Cumulative Damage Power

=Cumulative Damage Power Relationship= This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and the life-stress relationship is based on the power relationship. Given a time-varying stress $$x(t)\,\!$$ and assuming the power law relationship, the life-stress relationship is given by:


 * $$L(x(t))={{\left( \frac{a}{x(t)} \right)}^{n}}\,\!$$

In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the power law relationship:


 * $$L(x(t))={{e}^{{{\alpha }_{0}}+{{\alpha }_{1}}\ln \left( x(t) \right)}}\,\!$$

Therefore, instead of displaying $$a\,\!$$ and $$n\,\!$$ as the calculated parameters, the following reparameterization is used:


 * $$\begin{align}

{{\alpha }_{0}}=\ & \ln ({{a}^{n}}) \\ {{\alpha }_{1}}=\ & -n \end{align}\,\!$$

Cumulative Damage Power - Exponential
Given a time-varying stress $$x(t)\,\!$$ and assuming the power law relationship, the mean life is given by:


 * $$\frac{1}{m(t,\,x)}=s(t,\,x)={{\left( \frac{x(t)}{a} \right)}^{n}}\,\!$$

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


 * $$R(t,\,x(t))={{e}^{-I(t,\,x)}}\,\!$$

where:


 * $$I(t,\,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int{}_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du\,\!$$

Therefore, the pdf is:


 * $$f(t,\,x)=s(t,\,x){{e}^{-I(t,\,x)}}\,\!$$

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 (e.g., mean life, failure rate, etc.) can be obtained utilizing the statistical properties definitions presented in previous chapters. The log-likelihood equation is as follows:


 * $$\begin{align}

& \ln (L)= & \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [s({{T}_{i}},\,{{x}_{i}})]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\left( I({{T}_{i}},\,{{x}_{i}}) \right) -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }\left( I(T_{i}^{\prime },\,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}\,\!$$

where:


 * $$\begin{align}

& R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },\,x_{i}^{\prime \prime })= & {{e}^{-I(T_{Li}^{\prime \prime },\,x_{i}^{\prime \prime })}} \\ & R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },\,x_{i}^{\prime \prime })= & {{e}^{-I(T_{Ri}^{\prime \prime },\,x_{i}^{\prime \prime })}} \end{align}\,\!$$

and:
 * $${{F}_{e}}\,\!$$ is the number of groups of exact times-to-failure data points.


 * $${{N}_{i}}\,\!$$ is the number of times-to-failure in the $${{i}^{th}}\,\!$$ time-to-failure data group.


 * $${{T}_{i}}\,\!$$ is the exact failure time of the $${{i}^{th}}\,\!$$ group.
 * $$S\,\!$$ is the number of groups of suspension data points.


 * $$N_{i}^{\prime }\,\!$$ is the number of suspensions in the $${{i}^{th}}\,\!$$ group of suspension data points.


 * $$T_{i}^{\prime }\,\!$$ is the running time of the $${{i}^{th}}\,\!$$ suspension data group.
 * $$FI\,\!$$ is the number of interval data groups.


 * $$N_{i}^{\prime \prime }\,\!$$ is the number of intervals in the $${{i}^{th}}\,\!$$ group of data intervals.


 * $$T_{Li}^{\prime \prime }\,\!$$ is the beginning of the $${{i}^{th}}\,\!$$ interval.


 * $$T_{Ri}^{\prime \prime }\,\!$$ is the ending of the $${{i}^{th}}\,\!$$ interval.

