What are the advantages of a variable-incidence wing as used in F-8?
Do weight and other issues outweigh the advantages and that's why it's not found in many (any?) other aircraft?
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The main reason for having variable incidence wing in the F-8 was that it allowed the production of increased lift due to a greater angle of attack without reducing pilot visibility as the fuselage stayed level.
For aircraft with high angle of attack TO/landings (like delta winged aircraft), pilot visibility is a problem in low speed landings, as the nose will obstruct the runway (or carrier, for that matter) in extreme cases. There are two ways to solve this problem:
Droop the nose, like the Concorde or
Make the wings variable incidence, like the F-8.
The first option is usually preferred (same thing was done in Tu-144) as the weight penalty is (significantly) less. However, this wasn't possible in the F-8 as the engine air intake was in the nose; so the incidence of wings was changed. The variable incidence wing had the added advantage of reducing the length of landing gear required and improved the low speed characteristics.
In case of Martin XB-51, the variable incidence wings were used to reduce take off distance as it was expected to be operated from forward airfields.
The Vought F-8 Crusader had a variable incidence wing because of the role it was designed to fulfill.
Higher take-off, approach and landing speed were not an issue on the F-104, so the angle of attack (AoA) was not critical. On landing the '104 could also make use of a drogue chute.
On the other hand on the F-8 the AoA was critical because of the short, cable-arrested landings on the deck of the aircraft carriers.
Considering the relatively small clearance under the fuselage of the F-8, an high AoA during landing could lead to an in-flight engagement of the hook (something definitely undesirable), hence the necessity to keep the fuselage as horizontal as possible.
Tail clearance on take-off and landing and visibility from cockpit.
The aircraft needed rather high angle of attack on take-off and landing. The resulting attitude would:
So they chose this method of reducing the pitch.