.. 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_01_coulomb_power_screw.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_01_coulomb_power_screw.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_01_coulomb_power_screw.py:


Coulomb's friction and the power screw (1785)
=============================================

Charles-Augustin de Coulomb's prize memoir *Théorie des machines simples*
(1781, published 1785) measured sliding friction and found it proportional
to the normal load and nearly independent of contact area and speed:
:math:`F = \mu N`. A screw thread is an inclined plane wrapped round a
cylinder. Unwrapping one turn gives a slope of lead angle
:math:`\tan\lambda = L/(\pi d_m)`, and Coulomb's law on that slope gives the
torque to raise a load,
:math:`T = \tfrac12 W d_m\tan(\phi + \lambda)` with :math:`\tan\phi = \mu`
(:func:`~engineeringkit.mechanical_engineering.screw_torque_raise`). The
example solves the force balance on the unwrapped thread directly and
shows why a self-locking screw is always less than 50% efficient.

.. GENERATED FROM PYTHON SOURCE LINES 19-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.mechanical_engineering import screw_torque_raise








.. GENERATED FROM PYTHON SOURCE LINES 25-32

The unwrapped thread
--------------------

A block carrying the axial load :math:`W` is pushed up a slope
:math:`\lambda` by a horizontal force :math:`P`, the thread force at the
mean radius. Balance along and normal to the slope, with friction
:math:`\mu N` opposing the motion, is linear in :math:`P` and :math:`N`.

.. GENERATED FROM PYTHON SOURCE LINES 32-43

.. code-block:: Python


    W, dm, lead = 20e3, 0.032, 0.006
    lam = np.arctan(lead / (np.pi * dm))
    for mu in (0.0, 0.08, 0.15):
        A = np.array([[np.cos(lam), -mu], [-np.sin(lam), 1.0]])
        P, N = np.linalg.solve(A, [W * np.sin(lam), W * np.cos(lam)])
        T = screw_torque_raise(W, dm, lead, mu)
        print(f"mu = {mu:.2f}: force balance T = {P * dm / 2:7.3f} N m, formula T = {T:7.3f} N m")
        assert np.isclose(P * dm / 2, T)
    assert np.isclose(2 * np.pi * screw_torque_raise(W, dm, lead, 0.0), W * lead)





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

 .. code-block:: none

    mu = 0.00: force balance T =  19.099 N m, formula T =  19.099 N m
    mu = 0.08: force balance T =  44.913 N m, formula T =  44.913 N m
    mu = 0.15: force balance T =  67.705 N m, formula T =  67.705 N m




.. GENERATED FROM PYTHON SOURCE LINES 44-51

Efficiency and self-locking
---------------------------

Efficiency is the lifting work per turn over the input work,
:math:`\eta = W L / (2\pi T) = \tan\lambda / \tan(\phi + \lambda)`. A screw
holds its load without a brake when :math:`\phi \ge \lambda`; at that
limit :math:`\eta = (1 - \tan^2\lambda)/2`, below one half.

.. GENERATED FROM PYTHON SOURCE LINES 51-69

.. code-block:: Python


    leads = np.linspace(0.5e-3, 40e-3, 200)
    fig, ax = plt.subplots(figsize=(6, 3.8))
    for mu in (0.08, 0.15, 0.25):
        eta = W * leads / (2 * np.pi * screw_torque_raise(W, dm, leads, mu))
        locking = np.tan(np.arctan(leads / (np.pi * dm))) <= mu
        ax.plot(np.degrees(np.arctan(leads / (np.pi * dm))), eta, label=f"mu = {mu}")
        assert np.all(eta[locking] < 0.5)
    ax.axhline(0.5, color="k", ls=":")
    ax.set(xlabel="lead angle [deg]", ylabel="efficiency", title="Square-thread power screw")
    ax.legend()
    fig.tight_layout()

    limit = W * lead / (2 * np.pi * screw_torque_raise(W, dm, lead, np.tan(lam)))
    print(f"lead angle {np.degrees(lam):.2f} deg; efficiency at the self-locking limit {limit:.4f}")
    assert np.isclose(limit, (1 - np.tan(lam) ** 2) / 2)

    plt.show()



.. image-sg:: /api/gallery/mechanical_engineering/machine_elements/images/sphx_glr_plot_01_coulomb_power_screw_001.png
   :alt: Square-thread power screw
   :srcset: /api/gallery/mechanical_engineering/machine_elements/images/sphx_glr_plot_01_coulomb_power_screw_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    lead angle 3.42 deg; efficiency at the self-locking limit 0.4982





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

   **Total running time of the script:** (0 minutes 0.030 seconds)


.. _sphx_glr_download_api_gallery_mechanical_engineering_machine_elements_plot_01_coulomb_power_screw.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_01_coulomb_power_screw.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_coulomb_power_screw.ipynb <plot_01_coulomb_power_screw.ipynb>`

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

      :download:`Download Python source code: plot_01_coulomb_power_screw.py <plot_01_coulomb_power_screw.py>`

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

      :download:`Download zipped: plot_01_coulomb_power_screw.zip <plot_01_coulomb_power_screw.zip>`


.. only:: html

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

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