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adcats et byu edu conference Presentations Je

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  • pg 1
									                  Including GD&T Tolerance
                    Variation in a Commercial
                       Kinematics Application


Jeff Dabling
Surety Mechanisms & Integration
Sandia National Laboratories

   Research supported by:
 Summary
 Variation Propagation
 Obtaining Sensitivities
 Variation/Velocity Relationship
 Equivalent Variational Mechanisms in 2D
 EVMs in 3D
 Example in ADAMS
 3 Sources of Variation in Assemblies



DA      q                           q                R
                        R +DR                    R
 + A   q+           R           A
 A     Dq
            U                           U
            U +DU                       U + DU

        Dimensional and                  Geometric
           Kinematic
DLM Vector Assembly Model
    C
    L
                RL                                Gap

              Open Loop
                 RT
                                      e       i
                                  r
            Plunger                       
                          u                             Pad
                              q       Arm          g
        a                                                     Reel
b           Base      Closed Loop
    h        RL
How Geometric Variation Propagates
                      Y
  3D cylindrical                                    Rotational
  slider joint                                      Variation
                              X
                                                                  Flatness
                Z                                                 Tolerance
                                                                  Zone


                              Cylindricity   View normal to the cylinder axis
         Nominal
         Circle               Tolerance
                              Zone
                                             The effect of feature variations
Translational                  Flatness
                                             in 3D depends upon the joint
Variation                      Tolerance
                               Zone          type and which joint axis you
                                             are looking down.

       View looking down the cylinder axis
3D Propagation of Surface Variation
                     K Kinematic Motion
                     F Geometric Feature Variation
                 F                                   F

         K                                  K
                 y                                   y
                         x       K                       x       K

         z                   F              z                F

 K                                   K          F
             K
 Cylindrical Slider Joint                Planar Joint
Variations Associated with Geometric
Feature – Joint Combinations
                                                                                                                (Gao 1993)
       Geom
 Joints Tol
 Planar         R x Rz   Rx Rz                            Rx R z    R x Rz   Rx Rz    Rx R z R xRzT y

 Revolute                R x Rz     Rx Rz       Rx Rz     Rx R z    R x Rz   Rx Rz    Rx R z      Rx Rz     T x Tz   Tx T z

 Cylindrical             R x Rz     R x Rz      Rx Rz     Rx R z    R x Rz   Rx Rz    Rx R z      Rx Rz     T x Tz   Tx T z

 Prismatic     R xR yR z R xR yRz                        R xRy Rz R xR yR z Rx RyR z R xR yR z

 Spherical                          T x T yTz            Tx T yTz                                T x TyT z Tx T yT z Tx TyT z

 CrsCyl                    Ty         Ty          Ty        Ty                                      Ty                 Ty

 ParCyl                  Ty R x     T y Rx      Ty R x Ty R x        Rx        Rx      Rx         Ty R x               Ty

        E                T y Rx                           Ty R x    Ty R x   Ty R x   Ty R x                           Ty
 EdgSli                                                   Ty R x    Ty R x   Ty R x   Ty R x      Ty R x
        P T y Rx         Ty R x
                         T y Rx     Ty R x      T y Rx    Ty R x    Ty R x   Ty R x   Ty R x      Ty R x               Ty
 CylSli C
        P T y Rx         Ty R x                           Ty R x    Ty R x   Ty R x   Ty R x      Ty R x
        Pt                                                                                                             Ty
 PntSli     Ty             Ty                               Ty       Ty       Ty        Ty          Ty
        P
        S                             Ty                    Ty                                      Ty                 Ty
 SphSli     Ty             Ty                               Ty       Ty       Ty        Ty          Ty
        P
    Including Geometric Variation
   Variables used have nominal values of zero
   Variation corresponds to the specified tolerance value
         Rotational variation due to flatness                    Rotational = ±D
          variation between two planar surfaces:                  Variation
                                                                              Flatness
                          Flatness Tolerance Zone                           Tolerance = a
             D  tan 1 
                          Characteristic Length  
                                                                              Zone
                                                  


         Translational     variation     due     to   flatness    Translational =±a/2
          variation:                                               Variation
                                        a
            Translatio Variation  
                      nal                                                Flatness
                                         2                               Tolerance =   a
                                                                         Zone
           Geometric Variation Example
          Translational: additional vector with                                              f

           nominal value of zero. (a3, a4)                                                                             .01
                                                                                    U2
          Rotational: angular variation in the joint                                     q
           of origin and propagated throughout the                                            R                    H

           remainder of the loop. (1, 2)                                   A                         .02
                                                                                                             .01

                                                                                     U1                                      .01


H x  U1 cos(0) + R1 cos(0) + H cos(90 + 1+2) + R2 cos(90 + f + 1 2) +
                                                                   +                              f
                                                                                 (a3, a4)
    R3 cos(90 + f + 1+2 ) +a3cos(90 + 1 2) +a4cos(90 + 1+2) +
                                         +                                                            R2
                                                                                    U2
            U 2 cos(180 + f + 1+2) + A cos(270 + 1+ 2)  0                                R3
                                                                                          q
                                                                                                                   H

                                                                             A
                                                                                              R1

                                                                                    U1                        (1, 2)
Sensitivities from Traditional 3D
Kinematics
Sandor,Erdman 1984:
   3D Kinematics using 4x4 transformation matrices [Sij] in a
    loop equation    [ S ]  [ S ][S ][ S ][S ]  [ I ]
                        00       01       23       ( n 1) n   n0


