two_forces_on_a_rectangle.ipynb

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  "name": "two_forces_on_a_rectangle"
 },
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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "Two forces on a rectangle"
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Introduction"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Consider an object with two or more forces acting on it. The forces are fixed to the object. What would happen to the object?\n\nAccording to Newton's second law a net force results in an acceleration of the object and thus translation. If a net torque is present, then this will result in an angular acceleration and therefore rotation.\n"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Let's consider a more specific case. We have a rectangle of size 4 x 4 meter, and which weights 400 gram. The height is infinitesimal.\n\nTwo fixed forces act on the rectangle. The forces are assumed to be parallel to each other and parallel to the initial orientation of the rectangle. The first force is 5 newton and acts on coordinate (-2,-2, 0), and the second force is 3 newton and acts on (+1,-1, 0). The reference point is in this case the center of the rectangle."
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Theory"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Force and torque"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Newton's second law, $\\vec{F}=m\\vec{a}$, states that a net force results in an acceleration $\\vec{a}$ of an object with a mass $m$. The same can be written for torque, $\\vec{\\tau} = I \\vec{\\alpha}$, where $I$ is the mass moment of inertia and $\\vec{\\alpha}$ the angular acceleration.\n \n"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The net force on the object is given by the vector summation over all the forces acting on the object. $\\vec{F}_{net} = \\displaystyle{\\sum_{i=1}^n} \\vec{F}_i=\\vec{F}_1 + \\vec{F}_2$.\n"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Torque, or moment of force, is denoted as $\\mathbf{\\tau}$ and is defined as the outer product between the length of the lever arm and the force, $\\vec{\\tau} = \\vec{r} \\times \\vec{F}$. The magnitude of the torque can be calculated by $\\tau = \\|\\vec{r}\\| \\|\\vec{F}\\| \\cos{\\theta}$.\n\n\n"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The net torque is given by $\\tau_{net} = \\displaystyle{\\sum_{i=1}^n} \\tau_i = \\tau_1 + \\tau_2$. The magnitude of the torque can also be calculated by $\\tau = - \\|\\vec{r}_1\\| \\|\\vec{F_1} \\cos{\\theta_1} + \\|\\vec{r_2} \\|\\vec{F_2} \\cos{\\theta_2}$. Note the minus sign."
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Torque lever arms"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "A force applied at a certain distance from the center of rotation causes a torque."
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Mass and moment of inertia"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The moment of inertia is obtained by integrating over the mass (volume) density $\\rho_V(\\vec{r})$ taking into account the mass its distance from the center of rotation. This integral is written as $I = \\displaystyle{\\int_V} \\rho_V(\\vec{r}) \\vec{r}^2 \\mathrm{d} V$.\n\nConsidering a two-dimensional world, the integral is written as $I = \\displaystyle{\\int_A} \\rho_A(\\vec{r}) \\vec{r}^2 \\mathrm{d} A$, where $\\rho_A(\\vec{r})$ is the area density.\n"
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": "Translation and rotation"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "From Newton's second law it follows that the acceleration of the rectangle is given by $\\vec{a} = \\vec{F}/m$ and the angular acceleration by $\\vec{\\alpha}=\\vec{\\tau}/I$.\n\nThe change in velocity is given by $\\Delta \\vec{v} (t) = \\vec{a} \\Delta t$. Similarly, the change in angular velocity is given by $\\Delta \\vec{\\omega} (t)=\\vec{\\alpha} \\Delta t$. In these equations $\\Delta t$ represents the timestep.\n\nThe change in position is calculated as $\\Delta x(t) = \\vec{v}(t)t + 0.5 \\vec{a}(t) \\Delta t^2$, where $v(t)$ is the velocity at time $t$. Similarly, the change in orientation is given by $\\Delta \\theta (t) = \\vec{\\omega}(t)t + 0.5 \\vec{\\alpha}(t) \\Delta t^2$, where $\\omega (t)$ is the angular velocity at time $t$. "
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Python model"
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Now that we have a description of the physics involved, let's see whether we can simulate this."
