mechanical_vibrations.ipynb

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   "source": [
    "The following diagram represents the problems being solved below:  \n",
    "source: http://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx"
   ]
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    "<img src=\"img/hanging_mass.png\" height=\"500\" width=\"400\">"
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    "\\begin{align}\n",
    "mu'' + γu' + ku = F(t)\n",
    "\\end{align}\n",
    "\n",
    "\n",
    "\\begin{align}\n",
    "mg=kL\n",
    "\\end{align}\n",
    "\n",
    " - $\\ mu'' \\$ corresponds to mass x acceleration\n",
    " \n",
    " - $\\ γu' \\$ corresponds to the damping factor x velocity  \n",
    " \n",
    " - $\\ ku \\$ corresponds to the spring constant x position  \n",
    " \n",
    " - F(t) is the forcing function  "
   ]
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   "cell_type": "markdown",
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   "source": [
    "# Free, Undamped Vibrations\n",
    "\n",
    "Example 1 A 16 lb object stretches a spring 8/9 ft by itself. There is no damping and no external forces acting on the system. The spring is initially displaced 6 inches upwards from its equilibrium position and given an initial velocity of 1 ft/sec downward. Find the displacement at any time t, u(t).\n",
    "\n",
    "\\begin{align}\n",
    "mu'' + ku = 0\n",
    "\\end{align}"
   ]
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       "  938.582,751.23 940.795,754.444 943.008,757.643 945.221,760.823 947.435,763.98 949.648,767.113 951.861,770.217 954.074,773.291 956.287,776.332 958.5,779.336 \n",
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       "  982.844,809.261 985.057,811.634 987.27,813.939 989.484,816.175 991.697,818.339 993.91,820.429 996.123,822.443 998.336,824.378 1000.55,826.236 1002.76,828.012 \n",
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       "  1049.24,844.453 1051.45,844.182 1053.66,843.814 1055.88,843.349 1058.09,842.789 1060.3,842.132 1062.52,841.381 1064.73,840.535 1066.94,839.596 1069.16,838.564 \n",
       "  1071.37,837.44 1073.58,836.225 1075.79,834.919 1078.01,833.525 1080.22,832.044 1082.43,830.476 1084.65,828.823 1086.86,827.086 1089.07,825.267 1091.29,823.368 \n",
       "  1093.5,821.391 1095.71,819.336 1097.93,817.206 1100.14,815.003 1102.35,812.729 1104.56,810.385 1106.78,807.975 1108.99,805.5 1111.2,802.963 1113.42,800.366 \n",
       "  1115.63,797.712 1117.84,795.002 1120.06,792.241 1122.27,789.429 1124.48,786.57 1126.7,783.665 1128.91,780.718 1131.12,777.731 1133.34,774.707 1135.55,771.649 \n",
       "  1137.76,768.559 1139.97,765.44 1142.19,762.295 1144.4,759.127 1146.61,755.938 1148.83,752.732 1151.04,749.51 1153.25,746.277 1155.47,743.035 1157.68,739.786 \n",
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       "  1624.64,733.973 1626.86,730.721 1629.07,727.476 1631.28,724.238 1633.5,721.012 1635.71,717.799 1637.92,714.604 1640.14,711.428 1642.35,708.275 1644.56,705.147 \n",
       "  1646.78,702.048 1648.99,698.979 1651.2,695.944 1653.41,692.945 1655.63,689.986 1657.84,687.068 1660.05,684.194 1662.27,681.367 1664.48,678.589 1666.69,675.864 \n",
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       "  1691.04,649.99 1693.25,648.066 1695.46,646.222 1697.68,644.459 1699.89,642.779 1702.1,641.184 1704.32,639.675 1706.53,638.253 1708.74,636.92 1710.96,635.677 \n",
       "  1713.17,634.525 1715.38,633.465 1717.59,632.498 1719.81,631.625 1722.02,630.846 1724.23,630.163 1726.45,629.575 1728.66,629.084 1730.87,628.689 1733.09,628.392 \n",
       "  1735.3,628.191 1737.51,628.089 1739.73,628.083 1741.94,628.176 1744.15,628.365 1746.37,628.651 1748.58,629.035 1750.79,629.514 1753,630.09 1755.22,630.762 \n",
       "  1757.43,631.529 1759.64,632.39 1761.86,633.345 1764.07,634.394 1766.28,635.534 1768.5,636.766 1770.71,638.088 1772.92,639.499 1775.14,640.997 1777.35,642.582 \n",
       "  1779.56,644.252 1781.77,646.005 1783.99,647.841 1786.2,649.756 1788.41,651.75 1790.63,653.821 1792.84,655.966 1795.05,658.184 1797.27,660.472 1799.48,662.829 \n",
       "  1801.69,665.251 1803.91,667.738 1806.12,670.285 1808.33,672.891 1810.55,675.555 1812.76,678.274 1814.97,681.045 1817.18,683.866 1819.4,686.734 1821.61,689.646 \n",
       "  1823.82,692.601 1826.04,695.595 1828.25,698.626 1830.46,701.69 1832.68,704.787 1834.89,707.911 1837.1,711.062 1839.32,714.236 1841.53,717.43 1843.74,720.641 \n",
       "  1845.95,723.867 1848.17,727.105 1850.38,730.351 1852.59,733.603 1854.81,736.858 1857.02,740.112 1859.23,743.363 1861.45,746.608 1863.66,749.843 1865.87,753.066 \n",
