ESD.33 -- Systems Engineering
Session #9Critical Parameter Management &
Error Budgeting
Dan Frey
Plan for the Session• Follow up on session #8• Critical Parameter Management• Probability Preliminaries• Error Budgeting
– Tolerance– Process Capability– Building and using error budgets
• Next steps
S - CurvesAtish Banergee –
We first studied S-curves in technology strategy…The question remained why the S-curve has the peculiar shape. Well I found the answer in system dynamics. It is a general phenomenon and not restricted to technology.
It can be thought of as two curves:1. The lower part of the curve is growth with acceleration....2. The upper part of the s-curve is called a goal-seeking curve and can be thought of as growth with deceleration…
Trends in Compressor Performance
Wisler, D. C., 1998, Axial Flow Compressor and Fan Aerodynamics”, Handbook of Fluid Dynamics, CRC Press., ed. R. Johnson.
Evolution of Jet Engine Performance
1.05Ghost
JT3C
Cruise thrust specific fuelconsumption
JT8D-17
P&W
RR
GE
deHavilland
JT8D-209 JT8D-15
JT8D-17AJT8D-219
TAY
CFM56-3RB211-535C
RB211-524D4 UPJT9D-7R4H1 CFM56-5B
4460V2500
RB211-524D4DCFM56-5C2
4083416843804156
4056CF6-80C22037
RB211-535E4CF6-80A/A1 CFM56-5A1JT9D-7R4D (B)
RB211-524D4CF6-50E2/C2
CFM56-2C1RB211-524C2
JT9D-7AJT9D-59A
JT8D-217AJT9D-7JT9D-3
JT8D-9
JT9D-3A
CF6-50E/CCF6-6D
JT3DCJ805
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
1945
1950
1955
1960
1965
1970
Certification date
1975
1980
1985
1990
1995
2000
2005
( (lb fuel/hrlb thrust
Adapted from Koff, B. L. "Spanning the World Through Jet Propulsion.” AIAA Littlewood Lecture. 1991.
Plan for the Session• Follow up on session #8• Critical Parameter Management• Probability Preliminaries• Error Budgeting
– Tolerance– Process Capability– Building and using error budgets
• Next steps
Critical Parameter Management
• CPM provides discipline and structure
• Produce critical parameter documentation– For example, a critical parameter drawing
• Traces critical parameters all the way through to manufacture and use
• Determines process capability (Cp or Cpk)
• Therefore, requires probabilistic thinking
Plan for the Session• Follow up on session #8• Critical Parameter Management• Probability Preliminaries• Error Budgeting
– Tolerance– Process Capability– Building and using error budgets
• Next steps
Probability Definitions
• Sample space – a list of all possible outcomes of an experiment– Finest grained– Mutually exclusive – Collectively exhaustive
• Event - A collection of points in the sample space
Concept Question
• You roll 2 dice
• Give an example of a single point in the sample space?
• How might you depict the full sample space?
• What is an example of an “event”?
Probability Measure
• Axioms– For any event A,
– P(U)=1
– If A∩B=φ, then P(AUB)=P(A)+P(B)
0)( ≥AP
For the case of rolling two dice: A = rolling a 7 and B = rolling a 1 on at least one dieIs it the case that P(A+B)=P(A)+P(B)?
Discrete Random Variables
• A random variable that can assume any of a set of discrete values
• Probability mass function– px(xo) = probability that the random variable x
will take the value xo
• Let’s build a pmf for rolling two dice– random variable x is the total
px(x)
x x=10
Continuous Random Variables
• Can take values anywhere within continuous ranges
• Probability density functions obey three rules
–
–
–
xxfx allfor )(0 ≤
1d)( =∫∞
∞−
xxfx
{ } xxfUxLPU
Lx d)(∫=≤<
L U x
fx(x) { }UxLP ≤<
Measures of Central Tendency
• Expected value
• Mean µ = E(x)
• Arithmetic average
• Median• Mode
∫=b
ax xxfxgxgE d)()())((
∑=
n
iix
n 1
1
x
fx(x)
Measures of Dispersion
• Variance
• Standard deviation
• Sample variance
• nth central moment
• Covariance
)))((()( 22 xExExVAR −== σ
)))((( 2xExE −=σ
∑−
−−
=n
ii xx
nS
1
22 )(1
1
)))((( nxExE −
)))())(((( yEyxExE −−
Sums of Random Variables• Average of the sum is the sum of the
average (regardless of distribution and independence)
• Variance also sums iff independent
• This is the origin of the RSS rule– Beware of the independence restriction!