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "To keep track of the position of the rectangle, we first define a `Point` class."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "class Point(object):\n    \"\"\"Point or coordinate.\"\"\"\n    def __init__(self, x, y):\n        self.x = x\n        self.y = y\n        \n    def __add__(self, other):\n        return self.__class__(self.x+other.x, self.y+other.y)\n    def __sub__(self, other):\n        if isinstance(other, Point):\n            return Vector(self.x-other.x, self.y-other.y)\n        else:\n            return self.__class__(self.x-other.x, self.y-other.y)\n        \n    def __repr__(self):\n        return \"Point({:.1f},{:.1f})\".format(self.x, self.y)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "We will also define a `Vector` class. Instance of this class we will use for vectors like forces and lever arms."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "class Vector(object):\n    \"\"\"Vector.\"\"\"\n    \n    def __init__(self, x, y):\n        self.x = x\n        self.y = y\n    \n    def __add__(self, other):\n        return self.__class__(self.x+other.x, self.y+other.y)\n\n    def __sub__(self, other):\n        return self.__class__(self.x-other.x, self.y-other.y)\n    \n    def __mul__(self, other):\n        try:\n            return self.__class__(self.x*other.x, self.y*other.y)\n        except AttributeError:\n            return self.__class__(self.x*other, self.y*other)\n                \n    def __div__(self, other):\n        return self.__class__(self.x/other, self.y/other)\n    \n    def __pow__(self, other):\n        return self.__class__(self.x**other, self.y**other)\n    \n    def __rmul__(self, other):\n        return self.__class__(self.x*other, self.y*other)\n    \n    @property\n    def norm(self):\n        \"\"\"Vector norm.\"\"\"\n        return (self.x**2.0 + self.y**2.0)**(0.5)\n    \n    def cross(self, other):\n        \"\"\"Cross product.\"\"\"\n        return self.x * other.y - other.x * self.y\n    \n    def rotate_by(self, rotation):\n        \"\"\"Rotate vector by a given amount of radians.\"\"\"\n        x = self.x * math.cos(rotation) - self.y * math.sin(rotation)\n        y = self.x * math.sin(rotation) + self.y * math.cos(rotation)\n        return Vector(x, y)\n        \n        \n    def __repr__(self):\n        return \"Vector({:.1f},{:.1f})\".format(self.x, self.y)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "We will also define a `Torque` class. A torque requires an `arm` and a `force`, both instances of `Vector`. The torque is calculated using the cross or outer product."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "class Torque(object):\n    \n    def __init__(self, arm, force):\n        self.force = force\n        self.arm = arm\n\n    def __add__(self, other):\n        return float(self._value + other)\n        \n    def __radd__(self, other):\n        return float(other + self._value)\n\n    def __repr__(self):\n        return \"Torque({:.1f}\".format(self._value)\n    \n    @property\n    def _value(self):\n        return self.arm.cross(self.force)\n    \n    ",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "And of course we need a definition for our `Rectangle`. \n\n* The attribute `position` represents the center of mass of the rectangle and is an instance of `Point`. \n\n* The attribute `velocity` represents the velocity of the rectangle and is an instance of `Vector`. \n\n* Since we have only rotation in one plane, the angular velocity is a scalar and therefore stored as a float as `velocity_angular`.\n\n* The attribute `angle` represents the orientation of the rectangle around the center of mass, and is a float.\n\n* The attributes `forces` and `torques` are iterables, like lists."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "class Rectangle(object):\n    \"\"\"The moving rectangle.\"\"\"\n    \n    def __init__(self, position, angle, velocity, velocity_angular, mass, moment_of_inertia, force, torque):\n        self.position = position\n        self.velocity = velocity\n        self.velocity_angular = velocity_angular\n        self.angle = angle\n        self.mass = mass\n        self.moment_of_inertia = moment_of_inertia\n        self.force = force\n        self.torque = torque\n\n    @property\n    def acceleration(self):\n        return self.force / self.mass\n        \n    @property\n    def acceleration_angular(self):\n        return self.torque / self.moment_of_inertia",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The angular acceleration depends on the moment of intertia of the rectangle. Therefore, in order to create an instance of `Rectangle` we need the moment of inertia, for which we need to know the center of rotation. The center of rotation depends on the forces and their lever arms."