10.1. Wear-out model#

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10.1.1. Model description#

Wear-out can occur at component level and the associated failure rates are, in general, not constant. To calculate the reliability of components prone to wear-out, three approaches are introduced in the following for the failure rate \(\lambda\) and the reliability \(R\):

  • Model 1

\[\begin{split} \lambda(t) &= \lambda_{R} + \left( \frac{\beta}{\alpha} \right)\left( \frac{t}{\alpha} \right)^{\beta - 1}\\ R(t) &= {\exp\left\lbrack - \left( \lambda_{R} + \left( \frac{t}{\alpha} \right)^{\beta} \right) \right\rbrack} \end{split}\]
  • Model 2

\[\begin{split} \lambda(t) &= \begin{cases} \lambda_{R}, &\text{if} \qquad t \le \tau\\ \lambda_{R} + \left( \frac{\beta}{\alpha} \right)\left( \frac{t - \tau}{\alpha} \right)^{\beta - 1}, &\text{if} \qquad t > \tau \end{cases}\\ R(t) &= \begin{cases} \exp\left( - \lambda_{R} t \right), &\text{if} \qquad t \le \tau\\ \exp\left( - \left( \lambda_{R} + \left( \frac{t - \tau}{\alpha} \right)^{\beta} \right) \right)&\text{if} \qquad t > \tau \end{cases} \end{split}\]
  • Model 3

\[\begin{split} \lambda(t) &= \begin{cases} \lambda_{R}, &\text{if} \qquad t \le \tau\\ \left( \frac{\beta}{\alpha} \right)\left( \frac{t}{\alpha} \right)^{\beta - 1}, &\text{if} \qquad t > \tau \end{cases}\\ R(t) &= \begin{cases} \exp\left( - \lambda_{R}t \right), &\text{if} \qquad t \le \tau\\ \exp\left( - \left( \frac{t}{\alpha} \right)^{\beta} \right) &\text{if} \qquad t > \tau \end{cases} \end{split}\]

This expressions contain a set of parameters that influence the failure rate and reliability evolution (\(\tau, \lambda_R, \alpha, \beta\)). A summary of these parameters and their meaning is given in Table 10.1.1.

Table 10.1.1 Input variables for reliability analysis#

Name

Description

Unit

\(\tau\)/tau

End time for Model 1 and Model 2 time-invariant behaviour

\(h\)

\(\lambda_R\)/lambda_r

Time-invariant failure rate

\(1/h\)

\(\alpha\)/alpha

Model scale parameter

\(h\)

\(\beta\)/beta

Model shape parameter

\(-\)

10.1.2. Interactive failure rate and reliability prediction#

This page offers an interactive failure rate and reliability prediction tool that lets the user specify the properties of all variables listed in Table 10.1.1. Additionally, a drop-down menu lets the user select the model type and a slider lets the user choose the maximum simulation time \(t_{\text{max}}\)/t_max so that \(t\in[0,t_{\text{max}}]\).

Note

To run the interactive reliability prediction on this page, click the –> Live Code button on the top of the page. Wait a few seconds until the Kernel has loaded and run the cell below with Run.

from nrpmint.booktools import wear_out

# start the web user-interface
wear_out.web_ui()