Tolerance Design
1
ME222/424
MSC 424
TME 424
PD Funkenbusch (ME 222/424, TME 424, MSC 424)
Tolerance Design
2
 Where do tolerances come from?


History
Absolute vs. Statistical  tolerance as acceptable range
 Summing tolerances



Worst-case
Statistical
More complex systems
 Tolerance Design methodology

Tolerancing based on variance control
PD Funkenbusch (ME 222/424)
Interchangeable parts
3
 Inventor? Depends on your
interpretation. But maybe as
early as 1040’s AD (Chinese
moveable type).
 Eli Whitney helped popularize
http://p2.img.cctvpic.com
the idea for the manufacture of
guns.
 “Took off” in the 1800’s as a
cornerstone of mass production.
 Tolerances make sure parts will
fir together.
http://3.bp.blogspot.com
PD Funkenbusch (ME 222/424)
Tolerance as acceptable range
4
 Set maximum and minimum values for some
characteristic (e.g. part length).


LSL = Lower Specification Limit
USL = Upper Specification Limit
 For simplicity we will assume that the average value (m)
is midway between the LSL and USL.
 Tolerance = D


LSL = m - D
USL = m + D
 Range of values  m ± D
PD Funkenbusch (ME 222/424)
m±D
5
 User (customer) friendly
 Immediate sense of the values likely to be encountered.
 Quick measure of “quality”
 Small D values are immediately impressive
 But what does the tolerance actually mean?
 Two common methods to specify
 Absolute
 Statistical
PD Funkenbusch (ME 222/424)
Absolute (method one)
6
 Inspect all components
 Reject (discard) all those outside of the tolerance
(m ± D)
 All parts (that pass) will be within the tolerance
 Absolute limit
PD Funkenbusch (ME 222/424)
Statistical (method two)
7
 Analyze a sample of components. Determine average (m) , standard
deviation (s), and distribution.
 Set tolerance so that only a small, specified fraction of components
will be outside of the range.
 For simplicity here, we will assume that all distributions are normal.
 Examples:
 D = 1 s  68.3 % in tolerance  average ± one standard deviation
 D = 3 s  99.73 % in tolerance
 D = 6 s  99.9999998% in tolerance


D = 4.5s  99.99966% in tolerance
Note: D = 4.5s (evaluated over an extended time period) is sometimes used as the
cut-off point for “Six Sigma Quality”  less than 3.4 DPMO (defects per million
opportunities)  “rule of thumb” to adjust for drift in product mean
PD Funkenbusch (ME 222/424)
Summing tolerances
8
 Often concerned about how
to sum tolerances.
m2 ± D2
m1 ± D1
 Determine final tolerance
based on component
tolerances
 Determine how to adjust
component tolerances to
achieve a desired final
tolerance

?±?
PD Funkenbusch (ME 222/424)
Add in information on costs
to find the most cost
effective approach
Example
9
m1 ± D1
m2 ± D2
m3 ± D3
m4± D4
 Fit three components
end to end into a slot
machined into a fourth
component.
 Want to determine the
“gap” remaining.
mg ± Dg


m2 ± D2
PD Funkenbusch (ME 222/424)
Average length of gap
Tolerance on the gap
Gap
10
 Average

mg ± Dg
 add & subtract based on
geometry
 mg = m4 - m1 - m2 - m3
 Tolerance


 always sum
Uncertainty increases with each
component included
 Different ways of summing


