I am looking for an aeronautical equation given as a simple expression
(I need to be able to do this in Excel) to determine the altitude
required to recover from a dive given the following variables:
Altitude, Dive angle(radians), velocity(feet/sec), maximum pullout
G loading(G's), Corner velocity(if required).

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Clarification of Question by varmint007-ga on 13 Sep 2004 02:20 PDT

I should clarify that I'm not looking for the altitude to "safely"
recover from the dive, just the minimum possible altitude assuming a
theoretically instant response. Other variables such as pilot response
time, time to roll wings level, G-onset rate, ect. can be ignored.

This is something I need to simulate a basic Ground Proximity Warning System.

Thanks,
Varm

Assuming the recovery follows the arc of a circle radius r, centre C,
then the dive path of the plane is tangent to the circle at a point X
and subtends an angle A with the horizontal.
Without loss of generality we can consider the case where the plane
just manages to avoid hitting the ground, which forms a horizontal
tangent to the circle at Y.
The required altitude is the difference in altitudes of X and Y. 

By elementary geometry, the radius CX subtends the dive angle A with
the radius CY. The required altitude is thus r - r cos A.

If I remember my physics correctly, the maximum pullout (G) is equal
to v squared divided by the radius, where v is the corner velocity. r
is therefore
v * v / G.

(The elementary geometry used can be easily seen by projecting the
dive path D and the line CY until they intersect at Z. D intersects
the horizontal at W. Then the angle YWZ is clearly A, hence the angle
at Z must be 90 - A (degrees).
The triangle CXZ is also right-angled, so the angle at C must be 90 -
(90 - A) = A).

The triangle CXZ is a right-angled triangle,

Thank you for your comments. I think one thing that must be considered
is the gravity influence on the load factor at any given point along
the circle. As the aircraft's dive continues to get more shallow,
gravity has an increasing negative influence on the lift vector. That
is to say, we're not actually talking about a circle, but a parabola.
So there must be more components to the R=V^2/G equation. R=V^2/G*?

Lets assume we have an aircraft traveling at 600 fps, at a 75 deg
angle. We wish to pull out of the dive at 4Gs. How much altitude would
be required to complete the pullout?

Insert equation here:)______________________

Thanks
Varm

I think a parabola would be appropriate if the entire acceleration
could be applied upward (decelerating the downward motion, then
reversing it).  But obviously the wing's lift is more "normal" to the
dive path than it is to the ground.  Combining the wing's drag, the
force of gravity, and the lift would appear to produce a curve
intermediate between the circular path suggested by aldante-ga and the
parabolic path suggested by varmint007-ga.

I'll have to sleep on this one!

regards, mathtalk-ga

To answer this question I think the following information will be needed.

Initial conditions - which you have given.
Aircraft Mass.
Drag coefficient of said aircraft
Lift coefficient of said aircraft
Thurst of engine/engines for this aircraft

These are the MOST general things to take into account.
Naturally the coefficients will give us the drag and lift as a
function of aircraft velocity and density (assume density does not
fluctuate if slow moving aircraft and small altitude changes).

So now we can form a force vector from drag, lift, thrust, and
mass*gravity at any instant of time.  Now take the initial conditions
and integrate through time to see if the airplane crashes or flys. 
The only thing missing is the moment on the aircraft due to
aerodynamics forces which will dictate its rate of rotation.  This
information should also be obtained from the same source you got
coefficients of lift and drag from.

That is at least the minimum that one needs to solve this problem to
the most elementary of levels - imho.

I have a feeling that mathtalk will give you a nice approximation though :)

Best,
Steve.

"Assume a spherical aircraft, in simple harmonic motion..."

;-)

Thank you all again,

I wasn't sure if the CL/CD would be required or not; I can calculate
any coeficient required and feed that into the equation. As mentioned,
coefficients will change with dynamic pressure(q), and mass will
change with fuel load, but I agree these factors can probably be
considered static at the initial point for the relatively small
altitude and time envelopes we're talking about.

I would assume the CL used would have to be between Vc and L/Dmax at
the initial dynamic pressure.

Varm

Mathtalk - that is the funniest thing I heard today - and will no
doubt use it in all my future work. :)  Your a genius!

varmint007 - Just to clarify on Lift coefficient - 

Lift Coefficient = (Lift)/((1/2)*density*(velocity^2)*D^2)

For this problem the density is the fairfield density - lets assume
its 1.229 kg/m^3.  For velocity - also the farfield value - so just
use airspeed - which is the indepdendent variable.  As for D, this
never changes because the size of the aircraft is not changing - just
use the correct D - should be given from whoever found the coefficient
in the first place. :)  Finally the Cofficient of lift doesn't change
- its a constant for any single flow situation.

So in conclusion set up the above equation like this:
Just assume density is constant - after all you did want ballpark?
Lift = function(velocity)
Lift = ((1/2)*density*(velocity^2)*D^2)*coefficientlift

The same can be done for drag:
drag = function(velocity)
drag = ((1/2)*density*(velocity^2)*D^2)*dragcoefficient

I'm hoping that the angle of attack - alpha - of the fluid on the
aircraft does not stray to much from zero - or this whole analysis
would be truely useless.

saem_aero,

It's been my understanding the the only component of CL that's static
is the CLa, or lift due to AoA rate.

The CL is actually the CLa*AoA-Alpha0. where Alpha0 is the AoA where
CL=0. Since AoA is never constant, neither is CL. Lift is therefore
CL*q*S where q is dynamic pressure (1/2rho*V^2), and S is the wing
planform area.

Further, CD is also constantly changing based on the CL's influence on
the induced component. CD = CDo(constant)+CDi CDm is not important for
these purposes, but CDi is 1/[Pi*AR*e]*CL^2.

At any rate, finding the values for lift and drag at any condition is
not really a problem. In fact, using a static CL/CD in the equation
would suffice so long as it was representative of a value near Vc.

Varm.