|
|
| (5 intermediate revisions by 3 users not shown) |
| Line 1: |
Line 1: |
| ==Eyring-Exponential==
| | #REDIRECT [[Eyring_Relationship#Eyring-Exponential]] |
| | |
| <br>
| |
| The <math>pdf</math> of the 1-parameter exponential distribution is given by:
| |
| | |
| <br>
| |
| ::<math>f(t)=\lambda \cdot {{e}^{-\lambda \cdot t}}</math>
| |
| | |
| <br>
| |
| It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:
| |
| | |
| <br>
| |
| ::<math>\lambda =\frac{1}{m}</math>
| |
| | |
| <br>
| |
| :thus:
| |
| | |
| <br>
| |
| ::<math>f(t)=\frac{1}{m}\cdot {{e}^{-\tfrac{t}{m}}}</math>
| |
| | |
| <br>
| |
| The Eyring-exponential model <math>pdf</math> can then be obtained by setting <math>m=L(V)</math> in Eqn. (eyring):
| |
| | |
| <br>
| |
| ::<math>m=L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
| |
| | |
| <br>
| |
| and substituting for <math>m</math> in Eqn. (pdfexpm2):
| |
| | |
| <br>
| |
| ::<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>
| |
| <br>
| |
| ===Eyring-Exponential Statistical Properties Summary===
| |
| ====Mean or MTTF====
| |
| <br>
| |
| The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) for the Eyring-exponential is given by:
| |
| | |
| <br>
| |
| ::<math>\begin{align}
| |
| & \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t,V)dt=\mathop{}_{0}^{\infty }t\cdot V{{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-tV{{e}^{\left( A-\tfrac{B}{V} \right)}}}}dt \\
| |
| & = & \frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}
| |
| \end{align}</math>
| |
|
| |
| ====Median====
| |
| <br>
| |
| | |
| The median, <math>\breve{T},</math>
| |
| for the Eyring-exponential model is given by:
| |
| | |
| <br>
| |
| | |
| ::<math>\breve{T}=0.693\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
| |
| | |
| <br>
| |
| | |
| ====Mode====
| |
| <br>
| |
| The mode, <math>\tilde{T},</math>
| |
| for the Eyring-exponential model is <math>\tilde{T}=0.</math>
| |
| <br>
| |
| ====Standard Deviation====
| |
| <br>
| |
| The standard deviation, <math>{{\sigma }_{T}}</math>, for the Eyring-exponential model is given by:
| |
| <br>
| |
| ::<math>{{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
| |
| <br>
| |
| ====Eyring-Exponential Reliability Function====
| |
| <br>
| |
| The Eyring-exponential reliability function is given by:
| |
| | |
| <br>
| |
| ::<math>R(T,V)={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
| |
| | |
| <br>
| |
| This function is the complement of the Eyring-exponential cumulative distribution function or:
| |
| | |
| <br>
| |
| ::<math>R(T,V)=1-Q(T,V)=1-\mathop{}_{0}^{T}f(T,V)dT</math>
| |
| | |
| <br>
| |
| :and:
| |
| | |
| <br>
| |
| ::<math>R(T,V)=1-\mathop{}_{0}^{T}V{{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}dT={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
| |
| | |
| ====Conditional Reliability====
| |
| <br>
| |
| The conditional reliability function for the Eyring-exponential model is given by:
| |
| <br>
| |
| ::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
| |
| <br>
| |
| ====Reliable Life====
| |
| <br>
| |
| For the Eyring-exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R,}}</math> is given by:
| |
| <br>
| |
| ::<math>R({{t}_{R}},V)={{e}^{-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
| |
| <br>
| |
| ::<math>\ln [R({{t}_{R}},V)]=-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>
| |
| <br>
| |
| :or:
| |
| <br>
| |
| ::<math>{{t}_{R}}=-\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\ln [R({{t}_{R}},V)]</math>
| |
| | |
| ===Parameter Estimation===
| |
| <br>
| |
| ====Maximum Likelihood Estimation Method====
| |
| <br>
| |
| | |
| The complete exponential log-likelihood function of the Eyring model is composed of two summation portions:
| |
| | |
| <br>
| |
| ::<math>\begin{align}
| |
| & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ {{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}{{e}^{-{{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}\cdot {{T}_{i}}}} \right] \\
| |
| & & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\cdot {{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}\cdot T_{i}^{\prime }+\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>
| |
| | |
| <br>
| |
| :where:
| |
| | |
| <br>
| |
| ::<math>R_{Li}^{\prime \prime }={{e}^{-T_{Li}^{\prime \prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}}}}</math>
| |
| | |
| <br>
| |
| ::<math>R_{Ri}^{\prime \prime }={{e}^{-T_{Ri}^{\prime \prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}}}}</math>
| |
| | |
| <br>
| |
| :and:
| |
| <br>
| |
| • <math>{{F}_{e}}</math> is the number of groups of exact times-to-failure data points.
| |
| <br>
| |
| • <math>{{N}_{i}}</math> is the number of times-to-failure in the <math>{{i}^{th}}</math> time-to-failure data group.
| |
| <br>
| |
| • <math>{{V}_{i}}</math> is the stress level of the <math>{{i}^{th}}</math> group.
| |
| <br>
| |
| • <math>A</math> is the Eyring parameter (unknown, the first of two parameters to be estimated).
| |
| <br>
| |
| • <math>B</math> is the second Eyring parameter (unknown, the second of two parameters to be estimated).
| |
| <br>
| |
| • <math>{{T}_{i}}</math> is the exact failure time of the <math>{{i}^{th}}</math> group.
| |
| <br>
| |
| • <math>S</math> is the number of groups of suspension data points.
| |
| <br>
| |
| • <math>N_{i}^{\prime }</math> is the number of suspensions in the <math>{{i}^{th}}</math> group of suspension data points.
| |
| <br>
| |
| • <math>T_{i}^{\prime }</math> is the running time of the <math>{{i}^{th}}</math> suspension data group.
| |
| <br>
| |
| • <math>FI</math> is the number of interval data groups.
| |
| <br>
| |
| • <math>N_{i}^{\prime \prime }</math> is the number of intervals in the i <math>^{th}</math> group of data intervals.
| |
| <br>
| |
| • <math>T_{Li}^{\prime \prime }</math> is the beginning of the i <math>^{th}</math> interval.
| |
| <br>
| |
| • <math>T_{Ri}^{\prime \prime }</math> is the ending of the i <math>^{th}</math> interval.
| |
| <br>
| |
| The solution (parameter estimates) will be found by solving for the parameters <math>\widehat{A}</math> and <math>\widehat{B}</math> so that <math>\tfrac{\partial \Lambda }{\partial A}=0</math> and <math>\tfrac{\partial \Lambda }{\partial B}=0</math> where:
| |
| | |
| | |
| <br>
| |
| ::<math>\begin{align}
| |
| & \frac{\partial \Lambda }{\partial A}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( 1-{{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}{{T}_{i}} \right)-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}T_{i}^{\prime } \\
| |
| & & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\left( T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime } \right){{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}}}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
| |
| \end{align}</math>
| |
| | |
| <br>
| |
| ::<math>\begin{align}
| |
| & \frac{\partial \Lambda }{\partial B}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left[ {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}{{T}_{i}}-\frac{1}{{{V}_{i}}} \right]+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}T_{i}^{\prime } \\
| |
| & & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\left( T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime } \right){{e}^{A-\tfrac{B}{{{V}_{i}}}}}}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
| |
| \end{align}</math>
| |
| | |
| <br>
| |