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# Some comments about Tyrannosaurus rex speed.

`I tried to do an exercise and get an upper limit of the speed range of
``Tyrannosaurus rex using published data. After analyzing the literature on
``the speed of this dinosaur I put the more favorable to fast running T. rex
``values on the parameters of the equation 1 of Hutchinson (2004a) to
``calculate the highest possible G (the ratio of Ground Reaction Force to body
``weight) and therefore the highest possible speed.
`
mi=(100·G·g·R·L·d)/(cos p·FPUA·c·r)
See below for significances.

`Most of published data are about MOR 555, so I used this T. rex to begin.
``First I assumed that ankle extensor muscle masses are the critical limit on
``running capacity (Hutchinson et al. 2011).
`

`I took from Gatesy et al. (2009) the highest result of G (1.87), with an
``ankle extensor muscle mass (mi) of 5% of body mass. This implies the best
``limb posture for fast running I found published. Other results can be found
``in Hutchinson and Garcia (2002) and Hutchinson (2004b).
`

`I took two values of muscle parameters within the range of the sensitivity
``analysis did in Bates et al. (2010). These were Force per unit of muscle
``area (FPUA) of 400000 Nm-2 (instead of 300000 Nm-2), cited in literature and
``muscle fascicle length (L) 15% lesser.
`

`Finally, for getting the largest possible ankle extensor muscle mass I
``decided to use the maximal estimation of shank mass with the minimal
``estimated masses of the other parts of MOR 555 from Hutchinson et al.
``(2011). Body mass 6553 kg, ankle extensor muscle mass relative to body mass
``(mi) 4.91%.
`

`Then I calculated the maximum G using these values. I got, for a 6553 kg T.
``rex, with a mi of 4.91%, a maximum G of 2.89. This implies a Duty Factor
``(Df) of about 0.27, a Froude number (Fr) of about 10.88 and a speed of about
``16.34 ms-1 (36,47 mph, 58.82 kmph). G of 2.89 is over the value of 2.5,
``considered the limit for a high-speed running T. rex (Hutchinson and Garcia
``2002). This maximum theoretical speed is consistent with that of Paul
``(1998), but it still below 20 ms-1 and it is only the upper limit of the
``possible range of maximum speed and should be considered that actual maximum
``speed may well be lower.
`

`I acknowledge that I have taken the parameters to their maximum limits and
``that it is possible that the actual values were lower and therefore the
``speed were lower, but I think the fast-running hypothesis can not be ruled
``out in a six ton T. rex (see below for a heavier one). Besides these
``calculations must be validated using the models and taking into account COM
``position and should be considered preliminary. However, one or more of these
``high values must be accurate because if we apply none of them and use the
``ankle extensor muscle values from Hutchinson et al. (2011) we get a value of
``G lesser than 1 and this would imply that T. rex could not even walk.
`

`There are much less published information about FMNH PR 2081 (Sue) so I
``assumed that the ankle effective mechanical advantage (EMA=r/R) was the same
``as MOR 555 and adjusted ankle extensor muscle fascicle length (L)
``proportional to limb length. It was a 9-10 tons T. rex with only slightly
``longer limbs than MOR. Equation 1 is Body mass independent but greater body
``mass and nearly equal limbs implies a lower mi (that is a percentage of body
``mass) and therefore reduced G and speed.
`

`In fact, the maximum mi I got for Sue using the maximal estimated shank mass
``with the minimal estimated masses of the other parts from Hutchinson et al.
``(2011) was 3.58% of a body mass of 10346 kg. This was below the mi of about
``5% that I calculated to get a G of 2.5 and so this did not support a
``fast-running ability. Applying the maximum values to the equation
``parameters, with a mi of 3.58%, I obtained a speed about 8.41 ms-1 (18.77
``mph,30.28 kmph)[in the middle of the range published in Hutchinson et al.
``2011(5-11 ms-1), but still a running gate (Hutchinson and Garcia 2002)] and
``not applying these values I obtained G lesser than 1 (no walking ability)
``again.
`
Conclusions.
Assuming these body masses of MOR and Sue are correct

`1- Fast-running hypothesis can not be ruled out in a 6 ton tyrannosaur, but
``should be discarded in a 10 ton T. rex, taking into account the data we have
``(of course, it must be said that data do not rule out a slower maximum speed
``in MOR either).
`