Cumulative Damage Power - Weibull
Given a time-varying stress $$x(t)\,\!$$ and assuming the power law relationship, the characteristic life is given by:


 * $$\frac{1}{\eta (t,x)}=s(t,x)={{\left( \frac{x(t)}{a} \right)}^{n}}\,\!$$

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


 * $$R(t,x(t))={{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}\,\!$$

where:


 * $$I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du\,\!$$

Therefore, the pdf is:


 * $$f(t,x)=\beta s(t,x){{\left( I(t,x) \right)}^{\beta -1}}{{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}\,\!$$

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:


 * $$\begin{align}

& \ln (L)= \Lambda =\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}\ln [\beta s({{T}_{i}},{{x}_{i}}){{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta -1}}]-\overset{Fe}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,{{N}_{i}}{{\left( I({{T}_{i}},{{x}_{i}}) \right)}^{\beta }} -\overset{S}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime }{{\left( I(T_{i}^{\prime },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}\,\!$$

where:


 * $$\begin{align}

& R_{Li}^{\prime \prime }(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })= & {{e}^{-{{\left( I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime }) \right)}^{\beta }}}} \\ & R_{Ri}^{\prime \prime }(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })= & {{e}^{-{{\left( I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime }) \right)}^{\beta }}}} \end{align}\,\!$$

and:


 * $${{F}_{e}}\,\!$$ is the number of groups of exact times-to-failure data points.


 * $${{N}_{i}}\,\!$$ is the number of times-to-failure in the $${{i}^{th}}\,\!$$ time-to-failure data group.


 * $${{T}_{i}}\,\!$$ is the exact failure time of the $${{i}^{th}}\,\!$$ group.


 * $$S\,\!$$ is the number of groups of suspension data points.


 * $$N_{i}^{\prime }\,\!$$ is the number of suspensions in the $${{i}^{th}}\,\!$$ group of suspension data points.


 * $$T_{i}^{\prime }\,\!$$ is the running time of the $${{i}^{th}}\,\!$$ suspension data group.


 * $$FI\,\!$$ is the number of interval data groups.


 * $$N_{i}^{\prime \prime }\,\!$$ is the number of intervals in the $${{i}^{th}}\,\!$$ group of data intervals.


 * $$T_{Li}^{\prime \prime }\,\!$$ is the beginning of the $${{i}^{th}}\,\!$$ interval.


 * $$T_{Ri}^{\prime \prime }\,\!$$ is the ending of the $${{i}^{th}}\,\!$$ interval.

Cumulative Damage-Power-Weibull Example

Using the simple step-stress data given here, one would define $$x(t)\,\!$$ as:


 * $$\begin{align}

x(t)=\ & 2,\text{   }0<t\le 250 \\ =\ & 3,\text{   }250<t\le 350 \\ =\ & 4,\text{   }350<t\le 370 \\ =\ & 5,\text{   }370<t\le 380 \\ =\ & 6,\text{   }380<t\le 390 \\ =\ & 7,\text{   }390<t\le +\infty \end{align}\,\!$$

Assuming a power relation as the underlying life-stress relationship and the Weibull distribution as the underlying life distribution, one can then formulate the log-likelihood function for the above data set as,


 * $$\begin{align}

& \ln (L) = \Lambda =\overset{F}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,\ln \left\{ \beta {{\left[ \frac{x(t)}{a} \right]}^{n}}{{\left[ \int_{0}^{{{t}_{i}}}{{\left[ \frac{\left[ x(u) \right]}{a} \right]}^{n}}du \right]}^{\beta -1}} \right\} -\overset{F}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,\left\{ {{\left[ \int_{0}^{{{t}_{i}}}{{\left[ \frac{\left[ x(u) \right]}{a} \right]}^{n}}du \right]}^{\beta }} \right\} \end{align}\,\!$$

where:


 * $$F\,\!$$ is the number of exact time-to-failure data points.


 * $$\beta \,\!$$ is the Weibull shape parameter.


 * $$a\,\!$$ and $$n\,\!$$ are the IPL parameters.


 * $$x(t)\,\!$$ is the stress profile function.


 * $${{t}_{i}}\,\!$$ is the $${{i}^{th}}\,\!$$ time to failure.

