Exercise Quality Management WS 2009/2010

Transcription

1 Exercise Quality Management WS 2009/ Quality management in field data evaluation Prof. Dr.-Ing. Robert Schmitt Quality management in field data evaluation L5 Seite 0

2 Content Weibull-analysis Median-Rank-technique Extrapolation in the Weibull-diagram Seite 1 Quality management in field data evaluation L5 Seite 1

3 Weibull analysis Motivation statistical analysis of the failure behaviour of parts failure probability failure mechanism Advantage no representative samples are required as input data the first failures in a total population are sufficient (the manufacturer can usually access this information easily Seite 2 Weibull - analysis The reliability of a product is influenced by various parameters. Stress conditions (dependent on particular application-cases as well as the product s quality are decisive for the lifespan of the product. By the means of the statistics it is also possible to describe random failures of products. Quality management in field data evaluation L5 Seite 2

4 Task Typical Task : Company wishes to know how many of the produced vehicles will break down before they have travelled km. Problem: There is no experience with vehicles of this age. The oldest vehicles have run only km. Sources of data: failure statistics from garages distribution of mileage (distribution of the number of km travelled after one year from market analyses or questioning Seite 3 Task The failure behaviour of the production amount of a certain production space of time is to be examined. In this example the production amounts ntotal = 3780 produced vehicles. The example presented here is based on the directive "reliability-assurance in automobile industry" of the organisation of the automobile industry (VDA. Quality management in field data evaluation L5 Seite 3

5 Approach 1. Classify in categories of mileage 2. Read off the cumulative frequency per category of mileage H S (percentage of vehicles which have not exceeded a certain distance. 3. Determine the individual frequency per category of mileage H E (percentage of vehicles in the individual distance categories. 4. Calculate the number of parts which are not damaged per category of mileage (Number: not damaged and in the corresponding distance category 5. Wanted: Cumulative frequency for the failure H j Problem: Solution: Life expectancy characteristic (km does not correspond to life expectancy (t Vehicles with still low mileage will break down in the near future. Account must be taken of these units which did not break down, in statistical evaluations. Take account of the units which did not break down in each category. Use the mean ordinal number j (t j Seite 4 Approach cumulative frequency per category of mileage HS := percentage of vehicles which have not exceeded a certain distance. individual frequency per category of mileage HE := percentage of the vehicles in the individual distance categories. Quality management in field data evaluation L5 Seite 4

6 Approach 6. Calculate the cumulative frequency for the failure distribution H j (Median-Rank-Technique 7. Enter the failure probability in the Weibull grid 8. Interpretation 9. Extrapolation 10. Read off the results Seite 5 Quality management in field data evaluation L5 Seite 5

7 Formulae (1 Calculating the mean ordinal number j(t j j( tj = j( tj + [ n ( tj N ( t ] 1 damaged j (1 j(t j j(t j-1 n damaged (t j N(t j mean ordinal number previous ordinal number number of damaged parts (failures increase Seite 6 Formulae The mean ordinal number j(tj considers the still undamaged parts per category of distance. Quality management in field data evaluation L5 Seite 6

8 Formulae (2 The increase N(t j calculates as follows: n + 1 j( tj 1 N( t j = 1+ n number of previous parts (2 applies: j(t j-1 =j(t 0 =0 n = Total population (sum of all parts in all categories of use, damaged and not damaged Number of previous parts = j i= 1 j 1 n undamaged + n damaged i= 1 Seite 7 Quality management in field data evaluation L5 Seite 7

9 Formulae (3 Calculating the cumulative frequency H j The cumulative frequencies H j are calculated using the approximation equation in the Median-Rank-Technique. j( tj 0.3 Hj( tj = n (3 Seite 8 Median-Rank-Technique Quality management in field data evaluation L5 Seite 8

10 Sources of data mileage category t j [km] Number of damaged parts n damaged (t j table 1 : failure statistics for light trucks Number of all vehicles involved: n=3780 Seite 9 Sources of data The failure statistics are known from an evaluation of guarantee cases. Quality management in field data evaluation L5 Seite 9

11 Distribution of mileage cumulative frequency H (% figure 1: distribution of mileage < 16 distance [1000km] Seite 10 Also known to the manufacturer by comparison with a similar truck is the distribution of mileage for oneyear-old trucks. For example 1.2 % of all vehicles have travelled 10,000 km or less. 50% of all vehicles have travelled 30,000km or less. Quality management in field data evaluation L5 Seite 10

