One G rolls: a physicist's view

As a follow-on to the discussion of "one G rolls" in this NG some time
back, an analysis of a simplified "physicist's version" of the one G
roll problem, with formulas and graphs, is available at

http://www.stanford.edu/~siegman/one_g_roll.html

I'm not sure why I was driven to do this -- maybe nothing better to do
with my time these days? -- but I was, so here it is.

I'll give a quick summary of results first, then a more detailed summary
of the assumptions behind them. Any comments on this will be welcome
(but on the NG, please; not by private email). And before anyone jumps
too hard on my approach or results, please look at the assumptions to
see what I'm claiming and what I'm not.

RESULTS:

Unless I've made some dumb mistake (always a possibility), if an
aircraft starts out in level flight and rolls through 360 degrees
clockwise about its forward axis in 10 seconds, while maintaining 1 G
force on the aircraft and the pilot, and a constant forward velocity, it
will perform a circular corkscrew "barrel roll" type of motion with a
radius of about 80 feet around a curved "guiding axis" (although the
actual orbit will not look much like a corkscrew), and will end the
maneuver displaced about 500 feet to the right, with an altitude loss of
about 1600 feet, and a screaming final downward velocity component of
300 feet/second.

If on the other hand it can enter the roll maneuver *inverted* and with
an initial upward velocity of 150 feet/second (i.e., an initial roll
angle of 180 degrees), and again do a 360 degree, one G roll starting
from and ending up back at that roll angle, it can end up with no final
elevation loss; only a 200 foot maximum vertical elevation rise and fall
during the maneuver; the same 500 foot displacement to the right; and
only half the downward final velocity of the first case.

Those are the theoretical predicitions -- I'll leave it to the pilots on
the group to do the experiments . . .

ASSUMPTIONS:

1) The basic assumption here is that some object up in the sky is going
to be acted on by a constant 1 G transverse force, due to lift and other
aerodynamic foces, gas jet thrustors, attached wires, whatever).

The direction of that force is then rotated around through 360 degrees
in a vertical plane called the transverse plane, with sideways and
vertical coordinate axes x and y; and we ask how the object moves in
those x and y directions as a result of this rotating applied force.

2) At the same time the object may also be moving forward in a "z"
direction perpendicular to that x, y plane -- in fact, if it's an
airplane it is surely doing so. A second assumption is that during the
one G roll maneuver the object's forward velocity in that z direction
(i.e.e, over the ground) remains unchanged.

Note that if this forward velocity were to change during the maneuver,
then that would mean that some additional net force must have been
applied on the object along the z direction, and this would then become
a more complicated problem, and maybe no longer a "one G problem".

3) Drag effects associated with the sideways and vertical motions are
assumed to be either negligible or absorbed in the applied 1 G force.
Angular momentum effects associated with the roll are ignored, because
they can be.

4) Finally, I'm not a pilot. I don't claim to know what pilots believe
is a one G roll; and I have no idea whether a real plane could move in
the way described in the analysis, or what control inputs would be
needed to make it do so. The analysis nonetheless seems to to match up
with what some of the people in the earlier discussion described as a
one G roll; and it gives at least a starting point for understanding
what sort of motions will be required to make that kind of one G roll.