       "  1868.09,756.273 1870.3,759.462 1872.51,762.631 1874.73,765.777 1876.94,768.895 1879.15,771.985 1881.36,775.042 1883.58,778.065 1885.79,781.05 1888,783.994 \n",
       "  1890.22,786.895 1892.43,789.751 1894.64,792.558 1896.86,795.314 1899.07,798.018 1901.28,800.665 1903.5,803.255 1905.71,805.785 1907.92,808.252 1910.14,810.654 \n",
       "  1912.35,812.99 1914.56,815.256 1916.77,817.452 1918.99,819.575 1921.2,821.623 1923.41,823.594 1925.63,825.486 1927.84,827.298 1930.05,829.028 1932.27,830.674 \n",
       "  1934.48,832.236 1936.69,833.711 1938.91,835.1 1941.12,836.399 1943.33,837.609 1945.54,838.727 1947.76,839.753 1949.97,840.685 1952.18,841.523 1954.4,842.267 \n",
       "  1956.61,842.915 1958.82,843.466 1961.04,843.921 1963.25,844.278 1965.46,844.538 1967.68,844.7 1969.89,844.764 1972.1,844.731 1974.32,844.599 1976.53,844.37 \n",
       "  1978.74,844.043 1980.95,843.619 1983.17,843.098 1985.38,842.482 1987.59,841.769 1989.81,840.962 1992.02,840.061 1994.23,839.067 1996.45,837.981 1998.66,836.803 \n",
       "  2000.87,835.535 2003.09,834.178 2005.3,832.733 2007.51,831.201 2009.72,829.583 2011.94,827.882 2014.15,826.098 2016.36,824.234 2018.58,822.29 2020.79,820.268 \n",
       "  2023,818.171 2025.22,816 2027.43,813.756 2029.64,811.443 2031.86,809.061 2034.07,806.614 2036.28,804.103 2038.5,801.531 2040.71,798.9 2042.92,796.212 \n",
       "  2045.13,793.47 2047.35,790.677 2049.56,787.835 2051.77,784.947 2053.99,782.016 2056.2,779.044 2058.41,776.034 2060.63,772.988 2062.84,769.909 2065.05,766.8 \n",
       "  2067.27,763.663 2069.48,760.502 2071.69,757.32 2073.9,754.119 2076.12,750.902 2078.33,747.672 2080.54,744.433 2082.76,741.185 2084.97,737.934 2087.18,734.681 \n",
       "  2089.4,731.429 2091.61,728.182 2093.82,724.942 2096.04,721.712 2098.25,718.495 2100.46,715.293 2102.68,712.111 2104.89,708.95 2107.1,705.813 2109.31,702.704 \n",
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       "  2177.92,633.669 2180.13,632.681 2182.35,631.786 2184.56,630.985 2186.77,630.279 2188.99,629.667 2191.2,629.152 2193.41,628.734 2195.63,628.413 2197.84,628.189 \n",
       "  2200.05,628.063 2202.26,628.034 2204.48,628.104 2206.69,628.272 2208.9,628.537 2211.12,628.9 2213.33,629.36 2215.54,629.917 2217.76,630.571 2219.97,631.32 \n",
       "  2222.18,632.165 2224.4,633.103 2226.61,634.135 2228.82,635.26 2231.04,636.476 2233.25,637.782 2235.46,639.177 2237.67,640.659 2239.89,642.228 2242.1,643.881 \n",
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       "  628.748,345.688 630.961,330.277 633.174,315.232 635.387,300.567 637.6,286.296 639.813,272.432 642.026,258.985 644.239,245.966 646.452,233.389 648.665,221.264 \n",
       "  650.879,209.605 653.092,198.42 655.305,187.721 657.518,177.518 659.731,167.82 661.944,158.635 664.157,149.973 666.37,141.84 668.583,134.244 670.797,127.191 \n",
       "  673.01,120.689 675.223,114.743 677.436,109.357 679.649,104.538 681.862,100.288 684.075,96.6114 686.288,93.5117 688.501,90.9913 690.714,89.0523 692.928,87.6965 \n",
       "  695.141,86.9248 697.354,86.7373 699.567,87.1335 701.78,88.1135 703.993,89.6768 706.206,91.8222 708.419,94.5484 710.632,97.853 712.846,101.733 715.059,106.187 \n",
       "  717.272,111.209 719.485,116.796 721.698,122.942 723.911,129.643 726.124,136.893 728.337,144.685 730.55,153.012 732.763,161.867 734.977,171.242 737.19,181.128 \n",
       "  739.403,191.516 741.616,202.396 743.829,213.758 746.042,225.592 748.255,237.886 750.468,250.629 752.681,263.807 754.894,277.409 757.108,291.422 759.321,305.836 \n",
       "  761.534,320.638 763.747,335.814 765.96,351.35 768.173,367.233 770.386,383.448 772.599,399.981 774.812,416.817 777.026,433.941 779.239,451.338 781.452,468.992 \n",
       "  783.665,486.888 785.878,505.01 788.091,523.342 790.304,541.868 792.517,560.57 794.73,579.432 796.943,598.438 799.157,617.569 801.37,636.808 803.583,656.139 \n",
       "  805.796,675.541 808.009,694.998 810.222,714.491 812.435,734.001 814.648,753.51 816.861,773.006 819.075,792.47 821.288,811.883 823.501,831.229 825.714,850.488 \n",
       "  827.927,869.644 830.14,888.679 832.353,907.576 834.566,926.317 836.779,944.887 838.992,963.267 841.206,981.443 843.419,999.397 845.632,1017.11 847.845,1034.58 \n",
       "  850.058,1051.77 852.271,1068.69 854.484,1085.3 856.697,1101.6 858.91,1117.57 861.123,1133.2 863.337,1148.47 865.55,1163.38 867.763,1177.89 869.976,1192.01 \n",
       "  872.189,1205.72 874.402,1219 876.615,1231.84 878.828,1244.24 881.041,1256.19 883.255,1267.67 885.468,1278.66 887.681,1289.17 889.894,1299.19 892.107,1308.69 \n",
