Do you use actual runway limited takeoff weights for this?
You betcha. So long as we're within the weight limit values for the runway, stopping shouldn't be a problem. Insofar as rejects in the simulator, we generally do max takeoff weight, but we do verify the stop distance against what's calculated (bearing in mind that it's a simulator and not the actual airplane).
In the real world, of course, we do the same; the runway-limited actual takeoff weight is used to calculate stopping distance from V1, and a stop margin is calculated off that. We anticipate stopping within the calculated distance, so long as we do our part.
I think I may have confused Angle of Attack and climb pitch. I thought they were the same thing. It appears not.
Angle of attack and pitch angle are never the same, except by coincidence. Angle of attack is the angle formed between the airstream as it arrives at the wing, and the mean aerodynamic chord of the wing (which changes with aircraft configuration. Angle of attack is different close to the ground ("ground effect) than in flight for a given pitch angle and airspeed, and angle of attack is different in flight for a given pitch angle and airspeed depending on flap and leading edge device position (aircraft configuration).
I understand where you're coming from; it's easier to envision angle of attack as a function of pitch, and to a certain extent that works for simple explanations. It doesn't work if you want to expand beyond a simple explanation, though.
Angle of attack isn't just the angle between the pitch of the airplane, and the free airstream; the pitch of the airplane isn't the same as the angle of incidence, which is the angle the wing is set relative to the longitudinal axis of the airplane. Even head-on into the wind, the airplane normally has a positive angle of attack of several degrees. That's because the chord line of the wing, specifically the aerodynamic chord line of the wing, is set at a different angle than the long axis of the airplane. This means that the chord line (the line between the leading edge of the wing and the trailing edge of the wing, to make it simple, is different than the line formed from the front end of the airplane to the back. That difference is the "angle of incidence," and it's significant here because as the airplane sits, without the pilot doing anything, the angle of attack is always different than the pitch angle.
The angle of attack is further modified because of the way air approaches the wing. Behind the wing air is forced down, the measure of which is lift. We call that "downwash," and ahead of the wing air is given an upward vector, called "upwash." This local flow, or upwash, affects the angle of attack. As you can imagine, air angled upward from ahead of and below the wing creates an even bigger angle to the chordline of the wing than the free airstream. This is more pronounced when slow and "dirty," when flaps and leading edge devices are extended.
Angle of attack, then, changes with aircraft configuration and airspeed. As you're imagining it, however, the most direct control over angle of attack is the pilot's pushing forward or pulling back on the control column or stick. Pitching up and down also influences angle of attack. To confuse you a little more, an airplane in cruise might have a power reduction made, and begin a descent, but see little change in angle of attack, even though the nose may be pointed down more; the angle of attack sometimes changes with pitch, but other times it doesn't, or doesn't change in proportion to pitch changes.
It may be easier to think in terms of pitch for now, however, relative to the questions you're asking.
So just so I have this clear, the corrective action you would take with a failure at rotation is: rudder input to keep the aircraft flying straight and lowering the nose to keep your airspeed from decaying and ensure you accelerate to V2?
Rudder input, yes. Lowering the nose, no. The crux of your question appears to be what to do if the nose is in the air (in other words, we've reached rotation speed, and have pulled back on the control column, raising the nose into the air, as we prepare to take off) and we experience an engine failure.
The short answer is we fly the airplane, or whatever portion of it is already flying. Rudder to keep the airplane going straight down the runway and under control, and keep the pitch attitude. Rotation speed and the climb pitch attitude has already calculated with an engine failure in mind. We don't need to touch the power, we don't need to drop the nose back down; we've already reached the rotation speed that we calculated earlier. We're there.
We have already rotated at that speed, and it's enough. In the airplane I fly, we stop the rotation at 10 degrees until we are off the ground and have a positive climb going. Then we can continue to pitch up.
We've reached V1. We've reached Vr. We've rotated, and we're waiting to come off the ground. We're not going to lower the nose back down. If we do that, we're going to dump lift. We don't want that. We want to go fly. We may be very runway-limited, and we're looking to get off the ground now, and get away from obstacles so we can deal with our new problem.
Our next concern is achieving and holding V2. This is our takeoff safety speed, and it's predicated on the engine-out climb at our previously calculated power settings, minus an engine. My employer's policy is to make the climb at V2; however, if we've passed through V2 (due to a slow rotation, perhaps, or we're just accelerating faster than we thought, or we had already achieved a higher speed when the engine failed), we can hold up to V2 plus 10 knots, as we climb.
At this point, we do control airspeed with pitch. We're looking for V2; if we're not getting there quickly, we'll hold our pitch attitude while we accelerate. If we pass through V2 and the airplane wants to keep accelerating, then we're going to increase our pitch to keep our speed at V2.
This is probably a daft question, but I wonder how you establish by how many degrees to lower the nose to maintain the airspeed and accelerate to V2?
None, Nicholas. There's no need. If we've already reached Vr and have rotated, then we're not going to put the nose back down; we've already met the speed requirement, and we know we can continue at this weight. There are rare, and unusual exceptions, such as windshear on the takeoff which might cause the airspeed increase to stagnate or fall back. We're past V1, and we're going flying. We're going to hold our pitch attitude and look for V2. The airplane will probably be off the ground before we reach that point, anyway.
It's not really a matter of lowering the nose as it is rotating a little more slowly. If the engine failure occurs just after V1, before Vr,then rotate slower. If the engine failure occurs during Vr, with that speed already met and the airplane being rotated to the precalculated altitude, we've already got the necessary speed. Time to go fly.
Is it a question of feel, experience on, knowledge of, and simulator training for your particular aircraft type, or do the aircraft's systems (e.g. Flight Director) provide guidance?
I'm not really looking at the flight director, at this point. I'm noting my airspeed and my pitch. I'm consciously rotating slower; if I normally rotated in six seconds, this time I might take nine. I'm noting my airspeed and my pitch attitude. I'm keeping a visual reference outside on the runway centerline and transitioning in and out to my instrumentation. Once Vr has arrived, I'm in and out of the cockpit, mostly in, looking to keep the airplane straight and put the "pipper" on the attitude indicator on top of my 10 degree nose-up line. I'll be stopping there until the airplane comes off the ground, and about that time I should be close to V2. I'll continue increasing my pitch to hold V2, to the preplanned target pitch attitude established as part of the takeoff calculations. I'll vary that pitch just slightly to hold V2, as necessary.
From there, we'll be climbing to a pre-calculated engine-out altitude (again calculated as part of our departure preparations), where I'll be leveling off an cleaning up (flaps up) as I accelerate to my clean climb speed. Then climb will continue.
Big airplanes can be felt, a little, but they're largely numbers airplanes. Calculate the right numbers, and and most of the time the performance (at least in our equipment) comes out extremely close, if not on. Knowing how much to rotate and pitch is largely a matter of training and calculations; we have the numbers in front of us. The climb is mostly about flying the airplane (as opposed to letting it fly us) and using the numbers that we've calculated that will allow the airplane to do what it's certified to do.