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TABLE 1. Centers of mass of neck segments with varying tracheal diameters, and a head with a uniform assumed density and with air spaces. The dimensions are relative to the centers of the model's coordinate system. Theoretical models with no air spaces are designated Solid.

  Solid Tracheal diameter
6.768 cm 4 cm
Neck x y z x y z x Y z
C9 9.239 507.007 -532.839 9.243 506.981 -524.370 9.241 506.998 -529.694
C8 9.484 611.316 -476.928 9.446 611.329 -467.692 9.472 611.320 -473.537
C7 9.375 708.540 -420.619 9.372 708.534 -411.045 9.374 708.538 -417.114
C6 8.743 792.622 -377.677 8.720 792.606 -367.916 8.735 792.616 -374.124
C5 8.161 872.473 -351.192 8.163 872.476 -341.086 8.161 872.474 -347.503
C4 8.998 948.314 -337.605 9.048 948.316 -327.250 9.014 948.315 -333.842
C3 9.515 1025.218 -330.166 9.513 1025.215 -326.310 9.513 1025.215 -326.310
C2-1 11.729 1103.845 -328.743 11.520 1103.238 -314.045 11.655 1103.614 -323.328
Solid Air spaces
 Head x y z x   y z      
Head  11.040 1372.946 -344.703 11.100 1356.158 -347.813

TABLE 2. Inertial properties of neck divisions and the head of Allosaurus, estimated assuming no air spaces and the density αtissue=1060 kg/m^3 (that of muscle). For neck segments, Ixx is about a vertical x axis and primarily resists lateroflexion (yaw). Iyy is about a longitudinal y axis and resists roll. Izz is about the transverse z axis and resists dorsoventral movements (pitch). Note that segment C2-1 tapers anteriorly and behaves differently than other segments. For the head, the x axis is horizontal and Ixx resists pitch; the y axis is vertical and Iyy resists lateroflexion; and the z axis is longitudinal and Izz resists roll.

Segment Vol
(m^
3)
Mass
(kg)
Mass moments of inertia (kg m^2)
I xx I yy I zz I xy I xz I yz
C9 0.006359 6.741 0.02926 0.07513 0.06812 -0.01654 -0.000007 0.000036
C8 0.006403 6.787 0.02927 0.07689 0.07158 -0.01796 -0.000002 -0.000005
C7 0.005565 5.899 0.02308 0.06947 0.06425 -0.01603 0.000156 -0.000048
C6 0.004675 4.955 0.01739 0.05716 0.05148 -0.01206 0.000161 0.000079
C5 0.004786 5.074 0.01764 0.05782 0.05268 -0.01232 -0.000027 -0.000007
C4 0.004177 4.428 0.01441 0.05228 0.04768 -0.01121 0.00001 0.000043
C3 0.004905 5.200 0.01700 0.06200 0.05800 -0.01300 -0.000015 -0.000200
C2-1 0.003422 3.627 0.00900 0.03900 0.03900 -0.00900 0.000037 0.000142
Head 0.026120 27.69 0.92420 0.34680 1.16300 0.22170 -0.000522 -0.001461

  TABLE 3. Volumes and inertial properties of the head of Allosaurus, calculated for varying tissue densities. As in Table 2 values for the head, the x axis is horizontal and Ixx resists pitch; the y axis is vertical and Iyy resists lateroflexion; and the z axis is longitudinal and Izz resists roll.

ρtissue
(kg/m^
3)
Vol
(m^
3)
Mass
(kg)
Mass moment of inertia (kg m^2)
I xx I yy I zz I xy I xz I yz
1128 0.022 25.16 0.8425 0.3490 1.0810 0.2127 -0.00051 -0.00153
1231 0.022 27.46 0.9197 0.3809 1.1801 0.2322 -0.00056 -0.00167
1321 0.022 29.48 0.9873 0.4089 1.2668 0.2492 -0.00060 -0.00179
1347 0.022 30.06 1.0066 0.4169 1.2916 0.2541 -0.00061 -0.00183
1373 0.022 30.64 1.0259 0.4249 1.3163 0.2590 -0.00063 -0.00186

TABLE 4. Inertial properties of Allosaurus neck segments, estimated with average tracheal diameters of 6.768 cm (from a regression for all birds), and 4 cm (regression for chickens and relatives; Hinds and Calder, 1971). Mass moments of inertia are as in Table 2 values for neck segments. Ixx is about a vertical x axis and primarily resists lateroflexion (yaw). Iyy is about a longitudinal y axis and resists roll. Izz is about the transverse z axis and resists dorsoventral movements (pitch).

