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Table 2 The reliability characteristic functions of each subsystem

From: Reliability analysis of subway vehicles based on the data of operational failures

Subsystem

Failure density function

Cumulative distribution function

Running gear

\( f(t)=\frac{0.9124}{13.5450}{\left(t/13.5450\right)}^{\hbox{-} 0.0876}\exp \left[-{\left(t/13.5450\right)}^{0.9124}\right] \)

F(t) = 1 − exp[−(t/13.5450)0.9124]

Traction system

\( f(t)=\frac{0.9940}{10.6495}{\left(t/10.6495\right)}^{-0.006}\exp \left[-{\left(t/10.6495\right)}^{0.9940}\right] \)

F(t) = 1 − exp[−(t/10.6495)0.9940]

Brake system

\( f(t)=\left\{\begin{array}{l}\frac{1}{1.3003\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-2.3693\right)}^2}{2\times {1.3003}^2}},t>0\\ {}0,t=0\end{array}\right. \)

\( F(t)=\Phi \left(\frac{\ln t-2.3693}{1.3003}\right) \)

Control and diagnostic system

\( f(t)=\left\{\begin{array}{l}\frac{1}{1.3581\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-2.5573\right)}^2}{2\times {1.3581}^2}},t>0\\ {}0,t=0\end{array}\right. \)

\( F(t)=\Phi \left(\frac{\ln t-2.5573}{1.3581}\right) \)

Auxiliary system

\( f(t)=\left\{\begin{array}{l}\frac{1}{0.9349\sqrt{2\pi }t}{e}^{\frac{{\left(\ln t-1.4915\right)}^2}{2\times {0.9349}^2}},t>0\\ {}0,t=0\end{array}\right. \)

\( F(t)=\Phi \left(\frac{\ln t-1.4915}{0.9349}\right) \)