Copy of euler-balloon.ipynb

copy-of-euler-balloon.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/gist/vuddameri/a37556af181df4a8e8351ec083574d5a/copy-of-euler-balloon.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "1c98c236",
      "metadata": {
        "id": "1c98c236"
      },
      "source": [
        "<h1><font color='blue'><center> CVEN 2320 Differential Equations for Civil Engineers </center></h1>"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "cdee0e99",
      "metadata": {
        "id": "cdee0e99"
      },
      "source": [
        "<center><font color='teal'><h2> Venki Uddameri, Ph.D. P.E., </h2>"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "daa40516",
      "metadata": {
        "id": "daa40516"
      },
      "source": [
        "<hr>"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "7cf61277",
      "metadata": {
        "id": "7cf61277"
      },
      "source": [
        "<h4> Problem Statement:"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "b47f48cd",
      "metadata": {
        "id": "b47f48cd"
      },
      "source": [
        "A hot air balloon with a mass of 2200 lb$_m$ moves with upward with an accelration,a, of 34 ft/s$^2$.  The drag force on the balloon is proportional to the square of the velocity.  Assume the constant of proportionality is 0.5.  Assume the balloon is nearly at rest at the start, plot the velocity profile of the balloon.  Will the velocity reach a constant value in time?   Neglect the forces of buoyancy.  The acceleration of the gravity, g, equals 32.17 ft/s$^2$."
      ]
    },
    {
      "cell_type": "markdown",
      "id": "23e9567f",
      "metadata": {
        "id": "23e9567f"
      },
      "source": [
        "<h4> Mathematical Model:"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "2e4f77f6",
      "metadata": {
        "id": "2e4f77f6"
      },
      "source": [
        "$ F_n = F_l + F_g - F_d $\n",
        "\n",
        "$ m\\frac{dv}{dt} = ma - mg - kv^2 $\n",
        "\n",
        "$ \\frac{dv}{dt} = (a - g) - \\frac{k}{m}v^2 $\n",
        "\n",
        "Using the Euler's Method the velocty term can be written as:\n",
        "\n",
        "$ v_{t+1} = v_t + \\left[ (a-g) - \\frac{k}{m}v_t^2 \\right] $"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "a1a2f519",
      "metadata": {
        "id": "a1a2f519"
      },
      "source": [
        "<hr>"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "3efb3006",
      "metadata": {
        "id": "3efb3006"
      },
      "source": [
        "<h4> Code:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "d16437c5",
      "metadata": {
        "id": "d16437c5"
      },
      "outputs": [],
      "source": [
        "# Import libraries\n",
        "import numpy as np\n",
        "from matplotlib import pyplot as plt"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "f44fc4b5",
      "metadata": {
        "id": "f44fc4b5"
      },
      "outputs": [],
      "source": [
        "# Add data\n",
        "m =2000  #lbm\n",
        "a = 34  # acceleration upward ft/s^2\n",
        "g = 32.17 #acceleration due to gravity ft/s^2\n",
        "k=0.5 # drag coefficient\n",
        "vo = 0  # initial velocity\n",
        "delt = 1 # sec  # time-step\n",
        "tmax = 600  # total time'\n",
        "timex = np.arange(delt,tmax+delt,delt)  # times where calculations are made"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "1fad74a4",
      "metadata": {
        "id": "1fad74a4"
      },
      "outputs": [],
      "source": [
        "# Apply Euler's method\n",
        "v = [vo,]  # velocity vector\n",
        "vold = vo  # initial\n",
        "told = 0  # initial timestep\n",
        "for t in timex:\n",
        "    vnew = vold + ((a-g) - k/m*np.power(vold,2))*delt\n",
        "    v.append(vnew)\n",
        "    vold = vnew\n",
        "    told = t"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "d1be9e73",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 472
        },
        "id": "d1be9e73",
        "outputId": "147afc78-7dab-4c1d-cd3f-ebc7156fd73f"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "# Add the initial time and make plot\n",
        "timexa = np.append(0,timex) # add initial time\n",
        "plt.plot(timexa,v,'o', mfc='none')\n",
        "plt.xlabel('time (s)')\n",
        "plt.ylabel('velocity ft/s')\n",
        "plt.title('Velocity Profile of a Balloon')\n",
        "plt.grid()\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "id": "f2fe7995",
      "metadata": {
        "id": "f2fe7995"
      },
      "outputs": [],
      "source": []
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3 (ipykernel)",
      "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": "3.10.12"
    },
    "colab": {
      "provenance": [],
      "include_colab_link": true
    }
  },
  "nbformat": 4,
  "nbformat_minor": 5
}