.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/aerodynamics/forces/plot_03_lilienthal_coefficients.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_aerodynamics_forces_plot_03_lilienthal_coefficients.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_aerodynamics_forces_plot_03_lilienthal_coefficients.py:


Lilienthal's lift and drag coefficients and the polar diagram
=============================================================

Otto Lilienthal's *Der Vogelflug als Grundlage der Fliegekunst* (1889)
tabulated the forces on cambered wings measured on a whirling arm, reduced to
dimensionless coefficients and plotted against each other in a *polar
diagram*. Dividing a measured force by :math:`qS` removes the dependence on
speed and size, so tests at one speed predict forces at another. This example
reduces synthetic whirling-arm measurements with
:func:`~engineeringkit.aerodynamics.lift_coefficient` and
:func:`~engineeringkit.aerodynamics.drag_coefficient` and reads the best glide
ratio off the polar with :func:`~engineeringkit.aerodynamics.glide_ratio`.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.aerodynamics import drag_coefficient, dynamic_pressure, glide_ratio, lift, lift_coefficient








.. GENERATED FROM PYTHON SOURCE LINES 23-28

Synthetic measurements at two arm speeds
----------------------------------------

A cambered plate with an assumed underlying behaviour is "measured" at
8 m/s and 12 m/s; the raw forces differ by a factor of 2.25.

.. GENERATED FROM PYTHON SOURCE LINES 28-46

.. code-block:: Python


    rng = np.random.default_rng(1889)
    alpha = np.radians(np.arange(-3, 16, 1.5))
    true_cl = 0.3 + 4.6 * alpha - 6.0 * np.clip(alpha - np.radians(10), 0, None)
    true_cd = 0.045 + 0.07 * true_cl**2
    rho, S = 1.225, 0.3
    fig, axes = plt.subplots(1, 2, figsize=(9, 3.8))
    for V, marker in [(8.0, "o"), (12.0, "s")]:
        q = float(dynamic_pressure(rho, V))
        L = lift(q, S, true_cl) * (1 + 0.02 * rng.standard_normal(alpha.size))
        D = q * S * true_cd * (1 + 0.02 * rng.standard_normal(alpha.size))
        axes[0].plot(np.degrees(alpha), L, marker, label=f"lift at {V:.0f} m/s")
        cl = lift_coefficient(L, rho, V, S)
        cd = drag_coefficient(D, rho, V, S)
        axes[1].plot(cd, cl, marker, label=f"{V:.0f} m/s")
        best = int(np.argmax(glide_ratio(cl, cd)))
        print(f"{V:4.0f} m/s: best L/D = {glide_ratio(cl[best], cd[best]):.1f} at alpha = {np.degrees(alpha[best]):.1f} deg")




.. image-sg:: /api/gallery/aerodynamics/forces/images/sphx_glr_plot_03_lilienthal_coefficients_001.png
   :alt: plot 03 lilienthal coefficients
   :srcset: /api/gallery/aerodynamics/forces/images/sphx_glr_plot_03_lilienthal_coefficients_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

       8 m/s: best L/D = 9.1 at alpha = 6.0 deg
      12 m/s: best L/D = 9.1 at alpha = 4.5 deg




.. GENERATED FROM PYTHON SOURCE LINES 47-49

Coefficients collapse both runs onto one polar
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-57

.. code-block:: Python


    axes[0].set(xlabel="angle of attack [deg]", ylabel="measured lift [N]", title="Raw forces depend on speed")
    axes[0].legend()
    axes[1].set(xlabel="$C_D$", ylabel="$C_L$", title="Lilienthal's polar diagram")
    axes[1].legend()
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.127 seconds)


.. _sphx_glr_download_api_gallery_aerodynamics_forces_plot_03_lilienthal_coefficients.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/aerodynamics/forces/plot_03_lilienthal_coefficients.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_lilienthal_coefficients.ipynb <plot_03_lilienthal_coefficients.ipynb>`

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

      :download:`Download Python source code: plot_03_lilienthal_coefficients.py <plot_03_lilienthal_coefficients.py>`

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

      :download:`Download zipped: plot_03_lilienthal_coefficients.zip <plot_03_lilienthal_coefficients.zip>`


.. only:: html

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

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