fluid loaded structure

ACOUSTICS AND HYDRODYNAMICS OF FLUID-STRUCTURE

INTERACTION IN A SUBMERGED ELASTIC DUCT

A Dissertation

Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

by

Wenlong Zhang,

Hafiz M. Atassi, Director

Graduate Program in Aerospace and Mechanical Engineering

Notre Dame, Indiana

April 2010

ACOUSTICS AND HYDRODYNAMICS OF FLUID-STRUCTURE

INTERACTION IN A SUBMERGED ELASTIC DUCT

Abstract

by

Wenlong Zhang

This dissertation studied the interaction of nonuniform flows with propeller blades

in a submerged elastic duct. The acoustic radiation from the duct is calculated and cor-

related to the flow nonuniformities and the propeller and duct characteristics. First,

a benchmark problem is studied wherein the sound radiation from an infinite plate

with or without ribs is examined for different excitations sources: normal single force,

monopole, dipole, and vortex excitations. The investigation of the sound radiation from

a plate gives us a fundamental understanding of flexure waves.

Second, the case of a cylindrical duct with or without ribs is considered and the dis-

persion relation of the rib-stiffened duct modes is compared with that of a un-stiffened

duct. The dispersion relation of the stiffened duct has a periodic structure similar to

that of connected oscillators with large number of independent modes. Because of our

interest in the acoustic radiation from such a system, we focus our attention on the

flexure modes. The sound radiation is first tested with simple internal forces such as

monopoles and dipoles. The results for un-stiffened ducts show strong directivity as the

dipole radial location moves closer to the duct wall. For stiffened ducts, the magnitude

of the acoustic response as well as the directivity vary strongly and show large peaks

near the stiffened duct free modes.

Wenlong Zhang

Third, the scattering phenomena in a rigid duct and an elastic duct is investigated.

The effect of the impedance on the acoustic sources is also examined. The results show

that the impedance begin to have the significant effect on the unsteady lift when the

magnitude of the non dimensional impedance is the order of one. The main effect of

the elastic wall comes from the location of the blade and the upstream of the blade.

Finally, a model for flow-propeller interactions in a submerged elastic duct is devel-

oped. This model examines and quantifies the mechanism of flow-propeller interaction

in a flexible duct. The model couples the fluid motion with the elastic duct vibration

and yields the duct flexural displacement. This leads to the evaluation of the radiated

sound. The coupling between the elastic duct and the flow-propeller system is studied

by changing the Euler code which accounts for the rotor/stator interaction problem. The

results suggests that for different combination of rotor/stator blade counts, it is possi-

ble to have low circumferential mode number, which is an efficient radiator of acoustic

energy.

CONTENTS

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Structural Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Thin Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Thin Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Fluid-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . 15

CHAPTER 2: MATHEMATICAL FORMULATIONS . . . . . . . . . . . . . . 182.1 Mathematical Formulations of Elastic Shell . . . . . . . . . . . . . . . 18

2.1.1 Differential Equations of Isotropic Elastic Shell . . . . . . . . . 192.1.2 Modal Equations of Isotropic Elastic Shell . . . . . . . . . . . 212.1.3 Interior and Exterior Fluid Loading . . . . . . . . . . . . . . . 23

2.2 Mathematical Formulations of Fluid Motion . . . . . . . . . . . . . . . 272.3 Coupling between Shell Equations and Euler Equations . . . . . . . . 29

CHAPTER 3: SOUND RADIATION FROM THIN PLATES . . . . . . . . . . . 323.1 Dispersion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Single Force Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3 Monopole Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Dipole Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.5 Thin Plate with Ribs . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5.1 Dispersion Curves of Thin Plate with Ribs . . . . . . . . . . . . 453.5.2 Unit Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.5.3 Monopole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5.4 Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

ii

3.6 Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

CHAPTER 4: SOUND RADIATION FROM THIN SHELLS . . . . . . . . . . . 574.1 Dispersion Curves of Waves in an Elastic Shell . . . . . . . . . . . . . 58

4.1.1 Code Verification . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.2 In-vacuo Wave Numbers versus Frequency for Different Cir-

cumferential Mode Numbers . . . . . . . . . . . . . . . . . . . 604.1.3 Effects of Shell Thickness on Flexure Curves . . . . . . . . . . 684.1.4 Effects of Shell Material on Flexure Curves . . . . . . . . . . . 704.1.5 Water-filledWave Numbers versus Frequency for Different Cir-

cumferential Mode Numbers . . . . . . . . . . . . . . . . . . . 714.1.6 New Method for Obtaining Dispersion Curves . . . . . . . . . 81

4.2 Far Field Acoustic Radiation Excited by a Single Force . . . . . . . . . 844.3 Far Field Acoustic Radiation Excited by a Monopole . . . . . . . . . . 87

4.3.1 Sound Pressure versus Frequency in Response to a Monopole . 904.3.2 Pressure Directivity of a Monopole . . . . . . . . . . . . . . . 92

4.4 Far Field Acoustic Radiation Excited by Dipole Excitations . . . . . . . 934.4.1 Axial Dipole Excitation . . . . . . . . . . . . . . . . . . . . . 994.4.1.1 Sound Pressure versus Frequency in Response to an Axial

Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.4.1.2 Pressure Directivity of an Axial Dipole . . . . . . . . . . . . 1024.4.2 Radial Dipole Excitation . . . . . . . . . . . . . . . . . . . . . 1034.4.2.1 Sound Pressure versus Frequency in Response to a Radial

Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4.2.2 Pressure Directivity of a Radial Dipole . . . . . . . . . . . . 1104.4.3 Circumferential Dipole Excitation . . . . . . . . . . . . . . . . 1154.4.3.1 Sound Pressure versus Frequency in Response to a Circum-

ferential Dipole . . . . . . . . . . . . . . . . . . . . . . . . . 1154.4.3.2 Pressure Directivity of a Circumferential Dipole . . . . . . . 117

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

CHAPTER 5: SOUND RADIATION FROM THIN SHELLS WITH RIBS . . . . 1225.1 Dispersion Curves of Waves for the Water-Filled Shell with Ribs . . . . 1275.2 Dispersion Curves of Waves for the Air-Filled Shell with Ribs . . . . . 1295.3 Far Field Acoustic Radiation from the Stiffened Shell . . . . . . . . . . 132

5.3.1 Single Force Excitation . . . . . . . . . . . . . . . . . . . . . 1355.3.2 Monopole Excitation . . . . . . . . . . . . . . . . . . . . . . . 1355.3.3 Axial Dipole Excitation . . . . . . . . . . . . . . . . . . . . . 1365.3.4 Circumferential Dipole Excitation . . . . . . . . . . . . . . . . 1405.3.5 Radial Dipole Excitation . . . . . . . . . . . . . . . . . . . . . 142

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

iii

CHAPTER 6: SOUND RADIATION FROM A PROPELLER . . . . . . . . . . 1456.1 Rigid Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.1.1 Code Verification . . . . . . . . . . . . . . . . . . . . . . . . . 1496.1.2 Uniform Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.1.3 Swirling Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.2 Duct Elasticity as Impedance . . . . . . . . . . . . . . . . . . . . . . . 1536.2.1 Axially Homogenous Duct . . . . . . . . . . . . . . . . . . . . 1596.2.2 Effect of Discontinuities . . . . . . . . . . . . . . . . . . . . . 166

6.3 Coupled Propeller-Duct . . . . . . . . . . . . . . . . . . . . . . . . . . 1696.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

CHAPTER 7: CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 185

APPENDIX A: RIB MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188A.1 Strain energy of the T-rib . . . . . . . . . . . . . . . . . . . . . . . . . 188A.2 Kinetic energy of the T-rib . . . . . . . . . . . . . . . . . . . . . . . . 190A.3 Applied forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190A.4 Using Lagranges Equation . . . . . . . . . . . . . . . . . . . . . . . . 191A.5 Applying Fourier decomposition . . . . . . . . . . . . . . . . . . . . . 193

APPENDIX B: THIN SHELL THEORIES . . . . . . . . . . . . . . . . . . . . . 195

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

iv

FIGURES

1.1 Sound radiation from rotor blades interaction with turbulent and swirlingmotion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Model development scheme for elastic duct and fluid motion. . . . . . 16

2.1 Geometry of cylindrical shell excited by point force. . . . . . . . . . . 19

3.1 The geometry of an infinite thin plate of uniform thickness h. . . . . . . 33

3.2 Flexure waves of an infinite steel plate excited by a unit force. Thecolor scale represents the log10 displacement value in meters. . . . . . 38

3.3 Far field sound radiation from 1 cm steel plate with water loading onone side only. The excitation is a unit point force at the origin. . . . . . 39

3.4 Far field sound radiation from 1 cm steel plate with water loading onone side only. The excitation is a monopole at x0 = 0, y0 = 0 andz0 = 0:05m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 Far field sound radiation from 1 cm steel plate with water loading onone side only. The excitation is a dipole at x0 = 0, y0 = 0 and z0 =0:05m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.6 Flexure waves of an infinite steel plate with ribs excited by a unit force.The color scale represents the log10 displacement value in meters. . . . 46

