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 [ Note 1], 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 power law relationship:
Therefore, instead of displaying a and n as the calculated parameters, the following reparameterization is used:
Given a time-varying stress x(t) and assuming the power law 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 (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:
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 power law 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 power law relationship, the median life is given by:
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.
See Also:
Time-Varying Stresses: The Cumulative Damage Model
Cumulative Damage Arrhenius Relationship
Cumulative Damage Exponential Relationship
Cumulative Damage General Log-Linear Relationship
Time-Varying Stress Model Confidence Intervals
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