.. _sphx_glr_gallery_analyses_cyclic:

Cyclic-symmetry modal
=====================

Modal analysis of rotationally periodic structures — compute the full
rotor's spectrum from a single base sector plus phase constraints.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="A rotor with N identical sectors spanning 360° has a spectrum that factors by harmonic index k \in \{0, 1, \dots, N/2\}. Analysing one base sector plus the constraint">

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  .. image:: /gallery/analyses/cyclic/images/thumb/sphx_glr_example_rotor_sector_thumb.png
    :alt:

  :doc:`/gallery/analyses/cyclic/example_rotor_sector`

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      <div class="sphx-glr-thumbnail-title">Cyclic-symmetry modal on a rotor sector</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Drives the bundled bladed-rotor sector through the CyclicModel API end-to-end:">

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  .. image:: /gallery/analyses/cyclic/images/thumb/sphx_glr_example_full_rotor_mode_slides_thumb.png
    :alt:

  :doc:`/gallery/analyses/cyclic/example_full_rotor_mode_slides`

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      <div class="sphx-glr-thumbnail-title">Full-rotor mode-shape slides via CyclicModel</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A 12-sector annular disk&#x27;s modal spectrum decomposes per harmonic index k \in \{0, 1, \ldots, N/2\}.  Each harmonic&#x27;s lowest non-rigid mode picks out a distinct nodal diameter pattern: k = 0 is the breathing (axisymmetric) mode, k = 1 is the rigid-tilt-like mode, k = 2 is the diametrical-2 (two stationary radial nodes), k = 3 adds a third nodal diameter, and so on up to k = N / 2 = 6, the highest harmonic sustainable on a 12-sector rotor.">

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  .. image:: /gallery/analyses/cyclic/images/thumb/sphx_glr_example_cyclic_mode_family_thumb.png
    :alt:

  :doc:`/gallery/analyses/cyclic/example_cyclic_mode_family`

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      <div class="sphx-glr-thumbnail-title">Cyclic-symmetry mode family across every harmonic</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A 12-sector annular disk has its global eigenspectrum factored exactly by harmonic index k \in \{0, 1, \ldots, N/2\} (Thomas 1979, Wildheim 1979).  Each k produces a reduced complex-Hermitian eigenproblem on a single base sector; femorph-solver solves the real-symmetric 2n-augmentation form (Grimes, Lewis &amp; Simon 1994) so scipy.sparse.linalg.eigsh can return mode pairs at every harmonic.">

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  .. image:: /gallery/analyses/cyclic/images/thumb/sphx_glr_example_cyclic_modes_disk_thumb.png
    :alt:

  :doc:`/gallery/analyses/cyclic/example_cyclic_modes_disk`

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      <div class="sphx-glr-thumbnail-title">Cyclic-symmetry modal sweep — annular disk</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A bladed rotor&#x27;s first non-rigid harmonic produces a travelling-wave mode pair: two modes at the same frequency whose mode shapes differ by a 90° phase shift around the rotor.  The pair is what energizes engine-order excitation under operating speed — when the rotor&#x27;s harmonic index k matches the engine order, the travelling-wave amplitude grows resonantly (Crandall &amp; Mark 1963; Bathe §10.3.4).">

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  .. image:: /gallery/analyses/cyclic/images/thumb/sphx_glr_example_cyclic_modes_bladed_rotor_thumb.png
    :alt:

  :doc:`/gallery/analyses/cyclic/example_cyclic_modes_bladed_rotor`

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      <div class="sphx-glr-thumbnail-title">Cyclic-symmetry travelling-wave pair — bladed rotor</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /gallery/analyses/cyclic/example_rotor_sector
   /gallery/analyses/cyclic/example_full_rotor_mode_slides
   /gallery/analyses/cyclic/example_cyclic_mode_family
   /gallery/analyses/cyclic/example_cyclic_modes_disk
   /gallery/analyses/cyclic/example_cyclic_modes_bladed_rotor