   Uses Derivative Operator Matrices ([Qlm], [Dlm]) to eliminate
    need to numerically evaluate partial derivatives
                      [ Sij (qm )]
                                       [ Sij (qm )][Qlm ]
                         qm
   Equivalent to a small perturbation method; intensive
    calculations required for each sensitivity
Sensitivities from
Global Coordinate Method
                                                                         (Gao 1993)

 Uses 2D, 3D vector equations
 Derives sensitivities by evaluating effects of
  small perturbations on loop closure equations
    Length Variation Rotational Variation
         H x              H x
               cos a             3Y   2 Z
         Li               fi
         H y              H y
                  cos             1Z   3 X
          Li              fi
         H z              H z
               cos              2 X   3Y
         Li               fi
         Hq x             Hq x
                 0                1
          Li              f i
         Hq y             Hq y
                 0                 2
          Li              f i
         Hq z             Hqx
                 0              3               (taken from Gao, et. al 1998)
          Li              f i
Variation – Velocity Relationship
                                                                          (Faerber 1999)
Tolerance sensitivity solution                                                        24
                 da 2             da 2                          r3
                  dr               dr 
da 3              1              1                            23

         B A dr2   Si , j  dr2 
              1                       

da 4 
                                                                            r4
                  dr               dr                 r2
                    3
                                    3                        22
                  dr4 
                                   dr4 
                                                                         21
                                                                     r1

Velocity analysis of the equivalent mechanism
                   2               2 
                                                               r3



                  r 
                                    r 
                                        
 3              1                1
                
                   
                              
         B 1 A  r2   J i , j
                    
                                      
                                       
                                      r2                                       r4

 4 
                                                     r2

                  r 
                                    r 
                                       
                   3                3
                   r4 
                                   r4 
                                      
                                        When are the sensitivities the same?
2D Equivalent
Variational Mechanisms                                                              (Faerber 1999)
                                                                               24
                                                                                          r3
                                                             r3
    2D dimensional variations to a
    Add Kinematic Joints:
    kinematic model using kinematic                               23
    elements                                            r2
                                                                          r4
                                                                                     r2
                                                                                                 r4
                                             Parallel
   Converts kinematic analysis to
      Edge Slider Planar    Cylinder Slider Cylinders
                                                             22

    variation analysis                                                   21
     Equivalent Variational Joint:                            r1

   Extract tolerance sensitivities from
    velocity analysis
                                                                       Kinematic Assembly
   Even works for static assemblies (no    Parallel
      Edge Slider   Planar Cylinder Slider Cylinders
    moving parts)




                                                                         Static Assembly
            3D Equivalent
            Variational Mechanisms
                 3D Kinematic Joints:                                        Equivalent Variational Joints:




 Rigid (no motion)     Prismatic        Revolute        Parallel Cylinders




   Cylindrical         Spherical         Planar           Edge Slider




Cylindrical Slider    Point Slider   Spherical Slider   Crossed Cylinders
Geometric Equivalent
Variational Mechanisms
                                f
            f                                                                    Y                      d                   f
                                                     f       f                                     R1
                            f                                                        f                          f
                f                                                           f                X
                                        f                f              f                                                   d
            f                                                          Z                                    R2
                                                                                 f

            Rigid                           Prismatic                       Revolute              Parallel Cylinders

                                                                                         f
                            Y                                               f
                                    f            f                              f
        f
                                        X                        f
                    f
                                             f
        Z               f                                                                               f               f

       Cylindrical                          Spherical                           Planar              Edge Slider


                                                                                     d                      d           R1
                d       R
                                                 f                               f   R                              f


                                                                                                                                R2
                        f
            f
    Cylindrical Slider                      Point Slider             Spherical Slider            Crossed Cylinders
  Example Model: Print Head



                                                            Geometric EVM
                                                                          A
                                                                              f3
Pro/E model                            h
                                  f2                                          a2
                Inset A                                                            a3
                                           q1
                f3   j
                          i                     g
                                                        Inset B
                k                                           f     B
                                                              e           f
                                                            d
                                                    c                              e
                                       Z                                                a1
                              b
                                       a                              c                 d
                                           X                                       f1
     Print Head Results
 Results from Global Coordinate Method:
    A      B        D         E      G I J    K     L
C 0        0        0         0     0  1 1     0    0
f1  0   0.2410   0.2410    0.2410   0  0 0  0.2410 0
                                                     
F  1    0.0602  0.0602  0.0602  1 0 0 0.0602 1
f3  0
         0.2410  0.2410  0.2410 0 0 0  0.2410 0
                                                      



 Results from ADAMS velocity analysis:
    A      B       D          E      G I J    K     L
C 0        0        0         0     0  1 1     0    0
f1  0   0.2410   0.2410    0.2410   0  0 0  0.2410 0   3D GEVM in ADAMS
                                                     
F  1    0.0602  0.0602  0.0602  1 0 0 0.0602 1
f3  0
         0.2410  0.2410  0.2410 0 0 0  0.2410 0
                                                      
Research Benefits
   Comprehensive system for including geometric
    variation in a kinematic vector model
   More efficient than homogeneous transformation
    matrices
   Allows use of commercial kinematic software to
    perform tolerance analysis
   Allows static assemblies to be analyzed in addition to
    mechanisms
   Ability to perform variation analysis in more widely
    available kinematic solvers increases availability of
    tolerance analysis
Current Limitations
 Implementing EVMs is currently a manual
  system, very laborious
 Manual implementation of EVMs can be very
  complex when including both dimensional and
  geometric variation
 Difficulty with analysis of joints with
  simultaneous rotations
Questions?

								
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