{
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   "cell_type": "markdown",
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   "source": [
    "# Free Vibrations with Viscous Damping\n",
    "\n",
    "# Introduction\n",
    "\n",
    "So far we have learned about free vibrations of conservative single degree of freedom linear systems. In this lesson, we will explore how these systems behave when dissaptive forces are added to the system. Dissipation is present in all real systems. For example, the shock absorbers in a car dissipative energy as heat to control the characteristics of the vibration of the vehicle. Drag due to air resistance is also a dissaptive force.\n",
    "\n",
    "## Goals\n",
    "\n",
    "- Write the equations of motion of the system.\n",
    "- Derive the homogenous solution.\n",
    "- Investigate the time series solution for different system properties.\n",
    "- Be able to explain the differences in the three solutions this system.\n",
    "\n",
    "## Prerequisites\n",
    "\n",
    "- Free undamped vibrations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The System\n",
    "\n",
    "![Alt text](mass_spring_damper.svg)\n",
    "\n",
    "We are going to extend the homogenous mass spring damper system by adding an applied sinusoidal force. The quantities of interest are:\n",
    "\n",
    "- $x,\\dot{x},\\ddot{x}$: position, velocity, and acceleration\n",
    "- $m$: mass\n",
    "- $c$: viscous damping coefficient\n",
    "- $k$: spring constant"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Equations of Motion\n",
    "\n",
    "We first write down Newton's Second Law for this system by examining the free body diagram:\n",
    "\n",
    "$$\\sum F_x = m \\ddot{x} = -c \\dot{x} -kx$$\n",
    "\n",
    "which provides the second order differential equation in canonical form:\n",
    "\n",
    "$$m\\ddot{x} + c\\dot{x} + kx = 0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homogenous Solution\n",
    "\n",
    "To solve this second order homogenous system we can assume a solution of:\n",
    "\n",
    "$$x(t) = a e^{\\lambda t}$$\n",
    "\n",
    "If we differentiate twice and plug the resulting three equations into the differential equation we will get:\n",
    "\n",
    "$$(m \\lambda^2 + c \\lambda + k) a e^{\\lambda t} = 0$$\n",
    "\n",
    "$a e^{\\lambda t}$ is never zero so the unique solution follows when:\n",
    "\n",
    "$$m \\lambda^2 + c \\lambda + k = 0$$\n",
    "\n",
    "This is a quadratic equation in $\\lambda$ and using the quadratic equation we can find the roots of this so called *characteristic equation*.\n",
    "\n",
    "$$\\lambda_{1, 2} = -\\frac{c}{2m} \\pm \\frac{1}{2m} \\sqrt{c^2 - 4km}$$\n",
    "\n",
    "$\\lambda_{1, 2}$ can be real or complex depending on the discriminant: $c^2 - 4km$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is useful to define some new quantities:\n",
    "\n",
    "- $\\omega_n = \\sqrt{\\frac{k}{m}}$: undamped natural frequency\n",
    "- $c_c= 2m\\omega_n$: critical damping coefficient\n",
    "- $\\zeta = \\frac{c}{c_c}$: damping ratio\n",
    "- $\\omega_d = \\omega_n \\sqrt{1 - \\zeta^2}$: damped natural frequency\n",
    "\n",
    "The equation of motion can also be written in terms of these ratios by dividing through by the mass:\n",
    "\n",
    "$$\\ddot{x} + 2 \\zeta \\omega_n \\dot{x} + \\omega_n^2 x = 0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the roots can be written as:\n",
    "\n",
    "$$\\lambda_{1, 2} = -\\zeta \\omega_n \\pm \\omega_n\\sqrt{\\zeta^2 - 1}$$\n",
    "\n",
    "Depending on the value of $\\zeta$ the roots will be a complex conjugate pair, two distinct real roots, or a repeated real root. The solution $x(t)$ depends on these three cases: $\\zeta < 1$, $\\zeta > 1$, and $\\zeta = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Underdamped ($\\zeta < 1$)\n",
    "\n",
    "This value of $\\zeta$ will produce a complex conjugate pair of roots. We can pull out $i=\\sqrt{-1}$ and write the roots as:\n",
    "\n",
    "$$\\lambda_{1, 2} = -\\zeta \\omega_n \\pm i \\omega_d$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since there are multiple roots the solution to this system and the this is linear, the solution can be written as a linear combination of the solutions:\n",
    "\n",
    "$$x(t) = e^{-\\zeta \\omega_n t}(a_1 e^{i\\omega_dt} + a_2 e^{-i\\omega_dt})$$\n",
    "\n",
    "Using Euler's formula we can reduce this to:\n",
    "\n",
    "$$ x(t) = Ae^{-\\zeta \\omega_n t} sin(\\omega_d t + \\phi)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$A$ and $\\phi$ are determined by the initial conditions.\n",
    "\n",