3.7 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a unit point force between ribs. . . . . . . . . . . . . . . . 47

3.8 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a unit point force on a rib. . . . . . . . . . . . . . . . . . 48

3.9 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a unit point force between ribs. The mass of the ribs is fivetimes that of Figure 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.10 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a unit point force on a rib. The mass of the ribs is fivetimes that of Figure 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . 49

v

3.11 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a monopole between ribs. . . . . . . . . . . . . . . . . . . 49

3.12 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a monopole on a rib. . . . . . . . . . . . . . . . . . . . . 50

3.13 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a dipole between ribs. . . . . . . . . . . . . . . . . . . . . 51

3.14 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a dipole on a rib. . . . . . . . . . . . . . . . . . . . . . . 51

3.15 Broadside far field sound level of steel plate stiffened by ribs. Theexcitation is a vortex above the plate (z0 = 0:05m) . . . . . . . . . . . . 54

3.16 Far field (q = p=4) sound level of steel plate stiffened by ribs. Theexcitation is a vortex above the plate (z0 = 0:05m). . . . . . . . . . . . 55

4.1 Wavenumber versus frequency dispersion plot of a water filled steelcylinder surrounded by a vacuum, for circumferential harmonic modenumber m=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Wavenumber versus frequency dispersion plot of a vacuo steel cylindersurrounded by a vacuum, for circumferential harmonic m=1. . . . . . . 62

4.3 Ratio of displacement, jzx=zrj and jzq=zrj, versus frequency for branch0 with m=1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63









vi


4.13 Wavenumber versus frequency dispersion plot of a vacuo steel cylindersurrounded by a vacuum, for circumferential harmonic m=10. . . . . . 69

4.14 Ratio of displacement, jzx=zrj and jzq=zrj, versus frequency for branch1 with m=10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.15 Flexure curves of a vacuum steel cylindrical shell surrounded by a vac-uum for different thickness with m=1. . . . . . . . . . . . . . . . . . . 70

4.16 Flexure curves of a vacuum steel cylindrical shell surrounded by a vac-uum for different thickness with m=10. . . . . . . . . . . . . . . . . . 71

4.17 Flexure curves of a vacuum cylindrical shell surrounded by a vacuumfor different shell materials with m=1. . . . . . . . . . . . . . . . . . . 72

4.18 Flexure curves of a vacuum cylindrical shell surrounded by a vacuumfor different shell materials with m=10. . . . . . . . . . . . . . . . . . 72

4.19 Real part of exterior water loading on an infinite cylinder of radium 1m and axial wavenumber a = p=2. The negative sign indicates that itacts as a mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.20 Imaginary part of exterior water loading on an infinite cylinder of ra-dium 1 m and axial wavenumber a = p=2. The negative sign indicatesthat it acts as a resistance. . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.21 Real part of interior water loading on an infinite cylinder of radium 1 mand axial wavenumber a = p=2. . . . . . . . . . . . . . . . . . . . . 74

4.22 Wavenumber versus frequency dispersion plot of a water-filled steelcylinder surrounded by a vacuum, for circumferential harmonic m=1. . 76

4.23 Ratio of displacement, zx=zr and zq=zr, versus frequency for branch 0with m=1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76






vii



4.31 Ratio of displacement, jzx=zrj and jzq=zrj, versus frequency for branch1 with m=10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.32 Wavenumber versus frequency dispersion plot of a water-filled Alu-minum cylinder surrounded by a vacuum, for circumferential harmonicm=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.33 Flexure waves of a water-filled Aluminum cylinder surrounded by avacuum excited by a unit force, for circumferential harmonic m=0. . . . 83

4.34 Flexure waves of a water-filled Aluminum cylinder surrounded by wa-ter excited by a unit force, for circumferential harmonic m=0. . . . . . . 84

4.35 The schematic of the spherical coordinate system. . . . . . . . . . . . . 85

4.36 Far field sound radiation from a water filled steel shell excited by a unitradial point force at f = 90o. . . . . . . . . . . . . . . . . . . . . . . . 86

4.37 Far field sound radiation from a water filled steel shell excited by a unitradial point force at f = 70o. . . . . . . . . . . . . . . . . . . . . . . . 87

4.38 Far field sound pressure levels at f = p=2 of the aluminum cylindricalshell caused by a monopole. . . . . . . . . . . . . . . . . . . . . . . . 91

4.39 Far field sound pressure levels at f = p=3 of the aluminum cylindricalshell caused by a monopole. . . . . . . . . . . . . . . . . . . . . . . . 91

4.40 Pressure directivity of a monopole located at axis of cylinder for re-duced frequency w = 0:1. . . . . . . . . . . . . . . . . . . . . . . . . 93

4.41 Pressure directivity of a monopole located at axis of cylinder for re-duced frequency w = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 94



4.44 Pressure directivity of a monopole located at r0 = 0:5, q0 = 0, and x0 =0 for reduced frequency w = 0:1. . . . . . . . . . . . . . . . . . . . . 95

4.45 Pressure directivity of a monopole located at r0 = 0:5, q0 = 0, and x0 =0 for reduced frequency w = 1. . . . . . . . . . . . . . . . . . . . . . 96


viii


4.48 The schematic of the dipole excitation. . . . . . . . . . . . . . . . . . . 98

4.49 Far field sound pressure levels at f = p=3 of the aluminum cylindricalshell caused by an axial dipole. . . . . . . . . . . . . . . . . . . . . . . 100

4.50 Far field sound pressure levels at f = p=4 of the aluminum cylindricalshell caused by axial dipoles. . . . . . . . . . . . . . . . . . . . . . . . 101

4.51 Far field sound pressure levels of the aluminum cylindrical shell causedby axial dipoles at f = p=4 with different azimuthal angles. . . . . . . 102

4.52 Pressure directivity of an axial dipole located at axis of cylinder forreduced frequency w = 0:1. . . . . . . . . . . . . . . . . . . . . . . . 103

4.53 Pressure directivity of an axial dipole located at axis of cylinder forreduced frequency w = 1. . . . . . . . . . . . . . . . . . . . . . . . . 104



4.56 Pressure directivity of an axial dipole located at r0 = 0:5, q0 = 0 forreduced frequency w = 0:1. . . . . . . . . . . . . . . . . . . . . . . . 105

4.57 Pressure directivity of an axial dipole located at r0 = 0:5, q0 = 0 forreduced frequency w = 1. . . . . . . . . . . . . . . . . . . . . . . . . 106



4.60 Far field sound pressure levels at f = p=2 of the aluminum cylindricalshell caused by radial dipoles. . . . . . . . . . . . . . . . . . . . . . . 108

4.61 Far field sound pressure levels at f = p=4 of the aluminum cylindricalshell caused by radial dipoles. . . . . . . . . . . . . . . . . . . . . . . 108

4.62 Derivative of Bessel function J0jmj(r0k sinf). . . . . . . . . . . . . . . . 109

4.63 Far field sound pressure levels of the aluminum cylindrical shell causedby radial dipoles at f = p=4 with different azimuthal angles. . . . . . . 110

4.64 Pressure directivity of a radial dipole located at axis of cylinder forreduced frequency w = 0:1. . . . . . . . . . . . . . . . . . . . . . . . 111

4.65 Pressure directivity of a radial dipole located at axis of cylinder forreduced frequency w = 1. . . . . . . . . . . . . . . . . . . . . . . . . 112

ix



4.68 Pressure directivity of a radial dipole located at r0 = 0:5, q0 = 0 forreduced frequency w = 0:1. . . . . . . . . . . . . . . . . . . . . . . . 113

4.69 Pressure directivity of a radial dipole located at r0 = 0:5, q0 = 0 forreduced frequency w = 1. . . . . . . . . . . . . . . . . . . . . . . . . 114



4.72 Far field sound pressure levels at f = p=2 of the aluminum cylindricalshell caused by circumferential dipoles. . . . . . . . . . . . . . . . . . 116

4.73 Far field sound pressure levels at f = p=4 of the aluminum cylindricalshell caused by circumferential dipoles. . . . . . . . . . . . . . . . . . 117

4.74 Far field sound pressure levels of the aluminum cylindrical shell causedby circumferential dipoles at f = p=4 with different azimuthal angles. . 118

4.75 Pressure directivity of a circumferential dipole located at r0 = 0:5 forreduced frequency w = 0:1. . . . . . . . . . . . . . . . . . . . . . . . 119

4.76 Pressure directivity of a circumferential dipole located at r0 = 0:5 forreduced frequency w = 1. . . . . . . . . . . . . . . . . . . . . . . . . 119



5.1 Schematic of a rib-stiffened cylindrical shell. M is the observer point inthe spherical coordinate system. R is the distance between the sourceand observer, q is the polar angle, and f is the azimuthal angle. . . . . . 123

