Including GD&T Tolerance Variation in a Commercial Kinematics Application

1. Including GD&T Tolerance Variation in a Commercial Kinematics Application Jeff Dabling Surety Mechanisms & Integration Sandia National Laboratories Research supported by:

2. Summary • Variation Propagation • Obtaining Sensitivities • Variation/Velocity Relationship • Equivalent Variational Mechanisms in 2D • EVMs in 3D • Example in ADAMS

3. 3 Sources of Variation in Assemblies q R q D D A R + R R A R q + Dq A + A U U D U U + D U + U Geometric Dimensional and Kinematic

4. C L R L R T Arm R L DLM Vector Assembly Model Gap Open Loop i e r  Plunger Pad u g  Reel a b Base Closed Loop h

5. How Geometric Variation Propagates Y 3D cylindrical slider joint RotationalVariation X FlatnessToleranceZone Z View normal to the cylinder axis CylindricityToleranceZone Nominal Circle The effect of feature variations in 3D depends upon the joint type and which joint axis you are looking down. FlatnessToleranceZone Translational Variation View looking down the cylinder axis

6. 3D Propagation of Surface Variation K Kinematic Motion F Geometric Feature Variation F F K K y y K K x x F z F z K F K K Cylindrical Slider Joint Planar Joint

7. G e o m T o l J o i n t s R R R R R R R R y z x y z x y z T T T T T T T T T T T T T T T x y z x y z x y z x y z x y z T T T T T T y y y y y y T R T R T R T R R R R T R T y x y x y x y x x x x y x y T R T R T R T R T R T E y x y x y x y x y x y P T R T R T R T R T R T R T R y x y x y x y x y x y x y x T R T C T R T R T R T R T R T y x y x y x y x y x y x y P T R T R T R T R T R T R T R y x y x y x y x y x y x y x P t P S P Variations Associated with Geometric Feature – Joint Combinations (Gao 1993) P l a n a r R R R R R R R R R R R R R R T x z x z x z x z x z x z x z y R e v o l u t e R R R R R R R R R R R R R R R R T T T T x z x z x z x z x z x z x z x z x z x z C y l i n d r i c a l R R R R R R R R R R R R R R R R T T T T x z x z x z x z x z x z x z x z x z x z P r i s m a t i c R R R R R R R R R R x y z x y z x y z x S p h e r i c a l C r s C y l P a r C y l E d g S l i R T R y x y x C y l S l i T y P n t S l i T T T T T T T y y y y y y y T T T T y y y y S p h S l i T T T T T T T y y y y y y y

8. Rotational variation due to flatness variation between two planar surfaces: Rotational Variation = ±Db Flatness Tolerance = Zone a Characteristic Length • Translational variation due to flatness variation: =±a/2 Translational Variation Flatness Tolerance = Zone a Including Geometric Variation • Variables used have nominal values of zero • Variation corresponds to the specified tolerance value

9. f .01 U 2 q R H .02 A .01 b1 b2 b1 b2 f U .01 1 (a3, a4) b1 b2 b1 b2 b1 b2 R2 U2 b1 b2 b1 b2 R3 q H A R1 U1 (b1, b2) Geometric Variation Example • Translational: additional vector with nominal value of zero. (a3, a4) • Rotational: angular variation in the joint of origin and propagated throughout the remainder of the loop. (b1, b2) a3 a4

10. Sensitivities from Traditional 3D Kinematics Sandor,Erdman 1984: • 3D Kinematics using 4x4 transformation matrices [Sij] in a loop equation • Uses Derivative Operator Matrices ([Qlm], [Dlm]) to eliminate need to numerically evaluate partial derivatives • Equivalent to a small perturbation method; intensive calculations required for each sensitivity

11. 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 (taken from Gao, et. al 1998)

12. 2 4 r 3 2 3 r 4 r 2 2 2 2 1 r 1 Variation – Velocity Relationship (Faerber 1999) Tolerance sensitivity solution Velocity analysis of the equivalent mechanism When are the sensitivities the same?

13. 2D Kinematic Joints: Kinematic Assembly Equivalent Variational Joint: Parallel Cylinders Edge Slider Planar Cylinder Slider Static Assembly 2D Equivalent Variational Mechanisms (Faerber 1999) • Add dimensional variations to a kinematic model using kinematic elements • Converts kinematic analysis to variation analysis • Extract tolerance sensitivities from velocity analysis • Even works for static assemblies (no moving parts)

14. 3D EquivalentVariational 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

15. R Geometric EquivalentVariational Mechanisms f Y d f f f R1 f f f f f X f f d f f R2 Z f f Parallel Cylinders Revolute Prismatic Rigid f Y f f f f X f f f f Z f f f Spherical Planar Edge Slider Cylindrical d d R1 d f f f R R2 f f Crossed Cylinders Spherical Slider Point Slider Cylindrical Slider

16. A f3 h f a2 2 a3 Inset A q1 j f g i 3 Inset B k B f e f d e c a1 Z b d a c f1 X Example Model: Print Head Geometric EVM Pro/E model

17. A B D E G I J K L A B D E G I J K L C C f1 f1 F F f3 f3 Print Head Results Results from Global Coordinate Method: Results from ADAMS velocity analysis: 3D GEVM in ADAMS

18. 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

19. 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

20. Questions?