Mechanical Engineering 101

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Sherman Beasley
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1 Mechanical Engineering 101 University of California, Berkeley Lecture #1 1

2 Today s lecture Statistical Process Control Process capability Mean shift Control charts eading: pp

3 .Precision 3

4 Process variation assignable causes you know what caused variability fix these! natural causes inherent variability or assignable, but cost/benefit doesn t merit fixing 5

5 Scrap rate q from natural variability of process acceptable upper and lower spec. limits design specs deviation of actual from desired mean defects - good parts + defects L nominal dimension U 6

6 Process capability c p summary statistic for comparing design tolerance to variation of (centered) process c p U L 6 7

7 Process capability Design spec is can make setpoint (i.e. = 1) =.00 c p =? c p U L 6 8

8 .Process capability If c p = 1, how many defective ppm? (ppm = parts per million) c p U L 6 10

9 .Defect rates from c p C p Tolerance Defect Defects (no. of std. dev.) rate, % ppm ± ,400 acceptable ± , ± 3 0.7, ± ± ±

10 Today s lecture Statistical Process Control Process capability Mean shift Control charts 15

11 .Mean shift process can t make setpoint at nominal value E.g. design spec is process = 1.001, =.00 fraction defective q=? 16

12 .Mean shift process can t make setpoint at nominal value design spec is process = 1.001, =.00 fraction defective q=? upper limit +z = ( )/.00 = + std devs F() =.9773 lower limit -z = ( )/.00 = -3 std devs F(3) = setpoint + 18

13 Mean shift c p doesn t reflect mean shift c p U L 6 0

14 Mean shift c p doesn t reflect mean shift c p U L 6 if can t make setpoint, summary statistic is c pk calculated for side that s worse c pk U min, 3 L 3 1

15 .Mean shift c pk U L min, 3 3 Design spec is process = 1.001, =.00 c pk?? min,??

16 .Mean shift Design spec is process = 1.000, =.00 c p = 5/6 process = 1.001, =.00 c pk = /3 lower -- we lost capability with mean shift 4

17 Mean shift Some mean shift is expected mean shift of 1.5 typical Set tolerances, reduce process variance accordingly for tolerance of 6, mean shift of 1.5 c pk = 1.5, for a defect rate of 3.4 ppm 6

18 Mean shift Source: Making war on defects, IEEE Spectrum, September 1993, pp

19 Today s lecture Statistical Process Control Process capability Mean shift Control charts X-bar and control charts Attribute control charts 9

20 Learning curve, defects eduction in manufacturing defects with time for two products Testing if process in control only works at steady state steady state value of defect rates ef:. Mahoney, High Mix Low Volume Manufacturing, Prentice-Hall, 1997, p

21 Statistical Process Control Especially once you ve gotten defects low, may not be cost-effective to test all parts Continuous hypothesis testing hypothesis: we re (still) doing ok Two stages determine capability of process ensure process remains in control 31

22 Basic control charts (UCL) e.g.+3 (LCL) e.g.-3 e.g.+3 e.g.-3 3

23 Control charts each sample on x-axis a group of n samples rational subgroup taken under same conditions hypothesis: we re sampling a stationary process so mean, variance of subgroups should be similar 33

24 Control chart types For quantitative, continuous variables (e.g. diameter or other dimension) X chart (x-bar) plots average of series of sample groups indicates how process mean varies chart plots range of series of sample groups indicates how variability of process changes 34

25 X bar chart m subgroups, each of size n Calculate X and s for each mean value all subgroups = X process standard deviation estimate = s 35

26 Averages and sums of random variables Assuming all variables have same variance, Sum of n values Variance n Since variance for X+Y = x + Y Average of n values Variance /n 36

27 Correction for mean say we knew true V(X i )= is standard deviation of variable but we need standard deviation of mean of n values V ( X UCL LCL i ) n 3 n 3 n 37

28 Correction for small n for estimates from samples correction factor from table n LCL n UCL 3 3 n n c s X LCL n n c s X UCL ) ( 3 ) ( ) ( ) ( ) ( 4 n c s s E s E

29 Small subgroup sizes sample range almost as much info as std dev easier to calculate correlation with std dev known -chart: X-bar chart: UCL D 4 ( n)* UCL X X A ( n)* LCL D 3 ( n)* LCL X X A ( n)* 39

30 Small subgroup sizes sample range almost as much info as std dev easier to calculate correlation with std dev known -chart: X-bar chart: UCL D 4 ( n)* UCL X X A ( n)* LCL D 3 ( n)* LCL X X A ( n)* 40

31 Suspicious patterns UCL (~3) quality measure 1-1 Sample groups

32 .Example: is process in control? Find center, UCL, LCL for X-bar and charts in control? subgroup (n=5) x-bar UCL D 4 ( n)* UCL X X A ( n)* LCL D 3 ( n)* LCL X X A ( n)* 43

33 Example: is process in control? Find center, UCL, LCL for X-bar and charts in control? 1) yes ) looks suspicious 3) no subgroup (n=5) x-bar UCL D 4 ( n)* UCL X X A ( n)* LCL D 3 ( n)* LCL X X A ( n)* 44

34 Example: is process in control? X-bar

35 Control limits If we take more samples (n increases), control limits should be 1. Wider. Narrower 3. Unchanged 51

36 Today s lecture Statistical Process Control Process capability Mean shift Control charts X-bar and control charts Attribute control charts 53

37 Control chart types For quantitative, continuous variables X chart (x-bar) plots average of series of sample groups indicates how process mean varies chart plots range of series of sample groups indicates how variability of process changes For discrete attributes (good/bad; go/no-go) p chart plots percentage defective in sample groups c chart plots number (count) of defects in sample groups 54

38 Attribute control charts Examples number of defects per automobile fraction of non-conforming parts in a sample presence or absence of flash in molded part number of flaws in a sheet product defects/flaws in a painted surface shorts or opens in a printed circuit board 55

39 P charts Expected proportion defective = p Sample size = n Exponential distribution for number defective in sample E(X)=np V(X)=np(1 p) UCL p 3 p(1 p) / n LCL p 3 p(1 p) / n 56

40 .P charts If average fraction defective = 1/6, samples of 4 units taken, UCL is 57

41 P charts derivation Proportion defective = p For underlying population Mean = p Variance = p(1-p) derivation below ( X n X ) E( X X ) E( X X ) p(1 p) (1 p)(0 p p p ( 1 p p ) (1 p) p p p(1 p) ) 60

42 Averages and sums of random variables Assuming all variables have same variance, Sum of n values Variance n Since variance for X+Y = x + Y Average of n values Variance /n 61

43 P charts derivation Proportion defective = p Sample size = n UCL p Underlying population LCL p Mean = p Variance = p(1 p) {see prev. derivation} Number defective in sample X Percentage defective X/n E(X/n)=expected percentage defective=p V(X/n)=p(1 p)/n 3 3 p(1 p(1 p) / n p) / n 6

44 Feedback loop ef: Groover, p

Control of Manufacturing Processes

Control of Manufacturing Processes Control of Processes David Hardt Topics for Today! Physical Origins of Variation!Process Sensitivities! Statistical Models and Interpretation!Process as a Random Variable(s)!Diagnosis of Problems! Shewhart