5.2 The schematic of the rib stiffened cylindrical shell. . . . . . . . . . . . 125

5.3 Flexure waves of a water-filled Aluminum cylinder surrounded by wa-ter for circumferential harmonic m=0, which is excited by a unit force. . 128

5.4 Flexure waves of a water-filled Aluminum ribbed cylinder surroundedby water for circumferential harmonic m=0, which is excited by a unitforce. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

x





5.9 Radial displacements of a air-filled ribbed shell surrounded by air forcircumferential harmonic m=10, which is excited by a unit force. . . . . 133

5.10 Circumferential displacements of a air-filled ribbed shell surrounded byair for circumferential harmonic m=10, which is excited by a unit force. 134

5.11 Axial displacements of a air-filled ribbed shell surrounded by air forcircumferential harmonic m=10, which is excited by a unit force. . . . . 134

5.12 Far field sound pressure levels at broadside (f = p=2;q = 0) of ribbedand unribbed cylindrical shells caused by a unit radial point force. . . . 136

5.13 Far field sound pressure levels at 45o off broadside (f = p=4;q = 0) ofribbed and unribbed cylindrical shells caused by a unit radial point force. 137

5.14 Far field sound levels at f = p=2 of ribbed cylindrical shell by a monopole.1385.15 Far field sound pressure levels at f = p=3 of the ribbed cylindrical shell

caused by a monopole. . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.16 Far field sound pressure levels at f = p=3 of the ribbed cylindrical shellcaused by axial dipoles. . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.17 Far field sound pressure levels at f = p=4 of the ribbed cylindrical shellcaused by axial dipoles. . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.18 Far field sound pressure levels at f = p=2 of the ribbed cylindrical shellcaused by circumferential dipoles. . . . . . . . . . . . . . . . . . . . . 141

5.19 Far field sound pressure levels at f = p=4 of the ribbed cylindrical shellcaused by circumferential dipoles. . . . . . . . . . . . . . . . . . . . . 141

5.20 Far field sound pressure levels at f = p=2 of the ribbed cylindrical shellcaused by radial dipoles. . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.21 Far field sound pressure levels at f = p=4 of the ribbed cylindrical shellcaused by radial dipoles. . . . . . . . . . . . . . . . . . . . . . . . . . 143

xi

6.1 Schematic of computational domain of a rigid duct. . . . . . . . . . . . 148

6.2 Comparison of the magnitude of the unsteady sectional lift coefficientalong the span with that of Atassi et al: (2004) . . . . . . . . . . . . . 150

6.3 Pressure magnitude distribution along the x-axis at the duct wall forreduced frequency w = 0:5. . . . . . . . . . . . . . . . . . . . . . . . 152

6.4 Unsteady sectional lift coefficient along the span for reduced frequencyw = 0:5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.5 Pressure magnitude distribution along the x-axis at the duct wall forreduced frequency w = 1. . . . . . . . . . . . . . . . . . . . . . . . . 154

6.6 Unsteady sectional lift coefficient along the span for reduced frequencyw = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.7 Pressure magnitude distribution along the x-axis at the duct wall forreduced frequency w = 3. . . . . . . . . . . . . . . . . . . . . . . . . 156

6.8 Unsteady sectional lift coefficient along the span for reduced frequencyw = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.9 Comparison of the magnitudes of the unsteady sectional lift coefficientalong the span for different mean flows. . . . . . . . . . . . . . . . . . 157

6.10 Schematic of computational domain of an elastic duct. . . . . . . . . . 158

6.11 Pressure magnitude distribution along the x-axis at the duct wall fordifferent impedances (q = 0) . . . . . . . . . . . . . . . . . . . . . . . 161

6.12 A close up for the peak part of the pressure magnitude distribution alongthe x-axis at the duct wall for different impedances (q = 0) . . . . . . . 161

6.13 A close up for the middle part of the pressure magnitude distributionalong the x-axis at the duct wall for different impedances (q = 0) . . . . 162

6.14 Pressure magnitude distribution along the x-axis at the duct wall fordifferent impedances (q = p=10) . . . . . . . . . . . . . . . . . . . . . 163

6.15 Comparison of the magnitudes of the unsteady sectional lift coefficientalong the span for different acoustic impedances . . . . . . . . . . . . . 163

6.16 Pressure magnitude distribution along the x-axis at the duct wall forreduced frequency w = 0:5 (impedance zi = 1:01:0i). . . . . . . . . 164

6.17 Unsteady sectional lift coefficient along the span for reduced frequencyw = 0:5 (impedance zi = 1:01:0i). . . . . . . . . . . . . . . . . . . 165

6.18 Pressure magnitude distribution along the x-axis at the duct wall forreduced frequency w = 1 (impedance zi = 1:01:0i). . . . . . . . . . 166

6.19 Unsteady sectional lift coefficient along the span for reduced frequencyw = 1 (impedance zi = 1:01:0i). . . . . . . . . . . . . . . . . . . . 167

xii

6.20 Pressure magnitude distribution along the x-axis at the duct wall forreduced frequency w = 3 (impedance zi = 1:01:0i). . . . . . . . . . 168

6.21 Unsteady sectional lift coefficient along the span for reduced frequencyw = 3 (impedance zi = 1:01:0i). . . . . . . . . . . . . . . . . . . . 169

6.22 Pressure magnitude distribution along the x-axis at the duct wall fordifferent impedances (q = 0) . . . . . . . . . . . . . . . . . . . . . . . 170

6.23 A close up for the peak part of the pressure magnitude distribution alongthe x-axis at the duct wall for different impedances (q = 0) . . . . . . . 170

6.24 A close up for the middle part of the pressure magnitude distributionalong the x-axis at the duct wall for different impedances (q = 0) . . . . 171

6.25 Pressure magnitude distribution along the x-axis at the duct wall fordifferent impedances (q = p=10) . . . . . . . . . . . . . . . . . . . . . 171

6.26 Comparison of the magnitudes of the unsteady sectional lift coefficientalong the span for different acoustic impedances . . . . . . . . . . . . . 172

6.27 Far filed sound pressure directivity for an aluminum duct. The noisesources are obtained from the rotor/stator duct system . . . . . . . . . . 173

6.28 Schematic of computational domain of the duct with impedance dis-continuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.29 Pressure magnitude distribution along the x-axis at the duct wall ofimpedance discontinuities (q = 0) . . . . . . . . . . . . . . . . . . . . 175

6.30 A close up for the peak part of the pressure magnitude distribution alongthe x-axis at the duct wall of impedance discontinuities (q = 0) . . . . . 175

6.31 A close up for the middle part of the pressure magnitude distributionalong the x-axis at the duct wall of impedance discontinuities (q = 0) . 176

6.32 Pressure magnitude distribution along the x-axis at the duct wall ofimpedance discontinuities(q = p=10) . . . . . . . . . . . . . . . . . . 176

6.33 Comparison of the magnitudes of the unsteady sectional lift coefficientalong the span of impedance discontinuities. . . . . . . . . . . . . . . 177

6.34 Imaginary part of exterior water loading on an infinite cylinder of ra-dium 1 m and axial wavenumber a = 0:5. The negative sign indicatesthat it acts as a resistance. . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.35 Imaginary part of exterior water loading on an infinite cylinder of ra-dium 1 m and axial wavenumber a = 5:0. The negative sign indicatesthat it acts as a resistance. . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.36 Real part of duct wall pressure in response to rotor stator interaction forrigid duct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

xiii

6.37 Imaginary part of duct wall pressure in response to rotor stator interac-tion for rigid duct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.38 Real part of duct wall pressure in response to rotor stator interaction forelastic duct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.39 Imaginary part of duct wall pressure in response to rotor stator interac-tion for elastic duct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.40 Comparison of unsteady lift coefficient between rigid duct and elasticduct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.41 Comparison of sound pressure directivity for an aluminum duct be-tween the rigid wall excitation and the elastic wall excitation. . . . . . . 183

xiv

TABLES

2.1 SCALING OF FORCES. . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 PROPERTIES OF DIFFERENT MATERIALS. . . . . . . . . . . . . . 61

6.1 IMPEDANCES OF THE ALUMINUM SHELL. . . . . . . . . . . . . 179

xv

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my advisor, Professor Hafiz M.

Atassi, for his tireless moral and academic guidance, and consistent inspiration during

my doctoral research. I would also like to thank the members of my dissertation com-

mittee, Professors Stephen Batill, Robert Nelson and Scott Morris for reviewing my

dissertation and providing valuable suggestions and comments. Special thanks for Dr.

William K. Blake for his insightful comments.

I would also like to thank the Office of Naval Research and the Center for Applied

Mathematics for the financial support.

