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Arrhenius-Weibull Statistical Properties Summary

Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} }[/math] (also called [math]\displaystyle{ MTTF }[/math] by some authors), of the Arrhenius-Weibull relationship is given by:

[math]\displaystyle{ \overline{T}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right) }[/math]

where [math]\displaystyle{ \Gamma \left( \tfrac{1}{\beta }+1 \right) }[/math] is the gamma function evaluated at the value of [math]\displaystyle{ \left( \tfrac{1}{\beta }+1 \right) }[/math] .

Median

The median, [math]\displaystyle{ \breve{T}, }[/math] for the Arrhenius-Weibull model is given by:

[math]\displaystyle{ \breve{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}} }[/math]

Mode

The mode, [math]\displaystyle{ \tilde{T}, }[/math] for the Arrhenius-Weibull model is given by:

[math]\displaystyle{ \tilde{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}} }[/math]

Standard Deviation

The standard deviation, [math]\displaystyle{ {{\sigma }_{T}}, }[/math] for the Arrhenius-Weibull model is given by:

[math]\displaystyle{ {{\sigma }_{T}}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}} }[/math]

Arrhenius-Weibull Reliability Function

The Arrhenius-Weibull reliability function is given by:

[math]\displaystyle{ R(T,V)={{e}^{-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}} }[/math]

If the parameter [math]\displaystyle{ B }[/math] is positive, then the reliability increases as stress decreases.

Behavior of the reliability function at different stress and constant parameter values.

The behavior of the reliability function of the Weibull distribution for different values of [math]\displaystyle{ \beta }[/math] was illustrated in Chapter 5. In the case of the Arrhenius-Weibull model, however, the reliability is a function of stress also. A 3D plot such as the ones shown in Fig. 8 is now needed to illustrate the effects of both the stress and [math]\displaystyle{ \beta . }[/math]

[math]\displaystyle{ }[/math]

Reliability function for [math]\displaystyle{ \Beta\lt 1 }[/math], [math]\displaystyle{ \Beta=1 }[/math], and [math]\displaystyle{ \Beta\gt 1 }[/math].

Conditional Reliability Function

The Arrhenius-Weibull conditional reliability function at a specified stress level is given by:

[math]\displaystyle{ R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}} }[/math]

or:

[math]\displaystyle{ R(T,t,V)={{e}^{-\left[ {{\left( \tfrac{T+t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }} \right]}} }[/math]

Reliable Life

For the Arrhenius-Weibull relationship, the reliable life, [math]\displaystyle{ {{t}_{R}} }[/math] , of a unit for a specified reliability and starting the mission at age zero is given by:

[math]\displaystyle{ {{t}_{R}}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left\{ -\ln \left[ R\left( {{t}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}} }[/math]

This is the life for which the unit will function successfully with a reliability of [math]\displaystyle{ R({{t}_{R}}) }[/math] . If [math]\displaystyle{ R({{t}_{R}})=0.50 }[/math] then [math]\displaystyle{ {\breve{T} }[/math], the median life, or the life by which half of the units will survive.

Arrhenius-Weibull Failure Rate Function

The Arrhenius-Weibull failure rate function, [math]\displaystyle{ \lambda (T) }[/math] , is given by:

[math]\displaystyle{ \lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}} }[/math]

Failure rate function for [math]\displaystyle{ \Beta\lt 1 }[/math], [math]\displaystyle{ \Beta=1 }[/math], and [math]\displaystyle{ \Beta\gt 1 }[/math].