“Statistical” (Probabilistic)
Tolerancing
(Section 5.4 of Vardeman and Jobe)
1
Piecing Together Information on
Variation (to Predict Overall)
• Sometimes geometry or physical theory
gives me an equation for a variable of
interest in terms of other, more basic
variables
U = g ( X , Y ,..., Z )
• If I have information on how the inputs
vary, I can sometimes infer how U varies
2
Car Door Example
p = ( − x sin φ , x cos φ )

π 
π 




q = p +  y cos  φ +  θ1 −   , y sin  φ +  θ1 −   
2 
2 





s = ( q1 + q2 tan(φ + θ1 + θ2 − π ),0 )
u = ( q1 + ( q2 + d ) tan(φ + θ1 + θ2 − π ), − d )
g1 = w − s1
g 2 = w − u1
Given nominals and “sigmas”
for x, y , w, φ ,θ1 and θ 2
what can I say about the
the nominals and “sigmas” for
the gaps, g1 and g 2 ?
3
Example 5.8
R2 R3
R = R1 +
R2 + R3
µ R1 = 100Ω and µ R2 = µ R3 = 200Ω
σ R1 = 2Ω and σ R2 = σ R3 = 4Ω
What about R?
4
Methods for Studying U
(for Independent X,Y,…,Z)
• Exact mean and variance for linear g
– Equations (5.23) and (5.24)
– Helpful for simple tolerance stack-up
calculations
• Approximations for mean and variance
(based on a linearization) for other g
– Equations (5.26) and (5.27)
– Easily done on something like Mathcad
– Variance involves both the input variances
AND the derivatives/rates of change
5
VarU and Derivatives
6
Example 5.9
• Nice little tolerance stack-up problem
U = Y − X1 − X2 − X 3 − X 4
σ U2 = 12 σ Y2 + ( −1) 2 σ X2 1 + ( −1) 2 σ X2 2 + ( −1) 2 σ X2 3 + ( −1) 2 σ X2 4
• U was head space in a carton designed to
hold 4 units of product
7
Mathcad for Example 5.8
R( R1 , R2 , R3)
R2. R3
R1
R2
R1
100
σ1
2
R2
200
σ2
4
R3
200
σ3
4
d
d R1
R( R1 , R2 , R3) =
1.00
d
R( R1 , R2 , R3) =
d R2
.25
d
.25
d R3
σR
σR =
R( R1 , R2 , R3) =
d
d R1
2.449
R3
2
2
R( R1 , R2 , R3) . σ1
d
d R2
2
2
R( R1 , R2 , R3) . σ2
d
d R3
2
2
R( R1 , R2 , R3) . σ3
8
Simulation … the Easiest Method
for Studying U
• Any decent statistical package will let you
simulate sets of X,Y,…,Z, compute U’s, and
then look at descriptive statistics for the
simulated values
9
Minitab for Example 5.8
MTB >
SUBC>
MTB >
SUBC>
MTB >
MTB >
Random 1000 c1;
Normal 100 2.
Random 1000 c2 c3;
Normal 200 4.
let c4=c1+((c2*c3)/(c2+c3))
Describe c4.
Calc>>Random Data>>Normal
Calc>>Random Data>>Normal
Calc>>Calculator
Stat>>Basic Statistics>>Display Basic
Descriptive Statistics
Variable
c4
N
1000
Mean
200.09
Median
200.19
TrMean
200.10
Variable
c4
Minimum
192.83
Maximum
207.13
Q1
198.50
Q3
201.69
StDev
2.45
SE Mean
0.08
10
More Mintab for Example 5.8
11
Workshop Door Exercise
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
SUBC>
MTB >
MTB >
MTB >
MTB >
MTB >
MTB >
MTB >
MTB >
c1=x, c2=y, c3=w, c4=phi
c5=theta1, c6=theta2, d=40
(units of cm and radians)
Random 1000 c1;
Normal 20 .01.
Random 1000 c2;
Normal 90. .01.
Random 1000 c3;
Normal 90.4 .01.
Random 1000 c4;
Normal 0 .001.
Random 1000 c5 c6;
Normal 1.570796 .001.
let c7=-c1*sin(c4)
let c8=c1*cos(c4)
let c9=c7+c2*cos(c4+(c5-1.570796))
let c10=c8+c2*sin(c4+(c5-1.570796))
let c11=c9+c10*tan(c4+c5+c6-3.141593)
let c12=c9+(c10+40)*tan(c4+c5+c6-3.141593)
let c13=c3-c11
let c14=c3-c12
12
More Door
MTB > Describe 'g1' 'g2'.
Descriptive Statistics
Variable
g1
g2
N
1000
1000
Mean
0.40053
0.40138
Median
0.40035
0.39775
TrMean
0.40040
0.40144
Variable
g1
g2
Minimum
0.31109
0.12174
Maximum
0.51216
0.75907
Q1
0.37877
0.34043
Q3
0.42168
0.46280
StDev
0.03078
0.09153
SE Mean
0.00097
0.00289
• What does this analysis show about the gaps?
• How could you study the difference in gaps,
g1-g2, and/or any relationship between gaps? 13