       "  894.32,1317.68 896.533,1326.14 898.746,1334.07 900.959,1341.45 903.172,1348.3 905.386,1354.58 907.599,1360.32 909.812,1365.48 912.025,1370.08 914.238,1374.11 \n",
       "  916.451,1377.56 918.664,1380.43 920.877,1382.72 923.09,1384.43 925.304,1385.55 927.517,1386.09 929.73,1386.05 931.943,1385.42 934.156,1384.2 936.369,1382.41 \n",
       "  938.582,1380.03 940.795,1377.07 943.008,1373.54 945.221,1369.44 947.435,1364.76 949.648,1359.52 951.861,1353.72 954.074,1347.36 956.287,1340.45 958.5,1332.99 \n",
       "  960.713,1325 962.926,1316.47 965.139,1307.42 967.353,1297.85 969.566,1287.77 971.779,1277.19 973.992,1266.13 976.205,1254.58 978.418,1242.57 980.631,1230.1 \n",
       "  982.844,1217.18 985.057,1203.82 987.27,1190.05 989.484,1175.86 991.697,1161.28 993.91,1146.32 996.123,1131 998.336,1115.32 1000.55,1099.3 1002.76,1082.95 \n",
       "  1004.98,1066.3 1007.19,1049.34 1009.4,1032.1 1011.61,1014.6 1013.83,996.85 1016.04,978.863 1018.25,960.659 1020.47,942.252 1022.68,923.659 1024.89,904.898 \n",
       "  1027.11,885.983 1029.32,866.933 1031.53,847.763 1033.75,828.492 1035.96,809.136 1038.17,789.712 1040.38,770.239 1042.6,750.734 1044.81,731.214 1047.02,711.698 \n",
       "  1049.24,692.204 1051.45,672.75 1053.66,653.354 1055.88,634.036 1058.09,614.813 1060.3,595.701 1062.52,576.715 1064.73,557.871 1066.94,539.187 1069.16,520.681 \n",
       "  1071.37,502.37 1073.58,484.271 1075.79,466.4 1078.01,448.774 1080.22,431.408 1082.43,414.318 1084.65,397.52 1086.86,381.027 1089.07,364.855 1091.29,349.019 \n",
       "  1093.5,333.532 1095.71,318.408 1097.93,303.661 1100.14,289.303 1102.35,275.347 1104.56,261.807 1106.78,248.694 1108.99,236.019 1111.2,223.796 1113.42,212.035 \n",
       "  1115.63,200.746 1117.84,189.941 1120.06,179.631 1122.27,169.821 1124.48,160.52 1126.7,151.736 1128.91,143.477 1131.12,135.752 1133.34,128.568 1135.55,121.933 \n",
       "  1137.76,115.852 1139.97,110.331 1142.19,105.376 1144.4,100.992 1146.61,97.1813 1148.83,93.9488 1151.04,91.2971 1153.25,89.2287 1155.47,87.7453 1157.68,86.8482 \n",
       "  1159.89,86.5382 1162.11,86.8153 1164.32,87.6792 1166.53,89.1287 1168.75,91.1624 1170.96,93.7782 1173.17,96.9732 1175.38,100.744 1177.6,105.088 1179.81,109.999 \n",
       "  1182.02,115.473 1184.24,121.504 1186.45,128.088 1188.66,135.22 1190.88,142.892 1193.09,151.099 1195.3,159.832 1197.52,169.085 1199.73,178.849 1201.94,189.115 \n",
       "  1204.15,199.876 1206.37,211.121 1208.58,222.84 1210.79,235.023 1213.01,247.66 1215.22,260.74 1217.43,274.25 1219.65,288.178 1221.86,302.513 1224.07,317.241 \n",
       "  1226.29,332.348 1228.5,347.821 1230.71,363.646 1232.93,379.808 1235.14,396.291 1237.35,413.08 1239.56,430.158 1241.78,447.51 1243.99,465.119 1246.2,482.972 \n",
       "  1248.42,501.053 1250.63,519.347 1252.84,537.836 1255.06,556.503 1257.27,575.331 1259.48,594.304 1261.7,613.404 1263.91,632.614 1266.12,651.917 1268.33,671.296 \n",
       "  1270.55,690.733 1272.76,710.213 1274.97,729.716 1277.19,749.227 1279.4,768.727 1281.61,788.2 1283.83,807.628 1286.04,826.993 1288.25,846.279 1290.47,865.467 \n",
       "  1292.68,884.541 1294.89,903.482 1297.11,922.273 1299.32,940.897 1301.53,959.336 1303.74,977.571 1305.96,995.586 1308.17,1013.36 1310.38,1030.89 1312.6,1048.16 \n",
       "  1314.81,1065.14 1317.02,1081.83 1319.24,1098.21 1321.45,1114.26 1323.66,1129.97 1325.88,1145.32 1328.09,1160.3 1330.3,1174.91 1332.51,1189.11 1334.73,1202.9 \n",
       "  1336.94,1216.28 1339.15,1229.22 1341.37,1241.71 1343.58,1253.75 1345.79,1265.33 1348.01,1276.43 1350.22,1287.04 1352.43,1297.15 1354.65,1306.76 1356.86,1315.85 \n",
       "  1359.07,1324.43 1361.29,1332.47 1363.5,1339.97 1365.71,1346.94 1367.92,1353.35 1370.14,1359.21 1372.35,1364.51 1374.56,1369.24 1376.78,1373.41 1378.99,1377 \n",
       "  1381.2,1380.02 1383.42,1382.45 1385.63,1384.3 1387.84,1385.57 1390.06,1386.25 1392.27,1386.34 1394.48,1385.85 1396.69,1384.77 1398.91,1383.1 1401.12,1380.84 \n",
       "  1403.33,1378 1405.55,1374.59 1407.76,1370.59 1409.97,1366.02 1412.19,1360.88 1414.4,1355.18 1416.61,1348.92 1418.83,1342.11 1421.04,1334.74 1423.25,1326.85 \n",
       "  1425.47,1318.42 1427.68,1309.46 1429.89,1300 1432.1,1290.02 1434.32,1279.55 1436.53,1268.6 1438.74,1257.16 1440.96,1245.25 1443.17,1232.89 1445.38,1220.08 \n",
       "  1447.6,1206.83 1449.81,1193.16 1452.02,1179.08 1454.24,1164.6 1456.45,1149.73 1458.66,1134.48 1460.87,1118.88 1463.09,1102.93 1465.3,1086.65 1467.51,1070.06 \n",
       "  1469.73,1053.15 1471.94,1035.97 1474.15,1018.51 1476.37,1000.79 1478.58,982.838 1480.79,964.661 1483.01,946.278 1485.22,927.708 1487.43,908.967 1489.65,890.074 \n",
