5. C modeling#

5.1. Characteristics of modeling#

Modeling:

18 Equally Distributed POU_D_E Beam Elements
19 Discrete Elements DIS_TR

5.2. Characteristics of the mesh#

The axis of the beam is oriented according to the vector \((\mathrm{cos}(\mathrm{\pi }/4),\mathrm{sin}(\mathrm{\pi }/4)\mathrm{,0})\).

Mesh:	Number of nodes: 19
	Number of meshes and types: 18 SEG2, 19 POI1

Node names:	Point \(A\) = \(N1\)
	Point \(B\) = \(N19\)

5.3. Mass of discrete elements#

In this modeling, the mass of the beam is not taken into account on the beam elements but on the discrete elements. The mass characteristics to be assigned to each element are calculated in order to be equivalent to the case where the mass is affected on the beam elements.

Item Length: \(e=\frac{L}{18}=0.05m\)

Characteristics of the discrete elements in the base \((t,v,z)\)

Knots \(\mathrm{N2}\) to \(\mathrm{N18}\)	Knots \(\mathrm{N1}\) and \(\mathrm{N19}\)
\(m=\rho e\pi \frac{{D}^{2}}{4}=0.7657\mathrm{kg}\)	\(m\text{'}=\rho \frac{e}{2}\pi \frac{{D}^{2}}{4}=0.3829\mathrm{kg}\)
\({I}_{\text{tt}}=m\text{.}\frac{{D}^{2}}{8}=\mathrm{2,}\text{393}\text{.}{\text{10}}^{-4}\text{kg}\text{.}{m}^{2}\)	\(I{\text{'}}_{\text{tt}}=m\text{'}\text{.}\frac{{D}^{2}}{8}=\mathrm{1,}\text{196}\text{.}{\text{10}}^{-4}\phantom{\rule{0.5em}{0ex}}\text{kg}\text{.}{m}^{2}\)
\({I}_{\text{vv}}={I}_{\text{zz}}=\frac{{I}_{\text{tt}}}{2}+m\text{.}\frac{{e}^{2}}{\text{12}}=\mathrm{2,}\text{791}\text{.}{\text{10}}^{-4}\text{kg}\text{.}{m}^{2}\)	\(I{\text{'}}_{\text{vv}}=I{\text{'}}_{\text{zz}}=\frac{I{\text{'}}_{\text{tt}}}{2}+m\text{'}\text{.}\frac{{e}^{2}}{3}=\mathrm{1,}\text{395}\text{.}{\text{10}}^{-4}\text{kg}\text{.}{m}^{2}\)

Table 5.3-1: Calculating the characteristics of discrete elements

Two solutions are possible to define the characteristics in the database:

or change the base from the local coordinate system from beam \((t,v,z)\) to the global coordinate system \((x,y,z)\). To do this, a base change must be made by rotating the \(z\) axis and the \(–45°\) value. We get:

\(\overline{\overline{I}}=\left[\begin{array}{ccc}{I}_{\mathit{xx}}& {I}_{\mathit{xy}}& 0\\ {I}_{\mathit{xy}}& {I}_{\mathit{yy}}& 0\\ 0& 0& {I}_{\mathit{zz}}\end{array}\right]\) with:

\({I}_{\mathit{xx}}={\mathrm{cos}}^{2}(\mathrm{\pi }/4){I}_{\mathit{tt}}+{\mathrm{sin}}^{2}(\mathrm{\pi }/4){I}_{\mathit{vv}}\)

\({I}_{\mathit{yy}}={\mathrm{sin}}^{2}(\mathrm{\pi }/4){I}_{\mathit{tt}}+{\mathrm{cos}}^{2}(\mathrm{\pi }/4){I}_{\mathit{vv}}\)

\({I}_{\mathit{xy}}=\mathrm{cos}(\mathrm{\pi }/4)\mathrm{sin}(\mathrm{\pi }/4)({I}_{\mathit{tt}}-{I}_{\mathit{vv}})\)

Characteristics of the discrete elements in the base \((x,y,z)\)