I would like to express my sincere thanks to my family and friends through all

weathers. I would like specially to thank my wife for her love, support and continuing

encouragement.

xvi

NOMENCLATURE

Latin Symbols

A Velocity amplitude (m=s)

A Monopole strength

a Shell radius

B Number of rotor blades

c Pressure intensity coefficient

c Chord length (m)

c Speed of sound (m=s)

E Youngs modulus

E Excitation

e Spectral excitation

e Direction vector

F Field quantity

f Spectral field quantity

H Hankel function

h Shell thickness

J Bessel function

k Wave number

L Domain length (m)

L linearized Euler operator

xvii

m Circumferential mode number

M Mach number

~n Normal direction

N Number of points

p Pressure (Pa)

P Power (Watts)

P Pressure eigenfunction

r Radial position (m)

R Inverse matrix of stiffness matrix

s Entropy (kJ=kg=K)

S Stiffness matrix

t Time (s)

u;v;w Radial, azimuthal and axial velocities (m=s)

U Velocity (m=s)

V Number of stator vanes

W Transverse displacement of thin plate (m)

x Axial position (m)

Z Shell displacement (m)

Greek Symbols

a Axial wave number (1=m)

d Dirac delta function

z Spectral displacement

h Damping factor (m)

q Angle in degrees (o)

q Circumferential position

xviii

J Angular spacing

L Convected eigenvalue

n Poissons ratio

r Density (kg=m3)

s Inter-blade phase angle

f Potential (m=s2)

c Stagger angle (o)

w Angular frequency (1=s)

Superscripts

R Vortical velocity

0 Perturbation Dimensionless value

Subscripts

h Evaluated at hub radius

m Circumferential mode number

m Evaluated at mean radius

n Radial counter

n Normal direction

o Steady flow quantity

r Radial direction

t Evaluated at tip radius

x Axial direction

q Circumferential direction

xix

CHAPTER 1

INTRODUCTION

The interactions between fluid and flexible structures give rise to numerous phys-

ical problems and phenomena. In these cases, the flexible structure is surrounded by

the fluid, and fluid flow exerts pressure on the solid structure causing it to deform. For

nonuniform turbulent flow, the structure vibrates in response to the unsteady fluid load-

ing and radiates sound in the surrounding fluid. When the interaction between nonuni-

form flow and a propeller in an elastic duct is studied, there are two main questions that

need to be answered: 1) How does the elastic duct change the fluctuating hydrodynamic

pressure along the propeller blades? 2) How does it affect the far field radiated sound?

To answer these questions and obtain a better understanding of fluid-structure interac-

tion phenomena, we need to bridge the gab between hydrodynamics which assumes

rigid ducts and blades and structural acoustics which assumes flexible structures. The

coupling between the fluid motion and the duct vibration need to be modeled, which

takes place at the duct boundary.

In this work, the coupled nonuniform flow interaction with a propeller in an elas-

tic duct is studied. Figure 1.1 schematically shows sound radiation from rotor blades

interaction with turbulent and swirling motion. As can be seen, the elastic duct with a

propeller is submerged and filled with water. The propeller is modeled as a rotor/stator

stage. For simplicity, the duct is modeled as an infinite thin cylindrical shell. The

propeller usually works in a nonuniform flow, and the non-uniformities are caused by

1

Figure 1.1. Sound radiation from rotor blades interaction with turbulent andswirling motion.

ingested turbulence. To control vibration and noise, while at the same time enhancing

structural integrity, the cylindrical shell is stiffened with ribs.

The ingested turbulence interacts with the blades, and the blade forces can be re-

garded as dipole sources. Due to the swirling flow, the centrifugal force will act on the

duct too. The forces excite flexure waves in the duct. The radiated sound is produced

by duct vibration and hydro-acoustic scattering. The present research is directed toward

investigating the coupling effects between the fluid motion and the duct vibration and

to control these undesirable effects.

In the traditional approach of studying the flow-propeller-duct interaction, the duct

is assumed to be rigid and the hydrodynamic forces give the equivalent dipole strength.

2

Moreover, a compact blocked dipole model is used to calculate the radiated sound in the

far field. This approach does not account for the coupling effect between the flow and

the elastic duct which affects the strength of the hydrodynamic forces and the modeling

of the radiated sound. It also ignores the spatially distributed nature of the flow dipoles.

Our study of the sound pressure level radiated from a dipole inside a duct shows strong

dependence of the sound pressure level on the radial position of the dipole and its

orientation. The difference of sound pressure level between two radial positions of

a dipole can reach more than 10 dB. This fact demonstrates the inadequate assumption

often used in structural acoustics wherein distributed dipole sources are treated as a

single blocked dipole when the acoustic transfer function is calculated.

The main source of noise radiated from a submerged flow-propeller-duct system is

the interaction of irregular inflow disturbances with the propeller/vanes blades and the

duct wall. These irregular inflow patterns arise from various flow phenomena, such

as inlet distortion and turbulence, guide vanes-rotor interaction and second flows, etc.

The interaction of inflow disturbances with the blades produces unsteady hydrodynamic

forces (dipole sources) which, coupled with the duct flexural modes, results in unwanted

vibration and noise. For ducts with axial discontinuities, such as control ribs and finite

ducts, the dipole sources and flow vortices may couple with duct modes to produce

additional sources of noise.

The main objectives of this dissertation are to identify and quantify the mecha-

nisms of the fluid-structure interaction relevant to noise radiation of submerged flow-

propeller-duct, and develop predictive methods and tools to quantify and control the

interaction processes. Studying this interdisciplinary problem requires using different

fields including structural acoustics, hydrodynamics, fluid-structure interaction, vibra-

tion and control, etc.

3

1.1 Structural Acoustics

Structural acoustics is concerned with the coupled dynamic response of elastic

structures in contact with fluids. For heavy fluids, such as water, the coupling is two-

way, since the structural response is influenced by the fluid response, and vice versa.

For lighter fluids like air, the coupling may be deemed as in just one-way, where the

structural vibration affects the fluid response, but not vice versa.

In this work, the structural acoustic problem of interest is the vibration of submerged

thin structures, such as thin plates and thin shells. A thin shell is defined as a shell with

a thickness which is small compared to its other dimensions. A primary difference

between a shell structure and a plate structure is that, in the unstressed state, the shell

structure has curvature as opposed to plates structures which are flat.

1.1.1 Thin Plate

One of the fundamental problems of structural acoustics is that of a fluid-loaded

elastic infinite plate excited by a line or point force of harmonic time variation. The

vibration of and sound radiation from plates has been studied for many years, since a

lot of industrial structures can be modeled as a thin plate, such as aircraft and marine

structures.

A comprehensive review of thin plates of models and results present in literature

before 1970s can be found in the book of Leissa [45]. The equation of motion for fluid-

loaded plate is the equation of motion for the vacuum plate plus a term corresponding

to the force excited by the fluid. Crighton [18, 19] gave the exact expressions for a

thin fluid-loaded elastic plate driven by a line or point force. Crighton [17, 20] also

investigated the free and forced waves on a fluid-loaded elastic plate. Maidanik [49]

studied the influence of fluid loading on the radiation from infinite plates for low fre-

4

quencies. DiPerna [23, 24] developed a new method to calculate the Greens function

for a fluid-loaded elastic plate, which is accomplished by introducing a rational function

approximation of the acoustic impedance. This method is numerically efficient and the

results agree very well with the calculations which utilized numerical integration pro-

cedures.

Structures, such as aircraft and submarines, are often stiffened with ribs (beams) for

structural integrity. There have been many studies of such structures, including that of

Evseev [27] who obtained a solution in closed form by using integral transformation.

Leppington [46] and Stepanishen [64] studied the sound radiation from thin plates on

which a plane sound wave is incident by using Fourier transforms. Crighton [21] gave

a comprehensive analytical description of acoustic and vibration phenomena associated

with the interaction between plane-wave and single rib on a fluid-loaded plate. Mace

[47, 48] developed a solution for the sound radiation from a point-excited infinite elas-

tic plate stiffened by two sets of parallel stiffeners, bulkhead and frame. It was found

that the effects of stiffeners on the far field radiated pressure depend primarily on the

number of frames and the point at which the excitation is applied. Eatwell [25] devel-

oped expressions for the response of a fluid-loaded plate with ribs to a general force

distribution. It was shown that a plate stiffened with equally spaced ribs tends to pro-

duce the pass-bands and stop-bands, dependent on frequency and angle. But for the

plate stiffened with unequally spaced ribs, the structure of pass- and stop-bands may

not exist.

The vibration of thin plates is a two-dimensional problem. The study of the sound

radiation from a thin plate gives us a fundamental understanding of structural acous-

tics. More complicated acoustic and vibration phenomena will be observed from the

vibration of thin shells, which is a complex three-dimensional problem.

5

1.1.2 Thin Shell

Developing an understanding of the structural acoustic response of ribbed cylindri-

cal shell is a crucial aspect in the acoustic design and diagnostic analysis of marine

vehicles, aircraft, and a number of components of system such as reinforced piping and

support structures. The infinite thin cylindrical shells with control ribs are good theo-

retical models to study to provide some guidance in the process of understanding the

structural acoustics.

