Tolerance Analysis of Assemblies Using Kinematically-Derived Sensitivities

1. Tolerance Analysis of Assemblies Using Kinematically-Derived Sensitivities Paul Faerber Motorola Corporation – Lawrenceville, GA Presented by: Jeff Dabling BYU – Graduate Student

2. Differences (Kinematic) Dimensions not allowed to vary Requires sensitivity to velocity inputs Multiple Position Analysis Tolerance Analysis Kinematic Analysis • Similarities • Both use vector loops • Both use kinematic joints • Both use sensitivities • Differences (Tolerance) • Dimensions allowed to vary • Requires sensitivity to dimensional variation • Single position analysis

3. Research Objectives • Model static assemblies with a kinematic modeler • Extract tolerance sensitivities from a kinematic solver • Perform tolerance analysis on a mechanism in multiple positions • Assess difficulty of adding tolerance analysis to commercial kinematic software

4. 3 3 2 4 3 3 4 2 2 4 2 4 1 1 1 1 Tolerance Analysis of Mechanisms Current Method: Solid Model New Position Solid Model New Position Solid Model New Position Original Solid Model Tolerance models for each position of interest

5. Merging Kinematic and Tolerance Analyses Tolerance models for each position Kinematic/ Dynamic Model Multiple Positions Sensitivities

6. q q q q ( ) ( ) ( ) ( ) i i i i + + + = r e r e r e r e 0 1 2 3 4 1 2 3 4 w q + w q + w q = r sin r sin r sin 0 2 2 2 3 3 3 4 4 4 w q + w q + w q = r cos r cos r cos 0 2 2 2 3 3 3 4 4 4 q é ù r sin 2 2 = A ê ú q r cos ë û 2 2 q q - q q é ù r (cos sin cos sin ) 2 2 4 4 2 ê ú w ì ü q q - q q r (cos sin cos sin ) { } { } [ ] w 3 ê ú - = - w = 1 3 4 3 3 4 í ý B A q q + q q r (cos sin cos sin ) w 2 2 ê ú î þ 2 2 3 3 2 4 ê ú q q - q q r (cos sin cos sin ) ë û 4 4 3 3 4 Kinematic Analysis 4-bar mechanism Vector loop equation 3 Resulting velocity equations 2 4 1 Matrix formulation Kinematic Solution OutputsInput

7. Tolerance Analysis 4-bar mechanism Vector loop equations Linearized equations

8. a ì ü d 2 ï ï dr ( ) ï ï é ù 1 a - q - q - q ì ü d ï ï r sin r sin r sin [ ] [ ] [ ] 3 3 3 4 4 4 4 = + = ê ú dr í ý í ý B A B 0 ( ) 2 a q + q q d r cos r cos r cos ê ú î þ ï ï ë û 4 3 3 4 4 4 4 dr ï ï 3 ï ï dr î þ 4 ( ) é ù - q - q - q q q q q r sin r sin r sin cos cos cos cos [ ] A 2 2 3 3 4 4 1 2 3 4 = ê ú ( ) q + q + q q q q q r cos r cos r cos sin sin sin sin ê ú ë û 2 2 3 3 4 4 1 2 3 4 ì ü ì ü da2 da2 ï ï ï ï dr1 dr1 ï ï ï ï [ ] ì ü da3 ï ï ï ï [ ] - 1 Si,j dr2 dr2 = - = í ý í ý í ý B A da4 î þ ï ï ï ï dr3 dr3 Outputs ï ï ï ï ï ï ï ï dr4 dr4 î þ î þ Inputs Tolerance Analysis, continued 4-bar mechanism Matrix Formulation Tolerance Analysis Solution (non-statistical)

9. Estimated Tolerance Accumulation 4-bar mechanism RSS RSS

10. Observations Kinematic Analysis Solution Tolerance analysis solution (non-statistical)

11. w ì ü w w ì ü ì ü 2 2 2 ï ï ï ï ï ï r & r r & & ï ï ï ï ï ï 1 1 1 [ ] w ì ü w ì ü ï ï ï ï ï ï [ ] 3 3 - + = 1 r r r A B 0 = - = & & & í ý í ý í ý í ý í ý B A J 2 2 2 w i , j w î þ î þ ï ï ï ï ï ï 4 4 r r & & r - q - q & é ù r sin r sin ï ï ï ï 3 3 ï ï 3 3 3 4 4 = B ê ú ï ï ï ï r r & & ï ï î þ î þ r q q & r cos r cos î þ 4 4 ë û 4 3 3 4 4 Kinematic Analysis of an Equivalent Variational Mechanism Vector loop equation (Dimension ri not constant) Resulting velocity equations Matrix formulation Kinematic analysis solution

12. Comparisons Tolerance analysis solution (non-statistical) Kinematic analysis of equivalent mechanism solution Are the sensitivities the same?

