BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

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26 GheorgheNegru i) costs for penalization regarding the downtime (operational losses, impact on quality, impact on security and environment). ii) costs for corrective maintenance: iii) manpower: direct costs related with the manpower in the event of an unplanned action; iv) materials and replacement parts: direct costs related to the consumable parts and the replacements used in the event of an unplanned action. In the technical literature (Adolfo, 2007) are presented three basic models related to the nonreliability cost within the life cycle cost analysis (LCCA): constant failures rate, deterministic failures rate and Weibull distribution failures rate. These models include in their evaluation processes the quantification of the impact that could cause the diverse failure events in the total costs of a operational asset. In the followings will be presented the Weibull distribution failure rate model and its application to the case of the specialized wheeled vehicle systems. 2. Weibull Distribution Failure Rate Model In terms of cost analysis structure, this model will estimate the nonreliability cost with failure frequencies calculated from a Weibull distribution function. The model proposes the following sequential flow: 1. Identify, for each alternative to evaluate, the main types of failures. This way for certain equipment there will be f=1… F failure modes. 2. Determine, for each failure mode, the “times to failure” (operational times). This information will be gathered, based on failure records, databases and/or experience of the maintenance and operations. 3. Calculate the cost of failures C f (in monetary units/failure). These costs include replacement of parts, manpower, production loss penalization and operational impact costs. 4. Determine the frequency of expected failures δ f using the Weibull distribution function. Will be used the following notations in Table 1 Variable ζ tf MTBF Γ α β Table 1 Significance Frequency of failures Time between failures Mean time between failures (inverse of the frequency) Gamma function Characteristic life Shape parameter
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 27 If we assume that the random variable t f is distributed according to a Weibull function of parameters α>0 and β>0, its density function is The mean μ is β− ( ) ( βt ) f t The variance is given by σ − β t f 1 α = e , for t ≥ 0. (1) f f f ⎛ 1 ⎞ μ= αΓ ⎜1+ ⎟ ⎝ β . (2) ⎠ ⎛ ⎛ 2 ⎞ ⎛ 2 ⎞ = α ⎜ Γ ⎜1+ ⎟−Γ ⎜1+ ⎟ ⎝ ⎝ β ⎠ ⎝ β ⎞⎟ . (3) ⎠⎠ 2 2 2 The MTBF is the expected value of the random variable t f , which is equal to the mean μ, ⎛ 1 ⎞ MTBF = μ= αΓ ⎜1+ ⎟ ⎝ β . (4) ⎠ The parameters α and β are calculated from the following expressions ⎛ F F ⎛ ⎞ ⎞ ⎜ − ln F ⎜∑ Afi ( tfi ) ⎟ ⎟ ⎜ i f A ⎝ = ⎠ ⎟ fi ⎜ ∑ ⎟ i= f α = exp⎜ ⎟, (5) F F ⎜⎛ ⎞ ⎛ ⎞ Afi ln ( tfi ) Afi ln ( t ⎟ ⎜⎜∑ − ⎟−⎜∑ fi ) ⎟⎟ ⎝ i= f ⎠ ⎝ i= f ⎠ ⎜ ⎟ ⎝ ⎠ where β = A fi F ∑ F ∑ i= f 1 ln( t ) i= f fi A fi ( 1− ln( α) ) , (6) ⎛ ⎛ ⎞⎞ ⎜ ⎜ 1 ⎟⎟ ln ⎜ln ⎜ f ⎟⎟ ⎜ ⎜1− ⎟ ⎟ ⎝ F + 1 = ⎝ ⎠⎠ . (7) ln( t ) fi
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