Fundamentals of Reliability Engineering and Applications

Transcription

1 Fundamentals of Reliability Engineering and Applications E. A. Elsayed Rutgers University Quality Control & Reliability Engineering (QCRE) IIE February 21,

2 Outline Part 1. Reliability Definitions Reliability Definition Time dependent characteristics Failure Rate Availability MTTF and MTBF Time to First Failure Mean Residual Life Conclusions 2

3 Outline Part 2. Reliability Calculations 1. Use of failure data 2. Density functions 3. Reliability function 4. Hazard and failure rates 3

4 Outline Part 3. Failure Time Distributions 1. Constant failure rate distributions 2. Increasing failure rate distributions 3. Decreasing failure rate distributions 4. Weibull Analysis Why use Weibull? 4

5 Outline Part 2. Reliability Calculations 1. Use of failure data a) Interval data (no censoring) b) Exact failure times are known 2. Density functions 3. Reliability function 4. Hazard and failure rates 5

6 Basic Calculations Suppose n 0 identical units are subjected to a test. During the interval (t, t + t ), we observed n f (t ) failed components. Let n s (t ) be the surviving components at time t, then we define: Failure density function Failure rate function n ˆ( ) f f t n n () ˆ( ) f t ht, n () t t s 0 () t t Reliability function ˆ n () () ( ) s t Rt PT r t n 0 6

7 )()Basic Definitions Cont d The unreliability F(t) is FtR (Example: 200 light bulbs were tested and the failures in 1000-hour intervals are Time Interval (Hours) t Failures in the interval Total 200 7

8 Calculations Time Interval Failure Density Hazard rate 4 f () t x 10 ht () x 10 4 Time Interval (Hours) Failures in the interval Total

9 Failure Density vs. Time x 10 3 Time in hours 9

10 Hazard Rate vs. Time Time in Hours 10

11 Reliability Calculations Time Interval (Hours) Failures in the interval Total 200 Time Interval Reliability R() t /5= /4.0=0.5 1/3.33= /10=

12 Reliability vs. Time x 10 3 Time in hours 12

13 Exponential Distribution Definition (t) ht () 0, t 0 Time f ( t) exp( t) R( t) exp( t) 1 F( t) 13

14 ()Exponential Model Cont d Statistical Properties T Mariaed2MiF1 nce 2 1Va nlifeln Failures/hr 0MTTF=200,000 hrs or 20 years Fa deviation 0Standard of MTTF is 200,000 hrs or 20 years Median life =138,626 hrs or 14 years ilures/hr14

15 Exponential Model Cont d T MProperties FStatistical Failu 0MTTF=200,000 hrs or 20 years res/hrit is important to note that the MTTF= (1/failure rate) is only applicable for the constant failure rate case (failure time follow exponential distribution. 15

16 Empirical Estimate of F(t) and R(t) When the exact failure times of units is known, we use an empirical approach to estimate the reliability metrics. The most common approach is the Rank Estimator. Order the failure time observations (failure times) in an ascending order: tt ttt.t i i i n t n16

17 Ft ( i ) Empirical Estimate of F(t) and R(t) is obtained by several methods i1. Uniform naive estimator n1i2. Mean rank estimator 3. Median rank estimator (Bernard) 4. Median rank estimator (Blom) n in / 1 4/in17

18 Empirical Estimate of F(t) and R(t) Assume that we use the mean rank estimator ˆ i Ft ( i ) n 1 n 1 i Rt ˆ( i ) ti t ti 1 i 0,1, 2,..., n n 1 Since f (t ) is the derivative of F(t ), then Ft ˆ( ) Ft ˆ( ) 1 ˆ( ) i i f ti ti ti 1 ti ti.( n 1) 1 fˆ( ti ) t.( n 1) i 18

19 )Empirical Estimate of F(t) and R(t) ˆ( 1 t ) i t.( n 1 i ) Ht ˆ( ) ln( Rt ˆ( ) Example: i i i Recorded failure times for a sample of 9 units are observed at t=70, 150, 250, 360, 485, 650, t855, 1130, Determine F(t), R(t), f(t),,h(t) (19

20 Calculations i t (i) t(i+1) F=i/10 R=(10-i)/10 f=0.1/ t =1/( t.(10 i)) H(t)

21 Reliability Function 1.2 Reliability Time 21

22 Probability Density Function Density Function Time 22

23 Constant Failure Rate Failure Rate Time 23

24 Exponential Distribution: Another Example Given failure data: Plot the hazard rate, if constant then use the exponential distribution with f (t), R (t) and h (t) as defined before. We use a software to demonstrate these steps. 24

25 Input Data 25

26 Plot of the Data 26

27 Exponential Fit 27

28 Exponential Analysis

29 Summary In this part, we presented the three most important relationships in reliability engineering. We estimated obtained estimate functions for failure rate, reliability and failure time. We obtained these function for interval time and exact failure times. 29