wave1d_pile.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false
      },
      "source": [
        "<div style=\"display: flex; align-items: center;\">\n",
        "    <img src=\"https://github.com/nagelt/Teaching_Scripts/raw/9d9e29ecca4b04eaf7397938eacbf116d37ddc93/Images/TUBAF_Logo_blau.png\" width=\"500\" height=\"auto\" height=\"auto\" style=\"margin-right: 100px;\" />\n",
        "    <div>\n",
        "        <p><strong>Prof. Dr. Thomas Nagel</strong></p>\n",
        "        <p>Chair of Soil Mechanics and Foundation Engineering<br>Geotechnical Institute<br>Technische Universität Bergakademie Freiberg.</p>\n",
        "        <p><a href=\"https://tu-freiberg.de/en/soilmechanics\">https://tu-freiberg.de/en/soilmechanics</a></p>\n",
        "    </div>\n",
        "</div>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Pfahl unter zyklischer Last\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Module laden"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [],
      "source": [
        "import sympy as sp\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from ipywidgets import widgets\n",
        "from ipywidgets import interact\n",
        "from sympy.interactive import printing\n",
        "import plot_functions.plot_settings\n",
        "%run plot_functions/wave1d_pile_interaction.ipynb\n",
        "printing.init_printing()\n",
        "%load_ext ipydex.displaytools"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "_Last line enables special comment **##:** for LaTeX formatting of output_ "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Stationärer Zustand"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "![wavebar_pile.png](https://github.com/nagelt/soil_dynamics/raw/master/Images/wavebar_pile.png)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Variablen für symbolische Rechnung deklarieren (sympy)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {},
      "outputs": [],
      "source": [
        "A, rho, E, kappa, omega, z, t, C1, C2, L, F, H = sp.symbols('A rho E kappa omega z t C1 C2 L F H')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Hilfsfunktion um die Koeffizienten ($C_1$, $C_2$) aus den Randbedingungen (Kraftverlauf oben, unverschieblich unten) zu bestimmen"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {},
      "outputs": [],
      "source": [
        "def solveBC(f):\n",
        "    dfdz = f.diff(z)\n",
        "    BCtop = dfdz.subs(z, 0) + F/(E*A)\n",
        "    BCbottom = f.subs(z, L) \n",
        "    BCsol = sp.solve([BCtop, BCbottom], C1, C2)\n",
        "    return BCsol    "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Aus dem Ansatz $w(z,t)=f(z)sin(\\omega t)$ folgt für $f(z)$ in einem Fall der Diskriminante"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "scrolled": true
