:orphan:

Analyses
========

Runnable examples organised by the *kind of analysis* rather than
which element happens to be used.  Each sub-gallery walks through the
full loop — build the model, solve, interpret the result — for one
analysis family.

.. grid:: 1 2 2 2
    :gutter: 3

    .. grid-item-card:: Static
        :link: static/index
        :link-type: doc

        :math:`K u = F`.  Cantilever deflections, plane-stress /
        plane-strain problems, single-element patch tests.

    .. grid-item-card:: Modal
        :link: modal/index
        :link-type: doc

        :math:`K \varphi = \omega^2 M \varphi`.  Plate modes, SDOF
        oscillators, beam cantilever frequencies.

    .. grid-item-card:: Transient
        :link: transient/index
        :link-type: doc

        :math:`M \ddot u + C \dot u + K u = F(t)` via Newmark-β.

    .. grid-item-card:: Cyclic
        :link: cyclic/index
        :link-type: doc

        Cyclic-symmetry modal on a base sector with phase constraints.


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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 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
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  :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
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  :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
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  :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>
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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
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  :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>
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Harmonic analysis
=================

Frequency-response sweeps via modal superposition —
:math:`(K - \omega^2 M + i\,\omega C)\,u(\omega) = F`.  The complex
receptance is back-substituted from a precomputed eigenbasis at
every excitation frequency.


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    <div class="sphx-glr-thumbcontainer" tooltip="A single-DOF mass-spring-damper has the analytical receptance">

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  .. image:: /gallery/analyses/harmonic/images/thumb/sphx_glr_example_sdof_frf_thumb.png
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  :doc:`/gallery/analyses/harmonic/example_sdof_frf`

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      <div class="sphx-glr-thumbnail-title">SDOF receptance — closed form vs modal superposition</div>
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Modal analysis
==============

Free-vibration modes — :math:`K \varphi = \omega^2 M \varphi`.  Plate
modes, SDOF oscillators, beam cantilever frequencies.


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    <div class="sphx-glr-thumbcontainer" tooltip="End-to-end modal-analysis example: build a 1 m × 1 m × 10 mm steel plate as a 2-through-thickness, 40 × 40 in-plane hex mesh in pyvista, wrap it in femorph_solver.Model, clamp the x = 0 edge, and extract the first 10 modes with femorph_solver.Model.modal_solve.">

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  .. image:: /gallery/analyses/modal/images/thumb/sphx_glr_example_plate_modes_thumb.png
    :alt:

  :doc:`/gallery/analyses/modal/example_plate_modes`

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      <div class="sphx-glr-thumbnail-title">HEX8 — cantilever-plate modal (2 × 40 × 40 hex mesh)</div>
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Static analysis
===============

:math:`K u = F` with Dirichlet BCs.  The workhorse analysis — one
factorisation, one back-solve.


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    <div class="sphx-glr-thumbcontainer" tooltip="Static analysis of a cantilever plate under a distributed tip load. This example walks through the full femorph_solver.Model.solve → StaticResult → femorph_solver.io.static_result_to_grid → pyvista rendering loop and checks static equilibrium via StaticResult.reaction.">

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  .. image:: /gallery/analyses/static/images/thumb/sphx_glr_example_cantilever_plate_thumb.png
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  :doc:`/gallery/analyses/static/example_cantilever_plate`

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      <div class="sphx-glr-thumbnail-title">HEX8 — cantilever plate static analysis</div>
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Transient analysis
==================

Time-domain simulations of the equation of motion
:math:`M \ddot u + C \dot u + K u = F(t)` using Newmark-β.


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    <div class="sphx-glr-thumbcontainer" tooltip="A single-DOF mass-spring-damper subjected to a step load shows off the Newmark-β transient integrator in solve_transient.  We integrate for long enough to watch the underdamped oscillation settle toward the new static equilibrium, then compare the result to the textbook second-order response.">

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  .. image:: /gallery/analyses/transient/images/thumb/sphx_glr_example_sdof_transient_thumb.png
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  :doc:`/gallery/analyses/transient/example_sdof_transient`

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      <div class="sphx-glr-thumbnail-title">SDOF transient — step-load response</div>
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.. toctree::
   :hidden:
   :includehidden:


   /gallery/analyses/cyclic/index.rst
   /gallery/analyses/harmonic/index.rst
   /gallery/analyses/modal/index.rst
   /gallery/analyses/static/index.rst
   /gallery/analyses/transient/index.rst


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  .. container:: sphx-glr-footer sphx-glr-footer-gallery

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

      :download:`Download all examples in Python source code: analyses_python.zip </gallery/analyses/analyses_python.zip>`

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

      :download:`Download all examples in Jupyter notebooks: analyses_jupyter.zip </gallery/analyses/analyses_jupyter.zip>`


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

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