       "  1491.86,871.043 1494.07,851.89 1496.28,832.632 1498.5,813.287 1500.71,793.874 1502.92,774.41 1505.14,754.912 1507.35,735.399 1509.56,715.888 1511.78,696.396 \n",
       "  1513.99,676.941 1516.2,657.54 1518.42,638.21 1520.63,618.969 1522.84,599.832 1525.05,580.819 1527.27,561.944 1529.48,543.225 1531.69,524.679 1533.91,506.322 \n",
       "  1536.12,488.171 1538.33,470.242 1540.55,452.552 1542.76,435.116 1544.97,417.952 1547.19,401.074 1549.4,384.5 1551.61,368.244 1553.83,352.323 1556.04,336.753 \n",
       "  1558.25,321.541 1560.46,306.702 1562.68,292.248 1564.89,278.195 1567.1,264.554 1569.32,251.34 1571.53,238.563 1573.74,226.236 1575.96,214.37 1578.17,202.976 \n",
       "  1580.38,192.064 1582.6,181.644 1584.81,171.725 1587.02,162.316 1589.23,153.426 1591.45,145.061 1593.66,137.231 1595.87,129.941 1598.09,123.198 1600.3,117.008 \n",
       "  1602.51,111.376 1604.73,106.308 1606.94,101.807 1609.15,97.8781 1611.37,94.5245 1613.58,91.7491 1615.79,89.5542 1618.01,87.9396 1620.22,86.9069 1622.43,86.4576 \n",
       "  1624.64,86.5926 1626.86,87.3123 1629.07,88.6166 1631.28,90.5046 1633.5,92.9753 1635.71,96.0267 1637.92,99.6564 1640.14,103.862 1642.35,108.639 1644.56,113.983 \n",
       "  1646.78,119.891 1648.99,126.358 1651.2,133.376 1653.41,140.941 1655.63,149.045 1657.84,157.68 1660.05,166.84 1662.27,176.515 1664.48,186.696 1666.69,197.374 \n",
       "  1668.91,208.539 1671.12,220.179 1673.33,232.284 1675.55,244.842 1677.76,257.84 1679.97,271.266 1682.19,285.107 1684.4,299.355 1686.61,313.997 1688.82,329.018 \n",
       "  1691.04,344.406 1693.25,360.147 1695.46,376.226 1697.68,392.629 1699.89,409.342 1702.1,426.35 1704.32,443.637 1706.53,461.189 1708.74,478.989 1710.96,497.023 \n",
       "  1713.17,515.274 1715.38,533.725 1717.59,552.362 1719.81,571.166 1722.02,590.121 1724.23,609.209 1726.45,628.414 1728.66,647.717 1730.87,667.101 1733.09,686.548 \n",
       "  1735.3,706.039 1737.51,725.555 1739.73,745.078 1741.94,764.59 1744.15,784.078 1746.37,803.525 1748.58,822.911 1750.79,842.22 1753,861.433 1755.22,880.532 \n",
       "  1757.43,899.501 1759.64,918.322 1761.86,936.977 1764.07,955.45 1766.28,973.726 1768.5,991.786 1770.71,1009.62 1772.92,1027.2 1775.14,1044.52 1777.35,1061.56 \n",
       "  1779.56,1078.32 1781.77,1094.76 1783.99,1110.88 1786.2,1126.67 1788.41,1142.1 1790.63,1157.17 1792.84,1171.86 1795.05,1186.16 1797.27,1200.05 1799.48,1213.53 \n",
       "  1801.69,1226.57 1803.91,1239.18 1806.12,1251.32 1808.33,1263 1810.55,1274.21 1812.76,1284.94 1814.97,1295.17 1817.18,1304.9 1819.4,1314.12 1821.61,1322.82 \n",
       "  1823.82,1330.99 1826.04,1338.62 1828.25,1345.7 1830.46,1352.24 1832.68,1358.22 1834.89,1363.63 1837.1,1368.48 1839.32,1372.76 1841.53,1376.46 1843.74,1379.59 \n",
       "  1845.95,1382.13 1848.17,1384.09 1850.38,1385.47 1852.59,1386.26 1854.81,1386.47 1857.02,1386.09 1859.23,1385.12 1861.45,1383.57 1863.66,1381.43 1865.87,1378.72 \n",
       "  1868.09,1375.43 1870.3,1371.56 1872.51,1367.13 1874.73,1362.12 1876.94,1356.56 1879.15,1350.43 1881.36,1343.76 1883.58,1336.53 1885.79,1328.77 1888,1320.47 \n",
       "  1890.22,1311.64 1892.43,1302.3 1894.64,1292.44 1896.86,1282.08 1899.07,1271.23 1901.28,1259.89 1903.5,1248.08 1905.71,1235.8 1907.92,1223.07 1910.14,1209.9 \n",
       "  1912.35,1196.3 1914.56,1182.29 1916.77,1167.87 1918.99,1153.06 1921.2,1137.88 1923.41,1122.33 1925.63,1106.44 1927.84,1090.22 1930.05,1073.68 1932.27,1056.84 \n",
       "  1934.48,1039.71 1936.69,1022.31 1938.91,1004.65 1941.12,986.752 1943.33,968.627 1945.54,950.294 1947.76,931.769 1949.97,913.069 1952.18,894.21 1954.4,875.209 \n",
       "  1956.61,856.084 1958.82,836.849 1961.04,817.524 1963.25,798.125 1965.46,778.669 1967.68,759.173 1969.89,739.655 1972.1,720.133 1974.32,700.623 1976.53,681.143 \n",
       "  1978.74,661.712 1980.95,642.346 1983.17,623.065 1985.38,603.885 1987.59,584.824 1989.81,565.902 1992.02,547.136 1994.23,528.545 1996.45,510.143 1998.66,491.942 \n",
       "  2000.87,473.961 2003.09,456.215 2005.3,438.722 2007.51,421.498 2009.72,404.559 2011.94,387.919 2014.15,371.595 2016.36,355.601 2018.58,339.951 2020.79,324.659 \n",
       "  2023,309.74 2025.22,295.206 2027.43,281.07 2029.64,267.345 2031.86,254.043 2034.07,241.176 2036.28,228.755 2038.5,216.791 2040.71,205.295 2042.92,194.278 \n",
       "  2045.13,183.749 2047.35,173.718 2049.56,164.193 2051.77,155.185 2053.99,146.702 2056.2,138.751 2058.41,131.336 2060.63,124.464 2062.84,118.141 2065.05,112.374 \n",
       "  2067.27,107.169 2069.48,102.531 2071.69,98.4639 2073.9,94.9728 2076.12,92.0608 2078.33,89.731 2080.54,87.9856 2082.76,86.8264 2084.97,86.2547 2087.18,86.271 \n",
