Degradation model

10.1. Degradation model#

Go to handbook

10.1.1. Model description#

Degradation can occur at component level and the associated failure rates are, in general, not constant. To calculate the reliability of components prone to degradation, 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()
Contents