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RE: supersonic sauropods
Yes, we modeled both with and without damping. It did not make any
qualititive difference in the results.
Modeling the damping is fairly easy, because the software package we used
has dashpots as a feature - you just drag and drop the dashpot icon and then
set the parameters. Since we don't have good knowledge of the details of
sauropod tail musclulature, I tried a wide variety of different damping
parameters. In addition, we modeled air resistance and joint friction too.
None of this changed the basic result.
The fundamental mechanism for transferring torque from the base of the tail
to the tip is momentum transfer. This depends on the vertebrae being
linked. Damping will only effect it if the action of pulling on a joint
between vertebrae (extension), or bending of the joint is greatly impeded by
muscles, and damped into heat.
There are two ways to model the tail. The approach that we took is to
assume that the primary motive power is confined to the base of the tail
(where the caudal ribs are), and thus the tail is a passive object. This
seems reasonable because of the size of muscle attachment points,
particularly in the double convex "whiplash" caudal vertebrae which make up
over half of the tail.
So, in this passive approach, we assume that any muscles which attach to the
joints are fairly small and weak. So damping does not matter, because these
small mass muscles are not going to damp enough to make this a problem.
Alternatively, you can assume that Apatosaurus had considerable musculature
all the way to the tip. In that case, there could be a lot of damping.
But, in that case these strong muscles could help with the cracking action!
What would the point of strong muscles which could only damp, but not assist
Again, simulations confirm this - a wide range of parameters produced
qualitatively similar results - including supersonic motion. Once you have
an exponential decrease in mass, with the extreme ratio found in
Apatosaurus, and you have highly flexible joints, you have the basis for
There are several other ways to see that damping would not be an important
In a dashpot model, the damping is proportional to velocity - in this case
the rotation velocity of joint flexion. However, most of the tail is very
slow! In the paper there is a graph of the maximum velocity achieved by
each vertebrae. It drops exponentially as you move away from the tip. The
typical period of rotation between maximum extension on one side to the
other for a joint is on the order of 1 to 2 seconds, which is a very
reasonable speed. We don't need very fast joint action for the effect to
Yet another way to look at this is to consider the range of joint action.
Strongly dissapative joints would have a dynamic effect similar to
restricting the range of motion (and also consuming more energy). In the
simulations I tried a range of joint constraints that was very large - all
the way down to just 2 degrees motion per joint at the base of the tail, and
only 15 degrees for the whiplash caudals at the tip. This range is very
conservative when compared to other measures. At worst, a highly
dissipative tail would be like a tail with limited joint motion, which also
took more energy due to damping. But the energy calculations show that our
assumptions are very conservative. If it took twice the torque or twice the
energy, it would still be reasonable. Even a factor of 4 in torque and
energy is not out of the question, when you compare the task of tail
cracking to things we know that sauropods did - like walking.
This said, I want to emphasize that I cannot PROVE that supersonic tail
motion. The goal is only to establish a feasibility argument that it could
have happened. That's hardly surprising because this is a behavorial
feature and behaviors don't fossilize very well!
Until we get more direct fossil evidence, this is just an interesting
conjecture. Obviously, I am in favor of it, but it is important not to get
too carried away with claims.
More generally, I think that computer simulation can offer us some
interesting new ways to check ideas about dinosaurs in a more rigourous
manner than you can with purely verbal arguments.
> -----Original Message-----
> From: John R. Hutchinson [SMTP:firstname.lastname@example.org]
> Sent: Monday, December 15, 1997 10:49 AM
> To: email@example.com
> Subject: supersonic sauropods
> Just got a copy of Mhyrvold and Currie 1997, Paleobiology 23(4): 393-409,
> "Supersonic sauropods? Tail dynamics in the diplodocids." An interesting
> A question for the authors if they're out there: How did you incorporate
> damping into your model? I don't see it anywhere. Bullwhips aside,
> muscle-tendon contractile units today are generally modeled using
> "spring-dashpot" theory as an abstraction (most basic muscle textbooks and
> physiology courses will discuss this). I think I can see the spring part,
> but was there any damping in the model?
> This really matters, IMHO. Applying 20-50 thousand N-m torque at the base
> of the tail doesn't mean it's transmitted 100% to the tip (or am I wrong?)
> -- damping due to energy dissipation in cartilages and other soft tissues
> will take its toll. Or maybe I'm missing something here, or have
> misremembered my comparative physiology classwork. I'll reread the paper
> but I couldn't find any discussion of this assumption in it.
> I thought I'd ask, with all due respect to the authors of course. It's
> always nice to see people testing functional hypotheses using biomechanics
> rather than telling functional stories using gestalt impressions of the
> inner workings of biology.
> John R. Hutchinson
> Department of Integrative Biology
> 3060 Valley Life Sciences Bldg.
> University of California - Berkeley
> Berkeley, CA 94720 - 3140
> Phone: (510) 643-2109
> Fax: (510) 642-1822