The parameter estimates for $$\hat{\beta }\,\!$$, $$\hat{a}\,\!$$ and $$\hat{n}\,\!$$ can be obtained by simultaneously solving, $$\tfrac{\partial \Lambda }{\partial a}=0\,\!$$ and $$\tfrac{\partial \Lambda }{\partial n}=0\,\!$$. Using ALTA, the parameter estimates for this data set are:


 * $$\begin{align}

\widehat{\beta }=\ & 2.67829 \\ \widehat{\alpha }=\ & 11.72208 \\ \widehat{n}=\ & 3.998466 \end{align}\,\!$$

Once the parameters are obtained, one can now determine the reliability for these units at any time $$t\,\!$$ and stress $$x(t)\,\!$$ from:


 * $$R\left( t,x\left( t \right) \right)={{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}}\,\!$$

or at a fixed stress level $$x(t)=2 \text{ V}\,\!$$ and $$t=300 \text{ hours}\,\!$$,


 * $$R\left( t=300,x(t)=2 \right)={{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}}=97.5%\,\!$$

The mean time to failure $$(MTTF)\,\!$$ at any stress $$x(t)\,\!$$ can be determined by:


 * $$MTTF\left( x\left( t \right) \right)=\int_{0}^{\infty }t\left[ \left\{ \beta {{\left[ \frac{x\left( t \right)}{a} \right]}^{n}}{{\left[ \int_{0}^{t}{{\left[ \frac{x\left( u \right)}{a} \right]}^{n}}du \right]}^{\beta -1}} \right\}{{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}} \right]dt\,\!$$

or at a fixed stress level $$x\left( t \right)=2 \text{ V}\,\!$$,


 * $$MTTF\left( x\left( t \right) \right)=1046.3 \text{ hours}\,\!$$

Any other metric of interest (e.g., failure rate, conditional reliability etc.) can also be determined using the basic definitions given in Appendix A and calculated automatically with ALTA.

Cumulative Damage Power - Lognormal
Given a time-varying stress $$x(t)\,\!$$ and assuming the power law relationship, the median life is given by:


 * $$\frac{1}{\breve{T}(t,x)}=s(t,x)={{\left( \frac{x(t)}{a} \right)}^{n}}\,\!$$

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


 * $$\begin{align}

R(t,x(t))=1-\Phi (z) \end{align}\,\!$$

where:


 * $$z(t,x)=\frac{\ln I(t,x)}{\sigma _{T}^{\prime }}\,\!$$

and:


 * $$I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du\,\!$$

Therefore, the pdf is:


 * $$f(t,x)=\frac{s(t,x)\varphi (z(t,x))}{\sigma _{T}^{\prime }I(t,x)}\,\!$$

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:


 * $$\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}\,\!$$

where:


 * $$\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}\,\!$$

and:


 * $${{F}_{e}}\,\!$$ is the number of groups of exact time-to-failure data points.


 * $${{N}_{i}}\,\!$$ is the number of times-to-failure in the $${{i}^{th}}\,\!$$ time-to-failure data group.


 * $${{T}_{i}}\,\!$$ is the exact failure time of the $${{i}^{th}}\,\!$$ group.


 * $$S\,\!$$ is the number of groups of suspension data points.


 * $$N_{i}^{\prime }\,\!$$ is the number of suspensions in the $${{i}^{th}}\,\!$$ group of suspension data points.


 * $$T_{i}^{\prime }\,\!$$ is the running time of the $${{i}^{th}}\,\!$$ suspension data group.


 * $$FI\,\!$$ is the number of interval data groups.


 * $$N_{i}^{\prime \prime }\,\!$$ is the number of intervals in the $${{i}^{th}}\,\!$$ group of data intervals.


 * $$T_{Li}^{\prime \prime }\,\!$$ is the beginning of the   interval.


 * $$T_{Ri}^{\prime \prime }\,\!$$ is the ending of the $${{i}^{th}}\,\!$$ interval.