1. GATEWAY
Tolerance Design
Department of Mechanical Engineering, The Ohio State
Sl. #1 University

2. GATEWAY
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
Sl. #2 University

3. GATEWAY
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
Sl. #3 University Source: Tolerance Design, p 10

4. GATEWAY
Affected Areas
Engineering
Tolerance
Product Design Quality Control
Manufacturing
Department of Mechanical Engineering, The Ohio State
Sl. #4 University

5. GATEWAY
Questions
• “Can customer tolerances be
accommodated by product?”
• “Can product tolerances be
accommodated by the process?”
Department of Mechanical Engineering, The Ohio State
Sl. #5 University

6. GATEWAY
Tolerance vs. Manufacturing
Process
• Nominal tolerances for
steel
• Tighter tolerances =>
increase cost $
Department of Mechanical Engineering, The Ohio State
Sl. #6 University

7. GATEWAY
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
Sl. #7 University

8. GATEWAY
Tolerance Types
• Size
• Form
• Location
• Orientation
Department of Mechanical Engineering, The Ohio State
Sl. #8 University

9. GATEWAY
Size Tolerances
Department of Mechanical Engineering, The Ohio State
Sl. #9 University

10. GATEWAY
Form Tolerances
Department of Mechanical Engineering, The Ohio State
Sl. #10University

11. GATEWAY
Location Tolerances
Department of Mechanical Engineering, The Ohio State
Sl. #11University

12. GATEWAY
Orientation Tolerances
Department of Mechanical Engineering, The Ohio State
Sl. #12University

13. GATEWAY
Tolerance Buildup
Department of Mechanical Engineering, The Ohio State
Sl. #13University

14. GATEWAY
Statistical Principles
• Measurement of central tendency
− Mean
− Median
− mode
• Measurement of variations
− Range LSL X USL
− Variance
− Standard deviation
3σ
tolerance
Department of Mechanical Engineering, The Ohio State
Sl. #14University

15. GATEWAY
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
Sl. #15University

16. GATEWAY
Tolerance SPC Charting
Department of Mechanical Engineering, The Ohio State
Sl. #16University Figure Source: Tolerance Design, p 125

17. GATEWAY
Tolerance Analysis Methods
• Worst-Case analysis
• Root Sum of Squares
• Taguchi tolerance design
Department of Mechanical Engineering, The Ohio State
Sl. #17University

18. GATEWAY
Initial Tolerance Design
Initial
Tolerance
Design
Department of Mechanical Engineering, The Ohio State
Sl. #18University Figure Source: Tolerance Design, p 93

19. GATEWAY
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
Sl. #19University Figure Source: Tolerance Design, p 92-93

20. GATEWAY
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
Sl. #20University

21. GATEWAY
Worst-Case Analysis
m
WCmax = ∑ (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 of Mechanical Engineering, The Ohio State
Department
University
Sl. #21 Source: “Six sigma mechanical design tolerancing”, p 13-14.

22. GATEWAY
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
Sl. #22University

23. GATEWAY
Worst Case Scenario Example
Department of Mechanical Engineering, The Ohio State
Sl. #23University Source: Tolerance Design, pp 109-111

24. GATEWAY
Worst Case Scenario Example
Department of Mechanical Engineering, The Ohio State
Sl. #24University Source: Tolerance Design, pp 109-111

25. GATEWAY
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
Sl. #25University Source: Tolerance Design, pp 109-111

26. GATEWAY
Non-Linear Tolerances
y = f (x1, x 2 , x 3 ,...x n )
∂f ∂f ∂f ∂f
Toly = tol1 + tol2 + tol3 + ...+ toln
∂x1 ∂x 2 ∂x 3 ∂x n
∂f ∂f ∂f ∂f
Nomy ≈ x1 + x2 + x 3 + ...+ xn
∂x1 ∂x 2 ∂x 3 ∂x n
Department of Mechanical Engineering, The Ohio State
Sl. #26University Wource: “Six sigma mechanical design tolerancing”, p 104

27. GATEWAY
Root Sum-of-Square
• RSS
• Assumes normal distribution
behavior
1 −(1/ 2)[x− μ )/σ ]2
f (x) = e
σ 2π
Department of Mechanical Engineering, The Ohio State
Sl. #27University Wource: “Six sigma mechanical design tolerancing”, p 16

28. GATEWAY
RSS method
• Assembly tolerance stack equation
f (x) = T + T + T + ...T 1
2
2
2
3
2
n
2
Department of Mechanical Engineering, The Ohio State
Sl. #28University Wource: “Six sigma mechanical design tolerancing”, p 128

29. GATEWAY
Pool Variance in RSS
Tol
σ adjusted =
3Cp
⎛ Te ⎞ ⎛ Tpi ⎞2 m 2
σ gap = ⎜ ⎟ + ∑⎜ ⎟
⎝ 3Cp ⎠ i=1 ⎝ 3Cpi ⎠
Department of Mechanical Engineering, The Ohio State
Sl. #29University Wource: “Six sigma mechanical design tolerancing”, p 128

