3. Modeling B — Direct Transient Analysis#

3.1. Problem and sketch of the calculation#

We now want to calculate the movement of the plate on its vibrating table. We start with the most natural calculation, the direct calculation on a physical basis. Two calculations are carried out: without and with Rayleigh damping type.

An essential question lies in modeling the loading on the plate. Here, we choose to use a force of gravity, intensity and direction corresponding to the settings of the vibrating table.

The main steps of the command file are:

  • start:* DEBUT *

  • reading the mesh:* LIRE_MAILLAGE *

  • model construction:* AFFE_MODELE *

  • definition of materials:* DEFI_MATERIAU *

  • assignment of materials to the different groups of the mesh:* AFFE_MATERIAU *

  • boundary conditions:* AFFE_CHAR_MECA *

  • load definition:* AFFE_CHAR_MECA (PESANTEUR =_F (GRAVITE =300. , DIRECTION =( -1,0.1)) *

  • construction of the mass and stiffness matrices and the second member vector:* ASSEMBLAGE *

  • definition of time modulation (sine of period \(15\times 2\mathrm{\pi }\) seconds):* FORMULE *

  • transitory analysis:* DYNA_VIBRA (TYPE_CALCUL =” TRAN “, BASE_CALCUL =” PHYS”) *

  • We study the response of the structure up to \(0.5s\) with a time step of \(0.002s\) (to be entered under the keyword* INCREMENT *)

  • recovery of the movement at a point:* RECU_FONCTION *

  • curve printing:* IMPR_FONCTION *

3.2. Results#

Let’s see the results curve:

_images/deplacement_P_DZ_transitoire_direct_sans_amor.png

Fig. 3.1 Vertical displacement (DZ) in P#

A sinusoidal response to the excitation frequency is observed. However, this sinusoidal response is strongly modulated by the response of the first natural mode of the plate.

3.3. Taking depreciation into account#

It is now proposed to take depreciation into account in the context of the calculation carried out previously. To do this, we consider the Rayleigh proportional damping model, with the following parameters:

  • \(\alpha = 0.00011575 s\): value to be entered in* AMOR_ALPHA in DEFI_MATERIAU *;

  • \(\beta = 1.142397 {s}^{-1}\): value to be entered in* AMOR_BETA in DEFI_MATERIAU *.

These parameters correspond to a target reduced damping of 2%, over the frequency range [5; 50] Hz.

Once these parameters have been entered in the DEFI_MATERIAU operator, it is necessary to assemble the damping matrix via the ASSEMBLAGE macro command (option AMOR_MECA). The transitory calculation with amortization can then be carried out via the DYNA_VIBRA operator, by entering the amortization matrix assembled via the MATR_AMOR option.

The movements obtained are then compared with those obtained without damping. The temporal evolution of the DZ component of displacements with and without Rayleigh damping is shown below:

_images/Compa-depl_P_DZ_transitoire_direct_avec_et_sans_amor.png

Fig. 3.2 Comparison of vertical displacement between two calculations: without and with Rayleigh damping#