Knots \(N2\) to \(N18\)	Knots \(N1\) and \(N19\)
\(m=\rho \text{.}e\text{.}\pi \text{.}\frac{{D}^{2}}{4}=\mathrm{0,}\text{7657}\phantom{\rule{0.5em}{0ex}}\text{kg}\)	\(m\text{'}=\rho \text{.}\frac{e}{2}\text{.}\pi \text{.}\frac{{D}^{2}}{4}=\mathrm{0,}\text{3829}\phantom{\rule{0.5em}{0ex}}\text{kg}\)
\(\begin{array}{c}{I}_{\text{xx}}={I}_{\text{yy}}=\frac{1}{2}\left(m\text{.}\frac{{D}^{2}}{8}+\frac{1}{2}m\text{.}\frac{{D}^{2}}{8}+m\text{.}\frac{{e}^{2}}{\text{12}}\right)\\ \phantom{\rule{7em}{0ex}}=\mathrm{2,}\text{592}\text{.}{\text{10}}^{-4}\phantom{\rule{0.5em}{0ex}}\text{kg}\text{.}{m}^{2}\end{array}\)	\(\begin{array}{c}I{\text{'}}_{\text{xx}}=I{\text{'}}_{\text{yy}}=\frac{1}{2}\left(m\text{'}\text{.}\frac{{D}^{2}}{8}+\frac{1}{2}m\text{'}\text{.}\frac{{D}^{2}}{8}+m\text{'}\text{.}\frac{{e}^{2}}{\text{12}}\right)\\ \phantom{\rule{7em}{0ex}}=\mathrm{1,}\text{296}\text{.}{\text{10}}^{-4}\phantom{\rule{0.5em}{0ex}}\text{kg}\text{.}{m}^{2}\end{array}\)
\({I}_{\text{zz}}=\frac{{I}_{\text{tt}}}{2}+m\text{.}\frac{{e}^{2}}{\text{12}}=\mathrm{2,}\text{792}\text{.}{\text{10}}^{-4}\phantom{\rule{0.5em}{0ex}}\text{kg}\text{.}{m}^{2}\)	\(I{\text{'}}_{\text{zz}}=\frac{I{\text{'}}_{\text{tt}}}{2}+m\text{'}\text{.}\frac{{e}^{2}}{\text{12}}=\mathrm{1,}\text{396}\text{.}{\text{10}}^{-4}\phantom{\rule{0.5em}{0ex}}\text{kg}\text{.}{m}^{2}\)
\(\begin{array}{c}{I}_{\text{xy}}={I}_{\text{yx}}=\frac{1}{2}\left\{m\text{.}\frac{{D}^{2}}{8}-\left(\frac{1}{2}m\text{.}\frac{{D}^{2}}{8}+m\text{.}\frac{{e}^{2}}{\text{12}}\right)\right\}\\ \phantom{\rule{7em}{0ex}}=-\mathrm{1,}\text{994}\text{.}{\text{10}}^{-5}\phantom{\rule{0.5em}{0ex}}\text{kg}\text{.}{m}^{2}\end{array}\)	\(\begin{array}{c}I{\text{'}}_{\text{xy}}=I{\text{'}}_{\text{yx}}=\frac{1}{2}\left\{m\text{'}\text{.}\frac{{D}^{2}}{8}-\left(\frac{1}{2}m\text{'}\text{.}\frac{{D}^{2}}{8}+m\text{'}\text{.}\frac{{e}^{2}}{\text{12}}\right)\right\}\\ \phantom{\rule{7em}{0ex}}=-\mathrm{9,}\text{971}\text{.}{\text{10}}^{-6}\phantom{\rule{0.5em}{0ex}}\text{kg}\text{.}{m}^{2}\end{array}\)

Table 5.3-2: Calculating the characteristics of discrete elements

or declare the characteristics in the local coordinate system of the beam and use the nautical angles to define the orientation of the local coordinate system. This method is used for modeling E.

3.3 Tested quantities and results

Rotor stopped (\(\Omega =0\)) (frequencies in \(\mathrm{Hz}\))

Identification	Reference type	Reference value	Tolerance
Mode 1	“ANALYTIQUE”	122.7475	0.1%
Mode 2	“ANALYTIQUE”	490.9899	0.1%
Mode 3	“ANALYTIQUE”	1104,7273	0.1%
Mode 4	“ANALYTIQUE”	1963,9596	0.1%

Calculation of natural frequencies using the Sorensen algorithm

Rotating rotor (\(\Omega ={10}^{4}{\mathrm{rd.s}}^{-1}\)), direct modes (frequencies in \(\mathrm{Hz}\))

Identification	Reference type	Reference value	Tolerance
Mode 2	“ANALYTIQUE”	125.8150	0.1%
Mode 4	“ANALYTIQUE”	503,2598	0.1%
Mode 6	“ANALYTIQUE”	1132,3346	0.1%
Fashion 9	“ANALYTIQUE”	2013.0393	0.1%

Rotating rotor (\(\Omega ={10}^{4}{\mathrm{rd.s}}^{-1}\)), retrograde modes (frequencies in \(\mathrm{Hz}\))

Identification	Reference type	Reference value	Tolerance
Mode 1	“ANALYTIQUE”	119.7548	0.1%
Mode 3	“ANALYTIQUE”	479.0191	0.1%
Mode 5	“ANALYTIQUE”	1077,7931	0.1%
Mode 7	“ANALYTIQUE”	1916.0765	0.1%