{
 "metadata": {
  "name": ""
 },
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 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from pylab import *\n",
      "%matplotlib inline\n",
      "from matplotlib import animation\n",
      "\n",
      "# nastavitve za izris grafov (http://matplotlib.org/1.3.1/users/customizing.html)\n",
      "rc('text', usetex=True)\n",
      "rc('font', size=12, family='serif', serif=['Computer Modern'])\n",
      "rc('xtick', labelsize='small')\n",
      "rc('ytick', labelsize='small')\n",
      "rc('legend', fontsize='medium')\n",
      "rc('figure', figsize=(5, 3))\n",
      "rc('lines', linewidth=2.0)\n",
      "rc('axes', color_cycle=['k'])\n",
      "rc('contour', negative_linestyle='solid')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Animacije"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Pripravimo animacijo utripanja dveh vzmetnih nihal z masama $m$ in koeficientoma vzmeti $k$, sklopljenih s \u0161ibkej\u0161o vzmetjo s koeficientom $k^\\prime$. Na za\u010detku je raztezek vzmeti prvega nihala A, vzmet drugega pa je neraztegnjena. Obe ute\u017ei na za\u010deku mirujeta. Re\u0161iti moramo sklopljen sistem diferencialnih ena\u010db\n",
      "\n",
      "$$m\\ddot{x_1}=-kx_1+k^\\prime(x_2-x_1),$$\n",
      "\n",
      "$$m\\ddot{x_2}=-kx_2-k^\\prime(x_2-x_1),$$\n",
      "\n",
      "z za\u010detnimi pogoji $x_1(0)=A$, $x_2(0)=0$, $\\dot{x_1}(0)=0$ in $\\dot{x_2}(0)=0$. Re\u0161itev je\n",
      "\n",
      "$$x_1(t)=\\frac{A}{2}\\left(\\cos(\\omega_1 t)+\\cos(\\omega_2 t)\\right),$$\n",
      "\n",
      "$$x_2(t)=\\frac{A}{2}\\left(\\cos(\\omega_1 t)-\\cos(\\omega_2 t)\\right),$$\n",
      "\n",
      "z $\\omega_1=\\sqrt{\\frac{k}{m}}$ in $\\omega_2=\\sqrt{\\frac{k+2k^\\prime}{m}}$. V animaciji <a href=\"http://burana.ijs.si/rovfx/10/utripanje.gif\">utripanje.gif</a> je prikazana ena perioda utripanja za $A=3\\,$cm, $\\omega_1=10\\pi\\,$s$^{-1}$ in $\\omega_2=12\\pi\\,$s$^{-1}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ((axa), (axb)) = subplots(2, 1, figsize=(6, 3))\n",
      "axa.set_xlim([0, 30])\n",
      "axa.set_ylim([-1, 1])\n",
      "vzmet1,=axa.plot([], [])\n",
      "vzmet2,=axa.plot([], [])\n",
      "vzmet3,=axa.plot([], [])\n",
      "utez1,=axa.plot([], [], 'or', markersize=40)\n",
      "utez2,=axa.plot([], [], 'ob', markersize=40)\n",
      "axa.set_xlabel(r'Polo\\v zaj (cm)')\n",
      "axa.set_yticks([])\n",
      "axa.grid()\n",
      "axb.set_xlim([0, 1])\n",
      "axb.set_ylim([-3, 3])\n",
      "axb.set_xlabel(r'\\v Cas (s)')\n",
      "axb.set_ylabel(r'Odmik (cm)')\n",
      "axb.grid()\n",
      "crta,=axb.plot([], [])\n",
      "\n",
      "A=3.0\n",
      "w1=5.0 * 2 * pi\n",
      "w2=6.0 * 2 * pi\n",
      "x1t=lambda t: 0.5 * A * (cos(w1 * t) + cos(w2 * t))\n",
      "x2t=lambda t: 0.5 * A * (cos(w1 * t) - cos(w2 * t))\n",
      "\n",
      "t=linspace(0, 1, 200)\n",
      "axb.plot(t, x1t(t), 'r', t, x2t(t), 'b')\n",
      "\n",
      "def init():\n",
      "    vzmet1.set_data([], [])\n",
      "    vzmet2.set_data([], [])\n",
      "    vzmet3.set_data([], [])\n",
      "    utez1.set_data([], [])\n",
      "    utez2.set_data([], [])\n",
      "    crta.set_data([], [])\n",
      "\n",
      "nf=200;    \n",
      "def animate(f):    \n",
      "    t=f / float(nf)\n",
      "    x1=x1t(t)\n",
      "    x2=x2t(t)\n",
      "    vzmet1.set_data([0, 10 + x1], [0, 0])\n",
      "    setp(vzmet1, lw=20 * 10 / (10 + x1))\n",
      "    vzmet2.set_data([10 + x1, 20 + x2], [0, 0])\n",
      "    setp(vzmet2, lw=10 * 10 / (10 - x1 + x2))\n",
      "    vzmet3.set_data([20 + x2, 30], [0, 0])\n",
      "    setp(vzmet3, lw=20 * 10 / (10 - x2))\n",
      "    utez1.set_data([10 + x1], [0])\n",
      "    utez2.set_data([20 + x2], [0])\n",
      "    crta.set_data([t, t], [-A, A])\n",
      "    fig.tight_layout(pad=0.3)\n",
      "\n",
      "anim=animation.FuncAnimation(fig, animate, init_func=init, frames=nf, interval=25, blit=True)\n",
      "anim.save('utripanje.gif', writer='imagemagick');\n",
      "close()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<img src=\"http://burana.ijs.si/rovfx/10/utripanje.gif\">"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Naloga"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<OL>\n",
      "<LI>Pripravi animacijo vodoravnega meta \u017eoge z maso $m$, na katero poleg sile te\u017ee ($g=9.81\\,$m/s$^2$) deluje tudi zaviralna sila $\\mathbf{F}_\\mathrm{u}$, za katero velja kvadratni zakon upora: $\\mathbf{F}_\\mathrm{u}=-cmv\\mathbf{v}$, $c=1\\,$m$^{-1}$. \u017dogo vr\u017eemo z vi\u0161ine $1\\,$m nad vodoravnimi tlemi z za\u010detno hitrostjo $1\\,$m/s. Od tal se \u017eoga odbija elasti\u010dno. Animacijo pripravi v formatu \"animated GIF\". Velikost oddane datoteke naj bo najve\u010d $5\\,$MB, uporabi\u0161 lahko program za stiskanje datotek v format RAR.  \n",
      "</OL>"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}