UNIVERSITY OF CINCINNATI
March 8, 2001
I,
Joon-Hyun Lee
,
hereby submit this as part of the requirement for the
degree of:
Ph.D.
in:
Mechanical Engineering
It is entitled:
DEVELOPMENT OF NEW TECHNIQUE FOR DAMPING
IDENTIFICATION AND SOUND TRANSMISSION
ANALYSIS OF VARIOUS STRUCTURES
Approved by:
Dr. Jay H. Kim
Dr. David L. Brown
Dr. Milind Jog
Dr. Yijun Liu
DEVELOPMENT OF NEW TECHNIQUE FOR DAMPING
IDENTIFICATION AND SOUND TRANSMISSION ANALYSIS OF
VARIOUS STRUCTURES
A dissertation submitted to the
Division of Research and Advanced Studies
of the University of Cincinnati
In partial fulfillment of the
requirements for the degree of
DOCTORATE OF PHILOSOPHY (Ph.D.)
in the Department of Mechanical, Industrial and Nuclear Engineering
of the College of Engineering
2001
by
Joon-Hyun Lee
B.S.M.E., Yonsei University, 1988
M.S.M.E., Yonsei University, 1990
Committee Chair: Dr. Jay H. Kim
Abstract
A new experimental method to identify damping characteristics of dynamic systems and
unique analytical techniques to study sound transmission characteristics of various
structures were developed in this dissertation work.
The damping identification method developed in this work identifies damping
characteristics of a dynamic system in matrix forms from measured frequency response
functions. A theoretical example was used to validate the method and study the noise
effect on the identification results. A unique set of experimental measurements was
devised to verify the practicality of the method in real engineering applications. Some
potential applications and possible improvements of the method were discussed.
While sound transmission characteristics of structures are important basic information
in noise control, the related analysis usually becomes a very difficult task because of the
complicated interactions between the structures and acoustic media. Solution techniques
were developed to study sound transmission characteristics of various cylindrical
structures with a single wall, double walls, and double walls lined with porous material
for the first time in this study. Generally the system was idealized as an infinitely long
circular cylinder subjected to a plane incident wave. The solution technique was extended
to solve the sound transmission problems of periodically stiffened panels and cylinders.
For all cases, exact solutions were obtained by using the full shell vibration equations
coupled with the acoustic wave equations using the mode superposition method.
ii
An approximate analysis technique was proposed to calculate sound transmission
through cylindrical walls lined with porous material. Because the porous material has
both solid and fluid phases, which makes the related analysis very complicated. The
unique approximate method was developed as a two-step analysis allowing relatively
easy calculation of the sound transmission in such structures.
An analysis method for stiffened plates and stiffened circular cylindrical shells was
also developed for the first time in this work. The space harmonic expansion method was
used to solve for the periodically stiffened structures.
In all cases of the sound transmission studies, analytical solutions were compared to
corresponding measured results except for the periodically stiffened structures.
effects of important design parameters are studied to obtain useful design guidelines.
iii
The
Acknowledgements
Bring Glory and Special Thanks to God!
I would like to remember and acknowledge the many individuals who gave me very
fruitful discussions and suggestions. In particular, I would like to express my deepest
gratitude to my advisor Dr. Jay H. Kim, whose excellent guidance, brilliant ideas and
dedication played more than a significant role in the success of this endeavor. Without his
great guidance, this dissertation would not be possible. I also wish to acknowledge Dr.
David L. Brown, Dr. Milind Jog, and Dr. Yijun Liu for being my dissertation committee
members providing valuable help and guidance. I would like to thank Dr. I-Chi Wang
and Dr. Randall J. Allemang for their great teaching and thoughtful concerns.
I am truly grateful for the members of UC SDRL, in no specific order, Dr. Allyn Phillips,
Bill Fladung, Dr. Doug Adams, Matt Witter, Dan Lazor, Eric Frenz, Jeff Hylok, Amit
Shukla, and Tom Terrell who have provided me with whatever I needed, which enabled
me to enjoy my graduate student life while pursuing my degree. Especially, I wish to
thank Bruce Fouts for English proofreading and his friendship. I thank Rhonda Christman
for her excellent secretarial help during my graduate student life at UC SDRL.
My special thanks go to Yeongho Lee, Heesuk Rho, Bangyong Keum, and Jinsoo Kim
who have always been there when I needed them and provided steady, constant and great
friendship over the last 3 years.
I also wish to acknowledge Dr. Xinyu Dou and Dr. Shaohai Chen at Motorola for helping
me to find the job opportunity at Motorola.
iv
Last, but by no means least, I am most grateful to my parents and sisters for their neverending love, encouragement, support, and understanding for the last 36 years, without
which this dissertation could not have been accomplished.
This work was funded by ArvinMeritor Industries. I wish to thank Robert T. Usleman,
Dr. Han-Jun Kim, Scott Grinker, Vasudeva Kothamasu, Deming Wan, Steve Fang, Dick,
and Rick among many others at ArvinMeritor Advanced Engineering not only for the
financial support but also for the personal friendship and encouragement.
v
To My Parents
vi
Table of Contents
TABLE OF CONTENTS ..............................................................................................................................................I
LIST OF FIGURES ......................................................................................................................................................5
LIST OF TABLES .......................................................................................................................................................12
CHAPTER 1 – INTRODUCTION..........................................................................................................................13
1-1. General....................................................................................................................... 13
1-2. Structure of the Dissertation...................................................................................... 14
CHAPTER 2 – IDENTIFICATION OF DAMPING MATRICES OF DYNAMIC SYSTEMS ............16
2-1. Introduction................................................................................................................ 16
2-2. Identification Theory................................................................................................. 18
2-3. Theoretical Validation of the Procedure.................................................................... 21
2-4. Study of Error due to the Noise in FRFs ................................................................... 23
2-4.1. Relative Magnitude of Different Damping Mechanisms ........................................................................ 23
2-4.2. Study of Noise Effect .................................................................................................................................... 27
2-5. Experimental Validation of Identification Theory .................................................... 38
2-5.1. Strategy for Experimental Validation......................................................................................................... 41
2-5.2. Necessary Measurement and Signal Processing Issues ........................................................................... 42
2-6. Experimental Results ................................................................................................. 52
2-7. Conclusions ............................................................................................................... 63
CHAPTER 3 – SOUND TRANSMISSION THROUGH SINGLE-WALLED CYLINDRICAL
SHELLS .........................................................................................................................................................................65
3-1. Introduction................................................................................................................ 65
3-2. Formulation of the Problem....................................................................................... 69
3-2.1. The System Model......................................................................................................................................... 69
3-2.2. Vibro-Acoustic Equations............................................................................................................................ 69
3-3. Solution Process ........................................................................................................ 72
3-3.1. Solution of the Equations.............................................................................................................................. 72
3-3.2. Transmission Loss (TL) ................................................................................................................................ 76
1
3-4. Convergence Checking.............................................................................................. 77
3-5. Comparison to an Equivalent 1-D Model.................................................................. 80
3-5.1. Formulation of Equations............................................................................................................................. 80
3-5.2. Solution Procedure......................................................................................................................................... 82
3-5.3. Comparison of the 1-D and 2-D Solutions................................................................................................ 85
3-6. Comparison to Experimental Measurements............................................................. 85
3-7. Parame ter Studies ...................................................................................................... 89
3-7.1. Effect of the Incidence Angle ...................................................................................................................... 89
3-7.2. Effect of Different Materials ........................................................................................................................ 90
3-7.3. Radius Effects................................................................................................................................................. 91
3-7.4. Thickness Effects ........................................................................................................................................... 92
3-8. Conclusions ............................................................................................................... 93
CHAPTER 4 - SOUND TRANSMISSION THROUGH DOUBLE-WALLED CYLINDRICAL
SHELLS .........................................................................................................................................................................95
4-1. Introduction................................................................................................................ 95
4-2. Analytical Solution Procedure ................................................................................... 97
4-2.1. Sound Transmission by Bending Waves in Shells ................................................................................... 97
4-2.2. Sound Transmission Analysis by Plane Wave Model.......................................................................... 109
4-2.3. Combined Solutions .................................................................................................................................... 111
4-3. Comparison to Experimental Results ...................................................................... 112
4-3.1. Measurement Setup ..................................................................................................................................... 112
4-3.2. Single Shell Measurement.......................................................................................................................... 114
4-3.3. Double Shell Measurement ........................................................................................................................ 115
4-4. Parameter Studies .................................................................................................... 119
4-4.1. Choice of Incident Angle in Analysis....................................................................................................... 119
4-4.2. Effect of the Double-Wall Construction.................................................................................................. 119
4-4.3. Effect of Thickness...................................................................................................................................... 120
4-4.4. Effect of the Airgap ..................................................................................................................................... 120
4-5. Conclusions ............................................................................................................. 123
CHAPTER 5 – SOUND TRANSMISSION THROUGH DOUBLE-PANELS LINED WITH
ELASTIC POROUS MATERIAL....................................................................................................................... 125
5-1. Introduction.............................................................................................................. 125
5-2. Development of Simplified Analysis Method ......................................................... 127
5-2.1. Review of the Full Theory.......................................................................................................................... 127
5-2.2. Development of Approximate Analysis Procedure................................................................................ 139
5-3. Comparison of the Solutions from the Simple and Full Analyses .......................... 147
5-3.1. Formulation of the Problem....................................................................................................................... 147
5-3.2. Comparison of the Approximate Solution to the Solution from the Full Theory ............................. 158
2
5-4. Conclusions ............................................................................................................. 162
CHAPTER 6 – SOUND TRANSMISSION THROUGH DOUBLE-WALLED CYLINDERS LINED
WITH ELASTIC POROUS CORE..................................................................................................................... 164
6-1. Introduction.............................................................................................................. 164
6-2. Description of the Problem...................................................................................... 166
6-3. Derivation of the System Equation.......................................................................... 167
6-3.1. Double Shell with Bonded-Bonded Porous Material Layer................................................................. 167
6-3.2. Double Shell with Bonded-Unbonded Porous Material Layer ............................................................ 176
6-4. Calculation of Transmission Loss (TL) ................................................................... 185
6-4.1. B-B Shell ....................................................................................................................................................... 187
6-4.2. B-U Shell ....................................................................................................................................................... 190
6-4.3. U-U Shell....................................................................................................................................................... 193
6-4.4. Discussions.................................................................................................................................................... 193
6-5. Conclusions ............................................................................................................. 196
CHAPTER 7 – SOUND TRANSMISSION THROUGH STIFFENED PANELS .................................. 197
7-1. Introduction.............................................................................................................. 197
7-2. Formulation of the System Equation....................................................................... 201
7-3. Solution Procedure .................................................................................................. 206
7-3.1. Solution of the Governing Equation......................................................................................................... 206
7-3.2. The Transmission Loss (TL) Obtained from the Solution.................................................................... 210
7-3.3. Convergence of the Solution...................................................................................................................... 213
7-4. Parameter Studies .................................................................................................... 214
7-4.1. Parameters Related to Modeling ............................................................................................................... 214
7-4.2. Parameter Related to Design..................................................................................................................... 216
7-5. Conclusions ............................................................................................................. 223
CHAPTER 8 – SOUND TRANSMISSION THROUGH STIFFENED CYLINDRICAL SHELLS . 225
8-1. Introduction.............................................................................................................. 225
8-2. Formulation of the System Equation....................................................................... 227
8-2.1. Assumed Solutions ...................................................................................................................................... 229
8-2.2. Boundary Conditions at the Structure-Acoustic Interfaces .................................................................. 230
8-2.3. Equations of Motion of the System .......................................................................................................... 232
8-3. Calculation of Transmission Loss ........................................................................... 247
8-4. Convergence of the Solution ................................................................................... 252
3
8-5. Parameter Studies .................................................................................................... 253
8-5.1. Parameters Related to Modeling ............................................................................................................... 254
8-5.2. Study of Design Parameters ....................................................................................................................... 256
8-6. Conclusions ............................................................................................................. 260
CHAPTER 9 – CONCLUSIONS.......................................................................................................................... 262
9-1. Summary.................................................................................................................. 262
9-2. Contributions ........................................................................................................... 262
9-3. Recommendations for Further Research ................................................................. 263
BIBLIOGRAPHY..................................................................................................................................................... 266
4
List of Figures
Figure 2.1. Three DOF lumped parameter model............................................................. 22
Figure 2.2. Viscous damping index .................................................................................. 25
Figure 2.3. Structural damping index ............................................................................... 26
Figure 2.4. FRF of the system with 0.1% viscous and structural damping and 0.1% noise
.................................................................................................................................... 26
Figure 2.5-a. Error ratio diagram of a system with 0.1% damping and 0.1% noise (a) the
identified viscous damping......................................................................................... 29
Figure 2.5-b. Error ratio diagram of a system with 0.1% damping and 0.1% noise (b) the
identified structural damping ..................................................................................... 30
Figure 2.6. FRF of the system with 0.1% viscous and structural damping and 0.5% noise
.................................................................................................................................... 30
Figure 2.7-a. Error ratio diagram of a system with 0.1% damping and 0.5% noise (a) the
identified viscous damping......................................................................................... 31
Figure 2.7-b. Error ratio diagram of a system with 0.1% damping and 0.5% noise (b) the
identified structural damping ..................................................................................... 31
Figure 2.8. FRF of the system with 0.1% viscous and structural damping and 1.0 % noise
.................................................................................................................................... 32
Figure 2.9-a. Error ratio diagram of a system with 0.1% damping and 1.0% noise (a) the
identified viscous damping......................................................................................... 32
Figure 2.9-b. Error ratio diagram of a system with 0.1% damping and 1.0% noise (b) the
identified structural damping ..................................................................................... 33
Figure 2.10. FRF of the system with 0.5% viscous and structural damping and 0.1% noise
.................................................................................................................................... 33
Figure 2.11-a. Error ratio diagram of a system with 0.5% damping and 0.1% noise (a) the
identified viscous damping......................................................................................... 34
Figure 2.11-b. Error ratio diagram of a system with 0.5% damping and 0.1% noise (b) the
identified structural damping ..................................................................................... 34
Figure 2.12. FRF of the system with 0.5% viscous and structural damping and 0.5% noise
.................................................................................................................................... 35
5
Figure 2.13-a. Error ratio diagram of a system with 0.5% damping and 0.5% noise (a) the
identified viscous damping......................................................................................... 35
Figure 2.13-b. Error ratio diagram of a system with 0.5% damping and 0.5% noise (b) the
identified structural damping ..................................................................................... 36
Figure 2.14. FRF of the system with 0.5% viscous and structural damping and 1.0% noise
.................................................................................................................................... 36
Figure 2.15-a. Error ratio diagram of a system with 0.5% damping and 1.0% noise (a) the
identified viscous damping......................................................................................... 37
Figure 2.15-b. Error ratio diagram of a system with 0.5% damping and 1.0% noise (b) the
identified structural damping ..................................................................................... 37
Figure 2.16-a. Experimental setup (a) clamped beam without a damper ......................... 39
Figure 2.16-b. Experimental setup (b) clamped beam with a damper .............................. 39
Figure 2.17-a. Test setup (a) schematic diagram .............................................................. 40
Figure 2.17-b. Test setup (b) geometry............................................................................. 40
Figure 2.18. Implication of identifying damping in the viscous and structural damping
matrices ...................................................................................................................... 44
Figure 2.19. A typical FRF in Bode plot .......................................................................... 44
Figure 2.20. Illustration to explain the mistake to combine C and D matrices identified
using different bands .................................................................................................. 45
Figure 2.21. Phase correction of FRF:.............................................................................. 46
Figure 2.22. Single DOF system....................................................................................... 47
Figure 2.23. Illustration of damping identification using Argand plot ............................. 48
Figure 2.24-a. Errors in identified damping as a function of the phase error ................... 49
Figure 2.24-b. Errors in identified damping as a function of the phase error ................... 50
Figure 2.25-a. Errors in other identified structural parameters as a function of the phase
error ............................................................................................................................ 50
Figure 2.25-b. Errors in other identified structural parameters as a function of the phase
