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 x(t) and assuming the exponential relationship, the life-stress relationship is given by:
In ALTA PRO, 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 C and b as the calculated parameters, the following reparameterization is used:
See the following sections:
Given a time-varying stress x(t) and assuming the exponential life-stress relationship, the mean life is given by:
The reliability function of the unit under a single stress is given by:
where:
Therefore, the pdf is:
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:
where:
and:
Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure in the ith time-to-failure data group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is
the number of suspensions in the ith group of
suspension data points.
is
the running time of the ith suspension
data group.
FI is the number of interval data groups.
is
the number of intervals in the ith group of
data intervals.
is
the beginning of the ith
interval.
is
the ending of the ith
interval.
Given a time-varying stress x(t) and assuming the exponential life-stress relationship, the characteristic life is given by:
The reliability function of the unit under a single stress is given by:
where:
Therefore, the pdf is:
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:
where:
and:
Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure in the ith time-to-failure data group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is
the number of suspensions in the ith group of
suspension data points.
is
the running time of the ith suspension
data group.
FI is the number of interval data groups.
is
the number of intervals in the ith group of
data intervals.
is
the beginning of the ith
interval.
is
the ending of the ith
interval.
Given a time-varying stress x(t) and assuming the exponential life-stress relationship, the median life is:
The reliability function of the unit under a single stress is given by:
where:
and:
Therefore, the pdf is:
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:
where:
and:
Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure in the ith time-to-failure data group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is
the number of suspensions in the ith group of
suspension data points.
is
the running time of the ith suspension
data group.
FI is the number of interval data groups.
is
the number of intervals in the ith group of
data intervals.
is
the beginning of the ith
interval.
is
the ending of the ith
interval.
Example
Using the simple step-stress data given in the Time-Varying Stress Model Formulation section, one would define x(t) as,
and the times-to-failure t as 280, 310, 330, 352, 360, 366, 371, 374, 378, 381, 385.
Assuming a power relation as the underlying life-stress relation and the Weibull distribution as the underlying life distribution, one can then formulate the log-likelihood function for the above data set as,
where:
F is the number of exact time-to-failure data points.
β is the Weibull shape parameter.
a and n are the IPL parameters.
x(t) is the stress profile function.
ti is the ith time-to-failure.
The parameter
estimates for 
, 
and
can be obtained by simultaneously solving 
= 0, 
= 0 and 
= 0.
Utilizing ALTA 7 PRO, the parameter estimates
for this data set are:
= 2.68
= 11.72
= 4.00
Once the parameters are obtained, one can now determine the reliability for these units at any time t and stress x(t) from:
or at a fixed stress level x(t) = 2V and t = 300,
The mean time to failure (MTTF) at any stress x(t) can be determined by:
or at a fixed stress level x(t) = 2V,
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 through ALTA 7 PRO.
See
Also:
Time-Varying Stresses: The Cumulative Damage Model
Cumulative Damage Power Relationship
Cumulative Damage Arrhenius Relationship
Cumulative Damage General Log-Linear Relationship
Time-Varying Stress Model Confidence Intervals
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