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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 R z Rx R z R x Rz Rx Rz Rx R z R xRzTy Revolute 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 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 Tx TyTz Tx TyTz T x TyTz Tx T yTz Tx TyT z CrsCyl Ty Ty Ty Ty Ty Ty ParCyl Ty R x Ty Rx T y R x Ty R x Rx Rx Rx Ty R x Ty T y Rx Ty R x Ty R x Ty R x Ty R x Ty EdgSli E Ty R x Ty R x Ty R x Ty R x Ty R x P Ty 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 Ty 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 Ty Ty Ty Ty SphSli S T Ty Ty Ty Ty Ty Ty P y Including Geometric Variation Variables used have nominal values of zero Variation corresponds to the specified tolerance value Rotational variation due to flatness Rotational = ±Db variation between two planar surfaces: Variation Flatness Tolerance = a Zone Translational variation due to flatness Translational =±a/2 variation: Variation Flatness 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. (b1, b2) A .02 .01 U1 .01 b1 b2 b1 b2 f (a 3, a 4) b1 b2 a3 b1 b2 a4 b1 b2 R2 U2 b1 b2 b1 b2 q R3 H A R1 U1 (b 1, b 2) 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 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) Variation – Velocity Relationship (Faerber 1999) Tolerance sensitivity solution 24 r3 23 r4 r2 22 21 r1 Velocity analysis of the equivalent mechanism When are the sensitivities the same? 2D Equivalent Variational Mechanisms (Faerber 1999) 2D dimensional variations to a Add Kinematic Joints: kinematic model using kinematic elements Converts kinematic analysis to variation analysis Equivalent Variational Joint: Extract tolerance sensitivities from velocity analysis Kinematic Assembly works for Cylinder Slider Parallel EvenSlider Planarstatic assemblies (no Edge 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 f R1 f f X f f f f f d f Z R2 f Rigid Prismatic Revolute Parallel Cylinders f Y f f f f X f f f f Z f f f Cylindrical Spherical Planar Edge Slider R d d R1 d 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 f1 F f3 Results from ADAMS velocity analysis: A B D E G I J K L C f1 3D GEVM in ADAMS F f3 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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