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 |
Segment | Vol (m^3) |
Mass |
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 (m3) |
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 |