Template:Cd exponential lognormal

Cumulative Damage Exponential - Lognormal

Given a time-varying stress [math]\displaystyle{ x(t) }[/math] and assuming the exponential life-stress relationship, the median life is:

[math]\displaystyle{ \frac{1}{\breve{T}(t,x)}=s(t,x)=\frac{{{e}^{-bx(t)}}}{C} }[/math]

The reliability function of the unit under a single stress is given by:

[math]\displaystyle{ R(t,x(t))=1-\Phi (z) }[/math]

where:

[math]\displaystyle{ z(t,x)=\frac{\ln I(t,x)}{\sigma _{T}^{\prime }} }[/math]

and:

[math]\displaystyle{ I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}du }[/math]

Therefore, the [math]\displaystyle{ pdf }[/math] is:

[math]\displaystyle{ f(t,x)=\frac{s(t,x)\varphi (z(t,x))}{\sigma _{T}^{\prime }I(t,x)} }[/math]

Parameter estimation can be accomplished via maximum likelihood estimation methods, and confidence intervals can be approximated using the Fisher matrix approach. Once the parameters are determined, all other characteristics of interest can be obtained utilizing the statistical properties definitions (e.g. mean life, failure rate, etc.) presented in previous chapters. The log-likelihood equation is as follows:

[math]\displaystyle{ \begin{align} & \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\frac{s({{T}_{i}},{{x}_{i}})\varphi (z({{T}_{i}},{{x}_{i}}))}{\sigma _{T}^{\prime }I({{T}_{i}},{{x}_{i}})}] \overset{S}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\ln \left( 1-\Phi (z(T_{i}^{\prime },x_{i}^{\prime })) \right)+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })] \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & z_{Ri}^{\prime \prime }= & \frac{\ln I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} \\ & z_{Li}^{\prime \prime }= & \frac{\ln I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} \end{align} }[/math]

and:
• [math]\displaystyle{ {{F}_{e}} }[/math] is the number of groups of exact times-to-failure data points.
• [math]\displaystyle{ {{N}_{i}} }[/math] is the number of times-to-failure in the [math]\displaystyle{ {{i}^{th}} }[/math] time-to-failure data group.
• [math]\displaystyle{ {{T}_{i}} }[/math] is the exact failure time of the [math]\displaystyle{ {{i}^{th}} }[/math] group.[math]\displaystyle{ S }[/math]
• is the number of groups of suspension data points.
• [math]\displaystyle{ N_{i}^{\prime } }[/math] is the number of suspensions in the [math]\displaystyle{ {{i}^{th}} }[/math] group of suspension data points.
• [math]\displaystyle{ T_{i}^{\prime } }[/math] is the running time of the [math]\displaystyle{ {{i}^{th}} }[/math] suspension data group.
• [math]\displaystyle{ FI }[/math] is the number of interval data groups.
• [math]\displaystyle{ N_{i}^{\prime \prime } }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}} }[/math] group of data intervals.
• [math]\displaystyle{ T_{Li}^{\prime \prime } }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.
• [math]\displaystyle{ T_{Ri}^{\prime \prime } }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.