Variation in Assemblies
• Goals of this class
– Understand how to represent variation at the assembly level
– Link models to models of variation at the part level
– Link models to nominal chain of frames
– Get an idea of statistical process control and statistical tolerancing vs worst case tolerancing var_assemblies 10/4/2004
© Daniel E Whitney
1

History of Tolerances in Assemblies
• Statistical process control and statistical tolerancing permit a bet on interchangeability
• Coordination makes a comeback as functional build
• Quality today is more than interchangeability
– The product almost certainly will “work”
– Real quality means
• Durability
• Reliability
• Low noise, vibration, etc
– All are associated with low running clearances, fine balance, etc., all requiring closer tolerances, robust design, etc. var_assemblies 10/4/2004
© Daniel E Whitney
2

Recent Gains in Productivity
• Wall Street Journal, Sept 28, 00, p 2:
– “Many analysts may have overlooked one of the main drivers of the productivity gains of the last five years: machine tools… more than half the increase has occurred outside the computer, software, and telecommunications sectors...
– Greater precision in machine tools has helped cut the energy use of air conditioners by 10% between 1990 and 1997 … using a new kind of compressor whose manufacture required
… precision down to 10 microns”
– One of the principal reasons for more reliable automobile transmissions ...
– Open architecture, digital programming, ceramic tool bits are part of the picture var_assemblies 10/4/2004
© Daniel E Whitney
3

KCs and Variation
• A KC is delivered when it achieves its desired value within some specified tolerance
• This puts the emphasis on behavior at the assembly level, not the part level
• To deliver the KC we must
– Design the chain and the location strategy at each joint in the chain
– Ensure that variation at the part level does not propagate to the assembly level and upset delivery of the KC
– This means understanding variation in each of the joints in the chain var_assemblies 10/4/2004
© Daniel E Whitney
4

Logic Tree of Control of Variation
Main Decision
We can’t meet assembly
KCs by controlling part variations
We can meet assembly
KCs by controlling part variations
Functional Build
Selective
Assembly
Belief in
Coordination
Adjustment var_assemblies 10/4/2004
© Daniel E Whitney
Build to Print
Belief in “systems”
Statistical
Process
Control
5

Two Kinds of Errors
• Change in process average (mean shift)
• Process variation around the mean (variance)
• These are different kinds of errors that require different approaches
– The screwdriver workstation-RCC story
– Whitehead’s reason for using kinematic design: to eliminate variation (“erratic errors”)
– Taguchi’s systematic approach
• Correct mean shift first, then reduce variation
• The other sequence makes sense, too var_assemblies 10/4/2004
© Daniel E Whitney
6

Basic Definitions
Mean
∝
Standard
Deviation
σ
-0.04
-0.03
-0.02
-0.01
0 0.01
0.02
0.03
0.04
±3
σ
=99.73% of all events if Gaussian var_assemblies 10/4/2004
© Daniel E Whitney
7

Precision vs Accuracy
Images removed for copyright reasons.
Source:
Figure 5-16 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
Consistent but consistently wrong var_assemblies 10/4/2004
© Daniel E Whitney
Right on the average
8

Best Situation
Image removed for copyright reasons.
Source:
Figure 5-16 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
9

Models of How Errors Accumulate in
Assemblies
• Worst case tolerancing essentially tracks the max error, treats it as mean shift, and ignores the variance
– Assumes all errors are at their extremes at the same time
– This is deterministic, not statistical
– Errors accumulate linearly with the number of parts
• Statistical tolerancing (AKA root sum square) tracks the variance and assumes zero mean shift
– Assumes errors are distributed randomly ± between limits
– Errors accumulate with SQRT of number of parts but only if the average part error = the desired nominal dimension!
(i.e., average part (& assy) error = 0.) var_assemblies 10 10/4/2004
© Daniel E Whitney

Stapler Variations
TOP VIEW
Images removed for copyright reasons.
Source: Figure 5-14
in [Whitney 2004] Whitney, D. E.
Mechanical Assemblies:
Their Design, Manufacture, and
Role in Product Development.
New York, NY: Oxford University
Press, 2004. ISBN: 0195157826.
FRONT VIEW
Images removed for copyright reasons.
Source:
Figure 5-15 in [Whitney 2004] Whitney, D. E.
Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development.
New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
Worst Case Statistical var_assemblies 10/4/2004
© Daniel E Whitney
11

