## ME 406- Manufacturing and Design

Mechanical Engineering Department

ME 406- Manufacturing and Design

Project A (Engineering Statistics and Manufacturing)

Title of the Project

Quality Control, Regression Analysis, and Models Fitting

Term 171

Class Section- xx

Project Team – Group X_X

xxxxxxxxx Name 1 (Group Coordinator)

xxxxxxxxx Name 2

xxxxxxxxx Name 3

xxxxxxxxx Name 4

Assigned on 13th November 2017

Due on 4th January 2018 (Before Exam starts)

Note Please Attempt each question as asked using the software where ever mentioned. Full Report generated by STATGRAPHICS Must be included in Problems 1 .All Excel sheets be included in PROBLEMS where Excel has been used such as in Q

Each problem should be started on separate page after pasting the the problem statement at the top of the page. Your Solution will be typewritten.

A hard copy as (WORD +EXCELL SHEETS) be provided along with a Hard copy in PDF format. Plus Software generated reports

A soft copy as a CD (WORD +EXCELL SHEETS) be provided along with a soft copy copy in PDF format. Plus Software generated reports. CD Must have mentioned that it is STSTISTICS PROJECT and it shoulf have All Project students Name and ID mentioned on CD and a on a text file inside the cD.

Project copies both word and PDF will be uploaded on WebCT as will be explained later.

PROBLEM # 1 (1.5 point)

For alpha distribution

Find

a) Plot F(t) versus t/C for =0.05 , 0.1.0.15,0.2 ,0.25 and 0.3 (all curves should be in one Figure for range of t/C varying from (0+=0.05) to 4). Hint use Normal Distribution Function(select True) of Excel subroutine for Z=-

b) Find pdf for t varying from -∞ to +∞ ,(note at t=0 function is undefined). Note

where

Z=-

c) Plot f(t) versus t/C for =0.05 , 0.1.0.15,0.2 ,0.25 and 0.3 .all curves in one Figure for range of t/C varying from (0+=0.05) to 4 . Hint use Normal Distribution Function (select False) of Excel subroutine for Z=-

d) Median value of T, t0.5

e) Quantile, tp which is the solution of F(tp)=p .

f) Mode of T, (value of t where

PROBLEM # 2 (1.0 point) Use EXCEL and attach all spreadsheet analysis with solution)

Fifty measurements of the ultimate tensile strength of wire are given in the accompanying table.

a) Group the data and make an appropriate normalized histogram (with total area of histogram be 1 ) to approximate the PDF.

b) Calculate and for the distribution from the ungrouped data.

c) Using and from part b, draw a normal distribution through the normalized histogram .histogram.

 Ultimate Tensile Strength 103,779 102,325 102,325 103,799 102,906 104,651 105,377 100,145 104,796 105,087 104,796 103,799 103,197 106,395 106,831 103,488 100,872 100,872 105,087 102,906 97,383 104,360 103,633 101,017 101,162 101,453 107,848 104,651 98,110 103,779 99,563 103,197 104,651 101,162 105,813 105,337 102,906 102,470 108,430 101,744 103,633 105,232 106,540 106,104 102,616 106,831 101,744 100,726 103,924 101,598

Source: Data from E. B. Haugen, Probabilistic Mechanical Design Wiley, New York, 1980

(c) Determine the mean, median, and mode from the ungrouped data.

(d) Determine the range and standard deviation from the ungrouped data

(e) Plot the cumulative frequency distribution on normal-probability paper, and determine the mean and standard deviation.

(f), for the data given in Table .what are the 95 percent confidence limits on the mean of the population?

PROBLEM # 3 (1.0 point) (Use EXCEL and attach all spreadsheet analysis with solution)

Three sets of identical twenty five fatigue specimens were tested at the three different level of stresses.. The number of cycles to failure. The results are expressed as log, were as follows.

