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| ::<math>{{F}_{2}}(t;{{S}_{2}})=1-{{e}^{-{{[KS_{2}^{n}((t-{{t}_{1}})+{{\varepsilon }_{1}})]}^{\beta }}}}\,\!</math> | | ::<math>{{F}_{2}}(t;{{S}_{2}})=1-{{e}^{-{{[KS_{2}^{n}((t-{{t}_{1}})+{{\varepsilon }_{1}})]}^{\beta }}}}\,\!</math> |
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| In other words, the probability that the units will fail at a time, <math>t\,\!</math>, while at <math>{{S}_{2}}\,\!</math> and between <math>{{t}_{1}}\,\!</math> and <math>{{t}_{1}}\,\!</math> is equivalent to the probability that the units would fail after accumulating <math>(t-{{t}_{1}})\,\!</math> plus some equivalent time, <math>{{\varepsilon }_{1}},\,\!</math> to account for the exposure the units have seen at <math>{{S}_{1}}\,\!</math>. | | In other words, the probability that the units will fail at a time, <math>t\,\!</math>, while at <math>{{S}_{2}}\,\!</math> and between <math>{{t}_{1}}\,\!</math> and <math>{{t}_{2}}\,\!</math> is equivalent to the probability that the units would fail after accumulating <math>(t-{{t}_{1}})\,\!</math> plus some equivalent time, <math>{{\varepsilon }_{1}},\,\!</math> to account for the exposure the units have seen at <math>{{S}_{1}}\,\!</math>. |
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| *The equivalent time, <math>{{\varepsilon }_{1}},\,\!</math> will be the time by which the probability of failure at <math>{{S}_{2}}\,\!</math> is equal to the probability of failure at <math>{{S}_{1}}\,\!</math> after an exposure of <math>{{t}_{1}}\,\!</math> or: | | *The equivalent time, <math>{{\varepsilon }_{1}},\,\!</math> will be the time by which the probability of failure at <math>{{S}_{2}}\,\!</math> is equal to the probability of failure at <math>{{S}_{1}}\,\!</math> after an exposure of <math>{{t}_{1}}\,\!</math> or: |
| | | |
| ::<math>\begin{align} | | ::<math>\begin{align} |
| {{F}_{1}}({{t}_{1}};{{S}_{1}})=\ & {{F}_{2}}({{\varepsilon }_{1}},{{S}_{2}}) \\ | | {{F}_{1}}({{t}_{1}};{{S}_{1}})=\ & {{F}_{2}}({{\varepsilon }_{1}};{{S}_{2}}) \\ |
| 1-{{e}^{-{{(KS_{1}^{n}{{t}_{1}})}^{\beta }}}}=\ & 1-{{e}^{-{{(KS_{2}^{n}{{\varepsilon }_{1}})}^{\beta }}}} \\ | | 1-{{e}^{-{{(KS_{1}^{n}{{t}_{1}})}^{\beta }}}}=\ & 1-{{e}^{-{{(KS_{2}^{n}{{\varepsilon }_{1}})}^{\beta }}}} \\ |
| S_{1}^{n}{{t}_{1}}=\ & S_{2}^{n}{{\varepsilon }_{1}} \\ | | S_{1}^{n}{{t}_{1}}=\ & S_{2}^{n}{{\varepsilon }_{1}} \\ |
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| where: | | where: |
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| ::<math>{{\varepsilon }_{i-1}}=({{t}_{i-1}}-{{t}_{i-2}}){{\left( \frac{{{S}_{i-1}}}{{{S}_{i}}} \right)}^{n}}+{{\varepsilon }_{i-2}}\,\!</math> | | ::<math>{{\varepsilon }_{i-1}}=({{t}_{i-1}}-{{t}_{i-2}}+{{\varepsilon }_{i-2}}){{\left( \frac{{{S}_{i-1}}}{{{S}_{i}}} \right)}^{n}}\,\!</math> |
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| *Once the ''cdf'' for each step has been obtained, the ''pdf'' can also then be determined utilizing: | | *Once the ''cdf'' for each step has been obtained, the ''pdf'' can also then be determined utilizing: |
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| ::<math>{{f}_{i}}(t,{{S}_{i}})=-\frac{d}{dt}\left[ {{F}_{i}}(t,{{S}_{i}}) \right]\,\!</math> | | ::<math>{{f}_{i}}(t,{{S}_{i}})=-\frac{d}{dt}\left[ {{F}_{i}}(t,{{S}_{i}}) \right]\,\!</math> |
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| Once the model has been formulated, model parameters (i.e., <math>K\,\!</math>, <math>n\,\!</math> and <math>\beta \,\!</math> ) can be computed utilizing maximum likelihood estimation methods. | | Once the model has been formulated, model parameters (i.e., <math>K\,\!</math>, <math>n\,\!</math> and <math>\beta \,\!</math>) can be computed utilizing maximum likelihood estimation methods. |
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| The previous example can be expanded for any time-varying stress. ALTA allows you to define any stress profile. For example, the stress can be a ramp stress, a monotonically increasing stress, sinusoidal, etc. This section presents a generalized formulation of the cumulative damage model, where stress can be any function of time. | | The previous example can be expanded for any time-varying stress. ALTA allows you to define any stress profile. For example, the stress can be a ramp stress, a monotonically increasing stress, sinusoidal, etc. This section presents a generalized formulation of the cumulative damage model, where stress can be any function of time. |
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| | <div class="noprint"> |
| {{Example:CD-GLL_Weibull}} | | {{Example:CD-GLL_Weibull}} |
| | </div> |
|
| |
|
| =Cumulative Damage Power Relationship= | | =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 <math>x(t)\,\!</math> and assuming the power law relationship, the life-stress relationship is given by:
| | See [[Cumulative Damage Power]] |
