bras.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3d455fc1-ef8b-4d9c-ae06-ca4d035c452b",
   "metadata": {},
   "source": [
    "Markdown macros\n",
    "\n",
    "$\\newcommand{\\I}{\\mathrm{i}} \n",
    "\\newcommand{\\E}{\\mathrm{e}} \n",
    "\\newcommand{\\D}{\\mathop{}\\!\\mathrm{d}}\n",
    "\\newcommand{\\bra}[1]{\\langle{#1}|}\n",
    "\\newcommand{\\ket}[1]{|{#1}\\rangle}\n",
    "\\newcommand{\\braket}[1]{\\langle{#1}\\rangle}\n",
    "\\newcommand{\\bm}[1]{\\boldsymbol{#1}}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a2f3c8d7-4502-49ac-9537-60914845c2ce",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# numpy\n",
    "import numpy as np\n",
    "# matplotlib\n",
    "import matplotlib\n",
    "import matplotlib.pylab as plt\n",
    "from IPython.display import Image\n",
    "# sympy\n",
    "import sympy as sy\n",
    "sy.init_printing(use_unicode = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "6ca3fce0-7a50-4c20-8b55-20c1f8dce425",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9c30484-252f-4efd-afee-9ea0ecd25a34",
   "metadata": {},
   "source": [
    "# Bras articulé\n",
    "\n",
    "On étudie le mouvement d'un bras constitué de deux éléments identiques de masse $m$ et de longueur $l$, raccordés à leur extemité; les deux bras peuvent tourner sans frottement autour des axes $O_1$ et $O_2$ (voir la figure).\n",
    "\n",
    "On choisit un repère fixe $(X,Y,Z)$ dans lequel la gravité a la direction $-\\hat{\\bm Y}$. Les repères mobiles sont attachés *aux centres de masse* de deux bras.\n",
    "\n",
    "On note 1 le premier bras et 2 le second.\n",
    "\n",
    "Le repère mobile $(x_1,y_1)$ résulte de la rotation autour de $\\hat{\\bm Z}$ du repère fixe d'un angle $\\phi_2$, une rotation supplémentaire d'angle $\\phi_2$, donne l'orientation du second bras. Par conséquent, le second bras fait un angle $\\phi = \\phi_1 + \\phi_2$ par rapport à l'horizontale $X$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "4965fb21-31a2-4470-a8e3-90de8da5bc2e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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QiAkhALTrQH09xuT5hGswUCUZQBKCEADqyDFqWz2gYvVGphymRYFIbBkFgCBWb1GzPMzw8Lt9fG8AxCAIAaAttwDbVV8vblh6vglb4zB6gFQEIQAk8PHPR+2n8Ht3aNr3efu18SBAPO+PP+rLsZ+qAsBhyxT8+t/X+Xd4eEL44q0Of31sMAiNBwESMiEEgOPuBoMtzAkzaaQGAUjLXUYB4KAc+bdllPf65jE5nkw7jAcB0jIhBIAjVmswyZbR16pEmvEgQFQmhACw2/nZ4JmL3g9fB3jsosR2GA8CJGdCCACteNtgZyLt8JTPeBAgMEEIAPskHA8+eltfx8aDZ24u2ggzOoAcBCEA7JDkRjK3PFt22t2tYu6SbD5jsEwNrj7VkWlRIDDXEALAVslvK7p6yt/1ep1ffteExWrQeBBgHIIQADZ5W4O7bjH6utPOz+XOzwaTPI0AtCgQmy2jAPBeXyfOP6vBLaM/40GAoQhCAAglyWyQmRYFwhOEAPBGR+PBkzW4ek0jAIG5hhAAXumuBqcmt30e5jB6gKxMCAHgqe01uOuOMjmcj0DjQYABCUIAWNfRbHCj7jLPeBAgN1tGAWBFdzWYMPa660YADjMhBIB73dXgeQ1edmg8CFCAIAQAfjMeBBiKIASAPxgPtsB4EKAMQQgAvx2rweq3GE3FeBBgNIIQAP414GxwMh4EGJsgBIBpGrUGl4wHJy0KjEcQAsC4NWg8CDA4QQgAGA9OkxYFhiQIARid8WA7JBlAYYIQgKGdr8EwtxgdnBYFxiQIARjXsLPB6c/xoP2iAMMShAAMauQabJPD6AHKE4QAjGjwGjQeBGAmCAGA+owHAaoQhAAMJ+F4sPc7yhgPAgxOEAIwlsE3i05Om2jmoQFaIAgBGIgaXDIeBEAQAjAKNTgZDzbz0ACNEIQADEEN3jEeBGAShAAwDuNBAO58q/0EAP6192C0y+VixMFGmcaD/d5i1H87kxYFmKbJhBBoxFyD1+t1/j71crm8HmXMf9rguIMG2Sw6a/C/F0kGUJ0gBOq71eD8W7MLElKDj/wnNmlRgP8IQqAJy+9Q384xbq/g+1peU4M3xoMArBKEQGUuBSQTNcgzWhTgRhACEJAaXNp7xyYAxiEIgZoet7E1uLENVvV7i9EWOIweoBGOnQAqezavePbyFxcQPsakYciYjAeXjAcBeEEQAjUl/Pb08VrE29kVvgkeihpsn/EgQDtsGQUacvj2oat3prm9xDbUcajBF/xkBIBHghDo3ov7lGrCoajBRw1+5hsPAjRFEAI9eTZCvO0OhTK6u6OM8SAAqwQh0ArHzUNaDf6UxHgQoDWCEACC80MWAJ5xl1Gge77ZhUfGgwBsYUIIdMOeUl77+t9Xd5f2FeC/l0mLAjwnCIHINOSAZOFkPAjAZraMAvGpwQHdmjDHWRQdBadP/kmLArxkQgj04cDEY34T3xAPbsCBYYPjQQCaJQiBVrw4RH75ko2BpwZZGjALm+K0CYBm2TIKNOR6vc4hd9eEt5dvpAZZNTdhjk2k7Tjw0xMARiYIgbac/xZWDUJTjAcBWmbLKBDKsxp0VRUz40EAWBKEQAc2nh5hNsgBe68tdC3iLsaDAI2zZRSoL8lpgWqQtx7Hg3PdZT2jogr/IQCwkSAEKnu7mXNLLt5ex9ZQzuj6rjMNfvIbDwK0z5ZRoGm7ahBeeDYefNT7GRXGgwBsZ0II1Pf2+9fz3+D6Fpm9+poWNvhjEeNBgC4IQqCy+YzB1WDbeFmg2OOt7ePB1Veb37yLsaH/HADYRRAC9T02YZLbzEAqjaeg8SAAhwlCoAlzEy5/W/HJEF7jgXeY/3AmLQqwkyAEWuF7WTLp5TrAY4wHATjDXUYBGIvxYGBaFGAvQQhAZMaDAPCCIAQAknHaBEBfBCEAYR0+baIL7sMEwHmCEABIw3gQoDuCEICYjAcB4C1BCAAkYDwI0CNBCEBAsceDS8aDAJwhCAGgMw2eNmE8CNApQQhANMaDALCRIAS69PHPR+wDx+EZ48FGHhoghm+1nwDAPssOvGvCqFMgdjEeBIDtBCEQxzglwLCMBwFISxACkUlEov6NGw9OWhQgBUEIjEUixhb7ylLjQQCSE4RAZ77+95X2m34XIgYW9W/TeHDSogCJCEKgP/N3+ZlmQUaI/TIeBIC9BCHQq+SjwmckIqxy2gRAAIIQ6FixJrxjl2mDYnf7cjxovygACQlCoG+1mnApdorAKuNBgBgEIdC9FprwjkQsLPYH3HgQgHwEIRBB1tvMJLF8bpFahTEZDwKE8VftJwCQTC+h1XK49ij2eHDJeBCA5AQhEErUEmBYDZ42YTwIEIkgBKLRhEMxHgSAMwQhEFDUJGA0xoONPDRAYIIQiKnlJmz5ufXFeBAAThKEQFhf//uKmgeMwHgQgAIEIRBca03Y2vOJJOrH1nhw0qIA2QhCIL6onTC42Kd3GA8CUIYgBIbQSBM28jRCivqxNR6ctChAToIQGEXUYBiT8SAAJCEIgYHUvc2MIqVfTpsAiEoQAsMRZr2LfdrEcjxovygAuQlCYETl+yFSsTAa40GAwAQhMCiF1injQQBISBAC44oUEpCJ8SBAbIIQGFqx28zEvitmMbHHg0vGgwCUIQgBCkXFxz8fspAXGjxtwngQIDxBCDBNBQdNsvAw40EASE4QAvyrZF3IQu4YDwJQhSAE+K3wxEkWbmc8OBQtClCMIAT4Q9rbzGx5b7IQ40EAahGEACuSNOHtncjC5IwHA9OiACUJQoB1yZNjexYqwzuxPyDGgwBUJAgBnjrThM/eduOWVFk4IOPBSYsCFCcIAV7JtDVRFm4X+3YyDY4HARiKIAR440B+bHwTWUiDHEYPMJRvtZ8AQAfmbMtUZbcmfP3+5z+NNBzbYpzxoP2iAFRhQgiw1fa53+H372ak1GU8CDAaQQiwQ4HxlCy8MR4EgNwEIcA+ZZrEGRWUZzwIMCBBCLDbi1RLm4sj33Um9nhwyXgQgIoEIcARG1Ot5GOFzMKoGjxtwngQYEyCEOC4u07LmohDZaHxIACU4dgJgFMKh4ozKgIwHgSgHSaEAF0a6makUcvWeHDSogC1CUKAjoXMwr6e7V7GgwA0xZZRgO7NTbhlE+nU4bStuye8kfHgpEUBGmBCCBBEjLvOtPzczjMeBKA1ghAglBhZGJ7x4KRFAdogCAEC6jQLY5820eB4EABcQwgQljMqeM1h9AAIQoD4tt91pmIWjjMeDLxf9NfPN6/w/UeR5wHAZoIQYBRdZCHFHJ7Rva2+fA8NQHKCEGAsbZ5RYTwIAFW4qQzAiDq96wypuHoQgJkgBBhXI1kYezy4FGM8ePg6QBcQAjRIEAKMrpEsjKrB0yaqzOjmGjQeBGiNIARgmv7LwrdlmDwLjQd7ZNYHEIYgBOAPVbIwqsDjwV1NKCABmiUIAVhRKwuNBwOzXxSgQZdgX6JuP4sNti6AijZW34GWi71ftMHTJpIn2a4zCb+uiR8dgPNMCAF4o9hdZyLV4FIjNZjDrr2gH5fPj8tnrqcCwCGCEIBNkmdh7KsQA189uMtjMcpCgKYIQgB2cEbFXoHHg7O3Q8LvP1Y2i2rCLf72UQLyE4QA7Hb+jIpxrh4cwbMmXL58tQllIUB1ghCA45xR0YuKd/i8PfTX9VMWArRGEAJw1t4sHGc8GH6/6M3jkPDZ2FAWAjTFsRMApHRsGCgIsyo2HlyeQjEH4euHfoxAR1PczBcQ/p8PCJCZCSEAKW2868zdm2R6MuU1WINVbDyOwqjwNTUIFGBCCEAu+U60b1aDQdjC1YNvrUbg4NPCvy+fghAowIQQgFy2TAsj1eBSIzXYi2cXFlZ4KgCDMSEEoJDVgWGkIDQeTPLQLiycXEAIFGRCCEAhjwPDSDW41EgNdsqFhTM1CJTxrfYTAGAsUSOwwcPoK44HT5qb8C4CPy4r20oBOMmEEABSMh6cErXo44WFY44KAbJyDSEAnOXqwayPPshVhX8/aV17R4GsBCEAnNXgV58ebyfzWtQsvHXgLfyWB048/ilAWraMAsAprh4sI+S5FLe7iT7rvdsfPZsfApwkCAEgjXbGgxVlbdFIVxX+ffncfraEJgTyEYQAxHR5kOlRcrxbXggwKnyxEfR19WlCIDlBCEBAI3davKsHH3U9Knx7WeDqy11DCGQiCAEI6LqQ71EavLnoUHocFbpJDNAaB9MDENloo8IRxoNLj0fYz7+OcQPSVct7kC5fePcSwQlsJAgBGELyCZ7xYDu+rvf7RT8u93tKW/B2PHjsEsHHRNx1xxpgcLaMAkAQo40Hl7q+qnBpb8KtDgwfjzEEeEYQAsApxoPt6PGqwo1Wp4urNXj3apoQeE0QAhBWvgsIG7w0ceTx4FKzo8JMt5O57Q4FOEYQAhBfviGe8WCbehwV6jqgCkEIAPsYD3ZhdVRY56ls9vZmM24SAyTnLqMAcJDx4NR8i97dgDTYoRT6EDjPhBCAmDLN8YwHu9PI9tHD8WY8CGQlCAEILtMcz3hw6qdFm9o+unqt4LMLCA/XoIwENhKEALCV8WDXqt999HWbPf7p+ahTg8BbghAAOKLHFm1k++iWG4qeqcH5bdUgsIUgBCCgHKO85fu0X7RfdbePbjwv/lZ0ahDI7RLsS9rtq3WwdQGwS44vBw0GocPoz3jswJJ3H10OAB/j8OQdaNQgsJ1jJwDgvQZrkJPuTqSYpunjcj88zOdxVHi+4tQgcIAtowDQJePB81q4++jcb/lqcMv1isDIBCEA0STfL2o8GFv1JjzPbBA4TBACQH+MB9OqeCLF+QmeGgTOcA0hAGxlPBhYxUsKz7TcrSdtDQWOMSEEgFccRt/IQxfQwiWFu4hA4DxBCEAo+c4fMh4cRMXtoznYSgq8ZssoADxlPDimYttHT474xB5wngkhALxnPDgN1qLFto+KOqAuQQhAfMcGfcaD9HVJIcABghCAODJdQGg8OA3colmb0HgQqE4QAsAK40FuzAmBwAQhAMFdLhcjvpO0qCYEohKEAHBvOR4Uk8w0IRBStB+a5jt+CoAuzF8Ibl8Fjo0HGwxCh9E34rEDcxxHAfncPod96jJzDiEAoVyv18vlcuYKwAZrkHbM30MvszDTEYWQnLE2q0wIAeAPDQah8WCD7r631oS0bDUFfdIycw0hAKxrpAZpk0sK6cXqPmc1yI0JIQD8ZjzYyEP3wpyQlhkMsoUJIQCsaKQGaZw5IW36uHwaDLKRCSEA/Mt4sJGH7o45Ie0wFWQvE0IAuNdIDdILc0IaYSrIAYIQAKbpz/FgI8zoOvLYhLKQwhySyTGCEAD+YDw4adFDHkcxmpBaDAbZzjWEANDn1YM/f/3x2x/fiz46z7mkEOiIIASAFr98rCfZXQQ+cy4O1eB5mhDohS2jAIyuj6sHf/7aWoPT5m4kG3tHgV4IQgBo3oHA2xWQC8aDqWhCoAuCEIChNXj14B+Odt3vN6ceTQi0zzWEAAytwSD8PaNLmHPbrio0HszB9YScNH8K+cwhExNCAMbVYA0SjzkhhznQkgIEIQA0JMt4cNt7Mx7Mx2yHvaQgxQhCAGhvPJjj2j/XE1a1bELf6PPa42eInymQjyAEYFB9nDYxwEOPQxPy1upgUA2SlSAEYHRDjAdzv2e20YQ88ywF1SC5CUIARmQ82MhDD0gTckcKUpcgBGBozY0HGYAm5EYKUp0gBGA4TY8Hc+/qtGu0DZoQlwvSiG+1nwAAVNP7ePDX9O9x89+n45lnv2gtX9ffPfBxMRQanU8AajEhBGAsTY8H97jV4N2v6Yg5IZM9otQmCAEYVO/jwSSMB6vThMOaO1AKUp0gBGAgYcaDRKUJgcIEIQAMSos24m5GpAmBkgQhAKNYjgftF6UpmhCoRRACQDUVZnQ/vld7aF7ShEAVghCAIXQzHvzhfqHj0oQBzKcL+rujI4IQAOqoOKMzHmyWJuyXDqRTghCAsTQ9HgQHlPdJCtIvQQhAfJ2dNpFv1+iP728emjY4nLAjq4NBVU9HBCEAAxl6POjqxK5owvY9S0E1SF8EIQDBdTYenGWLN+PBjmjCZklBIhGEAIyis/Fg2iY0HuyfJmyEFCQYQQhAZF2OB5NTg91y09GmuFyQkAQhAEPobDw4+/E9Qcst3oP9oj3ShI0wGCQqQQhAWEHGg4ebMElP0gDVUd1dDUpBIhGEAMTX5Xhw6UDaPby+8WDX3GCmrtvHXwoSz7faTwAAsggyHlyaG+/nr02vRjhf198XsH1cZElpPuBEJQgBoCu33vv55OVrjAfj0YRAEraMAhDQcjzY/X5R+I8bzADJCUIAKKHijM54MBJNCKQlCAGIxniQ2OwUBRIShACQnfEgabnpaBLzQfOrx83DOAQhAGEZDxKYJjzJBw1mghCAUAKeNtHnQ1OAJjzmcSRoFy4jE4QAxGQ8yGg04Vuru0PVIINzDiEAcRgPNvLQFLM8rZ4XVj9KUhAmE0IAQjIeZBw2jr61OhVUgzAThAAEYTzIsDThM/aIwlu2jAIQjfHgpEUZnj2isJEJIQARGA8yOEPCJXtEYTtBCEAoxoOTFh2VJlwlBeE1QQhA9xocD0J1IzfhrQClILwlCAEgPadNUIX+uTEYhI0EIQB9W44H7RcFG0eBXQQhACRmPEhdmhDYThAC0DHjQXhLEwIvCEIASMl4kBa4fA7YSBACEIHxINyJt3H04/I5/6/2E4FQBCEAvWrwtAnjQZoSpgl1IOQjCAHonvEgbNFjU0lByO1b7ScAAEcYD8IWX9deg2r1abs2EpIzIQSgb8aD8FqPG0cfn6eD5iETQQhAf4wHYZeOmnB1j6gUhHxsGQWgY8aDEIY9olCFCSEAnTEehANaHhI+mwqqQShAEALQK+NB2KXlJlySglCSIASgJw2OB4GEpCAUdgn249XbNwrB1gXAbBmEjfxTX2u/6K+ff/zWt9FstJwN+rQBTAgB6EaDNQjd6WXjKFCGIASA49xOBoCuCUIA+mA8CKkYEgI3ghAADqo4HjSZ5CRNCMwEIQCdMR6EXqweMAg0RRAC0IEGT5swHqR3WYeEUhB6IQgB6InxICSUowmlIPTlW+0nAABvGA828tDw2moHOuoQGmdCCEA3jAchuVRDwse3/bp+qkFonwkhAE0zHoTGGQxC10wIAeiD8eCkRcnj8JBw9XJBg0HoiyAEoF3Gg1DG3iaUghCGLaMAdMB4cNKiNGM1BSs8DyAFE0IAGtXgeBACO7xxVA1C1wQhAGzltAli29uE9ohCALaMAtCi5XjQflFoigiESEwIAWAT40FGkOpYQqAXghCA5hgPAkAZghAA3jMeZByGhDAUQQhAu4wHoQpXCcI43FQGgLY0eNqE8SAjuA0D5xrUhDAIQQhAo4wHoQz7QmFkghCAhhgPNvLQDEIKAoIQgBYZD0JujzVomygMSBAC0ArjQShjdTCoBmFMghCA5hgPTlqUPKQgcEcQAtAE40HISgoCqwQhAG0xHpy0KKm5XBB4RhACUF+D40GIwWAQeE0QAsAKp00QkhQE7vxV+wkAMLrleNB+UUjrVoBf1081CDwyIQSAe8aDRKIDgRdMCAGoyXgQACoShADwB+NBAMYhCAFogvEgAJQnCAGopsHTJowHARiKm8oAUJ/xIOyyPF3QPWOAMwQhAHUYD8IBqwfNAxxmyygAlRkPTlqUbR5r0HgQOMmEEIAKjAdhl9XBoBoEzhOEANRkPDhpUV6SgkBWghCA0owHYSN7RIHcBCEA1RgPTlqUJ6QgUIYgBKCoBseD0BR7RIGSBCEAo3MYPe0wGAQKE4QAlLMcD9ovCq9JQaAA5xACMDTjQZpyi0A1CJRhQghAIcaDsIUUBEoyIQRgXMaDAAxOEAJQmvEgADRCEAJQQoOnTRgPAoAgBKAo40EAaIebygCQnfEgI1seLeiGMUBrTAgBKMd4cNKig3k8aB6gKSaEAORlPMiYpCDQBUEIQCHGg5MWHcNqCtosCrRJEAKQkfEgo3msQSkItEwQAlCC8eCkRaOTgkCPBCEAuTQ4HoQc7BEF+iUIARiIw+hJSwoCvROEAGSxHA/aL8ogpCDQHecQAjAK40GyUoNAj0wIAUjPeJBBiECgdyaEAAzBeBAAHglCADIyHgSAlglCABJr8LQJ40EAWCUIAcjFeBAAGuemMgCkZDxIALfTBd0zBgjPhBCALIwHJy3ap9Wz5gGiMiEEIBnjQbomBYEBCUIA0jMenLRoV1ZT0H5RYASCEIA0jAfp1GMNSkFgHIIQgMSMByct2gmDQQBBCEACDY4H4QUpCDAThADE5DB6VklBgCVBCMBZy/Gg/aK0zOWCAHecQwhAQMaDbKEGAUwIATjFeJAeSUGAmSAEIBrjQZ7RgQB3bBkFIA3jQQDojiAE4LgGT5swHgSA7QQhAAkYDwJAjwQhAAcZDwJA79xUBoCzjAcnLVrV7XRB94wB2EsQAnCE8SAteDxoHoBdBCEApxgPTlq0BikIkIQgBGA340HqeqxBm0UBjhGEABxnPDhp0bJWB4NqEOAwQQjAPg2OBxmBFATIQRAC0D2H0ccmBQHyEYQA7LAcD9ovSgEuFwTIShAC0DfjwagMBgEKEIQAbGU8SEVSECCHv2o/AQA4zngwsFsBfl0/1SBAJiaEAOxmPEgZOhAgNxNCADZp8LQJ40EAOEkQArCP8SAAhGHL6Ij23hbicrn4/g8GZzwIACGZEA5n/q7uer3OjXe5XF5/nzf/aYPfCwJV+PHQpEUBCMSEcCy3Gpx/e71elR7wVoP/UEiyfi1PF3TPGIDqBOFwlj/df/tN3u0VzASAyT8F0zRp0RNWD5oHoC5bRgfiUkDgAONBknisQeNBgBaYEALAVlr0gNXBoBoEaIQgHMXjz/gb/Kk/0Jq9NyWGJSkI0D5BOJBn38w9e/nGCwjvblQDkJXD6LsgBQF6IQhHkSPYzBghNuNBjnG5IEBHBOGgztw+dPm2mhAoyXiwcQaDAN0RhOxmUACj8V89WxgMAvRIELLOCYQwuAbn/8aDHZGCAL1wDuGIxB6wnX8o2GiOwK/rpxoE6IggBOCe8WAjD90dKQjQHUEIwFPGgwAQmyBkhT2lMDLjQQAYhyAEYJ0fCU1aFIDoBCEAvxkP8ujj8rl6wCAAATh2gnsNfjsIlGc8OGnRJwfNAxCJCeGIbt/nPbbf8iW+HYTRNPjzIElWkRoEGIEJ4cc9lKUAACAASURBVKCu1+v8nd/d93+3lwMMbuQWlYIA4xCE4zIABJZsEGB6koJOFwQITBAC0CiH0ZckBQHGJAj5gxMIYUzGg4N7rEEpCDAIQTgWvQf0wniwDINBgMEJwoG8vVuMXAT85z8OKQjAJAi5UYMwrAbvLWw8WJ4