Logic Tree of Statistical and Worst-Case
Tolerancing
Tolerance method
Worst case Statistical
Errors grow with N
Errors grow with sqrt(N) IF mean = nominal
Throws away parts that stat tol keeps var_assemblies 10/4/2004
Process capability Statistical Process measures adherence control keeps of process mean to nominal
© Daniel E Whitney process on its mean
12

How Variations Accumulate
• Assumptions
– All variations are described by the same distribution
– All variations are independent of each other
– The SSN jackpot in Milwaukee
• Then:
– The mean or average of a sum of such variations is the sum of their individual means
E
i = 1 x i
=
N i = 1
E x i
= NE xi if all E [ x i
] are the same
• Actually this is true regardless of the above assumptions
– The variance of the sum of such variations is the sum of their individual variances
σ
2
N
i = 1 x i
N
= i = 1
σ
2 x i var_assemblies 10/4/2004
= N
σ
2
© Daniel E Whitney xi
13

Total Error
Total Accumulated Error =
Accumulated Mean Errors
+ Accumulated Standard Deviation Errors
=
N
meanshift i
+
N
(3
σ i
)
2 i
=
1 i
=
1
=
N * meanshift
+
N * 3 * stddev if all meanshifts and standard deviations are the same var_assemblies 10/4/2004
© Daniel E Whitney
14

What It Means
• Sums of zero-mean errors accumulate at the rate because the + and - errors tend to cancel
• Sums of non-zero-mean errors accumulate at the rate N because there are no cancellations
N
• Therefore, non-zero-mean errors accumulate much more rapidly than zero-mean errors
15 var_assemblies 10/4/2004
© Daniel E Whitney

Root Sum Square (RSS) Method
Assume tolerance band =
±
T
Assume T = 3
σ of error distribution
Then accumulated error from N of T i
=
Accumulated error =
N ∑
T i i =1
But this is really the entire accumulated error only if there is no meanshift. If there is meanshift then the RSS method will badly under-estimate the error. var_assemblies 10/4/2004
© Daniel E Whitney
16

Why the Mean Should Seek the Nominal
Image removed for copyright reasons.
Source:
Figure 5-21 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
How to survive Las Vegas var_assemblies 10/4/2004
© Daniel E Whitney
17

Statistical and Worst Case Compared
Image removed for copyright reasons.
Source:
Figure 5-22 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
18

Statistical Process Control
• SPC aims to get a process in statistical control
– The mean varies in a random way and stays within the mean control limits
– The range varies in a random way and stays within the range control limits
• X-bar and R charts plot the mean and range of samples taken from production parts
• Deviations from the mean, consistent errors, and excess variation can be found quickly and eliminated before bad parts are made
• If a process is in control, then there is a chance that it can deliver parts without mean shift
19 var_assemblies 10/4/2004
© Daniel E Whitney

Process Capability
• Every process has a mean and a “natural tolerance” range which should be tighter than the product needs
• Process capability index C pk tracks whether the process range is within the 3
σ tolerance band and the process mean is on the target nominal dimension
C pk
= min

USL
−
X LSL
−
X


3
σ
;
3
σ
USL
= upper spec (tolerance) limit
LSL
= lower spec (tolerance) limit
X = mean of the process
σ =
variance of the process
20 var_assemblies 10/4/2004
© Daniel E Whitney

Process Capability Cpk
Image removed for copyright reasons.
Source:
Figure 5-18 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies
# of
σ
1
2
3
4
10/4/2004
% within ±# of
σ
68.269%
95.450%
99.730%
99.994%
© Daniel E Whitney
% outside
31.731%
4.550%
0.270%
0.006%
21

Process Control & Capability
Image removed for copyright reasons.
Source:
Figure 5-20 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
22