The literature on vibration of shells is extremely wide. A comprehensive review

of models and results present in literature before 1970s can be found in the book of

Leissa [44]. For an infinite thin cylindrical shell, the material of the shell is assumed to

be linearly elastic, isotropic, and homogeneous. The displacements of the shell are as-

sumed to be small, hence they yield linear equations. The shell model has been studied

by many researchers, and reported in books, Leissa [44], Timoshenko & Woinowsky-

Krieger [65], Junger & Feit [41] and Skelton & James [63]. The classical theories of

thin shells are governed by eighth order systems of differential equations which, as

shown in chaper 2, take many terms, depending upon the different assumptions made

by different theories.

Structural acoustics focuses on assumed sources of noise. A source in the cylin-

drical shell will generate several types of waves that propagate along the shell with

associated disturbances in the fluid [62]. The fluid, even non-flowing, undergoes small

amplitude vibration relative to some equilibrium position. The source excitation can

be decomposed into components of different frequencies. By taking Fourier transform,

the differential shell equations can be simplified to spectral equations. The free prop-

agating modes are obtained by solving the algebraic equation, that is, by setting the

determinant of the coefficient of the spectral equation of motion for cylindrical shell

6

to zero. This algebraic equation is called the dispersion equation. By solving this dis-

persion equation, it can be found that there are three important propagating waves in

the shell which are corresponding to flexure wave, compression wave and torsion wave.

Dispersion curves are obtained by plotting wavenumber versus frequency, which can be

used to explain the acoustic radiation from the shell. In the paper of Brazier-Smith and

Scott [9], the authors used the winding number integrals to calculate the roots of the

dispersion equation. The results show that this method gives a high degree of accuracy,

and it is applicable to solution for the zeros of any analytic function.

In the present work, we have developed a general numerical approach to find the

roots of the dispersion relation. This is because the analytical method can not be used to

find the roots of the dispersion equation of the structure with complex geometry. In our

new approach, the displacements of the shell will be solved by applying a point force

to the system. The dispersion curves are obtained by plotting displacements versus

frequency and wavenumber.

Much research has been devoted to the understanding of wave propagation in shells

for two cases, in vacuo and in fluid filled. Fuller & Fahy [30] investigated wave prop-

agation in cylindrical elastic shells filled with fluid. Dispersion curves for waves of

circumferential number m= 0 and m= 1 were derived and branches in the real, imagi-

nary and complex planes were found. The behavior of free modes was found to depend

strongly on the thickness of the shell wall, and on the ratio of the density of the shell ma-

terials to the density of the contained fluid. Further analysis of free propagating waves

in an infinite elastic duct was presented by Scott [62]. In his paper, the free modes of

propagation of an infinite fluid-loaded thin cylindrical shell has been studied and the

general properties of the roots of the dispersion relation has been derived. The roots

were identified and described as the compressional root, shear root(torsional root), and

7

real root(flexural root).

Brevart and Fuller [10] studied the vibrational energy distribution in an infinite elas-

tic shell with an internal uniform flow. Dispersion curves for both upstream and down-

stream convection were obtained. Their results show that the effect of convection is

greatest near the cut-on frequencies and near coincidence points. Brevart and Fuller

[11, 12] also derived an expression for the radial displacement of an infinite cylindri-

cal shell subjected to an impulsive line force. They investigated the energy exchange

phenomena occurring between the fluid and the shell. Their results show that a smaller

amount of energy is delivered to the fluid for higher circumferential mode excitation.

In the shell with no ribs, those free modes whose wavelengths are greater than

acoustic wavelength, i.e. the modes associated with torsion and compression, are well

coupled to the fluid and control the far-field sound radiation arising from vibrations of

the shell [57]. The free flexural modes have shorter wavelength and do not radiate to

the far field. However, as the complexity of the shell system increases, i.e. the shell

is stiffened by ribs, the number of modes well coupled to the acoustic fluid increases

significantly, and the flexure modes begin to radiate to the far field.

Stiffened shell structures have been found in many applications in engineering struc-

tures such as steel chimneys, ship hulls, submarines and so on. Structures stiffened by

ribs can achieve great economy without compromising strength and durability. The

rib models have been studied by many researchers [15, 37, 47, 66]. When the shell is

stiffened by periodically spaced identical axisymmetric ribs, the ribs are assume to ex-

ert meridional moments on the shell [63]. In addition to the axial, circumferential and

radial stresses, the shell equations need to be expanded to include the moment excita-

tion. After taking Fourier transform, the dynamics of the rib is assumed to be modeled

by a 4 4 dynamic stiffness matrix, which relates the forces and displacements at the

8

cylinder attachment points.

If the mass of the rib is assumed to be much larger than that of the shell between

two ribs, then the shell stiffened with ribs can be simplified as an infinite lattice of

identical particles of the same mass. Leon Brillouin [13] studied the wave propagation

in periodic structures. It was found that if the elementary cell of the one-dimensional

lattice contains a system with N degrees of freedom, there will be N different waves

corresponding to each wavenumber, with N different frequencies. Hence, the number

of degrees of freedom inside an elenmentary cell equals the number of branches in the

dispersion curve.

The treatment of ribs can by simplified by choosing to determine the traveling waves

in the shell [37]. The transformation property of traveling waves in a periodic system

is a result known as Blochs theorem. It means for example that the axial variation

of the shell displacement x (x) can be factorized into eiqx f (x) where f (x) is periodic,

and q is the Bloch wavenumber. This in turn means that the x-dependence of each

xi can be expressed as a Fourier series containing axial wavenumbers q+ 2pn=d with

integer values of n. The Bloch wavenumber q is defined to be the wavenumber which

lies between p=d and p=d. Each of these Fourier components represents a diffractedwave produced by the grating of ribs.

Bernblit [7, 8] developed a formulation to calculate the radiated sound from an

infinite cylindrical shell, which is stiffened on the inside by periodic ribs. The effect of

the ribs is modeled by the action of forces, assuming the height of the rib is small in

comparison with the radius of the shell. Burroughs [14] studied the far field acoustic

radiation from a point-driven, fluid-loaded circular cylinder with doubly periodic ring

stiffeners. It was found that the effect of each of the two sets of rings was included in

separated terms that were added to the solution for the unsupported cylinder.

9

Hodges et al: [37, 38] developed a theory of vibration of a cylindrical shell stiffened

by circular T-section ribs inside the shell for low frequency. The governing equations

of the shell with ribs were derived by considering the potential and kinetic energies

of the shell and ribs. The total potential energy was represented by the contribution

from the shell and the ribs. And the rib potential energy was decomposed into the an-

tisymmetric and symmetric contributions. The kinetic energy was similarly calculated

as the potential energy. The vibratory behavior of the ribbed shell was studied. For

the frequency range from zero to three times the ring frequency, good agreement were

obtained between the theoretical modeling and the experiments.

For high frequency, Vasudevan[66] developed a general formulation for a shell stiff-

ened by T-section ribs inside the shell. The equations of motion of the shell were de-

veloped using the Love-Timoshenko strain relations. And the equations of motion of

the ribs were obtained by representing the properties of the ribs at the centroid of its

cross-sectional area. The numerical results of sound radiation showed that the disper-

sion curves depend on the properties of the shell/rib system and the spacing between

the ribs. Damping can also be introduced to control sound radiation which is produced

by the action of the flexural waves.

Cuschieri [22] developed a hybrid method to obtain the spatial domain solution

of the response of a ribbed cylindrical shell. The results show this method is very

efficient computationally, especially in the case of evaluation of the response of Greens

function. Choi et al: [16] used a normal mode approach to analyze the vibration of a

submerged periodic cylindrical shell with ribs. It was found that for the equally spacing

ribs, the frequencies near the natural frequencies of the ribs can be interpreted as a pass

band, and the remaining frequency range are essentially stop bands. If the irregularities

exist in the ribs, some stop bands may overlap with pass bands. Similar results were

10

also found by Photiadis et al: [39, 40, 58, 60].

Marcus et al: [5052] used finite-element modeling to study the radiated sound

from the cylindrical shell with ribs. A finite element program called SARA2d was used

to calculate far field radiated pressure. The authors found that for higher circumferen-

tial mode numbers (m> 10), two structural resonances dominate the vibratory response

of the shell. Rib thickness variations strongly affect the first pass band, while rib spac-

ing variations strongly affect the second pass band. The authors also illustrated in a

preliminary way the importance of non-axisymmetric structure in the radiation physics

of fluid-loaded framed cylindrical shells.

Besides theoretical and numerical results, several experiments have been carried out

for the cylindrical shell with ribs. Saijyou and Yoshikawa [61] experimentally studied

the relationship between the flexural wave velocity and the excited vibration mode of a

thin cylindrical shell without fluid loading.