UBBiTIBzL29jbUoOr3zu6AAkC8B/vmKQgwMgE4UDmMwZXv+Gbc27j94K+ZYRIjAcbeejydCAA0zT9VfsJUNTjufOXy2VXDQJR+UcAAAZkQjicuyb0LSCMzHgQAAYnCEckAoE7/lmYtCgAQ7JlFGBQxoMAgAkhwOiMB6dALXo7XdA9YwDYwoQQYETGgyGtnjUPAC8IQgBGF6NF1SAABwhCgOG41XB49osCsJFrCAGoz2H0qUhBAHYRhABjMR6M4XGDqBQE4ABBCEBlxoO7uFYQgIQEIcCgjAe7IwUBSM5NZQAG4rSJRh76ADUIQA6CkDc+/vmo/RSA9IwHu/Z1/XTFIABJCEJemWvw458PWQgBGA828tAnSUEAEnINIU/dReDHPx9f//uq9WSAhIwHe6QDAcjBhJAdzAmhX8aDAMAjQcg+mhB6Zzw4aVEA+I8g5KlnG0Q1IXTHeBAAWCUIOUITQqeMByctCgALgpCDNCH0wniwZR+Xz/l/tZ8IAIMShBynCYHutNOiOhCAFghCXnl7zoQmhMYtx4P2izZCCgLQDucQcpbzCYFdRj6MfrUDHTAIQEUmhCRgTghtMh5symMNfl0/1SAAdQlC3tg4/dOEwBZjjgdX94hKQQBaYMsoydg7Cs0yHqzFHlEAGicISUkTQjucNlH3oaUgAF2wZZTE7B2F1hgPludyQQB6IQh5b+/QTxNCdcaDjTz0TAoC0CxBSBaaEBphPFjFrQANBgFonCAkF00ItRgPtkAKAtAFQUhGmhDqMh6chmxRANhOELLJ4XuHakIozHgQANhOEJKdJoQqjAcnLQoA7whCStCEUEaD40EAoGWCkK1OnjivCWFMkU6b+Lh83v6X9j0DQC2CkHI0IWS1HA/aL5qWCAQgKkFIUZoQhhJgPLiags6TACAMQUhpmhByMB7MYTUF1SAAkQhCdjh5GeGNJoQRdD0eNBgEYBDfaj8BBvXxz0eqvIR4buO+A7M+48GTVq8VlIIARCUIqUYTwltzGb5tvAZPm+hxPCgFARiQLaPUZO8obHG5XDYmn/HgYS4XBGBMgpB9ks/0NCE8Wu26Z1loPHj+oV0uCMCwbBmlPntH4dGtCe9678UmUuPBJKQgAEMRhDRBE8Izc+atZmGbKo4HDxOBAAzLllFaYe8ovHC9Xl8PAI0Hpz5bFADqEoTslm+UpwnhtRdZ2MjMUJIBQF8EIW3RhPDWsyzcfjPSkLQoABwgCGmOJoQtZCEAcJ4g5IjcN4DRhLBRU1nY4GkTH5fP1ePmAYCZu4zSKPcdhe0OnFERng4EgC0EIe3ShLDX6zMqspZhO+NBKQgA29kyStPsHYUDXuwjLf9kCnusQWcMAsALgpCDis3uNCEc8/bowoRaGA+uXi6oBgHgNVtG6YC9o3DY6ibSYFb3iEpBANhCENIHTQhnBL560FQQAM6wZZRu2DsKLP36eT8b/Lp+qkEA2MWEkOO+/vcl0mBwVcaDv36uvFAKAsABghCALFazbZqm7z8SP5AUBIDDBCEAB62OB5914OMrnC9DKQgAJwlCuuGmMmR1dx/Ou7uwvP5TZm9TcPX1D2Th9x8172QDAJG4qQynlIm0r/99qUHyuVwul8vlujD9WYB3f3q9XmOf4rDRXZLtrcHzbwgAnGdCSNN0ILnNaXc38ZuTb+7A+f/vXn/6rxJLPtV2pErBu/ewfVRoPAgAqQhCGiUFKeZF172YBKpBACCAaD/hvn33FmxdjUt++IQapIwXU75n/5iM/I/MsVvI7LJlSKhIbx6PYazzPADomQkhbZGCFHPsOsABO3B63mDJL/87fJsZAOAYQUgrpCDlvR0P0to4rrXnAwC9c5dRgKcOzwPne9KkfTLlva4vdwcFgABMCEng639fyS8jhNwybf4M0IFTq4O4Np8VAHRNENKKj38+7BqlX8ubzXTdhBujK+t48NdPlxECQCGCkArm8DNUpE3H7iMa42YzLY/gWn5uANAvQUhRyxng40ZTQ0KoRW4BwJgEIWlsuYxwS+xpQnZZftb5zDnmQAqWv52MXgWATAQhJTz7Tt3daDjg2eeMnybspbIAAEFIXm+/QbdxlC2K/eBgywWEl8slwBWDahAAmAQhCT2mna7jpAYHyF3fQXTWXQp294QBoCOCkJSOFaAhIUuHI3DX58yxW4kefpNGKCsA4I4gBOo7PwlM8hOEt9O/fseDaVPw+49y95URsQCQlSCkCYaEA0q4HXTvp8qLrnt7rHyP40FNBQA8IwhplyYMKfllgQc+SW7Vt6y7u1vFPP526rAGe0/B3p8/ALRPENIKR1AElu9v9uSPDJbDwFvs3XLxblTYVw3mTqmsu0a//8j1ngGAO4KQhtg4GkmBvD/zufG67vpqvzthpmphFgIALROEQDIlZ7x+UrCqZERlGhIaDwJASYKQthgS9qj8Xl+fEo+CzdOCLQcAmiUI6YAmbFDFCz59Mtyp2E7Jh4TGgwBQmCCkOe4u06wW/l7U4J3qk7SETXirweqLAoBxCEL6YEhYUQsdOPM5sBSsmtQgAFQhCGmRIWF1DX78G6/B1ePsV4+1OK+1ZJpb7sycUA0CQC2Xru+u/uj27VewdY3psUkaT4IAGuzAmb/6WeO9dKwJ1eBhH5fP5W+/rp/rrwcAz5kQAv9Sg41rv5f2jgqXt5Bpf3UAEJIJIU0zJCypzSD0Nz51G0vPyvDxVqKdLrA6E0IAzjMhpDPuLjMUf9ddl9KWMyS6XiAABCAIaZq7y5TU2kd78BoMX0rhFwgAXbBllA7YOFpMO0E4+F9x7FiKvboz7raAZmV/KQAzE0Lgt0aGhCPXYPhYCr/AXnxcPjUhAJMgpAuPleJKwsCG/ZsNX0rhFwgAPRKEwB/qDgnHrMHwpRR+gQDQL0FIHwwJRzDmX2jsWIq9OgAIQBDSMU2YSZUh4YB/leFjKfwCu+YCQgBm7jJKT9xutKSSTTja32P4Ugq/wHyK3WhUEAIwMyGkJzaOhjTa32DsWIq9OgCIRxAC68psHB2qBsPHUvgFAkA8tozSHxtHi8kdhOP8xYUvpfALLKnArlH7RQG4MSEEnso6JBykBsOXUvgFAkBsgpD+uJKwjKzjwUH+vmLHUuzVAcAgBCFBaMKE7BQ9L3wshV9gYPaLArAkCOlSlYPyBqEGTwpfSuEXWN3X9bPY4RMAIAjplY2jybmn6EnhSyn8AgFgQIIQeJ+CSUayarBfsVcHACMThHTMkPC8LSmY5IEC/72Ej6XwCxyKCwgBuOMcQvq22jOB2yOt1zX4+GE8PCSM+jcSvpTCL7BlmS4jFIQA3DEhpG/uLnNMscFg2nfVlNixFHt1AMCNCSERPOZN1Ag572QK7s3vkH8RsWMp9uo6YkIIQBkmhDCQvXtEV19nexPGq8HwsRR+gYNTgwA8MiEkCEPC1xLuEd0YhME+/uFLKfwCe5R8SCgIAXhkQgjBJb9ccMuQUA12JPbqAIDXBCFBOILiUa1si/RhDx9LmRZ426wx2a8BAG2zZZQ4HEGxdP5ywWPvP8wHXArutYzAaZqu1+vjS9I+4ggS7hq1XxSAVSaExOEIilmZweDqRztGDUrBvZ79JG7528vlMr+aLASApghCQhl842jdS/tifJxj12CxFHx0GxheLtF2pgBA1wQhRFAlBZf5HaAGY6fgVLUGb6+mCWuxXxSAZ6J9VXYNIdN4R1DkvlwwPCl4wOF/bP0rvVeSywgFIQDPmBBCx0Y7/iE5KXiAO4gCQCSCkIBGuJJQCp4XuwYLrO5kDdo4CgAtEISMIkwTSsHzYqfglHOB5zd8Ph5HQW72iwLwgiAkpqhHULhc8CQpeIbL/6r4un4mPI0QAO4IQsIKtnHUYPC82DWYe3U5xnp2jQJAdYIQWicFz4udglPZBUo4AIgk2k9n7WjiTu9HUNgjepIUTCLtnUXdp3SvM1tGXUAIwGsmhNAog8GTpGAq+q06lxECkI8gJLgerySUgufFrsHYqwMAShKEjKjZJpSC54WPpd4X6MyJkuwXBeAtQUh8vRxB4XLBk3ovpbeqLDDrflEbULezaxSATAQhQ2h846jB4ElSEADgGEEINUnB82LHUuzVAQDVOXaCgTR1BIUUPC98LLWwwHwHTiR5h0PZu2XUBYQAbGFCyNBqbRx1ueBJLZRSVo0sMOsNYNTgXi4jBCAHQchAWri7jMHgSY2UUj7hF/jaY4LqRgDIShAylop3l5GC58WOpcZXl3a/6Op7u1zur2K4XC7zW8lCAMhEEEIJ9oie1HgsnRd+gW891uA0TdfrdQ7C1T/lha4vIHQ7AICSon2J9VWELUreXcZg8KTwpdTyAhPeUeb1u3rde/5hX9p4GeGWINwys717yZlX2/V8Vt+hcTFADiaEkIsUPK/lWDqv/dXdBnRT/hmd7/ULW37A5625yw/+41/H6qh246ttfDI3q+9w+RKfJwAJCUJGlPtKQil4XvuxdFL4BS6lPbuCVG5/F3P530Jr4/bdJLt8355EsvypxPKtfCIBpBLtn1Q7i9hotdmSdJrLBU8KX0p9LfB8y215D6//6fYP+523u0bf7hddvX/P9F99bflr2vhqb5/G8rdbtg1vfM8AbGdCyKByHEFhMHhSX6V0QI8LXJ3PHH5XB/6IAvZuDE4yoMt6yiUA2wlCxpVw46gUPK/HWNouxuoOZIDhHhttHw8CkJYghFOk4HkxYumF3hd4eEiYpAYl5V5b9os++6ONH2TjQYBIXEPI6M4cQeFywZN6L6W3Ii1w17+uyc+r8E/6nReXER6+gHB693FO+Gq7rgl0ASFAViaEcITB4EmRSmlVvAVuv3tkwh/MqcEcWqtBAOoShIxu75WEUvC8eLG0FHh1yyacnnfF4x8dowaZ1CNAfoIQVqw2oRQ8L3AszcIv8JZn87F1z/70PDV4zNv9ogBwRxDCpiMoXC54UvhSCr/AO1lTTQ2+9XX9fHsaYZtO/hzBZwVAcoIQpunlxlGDwZPCl1L4BRb2rAaTnH3HnfIXEALQGkEIT0nB82LHUuzVVWE2yJILCAEKEITwry0bR5evnPXJBBA+lsIvsDw1eFLdCwifxZvRLkDjBCHspgZfC19K4RdYxS0nDIU2avYywtf5d7tXbfL3DMAxghB+ezsklIJvxY6l2KurSARW4cpAACZBCBtJwbfCx1L4BTZOjTxaDgnbPHDizH5RPyYAKCPazn4/yOS8uyGhFHwrfCmFXyBjynGL0bvXeRaEd7H3+FarD+ErO0AOghBWzE0oBd8KX0rhF8jIcgfh6/Hg3gGgL+sAmQhC4KDYsRR7dTBtvq3r9ru/vp77bXmT5RseeG8AHCAIgd3Cx1L4BULjHkPRl3WATNxUBtghfCmFXyAAwJIgBDYJX0rhFwj9Mh4EyEcQAu/FjqXYq4OmuLIDoDWCEHilWCz9+vnvL77/KPOA/1KDUMzyysAX9yB1AiFASYIQWFerlOYyLJCFUhBKOpx5ZokAWQlCYEX1WPr1M2MTVl8dsMrNRQHKc+wE8IcqsXTbL/oobRZKQaho2XuPX6bVIEAVJoTAv9qMpYQ7SNtcIIzp9Q5SKQhQjCAEOiilkztI218gjGCZeeaBAI0QhDC6XmLp2Kiwl9XBaOQfQCMEIYyrx1jaNSrscYEAACUJQhhR16W0ZVTY9QIBAIoRhDCWMKX0bFQYZoEAAAUIQhhI7FiKvToAgBwEIQwhfCyFXyAAQA6CEIILWUrLzaIhFwgAUIYghLCiltKtBqMuEACgGEEIMYWMJSkIAJCWIIRoosaSGgQASE4QQhzhSyn8AgEACvur9hMA0ogXS99//HHzmPk8egAAEjIhhO6FTMHlr6UgAEAml+v1Wvs5pHS5XOZfBFsXrAqTgrfkW6bg9N8CPy6ft5d8XT8nAAASEYTQpTAp+MzdAm9NKAgBABKyZRT6E7sGX6/u4/KpCQEAUhGE0JPYKTgNsEAAgKbYMgp9CF9KbxfoSkIAgORMCKF1UhAAgExMCKFpsWNp7+oMCQEA0jIhhEbFTsFpgAUCALTPhBCaE76UzizQkBAAICETQmiIFAQAoCQTQmhF7FhKuDpDQgCAVEwIob7YKTgNsEAAgE4JQqjpvpR+/nr12j++53wuWeRIwa/r53JICADAYbaMQjV/xNLrFLzTQxlmnQraNQoAkIQghAqOp+BN201YYI/orQkFIQDAYYIQitq3R/St9rKw2OWChoQAAOcJQihkpZRO1uCsmSYsf+cYQ0IAgJMEIZSQqwZntZuw1k1EDQkBAE5yl1HIaz2WEtbg/N7qNaEjJQAA+iUIIZfwpVR9gc6fAAA4yZZRSO9NKaUdD94UHBJWT8Ebu0YBAM4QhJBYnRqc5W/CdlLwxq1lAAAOs2UUkmkwltJqfIEfl09NCACwiyCEBBovpfNaXqArCQEADhOEcNaOWMq6XzSPllPwkSEhAMAughCOay6Wkp4/0dzqAABITRDCEblj6df0u+u+TxXmin3VoF2jAADHCELYp3wp/Zq+l2zCvlLwkV2jAADbCULYofdYeq3r1RkSAgAcIAhhk65jaYtICzQkBADYSBDCG5FKaVX4BQIA8IwghKfCl1KwBdo1CgCw11+1nwA0KksspTsT4uT7//z8DFaDd5QhAMAWJoRwL3YpTaEXaEgIALCLIITfApfSLPwCl9xaBgDgLUEI01SylH58n37mOVTw5X7RcVLQkBAAYDtBCMFjKfbqAAA4w01lGFqde6vkuLXMk/c5eA0aFQIAvGZCyKAql1LajaNrNThyCto1CgCwkSBkRLFjKfbqAABIyJZRxtLQ+XupNo7++X5aWV0z6o4K/zaoBADaZkLIKFospbnlDu8dlYLP2TUKALCFICS+1kvp2PWEixpsfYEAALRKEBJcH7G0a1QoBXdyQj0AwDOCkLD6i6XlFtCfz//oP/0tsKzqu0ZdQAgAtE8QElD4Ugq/wORqDQn/z2QSAGibICSU8KUUfoEAAJQkCIkjdizFXl0m1XeNAgA0ThASQfhYCr/AAgrvGnUBIQDQBUFI35QSzXIBIQDQPkFIr6QgW9g1CgDwwl+1nwAcEbsGY6+uImUIAHDHhJDOhI+l8AsM7O66wdtv7R0FAJolCOlG+FIKv8Bacu8afQy/vy+fy1/PryALAYAGCUL6EDuWYq+uKcnvNfo29uY/mrNQEwIArRGEtC58LIVfYAtyDAl37Qj9v+unJgQAGiQIaVf4Ugq/wMBe1ODrEwg1IQDQFEFIi8KXUvgFxvZ2Nrj68nlImOkpAQAcIwhpTuxYir26Lpy8jNC9QwGASAQhDQkfS+EX2LJ2Tqi3axQAaIcgpAnhSyn8Agfxdjx4clOoAyoAgMIEIZWFL6XwC+xR8sMnlo7lnMsLAYAqBCE1xY6l2KvrUd1do6vTxeULNSEAUJ4gpI7wsRR+gQPKcTsZu0MBgLoEIaWFL6XwC+QZIz4AoDuCkHLCl1L4BUaS6TLCtzebMRIEAJoiCCkkdizFXl0k7Rw+AQDQAkFIduFjKfwCmR2+74vxIADQLEFIRuFLKfwCYzu8a3T1ZPlnragGAYCWCUJyiR1LsVcX25ldo6+HhC8qUQ0CAG0ShKQXPpbCL5C3VoeEj68z/0INAgDNEoSkFL6Uwi+Qt25DwtdNOL+OFAQAGicISSN8KYVf4JiOXUY4Z97fl89b9d0S8e51AAAaJwhJIHYsxV7dmJIcPnFLPltDAYB+CUJOCR9L4RfIefOEUA0CAD0ShBwUvpTCL5DZ4cMnAAACEITsFr6Uwi+QKdGu0dmx0+oBAFogCNkndizFXh352C8KAHRKELJV+FgKv0AAALgjCHkvfCmFXyAAAKwShLwRO5Zir46NztxX5uQFhKtv7jxDAKAYQchTsWMp9urYItV9Zc40m94DAOoShKwIH0vhFwgAAFsIQv4QvpTCL5DCjPgAgK4JQn6LHUuxV8dJjqcHAMYkCJmmAWIp/AI5JuHx9AAAPRKEowtfSuEXCAAAhwnCcYUvpfALBACAkwThoGLHUuzVAQBAKoJwOOFjKfwCycR9ZQCAAQnCgYQvpfALJAf3lQEARiYIhxC+lMIvEAAAchCE8cWOpdirAwCArARhZOFjKfwCKcxlhADAaARhTOFLKfwCAQCgAEEYUOxYir06qnBfGQBgWIIwlPCxFH6BAABQkiAMInwphV8gAACUJwi7F76Uwi+QprivDAAwFEHYt9ixFHt1NMVlhADAmARhr8LHUvgFAgBAdYKwP+FLKfwCAQCgEYKwJ+FLKfwCAQCgKYKwG7FjKfbqAACgTYKwA+FjKfwCAQCgTYKwaeFLKfwC6ZGTJwCAcQjCRoUvpfALpDtOngAABiQIWxQ7lmKvDgAAOiII2xI+lsIvEAAAOiIIWxG+lMIvEAAAuiMImxA7lmKvDgAA+iUIKwsfS+EXSEhuNAoADEIQVhO+lMIvkHjcaBQAGI0grCB8KYVfIAAAxCAIS4sdS7FXBwAAwQjCcsLHUvgFAgBAMIKwhPClFH6BAAAQkiDMK3wphV8gAAAEJggzih1LsVcHTp4AAEYgCLMIH0vhF8iwnDwBAAxFECYWvpTCLxAAAMYhCJMJX0rhFwgAAKMRhGnEjqXYqwMAgGEJwrPCx1L4BQIAwLAE4XHhSyn8AgEAYHCC8KDYsRR7dQAAwEwQ7hY+lsIvEAAAmAnCHcKXUvgFwi7OpgcAwhOEm4QvpfALhO2cTQ8AjEMQvhc7lmKvDgAAeEEQvhI+lsIvEAAAeEEQrgtfSuEXCAAAvCUI74UvpfALBAAANhKEf4gdS7FXBwAA7CUI/xU+lsIvEAAA2EsQxi+l8AsEAACOGT0IY8dS7NUBAAAnjRuEsWMp9uoAAIAkRgzC8LEUfoFQzMfl8+v6WftZAADkMlYQhi+l8AuEMr6unx+Xz9rPAgAgu4GCMHYsxV4dAACQwxBBGD6Wwi8QAADIIXgQhi+l8AsEAADyCRuE4Usp/AIBAIDcYgbhjx8/aj+FjKQgAACQRLQgjJ2CkxoEAADSiROE4Usp/AIBAIDCIgRh+FIKv0AAAKCK7oMwdizFXh0AAFBXx0G4Gku/fv7+9ffOLydUgwAAQFZdBuFdKS0j8NnL+4pDKQgAABTQXxAuY+lZCj6aX7P9LJSCAABAMX/VfgL73Hrp188dNXhz4E1KUoMAAEBJPU0IlzV4WJujQikIAACU100QJqnBm18/W2lCKQgAANTSTRDOEu75rN6EUhAAAKirj2sI53Zq/ArAXdQg9OLj8ln7KQAAPblcLpfLpfaz2KqzCWFaVYaEUhC68HX9lIIAwF63FLz94nq91ns673UQhFnHgyWbUAoCAMBoltPCBuOwgyAMQAoCAAANxmHrQRjg6kE1CAAA3GkkDlsPwgLy7RqVggAAwFsVLzgUhFlIQQAAYK/yY8MOgrCv/aJSEAAAOK9MHHYQhB35+fPnz59d9Sut+nv6Yx9zR0fZRHL7W/DxBwDqyheHgjANHQgAAHRHEJ4lBQEAgE4JwlM+Pz9dNEgOH5fP5W8bOaZmYJ+1nwAA0IfcV5ok/7awgyD8/qPF+8roQAAAoICss4EOgrA1UhAAAMjNsRPNkYIAAEA+Dqav4PuP968zqUEAACCDuneL+KviY28xZ9jGZsv3HNQgAACQynWh7jMxIXxFBwIAAKlUz79H3QRhpnuNPps9SkEAAOC8BiNwKf2W0ctCknd4a7PkG0fVIAAAkND1Qe1n9EaaCeFd+12v1/klt5e3/4GYSUEAAGAcl5OptiX5kmThLdVSbRy9Gw9KQZrycflc/vbr+rn+egAAcMLxINyVea014bIGpSANEoQAABRwMAgPBF7aJpyOZqHBIF0QhAAAFHAkCA+nXfUmNBikF4IQAGA0b+/KueUyvd1vuKvNlg9TsiQf7cpCU0G6IwgBAEaz5ZiGZyVVOghPbjQ9/B7uHEg7NUgXBCEAwMgez3HY+1Zb36TwRYA5DqLY0ng6kL4IQgCAkR2bpc1vtW/mt7c1CxxTAQhCAIDB7W3CAzU4TdNfe59KKjneJwAAwICO1eC0MQhvjPUAAAAKez1OO1yD05YgNMoDAAAobNfFfYdHd2+CMPl9QQEAAEjiZA1Oe7eMAgAAUMay9J7t3Dw5tysahHafAgAAJHG57DtVftWrIMy6X9QGVAAAgNeeDQmT1OBkyygAAEBfUtXgNE3fkrwXAAAAcjt/F5k75SaELiAEAADY6y7/0l589zQIs/bb2zVcLhcBCQAAcCdtKDV3DaEUBAAAWMp3S85N1xCef/i3Nyy9vcL1ehWEAAAAN/kSqZUJ4fU/tZ8IAABAQx5vJJOwD0sEYdbzDAEAAKJKflvRO0+DMFOAAgAAsMWLGkzVaNknhMaDAAAAez3WYI6eyhuEahAAAGCv3DtFb14FYcKHV4MAAABbvKjB5Ff2bZ0QHniw5UkSe98WAABgWMUa6k0QHn4eahAAAGCvy+XyuqHSDgnfTwhvj7fxwS6XixoEAADY620NJrdpy+j2JlymoBoEAADY6FgNnhwSftv4etfrdX6k1Qsc3U0UAADgsF01eKuz87YG4fTnnPDx4XUgAADAAanq7oAdQXij/QAAAM5LcvuVM1ceHglCAAAADnscCe46iX71zY81oSAEAAAo5PXu0Ldd9+LNjw0bBSEAAEAhJ6+/S3753qZjJwAAAIinlQnh6ujTaRYAAAD5tBKEeg8AAKAwW0YBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGwQBrGQAABV9JREFUJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQgBAAAGJQjh/9u7mxy1gSgKo1jqXWSz2WzW4QxQI2Sg8G+5yvecWdNu8CRSPr1XBgAAQglCAACAUIIQAAAglCAEAAAIJQgBAABCCUIAAIBQghAAACCUIAQAAAglCAEAAEIJQgAAgFCCEAAAIJQgBAAACCUIAQAAQglCAACAUIIQAAAglCAEAAAIJQgBAABCCUIAAIBQghAAACCUIAQAAAglCAEAAEIJQgAAgFCCEAAAIJQgBAAACCUIAQAAQglCAACAUIIQAAAglCAEAAAIJQgBAABCCUIAAIBQghAAACCUIAQAAAglCAEAAEIJQgAAgFCCEAAAIJQgBAAACCUIAQAAQglCAACAUIIQAAAglCAEAAAIJQgBAABCCUIAAIBQghAAACCUIAQAAAglCAEAAEIN4ziefQ8Q4c/wt84H/RsrfRAAAL0zIYQaqtVg5c8CAKBrghAAACCUIAQAAAglCAEAAEIJQrgaD5UBAGAmQQg1iDQAABokCAEAAEIJQgAAgFCCEC7FbioAAPMJQqhEqgEA0BpBCAAAEEoQAgAAhBKEUM/RW6O2UgEAWEQQAgAAhBKEAAAAoQQhAABAKEEIVR13zM8BQgAAlhKEAAAAoQQhAABAKEEIV2BfFACAFQQh1CbeAABohCAEAAAIJQgBAABCCULonh1UAADWEYRwAgkHAEALBCEAAEAoQQgAABBKEMI59toatX0KAMBqghAAACCUIAQAAAglCAEAAEIJQjjN9uN/DhACALCFIAQAAAglCAEAAEIJQuiVfVEAADYShHAmUQcAwIkEIQAAQChBCAAAEEoQQpfsmgIAsJ0ghJNJOwAAziIIAQAAQglCAACAUIIQzrd0a9SWKQAAuxCEAAAAoQQhAABAKEEInbEvCgDAXgQh7GP4te7PZR4AAPX9nH0DsMxzcY3jWL5g+2WL7mfyyv0N7z+ue3MAADiUIKQnz3F1H8c9h9Zret1fWXfZzJt5ePuGz69oQgAAWmNllM48sup5/nb7La5JdD1+XHpZ2WQ1tPCGk7+a8+ZlNksBANiRIKQbhSHbzPnbLmO6wmCw/PrXjxZ7AABUJgjp1WToN+f63WsQAAC6JghhvUJhSkcAANonCOlDoa9mzv26GA+Wt0btlAIAsC9BSDdae0rnovtp7eYBAOAmCOnFJKgm3/X3yY6XWQEFAOB6BCHsTz0CANAFQQht+XRQ0AFCAAB2Jwjhu9eJnwOEAABcgCCkP/UPEAIAwCUJQtjZEQcI7YsCAHAEQUi0T/F27lNh5B8AAHUIQrjdvu2LbtkmtYkKAECzBCGdCTwZaGAIAMBBhgv8d5koFZ4oMwxv/l1Mlkg/ve3G55ECAEBNJoQwq9km17yG3zAMvo8eAIC+/Jx9A9CWt+PBu3Ecn5NvTv4ZDwIA0DJBSJdmhtbXy+6N95x2c54u8zYFC78CAIA2OUMIu3GAEACAvjhDCAAAEEoQwlGMBwEAaJwzhPDdlb7VEAAAHkwI4YuZTxb1OBkAALojCKFkdeaZJQIA0D4ro7CVh4sCANApE0IomaTda+mpQQAA+mVCCAuUN0ilIAAAffHF9LCAeSAAAFciCAEAAEI5QwgAABBKEAIAAIQShAAAAKEEIQAAQKj/EUL9GuDRRZMAAAAASUVORK5CYII=",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 3,
