Re: Rex cornering overview (from T. rex mechanix)

["David Krentz" <david_krentz@fa.disney.com>]

Reply to:RE>Rex cornering overview (from T. rex mechanix)    10:23 AM
4/23/98

>I'm pretty sure there is very little the tail
could have done to contribute to rotating the total body mass in a turn.
Perhaps waving the tail in a sideways come-hither motion (eg. extend, swipe
left, retract, swipe right, repeat) could have contributed a tiny amount to
rotation as it would have had the effect of rotating the tip of the tail
about a vertical axis (and this rotation would generate a counter-rotation
in the body for as long as this motion continued); but the mass at the tip
of the tail would have been quite small compared to the overall mass of a
rex and the rotation speed rather low, so I think we can safely neglect this
strategy.<

  This makes a lot of sense, but if I'm understanding this correctly, are you
assuming that the creature is using the tip of it's tail to turn?  The
caudi-femoralis, or the MAJOR leg retracting muscle, runs from the great
trochantor on the femur and attaches all along the base of the tail( roughly
the first quarter or fifth of the tail length).  The rex could flex this
giant muscle, which added considerable weight just behind the centre of
gravity, at the push off part of it's stride.  The just planted foot could
perhaps twist in the direction that the tail causes it too.  I tried this
little test just now to see if I have a point:  I balanced on my right foot
and tipped forward with my body and left leg held horizontal.  I at first
threw my left foot( held out behind me) out towards the left.  eventually the
inertia turned me to the right, but not quick enough if I was a rex and need
that energy  to turn right away.  Then, I assumed the same pose and kicked
out my left leg to the left useing the muscles at my hips and my centre of
balance.  This caused me to pivot to right rather quickly.  So, what I'm
saying, is that if the rex used the big muscles at the base of it's
tail,rather than using the tip , it could turned quicker.  As the rest of the
tail followed through and down, it would continue the inertia for another (
decelerated ) stride and tilt the body upward.  In T-rex, I understand that
the base of the tail was more attune to lateral flexion, and the remaining
half of the tail was more up and down.
  I remember you mentioning to me in offlist discussions that the cheetah's
tail really didn't help it turn much, but the cheetah's tail certainly does
not have the same mass or muscle as the rex.
  Also, I realize that lateral flexion of a rex's dorsal spine was somewhat
possible, could this have aided our toothed friend in turning?

  Speedily writing
David Krentz
--------------------------------------
Date: 4/22/98 10:58 PM
To: David Krentz
From: ngear@gvtc.com
This is a summary of factors, discussed on and off list, proposed by me and
others, as having possibly contributed to help a running or fast-striding T.
rex (or similarly large theropod) negotiate a corner.  It is very long,
involved, and I couldn't avoid the math altogether.

    For the few of you still reading; to recap, the problem is that to make
a turn, an animal has to rotate its body to point in the new direction of
travel, and then halt that rotation at the conclusion of the turn.  The
problem is trivial for quadrupeds which have legs well fore and aft of their
centers of gravity (giving them ample leverage), and for bipeds which have a
compact or vertical body plan (and thus a low moment of inertia).  But for
theropods generally this would not have been so trivial because they had
long, horizontal bodies which would resist rotation, and their feet on
average were pretty much under their centers of gravity, giving them little
leverage with which to rotate.  And the story gets much worse for rex, and
similarly large theropods, because this resistance to rotation goes up,
roughly, by the square of the length.  I was of the opinion this would have
been a serious impediment to T. rex making quick turns.  Many disagreed.

So, in roughly ascending order of importance, here are the various
mechanisms which have been proposed to help a rex rotate, and my thoughts on
each.

1) Some action by the tail:  Proposals along these lines have been vague,
and I think with good reason.  I'm pretty sure there is very little the tail
could have done to contribute to rotating the total body mass in a turn.
Perhaps waving the tail in a sideways come-hither motion (eg. extend, swipe
left, retract, swipe right, repeat) could have contributed a tiny amount to
rotation as it would have had the effect of rotating the tip of the tail
about a vertical axis (and this rotation would generate a counter-rotation
in the body for as long as this motion continued); but the mass at the tip
of the tail would have been quite small compared to the overall mass of a
rex and the rotation speed rather low, so I think we can safely neglect this
strategy.
    About the only contribution I can imagine the tail could have made would
have been to delay the torque problem a tiny bit.  Swinging the tail inward
at the beginning of the turn might have allowed the pelvis to rotate a bit
sooner, and swinging the tail wide at the end of the turn might have allowed
the pelvis to stop rotating a bit sooner, but in both cases, conventional
torque would be needed to catch the tail up with the rest of the body.  The
problem here is that this only works if the tail has a large moment of
inertia, but if that is the case, then swinging the tail to one side would
have a large effect on moving the center of gravity off to that side, which
is not desirable, particularly at speed.  Admittedly, this could be largely
compensated for by swinging the head in the opposite direction (bringing
head, pelvis, and tail back into line), and this would also increase the
effectiveness of this delaying tactic by helping to rotate the pelvis, but
it would also mean that a big theropod trying to pursue prey through evasive
maneuvers would have go into a turn pointing its head out of the turn (and
away from its prey, presumably).  I think it more likely that even the large
theropods kept their heads pointed squarely at their targets during pursuit,
which means they would actually be pointing their heads inward going into a
turn; and to keep the center of gravity centered, that would require that
their tails swing *out* at this point, not in.