      },
      "outputs": [],
      "source": [
        "alpha = sp.sqrt(1-(kappa*H)**2)\n",
        "fa = C1*sp.cosh(alpha*z/H) + C2*sp.sinh(alpha*z/H)   # hyperbolic functions"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Die unbestimmten Koeffizienten $C_1$ und $C_2$ folgen aus den Randbedingungen"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {},
      "outputs": [],
      "source": [
        "BCsol_a = solveBC(fa)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Einsetzen in den ursprünglichen Ansatz"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "image/png": 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\n",
            "text/latex": [
              "$\\displaystyle - \\frac{F H \\left(\\sinh{\\left(\\frac{z \\sqrt{- H^{2} \\kappa^{2} + 1}}{H} \\right)} - \\cosh{\\left(\\frac{z \\sqrt{- H^{2} \\kappa^{2} + 1}}{H} \\right)} \\tanh{\\left(\\frac{L \\sqrt{- H^{2} \\kappa^{2} + 1}}{H} \\right)}\\right) \\sin{\\left(\\omega t \\right)}}{A E \\sqrt{- H^{2} \\kappa^{2} + 1}}$"
            ],
            "text/plain": [
              "     ⎛    ⎛     _____________⎞       ⎛     _____________⎞     ⎛     __________\n",
              "     ⎜    ⎜    ╱    2  2     ⎟       ⎜    ╱    2  2     ⎟     ⎜    ╱    2  2  \n",
              "     ⎜    ⎜z⋅╲╱  - H ⋅κ  + 1 ⎟       ⎜z⋅╲╱  - H ⋅κ  + 1 ⎟     ⎜L⋅╲╱  - H ⋅κ  +\n",
              "-F⋅H⋅⎜sinh⎜──────────────────⎟ - cosh⎜──────────────────⎟⋅tanh⎜───────────────\n",
              "     ⎝    ⎝        H         ⎠       ⎝        H         ⎠     ⎝        H      \n",
              "──────────────────────────────────────────────────────────────────────────────\n",
              "                                            _____________                     \n",
              "                                           ╱    2  2                          \n",
              "                                     A⋅E⋅╲╱  - H ⋅κ  + 1                      \n",
              "\n",
              "___⎞⎞          \n",
              "   ⎟⎟          \n",
              " 1 ⎟⎟          \n",
              "───⎟⎟⋅sin(ω⋅t) \n",
              "   ⎠⎠          \n",
              "───────────────\n",
              "               \n",
              "               \n",
              "               "
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "fasol = fa.subs([(C1, BCsol_a[C1]), (C2, BCsol_a[C2])])\n",
        "wasol = sp.simplify(fasol*sp.sin(omega*t))\n",
        "display(wasol)\n",