       "  2089.4,86.8754 2091.61,88.0673 2093.82,89.8458 2096.04,92.2089 2098.25,95.1546 2100.46,98.68 2102.68,102.782 2104.89,107.456 2107.1,112.697 2109.31,118.501 \n",
       "  2111.53,124.862 2113.74,131.774 2115.95,139.229 2118.17,147.22 2120.38,155.74 2122.59,164.781 2124.81,174.336 2127.02,184.397 2129.23,194.955 2131.45,206.001 \n",
       "  2133.66,217.523 2135.87,229.514 2138.08,241.961 2140.3,254.853 2142.51,268.181 2144.72,281.931 2146.94,296.091 2149.15,310.65 2151.36,325.594 2153.58,340.91 \n",
       "  2155.79,356.584 2158,372.602 2160.22,388.951 2162.43,405.614 2164.64,422.576 2166.86,439.823 2169.07,457.338 2171.28,475.105 2173.49,493.106 2175.71,511.326 \n",
       "  2177.92,529.746 2180.13,548.348 2182.35,567.121 2184.56,586.047 2186.77,605.109 2188.99,624.288 2191.2,643.568 2193.41,662.931 2195.63,682.358 2197.84,701.834 \n",
       "  2200.05,721.34 2202.26,740.858 2204.48,760.372 2206.69,779.865 2208.9,799.318 2211.12,818.715 2213.33,838.038 2215.54,857.271 2217.76,876.396 2219.97,895.397 \n",
       "  2222.18,914.256 2224.4,932.957 2226.61,951.482 2228.82,969.815 2231.04,987.939 2233.25,1005.84 2235.46,1023.49 2237.67,1040.89 2239.89,1058.01 2242.1,1074.84 \n",
       "  2244.31,1091.36 2246.53,1107.56 2248.74,1123.42 2250.95,1138.94 2253.17,1154.09 2255.38,1168.87 2257.59,1183.26 2259.81,1197.25 2262.02,1210.82 2264.23,1223.96 \n",
       "  2266.44,1236.66 2268.66,1248.91 2270.87,1260.69 2273.08,1272 2275.3,1282.83 2277.51,1293.16 2279.72,1302.99 2281.94,1312.31 2284.15,1321.11 2286.36,1329.39 \n",
       "  2288.58,1337.13 2290.79,1344.32 2293,1350.97 2295.22,1357.07 2297.43,1362.6 2299.64,1367.57 2301.85,1371.98 2304.07,1375.81 2306.28,1379.06 2308.49,1381.74 \n",
       "  2310.71,1383.83 2312.92,1385.34 2315.13,1386.27 2317.35,1386.61 2319.56,1386.36 2321.77,1385.53 2323.99,1384.11 2326.2,1382.11 2328.41,1379.52 2330.62,1376.35 \n",
       "  2332.84,1372.61 2335.05,1368.29 2337.26,1363.4 2339.48,1357.95 2341.69,1351.93 2343.9,1345.36 2346.12,1338.24 2348.33,1330.58 2350.54,1322.38 2352.76,1313.65 \n",
       "  \n",
       "  \"/>\n",
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       "M1941.79 312.204 L2280.76 312.204 L2280.76 130.764 L1941.79 130.764  Z\n",
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       "  1941.79,312.204 2280.76,312.204 2280.76,130.764 1941.79,130.764 1941.79,312.204 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip4200)\" style=\"stroke:#009af9; stroke-width:20; stroke-opacity:1; fill:none\" points=\"\n",
       "  1965.79,191.244 2109.79,191.244 \n",
       "  \"/>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2133.79, 208.744)\" x=\"2133.79\" y=\"208.744\">u1(t)</text>\n",
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       "  \"/>\n",
       "<g clip-path=\"url(#clip4200)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2133.79, 269.224)\" x=\"2133.79\" y=\"269.224\">u2(t)</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using DifferentialEquations\n",
    "using Plots\n",
    "\n",
    "g = 32.0 # ft/s\n",
    "weight = 16.0 # lbs\n",
    "m = weight / g\n",
    "L = 8.0 / 9.0\n",
    "\n",
    "k = m * g / L\n",
    "\n",
    "tspan = (0.0, 5.0)\n",
    "u₀ = [-0.5, 1.0] # u0 = 0.5 ft. and du = 1 ft/s\n",
    "\n",
    "function free_undamped(du, u, p, t)\n",
    "    y = u[1] # Position\n",
    "    dy = u[2] # Velocity\n",
    "    du[1] = dy # derivative of y is dy\n",
    "    du[2] = -k/m*y # a = -k/m*y\n",
    "end \n",
    "\n",
    "prob = ODEProblem(free_undamped, u₀, tspan)\n",
    "sol = solve(prob)\n",
    "\n",
    "plot(sol,linewidth=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " -0.5266671463615389\n",
       "  0.1221973859438604"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol(1.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.5266580709034284"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = 1.0\n",
    "0.52705 * cos(6.0 - 2.81984)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "18.0"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Free Damped Vibrations\n",
    "\n",
    "Take the spring and mass system from the first example and attach a damper to it that will exert a force of 12 lbs when the velocity is 2 ft/s. Find the displacement at any time t, u(t).\n",
    "\n",
    "Formula for damping force \\$ Fd = -γu' \\$\n",
    "\n",
    "Equation for system:\n",
    "\n",
    "\\begin{align}\n",
    "mu'' + γu' + ku = 0\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = 32.0 # ft/s\n",
    "weight = 16.0 # lbs\n",
    "m = weight / g\n",
    "L = 8.0 / 9.0\n",
    "\n",
    "k = m * g / L\n",
    "\n",
    "γ = 12.0 / 2.0 # Fd / v => the damping coefficient\n",