     "metadata": {
      "image/png": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='bras.png', width=400)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa42eda0-9ce8-478e-8587-79e0ac65ba3f",
   "metadata": {},
   "source": [
    "* Centre de masse\n",
    "\n",
    "La position des centres de masse est\n",
    "$$\\bm G_1 = \\frac{l}{2} \\hat{\\bm x}_1$$\n",
    "et\n",
    "$$\\bm G_2 = 2\\bm G_1 + \\frac{l}{2} \\hat{\\bm x}_2$$\n",
    "\n",
    "En utilisant la rotation $\\hat{\\bm x}_1 = R_z(\\phi_1) \\hat{\\bm X}$ on obtient\n",
    "$$\\hat{\\bm x}_1 = \\begin{pmatrix} \\cos\\phi_1 \\\\ \\sin\\phi_1 \\\\ 0\\end{pmatrix}$$\n",
    "De même on a \n",
    "$$\\hat{\\bm x}_2 = R_z(\\phi_2)R_z(\\phi_1) \\hat{\\bm X} = R_y(\\phi) \\hat{\\bm X}$$\n",
    "ce qui donne\n",
    "$$\\hat{\\bm x}_2 = \\begin{pmatrix} \\cos\\phi \\\\ \\sin\\phi \\\\ 0\\end{pmatrix}$$\n",
    "\n",
    "* Vitesse\n",
    "\n",
    "$$\\bm  V_1 = \\frac{l \\dot{\\phi}_1}{2}  \\begin{pmatrix} -\\sin\\phi_1 \\\\ \\cos\\phi_1 \\\\ 0\\end{pmatrix}$$\n",
    "et\n",
    "$$\\bm  V_2 = \\frac{l}{2}  \\begin{pmatrix} -2 \\dot{\\phi}_1\\sin\\phi_1 - (\\dot{\\phi}_1 + \\dot{\\phi}_2) \\sin(\\phi_1 + \\phi_2) \\\\ 2 \\dot{\\phi}_1\\cos\\phi_1 + (\\dot{\\phi}_1 + \\dot{\\phi}_2) \\cos(\\phi_1 + \\phi_2) \\\\ 0\\end{pmatrix}$$\n",
    "\n",
    "\n",
    "* Vitesse angulaire\n",
    "\n",
    "On a\n",
    "$$\\bm \\omega_1 = \\dot{\\phi}_1 \\hat{\\bm Z}$$\n",
    "et\n",
    "$$\\bm \\omega_2 = (\\dot{\\phi}_1 + \\dot{\\phi}_2) \\hat{\\bm Z} = \\dot{\\phi} \\hat{\\bm Z}$$\n",
    "\n",
    "Le moments d'inertie par rapport aux centres de masse sont (en supposant des cylindres de section négligeable)\n",
    "$$I = I _G = \\frac{ml^2}{12}, \\quad I_y = I_z = I$$\n",
    "pour les deux bras."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c19d3e17-5152-4be7-9321-887aa2d6d418",
   "metadata": {},
   "source": [
    "## Énergie cinétique\n",
    "\n",
    "L'énergie cinétique s'écrit\n",
    "$$K(\\phi_1,\\phi_2) = \\frac{m}{2}\\big( V_1^2 + V_2^2\\big) + \\frac{I}{2}\\big( \\omega_1^2 + \\omega_2^2 \\big)$$\n",
    "avec\n",
    "$$V_1^2 = \\frac{l^2}{4} \\dot{\\phi}_1^2$$\n",
    "et\n",
    "$$V_2^2 = \\frac{l^2}{4}\\Big[4\\dot{\\phi}_1^2 + (\\dot{\\phi}_1 + \\dot{\\phi}_2)^2 + 4\\dot{\\phi}_1(\\dot{\\phi}_1 + \\dot{\\phi}_2)\\cos\\phi_2  \\Big]$$\n",
    "La partie de vitesse angulaire est\n",
    "$$\\frac{I}{2}\\big(\\omega_1^2 + \\omega_2^2\\big) = \\frac{ml^2}{24}\\big( \\dot{\\phi}_1^2 + (\\dot{\\phi}_1 + \\dot{\\phi}_2)^2 \\big)$$\n",
    "On trouve finalement,\n",
    "$$K = \\frac{2ml^2}{3}\\dot{\\phi}_1^2 + \\frac{ml^2}{6}(\\dot{\\phi}_1 + \\dot{\\phi}_2)^2 + \\frac{ml^2}{2} \\dot{\\phi}_1(\\dot{\\phi}_1 + \\dot{\\phi}_2)\\cos\\phi_2 $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8492c4f2-a81e-4471-bab6-914cf2edf0eb",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "f1 = sy.Function(\"phi_1\", real = True)\n",
    "f2 = sy.Function(\"phi_2\", real = True)\n",
    "t = sy.symbols(\"t\", real = True)\n",
    "m, l, g = sy.symbols(\"m, ell, g\", positive = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "46674371-3853-4389-9785-260661f5ce9e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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QKlsTaujm8LIapX4p1MCU6i8pYwGFjgY93fja5v27YwjUqYELRfGwmi2UHroGjqKBbodt1C492rPJNghPHIufePHc0K3eMuwOzzMz5Tvq68KRcUDSpXXPn1ckg8pYeMzRndSFjq8XL1wG2BqYAbhNws6zmI/c2UqTpSdh8qLtQ/b1/frGwVBG/5xyfZroEh9TA90Oj6n9i6DNgSSebQ6DfiRV6V1SByobZAeS1d2SMEc3hYtnrKVVMXVqYFK41+bZozrzkcN7mjYTufrtWTTrByoexvOVDPb7F8+a1rZQr9c1YBrodmia6PFaDTB+qS7PLpMrQ+XnDuOsJenqzdGNkXv+4DPrg2pgYrwN7o0fzvy4YCtNyzAAK7+6WA3DBILZDE6T7VkCs5+lMy1Xsf90DazRQLfDNVrrdVIakC0xrm/iHFP0LG8JXcHO8lcDY7RbxaLJosmhU+wON90qYavMndLX9JA8p1f2sJlE2F62KWxu2yCHr+d3DazRQLfDNVrrdboG2muASYd73qr4PStNW01d9SpTEwYO/HBc+iPFm4SARlbXgon/0HfgpVQ2APXE5hqoaccGTDwXDuxxla00oN8ERY2uSnZdKmvCYEfSNTCvAVtMugXmjeBtpWkF8ygaQ6hj8KIul/FSRUHwvPBrLw1X1UkBgUf5LMOzAxT1PFwKxWxejobyf9c1yK00XwJhe3gSSmUT4AvNkA5O3lbE4+grMEubwtef2KPyq21lKc0c/D6ytJBDOLL9Icdzz+8aaKwB8wt8VMIdBHIJpT80JlSFTp2CE6qc2sVRDM6jqvLDRwiSDqay/k702YrlwE/Nu0qlDzBnSeZo+Hz2ya1RDAdfJMq9q1Qqs/oXGXudnIOtrLITGk0yJu3R5y+1lZEdCMeavrJKlsZyXK3Njxqw3xxLA+YbnX+6ERfmqA6+0lTHgrb7eLHiO93XOK5BcYJn+2r2AfJQIZ3AOdUOJrPHs9Mk3Mu6KRp3gp+8G+v1gD4YQEehVDYCvLAbyX1OtrLWTmi1nD0utpWECVhfTxRls9bK0kyOa7X5bIv0gkNrwHyj6z8807SB2bzpIRnicA0fQSYYHw93/lcdhqPSrMQe62LW+4fy+OYj8Lx/hHNx/9EZ5CGc+2CyYtK8wMt9KlC/eMhH5eDgmSd/zcPrJ/HKMIU3zBvR8PigCU624RhgXisOHSgTCAZK04+SQyiVDUAXljh5W/Htuo+d0GStbWWVGTSQpbUc12jzq9quV9pMA85H3Qg9zuhYgWc/OAsc4+QrEMrHaXysmL/RoZzVmmNc9/dKj5xdkIeTw7nZP5yTZjAbBeUBB55kUDnbYThmDku5jyQrZmWMk6sKKRrK47N8vMICbT6QzOstocMEN9uQuReSS2XUvcRwsraittvbTmgw4ZnYo/L2tZVFttBClo3kuEabX9R2KWDfFqmiTfNq6dbA1cBsKsxuZ7stzleGK82s89iCIa+IR8L9RBeft0rRZ/XLaVLg+Msy4MLt2NLqONzqBY4BKQ7kJXGIDo7yJ8XuE3qKbYXJwQQOZDBQvgehYvDg0PnngFiOLA3BQyPkU7dDAE+KZwBKZQOCS0l4/WIDJ2cr4q2VndBcW9hKtRnUyiI4JrO0B/yy+xP+K5BuN5HjqmweJe4b1E62S7d0Z2wv0gvpugWN6oTjeky/Biau0/Ie2yNg8+4gkPOe3Bw4MNjATPZj6FIkTonVFp2TFd9/lUeHnQ2CM0GBDdNhXZSQK2M1+SoEjtJOb6KDHFw5B5ek4eVgazZHP+nMPQ+lsojNi7g9ZVtpZSc0VDNbkV3xF0y/hpfwPwnvfTo86Vsli/DgJG0Xhx2T+D94m8kh3BZOyuYlP5NmdJw6q2A8Z2PVwxEwnoX6z8IvLRBedtbcLl1N3VZ8rKDrFg0l+iqbhamRsRUMK00LtgS1+61jnCErx9xKa6cyBst3it02pWIMjI7tVn+K9w3MwOjgqcCqxj0H9XzgwHdK4xwxeGtI8jmQBF+pkKOBbIYDHsDpaHgk0MnNEEtlvvpFRadsK63shAZrZiuypdGjC5ArDyfq+hL3iVAli+qF/Q9bvI9wNZMjwHsyNi8dwovbhVL6Ttew6xTwO5cEB/0+1OVcnapy8QPel4qX4F51SjpkaI6uL/9ddXhbYRjblLbduzdKx7bkSNTAhLw0To8mbDcB8iSzQXmzpBTAanE0Q/N5MQ0Ma3BGguG55kiAuELmHiOaBOHDSUEjFXBotqqlY+DgwQPfT1MVUnkFGhgpz2kIL3UNRuRyHvjKvYMKz7kyX/0yIunv1G2liZ3QWhvZyhJDqJJFfIa2Sv8cHbLbSI69bV580Y9brOzY9bJdqHiyW6Vv8YGuP1I8mdxUISgD0Saj8bUM7kpbLJrm6NKXacfUGA6/w1jvOJr+1MBMazXOuRG+pENpTGeEDoPRhcN6q5hnluy9pxTp6nmYF4pRqjMypeH7JwCUdgbi85zigzw6CTPoL5VnTpBqFtgepiFHQXkMBJy6BR8zI3gEF88t4X1JSNGAZ07jgpPDUPcRQlYEqZOzgJXKIjTnfSu9nLStiL+WdkJjtbaVagNYKovgsWEOseEA4tBajhY2z5ix13gnWT8XDhyl9c3VKzThiPt8rMO198+Ee3KwModMsIyLS8e0FLo5urQhdjGR2/NL/Wz71MCkmGqQN+L3NkCYdVoBTNOklFB6+LtTOYZpxjmirTIEiWe4qTyMp2RA5ognH2QXDZenGMde5HXE3PRmQkP4MNLkLFNlznAUhzN6h7VUNiV7OTmSu6h/lR/NVkS7lZ3QYM1sZU3r18oiOPjEYfJcPtU/mskh/Nn+sEbGPevQZ914Ir6YbJ/Ujo94wqlPxo0ZmdleLvavmfq7Sro45+R47vHD9zNdpfG6BsajaxYNvlFyPsJp2sCdmi0uoYow4BoILKl8LFgpgZUMq0dmjxNjUx7PHnFwyeDLMYbHuniOwOp5ZBi6L9JIILYBJ1E0DKqpslwebWvtnIM5VP5Z2gnKKbWjyva2kzkamQYq2UpcZWLfMYDnYU4WnksBg7OkCnhHg25JV1RIhJIcpbIEqm2yJBPOm75uE2wmDcl+pXz0AywTeef0FbPbxf/z4mzZ4iV2Y4Pina/D7hn5b3QhN06Q8JkuxpaRnl3J+IfV3OxY7ml9KVi2ZdlJU+ROQruJEDcLQ5Ku8KIDdIQOkAsHjewpWeAbPCWnWQMjFC60GvdszNqJ9/tboXZbm4opmFW2YyX9g0LABQ4M5WyCFMH/uPFNU96ZjHnnxGG2EVWGQ+UqGrPgSjQGXQmO7dqfFU8GuFLZgCCdwHBpm6wc6Wqb5J6tnaANtUGuHZvYyQyNUYMstQd4HyHI38zJUnXApKCrEeWSHKWyEZLD3NA3WVlz8I+0jZ0j6irDQTCe4ACGoHunf8X0bZwnk48h6J5xhHxWr491DStA5eGEedWNU8uTsUFlFngUleTLAFTflSs2fnZKcxiHvgmN0UEdqzcTJ+kKF/6AiQHO/w7cBTzIzVhVCjUwVr/VuGdj1k78v7kx7Ipp6KsNUobNHkc6UH4zR5OjMSKoGabgcpOXUlmEpt9upYFUO7a0E/hO0UjIs4k9tJSlgRybyJjQZU2WW01JJjewK87tQOFAmHjEY2rszD5kiOIUcTDDjlVAa86pQDOHdyc8rPKIzWGCz403yoMuaVa4S0ORrpChu5y+jBZ846BKoQamVH/vstu9MVwQAhnN/dbizNEolZfKtuBb9B4dmuYWcmyB8xB6maMxV76F3GtwzvFZKi+V5XhRHZxTyrmwetupPLXiZvU4OiORwM+AjlMZVoAJGPCzsmJw5z1M4HmP1d5rTVVJ5b1LZSovdsQxGDImxzHxwuoYvfwtqMSqMlwwUHfQncqgx3NGdlhKOwxZup4WOIdJgM+LI5z2nHw1MDHeZvfSwf2tsKEkGHUG1Qx7AZEI/1Uo7kULNSB9um/vLqxWBBdObIJOz2BS6ixFPPsWina3lX2VeIH1ZRdZm1dZyinulM8WIWcXio9SCurCabI9W1OfPvNSFzRZubFdyzezk7yp/BAB58jKnTE/Fz5VAY5pJzjS5kAZD1YF4aEuuos/ghHjm3O8wNfAxHj3vYfmEHCaBw9SYtbgD85MJ5jUgNqIwcHN6jB67pOAG2eKbreVjXXc0c9rQHZozuPVHLRgcRAfFOOk3CpOafcMVDFfEZrbppwjUSpnhZtzcPA1PEcVH8ANp3/9PXK6Fbfnk0kzjn8ulOg63QmP2wYGkdKpMQV+wFMKNTCl+nuX3QjDHJN7E+kIzlMDMmzeQQu3bs5TkM5118AeGlA/cCtUoXArsApUrNBwkkMQDs5GsD35ZMjcJgGPOMdUeK/M0KHizAZHpjQnd1kNz22jCmwSSnRHzzMDfcZIWNHN6bgGJsbb9P5W2O49xlCZTYmcEzI1KHpgW4Vgxvdc+aanh5Lr+eUDD2s60fVoaENJuz1uqNx61DyjZOeFSSRjAs5np3s+eEKZuycvCLx+hgMKxw3GltBJAU5eHHAMaxcz8MLrKanwXJkcBLLt5aFvi0/yWR2v3T4u0R0cnfAjb/hxCN0OgS3tlC4HACVqYEL4Fmlro3uQ4TStcT5ugf0CcPDsYTAcpTlUwJbG6i9/nKtOvIHzHlcPx9NAt8fj6d5RVj9wg6Vu2JqlPWxLM3aKxinwrCxfCNbyGF+pyxdxPlWaiblbdeoeh4VDwyGTTzlOmnxokm9jEs4YZ5c7tPRasNSbBNVhq5V30hnTkIEDTTxvxSnwut2cwxJYNmTpqga7VWxLu9W3YnPaMTJWviZnXGb3NTAG2yrG6ROcr8RpmkGguB4eDJ33JG1GiFFh/BjYPkZ1jrpF7pyBn6M858gzbdDt8QRaTu3AWDk3qO8Ex9hh48eEc5UzjqScXim/arcH3Lp2upLjFeXIoBhHjGNu8kodeHXtdE3oKo8tV7Zos0Ew8EP9kt5mYbIE9isw3+h85Y1wffD4zJvuh/78a9Mp3p2/GPtLIAPuDnN/Ne6Lodvjvhr8X30GPTfw/S/rIlNM9OecO6vZrINaqZUaujnUrEapXwo1MKX6a8tsF9b5SpymGdGjtRgvqZ4cRbzlgvGxpcIMrYeugYNqoNtjO3VLlzx/bLKyasdVe0xeRp4bupVZhgJbvHOHbjJV09mVdCeVPZ/wm22bGpgJ4nYZ5hudvtietf35vtKMlKyGYjbGyTkePvfQNXBUDXR7PKr6z404278828xti86eVfCOikXDnS6cGivBPxWXdqDm6ArFJPCMdW5lXAMzQdwowyYfzlfiNG22YQWN6Jw3Gm8wGAnfgrTV+HkL1bk/Ww10ezzbpjsK44xZunh2mfoHmp3yS47P8SwYfMOiV85Up0g3VobgOYEMn+aHYpBdDcykUtsMW1AOK01j1pagbcmdITY1EhMIvvbvZmn+fqfYdHWGUnWWz1UD3v66PZ5rAx6Jbz9ezTrH1uwtoSvYWf5qYFrLEOGzBaUb/2+8gA5GaSuM6lzPrdcBWwEckeYkGFu0zLbswNT1KKNLenQNdHs8ehN0BroGbEE5rDRRCTc4TK5rX03xTiZKIh6CBq+5PfcBtie6BhpqoNtjQ2V2VF0DSzTgJ61UYdvZPabjmSaBk6HmNF3Gtf5IMR9dq+xd7tPTQLfH02uTztFVacB2X4fF5I0X/62P+ynRq7KHLmzXQNdA10DXQEEDPJ4jDO+0mtO0dxC3/pjwA/n+2zXQNdA10DXQNXD6GvjMs2gLS/ftWfLsCzjmVU9flA051JaYfZmCd5n45qz7ZuSGJDvqroGugYQGel9MKKVnHVID5hPHK00ZJg843Z6t0raHe0jGToaW5OfgxVvFHPHnODQnZ3+9dr2cTAN1Rq5GA70vXk1Tn6Sgsj8OhOIPh0NAMGrbs6Ttg8B8/eEqg5TEV/gfKTZd7JRmQsE9r6H00DXQNXAADfS+eAAldxJzGjBfOKwyqRA6zV89htxnl+YIXEI5n4Cy50K9UOUAAAMTSURBVLuhPOxn36kjM/PooWuga2B7DfS+uL2OO4WyBr70xa9DsMFpyiHgTVlV8a3Vaw3MLFIfMbDjxjbzuFb9dLm7Bg6lgd4XD6XpTienATfeyzcOO48ADk7T13pDLCB7+OmzLz+SzDWryMeXr4kuYdfAcTXQ++Jx9d+pDz4QnzBymOgmdpp8FZ9gy9KHu+v4NYfIajsXahxrrm7P7xroGqjTQO+LdXrqUNtpwHYVR1uzkBs5Tc3wbIuWAzE9TDVgf0Y6Lek5XQNdA4fUQO+Lh9T29dFi4cip2dmVJqp5pYsTpOZpybuGkHqWaXLbzJf3NnvoGuga2FYDvS9uq9+OvaAB+b5PVcyV/FPs0UoTPKpgf9VyVR8ol9y2LZvagrU8OxCEqnroGuga2EADvS9uoNSOcokGXnpgFpCTMHGaHgLH+bmM15zFpOKFZrA9/UlCNltpjt7XScD1rK6BroE2Guh9sY0eO5YFGvA+jzdIfgkmbyMMOadpHtY87qjSBd9wECr1/V0+ZP8+p8QL1kcXrWvgWBroffFYmr9uunaehy/BJUPSaXrnQKWvvedNVr60TMnKHvYHxcO7ql7+Z8p/fmnydnm6Bk5VA70vnmrLXDxfLBR/lP1lH8Xd5lSgSt/r4rkmSLJeN1f/jPNZVfKBdr5uz8Ef4qe6T30pSEU9dA10DWykgd4XN1JsRzvVgMb4r5XLI8miv8s6TY+Syj8L2Q+6sp7Xw15EJDnvJchVHYK6iIbrQlycBnpfvLgmPVmBZGs4S/7d6itvd1lek9uzBq3KvKPCluUPltfjroGuga6BroGugQvTwE+S5zf5vORrJqGsRacJoJCw6vpEsT0gDev3dNdA10DXQNdA18DZakC+jW8ScPEnAbNhbnvWEIDs30L+Rhfbl6Xwh2Dico7vVjEUV+z3yzQgPf+hGp8sq3UU6G4nR1F7J9o1cHka2HPcYyf1C+GY821OcVVOU8h43QKnx/uKOcQ888w9C7yK56FOo8f/YV/+lEO3k1Nunc5b18B5amDVuCe/xrNMnmNWv4P//0+13rIaZv8VAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}\\frac{\\ell \\cos{\\left(\\phi_{1}{\\left(t \\right)} \\right)}}{2}\\\\\\frac{\\ell \\sin{\\left(\\phi_{1}{\\left(t \\right)} \\right)}}{2}\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\frac{\\ell \\cos{\\left(\\phi_{1}{\\left(t \\right)} + \\phi_{2}{\\left(t \\right)} \\right)}}{2} + \\ell \\cos{\\left(\\phi_{1}{\\left(t \\right)} \\right)}\\\\\\frac{\\ell \\sin{\\left(\\phi_{1}{\\left(t \\right)} + \\phi_{2}{\\left(t \\right)} \\right)}}{2} + \\ell \\sin{\\left(\\phi_{1}{\\left(t \\right)} \\right)}\\\\0\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡ell⋅cos(φ₁(t))⎤  ⎡ell⋅cos(φ₁(t) + φ₂(t))                 ⎤⎞\n",
