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Tolerance Design
Department of Mechanical Engineering, The Ohio State University
Sl. #1
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Design Specifications and Tolerance

Develop from quest for production quality and
efficiency

Early tolerances support design’s basic
function

Mass production brought interchangeability

Integrate design and mfg tolerances
Department of Mechanical Engineering, The Ohio State University
Sl. #2
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Definition
“The total amount by which a given
dimension may vary, or the difference
between the limits”
- ANSI Y14.5M-1982(R1988) Standard [R1.4]
Department of Mechanical Engineering, The Ohio State University
Sl. #3
Source: Tolerance Design, p 10
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Affected Areas
Engineering
Tolerance
Product Design
Quality Control
Manufacturing
Department of Mechanical Engineering, The Ohio State University
Sl. #4
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Questions

“Can customer tolerances be
accommodated by product?”

“Can product tolerances be
accommodated by the process?”
Department of Mechanical Engineering, The Ohio State University
Sl. #5
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Tolerance vs. Manufacturing Process

Nominal tolerances for
steel

Tighter tolerances =>
increase cost $
Department of Mechanical Engineering, The Ohio State University
Sl. #6
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Geometric Dimensions

Accurately communicates the function of part

Provides uniform clarity in drawing delineation
and interpretation

Provides maximum production tolerance
Department of Mechanical Engineering, The Ohio State University
Sl. #7
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Tolerance Types




Size
Form
Location
Orientation
Department of Mechanical Engineering, The Ohio State University
Sl. #8
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Size Tolerances
Department of Mechanical Engineering, The Ohio State University
Sl. #9
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Form Tolerances
Department of Mechanical Engineering, The Ohio State University
Sl. #10
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Location Tolerances
Department of Mechanical Engineering, The Ohio State University
Sl. #11
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Orientation Tolerances
Department of Mechanical Engineering, The Ohio State University
Sl. #12
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Tolerance Buildup
Department of Mechanical Engineering, The Ohio State University
Sl. #13
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Statistical Principles

Measurement of central tendency
 Mean
 Median
 mode

Measurement of variations
 Range
 Variance
 Standard deviation
LSL
USL
X
3s
tolerance
Department of Mechanical Engineering, The Ohio State University
Sl. #14
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Probability

Probability
 Likelihood of occurrence

Capability
 Relate the mean and variability of the
process or machine to the permissible
range of dimensions allowed by the
specification or tolerance.
Department of Mechanical Engineering, The Ohio State University
Sl. #15
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Tolerance SPC Charting
Department of Mechanical Engineering, The Ohio State University
Sl. #16
Figure Source: Tolerance Design, p 125
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Tolerance Analysis Methods



Worst-Case analysis
Root Sum of Squares
Taguchi tolerance design
Department of Mechanical Engineering, The Ohio State University
Sl. #17
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Initial Tolerance Design
Initial
Tolerance
Design
Department of Mechanical Engineering, The Ohio State University
Sl. #18
Figure Source: Tolerance Design, p 93
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References



Handbook of Product Design for Manufacturing: A Practical Guide to
Low-Cost Production, James C. Bralla, Ed. in Chief; McGraw-Hill, 1986
Manufacturing Processes Reference Guide, R.H. Todd, D.K. Allen & L.
Alting; Industrial Press Inc., 1994
Standard tolerances for mfg processes




Machinery’s Handbook; Industrial Press
Standard Handbook of Machine Design; McGraw-Hill
Standard Handbook of Mechanical Engineers; McGraw-Hill
Design of Machine Elements; Spotts, Prentic Hall
Department of Mechanical Engineering, The Ohio State University
Sl. #19
Figure Source: Tolerance Design, p 92-93
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Worst-Case Methodology

Extreme or most liberal condition of tolerance
buildup

“…tolerances must be assigned to the
component parts of the mechanism in such a
manner that the probability that a mechanism
will not function is zero…”
- Evans (1974)
Department of Mechanical Engineering, The Ohio State University
Sl. #20
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Worst-Case Analysis
m
WC max   N p i  Tp i 
i1
m
WCmin   N p i  Tp i 
i1

Ne + Te => Maximum assembly envelope
 Ne - Te => Minimum assembly envelope

Department of Mechanical Engineering, The Ohio State University
Sl. #21
Source: “Six sigma mechanical design tolerancing”, p 13-14.
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Assembly gaps
m
Gmax  N e  Te   N p i  Tp i 
i1
m
Gmin  N e  Te   N p i  Tp i 

i1
m
Gnom  N e   N p i 

i1
Department of Mechanical Engineering, The Ohio State University
Sl. #22
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Worst Case Scenario Example
Department of Mechanical Engineering, The Ohio State University
Sl. #23
Source: Tolerance Design, pp 109-111
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Worst Case Scenario Example
Department of Mechanical Engineering, The Ohio State University
Sl. #24
Source: Tolerance Design, pp 109-111
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Worst Case Scenario Example
• Largest => 0.05 + 0.093 = 0.143
• Smallest => 0.05 - 0.093 = -0.043
Department of Mechanical Engineering, The Ohio State University
Sl. #25
Source: Tolerance Design, pp 109-111
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Non-Linear Tolerances
y  f (x1, x2, x3,...xn )
f
f
f
f
Toly 
tol1 
tol2 
tol3  ...
toln

x1

x2

x3

xn

f
f
f
f
Nomy 
x1 
x2 
x 3  ...
xn
x1
x 2
x 3
x n
Department of Mechanical Engineering, The Ohio State University
Sl. #26
Wource: “Six sigma mechanical design tolerancing”, p 104
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Root Sum-of-Square