| | |
| ::<math>L(x(t))={{\left( \frac{a}{x(t)} \right)}^{n}}\,\!</math>
| |
| | |
| In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the power law relationship:
| |
| | |
| ::<math>L(x(t))={{e}^{{{\alpha }_{0}}+{{\alpha }_{1}}\ln \left( x(t) \right)}}\,\!</math>
| |
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| Therefore, instead of displaying <math>a\,\!</math> and <math>n\,\!</math> as the calculated parameters, the following reparameterization is used:
| |
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| ::<math>\begin{align}
| |
| {{\alpha }_{0}}=\ & \ln ({{a}^{n}}) \\
| |
| {{\alpha }_{1}}=\ & -n
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| \end{align}\,\!</math>
| |
| | |
| ==Cumulative Damage Power - Exponential==
| |
| Given a time-varying stress <math>x(t)\,\!</math> and assuming the power law relationship, the mean life is given by:
| |
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| ::<math>\frac{1}{m(t,\,x)}=s(t,\,x)={{\left( \frac{x(t)}{a} \right)}^{n}}\,\!</math>
| |
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| The reliability function of the unit under a single stress is given by:
| |
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| ::<math>R(t,\,x(t))={{e}^{-I(t,\,x)}}\,\!</math>
| |
| | |
| where:
| |
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| ::<math>I(t,\,x)=\underset{0}{\mathop{\overset{t}{\mathop{\mathop{}_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du\,\!</math>
| |
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| Therefore, the ''pdf'' is:
| |
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| ::<math>f(t,\,x)=s(t,\,x){{e}^{-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 (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:
| |
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| ::<math>\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}\,\!</math>
| |
| | |
| where:
| |
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| ::<math>\begin{align}
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| & 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}\,\!</math>
| |
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| and:
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|
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| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.
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| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
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| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
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|
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| *<math>S\,\!</math> is the number of groups of suspension data points.
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| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
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| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
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|
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| *<math>FI\,\!</math> is the number of interval data groups.
| |
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| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
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| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
| |
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| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
| |
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| ==Cumulative Damage Power - Weibull==
| |
| Given a time-varying stress <math>x(t)\,\!</math> and assuming the power law relationship, the characteristic life is given by:
| |
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| ::<math>\frac{1}{\eta (t,x)}=s(t,x)={{\left( \frac{x(t)}{a} \right)}^{n}}\,\!</math>
| |
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| The reliability function of the unit under a single stress is given by:
| |
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| ::<math>R(t,x(t))={{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}\,\!</math>
| |
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| where:
| |
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| ::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du\,\!</math>
| |
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| Therefore, the ''pdf'' is:
| |
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| ::<math>f(t,x)=\beta s(t,x){{\left( I(t,x) \right)}^{\beta -1}}{{e}^{-{{\left( I(t,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>\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}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>\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}\,\!</math>
| |
| | |
| and:
| |
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| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.
