Solution
2.3.9 The V-n Diagram
We now introduce a diagram that depicts the structural load limits for an aircraft as a function of airspeed and load factor. We first define the non-dimensional load factor, n, as
n≡ L
W (2.49)
where L and W are the aircraft lift and weight, respectively. The flight envelope, discussed in the previous section, applies to an aircraft in level flight, where the lift equals the weight, and thus the load factor, from Equation (2.49), equals one. In this situation, the inertia force acting on the aircraft is simply equal to its mass times the acceleration due to gravity. We commonly refer to this
k
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Introductory Concepts 145
as flight at 1 g or a load factor of one. The load factor, n, is non-dimensional, but when we refer to a load factor, we often specify it in terms of g’s. For instance, if an aircraft is flying such that the lift is three times greater than the weight, the aircraft is flying at 3 g or a load factor of 3. The aircraft load factor can be greater than or less than one if the aircraft is maneuvering, such as in a horizontal turn, pull-up, or pushover. Wind gusts encountered by the aircraft can also increase or decrease load factor, sometimes significantly.
The V-g or V-n diagram defines an allowable structural envelope that is a function of the load factor, n, and airspeed, V (Figure 2.32). Flight at airspeeds and load factors within the boundaries of the V-n diagram are within the structural operating limits of an aircraft. The limits shown on a V-n diagram are the aerodynamic (stall) limit on the left boundary, positive and negative structural load limits on the upper and lower boundaries, respectively, and a maximum airspeed limit on the right boundary. Each V-n diagram is valid for a specific type or model of an aircraft at a specific gross weight and altitude, in a specific aircraft configuration, and for a specific type of loading.
The aircraft configuration definition includes the position of the flaps, the landing gear position, etc. The type of loading is defined as either symmetrical or asymmetrical (rolling) loading.
There are two types of structural load factor boundaries that are shown on the V-n diagram, the limit load factor and the ultimate load factor. The limit load factor is the maximum load factor that an aircraft can be subjected to without any permanent structural deformation of the primary structure. Above the limit load factor, the aircraft primary structure may be permanently deformed or damaged, perhaps resulting in an unsafe flight condition. If the ultimate load factor is exceeded, the aircraft primary structure will fail. There are positive and negative boundaries for both the limit load factor and the ultimate load factor, corresponding to positive and negative g’s, respectively. As shown in the figure, the maximum negative limit and negative ultimate load factors are typically smaller in magnitude than the positive values for aircraft that are designed primarily to fly in a positive g flight regime. For some types of aerobatic aircraft, the maximum negative limit and negative ultimate load factors can be equal in magnitude to the positive values.
Positive lift limit (stall)
Negative lift limit (stall)
Limit airspeed (+)
(−) 1.0
Load factor, n
0 Airspeed, V∞
Normal flight envelope (shaded area) Positive limit load factor
Negative limit load factor Negative ultimate load factor
Positive ultimate load factor
1-g stall speed, Vs
Figure 2.32 The V-n diagram.
k k The left side of the V-n diagram is the aerodynamic lift boundary of an aircraft. The airspeed
along the left boundary is the aircraft aerodynamic stall speed as a function of the load factor. The aircraft is unable to maintain level flight at any airspeed–load factor combinations above this line.
The intersection of the lift limit line with a line corresponding to a load factor of one provides the 1 g stall speed of an aircraft, Vs,1g. There is a lift limit line for positive and negative load factors.
The negative load factors corresponds to negative lift flight, that is, the lift being produced due to negative angle-of-attack on the wing or inverted flight. For a given load factor magnitude, the negative load factor stall speed is typically higher than the positive load factor stall speed. This is primarily due to the shape of the wing airfoil section, which is designed to be more efficient at producing lift at positive, rather than negative angle-of-attack. The shape of the lift limit lines is not linear; rather they vary with the square of the airspeed. For an aircraft in steady flight at a load factor n, the lift is given by
L = nW = 1
2𝜌∞V∞2SCL (2.50)
The stall speed at a load factor n, Vs,n, is given by
Vs,n=
√ 2nW
𝜌∞SCL,max =√
n Vs,1g (2.51)
where the 1 g stall speed, Vs,1g, is defined by Equation (2.48). Hence, the stall speed at a load factor n, varies with the square root of the load factor multiplied by the 1 g stall speed.
The lift limit line and the positive limit load factor line intersect at a point called the maneuver point. The airspeed corresponding to the maneuver point is the called the corner speed, VA. The maneuver point and the corner speed have important implications relative to the turn performance and structural limitations of an aircraft. Below the corner speed, an aircraft stalls before it reaches the limit load. Above the corner speed, the limit load can be reached. The right side boundary of the V-n diagram is the limit airspeed, above which the aircraft will sustain structural damage or failure of the primary structure. The damage or failure may be due to exceeding structural loads in a critical gust situation, structural dynamic phenomena, or compressibility effects.