error ............................................................................................................................ 51
Figure 2.25-c. Errors in other identified structural parameters as a function of the phase
error ............................................................................................................................ 51
6
Figure 2.26. Phase mismatch found from the calibration................................................. 52
Figure 3.1. Schematic diagram of the single cylindrical shell: 2-D model....................... 70
Figure 3.2. Algorithm for identifying the optimum mode number ................................... 78
Figure 3.3. Mode Convergence Diagram for the single shell (Ri=1.0 m, hi=1.0 mm) at
1,000 Hz..................................................................................................................... 79
Figure 3.4. Mode Convergence Diagram for the single shell (Ri=1.0 m, hi=1.0 mm) at
10,000 Hz................................................................................................................... 79
Figure 3.5. Schematic diagram of an infinite beam model............................................... 81
Figure 3.6. TL curves of the infinite beam and single shell ............................................. 85
Figure 3.7. Experimental setup of TL measurement ......................................................... 88
Figure 3.8. Calculate TL averaged for random incident angles of the single shell
compared with measured TL ...................................................................................... 88
Figure 3.9. TL curves for the single shell with respect to incidence angle ....................... 90
Figure 3.10. TL curves for the single shell with respect to material ................................ 91
Figure 3.11. TL curves for the single shell with respect to radius .................................... 92
Figure 3.12. TL curves for the single shell with respect to thickness.............................. 93
Figure 4.1. Schematic description of the problem: 2-D model......................................... 98
Figure 4.2. TLs calculated from 2-D model at muffler condition .................................. 108
Figure 4.3. Schematic description of the problem: 1-D model....................................... 110
Figure 4.4. TLs calculated from 1-D model at muffler condition .................................. 110
Figure 4.5. Combined TLs from Figures 4.2 and 4.4. .................................................... 112
Figure 4.6. Experimental setup of TL measurement ....................................................... 113
Figure 4.7-a. Calculated TL compared with measured TL (a) single shell, source inside
.................................................................................................................................. 116
Figure 4.7-b. Calculated TL compared with measured TL (b) single shell, source outside
.................................................................................................................................. 116
Figure 4.7-c. Calculated TL compared with measured TL (c) single shell, TL averaged
for random incident angles ....................................................................................... 117
7
Figure 4.8-a. Calculated TL compared with measured TL (a) double shell, source inside
.................................................................................................................................. 117
Figure 4.8-b. Calculated TL compared with measured TL (b) double shell, source outside
.................................................................................................................................. 118
Figure 4.8-c. Calculated TL compared with measured TL (c) double shell, TL averaged
for random incident angles ....................................................................................... 118
Figure 4.9. Effect of the incidence angle in TLs of the double shell (Re ≅ Ri=0.1m,
hi=0.6mm, he=0.4mm).............................................................................................. 121
Figure 4.10. TLs of the single shell (R=0.1m, h=1.0mm) and double shell (Re ≅ Ri=0.1m,
hi=he=1.0mm) ........................................................................................................... 122
Figure 4.11. TLs of the double shell (Re ≅ Ri=0.1m) with respect to thickness
combination.............................................................................................................. 122
Figure 4.12. Effect of airgap in TLs of the double shell (Re ≅ Ri=0.1m, hi=0.6mm,
he=0.4mm) ................................................................................................................ 123
Figure 5.1. Illustration of wave propagation in the porous layer .................................... 130
Figure 5.2-a. Illustration of wave propagation in the B-B double-panel........................ 133
Figure 5.2-b. Illustration of wave propagation in the B-U double-panel........................ 134
Figure 5.2-c. Illustration of wave propagation in the U-U double-panel........................ 135
Figure 5.3. Detailed cross-sectional view of the open surface of a porous layer [48] .... 136
Figure 5.4. Detailed cross-sectional view of porous layer directly attached to a panel [48]
.................................................................................................................................. 137
Figure 5.5. Frame and shear wave contributions to the fluid and solid displacements in
the y-direction for the B-B double-panel (normalized by the strength of the airborne
wave):....................................................................................................................... 142
Figure 5.6. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the B-B double-panel (normalized by the energy of the airborne
wave):....................................................................................................................... 143
Figure 5.7. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the B-U double-panel (normalized by the energy of the airborne
wave):....................................................................................................................... 144
Figure 5.8. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the U-U double-panel (normalized by the energy of the airborne
wave):....................................................................................................................... 145
8
Figure 5.9. Simplified model of the B-B double-panel .................................................. 147
Figure 5.10. Simplified model of the B-U double-panel ................................................ 152
Figure 5.11. Simplified model of the U-U double-panel................................................ 156
Figure 5.12. Comparison of TLs of the B-B double-panel............................................. 160
Figure 5.13. Comparison of calculated TLs of the B-B double-panel............................ 160
Figure 5.14. Comparison of TLs of the B-U double-panel............................................. 161
Figure 5.15. Comparison of TLs of the U-U double-panel............................................. 161
Figure 6.1. Schematic diagram of the double shell with porous layer............................ 166
Figure 6.2. Cross-sectional view of the shell with a B-B porous layer .......................... 168
Figure 6.3. Cross-sectional view of the shell with a B-U porous layer .......................... 177
Figure 6.4. The incident, reflected, and transmitted waves of the B-U configuration in the
r-z plane .................................................................................................................... 178
Figure 6.5. Cross-sectional view of the shell with a U-U porous layer .......................... 181
Figure 6.6. The incident, reflected, and transmitted waves of the U-U configuration in the
r-z plane .................................................................................................................... 182
Figure 6.7. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the B-B double-panel configuration (normalized by the energy of
the airborne wave):................................................................................................... 188
Figure 6.8. Comparison of the calculated TLs of cylindrical double-walled shells in B-B
configuration ............................................................................................................ 189
Figure 6.9. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the B-U double-panel (normalized by the energy of the airborne
wave):....................................................................................................................... 191
Figure 6.10. Comparison of the calculated TLs of cylindrical double-walled shells in B-U
configuration ............................................................................................................ 192
Figure 6.11. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the U-U double-panel (normalized by the energy of the airborne
wave):....................................................................................................................... 194
Figure 6.12. Comparison of the calculated TLs of cylindrical double-walled shells in U-U
configuration ............................................................................................................ 195
9
Figure 6.13. Comparison of the calculated TLs of the three double-walled shells with
three porous layers ................................................................................................... 195
Figure 7.1. Schematic representation of a stiffened panel.............................................. 200
Figure 7.2. Comparison of the predicted averaged TLs between the stiffened and the
unstiffened panels..................................................................................................... 212
Figure 7.3. Comparison of the predicted TLs between the stiffened and the unstiffened
panels on which a plane wave is incident with an angle 45° ................................... 212
Figure 7.4. Coefficient convergence diagram for the stiffened panel (t=1.27mm) at 3,000
Hz ............................................................................................................................. 213
Figure 7.5. TL curves for the stiffened panel with respect to incidence angle ............... 215
Figure 7.6. TL curves for the stiffened panel with respect to phase attenuation angle .. 215
Figure 7.7. TL curves for the stiffened panel with respect to loss factor ....................... 216
Figure 7.8. TL curves for the stiffened panel with respect to stiffener mass (K t =3.6×109
N/m) ......................................................................................................................... 218
Figure 7.9. TL curves for the stiffened panel with respect to stiffener mass (K t =1.0×105
N/m) ......................................................................................................................... 218
Figure 7.10. TL curves for the stiffened panel with respect to plate material ................ 220
Figure 7.11. TL curves for the stiffened panel with respect to thickness of the panel ... 221
Figure 7.12. TL curves for the stiffened panel with respect to stiffener spacing ........... 221
Figure 7.13. TL curves for the stiffened panel with respect to rotational stiffness of the
stiffener..................................................................................................................... 222
Figure 7.14. TL curves for the stiffened panel with respect to translational stiffness of the
stiffener..................................................................................................................... 223
Figure 8.1. Schematic representation of a stiffened shell ............................................... 228
Figure 8.2. Comparison of the predicted averaged TLs between the stiffened and the
unstiffened shells ...................................................................................................... 251
Figure 8.3. Comparison of the predicted TLs between the stiffened and the unstiffened
shells on which a plane wave is incident with an angle 45° .................................... 251
Figure 8.4. TL convergence diagram for the stiffened cylindrical shell (R=0.1m, t=1.0
mm) at 3,000 Hz....................................................................................................... 253
10
Figure 8.5. TL curves for the stiffened shell with respect to incidence angle ................ 254
Figure 8.6. TL curves for the stiffened shell with respect to phase attenuation angle .... 255
Figure 8.7. TL curves for the stiffened shell with respect to loss factor......................... 256
Figure 8.8. TL curves for the stiffened shell with respect to shell material ................... 257
Figure 8.9. TL curves for the stiffened shell with respect to shell thickness.................. 258
Figure 8.10. TL curves for the stiffened shell with respect to stiffener spacing............. 259
Figure 8.11. TL curves for the stiffened shell with respect to translational stiffness of the
stiffener..................................................................................................................... 260
11
List of Tables
Table 2.1. Comparison of the identification methods: effect of noise on the identified
matrices ...................................................................................................................... 22
Table 2.2. Matrices of the 3 DOF lumped parameter system ........................................... 23
Table 2.3-a. Summary of experimental comparisons (a) summary of Tables 3 to 9 ........ 55
Table 2.3-b. Summary of experimental comparisons (b) purposes of comparisons......... 55
Table 2.4. Damping matrices identified using side band (350-440 Hz), neither phase
matched nor FRM conditioned................................................................................... 56
Table 2.5. Damping matrices identified using side band (350-440 Hz), phase matched but
not FRM conditioned ................................................................................................. 57
Table 2.6. Damping matrices identified using side band (350-440 Hz), phase matched and
FRM conditioned........................................................................................................ 58
Table 2.7. Damping matrices identified using low band (50-200 Hz), phase matched and
FRM conditioned........................................................................................................ 59
Table 2.8. Damping matrices identified using a different low band (50-120 Hz), phase
matched and FRM conditioned .................................................................................. 60
Table 2.9. Damping matrices identified using a different side band (300-480 Hz), phase
matched and FRM conditioned .................................................................................. 61
Table 2.10. Damping matrices identified using wide band (50-800 Hz), phase matched
and FRM conditioned................................................................................................. 62
Table 3.1. Physical dimensions and simulation conditions .............................................. 89
Table 3.2. Material properties........................................................................................... 90
Table 4.1. Parameters to calculate TLs of the double shell at the muffler condition ..... 108
Table 4.2. Parameters to calculate TLs of the double shell at the test condition ........... 114
Table 7.1. Dimensions of the panel and simulation conditions ...................................... 211
Table 7.2. Material properties of the stiffened panel...................................................... 219
Table 8.1. Dimensions of the cylindrical shell and simulation conditions ..................... 250
Table 8.2. Material properties of the stiffened shell ....................................................... 257
12
Chapter 1 – Introduction
1-1. General
One major accomplishment of this research is developing a new method to identify
damping characteristics of a dynamic system in damping matrices, which represent
spatial distributions of the damping as well as different damping mechanisms. Damping
parameters have been of relatively minor concern to engineers compared to other modal
parameters. Often damping characteristics are identified as modal damping ratios, which
represent equivalent effects of many different damping mechanisms without any spatial
information. This approach is only valid for small damping ratio, for low noise and low
modal density.
For these reasons, a new damping identification theory and an
experimental method are proposed in this research to identify damping characteristics of
a dynamic system in damping matrices.
The other major accomplishment of this dissertation is developing general analysis
methods for acoustic-structure interaction problems, especially for various cylindrical
structures.
Vibrations in acoustic media and elastic structures essentially involve
propagation of wave motions. In dealing with a vibro-acoustic problem based on the
wave description, the following three categories of practical problems are addressed in
this dissertation.
A single-walled or double-walled cylindrical shell is commonly used to confine
strong acoustic pressure filled within a cavity, therefore the shell must provide good noise
insulation over broad audio- frequencies.
It is intended to develop simple analytical
13
procedures for the relative comparison of design alternatives to complement detail
analysis tools based on a numerical analysis technique.
Vibro-acoustic analysis of sound transmission through structures lined with porous
material still remains to be a very difficult task in many practical applications due to
inherent complexity of the multi- wave propagation theory in elastic porous material. The
desire to find a simplified model served as the practical motivation of studying this topic.
In this dissertation, the simplified method is developed and then applied to predict the
sound transmission through a double-walled cylindrical shell sandwiching porous
material as a practical application.
For various reasons stiffened structures are often used as found in an aircraft and
building structures. Analysis for vibration characteristics of such structures is a
formidable task because of the need to model the interactions between stiffener and
structure. Vibro-acoustic analysis to study the sound transmission characteristics through
such structures obviously becomes more difficult, therefore are rarely found, mostly in
numerical work, in any reported studies. This research is substantiated by the scarcity of
the research associated with the vibro-acoustic analysis of stiffened structures.
1-2. Structure of the Dissertation
This dissertation is composed of nine chapters and organized in the following way. In
Chapter 2, a new experimental method to identify damping matrices is described. In
Chapter 3, a vibro-acoustic analysis to find sound transmission through a cylindrical
single-walled shell is presented along with the experimental results. In Chapter 4, sound
transmission through a double-walled shell is analyzed analytically and experimentally.
14
Chapters 5 and 6 present a simplified method to represent multi-wave theory in
describing an elastic porous material.
In Chapter 5, development of the simplified
method is explained and verified by comparing the TLs from the simplified method with
the reported results from the full model and measurements. In Chapter 6, the simplified
method is applied to calculate the TLs of a foam- lined cylindrical double-walled shell for
three different types of the porous core. Chapters 7 and 8 are about development of the
vibro-acoustic analysis method to calculate the sound transmission through a periodically
stiffened panel and a periodically stiffened cylindrical shell. In Chapter 7, a vibroacoustic analytical model for stiffened panels, which is supported by finite stiffness
elements at regular intervals, is developed by applying the principle of virtual work and
space harmonic expansion method. In Chapter 8, transmission of sound is analyzed for a
stiffened cylindrical shell, which is stiffened by equally spaced stiffening rings, by the
same methods as in the stiffened panel. In Chapter 9, overall contributions of this
investigation to the state of art in vibro-acoustic analysis as well as damping
identification are summarized, and future works that can be extended from this
investigation are addressed.
Because two major different studies (damping and vibro-acoustics) encompassing
seven different subjects are reported in this dissertation, the literature survey is included
in the introduction section of each chapter.
15
Chapter 2 – Identification of Damping Matrices of Dynamic Systems
2-1. Introduction
A viscous or structural damping model describes the energy loss mechanism in a
vibrating system in a simple mathematical form [1]. The modal damping or proportional
damping concept further uses an assumption that the spatial distribution of damping
follows the mode shape (modal damping) or the system geometry (proportional
damping). Such assumptions are obviously not always valid. For exa mple, when a
cantilever is assembled to its base structure, a relatively large energy loss mechanism will
exist along the interface. If the damping distribution of such a structure is known in more
detail, more accurate stress analysis of the structure will be possible, which will benefit a
high cycle fatigue (HCF) analysis of the structure (e.g., a turbine blade). In a high-speed
rotor system, different damping mechanisms have different effects on the system stability
[2-4].
Therefore, finding different damping mechanisms in respective matrices will
improve the quality of the simulation model of such a system.
In most past works, the damping matrix of a structure has been identified using FRFs
indirectly. Typically, modal parameters such as natural frequencies and modes are
extracted first, then the mass, stiffness and damping matrices using those identified
parameters [5-11]. Since damping matrices have much smaller effect on the system
responses compared to the mass and stiffness matrices, the damping matrices identified in
this manner become inaccurate. In a typical experimental modal analysis [10, 11], detail
information of the damping effect is usually not a main concern.
16
Over the past decade, extensive research activities have been made in the model
updating, in which the damping matrix is identified as a part of the result. For example,
an incremental least-squares method was used for the model updating by M. Dalenbring
[12] and H. G. Lee et al. [13]. The basic idea of model updating techniques is finding a
theoretical model whose response is best matched with the measured system response.
The damping matrices are identified to match the system response of the experimental
and theoretical models, but their uniqueness is not guaranteed.
Tsuei et al. [14-16] developed a method that works directly on FRFs to find the
damping matrices as the primary objectives of identification.
Lee and Kim [17-18]
conducted a theoretical validation of the method and related noise study, and also
attempted an experimental validation of the technique.
While working to conduct an experimental validation of the method proposed by
Tsuei, it was realized that a much simpler algorithm could be used. The method uses a
dynamic stiffness matrix (DSM), or the inverse of FRM. The method is very simple,
requiring far fewer steps of numerical operations compared to the previously used
method. Owing to this simplicity, the identification result is much less influenced by the
measurement errors and noises. A theoretical example is used to validate the algorithm,
study the noise effect on the identification results and demonstrate advantages of the new
method over the previously used method. A set of experimental measurements is devised
and conducted to validate the practicality of the method.
17
2-2. Identification Theory
The equation of motion of a n degrees of freedom (DOF) dynamic system subjected to a
harmonic input force is:
M&x&( t ) + Cx& (t ) + ( jD + K ) x( t ) = f (t ) = F (ω) e j ωt
(2.1)
where, M, C, D and K are the mass, viscous damping, structural damping and stiffness
matrices respectively, j = −1 , and x(t) and f(t) are the displacement and force vectors.
Letting x (t ) = X (ω) e j ωt , Equation (2.1) becomes:
( K − M ω 2 )+ j (ωC + D ) X (ω ) = F (ω )
(2.2)
The dynamic stiffness matrix (DSM) is defined as:
[ H ( ω ) C ]−1 = ( K − M ω 2 ) + j (ωC + D )
(2.3)
where, H C (ω ) is the frequency response matrix (FRM) defined as:
H C (ω ) = H ijC = X i / Fj
i , j = 1,2,3,...
(2.4)
In Equation (2.4), the superscript C is to indicate the variable is a complex quantity, and
H ijC is the frequency response function (FRF) measured between the nodes i and j.
Because FRM is much easier to measure than the DSM, the DSM is obtained by inverting
the measured FRM.
If the DSM is available, Equation (2.3) can be rewritten:
imag ( H C (ω ) −1 ) = ωC + D
(2.5)
real ( H C (ω ) −1 ) = K − ω 2 M
(2.6)
18
where, imag and real stand for the imaginary and real parts respectively. For example,
imag ( H C (ω ) −1) is the matrix composed of the imaginary part of the DSM matrix
H C (ω ) −1 . Equations (2.5) and (2.6) can be put into:
[I
D
ω ] = imag (H C (ω ) −1 )
C
(2.7)
where, I is an n × n identity matrix, and:
I
K
−ω 2 = real ( H C (ω )−1 )
M
(2.8)
Therefore, the system damping matrices C and D can be found by a pseudo-inverse
procedure of Equation (2.7) as follows:
+
D
C
2 n ×n
imag ( H C (ω1 ) −1 )
I ω1 I
I ω I
C
−1
2
imag ( H (ω2 ) )
= .
