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## Exact Constraint Design Using Tolerance Analysis Methods

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Title: Exact Constraint Design Using Tolerance Analysis Methods

1
Exact Constraint Design Using Tolerance Analysis
Methods
• Danny Smith
• Brigham Young University
• 15 June 2001

Special Acknowledgements to ADCATS Research NSF
Grant DMI 0084880
2
Presentation Outline
• Background
• Constraint Analysis and Screw Theory
• Tolerance Analysis
• Variation-based Constraint Analysis of Assemblies
(VCAA) Method
• Case Studies
• Conclusion

3
Why Analyze for Constraints?
• Key Definitions
• Degrees of Freedom
• Exact-constraint
• Overconstraint
• Underconstraint

4
Common Assembly Problems
• Overconstraint or Redundant DOF
• Underconstraint or Idle DOF

5
Current Constraint Methods
• Kinematic Constraint Pattern Analysis
• Blanding 1999
• Geometric Constraint Solving
• Hoffmann and Vermeer 1995
• Screw Theory-Based Constraint Analysis
Whitney 2001

6
Screws Twists and Wrenches
• Twist

Wrench
7
Fundamental Principles
• Reciprocity of twists and wrenches
• Screw coordinate representation
• Virtual coefficient
• Solve for twistmatrix and wrenchmatrix

8
Screw Theory Steps
• 1. Locate mating features on assembly using
transformation matrices.
• 2. Form Twist matrices for each mating feature
• 3. Use screw algorithms and linear algebra to
solve for Resultant Twist and Resultant Wrench
matrices

9
DOF Analysis Example
• Individual feature screw representation
• Algorithms
• Resultant Twistmatrix and Wrenchmatrix
• Interpretation

Taken From Adams and Whitney 2001
10
Example (Cont.)
• Assembly DOF and Constraint Solution

Taken From Adams and Whitney 2001
11
Tolerance Analysis Background
• Dimensional, Kinematic, and Geometric Variation
• Direct Linearization Method (DLM)
• Vector Loops
• Global Coordinate Method (GCM)

Please see Chase 1999 and Gao 1993 for
complete details
12
Direct Linearization Method
• Manufactured or Independent variables
• Assembly or Dependent variables
• Geometric Feature variables

13
Vector Loops and GCM
• A Matrix
• Independent Variable Sensitivity Matrix
• B Matrix
• Dependent Variable Sensitivity Matrix
• F Matrix
• Geometric Feature Variable Matrix
• Sensitivities are determined by the GCM

14
Development of the Variation-based Constraint
Analysis of Assemblies (VCAA) Method
• Variation analogies
• Velocity
• Force and moments
• GCM connection
• Employs screw theory
• Solves for under- and overconstraints

underconstraint information
B
overconstraint information
F
15
VCAA for Underconstraints
T
B

T
column
i
joint
DLM Tolerance Analysis
Transpose and Switch
Associate Dependent Variables to Joint Types
W
W
T
i
j
intermediate-joint
intermediate-part
j
Resultant-part
Reciprocal Operation
Reciprocal Operation
Union Matrices For Each Part
16
VCAA for Overconstraints
W

F

W
column
joint
i
DLM Tolerance Analysis
Transpose
Associate Geometric Feature Variables to Joint
Types
T
W

or
or
intermediate-part
j
Resultant-part
j

T
T
W
intermediate-joint
i
intermediate-loop
k
k
Resultant-loop
Union Matrices For Each Part or Loop
Reciprocal Operation
Reciprocal Operation
17
Case Studies of VCAA
• Case 1 - One-way Clutch Assembly in 2-D
• Case 2 - Stacked Blocks Assembly in 2-D
• Case 3 - Crank Slider Assembly in 3-D

18
Case 1 - One-Way Clutch Assembly
• Transmits torque in one rotational direction
• Assembly formed from Roller, Hub, and Ring
• Pressure Angle ?1 is the key dimension

19
Case 1 - Sensitivity Matrices
• Sensitivity Matrices Calculated using GCM

20
Case 1 - Underconstraint Analysis
• Form Joint Twists for each joint from B
• Perform intermediate steps
• Evaluate Resultant Twist for each part to
identify underconstraint information

21
Case 1 - Underconstraint Solution and Results
• Resultant Twists for each part show any
underconstrained degrees of freedom

22
Case 1 - Overconstraint Analysis
• Form Joint Wrenches for each joint from F
• Perform intermediate steps
• Evaluate Resultant Wrench for each part to
identify overconstraint information

23
Case 1 - Overconstraint Solution and Results
• Resultant Wrench for each set shows any
overconstrained degrees of freedom

24
Case 2 Stacked Blocks Assembly
• Theoretical assembly for tolerance analysis
• Assembly formed from Base, Block, and Cylinder
• Vertical placement A of cylinder is key dimension
• Three Vector Loops needed