Photiadis .el. [57] was among the first to experimentally study wave-number space

response of a near periodically ribbed shell. The experimental results show a clear

dispersion structure dominated primarily by the Bloch wave-number of flexural wave

q and its replications by scattering from the periodic array, q+ 2pn=d. The observed

Bloch wave number differs significantly from the flexure wave number of the unribbed

shell, particularly at the lower end of the frequency spectrum. Sizable frequency gaps

are typically a dominant feature of the results. The results show that there are three

passbands. Typically there is a fairly small frequency gap separating the two lower

bands, and a significant frequency gap separating the middle and the upper bands. The

physical mechanisms underlying most of these results can be interpreted as follows.

The two lower bands are most likely due to near resonant motion of the ribs coupled by

flexural motion of the shell. The lower band is associated with out-of-plane twisting rib

11

motion and the upper band is associated with in-plane flexural rib motion. The small

frequency gap arises from the rib transmission resonance, essentially phase matching

to the flexural wave in the rib. The upper passband involves primarily reverberant flex-

ural motion of the shell and, apart from the very significant band gaps, the Bloch wave

number in this and can be predicted with reasonable accuracy by the fluid loaded flex-

ural wave number of the unribbed shell scattering into the Brillioun zone by the near

periodic array.

Photiadis .el. [59] also studied the resonant response of complex shell structures.

It was found that as the system becomes less uniform, the resonances of the system

becomes more and more local, associated with particular locations along the structure,

rather than with particular wave numbers. Generally, the effects of the increasing com-

plexity are to increase the spatial localization of vibrational energy and to increase the

number of resonant modes which are well coupled to acoustic waves in the surrounding

fluid.

1.2 Hydrodynamics

Structural acoustics focuses on assumed sources of noise, such as, single force,

monopole excitation, and dipole excitation. When sound radiation from a propeller is

investigated, the source of noise is obtained from the interaction of the flow-propeller-

duct system. The propeller, which works in a nonuniform flow, is modeled as a ro-

tor/stator stage. The shed wakes from rotor blades convect and interact with stator

blades. The common approach of computing such interaction processes is to introduce

an incident gust generated by the rotor blade, then subject the stator blades to the gust.

Although turbomachinery blades have complicated geometries, some simplified

modes were developed to investigate the rotor/stator interaction. Atassi and Hamad [36]

12

studied discrete tone sound generation in a subsonic fan subject to a three-dimensional

gust, and they treated the rotor and stator as linear cascades of thin airfoils in a rectan-

gular duct. They developed a numerical code to solve the governing integral equations

and directly give the unsteady blade pressure and the duct propagating modes. For high

frequency, Peake [56] and Glegg [31] used the Weiner-Hopf technique to solve the in-

tegral equation. Fang and Atassi [29] developed numerical models to account for the

effect of the blade loading.

In this work, the pressure field in the duct is obtained by solving Eulers equations.

The intensity of flow nonuniformities is usually small in the absence of strong shock

waves [4], so the flow quantities can be treated as steady values plus small disturbances.

This will simplify the mathematical treatments of flow motions. Because of the rotation

of rotors, it is necessary to represent the propagation of upstream disturbances in a

swirling mean flow [6]. The representation of upstream disturbances is based on the

normal mode analysis.

Normal mode analysis for the wave propagating in a swirling flow has been studied

by Golubev and Atassi [33], Kousen [42, 43], and Ali et al. [2]. The normal mode

analysis shows that there exist two distinct sets of eigenvalues at moderate subsonic

Mach numbers. One set represents the nearly-sonic pressure dominated modes with

small vorticity associated with them. The other represents the nearly convected vor-

ticity dominated modes with small pressure content. The pressure dominated modes

contain most of the pressure, even at high Mach numbers, and are used to represent the

acoustic pressure. Ali [1] also examined the effects of mean flow swirl on the genera-

tion, evolution, and propagation of unsteady disturbances. The accuracy of his solution

is excellent. He investigated the coupling among the pressure, vorticity, and entropy in

the different types of modes. The results show that the coupling between pressure and

13

vorticity is very strong. The vorticity content in the acoustic modes increases as the

Mach number increases.

In the numerical simulation for the interaction problem of the rotor/stator system,

calculations are performed on truncated domains, so non-reflecting boundary condi-

tions are necessary to avoid non-physical reflections at the far field boundaries. Accu-

rate computations of rotor/stator interaction noise depend on the accuracy of the non-

reflecting boundary conditions.

For uniform flow, Fang and Atassi [28] used a normal mode analysis to derive a

non-reflecting boundary condition. Golubev and Atassi [34, 35], and Montgomery and

Verdon [53] developed non-reflecting boundary conditions for swirling flow in an annu-

lar duct. The boundary conditions developed by Ali et al. [3] require the knowledge of

the eigenmodes, since it uses a numerical filter. The boundary conditions developed by

Montgomery and Verdon need both left and right eigenfunctions from the normal mode

analysis for general swirling flows. Elhadidi [26] developed a new boundary condition

for high frequency calculations. He used Gram-Schmidt procedure and the inner prod-

uct to calculate the acoustic pressure coefficients, thus avoided matrix inversion which

is usually ill-conditioned.

Atassi et al: [3] has derived non-reflecting boundary conditions for acoustic and

vortical waves propagating in a duct. Ali [1] implemented this non-reflecting bound-

ary condition in a three-dimensional linearized Euler code to study the propagation of

waves in a rigid duct with swirling flows. He examined the effects of swirling flow on

the generation, propagation, and stability of unsteady disturbances. He also investigated

the problem of a rotor-wake interacting with unloaded stator vanes in swirling flows.

The numerical results of scattering vortical waves in an annular duct for a uniform

flow were validated by comparison with the lifting surface theories of Namba [54] and

14

Schulten [55]. The three-dimensional calculations were compared to the strip theory

approximation. The comparison showed that strip theory is not a good approximation

for predicting the acoustics of the three-dimensional scattering problem.

Elhadidi [26] investigated the interaction of high frequency, unsteady, three dimen-

sional incident disturbances with an annular cascade of loaded blades in a rigid duct

with swirling flows. An efficient numerical model was developed and the numeri-

cal scheme was made efficient by splitting the velocity field into nearly-acoustic and

nearly-convected vortical components. This leads to a coupled set of equations, which

can be solved iteratively. Numerical results showed that steady blade loading increases

the acoustic pressure compared to the unloaded blades in swirling flows. The results

also indicated that spanwise blade loading and blade twist excite higher order acoustic

modes and may contribute significantly to the sound level. He also investigated pas-

sive noise reduction techniques by increasing rotor/stator gap, applying blade lean and

sweep and mean flow acceleration. Numerical Results indicated that blade lean and

sweep are effective means for noise reduction. It was found that significant reduction

in unsteady lift and sound pressure is obtained by increasing the gap.

In the present work, the interaction of rotor-wakes with unloaded stator vanes in

an elastic duct will be studied. The new boundary conditions at the duct wall will be

derived using the approximation of a thin shell and implemented in a three-dimensional

linearized Euler code. The coupling between fluid motion and duct vibration will be

investigated.

1.3 Fluid-Structure Interaction

In this dissertation, the effects of an elastic duct on the sound radiation from a pro-

peller inside the duct will be examined. The coupling of the duct motion to the propeller

15

Coupling

Acoustic radiation

Shell model for elastic duct

Euler model for fluid

Ribs stiffener

Elastic duct vibration

Swirling centrifugalIngested turbulence,Blade dipoles,Fluid forces:

Figure 1.2. Model development scheme for elastic duct and fluid motion.

generated flow is implemented using Eulers equations and the shell equations, which

are coupled by the boundary condition between the fluid and the duct.

Figure 1.2 shows the model development scheme for the elastic duct vibration and

fluid motion. Several types of fluid forces act on the duct: the blade generated dipole

sources, ingested turbulence and swirling centrifugal force. The elastic duct deforms

in response to the forces. The fluid motion is modeled by the Euler equations and the

duct vibration is modeled by the shell modal equations. And the radiated sound is

investigated by coupling the Euler equations and shell equations at duct boundaries.

The dissertation is organized as follows. In chapter 2, the governing equations

of the shell and rib motions and linearized Euler equations of fluid motion will be

presented. The shell equation is described by Goldenveizer & Novozhilov [63]. The

Euler equations are linearized about the steady mean flow. Then the equations are

reorganized by splitting the disturbance velocity to vortical and potential parts. At

16

last, the boundary condition at the duct radius will be derived, which couples the shell

equations and the Euler equations. In chapter 3, the sound radiation from thin plates will

be investigated. Both the regular thin plate and the thin plate with ribs are considered.

The sound radiations from regular thin shells and rib stiffened shells will be studied in

chapter 4 and chapter 5, respectively. Different excitations sources are considered, such

as single radial point force, monopole excitation, and dipole excitations. In chapter

6, the scattering phenomena in a rigid duct will first be examined, then the scattering

phenomena in an elastic duct will be studied. The sound radiation from a coupled

propeller-elastic-duct system will then be investigated. Finally, conclusions will be

given in chapter 7.