       "⎜⎢──────────────⎥  ⎢────────────────────── + ell⋅cos(φ₁(t))⎥⎟\n",
       "⎜⎢      2       ⎥  ⎢          2                            ⎥⎟\n",
       "⎜⎢              ⎥  ⎢                                       ⎥⎟\n",
       "⎜⎢ell⋅sin(φ₁(t))⎥, ⎢ell⋅sin(φ₁(t) + φ₂(t))                 ⎥⎟\n",
       "⎜⎢──────────────⎥  ⎢────────────────────── + ell⋅sin(φ₁(t))⎥⎟\n",
       "⎜⎢      2       ⎥  ⎢          2                            ⎥⎟\n",
       "⎜⎢              ⎥  ⎢                                       ⎥⎟\n",
       "⎝⎣      0       ⎦  ⎣                   0                   ⎦⎠"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# centre de masse\n",
    "G_1 = (l/2) * sy.Matrix([sy.cos(f1(t)), sy.sin(f1(t)), 0])\n",
    "G_2 = 2*G_1 + (l/2) * sy.Matrix([sy.cos(f1(t) + f2(t)), sy.sin(f1(t) + f2(t)), 0])\n",
    "G_1, G_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "25e55d74-5695-4bf1-ab69-80ebf4e443b3",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}- \\frac{\\ell \\sin{\\left(\\phi_{1}{\\left(t \\right)} \\right)} \\frac{d}{d t} \\phi_{1}{\\left(t \\right)}}{2}\\\\\\frac{\\ell \\cos{\\left(\\phi_{1}{\\left(t \\right)} \\right)} \\frac{d}{d t} \\phi_{1}{\\left(t \\right)}}{2}\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\frac{\\ell \\left(\\left(- \\frac{d}{d t} \\phi_{1}{\\left(t \\right)} - \\frac{d}{d t} \\phi_{2}{\\left(t \\right)}\\right) \\sin{\\left(\\phi_{1}{\\left(t \\right)} + \\phi_{2}{\\left(t \\right)} \\right)} - 2 \\sin{\\left(\\phi_{1}{\\left(t \\right)} \\right)} \\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)}{2}\\\\\\frac{\\ell \\left(\\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)} + \\frac{d}{d t} \\phi_{2}{\\left(t \\right)}\\right) \\cos{\\left(\\phi_{1}{\\left(t \\right)} + \\phi_{2}{\\left(t \\right)} \\right)} + 2 \\cos{\\left(\\phi_{1}{\\left(t \\right)} \\right)} \\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)}{2}\\\\0\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡                d         ⎤  ⎡    ⎛⎛  d           d        ⎞                \n",
       "⎜⎢-ell⋅sin(φ₁(t))⋅──(φ₁(t)) ⎥  ⎢ell⋅⎜⎜- ──(φ₁(t)) - ──(φ₂(t))⎟⋅sin(φ₁(t) + φ₂(\n",
       "⎜⎢                dt        ⎥  ⎢    ⎝⎝  dt          dt       ⎠                \n",
       "⎜⎢──────────────────────────⎥  ⎢──────────────────────────────────────────────\n",
       "⎜⎢            2             ⎥  ⎢                                     2        \n",
       "⎜⎢                          ⎥  ⎢                                              \n",
       "⎜⎢                d         ⎥, ⎢     ⎛⎛d           d        ⎞                 \n",
       "⎜⎢ ell⋅cos(φ₁(t))⋅──(φ₁(t)) ⎥  ⎢ ell⋅⎜⎜──(φ₁(t)) + ──(φ₂(t))⎟⋅cos(φ₁(t) + φ₂(t\n",
       "⎜⎢                dt        ⎥  ⎢     ⎝⎝dt          dt       ⎠                 \n",
       "⎜⎢ ──────────────────────── ⎥  ⎢ ─────────────────────────────────────────────\n",
       "⎜⎢            2             ⎥  ⎢                                     2        \n",
       "⎜⎢                          ⎥  ⎢                                              \n",
       "⎝⎣            0             ⎦  ⎣                                     0        \n",
       "\n",
       "                   d        ⎞⎤⎞\n",
       "t)) - 2⋅sin(φ₁(t))⋅──(φ₁(t))⎟⎥⎟\n",
       "                   dt       ⎠⎥⎟\n",
       "─────────────────────────────⎥⎟\n",
       "                             ⎥⎟\n",
       "                             ⎥⎟\n",
       "                  d        ⎞ ⎥⎟\n",
       ")) + 2⋅cos(φ₁(t))⋅──(φ₁(t))⎟ ⎥⎟\n",
       "                  dt       ⎠ ⎥⎟\n",
       "──────────────────────────── ⎥⎟\n",
       "                             ⎥⎟\n",
       "                             ⎥⎟\n",
       "                             ⎦⎠"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# vitesse: calcul des dérivées\n",
    "V_1 = sy.diff(G_1, t)\n",
    "V_2 = sy.simplify( sy.diff(G_2, t) )\n",
    "V_1, V_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "93a7194c-6d72-4a67-8076-12401856b5f4",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\ell^{2} \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2}}{4}$"
      ],
      "text/plain": [
       "                2\n",
       "   2 ⎛d        ⎞ \n",
       "ell ⋅⎜──(φ₁(t))⎟ \n",
       "     ⎝dt       ⎠ \n",
       "─────────────────\n",
       "        4        "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# carrée des vitesses\n",
    "V2_1 = sy.simplify( V_1.T * V_1 )[0]\n",
    "V2_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "93af366d-1910-46f4-9818-4e051403c3cf",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\ell^{2} \\left(\\cos{\\left(\\phi_{2}{\\left(t \\right)} \\right)} \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2} + \\cos{\\left(\\phi_{2}{\\left(t \\right)} \\right)} \\frac{d}{d t} \\phi_{1}{\\left(t \\right)} \\frac{d}{d t} \\phi_{2}{\\left(t \\right)} + \\frac{5 \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2}}{4} + \\frac{\\frac{d}{d t} \\phi_{1}{\\left(t \\right)} \\frac{d}{d t} \\phi_{2}{\\left(t \\right)}}{2} + \\frac{\\left(\\frac{d}{d t} \\phi_{2}{\\left(t \\right)}\\right)^{2}}{4}\\right)$"
      ],
      "text/plain": [
       "     ⎛                                                                        \n",
       "     ⎜                                                             ⎛d        ⎞\n",
       "     ⎜                      2                                    5⋅⎜──(φ₁(t))⎟\n",
       "   2 ⎜           ⎛d        ⎞               d         d             ⎝dt       ⎠\n",
       "ell ⋅⎜cos(φ₂(t))⋅⎜──(φ₁(t))⎟  + cos(φ₂(t))⋅──(φ₁(t))⋅──(φ₂(t)) + ─────────────\n",
       "     ⎝           ⎝dt       ⎠               dt        dt                4      \n",
       "\n",
       "2                                    2⎞\n",
       "    d         d           ⎛d        ⎞ ⎟\n",
       "    ──(φ₁(t))⋅──(φ₂(t))   ⎜──(φ₂(t))⎟ ⎟\n",
       "    dt        dt          ⎝dt       ⎠ ⎟\n",
       "─ + ─────────────────── + ────────────⎟\n",
       "             2                 4      ⎠"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# carrée des vitesses\n",
    "V2_2 = sy.simplify( V_2.T * V_2 )[0]\n",
    "V2_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "87361806-7186-4cb3-917c-8f0c90eb2a4a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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dGh4RcwrDGBWRwxv+B7pWv1mAMT2zbcSbSLZtVMRIGROlIiAdYfJkOjQwQPbp7AGASkiWMPOZ5Ce6VucIuscZQFwbv2VwsZsNZXhrKbx9UZX16XYxBE4GAdOh9KG2t4/SsTpITgkzB8sNh+AZwei/UTorBAx/gxT/RRH8wKmx1dTIZA+GwAkgYDqUN8jmFPLwWjS3hJlVAIdiGPf2q6aXimsfNtf54zcMsd8x1PPYvSFw1AiYDuUPrzmFfMyWLPFOjeEY2AttU+M3CiHRO4+wtYTjwKm8H3AgobhdDYFjQ8B0KHNE/wNhY4HR39ersgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\ell^{2} m \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)} + \\frac{d}{d t} \\phi_{2}{\\left(t \\right)}\\right)^{2}}{24} + \\frac{\\ell^{2} m \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2}}{24}$"
      ],
      "text/plain": [
       "                              2                     2\n",
       "   2   ⎛d           d        ⎞       2   ⎛d        ⎞ \n",
       "ell ⋅m⋅⎜──(φ₁(t)) + ──(φ₂(t))⎟    ell ⋅m⋅⎜──(φ₁(t))⎟ \n",
       "       ⎝dt          dt       ⎠           ⎝dt       ⎠ \n",
       "─────────────────────────────── + ───────────────────\n",
       "               24                          24        "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# vitesse angulaire\n",
    "w_1 = sy.diff(f1(t), t)\n",
    "w_2 = sy.diff(f1(t), t) + sy.diff(f2(t), t)\n",
    "# wIw\n",
    "I = m*l**2/12\n",
    "k = I*w_1**2/2 + I*w_2**2/2\n",
    "k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "5b8ccd3b-ca2c-498e-92fa-ee9e3174085d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\ell^{2} m \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)} + \\frac{d}{d t} \\phi_{2}{\\left(t \\right)}\\right)^{2}}{24} + \\frac{\\ell^{2} m \\left(\\cos{\\left(\\phi_{2}{\\left(t \\right)} \\right)} \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2} + \\cos{\\left(\\phi_{2}{\\left(t \\right)} \\right)} \\frac{d}{d t} \\phi_{1}{\\left(t \\right)} \\frac{d}{d t} \\phi_{2}{\\left(t \\right)} + \\frac{5 \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2}}{4} + \\frac{\\frac{d}{d t} \\phi_{1}{\\left(t \\right)} \\frac{d}{d t} \\phi_{2}{\\left(t \\right)}}{2} + \\frac{\\left(\\frac{d}{d t} \\phi_{2}{\\left(t \\right)}\\right)^{2}}{4}\\right)}{2} + \\frac{\\ell^{2} m \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2}}{6}$"
      ],
      "text/plain": [
       "                                         ⎛                                    \n",
       "                                         ⎜                                    \n",
       "                              2          ⎜                      2             \n",
       "   2   ⎛d           d        ⎞       2   ⎜           ⎛d        ⎞              \n",
       "ell ⋅m⋅⎜──(φ₁(t)) + ──(φ₂(t))⎟    ell ⋅m⋅⎜cos(φ₂(t))⋅⎜──(φ₁(t))⎟  + cos(φ₂(t))\n",
       "       ⎝dt          dt       ⎠           ⎝           ⎝dt       ⎠              \n",
       "─────────────────────────────── + ────────────────────────────────────────────\n",
       "               24                                                             \n",
       "\n",
       "                                    2                                    2⎞   \n",
       "                         ⎛d        ⎞    d         d           ⎛d        ⎞ ⎟   \n",
       "                       5⋅⎜──(φ₁(t))⎟    ──(φ₁(t))⋅──(φ₂(t))   ⎜──(φ₂(t))⎟ ⎟   \n",
       " d         d             ⎝dt       ⎠    dt        dt          ⎝dt       ⎠ ⎟   \n",
       "⋅──(φ₁(t))⋅──(φ₂(t)) + ────────────── + ─────────────────── + ────────────⎟   \n",
       " dt        dt                4                   2                 4      ⎠   \n",
       "─────────────────────────────────────────────────────────────────────────── + \n",
       "               2                                                              \n",
       "\n",
       "                   \n",
       "                   \n",
       "                  2\n",
       "   2   ⎛d        ⎞ \n",
       "ell ⋅m⋅⎜──(φ₁(t))⎟ \n",
       "       ⎝dt       ⎠ \n",
       "───────────────────\n",
       "         6         "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = m*V2_1/2 + m*V2_2/2 + k\n",
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "b5b4d7d2-4314-4029-9839-eba24d78917e",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\frac{\\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2}}{2} + \\frac{\\frac{d}{d t} \\phi_{1}{\\left(t \\right)} \\frac{d}{d t} \\phi_{2}{\\left(t \\right)}}{2}\\right) \\cos{\\left(\\phi_{2}{\\left(t \\right)} \\right)} + \\frac{5 \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2}}{6} + \\frac{\\frac{d}{d t} \\phi_{1}{\\left(t \\right)} \\frac{d}{d t} \\phi_{2}{\\left(t \\right)}}{3} + \\frac{\\left(\\frac{d}{d t} \\phi_{2}{\\left(t \\right)}\\right)^{2}}{6}$"
