Free-free axial rod — natural frequencies
============================================

Companion to :doc:`axial_rod_nf` (fixed-free).  A uniform rod
with both ends free vibrates longitudinally with mode shapes
:math:`u_n(x) = \cos(n \pi x / L)` and natural frequencies

.. math::

    f_n = \frac{n}{2 L} \sqrt{\frac{E}{\rho}},
    \qquad n = 0, 1, 2, 3, \ldots

The :math:`n = 0` mode is rigid-body axial translation at zero
frequency.  The first elastic mode (:math:`n = 1`) is at
:math:`f_1 = \sqrt{E/\rho} / (2 L)` — twice the fixed-free
fundamental.

This is a useful sanity check for the eigen solver's handling
of near-zero / rigid-body modes: the solver should return one
near-zero rigid-body mode followed by the elastic family.

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)
    * - Young's modulus ``E``
      - 200 GPa
    * - Density ``ρ``
      - 7 850 kg/m³
    * - Expected ``f_1``
      - ``√(E/ρ) / (2L)`` = 2 523.77 Hz
    * - Expected ``f_2``
      - 2 f₁ = 5 047.54 Hz

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

Ran by :file:`tests/validation/test_free_free_rod_nf.py` using
the ``LINK180`` (TRUSS2) kernel on a chain of 2-node spar
elements.  Transverse DOFs (UY, UZ) pinned on every node so the
truss-element zero transverse stiffness doesn't admit arbitrary
side-sway modes.  Axial UX is fully unconstrained — the solver
correctly returns a rigid-body translation mode at f ≈ 0 Hz
followed by the elastic family.

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

    * - Refinement
      - Elements
      - ``f_1`` (Hz)
      - Error vs closed form
    * - Coarse
      - 20
      - 2 526.37
      - +0.10 %
    * - Medium
      - 40
      - 2 524.42
      - +0.03 %
    * - Refined
      - 80
      - 2 523.93
      - +0.006 %

Linear-Lagrange truss elements integrate the wave equation
with consistent mass and recover the elastic spectrum to
near-machine-precision on dense meshes.

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

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

    * - Source
      - Reported ``f_1`` (Hz)
      - Problem ID / location
    * - Closed form (wave equation)
      - 2 523.77
      - Rao 2017 §8.2 Table 8.1
    * - Meirovitch (2010) §6.6
      - 2 523.77
      - Free-free rod
    * - femorph-solver (refined)
      - 2 523.93
      - :file:`test_free_free_rod_nf.py`
    * - MAPDL Verification Manual
      - ≈ 2 524
      - VM-47 (free-free torsion analogue)
    * - Abaqus Verification Manual
      - ≈ 2 524
      - AVM 1.6.x free-free-rod NF family

Source
------

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

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