    "\n",
    "tspan = (0.0, 3.0)\n",
    "u₀ = [-0.5, 1.0] # u0 = 0.5 ft. and du = 1 ft/s\n",
    "\n",
    "function free_undamped(du, u, p, t)\n",
    "    y = u[1] # Position\n",
    "    dy = u[2] # Velocity\n",
    "    du[1] = dy # derivative of y is dy\n",
    "    du[2] = -(γ*dy + k*y) / m # a = -k/m*y\n",
    "end \n",
    "\n",
    "prob = ODEProblem(free_undamped, u₀, tspan)\n",
    "sol = solve(prob)\n",
    "\n",
    "plot(sol,linewidth=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.006196880441665897"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u(t) = -0.5 * exp(-6t) - 2t*exp(-6t)\n",
    "u(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " -0.006196372607920885\n",
       "  0.03222076505774027 "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Forced Undamped\n",
    "\n",
    "A 3 kg object is attached to spring and will stretch the spring 392 mm by itself. \n",
    "There is no damping in the system and a forcing function of the form\n",
    "\n",
    "\\begin{align}\n",
    "mu'' + ku = F(t)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
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       "  1779.56,796.097 1781.77,793.583 1783.99,791.054 1786.2,788.51 1788.41,785.953 1790.63,783.384 1792.84,780.806 1795.05,778.22 1797.27,775.627 1799.48,773.029 \n",
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       "  1890.22,681.967 1892.43,680.567 1894.64,679.225 1896.86,677.94 1899.07,676.715 1901.28,675.551 1903.5,674.447 1905.71,673.406 1907.92,672.428 1910.14,671.513 \n",
       "  1912.35,670.663 1914.56,669.878 1916.77,669.159 1918.99,668.507 1921.2,667.921 1923.41,667.403 1925.63,666.953 1927.84,666.572 1930.05,666.26 1932.27,666.016 \n",
       "  1934.48,665.843 1936.69,665.739 1938.91,665.705 1941.12,665.742 1943.33,665.849 1945.54,666.026 1947.76,666.274 1949.97,666.592 1952.18,666.98 1954.4,667.439 \n",
       "  1956.61,667.968 1958.82,668.568 1961.04,669.237 1963.25,669.975 1965.46,670.783 1967.68,671.659 1969.89,672.604 1972.1,673.617 1974.32,674.697 1976.53,675.845 \n",
       "  1978.74,677.06 1980.95,678.34 1983.17,679.687 1985.38,681.098 1987.59,682.573 1989.81,684.111 1992.02,685.713 1994.23,687.376 1996.45,689.1 1998.66,690.884 \n",
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       "  1159.89,575.396 1162.11,568.965 1164.32,562.636 1166.53,556.413 1168.75,550.299 1170.96,544.3 1173.17,538.419 1175.38,532.661 1177.6,527.029 1179.81,521.529 \n",
       "  1182.02,516.163 1184.24,510.937 1186.45,505.853 1188.66,500.917 1190.88,496.131 1193.09,491.5 1195.3,487.028 1197.52,482.718 1199.73,478.571 1201.94,474.591 \n",
       "  1204.15,470.782 1206.37,467.146 1208.58,463.688 1210.79,460.41 1213.01,457.315 1215.22,454.407 1217.43,451.687 1219.65,449.159 1221.86,446.825 1224.07,444.688 \n",
       "  1226.29,442.749 1228.5,441.01 1230.71,439.474 1232.93,438.143 1235.14,437.017 1237.35,436.099 1239.56,435.39 1241.78,434.891 1243.99,434.603 1246.2,434.528 \n",
       "  1248.42,434.665 1250.63,435.017 1252.84,435.583 1255.06,436.364 1257.27,437.36 1259.48,438.571 1261.7,439.998 1263.91,441.64 1266.12,443.497 1268.33,445.569 \n",
       "  1270.55,447.855 1272.76,450.355 1274.97,453.068 1277.19,455.992 1279.4,459.127 1281.61,462.472 1283.83,466.026 1286.04,469.787 1288.25,473.754 1290.47,477.926 \n",
       "  1292.68,482.3 1294.89,486.875 1297.11,491.649 1299.32,496.62 1301.53,501.785 1303.74,507.143 1305.96,512.69 1308.17,518.424 1310.38,524.342 1312.6,530.441 \n",
       "  1314.81,536.718 1317.02,543.17 1319.24,549.794 1321.45,556.585 1323.66,563.541 1325.88,570.657 1328.09,577.93 1330.3,585.356 1332.51,592.929 1334.73,600.647 \n",
       "  1336.94,608.504 1339.15,616.496 1341.37,624.618 1343.58,632.865 1345.79,641.232 1348.01,649.715 1350.22,658.309 1352.43,667.009 1354.65,675.811 1356.86,684.709 \n",
       "  1359.07,693.698 1361.29,702.773 1363.5,711.928 1365.71,721.157 1367.92,730.456 1370.14,739.818 1372.35,749.239 1374.56,758.712 1376.78,768.232 1378.99,777.793 \n",
       "  1381.2,787.39 1383.42,797.016 1385.63,806.666 1387.84,816.334 1390.06,826.015 1392.27,835.701 1394.48,845.388 1396.69,855.069 1398.91,864.738 1401.12,874.389 \n",
       "  1403.33,884.016 1405.55,893.613 1407.76,903.174 1409.97,912.691 1412.19,922.16 1414.4,931.573 1416.61,940.925 1418.83,950.208 1421.04,959.416 1423.25,968.544 \n",
       "  1425.47,977.584 1427.68,986.53 1429.89,995.379 1432.1,1004.13 1434.32,1012.76 1436.53,1021.29 1438.74,1029.69 1440.96,1037.96 1443.17,1046.1 1445.38,1054.11 \n",
       "  1447.6,1061.97 1449.81,1069.68 1452.02,1077.23 1454.24,1084.63 1456.45,1091.86 1458.66,1098.92 1460.87,1105.8 1463.09,1112.51 1465.3,1119.02 1467.51,1125.36 \n",
