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 x(t) and assuming the Arrhenius 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 Arrhenius 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 Arrhenius relationship, the mean life is:
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 Arrhenius 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:
here:
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 Arrhenius 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 Power Relationship
Cumulative Damage Exponential Relationship
Cumulative Damage General Log-Linear Relationship
Time-Varying Stress Model Confidence Intervals
Go
to Weibull.com
Go to ReliaSoft.com
©1998-2010. ReliaSoft Corporation. ALL RIGHTS RESERVED.