What happens during rotation at takeoff?

Hi,

I'm developing my own simulator so this is a generic simulation question I'm hoping
the collective experience here can shed some light on.

It seems so simple. You pull on the stick, rotate and you take off.

If you analyze the situation in the craft frame with origin for moments
at the rear wheels it's straighforward. The elevator pushes down and that
rotates the craft around the wheels and the CG rises.

But in general we put some other point as the moment origin (say the CG for simplicity) and
there things get tricky. The elevator pushes down and induces a positive moment around the CG.
Intuitively this torque should push down on the rear gears and lift the CG (do a "wheelie")
For the CG to lift there needs to be a force applied to it (as opposed to a kinematic
displacement due to rotation around the rear wheel).
Where does this force come from?

	You could posit a reaction force corresponding to the torque induced by the elvator
	but the problem of computing a force from a torque is mathematically
	ill-defined (there's no unique inverse of a cross product). There's also
	no way to decide on force distributions between three wheels reacting to the torque.

	And neither larcsim nor jsbsim pull any shenanigans like this.

So, presumably, the force raising the CG must implicitly come from ground interaction
reactions. Let's look at this in detail.

Initially the steady state is such that ground interaction produces
reaction forces that balance the crafts weight. Fleft + Fright + Fnose = M * g;

As you accelerate and pull back on the stick a positive pitch moment increases the pitch
angle, pushing the rear gear down and pulling the nose gear up. This causes
a redistribution of the reaction forces where the rear gear reacts more and the
nose gear reacts less but given that torque is equivalent to a couple these
reactions tend to cancel each other out.
As a result here is no net upward force to lift the CG.

	Ground reaction forces produce reaction moments and these moments also
	equally balance the moment induced by elevator torque.

Once the nosegear leaves the ground (i.e. its spring compression reaches 0)
then it can no longer compensate for increased reaction of the rear gears
and you should get a net upward force due to the rear gears reaction.

My problem is no matter how I play with gear positions and spring K's
I never get to the point where the nosegear departs. All pulling back
on the stick does is redistribute gear spring compressions w/o any rotation.
The harder you pull the harder the rear gear springs react and pitch angle
changes even less keeping the front gear from departing.

If anything, is the above way of analyzing this situation correct?


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