| |
| | |
| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
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| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
| |
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| *<math>S\,\!</math> is the number of groups of suspension data points.
| |
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| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
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| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
| |
| | |
| *<math>FI\,\!</math> is the number of interval data groups.
| |
| | |
| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
| |
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| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
| |
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| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
| |
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| '''Cumulative Damage-Power-Weibull Example'''
| |
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| Using the simple step-stress data given [[Time-Varying Stress Models#Model Formulation|here]], one would define <math>x(t)\,\!</math> as:
| |
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| ::<math>\begin{align}
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| x(t)=\ & 2,\text{ }0<t\le 250 \\
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| =\ & 3,\text{ }250<t\le 350 \\
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| =\ & 4,\text{ }350<t\le 370 \\
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| =\ & 5,\text{ }370<t\le 380 \\
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| =\ & 6,\text{ }380<t\le 390 \\
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| =\ & 7,\text{ }390<t\le +\infty
| |
| \end{align}\,\!</math>
| |
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| 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,
| |
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| ::<math>\begin{align}
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| & \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}\,\!</math>
| |
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| where:
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| *<math>F\,\!</math> is the number of exact time-to-failure data points.
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| *<math>\beta \,\!</math> is the Weibull shape parameter.
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| *<math>a\,\!</math> and <math>n\,\!</math> are the IPL parameters.
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| *<math>x(t)\,\!</math> is the stress profile function.
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| *<math>{{t}_{i}}\,\!</math> is the <math>{{i}^{th}}\,\!</math> time to failure.
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| The parameter estimates for <math>\hat{\beta }\,\!</math>, <math>\hat{a}\,\!</math> and <math>\hat{n}\,\!</math> can be obtained by simultaneously solving, <math>\tfrac{\partial \Lambda }{\partial a}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial n}=0\,\!</math>. Using ALTA, the parameter estimates for this data set are:
| |
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| ::<math>\begin{align}
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| \widehat{\beta }=\ & 2.67829 \\
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| \widehat{\alpha }=\ & 11.72208 \\
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| \widehat{n}=\ & 3.998466
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| \end{align}\,\!</math>
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| Once the parameters are obtained, one can now determine the reliability for these units at any time <math>t\,\!</math> and stress <math>x(t)\,\!</math> from:
| |
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| ::<math>R\left( t,x\left( t \right) \right)={{e}^{-{{\left[ \int_{0}^{t}{{\left[ \tfrac{x(u)}{a} \right]}^{n}}du \right]}^{\beta }}}}\,\!</math>
| |
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| or at a fixed stress level <math>x(t)=2V\,\!</math> and <math>t=300\,\!</math>,
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| ::<math>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%\,\!</math>
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| The mean time to failure (MTTF) at any stress <math>x(t)\,\!</math> can be determined by:
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| ::<math>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\,\!</math>
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| or at a fixed stress level <math>x\left( t \right)=2V\,\!</math>,
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| ::<math>MTTF\left( x\left( t \right) \right)=1046.3hrs\,\!</math>
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| Any other metric of interest (e.g., failure rate, conditional reliability etc.) can also be determined using the basic definitions given in [[Appendix A: Brief Statistical Background|Appendix A]] and calculated automatically with ALTA.
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| ==Cumulative Damage Power - Lognormal==
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| Given a time-varying stress <math>x(t)\,\!</math> and assuming the power law relationship, the median life is given by:
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| ::<math>\frac{1}{\breve{T}(t,x)}=s(t,x)={{\left( \frac{x(t)}{a} \right)}^{n}}\,\!</math>
| |
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| The reliability function of the unit under a single stress is given by:
| |
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| ::<math>\begin{align}
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| R(t,x(t))=1-\Phi (z)
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| \end{align}\,\!</math>
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| where:
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| ::<math>z(t,x)=\frac{\ln I(t,x)}{\sigma _{T}^{\prime }}\,\!</math>
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| and:
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| ::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,{{\left( \frac{x(u)}{a} \right)}^{n}}du\,\!</math>
| |
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| Therefore, the ''pdf'' is:
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| ::<math>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>\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:
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| ::<math>\begin{align}
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| & z_{Ri}^{\prime \prime }= & \frac{\ln I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} \\
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| & z_{Li}^{\prime \prime }= & \frac{\ln I(T_{Li}^{\prime \prime },x_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}
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| \end{align}\,\!</math>
| |
| | |
| and:
| |
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| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact time-to-failure data points.
