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					        CAD-Based Tolerance Analysis--
                       An Overview

    C
    L
               RL                                Gap
             Open Loop                                         by Ken Chase
                                                         Brigham Young University
                  RT
                                             i
                                     e
                                 r
            Plunger                      
                         u                             Pad
                                    Arm          g
        a                                                    Reel
            Base       Closed Loop
b
    h        RL
         Concurrent Engineering
                Product            Product
                Release            Release

Cost
 or
Effort




                Development Time

   When manufacturing considerations are
    included early in the design process,
      product development time may be
           significantly shortened
          Critical Link

Engineering   Tolerances   Manufacture
  Design                   Engineering



    Tolerance analysis is the link
  between design & manufacturing
Effects of Tolerances are Far-reaching

      Engineering            Manufacturing
       Design


   Resultant Dimensions    Production Cost
   Fit and Function        Process Selection
   Design Limits           Machine Tools
   Performance             Operator Skills
   Sensitivity             Tooling, Fixtures
   Design Intent           Inspection Precision
   Robust to Variation     Assemblability
   Customer Satisfaction   Scrap and Rework
     Tolerance Analysis
 Given                       Find
Component               Assembly
Tolerances              Tolerance




                        LL          UL

             Assembly
             Function


                        Acceptance
                         Fraction
         Tolerance Analysis
              Promotes
       Concurrent Engineering

 Assembly          Assembly      Component
 Tolerances        Tolerance     Tolerances
                    Analysis


Performance      Engineering     Production
Requirements       Model         Requirements


              Improved Performance
                 Decreased Cost
  3 Sources of Variation in Assemblies




                     R +∆R       
A+∆A                                          R
            R           A
   A


         U
                                      U
          U +∆U

   Dimensional and                Geometric
      Kinematic
The CATS System
          3-D CAD System

   CATS Application Interface


CATS                       CATS
Modeler                   Analyzer




                            Mfg
  CAD                     Process
Database
                          Database
    CATS Modeler Milestones
•   1986   CADAM 1-D graphical modeler
•   1987   GE Calma 1-D graphical modeler
•   1988   HP ME-10 2-D vector modeler
•   1989   Alpha 1 3-D solid modeler
           X Windows CATS interface
•   1990   AutoCAD 2-D modeler: AutoCATS
•   1991   Auto loop generation
•   1992   CATIA & Computervision 3-D modelers
•   1993   Assembly tolerance requirements models
•   1995   Pro/E 3-D parametric modeler
           Automatic joint recognition
• 2000     Variation modeling on ADAMS
            Vector Assembly Model
    C
    L
                  RL                                Gap
                Open Loop
                   RT
                                        e       i
                                    r
              Plunger                       
                            u                             Pad
                                       Arm          g
        a                                                       Reel
              Base      Closed Loop
b              RL
    h
                    3-D Kinematic Joints


                                                                  Parallel
Rigid (no motion)    Prismatic (1)         Revolute (1)         Cylinders (2)




 Cylindrical (2)      Spherical (3)         Planar (3)         Edge Slider (4)




   Cylindrical                                                   Crossed
    Slider (4)       Point Slider (5)   Spherical Slider (5)   Cylinders (5)
             Vector Path Through a Joint


                                                   
                                             U2
            Datum 1


                                                 Datum 2
Datum 2 U                              U1

                            Datum 1

            2-D Joint                 3-D Joint
Vector Path Across a Part




               DRF
 A                    R


  DRF
            U

 Passes through the DRF
  2-D Propagation of Surface Variation

          Translational                Rotational
            Variation                  Variation
Nominal                    Tolerance
Circle                     Zone

                                                      Tolerance
                                                       Zone
                           Tolerance
                           Zone




     Cylinder on a plane               Block on a plane
 3-D Propagation of Surface Variation
                 K Kinematic Motion
                 F Geometric Feature Variation
                 F                               F
                                          K
        K
                y
                y                                y
                      x        K                      x       K

        z                                z                F
                           F
K                                  K         F
            K
Cylindrical Slider Joint               Planar Joint
       Assembly Tolerance Specifications

                  DESIGN SPECIFICATIONS
    Component Tolerances            Assembly Tolerances
Parallelism                    Parallelism
                           A                              A
                                        Part B

       Part B

                                                    -A-
                  -A-                   Part C
                  DESIGN SPECIFICATIONS
Component Tolerances           Assembly Tolerances
Perpendicularity & Angularity       Perpendicularity & Angularity
          A                                                              A

                                A
                                                             ± d




    -A-                                                      -A-

Concentricity & Runout              Concentricity & Runout




                                                       -A-
                                A
                                                                         A



                                A

                                                                     A
    -A-
CE/TOL Modeler
         CATS Assembly Modeler Status
  Modeling Task                Graphical   Automation Level
                               Interface

1. Specify datums                 √        All graphical
2. Specify assembly specs         √        All graphical
3. Select and locate              √        All graphical
    assembly joints
4. Define datum paths             √        All graphical
5. Define closed vector loop      √        Auto loop generation
6. Define open vector loops       √        Auto loop generation
7. Specify geometric              √        Auto DOF check
    variations
                  CE/TOL Analyzer