       "  1469.73,1131.49 1471.94,1137.43 1474.15,1143.16 1476.37,1148.69 1478.58,1154.01 1480.79,1159.11 1483.01,1164 1485.22,1168.66 1487.43,1173.09 1489.65,1177.29 \n",
       "  1491.86,1181.26 1494.07,1185 1496.28,1188.49 1498.5,1191.74 1500.71,1194.75 1502.92,1197.5 1505.14,1200.01 1507.35,1202.27 1509.56,1204.27 1511.78,1206.01 \n",
       "  1513.99,1207.49 1516.2,1208.72 1518.42,1209.67 1520.63,1210.37 1522.84,1210.8 1525.05,1210.95 1527.27,1210.85 1529.48,1210.47 1531.69,1209.82 1533.91,1208.89 \n",
       "  1536.12,1207.7 1538.33,1206.24 1540.55,1204.5 1542.76,1202.5 1544.97,1200.22 1547.19,1197.67 1549.4,1194.85 1551.61,1191.77 1553.83,1188.42 1556.04,1184.8 \n",
       "  1558.25,1180.91 1560.46,1176.77 1562.68,1172.36 1564.89,1167.7 1567.1,1162.77 1569.32,1157.6 1571.53,1152.17 1573.74,1146.5 1575.96,1140.58 1578.17,1134.41 \n",
       "  1580.38,1128.01 1582.6,1121.37 1584.81,1114.5 1587.02,1107.4 1589.23,1100.07 1591.45,1092.52 1593.66,1084.76 1595.87,1076.78 1598.09,1068.59 1600.3,1060.19 \n",
       "  1602.51,1051.59 1604.73,1042.8 1606.94,1033.82 1609.15,1024.64 1611.37,1015.29 1613.58,1005.76 1615.79,996.063 1618.01,986.198 1620.22,976.173 1622.43,965.994 \n",
       "  1624.64,955.667 1626.86,945.197 1629.07,934.592 1631.28,923.857 1633.5,912.998 1635.71,902.023 1637.92,890.938 1640.14,879.749 1642.35,868.464 1644.56,857.09 \n",
       "  1646.78,845.634 1648.99,834.104 1651.2,822.507 1653.41,810.85 1655.63,799.138 1657.84,787.379 1660.05,775.579 1662.27,763.747 1664.48,751.889 1666.69,740.014 \n",
       "  1668.91,728.128 1671.12,716.24 1673.33,704.356 1675.55,692.485 1677.76,680.633 1679.97,668.809 1682.19,657.019 1684.4,645.271 1686.61,633.572 1688.82,621.929 \n",
       "  1691.04,610.351 1693.25,598.844 1695.46,587.415 1697.68,576.072 1699.89,564.822 1702.1,553.673 1704.32,542.631 1706.53,531.703 1708.74,520.897 1710.96,510.22 \n",
       "  1713.17,499.68 1715.38,489.282 1717.59,479.035 1719.81,468.945 1722.02,459.019 1724.23,449.265 1726.45,439.689 1728.66,430.299 1730.87,421.101 1733.09,412.102 \n",
       "  1735.3,403.305 1737.51,394.716 1739.73,386.341 1741.94,378.187 1744.15,370.259 1746.37,362.563 1748.58,355.106 1750.79,347.892 1753,340.927 1755.22,334.217 \n",
       "  1757.43,327.765 1759.64,321.578 1761.86,315.659 1764.07,310.013 1766.28,304.645 1768.5,299.558 1770.71,294.756 1772.92,290.243 1775.14,286.023 1777.35,282.099 \n",
       "  1779.56,278.474 1781.77,275.151 1783.99,272.133 1786.2,269.423 1788.41,267.023 1790.63,264.935 1792.84,263.162 1795.05,261.705 1797.27,260.566 1799.48,259.747 \n",
       "  1801.69,259.249 1803.91,259.073 1806.12,259.22 1808.33,259.692 1810.55,260.486 1812.76,261.604 1814.97,263.046 1817.18,264.812 1819.4,266.902 1821.61,269.317 \n",
       "  1823.82,272.056 1826.04,275.117 1828.25,278.501 1830.46,282.207 1832.68,286.232 1834.89,290.575 1837.1,295.236 1839.32,300.211 1841.53,305.498 1843.74,311.096 \n",
       "  1845.95,317.002 1848.17,323.212 1850.38,329.724 1852.59,336.534 1854.81,343.639 1857.02,351.035 1859.23,358.719 1861.45,366.686 1863.66,374.931 1865.87,383.451 \n",
       "  1868.09,392.24 1870.3,401.293 1872.51,410.605 1874.73,420.17 1876.94,429.984 1879.15,440.039 1881.36,450.33 1883.58,460.85 1885.79,471.593 1888,482.554 \n",
       "  1890.22,493.728 1892.43,505.107 1894.64,516.687 1896.86,528.458 1899.07,540.415 1901.28,552.551 1903.5,564.858 1905.71,577.33 1907.92,589.958 1910.14,602.736 \n",
       "  1912.35,615.657 1914.56,628.711 1916.77,641.893 1918.99,655.193 1921.2,668.605 1923.41,682.12 1925.63,695.73 1927.84,709.427 1930.05,723.203 1932.27,737.049 \n",
       "  1934.48,750.957 1936.69,764.918 1938.91,778.924 1941.12,792.966 1943.33,807.035 1945.54,821.122 1947.76,835.219 1949.97,849.315 1952.18,863.403 1954.4,877.472 \n",
       "  1956.61,891.513 1958.82,905.516 1961.04,919.473 1963.25,933.374 1965.46,947.214 1967.68,960.984 1969.89,974.675 1972.1,988.278 1974.32,1001.78 1976.53,1015.18 \n",
       "  1978.74,1028.47 1980.95,1041.63 1983.17,1054.65 1985.38,1067.54 1987.59,1080.27 1989.81,1092.85 1992.02,1105.26 1994.23,1117.5 1996.45,1129.56 1998.66,1141.42 \n",
       "  2000.87,1153.09 2003.09,1164.56 2005.3,1175.81 2007.51,1186.83 2009.72,1197.64 2011.94,1208.2 2014.15,1218.53 2016.36,1228.6 2018.58,1238.42 2020.79,1247.98 \n",
       "  2023,1257.27 2025.22,1266.28 2027.43,1275.01 2029.64,1283.45 2031.86,1291.59 2034.07,1299.43 2036.28,1306.97 2038.5,1314.19 2040.71,1321.1 2042.92,1327.69 \n",
       "  2045.13,1333.95 2047.35,1339.88 2049.56,1345.48 2051.77,1350.74 2053.99,1355.65 2056.2,1360.21 2058.41,1364.42 2060.63,1368.28 2062.84,1371.77 2065.05,1374.91 \n",
