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

Dimensional and Kinematic Geometric 3 Sources of Variation in Assemblies U    A A +  A U +  U R R +  R U  A U +  U R R

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

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

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

Variations Associated with Geometric Feature – Joint Combinations (Gao 1993) Joints Geom Tol Prismatic R x R z R x R z R x R z R x R z R x R z R x R z R x R z T y R x R z R x R z R x R z R x R z R x R z R x R z R x R z T x T z T x T z R x R z R x R z R x R z R x R z R x R z R x R z R x R z T x T z T x T z R x R y R z R x R z R x R z R x R y R z R x R y R z R x y R x T y R x T y T y T y T y T y T y T y T y T y T y T y T y T y T y T y T y T y T y T y Cylindrical Revolute Planar Spherical CrsCyl ParCyl EdgSli CylSli PntSli SphSli

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

Geometric Variation Example Translational: additional vector with nominal value of zero. (  3,  4 ) Rotational: angular variation in the joint of origin and propagated throughout the remainder of the loop. (  1,  2 ) A.01  U 1  R U 2 H U1U1  A H U2U2  R3R3 R2R2 R1R1 (  3,  4 ) (  1,  2 ) 11 11 11 11 11 11 11 22 22 22 22 22 22 22 33 44

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

Sensitivities from Global Coordinate Method Uses 2D, 3D vector equations Derives sensitivities by evaluating effects of small perturbations on loop closure equations (Gao 1993) Length Variation Rotational Variation (taken from Gao, et. al 1998)

Variation – Velocity Relationship Tolerance sensitivity solution Velocity analysis of the equivalent mechanism When are the sensitivities the same? r 2 r 3 r 4 r (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) 2D Equivalent Variational Mechanisms Kinematic Assembly Static Assembly (Faerber 1999) 2D Kinematic Joints: Equivalent Variational Joint: Edge SliderPlanarCylinder Slider Parallel Cylinders

3D Kinematic Joints: Equivalent Variational Joints: 3D Equivalent Variational Mechanisms Rigid (no motion)PrismaticRevolute Parallel Cylinders CylindricalSphericalPlanarEdge Slider Cylindrical SliderPoint SliderSpherical SliderCrossed Cylinders

Geometric Equivalent Variational Mechanisms R2R2 R1R1 f d Crossed Cylinders Spherical Slider f R d Point Slider f Cylindrical Slider f f R d Edge Slider f f Planar f f f Spherical f f f Cylindrical f f Y X Z f f Parallel Cylinders R2R2 R1R1 f f d d Revolute f f Y X Z f f Prismatic f f f Rigid f f f f f f

Example Model: Print Head Pro/E model Geometric EVM

Print Head Results 3D GEVM in ADAMS A B D E G I J K L C 11 F 33 Results from Global Coordinate Method: A B D E G I J K L C 11 F 33 Results from ADAMS velocity analysis:

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?