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Sensitivity Analysis

Often in engineering analysis, we are not only interested in predicting theperformance of a vehicle, product, etc, but we are also concerned with the

sensitivity of the predicted performance to changes in the design and/or errors inthe analysis. The quantification of the sensitivity to these sources of variability iscalled sensitivity analysis.

Lets make this more concrete using an example. Suppose we are interested inpredicting the take off distance of an aircraft. From Andersons, Intro. To Flight,an estimate for take-off distance is given by Eqn (6.104):

7

f max

244.1

L

LOSCg

WS

U

To make the example concrete, lets consider the jet aircraft CJ-1 described byAnderson. In that case, the conditions were:

0.1

19815

7300

/2.32

@/002377.0

318

max

2

3

2

f

LC

lbsW

lbsT

sftg

levelseaftslugs

ftS

U

Thus, under these nominal conditions:

73000.1318002377.02.32

1981544.12

LOS

ftSLO 3182 @ nominal conditions

Suppose that we were not confident of the value and suspected that itmight be 0.1 from nominal.

maxLC

That is,

0.9 1.1maxL

C

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Sensitivity Analysis

Also, lets suppose that the weight of the aircraft may need to be increased for ahigher load. Specifically, lets consider a 10% weight increase:

1) Linear sensitivity analysis

2) Nonlinear sensitivity analysis (i.e. re-evaluation)

They both have their own advantage and disadvantages. The choice is oftenmade based on the problem and the tools available. Well look at both options.

Linear Sensitivity Analysis

Linear sensitivity based on Taylor series approximations. Suppose we were

interested in the variation of with w &LOS maxLC

Then:

WW

SC

C

SCS

WWCCS

LOL

L

LOLLO

LLLO

'w

w'

w

w

#''

max

max

max

maxmax

)(

),(

That is, the change in is:LOS

WW

SC

C

SS LOL

L

LOLO '

w

w'

w

w{'

max

max

The derivativesW

S

C

S LO

L

LO

w

w

w

w&

max

are the linear sensitivities of to changes in

& , respectively.

LOS

maxLC jW

Returning to the example:

maxmaxmax

2

244.1

L

LO

LL

LO

C

S

TSCg

W

C

S

w

w

fU

W

S

TSCg

W

W

S LO

L

LO 244.12

max

w

w

fU

These can often be more information by looking at percent or fractional changes:

W

W

W

S

S

W

C

C

C

S

S

C

WW

S

SCC

S

SS

S

LO

LOL

L

L

LO

LO

L

LO

LO

L

L

LO

LOLO

LO

'

w

w

'

w

w

'w

w'w

w#

'

max

max

max

max

max

max

11

Fractional sensitivities

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Sensitivity Analysis

For this problem:

W

W

S

S

W

S

S

W

C

C

S

S

C

S

S

C

LO

LOLO

LO

L

L

LO

LO

L

L

LO

L

'|

'

w

w

'|

'

w

w

0.20.2

10.1

max

max

max

maxmax

LLO

LO'

|'

max

max

LO

LO '|'

.

Thus, a small fraction change in will have an equal but opposite effect on

the take-off distance.maxL

C

The weight change will result in a charge of which is twice as large and in

the same direction.LOS

Thus, is more sensitive to W than changes at least according to linear

analysis.

LOS maxLC

Example

We were interested in varying 0.1 which according to linear analysis

would produce variation in take-off distance:maxL

C

LOS1.0P

ftSSC

ftSSC

LOLOL

LOLOL

3181.01.1

3181.09.0

max

max

|'o

|'o

For a weight increase of 10% we find

ft

SSlbW LOLO

636

1.02198151.1

|

|'o

Nonlinear Sensitivity Analysis

For a nonlinear analysis, we simply re-evaluate the take off distance at thedesired condition (including the perturbations). So, to assess the impact of the

variations we find:maxL

C

ftCS

CS

LLO

LLO

3535)9.0(

)7300)(9.0)(318)(002377.0)(2.32(

)19815(44.1)9.0(

max

max

2

ftCS LLO 6.353)1.0( max ''

which agrees well with linear result

tSSC

tSC

L

LL

3181.01.1

3181.090

max

max

'o

'o

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Sensitivity Analysis

Similarly,

ftCS LLO 28921.1max

ftCS LLO 2901.0max '

Finally, a 10% W increase to 21796lbs gives:

ftWWS

ftlbWS

LO

LO

6681.0

385021796

''

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