.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/mechanical_engineering/transmission/plot_03_lewis_gear_tooth.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` 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_transmission_plot_03_lewis_gear_tooth.py: Lewis's gear-tooth bending formula (1892) ========================================= Wilfred Lewis, of William Sellers and Co. in Philadelphia, gave the first rational strength calculation for gear teeth in a talk to the Engineers' Club of Philadelphia on 15 October 1892. He treated the tooth as a cantilever with the whole load at its tip and noticed that a parabola drawn from the load point, tangent to the tooth flanks at the root, is a beam of uniform strength. The root section where they touch is therefore the weakest, and the stress there is :math:`\sigma = W_t/(F m Y)` (:func:`~engineeringkit.mechanical_engineering.lewis_bending_stress`), with a form factor :math:`Y` that depends only on the tooth shape and so on the number of teeth (:func:`~engineeringkit.mechanical_engineering.lewis_form_factor`). Carl Barth's factor raises the stress for the dynamic load at speed (:func:`~engineeringkit.mechanical_engineering.barth_velocity_factor`). The AGMA bending equation still starts from Lewis's. .. GENERATED FROM PYTHON SOURCE LINES 23-28 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from engineeringkit.mechanical_engineering import barth_velocity_factor, lewis_bending_stress, lewis_form_factor .. GENERATED FROM PYTHON SOURCE LINES 29-37 The parabola of uniform strength -------------------------------- A cantilever of width :math:`F` with thickness :math:`t(y) = 2\sqrt{c\,y}` at distance :math:`y` below the load has bending stress :math:`6Wy/(F t^2) = 3W/(2Fc)` at every section. If the tooth's root thickness :math:`t` is reached at depth :math:`l`, Lewis's :math:`x = t^2/4l` and :math:`Y = 2x/3m` give the same stress. .. GENERATED FROM PYTHON SOURCE LINES 37-49 .. code-block:: Python W, F, module = 1500.0, 0.02, 0.003 y = np.linspace(1e-4, 0.006, 50) c = 1.2e-3 thickness = 2 * np.sqrt(c * y) stress = 6 * W * y / (F * thickness**2) assert np.allclose(stress, stress[0]) t_root, depth = 0.0055, 0.0068 Y = 2 * (t_root**2 / (4 * depth)) / (3 * module) print(f"form factor of this tooth: Y = {Y:.3f}") assert np.isclose(lewis_bending_stress(W, F, module, Y), 6 * W * depth / (F * t_root**2)) .. rst-class:: sphx-glr-script-out .. code-block:: none form factor of this tooth: Y = 0.247 .. GENERATED FROM PYTHON SOURCE LINES 50-55 Sizing a pinion --------------- A 20-tooth, 3 mm module cut pinion transmits 5 kW at 1500 rpm with an allowable bending stress of 80 MPa. Find the face width. .. GENERATED FROM PYTHON SOURCE LINES 55-77 .. code-block:: Python teeth, power, rpm, allowable = 20, 5e3, 1500.0, 80e6 pitch_diameter = teeth * module V = np.pi * pitch_diameter * rpm / 60 Wt = power / V Kv = barth_velocity_factor(V, "cut") face = Kv * Wt / (module * lewis_form_factor(teeth) * allowable) print(f"V = {V:.2f} m/s, Wt = {Wt:.0f} N, Kv = {Kv:.3f}, face width = {face * 1e3:.1f} mm") assert np.isclose(lewis_bending_stress(Wt, face, module, lewis_form_factor(teeth), Kv), allowable) fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 3.8)) n = np.arange(12, 401) ax1.semilogx(n, lewis_form_factor(n)) ax1.set(xlabel="number of teeth", ylabel="Lewis form factor Y", title="Bigger gears have stronger teeth") v = np.linspace(0, 25, 200) for profile in ("cast", "cut", "hobbed", "ground"): ax2.plot(v, barth_velocity_factor(v, profile), label=profile) ax2.set(xlabel="pitch-line velocity [m/s]", ylabel="Kv", title="Barth's dynamic factor") ax2.legend() fig.tight_layout() plt.show() .. image-sg:: /api/gallery/mechanical_engineering/transmission/images/sphx_glr_plot_03_lewis_gear_tooth_001.png :alt: Bigger gears have stronger teeth, Barth's dynamic factor :srcset: /api/gallery/mechanical_engineering/transmission/images/sphx_glr_plot_03_lewis_gear_tooth_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none V = 4.71 m/s, Wt = 1061 N, Kv = 1.773, face width = 24.3 mm .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.177 seconds) .. _sphx_glr_download_api_gallery_mechanical_engineering_transmission_plot_03_lewis_gear_tooth.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/transmission/plot_03_lewis_gear_tooth.ipynb :alt: Launch JupyterLite :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_03_lewis_gear_tooth.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_03_lewis_gear_tooth.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_03_lewis_gear_tooth.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_