NDOF.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "9ab5efa7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For information about changes from previous versions of the io package, run 'news io'.\n"
     ]
    }
   ],
   "source": [
    "% pkg install -forge io\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6a3c9170",
   "metadata": {},
   "outputs": [],
   "source": [
    "pkg load io"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "15adb162",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "damping_compare"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "52693f90",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k =  34000000\n"
     ]
    }
   ],
   "source": [
    "[a,B] = IG(48e6*1.4,.7);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "acfba5ab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "semilogx(B,\"bo\")\n",
    "set (gca, 'xtick', a); \n",
    "xticks = get (gca, \"xtick\");\n",
    "xlabels = arrayfun (@(x) sprintf(\"%1.1E\",x), xticks, \"uniformoutput\", false);\n",
    "set (gca, 'xticklabel', xlabels);\n",
    "set (gca,'xminortick','off');\n",
    "grid on;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1d02ea97",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "020e3220",
   "metadata": {},
   "outputs": [],
   "source": [
    "clear IG"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "c320090e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a =\n",
      "\n",
      "     3.1250\n",
      "     3.9373\n",
      "     4.9606\n",
      "     6.2500\n",
      "     7.8745\n",
      "     9.9213\n",
      "    12.5000\n",
      "    15.7490\n",
      "    19.8425\n",
      "    25.0000\n",
      "    31.4980\n",
      "    39.6850\n",
      "    50.0000\n",
      "    62.9961\n",
      "    79.3701\n",
      "   100.0000\n",
      "   125.9921\n",
      "   158.7401\n",
      "   200.0000\n",
      "   251.9842\n",
      "   317.4802\n",
      "   400.0000\n",
      "   503.9684\n",
      "\n"
     ]
    }
   ],
   "source": [
    "a\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "a1e2e3ed",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M =\n",
      "\n",
      "   1   0   0   0\n",
      "   0   1   0   0\n",
      "   0   0   5   0\n",
      "   0   0   0   6\n",
      "\n",
      "b1 =\n",
      "\n",
      "  -2   3  -1\n",
      "  -1   2  -2\n",
      "\n",
      "b2 =\n",
      "\n",
      "  -0   0  -0\n",
      "  -0   0  -0\n",
      "\n",
      "K =\n",
      "\n",
      "   0   0  -1   0\n",
      "   0   0   0  -1\n",
      "   3  -2   0   0\n",
      "  -2   2   0   0\n",
      "\n",
      "F =\n",
      "\n",
      "   0   0   7   7\n",
      "\n",
      "Y0 =\n",
      "\n",
      "  -0.03571 - 0.00000i\n",
      "  -0.02950 - 0.00000i\n",
      "   0.00000 - 0.22435i\n",
      "   0.00000 - 0.18535i\n",
      "\n",
      "V =\n",
      "\n",
      " Columns 1 through 3:\n",
      "\n",
      "   1.00000 + 0.00000i   1.00000 - 0.00000i   0.76619 + 0.00000i\n",
      "  -0.63849 - 0.00000i  -0.63849 + 0.00000i   1.00000 + 0.00000i\n",
      "  -0.00000 - 0.92488i  -0.00000 + 0.92488i   0.00000 - 0.21390i\n",
      "   0.00000 + 0.59053i   0.00000 - 0.59053i   0.00000 - 0.27917i\n",
      "\n",
      " Column 4:\n",
      "\n",
      "   0.76619 - 0.00000i\n",
      "   1.00000 - 0.00000i\n",
      "   0.00000 + 0.21390i\n",
      "   0.00000 + 0.27917i\n",
      "\n",
      "D =\n",
      "\n",
      "Diagonal Matrix\n",
      "\n",
      " Columns 1 through 3:\n",
      "\n",
      "   0.00000 + 0.92488i                    0                    0\n",
      "                    0   0.00000 - 0.92488i                    0\n",
      "                    0                    0   0.00000 + 0.27917i\n",
      "                    0                    0                    0\n",
      "\n",
      " Column 4:\n",
      "\n",
      "                    0\n",
      "                    0\n",
      "                    0\n",
      "   0.00000 - 0.27917i\n",
      "\n",
      "ans =\n",
      "\n",
      "  -0.03571 - 0.00000i\n",
      "  -0.02950 - 0.00000i\n",
      "   0.00000 - 0.22435i\n",
      "   0.00000 - 0.18535i\n",
      "\n"
     ]
    }
   ],
   "source": [
    "NDOF_f2([1,2],[5,6],[0,0],[7,7],1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "59dcb975",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "M =\n",
      "\n",
      "   1   0\n",
      "   0   2\n",
      "\n",
      "K =\n",
      "\n",
      "   0.00000  -1.00000\n",
      "   1.00000   0.11000\n",
      "\n",
      "F =\n",
      "\n",
      "   0   1\n",
      "\n",
      "Y0 =\n",
      "\n",
      "  -1.2827e-002 - 1.1372e-004i\n",
      "  7.1451e-004 - 8.0592e-002i\n",
      "\n",
      "V =\n",
      "\n",
      "  -1.00000 + 0.00000i  -1.00000 - 0.00000i\n",
      "   0.02750 + 0.70657i   0.02750 - 0.70657i\n",
      "\n",
      "D =\n",
      "\n",
      "Diagonal Matrix\n",
      "\n",
      "   0.02750 + 0.70657i                    0\n",
      "                    0   0.02750 - 0.70657i\n",
      "\n",
      "ans =\n",
      "\n",
      "  -1.2827e-002 - 1.1372e-004i\n",
