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Chapter 4
Vibration of Structures in Quiescent Fluids—I the Hydrodynamic Mass

## Excerpt

The general case of coupled fluid-structural dynamic analysis involving the complete set of equations of fluid dynamics and structural dynamics can be prohibitively complicated. In spite of 25 years of intensive research, today many of the coupled fluid-structural dynamic problems are solved based on the “weak coupling” assumption. In this approach, the force induced on the fluid due to structural motion is assumed to be linearly superimposible onto the original forcing function in the fluid. Under this assumption, the effect of fluid-structure interaction can be completely accounted for by an additional mass term, called the hydrodynamic mass, and an additional damping term, called hydrodynamic damping. These hydrodynamic mass and damping terms can then be separately computed and input into standard structural analysis computer programs for dynamic analysis of the coupled fluid-structure system. This approach allows computer programs developed, and data obtained from tests, without taking into account the effect of fluid-structure interaction, to be used in coupled fluid-structural dynamic analyses. Although developed based on coaxial cylindrical shells coupled by an annular fluid gap, the results in this chapter shed light on the physics of fluid-structure interaction and can be used in other geometry in special cases as will be discussed in the following chapter.

As an alternative to the finite-element method, which may be quite complicated in the case of 3D structures, closed-form equations for computing the in-air natural frequencies of cylindrical shells with either both ends simply-supported or both ends clamped, are given in Section 4.2. These equations can be easily solved numerically with today's widely available standard office software.

The weakly coupled fluid-structural dynamics problem can be solved by two alternative formulations of the hydrodynamic mass method: the generalized hydrodynamic mass and the full hydrodynamic mass matrix methods. The first formulation is for estimating the coupled fluid-shell frequencies when the uncoupled, in-air natural frequencies of the individual cylindrical shells are known from measurement, from separate finite-element analysis or by numerical solution of the characteristic equation derived in Section 4.2. The 2x2 effective generalized hydrodynamic mass matrix is given by:

$M̑Hmna=11+n−2Σα(Cαma)2hαnaM̑Hmnab=11+n−2Σα(CαmaCαmb)hαnabM̑Hmnba=11+n−2Σα(CαmaCαmb)hαnbaM̑Hmnb=11+n−2Σα(Cαmb)2hαnb$

• Summary
• Nomenclature
• 4.1 Introduction
• Strongly Coupled Fluid-Structure System
• Weakly Coupled Fluid-Structure System
• The Hydrodynamic Mass and Damping Method
• 4.2 Free Vibration of Circular Cylindrical Shells in Air
• 4.3 Acoustic Modes of a Fluid Annulus Bounded by Rigid Walls
• The Ripple Approximation
• 4.4 Vibration of a Coupled Fluid-Cylindrical Shell System
• 4.5 The 2 × 2 Generalized Hydrodynamics Mass Matrix
• The Physics of Fluid-Structure Coupling
• Example 4.1: Coupled Shell-Mode Vibration
• Example 4.2: Beam-Mode Vibration inside an Annular Fluid Gap
• 4.6 Extension to Double Annular Gaps
• 4.7 The Full Hydrodynamic Mass Matrix
• Example 4.3: Beam Mode Vibration inside a Fluid Annulus
• 4.8 Forced Response of Coupled Fluid-Shells
• Turbulence-Induced Vibration
• The Loss-of-Coolant Accident Problem
• Seismic Responses
• References
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