    "$$A = \\sqrt{\\frac{(v_0 + \\zeta\\omega_n x_0)^2 + (x_0\\omega_d)^2}{\\omega_d^2}}$$\n",
    "\n",
    "$$\\phi=\\arctan\\left(\\frac{x_0\\omega_d}{v_0+\\zeta\\omega_nx_0}\\right)$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Overdamped ($\\zeta > 1$)\n",
    "\n",
    "This value of $\\zeta$ produces two distinct real roots and the solution takes this form:\n",
    "\n",
    "$$ x(t) = e^{-\\zeta\\omega_nt}\\left(a_1 e^{-\\omega_n\\sqrt{\\zeta^2-1}t} + a_2 e^{\\omega_n\\sqrt{\\zeta^2-1}t}\\right) $$\n",
    "\n",
    "where:\n",
    "\n",
    "$$ a_1 = \\frac{-v_0 + (-\\zeta + \\sqrt{\\zeta^2 - 1})\\omega_n x_0}{2 \\omega_n \\sqrt{\\zeta^2-1}} $$\n",
    "\n",
    "$$ a_2 = \\frac{v_0 + (\\zeta + \\sqrt{\\zeta^2 - 1})\\omega_n x_0}{2 \\omega_n \\sqrt{\\zeta^2-1}} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Critically Damped ($\\zeta = 1$)\n",
    "\n",
    "Finally, the special case gives two repeated roots and the solution takes the form:\n",
    "\n",
    "$$ x(t) = (a_1 + a_2t)e^{-\\omega_n t}$$\n",
    "\n",
    "where:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "a_1 &= x_0 \\\\\n",
    "a_2 &= v_0 + \\omega_n x_0\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulation of the Motion\n",
    "\n",
    "Let's explore the solution to the system starting with these parameters:\n",
    "\n",
    "- $m=0.049$ kg\n",
    "- $k = 857.8$ N/m\n",
    "- $c=0.11$ kg/s\n",
    "\n",
    "Depending on the value of $\\zeta$ the solution will described by three qualitively different motions: over-damped $(\\zeta > 1)$, critically damped $(\\zeta = 1)$, or under-damped $(\\zeta < 1)$ motion. Use your code to discover what these motions look like and how the parameters of the system affect the behavior."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "# Exercise\n",
    "\n",
    "Create a function that accepts $m$, $c$, and $k$ as parameters and returns $\\omega_n$, $\\zeta$, and $c_c$.\n",
    "\n",
    "**Note**: we don't compute $\\omega_d$ in case the system is overdamped ($\\omega_d$ doesn't exist)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def vibration_properties(m, c, k):\n",
    "    # natural frequency [rad/s]\n",
    "    omega_n = np.sqrt(k / m)\n",
    "    # critical damping coefficient [kg/s]\n",
    "    cc = 2 * m * omega_n\n",
    "    # damping ratio\n",
    "    zeta = c / cc\n",
    "    return omega_n, zeta, cc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise\n",
    "\n",
    "Create a function that accepts $m$, $c$, $k$, $x_0$, $v_0$ and a simulation duration $d$ and returns an array of 1000 time values and an array of corresponding $x(t)$ values. Make use of the `vibration_properties` function inside this one.\n",
    "\n",
    "We need to handle underdamped, critically damped, and overdamped cases, so we'll use a series of `if` statements to compute the response in each case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def simulate(m, c, k, d, x0=0.0, v0=0.0):\n",
    "    # get our vibration properties\n",
    "    omega_n, zeta, cc = vibration_properties(m, c, k)\n",
    "    \n",
    "    # time values from 0 to d (duration) with 1000 points\n",
    "    time_array = np.linspace(0, d, num=1000)\n",
    "    \n",
    "    # underdamped\n",
    "    if zeta < 1:\n",
    "        # damped natural frequency [rad/s]\n",
    "        omega_d = omega_n * np.sqrt(1 - zeta**2)\n",
    "        # amplitude\n",
    "        A = np.sqrt(((v0 + zeta*omega_n*x0)**2 +\n",
    "                     (x0*omega_d)**2) / omega_d**2)\n",
    "        # phase\n",
    "        phi = np.arctan2(x0*omega_d, v0 + zeta*omega_n*x0)\n",
    "        # total response\n",
    "        x_array = A * np.exp(-zeta*omega_n*time_array) * \\\n",
    "            np.sin(omega_d*time_array + phi)\n",
    "    \n",
    "    # overdamped\n",
    "    elif zeta > 1:\n",
    "        # amplitude of first term\n",
    "        a1 = ((-v0 + (-zeta + np.sqrt(zeta**2-1))*omega_n*x0) /\n",
    "              (2*omega_n*np.sqrt(zeta**2-1)))\n",
    "        # amplitude of second term\n",
    "        a2 = ((v0 + (zeta + np.sqrt(zeta**2-1))*omega_n*x0) /\n",
    "              (2*omega_n*np.sqrt(zeta**2-1)))\n",
    "        # exponent\n",
    "        exp = omega_n * np.sqrt(zeta**2 - 1) * time_array\n",
    "        # total response\n",
    "        x_array = np.exp(-zeta*omega_n*t) * \\\n",
    "            (a1*np.exp(-exp) + a2*np.exp(exp))\n",
    "        \n",
    "    # critically damped\n",
    "    else:\n",
    "        # amplitude of first term\n",
    "        a1 = x0\n",
    "        # amplitude of second term\n",
    "        a2 = v0 + omega_n*x0\n",
    "        # total response\n",
