3. Modeling A

3.1. Characteristics of modeling

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

Node \(\mathrm{NO1}\) is embedded and subjected to an imposed acceleration \(\gamma (t)\).

3.2. Characteristics of the mesh

Number of knots: 4

Number of meshes and types: 3 DIS_T

3.3. Tested sizes and results

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

Mode Number	Code_Aster
1	0.94853
2	2.53344
3	5,30513

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 modes neglected by the static correction.

The static correction is taken into account according to the following various possibilities:

static correction*a posteriori;

static correction*a priori, by completing the base of dynamic modes with static modes at imposed force, with re-orthogonalization of the base thus completed;

static correction*a priori, by completing the dynamic mode base by static modes with imposed acceleration (pseudo-mode), with re-orthogonalization of the base thus completed;

static correction*a priori, by completing the base of dynamic modes by static modes with imposed force and static modes with imposed acceleration (pseudo-mode), with re-orthogonalization of the base thus completed.

Values for the relative displacement of node \(\mathit{NO}4\) at time \(t=\mathrm{19,4}s\) (m):

modal basis	Reference	Tolerance
truncated base	-0.011301	0.1%
full base	-0.011301	1.e -04%
static correction*a posteriori*	-0.011301	0.1%
static correction*a priori*, pseudo-mode, with re-orthogonalization	-0.011301	1.e -04%
static correction*a priori*, static force-imposed modes, with re-orthogonalization	-0.011301	1.e -04%
static correction*a priori*, pseudo-mode, static force-imposed modes, with re-orthogonalization	-0.011301	1.e -04%

Values for the relative displacement of node \(\mathit{NO}2\) at time \(t=\mathrm{19,4}s\) (m):

modal basis	Reference	Tolerance
truncated base	7.33162E-04	123.0%
full base	7.33162E-04	1.0E -04%
static correction*a posteriori*	7.33162E-04	19.5%
static correction*a priori*, pseudo-mode, with re-orthogonalization	7.33162E-04	1.0E -04%
static correction*a priori*, static force-imposed modes, with re-orthogonalization	7.33162E-04	1.0E -04%
static correction*a priori*, pseudo-mode, static force-imposed modes, with re-orthogonalization	7.33162E-04	1.e -04%

The value of static correction has been illustrated above: as expected when reading modal deformations, static correction is not visible for \(\mathit{NO4}\) but plays an important role for \(\mathrm{NO2}\). Without static correction in \(\mathit{NO}2\), the displacement is out of phase and its amplitude is reduced by \(\text{50\%}\). In \(\mathit{NO}2\), with static correction a posteriori, the error remains visible (less than \(\text{20\%}\)), but the calculation remains realistic; with static correction a priori, regardless of the nature of the static modes considered (with acceleration or with imposed force), the modal truncation error is completely compensated for, making it possible to find exactly the reference displacement. On a less caricatural calculation, the error will be less noticeable. We also note that the magnitude of the displacement at node \(\mathrm{NO2}\) is two orders smaller than that at node \(\mathrm{NO4}\).