Template:Cd gll exponential

Cumulative Damage General Log-Linear - Exponential

Given [math]\displaystyle{ n }[/math] time-varying stresses [math]\displaystyle{ \underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t)) }[/math] , the life-stress relationship is:

[math]\displaystyle{ \frac{1}{m\left( t,\overset{\_}{\mathop{x}}\, \right)}=s(t,\overset{\_}{\mathop{x}}\,)={{e}^{-{{a}_{0}}-\underset{j=1}{\mathop{\overset{n}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{a}_{j}}{{x}_{j}}(t)}} }[/math]

where [math]\displaystyle{ {{\alpha }_{0}} }[/math] and [math]\displaystyle{ {{\alpha }_{j}} }[/math] are model parameters. This relationship can be further modified through the use of transformations and can be reduced to the relationships discussed previously (power, Arrhenius and exponential), if so desired. The exponential reliability function of the unit under multiple stresses is given by:

[math]\displaystyle{ R(t,\overset{\_}{\mathop{x}}\,)={{e}^{-I(t,\overset{\_}{\mathop{x}}\,)}} }[/math]

where:

[math]\displaystyle{ I(t,\overset{\_}{\mathop{x}}\,)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{du}{{{e}^{^{^{{{\alpha }_{0}}+\overset{n}{\mathop{\underset{j=1}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}(t)}}}}} }[/math]

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

[math]\displaystyle{ f(t,\overset{\_}{\mathop{x}}\,)=s(t,\overset{\_}{\mathop{x}}\,){{e}^{-I(t,\overset{\_}{\mathop{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 [s({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}})]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( I({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}}) \right) -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\left( I(T_{i}^{\prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime }) \right)+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }] \end{align} }[/math]

where:

[math]\displaystyle{ \begin{align} & R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })= & {{e}^{-I(T_{Li}^{\prime \prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })}} \\ & R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })= & {{e}^{-I(T_{Ri}^{\prime \prime },\overset{\_}{\mathop{x}}\,_{i}^{\prime \prime })}} \end{align} }[/math]

and:
• [math]\displaystyle{ {{F}_{e}} }[/math] is the number of groups of exact time-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.