17

CHAPTER 2

MATHEMATICAL FORMULATIONS

The aim of this research is to investigate the interaction of nonuniform flows with

propeller blades in a submerged elastic duct. The fluid motion is governed by Euler

equations, and the duct vibration is modeled by shell equations. The fluid motion and

the duct vibration are coupled at the duct boundaries.

2.1 Mathematical Formulations of Elastic Shell

In this work, a model for fluid-structure interaction in a submerged elastic duct

is developed. For simplicity, the duct is modeled as an infinite thin cylindrical shell,

which is surrounded by water in a cylindrical coordinate system (x;q ;r). The geometry

is shown in Figure 2.1.

The material of the shell is assumed to be linearly elastic, isotropic, and homoge-

neous. The displacements of the shell are assumed to be small, hence they yield linear

equations. Shear deformation and rotary inertia effects are neglected. The thickness of

the shell is taken to be constant.

In the classical theory of small displacements of thin shells, the following assump-

tions were made by Love [44], which is called first approximations.

(1) The thickness of the shell is small compared with the other dimensions, for

example, the smallest radius of curvature of the middle surface of the shell.

18

ra

x

h

Fluid

Fluid

Force

Figure 2.1. Geometry of cylindrical shell excited by point force.

(2) Strains and displacements are sufficiently small so that the quantities of second

order and higher order magnitude in the strain-displacement relations may be neglected

in comparison with the first order terms.

(3) The transverse normal stress is small compared with the other normal stress

components and it can be neglected.

(4) Normals to the undeformed middle surface remain straight and normal to de-

formed middle surface suffer no extension. This means that the strains are zero in

normal direction.

2.1.1 Differential Equations of Isotropic Elastic Shell

The equations of shell motion have been studied by various researchers. In this

work, the equations of Goldenveizer & Novozhilov (which are also those of Arnold

& Warburton) are used [63]. For an isotropic cylinder, in which rotatory inertia and

19

transverse shear effects are omitted, the equations for the shell motion are presented

here as,

0BBBB@L11 L12 L13

L21 L22 L23

L31 L32 L33

1CCCCA0BBBB@

Zx

Zq

Zr

1CCCCA=0BBBB@

Ex

Eq

Er

1CCCCA ; (2.1)where,

L11 =E1 2

x2+

1n2a2

2

q 2

+rsh

2

t2;

L12 =E11+n2a 2

xq;

L13 =E1nax

;

L21 = L12;

L22 =E11n2

2

x2+

1a2

2

q 2+2b 2(1n)

2

x2+b 2

a2 2

q 2

+rsh

2

t2;

L23 =E11a2

q

b 2(2n) 3

qx2 b

2

a2 3

q 3

;

L31 =L13;

20

L32 =L23;

L33 = E1

1a2

+b 2a2 4

x4+b 2

a2 4

q 4+2b 2

4

x2q 2

+rsh

2

t2:

In these equations, Zx, Zq and Zr are the axial, tangential and radial displacements at

the cylinders mid-surface, being positive when acting in the positive directions of the

coordinates axes. Ex, Eq and Er are the axial, tangential and radial excitations, which

are the mechanical traction(forces per unit area). E1 is defined as,

E1 =Eh

1n2 ;

where, E is the Youngs Modulus, n is the Poissons ratio. And rs is shells density, h

is the thickness of the cylindrical shell, a is shells mean radius, and b 2 = h2

12a2 .

2.1.2 Modal Equations of Isotropic Elastic Shell

The shell equations are a system of three linear equations, the first equation is of

order 2, the second one is of order 3, and the third one is of order 4. We consider a

single harmonic excitation, eiwt , of frequency w , and use Fourier transform,

F(r;q ;x) =12p

m=+

m=

eimqZ +

f (r;m;a)eiaxda; (2.2)

f (r;m;a) =12p

Z 2p0

eimqZ +

F(r;q ;x)eiaxdxdq ; (2.3)

where m is the circumferential mode number and a is the axial wave number. F(r;q ;x)

is the field quantity and f (r;m;a) is the spectral field quantity. Thus the differential

equations of the shells motions, Eq. (2.1), are reduced to the spectral equations of shell

21

motions,

0BBBB@S11 S12 S13

S21 S22 S23

S31 S32 S33

1CCCCA0BBBB@

zx(m;a)

zq (m;a)

zr(m;a)

1CCCCA=0BBBB@

ex(m;a)

eq (m;a)

er(m;a)

1CCCCA ; (2.4)where, the amplitudes zx(m;a), zq (m;a), zr(m;a) are the spectral displacements; the

amplitudes ex(m;a), eq (m;a), er(m;a) are the spectral excitations. The elements in

the stiffness matrix S are given by,

S11 = E1

a2+m2

1n2a2

w2rsh;

S12 = E1(1+n)ma2a

;

S13 =iE1n aa ;

S21 = S12;

S22 = E1

(1n)a2

2+m2

a2+2a2b 2(1n)+ b

2m2

a2

w2rsh;

S23 =iE1ma2

+b 2(2n)a2m+ b2m3

a2

;

22

S31 =S13;

S32 =S23;

S33 = E1

1a2

+b 2m4

a2+a4b 2a2+2b 2m2a2

w2rsh:

2.1.3 Interior and Exterior Fluid Loading

The cylindrical shell is submerged and filled with water, and the interior and exterior

water loading are important for higher frequencies. In order to derive the formulations

for fluid loading, we consider a mechanical point force vector, F= (Fx;Fq ;Fr), located

on the cylinders surface at the coordinates (x0;q0;a). The excitation vector of Eq. (2.1)

is given by,

0BBBB@Ex(q ;x)

Eq (q ;x)

Er(q ;x)

1CCCCA=0BBBB@

Fxd (x x0)d (q q0)=aFqd (x x0)d (q q0)=a

Frd (x x0)d (q q0)=a pe(a;q ;x)+ pi(a;q ;x)

1CCCCA ; (2.5)

where pe(a;q ;x) is the exterior fluid pressure and pi(a;q ;x) is the interior fluid pres-

sure. Its spectral form is,

0BBBB@ex(m;a)

eq (m;a)

er(m;a)

1CCCCA=0BBBB@

fxei(mq0+ax0)=2pa

fqei(mq0+ax0)=2pa

frei(mq0+ax0)=2pa pe(a;m;a)+ pi(a;m;a)

1CCCCA : (2.6)

23

The governing equation for the fluid loading is given by,

1c20

D20Dt2

2p= 0; (2.7)

where p is the fluid loading, and c0 is the speed of the sound.D0Dt is the material deriva-

tive defined by,D0Dt

t

+U(x) : (2.8)

Generally, the mean swirling flow U(x) is vortical and can be assumed to be axisym-

metric, of the form,

U(x) =Ux(r)ex+Uqeq ; (2.9)

where Ux and Uq are the mean velocity components in the axial and circumferential

directions, respectively; ex and eq represent unit vectors in the axial and circumferential

directions, respectively.

Using Fourier transform Eqs.(2.2) and (2.3),

D0Dt

p= i(w+Uxa+Uqm=r)p; (2.10)

where p is the spectral form of the pressure.

Let

Lm =w+aUx+ mUqr ; (2.11)

then we obtain the reduced wave equation,

L2mc20

+2p= 0: (2.12)

24

L2mc20

=

wc0

+Mxa+mMqr

2: (2.13)

In water,Mx andMq usually are very small, so the termsMxa and mMqr can be neglected

unless the wavenumber or the mode number is very large.

