Reference solution
=====================

Calculation method used for the reference solution
--------------------------------------------------------


* Vibration modes in the plane

For these vibration modes, the equation for bending curved beams by V. Boussinesq (1883), without extending the neutral fiber, leads to:

.. math::
 :label: eq-1

    {f} _ {i} =\ frac {1} {1} {2} {2\ pi}\ sqrt {(\ frac {{i} ^ {2} -1)} ^ {2} -1)} ^ {2}}} {{i}} {{i}} {i} ^ {2} +1}\ times\ frac {{2} +1}\ times\ frac {{i} {\ mathit {EI}} {\ rho {\ mathit {AR}}} {{i}}} {{i}} {{i}} {{i}}} {{i}} {{i}} {{i}}} {{i}} {{i}} {{i}} {{i}}} {{i}} {{i}} {{i}} 4}})}

The reference solution is established for thin arcs such as :math:`\alpha R\ge 100\sqrt{(\frac{{I}_{z}}{A})}` with :math:`\alpha`, which is the center angle in radians.


* Off-plane vibration modes

For transverse vibration modes with rectangular section, the solution was established from the results of two calculation codes, using different formulations.


+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|Natural modes in the plane (polar coordinates :math:`(i,\theta )`)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |Off-plane clean modes                                                                                                                                                                    |
+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+
|Symmetric :math:`{u}_{i}^{\text{'}}=i\mathrm{cos}(i\theta )` :math:`{v}_{i}^{\text{'}}=\mathrm{sin}(i\theta )`                                                                                                                                                                                                          |:math:`i=0` Set rotation :math:`i=1` set translation :math:`i=2` :math:`i=3`|Antisymmetric :math:`{u}_{i}^{\text{'}}=i\mathrm{sin}(i\theta )` :math:`{v}_{i}^{\text{'}}=-\mathrm{cos}(i\theta )` :math:`{\theta }_{i}^{\text{'}}=-\frac{1-{i}^{2}}{R}\mathrm{cos}(i\theta )`                                                                                                                              |:math:`i=0` :math:`i=1` :math:`i=2` :math:`i=3` :math:`i=4`|                                                                                                                             |
+ .. image:: images/10000200000000A1000001E40D47DF5F899015DD.png                                                                                                                                                                                                                                                         +                                                                            + .. image:: images/10000200000000A8000001F385C32AFB0B5097BF.png                                                                                                                                                                                                                                                              +                                                           + .. image:: images/10000200000000AB000002080926F5C1B4FB4007.png                                                              +
|    :width: 0.7646in                                                                                                                                                                                                                                                                                                    |                                                                            |    :width: 0.8366in                                                                                                                                                                                                                                                                                                         |                                                           |    :width: 1.0665in                                                                                                         |
+    :height: 2.3673in                                                                                                                                                                                                                                                                                                   +                                                                            +    :height: 2.5445in                                                                                                                                                                                                                                                                                                        +                                                           +    :height: 3.2429in                                                                                                        +
|                                                                                                                                                                                                                                                                                                                        |                                                                            |                                                                                                                                                                                                                                                                                                                             |                                                           |                                                                                                                             |
+ :math:`{\theta }_{i}^{\text{'}}=-\frac{1-{i}^{2}}{R}\mathrm{sin}(i\theta )`                                                                                                                                                                                                                                            +                                                                            +                                                                                                                                                                                                                                                                                                                             +                                                           +                                                                                                                             +
|                                                                                                                                                                                                                                                                                                                        |                                                                            |                                                                                                                                                                                                                                                                                                                             |                                                           |                                                                                                                             |
+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+


Reference quantity
---------------------


* :math:`\mathrm{FREQ}`: frequency


Reference quantity and result
---------------------------------

+-----------------+---------------------+--------------------+----------------------+-------------------------------+
|                 |Component            |Nature of clean mode                       |Reference :math:`(\mathrm{Hz})`|
+                 +                     +--------------------+----------------------+                               +
|                 |                     |:math:`i`           |order                 |                               |
+-----------------+---------------------+--------------------+----------------------+-------------------------------+
|Modes in the plan|:math:`\mathrm{FREQ}`|:math:`2`           |:math:`\mathrm{4,5}`  |:math:`318.36`                 |
+                 +                     +--------------------+----------------------+-------------------------------+
|                 |                     |:math:`3`           |:math:`\mathrm{6,7}`  |:math:`900.46`                 |
+                 +                     +--------------------+----------------------+-------------------------------+
|                 |                     |:math:`4`           |:math:`\mathrm{8,9}`  |:math:`1726.55`                |
+                 +                     +--------------------+----------------------+-------------------------------+
|                 |                     |:math:`5`           |:math:`\mathrm{10,11}`|:math:`2792.21`                |
+-----------------+---------------------+--------------------+----------------------+-------------------------------+
|Transverse modes |:math:`\mathrm{FREQ}`|:math:`2`           |:math:`\mathrm{4,5}`  |:math:`511.`                   |
+                 +                     +--------------------+----------------------+-------------------------------+
|                 |                     |:math:`3`           |:math:`\mathrm{6,7}`  |:math:`1590.`                  |
+                 +                     +--------------------+----------------------+-------------------------------+
|                 |                     |:math:`4`           |:math:`\mathrm{8,9}`  |:math:`3184.`                  |
+-----------------+---------------------+--------------------+----------------------+-------------------------------+


Bibliographical references
---------------------------

[:ref:`1 <1>`] Structural Calculation Software Validation Guide: SFM, AFNOR technique, ISBN: 2-12-486611-7