.
.
.
.
.
I ω k I kn n imag ( H C (ω k ) −1 )
×2
kn× n
(2.9)
where, + means the pseudo-inverse of the matrix. If necessary, the stiffness and mass
matrices can also be found:
K
M
2 n× n
I
I
= .
.
I
+
−ω12 I
real ( H C (ω1 )−1 )
C
−1
−ω 22 I
real ( H (ω 2 ) )
.
.
.
.
2
C
−1
−ω k I kn ×2 n real ( H (ω k ) ) kn ×n
(2.10)
Equations (2.9) and (2.10) have to be set up at least at two frequencies (k=2) to make the
equations solvable. Usually the equations are over-determined by using more frequencies
than needed.
19
As it was shown, the procedure itself is surprisingly simple, looking almost like an
obvious identity. However the author could not find any previous works that used this
relationship to find damping matrices. The procedure proposed by Tsuei et al. [14-16],
which also finds the damping matrices from measured FRFs, may be compared to the
proposed method. In the method, C and D matrices are found by solving the following
equation.
C
ω H N (ω ) H N (ω ) = G(ω )
D
(2.11)
where, H N (ω ) the normal FRF, which is defined as:
[
H N (ω ) = K − Mω 2
]
−1
(2.12)
The normal FRF is obtained as:
H N (ω ) = H RC ( ω ) + H CI (ω ) H RC (ω ) HIC (ω )
−1
(2.13)
where, subscripts I and R stand for the imaginary and real parts, respectively, and
H RC (ω ) is the inverse of the real part of the FRM, i.e. ( real ( H C ) ) .
−1
−1
G (ω ) is defined as:
G (ω ) = − H IC ( ω ) H RC (ω )
−1
(2.14)
The above method is obviously more involved. The objectives of the identification,
elements of the damping matrices, have physically small effect on the FRM, therefore
each of these extra steps amplifies the effect of measurement errors or noises.
If only the viscous damping is used in the modeling, an equivalent viscous damping
matrix Ceq, that represents the entire energy loss mechanism system, can be obtained by
solving:
20
+
Ceq n× n
imag ( H C (ω1 ) −1 )
ω1 I
ω I
C
−1
imag ( H (ω2 ) )
2
= .
.
.
.
ω k I
imag ( H C (ω k ) −1 )
kn ×n
kn ×n
(2.15)
If only the structural damping is used in the model, an equivalent structural damping
matrix Deq can be obtained by solving:
+
Deq n ×n
I imag ( H C (ω1 ) −1 )
I imag ( H C (ω ) −1 )
2
= .
.
.
.
I imag ( H C (ω k ) −1 )
kn ×n
kn ×n
(2.16)
2-3. Theoretical Validation of the Procedure
A 3 DOF system shown in Figure 2.1 was used to compare the proposed method with the
Tsuei’s method. The 3 DOF system shown in Figure 2.1 is defined by the lumped masses
m1 , m2 and m3 of 10 kg, 14 kg and 12 kg, and the spring constants k 1 , k 2 , and k 3 of 2,000
N/m, 3,000 N/m and 2,500 N/m, the viscous damping coefficients c1 , c2 , and c3 of 20
N⋅s/m, 30 N⋅s/m and 25 N⋅s/m, and the structural damping coefficients d1 , d2 and d3 of
100 N/m, 150 N/m and 200 N/m respectively. Table 2.1 compares the identified results
from the two methods when 0.5% random noises are mixed in FRFs for two cases, when
the FRM is conditioned and not conditioned. Conditioning FRM involves making the
matrix symmetric, to utilize the fact that the FRM is theoretically symmetric. Section 25.2.5 can be referred to for the effect of this conditioning. The comparison shows that
the result from the new method is much less sensitive to the measurement noise, giving
much better identification results. If the FRM is conditioned, the new method identifies
21
the matrices in symmetric forms, however the other method does not, which indicates
that extra steps in the latter deteriorate the accuracy.
K2
K1
C1
D1
M1
K3
C2
D2
x1 (t)
M2
C3
M3
D3
x2 (t)
x3 (t)
Figure 2.1. Three DOF lumped parameter model
Table 2.1. Comparison of the identification methods: effect of noise on the identified
matrices
Estimation of Damping Matrices
From the Theoretical Data with 0.5% Noise
Estimation
Viscous Damping [C]
Structural Damping [D]
Method
50
-30
0
250
-150
0
Theoretical
-30
55
-25
-150
350
-200
Matrix
0
-25
25
0
-200
200
38.8
-19.1
-6.4
544.5
-431.4
148.3
Tsuei’s Method
-15.7
39.4
-17.4
-608.3
816.9
-431.0
(Unconditioned)
-13.2
-10.9
17.8
263.3
-475.0
337.3
49.6
-29.3
1.3
254.0
-158.8
-21.3
New Method
-30.6
52.3
-21.2
-141.9
387.3
-260.7
(Unconditioned)
-0.1
-26.8
22.4
4.5
-175.9
239.1
45.0
-21.9
-1.1
374.5
-361.6
19.6
Tsuei’s Method
-23.7
42.5
-23.9
-361.7
716.0
-248.3
(Conditioned)
-6.3
-13.9
23.5
129.7
-418.8
236.3
49.7
-30.0
0.6
253.4
-150.5
-9.2
New Method
-30.0
52.6
-24.0
-150.5
381.6
-218.5
(Conditioned)
0.6
-24.0
22.8
-9.2
-218.5
232.6
22
2-4. Study of Error due to the Noise in FRFs
The 3 DOF system shown in Figure 2.1 is defined by the viscous damping coefficients c1 ,
c2 , and c3 of 2 N⋅s/m, 3 N⋅s/m and 2.5 N⋅s/m with the same lumped masses, the stiffness
constants and the structural damping coefficients as in Section 2-3. The elements of the
mass, viscous damping, stiffness and structural damping matrices of the system are
calculated as in Table 2.2.
Table 2.2. Matrices of the 3 DOF lumped parameter system
Mass Matrix
10
0
0
(kg)
[M]
0
14
0
0
0
12
Viscous Damping
Matrix
(N⋅s/m)
[C]
5
-3
0
-3
5.5 -2.5
0
-2.5 2.5
Stiffness Matrix
5000
-3000
0
(N/m)
[K]
-3000
5500
-2500
0
-2500
2500
Structural Damping
Matrix
(N/m)
[D]
250
-150
0
-150
350 -200
0
-200 200
2-4.1. Relative Magnitude of Different Damping Mechanisms
It is expected that the effect of the noise on the accuracy of the damping identification
will be dependent on not only the noise level but also the magnitude of the damping. For
example, if the structure is heavily damped, the objects of the identification (elements of
the damping matrices) are large, therefore the accuracy of the identification will be less
sensitive to the noise level. Therefore, it is necessary to cross-compare the magnitudes of
the damping and the noise level. Comparing the noise level with the damping ratio
makes sense because both are non-dimensional parameters. For exa mple, we may say
that a 2% noise is large compared to a 1% damping ratio. Since the damping ratio is the
concept based on the proportional viscous damping that does not have any spatial
23
information, it is necessary to define a new concept to assess the relative magnitude of
the elements of general damping matrices.
The damping forces induced by the viscous and structural damping mechanisms
associated with specific degrees of freedom i and j can be considered as ω Cij and Dij .
Therefore, if one defines a frequency matrix such as:
D
ω ij = ij
Cij
(2.17)
Each element of this matrix represents the frequency below which the structural damping
effect is bigger than that of the viscous damping. For example, ω 11 of the system in our
example is found to be 50 rad/s, which means that below 50 rad/s the effect of the
structural damping is a more dominant energy dissipation mechanism for the motion
induced at the node 1 by the excitation force applied at the node 1. Figures 2.2 and 2.3
show interesting concepts: the viscous damping index and the structural damping index,
which are calculated for the system being studied. Figure 2.2 (Figure 2.3) plots the
percentage of the elements of the matrix [ ωij ], whose values are lower (higher) than the
frequency used as the abscissa. Therefore, for instance, the structural (viscous) damping
index is 45 % (55 %) at 10 Hz in Figure 2.3, which can be interpreted as approximately
45 % (55 %) of the system DOFs (4 out of 9 elements in this case) is damped more
structurally than viscously. Therefore, from Figures 2.2 and 2.3, it can be said that at or
above 11 Hz the system is damped primarily by the viscous damping mechanism, and at
or below 8 Hz by the structural damping. The frequency range in between may be
considered as the transition range.
24
Now, the concept of the damping ratio [1] is generalized. The elements of the
viscous and structural damping matrices are defined in terms of the damping ratio as
follows.
cij = 2 × ζ ij × mij × kij , i , j = 1,2,3.
(2.18)
dij = cij × ω ij , i, j = 1,2,3.
(2.19)
The above equations are used to relate the general damping matrices to the damping ratio.
For example, 1% viscous damping matrix can be determined by Equation (2.18) using ζij
= 0.01.
Then, using the viscous damping elements derived as such, 1% structural
damping elements are determined by Equation (2.19). Figure 2.4 compares H22 obtained
for the system with 0.1% structural and viscous damping defined in this way with 0.1 %
noise and 0 % noise.
Pecentage of Viscous damped Elements (%)
100
90
80
70
60
50
40
30
20
10
6
7
8
9
10
11
Frequency(Hz)
Figure 2.2. Viscous damping index
25
12
13
Pecentage of Structural damped Elements (%)
90
80
70
60
50
40
30
20
10
0
6
7
8
9
10
11
12
13
Frequency(Hz)
Figure 2.3. Structural damping index
10
10
Amplitude
10
10
10
10
10
0
-1
-2
-3
-4
-5
-6
0
10
20
30
40
50
Frequency (rad/sec)
Figure 2.4. FRF of the system with 0.1% viscous and structural damping and 0.1% noise
------, without noise ;
, 0.1% noise
26
2-4.2. Study of Noise Effect
The 3 DOF systems with the same M and K matrices but with two different levels of
damping ratios, 0.1% and 0.5%, are considered. For each case, 9 FRFs are calculated, to
which three levels of the random noises, 0.1%, 0.5% and 1%, are added. This results in 6
combinations of different noise levels relative to the damping levels. For example, the
combination of 0.1 % damping and 1 % noise represents a system, which has very small
damping, and whose measured FRFs are significantly tainted with noises. In this case an
accurate identification is not expected. Further, an error vector is defined, whose each
element is the difference between the exact value and the identified value divided by the
largest element of the damping matrix. For this example, which has 9 matrix elements,
the error vector is defined in % unit:
E1 ( c11 N
E ( c
2 12 N
E3 (c13 N
E4 ( c21 N
E5 = ( c22 N
E ( c
6 23 N
E7 ( c31 N
E ( c
8 32 N
E9 ( c33 N
− c11 )
− c12 )
− c13 )
− c21 )
100
×
− c22 )
max([ cij ])
− c23 )
− c31 )
− c32 )
− c33 ) Extracted
(2.20)
where, subscript N indicates the exact value.
Figure 2.5 shows the identification error defined as above for the case with 0.1 %
damping and 0.1 % noise. When Equation (2.9) is applied to identify damping matrices,
one can use FRFs at frequencies over the entire range or at frequencies around the
resonance peaks. The result obtained using the entire frequency range is marked as
“entire frequency range” and that obtained using the freque ncies near resonance
27
frequencies is marked as “peak band” in Figure 2.5 and other figures to follow.
Considering the fact that the damping effect is more pronounced at frequencies near the
resonance peaks, it was expected that peak band cases would provide better results. This
turns out to be not true for the structural damping, because its effect is predominantly in
the low frequency range.
Figure 2.5 shows that the maximum error as we defined is about 100 % when both the
damping ratio and noise are small but the same order (0.1 % and 0.1 %). Figures 2.6 and
2.7 show the case with 0.1 % damping ratio and 0.5 % noise level. Figures 2.8 and 2.9
show the case with 0.1 % damping and 1 % noise. As one can see from very large errors
in these figures, if the noise is significantly larger than the damping ratio, 5 to 1 and 10 to
1 in these cases, the identification result becomes useless.
Figures 2.10 and 2.11 are when the damping is large compared to the noise, 0.5%
damping and 0.1% noise. In this case, accurate results are obtained with maximum error
less than 10%. Figures 2.12 and 2.13 are for the case when both the damping and noise
are large but at the same level: 0.5% and 0.5%. The estimation error becomes almost the
same order (maximum is about 100 %) as in the case in Figure 2.5. Figures 2.14 and 2.15
are for the case when the damping is relatively large (0.5%), but the noise is even larger
(1%). Again, Figure 2.15 shows that the identification result contains too large errors to
be useful. General conclusions can be made from the error study.
•
The accuracy of the direct damping identification algorithm developed in this
work depends on the magnitude of the noise relative to the damping magnitude.
The identification method works accurately if the noise level is the same as or
lower than the damping ratio.
28
•
Using the data from FRFs only around the resonance peak improves the accuracy
of the identified viscous damping matrix slightly but not the structural damping
matrix.
150
Difference Ratio(%)
100
50
0
-50
-100
-150
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.5-a. Error ratio diagram of a system with 0.1% damping and 0.1% noise (a) the
identified viscous damping
, peak band;
, entire frequency range
29
100
Difference Ratio(%)
50
0
-50
-100
-150
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.5-b. Error ratio diagram of a system with 0.1% damping and 0.1% noise (b) the
identified structural damping
, peak band;
, entire frequency range
10
10
Amplitude
10
10
10
10
10
0
-1
-2
-3
-4
-5
-6
0
10
20
30
40
50
Frequency (rad/sec)
Figure 2.6. FRF of the system with 0.1% viscous and structural damping and 0.5% noise
------, W/O noise;
, 0.5% noise
30
5
7
x 10
6
5
Difference Ratio(%)
4
3
2
1
0
-1
-2
-3
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.7-a. Error ratio diagram of a system with 0.1% damping and 0.5% noise (a) the
identified viscous damping
, peak band;
, entire frequency range
5
6
x 10
4
Difference Ratio(%)
2
0
-2
-4
-6
-8
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.7-b. Error ratio diagram of a system with 0.1% damping and 0.5% noise (b) the
identified structural damping
, peak band;
, entire frequency range
31
10
10
Amplitude
10
10
10
10
10
0
-1
-2
-3
-4
-5
-6
0
10
20
30
40
50
Frequency (rad/sec)
Figure 2.8. FRF of the system with 0.1% viscous and structural damping and 1.0 % noise
------, W/O noise;
, 1.0% noise
4
12
x 10
10
Difference Ratio(%)
8
6
4
2
0
-2
-4
-6
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.9-a. Error ratio diagram of a system with 0.1% damping and 1.0% noise (a) the
identified viscous damping
, peak band;
, entire frequency range
32
4
x 10
5
Difference Ratio(%)
0
-5
-10
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.9-b. Error ratio diagram of a system with 0.1% damping and 1.0% noise (b) the
identified structural damping
, peak band;
, entire frequency range
10
-1
-2
10
Amplitude
-3
10
-4
10
10
10
-5
-6
0
10
20
30
40
50
Frequency (rad/sec)
Figure 2.10. FRF of the system with 0.5% viscous and structural damping and 0.1% noise
------, W/O noise;
, 0.1% noise
33
15
10
Difference Ratio(%)
5
0
-5
-10
-15
-20
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.11-a. Error ratio diagram of a system with 0.5% damping and 0.1% noise (a) the
identified viscous damping
, peak band;
, entire frequency range
15
Difference Ratio(%)
10
5
0
-5
-10
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.11-b. Error ratio diagram of a system with 0.5% damping and 0.1% noise (b) the
identified structural damping
, peak band;
, entire frequency range
34
10
-1
-2
10
Amplitude
-3
10
-4
10
10
10
-5
-6
0
10
20
30
40
50
Frequency (rad/sec)
Figure 2.12. FRF of the system with 0.5% viscous and structural damping and 0.5% noise
------, W/O noise;
, 0.5% noise
80
Difference Ratio(%)
60
40
20
0
-20
-40
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.13-a. Error ratio diagram of a system with 0.5% damping and 0.5% noise (a) the
identified viscous damping
, peak band;
, entire frequency range
35
40
Difference Ratio(%)
20
0
-20
-40
-60
-80
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.13-b. Error ratio diagram of a system with 0.5% damping and 0.5% noise (b) the
identified structural damping
, peak band;
, entire frequency range
10
-1
-2
10
Amplitude
-3
10
-4
10
10
10
-5
-6
0
10
20
30
40
50
Frequency (rad/sec)
Figure 2.14. FRF of the system with 0.5% viscous and structural damping and 1.0% noise
-----, W/O noise;
, 1.0% noise
36
2500
Difference Ratio(%)
2000
1500
1000
500
0
-500
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.15-a. Error ratio diagram of a system with 0.5% damping and 1.0% noise (a) the
identified viscous damping
, peak band;
, entire frequency range
800
600
400
Difference Ratio(%)
200
0
-200
-400
-600
-800
-1000
-1200
1
2
3
4
5
6
7
8
9
Element Number
Figure 2.15-b. Error ratio diagram of a system with 0.5% damping and 1.0% noise (b)
the identified structural damping
, peak band;
, entire frequency range
37
2-5. Experimental Validation of Identification Theory
The fact that an experimental identification method is working in a theoretical problem is
meaningless unless it also works in real experimental cases. An experimental validation
will be necessary to prove the practicality of the method. However, the difficulty in this
case was finding a dynamic system whose exact (or theoretical) damping matrices are
known. If such a system existed, the validation can be done in a much similar way as the
theoretical validation discussed in the previous section, comparing experimentally
identified damping matrices with the theoretical matrices. Not knowing such a system,
an indirect, partial validation of the identification method was devised.
