The lower end of the graph represents piston engines; piston-engine nacelles can be slightly lighter in weight.

The second graph in Figure 8.3 shows the wing and empennage group weights. The pistonand gas-turbine engine lines are not clearly separated. FBW-driven configurations have a smaller wing and empennage (see Chapter 13), as shown in separate lines with lighter weight (i.e., A320 and A380 class). The newer designs have composite structures that contribute to the light weight.

Figure 8.3 shows consistent trends but does not guarantee accuracy equal to semi-empirical relations, which are discussed in the next section.

8.10 Semi-empirical Equation Method (Statistical)

Semi-empirical relations are derived from theoretical formulation and then refined with statistical data to estimate aircraft component mass. It is an involved process to capture the myriad detailed parts. Mass estimation using semi-empirical relations can be inconsistent until a proper one is established. Several forms of semi-empirical weight-prediction formulae have been proposed by various analysts, all based on key drivers with refinements as perceived by the proponent. Although all of the propositions have similarity in the basic considerations, their results could differ by as much as 25%. In fact, in [5], Roskam describes three methods that yield different values, which is typical when using semi-empirical relations. One of the best ways is to have a known mass data in the aircraft class and then modify the semi-empirical relation for the match; that is, first fine-tune it and then use it for the new design. For a different aircraft class, different fine-tuning is required; the relations provided in this chapter are amenable to modifications (see [5] and [6]).

For coursework, the semi-empirical relations presented in this chapter are from [2] through [7]; some have been modified by the author and are satisfactory for conventional, all-metal (i.e., aluminum) aircraft. The accuracy depends on how closely aligned is the design. For nonmetal and/or exotic metal alloys, adjustments are made depending on the extent of usage.

To demonstrate the effect of the related drivers on mass, their influence is shown as mass increasing by (↑) and decreasing by (↓) as the magnitude of the driver is increased. For example, L(↑) means that the component weight increases when the length is increased. This is followed by semi-empirical relations to fit statistical data as well as possible. Initially, the MTOM must be guesstimated from statistics as in Chapter 6. When the component masses are more accurately estimated, the MTOW is revised to the better accuracy.

8.10.1 Fuselage Group – Civil Aircraft

A fuselage is essentially a hollow shell designed to accommodate a payload. The drivers for the fuselage group mass are its length, L(↑); diameter, Dave (↑); shell area and volume, (↑); maximum permissible aircraft velocity, V(↑); pressurization, (↑); aircraft load factor, n(↑); and mass increases with engine and undercarriage installation. The maximum permissible aircraft velocity is the dive speed explained in the V-n diagram in Chapter 5. For a noncircular fuselage, it is the average diameter obtained by taking half the sum of the width and depth of the fuselage; for a

The author does not compare the equations here. As mentioned previously, the best method depends on the type – weight equations show inconsistency. Torenbeek’s equation has been used for a long time, and Equation 8.14 is the simplest one.

The author suggests using Equation 8.14 for coursework. The worked-out example appears to have yielded satisfactory results, capturing more details of the technology level.

MFcivil = cfus × ke × kp × kuc × kdoor × (MTOM × nult)x × (2 × L × Dave × VD0.5)y,

(8.15)

where cfus is a generalized constant to fit the regression, as follows:

cfus = 0.038 for small unpressurized aircraft (leaving the engine bulkhead forward)

=0.041 for a small transport aircraft (≤19 passengers)

=0.04 for 20 to 100 passengers

=0.039 for a midsized aircraft

=0.0385 for a large aircraft

=0.04 for a double-decked fuselage

=0.037 for an unpressurized, rectangular-section fuselage

All k-values are 1 unless otherwise specified for the configuration, as follows:

ke = for fuselage-mounted engines = 1.05 to 1.07

kp = for pressurization = 1.08 up to 40,000-ft operational altitude = 1.09 above 40,000-ft operational altitude

kuc = 1.04 for a fixed undercarriage on the fuselage

=1.06 for wheels in the fuselage recess

=1.08 for a fuselage-mounted undercarriage without a bulge

=1.1 for a fuselage-mounted undercarriage with a bulge

kVD = 1.0 for low-speed aircraft below Mach 0.3

=1.02 for aircraft speed 0.3 < Mach < 0.6

=1.03 to 1.05 for all other high-subsonic aircraft kdoor = 1.1 for a rear-loading door

