1. Surface Variation and Mating Surface Rotational Error in Assemblies Taylor Anderson UGS June 15, 2001

2.  Introduction  Periodicity in surface variation  Characterization of surfaces  Quantifying assembly variation  Conclusions outline

3. introduction  Every product manufacturer in the world is chasing the product quality “Holy Grail”  Effective Product Lifecycle Management must include variation analysis and tolerance management  ADCATS and others are working to make this as painless as possible

4. component variation  Size or location variation  Form or shape variation  Feature orientation variation  Surface roughness variation

5. real-world surface variation  All real surfaces contain SOME variation.  Surface variation can cause assembly variation.  Surface variation can propagate through assemblies.

6. assembly variation  Component size variation  Component feature location variation  Component form or shape variation

7. accumulation of variation Geometric variations propagate through an assembly as imperfect shapes and surfaces contact each other.

8. propagation of variation F F F K K K F F X X Y Y  Assembly joints (contacts) have:  Kinematic degrees of freedom  Feature variation degrees of freedom  Feature variation propagates along kinematically constrained degrees of freedom K

9. research objectives 1. Characterize surface variation 2. Correlate rotational error magnitude due to surface variation

10.  Many manufacturing processes are periodic Milling, turning, machined molds, etc.  Many factors affect periodicity Spindle speeds / feed rates Vibration and/or deflection of: cutting tool material being cut fixturing assemblies machine tool periodicity in surface variation

11. Surface variation can be characterized as a sum of several sinusoids. periodicity in surface variation Surface Profile

12. extracting periodic information  Sum of periodic variations appears in nature – Vibratory systems – Optics – Signal processing – Acoustics – others… time signal amplitude sampling interval signal processing distance surface variation amplitude sample length surface variation

13. t y y Fourier analysis method T  Fixed sampling interval  Fixed sampling rate  Store ( t, y ) pairs – Time coordinate – Amplitude coordinate Time VariationFrequency Spectrum

14. AutoSpectrum Surface Fourier analysis method

15. C.L. 1 C.L. 2 / C.L. wavelength is not enough…

16. Scalable when rotation is less than 5 degrees. (small angle theorem) C.L. 1 C.L. 2 <  max rotation depends on / C.L. C.L. 1 C.L. 2 dimensionless parameter

17. non-dimensionalizing rotation Characteristic Length = C.L. Tolerance Zone

18. non-dimensionalizing rotation Characteristic Length = C.L. Tolerance Zone   = actual rotational error  Tolerance Zone

19.  ArcTan ( ) ZoneC.L. ==== non-dimensionalizing rotation Characteristic Length = C.L. Tolerance Zone  = standardized rotational error

20.  non-dimensionalizing rotation Characteristic Length = C.L. Tolerance Zone    is dimensionless (standardized) (actual)

21. Video microscope Collect simulated surface data Collect real surface data Known sinusoidal inputs Manufactured surfaces Surface generation program Analyze rotational error Interpret results research methodology SimulationApplication

22. theoretical surface simulation Inputs Assembly Simulation Random sinusoidal inputs for: Form variation (wavelength, amplitude, phase) Waviness variation (wavelength, amplitude, phase) Roughness variation (wavelength, amplitude, phase) Simulated Surfaces 200 data points per sample 4000 samples per Monte Carlo simulation

23. manufactured surface analysis Raw Data Digital Enhancement Assembly Simulation

24. Wavelength / Characteristic Length Max Rotation Magnitude / Beta 0.51.010.0 / C.L. / C.L. max rotational error vs.  / C.L.

25. 0.51.010.0 longer wavelengths max rotational error vs.  / C.L. Wavelength / Characteristic Length Max Rotation Magnitude / Beta 0.25 0.50 0.66 0.80 1.00 1.20 3.00 5.00

26. Zone #1: / C.L. < 0.5 Zone #2: / C.L. > 0.5 and / C.L. > 1.0 Zone #3: / C.L. > 1.0 0.51.010.0 max rotational error vs.  / C.L. Wavelength / Characteristic Length Max Rotation Magnitude / Beta

27. phase distribution assumption  Probability that a given C.L. will encounter a given phase is uniformly distributed.  Goal is statistical understanding of the distribution of rotational errors for various values of /C.L. C.L.

28. Wavelength / Characteristic Length Max Rotation Magnitude / Beta 0.51.010.0 max rotational error vs.  / C.L.

29. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0

30. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0 ≤ 0.50

31. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0 0.50

32. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0 0.66

33. distribution for  / C.L. = 0.66 0%100%Phase Amplitude Frequency 0 66%=0 66% in spike 4 3 2 1 1 2 3 4 +0.72 Amplitude

34. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0 0.80

35. distribution for  / C.L. = 0.80 0%100%Phase Amplitude Frequency 0 4 2 1 2 3 Amplitude 4 3 1 25%=0 25% in spike +1.70

36. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0 1.00

37. distribution for  / C.L. = 1.00 0%100% Phase Amplitude Frequency 0 4 1 2 3 Amplitude +2.35 2 4 3 1

38. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0 1.20

39. distribution for  / C.L. = 1.20 0%100% Amplitude Frequency 0 4 1 2 3 Amplitude +2.32 2 4 3 1 Phase

40. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0 3.00

41. distribution for  / C.L. = 3.00 0%100% Amplitude Frequency 0 4 1 2 3 Amplitude +1.05 2 4 3 1 Phase

42. Max Rotation Magnitude / Beta Phase rotational error vs.  / C.L. vs. phase Wavelength / Characteristic Length 0.51.010.0 5.00

43. distribution for  / C.L. = 5.00 0%100% Amplitude Frequency 0 4 1 2 3 Amplitude +0.63 2 4 3 1 Phase

44. rotational error distributions  Distributions different at every / C.L.  Distributions are highly non-normal  Logical, gradual change in shape / C.L. < 0.5 / C.L. = 0.66 / C.L. = 0.8 / C.L. = 1.0 / C.L. = 1.2 / C.L. = 3.0 / C.L. = 5.0 / C.L. = 

45. / C.L. Max  /  0.51.010.0 conclusions  This graph describes an UPPER BOUND on rotational error at a given value of / C.L.  Given uniformly distributed phase, these distributions describe the STATISTICAL PROBABILITY of a given rotational error at a given value of / C.L. 0  1 3

46. conclusions  Only SOME values of / C.L. are relevant to assemblies  / C.L. greater than 0.5  / C.L. less than 4.0 (higher for some applications)  Translates to geometric form variations  Roughness and waviness may be neglected

47. conclusions  Characterization using a sum of sinusoids is sufficient Most easily sampled frequencies are most important Very high and very low frequencies are actually least relevant  Non-dimensionalized graphs are scalable May be used for any size geometry  Form variation will dominate rotational error  Variation amplitude and rotation magnitude are linearly correlated within realm of small angle theorem

48. contributions  Rigorous mathematical relationships between periodic surface variation and rotational errors in assemblies  Surface variation simulation model  Application of Fourier transform to surface periodicity extraction  Three regions of rotational behavior  Non-dimensionalized rotation graphs  Monte Carlo simulation of distributions  Small angle theorem applicability

49. recommendations  Model new distributions for use in CATS  Fine-tune the frequency spectra extraction  Characterize manufacturing processes  Specify geometric tolerances based on selection of a characterized manufacturing process

50. Thank You !