        "#print(wasol)   # for safety if display fails"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Jetzt noch mal das gleiche für den anderen Fall der Diskriminante um reell zu bleiben (*alternativ total komplex*)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "image/png": 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JN6cw4kyPovw+xdcafxB+Ab/12Ziu6MsDp3LHpzF6NKWFx9RdlQdwXUewA/ArATc2zGFhrFXYA/m5XtLHWhiLC9tBaZgXqaMIkx+B86Vr6H4IvvsJA7gpcBc8ykflp44ygkGn+PRiH1LBm5gIAK4pL6RrtU7Oq/pX06DjpYPMUm2eFN/Y12yZMIjz1ouRGTLas5GEb8pYk752EVs/TYrbwMIYCNhOxegtROcMHDYAhM+8KIwBJRXS8Da4cXko19My8Fv7m66BatrVnkccdsdTAFfbxnIVTjTeCLR0oHELDlhdJPUxMAK2Z37RxAyOjc0zdHxqGocu4SHj6OPYjalD+Cu1m8ekMMpe6WDblh04FiWY21iJ5g3T1PBCl3Cu8HC9yLWxWlGFUFYp/CZmeKkIMoaLTmFUjrAVmFZ0Dq38VsoIF2ss43e5YXVKLsoTZ0YhK2zhkwKV4hpPlZ/J/q0iu7Z/GvNeEEh9GBF0gBXzvjx0ENtI4MXzbWFg0zm4faGDdhJOPHoTfquDcO7gEcj3WEdqeZ9j4q/ilmhzSYcJTzDUzTBIi32mE9rZy7M000r8XDMIPFlEwE3Hf1Q5nOQas62FYpBQOM+LcvFzPZuucJFz65NOrittq1BmLPea12arPhdG7JGnQDKKq+pPuMGKN2MX1ybjlwlcYvzie2fEwSXGScZG0sNDeInfx0CBkMhe+UUTR3EsYjI0T+f4JK7NORez4wiXq8I1Qb1cBxhW4TM30oU5npth7IJbnVcXIthVYnwON9lyix08haeCrWT2QgjHYAOIVqMhzd3mH1KG0pQqbiurJ9wGh55k00RLZyYzm9BqhSqeTmrCriu8tC1VK7QecNU216ufLKRGvlgy4T9PVst1C2KiQl6fnWV+4/XXeX0pDTxqTdPH9QEtc54OAGnBJKO5Kk40jW0nhbeNb0xEHCZN+cl71DEQXPY4Dlp/j+aYMg7K0zc+dXCyjauDeUjZOk46iu1PGqxzbCcM8JIoHAOtaqQVaRTPWFnLVySQR2kCLnJL9sdNmmgDfibGzzeg55Qqbr7NIh13IQh3INx5pELcbRqwIf9j6cpKAhfuJUL+S8u4pP4p8jpPp0BxvjKm4up8GraXvHlu0bQdj4PWczkcy8lj9V3TZd7ihrf1pndiZbjxqc6Vp60ZbHzFmO3JBxNMkhPjO1txe2gzd5Vsm5XIHgcwW06eDcAZC8bY5E7tIhEOpbuoiwpbLrPzdDnsh9Q8CVeHVDRDmj1wC1j2Og5al+dwLCeP1Xc1V2M0L9nwCNeYPxDI0o86lJF6SvMlhd3PKnG5TLZ1BvGPIntoc9tzGzaAsUS8RUH/sW87b7GdQ3R2ng5Babk0W+bqHrhFz+91HDRW53AsJ4/Vd233M1XI85lzC3XUjDUq3ZTBJouTbSNWIzr3f2nYXmQnbW67KNsGsNV3n/qFNiF7WB07t+SCX+fpBeDNnHXrXN0Jt+jl3Y2DRt0cjuXksfqWcCMP+ZoBu0KzSCybOhoXMW5mqXXeQnlbw/aT561pPaVvts0iHkveSJNhwwDWeCcRcqz7B2OTh1EbL6x1qz6bds7T2aC9qOA9cHWz3KLndjwOGjFzOJaTx+pbxI3j/fdzVa7yO8ve1ApbBInlcZ5hsxWOubBbU7mbbLP6iDsRe3CSuwb25k9y+aArAzDC33uYUXcO2cYvr2TfbkPVq2npPL0a1KMq2gNXN8kteknj257HQSNiDsdy8lh9h3TvvXu3vf9R1gWAEfBELt8DOoQcsc1r7dg4APMdqvfWquNSejlPl0K+ud49cdW51dzHS4fmcCwnz9LtXEP9W1xhO6mzeeWVtyi2uDKT1e9HbHMWUNfJ9ELVfHGdqrZVi/N0df21G646t1bHLVMoh2M5eay+w7qbNNhib2G08Q8NR5IjtnlV/atJg+0Nvr32y6oUW5cyztMV9MdOuercWgG3TIUcjuXksfqO7m7WYFOnhy8Jy2Uf/BByxDavqWOF/wfShztDXu92aUHAedoCzBWD98pV59YVSdRTVQ7HcvL0qHGo6M0abLGX+FNr/vOOifQocsQ2L9634tgDKcE3176Q398M7e8R52k/RrOkOABXnVuzMGd4oTkcy8kzXKNjpNy0wSYC8F02Xg3mT4qZUHcvR2zzSjo1fFZA+PtW6IAOcZ4OAGm+JLvmqnNrPuKMKDmHYzl5Rqi0/6SbNtjoHl28rHawRXWYVbYjtpm+XkqENzcD38lt/UPfpXRbc73O0+v3zlG46ty6PresxhyO5eSx+ty9Q2CTn/W4U999joAj4Ag4Ao6AI+AI7B+Bza+w7b+LvIWOgCPgCDgCjoAjcHQE7n3zzTfb+3Lu0XvN2+8IOAKOgCPgCDgCh0LAt0QP1d3eWEfAEXAEHAFHwBHYIgK+JbrFXnOdHQFHwBFwBBwBR+BQCLjBdqju9sY6Ao6AI+AIOAKOwBYRuNmi0q6zI+AI7A8BvfrPfwP/sb+WLd8iYXtveS1cA0fAEbgEATfYLkHP8zoCjsCUCHynwj6WcfHnlIV6WY6AI+AI7AEBN9j20IveBkdg4wjE1bXT1MaayuOjx/z/K2If1+bvxfiXlGyZq9xshTyjI+AI7B4BN9h238XeQEegjICMDQwX/s7t43LMomcYVaywTS38Q8WXVqj8P8jPtuuHFpbpzlVupjqezRFwBPaOgBtse+9hb58jUEeA//TjebFOkXHzlRLwX72