`2- Results show there could be two T. rex morphotypes differentiated by the
``ability of locomotion, despite that the difference may be due to ontogeny.
``This may have paleobiological implications that could be discussed
``elsewhere.
`

`3-This model seems to be very good to compare locomotion abilities of
``extinct animals. It could be used not only in intraspecific but also
``interspecific comparisons (i. e. predator-prey or two different predators).
`

`4- To reconstruct very large tyrannosaur locomotion may be necessary to use
``high values (within the range known in extant animals) in one or more of the
``parameters of the model. Using only conservative values can lead to results
``such as T. rex could not walk.
`

`I want to say I only changed some values in the model of of John Hutchinson
``and collaborators and commented the results. I think that they have created
``a solid quantitative method to estimate and compare extinct animal
``locomotion with their papers. T. rex speed is just one outcome among all
``that their work have and not the most important.
`
Thanks and excuse me for my bad English.
Manuel Garrido.
Madrid, Spain.
Methods:
mi=(100·G·g·R·L·d)/(cos p·FPUA·c·r)

`mi is the extensor muscle mass, expressed as percentage of body mass,G is
``the ratio of Ground Reaction Force to body weight, g is the acceleration due
``to gravity, R is the moment arm of the Ground Reaction Force, L is the mean
``muscle fascicle length, d is the muscle density, cos p is the cosine of the
``mean angle of muscle fascicle pennation, FPUA is the Force per unit of
``muscle area,c is the fraction of maximum exertion by the muscles and r is
``the mean moment arm of the muscles.
`
To estimate speed from G, I used these equations:
To get the Duty Factor we can solve from Alexander et al. (1979).
Df=Pi/(4*G)

`To get the Froude Number and speed we solve from equations from Alexander
``and Jayes (1983).
`
Fr=(Df/0,53)^(-1/0,28)
s=(Fr*g*hh)^0,5

`hh is hip height. I used 2.5 m for MOR (Hutchinson and Garcia 2002) and
``estimated it proportional to limb length for Sue (2,7 m).
`

`Note these two papers are about mammal locomotion. It should be taken into
``account and speed values should be considered only indicative.
`
References:

`Hutchinson JR (2004a) Biomechanical modeling and sensitivity analysis of
``bipedal running ability. I. Extant taxa. Journal of Morphology 262: 421-440.
`

`Hutchinson JR, Bates KT, Molnar J, Allen V, Makovicky PJ (2011) A
``Computational Analysis of Limb and Body Dimensions in Tyrannosaurus rex with
``Implications for Locomotion, Ontogeny, and Growth. PLoS ONE 6(10):
``e26037.doi:10.1371/journal.pone.0026037
`

`Gatesy SM, Baeker M, Hutchinson JR (2009) Constraint-based exclusion of limb
``poses for reconstructing theropod dinosaur locomotion. Journal of Vertebrate
``Paleontology 29: 535-544.
`

`Hutchinson JR, Garcia M (2002) Tyrannosaurus was not a fast runner. Nature
``415: 1018-1021.
`

`Hutchinson JR (2004b) Biomechanical modeling and sensitivity analysis of
``bipedal running ability. I. Extant taxa. Journal of Morphology 262: 421-440.
`

`Bates, Karl T. , Manning, Phillip L. , Margetts, Lee and Sellers, William
``I.(2010) Sensitivity Analysis in Evolutionary Robotic Simulations of Bipedal
``Dinosaur Running, Journal of Vertebrate Paleontology, 30: 2, 458 - 466.
`

`Paul GS (1998) Limb design, function and running performance in
``ostrichmimics and tyrannosaurs. Gaia 15: 257-270.
`

`Alexander, R. M., Maloiy, G. M. O., Hunter, B., Jayes, A. S. and Nturibi, J.
``(1979), Mechanical stresses in fast locomotion of buffalo (Syncerus caffer)
``and elephant (Loxodonta africana). Journal of Zoology, 189: 135-144.
`

`Alexander, R. M. and Jayes, A. S. (1983), A dynamic similarity hypothesis
``for the gaits of quadrupedal mammals. Journal of Zoology, 201: 135-152.
``
`