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "force_1 = Vector(0.0, 5.0)\nforce_2 = Vector(0.0, 3.0)\n\npos_force_1 = Point(-2.0,-2.0)\npos_force_2 = Point(+1.0,-1.0)\n\ncenter_of_rotation = Point(0.0,0.0)\n\narm_1 = center_of_rotation - pos_force_1\narm_2 = center_of_rotation - pos_force_2\n\nprint(\"Arm 1: {}\".format(arm_1))\nprint(\"Arm 2: {}\".format(arm_2))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Arm 1: Vector(2.0,2.0)\nArm 2: Vector(-1.0,1.0)\n"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Now that we know the center of rotation of the rectangle we can calculate its moment of inertia. First we calculate the area density of the rectangle."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "mass = 0.400\nlength = 4.0\nwidth = 4.0\narea = length * width\ndensity = mass / area\nprint(\"Area density: {} kg/m2\".format(density))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Area density: 0.025 kg/m2\n"
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "In order to calculate the moment of inertia we have to integrate over the rectangle, and therefore we first have to discretize it. We define a grid resolution `dx`. "
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "spacing = 0.001\npos_x = np.arange(-length/2, +length/2, spacing)\npos_y = np.arange(-width/2, +width/2, spacing)\ngrid_x, grid_y = np.meshgrid(pos_x, pos_y)\nnodes = len(grid_x)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "For the summation we need the mass for each small volume. The total mass is therefore divided by the amount of nodes in the grid."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "mass_per_node = mass / nodes**2.0\n\nprint(\"Mass per node: {}\".format(mass_per_node))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Mass per node: 2.5e-08\n"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The moment of inertia is finally given by"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "moment_of_inertia = mass_per_node * ((grid_x - center_of_rotation.x)**2.0 + (grid_y - center_of_rotation.y)**2.0 ).sum()\n\nprint(\"Moment of inertia: {}\".format(moment_of_inertia))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Moment of inertia: 1.0666668\n"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Now that we know the moment of inertia, all we need are the torques."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "torque_1 = Torque(arm_1, force_1)\ntorque_2 = Torque(arm_2, force_2)\n\nprint(\"Torque 1: {}\".format(torque_1))\nprint(\"Torque 2: {}\".format(torque_2))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Torque 1: Torque(10.0\nTorque 2: Torque(-3.0\n"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "net_force = force_1 + force_1\nnet_torque = torque_1 + torque_2\n\nprint net_force\nprint net_torque",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Vector(0.0,10.0)\n7.0\n"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Now we have everything we need to instantiate the rectangle. Note the initial orientation of the rectangle. "
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "rect = Rectangle(position=Point(0.0,0.0),\n                 angle=0.0,\n                 velocity=Vector(0.0,0.0), \n                 velocity_angular=0.0, \n                 mass=mass, \n                 moment_of_inertia=moment_of_inertia,\n                 force = net_force,\n                 torque = net_torque)",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Since this is a simulation in the time domain we have to define a timestep ``dt`` in seconds. Let's run the simulation for 10 seconds."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "dt = 0.001",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "We're almost ready to run the simulation. First we have to define the simulation time. Together with the previously defined timestep this determines the amount of steps that are performed."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simulation_time = 1.0\nsteps = int(math.ceil(simulation_time / dt))",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Just to check..."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "print(\"Timestep: {}\".format(dt))\nprint(\"Steps: {}\".format(steps))",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "Timestep: 0.001\nSteps: 1000\n"
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "Finally we are ready to implement the update equations. Every timestep the velocities, position and orientation are recalculated. The rotation is also applied to the force vectors."