| |
| | |
| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
| | |
| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
| |
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| *<math>S\,\!</math> is the number of groups of suspension data points.
| |
| | |
| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
| | |
| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
| |
| | |
| *<math>FI\,\!</math> is the number of interval data groups.
| |
| | |
| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
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| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the interval.
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| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
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|
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|
| =Cumulative Damage Arrhenius Relationship= | | =Cumulative Damage Arrhenius 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 Arrhenius life-stress relationship. Given a time-varying stress <math>x(t)\,\!</math> and assuming the Arrhenius relationship, the life-stress relationship is given by:
| | See [[Cumulative Damage Arrhenius]] |
| | |
| ::<math>L(x(t))=C{{e}^{\tfrac{b}{x(t)}}}\,\!</math>
| |
| | |
| In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the Arrhenius relationship:
| |
| | |
| ::<math>L(x(t))={{e}^{{{\alpha }_{0}}+{{\alpha }_{1}}\tfrac{1}{x(t)}}}\,\!</math>
| |
| | |
| Therefore, instead of displaying <math>C\,\!</math> and <math>b\,\!</math> as the calculated parameters, the following reparameterization is used:
| |
| | |
| ::<math>\begin{align}
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| {{\alpha }_{0}}=\ & \ln (C) \\
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| {{\alpha }_{1}}=\ & b
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| \end{align}\,\!</math>
| |
| | |
| ==Cumulative Damage Arrhenius - Exponential==
| |
| Given a time-varying stress <math>x(t)\,\!</math> and assuming the Arrhenius relationship, the mean life is:
| |
| | |
| ::<math>\frac{1}{m(t,x)}=s(t,x)=\frac{{{e}^{\tfrac{-b}{x(t)}}}}{C}\,\!</math>
| |
| | |
| The reliability function of the unit under a single stress is given by:
| |
| | |
| ::<math>\begin{align}
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| R(t,x(t))={{e}^{-I(t,x)}}
| |
| \end{align}\,\!</math>
| |
| | |
| where:
| |
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| ::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{{{e}^{\tfrac{-b}{x(u)}}}}{C}du\,\!</math>
| |
| | |
| Therefore, the ''pdf'' is:
| |
| | |
| ::<math>\begin{align}
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| f(t,x)=s(t,x){{e}^{-I(t,x)}}
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| \end{align}\,\!</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>\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}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>\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}\,\!</math>
| |
| | |
| and:
| |
| | |
| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact time-to-failure data points.
| |
| | |
| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
| | |
| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
| |
| | |
| *<math>S\,\!</math> is the number of groups of suspension data points.
| |
| | |
| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
| | |
| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
| |
| | |
| *<math>FI\,\!</math> is the number of interval data groups.
| |
| | |
| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
| |
| | |
| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| ==Cumulative Damage Arrhenius - Weibull==
| |
| Given a time-varying stress <math>x(t)\,\!</math> and assuming the Arrhenius relationship, the characteristic life is given by:
| |
| | |
| ::<math>\frac{1}{\eta (t,x)}=s(t,x)=\frac{{{e}^{\tfrac{-b}{x(t)}}}}{C}\,\!</math>
| |
| | |
| The reliability function of the unit under a single stress is given by:
| |
| | |
| ::<math>R(t,x(t))={{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int{}^{}}}\,}}\,\frac{{{e}^{\tfrac{-b}{x(u)}}}}{C}du\,\!</math>
| |
| | |
| Therefore, the ''pdf'' is:
| |
| | |
| ::<math>f(t,x)=\beta s(t,x){{\left( I(t,x) \right)}^{\beta -1}}{{e}^{-{{\left( I(t,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>\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}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>\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}^{-{{(I(T_{Ri}^{\prime \prime },x_{i}^{\prime \prime }))}^{\beta }}}}
| |
| \end{align}\,\!</math>
| |
| | |
| and:
| |
| | |
| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact time-to-failure data points.
