
Joint Distribution ◽  
Failure Modes ◽  
Physical Models ◽  
Total Probability ◽  
Traditional Model ◽  
Reliability Model ◽  
Interference Theory ◽  
Calculation Results ◽  
Set Up ◽  
Total Probability Theorem

In this paper, a precise reliability model for gear with correlated failure modes is presented. Firstly, physical models of stress and strength are set up against two main failure modes of a gear respectively. Then, the corresponding performance functions to the two failure modes are obtained according to stress-strength interference theory regarding randomness of variables in physical models of stress and strength. Furthermore, joint distribution of the two performance functions is deduced by Total Probability Theorem considering the correlation of random variables. So, reliability of a gear with correlated failure modes can be computed based on the joint distribution. Finally, an example is given and in this example the precise reliability model based on joint distribution and the traditional reliability model without considering the correlation of failure modes, are respectively adopted to calculate reliability of a gear. The calculation results are compared with that by Monte Carlo simulation and the compared results show that the gear’s reliability obtained by considering correlation of failure modes based on joint distribution is more accurate than that by the traditional model without considering the correlation of failure modes.

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