 TABLE 1: FATIGUE LIFE DATA NUMBER OF CYCLES TO FAILURES No. S1 S2 S3 380 MPa 340 MPa 300 MPa 1 34200 125500 954000 2 37700 156900 959400 3 42000 173600 1194600 4 42300 176900 1240500 5 48200 179400 1250400 6 52500 188500 1285500 7 55900 195100 1410500 8 58300 208100 1495100 9 61700 211900 1518700 10 64700 224100 1544700 11 65000 226000 1551400 12 65500 253000 1585900 13 70400 255500 1639100 14 71000 259000 1683700 15 72400 274000 1926100 16 75200 292000 2011300 17 77400 300400 2171800 18 77800 302300 2391500 19 87800 308300 2569400 20 93400 406300 2674900 21 94000 420700 2921700 22 97200 428500 3046500 23 99600 664800 3105500 24 116700 776100 3523200 25 122500 793900 4311700

Assume that the data at each stress level (S1, S2 and S3) is lognormally distributed.

Using directly the data in table determine

· What is the mean fatigue life ( μ) and its standard deviation (σ )?

· What is the mean Ln of fatigue life ( μlnN) and its standard deviation Ln of fatigue life (σlnN)?

· What are the Parameters of Lognormal distributions at S1,S2and S3.

· Fit lognormal distribution to the data using linear regression model using Excel or Use Statgraphics to fit the lognormal models to data for each stress level and comment on how good the fit is.

PROBLEM # 4 (1.5 point)

Q4. (a) The lifetime of a mechanical switch produced by a company has been determined to have a population mean of μ = 2000 h and σ = 200 h. The temper of a phosphor bronze leaf spring in the switch is changed slightly by the supplier. To determine whether this has changed the product, a sample of 100 switches is tested to give the sample values and . Has there been a change in the product? (0.75 point)

Q4. (b) A vendor of steel wire advertises a mean breaking load of 10,000 lb. A sample of eight tests shows a mean breaking load of 9250 lb and a standard deviation of 110 lb. Do our tests support the vendor’s claim? (0.75 point)

PROBLEM # 5 Control Charts (1 point)

Problem 5-Refer your Text Book above ( See Ebook provided as text book)- -Solve with appropriate calculations, tables and charts

5.1-Problem 8.1 Parts (a) and (b) –P454

5.2 -Problem 8.28 Parts (a) and (b) –P470

PROBLEM # 6

Regression Analysis (MUST USE STATGRAPHICS_ ATTACH COMPLETE REPORT PLUS ONE PAGE SUMMARY OF EACH FITTED MODEL) 2 points

Developing Cutting Forces Empirical Models of a Counter Boring Process in Aluminum.

Counter boring is an operation to enlarge the hole made using drilling. Counter boring or finish boring is a deep hole drilling process that requires a work piece with a pre-existing bore. Counter boring is used to enlarge the drilled hole to the proper depth and machine a square shoulder on the bottom to secure maximum clamping action from the faster. The drilling used to produce a circular hole by removing solid metal. The counter bore tool has a guide, called a pilot, which keep it positioned correctly in the hole. Counter boring tools are often used on low power machines were a small diameter solid boring tool is used for the pre-bore and then a counter boring tool is used to finish the job. Counter boring is also used when there is a heat treat process required after the initial hole is drilled or if a stepped hole is required.

Pilot of diameter d , which is the predrilled hole size in the workpiece diameter of already drilled hole..

D Dia of the enlarged hole

Visit the link and download STATGRAPHICS FREE FOR 30 DAYS AND USE MULTIPLE REGRESSION MODULE (SEE EXAMPLES ON WEBSITE) TO DEVELOP FOLLOWING MODELS>

Regression Analysis https://www.statgraphics.com/regression-analysis

And Quality Control Module (PROICESS CAPABILITY BANALYSIS) from the link https://www.statgraphics.com/process-capability-analysis

Following are the results of Cutting Forces measurements experiments performed at KFUPM Workshop by Professor Anwar K Sheikh. The results are being shared for regression analysis learning objectives.

