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


Calculation method
-----------------

The reference is given by the code CADYRO [:ref:`bib1 <bib1>`] for beam elements with variable cross section.

Moreover, the values resulting from the modeling using beam elements with a variable section (Modeling A) are framed by those obtained by those obtained from:


*B modeling, approximating the exact solution by lower values. This is equivalent to assigning to each element the**constant circular section**with a diameter equal to the diameter of the rotor studied**in line with the initial node** of the element.


*C modeling, approaching the exact solution by higher values. This consists in assigning to each element the**constant circular section**with a diameter equal to the diameter of the rotor studied** in line with the final node** of the element.

The B and C models are then equivalent (in the sense of the limit) to the A modeling when the number of elements used by them tends to infinity. In the present case, 2000 beam elements with a constant circular cross section are used.


Reference quantities and results
-----------------------------------

The reference quantity is the frequency of the modes at rest and for a rotation speed of 4000 :math:`\mathrm{tr}/\mathrm{mn}`. The number of modes calculated is 12.


+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|**Frequency Number**|**Rotation speed** **(** :math:`\mathit{tr}/\mathit{mn}` **)**|**Cadyro calculation with a variable-section tree** :math:`F` **(** :math:`\mathit{Hz}` **)**|
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|1                   |0                                                             |332.14                                                                                       |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |331.51                                                                                       |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|2                   |0                                                             |332.14                                                                                       |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |334.10                                                                                       |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|3                   |0                                                             |1193.81                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |1196.30                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|4                   |0                                                             |1193.81                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |1202.05                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|5                   |0                                                             |1607.19                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |1607.19                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|6                   |0                                                             |2291.51                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |2304.73                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|7                   |0                                                             |2291.51                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |2312.08                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|8                   |0                                                             |2512.22                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |2512.22                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|9                   |0                                                             |3160.39                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |3160.39                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|10                  |0                                                             |3520.23                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |3516.43                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|11                  |0                                                             |3520.23                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |3524.02                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|12                  |0                                                             |4724.14                                                                                      |
+                    +--------------------------------------------------------------+---------------------------------------------------------------------------------------------+
|                    |4000                                                          |4724.14                                                                                      |
+--------------------+--------------------------------------------------------------+---------------------------------------------------------------------------------------------+


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


Code validation file CADYRO 2.0, finite element software for predicting the dynamic behavior of rotors, H-P60-1994-04877-FR.