Rex cornering overview addendum

[ngear@gvtc.com]

(For those of you who feel my posts are way too long-- this would be the
time to reach for that Delete key.)


Okay, it looks like the original "overview" post wasn't quite as
comprehensive as I could have hoped, so here are some loose ends that need
tying off.

Air drag:
Under the method where the foot steps off to one side of the center of
gravity, I neglected to include aerodynamic considerations.  Rex moved
through air and so of course experienced air drag, and when it banked around
a corner, the center of that drag would have been to the inside of the turn
from where it was applying thrust to the ground (ie. its feet); thus we have
a torque couple.  So once in the turn air drag would have helped to rotate
the rex body in the direction of the turn.  This is a real effect that
definitely would have happened and would not have required any special
action on the part of the rex to make use of it.  To see how strong this
effect would have been, we need to know the amount of drag force and the
amount of sideways displacement between the center of drag and the feet.

To get the drag force, we need to know the coefficient of drag for a rex and
its frontal area, and then pick a speed to evaluate it at.  To get the
frontal area, I got various portraits of rex shown from either a front or
rear view, traced the outline, and figured the area for a figure height of
12 feet, and the largest frontal area I got was 36 square feet.  I'm going
to arbitrarily pick 15 miles per hour for a speed even though there seems to
be a wide divergence of opinion on the max speed of rex.  This one seems
fairly reasonable, however, and I'll include the formula at the end so you
can plug in your own pet speed.  For help with the aerodynamics, I pestered
James Cunningham, whose knowledge in this area far surpasses my own.
(Thanks Jim.)  He was of the opinion that the coefficient of drag for rex
was likely in the mid- 0.4's and almost certainly no higher than 0.5, so
let's say rex had a coefficient of drag of 0.6 just to be sure we aren't
underestimating the strength of this effect.  Plugging these numbers into
the formula he supplied yielded a drag force of almost 25 pounds.
    To know the amount of sideways displacement, we need to know how far the
rex was leaned over, and the approximate center of drag.  The faster the rex
could have made a turn, the more it could lean over in the turn.  Of course,
how fast the rex could turn is what is under consideration here, but let's
set that aside for now and start from the assumption rex could turn quickly
and give it a turning radius of 50 feet at this speed.  That means an inward
acceleration of .3 G's meaning a lean angle of 17 degrees off vertical.  I
previously estimated the running center of gravity at five feet up, and
since the legs would have added drag out of proportion to their mass, the
center of drag should be no higher than this; but the higher the center of
drag is, the stronger this effect is, so let's say it was six feet up.  At
17 degrees off vertical, the horizontal displacement would be 1.75 feet (21
in.).
    So 25 pounds drag at 1.75 feet displacement gives us 43.75 foot pounds
torque with which to rotate the rex body.  If we assume a seven ton rex,
that would mean .003 G's acting on a one-foot lever.  To translate this to
the beam and spool analogy I used last time, the spools on the 30-foot beam
would have to be .075 inches in diameter (about the thickness of a nickel).
This is, of course, an optimistic figure as to the strength of this effect,
and the reality was probably much less.  I would also remind everyone that
this effect only contributes to rotation and does nothing to halt rotation,
but seeing how tiny this effect is, there hardly seems any point.
    I was originally going to figure how much the tail could have
contributed to aero-rotation by sticking out sideways, but even if it
doubled the drag and moved the center of drag three feet to the side (both
values way too high) that would still only bump the spool sizes up to .4
inches in diameter, which would still yield a glacial turn rate.  (Nevermind
that if the rex stuck its tail out sideways, the front end and the tail
would no longer counterbalance each other and the rex would fall over.)

   Incidentally, in case any of you were wondering how much the rex had to
lean forward at 15 mph to offset 25 pounds air drag, that works out to about
one hundredth of a degree for a seven ton rex--or less than two-tenths of an
inch at the hips.

