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Sensitivity AnalysisOften in engineering analysis, we are not only interested in predicting theperformance of a vehicle, product, etc, but we are also concerned with thesensitivity 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.Let’s make this more concrete using an example. Suppose we are interested inpredicting 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): 1.44W 2 S LO g SC LmaxTo make the example concrete, let’s consider the jet aircraft CJ-1 described byAnderson. In that case, the conditions were: S 318 ft 2 0.002377 slugs / ft 3 @ sea level g 32.2 ft / s 2 T 7300 lbs W 19815 lbs C Lmax 1.0Thus, under these “nominal” conditions: 2 1.44 19815 S LO 32.2 0.002377 318 1.0 7300 S LO 3182 ft @ nominal conditionsSuppose that we were not confident of the C L max value and suspected that itmight be ±0.1 from nominal.That is, 0.9 C Lmax 1.1
Sensitivity AnalysisAlso, let’s suppose that the weight of the aircraft may need to be increased for ahigher 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 oftenmade based on the problem and the tools available. We’ll look at both options.Linear Sensitivity AnalysisLinear sensitivity based on Taylor series approximations. Suppose we wereinterested in the variation of S LO with w & C LmaxThen: S LO (C Lmax C Lmax ,W W) S LO S LO S LO (C Lmax ) C Lmax W C Lmax WThat is, the change in S LO is: S LO S LO S LO C Lmax W C Lmax W S LO SThe derivatives & LO are the linear sensitivities of S LO to changes in C Lmax WC Lmax & Wj , respectively.Returning to the example: S LO 1.44W 2 S LO 2 C Lmax g SC Lmax T C Lmax S LO 2 1.44W S LO 2 W g SC Lmax T WThese can often be more information by looking at percent or fractional changes: S LO 1 S LO 1 S LO C Lmax W S LO S LO C Lmax S LO W C Lmax S LO C Lmax W S LO W S LO C Lmax C Lmax S LO W W Fractional sensitivities16.100 2
Sensitivity AnalysisFor this problem: C Lmax S Lmax S LO C Lmax 1 .0 1 S LO C Lmax S LO C Lmax W S LO S LO W 2 .0 2 .0 S LO W S LO WThus, a small fraction change in C Lmax will have an equal but opposite effect onthe take-off distance.The weight change will result in a charge of S LO which is twice as large and inthe same direction.Thus, S LO is more sensitive to W than C Lmax changes at least according to linearanalysis.ExampleWe were interested in C Lmax varying ± 0.1 which according to linear analysiswould produce 0.1 S LO variation in take-off distance: C Lmax 0.9 S LO 0.1S LO 318 ft C Lmax 1.1 S LO 0.1S LO 318 ftFor a weight increase of 10% we find W 1.1 19815lb S LO 2 0.1 S LO 636 ftNonlinear Sensitivity AnalysisFor a nonlinear analysis, we simply re-evaluate the take off distance at thedesired condition (including the perturbations). So, to assess the impact of theC Lmax variations we find: 1.44(19815) 2 S LO (C Lmax 0.9) (32.2)(0.002377)(318)(0.9)(7300) S LO (C Lmax 0.9) 3535 ft S LO ( C Lmax 0.1) 353.6 ft which agrees well with linear result16.100 3
Sensitivity AnalysisSimilarly, S LO C Lmax 1.1 2892 ft S LO C Lmax 0 .1 290 ftFinally, a 10% W increase to 21796lb’s gives: S LO W 21796lb 3850 ft S LO W 0.1W 668 ft16.100 4