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Surface Variation and Mating Surface Rotational Error in Assemblies
Taylor AndersonTaylor AndersonUGSUGS
June 15, 2001June 15, 2001
Introduction
Periodicity in surface variation
Characterization of surfaces
Quantifying assembly variation
Conclusions
outline
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
component variation
Size or location variation
Form or shape variation
Feature orientation variation
Surface roughness variation
real-world surface variation
All real surfaces contain SOME variation.
Surface variation can cause assembly variation.
Surface variation can propagate through assemblies.
assembly variation
Component size variation
Component feature location variation
Component form or shape variation
accumulation of variation
Geometric variations propagate through an assembly as imperfect shapes and surfaces contact each other.
propagation of variation
F
F
FK
K
K
F
F
XX
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
research objectives
1. Characterize surface variation
2. Correlate rotational error magnitude due to surface variation
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
Surface variation can be characterized as a sum of several sinusoids.
periodicity in surface variation
Surface Profile
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
t
y y
Fourier analysis method
T
Fixed sampling interval
Fixed sampling rate
Store ( t , y ) pairs
– Time coordinate
– Amplitude coordinate
Time Variation Frequency Spectrum
AutoSpectrum
Surface
Fourier analysis method
C.L.1 C.L.2
/ C.L.
wavelength is not enough…
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
dimensionlessparameter
non-dimensionalizing rotation
Characteristic Length = C.L.Characteristic Length = C.L.
Tolerance ZoneTolerance Zone
non-dimensionalizing rotation
Characteristic Length = C.L.Characteristic Length = C.L.
Tolerance ZoneTolerance Zone
= actual rotational error
Tolerance ZoneTolerance Zone
ArcTanArcTan (( ))ZoneZone
C.L.C.L. ==
non-dimensionalizing rotation
Characteristic Length = C.L.Characteristic Length = C.L.
Tolerance ZoneTolerance Zone
= standardized rotational error
non-dimensionalizing rotation
Characteristic Length = C.L.Characteristic Length = C.L.
Tolerance ZoneTolerance Zone
Tolerance ZoneTolerance Zone
is dimensionless
(standardized)
(actual)
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
Simulation Application
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
manufactured surface analysis
Raw Data
Digital Enhancement
Assembly Simulation
Wavelength / Characteristic Length
Max
Rot
atio
n M
agni
tude
/ B
eta
0.5 1.0 10.0
/ / C.L.C.L.
max rotational error vs. / C.L.
0.5 1.0 10.0
longer wavelengths
max rotational error vs. / C.L.
Wavelength / Characteristic Length
Max
Rot
atio
n M
agni
tude
/ B
eta
0.25
0.50
0.66
0.80
1.00
1.20
3.00
5.00
Zone #1: / C.L. < 0.5Zone #2: / C.L. > 0.5 and / C.L. > 1.0 Zone #3: / C.L. > 1.0
0.5 1.0 10.0
max rotational error vs. / C.L.
Wavelength / Characteristic Length
Max
Rot
atio
n M
agni
tude
/ B
eta
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.
C.L.
C.L.
C.L.
Wavelength / Characteristic Length
Max
Rot
atio
n M
agni
tude
/ B
eta
0.5 1.0 10.0
max rotational error vs. / C.L.
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
≤ 0.50
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
0.50
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
0.66
distribution for / C.L. = 0.66
0% 100%Phase
Amplitude
Fre
qu
ency
0
66%=0
66% in spike
4
3
2
1
1
2
3
4
+0.72
+0.72
Am
pli
tud
e
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
0.80
distribution for / C.L. = 0.80
0% 100%Phase
Amplitude
Fre
qu
ency
0 4
2
1
2
3
Am
pli
tud
e
4
3
125%=0
25% in spike
+1.70
+1.70
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
1.00
distribution for / C.L. = 1.00
0% 100%Phase
Amplitude
Fre
qu
ency
0 4
1
2
3
Am
pli
tud
e
+2.35
2
4
3
1
+2.35
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
1.20
distribution for / C.L. = 1.20
0% 100%
Amplitude
Fre
qu
ency
0 4
1
2
3
Am
pli
tud
e
+2.32
2
4
3
1
Phase
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
3.00
distribution for / C.L. = 3.00
0% 100%
Amplitude
Fre
qu
ency
0 4
1
2
3
Am
pli
tud
e
+1.05
2
4
3
1
Phase
Max
Rot
atio
n M
agni
tude
/ B
eta
Phase
rotational error vs. / C.L. vs. phase
Wavelength / Characteristic Length
0.5 1.0 10.0
5.00
distribution for / C.L. = 5.00
0% 100%
Amplitude
Fre
qu
ency
0 4
1
2
3
Am
pli
tud
e
+0.63
2
4
3
1
Phase
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. =
/ C.L.
Max
/
0.5 1.0 10.0
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
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
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
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
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
Thank You !