Foot-sweep:
    In the original post, I mentioned two-foot torque, which at full stride
could only have occurred when both feet were loaded.  I mentioned how this
would have effectively increased the length of the contact patch with the
ground, giving the rex considerably more leverage with which to rotate its
body.  The only problem I saw was that the interval of time during which
both feet were firmly planted would have been very short at full stride.
Well Richard Keatinge has been playing guinea pig on himself, and has
brought to my attention another limiting factor I failed to consider.  When
the rex was standing, it could have employed the big muscles in its legs for
two-foot torque because the legs swing forward and backward to rotate in
this stance.  But when the legs are in the foremost or rearmost position,
the muscles which come into play for rotation are the ones responsible for
swinging the leg from side to side.  The muscles that pull the leg fore and
aft had attachment points well in front of and behind the hips, so they had
reasonably good mechanical advantage.  But from what I can tell (you rex
anatomists out there, feel free to jump in here) the muscles that pull the
leg out to the side wrapped close to the hip, so they were in a terrible
position for imparting side thrust and would have had very great mechanical
disadvantage when compared to the fore-aft thigh muscles.  (And I suspect
they were smaller muscles to begin with.)  So this would have been another
factor seriously limiting how much rex could have made use of two-foot
torque when striding at speed.  (Sorry about the sore muscles Richard.)
    Richard also points out that my list of factors isn't *quite* complete
if I neglect one-foot-sweep, and he is of course right.  That is where one
foot sweeps to one side while it is *near* the foremost or rearmost
position.  This would occur either just before or after the transition of
weight from the rear foot to the front.  If performed on the front foot
before load transition, the rex would be rotating its whole mass against the
inertia of its lower leg, so the effect here would have been extremely tiny
(and would have meant the foot was traveling sideways as it met the ground,
which is a good way to curl a toe under).  But the picture improves a bit if
the front foot is swept to the side after it is loaded, or if the rear foot
sweeps to the other side before it is unloaded.  The effect will still be
small, but the idea here is to look at all physically possible factors, and
this certainly qualifies.
    The reason it will be small is because of a factor I haven't had to
bring up before, and that is misdirection of energy.  This happens whenever
there is a path of lesser resistance for the energy you are putting into a
system to take.  (Any of you without posi-track who have ever had your car
get stuck because one wheel was on slick ice know how maddening this can
be.)  Consider the case of a 25 pound weight.  If you attach a rope to it,
lifting it is no big deal.  But if you run that rope over a pulley down to a
five pound weight and then try to lift the big weight by yanking up on the
pulley, you will find that almost all your effort will go into lifting the
little weight and hardly any of it into lifting the big one.  The reason
this applies to foot sweep has to do with the position of the foot.  When it
is in (almost) the foremost position, it might be four feet forward of the
center of gravity, so we have a four foot lever acting to rotate the rex
about its vertical axis.  But it is also five feet down from its center of
gravity, so you have a five foot lever acting to rotate the rex about its
longitudinal (fore-aft) axis.  The problem is not only the difference in
lever lengths, it is also in the different resistances to rotation.  There
is much more resistance to rotation about the vertical axis than the long
axis.  So how would these two have played out against each other?
    To know the resistance to rotation, we need to know the mass and the
radius of gyration (the radius at which all the mass could be concentrated
without affecting the moment of inertia).  For the sake of simplicity, let's
say the mass being rotated about each axis is the same, so we only have to
worry about comparing radii of gyration.  On a uniform body, depending on
the shape, the radius of gyration is usually between two-thirds and
three-quarters of the way out (because the momentum of a point increases as
the square of the distance from the axis of rotation, and we are averaging
for all points) but the rex was thicker in the middle, so let's chop it down
some and estimate the radius of gyration at 10 feet.  Around the long axis,
let's give it a generous two-foot radius of gyration (that would be for an
*average* diameter of 5.6 feet--and for most of its length it wasn't nearly
that).  We square the radius of gyration when we want inertia, so we can
find out the relative resistances about these two axes by dividing inertial
resistance by lever length.  So the proportion of resistance about the long
axis is 4/5 and the proportion about the vertical axis is 100/4.  (Meaning
rotation about the long axis offers less than one thirtieth the resistance
that rotation about the vertical axis does.)  Going back to the pulley and
weights, this would be like trying to lift a 25 pound weight with the pulley
when it is coupled to a .8 pound weight, which means 97 percent of the
energy would be misdirected to rotation about the long axis (see the end of
this post for an elaboration on this).
    However, although this method is three-percent efficient toward the very
front and rear of the stride, when the foot is closer to the center of
gravity, the lever length decreases, and when the foot is alongside the
center of gravity, sweeping the foot from side to side has zero effect on
rotation about the vertical axis.  So if we average foot sweep over the
length of the stride, it drops to one and a half percent efficiency.
    Ignoring what I said before about the muscles for sweeping the foot
being smaller and not having as much mechanical advantage, if we assume the
foot could have matched the acceleration of that massive braking stride I
mentioned in the original overview and say it could have imparted a full .25
G's sideways acceleration to one side for the first half of the stride and
reversed it to the other side for the latter half, then how would this have
translated to the beam and spool setup?  Since the lever to the front
position was four feet instead of two feet, we need to multiply the spool
size by two, and then multiply by the efficiency, which is one and a half
percent.  So the adjusted size of the spools should have been under three
quarters of an inch in diameter.  Of course, there is no way the foot could
have sustained a quarter G acceleration for a full half stride because
almost all of that would have gone into rotation about the long axis and
there is no way the rex body could have rotated that far from side to side
about its long axis.
    To flog this completely to death, I should also point out that for
maximum effectiveness, the rex should use its suspended leg to resist
rotation about the long axis.  This would mean swinging the leg in the same
direction as the body is trying to roll.  When the loaded foot sweeps right
from the centerline of the rex, the suspended foot should sweep left.  This
would mean this strategy would work well for only one leg.  If the right
foot sweeps right in front, the left foot should swing left as it comes
forward and they would pass with greater than normal distance between them
(I've seen roller skaters use this leg motion for propulsion).  But if the
left foot sweeps right in front, the right foot should sweep left, and the
two would collide rather than pass.  One final minor, and mostly neutral,
point is that sweeping the foot to one side against the inertia of the body
will push the center of gravity off its path, so to compensate, the rex
would have to step a bit to the wrong side of the centerline, sweep to the
normal side at mid-stride, and back to the wrong side again at the end of
the stride.  So basically, this method involves large amounts of vigorous
energy, almost none of which goes to rotating the rex about its vertical
axis, and thus is highly improbable.  But at least we've covered it now.