Cutting Force Fz as a Function of V,D,d and f

 Fz Newton Speed , V mm/minutes Feed , f Mm/revolution d mm D mm Ln (FZ) Ln(Speed) Ln(feed) Ln(D) 52 2463.007 0.03 3.5 6.5 3.951244 7.809138 -3.50656 1.252763 78 2463.007 0.05 3.5 6.5 4.356709 7.809138 -2.99573 1.252763 104 2463.007 0.08 3.5 6.5 4.644391 7.809138 -2.52573 1.252763 130 2463.007 0.12 3.5 6.5 4.867534 7.809138 -2.12026 1.252763 65 3903.426 0.03 3.5 6.5 4.174387 8.26961 -3.50656 1.252763 78 3903.426 0.05 3.5 6.5 4.356709 8.26961 -2.99573 1.252763 104 3903.426 0.08 3.5 6.5 4.644391 8.26961 -2.52573 1.252763 130 3903.426 0.12 3.5 6.5 4.867534 8.26961 -2.12026 1.252763 65 4948.004 0.03 3.5 6.5 4.174387 8.50674 -3.50656 1.252763 78 4948.004 0.05 3.5 6.5 4.356709 8.50674 -2.99573 1.252763 117 4948.004 0.08 3.5 6.5 4.762174 8.50674 -2.52573 1.252763 130 4948.004 0.12 3.5 6.5 4.867534 8.50674 -2.12026 1.252763 65 6157.516 0.03 3.5 6.5 4.174387 8.725429 -3.50656 1.252763 78 6157.516 0.05 3.5 6.5 4.356709 8.725429 -2.99573 1.252763 104 6157.516 0.08 3.5 6.5 4.644391 8.725429 -2.52573 1.252763 130 6157.516 0.12 3.5 6.5 4.867534 8.725429 -2.12026 1.252763 72 7806.851 0.03 3.5 6.5 4.276666 8.962757 -3.50656 1.252763 85 7806.851 0.05 3.5 6.5 4.442651 8.962757 -2.99573 1.252763 111 7806.851 0.08 3.5 6.5 4.70953 8.962757 -2.52573 1.252763 130 7806.851 0.12 3.5 6.5 4.867534 8.962757 -2.12026 1.252763 78 2463.007 0.03 5.5 10 4.356709 7.809138 -3.50656 1.704748 104 2463.007 0.05 5.5 10 4.644391 7.809138 -2.99573 1.704748 130 2463.007 0.08 5.5 10 4.867534 7.809138 -2.52573 1.704748 182 2463.007 0.12 5.5 10 5.204007 7.809138 -2.12026 1.704748 84 3903.426 0.03 5.5 10 4.430817 8.26961 -3.50656 1.704748 110 3903.426 0.05 5.5 10 4.70048 8.26961 -2.99573 1.704748 143 3903.426 0.08 5.5 10 4.962845 8.26961 -2.52573 1.704748 182 3903.426 0.12 5.5 10 5.204007 8.26961 -2.12026 1.704748 91 4948.004 0.03 5.5 10 4.51086 8.50674 -3.50656 1.704748 117 4948.004 0.05 5.5 10 4.762174 8.50674 -2.99573 1.704748 130 4948.004 0.08 5.5 10 4.867534 8.50674 -2.52573 1.704748 182 4948.004 0.12 5.5 10 5.204007 8.50674 -2.12026 1.704748 78 6157.516 0.03 5.5 10 4.356709 8.725429 -3.50656 1.704748 104 6157.516 0.05 5.5 10 4.644391 8.725429 -2.99573 1.704748 143 6157.516 0.08 5.5 10 4.962845 8.725429 -2.52573 1.704748 195 6157.516 0.12 5.5 10 5.273 8.725429 -2.12026 1.704748 91 7806.851 0.03 5.5 10 4.51086 8.962757 -3.50656 1.704748 117 7806.851 0.05 5.5 10 4.762174 8.962757 -2.99573 1.704748 143 7806.851 0.08 