      ],
      "text/plain": [
       "⎛           2                      ⎞                           2              \n",
       "⎜⎛d        ⎞    d         d        ⎟                ⎛d        ⎞    d         d\n",
       "⎜⎜──(φ₁(t))⎟    ──(φ₁(t))⋅──(φ₂(t))⎟              5⋅⎜──(φ₁(t))⎟    ──(φ₁(t))⋅─\n",
       "⎜⎝dt       ⎠    dt        dt       ⎟                ⎝dt       ⎠    dt        d\n",
       "⎜──────────── + ───────────────────⎟⋅cos(φ₂(t)) + ────────────── + ───────────\n",
       "⎝     2                  2         ⎠                    6                   3 \n",
       "\n",
       "                      2\n",
       "           ⎛d        ⎞ \n",
       "─(φ₂(t))   ⎜──(φ₂(t))⎟ \n",
       "t          ⎝dt       ⎠ \n",
       "──────── + ────────────\n",
       "                6      "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.collect(sy.expand(K/m/l**2), sy.cos(f2(t)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac803491-acfe-4d54-bfd3-5e4e508d14d6",
   "metadata": {},
   "source": [
    "## Équation de mouvement\n",
    "\n",
    "Méthode dite analytique (voir [Wikipedia](https://en.wikipedia.org/wiki/Lagrangian_mechanics)).\n",
    "\n",
    "On utilise la formule, valable pour le cas conservatif:\n",
    "$$\\frac{\\D}{\\D t}\\frac{\\D \\mathcal{L}}{\\D \\dot\\phi_1} - \\frac{\\D \\mathcal{L}}{\\D \\phi_1} = 0, \\quad\n",
    "\\frac{\\D \\mathcal{L}}{\\D \\dot\\phi_2} - \\frac{\\D \\mathcal{L}}{\\D \\phi_2} = 0$$\n",
    "avec\n",
    "$$\\mathcal{L} = K - U$$\n",
    "où $K = K(\\phi_{1,2}, \\dot\\phi_{1,2})$ est considérée comme une fonction des angles et des vitesses, et $U(\\phi_{1,2})$ est l'énergie potentielle, due à la gravité $-mgZ$:\n",
    "$$U = \\frac{3mgl}{2}\\sin(\\phi_1) + \\frac{mgl}{2}\\sin(\\phi_1 + \\phi_2)$$\n",
    "($Z = l\\sin\\phi_1/2$, etc. c'est la hauteur du centre de masse). La fonction $\\mathcal{L}$ s'appelle le \"lagrangien\", et c'est la différence de l'énergie cinétique et de l'énergie potentielles vues comme fonctions de positions et vitesses de tous les dégrés de liberté du système mécanique."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "b416ee13-2e95-4940-808f-77b205305ae5",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\ell g m \\sin{\\left(\\phi_{1}{\\left(t \\right)} + \\phi_{2}{\\left(t \\right)} \\right)}}{2} + \\frac{3 \\ell g m \\sin{\\left(\\phi_{1}{\\left(t \\right)} \\right)}}{2}$"
      ],
      "text/plain": [
       "ell⋅g⋅m⋅sin(φ₁(t) + φ₂(t))   3⋅ell⋅g⋅m⋅sin(φ₁(t))\n",
       "────────────────────────── + ────────────────────\n",
       "            2                         2          "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U = 3*m*g*l/2*sy.sin(f1(t)) + m*g*l/2*sy.sin(f1(t) + f2(t))\n",
    "U"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "3bb29ba9-874a-46bc-a985-c17dbfc2fb48",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "lag = K - U\n",
    "f1t = sy.diff(f1(t), t)\n",
    "f2t = sy.diff(f2(t), t)\n",
    "Eq1 = sy.diff(sy.diff(lag, f1t), t) - sy.diff(lag, f1(t))\n",
    "Eq2 = sy.diff(sy.diff(lag, f2t), t) - sy.diff(lag, f2(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "1cbbd59a-2d8d-429d-88b2-0356bf74c85a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left(\\frac{d^{2}}{d t^{2}} \\phi_{1}{\\left(t \\right)} + \\frac{\\frac{d^{2}}{d t^{2}} \\phi_{2}{\\left(t \\right)}}{2}\\right) \\cos{\\left(\\phi_{2}{\\left(t \\right)} \\right)} - \\sin{\\left(\\phi_{2}{\\left(t \\right)} \\right)} \\frac{d}{d t} \\phi_{1}{\\left(t \\right)} \\frac{d}{d t} \\phi_{2}{\\left(t \\right)} - \\frac{\\sin{\\left(\\phi_{2}{\\left(t \\right)} \\right)} \\left(\\frac{d}{d t} \\phi_{2}{\\left(t \\right)}\\right)^{2}}{2} + \\frac{5 \\frac{d^{2}}{d t^{2}} \\phi_{1}{\\left(t \\right)}}{3} + \\frac{\\frac{d^{2}}{d t^{2}} \\phi_{2}{\\left(t \\right)}}{3} + \\frac{g \\cos{\\left(\\phi_{1}{\\left(t \\right)} + \\phi_{2}{\\left(t \\right)} \\right)}}{2 \\ell} + \\frac{3 g \\cos{\\left(\\phi_{1}{\\left(t \\right)} \\right)}}{2 \\ell}$"
      ],
      "text/plain": [
       "⎛               2       ⎞                                                     \n",
       "⎜              d        ⎟                                                     \n",
       "⎜             ───(φ₂(t))⎟                                                     \n",
       "⎜  2            2       ⎟                                               sin(φ₂\n",
       "⎜ d           dt        ⎟                         d         d                 \n",
       "⎜───(φ₁(t)) + ──────────⎟⋅cos(φ₂(t)) - sin(φ₂(t))⋅──(φ₁(t))⋅──(φ₂(t)) - ──────\n",
       "⎜  2              2     ⎟                         dt        dt                \n",
       "⎝dt                     ⎠                                                     \n",
       "\n",
       "                        2            2                                        \n",
       "                2      d            d                                         \n",
       "     ⎛d        ⎞    5⋅───(φ₁(t))   ───(φ₂(t))                                 \n",
       "(t))⋅⎜──(φ₂(t))⎟        2            2                                        \n",
       "     ⎝dt       ⎠      dt           dt           g⋅cos(φ₁(t) + φ₂(t))   3⋅g⋅cos\n",
       "───────────────── + ──────────── + ────────── + ──────────────────── + ───────\n",
       "     2                   3             3               2⋅ell               2⋅e\n",
       "                                                                              \n",
       "\n",
       "       \n",
       "       \n",
       "       \n",
       "       \n",
       "(φ₁(t))\n",
       "───────\n",
       "ll     \n",
       "       "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.collect( sy.simplify(Eq1/m/l**2), sy.cos(f2(t)) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "dc1b0530-4b56-435c-bd83-7809a7b17722",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sin{\\left(\\phi_{2}{\\left(t \\right)} \\right)} \\left(\\frac{d}{d t} \\phi_{1}{\\left(t \\right)}\\right)^{2}}{2} + \\frac{\\cos{\\left(\\phi_{2}{\\left(t \\right)} \\right)} \\frac{d^{2}}{d t^{2}} \\phi_{1}{\\left(t \\right)}}{2} + \\frac{\\frac{d^{2}}{d t^{2}} \\phi_{1}{\\left(t \\right)}}{3} + \\frac{\\frac{d^{2}}{d t^{2}} \\phi_{2}{\\left(t \\right)}}{3} + \\frac{g \\cos{\\left(\\phi_{1}{\\left(t \\right)} + \\phi_{2}{\\left(t \\right)} \\right)}}{2 \\ell}$"
      ],
      "text/plain": [
       "                                       2            2            2            \n",
       "                      2               d            d            d             \n",
       "           ⎛d        ⎞    cos(φ₂(t))⋅───(φ₁(t))   ───(φ₁(t))   ───(φ₂(t))     \n",
       "sin(φ₂(t))⋅⎜──(φ₁(t))⎟                 2            2            2            \n",
       "           ⎝dt       ⎠               dt           dt           dt           g⋅\n",
       "─────────────────────── + ───────────────────── + ────────── + ────────── + ──\n",
       "           2                        2                 3            3          \n",
       "\n",
       "                  \n",
       "                  \n",
       "                  \n",
       "                  \n",
       "cos(φ₁(t) + φ₂(t))\n",
       "──────────────────\n",
       "     2⋅ell        "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.simplify(Eq2/m/l**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "610880bb-a637-4862-9f46-6bfb2e91a4c9",
   "metadata": {},
   "source": [
    "On obtient finalement,\n",
    "$$\\left( \\frac{4}{3} + \\cos\\phi_2 \\right) \\ddot\\phi_1 +\n",
    "\\left( \\frac{1}{3} + \\frac{1}{2}\\cos\\phi_2 \\right) \\ddot\\phi_2 -\n",
    "\\dot\\phi_2 (\\dot\\phi_1 + \\dot\\phi_2) \\sin \\phi_2 + \n",
    "\\frac{3g}{2l} \\cos\\phi_1 + \\frac{g}{2l} \\cos(\\phi_1 + \\phi_2) = C_1$$\n",
    "et\n",
    "$$\\left( \\frac{1}{3} + \\frac{1}{2}\\cos\\phi_2 \\right) \\ddot\\phi_1 + \n",
    "\\sin\\phi_2 \\frac{\\dot\\phi_1^2}{2} + \n",
    "\\frac{\\ddot\\phi_2}{3} + \\frac{g}{2l} \\cos(\\phi_1 + \\phi_2) = C_2$$\n",
    "où nous avons ajouté deux moteurs en $O_{1,2}$, dont les couples respectifs sont $C_{1,2}$.\n",
    "\n",
    "Le cas du *double pendule* s'obtient en faisant $C_{1,2}= 0$ et en inversant la gravité $g \\rightarrow -g$ (pour avoir a $\\phi_{1,2} =  0$ le point d'équilibre)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba00f76b-e478-46f7-b3a8-12d8f05fb121",
   "metadata": {},
   "source": [
    "## Application: décollage\n",
    "\n",
    "On se propose de déterminer la vitesse angulaire $\\Omega$ nécessaire pour faire décoller le bras (2) du système de la figure ci-dessous."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e7fe5896-ef88-4d44-8df0-7ed6befc53ed",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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s10cQHjThxG0BakIAAGCD4+l0aj3DCjMbZUn+aO0HAADspZsdwrOE+4RqEAAAKKSzIDwka0I1CAAAlNNfEB5yNOH78YsaBAAAiuoyCA+jN2HD+xUBAIA8eg3Cw6BNeLUxeFCDAABAMR0H4eFeE3adhbfDq0EAAKCcT60HeNU5mX653e74pbuOkoIAAEB9fe8QXnS9VagGAQCAJgYJwsO9iOoiC2/vGFSDAABAHeME4eFBTYVtQufHAAAAbXV/D+Gtu3cVTn+prUeBGmE2AAAglQGD8Oz99O3OvXnHL4dG6TWzUSkFAQCAJoYNwsO9rcIfP193w1AKAgAAMY0chGePsvBQuAx1IAAAENz4QXh2abD5Mjy8UGtLTq+RggAAQBzH0+nUeoYGKh89qgMBAICAkgbhVNE4lIIAAEBYgvAXL8ah/AMAADoiCBe5f+eh/AMAAHr2j9YD9OG2/dQgAADQO0EIAACQlCAEAABIShACAAAkJQgBAACSEoQAAABJCUIAAICkBCEAAEBSghAAACApQQgAAJCUIAQAAEhKEAIAACQlCAEAAJIShAAAAEkJQgAAgKQEIQAAQFKCEAAAIClBCAAAkJQgBAAASEoQAgAAJCUIAQAAkhKEAAAASQlCAACApAQhAABAUoIQAAAgKUEIAACQlCAEAABIShACAAAkJQgBAACSEoQAAABJCUIAAICkBCEAAEBSghAAACApQQgAAJCUIAQAAEhKEAIAACQlCAEAAJIShAAAAEkJQgAAgKQEIQAAQFKCEAAAIClBCAAAkJQgBAAASEoQAgAAJCUIAQAAkhKEAAAASQlCAACApAQhAABAUoIQAAAgKUEIAACQlCAEAABIShACAAAkJQgBAACSEoQAAABJCUIAAICkBCEAAEBSghAAACApQQgAAJCUIAQAAEhKEAIAACQlCAEAAJIShAAAAEkJQgAAgKQEIQAAQFKCEAAAIClBCAAAkJQgBAAASEoQAgAAJPWp9QAPHY/H+Tc4nU6vv6vl7wQAAGAw4+8QPg1LAACAnOLuEF727q6Kbu2e3vntp+/EriAAAMChix3CHfvtdDqpQQAAgLMOgvDK5ktApSAAAMBUH0H4Yssdj8d9a/D99G3H9wYAANDEzqVUzuabAHevQeB1n79+vvz4+5/fl7z9kjcDAGCVbmJp29Ey59/VyxohiXMNngPvUoYzvbfkbQAA2KCPS0YPm6JODUJA0xo8aDwAgKa6CcK11CCENY3A6bWjd9keBAAop6cgnNbd/FmjahBicisgAEAoPQXhKmoQAABgXsdB+GiT0LGiENPt1aFPrxcFAKCozoLwaempQYjs0fWij37eDYQAAEV1FoTz1CBEJuoAAKLpOwinV43OHzMDRGP3DwCguf6C8O4eoGNFAQAA1uovCG+pQRiSLUQAgNK6DMLbFyRUg9AXsQcAEEGXQXhFDQIAAGwwQhACAACwQfdBaHsQhuSaUgCACroMQq8wAQAA8LougxAAAIDXCUIgnMv1ogAAFNV3ELqBEDp1uTPwtv2mP+MGQgCAoo7dNdX0BsLuhgem7u4Efv/zuxNlAADq+NR6ACAvvQcA0Fbfl4wCAACwWcdB6HpRGJLrRQEAqunsklGvQAi903sAAHH0tEN4VYPiELrz9PUk5CIAQE3Rdwjnq+/yqy4fhQGoQQCAyqIHodKDwTyNPTUIAFBNT5eMAr2bvsbglfPPq0EAgJr6e2F6oHefv36ehp8rRQEAWhGEQAPTfUIdCADQiiAEAABIyj2EAAAASQlCAACApAQhAABAUoIQAAAgKUEIAACQlCAEAABIShACAAAkJQgBAACS+tR6gJ4cj8fzD06nU9tJgKkvX7+cf/Dtz29tJwEA6IsgXOpSg0BY5zKUhZDE56+fp//5/c/vy38VgLOjza6FpkHogwahXHYILzQhjO0ce9PGu/qZz18/3/ahJgS45R5CYEC3iQgM47YGL/95/qWr9vv89fPl56sOCtADQbiFy0chlLv7gV++fpGFMKqZvb6Z6rNDCHBLEC6iAKFTmhAGs+TKT+EHsJxDZYARfPvz26P2c9IMDGPbNZ/6EGCGHUIgBVuFMIZHdef+QIBtBOFGLiKFaJ7uAbqrEDKwHwiwiiB8TvvBSDQh9EvsAexOEALjWHijoCYEADgThNvZOYR+uXwURnK5gdAWIsBagvAJ1Qd9WXWaqCYEAJIThEBqtgoBgMwEITCaDS85qAkBgJwE4UtcUArDsFUInVpyA6FXKQR4RBDO0XvQqQ2bhGeaEMajBgFmCEKAX2hCiOnz18/n/234vU4fBXhEEAJc04TQi6d9aHsQYJ4gfJXLSiGmzVeNAgHNdN3T3T/bgwAzBOFDSg8AgrhE3bTuPn/9fPWf099yvr5UDQLMO55Op9YzBLU8CH0MIawNF3/aWoSYbjcJZ2rw9g0AuEsQPrRqh9CHEWJaG4RqEABIxSWj97leFMawKvDUIACQjSAEOBzUIACQkiAEBrew9LzUBACQkCDch0tMYQCaEADIRhDeoe5gMI82Cb/9+e3qlzQhAJDKp9YDALRxScFvf37TgdDE9FPPfbwATXjZiTu27RD6SEJw8089PTGFau5+C8bnHUATLhndjQtNoRd3n3dOf9KGIRTy5esXNQgQih3COzannQ8mdO3qeapnqLCjme+z+FwDaMgO4W7UIPTOATNQwqNdQQAisEO4wmXn0AcNBmafEPaypAN9igG0ZYcQ4BeensLrFu4K+nQDaE4QAlxzwAxs5gJRgL64ZHQFl4xCKl6IApbbEIE+rQAisEMI8JwdD3hk25agGgQIQhAC3OfQUZjn6lCAAbhkdAWXjEJCDh2FWy92oM8jgDjsEALM8cwVpl7fFfQ5BRCKHcIV7BBCWg6YgV2uDvXpAxCNHUKA57wQBQAwJEEIsIgmJLnXN/dsDwIEJAgBttCEJPRK0alBgJgEIcBSXogCABiMIARYwS4HyW37FPCJAxCWIARYx82EJLe27tQgQGSCEGA1TQgAjEEQAmyhCUlr1T9424MAwQlCgB1oQpJQgwCDEYQAGzl0lGyu/pHP954aBOiCIATYThOSx90aVH0AvROEAC/xhJjhffn6ZWZv8O6ngM8LgF4IQoBXOWCGgd3+k76NvaufUYMAHRGEADvQhAzpdmPwUeyJQIBOHU+nU+sZunE8Hs8/8EEDbq06bwPi808aIAM7hAD7cMAMI1GDAEnYIVzBDiHwlKfR9G7JTYMADMMOIcCePHWma2oQIBs7hCvYIQQWmj6r9nyaXtjfBkjIDiHA/hw6SnfUIEBOghCgOE1IcGoQIC1BCFCEQ0fphRoEyMw9hCu4hxBYy1NtInOEDAB2CAEK8vSasG6/W+GfK0BCghCgLAfMEJC9awDOBCFAcZqQUNQgABeCEKAGTUgQahCAKUEI0IAmpL4vX7+oQQCuOGV0BaeMAi/ydJxW/NsD4C47hDC+4z2th0rKs3CaUIMAPCIIAapyMyGVqUEAZghCGN/pp9aD8IMmpBo1CMA89xCu4B5CeuffcCjTZ+qeprO72+81+GcGwC07hADt2SdkX2oQgIXsEK5gd4Xe+Tccjcv5KMG/KwCWs0MI0Ixn6uxODQKwiiCELGwPxuSAGXakBgFYyyWjK3g+Tdf8A47MATO8yE2DAGxjhxCgPfuEvEINArCZIAQIRxOy3O1lomoQgOUEIaTgetH4rp7Ea0KWcNMgAC8ShDCU469aj8M6mpBV1CAAr/vUegBgN+cCnO4BHo8OjurMtz+/6UCectMgAHuxQwgjuOwHXuXf6XSyT9gdB8wwTw0CsCO7Byu4C4uY5v9lToPQP92OeCEK7nKZKAD7skMIfXv6fQoROAD7hJypQQB2JwgBInLADFfUIAAlCELomMuYx6YJuVCDABQiCKFXC0+LEY1d87yfL1+/qEEAynGozAqeWBPKwn+Q/t0OwAEzaUlBAEqzQwgQnReiyEkNAlCBIIQueXXBzDRhBmoQgDoEIYzM9aLDcMBMKmoQgGrcQ7iC59bE4QbCnHTC8G5T398yAEXZIQTohjYY21UNfv+f79//53urYQBIQhBCOu4/7JoDZkZ1W4OtJgEgFUEIw3K96Kg04XjUIACtCEKA/mjCkczUoO/mAFCaIIQuPX2a+Oi60OPRUVID0oSd+vL1ixoEoC1BCANyl2AGXoiid7d/ZZcaPJ1OahCAOgQh9Or8fPG2/c4/c/cJpe3BwXR96Ojxp9aDtDFzoKhPUgBq8uxwBUd0ENDt8+mrf59Xb+Bf73imadFRImZ+RHWZKABxCMIVMj99ASLrsQnTPqKqQQBCEYQrpH36AsTXXRMmfESduWnwkOnjAEAo7iEEGI0DZgJSgwDEZIdwhYTfzwY6cpUcwfcJUz2izr/ufIaPAABh2SEEGIQXoohpvgYPXicGgKYEIcA4gu8KJvS0Bs80IQCtCEKAoUyb0CZhW/MHirpSFIAI3EO4Qqo7XoCuxT90dOxH1OVHyEz3Bof8UAAQnB1CgMHZJ6zsdmPwUoO3G4OP4hAA6hCEAANywEwrG153XhMC0JAgBBiTJqxvQw3e0oQA1CQIAYYV8+7BUb1Sg1dvoAkBqMahMiuMfQQCMKqYB8yM9Ii6/AiZeVcdOMBHBoD4BOEKvmULdOrzvz5ffvzopfBa6f3L0FUNfvvz2yut69BRACpzyShALtM45EW3NXj4eZTotpxzwAwAlQlCgPFd7Qpqwl3crcHXaUIAanLJ6Aoj3fECJFQoYLbp/RG16AfTzYQAVGOHECCLOCfKdO3L1y+l09qhowBUY4dwhd6/nw1wCHPoaKePqDV3We0TAlCBHUKAXKYN49XqV6l8za0CBKACQQiQmiZcqMkdmA6YAaA0QQiQzlXMaMKnGp7HowkBKMo9hCt0escLwF1tDx3t5RH1tpbr33jpZkIAyrFDCJCUQ0efilCDB4eOAlCSHcIVevl+NsByrQ4djf+IGuplGw/2CQEoww4hQGoOHb0rWg0eFCAAZQhCgOw04ZWANXjmgBkAdicIAfhF5ib88vVL2Bo804QA7EsQAuCFKA6HMEfIrKIJAXiRIATgcOghfoq63RgM+wFx6CgAOxKEAPxQ/2bCIAelBL9M9JYmBGAvghCAv1VrwtPppAZfEeSjB0DvBCEAv0h16GinNXjmgBkAXicIAZgzahPGP1B0CU0IwIuOrjlZ7vK11gcNGN4AsTRjsNVNU9BXKABWsUMIwB0DvxDFYDV4xT4hAKsIQgDuG6yUzoasQYeOArCZIATgocEOmBmyBs9cKQrANu4hXME9hEBO047qNKJua7bThcxzMyEAa9khBGCFHvcJbzcGh6zBg0NHAVhPEALwRNcHzAx8mehdmhCAVQQhAM912oTZavCWJgRgniAEYJHuaiptDTp0FIDlHCqzgkNlALo4YCbJETLzrjrQVy4A7rJDCMAK8V+IQg2eKUAAlhCEAGwXrQnzHCi6hANmAHhKEAKwTtgDZtLeNDhDEwIwTxACsFrAJlSDS2hCAK4IQgC2iFNcX75+UYMzHDoKwAxBCMBGEQ6YcYTMEpoQgEe87MQKXnYC4FbDF6KwMbjKtAN9IQPgzA4hAC9ptU+oBtdywAwAtwQhAHuq04RqcBtNCMCVQS4ZLXcZzPv7++XHf/zxx/kHv//++45/BMAA/vrfv6b/+dt//bbhnUwfch9x0+CLrjpwjKcBAGz2qfUA291+STv/zC53+i15UgLAxW//9dtVE25weex99CCsBl93+XJ5djwO8q1hALbp8svAkuTbnIVXz0L++uPvH38//PiPz4e/dwh/s1kI8NO0CW83CaePqHdNH1Fvm9BlojuyTwjAWWdBuCrzNjTh5fnH3Wctd4PwTBYCnN024dMOvHI3C9Xg7hw6CsChryDcEHjLf8v0W9GPnrvMBOFBEwL89Mu1o/+z5WbCqyZUg4VoQgC6CcLNl4Au+Y3zG4MX80F4JgsBrm8m3NSEh5+PqFfvTQ3uSxMCJNfBy04cj8c6rwi/9qKmcu8EoGvXdw/+a+NhM3/9719qsDQvRAGQXAdBePFiDT76Orf7gaKaEGDzruDf/nX9OhZqsAJNCJBN9CB8fW9w/jcuvFgUgOV+PKJOm3DtJuFNDR68IFAxV18oNSFAKqGDsM6VoocCNSgvgbR+eQDc1oT/2ucWRJZz9yBAWnGDsMR3KK/e5/mbzYXiTRMCHA7rm/BeDV4eUW0SluNmQoCc4gbhhW9bAvTi+ffCZprwX38t2RvUhOVoQoCEggbhGF+HbBICHA43XXe3CW9T8NffNX1E1YTlaEKAbCIGYZ3XRCp6vShAQnOPqPNNuP6mQU1YhyYEGF7EIARgQI8yb+sRMpqwEIeOAqQyfhD6SgYQxe0BM2tq0DUd1bh7HyCPcEFY9HrR+l/hPH0B+MXMoaPrX17CJmE5biYESOIfx8JaLxCAHmx9sUFNWI4mBMgg3A4hAIP7tf3+/d//fvGl5zVhOZoQYHiDB6GvXgAB/fu//331gxdpwjp8VQUYzz9Oa5x/z4bfslDRrzRukQeI49///e+9avBMExbi0FGAsQ2+Q9jcb7+3ngBgFB5RW9GEAAOLG4Svb+jVeYF7AIKwSViOL6MAo4obhHX4fjPAXiI8omrCchwwAzCkYYPw6fZghScNEZ4bAYxh+SOqJixHEwKMJ1YQ+koDwOs0YR2+UgMMIFYQ7mXh3YPnZwyF9vFsDwIJxXlE1YSFOGAGYDADBuGGs2TEG8BePKIOTxMCjCRcEO54jtmSd1XoW8ieDwHsZfMjqk3Cchw6CjCMcEE4teGbjpffsvxr1e4XjqpBILk4j6iasBy3/QOMIWIQbv6+44YanNrlGYwaBDhEekTVhOVoQoABRAzCw+RrzMIvMMfj8ZUavDxd+O33l55/qEGAiziPqJqwDk0I0KPjqnw6P9ZXu3NgYeO9uDE4dXnS8Ncfd371++HHz34+XD9PkYIAj9w+ov77//1z+p//9//8Z/qfhR5RZWEhVx3o9kKAvnxqPcCc0+l0/jJzN0Q3nCb61Pv7+/SWwrtZeEUKAszziDq2yxfrs+Nx3feaAWgr9A7h7R99pdwkt99I/uuPX3YIPWsB2Oyf//xlh/A///nPo7fcl03Cckp8lxaACoLeQ3jrdE+5P+6yVXgxLUA1CNAjQViOA2YAOhX6ktHmpk8dPI0AGMDt9/vYy/TaUReOAvSim0tGI9jx9BqAPG4DrNUloxeasBzXjgL0pZtLRgHoVMD6CjjSkFw7ChCfIATgh+NPT99s7XsOGGABRxrDzJHgAAQkCAHyOv7q9ufv/pZtf5YAy8OVogAdEYQAGV313qOTnK/yb7A7yTVqOQ4dBeiFIATIZZqC86/ic/6l6bmRL/7RAQMs4EjD0IQAXRCEAIlsOAFy2oSvCxhgAUcahiYEiE8QAmRxtTG47fe+LmCABRxpSJoQICBBCJDCK6+kOv0tw9xAeEUTFuLQUYDgBCHA+F6pQXiRJgSITBACDE4NLmSTsBz/9gDCEoQAI7Mbs4omLMcBMwAxCUKAYW04U3Rehn0eTViOJgQISBACjC9DyO1IE9ahCQEiEIQAY+ro2XbAAAs40hgcMAMQjSAEGFB3z7MFWB6aECAUQQgwuL3uHix03enb21uJd/s6jVqOa5gB4jiuelA+fxsv7eO4o9uBXux+nEwdARss4EjD6PRfKcBg7BACjKbf59kB6yvgSMNw6ChABIIQgEACBljAkYakCQGaEIQAxBIwwAKONAYHzAA0JwgBCEeA5aEJAdoShABDGeb5dLQmjDbPSPq60xVgMIIQYFieZ+9LE5bjgBmAVgQhAHMaPjsPGGABRxqGJgRoQhACEFfAAAs40pA0IUAdghBgKONdJhowwAKONAYHzADUJwgBeOh4PEYozIABFnCkMWhCgMoEIcCwXnwyHeq5uADLI8L3IADyEIQAPOSp+QyNWo4DZgCqEYQAo9ml4oJcLDoVMMACjjQMTQhQhyAEGNm2Z9Jhn38HDLCAIw1DEwJUIAgBBvTKM+nz20fbHrwIGGABRxqSJgQoQRACjGlbE56vFA1bg2cBAyzgSGNw6ChAaYIQYFirmvB4PAa8b/ARAZZHL/8mATolCAFGNt3uO/50+dXjRPyNweA0ajluJgQoZ903g4PfWFLa5YtQ2o8A0LVHz6T7fUwL2GABRxrG9B9wv/9oAaIRhCsIQoBoAgZYwJGGoQkBdueSUQA6FrC+Ao40JNeOAuxCEALQt4ABFnCkMTh0FGB3ghCA7gmwPDQhwL4EIQDsT6OW4+5BgB0JQgBGEDDAAo40DC9EAbAXQQjAIAIGWMCRhqEJAXYhCAEYR8AACzjSkDQhwDaCEIAGPj4+Pj4+SrzngAEWcKQxOGAG4HWCEIDaLimYpwkpRBMCvEgQAtBSoSaMRqOW49BRgFcIQgAGFDDAAo40DAfMAGwmCAEYU8AACzjSMDQhwDaCEIBhBQywgCMNSRMCLCQIARhZwAALONIYHDADsIEgBGBwAiwPTQiwliAEYHzRmjDaPCNx6CjAKoIQABrQhOU4YAZgOUEIQAoBAyzgSMPQhAALCUIAsggYYAFHGoYmBFhCEAKQSMAACzjSkDQhwF2CEIBcAgZYwJHG4NBRgKcEIQDpCLA8HDoKME8QAkB7GrUcNxMCzBCEAGQUMMACjjQMTQjwiCAEIKmAARZwpGFoQoC7BCEAeQUMsIAjDUkTApwJQgBSCxhgAUcag0NHAW4JQgCyE2B5OHQU4IogBIBwNGo5biYEmBKEABAxwAKONAxNCHAhCAHgcAgZYAFHGoYmBDgThADwQ8AACzjSkDQhkJYgBIC/BQywgCONwaGjAAdBCABXBFgemhBAEAJAdBq1HC9EASQnCAHgWsAACzjSMBwwA2QmCAHgjoABFnCkYWhCIC1BCAD3BQywgCMNSRMCeQhCAHgoYIAFHGkMDpgBchKEADBHgOWhCYGEBCEAdEajluPQUSAbQQgATwQMsIAjDcMBM0AqghAAngsYYAFHGoYmBPIQhACwSMAACzjSkDQhMDBBCABLBQywgCONwQEzQBKCEIDa3t7e7v64CwEDLOBIY9CEQAbHVadpnR8K0x7AdflKkPYjAMBZtAaLNs9Iph3oCQAwHjuEANA9QViOA2aAsQlCAFgtYIAFHGkYmhAYmCAEgC0CBljAkYahCYFRCUIA2ChggAUcaUiaEBiGIPGS7z8AACAASURBVASA7QIGWMCRxuDQUWBIghAAXiLA8uj6lNHj8ShigVuCEABGo1HL6fFmwnMKnk6n0+nUy8xANYIQAF4VMMACjjSMvprw6kWkNSFwRRACwA4CBljAkYbRVxN2faUrUJogBIB9BAywgCMNKWwThh0MiEMQAsBuAgZYwJHGEP/Q0auLRQHuEoQAsCcBlkf8JrzVxZBATYIQAAanUcsJu/82E35hZwaaEIQAsLOAARZwpGFEPmBG+wFPCUIA2F/AAAs40jCiNWGEGYBeCEIAKCJggAUcaUhBeux2e/D88vRNhgHCEoQAUErAAAs40hjiHDATJEeBXghCAChIgOURpwnvsj0I3CUIASAXjVpOnOKKMwkQnCAEgLICBljAkYbR9oCZR3+i7UHgEUEIAMUFDLCAIw3DoaNARwQhANQQMMACjjSkCD1mexB4RBACQCUBAyzgSGNocsDM5U+52qJUg8AMQQgA9QiwPCIcOqoGgacEIQBUFa0Jo80zkrYxFuFSVSA+QQgA2WnCclodMHP+s2wPAk8JQgCoLWCABRxpGK2aUA0CSwhCABr4+Pj4+PhoPUVLAQMs4EhDKt2Ep9PJrYPAcoIQgNouKagJW49wLeBIY6hwwMzlfapBYBVBCEBLmrD1CNcCjjSGaoeOqkFgFUEIAC0JsDykGhCQIAQAfqFRy2l1wAzAI4IQABoLGGABRxpGiSZ03yCwmSAEgPYCBljAkYaxbxOqQeAVghAAQggYYAFHGtIrTbiqBl2kCtwShAAQRcAACzjSGPY6dFQNAi8ShAAQiADLY/frPGfe4bkGXVkK3BKEAMAcjVpOoQNmbn/mdDqpQeAuQQgAsQQMsIAjDWPfJjy/t8v7OR6PjpwB5glCAAgnYIAFHGkYuzfh6XS6pKAaBOat+6ZR8gvQL4/RaT8CALv4+PiY/ufb21urSYLb1mB//XH/53/7/ZVZfpCFhVx1oGcaQDWCcAVBCLALQbjcfIA9ar+7dgnCgyYsRhMCTXxqPQAA8ND7+/tVgK2KQDpyvs7z8p/u/QPqcA8hALCCHcJyFCBQnyAEgNB2CbC9rhc904TllHghCoAZghAAogsYYAFHGoYmBGoShADQgUuAbdjr23d78EIT1qEJgaIEIQD04ZUmLEQTFnJ1M6EmBMoRhADQjQ1NGKceWUUTAnUIQgBgO5uE5Th0FKhAEAJATwIGWMCRhuGAGaA0QQgAnfnrj6UvT1/telFNWI4mBIoShADQky/H99Yj3KcJ69CEwL4EIQD04cvx/aoGf/t9bg+w/nEymrAQB8wA5QhCAIjuNgUPh8O30/s5wJwjmoEmBAoRhAAQ2t0U/Hb68ZOPmrBVJdokLMeho0AJghAAgnq0MdhglDU0YTkOmAF2JwgBIJxHKXi3BgNeOKoJy9GEwL4EIQAEsioFL66aMEIcasJyNCGwo+Oq69HPDzppL2G/POam/QgA7OLj42P6n29vb60miebFC0QDNljAkcZw1YGemQCb2SEEgPZ2uV0wYH0FHGkMDh0F9iIIAaClbdeIPiLA8rArCOxCEAJAG/umYFgatRw3EwKvE4QA0EC5FAwYYAFHGoYmBF7kUJkVHCoDsIvpoTLZTpS57cCz3XcFAzZYwJGGMU1Bz1KAVQThCoIQYC+XJswThNVS8CJggAUcaQwOHQU2E4QrCEIANqifghcBAyzgSGPQhMA2n1oPAABjetSBhyopePb+/i7AkjidTu4hBDYQhACws4ZbgvFp1HKmTXg8rrsKDEjLJaMruGQUgBkRtgTvChhgQUZa+JW9r7hywAywiiBcQRACcFf8LcEgATbVZKSnF1XefonvqwbPNCGwnCBcQRACMBV2S/Cu5E245MyV2+c5nT7zccAMsJwgXEEQAnDWVwpe5GzCtdtlly3Brr/ua0JgIYfKAMAK8a8OnZHwQJcNF0+ej2bpvaCuDh0dYEVAIYIQAJ7rdEswvqKN+soWX9fbg2deiAJYwiWjKwzwtQGA5WYi8NBzBwbcJCwx0otftYf5ou+AGWCeIFxhmK8NADwyH4Fn/abgxfBN+PqX7JG+6GtCYIYgXGGkrw0AXCyJwMMQHTg1cBPu8vV6pC/6DpgBZgjCFUb62gCQXM4IvDJkE+61GzbYF31NCDziUBkAshCBV8Y7dHT3ayOHCSeHjgKPCEIAhrWwAA+ZIjC4vRpV7dxy6Chwl0tGVxjs6hGAIYnAtQJuEm4bad/twVGf8zhgBrgiCFcQhAARLE++WyLwrgGacPd75AZ+zqMJgSlBuIIgBKjmleq7IgKX6L0JRc5yDpgBpgThCoIQYC879t4tBbhN100oCFfRhMCFIFxBEALs5cUglHyFdNqEanADTQicOWUUgKBUX33jvRAFjzh0FDizQ7iCHUIAhhcwCJ+OZIdwMx864B+tBwCAFD6XvG1yRz0GIZtNI9CGIeQkCAGAXwQMsJmRZMyLNCEkJwgBgGt9NeHF2hthltg+cZ8SLhmSE4QAwB2dNuFyp5/m32DHPzGsq2VqQkhFEAJAcb3cQHglYBOWcBt+eVLwItt6gQtBCAA1fPcqGnu4bVQlsxc3E0JOghCABj4+Pj4+PlpPwXMBNwkDjjQMTQgJCUIAarukoCbsQsAAezTSLg2TfMtRE0I2ghCAljI0Yac3EE511ITbKJ9HfGRgeMe1BzQfXvvO2d2HlV6+FXcZvpeBAWK6isC3t7dWk9Tx+fg+xg2ElbLwj7/u//zvv93+3GWkF79GT5+f+Cp/uHnC5mMCA/tU8w/rugYBILn39/dSTfgoAu++zb0yvDge132zm7tOp9P0aZuPKgys6iWjp4mafy4AENeSGnzw9pdAfeXON9uDd/lQQBK1Lxm9eld7vbc6epwZIKAMl4w+um9wgGtH99wkXJuCV35uFd5eOHpY/MXatZHz1DIMr00QdvrgIggBdjFwEF468BJ+0xsIb3+1U/s04Ys1ePZCE149q9nxu96D6fRpG7BQ41NGPawAMIxz730/PTxC5vJLvZ87ukMQ7lKDk/czvXZ02nhn098x/cnp8xDPSZZw6CiMx8tOAMCrPh/fLzX49I014c5umvDwaxYeJhF46cC7JxpowruuPiyaEAYjCAHgJTMXgs5XX94m3Gt78JnTA3X+9JFoQhhYgyD0IALAMJ7eFnj353u/h/AlJWrw3iYh+xLSMKqWO4QeWQDo2jCHxGy2OsDK7Q1qwvJeeW0PICyXjAJAM71fNXoIGWABRxqGJoTxCEIA2OLp9uAAsbdQwAALONKQNCEMoHYQ3n3gON6oPBUAlJDnUtIQAfbr9aghRhqRA2ZgMM12CM+PJuf8uz34y4MLAAMb8ubD5wFW63BRStOEMJKWl4xeUnD6k5oQgPiGLLrXRduUizbPSBwNCMNoFoTnGmz1pwNAUXluICzkj8P38/9efD+asBwHzMAYqgbh9MFCDQIwtqeHzQy5uxgwwAKONAxNCANwyigAsKeAARZwpCFpQuhRmyC0PQhA1zZv7o29PXgRMMACjjQGB8xA7+wQAsB2d+8VfHQDYZIaPAsYYAFHGoMmhK7VC8KFjw4eRADownzU3f5qqho8uw6w339rMwflufgL+tVgh3DhQ4ZHFgC6sORA0YQ12MCC4LRJWI4DZqBTLhkFgI0udTffhOdf/X56T1iDAQMs4EjD0ITQo3UvBnj+3N62d3d5XJj/7QvfrInIswF05OPjY/qfb29vrSbZy3QD8DYOE3bglV8a7I+/lvyWyysQ/n74vPSPWXNJqiwsx8uMQV8+1flj1t5A6OEDgI7cbhWKwKn39/fiAbbyBsUaI3E4HI/r9h6A+mpfMupBAYCBnTtQDd76u75KHC2z6X0KwkIcOgp96eAeQo8jADCAsk1IJDYAoCOBgtD1ogD0bsmJo+zshby0SViOA2agFzWC0KMAAHm4XnTG/puEL78fTViOJoQuVN0h3LD1515kABjJL034Ys7tVJWasBxNCPEFumT0TP4BwNh+CbBtUfd6TP5KE9ahCSGgcEE4ZXsQgI64gXC56yZcVXdlzqTRhIU4dBSCq/Q6hEucTqfj8ejFTAHolxsIl7t+JcBL5t195XoHk/bs/Bzv8p++4w+h1NghPP20/C0Xvj0AMJqrDcO9rw6dYZOwHM/rIKzQl4wCMLy3t7fWI+zG9uBaAQMs4EjDcMAMxCQIAajtEoEj1SDbBAywgCMNQxNCQIIQgAbe3t7UIGcBAyzgSEPShBCBIAQAGgsYYAFHGoNDRyEaQQgAtCfA8tCEEIogBAC4Q6OW49BRiEMQAgAhBAywgCMNwwEzEIQgBACiCBhgAUcahiaECAQhABBIwAALONKQNCE0IQgBgFgCBljAkcbggBloThACANCMJoS2BCEAwHM2Cctx6Cg0JAgBABbRhOU4YAZaEYQAAEtpwnI0ITQhCAGAuAIGWMCRhqEJoT5BCACEFjDAAo40JE0IFQhCACC6gAEWcKQxOHQUKhOEAEAHBFgeDh2FmgQhAMAWGrUcNxNCNYIQAOhDwAALONIwNCHUIQgBgG4EDLCAIw1DE0IFghAA6EnAAAs40pA0IZQgCAGAzgQMsIAjjcGho1CaIAQA+iPA8nDoKBQlCAEAdqBRy3EzIZQjCAGALgUMsIAjDUMTQiGCEADoVcAACzjSMDQhlCAIAYCOBQywgCMNSRPCLgQhANC3gAEWcKQxOHQUdicIAYDuCbA8NCHsSxAC0MDHT60HgVI0ajleiAJ2JAgBqG3agZqQvQQMsIAjDcMBM7AXQQgADCJggAUcaRiaEHYhCAGAcQQMsIAjDUkTwjaCEAAYSsAACzjSGBwwA68ThADAaARYHpoQXiQIAQCK06jlOHQUXiEIAYABBQywgCMNwwEzsJkgBADGFDDAAo40DE0I