| |
| | |
| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
| | |
| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
| |
| | |
| *<math>S\,\!</math> is the number of groups of suspension data points.
| |
| | |
| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
| | |
| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
| |
| | |
| *<math>FI\,\!</math> is the number of interval data groups.
| |
| | |
| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
| |
| | |
| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| ==Cumulative Damage Arrhenius - Lognormal==
| |
| Given a time-varying stress <math>x(t)\,\!</math> and assuming the Arrhenius relationship, the median life is given by:
| |
| | |
| ::<math>\frac{1}{\breve{T}(t,x)}=s(t,x)=\frac{{{e}^{\tfrac{-b}{x(t)}}}}{C}\,\!</math>
| |
| | |
| The reliability function of the unit under a single stress is given by:
| |
| | |
| ::<math>\begin{align}
| |
| R(t,x(t))=1-\Phi (z)
| |
| \end{align}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>z(t,x)=\frac{\ln I(t,x)}{\sigma _{T}^{\prime }}\,\!</math>
| |
| | |
| and:
| |
| | |
| ::<math>I(t,\,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{{{e}^{\tfrac{-b}{x(u)}}}}{C}du\,\!</math>
| |
| | |
| Therefore, the ''pdf'' is:
| |
| | |
| ::<math>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>\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>\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>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.
| |
| | |
| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
| | |
| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
| |
| | |
| *<math>S\,\!</math> is the number of groups of suspension data points.
| |
| | |
| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
| | |
| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
| |
| | |
| *<math>FI\,\!</math> is the number of interval data groups.
| |
| | |
| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
| |
| | |
| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
| |
|
| |
|
| =Cumulative Damage Exponential Relationship= | | =Cumulative Damage Exponential 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 exponential relationship. Given a time-varying stress <math>x(t)\,\!</math> and assuming the exponential relationship, the life-stress relationship is given by:
| | See [[Cumulative Damage Exponential]] |
| | |
| ::<math>\begin{align}
| |
| L(x(t))=C{{e}^{bx(t)}}
| |
| \end{align}\,\!</math>
| |
|
| |
| In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the exponential relationship:
| |
| | |
| Therefore, instead of displaying <math>C\,\!</math> and <math>b\,\!</math> as the calculated parameters, the following reparameterization is used:
| |
| | |
| ::<math>\begin{align}
| |
| {{\alpha }_{0}}=\ & \ln (C) \\
| |
| {{\alpha }_{1}}=\ & b
| |
| \end{align}\,\!</math>
| |
| | |
| ==Cumulative Damage Exponential - Exponential==
| |
| Given a time-varying stress <math>x(t)\,\!</math> and assuming the exponential life-stress relationship, the mean life is given by:
| |
| | |
| ::<math>\frac{1}{m(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>\begin{align}
| |
| R(t,x(t))={{e}^{-I(t,x)}}
| |
| \end{align}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}du\,\!</math>
| |
| | |
| Therefore, the ''pdf'' is:
| |
| | |
| ::<math>\begin{align}
| |
| f(t,x)=s(t,x){{e}^{-I(t,x)}}
| |
| \end{align}\,\!</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>\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}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>\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}\,\!</math>
| |
| | |
| and:
| |
| | |
| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact time-to-failure data points.
| |
| | |
| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
| | |
| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
| |
| | |
| *<math>S\,\!</math> is the number of groups of suspension data points.
| |
| | |
| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
| | |
| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
| |
| | |
| *<math>FI\,\!</math> is the number of interval data groups.
