Clamped-clamped beam — first three bending natural frequencies
================================================================

Companion to the simply-supported beam modes
(:doc:`ss_beam_modes`) and cantilever higher-mode problems.
Both ends fully clamped — the characteristic equation differs
from the SS beam:

.. math::

    1 - \cos(\beta L) \cosh(\beta L) = 0

Roots (Rao 2017 §8.5 Table 8.1):

+-----+------------------+
| n   | ``β_n L``        |
+=====+==================+
| 1   | 4.730040745      |
+-----+------------------+
| 2   | 7.853204624      |
+-----+------------------+
| 3   | 10.99560784      |
+-----+------------------+

Natural frequencies follow the same scaling as other beam
modal cases:

.. math::

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

Problem
-------

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

    * - Parameter
      - Value
    * - Length ``L``
      - 1.0 m
    * - Cross-section
      - 0.05 m × 0.05 m
    * - Young's modulus ``E``
      - 200 GPa
    * - Density ``ρ``
      - 7 850 kg/m³
    * - Expected ``f_1``
      - 259.42 Hz
    * - Expected ``f_2``
      - 715.11 Hz
    * - Expected ``f_3``
      - 1 401.90 Hz

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

Ran by :file:`tests/validation/test_cc_beam_modes.py` on an
SOLID185 enhanced-strain hex mesh with both end faces fully
clamped.  Mode identification uses the same UZ-dominant +
expected-antinode filter as :class:`SimplySupportedBeamModes`
to pick the x-z bending modes from the interleaved bending /
torsion / axial families a square cross-section admits.

.. list-table::
    :header-rows: 1
    :widths: 25 25 25 25

    * - Refinement
      - Mesh
      - ``f_1`` (Hz)
      - Error vs Rao
    * - Coarse
      - 20 × 3 × 3
      - see test
      - within 5 %
    * - Medium
      - 40 × 3 × 3
      - 258.02
      - −0.54 %
    * - Refined
      - 80 × 3 × 3
      - see test
      - tighter

For the higher modes:

* ``f_2`` lands at 699.4 Hz on the reference mesh (−2.20 %).
* ``f_3`` lands at 1 342.4 Hz (−4.24 %).

Same downward Timoshenko shear correction the other beam-modal
problems exhibit at higher modes — solid-mesh frequencies sit
slightly below the Euler-Bernoulli closed form.  All three
modes pass their published tolerances comfortably.

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

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

    * - Source
      - Reported ``f_1`` (Hz)
      - Problem ID / location
    * - Closed form (Euler–Bernoulli)
      - 259.42
      - Rao 2017 §8.5 Table 8.1
    * - Timoshenko (1974) §5.3
      - 259.42
      - Clamped-clamped char. eq.
    * - femorph-solver (refined)
      - 258.02
      - :file:`test_cc_beam_modes.py`
    * - MAPDL Verification Manual
      - ≈ 259
      - VM-89 family (clamped variant)
    * - Abaqus Verification Manual
      - ≈ 259
      - AVM 1.6.x clamped-beam-NF family
    * - NAFEMS FV-3
      - 259.42
      - Clamped-clamped beam fundamental + higher modes

Source
------

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

Backing regression test:
:file:`tests/validation/test_cc_beam_modes.py`.