Tail action:
I'm not sure what to say here.  I mentioned in the original post that I
couldn't find the torque couple and I couldn't think how the tail could have
assisted in whole-body rotation, but failure to find an effect does not
constitute proof of the lack of existence of that effect, and I am still
getting messages intimating that the tail must have assisted somehow, so I
feel like I should at least make another effort.  I will give it two more
whacks.

>From typical use:
    Some theropods had stiffened tails.  From the standpoint of conservation
of energy (ie. reducing the amount of muscular energy required to hold a
tail out horizontally all the time) this was not strictly necessary.  Many
theropods managed just fine by having the spine tensioned to offset the
weight load without losing much side-to-side flexibility.  The main reason
for stiffening a tail so that it can't flex from side to side is if you want
to use it for a dynamic inertial stabilizer.  This is what a tightrope
walker uses a pole for.  If he starts to fall to one side, he can lever
against the pole's long moment of inertia (ie. resistance to rotation) to
push his center of gravity back to the correct side of the rope.  The longer
and stiffer the tail, the better theropods could lever themselves against
it.  (If the tail weren't stiffened, a massive pull to one side at the base
of the tail would either only pull the base of the tail over or would
require muscular effort along its entire length to keep it in line with the
base.)  Inertial stabilizers like a simple pole, or a stiffened tail, only
work in two axes.  In the case of the tightrope walker's pole, it resists
rotating the ends up and down or fore and aft, but the walker can easily
rotate the pole around its long axis.  Same for the stiffened theropod tail.
It resists swinging up and down and side to side, but can rotate easily
around its long axis.  Vertical tail rigidity could have come in handy for
head thrusts up or down, or could have assisted in a leap (both in adding to
take-off thrust and in pitching the body forward at the landing if the leap
fell short--or if the landing surface were steeply inclined.)  Side-to-side
rigidity would help with quick or violent head motions from side to side.
But here's the thing, a rigid tail would have made a particularly good
inertial stabilizer precisely because it resisted rotation about the
vertical axis (as well as lateral), so rather than being a help, the tail
had to have been a hindrance to making quick turns.  To generalize, I think
all theropod tails were used as dynamic stabilizers (in addition to
counterbalancing the front).  The stiffened ones were merely specialized for
particularly quick or violent motions.  I only mentioned the stiffened tails
as evidence that theropod tails did in fact function as dynamic stabilizers.
(I believe Ostrom has already covered this.)  Rex didn't need a stiffened
tail because its motions were not so quick, so muscular action along the
length of the tail--so that it could move as a unit--would have been
sufficient.  And if it is true rex occasioned woodlands, I can see how it
might be handy to have a tail that can bend if you want to be able to turn
around.

>From exhaustive elimination:
Swinging the tail from side to side only levers the body from side to side.
There will be no net rotation (because of conservation of angular momentum).

Raising and lowering the tail raises and lowers the head, but all action is
at right angles to rotation and thus would not contribute to it.