    "        x_array = (a1 + a2*time_array)*np.exp(-omega_n*time_array)\n",
    "        \n",
    "    return time_array, x_array"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise\n",
    "\n",
    "Using your functions choose various values of $c$ such that you get three different values for $\\zeta$ where $(\\zeta > 1)$, $(\\zeta = 1)$, and $(\\zeta < 1)$ and plot the resulting motions for these values. In a text box describe what you observe.\n",
    "\n",
    "First, we can use our `vibration_properties` function to see what changing $c$ does to $\\zeta$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c =  0.00, zeta = 0.00\n",
      "c =  2.22, zeta = 0.17\n",
      "c =  4.44, zeta = 0.34\n",
      "c =  6.67, zeta = 0.51\n",
      "c =  8.89, zeta = 0.69\n",
      "c = 11.11, zeta = 0.86\n",
      "c = 13.33, zeta = 1.03\n",
      "c = 15.56, zeta = 1.20\n",
      "c = 17.78, zeta = 1.37\n",
      "c = 20.00, zeta = 1.54\n"
     ]
    }
   ],
   "source": [
    "m = 0.049 # mass [kg]\n",
    "k = 857.8 # spring stiffness [N/m]\n",
    "\n",
    "for c in np.linspace(0, 20, 10):\n",
    "    omega_n, zeta, cc = vibration_properties(m, c, k)\n",
    "    print(\"c = {:5.2f}, zeta = {:.2f}\".format(c, zeta))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see from the \"table\" printout that as $c$ approaches the critical value $c_c$, $\\zeta$ approaches 1.\n",
    "\n",
    "Now we'll just select some values for $c$ which result in appropriate values of $\\zeta$ for each case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f4fc0c4b710>"
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     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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TDtWBrU+UbQJqPO/Etm3bZrjztm3b0rp1a+bOnYu9vT0rVqwAHi17LFSoUJZv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\nlNCIiIhIxFNCIyIiIhFPCY2IiIhEvP8fD/AERFYcFjAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f4fc0c4b6d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "# some values of c and plot properties\n",
    "# note: we can use cc from the above loop\n",
    "c_values = [3, cc, 20]\n",
    "labels = ['underdamped', 'critically damped', 'overdamped']\n",
    "alphas = [0.4, 0.7, 1]\n",
    "\n",
    "x0 = 1   # initial position [m]\n",
    "v0 = 100 # initial velocity [m/s]\n",
    "\n",
    "for c, label, alpha in zip(c_values, labels, alphas):\n",
    "    t, x = simulate(m, c, k, 0.15, x0=x0, v0=v0)\n",
    "    plt.plot(t, x, alpha=alpha, label=label, color='k')\n",
    "    \n",
    "plt.xlabel(r'$t$ [s]')\n",
    "plt.ylabel(r'$x(t)$ [m]')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise\n",
    "\n",
    "A spring-mass-damper system has a mass of 150 kg, stiffness of 1500 N/m, and damping coefficient of 200 kg/s. Calculate the undamped natural frequency, the damping ratio, and the damped natural frequency. Is the system overdamped, underdamped, or critically damped? Does the solution oscillate? Show the motion in a plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "undamped natural frequency: 3.16 rad/s\n",
      "damping ratio: 0.21\n",
      "damped natural frequency: 3.09 rad/s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7f4fb4feb5f8>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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tJaIUpz1/9BHMmhXfeJZMBx4YFra75Za4KxERKV8KLSWiFEPLU0+FFpZieNKy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\noWW//cLsoXIxalSYSXTttTB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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f4fb4fd64e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# system parameters\n",
    "m = 150  # mass [kg]\n",
    "k = 1500 # stiffness [N/m]\n",
    "c = 200  # damping coefficient [kg/s]\n",
    "\n",
    "# make up some initial conditions\n",
    "x0 = 1  # initial position [m]\n",
    "v0 = 10 # initial velocity [m/s]\n",
    "\n",
    "# compute the vibration properties\n",
    "omega_n, zeta, cc = vibration_properties(m, c, k)\n",
    "omega_d = omega_n * np.sqrt(1 - zeta**2)\n",
    "\n",
    "print(\"undamped natural frequency: {:.2f} rad/s\".format(omega_n))\n",
    "print(\"damping ratio: {:.2f}\".format(zeta))\n",
    "print(\"damped natural frequency: {:.2f} rad/s\".format(omega_d))\n",
    "\n",
    "# simulate and plot\n",
    "t, x = simulate(m, c, k, 10, x0=x0, v0=v0)\n",
    "plt.plot(t, x)\n",
    "plt.xlabel(r'$t$ [s]')\n",
    "plt.ylabel(r'$x(t)$ [m]')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since the damping ratio is less than one, the system is **underdamped** and it oscillates."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}