PD Funkenbusch (ME 222/424)
“Worst case”
“Statistical”
“Worst-case” summation
11
 Tolerance  sum D’s
 Dg = D1 + D2 + D 3 + D4
mg ± Dg
PD Funkenbusch (ME 222/424)
 With absolute tolerances,
this ensures that all the
gaps will be within the
tolerance (i.e. mg  Dg )
 Can also use with
statistical tolerances. In
this case it is possible to
have product outside of
tolerance  but the
probability will be small.
“Statistical” summation
12
 Tolerance  add
D2
‘s
 Most meaningful with
statistical tolerances.
 Dg2 = D12 + D22 + D32 + D42  Assuming all distributions
mg ± Dg
PD Funkenbusch (ME 222/424)
are normal and that the
component tolerances are
set to the same number of
s’s (e.g. D = 3s for all
components), then the
product tolerance should
correspond to that of the
components.
Where does this (summing squares) come from?
13
 From statistics, know that uncorrelated variances
can be added:
 stotal2 = s12 + s22 + s32 + s42 …
 Multiply thru by a constant squared, n2:
 n2stotal2 =n2s12 +n2s22 +n2s32 +n2s42 …
 But D = ns is how we defined the statistical tolerance,
so:
 Dproduct2 = D12 + D22 + D32 + D42 …
PD Funkenbusch (ME 222/424)
Numerical example
14
m1 ± D1
m2 ± D2
m3 ± D3
m4± D4
 Fit three components
end to end into a slot
machined into a fourth
component.
Comp.
mg ± Dg
m2 ± D2
PD Funkenbusch (ME 222/424)
m (mm) D (mm)
1
10
0.1
2
30
0.3
3
20
0.3
4
61
0.2
Average gap
15
Comp.
m (mm) D (mm)
1
10
0.1
2
30
0.3
3
20
0.3
4
61
0.2
mg ± Dg
PD Funkenbusch (ME 222/424)
 Add & subtract based on
geometry
 mg = m4 - m3 - m2 - m1
 mg
= 61 – 20 – 30 – 10
= 1 mm
Tolerance on gap
16
Worst case
Statistical
 Dg = D1 + D2 + D3 + D 4
 Dg2 = D12 + D22 + D32 + D42
=0.1 + 0.3 + 0.3 + 0.2 = 0.9 mm
 mg  Dg  0.1 to 1.9 mm
= (0.1)2 + (0.3)2 + (0.3)2 + (0.2)2
= 0.23
 Dg = 0.5 mm
 mg  Dg  0.5 to 1.5 mm
PD Funkenbusch (ME 222/424)
Tolerance type vs. summation method
17
Tolerance type
Summation
method
Absolute
Statistical
Worst-case
(add D’s)
All product within
tolerance
Some product out of
tolerance (generally
small %)
Statistical
(add D2 ’s)
Difficult to estimate
(“depends”)
Fraction in tolerance
related to components’
fraction in tolerance
PD Funkenbusch (ME 222/424)
Complications
18
 Simple approach works well for a linear stacking of components as
shown in the example.
 May not apply to:
 components with other “features” shape imperfections, roughness…
 more complicated geometries  3-D, rotations…
 properties other than length
 variability caused by other sources (e.g. environmental conditions)
 etc.
 As examples,
 variability in the output voltage of an electrical circuit because of differences in
component properties (resistances, capacitances, etc.) and geometry/assembly
 variability in engine performance due to component wear, ambient temperature,
fuel quality, etc.
PD Funkenbusch (ME 222/424)
Approaches to dealing with more complex systems
19
 Modeling



Need a good mathematical model
May be able to solve analytically, depending on complexity
Alternatively use the model to “test” different combinations of
component /environment values
Monte Carlo random sampling based on frequency of occurrence.
Generally requires large numbers of samples (1,000s or 10,000s) 
must be practical with the model
 Various systematic approaches, e.g. Tolerance design

 Experimental

Need to be able to identify and monitor or adjust sources of
variability
Analyze data collected “in the field”  is suitable data available?
 Systematic testing, e.g. Tolerance design

