3. Modeling A

3.1. Characteristics of modeling

The springs and point masses are modelled by discrete elements with 3 degrees of freedom DIS_T:

Node \(\mathit{NO1}\) is embedded and subjected to an imposed acceleration \(\gamma (t)\). We calculate the relative displacement of node \(\mathit{NO4}\).

Modal synthesis calculations

The complete basis of clean modes is considered. The time integration is carried out with Newmark, Euler and Devogelaere algorithms with a time step of \(\mathrm{0,001}s\). The calculations are archived every time step.

We consider reduced damping \({\xi }_{i}\) to zero for all the calculated modes.

The load is taken into account as a vector projected on the modal basis EXCIT :( VECT_GENE) or as a modal component EXCIT :( NUME_MODE) or both at the same time.

Direct calculations

The time integration is carried out either with the Newmark algorithm or with the explicit algorithm of centered differences with a time step of \(\mathrm{0,001}s\). The calculations are archived every ten steps.

Note:

Since the centered differences scheme can only be used with a diagonal mass matrix, elementary matrices are calculated with the option MASS_MECA_DIAG in the CALC_MATR_ELEM operator. *

Taking into account an initial status

In both types of calculation, it is verified that the relative displacement obtained from a calculation carried out at once is identical to that obtained in several times, that is to say by considering as the initial state, the result of the last calculated time step:

ETAT_INIT =_F (RESULTAT...) for a calculation by modal synthesis;

ETAT_INIT =_F (DEPL...

VITE..) for a direct calculation.

Taking into account the modes overlooked by static correction:

A modal base consisting of the first two natural modes is considered and completed by a mode corresponding to the static response of the studied system to a unit loading of the force imposed in the \(–x\) direction (keywords MODE_CORR and CORR_STAT in the operator DYNA_VIBRA).

3.2. Characteristics of the mesh

Number of knots: 4

Number of meshes and types: 3 DIS_T

3.3. Tested features

In particular, the consideration of an initial state and static corrections are tested.

3.4. Tested sizes and results

Natural frequencies (in \(\mathrm{Hz}\)) of the system:

Mode Number	Analytic
1	2,239
2	6,275
3	9.069

Relative displacement values for node \(\mathrm{NO4}\) for various times:

Transient calculation by modal synthesis

We test the consideration of a load in the form of a vector projected on a modal basis, in the form of a modal component, in the form of a projected vector and a modal component simultaneously as well as the taking into account of the neglected modes.

Time \((s)\)	Reference	Code_Aster
Generalized vector load
Newmark algorithm »	Relative error %	Code_Aster	Code_Aster
Modal component load
Euler Algorithm »	Relative error %
0.02	—2,700E—03	—2,680E—03	—2,680E—03	—2,660E—03	—1,481
0.04	—4,260E—02	—4,272E—02	—4,272E—02	—4,264E—02	0.091
0.05	—1.041E—01	—1.042E—01	0.134	—1.041E—01	0.015
0.06	—2,158E—01	—2,161E—01	—2,121	—2,159E—01	,	0.038
0.08	—6,813E—01	—6,819E—01	—6,813E—01	—6,816E—01	0.049
0.10	—1.658E+00	—1.659E+00	0.082	—1.659E+00	0.055	0.055

Load type	Time \((s)\)	Reference	Code_Aster	Relative error %
	0.02	—5,400E—03	—5,320E—03	—1,482
Generalized vector	0.04	—8.520E—02	—8.528E—02	0.091
and	0.05	—2.082E—01	—2.082E—01	0.015
modal component	0.06	—4.316E—01	—4.318E—01	0.038
simultaneously	0.08	—1.363E+00	—1.363E+00	0.049
(Euler)	0.10	—3,316E+00	—3,318E+00	0.055
	0.02	—4,000E—03	—3,985E—03	—0,373
Generalized vector	0.04	—4.640E—02	—4.640E—02	0.01
Devogelaere	0.05	—1.085E—01	—1.086E—01	0.084
(plus correction	0.06	—2.203E—01	—2.204E—01	0.039
static)	0.08	—6,842E—01	—6,843E—01	0.021
	0.10	—1.659E+00	—1.659E+00	0.026

Results with an incomplete modal basis without static correction are not tested. The benefits of static correction are illustrated below:

Displacement of node \(\mathrm{NO4}\) (in meters) as a function of time
	Full base		Incomplete database without static correction		Incomplete database with static correction

A non-regression test is done to ensure the correct functioning of the POST_GENE_PHYS command following the correction of a bug.

Time \((s)\)	Identification	Reference type	Reference value	Precision
0, 1	Field DEPL, DX Component, DX Component, Node N02	“NON_REGRESSION”		1E-6

Direct transient calculation

The displacements calculated at node \(N04\) are compared according to various integration schemes:

Time \((s)\)	Reference	Code_Aster
Newmark schema »	Relative error %	Code_Aster
Diagram of centered differences »	Relative error %
0.02	—2,700E—03	—2,680E—03	—2,680E—03	—2,660E—03	—1,482
0.04	—4,260E—02	—4,272E—02	—4,272E—02	—4,264E—02	0.091
0.05	—1.041E—01	—1.042E—01	0.134	—1.041E—01	0.015
0.06	—2,158E—01	—2,161E—01	—2,121	—2,159E—01	,	0.038
0.08	—6,813E—01	—6,819E—01	—6,819E—01	—6,745E—01	—1,004
0.10	—1.658E+00	—1.659E+00	0.082	—1.645E+00	—0.803

Taking into account an initial state:

As expected, the relative movements calculated at once are strictly identical to those obtained by considering the result of the last calculated time step as the initial state.

Non-regression tests on the energy balance are also carried out.