2whCAGBYAQMs4EhD0oSwkCAEAEYWMMACjjQGB8zABoIQABhcwAALONIYNCGsJQgBgPEJsDwcOgqrCEIAgAY0ajkOmIHlBCEAkELAAAs40jA0ISwkCAGALAIGWMCRhqEJYQlBCAAkEjDAAo40JE0IdwlCACCXgAEWcKQxOHQUnhKEAEA6AiwPh47CPEEIANCeRi3HzYQwQxACABkFDLCAIw1DE8IjghAASCpggAUcaRiaEO4ShABAXgEDLOBIQ9KEcCYIAYDUAgZYwJHG4NBRuCUIAYDsBFgemhCuCEIAgHA0ajleiAKmBCEAQMQACzjSMBwwAxeCEADgcAgZYAFHGoYmhDNBCADwQ8AACzjSkDQhaQlCAIC/BQywgCONwQEzcBCEAABXBFgemhAEIQBAdBq1HIeOkpwgBAC4FjDAAo40DAfMkJkgBAC4I2CABRxpGJqQtAQhAMB9AQMs4EhD0oTkIQgBAB4KGGABRxqDA2bISRACAMwRYHloQhIShAC09Pb21noEeC5aE0abZyQOHSUbQQhAbZcIVIOwmSYsxwEzpHJc9V2Q86dE2m+cXB4R0n4EAKCOmF9zAzZYwJGGMU3BUP8OYV+CcIWYX5wAYDxhv+YGDLBCIy3ZGVv+t7Nwny3UX/fVzKFmgx25ZBQAYKk8QbjE2JdTOmCGJAQhAMAKSZpw3w2xJe8t4BacJiQDQQgAsE7AJizh9NPM2yxvpNPE7c8ErMGzsIPBXgQhAED3kjRqEw4dZWyCEABgtYABVnqkR/t4axupx6bShAxMEAIAbJGwCR/J0EiakFEJQgCAjTI0YaFXX+j93jxNyDAEIQDAdhmacIkMgeTQUYYkCAEAXhKwCXc3baFXNvd6j6jeNzbhliAEABhNhQNmbn+y99hbyM2EDEYQAgC8KuAm4S4jFTpBtPd9Nk3ISAQhAMAORm3CR3qPuhdpQoYhCAEA9jFqE65qv5k6GjicBl4awxOEAAC7GbUJ77JJOP1PTUinBCEAwJ4CNuE2Swpnw+kye2Xk8Xhs3mCakAEIQgCAwdVv1NJpdHn/zRss+TYpAxCEAAA7C7hJuHmkp8GzsIial1s5Dpiha4IQAGB/IzXhNqnSSBPSL0EIAFBE1024qmqWXza5ywWW8Ysr/oRwIQgBAErpuglfV6eLgtSXA2bolCAEACgoYBMu9/rW37mLktSRJqRHghAAIJdCjeq8zYMPAh0ShAAAZQXcJJwfad+trct7K3cDYai9OAfM0BdBCABQXHdNuI39sTNNSEcEIQBADd01obrbiyYkMkEIAFBJd024gYw8c8AMvfjUegAA0vn4+Lj8+O3treEkUN/7+3vALJx6PV1Op9PtOyn9CoTH4zFai159HAJOCAc7hAC0NY1DSCJaEEabZyQKkPgEIQBAdrdN+GLJCKELB8wQnCAEAKgt4Kbc+/v78Xi8FIt02ZEmJDL3EAIANBDkZsI//vjj0S9N02XDjt/0DrrSNxBe3iDszuT0oxF5ThKyQwgA0EaEICxK9jxin5A41n1/4vxvN+3n9r7f5QJI6+ogGQeNcivV19yAWRhwpMOyiAr+D+ZqCcGnJQk7hAAALcWsr2gWbqkF33lTgAQkCAEA+IVGLccBM0QjCAEAGgsYYAFHGoYmJBRBCADQXsAACzjSMDQhcQhCAIAQAgZYkJFWJVOPfdXjzAxDEAIARBEkwKYCjjSGqwNmNCGtCEIAgEAEWB4OHSUCQQgAwByNWo6bCWlOEAIAxBIwwBqOtCGT+iorTUhbghAAIBxNmIompCFBCAAQUcAACzjSkDQhNQlCAICgAgZY5ZHypJFDR2lFEAIAxBWwCbvQY1BpQpoQhAAArFCzUTe/MEOnr+jQ6dh0TRACAIQWcJOwchPemv/VrrPKATNUJggBAKJL3oTZaEJqEoQAAB0IGGABRxqSJqQoQQgA0IeAARZwpDG8fsDM8af9hmJMghAAoBsCLI/NTagDWUUQAgD0JFoTRptnJBtOx5GCrCUIAQB4iSYsZ9UBM1dv0PuBq9QhCAEAOhMwwAKONIyFTXhbgwVnYiCCEACgPwEDLOBIQ7rbhGqQzQQhAECXAgZYwJHGMH/AjBrkFYIQAKBXAQMs4EhjuNuEtweKqkHWEoQAAB0TYHnM7xM6QoZtBCEAAHvSqOU8Sj4pyGaCEACgbwEDLOBIw7htPzXIKwQhAED3AgZYwJHG4KZB9iUIAQBGEDDAAo7UNUfIUIIgBKClt7e31iPAOAIGWMCROuUIGQo5rvqXdP6HmPYf3+XzMO1HAADq8DX3FdEarMQ82f6F2BikHDuEAAAUFC1Qu6MGKUoQAgAMJWCABRypF2qQ0gQhAMBoAgZYwJGCc4QMdQhCAIABBQywgCOFdZWCBzVIMQ6VWSHb7csAUNrts96FfC1eKGCD7TLS2M/KbAxSkx1CAIBhBQxCVlGDlCYIAQCoR6MupwapQBACAIwsYIAFHCmU00+tByEFQQgANOMpbx0BAyzgSJCTIAQAOiMjNwgYYAFHgoQEIQBACgEDLOBIkI0gBADIQoABVwQhANDS2us/XS86GI0KbQlCAIBEAgZYwJEgD0EIAJBLwAALOBIkIQgBgMZcBVpfwAALOBJkIAgBgG5Ixx0FDLCAI8HwBCEAQFICDBCEAAB5RWvCaPPA8AQhANCea0G50IRQkyAEAF51/KnonyIaCwkYYAFHglEdVz22nh/o0z4cX77Opf0IANDWtLjufjG6TbJX3mzVPHff4apnDk970tffogI22MxInpXBXgThCh56AGho+lX49ivyo5/Z9mYLh7m4+w4fvcHC97ntnfCKjprQszLYi0tGAaAbl+e+0yw8/+B0Ol09M77859o3m3d1aejMO7z6XU/f8/wze8/7K+goCIG9CEIA6MC55db+0oY3e/pOpv/56B3e/ryc64UAg2wEIQD052rTb8nb716DUIdGhaIEIQCwxUxhSseuBQywgCPBMAQhAET39DDPp3rZHlx+DSpFBQywgCPBGJwyuoLzrAB28fHxcfnx29tbw0l6cXvB58IvSTu+2cK7Bze88fzv3fAe2EvABruM5FkZ7MUOIQAtTeOQR6LVIElEDkJgL4IQANjTi/VowycUAQbDE4QAQGgSkSmNCvsShADAnNsdv1WFJud6FzDAAo4E/RKEANCT+jcQggCDgQlCAGA3uxw/o04D0oQwKkEIAEk9irdQZ4qKwzg0IQxJEAJAdvPR9UqSybnBaEIYjyAEgG7kuTPw9FPrQYhOo8KLBCEA8Lfj8bg5w0Jda0ohAQMs4EjQEUEIAKktyb+rt7kNv+PxqAbzCBhgAUeCXghCAOCHme3B2yacWvJbGEnAAAs4EnRBEAJAZxaG1tM3O7/BtOueni4zk4vyL5uAARZwJIhPEAJANxZ21/I8O/1qw2+5/EaXjCYkwGAAghAAgEFoVFhLEAIA+3MFaRIBAyzgSBCZIAQA5syfHAMBAyzgSBCWIAQAHpp24EwTysXkAgZYwJEgJkEIANy3OfNcL5pQwAALOBLMOz6z+ffO/EZBCABsd/tUQw2mJcCgtBKXYwhCAOC+q7S7Lb3bpyZqMLloTRhtHpg38zpAT18f6PZXF76qkCAEABaZvwbJa9MTkyZkAGsfXVc9IAtCAOCh+W8wr31RezIIGGABR4Knpo+rC68UPb/Z2gdkQQgALHK60XoiggoYYAFHgn1tq8GDIAQAYHcBAyzgSLDc0yNGD1vv4haEAADsT4BRwdPXVOjawsB7pQYPghAAgEI0IUVdUnDhC+4N6cUaPAhCAACSEKgZDFaGS46WefGObkEIAEApGoxWMmwbHo/H18/3EoQAABQUqglDDUM1Xcfho03CXWrwIAgBAChNhhFH13F4sVcNHgQhAAAVaEIC6rEMzwPv+EqwghAAgBo0IWEF3za8yr8da/BwOBRf8L7jthXz3wcAAFBCnJa5KhE7hAAAAFmUS1NBCAAAEFq5axX3vB8xg33v4Iyp3H50TNbbapI6rLfVJPM+Pj4uP357e9v8fnpZ716st9UkdSRf7++//17tj25yK2Pyv99C6y19P1ecv6bzSk+n03TJe433aZf3kkecfxaFeLRqNUkd1ttqkjo6Wu8rEXjR0Xp3Yb2tJqkj+Xpr1mATyf9++1pvwGkvNVjo/QtC/tb1Z+8G1ttqkjqst9UkdVhvq0nqsN5Wk9ShBltNUkeP64085EwN7nXpoiDkhx4/e19hva0mqcN6W01Sh/W2mqQO6201SR1qsNUkdfS13uDjHe7V4NVVo7sQhBwOvX32vs56W01Sh/W2mqQO6201SR3W22qSOu6ut/IdfTX/OH+/rSaZEXOqu0pfKXrhlFH6+OzdkfW2mqQO6201SR3W22qSOqy31SR1PFrvqK9W7++31SS3ThOtZ1lqpganP7nLbqEgzC7yZ28J1ttqkjqst9UkdVhvq0nqsN5Wk9Qxv97xmtDfb6tJpjN0F4FXqk0uCFML+NlblPW2mqQO6201SR3W22qSOqy31SR1LFnvSE3o77fVJIdJBzac4XVPT4vZd5Nw/FfV45FQn70VWG+rSeqw3laT1GG9rSapw3pbTVLHqvWWzsIK2envt9Ukw1h4duiOL0hohzCpbJ+91ttqkjqst9UkdVhvq0nqsN5Wk9QRar1qcHfZ1lvBtleSeHGTUBBmlO2z13pbTVKH9baapA7rbTVJHdbbapI6Nqy3XLOpwd1lW28Fq2pwxw+4IEwn22ev9baapA7rbTVJHdbbapI6rLfVJHVsXm+nNxP6+201yTB2f3XB5QRhLtk+e6231SR1WG+rSeqw3laT1GG9rSap48X17t6EpSPT32+rScZwPB5ff8nBV3rSoTKJZPvstd5Wk9Rhva0mqcN6W01Sh/W2mqSOHde7S8ipwX1lW285jxJu4Yf07m/f9tchCLPI9tlrva0mqcN6W01Sh/W2mqQO6201SR27r/eVnHPf4O6yrbeQp7t58x/YJZuBq/5qBGEK2T57rbfVJHVYb6tJ6rDeVpPUYb2tJqmj3HrXpl2dGxH9/baahH0JwvFl++y13laT1GG9rSapw3pbTVKH9baapI46631aetXOpPH322oSdicIB5fts9d6W01Sh/W2mqQO6201SR3W22qSOqy31SR1ZFtvNk4ZHVm2z17rbTVJHdbbapI6rLfVJHVYb6tJ6rDeVpPUkW29CQnCYWX77LXeVpPUYb2tJqnDeltNUof1tpqkDuttNUkd2dabkyAcU7bPXuttNUkd1ttqkjqst9UkdVhvq0nqsN5Wk9SRbb1pCcIBZfvstd5Wk9Rhva0mqcN6W01Sh/W2mqQO6201SR3Z1puZIBxNts9e6201SR3W22qSOqy31SR1WG+rSeqw3laT1JFtvckJwqFk++y13laT1GG9rSapw3pbTVKH9baapA7rbTVJHdnWiyAcR7bPXuttNUkd1ttqkjqst9UkdVhvq0nqsN5Wk9SRbb0cBOEwsn32Wm+rSeqw3laT1GG9rSapw3pbTVKH9baapI5s6+VMEI4g22ev9baapA7rbTVJHdbbapI6rLfVJHVYb6tJ6si2Xi4EYfeyffZab6tJ6rDeVpPUYb2tJqnDeltNUof1tpqkjmzrZUoQ9i3bZ6/1tpqkDuttNUkd1ttqkjqst9UkdVhvq0nqyLZergjCjmX77LXeVpPUYb2tJqnDeltNUof1tpqkDuttNUkd2dbLLUHYq2yfvdbbapI6rLfVJHVYb6tJ6rDeVpPUYb2tJqkj23q5SxB2Kdtnr/W2mqQO6201SR3W22qSOqy31SR1WG+rSerItl4eEYT9yfbZa72tJqnDeltNUof1tpqkDuttNUkd1ttqkjqyrZcZgrAz2T57rbfVJHVYb6tJ6rDeVpPUYb2tJqnDeltNUke29TJPEPYk22ev9baapA7rbTVJHdbbapI6rLfVJHVYb6tJ6si2Xp4ShN3I9tlrva0mqcN6W01Sh/W2mqQO6201SR3W22qSOrKtlyUEYR+yffZab6tJ6rDeVpPUYb2tOhq5OQAABe5JREFUJqnDeltNUof1tpqkjmzrZSFB2IFsn73W22qSOqy31SR1WG+rSeqw3laT1GG9rSapI9t6WU4QRpfts9d6W01Sh/W2mqQO6201SR3W22qSOqy31SR1ZFsvqwjC0LJ99lpvq0nqsN5Wk9Rhva0mqcN6W01Sh/W2mqSObOtlLUEYV7bPXuttNUkd1ttqkjqst9UkdVhvq0nqsN5Wk9SRbb1sIAiDyvbZa72tJqnDeltNUof1tpqkDuttNUkd1ttqkjqyrZdtBGFE2T57rbfVJHVYb6tJ6rDeVpPUYb2tJqnDeltNUke29bKZIAwn22ev9baapA7rbTVJHdbbapI6rLfVJHVYb6tJ6si2Xl4hCGPJ9tlrva0mqcN6W01Sh/W2mqQO6201SR3W22qSOrKtlxcJwkCyffZab6tJ6rDeVpPUYb2tJqnDeltNUof1tpqkjmzr5XWCMIpsn73W22qSOqy31SR1WG+rSeqw3laT1GG9rSapI9t62YUgDCHbZ6/1tpqkDuttNUkd1ttqkjqst9UkdVhvq0nqyLZe9iII28v22Wu9rSapw3pbTVKH9baapA7rbTVJHdbbapI6sq2XHQnCxrJ99lrv/2/PjnHdhqIghiL733SaAEmTwvoS+azhWcEQ1i0ebC1h1GstYdRrLWHUay1h1GstYaz15l49CE1r11uvtYRRr7WEUa+1hFGvtYRRr7WEsdab2/Ug1Kxdb73WEka91hJGvdYSRr3WEka91hLGWm+e0IPQsXa99VpLGPVaSxj1WksY9VpLGPVaSxhrvXlID0LB2vXWay1h1GstYdRrLWHUay1h1GstYaz15jk9CGlr11uvtYRRr7WEUa+1hFGvtYRRr7WEsdabR/UgRK1db73WEka91hJGvdYSRr3WEka91hLGWm+e1oOQs3a99VpLGPVaSxj1WksY9VpLGPVaSxhrvQH0IISsXW+91hJGvdYSRr3WEka91hJGvdYSxlpvGD0ICWvXW6+1hFGvtYRRr7WEUa+1hFGvtYSx1htMD8LHrV1vvdYSRr3WEka91hJGvdYSRr3WEsZab0g9CJ+1dr31WksY9VpLGPVaSxj1WksY9VpLGGu9gfUgfNDa9dZrLWHUay1h1GstYdRrLWHUay1hrPWG14PwKWvXW6+1hFGvtYRRr7WEUa+1hFGvtYSx1htFD8JHrF1vvdYSRr3WEka91hJGvdYSRr3WEsZabyw9CO+3dr31WksY9VpLGPVaSxj1WksY9VpLGGu9EfUgvNna9dZrLWHUay1h1GstYdRrLWHUay1hrPXG1YPwTmvXW6+1hFGvtYRRr7WEUa+1hFGvtYSx1htdD8LbrF1vvdYSRr3WEka91hJGvdYSRr3WEsZab07Qg/Aea9dbr7WEUa+1hFGvtYRRr7WEUa+1hLHWm0P0ILzB2vXWay1h1GstYdRrLWHUay1h1GstYaz15hw9CH9q7XrrtZYw6rWWMOq1ljDqtZYw6rWWMNZ6c5QehD+ydr31WksY9VpLGPVaSxj1WksY9VpLGGu9OU0PwuvWrrdeawmjXmsJo15rCaNeawmjXmsJY7w3J+hBeNH49db7MvVaSxj1WksY9VpLGPVaSxjjvTlED8Irxq+33pep11rCqNdawqjXWsKo11rCGO/NOXoQfmz8eut9mXqtJYx6rSWMeq0ljHqtJYzx3hylB+Fnxq+33pep11rCqNdawqjXWsKo11rCGO/NaXoQXrd2vfW+TL3WEka91hJGvdYSRr3WEka91pL8Tw/Ci17/Na9db73WEka91hJGvdYSRr3WEka91hLGWu+X6kF4xeu/5rXrrddawqjXWsKo11rCqNdawqjXWsJY6/1ePQg/9vqvee1667WWMOq1ljDqtZYw6rWWMOq1ljDWer/ar36e/Gvteuu1ljDqtZYw6rWWMOq1ljDqtZYw1nq/Xf8Q5q+1663XWsKo11rCqNdawqjXWsKo11rCWOt9gR6E+WPteuu1ljDqtZYw6rWWMOq1ljDqtZYw1nrf4Tcp5k7jzZx36gAAAABJRU5ErkJggg==",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 8,
     "metadata": {
      "image/png": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='decollage.png', width=400)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "90aec862-edc3-436f-91a4-a28423485d6e",
   "metadata": {},
   "source": [
    "Le bras effectue une rotation rigide $\\Omega$ autour de l'axe $Y$; le point $O$ auquel est attaché le bras horizontal est fixe (à la hauter $h$), par contre le point $C$ de l'articulation, situé à une distance $d$ de $O$, permet au deuxième bras de tourner dans le plan mobile $(x,y)$. La position du bras mobile (dans le repère tournant) est déterminée par l'angle $\\varphi$. Le bras mobile est de longueur $\\ell$ et de section négligeable; sa masse est $m$.\n",
    "\n",
    "Comme on ne s'intéresse qu'au décollage du point $P$, on suppose $\\varphi$ *constant*: notre but est de trouver la vitesse angularie $\\Omega$ pour laquelle la force normale en $P$, la réaction $\\bm N$, s'annule.\n",
    "\n",
    "### Cinématique\n",
    "\n",
    "On choisit le point $C$ comme origine des axes mobiles, et on y décrit le mouvement de rotation. Le aposition du centre de masse y est fixe\n",
    "$$\\bm G = -\\frac{\\ell}{2} \\hat{\\bm y}.$$\n",
    "\n",
    "Le moment d'inertie est $I_C = m\\ell^2/3$.\n",
    "\n",
    "\n",
    "* Coordonnées du repère fixe dans celles du mobile $(x,y,z)$:\n",
    "$$\\hat{\\bm X} = C_z(\\varphi) \\hat{\\bm x} = \\begin{pmatrix} \\cos\\varphi \\\\ -\\sin\\varphi \\\\ 0 \\end{pmatrix}$$\n",
    "et\n",
    "$$\\hat{\\bm Y} = C_z(\\varphi) \\hat{\\bm y} = \\begin{pmatrix} \\sin\\varphi \\\\ \\cos\\varphi \\\\ 0 \\end{pmatrix}.$$\n",
    "\n",
    "* Vitesse et accélération angulaire\n",
    "$$\\bm \\omega = \\Omega \\hat{\\bm Y} = \\Omega \\big( \\sin\\varphi \\hat{\\bm x} + \\cos\\varphi \\hat{\\bm y} \\big)$$\n",
    "et\n",
    "$$\\dot{\\bm \\omega} = \\dot\\Omega \\big( \\sin\\varphi \\hat{\\bm x} + \\cos\\varphi \\hat{\\bm y} \\big)$$\n",
    "\n",
    "### Dynamique\n",
    "\n",
    "* Moment par rapport à $C$. \n",
    "\n",
    "Le repère $C$ tourne à la vitesse angulaire $\\Omega$, $$\\frac{\\D_C  \\bm L}{\\D t} = m \\bm G \\wedge \\bm a + I_C \\dot{\\bm \\omega} + \\bm \\omega \\wedge (I_C \\bm \\omega)  = \\bm M_C$$\n",
    "avec l'accélération\n",
    "$$\\bm a = \\bm \\omega \\wedge (\\bm \\omega \\wedge \\bm d) = d \\Omega^2 \\begin{pmatrix} -\\cos\\varphi \\\\ \\sin\\varphi \\\\ 0 \\end{pmatrix}$$\n",
    "($\\bm d = d \\hat{\\bm X}$), et $\\bm M_C$ le moment de forces.\n",
    "\n",
    "On se rappelle que la variation d'un vecteur $\\bm v$ dans un repère \"A\" est \n",
    "$$\\frac{\\D_A \\bm v}{\\D t} = \\dot{\\bm v} + \\bm \\omega \\wedge  \\bm v$$\n",
    "exprimée dans un repère \"B\", lequel tourne à vitesse angulaire $\\bm \\omega$. La formule de $\\bm a$ résulte de la formule de la vitesse de rotation d'un point $\\bm r$, $\\bm v = \\omega \\wedge \\bm r$. Par conséquent, $\\bm G \\wedge (m\\bm a)$ est le moment dû à l'accélération (centripète) resentie au repère tournant.\n",
    "\n",
    "Le moment dû à l'accélération est donc\n",
    "$$m \\bm G \\wedge \\bm a = -\\frac{m\\ell d}{2}  \\Omega^2 \\cos\\varphi.$$\n",
    "\n",
    "Le terme d'inertie est\n",
    "$$\\bm \\omega \\wedge (I_C \\bm \\omega) = -\\frac{m\\ell^2\\Omega^2}{6}\\sin(2\\varphi).$$\n",
    "\n",
    "* Moment de forces\n",
