ESD.33 -- Systems Engineering Session #1 Course ... · 2. The upper part of the s-curve is called a...

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.



• 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



• 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)()()(

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



• 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



• 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

ESD.33 -- Systems Engineering Session #1 Course ... · 2. The upper part of the s-curve is called a...

Documents

Transcript of ESD.33 -- Systems Engineering Session #1 Course ... · 2. The upper part of the s-curve is called a...