Failure Infant Useful Life Period Wear-Out Phase FIGURE 93.6 Bathtub curve relationship between the failure rate function and time of the system, and the failure rate function is assumed to have a value of n during that period. i is referred to as the failure rate and is normally expressed in units of failures per hour. The reliability can be expressed in terms of the failure rate function as a differential equation of the form dr(t) Z(t)R(t) dt The general solution of this differential equation is given by R(r)=e-z(tdt If we assume that the system is in the useful-life stage where the failure rate function has a constant value 入 ition to the differential equation is an exponential function of the parameter A given by where n is the constant failure rate. The exponential relationship between the reliability and time is known as the exponential failure law, which states that for a constant failure rate function, the reliability varies exponen ally as a function of In addition to the failure rate, the mean time to failure(mttF)is a useful parameter to specify the quality of a system. The MTTF is the expected time that a system will operate before the first failure occurs. The mTTF can be calculated by finding the expected value of the time of failure From probability theory, we know that the expected value of a random variable, X, is EX1=「xf(x)dx e 2000 by CRC Press LLC© 2000 by CRC Press LLC of the system, and the failure rate function is assumed to have a value of l during that period. l is referred to as the failure rate and is normally expressed in units of failures per hour. The reliability can be expressed in terms of the failure rate function as a differential equation of the form The general solution of this differential equation is given by R(t) = e –*z(t)dt If we assume that the system is in the useful-life stage where the failure rate function has a constant value of l, the solution to the differential equation is an exponential function of the parameter l given by R (t) = e –lt where l is the constant failure rate. The exponential relationship between the reliability and time is known as the exponential failure law, which states that for a constant failure rate function, the reliability varies exponen￾tially as a function of time. In addition to the failure rate, the mean time to failure (MTTF) is a useful parameter to specify the quality of a system. The MTTF is the expected time that a system will operate before the first failure occurs. The MTTF can be calculated by finding the expected value of the time of failure. From probability theory, we know that the expected value of a random variable, X, is FIGURE 93.6 Bathtub curve relationship between the failure rate function and time. Constant Failure Rate Failure Rate Function Infant Mortality Phase Useful Life Period Time Wear-Out Phase dR t dt ztRt ( ) = – () () E X xf x dx [] () – = • • Ú