      "  7.1451e-004 - 8.0592e-002i\n",
      "\n"
     ]
    }
   ],
   "source": [
    "NDOF_f2(1,2,.11,1,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "54c7b18b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "  -2.5266e+003 - 2.5266e+002i  0.0000e+000 + 0.0000e+000i\n",
      "  0.0000e+000 + 0.0000e+000i  -2.5266e+003 - 2.5266e+002i\n",
      "\n"
     ]
    }
   ],
   "source": [
    "D.*D"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96dded6a",
   "metadata": {},
   "source": [
    "# `NDOF` revisited for the $N$th time\n",
    "\n",
    "Stiffness matrix:\n",
    "\n",
    "$\\begin{bmatrix}\n",
    "-(k_1+k_2) & k_2 & 0 & \\cdots  & 0\\\\ \n",
    "k_2 & -(k_2+k_3) & k_3 & \\cdots & \\vdots\\\\ \n",
    "0 & k_3 & \\ddots & \\ddots & 0\\\\ \n",
    "\\vdots & 0 & \\ddots & -(k_{n-1}+k_n)  & k_n\\\\ \n",
    "0 & \\cdots & 0 & k_n & -k_n\n",
    "\\end{bmatrix}$\n",
    "\n",
    "can include damping into the stiffness matrix by the transformation:\n",
    "\n",
    "$k \\rightarrow k + i \\omega c $ for \"traditional\" viscous damping.\n",
    "$k \\rightarrow k + i c $ for \"traditional\" viscous damping.\n",
    "need to vet the above\n",
    "\n",
    "Mass matrix:\n",
    "$-\\omega^2 \\begin{bmatrix}\n",
    "m_1 & 0 & 0 & \\cdots  & 0\\\\ \n",
    "0 & m_2 & \\ddots & \\ddots & \\vdots\\\\ \n",
    "0 & \\ddots & \\ddots & \\ddots & 0\\\\ \n",
    "\\vdots & 0 & \\ddots & m_{n-1}  & 0\\\\ \n",
    "0 & \\cdots & \\cdots & 0 & m_n\n",
    "\\end{bmatrix}$\n",
    "\n",
    "(the matrix typography above needs a bit of work)\n",
    "\n",
    "You would think this sufficient for describing the system, but the implementation in `NDOF` has separate velocities and displacements (even though those are not indpenedent?). I can not remember at the moment why I did it this way. Why was it necessary? Is it actually less efficient? In any case, to verify and document, I'll give it a shot:\n",
    "\n",
    "Another way to explain it: For $N$-d.o.f. system, we actually form a set of $2N$ independent equations, for all displacements and velocities. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "abf69ce5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "  -2\n",
      "  -3\n",
      "  -1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "circshift(-[1 2 3]',-1')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "adb44342",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "k =\n",
      "\n",
      "   1   2   3\n",
      "\n"
     ]
    }
   ],
   "source": [
    "k=[1 2 3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "67ea6a7f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "b1 =\n",
      "\n",
      "  -2   3  -1\n",
      "  -3   5  -2\n",
      "  -1   3  -3\n",
      "\n"
     ]
    }
   ],
   "source": [
    "b1 = [ circshift(-k',-1)  k' + [k(2:end)' ; 0]  -k' ]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "085178e0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "   3  -2   0\n",
      "  -2   5  -3\n",
      "   0  -3   3\n",
      "\n"
     ]
    }
   ],
   "source": [
    "full(spdiags(b1,-1:1,3,3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "09241354",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "d955d790",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "K =\n",
      "\n",
      "   0  -1\n",
      "   4   0\n",
      "\n"
     ]
    }
   ],
   "source": [
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "9cc83fc7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =  4\n"
     ]
    }
   ],
   "source": [
    "eig(k,m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "2e118677",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "v =\n",
      "\n",
      "  -0.12500 + 0.48412i  -0.12500 - 0.48412i\n",
      "   1.00000 + 0.00000i   1.00000 - 0.00000i\n",
      "\n",
      "d =\n",
      "\n",
      "Diagonal Matrix\n",
      "\n",
      "   0.5000 + 1.9365i                  0\n",
      "                  0   0.5000 - 1.9365i\n",
      "\n"
     ]
    }
   ],
   "source": [
    "[v,d]= eig(K,m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "05621cc7",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "Diagonal Matrix\n",
      "\n",
      "   1.9365        0\n",
      "        0  -1.9365\n",
      "\n"
     ]
    }
   ],
   "source": [
    "imag(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc2c067d",
   "metadata": {},
   "source": [
    "## "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "ad348fc3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "m =\n",
      "\n",
      "   1   0\n",
      "   0   1\n",
      "\n",
      "K =\n",
      "\n",
      "   0  -1\n",
      "   4   1\n",
      "\n",
      "F =\n",
      "\n",
      "   0   1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "m = [1 0 ; 0 1]\n",
    "K = [0 -1; 4 1 ]\n",
    "F = [0 force_vec]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f9be3ca7",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "hide_input": false,
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "4.2.1"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}