• Predicted rejects

• Quality level

• Skewed distributions

• Statistical algorithms
  built-in

• No equations to type
      CATS Analyzer Milestones
• 1984 1-D stackup with cost optimization
• 1986 Cost optimization with process selection
       CATS 1-D Analyzer v1
• 1987 Estimated Mean Shift Method
• 1988 CATS 2-D vector loop analysis; 2-D kinematic joints
       Linearized solution of implicit assembly functions
• 1989 2-D analysis with GD&T form variation
• 1990 Mating hole pattern statistical analysis
• 1991 3-D vector loop analysis; 3-D joints; 3-D GD&T
• 1993 Analysis of library of assembly tolerance specs
• 1994 Nonlinear tolerance analysis by MSM
• 1995 Variation Polygon representation
• 1996 Yield prediction for multiple assembly tolerance specs
• 1997 Effect of surface waviness on GD&T form variation
• 2000 Tolerance analysis on ADAMS
Complex Assembly Functions
Explicit          y = f(x)
            ac
   arccos ec
             
             
             




Implicit          f(x, y) = 0
 h x  b  c  cos(90  1 )  e  cos(270  1 )  0
 h y  b  c  sin(90  1 )  e  sin(270  1 )  0
 h   90  90  90  1  180   2  90  0
          b                    38%

          x              19%

          y          15%

          c        10%

          a        10%

          •   5%

          •


CATS determines the % contribution by
 each component tolerance to overall
         assembly variation
         Tolerance Allocation
     Given                   Find
 Assembly                  Component
 Tolerance                 Tolerances




LL       UL

              Allocation
               Scheme


Acceptance
 Fraction
           Accuracy / Efficiency
                    Solution         Accuracy          Efficiency

                       linear       Equiv. to 30,000     0.07 sec
CATS Linear          algebraic      MC simu lations


                    1 nonlinear    Equiv. to 100,000     0.8 sec
CATS Nonlinear      solution per    MC simu lations
                      variable


                    1 nonlinear    5,000 simulations     268 sec
Monte Carlo         solution per        typical
                                                          (30K
                     assembly
                                                       simulations)

         3 part assembly, 6 dimensions, 9 equations
       CATS Tolerance Analysis Status
  Analysis Task                       Graphical    Automation Level
                                      Interface
1. Generate assembly                              Automatic
   equations and sensitivities.
2. Set up matrices and solve                      Automatic
3. Calculate assembly variation          √        Built-in
   and percent rejects
4. Calculate and plot sensitivities      √        Built-in
   and percent contribution
5. Plot assembly distribution            √        Automatic
6. Perform tolerance synthesis           √        Built-in algorithms
7. Perform design iteration              √        Interactive graphical
                                                     interface
     Current Research
     r3




r2
     Equivalent Variational Mechanisms
       r3
                 • Add dimensional variations to
                   a kinematic model
                 • Modify the input and output
                   variables
                 • Extract the tolerance
                   sensitvities from a velocity
r2
                   analysis
                 • Converts a kinematic analysis
                   to a tolerance analysis
                 • Even works for static
                   assemblies (no moving parts)
   Tolerance Analysis
of Compliant Assemblies
Simple lap joint of two thin plates
Multiple Cases - Single FEA Model


y                               y
            z                               z

    x                               x
                Uniform X Gap                                  Twisted Gap




y                               y
                                                Rotated Interference / gap
        z                               z

    x                               x
                Uniform Y Gap
 Compliant Assembly Milestones
• 1996 Statistical FEA for compliant assemblies
       Material covariance due to elastic coupling
• 1998 Geometric covariance - surface continuity
       Geometric covariance from Bezier curves
• 1999 Geometric covariance by spectral analysis
       Wavelength effects on assembly variation
       Spectral characterization of surface variation
• 2000 Geometric covariance by polynomial analysis
       Statistical FEA Solution
FEA Solution:
    Closure Force           Fa  K a a

Statistical FEA Solution:
     Mean Closure Force     F  K eq  o
     Closure Force Cov       F  K eq  oK eqT
   where:
                            K eq  K a  a  K b  K b
                                                  1
     Equivalent Stiffness              K
Curve Fit Polynomial Covariance
Association for the Development of Computer-Aided Tolerancing Systems


       Dr. Ken Chase              Tel: (801) 378-6541
       Brigham Young University   FAX: (801) 378-5037
       435 CTB                    email: chasek@byu.edu
       Provo, Utah 84602          website: adcats.et.byu.edu
Tolerance Analysis of Compliant Assemblies


   Airplane skin panels




   Automotive body panels
           Material Covariance




• Displacing one node affects the displacement
   of surrounding nodes
• Described by the stiffness matrix of the part
        Geometric Covariance




• Nodal variations are not independently random
• Part surfaces are continuous
• Random surfaces must be used to include covariance
  effects in statistical analyses
          Covariance from Spectral Analysis
          Frequency
           spectrum
1.5                            2
                                               Auto-
 1
                                             spectrum            IFFT
                                   1.5
0.5

                                    1
                                                                          Auto-
 0
      0    20   40   60   80                                            correlation
           frequency               0.5                             2

                                                                   1
                                    0
                                         0   20   40   60   80
                                                                   0
                                              frequency
                                                                  -1

                                                                  -2
                                                                   -1   -0.5   0   0.5   1
                                                                   separation distance
               Comparison of Results
               Standard deviation of closure force
          Monte Carlo                                     FASTA
300                                        300

250                                        250

200                                        200

150                                        150

100                                        100

50                                         50

 0                                          0
      5   10   15   20 25   30   35   40         5   10   15   20 25   30   35   40
                    node                                       node

•Large sample size required •Very similar results
   for accuracy             • Smaller sample size
•5,000 FE solutions             required for accuracy
•Slower                     • 2 FE solutions

				
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