2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Direct modal calculation (without dynamic substructuring)
Calculation of natural modes in air:
With the option “BANDE” of the operator CALC_MODES the first 4 natural frequencies of the air system (mass-spring system) are calculated:
mode 1: |
vibration of the two cylinders in phase according to \(\mathrm{Ox}\) |
mode 2: |
vibration of cylinder No. 2 according to \(\mathrm{Oy}\) (on the right) |
mode 3: |
vibration of the two cylinders in phase opposition according to \(\mathrm{Ox}\) |
mode 4: |
vibration of cylinder No. 1 according to \(\mathrm{Oy}\) (on the left) |
These modes can be determined analytically [bib1].
The Code_Aster calculation provides for natural frequencies in air:
mode 1: |
\({f}_{1}=17.3555\mathrm{Hz}\) |
mode 2: |
\({f}_{2}=18.2034\mathrm{Hz}\) |
mode 3: |
\({f}_{3}=42.6760\mathrm{Hz}\) |
mode 4: |
\({f}_{4}=57.5418\mathrm{Hz}\) |
Calculation of the mass matrix added on a modal basis:
On this modal basis, we calculate the 4th order added mass matrix with the operator CALC_MATR_AJOU [U4.55.10] option “MASS_AJOU” keyword MODE_MECA (lower triangular terms):
\(\mathrm{m11}=300.67\mathrm{kg}/m\) |
|
\(\mathrm{m13}=269.98\mathrm{kg}/m\) |
|
\(\mathrm{m22}=269.98\mathrm{kg}/m\) |
|
\(\mathrm{m24}=31.05\mathrm{kg}/m\) |
|
\(\mathrm{m34}=–0.011\mathrm{kg}/m\) |
|
Adding this matrix to the generalized mass matrix:
The matrix thus determined is added to the generalized mass matrix (operator COMB_MATR_ASSE [U4.53.01]) and then the natural frequencies of the immersed structure are calculated with the operator CALC_MODES option “PLUS_PETITE” [U4.52.02].
The calculation finds the following natural frequencies:
mode 1: |
\(f{\text{'}}_{1}=15.8782\mathrm{Hz}\) |
mode 2: |
\(f{\text{'}}_{2}=16.7811\mathrm{Hz}\) |
mode 3: |
\(f{\text{'}}_{3}=39.0389\mathrm{Hz}\) |
mode 4: |
\(f{\text{'}}_{4}=53.0488\mathrm{Hz}\) |
2.2. Benchmark results#
Natural frequencies determined by Code_Aster in a direct calculation.
2.3. Bibliographical references#
J GIBERT - Vibrations of Structures. Interactions with fluids. Eyrolles (1988).