Benchmark solution
=====================


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

The calculation is based on the dynamic equilibrium and behavioral relationships [:ref:`bib2 <bib2>`] as set out below:


.. csv-table::

    ":math:`\frac{\partial {M}_{y}}{\partial x}+{V}_{z}=⟨\rho I⟩\frac{{\partial }^{2}{\theta }_{y}}{\partial {t}^{2}}` "," :math:`\frac{\partial {V}_{z}}{\partial x}=⟨\rho S⟩\frac{{\partial }^{2}{v}_{z}}{\partial {t}^{2}}`"
    ":math:`{M}_{y}=⟨EI⟩\frac{\partial {\theta }_{y}}{\partial x}` "," :math:`{V}_{z}=k⟨GS⟩\left(\frac{\partial {v}_{z}}{\partial x}-{\theta }_{y}\right)`"


These relationships make it possible to write the equation of transverse dynamic flexure motion :math:`v(x,t)`. The natural frequency equation is obtained after combining the boundary conditions.

The natural frequency equation is written as:

:math:`\mathrm{sin}({X}_{2})=0` with :math:`{X}_{2}={\left[{\stackrel{ˉ}{\omega }}^{2}\frac{(1+a)}{2}+\sqrt{{\stackrel{ˉ}{\omega }}^{2}({\stackrel{ˉ}{\omega }}^{2}{(\frac{1-a}{2})}^{2}+\frac{1}{{r}^{2}})}\right]}^{1/2}`

and

:math:`{\overline{\omega }}^{2}=\frac{⟨\rho I⟩{\omega }^{2}{L}^{2}}{⟨EI⟩}`; :math:`{\overline{r}}^{2}=\frac{⟨\rho I⟩}{⟨\rho S⟩{L}^{2}}`; :math:`a=\frac{⟨\rho S⟩⟨EI⟩}{k⟨\rho I⟩⟨GS⟩}`

The solutions to the natural frequency equation are then written as: :math:`{X}_{2}=n\pi (n=\mathrm{1,2}\mathrm{,3},\mathrm{...})`


Benchmark results
----------------------

5 first frequencies and associated natural modes of bending.


+-------------------------+--------------------------+----------------------------------------------------------------------------------+
| - Mode 1 frequency:     |:math:`64.476\mathrm{Hz}` |                                                                                  |
+                         +                          + .. image:: images/Object_23.svg                                                  +
|                         |                          |    :width: 89                                                                    |
+                         +                          +    :height: 25                                                                   +
|                         |                          |                                                                                  |
+                         +                          +                                                                                  +
|                         |                          |                                                                                  |
+-------------------------+--------------------------+----------------------------------------------------------------------------------+
| - Mode 2 frequency:     |:math:`131.918\mathrm{Hz}`|                                                                                  |
+                         +                          + .. image:: images/Object_24.svg                                                  +
|                         |                          |    :width: 89                                                                    |
+                         +                          +    :height: 25                                                                   +
|                         |                          |                                                                                  |
+                         +                          +                                                                                  +
|                         |                          |                                                                                  |
+-------------------------+--------------------------+----------------------------------------------------------------------------------+
| - Mode 3 frequency:     |:math:`198.734\mathrm{Hz}`|                                                                                  |
+                         +                          + .. image:: images/Object_25.svg                                                  +
|                         |                          |    :width: 89                                                                    |
+                         +                          +    :height: 25                                                                   +
|                         |                          |                                                                                  |
+                         +                          +                                                                                  +
|                         |                          |                                                                                  |
+-------------------------+--------------------------+----------------------------------------------------------------------------------+
| - Mode 4 frequency:     |:math:`265.383\mathrm{Hz}`|                                                                                  |
+                         +                          + .. image:: images/Object_26.svg                                                  +
|                         |                          |    :width: 89                                                                    |
+                         +                          +    :height: 25                                                                   +
|                         |                          |                                                                                  |
+                         +                          +                                                                                  +
|                         |                          |                                                                                  |
+-------------------------+--------------------------+----------------------------------------------------------------------------------+
| - Mode 5 frequency:     |:math:`331.963\mathrm{Hz}`|                                                                                  |
+                         +                          + .. image:: images/Object_27.svg                                                  +
|                         |                          |    :width: 89                                                                    |
+                         +                          +    :height: 25                                                                   +
|                         |                          |                                                                                  |
+                         +                          +                                                                                  +
|                         |                          |                                                                                  |
+-------------------------+--------------------------+----------------------------------------------------------------------------------+


Uncertainties about the solution
----------------------------

The reference solution is calculated using the assumptions of girder theory [:ref:`bib2 <bib2>`]: :math:`{\sigma }_{y}={\sigma }_{z}=0`.


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


1. VPCS: Software for calculating composite structures; Validation examples. Review of Composites and Advanced Materials, Volume 5 - special issue/ 1995 - Hermes edition.


2. CIEAUX J.M.: Dynamic flexure of composite beams with orthotropic phases; Validity of the quasistatic domain. Thesis from Paul Sabatier University Toulouse III, 1988.