Figure 2.16 shows two systems used in this experiment, a beam configured in two
different ways. The system shown in Figure 2.16-a is a uniform width beam whose ends
are clamped. The system in Figure 2.16-b is obtained by attaching a viscous damper to
the beam shown in Figure 2.16-a.
Four nodal points are used to define the system as
shown in Figure 2.17, which means that the damping matrices will be identified as 4 by 4
matrices. The viscous damper in the latter system was attached between the nodes 3 and
4 as shown in the figure. Accelerations are measured at four nodal points, which are
integrated twice to make the FRFs in terms of compliances. The multi-reference- impacttesting (MRIT) scheme [19] was used to obtain FRFs.
Roving the excitation to each
nodal point, 16 FRFs are obtained, which comprise a 4 × 4 FRM. The FRM is inverted to
obtain the DSM.
38
Figure 2.16-a. Experimental setup (a) clamped beam without a damper
Viscous Damper
Figure 2.16-b. Experimental setup (b) clamped beam with a damper
39
102
Amplitude
101
1
2
3
101-
Computer
Impact Hammer
100
102- 0
100
200
300
400
500
Frequency (rad/sec)
600
700
800
4
HP-VXI
Data-Acquisition
Main Frame
Clamped Beam
Figure 2.17-a. Test setup (a) schematic diagram
L=270 mm
d=54 mm
t=6 mm
Accelerometer
1
2
3
4
Viscous Damper
Figure 2.17-b. Test setup (b) geometry
40
W=19 mm
2-5.1. Strategy for Experimental Validation
The validation strategy is essentially to observe if the identified damping matrices
properly reflect the underlying physics and the configurations of the two models,
especially if the following facts are observed.
1. The diagonal elements of the damping matrices are positive.
2. The system with the damper (Figure 2.16-b) shows larger damping matrices,
especially viscous damping matrix, than the system without a damper.
3. The elements of the damping matrices of the system with a damper
corresponding to the nodes 3 and 4 are relatively large.
Satisfying above conditions is only a partial validation of the identification theory by
itself. However, because the identification algorithm itself was validated theoretically,
this partial validation is considered enough from the practical standpoint.
Besides the above three conditions, one may be tempted to use the symmetry of the
damping matrices as another observation point. Damping matrices will be identified in
symmetric forms if the FRM is symmetric. The FRM, which is theoretically a symmetric
matrix, is measured slightly non-symmetric. This deviation from the symmetry can be
considered the reflection of the quality of the measurement. Therefore, the symmetry of
the damping matrices may be useful to evaluate the quality of the measurement but not
the quality of the identification. Even for that purpose, using the FRM will be a better
option. The FRM may be conditioned to a symmetric form, which seems to improve the
identification result significantly as will be explained in Section 2-5.2.5.
41
2-5.2. Necessary Measurement and Signal Processing Issues
Several measurement and signal processing issues, some of whic h are not important in a
conventional modal analysis, were realized to be critical in damping matrices
identification after many trial and errors during the experiment. These technical issues
will be explained one by one.
2-5.2.1. DOFs of the Experimental Model
The dimension of damping matrices to be identified is determined by the DOFs of the
experimental model. For example, if FRFs are measured at four nodes as shown in
Figure 2.17, the matrices are identified as 4 × 4 matrices. Using more DOFs would
provide better spatial resolution of the damping information, however at the cost of
increased experimental effort. Also, more parameters (elements of damping matrices) to
be found will require higher accuracy in the measurement. DOFs of the experimental
model will have to be determined considering the necessary spatial resolution and
practical limitations.
2-5.2.2. Selection of Frequency Range
Modeling the system equation using C and D matrices implies that the damping force is
modeled as a linear function of frequency. As Figure 2.18 illustrates, the identification
process can be considered trying to find a best fitting straight line from scattered
experimental data points representing the damping force. From the figure, it is easy to
see tha t the matrix D will be found more accurately if the FRM data are taken from the
low frequency range to form the identification equation (Equation (2.9)). However,
accelerometers generally have poor accuracy in the low frequency range, which is further
42
deteriorated when integrating acceleration to displacement. Figure 2.19 is one of the
measured FRFs, which shows that the data below 50 Hz are not accurate.
The damping effect on the system response is more pronounced around the resonance
frequency (about 383 Hz in this case as seen in Figure 2.19). Therefore, the measured
data have effectively higher signal to noise ratios around the resonance frequency. This
is why side bands (frequency ranges between half power points) have been used in
damping identifications. Considering these facts, the frequency range was chosen as
follows in this work.
•
Data below 50 Hz are discarded.
•
The low frequency range is defined as 50 Hz to 200Hz. Using data from this
range is expected to provide a more accurate D matrix.
The side band was observed as 378 Hz to 389 Hz for the undamped beam and 374 Hz to
421 Hz for the damped beam. As a compromise (and also needing over-determination of
the identification equation), the side band in this experiment is defined as the range 350
Hz to 440 Hz. Using this band is expected to provide a more accurate C matrix. Using
the frequency range of interest of the particular problem may also be an option, especially
in practical situations.
One may be tempted to combine the C matrix identified by using the side band and
the D matrix identified using the low frequency band. Figure 2.20 illustrates the problem
in this approach, which will overestimate or underestimate the damping force.
43
Fd
Best fitting line: Deq
C
1
Deq
Best fitting line: Cω+D
D
Ceq
1
Best fitting line: Ceqω
ω
Figure 2.18. Implication of identifying damping in the viscous and structural damping
matrices
Amplitude
10
-2
-4
Bad Response Signal
10
-6
10
-8
10
0
100
200
300
400
500
600
700
800
500
600
700
800
Frequency (Hz)
Phase Angle(degree)
0
-50
Bad Phase
Signal
-100
-150
-200
0
100
200
300
400
Frequency (Hz)
Figure 2.19. A typical FRF in Bode plot
44
Imag(HC(ω)-1 )
C2
=Cω
ω +D
1
1
D1
C1
C2
1
D2
ω
Figure 2.20. Illustration to explain the mistake to combine C and D matrices identified
using different bands
, identified using low band; -----, identified using side band;
,
combined model
2-5.2.3. Sign Convention of FRFs
The sign convention of FRFs is not important in a typical modal analysis as long as it is
used consistently. In other words, consistent use of either -X/F or X/F would not cause
any problem in finding natural frequencies and mode shapes.
However, in the
identification procedure explained in Section 2-2, using –X/F will reverse the sign of the
matrices to be identified.
Especially, mixed use of sign conventions will make the
identification result invalid. The problem can be avoided by making the acceleration and
excitation directions the same at all measurement points. For a point where this is not
possible, the phase angle of the corresponding FRF has to be corrected numerically. In
practice, it will be prudential to check all FRFs and make sure that they all start with zero
phase angle at the low frequency range by adding or subtracting 180o if necessary. Figure
2.21 shows such a correction that we did for one of the measured FRFs.
45
Amplitude
10
-2
-4
10
10
10
-6
-8
0
100
200
300
400
500
Frequency (Hz)
600
700
800
0
100
200
300
600
700
800
Phase Angle(degree)
200
100
0
-100
-200
400
500
Frequency (Hz)
Figure 2.21. Phase correction of FRF:
, before correction; ----- , after correction
2-5.2.4. Phase Matching between the Force and Motion Transducers
Because the identification uses the imaginary part of the DSM, the FRFs have to be
obtained with accurate phase angle, which requires an accurate phase matching between
the force and motion transducers. Initially the importance of the phase matching was not
realized because it seldom becomes an important issue in conventional modal testing.
This problem can be best explained by a single DOF example shown in Figure 2.22. The
FRF of the system is:
X ( K − ω 2 M ) − j (ω C + D)
H (ω ) = =
F ( K − ω 2 M ) 2 + (ω C + D )2
(2.21)
Figure 2.23 shows the Argand plot [1] of this FRF. As the frequency increases, the plot
starts from 1/K in the real axis and crosses the imaginary axis at point P, whose
coordinate, −
1
, is used to find the equivalent viscous damping Ceq = ω C + D .
(ω C + D)
46
Now, suppose that there is a phase angle error of φ radian between the force and
displacement signals. The FRF will be measured as:
H (ω ) =
( K − ω 2 M ) − j (ωC + D ) jφ
e
( K − ω 2 M ) 2 + (ωC + D) 2
( K − ω 2 M ) + φ (ωC + D) + j [( K − ω 2 M )φ − (ωC + D) ]
≅
( K − ω 2 M ) 2 + (ω C + D) 2
(2.22)
The Argand curve crosses the imaginary axis when the real part becomes zero, therefore:
K − ω 2 M = − φ (ω C + D)
(2.23)
By substituting this into Equation (2.22), it is realized that the coordinate of this point
remains the same because:
H (ω ) =
−(1 + φ 2 )(ωC + D)
1
=−
2
2
(1 + φ ) (ωC + D)
(ωC + D)
(2.24)
K1 =5000 N/m
M 1 =10 kg
C1 =20 N⋅s/m
D1 =250 N/m
x1 (t)
Figure 2.22. Single DOF system
47
1/K
Real
H(ω)
−
1
(ω C + D)
P
Imaginary
Figure 2.23. Illustration of damping identification using Argand plot
Therefore, it is seen that the phase mismatch would not affect the damping parameter in
this method.
To see the effect of the phase mismatch on the damping matrices identified by the
proposed method, let the parameters M, K, C and D of the system in Figure 2.22 be 10
Kg, 5,000 N/m, 20 N⋅s/m and 250 N/m respectively. Then the proposed method
(Equation (2.9)) is applied to find C, D, M and K for various phase mismatches (φ
radian). Figures 2.24-a and b show the errors in the identified C and D matrices as
functions of the phase angle error in percentage, i.e., error /exact value × 100. Figures
2.25-a,b,c represent the errors in the identified M, K, and the natural frequency. As it is
shown, the phase mismatch causes much larger errors in C and D compared to other
modal parameters.
48
Figure 2.26 shows the phase angle between the signals from the force transducer and
one of the accelerometers using the ratio calibration setup [19]. The phase mismatch in
Figure 2.26 is compensated numerically at each fr equency to correct 4 FRFs obtained
from this set of the accelerometers and force transducer. All 16 FRFs are reconstructed
in this way before they are used to identify the damping matrices.
800
600
Error Ratio(%)
400
200
0
-200
-400
-600
-800
-20
-15
-10
-5
0
5
10
15
20
Phase Shift (Degree)
Figure 2.24-a. Errors in identified damping as a function of the phase error
-----, exact value;
, identified value, (a) viscous damping C
49
1500
1000
Error Ratio(%)
500
0
-500
-1000
-1500
-20
-15
-10
-5
0
5
10
15
20
Phase Shift (Degree)
Figure 2.24-b. Errors in identified damping as a function of the phase error
-----, exact value;
, identified value, (b) structural damping D
1
0
-1
Error Ratio(%)
-2
-3
-4
-5
-6
-7
-8
-20
-15
-10
-5
0
5
10
15
20
Phase Shift (Degree)
Figure 2.25-a. Errors in other identified structural parameters as a function of the phase
error
-----, exact value;
, identified value, (a) mass M
50
1
0
-1
Error Ratio(%)
-2
-3
-4
-5
-6
-7
-8
-9
-20
-15
-10
-5
0
5
10
15
20
Phase Shift (Degree)
Figure 2.25-b. Errors in other identified structural parameters as a function of the phase
error
-----, exact value;
, identified value, (b) stiffness K
0.1
Error Ratio(%)
0.05
0
-0.05
-0.1
-0.15
-20
-15
-10
-5
0
5
10
15
20
Phase Shift (Degree)
Figure 2.25-c. Errors in other identified structural parameters as a function of the phase
error
-----, exact value;
, identified value, (c) natural frequency ωn
51
5
4
3
Phase (degree)
2
1
0
-1
-2
-3
-4
-5
0
100
200
300
400
500
600
700
800
Frequency (Hz)
Figure 2.26. Phase mismatch found from the calibration
2-5.2.5. Conditioning of the FRM
A FRM (or DSM) is always measured slightly non-symmetric, while it is theoretically
symmetric. The FRM can be made symmetric by averaging two FRFs, using (Hij + Hji)/2
for both Hij and Hji. It was found that this conditioning not only makes the identified
matrices symmetric but also improves the quality of the identification results, perhaps
because of the averaging effect. Interestingly this conditioning did not work well with
Tsuei’s method. The method identifies non-symmetric damping matrices despite using a
conditioned FRM, which may have been caused by accumulation of numerical errors due
to extra steps of the method.
2-6. Experimental Results
Table 2.3 summarizes all the subsequent tables, how they were obtained and compared to
one another. For example, the table shows that the damping matrices in Table 2.4 were
identified using the FRM neither phase matched nor conditioned and using the side band
52
(350 Hz and 440 Hz). Table 2.3-b summarizes the purposes of the comparisons made.
All tables show the damping matrices in the same format, listing C, D, Ceq and Deq
matrices for two systems. Generally, the different configurations of the two systems are
reflected reasonably well in all cases. For example, damping matrices of the system with
a damper have larger damping matrices in all tables.
The effect of the phase matching can be seen by comparing Tables 2.4 and 2.5, and
the effect of the FRM conditioning can be seen by comparing Tables 2.5 and 2.6. Phase
matching improves the result in general, especially judging from the equivalent matrices
identified, whose diagonal elements become nearly all positive in Table 2.5 as the phase
is matched. Comparing Tables 2.5 and 2.6 shows that the effect of the FRM conditioning
improves the identification results in an overall sense from the three observation point of
views described in Section 2-5.1. It is believed that the averaging effect of the FRM
conditioning improves the result in addition to the obvious effect of making the matrices
symmetric.
By comparing Tables 2.6 and 2.7, it can be seen that the structural damping matrix
obtained are of higher quality if the lower frequency band data is used. Substantially
different matrices are obtained depending on whether the side band or the low band is
used. This indicates that the actual damping mechanism of the system is not a linear
function but a higher order function of frequency. For example, the small damper used in
the experiment will be neither viscous nor constant. Considering this, one may use the
frequency band of interest to identify the damping matrices. If the range is too wide, a
piecewise linear model or a non- linear damping model may have to be used. For the
latter, the identification method will have to be modified to include higher order terms.
53
Comparisons of Tables 2.7 and 2.8, Tables 2.6 and 2.9 show that a small change in
the frequenc y range results in also small changes in the identified matrices.
This
indicates that the large differences between the results in Tables 2.6 (obtained using the
side band) and 2.7 (obtained using the low band) were not caused by a numerical problem
but by the nature of the system.
Table 2.10 is the identification results obtained using a wide frequency range (50-800
Hz), which includes both the side band and low band. Such a result may be used to
represent the system in an average sense for a wide frequency range of interest as an
alternative to a piecewise or non-linear model.
Another interesting observation is that while the beam without a damper has a
symmetric geometry, the identified result does not reflect the symmetry (e.g., D11 is quite
different from D44 in Table 2.6). However, the geometric symmetry is better represented
in the C matrix when the side band is used (see Table 2.6), and in D matrix when the low
band is used (see Table 2.7), which is consistent with the previous discussions. During
our experiment, it was observed that even a small distortion of the system results in
substantially different damping matrices, which also do not show any geometric
symmetry.
This may be explained by the fact that a variation of the geometry or
clamping conditions, even if they are small, can cause significant changes in the energy
loss mechanism. This feature may be exploited for a positive purpose. For example
identified damping matrices may be used to inspect the quality of the assembly of high
precision equipment.
54
Table 2.3-a. Summary of experimental comparisons (a) summary of Tables 2.3 to 2.9
Table No.