5.5 10 4.962845 8.962757 -2.52573 1.704748 195 7806.851 0.12 5.5 10 5.273 8.962757 -2.12026 1.704748 117 2463.007 0.03 7.5 15 4.762174 7.809138 -3.50656 2.014903 143 2463.007 0.05 7.5 15 4.962845 7.809138 -2.99573 2.014903 195 2463.007 0.08 7.5 15 5.273 7.809138 -2.52573 2.014903 234 2463.007 0.12 7.5 15 5.455321 7.809138 -2.12026 2.014903 123 3903.426 0.03 7.5 15 4.812184 8.26961 -3.50656 2.014903 156 3903.426 0.05 7.5 15 5.049856 8.26961 -2.99573 2.014903 208 3903.426 0.08 7.5 15 5.337538 8.26961 -2.52573 2.014903 234 3903.426 0.12 7.5 15 5.455321 8.26961 -2.12026 2.014903 123 4948.004 0.03 7.5 15 4.812184 8.50674 -3.50656 2.014903 169 4948.004 0.05 7.5 15 5.129899 8.50674 -2.99573 2.014903 208 4948.004 0.08 7.5 15 5.337538 8.50674 -2.52573 2.014903 247 4948.004 0.12 7.5 15 5.509388 8.50674 -2.12026 2.014903 130 6157.516 0.03 7.5 15 4.867534 8.725429 -3.50656 2.014903 169 6157.516 0.05 7.5 15 5.129899 8.725429 -2.99573 2.014903 208 6157.516 0.08 7.5 15 5.337538 8.725429 -2.52573 2.014903 247 6157.516 0.12 7.5 15 5.509388 8.725429 -2.12026 2.014903 143 7806.851 0.03 7.5 15 4.962845 8.962757 -3.50656 2.014903 182 7806.851 0.05 7.5 15 5.204007 8.962757 -2.99573 2.014903 208 7806.851 0.08 7.5 15 5.337538 8.962757 -2.52573 2.014903 247 7806.851 0.12 7.5 15 5.509388 8.962757 -2.12026 2.014903 156 2463.007 0.03 9.5 18 5.049856 7.809138 -3.50656 2.251292 208 2463.007 0.05 9.5 18 5.337538 7.809138 -2.99573 2.251292 260 2463.007 0.08 9.5 18 5.560682 7.809138 -2.52573 2.251292 338 2463.007 0.12 9.5 18 5.823046 7.809138 -2.12026 2.251292 208 3903.426 0.03 9.5 18 5.337538 8.26961 -3.50656 2.251292 234 3903.426 0.05 9.5 18 5.455321 8.26961 -2.99573 2.251292 260 3903.426 0.08 9.5 18 5.560682 8.26961 -2.52573 2.251292 364 3903.426 0.12 9.5 18 5.897154 8.26961 -2.12026 2.251292 208 4948.004 0.03 9.5 18 5.337538 8.50674 -3.50656 2.251292 234 4948.004 0.05 9.5 18 5.455321 8.50674 -2.99573 2.251292 260 4948.004 0.08 9.5 18 5.560682 8.50674 -2.52573 2.251292 364 4948.004 0.12 9.5 18 5.897154 8.50674 -2.12026 2.251292 208 6157.516 0.03 9.5 18 5.337538 8.725429 -3.50656 2.251292 260 6157.516 0.05 9.5 18 5.560682 8.725429 -2.99573 2.251292 286 6157.516 0.08 9.5 18 5.655992 8.725429 -2.52573 2.251292 338 6157.516 0.12 9.5 18 5.823046 8.725429 -2.12026 2.251292 234 7806.851 0.03 9.5 18 5.455321 8.962757 -3.50656 2.251292 286 7806.851 0.05 9.5 18 5.655992 8.962757 -2.99573 2.251292 312 7806.851 0.08 9.5 18 5.743003 8.962757 -2.52573 2.251292