RSS
Assumes normal distribution behavior
1
(1/ 2)[x  )/s ]2
f (x) 
e
s 2

Department of Mechanical Engineering, The Ohio State University
Sl. #27
Wource: “Six sigma mechanical design tolerancing”, p 16
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RSS method

Assembly tolerance stack equation
f (x)  T  T  T  ...T
2
1
2
2
2
3
2
n
Department of Mechanical Engineering, The Ohio State University
Sl. #28
Wource: “Six sigma mechanical design tolerancing”, p 128
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Pool Variance in RSS
Tol
s adjusted 
3Cp
2
 Tpi 
 Te 
s  

  
gap
3Cp  i1 3Cpi 
2

m
Department of Mechanical Engineering, The Ohio State University
Sl. #29
Wource: “Six sigma mechanical design tolerancing”, p 128
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Probability
ZQ 

ZQ 
Q  Gnom
s gap
m


Q  N e   N pi 


i1
2


 Te 
Tpi


  
3Cp  i1 3Cpi 
2
m
Department of Mechanical Engineering, The Ohio State University
Sl. #30
Wource: “Six sigma mechanical design tolerancing”, p 128
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Probability for Limits
ZG min 

ZG max 
Gmin  Gnom
2
 Te 2 m  Tpi 


  
3Cp  i1 3Cpi 
Gmax  Gnom
2
 Te 2 m  Tpi 


  
3Cp  i1 3Cpi 
Department of Mechanical Engineering, The Ohio State University
Sl. #31

Wource: “Six sigma mechanical design tolerancing”, p 128
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Dynamic RSS
ZG min 

ZG max 
Gmin  Gnom
2
 Te 2 m  Tpi 


  
3Cpk  i1 3Cpki 
Gmax  Gnom
2
 Te 2 m  Tpi 


  
3Cpk  i1 3Cpki 
Department of Mechanical Engineering, The Ohio State University

Sl. #32
Wource: “Six sigma mechanical design tolerancing”, p 128
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Nonlinear RSS
f 2 2 f 2 2 f 2 2
 f 2
Toly    tol1    tol 2    tol3  ...   toln
x1 
x 2 
x 3 
x n 
Toli
s adjusted 
3Cpki
Department of Mechanical Engineering, The Ohio State University
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Sl. #33
Wource: “Six sigma mechanical design tolerancing”, p 128
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RSS Example
• Largest => 0.05 + 0.051 = 0.101
• Smallest => 0.05 - 0.051 = -0.001
Department of Mechanical Engineering, The Ohio State University
Sl. #34
Wource: “Six sigma mechanical design tolerancing”, p 128
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Taguchi Method
Input from the voice of the customer and QFD processes
Select proper quality-loss function for the design
Determine customer tolerance values for terms
in Quality Loss Function
Determine cost to business to adjust
Calculate Manufacturing Tolerance
Proceed to tolerance design
Department of Mechanical Engineering, The Ohio State University
Sl. #35
Wource: “Six sigma mechanical design tolerancing”, p 21
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Taguchi



Voice of customer
Quality function deployment
Inputs from parameter design
 Optimum control-factor set points
 Tolerance estimates
 Initial material grades
Department of Mechanical Engineering, The Ohio State University
Sl. #36
Wource: “Six sigma mechanical design tolerancing”, p 22
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Quality Loss Function


Identify customer costs for intolerable performance
Quadratic quality loss function
Ao
L(y)  k(y  m) 
(y  m) 2
o
2

Department of Mechanical Engineering, The Ohio State University
Sl. #37
Wource: “Six sigma mechanical design tolerancing”, p 208
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Cost of Off Target and Sensitivity


Cost to business to adjust off target
performance
Sensitivity, b

Ao
A
Ao
2
A  [b (x  m)]



Department of Mechanical Engineering, The Ohio State University
Sl. #38
Wource: “Six sigma mechanical design tolerancing”, p 226-227
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Manufacturing Tolerance


Sl. #39
Ao  o 
 
A  b 
Department of Mechanical Engineering, The Ohio State University
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Summary
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
Importance of effective tolerances
Tolerance Design Approaches
 Worst-Case analysis
 Root Sum of Squares
 Taguchi tolerance method


Continual process
Involvement of multi-disciplines
Department of Mechanical Engineering, The Ohio State University
Sl. #40
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Credits

This module is intended as a supplement to design classes in
mechanical engineering. It was developed at The Ohio State
University under the NSF sponsored Gateway Coalition (grant
EEC-9109794). Contributing members include:

Gary Kinzel…………………………………. Project supervisor
Phuong Pham.……………. ………………... Primary author

Reference:
“Six Sigma Mechanical Design Tolerancing”, Harry, Mikel J. and Reigle
Stewart, Motorola Inc. , 1988.
Creveling, C.M., Tolerance Design, Addison-Wesley, Reading, 1997.
Wade, Oliver R., Tolerance Control in Design and Manufacturing,
Industrial Press Inc., New York, 1967.
Department of Mechanical Engineering, The Ohio State University
Sl. #41
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Department of Mechanical Engineering, The Ohio State University
Sl. #42