Kinematics_ Simple Harmonic Oscillator.ipynb

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    "# A Simple Harmonic Oscillator\n",
    "\n",
    "A simple harmonic oscillator comes up often in many areas of physics. This ranges from simple spring-mass set-ups, to small displacements of pendulums, to vibrations in quantum field theory.\n",
    "\n",
    "![image.png](data:image/png;base64,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    "\n",
    "A harmonic oscillator is typically governed by a quadratic potential. Examples include elastic potential energy ($\\frac{1}{2}kx^2$), capacitance charging energy ($\\frac{1}{2}CV^2$) or inductance in magnetic fields ($\\frac{1}{2}LI^2$). We can write the motion of a harmonic oscillator in terms of Newtonian forces, such that:\n",
    "\n",
    "$$\n",
    "F_\\textrm{SHO} = ma = - kx\n",
    "$$\n",
    "\n",
    "We can see that the force $F_\\textrm{SHO}$ (proportional to mass $m$ and acceleration $a=\\frac{d^2x}{dt^2}$) is written as being proportional (to a factor $k$) to the displacement $x$ and directed in the opposite direction to the displacement. This force is known as the <u>restoring</u> force\n",
    "\n",
    "However, this set-up only considers a very simple system. For typical harmonic oscillators, there are typically two additional terms contributing to the force: (1) a damping term that dissipates energy, and (2) a driving term that adds energy. \n",
    "\n"
   ]
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   "source": [
    "## Damping Term\n",
    "\n",
    "In real world oscillators, there is often some resistance to the motion of the oscillator that dissipates energy as heat. For example, this could be friction, air resistance, or electrical resistance. This term is generally proportional to velocity, and acts to oppose the force. This can be written as\n",
    "\n",
    "$$\n",
    "F_\\mathrm{tot} = F_\\mathrm{SHO} + F_\\mathrm{damp}\n",
    "$$\n",
    "\n",
    "where $F_\\mathrm{damp} = - cv = -c\\frac{dx}{dt}$, $c$ is a positive constant. \n"
   ]
  },
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    "\n",
    "## Driving Term\n",
    "\n",
    "Another feature of real world oscillators is the prescence of a driving term. This could be a periodic force on a system, such as shaking a pendulum set-up, or radio waves for a oscillating circuit. This term can take on any form, depending on the drive applied, but is generally taken to be a sinusoidal force.\n",
    "\n",
    "For this simulation, we will not consider this term."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
    "id": "V6smPkp-tutX"
   },
   "source": [
    "## Numerical Integration\n",
    "\n",
    "The idea behind numerical integration is to take small steps in time ($\\delta t$) and determine the path the object will follow. This is different from reality as time flows smoothly and continuously. As such, we can deduce that the smaller time steps we take, the closer we get to a continuous flow of time, and the more accurate the simulation (At the cost of computational power).\n",
    "\n",
    "To do numerical integration, we must first understand (1) position $r$, (2) velocity $v$, and (3) acceleration $a$. In physics, velocity is defined as change in position over time $t$, and acceleration is change in velocity over time $t$. As simple way to express this is\n",
    "\n",
    "$$\n",
    "v \\equiv \\frac{dr}{dt} \\approx \\frac{r_f - r_i}{t} \\\\\n",
    "a \\equiv \\frac{dv}{dt} \\approx \\frac{v_f - v_i}{t}\n",
    "$$\n",
    "\n",
    "The time differentials ($d/dt$) indicate a change in time with respect to a variable. We can see that it is essentially taking the difference between the final and inital values (subscript $f$ and $i$), and dividing by the time difference $t$.\n",
    "\n",
    "We can use this equation to find the final velocity/position, given that we know the acceleration/velocity and the time taken. By choosing a small time step $\\delta t$, we can find the position at the next instance in time:\n",
    "\n",
    "$$\n",
    "v = \\frac{r_f - r_i}{\\delta t} \\\\\n",
    "r_f = r_i + v\\cdot\\delta t\n",
    "$$\n",
    "\n",
    "Likewise for velocity\n",
    "\n",
    "$$\n",
    "a = \\frac{v_f - v_i}{\\delta t} \\\\\n",
    "v_f = v_i + a\\cdot\\delta t\n",
    "$$\n",
    "\n",
    "Thus, we can sequentially determine the velocity and position of an object step-by-step, given that we know the acceleration (proportional to force, $F=ma$) on the object. "