PD Funkenbusch (ME 222/424)
Random sampling (Monte Carlo) example
20
P
trabecular
t
cortical
D
 Simple model for axial loading of a long bone. [This
is taken from a “Case Study” in Bartel, Davy, and
Keaveny’s Orthopedic Biomechanics textbook.
However, it is just an isostrain model for uniaxial
compression.]
PD Funkenbusch (ME 222/424)
Random sampling (Monte Carlo) example
21
 The load supported on the cortical portion of the bone,
Pc, is given by:
 Where P is the total applied load, t is the thickness of the
cortical shell, D is the diameter of the trabecular
centrum, and Ec and Et are the modulus of the cortical
and trabecular bone, respectively.
 Because of each of the terms in this equation will vary
from person to person, the load on the cortical bone will
also vary.
PD Funkenbusch (ME 222/424)
22
 The table below gives some information on each of the
parameters.
Parameter
Average value
Coefficient of variance
P (N)
1.5e3
20%
Et (Pa)
3.0e8
20%
Ec (Pa)
1.7e10
10%
t(m)
3.5e4
25%
D (m)
3.0e-2
25%
 Average values (except for P) are taken from Bartel et al.
and are nominally for vertebra. The rest of the values are
“stand-ins” (more or less made up), just to illustrate.
PD Funkenbusch (ME 222/424)
Results (different each time!)
23
Mean Square Error
Average
1.20E+03
1.40E+05
1.20E+05
1.15E+03
1.10E+03
MSE (N*N)
Load (N)
1.00E+05
1.05E+03
8.00E+04
6.00E+04
4.00E+04
1.00E+03
2.00E+04
9.50E+02
0.00E+00
1
10
PD Funkenbusch (ME 222/424)
100
N
1000
10000
1
10
100
N
1000
10000
Compare with a tolerance design (8 TC)
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MSE  7.09 e4
Average  1.09 e3
1.20E+03
1.40E+05
1.20E+05
1.15E+03
1.10E+03
MSE (N*N)
Load (N)
1.00E+05
1.05E+03
8.00E+04
6.00E+04
4.00E+04
1.00E+03
2.00E+04
9.50E+02
0.00E+00
1
10
PD Funkenbusch (ME 222/424)
100
N
1000
10000
1
10
100
N
1000
10000
Tolerance Design
25
TOLERANCE AS VARIABILITY
PD Funkenbusch (ME 222/424)
Taguchi’s approach to quality
26
 Genichi Taguchi (1924 – 2012)



quality “guru”, responsible for many innovations
developed many tools and methods
Tolerance design was Taguchi’s last resort method for improving
quality
 Taguchi’s concept of quality



Taguchi equated “quality” with reducing the variance (s2) in the final
product
Didn’t believe in using fixed “tolerances” (i.e. cutoff values)
So Tolerance design focuses on reducing s2 , without considering %
in/out of tolerance

Can be applied to non-normal distributions, but need to be cautious
about converting to a “D” and estimating % in tolerance
PD Funkenbusch (ME 222/424)
Tolerance Design  concept
27
 Assume that proportionality between variance in
components and final (product) variance still holds, but
with a proportionality constant (sensitivity) added
 stotal2 = h1s12 + h2s22 + h3 s32 + h4 s42 …
 Experiment



estimate variance for the product
determine contribution of each component variance to the total 
decide how to best improve tolerance (i.e. reduce variance) as needed
h values show sensitivity of final product variance to tolerance
(variance) of each component  think about the units…
PD Funkenbusch (ME 222/424)
Tolerance design experiment
28
 For each component, input
specific values
TC
measured
A
B
C
...
response


1
-1
-1
-1
Y1
2
-1
-1
+1
Y2
3
-1
+1
-1
Y3
4
-1
+1
+1
Y4
...
+1
-1
-1
…
match variance of component
(“levels” -1 ,+1)  m ± s
 Experiment tests different
combinations of component
levels
 Measure the response of the
TC = Treatment condition, one
“run” of the experiment
A, B, C = different components
-1, +1 = represent two different
component values to be used in
experimentation
PD Funkenbusch (ME 222/424)
product


 variation in these values
provides estimate of the total
product variance.
 also determine
contribution of each
component to total
Matrix selection
29
 Design of matrix is important
 Design Of Experiments (DOE)
 Usually 2-level
 Can include other (non-component) sources
 Matrix size (# of TC)
 Between ~ (n+1) and 2n
 n = number of components
 Much smaller than Monte Carlo style methods
 Large matrix provides more/better data (rare)
 But smaller sizes are still useful (common)
PD Funkenbusch (ME 222/424)
Example (Throttle handle)
 From “Designing experiments
for tolerancing assembled
products”, Soren Bisgaard.
Technometrics (1997), 142152
30
 Components



