3. Modeling A

3.1. Characteristics of modeling

We use the Euler Bernouilli POU_D_E beam element

3 beams:

\(\mathrm{ABC}\), \(\text{DEF}\), \(\mathrm{HGI}\) each cut into 10 meshes SEG2 Nodes \((B,H)\) and \((E,I)\) have the same coordinates.

Boundary conditions:
beams \(\mathrm{ABC}\) and \(\text{DEF}\)
DDL_IMPO: beam \(\mathrm{HGI}\) knots ends	( GROUP_NO: (PABC, PDEF) DX: 0., DY: 0., DRY: 0. ) ( GROUP_NO: (PHGI) DX: 0., DY: 0., DRX: 0. ) ( GROUP_NO: (NACDF) DZ: 0. )
DDL_link: nodal_force:	\({\mathrm{DZ}}_{B}–{\mathrm{DZ}}_{H}=0.\) and \({\mathrm{DZ}}_{E}–{\mathrm{DZ}}_{I}=0.\) Node: \(G\) \(\mathrm{Fz}\): \(–1.E5\)

Node names:	\(A=\mathrm{N1}\)	\(B=\mathrm{N6}\)	\(C=\mathrm{N11}\)
	\(D=\mathrm{N21}\)	\(E=\mathrm{N26}\)	\(F=\mathrm{N31}\)
	\(H=\mathrm{N41}\)	\(G=\mathrm{N46}\)	\(I=\mathrm{N51}\)

3.2. Characteristics of the mesh

Number of knots:	33
Number of meshes and types:	3*10 = 30 SEG2

3.3. notes

The blocking of the degrees of freedom \(\mathrm{DX}\) and \(\mathrm{DY}\) at all the nodes makes it possible to select only the transverse flexure modes (in the « vertical » plane).

3.4. Tested sizes and results

Frequency ( \(\mathrm{Hz}\) )

Clean mode order	Reference	Aster	% difference
1 2	16.456 38.165	16.4190 38.0468	—0.22 —0.31

Clean mode: value of \({W}_{B}/{W}_{G}\)

Symmetric eigenmode order	Reference	Aster*	% difference
1 2	1.213 —0.412	1.213 —0.412	0. 0.

	\({W}_{B}=\mathrm{DZ}\) in \(B\) (\(\mathrm{N6}\))	\({W}_{G}+{W}_{B}=\mathrm{DZ}\) in \(G\) (\(\mathrm{N46}\))
mode 1:	\({W}_{B}=0.5480\)	\({W}_{G}+{W}_{B}=1.\)
mode 2:	\({W}_{B}=–0.6698\)	\({W}_{G}+{W}_{B}=0.9559\)

Harmonic response:

Point	Value type ( \(m\) )	Reference	Aster	% difference
\(B,E\) \(G\) \(G\)	\({W}_{B}\mathrm{max}\) \({W}_{G}\mathrm{max}\ast\) \({W}_{B}+{W}_{G}\mathrm{max}\)	—0.098 —0.125 —0.227	—0.1003 —0.1271 —0.2274	2.45 1.60 0.18

3.5. notes

Calculations made by:

CALC_MODES

OPTION = “PLUS_PETITE” CALC_FREQ =_F (NMAX_FREQ = 3) SOLVEUR_MODAL =_F (METHODE = “TRI_DIAG”)

An antisymmetric mode is obtained for a frequency \(f=22.5676\mathrm{Hz}\). This natural frequency depends on the torsional constant provided; this is not defined in the reference data.

The \({W}_{B}/{W}_{G}\) values are not verified in the test but are obtained manually from \({W}_{B}\) and \({W}_{G}+{W}_{B}\).

The (WG) max value is not checked in the test. We only have access to \({W}_{B}\mathrm{max}\) and \(({W}_{B}+{W}_{G})\mathrm{max}\). \({W}_{G}\mathrm{max}\) is obtained manually by difference.

Contents of the results file:

First 3 natural frequencies, displacement of the \(B,E,G\) nodes in harmonic response.