3. PROBABILITY DISTRIBUTION
There are many probability distributions that can be utilized to express the fittest distribution of a set of TTF data. The probability distributions that will be selected to model the failure pattern of the diesel engines are normal distribution, exponential distribution, and Weibull distribution.
The normal distribution is chosen based upon central limit theorem assumption. The exponential distribution is selected due to its failure characteristics represent the useful life period in a bath-up curve. While the Weibull distribution is selected due to the flexibility of the determination of the parameter that can be well described the failure pattern of the data that might be laid either on useful life period or wear out period.
3.1. Exponential Distribution
If time to failure T of a component is exponentially distributed with parameter is λ and γ, then the pdf of the T is given by
f(t) = λe-λ(t-γ) (9)
where
λ = failure rate
γ = shape parameter
The reliability function of exponential distribution becomes
R(t) = e-λ(t-γ) (10)
while the failure rate function of the exponential distribution is given by
z(t) = λ (11)
Equation (9) is recognized as two-parameter exponential distribution. A one-parameter exponential distribution is obtained by letting shape parameter, γ, equals to zero.
3.2. Weibull Distribution
The Weibull distribution is one of the most widely used distributions in reliability engineering. It is because the distribution is able to model a great variety of data by adjusting the value of shape parameter β The pdf of three-parameter of Weibull distribution is given by
4. DATA ANALYSIS
The ships chosen as research objects are small cargo ships, which were frequently used to transport cargoes in Indonesia. The general information of ships that are chosen as research objects are shown in Table 1. As we can see in table 1, the ships were built at the various years, with the oldest one was built in 1965 and the newest one was built in 1987. One ship represents a ship that was built at the 60's, three ships were built at the 70's and three others were built at the 80's. The ships have length between 48 m to 73 m, breadth between 8 m to 11 m, and height between 5 m to 6m.
All ships have single diesel engines, which delivered power about 1000 hp each, installed and propelled the ships at the speed around 12 to 13 knots. The diesel engines are four-cycle diesel engines with the revolution of the engine between 345 rpm to 420 rpm, and the numbers of cylinders are between 6 to 8 cylinders.
The failure modeling for the diesel engine is expressed into one of continuous probability distribution, they are normal distribution, one and two-parameter exponential distribution, and two or three-parameter Weibull distribution. The normal distribution is chosen based upon central limit theorem. The exponential distribution is selected due to its failure characteristics represent the useful life period in a bath-up curve.
