Lesson 7 · 35 min

Sensitivities and Trade Studies

The business jet weighs 6943 kg, to four figures. Not one of its inputs is known to four figures. The most useful thing a first estimate can tell a design team is not the number but how it moves: which inputs matter, which do not, and what each requirement and each technology is worth in takeoff weight.

Learning objectives

Compute the sensitivity of \(\Wo\) to payload, range, \(L/D\), \(C\) and the empty-weight trend by resizing.
Explain why the true growth factor is smaller than the fixed-fraction one.
Read a payload–range trade study and use it to discuss requirements.
State what the first estimate is used for, and what replaces it later.

Resize, do not differentiate

The cleanest way to find a sensitivity is to change one input, resize completely, and compare. For the business jet:

Sensitivities of the business jet (baseline \(\Wo = 6943\ \text{kg}\))
Change	New \(\Wo\) (kg)	Change in \(\Wo\)
Payload +100 kg (one more passenger)	7508	+8.2%
Range +100 km	7105	+2.3%
Range +10% (+460 km)	7745	+11.6%
\(\LDmax\) +10%	6264	−9.8%
\(C\) −10% (cruise and loiter)	6202	−10.7%
Empty-weight trend −10% (technology factor 0.9)	5179	−25.4%
Empty-weight trend +10% (factor 1.1)	10 032	+44.5%

Three lessons hide in that table. The empty-weight trend is by far the most powerful input, which is why Lesson 2's choice of class and technology factor matters so much. Aerodynamics and propulsion are worth about the same, about 1% of \(\Wo\) for each 1% of \(L/D\) or \(C\). And the effects are not symmetric: making things worse costs more than making them better gains, because the denominator of the sizing equation shrinks toward zero.

Figure 7.1 A tornado chart for the business jet: the change in \(\Wo\) when one input at a time is changed by the chosen percentage, everything else held fixed. Better means the direction that makes the aircraft lighter, which is not always an increase: a lower \(C\) or empty-weight trend, a higher \(L/D\), a shorter range or less payload. Green bars to the left show the input made better; orange bars to the right, the input made worse. The rows are in order of importance, and the longest bars are the inputs to get right first. For the empty-weight trend, \(L/D\), \(C\) and range, worse costs more than better gains, because the denominator of the sizing equation shrinks toward zero (if it reaches zero, the sizing does not close); payload acts almost in proportion.

The true growth factor

Module 2 defined the growth factor with the fractions held fixed, \(1/(1 - \Wf/\Wo - \We/\Wo)\). For the business jet that is \(1/(1 - 0.2867 - 0.5721) = 7.08\): each kilogram of payload would add 7.08 kg to \(\Wo\). Resizing gives a smaller number:

Example 7.1 — One more passenger

Adding a passenger (\(100\ \text{kg}\)) raises the business jet's \(\Wo\) from \(6943\) to \(7508\ \text{kg}\). Find the true growth factor and explain the difference from 7.08.

Show solution

\[ \frac{\Delta\Wo}{\Delta W_{\text{payload}}} = \frac{7508 - 6943}{100} = 5.65 \]

The heavier aircraft has a slightly lower empty-weight fraction (the trend falls with weight), which takes back about a fifth of the growth. Fixed fractions overestimate the snowball; the full resize is the honest number. Both say the same thing: every kilogram added anywhere costs five to seven kilograms at takeoff.

Trade studies: payload and range

A trade study resizes the aircraft over a grid of requirements, so that the customer and the designers can see what each combination costs. Figure 7.2 does it for the business jet's two main requirements.

Figure 7.2 Takeoff weight against design range for the business jet with 4 to 12 passengers. Each line is a family of designs; the marked point is the chosen requirement. Change the technology to see the whole chart shift.

Example 7.2 — Which is cheaper?

A customer asks for either two more passengers (10 instead of 8) or 1000 km more range (5600 instead of 4600 km). Using the trade study, which costs more takeoff weight?

Show solution

From the sizing (Figure 7.2): 10 passengers at 4600 km gives \(\Wo = 8066\ \text{kg}\), \(+1123\ \text{kg}\) (+16%); 8 passengers at 5600 km gives \(\Wo = 8891\ \text{kg}\), \(+1949\ \text{kg}\) (+28%). The range costs nearly twice as much, because at this range the design is already climbing the steep part of its curve. The answer would be different for a short-range design: trade studies are specific to the design point.

What the first estimate is for

The first estimate is a starting point, not a result. It sets the scale of everything that follows:

Module 4 chooses the airfoil and wing geometry; Module 5 chooses the wing loading and thrust-to-weight ratio, which with \(\Wo\) give the wing area and engine thrust.
With those, a real layout can be drawn (Module 6), its drag (Module 9) and component weights (Module 10) estimated, and the aircraft resized with numbers from its own drawing instead of from history: refined sizing.
The sensitivities found here say where that later effort pays: for the business jet, the empty weight first, then \(L/D\) and \(C\).

Check your understanding

Key takeaways

Find sensitivities by changing one input and resizing.
The empty-weight trend is the most powerful input; \(L/D\) and \(C\) are worth about 1% of \(\Wo\) per 1%.
Worse inputs cost more than better ones gain: the denominator shrinks toward zero.
The true growth factor (about 5.7 for the business jet) is smaller than the fixed-fraction one (7.1).
Trade studies over payload and range show customers the price of their requirements.
This completes the module. Try the Practice Lab and the Self-Check Quiz; Module 4 begins the wing.