2. Reference solution#
The objective of the test case is to show the effect of modal truncation on a very simple example of seismic calculation and to illustrate the value of static correction, whether it is carried out a priori by adding static modes at the base of dynamic modes, or a posteriori. The reference solution is the transient calculation on a complete modal basis carried out with the operator DYNA_VIBRA.
The spring mass system has three degrees of freedom. It is therefore associated with three modes whose calculated frequencies are:
\({f}_{1}=\mathrm{0,946}\mathrm{Hz}\), \({f}_{2}=\mathrm{2,533}\mathrm{Hz}\), and \({f}_{3}=\mathrm{5,305}\mathrm{Hz}\).
The excitation frequency of the harmonic seismic signal was chosen at \({f}_{\mathrm{ex}}=2\mathrm{Hz}\) to keep only the first two modes in the modal base. The rule of retaining modes up to twice the maximum excitation frequency is respected.
The table of effective modal unit masses gives interesting information:
Effective unit mass |
|||
NUME_MODE |
FREQUENCE |
MASS_EFFE_UN_DX |
CUMUL_DX |
1 |
9.48538E-01 |
6.82972E-01 |
6.82972E-01 |
2 |
2.53344E+00 |
5.03369E-02 |
7.33309E-01 |
3 |
5.30513E+00 |
2.66691E-01 |
1.00000E+00 |
It is observed that the third mode will have a negligible dynamic response because its natural frequency is equal to \({f}_{3}\mathrm{=}\mathrm{5,305}\mathit{Hz}\), beyond \(2\mathrm{\times }{f}_{\mathit{ex}}\mathrm{=}\mathrm{4,0}\mathit{Hz}\). On the other hand, its effective unit modal mass in the direction \(x\) is equal to \(\text{26,7 \%}\). It is therefore not negligible and this mode can have an influence on the response of the mass-spring system through its quasi-static contribution. It is the aim of static correction to take it into account.
If we now look at the geometry of mode 3, we can see that it is mainly on node \(\mathrm{N02}\) that we will be able to observe the effect of the mode and therefore the effect of static correction.
Mode 1 |
Mode 2 |
Mode 3 |
||
Knot |
\(\mathrm{DX}\) |
|
|
|
\(\mathrm{N01}\) (point \(A\)) |
0.00000E+00 |
0.00000E+00 |
0.00000E+00 |
0.00000E+00 |
\(\mathrm{N02}\) (point \(B\)) |
5.08430E-02 |
9.84653E-02 |
9.93841E-01 |
|
\(\mathrm{N03}\) (point \(C\)) |
5.41213E-01 |
8.33623E-01 |
-1.10279E-01 |
-1.10279E-01 |
\(\mathrm{N04}\) (point \(D\)) |
8.39347E-01 |
-5.43487E-01 |
1.09069E-02 |
Indeed, upon reading the table of modal components, it appears that the mode 3 component on all the nodes is in the minority compared to the other modes, except for node \(\mathrm{N02}\).
2.1. Benchmark results#
The results given by DYNA_VIBRA with the complete modal base at nodes \(\mathrm{N02}\) and \(\mathrm{N04}\) at time \(t=\mathrm{19,4}s\) are taken as reference results.
2.2. Uncertainty about the solution#
Clarification on the integration of time in DYNA_VIBRA