mm_hw_ex.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "view-in-github"
   },
   "source": [
    "<a href=\"https://colab.research.google.com/github/szazs89/jupyter_ex/blob/master/mm_hw_ex.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "9U-3oNsHY8fs"
   },
   "source": [
    "# Homework example\n",
    "\n",
    "Created by: Zsolt Szabó"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "AbP2Sg80Zeto"
   },
   "source": [
    "This is an example document, which shows how to elaborate a homework in _Jupyter_ notebook."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "4mgO-ErDaCfF"
   },
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "ilMFlE8jaNMx"
   },
   "source": [
    "The task is to indentify the parameters of a one-degree-of-freedom (1 D.o.F) damped oscillator, which moves on an inclined plane of angle $\\alpha$. The oscillator consists of a mass $m$, a spring of stiffness $k$, a damper of parameter $c$. The spring and the damper connect the mass to the wall. The friction coefficient between the mass and the plane is  $\\mu$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 208
    },
    "colab_type": "code",
    "id": "M3bNO2OTciES",
    "outputId": "7e4d2d8c-e9fb-44ee-e05c-a3999d7f42a8"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f148e738080>,\n",
       " <matplotlib.lines.Line2D at 0x7f148e7382b0>,\n",
       " <matplotlib.lines.Line2D at 0x7f148e7385f8>]"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x144 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import scipy as SP\n",
    "import matplotlib.pyplot as PT\n",
    "#ty = SP.transpose([ [i*.1, SP.sin(i*.1)] for i in range(201) ])\n",
    "t = SP.transpose(range(201))*.1\n",
    "y = SP.sin(t)\n",
    "#PT.plot(t,y,'r.')\n",
    "def MASS(p,w,h2):\n",
    "  return SP.transpose(SP.transpose([p,p+[0,h2],p+[w,h2],p+[w,-h2],p+[0,-h2],p]))\n",
    "def SPRG(p,w,h2):\n",
    "  d = w/7\n",
    "  return SP.transpose(SP.transpose(\n",
    "      [p,p+[d*1.5,0],p+[2*d,h2],p+[3*d,-h2],p+[4*d,h2],\n",
    "       p+[5*d,-h2],p+[5.5*d,0],p+[7*d,0]]))\n",
    "def DMPR(p,w,h2):\n",
    "  d = w/7\n",
    "  return SP.transpose(SP.transpose(\n",
    "      [p,p+[d*5.5,0],p+[5.5*d,h2],p+[1.5*d,h2],p+[1.5*d,0],\n",
    "       p+[5.5*d,0],p+[5.5*d,h2],p+[7*d,h2]]))\n",
    "mass = SP.matrix(MASS(SP.matrix([5,2]),3,2))\n",
    "sprg = SP.matrix(SPRG(SP.matrix([0,3]),5,1))\n",
    "dmpr = SP.matrix(DMPR(SP.matrix([0,1]),5,.2))\n",
    "PT.figure(1,figsize=[5,2])\n",
    "PT.plot(mass[:,0],mass[:,1],'b',sprg[:,0],sprg[:,1],'r',dmpr[:,0],dmpr[:,1],'r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "nFP_B2N-7HQD"
   },
   "source": [
    "## Equation of motion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "BHx2q1yy7NCs"
   },
   "source": [
    "The equation of motion is according to Newton's 2nd Law:\n",
    "$$m\\ddot{x} = -kx -c\\dot{x}-\\mu N\\mathop{\\rm sgn}(\\dot{x}),$$\n",
    "where $N=mg\\cos\\alpha$ is the normal force acting on the mass from the inclined plane.\n",
    "\n",
    "The normalized form of the equation of motion is:\n",
    "$$\\ddot{x}+2\\zeta\\omega_n\\dot{x}+\\omega_n^2 x=-\\mu g\\cos\\alpha\\mathop{\\rm sgn}(\\dot{x}),$$\n",
    "where the natural angular frequency and the relative damping respectively are\n",
    "$$ \\omega_n=\\sqrt{\\frac{k}{m}},\\quad \\zeta=\\frac{c}{2m\\omega_n}. $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "JFwdWqHUkSYn"
   },
   "source": [
    "Analytical solution can be obtained separating the cases w.r.t. the sign of the non-homogenous term (r.h.s.):\n",
    "$$\n",
    "x(t) = e^{-\\beta t}(C_1\\cos\\gamma t+C_2\\sin\\gamma t)-\\frac{\\mu g\\cos\\alpha}{\\omega_n^2}\\begin{cases}\n",
    "1, & \\dot{x}>0\\\\\n",
    " -1, & \\dot{x}<0 \n",
    "\\end{cases},\n",
    "$$\n",
    "where $\\gamma=\\omega_n\\sqrt{1-\\zeta^2}$ and $\\beta=\\zeta\\omega_n$ are the damped angular frequency and the decay, respectively.\n",
    "\n",
    "Using the initial condition $\\;x(0)=x_0,\\; \\dot{x}(0)=0\\;$ one has to solve the following equations in order to obtain the constants $C_1$ and $C_2$:\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "x_0 &=& C_1+\\frac{\\mu g}{\\omega_n^2}\\cos\\alpha\\\\\n",
    "0 &=& -\\beta C_1+\\gamma C_2\n",
