1. Tolerance Analysis of Flexible Assemblies Michael Tonks

2. Outline  Background  Method  Example  Geometric Covariance  Current Research

3. Introduction  Statistical Tolerance Analysis (STA)… statistically predicts critical assembly variations  Why STA? Part variations propagate through assemblies Predicting effects of tolerance stack-up allows designer to avoid assembly problems

4. Flexible Parts in Assemblies  Flexible parts deform when assembled due to warping and dimensional variations  Produces internal stresses and deflections  Rigid body STA cannot predict deformations and stresses in flexible parts  A method for analyzing flexible parts is needed

5. Where is Flexible STA Needed? Aircraft Skin Panels Plastic Assemblies Automotive Body Panels

6. What is Flexible STA? Analysis method combines STA with finite element analysis (FEA)  Rigid body STA gives part misalignment  FEA used to model parts as flexible bodies, rather than rigid  Statistically predicts forces, stresses, and deformations from assembling flexible parts

7. The Flexible STA Method Flexible STA Method Steps 1.Determine misalignment 2.Model the compliant parts using FEA 3.Calculate covariances A.Material B.Geometric 4.Statistical FEA Solution for assembly forces, stresses, and deformations

8. 1. Determine Misalignment Rigid body STA is used to find the misalignment. A mean and standard deviation of the gap is found. Gap Vectors: V A, V B = Variation of parts A and B from nominal  A,  B = Equilibrium deflection after assembly

9. 2. Model Parts using FEA Parts are meshed in an FEA Program:  Create geometry  Mesh each compliant part, place nodes at fastener locations  Apply displacement boundary conditions  Output the global stiffness matrices  Create Super Elements  Create Equivalent Stiffness Matrix MSC.Nastran MSC.Marc

10. Create Super Element Matrices  The equivalent stiffness matrix is condensed by eliminating interior nodes that have no displacement  Matrix reduced to create equivalent super-element matrix Small and easy to analyze Contains all important information

11. Calculate equivalent stiffness The Super-element matrices from the mating parts are combined to form an equivalent stiffness matrix

12. Problems with STA Analysis  Some STA methods treat all nodes as independent from all other nodes  With real parts, nodal variations are NOT independent  Problem solved using covariance

13.  Covariance describes the interdependence between random variables  There are two sources of covariance in flexible assemblies: Material Covariance Geometric Covariance 3. Calculate Covariances Uncorrelated Partially Correlated Fully correlated

14. 3A. Material Covariance  Displacing one node affects the displacement of surrounding nodes  Described by the stiffness matrix of the part

15. 3B. Geometric covariance  Nodal variations are not independently random  Part surfaces are continuous  Random surfaces must be used to include covariance effects in statistical analyses

16. 4. Statistical FEA Solution  Mean FEA Solution F=K eq (  )  Variance FEA Solution Geometric Covariance   cov = S   S T Material Covariance  Fcov = K eq   cov K eq T Force Covariance Matrix Completely Characterizes surface with 2 FEA solutions

17. Example Sample problem analyzed using FASTA method and Monte-Carlo Method  Simple lap joint of two thin plates  Monte-Carlo method used with covariance  Results compared from two methods

18. Example Monte Carlo FASTA Standard deviation of closure force 5,000 FE solutions Large sample size required for accuracy Slow 2 FE solutions Very similar results Very Fast

19. Method Comments  Covariance matrix critical for accurate results  Material covariance method well established  Best geometric covariance method not yet determined Best way to characterize surface is difficult question

20. Geometric Covariance Methods  Three surface modeling methods have been developed to calculate Geometric Covariance Bezier Curves Polynomial Curves Spectral Analysis

21.  Sensitivity matrix found in terms of neighboring points  Sensitivity matrix gives continuity conditions used in finding the geometric covariance curve Polynomial Method Variation of points is constrained to be a polynomial curve Surface Profile

22. Polynomial Covariance Matrix  Matrix shows interdependence between any two nodes  Interdependence created with 3 rd order polynomial

23. Polynomial Method  Advantages Simple to apply Can handle long wavelengths  Disadvantages Does not consider frequency spectrum Must be matched to wavelength of surfaces

24. Spectral Analysis Method FFT 2 Models surface variation as a finite sum of discrete sinusoids

25. Create Covariance Matrix  Because the covariance is assumed to be independent of location, every row of the covariance matrix is similar  Variance, the diagonal terms of the covariance matrix, is the autocorrelation function at zero separation  The autocorrelation function can be shifted to form each row of the geometric covariance matrix: Autocorrelation Geometric Covariance Matrix

26. Wavelength Effect Wavelengths of surface variation effects stress in part Standard deviation of assembly stress  x \L=1 \L=1/3 \L=1/4 \L=1/2

27. Spectral Analysis Method  Advantages Models surface variations Takes in account wavelengths of variations  Disadvantages Cannot model wavelengths longer then part lengths

28. Current Limitations  Results of three Geometric Covariance methods have never been fully compared  Spectral Analysis Method cannot analyze variations with wavelengths greater then surface length  Never applied to production assemblies

29. Flexible STA Research Sponsor: Boeing Phantom Works 3 Year Project Goal: Create software analysis package for aircraft assemblies

30. Research Plans  Research to be conducted at BYU  First Year Plans: Two Masters Research Projects Develop measurement techniques Preparation for creation of analysis software

31. Application to Production Assemblies Partnership with Boeing/Salt Lake  Test Assembly Requirements: 1.Small enough to be measured on the BYU CMM machines. 2.Not too complex. Not too many parts 3.All metal parts. Joined by metal fasteners. 4.Assembly force required to close gaps between mating parts. 5.Available in small quantities for measurement purposes. Flexible Skin Rigid Spars Assembly Leading Edge Assembly

32. Project 1 Goal: To create a method that combines the strengths of current methods  Analyze simple 3-D problem with all current geometric covariance methods and compare results  Create hybrid method that can analyze surface wavelengths longer than mating surfaces

33. Project 2 Goal: Apply Flexible STA methods to a real assembly  Use rigid body STA to find gap variation  Measure real parts  Characterize variation statistically  Use hybrid flexible STA method to predict resultant deflections, stresses, and forces when assembled

34. Conclusion  Flexible assembly STA research at BYU moving from theoretical to real applications  Will be important tool for: Aerospace Automotive Electronic Many More!

35. Questions?