Modeling Surface Variations
For
Flexible Assemblies
by
Shrinivas Soman
Unigraphics Solutions
1
Analysis of Flexible Assemblies
Mean Assembly Force
{F} = [Keq]{ 0}
Assembly Force Variance
[ F] = [Keq][ 0][Keq]T
2
Functional Surface Characterization
• Relates Surface Topography to Tolerance Analysis
• Characterizes Profile of the Surface
• Discards Short Wavlength Variations
• Characterizes a Population of Surfaces using Two Functions -
– Mean Profile
– Average Autospectrum
• Computes Covariance using Average Autospectrum
3
Functional Surface Characterization Procedure
• Data Acquisition
• Part Set-up, Fixture Design
• Establishment of Coordinate System
• Sampling Issues - Alias, Frequency, Filtration,
Leakage
• Data Analysis
• Separation of Random and Non-random Variations
• Statistical Characterization of Random Variations
• Computing Autocorrelation (Covariance)
4
Sampling Issues
Alias in Frequency -
• Sampling rate must be higher than twice the highest frequency
Profile represented by sinusoid frequency of 10 Profile Sampling by sampling frequency of 100 and 12
1 1
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
profile height
profile height
0 0
-0.2 -0.2
-0.4 -0.4
-0.6
-0.6
-0.8
-0.8
-1
-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
spatial distance
spatial distance
5
Effect of Alias in Frequency
-3
x 10
4
3.5
3
Fs = 32
2.5
amplitude, inch
2
Fs = 128
1.5
Fs = 512
1
0.5
0
0 5 10 15 20 25
spatial frequency, cycles/profile length
• Frequency Alias limits the number of data points in a part length.
• Number of data points describing surface variations may not be
equal to number of nodes in the Finite Element Model.
6
Leakage and Frequency Filtration
Figure 4.7A Waveform with f=105cycles/unit length
1
0.5
• Leakage -
waveformheight
0
Frequencies which are not multiples of
-0.5 fundamental frequency, leak into
-1
adjacent frequencies.
0 0.02 0.04 0.06 0.08 0.1
spatial distance
Figure 4.7B Frequency Spectrum
0.8
• Frequency Filtration -
0.6
Finite tip of stylus acts as a low pass
amplitude
0.4
filter.
0.2
0
0 20 40 60 80 100 120 140
spatial frequency, cyles/unit length
7
Profiles Records in Spatial Distance Domain
0.665
0.66
0.655
Profile height
0.65
profile height, inch
0.645
0.64
0.635
0.63
0.625
0.62
0.615
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
spatial distance, inch
Spatial distance
8
Profile Records After Removal of Non-characteristic
Variations
x 10 -3
Profile height
Z Stylus
prof ile height
Leg Height
Surface to be Characterized
Plane of Scanning -5
(0,0) Y
-10
-15
0 0.5 1 1. 5 2 2.5 3 3.5 4 4.5 5
s patial dis tanc e
Spatial distance
9
Mean Surface Profile
-3
x 10 Mean Profile
6
2
Profile height
0
profile height
-2
-4
-6
-8
-10
-12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
spatial distance
Spatial distance
10
Random Variations, FFT’s and Autospectra
x 10-3 RandomVariations x 10-3 FFTs x 10-8 Autospectra
5
5
0 5
-5
0 0
0
x 10-3 2 4 0x 10-3 10 20 0x 10-8 10 20
5
5
0 5
-5
0 0
0
x 10-3 2 4 0x 10-3 10 20 0x 10-8 10 20
5
5
0 5
-5
0 0
0
x 10-3 2 4 0x 10-3 10 20 0x 10-8 10 20
5
5
0 5
-5
0 0
0
x 10-3 2 4 0x 10-3 10 20 0x 10-8 10 20
5
5
0 5
-5
0 0
0
x 10-3 2 4 0x 10-3 10 20 0x 10-8 10 20
5
5
0 5
-5
0 0
0 2 4 0 10 20 0 10 20
11
Average Autospectrum and Spatial Frequency
Classification
LONG WAVELENGTH REGION
MEDIUM WAVELENGTH REGION
x 10
-8
SHORT WAVELENGTH
Average Autospectrum REGION
3.5
2.5
autospectrum
1.5
0.5
0
0 5 10 15 20 25
spatial frequency
12
Stationary Process and Autocorrelation
(x )
(x)
(x )
n
• Autocorrelation Function R yy (x1, d) = (1 / n) YK(x1) YK(x1 + d)
k=1
• For Stationary Process, R yy (x1, d) = R yy (x2, d) = R yy (x1, d) = -------= R yy (xi, d)
• Averaging the Autospectrum averages variations across
the population.
• Stationarity assumption averages the variations along
the surface.
13
Circular and Zero Padded Autocorrelation
Autocorrelation Profile Length
Circular
Zero Padded
Biased
Zero Padded Scale Zero Padded Biased
Unbiased
14
Autocorrelation for Population of Surface Profiles
x 10-6
6
4
circular AC
2
zero padded biased AC
0
Autocorrelation
-2
-4
-6
unbiased AC
-8
-10
-12
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
spatial distance
15
Removing Effect of Long Wavelengths
x 10-3 Profiles x 10-3 Polynomial Curves x 10-3 Random Variations
2
5 5
0
0 0
-5 -5 -2
0 2 4 0 2 4 0 2 4
x 10-3 x 10-3 x 10-3
2
5 5
0
0 0
-5 -5 -2
0 2 4 0 2 4 0 2 4
x 10-3 x 10-3 x 10-3
2
5 5
0
0 0
-5 -5 -2
0 2 4 0 2 4 0 2 4
x 10-3 x 10-3 x 10-3
2
5 5
0
0 0
-5 -5 -2
0 2 4 0 2 4 0 2 4
x 10-3 x 10-3 x 10-3
2
5 5
0
0 0
-5 -5 -2
0 2 4 0 2 4 0 2 4
x 10-3 x 10-3 x 10-3
2
5 5
0
0 0
-5 -5 -2
0 2 4 0 2 4 0 2 4
16
Improved Autocorrelation
x 10-8
6
circular AC
4
zero padded AC
2
Autocorrelation Function
-2
-4
unbiased AC
-6
-8
0 1 2 3 4 5 6
spatial distance
17
Conclusion
• Spectrum Model is suitable for surfaces having dominant medium
wavelengths.
• Long wavelengths distort Spectrum Model.
• Polynomial Curves for removing long wavelengths?
18