| |
| | |
| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
| |
| | |
| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| ==Cumulative Damage Exponential - Weibull==
| |
| Given a time-varying stress <math>x(t)\,\!</math> and assuming the exponential life-stress relationship, the characteristic life is given by:
| |
| | |
| ::<math>\frac{1}{\eta (t,x)}=s(t,x)=\frac{{{e}^{-b\cdot x(t)}}}{C}\,\!</math>
| |
| | |
| The reliability function of the unit under a single stress is given by:
| |
| | |
| ::<math>R(t,x(t))={{e}^{-{{\left( I(t,x) \right)}^{\beta }}}}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int_{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}du\,\!</math>
| |
| | |
| Therefore, the ''pdf'' is:
| |
| | |
| ::<math>f(t,x)=\beta s(t,x){{\left( I(t,x) \right)}^{\beta -1}}{{e}^{-{{\left( I(t,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>\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}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>\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}\,\!</math>
| |
| | |
| and:
| |
| | |
| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact time-to-failure data points.
| |
| | |
| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
| | |
| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
| |
| | |
| *<math>S\,\!</math> is the number of groups of suspension data points.
| |
| | |
| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
| | |
| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
| |
| | |
| *<math>FI\,\!</math> is the number of interval data groups.
| |
| | |
| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
| |
| | |
| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| ==Cumulative Damage Exponential - Lognormal==
| |
| Given a time-varying stress <math>x(t)\,\!</math> and assuming the exponential life-stress relationship, the median life is:
| |
| | |
| ::<math>\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>\begin{align}
| |
| R(t,x(t))=1-\Phi (z)
| |
| \end{align}\,\!</math>
| |
| | |
| where:
| |
| | |
| ::<math>z(t,x)=\frac{\ln I(t,x)}{\sigma _{T}^{\prime }}\,\!</math>
| |
| | |
| and:
| |
| | |
| ::<math>I(t,x)=\underset{0}{\mathop{\overset{t}{\mathop{\int{}^{}}}\,}}\,\frac{{{e}^{-bx(u)}}}{C}du\,\!</math>
| |
| | |
| Therefore, the ''pdf'' is:
| |
| | |
| ::<math>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>\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>\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>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.
| |
| | |
| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
| |
| | |
| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.<math>S\,\!</math>
| |
| | |
| *<math>S\,\!</math> is the number of groups of suspension data points.
| |
| | |
| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
| |
| | |
| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
| |
| | |
| *<math>FI\,\!</math> is the number of interval data groups.
| |
| | |
| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
| |
| | |
| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
| |
| | |
| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
| |
|
| |
|
| =Cumulative Damage General Log-Linear Relationship= | | =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.
| | See [[Cumulative Damage General Loglinear]] |
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| ==Cumulative Damage General Log-Linear - Exponential==
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| Given <math>n\,\!</math> time-varying stresses <math>\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))\,\!</math>, the life-stress relationship is:
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| ::<math>\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>
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|
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| where <math>{{\alpha }_{0}}\,\!</math> and <math>{{\alpha }_{j}}\,\!</math> are model parameters.
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| 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.
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| The exponential reliability function of the unit under multiple stresses is given by:
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| ::<math>R(t,\overset{\_}{\mathop{x}}\,)={{e}^{-I(t,\overset{\_}{\mathop{x}}\,)}}\,\!</math>
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| where:
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| ::<math>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>
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| Therefore, the ''pdf'' is:
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| ::<math>f(t,\overset{\_}{\mathop{x}}\,)=s(t,\overset{\_}{\mathop{x}}\,){{e}^{-I(t,\overset{\_}{\mathop{x}}\,)}}\,\!</math>
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| 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:
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| ::<math>\begin{align}
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| & \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 }]
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| \end{align}\,\!</math>
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| where:
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| ::<math>\begin{align}
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| & 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 })}} \\
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| & 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 })}}
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| \end{align}\,\!</math>
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| and:
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| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact time-to-failure data points.
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| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
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| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
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| *<math>S\,\!</math> is the number of groups of suspension data points.
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| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
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| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
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| *<math>FI\,\!</math> is the number of interval data groups.
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| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
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| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
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| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
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| ==Cumulative Damage General Log-Linear - Weibull==
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| Given <math>n\,\!</math> time-varying stresses <math>\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))\,\!</math>, the life-stress relationship is given by:
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| ::<math>\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>
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| where <math>{{\alpha }_{j}}\,\!</math> are model parameters.