12 Solution (1 Determine the cumulative frequency for each category of mileage H S (t j c.f. figure 1: distribution of travelled distance Determine the individual frequency of each category of mileage H E (t j H E (t j = H S (t j - H S (t j -1 for mileage category t 1 = km: H E (t 1 = H E (t 1 = for mileage category t 2 = km: H E (t 2 = Seite 11 Solution The distribution of mileage is a cumulative function. This means that the travelled distances of the respective vehicles are accumulated. The distribution of mileage specifies, which fraction of the vehicles has not exceeded a certain distance. According to the classification of the failure statistics the percentual fraction of the vehicles in the respective distance categories must be determined. In the next step the individual frequencies of the distance categories are determined. For the further solution the allocation of the number of the non defective vehicles to the corresponding distance category tj is required. Therefore, first the percentual fraction of the non defective vehicles in the distance category tj is determined. This percentage is called individual frequency distance category HE(tj. To their calculation the cumulative frequency of the previous distance category tj-1 is subtracted from the cumulative frequency of the distance category tj. Quality management in field data evaluation L5 Seite 11

13 Solution (2 Determine the number of undamaged parts per category n undamaged (t j : n undamaged (t j = H E (t j * n undamaged n undamaged = n - n damaged = for mileage category t 1 = km: n undamaged (t 1 = H E (t 1 * n undamaged = for mileage category t 2 = km: n undamaged (t 2 = H E (t 2 * n undamaged = Seite 12 From the distribution of mileage the number of undamaged vehicles nundamaged in each individual category can be calculated. This calculation is based on the whole amount of all vehicles n total. The number of the damaged vehicles in the distance categories must be subtracted from n total. Quality management in field data evaluation L5 Seite 12

14 Solution (3 Interim result Category of mileage t j [km] 1. Cumulative frequency per category of mileage H (t j [%] 2. Individual frequency per category of mileage H E (t j [%] 3. Number of parts not damaged per category of mileage n undamaged (t j table 2 : Number of undamaged parts per category of mileage Seite 13 Quality management in field data evaluation L5 Seite 13

15 Solution (4 Calculating the mean ordinal number j(t j : c.f. equations (1 and (2 for category of mileage t 1 = km: n + 1 j( t 0 N(t 1 = = 1+ n number of previous parts j(t 1 = = j( t + ndamaged( t 1 N( t 1 0 Seite 14 Evaluating field failures, it is required to consider the vehicles which still show no damage on account of their very low travelled distance as well. However, these vehicles with low travelled distance can still break down and then would have to be considered in the failure statistics stated above. If for example, a vehicle which travelled a distance of 7,000 km was registered in the distance distribution, this vehicle can absolutely break down, before it has reached the travelled distance of the vehicle with the highest travelled distance. The problem occurs, if the lifespan characteristic feature (here: travelled distance, km does not correspond to the lifetime t. However, the not broke down vehicles contain information which must be considered with the statistic processing of the data. Quality management in field data evaluation L5 Seite 14

16 Solution (5 For category of mileage t 2 = km: N(t 2 = j(t 2 = Seite 15 Quality management in field data evaluation L5 Seite 15

17 Solution (6 Determining the cumulative frequency H j (t j for the failure distribution: c.f. equation (3 for category of mileage t 1 = km: j( t H 1 (t 1 = = n for category of mileage t 2 = km: H 2 (t 2 = Seite 16 The cumulative frequency Hj(tj for the failure distribution is calculated using the approximation formula for the Median-Rank-Technique. Quality management in field data evaluation L5 Seite 16

18 Solution (7 Interim result Category of distance travelled t j [km] Number of damaged parts n damaged (t j Number of undamaged parts per category of distance travelled n not damaged (t j Increase N(t j Mean ordinal number j(t j Cumulative frequency, breakdown H j (t j [%] Table 3: Cumulative frequencies per category of distance travelled for the breakdown distribution Seite 17 Quality management in field data evaluation L5 Seite 17

19 Solution (8 Entering the cumulative frequency of the categories of mileage H j (t j in the life expectancy grid (figure 2 lifetime curve composed of two branches 1. part: failure due to early failure 2. part: failure due to random failure and wear failure long-run behaviour is affected by wear failure Extrapolation extrapolation of the second straight line intersection point with t = km cumulative frequency H j (10 5 km = n damaged (10 5 km = Seite 18 Now the determined cumulative frequencies of the categories of mileage Hj(tj can be plotted over the respective upper limits of the categories of mileage. Quality management in field data evaluation L5 Seite 18

20 Weibull-diagram 99, cumulative frequency H (% 63, figure 2: Weibull-diagram mileage [1000km] Seite 19 Quality management in field data evaluation L5 Seite 19

21 Extrapolation 99, cumulative frequency H (% 63, figure 3: Extrapolation mileage [1000km] Seite 20 Extrapolation To estimate the expected failure behaviour, the curve must be extrapolated. However, prerequisite for such prognosis is that thorough experiences with the same or similar products are present. With the extrapolation of the straight lines must be considered that a breaking off of the curve is possible on account of increasing wear failures. Quality management in field data evaluation L5 Seite 20