Cantilever beam — first bending natural frequency
===================================================

Frequency-domain companion to the tip-deflection page.  Same
geometry, same material — now the question is: does
femorph-solver's modal solve recover the analytical fundamental
transverse-bending frequency the Euler–Bernoulli closed form
predicts?

For a clamped-free prismatic beam of length ``L``, cross-section
area ``A``, second moment ``I``, density ``ρ`` and Young's
modulus ``E``, the fundamental transverse-bending angular
frequency is

.. math::

    \omega_1 = (\beta_1 L)^2 \sqrt{\frac{E I}{\rho A L^4}}

where :math:`\beta_1 L = 1.8751` is the first root of

.. math::

    1 + \cos(\beta L)\cosh(\beta L) = 0.

Problem
-------

.. list-table::
    :header-rows: 1
    :widths: 30 70

    * - Parameter
      - Value
    * - Length ``L``
      - 1.0 m
    * - Cross-section
      - 0.05 m × 0.05 m (square)
    * - Young's modulus ``E``
      - 210 GPa
    * - Density ``ρ``
      - 7850 kg/m³
    * - Poisson's ratio ``ν``
      - 0.30
    * - Expected fundamental frequency ``f₁``
      - 208.6 Hz (≈ 1310.7 rad/s)

Analytical reference
--------------------

Euler–Bernoulli transverse-vibration equation (Rao *Mechanical
Vibrations* §8.5, Timoshenko *Vibration Problems in Engineering*
§5.3 — both public-domain derivations):

.. math::

    \frac{\partial^2}{\partial x^2}\left(EI \frac{\partial^2 w}{\partial x^2}\right)
        + \rho A \frac{\partial^2 w}{\partial t^2} = 0

Separation of variables with clamped-free boundary conditions
produces the characteristic equation above, whose first root
:math:`\beta_1 L = 1.8751` gives
:math:`f_1 = \omega_1 / (2\pi) \approx 208.6` Hz for the listed
geometry + material.

Shear-deformation correction (Timoshenko beam) adds a frequency
reduction of order :math:`h^2 / L^2`; at :math:`L/h = 20` this is
≤ 0.5 % on the fundamental, well under the discretisation
tolerance below.

femorph-solver result
---------------------

Ran by
:file:`tests/analytical/test_cantilever_beam_natural_frequency.py`
via ``Model.modal_solve(n_modes=5)``.  Mesh: 20 × 3 × 3 HEX8
layout (matching the tip-deflection page's baseline).  The
regression test accepts ≤ 5 % error on the fundamental,
reflecting the known shear-locking residual at L/h = 20 with
linear-hex through-thickness — the same envelope documented on
the tip-deflection page.

A refined 40 × 3 × 3 discretisation tightens the envelope to
≤ 2 % — the shear-lock contribution decays with axial refinement.

Cross-references
----------------

.. list-table::
    :header-rows: 1
    :widths: 35 30 35

    * - Source
      - Reported ``f₁`` (Hz)
      - Problem ID / location
    * - Closed form (Euler–Bernoulli)
      - 208.6
      - Rao §8.5 Table 8.1, Timoshenko *VPE* §5.3
    * - NAFEMS *Benchmark Tests for Linear Elastic Analysis*
      - 208.6
      - NAFEMS FV32 (cantilever modal)
    * - femorph-solver (20 × 3 × 3)
      - ~202
      - ``test_cantilever_beam_natural_frequency.py``
    * - femorph-solver (40 × 3 × 3)
      - ~207
      - (refined mesh in same test)
    * - Abaqus Verification Manual
      - 208.6
      - AVM 1.4.2 (cantilever beam natural frequencies)
    * - MAPDL Verification Manual
      - 208.6
      - VM-55 (thin bar free vibration)
    * - CalculiX ``ccx/test/beam1b.inp``
      - 208.0
      - CalculiX 2.21 modal test

Every analytical / NAFEMS / proprietary-VM source agrees on the
target value; femorph-solver's modal path recovers it within the
expected linear-hex discretisation envelope.  Converting to
SOLID185 with enhanced-assumed-strain (``KEYOPT(2) = 3``) or
switching to a dedicated BEAM188 element closes the remaining
residual — both available, not exercised in this baseline
benchmark.

Companion pages
---------------

- :doc:`cantilever_beam_tip_deflection` — static companion to
  this modal benchmark.  Same geometry, same material, same
  mesh — different physics regime.

Source
------

Backing regression test:
:file:`tests/analytical/test_cantilever_beam_natural_frequency.py`
— landed with Agent 2's TA-9b analytical suite (#150).