1. Geometric Covariance in Compliant Assembly Tolerance Analysis Jeffrey B. Stout Brigham Young University ADCATS June 16, 2000

2. Presentation Outline P Introduction P Background P Compliant Assembly Tolerance Analysis P Geometric Covariance P Simulations, Results, and Verification P Conclusions and Recommendations

3. Introduction Traditional Statistical Assembly Tolerance Analysis Methods P Traditional tolerance analysis methods predict the accumulation of manufacturing variations in assemblies. P Based on the assumptions that the variations are random and all of the parts are rigid. Loop 1 Loop 2 Two closed loops of a stacked block problem P Vector Loop Method (CE/TOL) P Monte Carlo Analysis (VSA)

4. Introduction A new method must be used to analyze assemblies with compliant parts. FEA and STA can be combined to analyze assembly tolerances with compliant parts.

5. Background Compliant Assembly Tolerance Analysis - Related Research P Variation Modeling in Vehicle Assembly by Hsieh and Oh P Tolerance Analysis of Flexible Sheet Metal Assemblies by Liu, Hu, and Woo P FEA/STA Method by Merkley and Chase, contributions by Soman, Bihlmaier, and Stout

6. Compliant Assembly Tolerance Analysis Method Outline - Combining STA with FEA P Find Misalignment of Rigid Parts P Model the Flexible Parts with FEA P Obtain the Condensed Stiffness Matrices P Apply Boundary Conditions P Solve for Closure Forces < Mean force < Variation of force about the mean

7. Find Misalignment of Rigid Parts P Create closed vector loop. Determine mean gap/interference. P Estimate variation due to stack-up. P Include dimensional variation of both rigid and flexible parts. Compliant Assembly Tolerance Analysis BD A EC A B CDE t = T A 2 + T B 2 +T C 2 +T D 2 +T E 2 t

8. Model the Flexible Parts with FEA P Create geometry P Mesh each compliant part, place nodes at fastener locations P Apply displacement boundary conditions P Output the global stiffness matrix Compliant Assembly Tolerance Analysis

9. Obtain Condensed Stiffness Matrices The Global Stiffness Matrix P Sum of element stiffnesses P Includes all model degrees of freedom P Symmetric P Very large and sparse KaKa 1 2 3 4 5 6 7 8 9 10 K a = Compliant Assembly Tolerance Analysis

10. Obtain Condensed Stiffness Matrices Matrix Partitioning K a = = Boundary DOF K bb Coupled DOF K bi Coupled DOF K ib Interior DOF K ii KbbKbb K ib K bi K ii P Sort the stiffness matrix for each part to group the boundary node DOF and interior node DOF P Partition the matrix Compliant Assembly Tolerance Analysis

11. Obtain Condensed Stiffness Matrices Create Super-Element. Condense the stiffness matrix for each part to include only the boundary nodes. KbbKbb K ib K bi K ii F b = K bb ** b + K bi ** i FbFb FiFi = *b*b *i*i 0 = K ib ** b + K ii ** i No forces on interior nodes, F i =0 Solve second equation for * i : * i = -K ii -1 *K ib ** b Substitute this into the first equation F b = K bb ** b - K bi *K ii - 1 *K ib ** b = [K bb - K bi *K ii - 1 *K ib ]** b K se = K bb - K bi *K ii -1 *K ib Super-element stiffness matrix: º F b and  i are unknown Compliant Assembly Tolerance Analysis

12. Obtain Condensed Stiffness Matrices Advantages of Super-Elements P Super-elements are well suited for assembly analysis < Each part can be represented by a super-element P Reduces part stiffness matrix DOF to boundary node DOF P Simplify matrix algebra, and reduce computation time P Provide sensitivities for statistical analysis Compliant Assembly Tolerance Analysis

13. Apply Boundary Conditions P Compliant members A and B are to be joined. The dimensional variations v a and v b of each part are given. P At the point of equilibrium, the assembly forces are equal and opposite. P Solve for * a and * b. These are the deflections required to assemble the parts at equilibrium. P Solve for the assembly closure forces from the part stiffness matrices. *a*a *b*b KaKa VbVb KbKb VaVa equilibrium position nominal position Compliant Assembly Tolerance Analysis