   ]
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   "source": [
    "# Simulation\n",
    "\n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
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    "id": "9DVSBvFCurRX"
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   "source": [
    "## Importing Libraries\n",
    "\n",
    "First, we have to import the libraries required. We will need\n",
    "\n",
    "1. `matplotlib.pyplot` for plotting our results\n",
    "2. `numpy` for making the math"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "zlMJTwk3nf8l"
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
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   "metadata": {
    "id": "auO967V9u8Wb"
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   "source": [
    "## Variables\n",
    "\n",
    "Consider the variables we need. For a spring-mass system, we can consider three variables: \n",
    "\n",
    "1.   Mass of Oscillator $m$\n",
    "2.   Spring Constant $k$\n",
    "3.   Viscous Damping Coefficient $c$\n",
    "\n",
    "And for a given oscillator, we can define initial conditions for the position, velocity. For this example, we will take initial velocity to be $0$, and consider only initial displacement.\n",
    "\n",
    "For multiple oscillators, we should consider this set of variables again for each oscillator. For this example, we will consider just one oscillator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "_mOPFfyivdQe"
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   "outputs": [],
   "source": [
    "m = 10                  # Mass of oscillator\n",
    "k = 2                   # Spring constant\n",
    "c = 1                   # Viscous damping coefficient\n",
    "\n",
    "x0 = 5                  # Initial position offset\n",
    "v0 = 0                  # Initial velocity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "kyc2VetwwBku"
   },
   "source": [
    "For numerical methods, we also need to define a time step $\\delta t$ that we will take at every iteration, as well as the total run time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "i6zBUACxwPq0"
   },
   "outputs": [],
   "source": [
    "timestep = 0.1\n",
    "runtime = 100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "LQ7-w8MHvz-b"
   },
   "source": [
    "For plotting later on, we also want something to keep track of the history of the motion of the oscillator. We will use a list, and append the position and velocity at every instant in time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "mg44V940wUBb"
   },
   "outputs": [],
   "source": [
    "pos_hist = [x0]\n",
    "vel_hist = [v0]\n",
    "time_hist = [0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "10p7odebwfhC"
   },
   "source": [
    "## Define the force\n",
    "\n",
    "We can write a function that will apply the force for us. It should take in the inputs of a specific oscillator (mass, spring constant, viscous damping coefficient) and its current state (position and velocity), and return us the state at the next instant in time (position and velocity). It should also update the history of the oscillator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "_1OdEHFgw1Fc"
   },
   "outputs": [],
   "source": [
    "def ApplyForce(mass, springk, dampc, pos, vel):\n",
    "    force = - springk*pos - dampc*vel                   # From the formula given above\n",
    "    accel = force/mass                                  # From Newton's equation\n",
    "\n",
    "    new_pos = pos + vel*timestep                        # From the equation for numerical integration\n",
    "    new_vel = vel + accel*timestep\n",
    "\n",
    "    pos_hist.append(new_pos)                            # Add the new position and velocity to the history\n",
    "    vel_hist.append(new_vel)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "sZWn4k_rxg6H"
   },
   "source": [
    "## Running the simulation\n",
    "\n",
    "We can now run the simulation. We will need to keep track of the current time, position and velocity of the oscillator. A neat way to do that is to simply consider the last element inside the history lists defined earlier.\n",
    "\n",
    "The iteration itself will be done using a *while* loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "nx8vMDWox_W1"
   },
   "outputs": [],
   "source": [
    "while time_hist[-1] < runtime:                                          # While the time is less than the total runtime, \n",