)()()( yExEyxE +=+
222 )()()( yxyx σσσ +=+
Concept Test
• A bracket holds a component as shown. The dimensions are independent random variables with standard deviations as noted. Approximately what is the standard deviation of the gap?A) 0.011”B) 0.01”C) 0.001”
"001.0=σ"01.0=σ
gap
Uniform Distribution
• A reasonable (conservative) assumption when you know the limits of a variable but little else
32)( LU −=σ
L U
Basic Applicationx=result of blue spinner
• I have two spinners
xxfx allfor )(0 ≤1d)( =∫∞
∞−
xxfx{ } xxfbxaPb
ax d)(∫=≤<
0
1.0
y=result of red spinnerz=x+y
0
0.5
0.75 1.5 0.50.25
• What are the pdfs for variables x, y, and z?
Simulation Can Quickly Answer the Question
trials=10000;nbins=trials/1000;x= random('Uniform',0,1,trials,1);y= random('Uniform',0,2,trials,1);z=x+y;subplot(3,1,1); hist(x,nbins); xlim([0 3]); subplot(3,1,2); hist(y,nbins); xlim([0 3]);subplot(3,1,3); hist(z,nbins); xlim([0 3]);
Probability Distribution of Sums
• If z is the sum of two random variables x and y
• Then the probability density function of z can be computed by convolution
yxz +=
∫∞−
−=z
z yzxzp ζζζ d)()()(
Convolution
∫∞−
−=z
z yzxzp ζζζ d)()()(
Convolution
∫∞−
−=z
z yzxzp ζζζ d)()()(
Central Limit Theorem
The mean of a sequence of n iidrandom variables with
– Finite µ
–
approximates a normal distribution in the limit of a large n.
( ) 0 <)( 2 >∞− + δδii xExE
Normal Distribution2
2
2)(
21)( σ
µ
πσ
−−
=x
x exf
µ +3σ+1σ-1σ-3σ +6σ-6σ
68.3%99.7%
1-2ppb
Joint Normal Distribution
• The lines of constant probability density are ellipsoids
• If the matrix K is diagonal, then the variables are uncorrelated and independent
( )p m
T( ) exp ( ) ( )xK
x K x= − − −⎧⎨⎩
⎫⎬⎭
−1
2
12
1
πµ µ
correlateduncorrelated
Independence
• Random variables x and y are said to be independent iff
• Or, knowledge of x provides no information to update the distribution of y
f x y f x f yxy x y( , ) ( ) ( )=
Expectation Shift
x
y(x)
E(x)
y(E(x))
E(y(x))
S
fx(x)fy(y(x))
S=E(y(x))- y(E(x)) Under utility theory (DBD),S is a key differencebetween probabilistic and deterministic design
Plan for the Session• Follow up on session #8• Critical Parameter Management• Probability Preliminaries• Error Budgeting
– Tolerance– Process Capability– Building and using error budgets
• Next steps
Error Budgets
• A tool for predicting and managing variability in an engineering system
• A model that propagates errors through a system
• Links aspects of the design and its environment to tolerance and capability
• Used for tolerance design, robust design, diagnosis…
Engineering Tolerances
• Tolerance --The total amount by which a specified dimension is permitted to vary(ANSI Y14.5M)
• Every componentwithin spec addsto the yield (Y)
q
p(q)
L U
Y
y
p(y)
Tolerance on Position
W
>25%W
Lead
Land
Tolerance of Form
0.25
THIS ON A DRAWING MEANS THIS
0.25 wide tolerance zone
GD&T Symbols
Type of Tolerance Characteristic
Straightness
Flatness
Circularity (Roundness)
Cylindricity
Profile of a Line
Profile of a Surface
Angularity
Perpendicularity
Parallelism
Position
Concentricity
Circular Runout
Geometric Characteristic Symbols
Total Runout
Symbol
Form
Profile
Orientation
Location
Runout
Arrowhead(s) may be filled in.
For Individualor RelatedFeatures
For IndividualFeatures
For RelatedFeatures
Multiple Tolerances
• Most products have many tolerances• Tolerances are pass / fail• All tolerances must be met (dominance)
35
Variation in Manufacture
• Many noise factors affect the system• Some noise factors affect multiple
dimensions (leads to correlation)
36
Process Capability Indicesp(q)
37
• Process Capability Index
• Bias factor
• Performance Index
( )CU L
p ≡− / 23σ
C C kpk p≡ −( )1
k
U L
U L≡
−+
−
µ2
2( ) /
qL UU L+2
U L−2
µ
Concept Test
• Motorola’s “6 sigma” programs suggest
that we should strive for a Cp of 2.0. If this
is achieved but the mean is off target so
that k=0.5, estimate the process yield.
