3. Algorithm#

After calculating the system’s complex frequencies and modes, the “MODES” list of modes_meca_c is one of the macro’s input data. The algorithm is shown below:

  • Classification of “MODES” modes in bending, twisting and traction/compression:

    • Extraction of flexure modes

    • Normalize the bending modes

    • Extraction of torsional modes

    • Normalize torsional modes

    • Extraction of traction/compression modes

    • Normalize traction/compression modes

  • Extraction of frequencies

    • Bending frequencies

    • Torsional frequencies

    • Traction/compression frequencies

    • Initialization of the frequency connection tables, the filling of these tables corresponds to the “No TRI “mode tracking method, following the order of the mode as it is in the mode_meca_c concepts.

  • Calculation of the direction of direct or inverse precession [1] _

for the bending modes at each speed of rotation, done in two different ways depending on the choice of type of precession calculation:

  • PREC_GOR: Precession identification is based on the sign of the largest orbit in a mode (Direct Precession, Reverse Precession)

  • PREC_MOY: The identification of precession will be based on the sign of the sum of the signs of all the orbits

  • Filling in the precession meaning table SENS

  • Fashion tracking:

  • If the mode tracking type is TRI_PREC_MOD, i.e. sorting frequencies from near to near according to the direction of precession. It requires the calculation of the axes of the orbits and the evaluation of the direction of travel of the orbits. The identification of precession will be based on the sign of the largest orbit or the sign of the sum of the signs of all the orbits. The frequencies are then classified into two groups (direct precession and reverse precession). In each group the frequencies are arranged in ascending order for each speed of rotation. Each curve in the Campbell diagram connects the points according to the speed of rotation respecting the order of ranking in each group (\(i\) th frequency of group \(p\) (speed \({\Omega }_{p}\)) with the \(i\) th frequency of the group \(p+1\) (speed \({\Omega }_{p+1}\)) etc.).

    • Updated flexure connection chart

  • If the type of fashion tracking is TRI_FORM_MOD, that is, sorting by the shape of the modes. Sorting frequencies according to the shape of the modes requires the calculation of the correlation matrix MAC of the modes. The comparison of the modes is then done as follows: each mode (starting from the highest rotation speed) is compared to the other modes of the previous rotation speed by taking the maximum correlation. This comparison makes it possible to draw for each mode the curve of evolution of its frequency as a function of the speed of rotation to constitute the Campbell diagram.

    • Updated flexure connection chart

    • Updated twisting connection chart

    • Updated traction/compression connection table.

  • Plot of the Campbell diagram

    • Plot of the Campbell diagram of the bending modes

    • Plot of the Campbell diagram of torsional modes

    • Plot of the Campbell diagram of traction/compression modes.

  • Tracing slope lines \(S\) and determining the intersection points (critical speeds).

3.1. Classification of fashions#

The rotor is rotated along the \(Z\) axis (this is the condition for the correct use of IMPR_DIAG_CAMPBELL).

After calculating the rotating modes at different speeds, the results are obtained:

  • flexure modes characterized by translational movements \(\mathit{DX}\), \(\mathit{DY}\) and rotation \(\mathit{DRX}\) and \(\mathit{DRY}\)

  • torsional modes characterized by rotational movements \(\mathit{DRZ}\)

  • tensile/compression modes characterized by translational movements \(\mathit{DZ}\).

Initially, the modes are classified according to these three categories.

For each mode, the following calculations are made:

  1. Total standard \(\mathit{Ntot}\)

\(\mathrm{Ntot}=\sqrt{\sum {\mid \mathrm{DX}\mid }^{2}+{\mid \mathrm{DY}\mid }^{2}+{\mid \mathrm{DZ}\mid }^{2}+{\mid \mathrm{DRX}\mid }^{2}+{\mid \mathrm{DRY}\mid }^{2}+{\mid \mathrm{DRZ}\mid }^{2}}\)

  1. Bending ratio \(\mathit{Nflexion}\)

\(\mathit{Nflexion}\mathrm{=}\frac{\sqrt{\mathrm{\sum }{\mathrm{\mid }\mathit{DX}\mathrm{\mid }}^{2}+{\mathrm{\mid }\mathit{DY}\mathrm{\mid }}^{2}+{\mathrm{\mid }\mathit{DRX}\mathrm{\mid }}^{2}+{\mathrm{\mid }\mathit{DRY}\mathrm{\mid }}^{2}}}{\mathit{Ntot}}\)

