.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/mechanical_engineering/machine_elements/plot_02_petroff_bearing.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_mechanical_engineering_machine_elements_plot_02_petroff_bearing.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_mechanical_engineering_machine_elements_plot_02_petroff_bearing.py:


Petroff's law of bearing friction (1883)
========================================

Railway axle bearings of the 1880s ran on oil, and their friction did not
behave like Coulomb's: it grew with speed and with the oil's viscosity, and
it fell as the load rose. Nikolai Petroff (1883) explained this by treating
the oil film as a thin layer sheared between the journal and its sleeve.
With the journal centred, the shear stress is :math:`\mu\omega r/c` all
round, so the friction torque is :math:`T = 2\pi\mu\omega r^3 L/c`
(:func:`~engineeringkit.mechanical_engineering.petroff_friction_torque`) and
the friction coefficient follows
:math:`f = 2\pi^2 (\mu N/P)(r/c)`. The example checks Petroff's thin-film
torque against the exact flow between concentric cylinders.

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.mechanical_engineering import petroff_friction_torque








.. GENERATED FROM PYTHON SOURCE LINES 24-32

Thin film against exact Couette flow
------------------------------------

Between cylinders of radii :math:`r` and :math:`r + c` the exact viscous
torque is :math:`4\pi\mu L\omega r^2 (r + c)^2 / ((r + c)^2 - r^2)`. The
relative difference from Petroff's formula is about :math:`3c/2r`, so the
thin-film model is good to a fraction of a percent at the usual
:math:`c/r \approx 10^{-3}`.

.. GENERATED FROM PYTHON SOURCE LINES 32-41

.. code-block:: Python


    viscosity, r, L, omega = 0.03, 0.025, 0.05, 2 * np.pi * 25
    ratios = np.geomspace(1e-4, 0.3, 40)
    exact = np.array([4 * np.pi * viscosity * L * omega * r**2 * (r + c) ** 2 / ((r + c) ** 2 - r**2) for c in ratios * r])
    petroff = petroff_friction_torque(viscosity, r, L, ratios * r, omega)
    error = np.abs(petroff / exact - 1)
    print(f"c/r = 1e-3: Petroff {petroff[np.searchsorted(ratios, 1e-3)]:.5f} N m, exact {exact[np.searchsorted(ratios, 1e-3)]:.5f} N m")
    assert np.all(np.diff(error) > 0) and np.allclose(error[:10], 1.5 * ratios[:10], rtol=1e-2)





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

 .. code-block:: none

    c/r = 1e-3: Petroff 0.78775 N m, exact 0.78914 N m




.. GENERATED FROM PYTHON SOURCE LINES 42-48

The Petroff line
----------------

Divide the torque by load times radius. Plotted against the bearing
characteristic :math:`\mu N/P`, every load and speed falls on one line of
slope :math:`2\pi^2 r/c`.

.. GENERATED FROM PYTHON SOURCE LINES 48-66

.. code-block:: Python


    c = 25e-6
    loads = np.array([500.0, 1000.0, 2000.0])
    speeds = np.linspace(5, 60, 12)
    fig, ax = plt.subplots(figsize=(6, 3.8))
    for W in loads:
        P = W / (2 * r * L)
        f = petroff_friction_torque(viscosity, r, L, c, 2 * np.pi * speeds) / (W * r)
        characteristic = viscosity * speeds / P
        ax.plot(characteristic, f, "o", label=f"W = {W:.0f} N")
        assert np.allclose(f, 2 * np.pi**2 * characteristic * r / c)
    line = np.linspace(0, ax.get_xlim()[1], 2)
    ax.plot(line, 2 * np.pi**2 * line * r / c, "k--", lw=1, label="2 pi^2 (r/c) mu N / P")
    ax.set(xlabel="bearing characteristic mu N / P [-]", ylabel="friction coefficient f", title="Petroff's law")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/mechanical_engineering/machine_elements/images/sphx_glr_plot_02_petroff_bearing_001.png
   :alt: Petroff's law
   :srcset: /api/gallery/mechanical_engineering/machine_elements/images/sphx_glr_plot_02_petroff_bearing_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.033 seconds)


.. _sphx_glr_download_api_gallery_mechanical_engineering_machine_elements_plot_02_petroff_bearing.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/mechanical_engineering/machine_elements/plot_02_petroff_bearing.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_petroff_bearing.ipynb <plot_02_petroff_bearing.ipynb>`

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

      :download:`Download Python source code: plot_02_petroff_bearing.py <plot_02_petroff_bearing.py>`

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

      :download:`Download zipped: plot_02_petroff_bearing.zip <plot_02_petroff_bearing.zip>`


.. only:: html

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

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