14. Apply Boundary Conditions 2-D Assembly Closure Force -Derivation KaKa VbVb KbKb VaVa nominal position Misaligned Parts Define Gap Vector: * o = v b -v a *o*o *a*a *b*b KaKa KbKb equilibrium position Force-Closed Assembly Define Closure Displacements: * a - * b = * o Closure Forces: f a = -f b *o*o Compliant Assembly Tolerance Analysis NOTE: Displacements and stiffness for all degrees of freedom must be linear and elastic.

15. Solve for Closure Forces 2-D Assembly Closure Force -Derivation *a*a *b*b KaKa VbVb KbKb VaVa Equilibrium Nominal f a = -f b K a @* a = -K b @* b since * b = * a - * o, then K a @* a = K b @* o - K b @* a K a @* a + K b @* a = K b @* o (K a + K f a = -f b ) @* a = K b @* o *o*o Closure Deflections: * a = (K a + K b ) -1 @ K b @* o * b = * o - * a Closure Forces: f a = K a @* a f a = K a @ (K a + K b ) - 1 @ K b @* o Compliant Assembly Tolerance Analysis

16. Solve for Closure Forces P Four types of assembly solutions: < Single Solution - Individual assembly, measured dimensional variation < Average - mean of a production lot of assemblies < Worst Case - stack-up analysis assuming all dimensions are at size limits < RSS - statistical analysis using root-sum-squares stack-up, statistical variation. Compliant Assembly Tolerance Analysis

17. Solve for Closure Forces Summary of Key Equations f a = K ra @* o Use the following equations for single and mean cases Use the covariance equation for statistical cases Define the stiffness matrix ratio K ra = K a @ (K a + K b ) - 1 @ K b  fa = K ra @  o @ K ra T  fa = K ra @  o f a = K a @ (K a + K b ) - 1 @ K b @* o Compliant Assembly Tolerance Analysis

18. Solve for Closure Forces Derivation of Statistical FEA/STA Equations E[f a ] = E[K ra A     fa  ra A   E[(f a -  fa ) A (f a -  fa ) T ] = E[(K ra A    K ra A  o ) A (K ra A    K ra A  o ) T ]  fa = E [ K ra A (    o ) A [K ra (    o )] T ]  fa = E [ K ra A (    o ) A (    o ) T A K ra T ]  fa = K ra A  o A K ra T First statistical moment: mean Second statistical moment: variance E = statistical expectation operator f a = K ra @* o Compliant Assembly Tolerance Analysis

19. Create Super-Element Stiffness Matrices Two Plate Example P Plate are modeled using plane stress theory P 10 x 10 element grid giving 11 nodes along the coincident edges P The global stiffness matrix of each part contains 121 nodes with two DOF per node, giving 242 total DOF P The super-element of each part only contained 22 DOF

20. Geometric Covariance P The Need for a Covariant Solution P Material and Geometric Covariance Defined P Combining Geometric Covariance and FEA/STA P The Curve Fit Polynomial Method P Forms of Geometric Covariance

21. Material and Geometric Covariance Defined General Covariance A coupling of two variables Geometric Covariance y x y x y x Uncorrelated Partially Correlated Fully Correlated Random variables x and y Ellipses indicate constant probability

22. Material and Geometric Covariance Defined Material Covariance Geometric Covariance  x =  /E  y =  A   y =  A  x Strain is fully correlated  = K A f The stiffness matrix defines the correlation of neighboring point displacements when a single point is subjected to a force. Force Original Surface Deformed Surface

23. Material and Geometric Covariance Defined Geometric Covariance Geometric variation of each point is unlikely to be completely random Variation of points will be in the form of a continuous surface

24. Combining Geometric Covariance and FEA/STA Combination Called CoFEA/STA Method Geometric Covariance  fa = K ra @ G o @ K ra T Ga=Sa@a@SaTGa=Sa@a@SaT Define the part geometric covariance matrix to be a function of the displacement sensitivities and the part variation. Go=So@o@SoTGo=So@o@SoT The same rule applies to the whole gap as well. The geometric covariance term replaces the variance term to form the CoFEA/STA equation: The sensitivity matrix S a defines the sensitivity of each DOF ' s position with respect to all other DOF. S a is symmetric by nature.

