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  1. THE MATHEMATICS OF SOMA BODY MECHANICS
  2.  
  3. The formula for KE is;
  4.  
  5. KE = Velocity * Velocity = n1, n1 * Mass = n2, n2 * 0.5 = n3 = KE
  6.  
  7. or;
  8.  
  9. KE = V * V = n1, n1 * M = n2, n2 * 0.5 = n3 = KE
  10.  
  11. An object's structual strength can be expressed in two ways; tensile and compressive strength, both of which are measured in pascals/m2. Tensile strength is used when the object is pulled, swung or struck on it's sides. Compressive strength is used when the object is loaded with a weight on it's top, or pushed from it's bottom.
  12.  
  13. The surface area of an object has to be determined before we can determine it's strength in pascals. Since the circle is the best model of the shapes found in living beings, it's formula will be the first presented;
  14.  
  15. Area = Radius * Radius = n1, n1 * pi(3.14) = n2 = Area
  16.  
  17. To calculate strength in pascals;
  18.  
  19. Area(m2) / 1(m2) = n1, n1 * Pascals = n2 = nPa
  20.  
  21. If KE exceeds nPa, the object is broken. If nPa exceeds KE, the object resists destruction;
  22.  
  23. Destruction = KE = n+, nPa = n- = Destuction
  24. Survival = nPa = n+, KE = n- = Survival
  25.  
  26. This applies to both tensile and compressive strengths, but note the difference in definition, and note also that natural materials often differ in their tensile and compressive strengths.
  27.  
  28. The Soma body can be divided into certain key points where knowing the tensile and compressive strengths is important, most notably;
  29.  
  30. - Wrist (Weakest link in the arm)
  31. - Ankle (Weakest link in the leg)
  32. - Chest
  33.  
  34. Any stress that could be put on the rest of the body either transfers through these locations, or survives at least the KE the wrist or ankle is rated for by virtue of exceeding their areas in cross section.
  35.  
  36. Our model Soma body will be the Somamba, or 'Mother of All Soma;'
  37.  
  38. To find area;
  39.  
  40. Area = pi(3.14) * 2 = n1, C / n1 = R = n2, n2 * n2 = n3, pi(3.14) * n3 = n4 = Area
  41.  
  42. Where C is circumference and R is Radius.
  43.  
  44. Somamba's calculated values are;
  45.  
  46. Head A1, Eye A2, Hair A3
  47. Neck B1
  48. Chest C1
  49. Abdomen D1
  50. Upper arm E1, Lower arm E2, Hand E3
  51. Thigh F1, Calf F2, Ankle F3, Foot F4
  52. Height G1
  53. Weight H1
  54.  
  55. A1-0.5575m, 22", 0.0247332m2, Tensile 3,215,316,000N, Compressive 618,330,000N
  56. A2-0.0254m, 1", 5.13402E-5m2, Tensile 6,674,226N, Compressive 1,283,505N
  57. A3-1.7E-5m, 0.00066929", 2.29979E-11m2, Tensile 2.989727N, Compressive 0.5749475N
  58. B1-0.305m, 12.00785", 0.00740269m2, Tensile 962,349,700N, Compressive 185,067,250N
  59. C1-0.6096m, 23.999952", 0.029572m2, Tensile 3,844,360,000N, Compressive 739,300,000N
  60. D1-0.434m, 17.08658", 0.0149889m2, Tensile 1,948,557,000N, Compressive 374,722,500N
  61. E1-0.342m, 13.46454", 0.0093077m2, Tensile 1,210,001,000N, Compressive 232,692,500N
  62. E2-0.09906m, 3.8999922", 0.000780884m2, Tensile 101,514,920N, Compressive 19,522,100N
  63. E3-0.023706666666666657m, 0.933331466666666", 4.4723E-5m2, Tensile 5,813,990N, Compressive 1,118,075N
  64. F1-0.588m, 23.14956", 0.0275134m2, Tensile 3,576,742,000N, Compressive 687,835,000N
  65. F2-0.329m, 12.95273", 0.00861355m2, Tensile 1,119,761,500N, Compressive 215,338,750N
  66. F3-0.09906m, 3.8999922", 0.000780884m2, Tensile 101,514,920N, Compressive 19,522,100N
  67. F4-0.0254m, 0.999998", 5.13402E-5m2, Tensile 6,674,226N, Compressive 1,283,505N
  68. G1-1.422m
  69. H1-7.1344kg(49kg)
  70.  