2) Wide-stepping:  If rex could have stepped to one side of the point
directly below its center of gravity, and then applied thrust or braking,
this would have contributed to rotating the body.  First point; a running or
fast-striding rex probably could not have added much thrust, so we are
mainly looking at braking here (and thus, necessarily, a loss of speed).
Second point; the strength of this effect has two variables: the braking
force and the amount of displacement to the side, and here are the
considerations for each.
    Variable One--Braking force.  Braking (and accelerating) is a much
bigger problem for bipeds than for quadrupeds because their center of
gravity is usually almost directly over their feet.  A quadruped can slam on
the brakes with almost no preparation because the front feet are always
placed well forward of the center of gravity.  There are basically two
things a biped can do to scrub off forward speed.  One is to stiffen the leg
when it is at the very front of the stride.  This will convert a little bit
of forward momentum into upward momentum.  This approach is only effective
to the degree it raises the center of gravity higher than would have
occurred in a normal stride, which usually isn't much.  This way of braking
does not take much energy, but it is also pretty gradual (you can see
runners stiff-leg right after they cross the finish line).  The other way is
to move the angle from the average point of ground contact up through the
center of gravity as far back as possible.  (Let's call this the braking
angle.)  There are two components to this strategy.  One is to move the
average ground contact point forward.  A theropod at speed would probably
have already been reaching forward about as far as it could with each
stride, so that only leaves truncating the latter portion of the stride to
move the average ground contact point forward.  The other is to hunker down
to lower the center of gravity (and thus the braking angle) as much as
possible (so this part of the strategy is incompatible with stiff-leg
deceleration).  This approach takes a lot more energy, but it will slow a
biped down faster.  Exactly how fast depends entirely on the braking angle.
This is one area where theropods had a distinct advantage over us vertical
bipeds.  Our center of gravity is above our pelvis, which has the effect of
giving us a more vertical braking angle, and we have to compensate by
leaning our torso back when braking.  Since most theropods carried their
center of gravity below their hips, their braking angles would automatically
be laid back further.  This also means that for maximum braking they should
do the opposite of what we do, and lean forward when decelerating to move
their center of gravity rearward with respect to their hips.  (As depicted
in David Krentz's Gorgosaurus sculpture.)  However, this is the strategy for
general deceleration, and what we are talking about here is deceleration on
one foot only, so there would not have been time to rotate the body into a
braking posture.  There is also the problem of having a truncated stride on
just one foot, meaning it will have a quicker cadence than the other side,
producing a limping run.  Even so, we might as well figure what sort of
braking force we could be talking about here.  Let's look at a forty-foot
long rex with an eight-foot leg and a standing center of gravity six feet
up.  Let's give it a generous reach forward with its leg planting at an
angle of thirty degrees back from vertical (meaning stride length is about
the same as leg length).  This would lower the hips--and thus the center of
gravity--by a foot.  The angle from the foot plant back to the center of
gravity would thus be about 40 degrees from vertical, but the value we need
is the angle down to the *average* point of contact.  Assuming uniform
speed, if we truncate a full third of the stride, the average braking angle
would be 15 degrees off vertical--meaning .26 G's deceleration.  (The speed
is, of course, not uniform because this animal is braking, but that merely
means that more time is spent in the latter portions of the stride, which
moves the average point of ground contact rearward and thus decreases the
braking force even further, but I thought this would be a small effect and
safe to neglect.)
    Variable Two--Sideways Displacement.  This is the choker.  In order to
rotate the body into the turn, the rex had to be able to place it's foot
well to the inside of the track directly below the path its center of
gravity would have taken through the turn.  But once in the turn, the rex
would have needed to bank over, meaning it had to place its feet to the
*outside* of this path.  So this strategy almost certainly could not have
worked once the turn had begun.  Could the rex have used this strategy to
initiate rotation at the very beginning of the turn?  Here's the problem.
When you step well to one side of your center of gravity, you fall toward
your other foot and change your overall direction of travel.  For the foot
to have the greatest torque for rotating the body, it needs to be as far to
the side of the center of gravity as possible.  But the further to the side
it is, the faster the body falls, the less time there is to apply rotational
torque to the body, and the greater the change in the direction of momentum
*away* from the very direction you are trying to go.  To counter this, the
other foot has to land even *further* to the outside of the path the center
of gravity would take, not only to arrest the outward momentum imparted by
the braking foot, but also to get the body headed in the right direction for
the turn.  So the rex would have to do a very spraddle-legged step to employ
this strategy for even one stride.  A very dangerous and acrobatic move.
Even so, having come this far, we might as well see if one massive braking
stride could have imparted lots of rotation.
    Let's take an extreme case.  If a rex tried to step two feet wide of its
center of gravity it would experience more than a .4 G acceleration toward
the other foot, and that acceleration would increase as the rex fell over.
(That low center of gravity which was an asset for braking becomes a
liability here.)  In less than one third of a second, it would have picked
up three miles per hour sideways velocity and displaced itself two feet to
the side.  To step two feet further to the side than this would make for a
total spraddle width of six feet.  Better chop the braking time down to one
sixth of a second for a spraddle of five feet.  (We do, after all, have to
save something for the forward reach of the leg.)  Pushing through all these
problems, we now have .26 G's deceleration acting to rotate a 40 foot torso
at a distance of two feet from its center of gravity.  The math gets really
ugly here, so I'm going to try to do this by analogy.  Picture two long
cables hanging from, well, anything.  At the end of each cable is a spool
and each spool is affixed to either side of a beam.  The cable winds off the
same side of each spool, so under the force of gravity, the weight of the
beam unwinds the spools causing the beam to rotate end over end as it
descends.  What is happening here is that gravitational acceleration is
acting at the distance of the radius of the spool from the center of gravity
to rotate the beam.  This is an analog of the rex, where the beam is the
torso, the radius of the spool represents the displacement of the foot to
the side of the center of gravity, and gravitational acceleration takes the
place of braking deceleration.  How long is the beam?  The rex was 40 feet
long, but it was heavier in the middle, so lets shorten our uniform beam to
30 feet.  How big are the spools?  Well in our original model the lever
length was two feet, but there was also only a quarter-G braking force, but
this beam is hanging in one full G so we reduce the spool sizes by three
quarters to compensate, giving us a radius of six inches.  So now we have a
thirty foot beam suspended between two one-foot diameter spools.  We release
the beam and let it start descending and rotating--but only for one/sixth of
a second, remember.  The rotational speed of the beam at the end of this
interval would be an outside figure for how much a single massive offset
braking stride could have started the rex body rotating.  Considering the
dangerous maneuver this requires, the redirecting of momentum away from the
turn, as well as the loss of about 1 m.p.h., I think the likelihood the rex
used this strategy is approximately nil.


