Select Page
Your Perfect Assignment is Just a Click Away

100% Original, Plagiarism Free, Customized to your instructions!     ## Robustness.pdf

Robust Design forProducts & Processes

Variation influence on the Optimum

2

Input

Outp

ut

Upper Specification Limit

Save SideShow Bob’s life

2000 feet

Pool is 200 feet longCopyright the Simpsons

Engineering

• r = range

• V = velocity

• g = acceleration dueto gravity (32.2 ft/s2)

( ) g

2sinVr

=Objectives

• Target range (r) is 2000 feet

• Minimise variation of range(less than 100 feet)

Response Sensitivity

0 10 20 30 40 50 60 70 80 90

Angle

Dis

tan

ce

Distribution Moments



−== dxxxfxE )()(

  2222 )()()()( −=−= 

−dxxfxxExExV

Transformations

( ) dy

dwywfyg =)( (3.1)

where: f (x) is the probability density function for x,

y = u(x) is the transformation function, and

x = w(y) is the inverse of the transformation function, and has only one root.

Example y x= +4 12

x

y=

−12

4

dx

dy=

1

4

The exponential probability density function is

f x ex

( ) =−



, x > 0

Substituting into Equation 3.Error! Bookmark not defined. gives

( ) 



=

−−

4

1)(

4/)12( yeyg

 , y > 12

The exponential probability density function is

f x ex

( ) =−



, x > 0

Substituting into Equation 3.Error! Bookmark not defined. gives

( ) 



=

−−

4

1)(

4/)12( yeyg

 , y > 12

Transformations

( )2111

, xxuy = and ( )2122

, xxuy = . If the inverse of the transformation functions,

x w y y1 1 1 2= ( , ) and x w y y

2 2 1 2= ( , ) have single roots, the joint probability density

function for y1 and y2 is

( ) ( ) Jyywyywfyyg21221121

,,,),( = (3.1)

where J is the Jacobian and is defined as the determinant of the partial derivatives;

Jx y x y

x y x y=   

   

1 1 1 2

2 1 2 2

/ /

/ /

System characteristics• Properties of means and variances of functions of

several variables, where y = f(x1, x2, x3, …, xn).

• An approximation for the mean of a function of several variables is:

• An approximation for the variance of a function of several variables is:

2

x

2n

1i i

2

yix

yV(y) σσ 





= 

=

=



+=

n

1i

2

x2

i

2

yix

y

2

1yE(y) σμ

Circuit Example

P = V2/R

• Voltage (V) is Normally distributed• mean = 12

• standard deviation = 0.1

• Resistance (R) is Normally distributed• mean = 2

• standard deviation = 0.2

Where does input data come from?

• Current production• Supplier• Discuss how to obtain mean and standard deviation for

voltage and resistance

Circuit Example1. What does the power distribution look like?

• Specifications

• 72 ± 10

• Will this circuit design meet specifications?

2. Change standard deviation of voltage to 0.3 and standard deviation of resistance to 0.4

• What is the mean and standard deviation of power?

3. Change standard deviation of voltage to 0.5 and standard deviation of resistance to 0.6

Statistical Bias

0

100

200

300

400

500

600

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Resistance

Po

we

rResistance decreased from 1.0 to 0.5

Power increased from 144 to 288

Power change = 144

Resistance increased from 1.0 to 1.5

Power decreased from 144 to 96

Power change = 48

Statistical Bias

0

20

40

60

80

100

120

0 2 4 6 8 10 12 14 16 18 20

Length

We

igh

t

Length increases from 9 to 10

Weight increases from 58.7 to 64.57

Weight change = 5.87

Length decreases from 9 to 8

Weight decreases from 58.7 to 52.83

Weight change = 5.87

Circuit Example

• Use robustness approximations to compute• Statistical bias

• Power standard deviation

• How do these results compare to the simulation?

ExampleY = 25 + 200P – 2P2 + 14T

1. Determine bias of Y & STD of Y using robustness equations & verify with simulation

• P = 0• Y = 7000

2. Determine values of T & P that minimize the STD of Y using robustness equations & verify with simulation

• Y = 7000

T = Normal (std = 10)

P = Normal (std = 5)

Example – Optimise using derivatives• Y = 25 + 200P – 2P2 + 14T

• Determine T and P

• Y = 7000

• Minimise variation of Y

( ) ( ) 22222

2

T

2

2

P

2

140101454P200

T

Y

P

YV(y)

=+−=





+



= σσ

( ) ( )

( ) 50542

1

T

Y

P

Y

2

1Bias

2

2

T2

22

P2

2

−=−=







+

= σσ

4P200P

Y−=

( )4

P

Y2

2

−=

14T

Y=

( )0

T

Y2

2

=

1st Order Der. P =

1st Order Der. T =

2nd Order Der. P =

2nd Order Der. T =

Example – Optimise using derivatives

Excel > Tools > Solver

T & P optimised by Solverwith minimal StD(Y)