30. GATEWAY
Probability
Q − Gnom
ZQ =
σ gap
⎛ m ⎞
Q − ⎜ N e − ∑ N pi ⎟
⎝ ⎠
ZQ = i=1
⎛ Te ⎞ ⎛ Tpi ⎞ 2
2 m
⎜ ⎟ + ∑⎜ ⎟
⎝ 3Cp ⎠ i=1 ⎝ 3Cpi ⎠
Department of Mechanical Engineering, The Ohio State
Sl. #30University Wource: “Six sigma mechanical design tolerancing”, p 128

31. GATEWAY
Probability for Limits
Gmin − Gnom
ZG min =
⎛ Te ⎞ 2 m ⎛ Tpi ⎞
2
⎜ ⎟ + ∑⎜ ⎟
⎝ 3Cp ⎠ i=1 ⎝ 3Cpi ⎠
Gmax − Gnom
ZG max =
⎛ Te ⎞ 2 m ⎛ Tpi ⎞
2
⎜ ⎟ + ∑⎜ ⎟
⎝ 3Cp ⎠ i=1 ⎝ 3Cpi ⎠
Department of Mechanical Engineering, The Ohio State
Sl. #31University Wource: “Six sigma mechanical design tolerancing”, p 128

32. GATEWAY
Dynamic RSS
Gmin − Gnom
ZG min =
⎛ Te ⎞ 2 m ⎛ Tpi ⎞
2
⎜ ⎟ + ∑⎜ ⎟
⎝ 3Cpk ⎠ i=1 ⎝ 3Cpk i ⎠
Gmax − Gnom
ZG max =
⎛ Te ⎞ 2 m ⎛ Tpi ⎞
2
⎜ ⎟ + ∑⎜ ⎟
⎝ 3Cpk ⎠ i=1 ⎝ 3Cpk i ⎠
Department of Mechanical Engineering, The Ohio State
Sl. #32University Wource: “Six sigma mechanical design tolerancing”, p 128

33. GATEWAY
Nonlinear RSS
⎛ ∂f ⎞ 2 2 ⎛ ∂f ⎞ 2 2 ⎛ ∂f ⎞ 2 2 ⎛ ∂f ⎞ 2
Toly = ⎜ ⎟ tol 1 + ⎜ ⎟ tol 2 + ⎜ ⎟ tol3 + ...+ ⎜ ⎟ toln
⎝ ∂x1 ⎠ ⎝ ∂x 2 ⎠ ⎝ ∂x 3 ⎠ ⎝ ∂x n ⎠
Toli
σ adjusted =
3Cpki
Department of Mechanical Engineering, The Ohio State
Sl. #33University Wource: “Six sigma mechanical design tolerancing”, p 128

34. GATEWAY
RSS Example
• Largest => 0.05 + 0.051 = 0.101
• Smallest => 0.05 - 0.051 = -0.001
Department of Mechanical Engineering, The Ohio State
Sl. #34University Wource: “Six sigma mechanical design tolerancing”, p 128

35. GATEWAY
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
Sl. #35University Wource: “Six sigma mechanical design tolerancing”, p 21

36. GATEWAY
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
Sl. #36University Wource: “Six sigma mechanical design tolerancing”, p 22

37. GATEWAY
Quality Loss Function
• Identify customer costs for intolerable
performance
• Quadratic quality loss function
Ao
L(y) = k(y − m) = (y − m) 22
Δo
Department of Mechanical Engineering, The Ohio State
Sl. #37University Wource: “Six sigma mechanical design tolerancing”, p 208

38. GATEWAY
Cost of Off Target and
Sensitivity
• Cost to business to adjust off target
performance
• Sensitivity, β
Ao Ao
φ= A = [β (x − m)]2
A Δ
Department of Mechanical Engineering, The Ohio State
Sl. #38University Wource: “Six sigma mechanical design tolerancing”, p 226-227

39. GATEWAY
Manufacturing Tolerance
Ao ⎛ Δ o ⎞
Δ= ⎜ ⎟
A⎝β⎠
Department of Mechanical Engineering, The Ohio State
Sl. #39University

40. GATEWAY
Summary
• 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
Sl. #40University

41. GATEWAY
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
Sl. #41University

42. GATEWAY
Disclaimer
This information is provided “as is” for general educational
purposes; it can change over time and should be interpreted
with regards to this particular circumstance. While much effort
is made to provide complete information, Ohio State University
and Gateway do not guarantee the accuracy and reliability of
any information contained or displayed in the presentation. We
disclaim any warranty, expressed or implied, including the
warranties of fitness for a particular purpose. We do not
assume any legal liability or responsibility for the accuracy,
completeness, reliability, timeliness or usefulness of any
information, or processes disclosed. Nor will Ohio State
University or Gateway be held liable for any improper or
incorrect use of the information described and/or contain herein
and assumes no responsibility for anyone’s use of the
information. Reference to any specific commercial product,
process, or service by trade name, trademark, manufacture, or
otherwise does not necessarily constitute or imply its
endorsement.
Department of Mechanical Engineering, The Ohio State
Sl. #42University