Axial rod — fundamental natural frequency (fixed-free)
========================================================

A uniform slender rod clamped at one end and free at the other
vibrates longitudinally with mode shapes
:math:`u_n(x) = \sin\!\bigl((2n-1) \pi x / (2L)\bigr)` and natural
frequencies

.. math::

    f_n = \frac{2n - 1}{4 L} \sqrt{\frac{E}{\rho}}, \qquad
    n = 1, 2, 3, \ldots

The fundamental (``n = 1``):

.. math::

    f_1 = \frac{1}{4 L} \sqrt{\frac{E}{\rho}}.

Unlike the bending problems, :math:`f_1` depends only on the
material-wave speed :math:`\sqrt{E / \rho}` and the length —
cross-sectional area cancels out.  This makes the problem a
pure test of the mass-and-stiffness matrix scaling, independent
of geometric properties.

Problem
-------

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

    * - Parameter
      - Value
    * - Rod length ``L``
      - 1.0 m
    * - Cross-sectional area ``A``
      - 1 × 10⁻⁴ m² (cancels out of ``f_1``)
    * - Young's modulus ``E``
      - 200 GPa
    * - Density ``ρ``
      - 7 850 kg/m³
    * - Expected fundamental ``f_1``
      - ``(1/4L) √(E/ρ) = 1 261.886 Hz``

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

The 1D axial wave equation ``ρA ü = EA u''`` with fixed-free
BCs ``u(0) = 0`` and ``EA u'(L) = 0`` separates into the mode
shapes + frequencies shown above (Rao, *Mechanical Vibrations*
6th ed., §8.2; Meirovitch, *Fundamentals of Vibrations* §6.6).

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

Ran by :file:`tests/validation/test_axial_rod_nf.py` using a
``TRUSS2`` (``LINK180``) mesh — a straight line of 2-node spar
elements along the global x-axis.  Transverse translations
(``UY`` / ``UZ``) are suppressed at every node so the axial mode
family is the first and only surviving family (pure-truss
elements have zero transverse stiffness and would otherwise
admit rigid-body transverse modes at zero frequency ahead of
the axial fundamental).

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

    * - Discretisation
      - Elements
      - ``f_1`` (Hz)
      - Error vs closed form
    * - Coarse
      - 10
      - 1 263.184
      - +0.10 %
    * - Medium
      - 20
      - 1 262.211
      - +0.03 %
    * - Reference
      - 40
      - 1 261.967
      - +0.01 %
    * - Refined
      - 80
      - 1 261.906
      - +0.002 %

Linear-Lagrange TRUSS2 elements converge quickly on axial modes
— the consistent-mass / consistent-stiffness pair recovers the
exact cosine mode shape to within machine precision as
``n_elem → ∞``.

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

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

    * - Source
      - Reported ``f_1`` (Hz)
      - Problem ID / location
    * - Closed form (wave equation)
      - 1 261.886
      - Rao 2017 §8.2
    * - Meirovitch (2010) §6.6
      - 1 261.886
      - Fixed-free rod
    * - femorph-solver (refined)
      - 1 261.906
      - :file:`test_axial_rod_nf.py`
    * - MAPDL Verification Manual
      - 1 262 (VM-47 uses torsion — same Helmholtz eq)
      - VM-47 *Torsional freq of a uniform shaft*
    * - Abaqus Verification Manual
      - ≈ 1 261.9
      - AVM 1.6.x bar-NF family

Source
------

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

Backing regression test:
:file:`tests/validation/test_axial_rod_nf.py` — four-mesh
convergence sweep + monotonic-convergence guard.