.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/members/plot_02_coulomb_torsion.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_structural_engineering_members_plot_02_coulomb_torsion.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_structural_engineering_members_plot_02_coulomb_torsion.py:


Coulomb's torsion (1784)
========================

Charles-Augustin Coulomb (1784) hung a weight on a thin wire, twisted it and
timed the oscillations. The restoring torque was proportional to the angle
of twist, to the fourth power of the wire's diameter and inversely to its
length, which in modern form is :math:`\phi = TL/(GJ)` with
:math:`J = \pi d^4/32`
(:func:`~engineeringkit.structural_engineering.torsion_angle`). The same
torsion balance later weighed the electric force and, in Cavendish's hands,
the Earth. The example simulates Coulomb's pendulum and recovers the shear
modulus from the period, then shows the shear stress across the wire
(:func:`~engineeringkit.structural_engineering.torsion_shear_stress`).

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy.integrate import solve_ivp

    from engineeringkit.structural_engineering import torsion_angle, torsion_shear_stress








.. GENERATED FROM PYTHON SOURCE LINES 25-27

A torsion pendulum
------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-39

.. code-block:: Python


    L, d, G = 0.8, 0.5e-3, 40e9  # brass wire
    J = np.pi * d**4 / 32
    k = 1.0 / torsion_angle(1.0, L, G, J)  # torque per radian [N m/rad]
    I_disk = 0.5 * 0.2 * 0.05**2  # 200 g disk of radius 50 mm [kg m^2]
    sol = solve_ivp(lambda t, s: [s[1], -k * s[0] / I_disk], (0, 120), [0.5, 0.0], max_step=0.05, rtol=1e-9, atol=1e-12)
    crossings = sol.t[1:][np.diff(np.sign(sol.y[0])) > 0]
    period = float(np.mean(np.diff(crossings)))
    G_measured = 4 * np.pi**2 * I_disk * L / (period**2 * J)
    print(f"period {period:.2f} s, recovered G = {G_measured / 1e9:.2f} GPa")
    assert np.isclose(G_measured, G, rtol=1e-3)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    period 5.67 s, recovered G = 40.00 GPa




.. GENERATED FROM PYTHON SOURCE LINES 40-42

Coulomb's fourth-power law
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-57

.. code-block:: Python


    diameters = np.array([0.25e-3, 0.5e-3, 1e-3])
    torques = 0.1 / torsion_angle(1.0, L, G, np.pi * diameters**4 / 32)  # torque for 0.1 rad
    print("torque ratios for doubling the diameter:", np.round(torques[1:] / torques[:-1], 3))
    assert np.allclose(torques[1:] / torques[:-1], 16)

    r = np.linspace(0, d / 2, 50)
    fig, axes = plt.subplots(1, 2, figsize=(10, 3.5))
    axes[0].plot(sol.t, sol.y[0])
    axes[0].set(xlabel="time [s]", ylabel="twist [rad]", title="Torsion pendulum")
    axes[1].plot(1e3 * r, torsion_shear_stress(k * 0.5, r, J) / 1e6)
    axes[1].set(xlabel="radius [mm]", ylabel="shear stress [MPa]", title="Shear across the wire at 0.5 rad")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/members/images/sphx_glr_plot_02_coulomb_torsion_001.png
   :alt: Torsion pendulum, Shear across the wire at 0.5 rad
   :srcset: /api/gallery/structural_engineering/members/images/sphx_glr_plot_02_coulomb_torsion_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    torque ratios for doubling the diameter: [16. 16.]





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.082 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_members_plot_02_coulomb_torsion.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/structural_engineering/members/plot_02_coulomb_torsion.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_coulomb_torsion.ipynb <plot_02_coulomb_torsion.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_coulomb_torsion.py <plot_02_coulomb_torsion.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_coulomb_torsion.zip <plot_02_coulomb_torsion.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_