    "Le moment de la gravité est\n",
    "$$\\bm M_C^G = -\\frac{\\ell}{2} \\hat{\\bm y} \\wedge mg(-\\hat{\\bm Y}) = -\\frac{mg\\ell}{2} \\sin\\varphi \\hat{\\bm z},$$\n",
    "et celui de la force de réaction en $P$\n",
    "$$M_C^P = \\ell N \\sin\\varphi \\hat{\\bm z}.$$\n",
    "\n",
    "\n",
    "### Équation de moments\n",
    "\n",
    "On arrive à l'équation de bilan de moments (dans la direction $z$, notez que $I_C \\dot{\\bm \\omega}$ n'a pas de composante dans cette direction)\n",
    "$$\\frac{mg\\ell}{2} \\sin\\varphi - N\\ell\\sin\\varphi = \\frac{m\\ell d}{2}  \\Omega^2 \\cos\\varphi + \\frac{m\\ell^2\\Omega^2}{6}\\sin(2\\varphi)$$\n",
    "\n",
    "La condition de décollement est $N=0$:\n",
    "$$\\Omega^2_0 = \\frac{g \\tan\\varphi_0}{d + (2\\ell/3) \\sin\\varphi_0},$$\n",
    "avec $\\cos\\varphi_0 = h/\\ell$. Pour $\\Omega > \\Omega_0$ le bras mobile décolle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "d115a39b-dba4-436c-ad1b-51549ac2568b",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "phi = sy.Symbol(\"varphi\", real = True)\n",
    "o = sy.Symbol(\"Omega\", real = True)\n",
    "m, g, l, d, h = sy.symbols(\"m, g, ell, d, h\", positive = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "e02ea912-0916-415b-bd07-f2ef40ad0a6e",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def C_z(phi):\n",
    "        c = sy.cos(phi)\n",
    "        s = sy.sin(phi)\n",
    "        return sy.Matrix([[c, s, 0], [-s, c, 0], [0, 0, 1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "8e7f3569-3d91-45fa-9091-18132b260ecc",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "e1 = sy.Matrix([1, 0, 0])\n",
    "e2 = sy.Matrix([0, 1, 0])\n",
    "e3 = sy.Matrix([0, 0, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "a57da85c-c3ad-4be9-a9e1-8998cf80c228",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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3Am4tSUOgITAvAhhO/q528M1HVnkpgzIxdCajjbsJbtQQaAgcGAHpHqNehs2DVpciDCovkSKzvvfX3vbbEGgIHBIBKS5K6/6aRfej602jyqvMvFtiZfbVIQvceDcEGgIdAr/o7p10L/t6qEulm1HlJaGYsKjT+29V4ho1BBoC8yEgXWNvBRffJm+kodXmNCPMfhfz17pGTbnSfVCaND/L3ZMKlGY8d79w44DtEqYtTe6+se4hM0a4/A3pJh1zT5qkvGLGriOUj+2DQ4xZ3Bp69TK4YuZK0X7GELBD8sbSnDKuyb2P/tYyk35diQ3/ozR5a+7/AbxcT4++JdNMAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}\\cos{\\left(\\varphi \\right)}\\\\- \\sin{\\left(\\varphi \\right)}\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}\\sin{\\left(\\varphi \\right)}\\\\\\cos{\\left(\\varphi \\right)}\\\\0\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡cos(varphi) ⎤  ⎡sin(varphi)⎤⎞\n",
       "⎜⎢            ⎥  ⎢           ⎥⎟\n",
       "⎜⎢-sin(varphi)⎥, ⎢cos(varphi)⎥⎟\n",
       "⎜⎢            ⎥  ⎢           ⎥⎟\n",
       "⎝⎣     0      ⎦  ⎣     0     ⎦⎠"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = C_z(phi)*e1\n",
    "Y = C_z(phi)*e2\n",
    "X, Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "5c6eab9e-13bb-4561-b7e0-51346ddcce14",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}\\Omega \\sin{\\left(\\varphi \\right)}\\\\\\Omega \\cos{\\left(\\varphi \\right)}\\\\0\\end{matrix}\\right], \\  \\left[\\begin{matrix}d \\cos{\\left(\\varphi \\right)}\\\\- d \\sin{\\left(\\varphi \\right)}\\\\0\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛⎡Ω⋅sin(varphi)⎤  ⎡d⋅cos(varphi) ⎤⎞\n",
       "⎜⎢             ⎥  ⎢              ⎥⎟\n",
       "⎜⎢Ω⋅cos(varphi)⎥, ⎢-d⋅sin(varphi)⎥⎟\n",
       "⎜⎢             ⎥  ⎢              ⎥⎟\n",
       "⎝⎣      0      ⎦  ⎣      0       ⎦⎠"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = o*Y\n",
    "c = d*X\n",
    "w, c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "4fb6badc-a304-41c4-b93f-e28fb61d4c77",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\Omega^{2} d \\cos{\\left(\\varphi \\right)}\\\\\\Omega^{2} d \\sin{\\left(\\varphi \\right)}\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡  2              ⎤\n",
       "⎢-Ω ⋅d⋅cos(varphi)⎥\n",
       "⎢                 ⎥\n",
       "⎢ 2               ⎥\n",
       "⎢Ω ⋅d⋅sin(varphi) ⎥\n",
       "⎢                 ⎥\n",
       "⎣        0        ⎦"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = w.cross(c) # vitesse de rotation de C\n",
    "a = sy.trigsimp( w.cross(v) ) # accélération angulaire de C\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "06d02903-3aa3-423b-b84e-b0bf5f4970e8",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( - \\frac{\\ell g m \\sin{\\left(\\varphi \\right)}}{2}, \\  - \\frac{\\Omega^{2} d \\ell m \\cos{\\left(\\varphi \\right)}}{2} - \\frac{\\Omega^{2} \\ell^{2} m \\sin{\\left(\\varphi \\right)} \\cos{\\left(\\varphi \\right)}}{3}\\right)$"
      ],
      "text/plain": [
       "⎛                          2                        2    2                    \n",
       "⎜-ell⋅g⋅m⋅sin(varphi)     Ω ⋅d⋅ell⋅m⋅cos(varphi)   Ω ⋅ell ⋅m⋅sin(varphi)⋅cos(v\n",
       "⎜─────────────────────, - ────────────────────── - ───────────────────────────\n",
       "⎝          2                        2                              3          \n",
       "\n",
       "      ⎞\n",
       "arphi)⎟\n",
       "──────⎟\n",
       "      ⎠"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = -l/2 * e2\n",
    "Ic = m*l**2/3\n",
    "MG = m*G.cross(a)\n",
    "MI = Ic * w.cross(w[0]*e1)\n",
    "Mgrav = G.cross(-m*g*Y)\n",
    "Mgrav[2], MG[2] + MI[2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "b5f36b73-cb4b-41b8-9e14-cec257abf684",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ - \\sqrt{3} \\sqrt{g} \\sqrt{\\frac{\\tan{\\left(\\varphi \\right)}}{3 d + 2 \\ell \\sin{\\left(\\varphi \\right)}}}, \\  \\sqrt{3} \\sqrt{g} \\sqrt{\\frac{\\tan{\\left(\\varphi \\right)}}{3 d + 2 \\ell \\sin{\\left(\\varphi \\right)}}}\\right]$"
      ],
      "text/plain": [
       "⎡           _________________________            _________________________⎤\n",
       "⎢          ╱       tan(varphi)                  ╱       tan(varphi)       ⎥\n",
       "⎢-√3⋅√g⋅  ╱  ─────────────────────── , √3⋅√g⋅  ╱  ─────────────────────── ⎥\n",
       "⎣       ╲╱   3⋅d + 2⋅ell⋅sin(varphi)         ╲╱   3⋅d + 2⋅ell⋅sin(varphi) ⎦"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.solve(-Mgrav[2] + MG[2] + MI[2], o)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee4dfdd4-7cfa-4342-a407-9298708b8869",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "# Graphes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aadfe509-8115-44e5-adaf-db9430903315",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "## Bras articulé\n",
    "(Robot, double pendule, ...)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab319f7c-89cd-4346-ac6f-da8683fd509f",
   "metadata": {},
   "source": [
    "```c\n",
    "import settings;\n",
    "usepackage(\"bm\");\n",
    "render = 4;\n",
    "size(300);\n",
    "\n",
    "// couleurs\n",
    "pen cr = rgb(0.5, 0, 0.1);\n",
    "pen cg = rgb(0.1, 0.4, 0.1);\n",
    "pen cb = rgb(0.1, 0, 0.5);\n",
    "\n",
    "// géométrie\n",
    "real phi1 = 30*pi/180;\n",
    "real phi2 = 35*pi/180;\n",
    "real phi = phi1 + phi2;\n",
    "real l = 1;\n",
    "\n",
    "pair o1 = (0,0);\n",
    "pair o2 = (l*cos(phi1), l*sin(phi1));\n",
    "pair o3 = (l*cos(phi), l*sin(phi));\n",
    "\n",
    "// axe fixe XY\n",
    "draw((0,0)--(2,0), EndArrow);\n",
    "label(\"$X$\", (2,0), S);\n",
    "draw((0,0)--(0,3*2/4), EndArrow);\n",
    "label(\"$Z$\", (0,3*2/4), W);\n",
    "\n",
    "// bras\n",
    "draw(o1--o2--(o2 + o3), 9bp+gray(.5));\n",
    "draw(Label(\"$l$\", LeftSide), (o2 + (-0.1, 0.05))--(o2 + o3 + (-0.1, 0.05)), Bars);\n",
    "\n",
    "// axes mobiles 1 et 2\n",
    "real s = 0.3;\n",
    "pair t1 = (-0.08, 0.1);\n",
    "pair t2 = (-0.2, 0.12);\n",
    "pair p1 = shift(t1)*(o2/2);\n",
    "pair p2 = shift(t2)*(o2 + o3/2);\n",
    "pair x1 = p1 + s*o2;\n",
    "pair y1 = p1 + s*(-sin(phi1), cos(phi1));\n",
    "pair x2 = p2 + s*o3;\n",
    "pair y2 = p2 + s*(-sin(phi), cos(phi));\n",
    "draw(p1 -- x1, 1bp+cg, EndArrow);\n",
    "draw(p1 -- y1, 1bp+cg, EndArrow);\n",
    "draw(p2 -- x2, 1bp+cg, EndArrow);\n",
    "draw(p2 -- y2, 1bp+cg, EndArrow);\n",
    "label(\"$x_1$\", x1, E);\n",
    "label(\"$y_1$\", y1, N);\n",
    "label(\"$x_2$\", x2, E);\n",
    "label(\"$y_2$\", y2, N);\n",
    "\n",
    "// centre de masse\n",
    "dot(o1, 8bp+cb);\n",
    "label(\"$O_1$\", o1, SW);\n",
    "dot(o2/2, 8bp+cr);\n",
    "dot(o2, 8bp+cb);\n",
    "label(\"$O_2$\", o2, 3*S);\n",
    "dot(o2 + o3/2, 8bp+cr);\n",
    "\n",
    "// gravité\n",
    "draw(o2/2--(o2/2 - (0, 0.6)), 1bp+cb, Arrow);\n",
    "label(\"$m\\bm g$\", (o2/2 - (0, 0.6)), W);\n",
    "draw((o2 + o3/2)--(o2 + o3/2 - (0, 0.6)), 1bp+cb, Arrow);\n",
    "label(\"$m\\bm g$\", (o2 + o3/2 - (0, 0.6)), E);\n",
    "\n",
    "// angles 30 et 35\n",
    "draw(\"$\\phi_1$\", arc(o1, 2*l/3, 0, 30), cb, ArcArrow);\n",
    "draw(\"$\\phi_2$\", arc(o2, 2*l/3, 30, 30 + 35), cb, ArcArrow);\n",
    "draw(o2--(2*o2), cb+dashed);\n",
    "\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f62c2730-e17a-45db-873f-718f7cb5dafd",
   "metadata": {
    "jp-MarkdownHeadingCollapsed": true,
    "tags": []
   },
   "source": [
    "## Bras articulé: décollage"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fed78c5a-443f-4433-88f7-97d40793afee",
   "metadata": {
    "tags": []
   },
   "source": [
    "```c\n",
    "import settings;\n",
    "import patterns;\n",
    "usepackage(\"bm\");\n",
    "render = 4;\n",
    "size(300);\n",
    "\n",
    "add(\"hatch\",hatch());\n",
    "\n",
    "// couleurs\n",
    "pen cr = rgb(0.5, 0, 0.1);\n",
    "pen cg = rgb(0.1, 0.4, 0.1);\n",
    "pen cb = rgb(0.1, 0, 0.5);\n",
    "\n",
    "real l = 1;\n",
    "real d = 0.8;\n",
    "real phi = 40*pi/180;\n",
    "real h = l*cos(phi);\n",
    "\n",
    "pair o = (0, h);\n",
    "pair C = (d, h);\n",
    "pair G = (d + l/2*sin(phi), h - l/2*cos(phi));\n",
    "pair P = (d + l*sin(phi), 0);\n",
    "\n",
    "// bras et repère fixe\n",
    "draw(o--C--P, 9bp+gray(.5));\n",
    "fill((0,0)--(0, -0.15)--(1.7, -0.15)--(1.7, 0)--cycle, pattern(\"hatch\"));\n",
    "// axes, centre de masse, ...\n",
    "dot(o, 8bp+cb);\n",
    "label(\"$O$\", o, 1.3*W);\n",
    "dot(C, 8bp+cb);\n",
    "label(\"$C$\", C, 2.2*Sw);\n",
    "dot(G, 8bp+cr);\n",
    "label(\"$G$\", G, NE);\n",
    "// angle\n",
    "draw(C--(C - (0, h)), 1bp+dashed+gray(0.7));\n",
    "draw(\"$\\phi$\", arc(C, l/3, -90, -90 + phi*180/pi), cb, ArcArrow);\n",
    "// cotes\n",
    "draw(\"$d$\", shift((0, 0.1))*(C--o), Bars);\n",
    "draw(\"$\\ell$\", shift((0.15, 0.12))*(P--C), Bars);\n",
    "draw(\"$h$\", shift((-0.15, 0))*((0,0)--o), Bars);\n",
    "\n",
    "// axes mobiles\n",
    "draw(Label(\"$y$\", EndPoint), C--(C + 0.4*(-sin(phi), cos(phi))), cg, Arrow);\n",
    "draw(Label(\"$x$\", EndPoint), C--(C + 0.4*(cos(phi), sin(phi))), cg, Arrow);\n",
    "\n",
    "// vitesse angulaire\n",
    "draw(arc((0, 1.8*h), 0.1, 200, 340), cr, ArcArrow);\n",
    "draw((0, 1.5*h)--(0, 2.0*h), 1bp+cr, EndArrow);\n",
    "label(\"$\\Omega$\", (0.1, 1.8*h), E);\n",
    "\n",
    "// forces\n",
    "draw(G--(G - (0,l/3)), EndArrow);  // gravité\n",
    "draw(P--(P + (0,l/5)), EndArrow);  // normale\n",
    "label(\"$m \\bm g$\", (G - (0,l/3)), W);\n",
    "label(\"$\\bm N$\", (P + (0,l/5)), W);\n",
    "\n",
    "draw(Label(\"$X$\", EndPoint, 2*E), (0, 0)--(1.8, 0),  Arrow);\n",
    "draw(Label(\"$Y$\", EndPoint, 1.4*NW), (0, 0)--(0, 1.4*h), Arrow);\n",
    "```\n"
   ]
  }
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