B modeling ============== Characteristics of modeling ----------------------------------- It is a system of shafts rotating along the :math:`Z` axis with negative rotation speeds. To obtain the same results as modeling A (with the minus sign), you must put a minus sign on the cross terms of the stiffness and damping matrices. The characteristics of the bearings are indicated in the following table. .. csv-table:: "Level", ":math:`\mathrm{P1}` "," :math:`{K}_{\mathrm{yy}}={8.10}^{7}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}{1.10}^{8}N\mathrm{/}m`" "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}{1.10}^{7}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}{6.10}^{7}N\mathrm{/}m`" "", "", ":math:`{C}_{\mathrm{yy}}=8.{10}^{3}\mathrm{Ns}/m` "," :math:`{C}_{\mathit{xx}}\mathrm{=}1.2{10}^{4}\mathit{Ns}\mathrm{/}m`" "", "", ":math:`{C}_{\mathit{yx}}\mathrm{=}3.{10}^{3}\mathit{Ns}\mathrm{/}m` "," :math:`{C}_{\mathit{xy}}\mathrm{=}{3.10}^{3}\mathit{Ns}\mathrm{/}m`" "", "", "", "" "Level", ":math:`\mathrm{P2}` "," :math:`{K}_{\mathrm{yy}}=5.{10}^{7}N/m` "," :math:`{K}_{\mathit{xx}}\mathrm{=}7.{10}^{7}N\mathrm{/}m`" "", "", ":math:`{K}_{\mathit{yx}}\mathrm{=}2.{10}^{6}N\mathrm{/}m` "," :math:`{K}_{\mathit{xy}}\mathrm{=}4.{10}^{7}N\mathrm{/}m`" "", "", ":math:`{C}_{\mathrm{yy}}=6.{10}^{3}\mathrm{Ns}/m` "," :math:`{C}_{\mathit{xx}}\mathrm{=}8.{10}^{3}\mathit{Ns}\mathrm{/}m`" "", "", ":math:`{C}_{\mathit{yx}}\mathrm{=}1.5{10}^{3}\mathit{Ns}\mathrm{/}m` "," :math:`{C}_{\mathit{xy}}\mathrm{=}1.5{10}^{3}\mathit{Ns}\mathrm{/}m`" Therefore, the precessions of the modes are also reversed, that is, the direct modes become retrograde and vice versa. Characteristics of the mesh ---------------------------- The rotor is meshed in 12 finite shaft elements of type POU_D_T and includes 4 discrete elements of type DIS_TR for modeling disks and bearings. Number of knots: 13 Number and type of elements: 12 SEG2 4 POI1 .. image:: images/10000000000003B00000025688CF9E6548A77A18.png :width: 4.5811in :height: 2.9083in .. _RefImage_10000000000003B00000025688CF9E6548A77A18.png: **Figure 1-b: Characteristic of the finite element model under ROTORINSA** Tested sizes and results ------------------------------ Natural frequencies as a function of rotation speed ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The values of the first 8 bending frequencies for speeds :math:`0\mathrm{tr}/\mathrm{mn}` and :math:`-60000\mathrm{tr}/\mathrm{mn}`, for both software programs, are shown in the table below. +--------------------------+---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ |**Freq number in flexion**|**Rotation speed** **(** :math:`\mathrm{tr}/\mathrm{min}` **)**|**ROTORINSA** |**Aster_code** | + + +------------------------+------------------+------------------------------------------+----------------------------------+ | | |:math:`∣F∣(\mathrm{Hz})`|**Damping factor**|**Tolerances of** :math:`∣F∣(\mathrm{Hz})`|**Damping tolerances** **reduced**| +--------------------------+---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ |A1 |0 |2.16212E+02 |4.76544E-02 |1.E-3 |1.E-3 | + +---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ | |-60000 |1.85365E+02 |-5.17463E-02 |1.E-3 |1.1E-3 | +--------------------------+---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ |2 |0 |2.63539E+02 |7.87281E-02 |1.E-3 |6.E-3 | + +---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ | |-60000 |2.96078E+02 |1.55245E-01 |1.E-3 |5.E-3 | +--------------------------+---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ |3 |0 |3.83210E+02 |5.01438E-02 |1.E-3 |14.E-3 | + +---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ | |-60000 |3.24718E+02 |1.57489E-03 |1.E-3 |70.E-3 | +--------------------------+---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ |4 |0 |4.39642E+02 |6.02275E-02 |1.E-3 |12.E-3 | + +---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ | |-60000 |4.72541E+02 |1.59683E-01 |1.2E-3 |3.E-3 | +--------------------------+---------------------------------------------------------------+------------------------+------------------+------------------------------------------+----------------------------------+ **Table 2-a: Flexion-type natural frequencies for Code_Aster and ROTORINSA** The frequencies obtained are in perfect harmony with those of ROTORINSA. There is an instability of the first mode, which appears at :math:`-16760\mathrm{tr}/\mathrm{mn}`. In Code_Aster, we also observe frequencies and modes of torsion and modes of traction/compression. These modes are not calculated by ROTORINSA, as it only models bending behavior. The values of these frequencies are tested in NON_REGRESSION and only when stopped. In fact, the modes of twisting and pulling are, by definition, invariant with respect to the speed of rotation.