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I am trying to understand how a variable sweep wing changes the aerodynamic center of a plane, and its effect on the plane's maneuverability. I also would like to know the importance of the pivot point as it somehow affects the center of pressure.

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With so many questions I need to structure the answer:

Effect on performance

Moving the wing backwards will also move the aerodynamic center backwards, but the high sweep of a swing wing normally means that the wing's leading edge is still subsonic even at Mach 2. This means that the aerodynamic center of the wing ist still near the quarter chord, so the shift in the center of pressure is limited to the geometric shift from sweeping the wing. Generally, a swing wing needs to be combined with a generously sized horizontal tail to have enough trim authority for all sweep angles.

The first jet capable to change sweep angle in flight, the Bell X-5, therefore shifted the wing forward when sweep was increased. Later designs used fixed pivot points, however.

If no possibility exists to trim the aircraft by shifting weight (mostly fuel), swing wings cause high trim drag when swept back. The biggest source of drag, however, is the increased structural mass for the sweep mechanism which will cause an increase in induced drag. Further, the mechanism takes up valuable volume at the center of the aircraft which now cannot be used for fuel storage. In total, the substantial improvement in wing drag due to variable sweep is almost completely eaten up by the mass increase and the trim drag increase.

A clear benefit of variable sweep is improved low speed performance: Both the reduced sweep and the increase in wing span will reduce low speed drag substantially, so a swing-wring design will show superior loiter performance. Also, take-off and landing distances are reduced because the full wing span can be used for highly effective slotted flaps.

Effect on Maneuverability

Swing wings cannot have ailerons - the high sweep angle of the trailing edge at high speed flight makes control surfaces there almost useless. Therefore, a swing wing aircraft will use differential deflection on the horizontal tail in lieu of ailerons (tailerons). At low speed this reduces the possible roll rate -- the unswept wing contributes high roll damping while the tailerons will have limited effectivity due to their small lever arm. The F-14 uses wing spoilers to improve roll rate at lower sweep angles, but spoiler control is less efficient than ailerons.

At high speed (and sweep angles) the reduced aspect ratio, the increased sweep angle and the much reduced moment of inertia around the roll axis will combine to produce a very agile aircraft.

Pivot position

The further outboard the pivot point, the smaller the shift in the center of pressure. If the wing uses a highly swept root, the pivot can be positioned outboard without loss of efficiency at high speed. Examples are the Sukhoi 17 or the Tupolev Tu-22. An outside pivot position also allows to place external loads on the inner part of the wing (wing glove) and to avoid the complexity of moveable wing stations.

The Grumman F-14 also extends a small, triangular wing extension (glove vane) at the leading edge when the wing is swept back to reduce center of pressure travel. This feature, however, was disabled later to reduce weight and complexity.

The reduced top speed demands on modern fighters mean that variable sweep will no longer be attractive to achieve an optimum design. Variable sweep vanished with the Mach 2+ requirement in the 1980s.

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Check out this discussion, or this one from which I quote:

NASA conducted a large research program on swing-wing aircraft design. One design alternative that NASA developed was used the F-111 as well as in later aircraft, such as the F-14. Instead of placing the pivot point for the wings in the center of the fuselage, each wing was give its own pivot point, and the pivot point was placed several feet outboard of the fuselage. The pivot points were covered with an aerodynamic fairing and became part of the wing. This arrangement effectively divided each wing into two parts—a nonmoving wing glove section covering the wing pivot mechanism and a swinging section. As the swinging section swept back at higher speeds, it would lose aerodynamic effect. However, the wing glove section would increase its aerodynamic effect as speed increased. This lessened the shift in the center of pressure.

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