3) Banking:  It has been suggested that the act of banking around a turn
(ie. leaning toward the center of a turn) will automatically re-orient the
body in the direction of the turn.  This is actually true, but only to a
small degree.  What provides the rotation is the changing orientation of the
"vertical" axis (vertical relative to the rex, that is).  Any object can by
similarly rotated by a changing axis, whether that change is imparted from
below (as in the case of the rex) or above (as in the case of a pendulum
weight swung in a circular arc) or both (as in the examples I'm going to
use).  Rotation due to change of axis is unaffected by direction of travel,
so we can isolate this effect with a model that is not traveling.  The
cleanest demonstration of this effect I can think of is the wobble rotor.
    I'm going to digress a bit to describe a wobble rotor now.  I'll be
going into more detail than is really apt for this piece because of the
number of people who sincerely thought banking could have made a significant
contribution to rotation (so skip ahead if you know how a wobble rotor
works).
    Picture a vertical powered shaft centered in a circular room.  Midway up
that shaft is a dogleg that is canted some degrees off vertical (say,
something like 20 or 30 degrees, just for the sake of picturing it).  On
that dogleg is mounted a flywheel on a bearing.  When the shaft rotates, the
bearing prevents the shaft from imparting torque directly to the flywheel,
so it is only by virtue of the wobble action that the flywheel will turn.
To see why it turns, imagine there is a laser shining out of the equator of
the flywheel onto the surrounding wall and we can track the course of that
spot of light.  When we start the shaft rotating, the point of light will
start bobbing up and down and begin moving in the same direction as the
shaft is rotating.  If we eliminated any cumulative rotation with some
restraining friction, the laser point would describe a sine wave, and the
maximum slope, both rising and falling, would be at the same angle as the
dogleg.  But without the restraining friction, each point on the flywheel is
free to "surf" the sine wave, and rotation will start to accumulate.  As the
thrust of one cycle carries forward into the next, the dot begins moving
progressively faster around the room.  The faster it goes around the room,
the more it approaches the speed of the spindle, and the slower it bobs up
and down.  Since it is the bobbing action which imparts rotation to the
flywheel, the slower it bobs, the less power the spindle can put into the
flywheel and the slower the acceleration.  (In a friction-free wobble rotor,
the rotor would continue to accelerate indefinitely, and yet never reach the
spindle speed.)  You can demonstrate the wobble rotor effect to yourself
with a bicycle wheel (preferably one with good bearings).  Grasp both ends
of the axle and hold the axle slightly off vertical.  Rotate the top and
bottom ends in a horizontal circular motion (the same direction for each,
but with one end 180 degrees out of phase with the other).  After several
rotations, you will see the wheel starting to rotate, and it will gradually
pick up speed (until air and bearing resistance won't let it go any faster).

