FW: Centripetal Forces on a Horizontally Grazing Sauropoda

["Mike Milbocker" <mmilbocker@psdllc.com>]

Daniel Bensen wrote:

>A very interesting idea.  How much effort would it take to swing the
>head in this way as compared to pumping the heart?

Energy Considerations

The human heart pumps approximately 5 L/min to service 100 kgs. The
requirements for a reptilian heart would be less, so we treat this as an
upper bound. The average brachiosaurus weighed 50,000 kgs. This would
require moving 2500 L/min. Assuming a cycle time of 10 s, then 417 L/cycle.
The volume of a sphere is 4pr3/3, which gives a radius of r = 47 cm. For a
distributed system of 10 ventricles, each holding 41.7 L/cycle, the radius
of the ventricles would be 21 cm or 8.3 inches. For a mildly endothermic
animal, this radius could be considerably smaller.

The human heart generates 2 watts of mechanical power to pump 5 L/min of
blood. Scaling the power by the volume, this is a 1000 watts for
brachiosaurus.

Assuming the tail and head generate the same energy, and 1 cycle consists of
either a swing of the head or a swing of the tail, we calculate the energy
as follows. Using the carotid artery as a model, it's diameter is 1 cm with
a cross sectional area of 0.78 cm2. The brachiosaur, scaling cross section,
would be about 22 cm in diameter. Interesting this is only about 1/2 the
radius of the proposed ventricles.

So the mass of the blood column would be 390 cm2 X 1000 cm X 1 g/cc = 390
Kg. This is close to the alternative calculation of 417 L/cycle. The moment
of inertia

I = 1/3 ML2

Where M = 390 kg and L = 10 m gives I = 13,000

The kinetic energy is

K = 1/2 I v2 where v = v/L

v = 0.312 and K = 633 J or P = 63.3 Watts

This calculation assumes no flow until after the swing. With bladder we
model the system as 417 L in the head bladder and 390 L refilling the neck
arteries. Then

K= 1/2 I v2 + 1/2 mv2

Where I = 13,000, v = 0.312, m = 417 kg, v = 3.12 m/s

K = 633 + 2030 = 2663 J or 266 Watts

For a 5 second swing

I = 13,000, v = 0.624, m = 417 kg, v = 6.24 m/s

K =  2531 + 8119 = 10, 650 J or P = 2130 W.

This calculation is based on pure grazing, now how much energy is contribute
in a head raise.

Energy head Raise = potential energy of filled ventricle + potential energy
of venous side

P = mgh =       417 kg X 9.8 m/s2 X 10 m + 390 kg X 0.5 X 9.8 m/s2 X 10 m =
40,866 J + 19,110 = 59,976 J

This would create 1000 watts of power if the head were raised once every
minute.

-----Original Message-----
From: Mike Milbocker [mailto:mmilbocker@psdllc.com]
Sent: Tuesday, April 13, 2004 1:50 PM
To: 'Daniel Bensen'
Subject: RE: Centripetal Forces on a Horizontally Grazing Sauropoda


Considerably more effort, but much of this effort is mandated already by
eating and preservation behavior. There are other considerations, a heart is
a pressure source and that pressure drops the further the blood travels from
the heart. In order to pump to the distal extremes, the heart pressure would
need to be quite large. I believe this is the core of many arguments
suggesting sauropods did not raise their heads. In the case of centripetal
force blood pumping, the pressure of the blood need not exceed ambient since
every element of blood is "pushed" independently through the centripetal
force. The power required by a heart to generate these high pressures would
probably be in the range of about 1000 watts.

-----Original Message-----
From: Daniel Bensen [mailto:dbensen@bowdoin.edu]
Sent: Tuesday, April 13, 2004 12:43 PM
To: mmilbocker@psdllc.com
Subject: RE: Centripetal Forces on a Horizontally Grazing Sauropoda


A very interesting idea.  How much effort would it take to swing the
head in this way as compared to pumping the heart?

Dan

-----Original Message-----
From: owner-dinosaur@usc.edu [mailto:owner-dinosaur@usc.edu] On Behalf
Of Mike Milbocker
Sent: Tuesday, April 13, 2004 11:33 AM
To: dinosaur@usc.edu
Subject: Centripetal Forces on a Horizontally Grazing Sauropoda

Centripetal Forces on a Horizontally Grazing Sauropoda

It has been presented that sauropods kept their head and neck coplanar
with
the body and tail, and swung the neck, out-stretched in a circular path,
called the browse plane, centered on the shoulder region.

Assuming the neck length to be L, and the head makes one 180 degree
transit
in time t then the force, F, that would push blood toward the head is
given
by

F = mar = m V2/L   where V = pL/t

To give

F = mp2L2/t2

And

ar = p2L2/t2

Let L= 10m, then ar = 973/t2

For ar to equal the acceleration of gravity (ar = 9.8 m/s2)

t = 10 seconds

This is not an unreasonably short period of time if the head were to be
involved in warning the sauropod of danger.

If the sauropod were raising its head in a circular trajectory
perpendicular
to the browsing plane, this would be the maximum time allowed in order
for
gravity not to impede blood flow, i.e., at this rise rate the blood
would be
"weightless".

For faster motion, the head could actively fill with blood while rising.
The
table below gives the g's of force at various rise times:

Rise Time               g's of force

10 s                    0.99
9 s                     1.23
8 s                     1.55
7 s                     2.03
6 s                     2.76
5 s                     3.97
4 s                     6.21

Thus, the rapid motion of the head in a circular arc can create
sufficient
blood pumping forces.

If the head and tail were swept asynchronously, the animal could create
an
efficient blood pumping system, independent of a heart. Combining
feeding
and defensive strategies with blood pumping may have allowed this class
of
animals to achieve their great size.