25
Case 2 - Sensitivity Matrices
• Sensitivity Matrices Calculated using GCM

26
Case 2 - Sensitivity Matrices
• Sensitivity Matrices Calculated using GCM

27
Case 2 - Underconstraint Analysis
• Form Joint Twists for each joint from B
• Perform intermediate steps
• Evaluate Resultant Twist for each part to
identify underconstraint information

28
Case 2 - Underconstraint Solution and Results
• Resultant Twists for each part shows any
underconstrained degrees of freedom

29
Case 2 - Overconstraint Analysis
• Form Joint Wrenches for each joint from F
• Perform intermediate steps
• Evaluate Resultant Wrench for each part to
identify overconstraint information

30
Case 2 - Overconstraint Solution and Results
• Resultant Wrench for each set show any
overconstrained degrees of freedom

31
Case 3 Crank Slider Assembly
• Assembly formed from Base, Crank, Link, and
Slider
• Slider Position U is the key dimension
• One Vector Loop needed

32
Case 3 - Sensitivity Matrices
• Sensitivity Matrices Calculated using GCM

33
Case 3 - Sensitivity Matrices
• Sensitivity Matrices Calculated using GCM

34
Case 3 - Underconstraint Solution and Results
• Resultant Twists for each part show any
underconstrained degrees of freedom

35
Case 3 - Overconstraint Solution and Results
• Resultant Wrench for each set shows any
overconstrained degrees of freedom

36
Conclusions
• VCAA Method connects Constraint Analysis and
Tolerance Analysis
• Based on Screw Theory and the Global Coordinate
Method
• The VCAA Method can extract twist and wrench
matrices directly from the vector model
• Can perform a constraint analysis and a tolerance
analysis simultaneously

37
Bibliography
• Adams, Jeffrey D. Feature Based Analysis of
Selective Limited Motion in Assemblies. Master
of Science Thesis, Massachusetts Massachusetts
Institute of Technology, 1998.
• Adams, Jeffrey D. Whitney, Daniel E.
Application of Screw Theory to Constraint
Analysis of Mechanical Assemblies Joined by
Features. In Journal of Mechanical Design
Transactions of the ASME, Vol. 123, pp. 26-32,
March 2001.
• Blanding, Douglass L. Exact Constraint Machine
Design Using Kinematic Principles. New York
ASME Press, 1999.
• Chase, Kenneth W. Dimensioning Tolerancing
Handbook, ed. Paul J. Drake, Jr., New York
McGraw Hill, Multi_Dimensional Tolerance
Analysis., 1999.

38
Bibliography (cont.)
• Chase, Kenneth W. Gao, Jinsong Magelby,
Spencer Sorensen, Carl. Including Geometric
Feature Variations in Tolerance Analysis of
Mechanical Assemblies. In IIE (Institute of
Industrial Engineers) Transactions, Chapman
Hall Ltd., pp. 795_807, 10 Oct 1996.
• Gao, Jinsong Chase, Kenneth Magleby, Spencer.
Generalized 3-D Tolerance Analysis of Mechanical
Assemblies with Small Kinematic Adjustments. In
IIE (Institute of Industrial Engineers)
Transactions, Chapman Hall Ltd, pp. 367_377, 4
April 1998.
• Gao, Jinsong Chase, Kenneth Magleby, Spencer.
Global Coordinate Method for Determining
Sensitivity in Assembly Tolerance Analysis in
Proceedings of the ASME International Mechanical
Engineering Conference and Exposition, Anaheim,
California, 1998

39
Bibliography (cont.)
• Gao, Jinsong. Nonlinear Tolerance Analysis of
Mechanical Assemblies. A Doctor of Philosophy
Dissertation, Provo, Utah Brigham Young
University, August 1993.
• Hoffmann, Christoph Vermeer, Pamela. Computing
in Euclidean Geometry (2nd Edition), ed. Du,
Ding-Zhu Hwang, Frank, Singapore World
Scientific Publishing Co. Pte. Ltd., Geometric
Constraint Solving in U2 and U3., pp. 266-298,
1995.
• Konkar, Ranjit. Incremental Kinematic Analysis
and Symbolic Synthesis of Mechanisms. Doctor of
Philosophy Dissertation, Palo Alto, California
Stanford University, June 1993.
• Konkar, Ranjit Cutkosky, M. Incremental
Kinematic Analysis of Mechanisms. In Journal of
Mechanical Design, Vol. 117, pp. 589-596,
December 1995.

40
Bibliography (cont.)
• Roth, Bernard. Screws, Motors, and Wrenches
that Cannot be Bought in a Hardware Store. In
Robotics Research, Chapter 8, pp 679-693, 1984.
• Waldron, K. J. The Constraint Analysis of
Mechanisms. In The Journal of Mechanisms, Volume
1, pp 101-114. Great Britain Pergamon Press,
1966.