Table 2.4
Table 2.5
Table 2.6
Table 2.7
Table 2.8
Table 2.9
Table 2.10
Frequency Range(Hz)
350-440 (side band)
350-440 (side band)
350-440 (side band)
50-200 (low band)
50-120 (low band)
300-480
50-800
Phase Match
No
Yes
Yes
Yes
Yes
Yes
Yes
FRM conditioning
No
No
Yes
Yes
Yes
Yes
Yes
Table 2.3-b. Summary of experimental comparisons (b) purposes of comparisons
Comparison
Table 2.4 vs Table 2.5
Table 2.5 vs Table 2.6
Table 2.6 vs Table 2.7
Table 2.7 vs Table 2.8, Table 2.6 vs Table 2.9
Table 2.10 vs Table 2.6, Table 2.10 vs Table 2.7
55
Effect to discuss
Phase matching
FRM conditioning
Low frequency vs side band
General frequency dependence
Wide range vs low range vs high range
Table 2.4. Damping matrices identified using side band (350-440 Hz), neither phase
matched nor FRM conditioned
Beam without a Damper
1.1318
-0.5126
-0.0070
0.2272
-1.6250
0.6619
0.3994
-1.2311
0.8173
-0.5434
-0.2157
1.2964
-0.1858
0.2111
-0.1659
0.1436
0.4982
-0.2418
-0.0894
0.3649
-0.4772
0.1912
0.2885
-0.7875
0.5911
0.1017
-0.4898
0.6450
-0.1749
-0.0419
0.1393
-0.1690
-0.5715
0.0131
0.6240
-0.9954
0.1876
0.0420
-0.1682
0.2185
-0.0378
-0.0235
0.0424
-0.0412
-0.2076
0.0119
0.2174
-0.3598
Beam with a Damper
[C]
(×103 N⋅s/m)
0.6457
-0.8416
0.7226
-0.1760
-1.0081
1.4290
-0.2292
-1.2553
[D]
(×107 N/m)
0.2173
0.4681
-0.1926
0.0600
-0.3181
-0.5578
1.1489
0.5668
[C]eq (×103 N⋅s/m)
1.2781
0.4078
0.1620
-0.0157
-1.9338
-0.0598
3.1146
0.2576
[D]eq
(×107 N/m)
0.4303
0.1674
0.0458
-0.0024
-0.6506
-0.0506
1.0733
0.1212
56
-0.6190
1.1703
-1.2683
0.3481
1.5215
-2.8448
2.5515
-1.7293
1.8753
0.5616
-3.9828
4.7676
0.0902
-0.4265
0.5076
-0.2258
-0.4990
0.9336
-0.7204
0.4032
-0.7435
0.2835
0.3502
-0.2739
-0.3781
0.0319
0.0865
-0.2546
0.1897
-0.3530
0.6288
-0.6531
-0.1092
1.3183
-3.0479
4.0364
-0.1295
-0.0111
0.0574
-0.1022
0.0411
-0.0763
0.1854
-0.2107
-0.0778
0.4829
-1.0636
1.4185
Table 2.5. Damping matrices identified using side band (350-440 Hz), phase matched but
not FRM conditioned
Beam without a Damper
1.2635
-0.5573
-0.0603
0.2751
-1.7413
0.7724
0.3219
-1.2152
0.8112
-0.6528
-0.0627
1.1389
-0.2276
0.2235
-0.1476
0.1278
0.5288
-0.2707
-0.0690
0.3609
-0.4740
0.2198
0.2471
-0.7443
0.6012
0.0930
-0.4898
0.6472
-0.2023
-0.0156
0.1209
-0.1649
-0.5683
-0.0132
0.6566
-1.0274
0.1892
0.0396
-0.1675
0.2186
-0.0456
-0.0160
0.0371
-0.0400
-0.2064
0.0044
0.2264
-0.3687
Beam with a Damper
[C]
(×103 N⋅s/m)
0.6855
-0.5608
0.8460
-0.3672
-1.3295
1.4531
0.1514
-1.2022
[D]
(×107 N/m)
0.2080
0.3758
-0.2235
-0.0413
-0.2400
-0.0574
1.0581
0.2656
[C]eq (×103 N⋅s/m)
1.2909
0.4421
0.1954
-0.0413
-2.0278
-0.0574
3.2310
0.2656
[D]eq
(×107 N/m)
0.4341
0.1767
0.0555
-0.0082
-0.6785
-0.0501
1.1081
0.1232
57
-0.9803
1.5027
-1.4925
0.3792
1.6949
-3.1251
2.8084
-1.9071
1.7637
0.8403
-4.4144
5.1599
-0.4588
0.1057
0.0347
-0.2455
0.2367
-0.4282
0.6983
-0.7016
-0.1534
1.4173
-3.2005
4.1811
-0.4588
0.1057
0.0347
-0.2455
0.2367
-0.4282
0.6983
-0.7016
-0.1534
1.4173
-3.2005
4.1811
-0.1526
0.0100
0.0424
-0.0995
0.0553
-0.0989
0.2064
-0.2253
-0.0922
0.5145
-1.1123
1.4650
Table 2.6. Damping matrices identified using side band (350-440 Hz), phase matched and
FRM conditioned
Beam without a Damper
4.4470
-1.1606
-1.2195
0.1236
-1.1606
0.8093
0.0355
-0.8394
-1.2195
0.0355
1.0391
-1.1003
-1.3414
0.3753
0.1559
0.5187
0.3753
-0.2731
0.0199
0.2401
0.1559
0.0199
-0.0159
-0.3262
0.5431
-0.0684
-0.7658
1.6333
-0.0684
0.0144
0.0934
-0.1407
-0.7658
0.0934
0.9928
-2.0496
0.1256
-0.0076
-0.2464
0.5595
-0.0076
-0.0062
0.0316
-0.0368
-0.2464
0.0316
0.3269
-0.6891
Beam with a Damper
[C]
(×103 N⋅s/m)
0.1236
3.7244
-0.8394
-2.9250
-1.1003
-2.1366
4.6854
8.8274
[D]
(×107 N/m)
0.5187
-1.1006
0.2401
0.9069
-0.3262
0.6684
-0.2728
-2.8563
[C]eq (×103 N⋅s/m)
1.6333
0.7872
-0.1407
-0.5047
-2.0496
-0.3527
3.8914
1.2044
[D]eq
(×107 N/m)
0.5595
0.2216
-0.0368
-0.1315
-0.6891
-0.0901
1.2727
0.2774
58
-2.9250
3.6654
-0.6137
-5.7900
-2.1366
-0.6137
2.3144
-1.5113
8.8274
-5.7900
-1.5113
7.0014
0.9069
-1.2101
0.1545
2.1703
0.6684
0.1545
-0.2671
-0.5905
-2.8563
2.1703
-0.5605
-0.2532
-0.5047
0.4358
-0.2015
0.0022
-0.3527
-0.2015
1.6015
-3.0873
1.2044
0.0022
-3.0873
6.3257
-0.1356
0.0911
-0.0634
0.1149
-0.0901
-0.0634
0.5545
-1.1270
0.2774
0.1149
-1.1270
2.2323
Table 2.7. Damping matrices identified using low band (50-200 Hz), phase matched and
FRM conditioned
Beam without a Damper
0.0145
-0.0137
-0.0004
0.0288
-0.0137
0.0160
-0.0073
-0.0036
-0.0004
-0.0073
0.0259
-0.0475
0.3286
-0.1718
-0.0366
0.0348
-0.1718
0.2560
-0.2221
0.0836
-0.0366
-0.2221
0.2288
-0.1070
3.7456
-1.9648
-0.4160
0.4240
-1.9648
2.9224
-2.5287
0.9461
-0.4160
-2.5287
2.6242
-1.2622
0.3298
-0.1729
-0.0366
0.0370
-0.1729
0.2572
-0.2226
0.0833
-0.0366
-0.2226
0.2309
-0.1107
Beam with a Damper
[C]
(×103 N⋅s/m)
0.0288
0.0020
-0.0036
0.0021
-0.0475
-0.0045
0.0999
0.0063
[D]
(×107 N/m)
0.0348
0.4302
0.0836
-0.3631
-0.1070
0.2803
0.2799
-0.2731
[C]eq (×103 N⋅s/m)
0.4240
4.8866
0.9461
-4.1207
-1.2622
3.2087
3.2776
-3.0947
[D]eq
(×107 N/m)
0.0370
0.4304
0.0833
-0.3630
-0.1107
0.2827
0.2877
-0.2726
59
0.0021
-0.0054
0.0047
-0.0005
-0.0045
0.0047
0.0011
-0.0116
0.0063
-0.0005
-0.0116
0.0276
-0.3631
0.4231
-0.4152
0.3087
0.2830
-0.4152
0.4436
-0.3237
-0.2731
0.3087
-0.3237
0.3816
-4.1207
4.7976
-4.7092
3.5047
3.2087
-4.7092
5.0379
-3.6867
-3.0947
3.5047
-3.6867
4.3606
-0.3630
0.4226
-0.4148
0.3087
0.2827
-0.4148
0.4437
-0.3246
-0.2726
0.3087
-0.3246
0.3838
Table 2.8. Damping matrices identified using a different low band (50-120 Hz), phase
matched and FRM conditioned
Beam without a Damper
0.0160
-0.0151
-0.0004
0.0316
-0.0151
0.0176
-0.0081
-0.0040
-0.0004
-0.0081
0.0285
-0.0519
0.3285
-0.1718
-0.0366
0.0346
-0.1718
0.2559
-0.2220
0.0836
-0.0366
-0.2220
0.2287
-0.1067
5.8306
-3.0552
-0.6483
0.6449
-3.0552
4.5465
-3.9377
1.4769
-0.6483
-3.9377
4.0762
-1.9411
0.3294
-0.1726
-0.0364
0.0363
-0.1726
0.2568
-0.2225
0.0834
-0.0366
-0.2225
0.2302
-0.1095
Beam with a Damper
[C]
(×103 N⋅s/m)
0.0316
0.0020
-0.0040
0.0023
-0.0519
-0.0050
0.1091
0.0069
[D]
(×107 N/m)
0.0346
0.4302
0.0836
-0.3631
-0.1067
0.2803
0.2794
-0.2732
[C]eq (×103 N⋅s/m)
0.6449
7.6161
1.4769
-6.4245
-1.9411
5.0042
5.0535
-4.8275
[D]eq
(×107 N/m)
0.0363
0.4303
0.0834
-0.3630
-0.1095
0.2828
0.2852
-0.2728
60
0.0023
-0.0060
0.0052
-0.0006
-0.0050
0.0052
0.0012
-0.0127
0.0069
-0.0006
-0.0127
0.0303
-0.3631
0.4231
-0.4152
0.3087
0.2830
-0.4152
0.4436
-0.3236
-0.2732
0.3087
-0.3236
0.3815
-6.4245
7.4816
-7.3432
5.4635
5.0042
-7.3432
7.8524
-5.7404
-4.8275
5.4635
-5.7404
6.7819
-0.3630
0.4228
-0.4150
0.3087
0.2828
-0.4150
0.4437
-0.3243
-0.2728
0.3087
-0.3243
0.3831
Table 2.9. Damping matrices identified using a different side band (300-480 Hz), phase
matched and FRM conditioned
Beam without a Damper
3.3420
-1.0509
-0.8668
0.3595
-1.0509
0.7685
-0.0740
-0.5962
-0.8668
-0.0740
0.9520
-0.8996
-0.9383
0.3355
0.0267
0.4332
0.3355
-0.2587
0.0601
0.1513
0.0267
0.0601
0.0157
-0.3995
0.5395
-0.0486
-0.7869
1.6534
-0.0486
-0.0042
0.1057
-0.1442
-0.7869
0.1057
0.9992
-2.0930
0.1221
0.0021
-0.2483
0.5473
0.0021
-0.0149
0.0367
-0.0379
-0.2483
0.0367
0.3179
-0.6850
Beam with a Damper
[C]
(×103 N⋅s/m)
0.3595
3.1503
-0.5962
-2.3648
-0.8996
-2.1231
3.0084
7.6908
[D]
(×107 N/m)
0.4332
-0.9086
0.1513
0.7243
-0.3995
0.6557
0.3390
-2.4704
[C]eq (×103 N⋅s/m)
1.6534
0.7059
-0.1442
-0.4162
-2.0930
-0.3593
4.0211
1.0451
[D]eq
(×107 N/m)
0.5473
0.1998
-0.0379
-0.1077
-0.6850
-0.0914
1.2936
0.2357
61
-2.3648
3.1607
-0.5468
-4.8655
-2.1231
-0.5468
1.7178
-0.8123
7.6908
-4.8655
-0.8123
4.7971
0.7243
-1.0421
0.1324
1.8607
0.6557
0.1324
-0.0820
-0.7967
-2.4704
1.8607
-0.7967
0.4332
-0.4162
0.3573
-0.1905
0.1401
-0.3593
-0.1905
1.4972
-2.9555
1.0451
0.1401
-2.9555
5.9626
-0.1077
0.0700
-0.0599
0.1488
-0.0914
-0.0599
0.5224
-1.0825
0.2357
0.1488
-1.0825
2.1212
Table 2.10. Damping matrices identified using wide band (50-800 Hz), phase matched
and FRM conditioned
Beam without a Damper
Beam with a Damper
3
-4.435
4.433
-1.313
-3.273
4.433
-6.299
4.269
-0.269
-1.313
4.269
-5.396
5.293
1.7203
-1.6192
0.3256
1.5447
-1.6192
2.3095
-1.5800
0.1207
0.3256
-1.5800
2.3361
-2.6984
0.6770
-0.3787
-0.3454
1.3174
-0.3787
0.5641
-0.4263
0.0898
-0.3454
-0.4263
1.5462
-2.7253
0.5360
-0.4355
-0.0250
0.6708
-0.4355
0.6275
-0.4401
0.0489
-0.0250
-0.4401
0.8953
-1.2849
[C]
(×10 N⋅s/m)
-3.273
-3.5075
-0.269
0.2278
5.293
0.8480
-10.638
-0.5235
[D]
(×107 N/m)
1.5447
1.5154
0.1207
-0.2711
-2.6984
-0.4127
5.4880
0.6572
[C]eq (×103 N⋅s/m)
1.3174
0.9957
0.0898
-0.5779
-2.7253
-0.3783
5.6706
1.4296
[D]eq
(×107 N/m)
0.6708
0.5788
0.0489
-0.2103
-1.2849
-0.1862
2.6474
0.5175
62
0.2278
1.6700
-0.9352
-1.8994
0.8480
-0.9352
-0.4527
4.1053
-0.5235
-1.8994
4.1053
-8.7964
-0.2711
-0.3965
0.2042
0.7000
-0.4127
0.2042
0.8626
-2.7609
0.6572
0.7000
-2.7609
5.8292
-0.5779
0.4916
-0.3285
0.1808
-0.3783
-0.3285
2.1106
-4.0989
1.4296
0.1808
-4.0989
8.5258
-0.2103
0.0494
-0.0456
0.1928
-0.1862
-0.0456
0.7417
-1.6646
0.5175
0.1928
-1.6646
3.4803
2-7. Conclusions
A new algorithm was proposed for experimental identification of the damping matrices,
which identifies the viscous and structural damping matrices of the equation of motion of
a dynamic system. The new algorithm is very simple, therefore provides more accurate
and robust results compared to the method previously used [14]. Theoretical validation
of the method and the related noise study were conducted using a 3 DOF lumped
parameter system. A set of measurements were taken, which served as a qualitative,
experimental validation of the procedure. Important measurement techniques necessary
for correct implementation of the proposed method learned from the experiment were
also reported, which included phase matching of FRFs, conditioning of the FRM and the
selection of the frequency range.
The method identifies damping matrices, which carry a lot more information than
damping ratios, therefore will enable some interesting applications. The following are
considered potentially promising applications.
(1) FEA-experiment hybrid modeling of a dynamic system: The mass and stiffness
matrices are formulated theoretically, and the damping matrices are identified
experimentally, which are then combined to obtain the system model. Because the
actual spatial distribution of the damping is considered in the model, this will
provide a more accurate analysis.
(2) Identified damping matrices may be used as valuable information for design or
inspection. For example, comparison of the spatial distributions of damping of a
new design and an existing design may provide good insights to designers. Because
63
even a very small change in the system causes a significant change in identified
damping matrices, the matrices may also be used for an inspection purpose.
(3) The method will be very useful if it is extended and applied to modeling of rotor
systems. The ability of the method to distinguish different damping mechanisms will
be very helpful in rotor systems, because different damping mechanisms have
different effects on the system stability.
The accuracy of the identified damping matrices depends almost entirely on the accuracy
of the measured FRFs, especially their phase angles. Therefore the techniques used to
measure FRFs will be critical to the feasibility of the above applications.
64
Chapter 3 – Sound Transmission through Single-Walled Cylindrical
Shells
3-1. Introduction
The desire to develop a basic analysis tool to aid selection of the sidewall of an
automotive muffler served as the practical motivation behind this study. An automotive
muffler is essentially a device that confines strong acoustic pulsations of the exhaust gas
within its internal cavity, therefore its sidewall has to provide good noise insulation in a
broad frequency range. It was intended to develop a simple analytical procedure for the
relative comparison of design alternatives to complement detail analysis tools based on a
numerical analysis technique such as the finite element method (FEM) or the boundary
element method (BEM).
A few major assumptions are used in modeling the system to simplify the problem.
Firstly, the system is considered to be infinitely long, which enables both the acoustic
media and the shell to be described by traveling waves. Secondly, the incident wave to
the system is a plane wave from the outside space. Finally it is assumed that the internal
shell cavity is anechoic so that no reflected wave exists in the cavity.
The last
assumption is not derived from a physical consideration, but from the need to make the
problem a rough approximation of the inverse of a muffler running in an anechoic
chamber, which has a diffusive field inside and anechoic condition outside. Obviously
the real muffler has neither an infinite length nor a plane incident wave. Therefore, it is a
basic assumption that the theoretical model represents main characteristics of the problem
well enough for relative comparisons of design alternatives.
65
Besides the afore- mentioned simplifications, no further approximations are necessary
in the solution procedure itself. The vibro-acoustic response of the system is obtained by
solving the shell equations and acoustic waves simultaneously, therefore fully
considering coupling effects between the structure and acoustic media. Love’s equation,
arguably the most widely accepted classical thin shell equation, is used to describe the
shell motion. The solution is obtained in a series solution using the mode superposition
method, which therefore can be considered exact if enough number of modes is used to
satisfy the convergence.
For the purpose of a partial validation of the analysis procedure, theoretical solutions
are compared to the solutions of an equivalent flat system, the sound transmission
through a thin beam that divides two semi- infinite acoustic media. It is shown that the
shell solution becomes closer to the beam solution as the radius of the shell is increased.
Then, the theoretical solutions are compared to the experimental results measured for a
shell, which has same dimensions but the length.
Because of the finite length and non-
plane wave input condition of the experimental model, the experimental comparison has
to be considered as another partial validation of the theoretical model rather than a
rigorous one. Comparing the two results also serves as a preview of the intended use of
the theoretical model: whether it can be used for a first cut design of the sidewall of a
muffler.
