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Position Error in Assemblies and Mechanisms. Statistical and Deterministic Methods. By: Jon Wittwer. Outline. Position Error of Part Features Position Error in Assemblies Direct Linearization Deterministic Methods Statistical Methods Summary Questions. 0.06. B. A. A. y. B. x. - PowerPoint PPT Presentation
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Position Error in Assemblies and Mechanisms
Statistical and Deterministic Methods
By: Jon Wittwer
Outline
Position Error of Part FeaturesPosition Error in AssembliesDirect LinearizationDeterministic MethodsStatistical MethodsSummaryQuestions
Position Error of Part Features
x
y
A B0.06
B
A
A Norm al Dis tribution About a Target Mean
0
5
10
15
20
25
30
35
40
45
24.94 24.96 24.98 25 25.02 25.04 25.06
1-D Statistical ErrorTarget (Nominal) Dimension: 25.00 inTolerance: ±.03 in
Process Standard Deviation: = ±.01 inYield: 99.73%
3
24.97 25.03
2-D Position Tolerance
x
y
A B0.06
B
A
Tolerance Zone
0.06
IdealPosition
ActualPosition
Assuming Both x and y are normally distributed…
2-D Statistical Position Error
Contours of Equal Probability: CIRCLE
Case 1: If x = y
XY
Frequency Distribution
Tolerance Zone
Yield: 98.889%
R 3
R 4
R 6
IdealPosition
2-D Statistical Position Error
Contours of Equal Probability: ELLIPSE
If x ≠ y
XY
Frequency Distribution
Tolerance Zone
3
4
6
Yield: 65%
Position Error in Assemblies
x
y
Position Error in Assemblies
x
y
r2
r3
r4
P
a3
b3
r1
Pbarr
rrrr
4332
1432 0Closed Loop:
Open Loop:
Position Error in Assemblies
The x and y position error of the Coupler Point (P) are no longer independent.
Position Error in Assemblies
x
y
r1
Position Error in Assemblies
x
y
r1
Px
Py
Position Error in Assemblies
Methods
Deterministic (Worst-Case):• Involve fixed variables or constraints
that are used to find an exact solution.
Probabilistic (Statistical):• Involve random variables that result
in a probabilistic response.
Direct Linearization (DLM)
01432 rrrr
Closed Loop:
0sinsinsinsin
0coscoscoscos
11443322
11443322
rrrr
rrrrhx:hy:
Taylor’s Series Expansion:
]0[}]{[}]{[}{ UBXAH
{X} = {r1, r2, r3, r4} :primary random variables{U} = {3, 4} :secondary random variables }]{[}{ 1 XABU
Solving for Assembly Variation
Open Loop:
Px =
Pbar 432
)90sin(sinsin
)90cos(coscos
333322
333322
bar
bar
}]{[}]{[}{ 1 XSXAEBCP
}]{[}]{[}{ UEXCP
Taylor’s Series Expansion:
Solving for Position Variation:Sensitivity Matrix
Py =
Worst-Case vs. Statistical
jij XSP
2)( jij XSP
Worst Case:
Statistical (Root Sum Square):
Deterministic Methods:1. Worst-Case Direct
Linearization:• Uses the methods just discussed.
2. Vertex Analysis: • Finds the position error using all
combinations of extreme tolerance values.
3. Optimization:• Determines the maximum error using
tolerances as constraints.
Analogy for Worst-Case Methods
Tolx
Toly
Ideal Position
Ideal Position: Center of Room
Tolerance Zone: Walls
Analogy: Vertex AnalysisFinds Corners of the Room
Ideal Position
Tolx
Toly
Analogy: Worst-Case DLMFinds Walls of the Room
Ideal Position
Tolx
Toly
Analogy: OptimizationFinds way out of the room
Ideal Position
Tolx
Toly
Deterministic Results
Statistical Methods1. Monte Carlo Simulation
• Thousands to millions of individual models are created by randomly choosing the values for the random variables.
2. Direct Linearization: RSS• Uses the methods discussed previously.
3. Bivariate DLM• Statistical method for position error
where x and y error are not independent.
Bivariate Normal Position Error
2
2
2
2
2
2
2
i
i
i
Xi
y
i
xxy
Xi
yyy
Xi
xxx
X
P
X
PV
X
PV
X
PV
Variance Equations
The partial derivatives are the sensitivitiesthat come from the [C-EB-1A] matrix
yxy
xyx
VV
VV
Variance Tensor
21
22
r
Finding Ellipse Rotation:Mohr’s Circle
Vxy
Vx
Vy
Vxy1: Major Diameter2: Minor Diameter
2
V1=12
V2=22
Statistical Method Results
Coupler Point Error
Max. Perpendicular
Max. Normal Error
Maximum Normal Error vs. Crank Angle
0.04
0.042
0.044
0.046
0.048
0 60 120 180 240 300 360
Crank Angle (degrees)
Err
or
(in
ch
es
)
Benefits of Bivariate DLM
1. Accurate representation of the error zone.
2. Easily automated. CE/TOL already uses the method for assemblies.
3. Extremely efficient compared to Monte Carlo and Vertex Analysis.
4. Possible to estimate the yield for a given tolerance zone.
5. Can be used as a substitute for worst-case methods by using a large sigma-level
Summary
2-D Position error is not always a circle.Accurate estimation of position error in assemblies must include correlation.Where it is feasible, Direct Linearization is a good method for both worst-case and statistical error analysis.
Questions