  1. Torsional ratio \(\mathit{Ntorsion}\)

\(\mathit{Ntorsion}\mathrm{=}\frac{\sqrt{\mathrm{\sum }{\mathrm{\mid }\mathit{DRZ}\mathrm{\mid }}^{2}}}{\mathit{Ntot}}\)

  1. Traction/compression ratio \(\mathit{Nlongi}\)

\(\mathit{Nlongi}\mathrm{=}\frac{\sqrt{\mathrm{\sum }{\mathrm{\mid }\mathit{DZ}\mathrm{\mid }}^{2}}}{\mathit{Ntot}}\)

The classification of the modes is as follows:

  1. For flexure modes: \(\mathit{Nflexion}>\mathit{Ntorsion}\mathit{et}\mathit{Nflexion}>\mathit{Nlongi}\)

  1. For twisting fashions: \(\mathit{Ntorsion}>\mathit{Nflexion}\mathit{et}\mathit{Ntorsion}>\mathit{Nlongi}\)

  1. For traction/compression modes: \(\mathit{Nlongi}>\mathit{Nflexion}\mathit{et}\mathit{Nlongi}>\mathit{Ntorsion}\)

If the system includes a mass (supporting organ), it is possible that there are mixed modes (all three at the same time). In this case, the same method as above is applied.

From the starting modal base \(\mathit{MODES}\), three modal bases are therefore extracted:

  1. \(\mathit{MODEf}\): containing the flexure modes

  2. \(\mathit{MODEt}\): containing the twisting modes

  3. \(\mathit{MODEl}\): containing the tensile/compression modes

For the modal bases \(\mathit{MODEf}\) and \(\mathit{MODEl}\), the normalization is done with respect to the largest of the translation components \((\mathit{DX},\mathit{DY},\mathit{DZ})\).

For the modal base \(\mathit{MODEf}\), normalization is done with respect to the largest of the rotation components \((\mathit{DRX},\mathit{DRY},\mathit{DRZ})\).

The extraction and normalization of the modes of the structure are done through the operators EXTR_MODE and NORM_MODE, respectively.

3.2. Calculation of the direction of precession for flexure modes#

3.2.1. Notions#

The rotating modes are defined by the ellipses at each node.

_images/Object_11.svg

Figure 3.2-a: Ellipses defining modes

For a given mode and for each node, we have the translational movements \(\mathit{DX}\), \(\mathit{DY}\) with:

\(\mathit{DX}\mathrm{=}({X}_{R},{X}_{I}),{\mathrm{\mid }\mathit{DX}\mathrm{\mid }}^{2}\mathrm{=}{\mathrm{\mid }{X}_{R}\mathrm{\mid }}^{2}+{\mathrm{\mid }{X}_{I}\mathrm{\mid }}^{2}\)

\(\mathrm{DY}=({Y}_{R},{Y}_{I}),{\mid \mathrm{DY}\mid }^{2}={\mid {Y}_{R}\mid }^{2}+{\mid {Y}_{I}\mid }^{2}\)

4 scenarios are possible:

  1. If \({\mathrm{\mid }\mathit{DX}\mathrm{\mid }}^{2}+{\mathrm{\mid }\mathit{DY}\mathrm{\mid }}^{2}\mathrm{\approx }0\) there is no ellipse, then \(a=\mathrm{0,}b=\mathrm{0,}\alpha =\mathrm{0,}S=0\)

  2. If \({\mathrm{\mid }\mathit{DX}\mathrm{\mid }}^{2}\mathrm{\approx }0\) there is no ellipse, the trajectory is along the \(Y\) axis, then \(a\mathrm{=}0,b\mathrm{=}\mathrm{\mid }\mathit{DY}\mathrm{\mid },\alpha \mathrm{=}\mathrm{0,}S\mathrm{=}0\)

  3. If \({\mathrm{\mid }\mathit{DY}\mathrm{\mid }}^{2}\mathrm{\approx }0\) there is no ellipse, the trajectory is along the \(X\) axis then \(a\mathrm{=}\mathrm{\mid }\mathit{DX}\mathrm{\mid },b\mathrm{=}0,\alpha \mathrm{=}90°,S\mathrm{=}0\)

  4. If \({\mathrm{\mid }\mathit{DX}\mathrm{\mid }}^{2}\mathrm{\ne }0\) and \({\mathrm{\mid }\mathit{DY}\mathrm{\mid }}^{2}\mathrm{\ne }0\) then we can calculate the precession \(S\) for each node.