Trachea
Diameter
(cm)

Segment Vol
(m^
3)

Mass
(kg)

Mass moment of inertia (kg m^2)

I xx

I yy

I zz

I xy

I xz

I yz

6.768 cm C9 0.00599 6.349 0.0282 0.0676 0.0598 -0.1451 -3.00E-06 3.40E-05
C8 0.00603 6.388 0.0281 0.0680 0.0618 -0.0155 -3.90E-05 5.00E-06
C7 0.00524 5.554 0.0221 0.0611 0.0551 -0.0137 1.54E-04 -4.70E-05
C6 0.00439 4.658 0.0170 0.0498 0.0438 -0.0107 1.43E-04 -7.60E-05
C5 0.00449 4.764 0.0176 0.0578 0.0527 -0.0123 2.70E-05 -7.00E-06
C4 0.00392 4.159 0.0137 0.0453 0.0401 -0.0094 4.40E-05 3.30E-05
C3 0.00480 5.088 0.0165 0.0532 0.0483 -0.0108 -1.90E-05 -1.80E-05
C2-1 0.00311 3.302 0.0080 0.0313 0.0310 -0.0070 -7.20E-05 1.61E-03
Vol
(m
3)
Mass
(kg)
I xx I yy I zz I xy I xz I yz
4 cm C9 0.00623 6.604 0.0289 0.0721 0.0647 -0.0157 -5.000E-06 3.50E-05
C8 0.00627 6.647 0.0289 0.0734 0.0677 -0.0170 -1.500E-05 -2.00E-06
C7 0.00545 5.779 0.0221 0.0611 0.0551 -0.0137 1.540E-03 -4.70E-05
C6 0.00458 4.852 0.0174 0.0541 0.0485 -0.0118 1.540E-04 -8.00E-05
C5 0.00468 4.965 0.0174 0.0548 0.0494 -0.0115 2.800E-05 -7.00E-06
C4 0.00409 4.334 0.0142 0.0496 0.0447 -0.0105 2.200E-05 3.90E-05
C3 0.00480 5.088 0.0171 0.0584 0.0539 -0.0122 -1.600E-05 -2.00E-05
C2-1 0.00331 3.511 0.0084 0.0360 0.0360 -0.0080 -4.000E-06 1.49E-04

  TABLE 5. Dimensions, baseline specific tension ST of 55 N/cm^2 (O'Brien et al., 2010), and Fmax of reconstructed muscles of Allosaurus. Abbreviations: MTCP=musculus transversospinalis capitis; MTCR=musculus transversospinalis cervicis; MLCS=musculus longissimus capitis superficialis; MLCP= musculus longissimus capitits profundus; MRCA=musculus rectus capitis anterior/ventralis.

Semi-major
(cm)
Semi-minor
(cm)
Area
(cm^
2)
Area
(cm^
2)
ST
(N/cm^2)
Fmax
(N)
Muscle baseline robust baseline robust baseline robust baseline robust
MTCP 3.513 3.865 2.550 2.805 30.13 36.46 55 1657 2005
MTCR 2.929 3.222 2.366 2.603 23.31 28.21 55 1282 1551
MLCS 2.996 3.295 3.267 3.593 32.91 39.82 55 1810 2190
MLCP 4.531 4.984 1.753 1.928 26.70 32.31 55 1469 1777
MRCA 4.016 4.418 1.624 1.786 21.93 26.54 55 1206 1460

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