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Chapter 5
Vibration of Structures in Quiescent Fluids—II Simplified Methods

## Excerpt

In the special case when only one of the cylinders is flexible, the “in-water” natural frequencies of the shell can be obtained from the “in-air” frequencies by simple rationing:

$f̄n=fnairμμ+M̑H$

As shown in Chapter 4, one of the major tasks of calculating the hydrodynamic mass is the calculation of the hydrodynamic mass component h. In this chapter, simplified expressions for calculating h in special cases are given. Of these, the most commonly used in the industry is the “slender cylinder” approximation:

$hαna+ρanb2n+a2nb2n−a2n hαnab+−2ρanb2n+a2nb2n−a2nhαnba+−2ρanb2n+a2nb2n−a2nhαnb+ρbnb2n+a2nb2n−a2n$

However, for application to large-shell structures commonly encountered in the power and process industries, Equation (5.2) often overestimates the hydrodynamic masses by as much as a factor of two or more. In addition, these equations give the hydrodynamic mass components in the form of surface densities. The effective hydrodynamic mass still has to be computed based on Equation (4.50). Failure to observe this has lead to inconsistencies in the hydrodynamic mass formulation of coupled fluid-shell problems.

• Summary
• Nomenclature
• 5.1 Introduction
• 5.2 One Cylinder Flexible
• 5.3 The Slender Cylinder Approximation
• Example 5.1
• 5.4 Incompressible Fluid Approximation
• 5.5 Equation of Fritz and Kiss
• Example 5.2
• 5.6 The Ripple Approximation
• Example 5.3: PWR Beam-Mode Vibration
• 5.7 Single Cylinder Containing Fluid
• 5.8 Single Cylinder in Infinite Fluid
• 5.9 Consistency of the Hydrodynamics Mass Formulation
• Single Infinite Cylinder Containing Liquid Undergoing Rectilinear Motion
• Inertia Load due to Hydrodynamics Mass
• The Enclosed Cavity Paradox
• 5.10 Hydrodynamic Masses for Other Geometries
• Square Cylinders
• Sloshing Problems
• Hydrodynamic Mass for Tube Bundles
• 5.11 Hydrodynamic Damping
• Example 5.4: Hydrodynamic Damping in Nuclear Reactor Internal Component
• Example 5.5: Spent Nuclear Fuel Racks
• References
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