    "    time_hist.append( time_hist[-1] + timestep )                        # Update the newest time step, by adding to the previous time step\n",
    "    ApplyForce(m, k, c, pos_hist[-1], vel_hist[-1])                     # Run the fuction defined above"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "yntl7x1ayTZT"
   },
   "source": [
    "## Plotting the Results\n",
    "\n",
    "We can see the motion of our oscillator by plotting the results. Consider the position/time and velocity/time graphs:"
   ]
  },
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     "elapsed": 657,
     "status": "ok",
     "timestamp": 1639449856468,
     "user": {
      "displayName": "Kai Xiang Lee",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgBJUAJUYel5rdbTnU92LBd__AiaBgIY-k-dk5xLA=s64",
      "userId": "12776711516058684946"
     },
     "user_tz": -480
    },
    "id": "jNSItm49yeF1",
    "outputId": "4c8d8770-08b5-42f9-a4b5-d034ac025b6a"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# For position\n",
    "\n",
    "plt.plot(time_hist, pos_hist)\n",
    "plt.title(\"Position against Time\")\n",
    "plt.xlabel(\"Time (s)\")\n",
    "plt.ylabel(\"Displacement (m)\")\n",
    "plt.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 313
    },
    "executionInfo": {
     "elapsed": 611,
     "status": "ok",
     "timestamp": 1639449860880,
     "user": {
      "displayName": "Kai Xiang Lee",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgBJUAJUYel5rdbTnU92LBd__AiaBgIY-k-dk5xLA=s64",
      "userId": "12776711516058684946"
     },
     "user_tz": -480
    },
    "id": "Z1vHxk2UytaA",
    "outputId": "b6edb5c9-a882-410e-aff6-382f08d049d6"
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# For velocity\n",
    "\n",
    "plt.plot(time_hist, vel_hist)\n",
    "plt.title(\"Velocity against Time\")\n",
    "plt.xlabel(\"Time (s)\")\n",
    "plt.ylabel(\"Velocity (m/s)\")\n",
    "plt.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-5eCXhTH07qk"
   },
   "source": [
    "We can see that over time, the amplitude and maximum speed slowly decreases due to the dissipation term."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "wmCN-SJgy0Ll"
   },
   "source": [
    "We can also consider the *phase-space diagram*. To plot this, we consider two variables that define the state of the system and plot them against each other. In this case, we can plot position against velocity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 295
    },
    "executionInfo": {
     "elapsed": 981,
     "status": "ok",
     "timestamp": 1639449986778,
     "user": {
      "displayName": "Kai Xiang Lee",
      "photoUrl": "https://lh3.googleusercontent.com/a-/AOh14GgBJUAJUYel5rdbTnU92LBd__AiaBgIY-k-dk5xLA=s64",
      "userId": "12776711516058684946"
     },
     "user_tz": -480
    },
    "id": "DYVxxHJN1Mih",
    "outputId": "3093a424-2f1b-41f9-b4fc-bea50107f001"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(pos_hist, vel_hist)\n",
    "plt.title(\"Phase Space Diagram\")\n",
    "plt.ylabel(\"Velocity (m/s)\")\n",
    "plt.xlabel(\"Displacement (m)\")\n",
    "plt.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ENir6sMB33gG"
   },
   "source": [
    "We can see that over time, the line spirals inwards as the system loses energy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "zAzhnxzX1bSB"
   },
   "source": [
    "# Possible Extensions\n",
    "\n",
    "1. Determine the effects of the mass, spring constant, and viscous damping coefficient on the (1) Half-life of the oscillator, (2) Natural frequency of the oscillator\n",
    "2. Add a sinusoidal driving term, and determine its effects on the oscillator\n",
    "3. What if you have two oscillators attached together (e.g., by a weak spring)\n",
    "![image.png](data:image/png;base64,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)\n",
    "4. Double pendulum set-up\n",
    "![image.png](data:image/png;base64,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)\n",
    "5. Adding more degrees of freedom (More dimensions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "HqNzRNzr309w"
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "colab": {
   "authorship_tag": "ABX9TyPi7ojzBk6WMB+SthhFF7d2",
   "collapsed_sections": [],
   "name": "Kinematics: Simple Harmonic Oscillator.ipynb",
   "provenance": [],
   "toc_visible": true
  },
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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