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From the same paper as the previous 2 slides.
Figure 13 shows the ABAQUS predicted axial strain response at strain gage location B1 when using the coarse (#1) and refined (#2) acoustic meshes. The response curves are very close both in magnitude and phasing. The close correlation between the two analyses was also apparent at the other strain gage locations. This indicates that for the applied UNDEX loading the structural response times are long when compared to the reflected wave oscillations obtained in the acoustic tube validation analyses. This result was predictable when considering an eigenvalue analysis for the cylinder with no external fluid. The modes that have the greatest potential for producing damage have frequencies well below 1500 Hz, and will be further reduced when the cylinder is submerged due to the added mass effect. The cylinder modes have response periods that are significantly longer than the shock wave rise time or reflected wave pressure oscillations. Thus, for this particular example, using an acoustic mesh and solution time increment that reasonably captures the shock wave reflected pulse and can represent the scattered acoustic waves at the structural response frequencies is adequate for obtaining a good solution.
The response shown in Figure 13 is dominated by the fundamental beam bending mode of the cylinder, for which the dominant motion is transverse to the cylinder axis. At any point along the cylinder axis the motion is dominated by a translation of the cross section through the fluid, similar to the motion used in the infinite cylinder modeling study. The only damping mechanisms in the analyses were due to acoustic radiation and the /Explicit default values for element bulk viscosity. The acoustic model does not include any losses due to hydrodynamic drag (fluid viscosity) associated with the motions of the cylinder.
The effect of hydrodynamic drag on the late time response of the cylinder is clearly shown in Figure 14, where the predicted axial strain response is compared to the experimental data. The experimental data was digitized from a published curve (Kwon & Fox, 1993), and was shifted by 0.2 milliseconds in order to align the experimental and analysis time axes.
The solution designated as ALPHA = 0, represents the original analysis, whereas the analysis designated as ALPHA=750 utilized mass proportional damping (10% critical at 600 Hz) applied to the cylinder as an approximation for the effects of hydrodynamic drag. The application of ALPHA damping does not have an adverse effect on the solution critical time increment. ALPHA damping does not significantly affect the early time response (high frequency), but does significantly reduce the late time response (low frequency). This is consistent with what is observed with the experimental data. In any event, ignoring hydrodynamic drag in an UNDEX analysis will produce conservative (high) levels for the structural response, which is often a desirable trait when doing a design evaluation analysis.
CONCLUSIONS:
ABAQUS/Explicit provides an efficient means to evaluate the transient response of structural-acoustic systems loaded by external acoustic sources. This was illustrated with the analysis of a submerged cylinder acted upon by a shock wave generated by an underwater explosion. The modeling studies presented in this paper indicate that sufficient accuracy for a submerged structure’s response can be obtained when positioning the external absorbing boundary of the acoustic domain a distance from the structure of between 1/3 to 1/2 the longest characteristic structural wavelength. Modeling studies also indicated that the degree of refinement in the acoustic domain mesh can be tailored to the characteristics of the shock pulse and the nature of structural response, i.e., short vs. long response times as compared to the shock pulse transient.
Reference:
Kwon, Y.W. and P.K. Fox, “Underwater Shock Response of a Cylinder Subjected to a Side-On Explosion,” Computers and Structures, Vol. 48, No. 4, 1993.
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