Effect of Mean Shift
0.00167 shift in each part
3
σ
=.01732
Range
± 1 sigma
± 2 sigma
± 2.5 sigma
± 3 sigma var_assemblies
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04
% Outside
31.74
4.54%
1.24%
0.27%
.005 total shift in 3 parts=0.5
3
σ
=.03
2.5
σ
σ
1.24% will exceed 0.03
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04
C pk
=1.00 C pk
=.833
10/4/2004
© Daniel E Whitney
23

How Could Non-zero Mean Variations
Occur?
• For material removal processes, operators stop as soon as the part enters the tolerance zone from the high side
• For material-adding processes, operators stop as soon as the part enters the zone from the low side
Images removed for copyright reasons.
Source:
Figure 5-24 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
Removal Process var_assemblies 10/4/2004
© Daniel E Whitney
Additive Process
24

Statistical vs Worst Case - Summary
• Statistical is cheaper because looser tolerances are allowed at each part or feature (cost rises as tolerances are tightened)
• Worst case guarantees interchangeable parts, even at batch size of one
• Statistical achieves interchangeability with some probability < 1, enhanced if there is a big bin of parts to choose from. The likelihood of the worst combination actually being picked is pretty low and is usually easy to recover from. var_assemblies 25 10/4/2004
© Daniel E Whitney

Image removed for copyright reasons.
Source:
Figure 5-23 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
LOGIC OF SPC ETC
© Daniel E Whitney
26

Math Models of Variation
• There are tolerance standards for designing single parts (ANSI Y 14.5 xx)
• There is no standard for tolerancing assemblies
• People just extend the part standard or finesse
• The following slides show how to use the 4x4 matrix method for both nominal part-part location modeling and variation modeling var_assemblies 10/4/2004
© Daniel E Whitney
27

Nominal and Varied KC Relationships
Nominal Varied
A
B
A
B
C
KC
KC’
Each relationship, internal or external, nominal or varied, is described by a 4x4 transform var_assemblies 10/4/2004
Internal relationship
External relationship
Key Characteristic
© Daniel E Whitney
28

Inside a Car Engine
Images removed for copyright reasons.
Source:
Figure 5-2 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
29

Chain of Frames for Engine
Image removed for copyright reasons.
Source:
Figure 5-3 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
30

Valve and Lifter Use Selective Assembly
Statistical Tolerancing cannot deliver this KC
Image removed for copyright reasons.
Source:
Figure 6-42 in [Whitney 2004] Whitney, D. E.
Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development.
New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
Image removed for copyright reasons.
Source:
Figure 5-5 in [Whitney 2004] Whitney, D. E.
Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development.
New York, NY: Oxford University Press, 2004.
ISBN: 0195157826. var_assemblies 10/4/2004
© Daniel E Whitney
31

How Selective Assembly is Done
Normal Distribution
0.5
0.4
0.3
0.2
0.1
-4 -3 -2 -1
0
0 x
1 2 3 4
Shaft
Bearing Clearance is important
Pick and measure a shaft.
If it is a bit big, pick a big bearing to get clearance right.
If it is a bit small, pick a small bearing.
For this to work over a long stretch, there must be about the same number of big shafts as big bearings, and the same for small ones. var_assemblies 32 10/4/2004
© Daniel E Whitney

WORLD
ORIGIN
T A
T B
Assembly Process Tolerances
NOMINAL
SHOWING TOLERANCES
T C T C T B
T D T D
T4
T3
T2
T1
WORLD
ORIGIN
T A
T 0 T 0
T4
T3 + DT3 (PARALLEL)
T2 + DT2 (SIZE)
T1
T C T B
T A
T D
T4
T3
T2
T1 var_assemblies
T 0
10/4/2004
OK!
© Daniel E Whitney
Not OK!
33

Nominal Mating of Parts
Images removed for copyright reasons.
Source:
Figure 3-17 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
34

Varied Part Location Due to Tolerances
T
AF T
FB'
A
B'
T
AB'
T
BB'
The varied location of Part B can be calculated from the nominal location of Part A. This process can be chained to Part C, etc., including errors on
Part B. It uses the same math as the nominal model.
var_assemblies 10/4/2004
© Daniel E Whitney
35

Equations for Connective Models
Nominal
Image removed for copyright reasons.
Source:
Figure 3-19 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
Varied
Image removed for copyright reasons.
Source:
Figure 3-20 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
36

