Sensitivity Analysis
Often in engineering analysis, we are not only interested in predicting the
performance 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 in
the analysis. The quantification of the sensitivity to these sources of variability is
called sensitivity analysis.
Let’s make this more concrete using an example. Suppose we are interested in
predicting the take off distance of an aircraft. From Anderson’s, Intro. To Flight,
an estimate for take-off distance is given by Eqn (6.104):
S LO
1.44W 2
gU f SC Lmax 7
To make the example concrete, let’s consider the jet aircraft CJ-1 described by
Anderson. In that case, the conditions were:
S
318 ft 2
Uf
0.002377 slugs / ft 3 @ sea level
g
T
32.2 ft / s 2
7300 lbs
W
19815 lbs
C Lmax
1.0
Thus, under these “nominal” conditions:
2
S LO
1.4419815
32.20.002377 3181.07300
S LO
3182 ft @ nominal conditions
Suppose that we were not confident of the C
might be ±0.1 from nominal.
That is,
0.9 ” C Lmax ”1.1
L max
value and suspected that it
Sensitivity Analysis
Also, let’s suppose that the weight of the aircraft may need to be increased for a
higher load. Specifically, let’s 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 often
made based on the problem and the tools available. We’ll look at both options.
Linear Sensitivity Analysis
Linear sensitivity based on Taylor series approximations. Suppose we were
interested in the variation of S LO with w & C Lmax
Then:
S LO (C Lmax 'C Lmax ,W 'W ) #
S LO (C Lmax ) wS LO
wS
'C Lmax LO 'W
wC Lmax
wW
That is, the change in S LO is:
wS LO
wS
'S LO {
'C Lmax LO 'W
wC Lmax
wW
The derivatives
wS LO
wS
& LO are the linear sensitivities of S LO to changes in
wC Lmax
wW
C Lmax & Wj , respectively.
Returning to the example:
wS LO
1.44W 2
2
wC Lmax
gU f SC Lmax T
wS LO
wW
2 1.44W
gU f SC Lmax T
2
S LO
C Lmax
S LO
W
These can often be more information by looking at percent or fractional changes:
'S LO
1 wS LO
1 wS LO
'W
'C Lmax #
S LO
S LO wC Lmax
S LO wW
C Lmax wS LO 'C Lmax
W wS LO 'W
S LO wC Lmax C Lmax
S LO wW W
Fractional sensitivities
16.100
2
Sensitivity Analysis
For this problem:
C Lmax wS Lmax
S LO wC Lmax
W wS LO
S LO wW
1 .0 Ÿ
2 .0 Ÿ
'C Lmax
'S LO
| 1
S LO
C Lmax
'S LO
'W
| 2 .0
S LO
W
Thus, a small fraction change in C Lmax will have an equal but opposite effect on
the take-off distance.
The weight change will result in a charge of S LO which is twice as large and in
the same direction.
Thus, S LO is more sensitive to W than C Lmax changes at least according to linear
analysis.
Example
We were interested in C Lmax varying ± 0.1 which according to linear analysis
would produce P 0.1 S LO variation in take-off distance:
Ÿ
C Lmax
0.9 o 'S LO | 0.1S LO
318 ft
C Lmax
1.1 o 'S LO | 0.1S LO
318 ft
For a weight increase of 10% we find
W
1.119815lb o 'S LO
| 20.1S LO
| 636 ft
Nonlinear Sensitivity Analysis
For a nonlinear analysis, we simply re-evaluate the take off distance at the
desired condition (including the perturbations). So, to assess the impact of the
C Lmax variations we find:
Ÿ
S LO (C Lmax
0.9)
S LO (C Lmax
0.9)
'S LO ( 'C Lmax
1.44(19815) 2
(32.2)(0.002377)(318)(0.9)(7300)
3535 ft
0.1)
353.6 ft
which agrees well with linear result
16.100
3
Sensitivity Analysis
Similarly,
S LO C Lmax
1.1
Ÿ 'S LO C Lmax
2892 ft
0 .1
290 ft
Finally, a 10% W increase to 21796lb’s gives:
S LO W
21796lb 3850 ft
'S LO 'W
16.100
0.1W 668 ft
4