The \(u\) movements following \(X\) and \(v\) following \(Y\) of this node will be of the following form:

\(u\mathrm{=}{X}_{R}\mathrm{cos}(\omega t)\mathrm{-}{X}_{I}\mathrm{sin}(\omega t)\) , \(v\mathrm{=}{Y}_{R}\mathrm{cos}(\omega t)\mathrm{-}{Y}_{I}\mathrm{sin}(\omega t)\)

3.2.2. Determining the direction of travel#

During a period of vibration, the ellipse can be traveled in a positive or negative direction (direct or inverse precession).

  1. at \(\omega t\mathrm{=}0\) the rotor axis is at point \(P\)

  2. at \(\omega t\mathrm{=}\frac{\pi }{2}\) the rotor axis is at point \(Q\)

The preceding relationships allow the components of the vectors \(\overrightarrow{\mathit{OP}}\mathrm{=}(\begin{array}{c}{X}_{R}\\ {Y}_{R}\\ 0\end{array})\) and \(\overrightarrow{\mathit{OQ}}\mathrm{=}(\begin{array}{c}\mathrm{-}{X}_{I}\\ \mathrm{-}{Y}_{I}\\ 0\end{array})\) to be calculated. The vector product sign \(\overrightarrow{\mathit{OP}\mathrm{\wedge }\mathit{OQ}}\mathrm{=}(\begin{array}{c}0\\ 0\\ {Y}_{R}{X}_{I}\mathrm{-}{Y}_{I}{X}_{R}\end{array})\) indicates the direction of the path or precession (positive or negative).

For each node, we have the results in maximum amplitude, i.e. :

\(u\mathrm{:}\mathit{XMAX}\mathrm{=}\mathrm{\sum }{X}_{R}^{2}+{X}_{I}^{2}\) and \(v\mathrm{:}\mathit{YMAX}\mathrm{=}\mathrm{\sum }{Y}_{R}^{2}+{Y}_{I}^{2}\)

For a mode, there are two ways to determine the direction of precession:

  • PREC_GOR: Precession is identified according to the sign of the largest orbit in a mode (Direct Precession, Reverse Precession).

  • PREC_MOY: The identification of precession will be based on the sign of the sum of the signs of all the orbits.

At the output, table SENS is filled in for the directions of precession of all the modes at all the speeds of rotation. This table makes it possible to follow the flexure modes with method TRI_PREC_MOD. It also makes it possible to determine the direction of precession on the Campbell diagram according to the color code defined in the user manual for the IMPR_DIAG_CAMPBELL command.

3.3. Calculation of matrices MAC between bases of modes with two speeds of rotation#

To sort the modes by comparison of shape, the correlation coefficients (matrix MAC) between the modes of the two rotation speeds of successive indices are calculated (matrix). We use the MAC_MODES operator for

  • calculate the MAC matrix between two successive modal flexure bases.

  • calculate the MAC matrix between two successive modal torsion bases.

  • calculate the MAC matrix between two successive modal traction/compression bases.

3.4. Sorting according to the direction of precession#

It is a method for tracking modes by sorting frequencies from near to near with verification of the direction of precession. Modes are classified into two groups based on the direction of precession (direct precession mode, reverse precession mode). Monitoring is then carried out in each group according to the order of the modes. We start the follow-up with the last rotation speed of number \(\mathit{NBV}\) and the last mode \(\mathit{NFREQf}\) of this speed (cf. [Figure 3.4-a]).

_images/Object_44.svg

Figure 3.4-a .4-a: Sorting according to the direction of precession

3.5. Sorting by mode shape (MAC)#

This method of tracking modes by comparison of shape requires the calculation of the correlation matrix MAC of the modes of two successive speeds of rotation, see paragraph 3.3.

We start tracking with the last rotation speed of number \(\mathit{NBV}\). For each mode of a given rotation speed \(\mathit{IVI}\), we look for a mode of the previous rotation speed \(\mathit{IVI}\mathrm{-}1\) that has the maximum correlation coefficient. The connection table is filled in with the value of the number of the selected mode (cf. [Figure 3.5-a]).

_images/Object_45.svg

Figure 3.5-a: Sorting by mode shape

This monitoring is done separately for the three categories of flexure, torsional and traction/compression modes.