PYsLXctOVKVavMP4e6nit9KQdLcpHGR/I/a2aOerwWOFPYxxp3iqcv6M7yX0uh3qf6ECfEC/XVtGeKw3GqZWNUUjYRzou2Xqdq1yplye0STnpW7aV037KK9BzOQKOwKoQcINtVd3hyjgC8yKgiRzDp9dYQwul/V7O93L5uPafchsNMoWbYUe2HPlamThqEnU4yW3UQeH8v+uPcv+V+7fcqo6srmFoTi1zlTtKT7UXI/Wljjc6MKxt21deF0fAEdgTAm6w7ak3vS2OQD8CGDS/6HiqyZ5Vrb+7siielSvEVqjOZ+VfjKasLcZocDySW2xblosOhqMZmI06KC9GCobLzw150S0V6sGwu2R17aT8s5SbKjrELz1YSbPVRgxnw2pIdk/jCDgCG0Lg/oZ0dVUdAUfgAgQ0ubMdyGFGGkZOn9iK1a89Cf/qiW+LRp/btsgY/iy6bTqYUdkWH7Kr/RgzrDBO+uzeXOXGNrvjCDgCjkBAwA02J4IjcAAEZFSwCvWPXIy1f2KTCeuTYAwpX2l1S+es5gSRn1WeUnyM6nSUD4PxiVxW/LqkUYckQ1hh0nmrDqqDtmIcdj7fFXVKiu72Di23uxSPdQQcAUegH4Gb/iSewhFwBHaAwNcyLmzb0VbYHg5oF6tSJUNI5WBAYaQVorCcLcahb4bWdCgqPnt4dottzpJOlkbhGGu0P6wWxvOTXMPBkuK+UvhPOnh+jzQYlejJyw23Fi4/cWPKJYuLI+AIOALZCLjBlg2dZ3QEtoGADAsMrHS70AyVzufOYj4aybNuvH2IhFUxuX152Xrkbc6SsUcBifAcXV856I6kOpxD7n7Rqfb8GtEqH6OKT3lgsGH4IRiutZccFE85pAlxMT3GIPlLRuOYcpXXxRFwBByBixFwg+1iCL0AR2D1CHwmA8NW11D2bdQYY6ZL7Pk18ocVtGio/CrXjL5afsWxXcqqFGkanxeLaTCE+qSmQ5ohloNxmBqkaRK+uYYhVvrLK+VL8bD0n+NR3G86zOC0lwtMD0s7plzL464j4Ag4AtkIuMGWDZ1ndATWj4AMD4wnVqdS4+hh1BxDpkvC6pbyFtud8rP1WHrmTOcPdAQjUC4rURg57+v4inMdRf6ksi8V3rm6FtPWdEjKwGuGVONKnup4r5K+65SyWBXE+AOv1s+BjCw31Kk84P1KRx/uIX38KYzlNND9joAjcDwE3GA7Xp97iw+CQDQQTnLNqClarjC+a9a3wlbaBiwyn42ZcBrr4Pkue+bLVuJuFWYrbfZSgOUhfWpAhvCWnzYdLDkGHd+ICwajBWa6lMXWKtu/YMPzbKwS2kd45c2XqGPjimN+qZ7TEXAEjoLA/aM01NvpCBwQAf4+KRhSDW3HELGVtlq08oWVLUXUthqjEWN5ML5s29DCTkqDAUV4+N5bEXH28IxYLU8lDWW06kBaxWPMIY2ra+eoYb8qCwONla/wrwg652O8GFfoQBuDKGzM6phlc9cRcAQcgYsRcIPtYgi9AEdgfQjIsMDQKD23VdESg6rL+LBVuVZjSHVg5DyWS1lNwmc0kOIBf6Xl2bBfOvKEDPGnT4dOgy4taIA/lCW9utqL4dZq5A6ow5M4Ao6AI5CNgBts2dB5RkdgnQjI6MCQYqWoaxXrDdrHtE0NwbAivun5M8LNIGzd2lQaVvEwgJ7Lb8bhC52zXTpEOnVQAcGgU9mtRtaQSmIaymorh3YgH8Y2nc/W98tzg4gblWcc/NcR2BUC996941EWF0fAEdg6AtEo4vkrW3nC2PpE4cUKmPwYQbwhmabhrUg+e4FR9VIHBp9tN5ZeMFA4aR5Fl2fHOp/JUjz1sK3KKhuGz6cKa3pDU1HBQDQdcE1HdPhd+ew5OYxEdLR4DC3eXG3b/lV0tyhveNFAbqm9OgevZzrQ/Qedm/HWXeAVY6WTfXIFPMCNfkdPMOky2pXExRFwBLaCgBtsW+kp19MR2CgCMhrYmsXAwojgrcfVGT0bhdbVdgQcgQMhcP9AbfWmOgKOwDIIsAXKyg+fBHFjbZk+8FodAUdg4wi4wbbxDnT1HYG1IyAjjW1GtumKlw/WrrPr5wg4Ao7A2hD4P8kRAKkqLFh6AAAAAElFTkSuQmCC\n",
            "text/latex": [
              "$\\displaystyle \\frac{F H \\left(- \\sin{\\left(\\frac{z \\sqrt{H^{2} \\kappa^{2} - 1}}{H} \\right)} + \\cos{\\left(\\frac{z \\sqrt{H^{2} \\kappa^{2} - 1}}{H} \\right)} \\tan{\\left(\\frac{L \\sqrt{H^{2} \\kappa^{2} - 1}}{H} \\right)}\\right) \\sin{\\left(\\omega t \\right)}}{A E \\sqrt{H^{2} \\kappa^{2} - 1}}$"
            ],
            "text/plain": [
              "    ⎛     ⎛     ___________⎞      ⎛     ___________⎞    ⎛     ___________⎞⎞   \n",
              "    ⎜     ⎜    ╱  2  2     ⎟      ⎜    ╱  2  2     ⎟    ⎜    ╱  2  2     ⎟⎟   \n",
              "    ⎜     ⎜z⋅╲╱  H ⋅κ  - 1 ⎟      ⎜z⋅╲╱  H ⋅κ  - 1 ⎟    ⎜L⋅╲╱  H ⋅κ  - 1 ⎟⎟   \n",