    Now, how does this apply to T. rex?  Let's say it wanted to make a right
angle turn, and was banked over at 20 degrees.  To see how much rotation the
banking effect would have produced, take your bicycle wheel axle, cant it 20
degrees off vertical, and rotate the ends one quarter of a horizontal
circle.  You can see for yourself that the amount of rotation would have
been pitifully small.  One last point is that this effect can only
contribute to imparting rotation, and does nothing to help arrest rotation
at the conclusion of a turn.

4) Two-foot torque to ground:  With two feet firmly in contact with the
ground, a rex would have effectively increased the length of its contact
patch with the ground and would have had considerably more leverage for
rotating its body.  Thus, rotation when standing or moving slowly was
probably not a huge problem (though still a problem).  But a fast-striding
rex would have had both feet firmly in contact with the ground for only a
brief instant out of each stride, and if rex could run, it could not use
this strategy at all when running.  Despite the advantage of extra leverage
this strategy had, I think its effectiveness would have been seriously
curtailed by having to be compressed into a very short interval of time.  It
would be like trying to rotate the body with intermittent hammer blows
(transmitted through the pelvis).
    I hesitate to even mention it, but there is one way a biped can have
both feet in contact with the ground for an extended period of time, even at
a good rate of speed--and that is to hop.  It is almost unthinkable to me
that a rex could have done this on any occasion, but I mention this because
if any smaller theropod could get in a few hops in the corners, that would
have given them another maneuverability advantage over rex.  I don't know of
anything about the leg construction of any theropod that suggests any of
them did this, but I don't know of any evidence that definitely rules out
the possibility that the smaller theropods might occasionally have hopped
(particularly the ones with rigid tails--since that would have been handy
for conservation of energy).  My bet, though, is that even IF any smaller
theropod could hop, it would have served a function like pronking or maybe
some courtship display rather than a maneuvering strategy for evasion or
pursuit.


5) One-foot torque to ground:  Simply put, this strategy is just twisting
the foot that is in contact with the ground in order to rotate the body the
other direction, and I think it had to have been the overwhelmingly dominant
source of the torque needed to rotate the rex body in a turn.  This strategy
could be applied almost continuously, it would not have introduced
destabilizing side thrusts or increased the likelihood of tripping, it would
not have required a reduction of speed, and almost all the energy it
consumes is applied directly to rotation.  The only problem it has is that
the rex foot was thirty-something inches across, which makes for a very
small torque platform against which to rotate a forty-foot long body.
Without knowing how strong the rex was, it is tough to speculate what the
maximum rotation was, but the physics suggest the big theropods like rex
would have been among the slowest turning bipeds of all time.

Implications:  From what I can tell, the rex could have been easily
outmaneuvered by any quadruped which could match its top speed.  The rex
also could not have turned as fast as any smaller theropod which could match
its top speed--even if the smaller theropod had the exact same proportions
(and I suspect many smaller theropods were actually faster than rex).  In
fact, I am hard-pressed to think of any animal that would have been slower
in the turns for its speed than rex and any similarly sized theropods.
    I have a feeling that some people will see this as further evidence for
the rex-as-obligate-scavenger view, but I still think it is premature to go
to that interpretation.  I think what this tells us is that rex specialized
in ponderous prey, and that its attacks occurred at low speeds.

My $.015 worth.

Nicholas Wren