7. E modeling#

7.1. Characteristics of modeling#

Modeling:

  • 18 equally-distributed POU_D_E beam elements

  • 19 Discrete Elements DIS_TR

7.2. 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 \(B\) = \(N19\)

7.3. 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 7.3-1: Calculating the characteristics of discrete elements

In this modeling, unlike modeling C, the characteristics are declared in the local coordinate system of the beam and the nautical angles are used to define the orientation of the local coordinate system.

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%