Moment Mz as a Function of V,D,d and f

 Mz Newton-meter Speed , V mm/minutes Feed , f Mm/ revolution d mm D mm Ln (Mz) Ln(Speed) Ln(feed) Ln(D) 39 2463.007 0.03 3.5 6.5 3.663562 7.809138 -3.50656 1.252763 59 2463.007 0.05 3.5 6.5 4.077537 7.809138 -2.99573 1.252763 72 2463.007 0.08 3.5 6.5 4.276666 7.809138 -2.52573 1.252763 104 2463.007 0.12 3.5 6.5 4.644391 7.809138 -2.12026 1.252763 39 3903.426 0.03 3.5 6.5 3.663562 8.26961 -3.50656 1.252763 52 3903.426 0.05 3.5 6.5 3.951244 8.26961 -2.99573 1.252763 78 3903.426 0.08 3.5 6.5 4.356709 8.26961 -2.52573 1.252763 117 3903.426 0.12 3.5 6.5 4.762174 8.26961 -2.12026 1.252763 39 4948.004 0.03 3.5 6.5 3.663562 8.50674 -3.50656 1.252763 59 4948.004 0.05 3.5 6.5 4.077537 8.50674 -2.99573 1.252763 78 4948.004 0.08 3.5 6.5 4.356709 8.50674 -2.52573 1.252763 117 4948.004 0.12 3.5 6.5 4.762174 8.50674 -2.12026 1.252763 39 6157.516 0.03 3.5 6.5 3.663562 8.725429 -3.50656 1.252763 52 6157.516 0.05 3.5 6.5 3.951244 8.725429 -2.99573 1.252763 78 6157.516 0.08 3.5 6.5 4.356709 8.725429 -2.52573 1.252763 124 6157.516 0.12 3.5 6.5 4.820282 8.725429 -2.12026 1.252763 39 7806.851 0.03 3.5 6.5 3.663562 8.962757 -3.50656 1.252763 59 7806.851 0.05 3.5 6.5 4.077537 8.962757 -2.99573 1.252763 72 7806.851 0.08 3.5 6.5 4.276666 8.962757 -2.52573 1.252763 117 7806.851 0.12 3.5 6.5 4.762174 8.962757 -2.12026 1.252763 52 2463.007 0.03 5.5 10 3.951244 7.809138 -3.50656 1.704748 72 2463.007 0.05 5.5 10 4.276666 7.809138 -2.99573 1.704748 98 2463.007 0.08 5.5 10 4.584967 7.809138 -2.52573 1.704748 163 2463.007 0.12 5.5 10 5.09375 7.809138 -2.12026 1.704748 65 3903.426 0.03 5.5 10 4.174387 8.26961 -3.50656 1.704748 84 3903.426 0.05 5.5 10 4.430817 8.26961 -2.99573 1.704748 110 3903.426 0.08 5.5 10 4.70048 8.26961 -2.52573 1.704748 169 3903.426 0.12 5.5 10 5.129899 8.26961 -2.12026 1.704748 65 4948.004 0.03 5.5 10 4.174387 8.50674 -3.50656 1.704748 98 4948.004 0.05 5.5 10 4.584967 8.50674 -2.99573 1.704748 110 4948.004 0.08 5.5 10 4.70048 8.50674 -2.52573 1.704748 169 4948.004 0.12 5.5 10 5.129899 8.50674 -2.12026 1.704748 65 6157.516 0.03 5.5 10 4.174387 8.725429 -3.50656 1.704748 98 6157.516 0.05 5.5 10 4.584967 8.725429 -2.99573 1.704748 117 6157.516 0.08 5.5 10 4.762174 8.725429 -2.52573 1.704748 169 6157.516 0.12 5.5 10 5.129899 8.725429 -2.12026 1.704748 65 7806.851 0.03 5.5 10 4.174387 8.962757 -3.50656 1.704748 98 7806.851 0.05 5.5 10 4.584967 8.962757 -2.99573 1.704748 130 7806.851 0.08 5.5 10 4.867534 8.962757 -2.52573 1.704748 163 7806.851 0.12 5.5 10 5.09375 8.962757 -2.12026 1.704748 81 2463.007 0.03 7.5 15 4.394449 7.809138 -3.50656 2.014903 130 2463.007 0.05 7.5 15 4.867534 7.809138 -2.99573 2.014903 195 2463.007 0.08 7.5 15 5.273 7.809138 -2.52573 2.014903 260 2463.007 0.12 7.5 15 5.560682 7.809138 -2.12026 2.014903 104 3903.426 0.03 7.5 15 4.644391 8.26961 -3.50656 2.014903 143 3903.426 0.05 7.5 15 4.962845 8.26961 -2.99573 