As a performance index of the system, the transmission loss (TL) is used throughout
the study, which is defined as the ratio of the total sound power of the incoming wave to
that of the transmitted wave. Using the theoretical model developed, parameter studies
66
are conducted for major design variables such as thickness, density and radius of the
cylindrical shell.
Noise transmission through a cylindrical shell was studied by many researchers
including Smith [20], White [21], Koval [22-25], Blaise et al. [26] and Tang et al. [2728] for various purposes, notably for design study of aircraft.
Sound transmission
through isotropic, orthotropic, and laminated fiber-reinforced composite shells, and
through double walls were investigated by Grooteman et al., Mulholland et al., and
White and Beranek [29-33]. Sound transmission through the flat sandwich structure,
which has a lightweight and flexible cores between relatively stiff skins, was investigated
by C. L. Dym et al. [34].
Smith [20] presented a theoretical study of transmission of the sound energy through
a thin, isotropic elastic cylindrical shell from the oblique plane wave excitation. He
defined a cross-sectional absorption coefficient that is the ratio of the power absorbed to
the incident power per unit length. White [21] investigated sound transmission in the
cylindrical shells of infinite and finite lengths. He found two important frequencies: the
ring and coincidence frequencies at which TL becomes minima. Koval [22-25] extended
Smith’s work to present an analytical model to predict TL of isotropic, orthotropic, and
laminated fiber-reinforced composite shells. In his work, the effects of external airflow
were included, and the fluids are considered different in the inside and outside of the
shell. Blaise et al. [26] extended Kaval’s work to study an orthotropic shell excited by an
oblique plane sound wave with two independent incident angles. The transmission
coefficient was obtained assuming the acoustic field is diffusive and the fluids inside and
outside the shell are the same. Tang et al. [27-28] considered an infinite cylindrical
67
sandwich shell excited by an oblique plane sound wave with two independent incident
angles. The effects of external airflow and a negative pressure differential between the
inside and outside shell surfaces were included in their study for different fluids in the
inside and outside of the shell.
In many studies surveyed, either a simplified shell theory was used or a numerical
method was employed in the solution procedure. For example, only the equation of
motion in the transverse direction was used to describe the shell motion in references [2728], neglecting in-plane equations completely, which may be not valid in general. Also,
the convergence of the series solution was not considered in the work by Tang et al.,
which may have contributed to result in very large, seemingly erroneous transmission
losses [27-28]. If an insufficient number of modes is used in the solution procedure, TL
is over-estimated significantly which seems to be the case in the results in references [2728].
M. El- Raheb et al. [35] used a boundary element (BEM) and experimental methods to
obtain the transmission loss of sound across a 2-D truss- like periodic panel in contact
with an acoustic fluid on both its faces. P. Sas et al. [36] studied the active control of
sound transmission through a double-panel by using coupled finite element method
(FEM), boundary element methods (BEM) and an experimental approach. The boundary
element analysis (BEA) was used in conjunction with the four-pole method by T. W. Wu
et al. to obtain the TL between the inlet and the outlet ports of a muffler [37]. A coupled
structural-acoustic analysis has to be conducted using an indirect BEM formulation to
calculate the TL of an elastic wall, which is computational very demanding even in the
modern standard.
68
3-2. Formulation of the Problem
3-2.1. The System Model
Figure 3.1 illustrates the schematic of the problem studied in this work, in which a plane
wave is incident to a cylindrical shell of infinite length with an incidence angle γ1. The
incident wave is a plane wave whose rays are traveling on planes parallel to the x- z plane.
Shown are the rays only on the x-z plane. The shell is defined in terms of the radius Ri,
wall thickness of hi, in vacuo bulk mass density ρi, µi Poisson’s ratio and in vacuo bulk
Young’s modulus Ei. The acoustic media in the outside and the inside of the shell are
defined by density and speed of sound: {ρ1 , c1 } inside and {ρ3 , c3 } outside.
3-2.2. Vibro-Acoustic Equations
Acoustic Wave Equations
In the external space, the wave equation becomes [38],
(
)
c1∇ 2 p I + p1R +
∂ 2 ( p I + p1R )
=0
∂t 2
(3.1)
where, p I and p R1 are the acoustic pressures of the incident and reflected waves, and ∇2
is the Laplacian operator in the cylindrical coordinate system:
∇2 =
1 ∂ ∂ 1 ∂2
∂2
r
+
+
r ∂r ∂r r 2 ∂θ 2 ∂z 2
(3.2)
In the internal cavity, the acoustic pressure of the transmitted wave p3T satisfies the
acoustic wave equation [38]:
c3∇ 2 pT3 +
∂ 2 pT3
=0
∂t 2
(3.3)
69
Reflected wave
Incident plane wave
γ1
z
Transmitted wave
Ri
y
θ
x
Figure 3.1. Schematic diagram of the single cylindrical shell: 2-D model
Equation of Motion of the Shell
Let {u10 , v 10, w10 } be the displacements of the shell at the neutral surface in the axial,
circumferential, and radial directions respectively.
Love’s equation [39], classical
equations of motion of the thin shell, can be specialized for a cylindrical shell as follows.
L1 {u10 , v10, w10 } = ρi hi u&&10
{
(3.4)
}
L2 u10 , v10, w10 = ρi hiv&&10
(3.5)
L3 {u10 , v 10, w10 } + ( p I + p1R ) − p3T = ρi hi w
&&10
(3.6)
Equations (3.4), (3.5) and (3.6) are equations of motion of the shell in the axial, radial
and circumferential directions respectively. In the equations, ρi and hi are the density of
the shell material and the thickness of the shell, respectively. The differential operators
L1 , L2 and L3 in Love’s Equation of the cylindrical shell in Equations (3.4) - (3.6) are
shown as follows:
70
L1{ u01 , v 10 ,w10 }= Ki
L2 { u01 , v 10 , w10} =
∂ 2u10 Ki (1 + µi ) ∂ 2v10 K i (1 − µ i ) ∂ 2u10 K i µ i ∂w10
+
+
+
∂z 2
2Ri
∂z ∂θ
2Ri2
∂θ 2
Ri ∂z
Ki (1 − µi ) ∂ 2 v10 1 ∂ 2 u10
K ∂ 2 v10
∂w10
∂ 2u10
( 2 +
)+ i(
+
+
µ
)
i
2
∂z
Ri ∂z∂θ
Ri Ri∂θ 2 Ri∂ θ
∂z∂θ
D (1 − µ ) 2∂3 w0 ∂2 v0
D
∂3 w0
1 ∂2 v10
∂3 u10
+ i 2 i ( − 2 1 + 21 ) + i2 (− 2 1 3 − 2
−
µ
)
i
2 Ri
∂z ∂θ ∂z
Ri
Ri ∂θ
Ri ∂ θ 2
∂z 2 ∂θ
L3{ u01 , v 10 ,w10} = Di [−
Di (1 − µ i )
∂ 4 u10 µi ∂ 4 w10
∂ 3 v10
2∂ 4 w10
∂3v10
−
(
+
)]
+
(
−
+
)
∂z 4 Ri2 ∂z 2 ∂θ 2 ∂z 2∂θ
Ri 2
∂z 2∂θ 2 ∂z 2 ∂θ
D
∂ 4 w0
1 ∂3v10
∂ 4u1
Ki ∂v10 w10
∂u10
+ 2i ( − 2 1 4 − 2
−
µ
)
−
(
+
+
µ
)
i
i
Ri
Ri ∂θ
Ri ∂θ 3
∂z 2 ∂θ 2
Ri Ri ∂θ Ri
∂z
(3.7)
(3.8)
(3.9)
where, Ki and Di are the membrane and bending stiffness defined as follows.
Ki =
Ei hi
1 − µ i2
Di =
Ei hi3
12 1 − µi2
(
(3.10)
)
(3.11)
where, Ei is the Young’s modulus of the shell material.
Boundary Conditions
On the internal and external shell surfaces, the particle velocities of the acoustic media in
the normal direction have to be equal to the normal velocity of the shell, which results in
the following equations.
∂ ( p I + p1R )
∂r
= − ρ1
∂pT3
∂ 2 w10
= − ρ3
∂r
∂t 2
∂ 2 w10
∂t 2
at r = Ri
(3.12)
at r = Ri
(3.13)
71
3-3. Solution Process
3-3.1. Solution of the Equations
The cylindrical coordinate description of the harmonic plane wave pI incident from outside to the direction shown in Figure 3.1 can be expressed as [40]:
∞
p I ( r , z ,θ , t ) = p 0 ∑ ε n ( − j ) n J n ( k1 r r ) cos[nθ ]e j (ω t −k1 z z )
(3.14)
n =0
where, p0 is the amplitude of the incident wave, j = − 1 , n=0,1,2,3,……., Jn is the
Bessel function of the first kind of order n; ω is the angular frequency, ε n =1 for n = 0 and
εn =2 for n=1,2,3,…. , k1 z = k1 sin( γ 1 ) , k1 r = k1 cos(γ 1 ) , and k 1 =
ω
.
c1
It is easily seen
that k1r = k12 − k12z . The phase speed of the wave to the direction of the shell surface is
defined as:
c p1 =
ω
c1
=
k1 z sin( γ 1 )
(3.15)
The waves radiated from the shell to the outside and into the cavity, p1R and p 3T , can
be represented as:
p1R ( r , z , θ , t ) =
p ( r , z ,θ , t ) =
T
3
∞
∑
n =0
∞
∑
n =0
p1Rn H n2 (k 1r r ) cos [n θ ]e j ( ωt −k 1 z z )
(3.16)
p 3Rn H 1n ( k 3 r r ) cos [nθ ]e j ( ω t − k 3 z z )
(3.17)
where, H 1n and H n2 are the Hankel functions of the first and second kind of order n,
respectively. The former represents the incoming wave and the second the outgoing
wave.
72
In Equation (3.17), it is known that k 3 r = k 32 − k 23 z . Because the traveling waves in
the acoustic media and in the shell are driven by the incident traveling wave, the wave
numbers (or trace velocities) in the z direction should match throughout the system,
therefore k 3 z = k1 z . Hence, the three components of the shell displacement can be
expressed as:
∞
w10 ( z ,θ , t ) = ∑ w10n cos[nθ ]e j (ωt − k
1z z
)
(3.18)
n =0
∞
u ( z, θ , t ) = ∑ u10n cos[nθ ]e j ( ωt −k1 z z )
0
1
(3.19)
n =0
∞
v10 ( z ,θ , t ) = ∑ v10n sin [nθ ]e j (ωt − k
1z
z)
(3.20)
n =0
where, n=0,12,3,… are the circumferential mode numbers.
Substituting Equations (3.14) and
(3.16) - (3.20) into the three shell equations
(Equations (3.4), (3.5) and (3.6)) results in three equations of motion. Additionally,
substituting the same equations into the boundary conditions in the radial directions,
Equations (3.12) and (3.13), results in two more equations. These five equations involve
with six variables: the amplitudes of the outgoing and incoming waves in the exterior
cavity, the incident wave in the interior cavity, and three displacements of the shell
structure.
Therefore, the solutions can be obtained as the ratios to the one of the
variables, the pressure amplitude of the incoming wave in this case. The ratio of the
amplitudes of the input and transmitted waves obtained this way allows the transmission
loss to be obtained.
73
Substituting Equations (3.14) and (3.16) - (3.20) into three shell equations
(Equations (3.4), (3.5) and (3.6)), and utilizing the orthogonality between trigonometric
functions, the following three equations are obtained for each circumferential mode
number n=0,1,2,3…:
K (1 − µ ) K (1 + µi )
Kµ
u10n ρ i hi ω 2 − K i k12z − i 2 i n 2 − i
nk1 z v0In j − i i k1 z w0In j = 0
2 Ri
2 Ri
Ri
K i (1 − µi ) 2 K i 2 Di (1 − µ i ) 2
k1z − 2 n −
k 1z
−
2
Ri
2 Ri2
Ki
Di
0 K i (1 − µ i )
2
0
u1n
k1z nj +
µ i k 1z nj − 2 µ i k1 z n + v In
2
R
R
R
D
i
2
2
i
i
i
+ R 4 n + ρi hiω
i
(3.21)
(3.22)
K
D (1 − µ )
D
+ w 0In − 2i n − i 2 i k 12z n − 4i n 3 = 0
Ri
Ri
Ri
D
D
K
D (1 − µ )
D
K
u10n − Di k14z − 2i µi k12z n 2 + i µi k1z j + v10n 2i µi k12z n − i 2 i k12z n + 4i n 3 − 2i n
Ri
Ri
Ri
Ri
Ri
Ri
2D (1− µ )
Dµ
D
K
+ w10n − i 2 i k12z n 2 − i 2 i k12z n 2 − 4i n 4 − 2i + ρi hiω 2
Ri
Ri
Ri
Ri
(3.23)
+ p1Rn Hn2 (k 1r Ri ) − p 3Tn H n1(k 3 r Ri ) = − p0ε n ( − j)n Jn ( k1r Ri )
where, hi, Ri, µi, ρi, and k1z are the thickness, radius, Poisson’s ratio, in vacuo bulk
density of the shell, and wave numbers, respectively.
Substituting Equations (3.14) and (3.16) - (3.20) into the two boundary conditions
(Equations (3.12) and (3.13)), and using the orthogonal conditions of the trigonometric
functions again, the following two equations are obtained for each mode number
n=0,1,2,3,….
′
p1Rn H n2 ( k1r Ri ) k1 r − ρ1ω 2 w10n = − p0 ε n ( − j ) n J n ′ ( k1r Ri )k1 r
(3.24)
′
p T3n H 1n (k 3 r Ri ) k 3 r − ρ 3ω 2 w10n = 0
(3.25)
74
In Equations (3.21) - (3.23) and (3.24) - (3.25), we have five unknowns and five
equations for each mode n. These five equations can be rearranged in a form of a matrix
equation:
0 0
0 0
G H
L 0
0 N
A B
D E
I J
0 0
0 0
C p1Rn
F p3Tn
K u10n =
M v10n
O w10n
0
0
P
Q
0
(3.26)
where,
A = ρi hiω 2 − Ki k12z −
Ki (1 − µi ) 2
K (1 + µ i )
n ,B = − i
nk1 z j
2
2 Ri
2 Ri
C=−
Kiµ i
K (1 − µ i )
K
D
k1 z j , D = i
k1 z nj + i µ i k1z nj − 2i µ i k12z n
Ri
2 Ri
Ri
Ri
E=−
K i (1 − µ i ) 2 Ki 2 Di ( 1 − µi ) 2 Di 2
k1 z − 2 n −
k1k + 4 n + ρi hiω 2
2
2
Ri
2Ri
Ri
F =−
D (1 − µ )
Ki
D
n − i 2 i k12z n − i4 n3 , G = H n2 (k1 r Ri ) , H = −H 1n ( k 3 r Ri )
2
Ri
Ri
Ri
I = −Di k14z −
J=
Di
K
µ i k12z n 2 + i µ i k1 z j
2
Ri
Ri
D (1 − µ )
Di
D
K
µi k12z n − i 2 i k12z n + 4i n 3 − 2i n
2
Ri
Ri
Ri
Ri
K=−
Di µi 2 2 2 Di ( 1 − µ i ) 2 2 Di 4 K i
k1 z n −
k1 z n − 4 n − 2 + ρ i hiω 2
Ri2
Ri2
Ri
Ri
′
′
L = H n2 ( k1 r Ri ) k1 r , M = − ρ1ω 2 , N = H 1n ( k 3 r Ri )k 3 r , O = − ρ 3ω 2
′
P = − p 0 ε n ( − j ) n J n ( k1 r Ri ) , Q = − p 0 ε n ( − j ) n J n ( k1r Ri ) k1 r
75
Five unknown coefficients p1Rn , p3Tn , u1n0 , v1n0 , and w1n0 are obtained in terms of po by
solving Equation (3.26), which can be substituted back to Equations (3.16) - (3.20),
providing a set of exact solutions in series expressions for the solution of the system. The
series solution can be considered as an exact solution if the series converges. The
convergence issue will be discussed later.
3-3.2. Transmission Loss (TL)
It is convenient to represent the solution in TL for the design purpose. TL can be defined
as the ratio of the incoming and transmitted sound powers per unit length of the cylinder.
TL = 10log10
WI
(3.27)
∞
∑W
n= 0
T
n
where, WnT is the transmitted power flow per unit length of the shell:
WT =
2 π
1
Re ∫ p3T ⋅ ∂ ( w10 ) * Ri dθ
∂t
2
0
where r=Ri
(3.28)
where Re{.} and the superscript * represent the real part and the complex conjugate of
the argument, respectively.
Substitution of Equations (3.17) and (3.18) for p 3T and w10 into above Equation (3.28)
yields an expression for the components of WnT .
{
=
2π
} ∫ cos [nθ ]⋅ R dθ
1
W = Re p3Tn H1n (k3 r Ri )⋅ ( jω w10n )* ×
2
T
n
2
i
where r=Ri
0
π Ri
× Re{ p3Tn H 1n (k 3 r Ri )⋅ ( jω w10n )* }
2ε n
(3.29)
where, ε n =1 for n = 0 and ε n =2 for n=1,2,3,……
76
W I , the incident power flow per unit length of the shell is,
WI =
cos(γ 1 ) p02
× 2 Ri
ρ1 c1
(3.30)
Finally, the transmission loss can be obtained by substituting Equations (3.29) and (3.30)
into (3.27)
{
}
Re p3Tn × H n1 (k3r Ri ) × ( jωw10n ) × ρ1c1π
TL = −10 log10 ∑
4ε n cos(γ 1 ) p02
n =0
∞
*
(3.31)
where, ε n =1 for n = 0 and ε n =2 for n=1,2,3,…..
