公差分析和尺寸链方法

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Simple Variation Simulation Example
100 +/- 0.1 (C)
49.9 +/-0.1
(A)
49.9 +/-0.1
(B)
? +/- ?
Given component tolerances, determine the variation in the measured dimension (the gap between the blocks and the base).
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Worst Case Analysis
In worst case analysis, it is assumed that the contributing dimensions are always within tolerance. By making this assumption worst case limits can be found within which the measured dimension must always fall.
Table 4 Results of Monte Carlo Simulation
Build #
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
(A) A
405 49.92 109 49.84 399 49.92 370 49.92 348 49.92 426 49.96
Frequency 7 18 78 147 143 81 25 1
Cumul. Freq. 0-7 8 - 25 26 - 103 104 - 250 251 - 393 394 - 474 475 - 499 500
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1
x
± 1
a
2
x2± b
2
3
1
y ±t
3
x
± 3
c
But the method can applied more widely than mechanical assembly. The general form is: given a function Y=f(x1, x2, …), and the distributions of xi, What is the distribution of y ?
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Monte Carlo Simulation
Table 1 Distribution for dimension A
Dimen. Range 49.74 - 49.78 49.78 - 49.82 49.82 - 49.86 49.86 - 49.90 49.90 - 49.94 49.94 - 49.98 49.98 - 50.02 50.02 - 50.06
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Part I 1-D Tolerance analysis
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Deviation
Characterizing the performance of a process
Frequency 1 7 58 134 182 90 25 3
Cumul. Freq. 0-1 2-8 9 - 66 67 - 200 201 - 382 383 - 472 473 - 497 498 - 500
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Root Sum Squares (RSS):
(Spotts, 1978, Lee and Woo, 1990)
Monte Carlo Simulation:
(Craig, 1989)
y x1 x2 x3
t abc t a2 b2 c2
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Variation vs. Tolerance
LSL
US L
Variation: • is what the process gives us, • may be quite different from the tolerance.
Tolerance or Specification is • the allowable level of variation, • based on functional consideration, • used to establish a part's conformability to design.
Assembly Model
Distribution of Tolerance
Explicit: Linearized Sensitivity Mechanistic Model Non-linear Model
Implicit:
Worst Case
Statistical: Root sum squares Monte Carlo
But the techniques for predicting variation or tolerance for an assembly is the same.
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Variation Simulation
Given individual part dimensions and their distribution, what are the assembly dimensions ?
100 +/- 0.1 (C)
49.9 +/-0.1
(A)
49.9 +/-0.1
(B)
? +/- ?
x (100 0.1) (49.9 0.1) (49.9 0.1) 0.1 x (100 0.1) (49.9 0.1) (49.9 0.1) 0.5
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Frequency 3 20 97 180 118 77 5 0
Cumul. Freq. 0-3 4 - 23 24 - 120 121 - 300 301 - 418 419 - 495 496 - 500 N/A
Table 2 Distribution for dimension B
Dimen. Range 49.74 - 49.78 49.78 - 49.82 49.82 - 49.86 49.86 - 49.90 49.90 - 49.94 49.94 - 49.98 49.98 - 50.02 50.02 - 50.06
4
3
2
1
0
-1
-2
-3
-4 -3 -2 -1 0 1 2 3 4
-4
Central tendency or mean: Spread or variation
x E[x] x1 x2 ... xn n
n
(xi x)2
Var[x] i1 n 1
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C
485 100.10 0.26
444 100.06 0.26
460 100.06 0.22
193 99.98 0.14
493 100.10 0.28
164 99.98 0.18
426 100.06 0.34
114 99.98 0.18
493 100.10 0.34
198 99.98 0.22
475 100.10 0.26
52 49.84 98 49.88 238 49.92 98 49.88 191 49.88 313 49.92 239 49.92 491 50.00 17 49.84 128 49.88 343 49.92 131 49.88 269 49.92 232 49.92 379 49.92
(C)
(C-A-B)
Table 3 Distribution for dimension C
Dimen. Range 99.84 - 99.88 99.88 - 99.92 99.92 - 99.96 99.96 - 100.00 100.00 - 100.04 100.04 - 100.08 100.08 - 100.12 100.12 - 100.16
Mean and variance of some linear functions
E[ax1 bx2 ] aE[x1] bE[x2 ]
Var[ax1 bx2 ] a2Var[x1] b2Var[x2 ] 2ab cov[x1x2 ]
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49.9 +/-0.1
(A)
49.9 +/-0.1
(B)
? +/- ?
y C A B
Var(y) Var(C) Var(A) Var(B)
toly tolC2 tolA2 tolB2 3 0.12 0.173
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Root Sum Squares
The idea behind RSS is to treat a tolerance as a normal distribution with certain process capability, and use random assembly.
100 +/- 0.1 (C)
➢ Auto-body Dimension Chain Method
• Integration of Key Characteristic of Product & Process
• Assembly Constraints • Tolerance and Solving of Dimension Chain
29 49.84 391 49.92 173 49.88 269 49.88 407 49.92 238 49.88
51 49.84 494 49.96 235 49.88 257 49.88
74 49.84 138 49.88 237 49.88
63 49.84
(B) B
362 49.92 413 49.96 371 49.92 356 49.92 331 49.92
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Commonly Used Variation Simulation Methods
1
x
± 1
a
2
x2± b
2
3
1
y ±t
3 x ±c
3
Worst Case:
(Conway, 1948; Chase and Parkinson, 1991)
85 99.94 0.14
361 100.02 0.18
328 100.02 0.22
339 100.02 0.26
284 100.02 0.22
19 99.90 0.18
289 100.02 0.22
124 99.98 0.18
78 99.94 0.18
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Variation Simulation Methods
Two things are essential in order to perform variation analysis. One is an assembly model or input-output model. The other is the distribution of variables.
车身质量控制系列讲座
公差分析和车身尺寸链方法
Tolerance Analysis & Auto-body Dimension Chain Method
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Contents
➢ 1-D Tolerance analysis
• Variation vs. Tolerance • Variation Simulation Methods • Advantages and Limitations of Each Method