Error Transform (Linearized)
DT =
1
–
δ.
δ.
z y
–
δ.
δ.
z
1 –
δ.
δ.
x
1 y x
0 0 0 dx dy dz
1 d
.
T’
T
DT var_assemblies
T’ = T * DT
10/4/2004
© Daniel E Whitney
37 dp

Angle Errors Are Important
Image removed for copyright reasons.
Please see:
Kedrosky, Paul S.
"In the Fray: What the Duffers Can Teach Tiger Woods."
The Wall Street Journal (April 14, 2004): D.16. var_asse mblies 10/4/2004
© Daniel E Whitney
38

Transform Order is Important
This is a frame 1 event
DT 22’
T= DT
11’
T
12
2
DT
11’
T
12
This is a frame 2 event
1
Tells us what everything looks like from frame
1’s pov var_assemblies 10/4/2004
T=T
12
DT
22’
Moves the focus to frame 2 and converts frame 2 events
© Daniel E Whitney to frame 1 coords.
39

Peg Position Error
Part A
Z
Y
A
TAD'
TAD
3
Y'
D'
Y
Z'
Z
D
X'
X var_assemblies
X
10/4/2004 errt
4
Z
X
2
Z'
Z
TAD'
Y'
D'
Y errt
D
X'
TAD
X
4
>>
T
AD
= trans (3,2,4) * roty ( dtr (90))
% dtr converts degrees to radians
=

0 0 1


 −
0 1
1 0
0
0
3
2
4





0 0 0 1
A
Y
© Daniel E Whitney
% First method
>>
DZ
= errt
>> DT
AD1
= trans (0,0, DZ )
T
AD' 1
=
DT
AD1
T
AD
% Second method
>>
DX
= errt
>>
DT
AD 2
= trans (
−
DX ,0,0)
>>
T
AD' 2
=
T
AD
DT
AD 2
>>
T
AD' 2
=
T
AD
T
DD’
40

Part A
Z
Y
A errt
TAD”
TAD
3
Y
Y”
D”
Z
D
X”
X
4
Z”
Peg Length Error
Z
2
Z”
Y”
D”
Z
TAD”
D
X”
Y errt
TAD
X
4
>>
T
AD
= trans (3,2, 4) * roty ( dtr (90))
% dtr converts degrees to radians
=




0 0 1 3 
0 1 0 2


−
1 0 0 4


0 0 0 1 
» DTpp=trans(0,0,randn*.003)
X
X
A
Y
41 var_assemblies 10/4/2004
© Daniel E Whitney

Part B
1 var_assemblies
1
2
TEF
X X'
Hole Angle Error
F
F
TEF
X
7
>>
T
EF
= trans (6,0,1)
=






1 0 0 6
0 1
0 0
0 0
0 0 





>>
DT
E E’
= roty ( dtr ( erra ))
>>
T
E ' F
=
DT
E ' E
T
EF E
E'
Z'
Z
Y
2
42 10/4/2004
© Daniel E Whitney

Final Variation at Assembly Level
2 Where does this point go?
F
Part B'
Part A'
TEF
T AD'
X
X’’
E'
E
D'
Z'
X'
Z
Z’’
>>
T
DE
= rotz ( dtr (180))
=




 −
1 0 0
0
−
1 0
0
0


0
0
0
0
1
0
0



1 
>>
T
AF
=
T
AD
T
DE
T
EF
=
 0




1
 0
>>
T '
AF
% T
D' E '
=
T
DE
−
0 1 4 
0
0
1 0 2
0
0
=
T
AD '
T
D ' E '
T
E ' F
10
1



Z var_assemblies
Y
A
10/4/2004
X
© Daniel E Whitney
43

MATLAB for Example
»TAD=trans(3,2,4)*roty(dtr(90))
»TAF=TAD*TDE*TEF
TAD =
TAF =
0 0 1 3
0
-1
0
1
0
0
0
0
0
2
4
1
»TDE=rotz(dtr(180))
0 0 1 4
0 -1 0
1 0 0
2
-2
0 0 0 1
TAF(1,4)
TDE =
»TEF=trans(6,0,1)
TEF =
1 0 0 6
0 1 0 0
0 0 1 1
0 0 0 1 var_assemblies 10/4/2004
-1 0 0 0
0 -1 0 0
0 0 1 0
0 0 0 1 for i=1:10000
DTad=trans(randn*.003,0,0);
DTdpp=trans(0,0,randn*.003);
DTee=roty(dtr((randn*.003)));
TAF=TAD*DTad*DTpp*TDE*DTee*TEF; m(i)=TAF(1,4); end
»hist(m,200)
© Daniel E Whitney
44