    "\\end{eqnarray}\\Rightarrow\n",
    "$$\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "C_1 &=& x_0-\\frac{\\mu g}{\\omega_n^2}\\cos\\alpha\\\\\n",
    "C_2 &=& \\frac{\\beta x_0}{\\gamma} - \\frac{\\beta}{\\gamma\\omega_n^2}\\mu g\\cos\\alpha\n",
    "\\end{eqnarray}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "O4xLfxe7Y8f1"
   },
   "source": [
    "## Numerical simulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "velb0o-7Y8f4"
   },
   "source": [
    "For numerical simulation we use ODEINT from python's _scipy_ module. Let us rewrite the differential equation in Cauchy form:\n",
    "$$\\begin{eqnarray}\n",
    "\\dot{y_1} &=& y_2,\\\\\n",
    "\\dot{y_2} &=& -\\omega_n^2 y_1-2\\beta y_2 \\mp \\mu g\\cos\\alpha.\n",
    "\\end{eqnarray}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "U0kDj1GS2UEP"
   },
   "source": [
    "ODEINT cannot solve the non-smooth equation, therefore $\\mu$ assumed to be zero. The following parameter values are used:\n",
    "$$\n",
    "\\omega_n = 62.832\\,\\text{rad/s},\\quad \\zeta=0.1\\quad\\Rightarrow\\quad\n",
    "\\gamma=\\omega_n\\sqrt{1-\\zeta^2}=62.517\\,\\text{rad/s}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 261
    },
    "colab_type": "code",
    "id": "HUd-8pLrY8gC",
    "outputId": "e54998f1-3a46-4d22-a71d-d59ffcd6c43a"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9.9498743710662\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f9b8a4eb2b0>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.integrate import odeint\n",
    "\n",
    "def model(y,t):\n",
    "  al = 20*SP.pi\n",
    "  D2a, mug = 2*.1*al, .2*0\n",
    "  dy0 = y[1]\n",
    "  dy1 = -al**2*y[0] - D2a*y[1] - mug*SP.sign(y[1])\n",
    "  return [dy0,dy1]\n",
    "\n",
    "print(10*SP.sqrt(1-.01))\n",
    "y0 = [.1,0]\n",
    "n2, tmax = 250, 1\n",
    "t = SP.linspace(0,tmax,2*n2+1)\n",
    "y = odeint(model,y0,t)\n",
    "\n",
    "PT.figure(1,figsize=[10,3])\n",
    "PT.subplot(121)\n",
    "PT.plot(t,y[:,0],':',label=r\"$x(t)$\")\n",
    "PT.xlabel(\"$t$[s]\")\n",
    "PT.legend(loc=\"best\")\n",
    "PT.subplot(122)\n",
    "PT.plot(y[:,0],y[:,1],'r--',linewidth=1)\n",
    "PT.xlabel(\"$x$\")\n",
    "PT.ylabel(\"$\\dot{x}$\")\n",
    "PT.title(\"Phase portrait\")\n",
    "PT.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "lbUtnEeB3oRQ"
   },
   "source": [
    "## Fourier analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "L-gV8xnjY8gX"
   },
   "source": [
    "According the Shannon's Law the highest detectable frequency is the half of the sampling frequency:\n",
    "$$\\nu_{max}=\\frac1{2\\Delta t}.$$\n",
    "The Fast Fourier Transform of $x(t)$ and its plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 283
    },
    "colab_type": "code",
    "id": "fvhLNIgFY8gk",
    "outputId": "60f7a41b-99b4-4436-fe00-90df043d47c0"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f9b8a52c518>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Y = SP.fft(y[:,0])\n",
    "Y = abs(Y[:n2+1])/n2\n",
    "f = SP.linspace(0,1/t[1]/2,n2+1)\n",
    "PT.bar(f[:50],Y[:50])\n",
    "PT.xlabel(r\"$\\nu$[Hz]\")\n",
    "PT.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "3d181Y3K8IzH"
   },
   "source": [
    "## Conclusions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "TqFAlD3A8sin"
   },
   "source": [
    "The simulations confirm the expected behaviour. (The known exponential solution function could also be plotted for reference.)\n",
    "The effect of Coulomb friction could not be investigated via simulations because ODEINT cannot handle the assumed non-smooth differential equation.\n",
    "The FFT plot shows the damped frequency around 10 Hz."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "bkOMpyPj8DUr"
   },
   "outputs": [],
   "source": []
  }
 ],
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  "colab": {
   "collapsed_sections": [],
   "include_colab_link": true,
   "name": "mm_hw_ex.ipynb",
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   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
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   "hotkeys": {
    "equation": "Ctrl-E",
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 "nbformat": 4,
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