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| The Weibull reliability function of the unit under multiple stresses is given by:
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| ::<math>R(t,\overset{\_}{\mathop{x}}\,)={{e}^{-{{\left( I(t,\overset{\_}{\mathop{x}}\,) \right)}^{\beta }}}}\,\!</math>
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| where:
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| ::<math>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>
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| Therefore, the ''pdf'' is:
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| ::<math>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>
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| 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:
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| ::<math>\begin{align}
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| & \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 }]
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| \end{align}\,\!</math>
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| where:
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| ::<math>\begin{align}
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| & 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 }}}} \\
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| & 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 }}}}
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| \end{align}\,\!</math>
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| and:
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| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact time-to-failure data points.
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| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
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| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
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| *<math>S\,\!</math> is the number of groups of suspension data points.
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| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
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| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
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| *<math>FI\,\!</math> is the number of interval data groups.
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| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
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| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
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| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
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| ==Cumulative Damage General Log-Linear - Lognormal==
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| Given <math>n\,\!</math> time-varying stresses <math>\underline{X}=({{X}_{1}}(t),{{X}_{2}}(t)...{{X}_{n}}(t))\,\!</math>, the life-stress relationship is given by:
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| ::<math>\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>
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| where <math>{{\alpha }_{j}}\,\!</math> are model parameters.
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| The lognormal reliability function of the unit under multiple stresses is given by:
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| ::<math>R(t,\bar{x})=1-\Phi (z(t,\bar{x}))\,\!</math>
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| where:
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| ::<math>z(t,\bar{x})=\frac{\ln I(t,\bar{x})}{\sigma _{T}^{\prime }}\,\!</math>
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| and:
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| ::<math>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>
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| Therefore, the ''pdf'' is:
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| ::<math>f(t,\bar{x})=\frac{s(t,\bar{x})\varphi (z(t,\bar{x}))}{\sigma _{T}^{\prime }I(t,\bar{x})}\,\!</math>
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| 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:
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| ::<math>\begin{align}
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| & \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 })]
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| \end{align}\,\!</math>
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| where:
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| ::<math>\begin{align}
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| & z_{Ri}^{\prime \prime }= & \frac{\ln I(T_{Ri}^{\prime \prime },\bar{x}_{i}^{\prime \prime })}{\sigma _{T}^{\prime }} \\
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| & z_{Li}^{\prime \prime }= & \frac{\ln I(T_{Li}^{\prime \prime },\bar{x}_{i}^{\prime \prime })}{\sigma _{T}^{\prime }}
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| \end{align}\,\!</math>
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| and:
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| *<math>{{F}_{e}}\,\!</math> is the number of groups of exact time-to-failure data points.
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| *<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.
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| *<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.
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| *<math>S\,\!</math> is the number of groups of suspension data points.
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| *<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.
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| *<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.
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| *<math>FI\,\!</math> is the number of interval data groups.
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| *<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.
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| *<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.
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| *<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.
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| =Confidence Intervals= | | =Confidence Intervals= |
| Using the same methodology as in previous sections, approximate confidence intervals can be derived and applied to all results of interest using the Fisher Matrix approach discussed in [[Appendix A: Brief Statistical Background|Appendix A]]. ALTA utilizes such intervals on all results. The formulas for such intervals are beyond the scope of this reference and are thus omitted. Interested readers can contact ReliaSoft for internal document ALTA-CBCD, detailing these derivations. | | Using the same methodology as in previous sections, approximate confidence intervals can be derived and applied to all results of interest using the Fisher Matrix approach discussed in [[Appendix A: Brief Statistical Background|Appendix A]]. ALTA utilizes such intervals on all results. |
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| =Notes on Trigonometric Functions= | | =Notes on Trigonometric Functions= |
| Trigonometric functions sometime are used in accelerated life tests. However ALTA does not include them. In fact, a trigonometric function can be defined by its frequency and magnitude. Frequency and magnitude then can be treated as two constant stresses. The GLL model discussed in [[General Log-Linear Relationship]] then can be applied for modeling. | | Trigonometric functions sometime are used in accelerated life tests. However ALTA does not include them. In fact, a trigonometric function can be defined by its frequency and magnitude. Frequency and magnitude then can be treated as two constant stresses. The GLL model discussed in [[General Log-Linear Relationship]] then can be applied for modeling. |
