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


Calculation method used for reference solutions
----------------------------------------------------------

The reference solution is a solution obtained with Python. Indeed, two calculation methods were used to determine the frequencies. Each of the methods uses the stiffness, mass, gyroscopic, and damping matrices calculated by*Code_Aster*. To find the frequencies of the quadratic modal problem, we use:


* the numpy python mathematical library (search for eigenvalues)


* the CALC_MODES command

Therefore, strictly speaking, it is not a non-regression. On the other hand, for the validation of

beam elements and in the absence of elements for comparison, it is indeed a question of non-regression.


Reference quantities
----------------------


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


* :math:`\mathrm{AMOR}\text{\_}\mathrm{REDUIT}`: reduced amortization


Benchmark result
----------------------

As an indication, the reference results for the right Euler beam are given below.


+----------+----------------------------------+
|**numpy**                                    |
+----------+----------------------------------+
|:math:`N°`|:math:`\mathrm{FREQ}(\mathrm{Hz})`|
+----------+----------------------------------+
|:math:`1` |:math:`123.915`                   |
+----------+----------------------------------+
|:math:`2` |:math:`124.546`                   |
+----------+----------------------------------+
|:math:`3` |:math:`497.033`                   |
+----------+----------------------------------+
|:math:`4` |:math:`499.575`                   |
+----------+----------------------------------+


+--------------+----------------------------------+---------------------------------------------+
|**CALC_MODES**                                                                                 |
+--------------+----------------------------------+---------------------------------------------+
|:math:`N°`    |:math:`\mathrm{FREQ}(\mathrm{Hz})`|:math:`\mathrm{AMOR}\text{\_}\mathrm{REDUIT}`|
+--------------+----------------------------------+---------------------------------------------+
|:math:`1`     |:math:`123.915`                   |:math:`0.0`                                  |
+--------------+----------------------------------+---------------------------------------------+
|:math:`20`    |:math:`7971.6`                    |:math:`0.0`                                  |
+--------------+----------------------------------+---------------------------------------------+
|:math:`40`    |:math:`21163.265`                 |:math:`0.0`                                  |
+--------------+----------------------------------+---------------------------------------------+
|:math:`60`    |:math:`37289.789`                 |:math:`0.0`                                  |
+--------------+----------------------------------+---------------------------------------------+
|:math:`80`    |:math:`74712.423`                 |:math:`0.0`                                  |
+--------------+----------------------------------+---------------------------------------------+


Uncertainty about the solution
---------------------------

Digital solution