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "positions = list()\nvelocities = list()\nforces = list()\nthetas = list()\n\nfor i in range(steps):\n    \n    rect.velocity += rect.acceleration * dt\n    rect.velocity_angular += rect.acceleration_angular * dt\n    \n    dx = rect.velocity + 0.5 * rect.acceleration**2.0\n    dtheta = rect.velocity_angular + 0.5 * rect.acceleration_angular**2.0\n    rect.position += dx\n    rect.angle += dtheta\n    \n    rect.force = rect.force.rotate_by(dtheta)\n    #for force in rect.forces:\n    #    force.rotate_by(dtheta) \n    #for torque in rect.torques:\n    #    torque.arm.rotate_by(dtheta)\n        \n    #print(\"Time: {:.1f} - Position: {} - Angle: {:.1f}\".format(i*dt, rect.position, rect.angle))\n    positions.append(rect.position)\n    velocities.append(rect.velocity)\n    forces.append(rect.force)\n    thetas.append(rect.angle%(2.0*math.pi))",
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "The graph clearly shows the rotation of the object."
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "plot([s.x for s in positions], [s.y for s in positions])\nxlabel(r'$x$ in m')\nylabel(r'$y$ in m')\ntitle(\"Position of the object\")\n",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": "<matplotlib.text.Text at 0x20b8790>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ybhO7rZopj0fE5RIpKREZOFBk9myzI1Lql/PltdO0FRjatm2Ly+VCRHC5XLRu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\n",
       "text": "<matplotlib.figure.Figure at 0x1fac250>"
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "plot([v.norm for v in velocities])\nxlabel(r'$t$')\nylabel(r'$\\|\\vec{v}\\|$ in m/s')\ntitle(\"Magnitude of velocity as function of time\")",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": "<matplotlib.text.Text at 0x226ef50>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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GkRV7VEmjSRM51qKkW4oTmQOTBpEVe1TSaN1a7qPcWZbI3Jg0iKxUYaG8bbnh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\nEBGRyf4/c0RmtzqkMEUAAAAASUVORK5CYII=\n",
       "text": "<matplotlib.figure.Figure at 0x2266950>"
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "plot([v.x for v in velocities], [v.y for v in velocities])\nxlabel(r'$x$ in m')\nylabel(r'$y$ in m')\ntitle(\"Velocity components of the object\")",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": "<matplotlib.text.Text at 0x287a510>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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soFSM3L4N/PorP65dA95+m6v2zs58B5bXwKCLlBRepSw8nPPihIdzugNXV/4S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\nhNCKBAwhhBBa+T/O+T1wEygBDQAAAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x2864fd0>"
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "plot([f.x for f in forces], [f.y for f in forces])\nxlabel(r'$x$ in m')\nylabel(r'$y$ in m')\ntitle(\"Force components of the object\")",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": "<matplotlib.text.Text at 0x24d7190>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ZWXUkb162ctmz09+yZeX7kicHmD8f4NAhtnMtXAiwdStbWam0bElMpyzy0kvs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\nGIZhGDvYODAMwzB2/B/UTHRj4VQh2gAAAABJRU5ErkJggg==\n",
       "text": "<matplotlib.figure.Figure at 0x2266d90>"
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "plot(thetas)\nxlabel(r\"$t$\")\nylabel(r\"$\\theta$\")\ntitle(\"Rotation angle as function of time\")",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": "<matplotlib.text.Text at 0x2a01390>"
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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smQD3+ysVub7WrpUVhcQEbCN9qMVP01EG8qtfRa4eG/g97rgD+J//KV2niorn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\nwaDbMm0HtA5ZMAFXHls5l1wS774UdFWBtBQLUINKQCMtJZD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       "text": "<matplotlib.figure.Figure at 0x1fa9310>"
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": "It's also interesting to experiment with changing the positions and sizes of the forces."
    }
   ],
   "metadata": {}
  }
 ]
}