2.014903 195 3903.426 0.08 7.5 15 5.273 8.26961 -2.52573 2.014903 260 3903.426 0.12 7.5 15 5.560682 8.26961 -2.12026 2.014903 117 4948.004 0.03 7.5 15 4.762174 8.50674 -3.50656 2.014903 156 4948.004 0.05 7.5 15 5.049856 8.50674 -2.99573 2.014903 221 4948.004 0.08 7.5 15 5.398163 8.50674 -2.52573 2.014903 260 4948.004 0.12 7.5 15 5.560682 8.50674 -2.12026 2.014903 130 6157.516 0.03 7.5 15 4.867534 8.725429 -3.50656 2.014903 156 6157.516 0.05 7.5 15 5.049856 8.725429 -2.99573 2.014903 195 6157.516 0.08 7.5 15 5.273 8.725429 -2.52573 2.014903 286 6157.516 0.12 7.5 15 5.655992 8.725429 -2.12026 2.014903 130 7806.851 0.03 7.5 15 4.867534 8.962757 -3.50656 2.014903 169 7806.851 0.05 7.5 15 5.129899 8.962757 -2.99573 2.014903 221 7806.851 0.08 7.5 15 5.398163 8.962757 -2.52573 2.014903 299 7806.851 0.12 7.5 15 5.700444 8.962757 -2.12026 2.014903 221 2463.007 0.03 9.5 18 5.398163 7.809138 -3.50656 2.251292 260 2463.007 0.05 9.5 18 5.560682 7.809138 -2.99573 2.251292 357 2463.007 0.08 9.5 18 5.877736 7.809138 -2.52573 2.251292 487 2463.007 0.12 9.5 18 6.188264 7.809138 -2.12026 2.251292 227 3903.426 0.03 9.5 18 5.42495 8.26961 -3.50656 2.251292 325 3903.426 0.05 9.5 18 5.783825 8.26961 -2.99573 2.251292 357 3903.426 0.08 9.5 18 5.877736 8.26961 -2.52573 2.251292 487 3903.426 0.12 9.5 18 6.188264 8.26961 -2.12026 2.251292 195 4948.004 0.03 9.5 18 5.273 8.50674 -3.50656 2.251292 292 4948.004 0.05 9.5 18 5.676754 8.50674 -2.99573 2.251292 357 4948.004 0.08 9.5 18 5.877736 8.50674 -2.52573 2.251292 520 4948.004 0.12 9.5 18 6.253829 8.50674 -2.12026 2.251292 292 6157.516 0.03 9.5 18 5.676754 8.725429 -3.50656 2.251292 445 6157.516 0.05 9.5 18 6.098074 8.725429 -2.99573 2.251292 650 6157.516 0.08 9.5 18 6.476972 8.725429 -2.52573 2.251292 747 6157.516 0.12 9.5 18 6.616065 8.725429 -2.12026 2.251292 357 7806.851 0.03 9.5 18 5.877736 8.962757 -3.50656 2.251292 487 7806.851 0.05 9.5 18 6.188264 8.962757 -2.99573 2.251292 650 7806.851 0.08 9.5 18 6.476972 8.962757 -2.52573 2.251292

Using STATGRAPHICS -MULTIPLE LINEAR REGRESSION MODULE develop the empirical model for cutting force Fz and Torque (Moment) Mz can be writing as following

Proposed Model 1 Force (Model 1 F)
Proposed Model 1 Moments (Model 1 M)

Proposed Model 2 Force (Model 1 F) .In spread sheet create a new column of val;ues.
Proposed Model 2 Moments (Model 1 M). In spread sheet create a new column of values.

Find A,f ,b and c d e etc in Fz model using first data table ,and Find B,d,e,f d using second Data table for each of the odel.. Write one page summary based upon completer regression report generated by STATGRAPHICS for each fitted model .Its goodness of fit as measured by R2 values and other important coefficients tabulated and plotted in the report. (Attach PDF copy of each full report of STATGRAPHICS output and your EXCEL File used as input data.

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