Cumulative Damage General Loglinear

Cumulative Damage General Log-Linear Relationship

This section presents a generalized formulation of the cumulative damage model where multiple stress types are used in the analysis and where the stresses can be any function of time.

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 pdf 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.

Cumulative Damage General Log-Linear - Weibull

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 given by:

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

where [math]\displaystyle{ {{\alpha }_{j}}\,\! }[/math] are model parameters.

The Weibull reliability function of the unit under multiple stresses is given by:

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

where:

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

Therefore, the pdf is:

[math]\displaystyle{ f(t,\overset{\_}{\mathop{x}}\,)=\beta s(t,\overset{\_}{\mathop{x}}\,){{\left( I(t,\overset{\_}{\mathop{x}}\,) \right)}^{\beta -1}}{{e}^{-{{\left( I(t,\overset{\_}{\mathop{x}}\,) \right)}^{\beta }}}}\,\! }[/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 [\beta s({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}}){{\left( I({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}}) \right)}^{\beta -1}}]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( I({{T}_{i}},{{\overset{\_}{\mathop{x}}\,}_{i}}) \right)}^{\beta }} -\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)}^{\beta }}+\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 },\bar{x}_{i}^{\prime \prime })= & {{e}^{-{{\left( I(T_{Li}^{\prime \prime },\bar{x}_{i}^{\prime \prime }) \right)}^{\beta }}}} \\ & R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },\bar{x}_{i}^{\prime \prime })= & {{e}^{-{{\left( I(T_{Ri}^{\prime \prime },\bar{x}_{i}^{\prime \prime }) \right)}^{\beta }}}} \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.

Cumulative Damage General Log-Linear - Lognormal

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 given by:

[math]\displaystyle{ \frac{1}{\breve{T}(t,\bar{x})}=s(t,\overset{\_}{\mathop{x}}\,)={{e}^{^{^{-{{a}_{0}}-\overset{n}{\mathop{\underset{j=1}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}(t)}}}}\,\! }[/math]

where [math]\displaystyle{ {{\alpha }_{j}}\,\! }[/math] are model parameters.

The lognormal reliability function of the unit under multiple stresses is given by:

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

where:

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

and:

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

Therefore, the pdf is:

[math]\displaystyle{ f(t,\bar{x})=\frac{s(t,\bar{x})\varphi (z(t,\bar{x}))}{\sigma _{T}^{\prime }I(t,\bar{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}},{{{\bar{x}}}_{i}})\varphi (z({{T}_{i}},{{{\bar{x}}}_{i}}))}{\sigma _{T}^{\prime }I({{T}_{i}},{{{\bar{x}}}_{i}})}] \overset{S}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\ln \left( 1-\Phi (z(T_{i}^{\prime },\bar{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 },\bar{x}_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} \\ & z_{Li}^{\prime \prime }= & \frac{\ln I(T_{Li}^{\prime \prime },\bar{x}_{i}^{\prime \prime })}{\sigma _{T}^{\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.