25. The Curve Fit Polynomial Method Geometric Covariance Variation of points is constrained to be in the form of a polynomial curve i i+1 i-1 i-2 x y y i = c o + c 1 A x i + c 2 A x i 2 + c 3 A x i 3 Find sensitivity of other y values with respect to y i y i =... + s i-2 A y i-2 + s i-1 A y i-1 + s i A y i + s i+1 A y i+1 +... P The set of y ' s to be considered can be either a local band or all the points on the mating edge (banded or truncated). P For curve fit polynomials, the s terms turn out to be purely a function of the x-spacing between the nodes. P The s terms fill the sensitivity matrix S; each y i has a set of s terms that fill a row or column of S.

26. Forms of Geometric Covariance P Zero Covariance < Independent variation, see previous example P Total Covariance < Mating surfaces constrained to displace as a unit P Curve Fit Polynomial Covariance < Mating surfaces constrained to be in form of a polynomial < Can be any order of polynomial including a 1st order line fit P Sinusoidal Covariance < Applied constraints of varying amplitudes and wavelengths < Capable of higher order waves < Fourier series = ability to simulate almost any surface Geometric Covariance

27. Simulations, Results, and Verification P Use of Monte Carlo Simulations P Zero Covariance Case P Full Covariance Case P Curve Fit Polynomial Covariance Case P Comparison of Results

28. Two Plate Example A y z x *o*o B P* o is a vector of gaps between the nodes along the mating surfaces. P* a and * b are vectors of the displacements at each node required to assemble the parts in equilibrium -to close the gap. P Tolerance on each mating edge = 0.05". Uniform distribution gives * o = {0.00167 0 0.00167 0...} in 2 part A part B 10.00" 0.10" thick alum. typ. fixed edge, typ. mating edge x y Simulations, Results, and Verification

29. Use of Monte Carlo Simulations Method to Verify CoFEA/STA P A Monte Carlo simulation can be used as an alternate method to analyze the statistical closure forces of a compliant assembly. P Large populations of individual assemblies with random variations are solved. P Results are obtained by examination of the mean and variance of the population of solutions. Simulations, Results, and Verification

30. Zero Covariance Case Two Plate Problem with 40 Elements Along Part Edges P Gap variation: each node on the mating edge is allowed to randomly vary within the tolerance zone. P Geometric covariance matrix has the constant variance magnitude down the diagonal, 0.00167in 2 P Solved using both CoFEA/STA and Monte Carlo simulation Simulations, Results, and Verification

31. Closure Force Covariance Matrix Solution to Monte Carlo simulation Zero Covariance Case Simulations, Results, and Verification

32. Force Covariance Solution Solution using CoFEA/STA Two Plate Example Zero Covariance Case

33. CoFEA/STA and MCS Solution - Closure Force Standard Deviation 510152025303540 0 2000 4000 6000 8000 10000 12000 Standard Deviation of Closure Forces - Using FEA/STA node number force (lb) max force = 12170.2 lb Simulations, Results, and Verification

34. Full Covariance Case Two Plate Problem with 40 Elements Along Part Edges P Gap variation: all nodes along the mating edge displace the same magnitude P Geometric covariance matrix is populated entirely by the gap variance magnitude, 0.00167in 2 P Solved using both CoFEA/STA and Monte Carlo simulation Simulations, Results, and Verification

35. Full Covariance Case Solution - Closure Force Covariance Matrix Simulations, Results, and Verification

36. Full Covariance Case Solution - Closure Force Standard Deviation 510152025303540 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Standard Deviation of Closure Forces - Using CoFEA/STA node number force (lb) max force = 521.8 lb Simulations, Results, and Verification