Cp and k Determine Yield • By definition
• If Gaussian
q
p(q)
L U
YY p q qFTL
U
= ∫ ( )d
Y C k C kFT p p= −⎛
⎝⎜
⎞
⎠⎟ + +
⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
12
3 22
1 3 22
1erf erf( ) ( )
This function to maps Cp and k to yield39
Cp and k Determine Quality Loss
[ ]
2
2 22/)(LossQuality ⎟
⎠⎞
⎜⎝⎛ +
−−
=LUd
LUAo
⎟⎟⎠
⎞⎜⎜⎝
⎛+= 2
2
91Loss)E(Quality
po C
kAq
L(q)
L UU L+2
Taguchi's quality loss function
ANSI's implied quality loss function
Ao
Crankshafts
• What does a crankshaft do?• How would you define the tolerances?• How does variation affect performance?
Printed Wiring Boards
• What does the second level connection do?
• How would you define the tolerances?• How does variation affect performance?
Cp and k for the System
0
50
100
150
200
250
freq
uenc
y
LL UL
Cp = 0 82.
k = 008.
YFT = 983%.
4
Producibility Analysis
• Rolled throughput yield (YRT)--The probability that all tolerances are met
• Motorola’s approach
• Assumes probabilistic independence
Y YRT FT ii
m
==
∏1
Motorola’s formulaYRT = 66 7%.
Hughes’ dataYRT = =0 983 0 2%368. .
5
Surface Mount Data
20 40 60 80 100 120
Side #1
Lead number
Side
to si
de e
rror
20 40 60 80 100 120
Side #3
Lead number
Side
to si
de e
rror
20 40
Side #2
Lead number
Side
to si
de e
rror
20 40
Side #4
Lead number
Side
to si
de e
rror
6
Plan for the Session• Follow up on session #8• Critical Parameter Management• Probability Preliminaries• Error Budgeting
– Tolerance– Process Capability– Building and using error budgets
• Next steps
Error Sources
• Kinematic errors– Straightness– Squareness– Bearings
• Drive related errors• Thermal errors• Static loading• Dynamics
Errors in a Linear DriveLe
ad d
evia
tion
(µm
)
Cumulative lead error (µm/mm)
Once per revolution lead error (µm)
1 revolution
Nominal travel (mm)
Angular Errorsεy
X
x
y
z
x
y
z
εy
X
x
y
z
εy
OK, so you put the error in the model. Now what will happen when the machine moves?
A Model of a Robot
Θz2
Z
xy
z
1000 mm
500 mm 400 mm
300 mm
Base
Θz1
Point p
60 mm
Errors in the Robot
Error Description µ σ
εz1 Drive error of joint #1 0 rad 0.0001 rad
εz2 Drive error of joint #2 0 rad 0.0001 rad
δz3 Drive error of joint #3 Z ·0.0001 0.01mm
εx3 Pitch of joint #3 0 rad 0.00005 rad
εy3 Yaw of joint #3 0 rad 0.00005 rad
xp2 Parallelism of joint 2 in the x direction 0.0002 rad
0.0001 rad
A Model of a Robot
• The matrices describe the intended motions and the errors
• Can be applied to any point on the end effector
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡−
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡+ΘΘΘ−+Θ
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
10006010010
500001
1000010000)cos(sin00sin)cos(
100001000010
1000001
2
2111
111
30
mmxpmmxpmm
mmmmmm
mmmm
mm
zzz
zzz
εε
T
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−−−
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡+ΘΘ
Θ−+Θ
⋅
10001
010001
100001000010
400001
1000010000)cos(sin00sin)cos(
333
3
3
222
222
zxy
x
y
zzz
zzz
Zmmmm
mmmm
mm
mmmmmm
δεεε
εε
ε
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
−=
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
130000
1'''
30 T
z
y
x
ppp
NOTE: These two should be swapped
Homework #5
• Short answers on TRIZ and probability• Error budgeting
– Two tasks are to be done with the robot– Analyze the tasks– Discuss changes to the system
• A Matlab file is available in the HW folder just so you don’t have to re-type the matrices
Next Steps
• You can download HW #5 Error Budgetting– Due 8:30AM Tues 13 July
• See you at Thursday’s session – On the topic “Design of Experiments”– 8:30AM Thursday, 8 July
• Reading assignment for Thursday– All of Thomke– Skim Box– Skim Frey
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