  71. Obviously, the compressive strength is the absolute limit for any limb. Using two limbs doubles the force in Newtons which can be survived.
  72.  
  73. Aerographene can compress by 90% - meaning that a 1m3 object could reduce it's volume to 0.1m3. Aerographene weighs 0.16mg/cm3, meaning a 1m3 object would weigh 160g. Aerographene can hold 900 times it's own weight in oil, meaning that 160g of AG could hold 144kg of gasoline - which holds 37.4MJ/L, meaning that this object could hold 37,400MJ.
  74.  
  75. Somamba has a body volume of 49,000cm3;
  76.  
  77. 490,000cm3 AG @ 90% Compression * 0.16mg/cm3 = 78,400mg = 78.4g/49,000cm3
  78.  
  79. 37.4MJ/L * 49,000 / 1,000 = 1,832.6MJ/49L/7.056kg
  80.  
  81. 78.4g + 7.056kg = 7.1344kg
  82.  
  83. 1N moves 1kg 1m;
  84.  
  85. 1N/1kg/1m
  86.  
  87. KE = V * V = n1, n1 * M = n2, n2 * 0.5 = n3 = KE
  88.  
  89. KE(559.8500724396445) = 559.8500724396445m/s * 559.8500724396445m/s = 313,432.103610675 (n1), n1 * 7.1344kg = 2,236,150 (n2), n2 * 0.5 = 1,118,075 (n3) = KE(559.8500724396445)
  90.  
  91. KE(559.8500724396445) = Max Survivable Velocity
  92.  
  93. The foot limits the maximum lift, and two feet doubles it;
  94.  
  95. 1,118,075kg(1.118075Mt) * 2 = 2,236,150kg(2.23615Mt)
  96.  
  97. Muscles can be 95% efficient, so;
  98.  
  99. 100% effciency * 95% = 1.052631578947368(n1), n1 = KE(2,367)NeN(1.05), MtNeN(1.05)
  100.  
  101. KE(559.8500724396445) * (1.05) = 1,176,921.052631578N, 1.118075MtNeN * (1.05) = 1,176,921.052631578N, 2.23615MtNeN * (1.05) = 2,353,842.105263157N
  102.  
  103. Power;
  104.  
  105. Volume of a cylinder;
  106.  
  107. Vol = R * R = n1, pi(3.14) * n1 = n2, n2 * H = Vol,
  108.  
  109. 0.000013571680264m3 = 0.06 * 0.06 = 0.0036, pi(3.14) * 0.0036 = 0.011309733552923, 0.011309733552923 * 0.0012 = 0.000013571680264m3
  110.  
  111. Surface area of the cylinder;
  112.  
  113. Surface area = R * R = n1, pi(3.14) * n1 = n2 = Surface area,
  114.  
  115. 0.011309733552923m = 0.06 * 0.06 = 0.0036, pi(3.14) * 0.0036 = 0.011309733552923m
  116.  
  117. To calculate Rotational energy;
  118.  
  119. Angular momentum = Radius * Radius = n1, n1 * Mass = n2 = Angular momentum,
  120.  
  121. Angular velocity = pi(3.14) * 2 = n1, n1 / 0.0175 * RPM = Angular velocity,
  122.  
  123. Moment of inertia = Angular momentum / Angular velocity = n1 = Moment of inertia,
  124.  
  125. Rotational Energy = AV * AV = n1, MOI * n1 = n2, n2 * 2 = n3 = Rotational Energy
  126.  
  127. One disk = 0.00217g, Radius 0.06m, Thickness 0.0012m, TensilePa 1,470,261,000, RPM 1,400,000,000, Rotational Energy 7,853,478.979149691N
  128.  
  129. 50 Disks = 0.1085g, Radius 0.06m, Thickness 0.06m, Rotational Energy 392,673,948.95748N
  130.  