13. a q d d dr dr dr dr 3 = - 2 + 1 + 2 + 3 + 4 ( J 1 ) J J J J 1 , 1 1 , 2 1 , 3 1 , 4 1 , 5 dt dt dt dt dt dt d d dr dr dr dr = - + - + - + - + - 4 2 1 2 3 4 ( J J ) ( J J ) ( J J ) ( J J ) ( J J ) 2 , 1 1 , 1 2 , 2 1 , 2 2 , 3 1 , 3 2 , 4 1 , 4 2 , 5 1 , 5 dt dt dt dt dt dt Transformation to Relative Angles Relative rotations Differential rotations Psuedo-velocities Kinematic analysis solution Tolerance analysis solution (after substitution) a q

14. Equivalent Variational Mechanisms • Uses kinematic elements to represent dimensional variations in a kinematic model of the assembly • ri are kinematic inputs • ri are proportional to dimensional tolerances

15. Equivalent Variational Mechanisms • Vector loops identical for both types of analyses • Use tolerance analysis techniques to develop vector loops for assemblies • Use these vector loops as a starting point in developing EVM

16. The Stack Blocks Assembly Gap Cylinder Block r p n 2 e d c a Frame b

17. Creating Vector Loop Assembly Models NetworkGraph Kinematic Joints Part and Feature Reference Frames

18. Dimensional Variations Each vector represents a link in the EVM Stacked Blocks Model Dimensional Variations Angular Variations Linear Variations

19. Equivalent Kinematic Joints Stacked Blocks Model Dimensional Variations Kinematic Variations

20. EVM Stack Blocks Assembly Stacked Blocks Model (completed) Dimensional Variations Kinematic Variations Fixed Joints Pin Joints

21. EVM Modeling Techniques a4 r = 0 a3 3 • Extracting Sensitivities from Kinematic Solver • Unit velocities are applied to each independent joint, one at a time. Resulting dependent variables represent the row of the tolerance sensitivity matrix corresponding to that joint. r = 1 r = 0 2 4 a2 q2 Independent variables: r2, r3, r4, q2 Dependent variables: a2, a3, a4

22. Variation Results for Stack Blocks Gap Sensitivities

23. Tolerance Analysis Using Equivalent Variational Mechanisms • Variations in link lengths are allowed by including slider elements in each link. • Sensitivities are used to form RSS expressions used in statistical tolerance analysis. • Kinematic modeler moves to the next location, and the process is repeated. Independent variables: r2, r3, r4, q2 Dependent variables: a2, a3, a4

24. Modeling Techniques for Mechanisms • Apply independent velocities as a reciprocating time function • Each new time step (second) is a new analysis point at the nominal link-lengths • Controlling the period of the function controls the resolution of the analysis, which affects the accuracy of the analysis • Cosine Function • Full magnitude at time = 0 • Period equal to the time step

25. Variation Results for Four-bar Mechanism Sensitivities vary with position (r1, q1 can also vary) Greatest sensitivity at 270º

26. Contributions • Defined relationship between kinematic and tolerance analyses • Developed method for creating and analyzing equivalent variational mechanisms (EVM) • Equivalent 2-D kinematic joints presented • Demonstration of method on static assemblies, as well as kinematic • Demonstrated using commercial kinematic software, ADAMS • Method for extracting kinematic sensitivities • Method for returning the model to its nominal dimensions at each time step

27. Recommendations for Future Research • Investigate relationship between the higher kinematic derivatives (acceleration and jerk) and the higher statistical moments (skewness and kurtosis) • Integrate with commercial kinematics CAD applications • Develop a user interface • Study degree of freedom problems • Extend into three-dimensional assemblies • Include form tolerances in this method