AIAA JOURNAL Vol. 43, No. 11, November 2005 Technical Notes TECHNICAL NOTES are short manuscripts describing new developments or important results of a preliminary nature. These Notes should not exceed 2500 words (where a figure or table counts as 200 words). Following informal review by the Editors, they may be published within a few months of the date of receipt. Style requirements are the same as for regular contributions (see inside back cover). Impedance Modeling Technique for a Fluid-Loaded Structure C. C. Cheng ∗ and P. W. Wang † National Chung Cheng University, Chia-Yi 621, Taiwan, Republic of China I. Introduction F OR a fluid-loaded structure, how to calculate the coupled mo- tions between the elastic structure and the associated medium that surrounds it has attracted many researchers for many years. 1−8 Different methods that include analytical approaches (e.g., the com- plex integral transform, asymptotic technique), and the numerical techniques [e.g. the finite element method (FEM), the boundary element method (BEM)], etc., are available, and each has its own advantages. Most of them derive the equation of motion of the fluid- loaded structure first and then find the associated vibro-acoustic response. However, great inherent difficulties in the interaction be- tween the fluid and the structure are encountered when the structure either has an irregular shape or is subject to boundary conditions that are so complicated even the numerical technique, for example, FEM or BEM, has trouble in modeling them. This Note proposes an impedance technique 9 to obtain the dy- namic response of a fluid-loaded structure. The structural surface is divided into multiple segments, and then the impedance of each structural segment and the fluid are investigated individually. By applying the conditions of force equilibrium and response compat- ibility between the structure and the fluid, the impedance of the fluid-loaded structure can be expressed as the impedance couplings between the fluid and the structure. For systems where either the equations of motion or the associated boundary conditions are dif- ficult to be modeled, the impedances of the structure and fluid can be obtained from experimental measurements. The impedance of a mechanical component expressed using the frequency response function (FRF) links the associated analytical model to practical measurements, which has been demonstrated by traditional modal testing. Therefore, it provides an alternative to obtaining the vibroa- coustic response of a fluid-loaded structure. Nevertheless, for a fluid-loaded structure, how to model the fluid- loading effect and then couple it with the impedance of the structure in vacuo is still an issue that must be answered first. It is well known that the fluid-loading effect on a structure can be modeled using Fourier transform as an acoustic wave impedance in the wave num- ber domain. However, the impedance of the structure in vacuo is expressed using the frequency response function and thus it is re- Received 11 April 2005; revision received 16 June 2005; accepted for publication 16 June 2005. Copyright c  2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rose- wood Drive, Danvers, MA 01923; include the code 0001-1452/05 $10.00 in correspondence with the CCC. ∗ Professor, Department of Mechanical Engineering, 160, San-Hsing, Ming-Hsiung; imeccc@ ccu.edu.tw. † Graduate Research Assistant, Department of Mechanical Engineering, 160, San-Hsing, Ming-Hsiung. quired to express the fluid loading in terms of impedance in the same FRF model as well. In this Note, a methodology of expressing the fluid loading as a radiation impedance on the structure is de- veloped. A formulation that assembles the radiation impedance and the structural impedance is introduced in order to demonstrate how a fluid-loaded structural impedance model is constructed by using the impedance of structure in vacuo and the radiation impedance. II. Mechanical Mobility of Structures Although the definition of mechanical mobility is well known, for the purpose of completeness, the concept of structural mobility will be introduced here using a simple beam as an example. 10,11 Consider a beam of length L , thickness b, width w, and density ρ 0 , whose surface is divided into N segments as shown in Fig. 1. When a harmonic force with a frequency ω acts on the midpoint of the nth segment, the equation of motion for the beam is expressed as EI ∂ 4 w ∂ x 4 + ρ A ∂ 2 w ∂ t 2 = F v δ(x − x n )e i ωt (1) where δ(x ) is the Dirac delta function, w the beam displacement, EI the bending rigidity, ρ A the mass per unit length, F v the concentrated force, and x n the coordinate corresponding to the midpoint of the nth segment. Based on the information of excitation at x n and the response at x m , the mobility of the beam m m,n is defined as m m,n = V v (x m , t ) F v e i ωt = ∞  k = 1 i ωφ k (x n )φ k (x m ) ( ω 2 k − ω 2 ) (2) where V v (x m , t ) is the transverse velocity of the mth beam seg- ment, ω k is the k th natural frequency of the beam, and φ k is the corresponding mass normalized shape function that satisfies the as- sociated boundary conditions. The first subscript of m indicates the response segment, and the second denotes the excitation segment, respectively. Based on the concept of frequency response function, 12 the force-velocity relation of the beam can be rewritten as V v = MF (3) where V v −{V v1 , V v2 , V v3 ,..., V v N } T is the beam transverse veloc- ity represented discretely by the velocity of each beam segment, F ={ F 1 , F 2 , F 3 ,..., F N } is the external force acting on each beam segment, for example, F = 0, 0, 0,..., F n = F v ,..., 0} in this ex- ample, and M is the structural mobility expressed as M =     m 1,1 m 1,2 .. m 1, N m 2,1 m 2,2 .. .. .. .. m m,n .. m N,1 .. .. m N, N     N × N (4) Fig. 1 Beam subjected to a concentrated harmonic force at the nth segment. 2454 Downloaded by NATIONAL CHENG KUNG UNIVERSITY on May 7, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.17131