Reliability-Based Design Optimization of Load Sharing Systems Using SSI-Markov Models
Abstract
:1. Introduction
Assumptions
- Only the brittle failures are considered in our approach. The plastic deformation phase of the system/components with respect to time is not considered in addition to other material related phenomena such as corrosion, change in material properties over time, etc. However, these changes could be incorporated into our RBDO formulation when the material’s deterioration rate is known. In such cases, the system cost will vary from the cost obtained from this approach since we add more information about the failure into the model. When the strength degrades with respect to time, the cost of the components increases in order to meet the reliability requirement.
- The system is assumed to have zero-simultaneous failure probability. This means that all the components in the system cannot fail simultaneously from a single load application.
- The loading duration on a component and load redistribution time after a component failure are assumed to be constant and occur in a very short interval of time. Hence, they are considered negligible.
- The component repair time is not considered, i.e., once a component fails, it is removed from operation and not put back into operation. However, the component repair process can be easily included in the transition probability/rate matrix in SSI-DTMC/CTMC to find the optimum design.
2. Materials and Methods
2.1. SSI–DTMC Load Sharing Model
2.2. SSI–CTMC Load Sharing Model
2.3. Matrix Decomposition Approach
Eigenvalue Method
3. Case Study: System with Two Identical Components
3.1. SSI–DTMC Formulation
3.2. SSI–CTMC Formulation
4. Results and Discussion
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
f(d, ) | Cost function with design parameter “d” and mean () of random design variable “X”. |
P(X) | Probability of random variable X = x, where x is any real number. |
Most Probable Point (MPP) obtained by transforming the random design variable “X” to | |
standard normal space. | |
Limit state function for component “i” in terms of MPP. | |
T | Total number of components. |
L | Load acting on the system. |
Yield strength of the component. | |
Loading rate at time t. | |
Failure rate at time t. | |
Reliability index for each limit state i. | |
Negative normalized gradient vector of limit state i. | |
Z | Transition probability matrix consisting of state of each components. |
n | Number of steps (DTMC). |
t | Time (CTMC). |
Probability of failure at time t. | |
D | Matrix of eigenvalues of Z. |
M | Matrix of eigenvectors of Z. |
M | Inverse Matrix of eigenvectors of Z. |
System reliability target. | |
J | Set of all states of components that keeps the system in operation. |
Probability of system in state “j”, . | |
Y | Set of all states of components that represents system failure. |
Probability of system in state “y”, ∀ y ∈ Y. | |
Z | State of the system at n = 0 or t = 0. |
K | Set consisting the condition of components at n = 0 or t = 0. |
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State | Component A | Component B |
---|---|---|
AB | working | working |
A’B | failed | working |
AB’ | working | failed |
A’B’ | failed | failed |
Type | Method | Duration | cm | cm | Cost ($) | ||||
---|---|---|---|---|---|---|---|---|---|
Static | 95 | 93.23 | - | 41.32 | 0.92 | 224.70 | 94.21 | 96.12 | |
Single loop | DTMC | 95 | 96.48 | 100 steps | 36.71 | 1.21 | 260.65 | 97.56 | 98.39 |
CTMC | 95 | 99.62 | 25 years | 50.22 | 1.14 | 338.30 | 99.28 | 99.99 |
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Subramaniyan, A.B.; Pan, R.; Du, X. Reliability-Based Design Optimization of Load Sharing Systems Using SSI-Markov Models. Designs 2019, 3, 34. https://doi.org/10.3390/designs3030034
Subramaniyan AB, Pan R, Du X. Reliability-Based Design Optimization of Load Sharing Systems Using SSI-Markov Models. Designs. 2019; 3(3):34. https://doi.org/10.3390/designs3030034
Chicago/Turabian StyleSubramaniyan, Arun Bala, Rong Pan, and Xiaoping Du. 2019. "Reliability-Based Design Optimization of Load Sharing Systems Using SSI-Markov Models" Designs 3, no. 3: 34. https://doi.org/10.3390/designs3030034