.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/structural_engineering/cables/plot_02_catenary.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_cables_plot_02_catenary.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_cables_plot_02_catenary.py:


The catenary (1691)
===================

Galileo thought a hanging chain was a parabola. In 1690 Jacob Bernoulli
challenged the readers of *Acta Eruditorum* to find its true shape, and the
next year three solutions appeared there, by his brother Johann, by
Leibniz and by Huygens: the curve :math:`y = a(\cosh(x/a) - 1)`, with
:math:`a = H/w` the ratio of the horizontal tension to the weight per unit
length (:func:`~engineeringkit.structural_engineering.catenary_cable`). A
cable carrying a deck uniform along the span instead hangs in a parabola
(:func:`~engineeringkit.structural_engineering.parabolic_cable`), which is
why suspension bridges are designed with parabolic cables. The example
compares the two shapes and shows where the parabola is a good enough
approximation for a cable under its own weight.

.. GENERATED FROM PYTHON SOURCE LINES 19-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.structural_engineering import catenary_cable, parabolic_cable








.. GENERATED FROM PYTHON SOURCE LINES 25-27

A power line between two towers
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-37

.. code-block:: Python


    w, L, f = 15.0, 300.0, 9.0  # conductor weight [N/m], span and sag [m]
    H, T, length = catenary_cable(w, L, f)
    Hp, Tp, length_p = parabolic_cable(w, L, f)
    a = H / w
    print(f"catenary: H = {H / 1e3:.2f} kN, T_max = {T / 1e3:.2f} kN, length {length:.3f} m")
    print(f"parabola: H = {Hp / 1e3:.2f} kN, T_max = {Tp / 1e3:.2f} kN, length {length_p:.3f} m")
    assert np.isclose(a * (np.cosh(L / (2 * a)) - 1), f)
    assert np.isclose(T, H + w * f)





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

 .. code-block:: none

    catenary: H = 18.77 kN, T_max = 18.91 kN, length 300.719 m
    parabola: H = 18.75 kN, T_max = 18.88 kN, length 300.718 m




.. GENERATED FROM PYTHON SOURCE LINES 38-40

Deep cables: the shapes part company
------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-60

.. code-block:: Python


    ratios = np.linspace(0.02, 1.0, 60)
    difference = [catenary_cable(w, L, k * L)[0] / parabolic_cable(w, L, k * L)[0] - 1 for k in ratios]
    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    x = np.linspace(-L / 2, L / 2, 201)
    for k in (0.1, 0.5):
        Hk = catenary_cable(w, L, k * L)[0]
        ak = Hk / w
        axes[0].plot(x, ak * (np.cosh(x / ak) - np.cosh(L / (2 * ak))), label=f"catenary, sag/span {k}")
        axes[0].plot(x, k * L * (4 * x**2 / L**2 - 1), "--", label=f"parabola, sag/span {k}")
    axes[0].set(xlabel="x [m]", ylabel="y [m]", title="Same span and sag")
    axes[0].set_aspect("equal")
    axes[0].legend(fontsize="small")
    axes[1].plot(ratios, 100 * np.array(difference))
    axes[1].set(xlabel="sag / span", ylabel="catenary H vs parabola [%]")
    fig.tight_layout()
    print(f"at sag/span 0.1 the horizontal tensions differ by {100 * difference[np.argmin(np.abs(ratios - 0.1))]:.1f}%")
    assert abs(difference[0]) < 0.01 * difference[-1]

    plt.show()



.. image-sg:: /api/gallery/structural_engineering/cables/images/sphx_glr_plot_02_catenary_001.png
   :alt: Same span and sag
   :srcset: /api/gallery/structural_engineering/cables/images/sphx_glr_plot_02_catenary_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    at sag/span 0.1 the horizontal tensions differ by 1.4%





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

   **Total running time of the script:** (0 minutes 0.118 seconds)


.. _sphx_glr_download_api_gallery_structural_engineering_cables_plot_02_catenary.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/cables/plot_02_catenary.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_catenary.ipynb <plot_02_catenary.ipynb>`

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

      :download:`Download Python source code: plot_02_catenary.py <plot_02_catenary.py>`

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

      :download:`Download zipped: plot_02_catenary.zip <plot_02_catenary.zip>`


.. only:: html

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

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