  131. 392,673,948.95748 / 1,176,921.052631578 = 333.6451056589283 Max V's
  132. 392,673,948.95748 / 1,176,921.052631578 = 333.6451056589283 One-hand lift
  133. 392,673,948.95748 / 2,353,842.105263157 = 166.8225528294641 Two-hand lift
  134.  
  135. Calculating parabolic arcs;
  136.  
  137. To calculate the vertical(Vy) and horizontal(Vx) velocities;
  138.  
  139. angle(cos) = n1, V * n1 = n2 = Vx
  140. angle(sin) = n1, V * n1 = n2 = Vy
  141.  
  142. To calculate time(Ty) spent reaching Max height(Dy);
  143.  
  144. Ty = 0 - Vy = n1, n1 * -G = n2 = Ty
  145.  
  146. To calculate Max height(Dy);
  147.  
  148. Dy = Vy * Ty = n1, 0.5 * -G = n2, n2 * Ty = n3, n2 * n3 = n4 = Dy
  149.  
  150. To calculate time spent in the air(Ty2);
  151.  
  152. Ty * 2 = Ty2
  153.  
  154. To calculate range(Dx);
  155.  
  156. Dx = Vy * Ty2 = n1 = Dx
  157.  
  158. Graphene-Laser-Electron Propulsion(GLEPnir);
  159.  
  160. >http://arxiv.org/pdf/1505.04254.pdf
  161. >pp13
  162. >Figure d, Kinetic energy distribution spectrum of electrons emitted from graphene sponge under laser (450 nm) illumination showed a broad energy distribution
  163. >4.25E4 mW cm-2
  164.  
  165. 42,500mW = 42.5W, cm-2 = 0.01cm, so 42.5W/0.01cm. 1cm / 0.01cm = 100(1)cm, 42.5 * 100 = 4,250W of thrust per square centimeter.
  166.  
  167. 1W = 1N, so one single square centmeter of graphene can produce 4,250N of thrust. 1N lifts 1kg.
  168.  
  169. 1eV = 1.60217656535 x 10-19N
  170. 1g of Carbon = 1.3351667e+23 = nElectron
  171. eV = neV * 1.60217656535 x 10-19N
  172. 75 * 1.60217656535 x 10e-19N = 1.2016324e-16N
  173. 1.2016324e-16N * nElectron(1.3351667e+23) = 16,043,795.6512N
  174. cm-2 = 0.1mm = n1, n1 * n1 = 0.01mm2
  175. Cube root of 16,043,795.6512 = 252.2139132846656
  176. 252.2139132846656 * 252.2139132846656 = 63,611.85805436484N/0.01mm2
  177. 63,611.85805436484N/0.01mm2
  178. 84W Laser = 42.5N = 50.5952380952381% Conversion Efficiency
  179. 63,611.85805436484N/0.01mm2 @ 125,726.9665074505N
  180.  
  181. BSA = Mass * 0.425 = n1, n1 * 0.007184 = n2, Height * 0.725 = n3, n2 * n3 = n4, n4 * 0.1 = n5 = BSA
  182.  
  183. Calculating Somamba's BSA;
  184.  
  185. BSA = (114lbs) * 0.425 = 48.45, 48.45 * 0.007184 = 0.3480648, 56" * 0.725 = 40.6, 0.3480648 * 40.6 = 14.13143088, 14.13143088 * 0.1 = 1.413143088 = BSA
  186.  
  187. 1m2 = 10,000cm2,
  188.  
  189. 1.413143088m2 = 14,131.43088cm2,
  190.  
  191. 14,131.43088cm2 * 4,250N = 60,058,581.24N,
  192.  
  193. KE = V * V = n1, n1 * M = n2, n2 * 0.5 = n3 = KE
  194.  
  195. KE = 4,103.210568588052 * 4,103.210568588052 = 16,836,336.97017268, 16,836,336.97017268 * 7.1344kg = 120,117,162.48, 120,117,162.48 * 0.5 = 60,058,581.24N = KE
  196.  
  197. Somamba BSA = 1.413143088m2 = 1,413,143.088mm2 / 0.01 = 141,314,308.8mm2
  198. 141,314,308.8mm2 * 63,611.85805436484N = 8,989,265,800,000N @ 4.5481404345238e+12N
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