{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import pyvista\npyvista.OFF_SCREEN = True\npyvista.set_jupyter_backend('static')"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\n# Free-free axial rod \u2014 natural frequencies\n\nCompanion to\n`sphx_glr_gallery_verification_example_verify_axial_rod_nf.py`\n(fixed-free).  A uniform rod with both ends free vibrates\nlongitudinally with cosine mode shapes\n$u_n(x) = \\cos(n \\pi x / L)$ and the canonical\nfree-free natural frequencies\n\n\\begin{align}f_n = \\frac{n}{2 L} \\sqrt{\\frac{E}{\\rho}},\n    \\qquad n = 1, 2, 3, \\ldots\\end{align}\n\nThe mode at $n = 0$ is the rigid-body axial translation\n\u2014 a zero-eigenvalue mode produced by the unconstrained boundary\ncondition.  Every elastic frequency $f_n, n \\ge 1$ follows\nthe formula above.\n\n## Implementation\n\nDrives the existing\n:class:`~femorph_solver.validation.problems.FreeFreeRodModes`\nproblem class on a 40-element TRUSS2 mesh \u2014 the cleanest 1D\ndiscretisation of the axial-wave operator\n(:doc:`/reference/elements/truss2`).  Transverse DOFs at every\nnode are pinned to UY = UZ = 0 so the element's zero transverse\nstiffness doesn't admit arbitrary side-sway modes; the axial DOF\nis unconstrained at both ends, producing the rigid-body /\nelastic spectrum the closed form predicts.\n\nThe extraction filter walks the spectrum starting at 1 Hz to\nskip the (analytically) zero-frequency rigid-body mode that the\nsolver returns at machine-precision noise level.\n\n## References\n\n* Rao, S. S. (2017) *Mechanical Vibrations*, 6th ed., Pearson,\n  \u00a78.2 Table 8.1 (axial wave equation eigenvalues).\n* Meirovitch, L. (2010) *Fundamentals of Vibrations*, Long Grove,\n  \u00a76.6 (free-free rod).\n* Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J. (2002)\n  *Concepts and Applications of Finite Element Analysis*, 4th\n  ed., Wiley, \u00a711 (modal analysis with rigid-body modes).\n\n## Vendor cross-references\n\n======================================  =====================  ============================================\nSource                                   Reported f_1 [Hz]      Problem ID / location\n======================================  =====================  ============================================\nClosed form (wave equation)              2 523.77               Rao 2017 \u00a78.2 Table 8.1\nMeirovitch (2010) \u00a76.6                   2 523.77               free-free rod\nMAPDL Verification Manual                \u2248 2 524                VM-47 (free-free torsion analogue)\nAbaqus Verification Manual               \u2248 2 524                AVM 1.6.x free-free rod NF family\n======================================  =====================  ============================================\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "from __future__ import annotations\n\nimport numpy as np\nimport pyvista as pv\n\nfrom femorph_solver.validation.problems import FreeFreeRodModes"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Build the model from the validation problem class\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "problem = FreeFreeRodModes()\nm = problem.build_model()\nprint(\n    f\"free-free rod mesh: {m.grid.n_points} nodes, {m.grid.n_cells} TRUSS2 cells; \"\n    f\"L = {problem.L} m, E = {problem.E / 1e9:.0f} GPa, \u03c1 = {problem.rho:.1f} kg/m\u00b3\"\n)\n\nc = np.sqrt(problem.E / problem.rho)\nf1_published = c / (2.0 * problem.L)\nf2_published = 2.0 * f1_published\nprint()\nprint(f\"Wave speed c = \u221a(E/\u03c1)        = {c:.3f} m/s\")\nprint(f\"f_1 = c / (2 L) (published)  = {f1_published:.3f} Hz\")\nprint(f\"f_2 = 2 f_1     (published)  = {f2_published:.3f} Hz\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Modal solve + frequency extraction\n\nA free-free rod exhibits one rigid-body axial-translation mode\nat $f = 0$ (machine-precision noise after numeric\ndiscretisation).  ``problem.extract`` skips it via a 1 Hz floor;\nthe next two finite frequencies are the analytical $f_1,\nf_2$.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "res = m.solve_modal(n_modes=problem.default_n_modes if hasattr(problem, \"default_n_modes\") else 6)\nfreqs = np.asarray(res.frequency, dtype=np.float64)\n\nprint()\nprint(\"  All recovered frequencies (lowest first):\")\nfor k, f in enumerate(freqs):\n    label = \"rigid-body\" if f < 1.0 else \"elastic\"\n    print(f\"    mode {k + 1}: f = {f:>10.4f} Hz   ({label})\")\n\nf1_computed = problem.extract(m, res, \"f1_axial\")\nf2_computed = problem.extract(m, res, \"f2_axial\")\nerr1 = (f1_computed - f1_published) / f1_published * 100.0\nerr2 = (f2_computed - f2_published) / f2_published * 100.0\nprint()\nprint(\n    f\"  f_1 computed = {f1_computed:.3f} Hz   published = {f1_published:.3f} Hz   err = {err1:+.4f} %\"\n)\nprint(\n    f\"  f_2 computed = {f2_computed:.3f} Hz   published = {f2_published:.3f} Hz   err = {err2:+.4f} %\"\n)\n\nassert abs(err1) < 2.0, f\"f_1 deviation {err1:.3f}% exceeds 2 %\"\nassert abs(err2) < 5.0, f\"f_2 deviation {err2:.3f}% exceeds 5 %\""
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Render the first elastic axial mode\n\nThe first elastic mode has $u(x) = \\cos(\\pi x / L)$ \u2014\nantisymmetric about the rod's midspan, with the two ends moving\nin opposite directions and a node (zero-displacement) at\n$x = L/2$.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "shapes = np.asarray(res.mode_shapes).reshape(-1, 3, len(freqs))\nelastic_idx = next((k for k, f in enumerate(freqs) if f > 1.0), 0)\nphi = shapes[:, :, elastic_idx]\n\nscale = 0.05 * problem.L / max(np.abs(phi).max(), 1e-30)\nwarped = m.grid.copy()\nwarped.points = m.grid.points + scale * phi\nwarped[\"|phi|\"] = np.linalg.norm(phi, axis=1)\nwarped[\"ux\"] = phi[:, 0]\n\nplotter = pv.Plotter(off_screen=True, window_size=(720, 280))\nplotter.add_mesh(m.grid, style=\"wireframe\", color=\"grey\", opacity=0.3)\nplotter.add_mesh(warped, scalars=\"ux\", cmap=\"coolwarm\", line_width=4, show_edges=False)\nplotter.add_text(\n    f\"first elastic mode  \u2014  f = {freqs[elastic_idx]:.2f} Hz\",\n    position=\"upper_left\",\n    font_size=11,\n)\nplotter.view_xy()\nplotter.camera.zoom(1.05)\nplotter.show()"
      ]
    }
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