Result for 10000 Samples
200
180
160
140
120
100
80
60
40
20 va r_assemblies
0
3.985 3.99
10/4/2004
3.995 4 4.005 4.01
© Daniel E Whitney
Nominal Value
4.015
45

Histogram of 100 Samples
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5 var _assemblies
0
3.992 3.994
10/4/2004
3.996 3.998 4 4.002 4.004 4.006 4.008 4.01
46

Constructing Matrix Representations of
GD&T var_assemblies 10/4/2004
© Daniel E Whitney
47

Position and Angle are Coupled by Rule #1
Images removed for copyright reasons.
Source:
Figure 6-15 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
f opt is derived by simulation to give the best Gaussian fit to the actual diamond var_assemblies 48 10/4/2004
© Daniel E Whitney

2T
S
Tolerance Zones as 4x4 Transforms
Two dimensional example:
F
2
Z
F
1
....
1
= 2 T /L
S
F
0
T
0,1
=
1 0 0 0
0 1 0 0
0 0 1 D
0 0 0 1
T
1,2
=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
D
DT
1 , 2
: dX = 0 dY = 0 d Z = f o p t
x T s
δδδδ....
δδδδ....
x
= f o p t
x 2 T s
/ L y y
= f o p t
x 2 T s
/ L x
δδδδ....
z
= 0 f = 0.95
o p t
T
0,2
= T
0,1
* T
1,2
* DT
1,2 var_assemblies 49 10/4/2004
© Daniel E Whitney

Table removed for copyright reasons.
Source:
Table 6-1 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826. var_assemblies 10/4/2004
© Daniel E Whitney
50

Two Ways to Use This Model
• Direct Monte Carlo simulation (can use Matlab)
– randomly select dX, dY,
#1) (use f opt
δ.
x
, etc (eliminate parts that violate Rule
= 1)Assume that T s
=
3
σ dX
= randn * T s
/ 3;
δ.
x
= randn *2 * T s
– multiply out all the Transforms
/ (3* LY )
– repeat many times, collect histograms
• Plug into a closed form solution (use f opt
Rule #1) to approximate
– requires assuming a Gaussian distribution for each of the errors
– delivers ellipsoids of location and angle error at the end of the transform chain
51 var_assemblies 10/4/2004
© Daniel E Whitney

Logic Tree of Control of Variation
Main Decision
We can’t meet assembly
KCs by controlling part variations
We can meet assembly
KCs by controlling part variations
Functional Build
Selective
Assembly
Belief in
Coordination
Adjustment var_assemblies 10/4/2004
© Daniel E Whitney
Build to Print
Belief in “systems”
Statistical
Process
Control
52

Functional Build
• Accept the fact that the mean cannot be brought to the desired value:
– Too much trouble, time, or cost
– Too much variation in the mean
– Some other mean may be shifted the other way
• Coordinate the parts involved
• Keep an eye on things and make adjustments
• This is often done on tools and dies if there is one source
(one die) for each part involved
• Then the focus is to control variation around the mean
• The SPC metric is then Cp
C p
=


USL
−
LSL

6
σ  var_assemblies 10/4/2004
© Daniel E Whitney
53

C p
Ignores Mean Shift
Image removed for copyright reasons.
Source:
Figure 5-19 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
54

Logic Tree for Tolerancing
Image removed for copyright reasons.
Source:
Figure 6-48 in [Whitney 2004] Whitney, D. E. Mechanical Assemblies: Their Design, Manufacture,
and Role in Product Development . New York, NY: Oxford University Press, 2004. ISBN: 0195157826.
var_assemblies 10/4/2004
© Daniel E Whitney
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