When Mxa and mMqr are neglected, due to the cylinder motion, the interior spectral

pressure must be a solution of the reduced wave equation which is finite at the origin

[63]. Thus,

pi(r;m;a) = Am(a)Jjmj(g1r);

where g1 = +q

(k21a2), k1 = w=c1 and c1 is the interior fluid sound speed. Byapplying the boundary condition,

pi(r;m;a) r

= r1w2zr(m;a); at r = a;

where, r1 is interior fluid density, thus, the interior spectral pressure is obtained as,

pi(r;m;a) = r1w2zr(m;a)Jjmj(g1r)g1J0jmj(g1a)

: (2.14)

Due to the cylinder motion, the exterior spectral pressure must be a solution of the

reduced wave equation which satisfies the radiation condition at infinity. [63]. Thus,

pe(r;m;a) = Bm(a)Hjmj(g2r);

where g2 = +q

(k22a2), k2 = w=c2 and c2 is the exterior fluid sound speed. By

25

applying the boundary condition,

pe(r;m;a) r

= r2w2zr(m;a); at r = a;

where, r2 is exterior fluid density, thus, the exterior spectral pressure is obtained as,

pe(r;m;a) = r2w2zr(m;a)Hjmj(g2r)g2H 0jmj(g2a)

: (2.15)

Plug Eq. (2.6), Eq. (2.14) and Eq. (2.15) into Eq. (2.4), the spectral equations of

motion of the cylindrical shell are obtained as,

0BBBB@S11 S12 S13

S21 S22 S23

S31 S32 S33+FL

1CCCCA0BBBB@

zx(m;a)

zq (m;a)

zr(m;a)

1CCCCA=0BBBB@

fxei(mq0+ax0)=2pa

fqei(mq0+ax0)=2pa

frei(mq0+ax0)=2pa

1CCCCA ; (2.16)

where, FL is the fluid loading, which is given by,

FL= r2w2Hjmj(g1a)g2H

0jmj(g2a)

r1w2Jjmj(g1a)g1J

0jmj(g1a)

: (2.17)

From the spectral equations of motion of the cylindrical shell, it can be seen thatthere are three force terms: (1) elastic term; (2) duct inertia term; (3) fluid loading term.The scales of these force terms are shown in Table 2.1. To compare these force terms,a steel cylindrical shell is considered. The shell is submerged in water with a radiusa= 1m, and a thickness h= 0:01m. In table 2.1, frequency is normalized, w =wa=c0,where c0 is the sound speed in water. rs is the density of the shell, and rw of water. Theduct inertia term and fluid loading term are normalized by the elastic term. From table2.1, it can be seen that the elastic term is the dominant term when the reduced frequencyis very small (w 1). When the reduced frequency is larger than 1, the duct inertiaterm and the fluid loading term become the dominant ones. This table shows that thefluid loading term is very important for lager frequencies (w > 1).

26

TABLE 2.1

SCALING OF FORCES.

Reduced frequency Elastic Duct inertia Fluid loading

w = wa=c0 Eh=a2 rsw2h rwwc0

Normalization 1 (rsc20=E)w2 (rwc20=E)(a=h)w

w 1 1 (dominant term) 0:1w2 2w

w 1 1 0:1 2(dominant term)w 1 1 0:1w2 2w (dominant term)

2.2 Mathematical Formulations of Fluid Motion

The fluid motion is governed by the conservation laws of mass, momentum, and

energy, and we use the Euler equations as the governing equations[1, 26],

D0rDt

+r U= 0; (2.18)

rD0UDt

=p; (2.19)

where r , U, and p are the density, velocity, and pressure of the fluid, respectively.

The governing equations are linearized about the steady mean flow quantities,

U(x; t) = U0(x)+u(x; t); (2.20)

p(x; t) = p0(x)+ p0(x; t); (2.21)

r(x; t) = r0(x)+r 0(x; t); (2.22)

27

where x stands for the position vector, t for time, and r0, U0, and p0 are the steady

density, velocity, and pressure of the fluid, respectively, and r 0, u, and p0 are the corre-

sponding unsteady perturbation quantities such that

ju(x; t)j jU0(x)j; (2.23)

jp0(x; t)j jp0(x)j; (2.24)

jr 0(x; t)j jr0(x)j: (2.25)

Thus, the first-order continuity and momentum equations resulting from the lin-

earization are given,

D0r 0

Dt+(u )r0+r0 u+r 0 U0 = 0; (2.26)

r0D0uDt

+(u )U0+r 0(U0 U0) =p0: (2.27)

We used a velocity splitting to describe the basic modes of the disturbance [5, 32].

The disturbance velocity is decomposed into vortical and potential parts,

u= u(R)+f : (2.28)

The governing equations become,

D0Dt

1c20

D0fDt

(f) = (u(R)); (2.29)

D0u(R)

Dt+(u(R) )U0 =(U0)f : (2.30)

28

2.3 Coupling between Shell Equations and Euler Equations

For the interactions between the nonuniform flows and the propeller blades and

the elastic duct, the duct vibration is modeled by the shell modal equations and the

fluid motion is modeled by the linearized Euler equations. In order to solve this fluid-

structure interaction problem, we need to couple the shell modal equations and the

linearized Euler equations by the boundary condition at the duct radius.

LetL be the linearized Euler operator,

L (u; p0;r 0) = 0: (2.31)

In the normal mode analysis, the flow disturbances are represented as the sum of lin-

early independent modes. This can be represented as a Fourier expansion,

fu; p0;r 0g(x;r;q ; t) =Z

m=

n=1

fumn(r); pmn(r);rmn(r)gei(kmnx+mqwt)dw :(2.32)

where m and n are integer modal numbers characterizing the circumferential and radial

eigenmodes. w is the frequency, and k is the wavenumber. Thus, Eq. (2.31) reduces to,

Lr

0BBBB@umn(r)

pmn(r)

rmn(r)

1CCCCA= 0; (2.33)

with boundary condition at the hub,

ur r

rh

= 0: (2.34)

29

At the duct tip, the surface is described by the following function,

r = a+zr;

where a is the mean radius of the duct and zr is the displacement of the duct. By

applying material derivative,

D0Dt

(razr) =D0zrDt +ur = 0:

ur =D0zrDt

: (2.35)

By applying Eq. (2.32) to Eq. (2.35), the boundary condition at the duct tip is ob-

tained as,

urmn = iLmnzrmn; (2.36)

where, Lmn is an eigenvalue of the convected operator D0Dt , defined by,

Lmn =w+aUx+ mUqr : (2.37)

From the shell equation, the displacements of the shell can be calculated by,

zmn = RFmn;

where matrix R= fRi jg is the inverse matrix of stiffness matrix S, and Fmn is the exci-tation which is given by,

Fmn = f0;0; pmng:

30

Thus, the radial displacement of the shell can be obtained as,

zrmn = R33pmn;

where p is the pressure field calculated from the Euler equations.

Finally, we obtain the boundary condition at the duct radius,

urmn = iLmnR33pmn: (2.38)

31

CHAPTER 3

SOUND RADIATION FROM THIN PLATES

As stated in the previous chapters, the main objective of this research is to study the

interaction between the vibrating structures and the fluids in which the structures are

submerged. The knowledge of both the structural acoustics and the fluid mechanics is

required for this purpose. Before the vibrations of thin cylindrical shells are studied,

the vibrations of thin plates are first examined, which is a two-dimensional problem,

rather than the complex three-dimensional one as the former cases.

The sound radiation from an infinite thin plate has been studied by many researches.

The geometry of an infinite thin plate of uniform thickness h can be shown in Figure

3.1. In this chapter, the governing equation of the transverse displacement of the thin

plate, without moment loading, is given by [63],

D 4W (x;y)

x4+2

4W (x;y)x2y2

+ 4W (x;y)

y4

+rsh

2W (x;y) t2

=

Sz pu(x;y;0)+ pl(x;y;0); (3.1)

whereW (x;y) is the transverse displacement, Sz is the transverse stress traction, pu is

the pressure in the upper fluid halfspace, pl is the pressure in the lower fluid halfspace,

and rs is the plates density. D is the bending stiffness,

D= Eh3=(12(1n2));

32

My

x

z

P

h

Figure 3.1. The geometry of an infinite thin plate of uniform thickness h.

where E is the Youngs modulus and n is the Poissons ratio.

The governing equation of the displacement can be simplified by Fourier transform

in the x and y coordinates,

F(x;y;z) =1

4p2Z +

Z +

f (a;b ;z)ei(ax+by)dadb ; (3.2)

f (a;b ;z) =Z +

Z +

F(x;y;z)ei(ax+by)dxdy; (3.3)

where F(x;y;z) is the field quantity and f (a;b ;z) is the spectral field quantity. a and

b are the wavenumbers in x and y directions, respectively.

Then the time-harmonic equation of motion of the isotropic plate, without moment

loading, is,

S(a;b )W (a;b ) = Sz(a;b ) pu(a;b ;0)+ pl(a;b ;0); (3.4)

33

where S(a;b ) is the plate stiffness, which is given by,

S(a;b ) = D(a2+b 2)2w2rsh: (3.5)

3.1 Dispersion Equation

The dispersion relation is the relation between the frequency and wavenumber, at

which the resonance occurs. For an infinite plate in vacuum, the dispersion equation is,

D(a2+b 2)2w2rsh= 0: (3.6)

By solving this equation, we can always find real values of wavenumbers( a and b ) for

a given frequency(w), which means that there are always propagating modes.

When the plate is submerged in fluid, the dispersion equation becomes,

D(a2+b 2)2w2rsh iruw2

gu irlw

2

gl= 0; (3.7)

where gu = +pk2ua2b 2, ku = w=cu, where cu is sound speed in the upper halfs-

pace whose density is ru. gl =+qk2l a2b 2, kl = w=cl , where cl is sound speed in

the lower halfspace whose density is rl . When k2u and k2l are less than a2+b 2, we can

find the real roots of wavenumbers for a given frequency, which means that the fluid

acts as a mass. When k2u or k2l is less than a

2+b 2, we can not find the real roots of

wavenumbers for a given frequency, which means that the fluid acts as a resistance.

34

3.2 Singl

fluid loaded structure

Documents