Transcript CATS 1-D One-Dimensional Tolerance Stackup Spreadsheet XL

CATS 1-D XL
One-Dimensional Tolerance
Stackup Spreadsheet
by Ken Chase
and
Jonathan Wittwer
Brigham Young University
Complex Assemblies
Process Variation
Mean
Standard Deviation
-1s
+1s
Rejects
LL
-3s
+3s
3s
Capability
A frequency plot shows how a
process varies from the mean
UL
Models for Predicting
Assembly Tolerance Stackup
 Worst Case (WC)
TASM   Ti
2
 Statistical (RSS)
s ASM
Ti 

 
3 
2
 Six Sigma (6s)
 Measured Data (Meas)
s ASM
 Ti 

  
3CPK 
s ASM 
 si2
Models for Predicting
Assembly Tolerance Stackup
Model
Worst
Case (WC)
Statistical
(RSS)
Six Sigma
(6s)
Measured
Data
(Meas)
Stack Formula
TASM 
 Ti
s ASM 

s ASM 
s ASM 

2
Ti 
 3 
 Ti 2


3C PK 
 si2
Predicts
Extreme limits of
variation.
Not statistical.
Probable variation.
Percent rejects
Long term variation
Percent rejects.
Variation using
existing parts.
Percent rejects.
Application
Critical sys tems.
No rejects permitted.
Most costly.
Reasonable estimate.
Some rejects allowed.
Less cos tly.
Drift in mean over time
is expected. High
quality levels desired.
After parts are made.
“What-if?” studies.
Centering the Mean
Center
Rejects
LL
Xmean
Centering the process in the LL/UL
window minimizes rejects
UL
Decrease Variation
Decrease Standard Deviation
Reduce rejects
LL
-3s
+3s
3s
Capability
UL
Higher s Values Mean
Higher Quality
LL
UL
-6s-5s-4s-3s-2s-1s0+1s+2s+3s+4s+5s+6s
3s
4.5s
6s
Controlling Variation Leads to
Higher Yields and Fewer Defects
Limits
±3s
Yield
99.73%
±4.5s
99.99966%
±6s
99.9999998%
Defects
2700 / Million
3.4 / Million
2 / Billion
Controlling Variation Leads to
Higher Yields and Fewer Defects
Limits
±3s
Yield
99.73%
±4.5s
99.99966%
±6s
99.9999998%
Defects
2700 / Million
3.4 / Million
2 / Billion
Mean Shifts Happen
Mean Shift
Mean variation is time-dependent
due to tool wear, temperature, etc.
Accounting for Mean Shifts
1.5s
LL
4.5s
UL
The Six Sigma Model allows the engineer to model
assembly variation due to mean shifts
Measures of Process Capability
Process Capability Index:
CP 
3s
UL - LL
6s
Cp adjusted for k:
UL
LL
k
UL
LL
Cpk= Cp (1-k)
 6s
6s Variation Defined
UL
LL
2
RSS: s ASM
T
   i 
3 
3s
k
2
6s: s ASM
 Ti 

  
3CPK 
where Cpk= Cp (1-k)
and
CP 
UL - LL
6s
UL
LL
 6s
Six Sigma
Dynamic Mean Shift
99.73%
(Short Term)
95.45%
(Long Term)
LSL
-3s
+3s
3s
USL
Capability
Comparing Short and Long Term Variation
Requirements for High Quality
99.9999998%
(Short Term)
LL
UL
99.99932%
(Long Term)
-6s
6s Capability
+6s
A goal of 4.5s long term requires
a 6s process in the short term
Excel Statistical Functions
NORMDIST(x,mean,stand_dev,T/Fflag)
=area under the Normal distrib at point x
NORMSDIST(z)
=area under the Standard Normal distrib
STANDARDIZE(x,mean,stand_dev)
=(x - mean)/stand_dev
NORMSINV(Probability)
=z corresp to a given probability (for z <5)
SUMSQ(G1:G25)^0.5
=square root of the sum of squares of si
Statistical Function Accuracy
a Yield
(1- a) Rejects
Z
Source
15 Place Math Tables
6th Order Polynomial Fit
Excel
Source
15 Place Math Tables *
6th Order Polynomial Fit
Excel
Yield Fraction a
3s
6s
0.998650101968370000
0.999999999013000000
0.998650187471804000
0.999999998751939000
0.998650032776765000
0.999999999009878000
Reject Fraction (1-a)
3s
0.001349898031629990
0.001349812528196100
0.001349967223235000
6s
0.000000000987000000
0.000000001248061000
0.000000000990122000
* the 6s value is from an alternate source, accurate to 12 places.
CATS 1-D Spreadsheet
CATS 1-D Inputs
Mean and Variance
Comparison
Yield and Rejects
Calculated in s units, %, or parts-per-million
Standard Normal Distribution
ZL
ZU
Transformation
x- x
Z
sx
Mean
X=0
Standard Deviation
Z (s units)
sx = 1.0
Used to determine % rejects from standard tables.
Does not show mean shift.
Transformed data all looks alike.
Modified Normal Distribution
LL
UL
Shows the mean shifts
Shows the quality levels
Allows all 3 curves to be plotted for comparison
Plot Controls
What can CATS 1-D users expect?
Probably won’t see:
An end to poverty and misery
All men treating each other as brothers
World peace
But, you might notice:
An increased understanding of the role of statistics in design
Fewer problems on the factory floor
Engineering and production talking to each other without shouting.