3-4. Convergence Checking
As one can see in Equations (3.14) and (3.16) - (3.20), the solutions, the complex
amplitudes p1Rn , p3Tn , u1n0 , v1n0 and w1n0 , are obtained in series forms. Therefore, an enough
number of modes has to be included in the analysis to make the solution converge. When
insufficient modes are used in the calculation, the resulting TL becomes overestimated
because the shell response will be underestimated. Very high TLs for a relatively thin
shell reported in the work by Tang et al. [27-28] are apparently caused by such a nonconverged solution. Once the solution converges at a given frequency, it can be assumed
to converge in all frequencies lower than that, because more terms are necessary to be
used in the calculation for a higher frequency. Therefore, the necessary number of modes
has to be determined at the highest frequency of interest.
Figure 3.2 shows a simple algorithm used in this work to ensure the convergence of
the solution. The transmission loss is calculated at the highest frequency of interest,
applying the analysis procedure discussed earlier adding one mode at a time. When the
77
TLs calculated at two successive calculations are within a pre-set error bound (1×10-5 dB
in this work), the solution is considered to have converged. The number of modes found
this way is used to calculate TL at all other frequencies. Figure 3.3 shows how the
calculated TL changes as the number of modes increases for the case of a shell of 1 m
radius and 1 mm thickness driven at 1,000 Hz. From the figure, it is known that at least
21 modes (n = 0 to 20) have to be used to obtain a converged solution at 1,000 Hz.
Figure 3.4 shows the convergence trend for the same case but at 10,000 Hz, which
indicates that 128 or more modes will have to be used.
Mode Number
n = n+1
Calculation of TL
at Maximum Frequency
No
ABS(TL(n+1 )-TL(n))<1×10-5
Yes
Find: Optimum Mode Number
Figure 3.2. Algorithm for identifying the optimum mode number
78
54
52
TL (dB)
50
48
46
44
42
40
0
20
40
60
80
100
Mode Number
Figure 3.3. Mode Convergence Diagram for the single shell (Ri=1.0 m, hi=1.0 mm) at
1,000 Hz
105
100
TL (dB)
95
90
85
80
75
0
50
100
150
200
Mode Number
Figure 3.4. Mode Convergence Diagram for the single shell (Ri=1.0 m, hi=1.0 mm) at
10,000 Hz
79
3-5. Comparison to an Equivalent 1-D Model
If the radius of the shell becomes very large, it is expected that the above problem
becomes geometrically similar to the sound transmission through a flat panel of infinite
length dividing two acoustic media.
If the structural wave in one direction is ignored,
the flat plate model can be modeled as a one-dimensional (1-D) problem, in which the
beam motion represent the motion of the panel. Even for a very large radius, the TL
curve of the shell will not become the same as that of the beam in the low frequency
range because the shell response is mainly determined by the membrane stiffness in the
low frequency range, which is not included in the beam model. However, their envelopes
are expected to become closer to each other in the high frequency range as the radius of
the shell increases. Therefore, comparing the solutions from the two models serves as a
partial validation of the shell model, screening out some of possible mistakes.
For
example, the convergence requirement of the shell solution was realized in this work
from such a comparison.
3-5.1. Formulation of Equations
In Figure 3.5, a unit-width beam of infinite lengt h is shown that divides two different
acoustic media. The system can be considered as a one-dimensional version of the flat
plate, in which the bending wave is traveling only in one direction.
The mass density and sound speed of the acoustic media in the incident and
transmitted sides are {ρ1 , c1 } and {ρ3 , c3 }. The in vacuo bulk mass density and Young’s
modulus of the beam are given as {ρb, Eb}. An oblique wave pI is incident with an angle
γ1 (measured from the normal axes to the beam) from one side, which induces the
80
traveling transverse wave in the beam, and the reflected acoustic wave p1R , and
transmitted wave p 3T . An anechoic condition is assumed in the transmitted side.
Incident
Wave
z
x
Transmitted Wave
γ1
1m
y
Internal
Air
External
Air
x
Reflected Wave
hi
Figure 3.5. Schematic diagram of an infinite beam model
81
In the incident side, the wave equation becomes [38]:
(
)
c1∇ 2 p I + p1R +
∂ 2 ( p I + p1R )
=0
∂t 2
(3.32)
where ∇2 the Laplacian operator in the rectangular coordinate. In the transmitted side,
the pressure p3 = pT3 , where p3T is transmitted wave, satisfies the acoustic wave equation
[38] as:
c3∇ 2 pT3 +
∂ 2 pT3
=0
∂t 2
(3.33)
Equation of motion of the beam is given as [39]:
− Ei I i
∂ 4 w10 I
∂ 2 w10
R
T
+
p
+
p
−
p
=
ρ
h
(
1 )
3
i i
∂x 4
∂t 2
(3.34)
where w10 is the transverse displacement of the beam Ei is the Young’s modulus of the
beam material, hi is the thickness of the beam and Ii is the area moment of inertia of the
beam given as:
Ii =
hi3
12
(3.35)
Boundary conditions are given as:
∂ ( p I + p1R )
∂z
= − ρ1
∂p3T
∂ 2 w0
= −ρ 3 21
∂z
∂t
∂ 2 w10
∂t 2
at z = 0
(3.35)
at z = 0
(3.37)
3-5.2. Solution Procedure
The harmonic incident wave is expressed as:
82
p I ( x, z , t ) = p0 e j (ω t −k1 x x − k1 z z )
(3.38)
where, p0 is the effective amplitude of the incident wave; ω is the angular frequency; and
k1 =
ω
,
c1
k1 x = k1 sin( γ 1 ) , k1 z = k1 cos( γ 1 )
(3.39)
k3 =
ω
,
c3
k 3 x = k1 x ,
(3.40)
k 3 z = k 32 − k 32x
p1R and p 3T which satisfy Equations (3.32) to (3.33) and displacement w10 can be
expressed as follows.
p1R ( x, z , t ) = p1Rc e j (ω t − k1 x x + k1 z z )
(3.41)
p3T ( x , z , t ) = p3Tc e j (ωt − k1 x x − k 3 z z )
(3.42)
w10 ( x , t ) = w10c e j (ω t −k1 x x )
(3.43)
Substituting Equations (3.38), (3.41) to (3.43) into Equations (3.34), (3.36) and
(3.37), which are equation of motion of the beam and two boundary conditions, three
equations are obtained for 3 unknown complex amplitudes p1Rc , p3Tc and w1c0 :
[
]
w10c ρ i bi hi ω 2 − Ei I i k14x + p1Rc − p3Tc = − p0
(3.44)
p1Rc k1 z j − ρ1ω 2 w10c = p0 k1 z j
(3.45)
− p3Tc k 3 z j − ρ 3ω 2 w10c = 0
(3.46)
Putting these three equations into a matrix format:
R
A B C p1 c H
D 0 E pT = I
3c
0 F G w10c 0
(3.47)
where,
83
A = 1 , B = −1 , C = ρ i bi hi ω 2 − Ei I i k14x , D = k1 z j , E = − ρ1ω 2 , F = −k 3 z j
G = − ρ 3ω 2 , H = − p0 , I = p 0 k1 z j
Again, solutions p1Rc , p3Tc , w1c0 are found by solving Equation (3.47).
TL is defined
in a similar way to that defined for the shell system. On the beam surface, z = 0,
therefore,
{
}
1
*
W = Re p3T ⋅ w& 10 or
2
T
2
1 p3T
2 ρ 3 c3
(3.48)
where Re{.} and the superscript * represent real part and the complex conjugate of the
argument, respectively. Substituting Equations (3.42) and (3.43) for p3T and w10 into
above Equation (3.48) yields an expression for the components of W T
WT =
{
1
*
Re p T3 c × ( jωw10c )
2
}
(3.49)
The transmission loss is, therefore, defined by
TL = −10 log 10
WT
WI
(3.50)
where W I is the incident sound power per unit length of the beam
WI =
cos(γ 1 ) p02
ρ1 c1
(3.51)
Then, the transmission loss can be obtained by substituting Equations (3.49) and (3.51)
into (3.50)
{
}
Re p3Tc × ( j ωw10c )* × ρ1c1
TL = −10 log 10
2
2
cos
(
γ
)
p
1
0
(3.52)
84
3-5.3. Comparison of the 1-D and 2-D Solutions
TLs calculated for the cylindrical shell of 1 mm thickness and 0.1 m, 0.5 m and 1.0 m
radius are compared those calculated for the corresponding beam cases in Figure 3.6. As
expected, the envelops of the TL curves become closer to each other as the radius of the
shell increases.
A relatively larger difference in the low frequency range may be
attributed to the circumferential membrane effect in the shell, which is more significant in
the lower shell modes, however completely ignored in the 1-D model.
80
70
60
TL (dB)
50
40
30
20
10
0
0
200
400
600
800
1000
Frequency (Hz)
Figure 3.6. TL curves of the infinite beam and single shell
, beam model;
, shell model (Ri=0.1 m);
shell model (Ri=0.5 m);
, shell model (Ri=1 m)
,
3-6. Comparison to Experimental Measurements
A cylindrical shell is made of steel, which has the radius 0.1 m, length 1 m and thickness
1 mm. The end faces of the shell are closed using thick end caps, which are snug- fitted
and sealed by tapes, to eliminate the direct sound transmission from the inside when the
85
TL is measured.
Because of the infinite length and plane wave conditions in the
theoretical model, the experimental comparison in this work could be only an indirect
one.
Figure 3.7 shows the experimental setup to measure TLs of the shell in an anechoic
chamber. Two pairs of microphones are installed at the same position on the inside and
outside surfaces of the shell (only the outside pair is shown) to measure the sound
intensities, from which TL is calculated. The measurement was conducted once using an
internal sound source, and the other time using an external source. The internal source
was a 3- inch speaker and the external source was a B&K (Type 4296) decahedral
speaker. A white noise with the maximum frequency of 6,400 Hz was used to drive the
sound sources. The internal sound source setup is considered to represent the theoretical
model better because it has an anechoic condition in the outside and a reverberant
condition in the inside, therefore can be considered as a rough reciprocal of the
theoretical model. The external sound source setup has incident and reflected waves both
in the inside and outside spaces.
The measured TLs are compared to the calculated TLs for the corresponding
theoretical model, which has the same dimensions as the experimental model but the
length. The Young’s modulus E and Poisson’s ratio µ are taken as 1.9 × 1011 Pa and 0.3,
and the temperature is taken as 20o C in the calculation. Calculated TLs obviously
depend on the choice of the incident angle γ1 of the plane wave used in the analysis. This
dependency can be removed by averaging TL over all possible incident angles.
86
According to the Paris formula [41], the average power transmission coefficient τ is
given as:
γ 1max
τ =2
∫
τ (γ 1 )sin γ 1 cos γ 1 dγ 1
(3.53)
0
where, τ (γ 1 ) is the power transmission coefficient calculated for the incident angle γ 1 ,
and γ1max is the maximum incident angle, which is chosen as 80o according to the
suggestion by Mulholland et al. [30]. Then, the average TL is obtained as:
TLavg = 10log
1
τ
(3.54)
The integration in Equation (3.53) is conducted numerically by the Simpson’s rule using
an integration step-size of 2o .
Figure 3.8 compares the calculated TLavg to the two measured TLs (one using the
source inside, the other using the source outside) in a third octave format. It is shown
that the TLavg compares fairly well in the frequency range above 300 Hz. A relatively
large difference between the curves in the low frequency range is attributed to the fact
that the effect of the boundary condition associated with the finite length of the
experimental model is more significant in the lower modes.
Effects of other
simplifications, modeling the shell motion only by the bending wave and assuming the
plane wave incident condition also must have contributed to the discrepancy, however
not necessarily only in the low frequency range.
The comparison shows that the theoretical solution will represent the characteristics
of the finite shell reasonably well, enough for it to be used as a design tool.
87
Figure 3.7. Experimental setup of TL measurement
80
70
60
TL (dB)
50
40
30
20
10
0
-10
10
1
10
2
10
3
10
4
Frequency (Hz)
Figure 3.8. Calculate TL averaged for random incident angles of the single shell
compared with measured TL
, calculated;
(source outside)
, measured (source inside);
88
, measured
3-7. Parameter Studies
With proper qualitative interpretations, the theoretical model developed can be used very
effectively in the basic design stage of cylinder-shape vibro-acoustic systems. As a
demonstration of such applications, design parameter studies are conducted. The basic
shell dimensions and simulation conditions used in the study are listed in Table 3.1.
Table 3.1. Physical dimensions and simulation conditions
Material (Fluid)
Density (kg/m3 )
Poisson’s Ratio
Young’s Modulus (Pa)
Radius (m)
Thickness (mm)
Temperature (°C)
Sound Speed (m/s)
Incidence Angle (degree)
Shell
Steel
7,750
0.3
2.1×1011
0.1
1.0
6,100
Cavity
Air
0.9389
103
388
45
Ambient
Air
1.21
20
343
3-7.1. Effect of the Incidence Angle
TLs calculated for three different incident angles (30o , 45o , 60o ) are plotted in Figure 3.9,
which indicates that the transmitted power decreases (TL increases) as the incidence
angle γ 1 increases. The incident angle of the incoming wave was chosen somewhat
arbitrarily, as 45o in the preceding analysis. TLs can be averaged for the incident angle as
it was done in Section 3-6 for a diffusive field, which however substantially increases the
related computation time. Because the main characteristics of the TL curve do not
change drastically for different incident angles as seen in Figure 3.9, using an arbitrarily
chosen incident angle, e.g. 45o , may serve as a better option in design applications.
89
100
80
TL (dB)
60
40
20
0
-20
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 3.9. TL curves for the single shell with respect to incidence angle
, γ1 = 30°;
, γ1 = 45°;
, γ1 = 60°
3-7.2. Effect of Different Materials
Figure 3.10 shows the effect of the shell material. Materials chosen for the comparison
are steel, aluminum and brass as shown in Table 3.2. The figure shows that the brass is
the most effective in the high frequency range. This is as expected because the density of
the brass is the largest, which makes it most effective in the mass controlled high
frequency range.
The figure also shows that the aluminum, which has the lowest
stiffness, is the least effective in the low frequency range, which is again as expected
because the low frequency range is controlled by the stiffness.
Table 3.2. Material properties
3
Density (kg/m )
Young’s Modulus (Pa)
Poisson’s Ratio
Steel
7,750
1.9×1011
0.3
Aluminum
2,700
0.71×1011
0.33
90
Brass
8,500
1.04×1011
0.37
80
70
60
TL (dB)
50
40
30
20
10
0
-10
1
2
10
3
10
4
10
10
Frequency (Hz)
Figure 3.10. TL curves for the single shell with respect to material
, steel;
, aluminum;
, brass
3-7.3. Radius Effects
As shown in Figure 3.11, a smaller radius makes the TL higher, which is caused by the
curvature effect of the shell on its stiffness. A large size muffler will have a sidewall of
smaller stiffness, which also will act as an effective radiator because of the large surface
area. Therefore the sidewall of such a large muffler will have to be designed with more
care.
91
80
70
60
TL (dB)
50
40
30
20
10
0
-10
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 3.11. TL curves for the single shell with respect to radius
, Ri=0.1 m;
, Ri=0.15 m;
, Ri=0.2 m
3-7.4. Thickness Effects
As it is seen in Figure 3.12, changing the thickness has a broadband effect on TL over the
entire range of the frequency. In general, TL increases 6 dB as the thickness doubles,
which is well anticipated. In a practical situation, the shell will have to be designed only
as thick as necessary because of the weight constraint. The type of analysis developed in
this work will be very useful in such a situation. For example, if the target TL is known
from the consideration of the noise level, a proper thickness of the shell can be easily
calculated.
92
90
80
70
60
TL (dB)
50
40
30
20
10
0
-10
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 3.12. TL curves for the single shell with respect to thickness
, hi=0.5 mm;
, hi=1.0 mm;
, hi=2.0 mm
3-8. Conclusions
An analytical model was developed to calculate the noise transmission through a thin
cylindrical shell with an intention to apply it to the design of the sidewall of automotive
mufflers. Three simplifications were made to idealize the problem: (1) the system is
infinitely long, (2) the input sound is a plane wave incident obliquely from the outside of
the shell, and (3) the inside of the shell is an anechoic cavity. Practical implications of
these simplifications were discussed.
The behavior of the simplified model was described by a set of vibro-acoustic
equations, which were solved exactly by the mode superposition method. Motions of the
shell in all three directions and the vibro-acoustic coupling effect were fully considered.
A scheme to ensure the convergence of the solution was included in the analysis
procedure, therefore the series solution could be considered as the exact solution of the
93
simplified problem. As a partial validation of the solution procedure, the theoretical
results were compared to the solution obtained from a simpler system: one dimensional
beam model. The transmission loss of a finite shell was measured experimentally, and
was compared to the corresponding analytical solutions. This experimental comparison
served as another partial validation of the theoretical solution and a preview of applying
the simplified theoretical model to design situations.
Taking the advantage of having an exact solution procedure, the performance of the
cylindrical shell as an acoustic barrier was studied. For a typical configuration of the
muffler used for a passenger car, the design parameter study was conducted to understand
the effects of the material, curvature and thickness of the shell on the performance of the
muffler.