• Friction factor (f) 0.01 < f < 0.05

• Pipe diameter (d) 0.5 < d < 10.0

• Flow rate (v) 7 < v < 50

• Pipe length (L) 2000

• Discharge level below reservoir surface (D) 25 < D < 500

• Specification 5 < HP < 45

• Friction factor (f) StDev = 0.002

• Pipe diameter (d) StDev = 0.1

• Flow rate (v) StDev = 2

• Pipe length (L) StDev = 1

• Discharge level below reservoir surface (D) StDev = 3

• HorsePower within specifications and minimise variance of HorsePower

Robustness example – Reservoir flow

D−+=2323

0006188.0000115.000000961.0 vddfLvdvHP

Class exercise – Salt tank mixing

• Q = grams of salt after t minutes

• Tank volume (V): 500 litres max

• Salt density (d): 260 grams/litre max

• Water flow (f) litres/min: (V/5) maximum, 0.1 minimum

• Fresh water in, uniform stirring, mixed water out

• Required

• More than 90 g of salt after 2 minutes

• Between 70 g and 80 g of salt after 5 minutes

• Inputs are Normally distributed with standard deviation below:

• StD(V) = 2StD(d) = 0.04StD(f) = 0.3

V

tf

dVeQ−

=

Better Estimate of System Variance

22

22

2

2

1

)()(2

1 )(

jii xx

n

i

n

j ji

x

n

i i

Yσσ

xx

x

Yσ  



+





=

= 

Example

Y = 25 + 200P – 2P2 + 14T

• Determine values of T & P that minimize the STD of Y using robustness equations & verify with simulation

– Y = 7000

T = Normal (std = 10)

P = Normal (std = 5)

Large Scale Systems

End

## How it Works

1. Clіck оn the “Place оrder tab at the tоp menu оr “Order Nоw” іcоn at the bоttоm, and a new page wіll appear wіth an оrder fоrm tо be fіlled.
2. Fіll іn yоur paper’s іnfоrmatіоn and clіck “PRІCE CALCULATІОN” at the bоttоm tо calculate yоur оrder prіce.
3. Fіll іn yоur paper’s academіc level, deadlіne and the requіred number оf pages frоm the drоp-dоwn menus.
4. Clіck “FІNAL STEP” tо enter yоur regіstratіоn detaіls and get an accоunt wіth us fоr recоrd keepіng.
5. Clіck оn “PRОCEED TО CHECKОUT” at the bоttоm оf the page.
6. Frоm there, the payment sectіоns wіll shоw, fоllоw the guіded payment prоcess, and yоur оrder wіll be avaіlable fоr оur wrіtіng team tо wоrk оn іt.

Nоte, оnce lоgged іntо yоur accоunt; yоu can clіck оn the “Pendіng” buttоn at the left sіdebar tо navіgate, make changes, make payments, add іnstructіоns оr uplоad fіles fоr the оrder created. e.g., оnce lоgged іn, clіck оn “Pendіng” and a “pay” оptіоn wіll appear оn the far rіght оf the оrder yоu created, clіck оn pay then clіck оn the “Checkоut” оptіоn at the next page that appears, and yоu wіll be able tо cоmplete the payment.

Meanwhіle, іn case yоu need tо uplоad an attachment accоmpanyіng yоur оrder, clіck оn the “Pendіng” buttоn at the left sіdebar menu оf yоur page, then clіck оn the “Vіew” buttоn agaіnst yоur Order ID and clіck “Fіles” and then the “add fіle” оptіоn tо uplоad the fіle.

Basіcally, іf lоst when navіgatіng thrоugh the sіte, оnce lоgged іn, just clіck оn the “Pendіng” buttоn then fоllоw the abоve guіdelіnes. оtherwіse, cоntact suppоrt thrоugh оur chat at the bоttоm rіght cоrner

NB

Payment Prоcess

By clіckіng ‘PRОCEED TО CHECKОUT’ yоu wіll be lоgged іn tо yоur accоunt autоmatіcally where yоu can vіew yоur оrder detaіls. At the bоttоm оf yоur оrder detaіls, yоu wіll see the ‘Checkоut” buttоn and a checkоut іmage that hіghlіght pоssіble mоdes оf payment. Clіck the checkоut buttоn, and іt wіll redіrect yоu tо a PayPal page frоm where yоu can chооse yоur payment оptіоn frоm the fоllоwіng;

1. Pay wіth my PayPal accоunt‘– select thіs оptіоn іf yоu have a PayPal accоunt.
2. Pay wіth a debіt оr credіt card’ or ‘Guest Checkout’ – select thіs оptіоn tо pay usіng yоur debіt оr credіt card іf yоu dоn’t have a PayPal accоunt.
3. Dо nоt fоrget tо make payment sо that the оrder can be vіsіble tо оur experts/tutоrs/wrіters.

Regards,

Custоmer Suppоrt