 Friction in a throttle handle
of outboard motors  too
much or too little  need to
improve the tolerance
 Matrix size


 Tracked friction by measuring
torque to turn the handle
 Three components in the
assembly
PD Funkenbusch (ME 222/424)
Knob, handle, and tube
But multiple dimensions on
the knob (three), and handle
(three)
Total of seven dimensions to
tolerance

Minimum size  8
Maximum size  128
Chose to use 64

Relatively
conservative/expensive
Throttle handle (experimental details)
31
 Knobs (dimensions A, B, C)
 m - s and m + s for each dimension
 Eight possible combinations
 Manufactured all eight combinations
 Handles (dimensions D, E, F)
 m - s and m + s for each dimension
 Eight possible combinations
 Manufactured four of the combinations
 Tube (dimension G)
 Manufactured two tubes, one with m - s and one with m + s
 Tested all combinations of these components
 8 x 4 x 2 = 64 combinations  64 TC
 Treatment condition  assembled one combination of components and
measured torque
PD Funkenbusch (ME 222/424)
Throttle handle (key results)
Source
s
%
A
Variance
(contrib.
to stotal2)
0.0028
29.50
B
0.0023
56.91
8.40
C
0.0024
10.04
1.48
D
0.0037
107.33
15.83
E
0.0030
45.16
6.67
F
0.0043
86.27
12.74
G
0.0040
342.25
50.52
Total
--
677.46 100.00
4.36
32
 A, B, C  knob
dimensions
 D, E, F  handle
dimensions
 G  tube dimension
 G is main contributor to
variance (>50%)


Part tolerances
(length)
Variance of torque
(force-length)2
PD Funkenbusch (ME 222/424)
Best bet to improve
performance
But also depends on
relative costs
Throttle handle (predicting improvement)
Source
s
A
Variance
(contrib.
to D2)
0.0028
29.50
B
0.0023
56.91
13.52
C
0.0024
10.04
2.38
D
0.0037
107.33
25.49
E
0.0030
45.16
10.73
F
0.0043
86.27
20.49
G
0.0040
0.0020
--
342.25
85.56
677.46
420.77
Total
%
7.01
20.32
99.95
33
 Consider the effect of
halving the tolerance (i.e.
s) for G



Variance of G (s2 ) will be
reduced to ¼
Contribution from G to
total with, therefore also
be reduced to ¼
Total Variance for the
throttle torque should be
reduced to ~ 421

PD Funkenbusch (ME 222/424)
677 – ¾ (342) = 421
Predictive equation
Source
s
A
Variance
(contrib.
to D2)
0.0028
29.50
B
0.0023
56.91
13.52
C
0.0024
10.04
2.38
D
0.0037
107.33
25.49
E
0.0030
45.16
10.73
F
0.0043
86.27
20.49
G
0.0040
0.0020
--
342.25
85.56
677.46
420.77
Total
Part tolerances
(length)
%
7.01
20.32
99.95
Variance of torque
(force-length)2
PD Funkenbusch (ME 222/424)
34
stotal2 = hA sA2 + hBsB2 + hCsC2
+ h D s D2 …
 hG sG2
= contribution of G to total =
SSG = 342.25
 hG
= 342.25/ sG2
= 342.25/(0.0040) 2
= 2.14 x 10 7
 Reduce sG to 0.0020
SSG= hG sG2
=2 .14 x 10 7 x (0.0020) 2
= 85.56
Summary
35
 Definitions of tolerance



Based on % in/out of tolerance
Absolute  all in tolerance
Statistical  known % out of tolerance
 Summation of tolerances


“worst-case”  summation of tolerances
“statistical” summation of the squares
 Tolerance design



Based on reducing the product variance
Assumes product variance is proportional to component variances
DOE to estimate total product variance, component contributions,
and the effects of changing component tolerances
PD Funkenbusch (ME 222/424)