We now look at the V-n diagrams of a subsonic aircraft, the Beechcraft T-34A Mentor (Figure 2.33), and a supersonic aircraft, the Lockheed F-104 Starfighter (Figure 2.34). The
Figure 2.33 Beechcraft T-34A Mentor. (Source: Courtesy of the author.)
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Introductory Concepts 147
Figure 2.34 Lockheed F-104 Starfighter Mach 2 supersonic interceptor aircraft. (Source: NASA.)
Beechcraft T-34A aircraft is a single-engine, two-place ex-military trainer with a straight, low-mounted wing, retractable landing gear, tandem cockpit, and bubble canopy. The T-34 is powered by a single, 225 hp (168 kW), air-cooled, six-cylinder piston engine. The first flight of the T-34A was in 1948. The T-34A is still flying today as a civilian general aviation airplane.
Designed and manufactured by the Lockheed Skunk Works, the F-104 is a Mach 2-class, super-sonic jet aircraft designed for high dash speeds to intercept enemy aircraft. The F-104 was the first military jet capable of sustained flight at Mach 2. The F-104 has a slender, pointed fuselage, a mid-mounted, low aspect ratio wing with a trapezoidal planform, and an aft-mounted T-tail. Pow-ered by a single General Electric J79 turbojet engine with afterburner, the lightweight F-104 has excellent climb and acceleration capabilities. The F-104 set many world speed and altitude records during its time in service. The first flight of the Lockheed F-104 Starfighter was on 17 February 1956.
The V-n diagrams of the subsonic T-34A and the supersonic F-104 are shown in Figure 2.35 and Figure 2.36, respectively. Load factor is plotted versus indicated airspeed in these figures.
Figure 2.35, for the T-34, is based on an aircraft gross weight of 2900 lb (1315 kg) or less. The stall speed of the T-34 is 53 knots (61 mph, 98 km/h) at 1 g and increases with increasing load factor. The T-34 has a positive limit load factor of 6 g and a positive ultimate load factor of 9 g.
The maximum dive airspeed of the T-34 is 243 knots (280 mph, 450 km/h). For the F-104, the 1 g stall speed is close to 200 knots (230 mph, 370 km/h). The positive limit and negative limit load factors are 7.33 g and −3.0 g, respectively. The F-104 maximum dive speed is 750 knots (860 mph, 1390 km/h) at sea level. The F-104 is limited to a maximum Mach number of 2.0 at an altitude of 40,000 ft (12,200 m) and above.
Example 2.9 Stall Speed at Elevated Load Factor A Beechcraft T-34 has a 1 g stall speed of 53 knots (1 knot equals 1 nautical mile, nm, per hour). Calculate the maximum lift coefficient, CL,max, and the stall speed at a load factor of 3. Assume a weight, W, of 2900 lb, a wing area, S, of 177.6 ft2, and sea level conditions.
k k
Maximum dive airspeed (243 knots)
Limit load factor - 6G
Ultimate load factor - 9G 10
9 8 7 6 5 4 3 2 1
−1
−2
−3
−4
−5 50
Normal operating conditions Based on gross weight of 2900 lbs or less Load factor - g
Operation with flaps down Dangerous operating range
Prohibited operation Indicated air speed knots
Accelerated stall
Operation with gear down and locked
100 109 150 165 200 243
0
Ultimate negative load factor Limit negative
load factor - 3g
Figure 2.35 Beechcraft T-34A Mentor V-n diagram. (Source: US Air Force, T-34A Flight Manual. T.O.
IT-34A-1S, 1 July 1960.)
Solution
Convert the 1 g stall speed into consistent units.
Vs,1g= 53 nm
h ×6076 ft 1 nm × 1 h
3600 s = 89.45ft s
Using Equation (2.48), the maximum lift coefficient is obtained from the 1 g stall speed as
Vs,1g=
√ 2W 𝜌∞SCL,max
CL,max= 2W
𝜌∞S(Vs,1g)2
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Introductory Concepts 149
Figure 2.36 Lockheed F-104 Starfighter V-n diagram. (Source: US Air Force, F/RF/TF-104G Flight Manual, T.O. 1F-104G-1, 31 March 1975.)
k k
CL,max= 2(2900 lb)
(
0.002377slugft3
)
(177.6 ft2) (
89.45fts)2 = 1.717 Using Equation (2.51), the stall speed at a load factor of 3 is
Vs,n=√
n Vs,1g=√
3 (53 knots) = 91.8knots