1 Vane Opening

EXAMPLE 15-1 Vane Opening

  A component part for a jet aircraft engine is manufactured

  UCL D 4 r 12.115215.82 12.27

  by an investment casting process. The vane opening on this

  LCL D 3 r

  casting is an important functional parameter of the part. We

  will illustrate the use of X and R control charts to assess the

  statistical stability of this process. Table 15-1 presents 20

  The and R control charts with these trial control limits

  samples of five parts each. The values given in the table have

  are shown in Fig. 15-8. Notice that samples 6, 8, 11, and 19 are

  been coded by using the last three digits of the dimension;

  out of control on the X chart and that sample 9 is out of

  that is, 31.6 should be 0.50316 inch.

  control on the R chart. (These points are labeled with a “1”

  The quantities x 33.3 and r 5.8 are shown at the

  because they violate the first Western Electric rule.)

  foot of Table 15-1. The value of A 2 for samples of size 5 is

  For the S chart, the value of c 4 0.94. Therefore,

  A 2 0.577. Then the trial control limits for the chart are X

  3s

  x A 2 r 33.32 2 10.577215.82 33.32 3.35 c 4 21 c

  and the trial control limits are

  For the R chart, the trial control limits are

  LCL 2.345 2.553 0.208

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  16-5 X AND R OR S CONTROL CHARTS

  Table 15-1 Vane-Opening Measurements Sample

  33.32 r 5.8 s 2.345

  38 UCL = 36.67 36 x= 34 33.32

  Figure 15-8 0 The and X R

  control charts for vane opening.

  Subgroup

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  CHAPTER 15 STATISTICAL QUALITY CONTROL

  6 5 UCL = 4.899 4 3 S = 2.345 deviation 2

  Figure 15-9 The S

  Sample standard

  0 LCL = 0.00

  control chart for vane

  The LCL is set to zero. If is used to determine the control s

  and for the R chart,

  limits for the X chart,

  LCL D 3 r 10215.02 0

  and this result is nearly the same as from . The S chart is shown r

  The revised control charts are shown in Fig. 15-10.

  in Fig. 15-9. Because the control limits for the X chart calculated

  Practical Interpretation: Notice that we have treated the

  from are nearly the same as from , the chart is not shown. s r

  first 20 preliminary samples as estimation data with which to

  Suppose that all of these assignable causes can be traced

  establish control limits. These limits can now be used to judge

  to a defective tool in the wax-molding area. We should discard

  the statistical control of future production. As each new

  these five samples and recompute the limits for the

  and R

  sample becomes available, the values of x and r should be

  charts. These new revised limits are, for the X chart,

  computed and plotted on the control charts. It may be desir- able to revise the limits periodically, even if the process

  UCL x A 2 r 33.21 10.577215.02 36.10

  remains stable. The limits should always be revised when

  LCL x A 2 r 33.21 10.577215.02 30.33

  process improvements are made.

  42 40 = not used in computing

  control limits 38

  Revised UCL = 36.10

  Revised LCL = 30.33

  Sample number

  Estimation of limits X Chart 16

  14 = not used in computing control limits

  Revised UCL = 10.57

  4 Figure 15-10 The X 2

  and R control charts for

  Sample number

  vane opening, revised

  Estimation of limits

  limits.

  R Chart

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  16-5 X AND R OR S CONTROL CHARTS

  Table 15-2 Summary Report from Minitab for the Vane-Opening Data Test Results for Xbar Chart

  TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 6 8 11 19

  Test Results for R Chart TEST 1. One point more than 3.00 sigmas from center line. Test Failed at points: 9