2. Benchmark solution#
2.1. Calculation method used for the reference solution#
The vibratory behavior of fluid-filled pipes is studied. The pipe is embedded at one end and free at the other end. The pipe cross section is constant and circular. We are interested in the low frequencies of the longitudinal behavior of pipes.
We define:
tube length: |
\(L\) |
Young’s modulus of the pipe: |
\(E\) |
pipe outer diameter: |
\(D\) |
wall thickness: |
\(\mathrm{ep}\) |
area of solid section: |
\({S}_{s}\) |
area of the fluid section: |
\({S}_{f}\) |
speed in the pipe (structure): |
\({c}_{s}\) |
speed in the fluid: |
\({c}_{f}\) |
The characteristics of the fluid and of the pipe were chosen so as to have the following relationship:
\({c}_{f}\mathrm{=}{c}_{s}\mathrm{=}\sqrt{\frac{E}{{\rho }_{s}}}\mathrm{=}c\mathrm{=}1000m\mathrm{/}s\)
In this specific case of equality of velocities, we show [bib2] that the first natural frequency of the coupled problem is such that:
\(\mathrm{tg}(\frac{\omega L}{{c}_{s}})=\sqrt{\frac{{S}_{s}}{{S}_{f}}\mathrm{.}\frac{E}{{\rho }_{f}{c}^{2}}}\)
In this case it is valid: \(f=\mathrm{157,94}\mathrm{Hz}\)
2.2. Benchmark results#
Only one model is used. The calculation of the modes is in formulation \(u,p,\varphi\). There is no reference solution in the case where the cross section is variable (modeling C).
2.3. Uncertainty of the solution#
Analytical solution.
2.4. Bibliographical references#
WAECKEL F., DUVAL C.: Note of principle and use of the pipe elements implemented in the*Code_Aster*. Internal R&D note HP-61/92.138
DUVAL C.: Dynamic response under random excitation in the*Code_Aster*. Internal R&D note HP-61/92.148