37. Curve Fit Polynomial Covariance Case Two Plate Problem with 40 Elements Along Part Edges P Mating edge constrained to conform to 3rd order polynomial P Set of points which affect geometric covariance include all points along the mating edge (truncated ends algorithm) P Solved using both CoFEA/STA and Monte Carlo simulation Simulations, Results, and Verification

38. Curve Fit Polynomial Covariance Case Gap Covariance Matrix -CoFEA/STA Method Simulations, Results, and Verification

39. Curve Fit Polynomial Covariance Case Closure Force Covariance Matrix - CoFEA/STA Method Simulations, Results, and Verification

40. Curve Fit Polynomial Covariance Case Closure Force Standard Deviation - CoFEA/STA Method 510152025303540 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Standard Deviation of Closure Forces - using CoFEA/STA node number force (lb) Simulations, Results, and Verification

41. Curve Fit Polynomial Covariance Case Gap Covariance Matrix - Monte Carlo Simulation Simulations, Results, and Verification

42. Curve Fit Polynomial Covariance Case Gap Histogram - Monte Carlo Simulation Simulations, Results, and Verification

43. Curve Fit Polynomial Covariance Case Closure Force Covariance Matrix - Monte Carlo Simulation Simulations, Results, and Verification

44. Curve Fit Polynomial Covariance Case Closure Force Histogram - Monte Carlo Simulation Simulations, Results, and Verification

45. Curve Fit Polynomial Covariance Case Closure Force Standard Deviation - Monte Carlo Simulation 510152025303540 0 200 400 600 800 1000 1200 1400 1600 1800 2000 node number force (lb) max force = 862.7 lb Simulations, Results, and Verification

46. Comparison of Results 510152025303540 0 200 400 600 800 1000 1200 1400 1600 node number 1.12 1.14 1.16 1.18 1.2 1.22 1.24 x 10 4 Standard Deviation of Closure Forces - using CoFEA/STA force (lb) NO COVARIANCE 5TH ORDER POLYNOMIAL COVARIANCE 3RD ORDER POLYNOMIAL COVARIANCE TOTAL COVARIANCE force (lb) P CoFEA/STA results compare almost perfectly with Monte Carlo simulations when boundary conditions are set up properly P Inclusion of geometric covariance dramatically reduces the variance of the closure forces Simulations, Results, and Verification

47. Conclusions P The FEA/STA method was introduced and found to be incomplete without consideration of surface continuity P Geometric covariance was introduced, derived, and incorporated into FEA/STA (CoFEA/STA) P The effects of three forms of geometric covariance were investigated using the CoFEA/STA method < Zero covariance, total covariance, and curve fit polynomial covariance P The CoFEA/STA method was demonstrated and verified using Monte Carlo simulations

48. Conclusions A variety of gap cases can be analyzed using the same FEA model, no additional iterations necessary. KaKa KbKb *o*o y z x KaKa KbKb 2 y z x KaKa KbKb *o*o y z x KaKa KbKb y z x 2 Uniform X-GapTwisted Offset Uniform Y-Gap Rotated Gap/Interference

49. Limitations Drawbacks of the CoFEA/STA method P Small deformation theory applies, elastic behavior P Assemblies and parts can only be analyzed in their linear range P Need access to full stiffness matrices of compliant parts P Need to be able to manipulate the stiffness matrices P Need to include effects of covariance

50. Applications P The CoFEA/STA theory and method of tolerance analysis for assemblies containing compliant parts allows: < Prediction of mean and variance closure forces in assemblies with compliant parts < Prediction of statistical deformations and equilibrium position of compliant parts after assembly < Sensitivity analysis of assemblies containing compliant parts which allows improvement of design and increased robustness < Modeling and improvement of manufacturing process variation

51. Future Work P Refine CoFEA/STA for plates and shells P Expand capabilities to models with more DOF per node - define geometric covariance with rotational DOF P Pursue geometric covariance using sinusoids P Create database of manufacturing variation that can be used to characterize geometric covariance for common processes P Combine CoFEA/STA with optimization tools to find best order of assembly