Temperature-Nonthermal (TNT)-Weibull Model

This example validates the calculation of the temperature-nonthermal relationship for Weibull distribution.

Data is from Table 7.10 on page 300 in book Life Cycle Reliability Engineering by Dr. Guangbin Yang, John Wiley & Sons, 2007.

Temperature and switching rate are the two stresses used in the accelerated life test for a type of 18-V compact electromagnetic relays. The cycles to failure are provided next.

The following temperature non-thermal life stress relationship is used:


 * $$\,\!L\left ( f,T \right )=Af^{B}e^{\left ( \frac{E_{a}}{kT} \right )}$$

where $$\,\!f$$ is the switching rate, $$\,\!T$$ is temperature. $$\,\!L\left ( f,T \right )$$ is the life characteristic affected by the two stresses. This relationship is called temperature non-thermal model in ALTA.

This relationship also can be expressed as the following:


 * $$\,\!ln\left ( L\left ( x_{1},x_{2} \right ) \right )=\alpha _{0}+\alpha _{1}x_{1}+\alpha _{2}x_{2}$$

where $$\,\!x_{1}=\frac{1}{T}$$ and $$\,\!x_{2}=ln\left ( f \right )$$. This is the General log-linear model with the proper stress transformation in ALTA.



The failure time distribution is a Weibull distribution. The book has the following results:


 * The maximum likelihood estimation (MLE) results for the parameters are: $$\,\!\alpha _{0}=0.671$$, $$\,\!\alpha _{1}=4640.1$$ , $$\,\!\alpha _{2}=-0.445$$ and $$\,\!\beta =1.805$$.


 * The eta parameter in the Weibull distribution at temperature of 30°C (303.15 K) and switching rate of 5 cycles/minute is estimated as $$\,\!4.244\times 10^{6}$$.


 * The estimated reliability at 200,000 cycles and temperature of 30°C (303.15 K) and switching rate of 5 cycles/minute is 0.996. Its one-sided lower 90% confidence bound is 0.992.
 * The two-sided 90% confidence interval for parameter $$\,\!\alpha _{2}$$ is [-0.751, -0.160].


 * The maximum likelihood estimation (MLE) results for the parameters are:



These results (especially $$\,\!\alpha _{2}$$) are slightly different from the one given in the book. If we use the results in the book, then the calculated log likelihood value is -710.354601 as given below.



This likelihood value is slightly smaller than the value given in ALTA which is -710.268519. Therefore, the result in ALTA is better in terms of maximizing the log likelihood value.




 * The η parameter in the Weibull distribution at temperature of 30°C (303.15 K) and switching rate of 5 cycles/minute is estimated as $$\,\!4.172\times 10^{6}$$.




 * The estimated reliability at 200,000 cycles and temperature of 30°C (303.15 F) and switching rate of 5 cycles/minute is 0.996. Its one-sided lower 90% confidence bound is 0.992.




 * The two-sided 90% confidence interval for parameter $$\,\!\alpha _{2}$$ is [-0.751, -0.160].



If the temperature-Nonthermal relationship is used directly, the same results will be obtained. The following is the estimated model parameters for the temperature-nonthermal model in ALTA. By doing the right transformations for the stresses, a general log linear model can become a temperature-nonthermal model.