       "  2067.27,1377.67 2069.48,1380.07 2071.69,1382.1 2073.9,1383.75 2076.12,1385.03 2078.33,1385.94 2080.54,1386.46 2082.76,1386.61 2084.97,1386.37 2087.18,1385.76 \n",
       "  2089.4,1384.76 2091.61,1383.39 2093.82,1381.63 2096.04,1379.49 2098.25,1376.97 2100.46,1374.08 2102.68,1370.8 2104.89,1367.15 2107.1,1363.12 2109.31,1358.72 \n",
       "  2111.53,1353.95 2113.74,1348.81 2115.95,1343.31 2118.17,1337.44 2120.38,1331.21 2122.59,1324.62 2124.81,1317.68 2127.02,1310.38 2129.23,1302.74 2131.45,1294.76 \n",
       "  2133.66,1286.43 2135.87,1277.77 2138.08,1268.78 2140.3,1259.46 2142.51,1249.82 2144.72,1239.87 2146.94,1229.6 2149.15,1219.03 2151.36,1208.16 2153.58,1197 \n",
       "  2155.79,1185.55 2158,1173.82 2160.22,1161.81 2162.43,1149.54 2164.64,1137.01 2166.86,1124.23 2169.07,1111.2 2171.28,1097.93 2173.49,1084.44 2175.71,1070.72 \n",
       "  2177.92,1056.79 2180.13,1042.66 2182.35,1028.33 2184.56,1013.82 2186.77,999.133 2188.99,984.277 2191.2,969.261 2193.41,954.093 2195.63,938.783 2197.84,923.339 \n",
       "  2200.05,907.772 2202.26,892.092 2204.48,876.308 2206.69,860.429 2208.9,844.466 2211.12,828.428 2213.33,812.325 2215.54,796.166 2217.76,779.963 2219.97,763.724 \n",
       "  2222.18,747.459 2224.4,731.179 2226.61,714.894 2228.82,698.612 2231.04,682.345 2233.25,666.103 2235.46,649.895 2237.67,633.731 2239.89,617.623 2242.1,601.579 \n",
       "  2244.31,585.611 2246.53,569.728 2248.74,553.94 2250.95,538.259 2253.17,522.693 2255.38,507.255 2257.59,491.953 2259.81,476.799 2262.02,461.802 2264.23,446.974 \n",
       "  2266.44,432.324 2268.66,417.864 2270.87,403.598 2273.08,389.535 2275.3,375.684 2277.51,362.056 2279.72,348.659 2281.94,335.503 2284.15,322.597 2286.36,309.949 \n",
       "  2288.58,297.569 2290.79,285.464 2293,273.643 2295.22,262.114 2297.43,250.885 2299.64,239.963 2301.85,229.355 2304.07,219.07 2306.28,209.113 2308.49,199.492 \n",
       "  2310.71,190.214 2312.92,181.284 2315.13,172.709 2317.35,164.496 2319.56,156.648 2321.77,149.174 2323.99,142.076 2326.2,135.362 2328.41,129.036 2330.62,123.102 \n",
       "  2332.84,117.566 2335.05,112.431 2337.26,107.702 2339.48,103.383 2341.69,99.4775 2343.9,95.9885 2346.12,92.9191 2348.33,90.272 2350.54,88.0498 2352.76,86.2547 \n",
       "  \n",
       "  \"/>\n",
       "<path clip-path=\"url(#clip6600)\" d=\"\n",
       "M1941.79 312.204 L2280.76 312.204 L2280.76 130.764 L1941.79 130.764  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<polyline clip-path=\"url(#clip6600)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  1941.79,312.204 2280.76,312.204 2280.76,130.764 1941.79,130.764 1941.79,312.204 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip6600)\" style=\"stroke:#009af9; stroke-width:20; stroke-opacity:1; fill:none\" points=\"\n",
       "  1965.79,191.244 2109.79,191.244 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip6600)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2133.79, 208.744)\" x=\"2133.79\" y=\"208.744\">u1(t)</text>\n",
       "</g>\n",
       "<polyline clip-path=\"url(#clip6600)\" style=\"stroke:#e26f46; stroke-width:20; stroke-opacity:1; fill:none\" points=\"\n",
       "  1965.79,251.724 2109.79,251.724 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip6600)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2133.79, 269.224)\" x=\"2133.79\" y=\"269.224\">u2(t)</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = 9.8 # m/s\n",
    "m = 3.0 # kg\n",
    "L = 0.392 # m\n",
    "\n",
    "k = m * g / L\n",
    "ω₀ = sqrt(k / m)\n",
    "\n",
    "\n",
    "tspan = (0.0, 5.0)\n",
    "u₀ = [0.2, -0.1] # u0 = 0.2 m and du = 1 m/s\n",
    "\n",
    "function free_undamped(du, u, p, t)\n",
    "    y = u[1] # Position\n",
    "    dy = u[2] # Velocity\n",
    "    du[1] = dy # derivative of dy/dt -> a\n",
    "    du[2] = (10cos(ω₀*t) - k*y) / m\n",
    "end \n",
    "\n",
    "prob = ODEProblem(free_undamped, u₀, tspan)\n",
    "sol = solve(prob)\n",
    "\n",
    "plot(sol,linewidth=5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " -0.24374303664802313\n",
       "  1.0837329519827101 "
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.24372467015540283"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u(t) = 0.200998cos(5t + 0.099699) + 1//3 * t * sin(5t)\n",
    "u(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " -0.01840427708528204\n",
       "  8.254060455452873  "
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.01969796387091091"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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