Recurving the tail shortens its moment of inertia and moves the center of
gravity forward.  To the degree recurving the tail makes rotation easier due
to the first effect, it compresses the stride length forward due to the
second effect.

Raising the tail very high has the same effect as recurving except that it
also increases resistance to rotation about the fore-aft axis.

Sweeping the tail around in a horizontal cone sweeps the head and trunk
around in a symmetrical cone, and generates a rotation about the fore-aft
axis which, if unresisted, will persist throughout the duration of this motion.

And those are all the basic motions I think the rex tail was capable of.
Using these as building blocks, the only combination of motions which I
could assemble into something which could have contributed anything at all
to rotation were swinging side-to-side and recurving--as I described in the
original overview post.  Other than that, the very best thing I can come up
with is that with the horizontal conical sweep, the torque generated might
have slowed the rate at which rex fell toward its other foot when employing
the massive braking stride strategy (and that is *really* reaching).



So the new revised list of factors in ascending order of importance is:

1) Tail action: extremely remote possibility that waving the tail in a
sideways come-hither could have contributed a tiny bit to rotation at a
large expenditure of energy.

2) Foot Sweep: Could have had a stronger effect than tail waving, but is at
least as unlikely.  See above.

3) Thrust vs. Air Drag:  A vanishingly tiny effect, but 100% probable that
rex used it.  (It had no choice.)  Contributes nothing to rotation braking.

4) Massive braking stride:  Strongest effect so far; much larger than Thrust
vs. Drag but still pretty damn small-- doesn't work once banking has
commenced, propels the body out of the turn, scrubs off lots of speed,
involves dangerous spraddle stride, high leg loads, and consumes a
tremendous amount of energy for very little gain.  Highly improbable.

5) Banking:  Another tiny effect, but bigger than Thrust vs. Drag and also
100% probable.  Contributes nothing to rotation braking.

6) Two-foot torque to ground:  Good leverage, but a very short interval in
which to accomplish it (impossible during a run); now downgraded as a
possible contributor due to the limited force the muscles could impart
sideways to the fore and aft leg positions.
   
7) One-foot torque to ground:  Still probably accounts for more than 95
percent of all rotation and rotation braking.  This is a bit aggravating for
me personally because I don't know how much power the rex could have put
into foot torque, which makes it a bit difficult to come up with actual
rotation rates.  I can say rex would have been slow to rotate; I just don't
have enough information to say exactly how slow.  Now, who can we go pester
to find out about rex leg strength?


Sorry I didn't get all this in last time, but it wouldn't have fit in one
message anyway.  There are probably still a few residual factors remaining
(I haven't looked into the Coriolis effect, for example) but they are
probably negligible and I think this hits all the major players.  If I can
get some figures on leg strength (and maybe soil compressibility), I'll try
to report back with real rotation rates and turning speeds.  In the
meantime, I think I'm going to go ponder how rex could have made good use of
its resistance to rotation about the lateral (side-to-side) axis to keep it
from falling forward.


This would be the missing half-cent to bring this up to my $.02 worth.
Nicholas


Addendum appendix

Air Drag:
The formula Jim supplied me for calculating drag in pounds force was:

Drag force = 0.0023769*(Coefficient of drag)*(Frontal area in square
feet)*(velocity in feet per second)^2

In the example I gave, coefficient of drag was 0.6; frontal area was 36
square feet; and 15 mph is 22 feet per second, so:

0.0023769 * 0.6 * 36 * 484 = 24.85



The Pulley problem (from the foot-sweep section):
If we relate resistance to rotation about the long and vertical axes, we get
the proportions of .8 about the long axis, to 25 about the vertical axis.
Imagine weights of these proportions floating in space, connected by a rope
which wraps around a pulley.  If you try to accelerate the weights by
pulling on the pulley, here's what happens.  The net resistance being acted
against is the combined mass of 25.8. and the magnitude of the imbalance
between the two masses is 24.2.  If the weights were equal in mass, every
unit you pulled the pulley would move the weights one unit.  But the
imbalance means that the small weight will move ahead by 24.2/25.8 units,
and the large weight will lag behind by the same proportion.
  So for every unit the pulley travels, the small weight will travel 1.94
units and the large weight will travel .06 units.  So this analogy shows why
three percent of the effect of foot-sweep would go to rotation about the
vertical axis, and 97 percent would be misdirected to rotation about the
fore-aft axis.




P.S.  RE: rex the ambush hunter; is there anything about rex to suggest its
enormous size and mass would have been particularly adaptive with respect to
either acceleration or concealment?  I think I don't understand this scenario.