              "F⋅H⋅⎜- sin⎜────────────────⎟ + cos⎜────────────────⎟⋅tan⎜────────────────⎟⎟⋅si\n",
              "    ⎝     ⎝       H        ⎠      ⎝       H        ⎠    ⎝       H        ⎠⎠   \n",
              "──────────────────────────────────────────────────────────────────────────────\n",
              "                                        ___________                           \n",
              "                                       ╱  2  2                                \n",
              "                                 A⋅E⋅╲╱  H ⋅κ  - 1                            \n",
              "\n",
              "      \n",
              "      \n",
              "      \n",
              "n(ω⋅t)\n",
              "      \n",
              "──────\n",
              "      \n",
              "      \n",
              "      "
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "beta = sp.sqrt(-1+(kappa*H)**2)\n",
        "fb = C1*sp.cos(beta*z/H) + C2*sp.sin(beta*z/H)    # trigonometric functions\n",
        "BCsol_b = solveBC(fb)\n",
        "fbsol = fb.subs([(C1, BCsol_b[C1]), (C2, BCsol_b[C2])])\n",
        "wbsol = sp.simplify(fbsol*sp.sin(omega*t))\n",
        "display(wbsol)\n",
        "#print(wbsol)   # for safety if display fails"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 8,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "application/vnd.jupyter.widget-view+json": {
              "model_id": "e7a586bae4cb4aad8feae0365732e41a",
              "version_major": 2,
              "version_minor": 0
            },
            "text/plain": [
              "interactive(children=(IntSlider(value=5, description=r'$f$ [Hz]:', max=101, min=1, step=5), IntSlider(value=101…"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "interactive_pile_plot()   "
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Wie ändert sich die Eigenfrequenz des Einmassenschwingers, wenn sich die Steifkeit erhöht (hohe Steifigkeit entspricht kurzer Bezugslänge $H$)? Vergleichen Sie diese Tendenz mit dem Einmassenschwinger."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Für große Bezugslängen nähert sich die Eigenfrequenz der des elastischen Stabs ohne äußere, verteilte Steifigkeit. Berechnen Sie die Periodendauer einer Eigenschwingung für geringe äußere Steifigkeit (großes $H$) und vergleichen Sie sie mit der Zeit, die eine Welle der Geschwindigkeit $c$ braucht um den Weg einer Pfahllänge ($L$) zurückzulegen."
      ]
    }
  ],
  "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": "3.9.13"
    },
    "widgets": {
      "application/vnd.jupyter.widget-state+json": {
        "state": {},
        "version_major": 2,
        "version_minor": 0
      }
    }
  },
  "nbformat": 4,
  "nbformat_minor": 4
}