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I was reading one of the documents linked on this question, particularly I was reading case study number 8 of this particular document.

In the conclusion the author notes that the aircraft flight characteristic calculations had been messed up to the point that all V-speeds were calculated for a 747 100 tons lighter than their actual Take Off Weight (TOW). Thus, they tried to rotate too early, couldn't get enough lift and the tail of the 747 struck the ground.

When reading that brief description, I had assumed that perhaps they were off by 5 to 8 knots, or something along those lines. So, I was surprised to see the author later note that Vr was actually off by close to 30 knots!

I'm really curious suddenly, how much can a 747's rotation speed change? I was told that rotation speed was generally somewhere around 130 knots for that class of plane, but if there is a 30 knot variance. Can it really be anywhere from 115 to 145 depending on the weight of the plane?

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As Henning Makholm's answer discusses, the weight difference between the lowest and highest weights is significant. The operating empty weight (OEW) of a 747-400 is 394,100 lb, while the maximum takeoff weight (MTOW) is 875,000 lb. This means there is a difference of about 400,000 lb between the lightest and heaviest weights at which a 747 may be taking off, making the MTOW about twice as heavy as the OEW.

The rotation speed Vr is the point at which the aircraft can pitch up and develop enough lift to climb. So at MTOW, the 747 must develop about twice as much lift than closer to OEW. Most of this force is lift from the wings.

Lift can be calculated with the following equation:

L = 1/2ρv2sCL

L = lift
ρ = air density
v = airspeed
s = wing area
CL = lift coefficient

The air density will certainly play a role. Higher altitudes and temperatures reduce the amount of lift. But this example pertains to the other variables.

An airliner can use flaps and slats to increase the CL and s of its wings.

The CL will also depend on the angle of attack α of the wings. An approximate curve for the 747 can be seen here (it's for a 747-200, but it should be close enough). When an aircraft rotates, it is changing the AOA, which increases the lift. It is very important when designing an aircraft to make sure it can pitch up enough while on the runway to achieve the pitch angle needed for takeoff. A 747-400 rotates to about 10 degrees on takeoff. Based on the approximate CL-α diagram, the lift coefficient changes from about 0.3 to 1.25 when α changes from 0 to 10 on rotation.

When the aircraft weight is increased, flaps may be increased from 10 to 20. However, flaps 20 may be standard because these guys said so, so lets assume that CL and s will also not change. This leaves the airspeed v as the only parameter left to increase lift.

In this study, figure A-7 shows a distribution of ground speed at liftoff that lies between 140 and 190 kts. These numbers will probably be a little higher than Vr due to the acceleration between Vr and liftoff, and an average headwind. This means that at weights at the lowest end, 130 kts would be a reasonable estimate for Vr.

So now lets make up some numbers. Lets take the weights 475,000 lb and 875,000 lb. Assume that the aircraft needs a lift 10% higher than its weight to take off. This gives us lift values of 522,500 lb and 962,500 lb. Using the lift equation and a low end speed of 130 kts to find the unknown portion.

522500 = 1/2ρ1302sCL

ρsCL = 61.8

Since we are assuming none of those vales will change, we can solve for v at the higher lift value.

962500 = 61.8/2v2

v = 176 kts

This is not too much lower than the high end liftoff value of 190 kts. So there you have it. Lift must be increased to account for the additional weight, and lift is proportional to the square of the airspeed.

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  • $\begingroup$ Man, I wish Terry would show up and tell us some actual numbers to throw into the formulas. I'm a little dubious about my 115kt take off speed for OEW... $\endgroup$ – Jay Carr Feb 14 '15 at 22:38
  • $\begingroup$ @JayCarr I agree. In the meantime I added some statistical values to go off of. $\endgroup$ – fooot Feb 14 '15 at 23:29
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It doesn't sound surprising to me.

There's more than a factor of 2 in difference between the empty weight and the MTOW of a 747-400, so one departing close to empty will be able to produce lift enough to climb away from the runway at a significantly lower speed than a fully-loaded one -- and therefore it should, because (all other things being equal) the quicker you get to altitude, the shorter a window is there for sudden problems to be immediately dangerous.

Since lift increases with the square of the airspeed, one would expect vr for an almost empty 747 to be less than 70% of that for the same aircraft at MTOW under similar conditions.

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Below are the lowest and highest Vr listed for 747-100 and -200 aircraft with the engines shown for a flaps 10 normal takeoff. They are from the Tower Air (now long defunct) QRH that pilots and engineers were issued.

JT9D-3A  110 to 174
JT9D-7A  114 to 177
JT9D-7F  115 to 179
JT9D-7J  118 to 180
JT9D-7Q  126 TO 182

The actual tables are lengthy, with the Vr varying according to takeoff weight, flap setting, temperature, pressure altitude, as well as at least one other thing I cannot offhand remember. Each cell of the table gives V1, Vr, V2, and the 3-engine rotation target. Lowest weight, of course, gives you the lowest Vr.

The f.e. grease-penciled the values he looked up on a plastic card, which he then passed to the f.o., who checked them. When the f.o. was satisfied they were correct, he propped the card against the instrument panel in front of the thrust levers. The captain could choose to accept them at face value or check them himself as well.

At the two 747 carriers I worked, there was a difference in culture as to how you were to rotate. At the first carrier, you rotated to 10 degrees nose up and held that attitude until you came off the ground, at which time you rotated to the 3-engine rotation target. This procedure was intended to prevent tail strikes. At Tower Air they taught the correct procedure (in my opinion) of rotating directly to the 3-engine rotation target.

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  • $\begingroup$ Why, in your opinion, is the latter method of rotating the correct one, if the former method comes with a reduced risk of a tailstrike? $\endgroup$ – Sean Jun 12 at 21:01
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    $\begingroup$ @Sean From memory and almost 30 years ago now, EIA my first 747 carrier, had stretch DC-8s, which apparently were prone to tail strikes (see quora.com/…). When they got 747s, their first captains were former DC-8 captains and they brought with them that culture. Boeing's procedure was to rotate directly to the rotation target, knowing that before you get there you will be off the ground. I was criticized and laughed at in the sim during my initial training for Tower Air, for the 10 degree procedure. $\endgroup$ – Terry Jun 13 at 4:53
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    $\begingroup$ @Sean I forgot one thing (and I was out of room in my previous comment). If you rotate to 10 degrees and then wait for the aircraft to lift off, your ground run will be greater than if you rotate directly to the 12 to 14 degree (usually) target. Also, when EIA got their 747s, they didn't hire any experienced 747 captains to the best of my knowledge. Tower Air, on the other hand, had only 747s and hired experienced 747 captains right off the bat as a startup. $\endgroup$ – Terry Jun 13 at 23:19
  • $\begingroup$ So, if you try to rotate directly to the target pitch attitude in a DC-8-60/70, you'll get a tailstrike, whereas a 747 will lift off before a tailstrike unless it's very severely overloaded? $\endgroup$ – Sean Jun 14 at 2:08
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    $\begingroup$ @Sean I think abetter statement would be that for max weight takeoffs, unless you rotate to the target attitude slowly, you're liable to get a tail strike. Tail strikes on 747s were just not a problem, even when overloaded. See talking-of-flying.com#overloaded. There was no tailstrike.The 747 sits higher and it has a body gear well aft of the wing gear. The angle from the aftmost gear to the tail is greater on the 747. Tailstands were typically used when loading stretch DC-8 freighters, but not typically used for 747 freighters. Also, 747 target rotation attitude can go as high as 18 degrees. $\endgroup$ – Terry Jun 14 at 3:50

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