Cantilever beam — higher bending natural frequencies
======================================================

Extends the fundamental-mode page
:doc:`cantilever_beam_natural_frequency` with the second and
third bending natural frequencies of a clamped-free prismatic
beam.

Cantilever bending frequencies follow

.. math::

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

where :math:`\beta_n L` are the roots of
:math:`1 + \cos(\beta L) \cosh(\beta L) = 0`:

+-----+------------------+
| n   | ``β_n L``        |
+=====+==================+
| 1   | 1.8751040687     |
+-----+------------------+
| 2   | 4.6940911330     |
+-----+------------------+
| 3   | 7.8547574382     |
+-----+------------------+
| 4   | 10.995540735     |
+-----+------------------+

For our reference geometry (``L = 1 m``, ``w = h = 0.05 m``,
``E = 200 GPa``, ``ρ = 7 850 kg/m³``) the first four frequencies
are ``f₁ ≈ 40.77 Hz``, ``f₂ ≈ 255.54 Hz``, ``f₃ ≈ 715.39 Hz``,
``f₄ ≈ 1 402 Hz``.

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
    * - Density ``ρ``
      - 7 850 kg/m³
    * - Expected ``f_2``
      - 255.54 Hz
    * - Expected ``f_3``
      - 715.39 Hz

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

Euler-Bernoulli separation-of-variables solution for a
clamped-free prismatic beam (Rao, *Mechanical Vibrations*
6th ed., §8.5 Table 8.1; Timoshenko, *Vibration Problems in
Engineering* §5.3).

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

Ran by :file:`tests/validation/test_cantilever_higher_modes.py`
on the same SOLID185 enhanced-strain hex mesh the fundamental-
mode problem uses.  Mode identification uses the UZ-dominant +
expected-antinode filter to pick the second and third
x-z-bending modes out of the interleaved bending / torsion /
axial families a square cross-section admits.

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

    * - Refinement
      - Mesh
      - ``f_2`` (Hz)
      - ``f_3`` (Hz)
    * - Coarse
      - 20 × 3 × 3
      - 255.654 (+0.06 %)
      - 711.883 (−0.49 %)
    * - Medium
      - 40 × 3 × 3
      - 253.709 (−0.70 %)
      - 700.449 (−2.09 %)
    * - Refined
      - 80 × 3 × 3
      - 253.092 (−0.94 %)
      - 697.282 (−2.53 %)

The small downward drift with refinement is the Timoshenko
shear + rotary-inertia correction the pure Euler-Bernoulli
reference omits — cantilever higher modes are more sensitive
to the correction than the fundamental.  Both values stay
comfortably under the published tolerances (6 % for ``f_2``,
12 % for ``f_3``).

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

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

    * - Source
      - Reported ``f_2`` (Hz)
      - Problem ID / location
    * - Closed form (Euler-Bernoulli)
      - 255.54
      - Rao 2017 §8.5 Table 8.1
    * - Timoshenko (1974) §5.3
      - 255.54
      - Cantilever char. eq.
    * - femorph-solver (refined)
      - 253.09
      - :file:`test_cantilever_higher_modes.py`
    * - MAPDL Verification Manual
      - ≈ 255
      - VM-57 family (cantilever-shaft modal)
    * - Abaqus Verification Manual
      - ≈ 255
      - AVM 1.6.x cantilever-bending family
    * - NAFEMS FV-2
      - 255.54
      - Cantilever transverse modes

Source
------

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

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