Simply-supported beam — first three bending natural frequencies
==================================================================

A prismatic slender beam of length :math:`L`, cross-section
:math:`w \times h`, simply supported at both ends, vibrates
transversely with mode shapes
:math:`w_n(x) = \sin(n \pi x / L)` and natural frequencies

.. math::

    f_n = \frac{n^{2}\pi}{2 L^{2}} \sqrt{\frac{E I}{\rho A}}.

For :math:`L / h = 20` the first few bending modes dominate the
low end of the spectrum; torsional + axial modes appear well
above them.

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``
      - 200 GPa
    * - Poisson's ratio ``ν``
      - 0.30
    * - Density ``ρ``
      - 7 850 kg/m³
    * - Expected ``f_1``
      - 114.44 Hz
    * - Expected ``f_2``
      - 457.76 Hz (4 × f₁)
    * - Expected ``f_3``
      - 1 030.0 Hz (9 × f₁)

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

Euler–Bernoulli separation-of-variables solution for a
simply-supported prismatic beam (Rao, *Mechanical Vibrations*
6th ed., §8.5; Meirovitch, *Fundamentals of Vibrations* §7.4).

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

Ran by :file:`tests/validation/test_ss_beam_modes.py` on an
SOLID185 enhanced-strain hex mesh with the same knife-edge
support convention as the static SS-beam problem.  Mode
identification uses a post-processing filter: pick the first
finite-frequency mode that is (a) dominated by UZ motion
(≥ 70 % of kinetic energy) and (b) has the expected number of
antinodes along the top-face centerline.  This isolates the
pure x-z-plane bending modes from the y-bending, torsional,
and axial families the square cross-section admits at
interleaved frequencies.

.. list-table::
    :header-rows: 1
    :widths: 15 15 20 20 20

    * - Refinement
      - Mesh
      - ``f_1`` (Hz)
      - ``f_2`` (Hz)
      - ``f_3`` (Hz)
    * - Coarse
      - 20 × 3 × 3
      - 113.62 (−0.7 %)
      - 438.60 (−4.2 %)
      - 1 072.41 (+4.1 %)
    * - Medium
      - 40 × 3 × 3
      - 113.33 (−1.0 %)
      - 433.35 (−5.3 %)
      - 1 042.17 (+1.2 %)
    * - Refined
      - 80 × 3 × 3
      - 113.23 (−1.1 %)
      - 431.12 (−5.8 %)
      - 1 031.49 (+0.2 %)

The persistent ~5 % shift on ``f_2`` is the combined rotary-
inertia + shear-deformation correction that Timoshenko beam
theory captures but the pure Euler–Bernoulli reference omits.
At ``L / h = 20`` the correction is small but not negligible
for the second bending mode.

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

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

    * - Source
      - Reported ``f_1`` (Hz)
      - Problem ID / location
    * - Closed form (Euler–Bernoulli)
      - 114.44
      - Rao 2017 §8.5
    * - Meirovitch (2010) §7.4
      - 114.44
      - SS beam transverse vibration
    * - femorph-solver (refined)
      - 113.23
      - :file:`test_ss_beam_modes.py`
    * - MAPDL Verification Manual
      - ≈ 114.4
      - VM-89 *Natural frequencies of a SS beam*
    * - Abaqus Verification Manual
      - ≈ 114.4
      - AVM 1.6.x SS beam natural freq family

Source
------

Problem class:
:class:`femorph_solver.validation.problems.SimplySupportedBeamModes`.

Backing regression test:
:file:`tests/validation/test_ss_beam_modes.py` — three-mesh
convergence sweep asserting each of ``f_1``, ``f_2``, ``f_3``
passes its published tolerance and that the three frequencies
are strictly ordered.