The value of index x depends on the aircraft size: 0 for aircraft with an ultimate load (nult) < 5 and between 0.001 and 0.002 for ultimate loads of (nult) >5 (i.e., lower values for heavier aircraft). In general, x = 0 for civil aircraft; therefore, (MTOM × nult)x = 1. The value of index y is very sensitive. Typically, y is 1.5, but it can be as low as 1.45. It is best to fine-tune with a known result in the aircraft class and then use it for the new design.

Then, for civil aircraft (nult <5), Equation 8.15 can be simplified to:

For the club-flying–type small aircraft, the fuselage weight with a fixed undercarriage can be written as:

If new materials are used, then the mass changes by the factor of usage. For example, x% mass is new material that is y% lighter; the component mass is as follows:

In a simpler form, if there is reduction in mass due to lighter material, then it is reduced by that factor. For example, if there is 10% mass saving, then:

MFcivil = 0.9 × MFcivil all metal

8.10.2 Wing Group – Civil Aircraft

The wing is a thin, flat, hollow structure. The hollow space is used for fuel storage in sealed wet tanks or in separate tanks fitted in; it also houses control mechanisms – accounted for separately. As an option, the engines can be mounted on the wing. Wing-mounted nacelles are desirable for wing-load relief; however, for small turbofan aircraft, they may not be possible due to the lack of ground clearances (unless the engine is mounted over the wing or it is a high-wing aircraft – few are manufactured).

The drivers for the wing group mass are its planform reference area, SW(↑);

aspect ratio, AR(↑); quarter-chord wing sweep, 1/ (↑); wing-taper ratio, (↑);

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mean-wing t/c ratio, (↓); maximum permissible aircraft velocity, V(↑); aircraft limit load, n(↑); fuel carried, (↓); and wing-mounted engines, (↓). The aspect ratio and

wing area give the wing span, b. Because the quarter-chord wing sweep, 1/ , is

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expressed in the cosine of the angle, it is placed in the denominator, as is the case with the t/c ratio because the increase in the t/c ratio decreases the wing weight by having better stiffness.

A well-established general analytical wing-weight equation published by SAWE [2] is as follows (others are not included):

MW = K(Mdg NZ)x1 SWx2 ARx3(t/c)x4(1 + λ)x5(cos 1/4)x6(B/C)tx7 SCSx8 (8.19) where C = wing-root chord, B = width of box beam at wing root, SCS = wingmounted control-surface reference area, and Mdg = MTOM.

The equation is modified for coursework. The term (MdgNZ)x1 in this book’s nomenclature is (MTOM × nult)0.48. The term (B/C)tx7 SCSx8 is replaced by the factor 1.005 and included in the factor K. The lift load is upward; therefore, mass carried by the wing (e.g., fuel and engines) would relieve the upward bending (like a bow), resulting in stress relief that saves wing weight. Fuel is a variable mass and when it is emptied, the wing does not get the benefit of weight relief; but if aircraft weight is reduced, the fixed mass of the engine offers relief. Rapid methods should be used to obtain engine mass for the first iteration.

Writing the modified equation in terms of this book’s notation, Equation 8.19 is replaced by Equation 8.20 in SI (the MTOM is estimated; see Chapter 6):

MW = cw × kuc × ksl × ksp × kwl × kre × (MTOM × nult)0.48 × SW0.78 × AR

×(1 + λ)0.4 × (1 − WFuel mass in wing/MTOW)0.4/[(Cos ) × t/c0.4] (8.20)

where cw = 0.0215 and flaps are a standard fitment to the wing. kuc = 1.002 for a wing-mounted undercarriage; otherwise, 1.0 ksl = 1.004 for the use of a slat