94
Chapter 4 - Sound Transmission through Double-Walled Cylindrical
Shells
4-1. Introduction
Double-walled flat or curved panels are found in many applications because they are
effective sound barriers that can increase the transmission loss (TL) over the equivalent
single-wall construction. Problems of sound transmission through panels of single or
double walls and sandwich panels have been investigated by many researchers, which are
mostly analytical efforts [21, 27-34, 42-44]. White [21] compared the analytical result
with the experimental measurement for a finite cylindrical single-walled shell for very
limited cases. In case of the single-walled shell case [45], the sound transmission through
a single-wall cylindrical shell was studied. In the work, an exact solution was obtained in
a series form using a classical shell vibration equation without ignoring any of three
directions of shell motions.
The desire to obtain design rules of thin, double-walled shells used as the side surface
of high-end automotive mufflers served as the practical motivation for this work. In such
applications, two thin plates are combined by spot welding, and rolled into a circular or
elliptical cylinder shape. Some simplifications of the problem are made so that it can be
solved exactly. First, the length of the system is assumed to be infinite, which eliminates
the need to consider the effect of the boundary conditions at the end of the shell. Second,
the incident acoustic wave is assumed to be a plane wave. Finally, an anechoic condition
is assumed in the interior cavity. The first and second simplifications reduce the problem
in consideration to a two-dimensional (2-D) problem, as it will be explained. The second
95
and third idealizations make the model a rough reciprocal of a muffler tested in an
anechoic chamber, which has the anechoic condition outside and a diffusive field inside.
Two different system models are combined to obtain TL of the system in the analysis.
The first model is to calculate the sound transmission due to the transverse bending
waves in the shells and the acoustic waves.
In this model, the system equation is
composed of two sets of shell vibration equations and three acoustic wave equations for
the external, gap and internal spaces. The exact solution to these equations is obtained in
series forms, and represented in terms of TL. The second model is to calculate the sound
transmission caused by the compression / rarefaction waves in the shell and the acoustic
space, in which the shells are treated like fluid. The model is described by a onedimensional (1-D) wave propagating through three layers of acoustic media, two shell
layers and one airgap layer. It is found that the TL from the 1-D model is lower than the
TL from the 2-D model in the low frequency range, and vice versa in the high frequency
range. Considering the definition of the TL (lower TL means higher transmitted sound),
it is easily realized that the lower TL curve should be taken to represent the system
response. Therefore, the system TL is obtained by combining TL curves from the 1-D
and 2-D models. This superposition concept is introduced for the first time in this work.
Sound transmissions through single and double shells are measured experimentally,
to which the theoretical solutions are compared. Single and double-walled shells used in
the experiment had the same dimensions and properties as the theoretical model except
their finite lengths. Thick end caps were used to close both ends of the cylinders to
eliminate the effect of the sound radiated from the end plates. TLs are measured in an
anechoic chamber, once with the sound source located inside the cylinder and the other
96
time with the sound source located outside of the cylinder.
If the reciprocity is
considered, it is realized that these setups are rough equivalents to each other. The main
purpose of the experimental comparison is to confirm that the formulation and
computation procedure of the problem are free of fundamental errors, which is always a
possibility in a numerical model without experimental validation.
Utilizing the
advantage of having a theoretical procedure validated experimentally, the effect of
important parameters such as thickness ratio and the size of the airgap are studied.
4-2. Analytical Solution Procedure
4-2.1. Sound Transmission by Bending Waves in Shells
Figure 4.1 shows a schematic of two concentric cylindrical shells of infinite length. Ri ,
Re, hi, and he indicate the radii and thickness of the shells, in which and in general the
subscripts i and e represent the inner and outer shells. As shown in the figure, a plane
wave is incident with an angle γ1 and reflected by the external surface of the cylinder.
Only the transmitted wave is considered in the internal cavity. The reflected waves are
ignored in the internal cavity to make the model approximate the reciprocal of a muffler
tested in an anechoic chamber, which has the incident and reflected waves internally and
only transmitted waves externally.
97
Reflected wave
γ1
Incident plane wave
z
Ri
Re
Transmitted wave
y
θ
x
Outer shell
r
tg
Ri
te
θ
ti
Re
Inner shell
Figure 4.1. Schematic description of the problem: 2-D model
98
4-2.1.1. Formulation of the Governing Equations
The fluid media in the external, in-between, and the internal space are defined by the
density and the speed of sound: {ρ1 , c1 }, {ρ2 , c2 } and {ρ3 , c3 }, respectively. Properties of
the shells are defined by the density, Young’s modulus, and the Poisson’s ratio: {ρi, Ei,,
µi}, {ρe, Ee, µe}. The incoming noise is idealized as a plane wave pI traveling in the x-z
plane incident with an angle γ1 as shown in the figure.
The acoustic pressure in the external space p1 = p I + p1R , where p I is the incident
wave and p R1 is the reflected wave, satisfies the wave equation [38]:
(
)
c1∇ 2 p I + p1R +
∂ 2 ( p I + p1R )
=0
∂t 2
(4.1)
where, ∇2 is the Laplacian operator. In the annular space between the shells, the pressure
is p2 = p2 T + p2R , where p T2 is the transmitted wave through the external shell and p 2R is
the reflected wave from the internal shell, which satisfies the acoustic wave equation:
(
)
c2 ∇ 2 pT2 + p2R +
∂ 2 ( pT2 + p2R )
=0
∂t 2
(4.2)
An anechoic condition is assumed in the internal cavity, therefore p3 = pT3 , which
satisfies:
c3∇ 2 pT3 +
∂ 2 pT3
=0
∂t 2
(4.3)
Love’s equations, classical thin shell equations, are used to describe the motions of the
two shells [39]. Equations of motion in the axial, radial and circumferential directions of
the inner shell are:
L1 {u10 , v10 , w10 } = ρ i hi u&&10
(4.4)
99
{
}
L2 u10 , v10 , w10 = ρ i hi &v&10
(4.5)
L3 {u10 , v 10 , w10 } + ( p2T + p2R ) − p3T = ρ i hi w
&&10
(4.6)
where, L1 , L2 and L3 are differential operators which can be found in reference [39],
{u , v , w } , i =1,2, represent the displacements of the shell of a point on the neutral
0
i
0
i
0
i
surface in the axial, circumferential, radial directions, where the subscript 1 and 2 denote
the variables associated with the inner shell and the outer shell.
For the outer shell, equations of motion in the direction of the axial, radial and
circumferential directions are:
L1 {u 20 , v20 , w20 } = ρ e he &u&02
{
(4.7)
}
L2 u 20 , v 20 , w02 = ρ e he &v&20
(4.8)
L3 {u20 , v 20, w20 } + ( p I + p1R ) − ( pT2 + p2R ) = ρ ehe w
&&20
(4.9)
At the interfaces between the shells and air, the following equations must be satisfied:
∂ ( p I + p1R )
∂r
∂ ( pT2 + p2R )
∂r
∂ ( pT2 + p2R )
∂r
= − ρ1
∂ 2 w02
at r = Re
∂t 2
(4.10)
= − ρ2
∂ 2 w02
at r = Re
∂t 2
(4.11)
= − ρ2
∂ 2 w10
at r = Ri
∂t 2
(4.12)
∂p3T
∂ 2 w0
= −ρ 3 21
∂r
∂t
at r = Ri
(4.13)
100
4-2.1.2. Solution Procedure
The solution to Equations (4.1) to (4.9) that satisfies four boundary conditions in
Equations (4.10) to (4.13) can be obtained using the mode superposition method. The
harmonic, plane incident wave pI can be expressed in the cylindrical coordinates as [40]:
∞
p I ( r, z ,θ , t ) = p0 ∑ ε n (− j ) n J n (k1r r ) cos[nθ ]e j (ωt −k 1z z )
(4.14)
n= 0
where po is the amplitude of the incident wave, n indicates the circumferential mode
number, ε n =1 for n = 0 and 2 for n=1,2,,3.., j = −1 , Jn is the Bessel function of the
first kind of order n, and ω is the angular frequency. Also, wave numbers in Equation
(4.14) are defined as:
k1 =
ω
,
c1
k1 z = k1 sin(γ 1 ) , k1r = k1 cos(γ 1 )
(4.15)
k2 =
ω
,
c2
k 2 z = k1 z ,
k 2 r = k 22 − k 22z
(4.16)
k3 =
ω
,
c3
k 3 z = k1 z ,
k 3 r = k32 − k32z
(4.17)
Because of the circular cylindrical geometry, the pressures p1R , p 2T , p 2R and p 3T are
expanded as:
∞
p ( r, z, θ , t ) = ∑ p1Rn H n2 ( k1r r ) cos[nθ ]e j (ωt −k1 z z )
R
1
(4.18)
n =0
∞
p T2 ( r , z , θ , t ) = ∑ p T2 n H 1n (k 2 r r ) cos[nθ ]e j ( ωt − k1 z z )
(4.19)
n =0
p 2R ( r , z , θ , t ) =
∞
∑p
n =0
R
2n
H n2 ( k 2 r r ) cos [n θ ]e j ( ωt −k 1 z z )
101
(4.20)
∞
p3T ( r, z, θ , t ) = ∑ p T3n H n1 (k 3 r r ) cos[nθ ]e j ( ωt −k1z z )
(4.21)
n =0
where H 1n and H n2 are Hankel functions of the first and second kind of order n. Notice
that the expressions satisfy the wave equations (Equations (4.1) to (4.3)) and the
directions of the traveling waves automatically.
Because the trace velocities of all traveling waves have to be the same, shell
displacements can be expressed as:
∞
w ( z , θ , t ) = ∑ w10n cos[nθ ]e j (ωt − k1z z )
0
1
(4.22)
n =0
∞
u10 ( z, θ , t ) = ∑ u10n cos[nθ ]e j ( ωt −k1 z z )
(4.23)
n =0
∞
v ( z ,θ , t ) = ∑v10n sin [nθ ]e j (ω t− k
0
1
1z
z)
(4.24)
n= 0
∞
w ( z ,θ , t ) = ∑ w20n cos[nθ ]e j ( ωt −k
0
2
1 zz)
(4.25)
n =0
∞
u 20 ( z, θ , t ) = ∑ u 20n cos[nθ ]e j ( ωt −k1 z z )
(4.26)
n =0
∞
v 20 ( z , θ , t ) = ∑ v20n sin[nθ ]e j (ωt − k1z z )
(4.27)
n =0
Reference [39] explains the choice of the displacement functions for the in-plane
displacements u and v in relation to those for w in the cylindrical shell.
Substituting the expressions in Equa tions (4.18) to (4.27) to six shell equations
(Equations (4.4) to (4.9)) and four boundary conditions (Equations (4.10) to (4.13))
provides 10 equations.
These equations can be used to solve for 10 unknowns:
102
0
p1Rn , p2Tn , p 2Rn , p3Tn , u1n0 , v1n0 w1n0 , u20 n , v 2n
and w20 n in terms of the amplitude of the incident
wave po . Six shell vibration equations are:
K (1 − µ ) K (1 + µi )
Kµ
u10n ρ i hi ω 2 − K i k12z − i 2 i n 2 − i
nk1 z v0In j − i i k1 z w0In j = 0
2 Ri
2 Ri
Ri
K i (1 − µi ) 2 K i 2 Di (1 − µ i ) 2
k1z − 2 n −
k 1z
−
2
Ri
2 Ri2
Ki
Di
0 K i (1 − µ i )
2
0
u1n
k1z nj +
µ k nj − 2 µ i k1 z n + v In
R i i 1z
Ri
Di 2
2
2R i
+ R 4 n + ρi hiω
i
(4.28)
(4.29)
K
D (1 − µ )
D
+ w 0In − 2i n − i 2 i k 12z n − 4i n 3 = 0
Ri
Ri
Rii
D
K
D
D (1 − µ )
D
K
u20n − Di k 14z − 2i µ i k12z n 2 + i µ i k1 z j + v 02n 2i µ i k12z n − i 2 i k12z n + 4i n 3 − 2i n
Ri
Ri
Ri
Ri
Ri
Ri
Dµ
2Di (1 − µi ) 2 2 Di 4 K i
+ w20n − i 2 i k12z n 2 −
k1z n − 4 n − 2 + ρi hi ω 2 + p2Tn H n1 (k 2r R i )
2
Ri
Ri
Ri
Ri
(4.30)
+ p 2Rn H n2 (k 2r R i ) − p T3n H n1 (k 3 r Ri ) = 0
K (1 − µ ) K (1 + µ e )
K µ
u 02 n ρ e h eω 2 − K ek12z − e 2 e n 2 − e
nk1 z v02 n j − e e k1 z w20n j = 0
2 Re
2 Re
Re
K e (1 − µ e ) 2 Ke 2 De (1 − µ e ) 2
k1z − 2 n −
k1z
−
2
Re
2 Re2
K e (1 − µ e )
Ke
De
0
2
0
u 2n
k1z nj +
µ e k1z nj − 2 µ e k1z n + v 2n
Re
Re
De 2
2
2R e
+ R 4 n + ρ e h eω
e
K
D (1 − µ )
D
+ w20n − e2 n − e 2 e k12z n − 4e n 3 = 0
Re
Re
Re
(4.31)
(4.32)
D (1 − µ )
De
µ e k12z n − e 2 e k12z n
2
R
Re
D
K
u20 n − De k14z − e2 µ e k12z n 2 + e µ e k1 z j + v 02 n e
Re
Re
De 3 K e
+ R 4 n − R 2 n
e
e
Dµ
2 De (1 − µ e ) 2 2 De 4 Ke
+ w02 n − e 2 e k12z n 2 −
k1 z n − 4 n − 2 + ρ e heω 2
2
Re
Re
Re
Re
R
2
T
1
R
2
n
+ p1 n H n ( k1r Re ) − p 2 n H n ( k 2 r Re ) − p 2 n H n ( k 2 r Re ) = − p0 ε n ( − j ) J n ( k1 r Re )
103
(4.33)
In above equations, the membrane stiffness and bending stiffness K and D are defined as
follows [39]:
Ki =
Eh
Ei hi
and K e = e e2
2
1 − µi
1− µe
Ei hi3
Di =
12 1 − µi2
(
)
(4.34)
E e he3
and De =
12 1 − µ e2
(
)
(4.35)
Four equations from the boundary conditions are:
′
p1Rn H n2 ( k1 r Re ) k1 r − ρ1ω 2 w20 n = − p 0 ε n ( − j ) n J n ′ ( k1r Re ) k1r
(4.36)
′
′
p T2 n H 1n (k 2 r Re ) k 2 r + p 2Rn H n2 ( k 2 r Re ) k 2 r − ρ 2 ω 2 w20 n = 0
(4.37)
′
′
p 2Tn H 1n ( k 2 r Ri )k 2 r + p 2Rn H n2 ( k 2 r Ri ) k 2 r − ρ 2 ω 2 w10n = 0
(4.38)
′
p T3 n H 1n ( k 3 r Re ) k 3 r − ρ 3 ω 2 w10n = 0
(4.39)
where, ( ) ' =
d
.
dr
Equations (4.28) to (4.39) constitute ten equations for ten unknowns, which can be
put into a matrix equation:
0
0 A B
0 0
0 0
0
0 D E
G H
I
0 J K
0
0 0 0
0 0
0 0
0
0 0 0
0 S T U 0 0
Y 0
0
0 0 0
0 A1 B1 0 0 0
0 D1 E1 0 0 0
0 G1 0 0
0 0
C 0
F 0
L 0
0 M
0 P
0 V
Z 0
C1 0
0 0
0 0
0
0
0
N
Q
W
0
0
0
0
0 p1Rn 0
0 p2Tn 0
0 p2Rn I 1
O p3Tn 0
R u 20n 0
=
X v 02 n 0
0 w20 n J1
0 u10n 0
F1 v10n 0
H1 w10n 0
104
(4.40)
where,
A = ρe heω 2 − Ke k12z −
K e (1 − µ e ) 2
K (1 + µe )
n ,B = − e
nk1 z j
2
2 Re
2 Re
C=−
K e µe
K (1 − µe )
K
D
k1 z j , D = e
k1 z nj + e µ e k1 z nj − e2 µe k12z n
Re
2 Re
Re
Re
E=−
K e (1 − µ e ) 2 K e 2 De (1 − µ e ) 2 De 2
k1 z − 2 n −
k1k + 4 n + ρ eheω 2
2
2
Re
2 Re
Re
F =−
D (1 − µ )
Ke
D
n − e 2 e k12z n − e4 n 3
2
Re
Re
Re
G = H n2 ( k1 r Re ) , H = −H 1n ( k 2 r Re ) , I = − H n2 ( k 2 r Re )
J = −De k14z −
K=
De
K
µe k12z n 2 + e µe k1z j
2
Re
Re
D (1 − µ )
De
D
K
µe k12z n − e 2 e k12z n + e4 n3 − e2 n
2
Re
Re
Re
Re
L=−
De µ e 2 2 2 De (1 − µ e ) 2 2 De 4 Ke
k1 z n −
k1 z n − 4 n − 2 + ρe heω 2
Re2
Re2
Re
Re
M = ρ i hiω 2 − Ki k12z −
N=−
P=
K i (1 − µ i ) 2
n
2 Ri2
K i (1 + µ i )
Kµ
nk1 z j , O = − i i k1 z j
Ri
2 Ri
Ki (1 − µi )
K
D
k1 z nj + i µ i k1z nj − 2i µ i k12z n
2 Ri
Ri
Ri
Q=−
Ki (1 − µ i ) 2 Ki 2 Di (1 − µi ) 2 Di 2
k1 z − 2 n −
k1k + 4 n + ρi hiω 2
2
Ri
2 Ri2
Ri
R=−
D (1 − µ )
Ki
D
n − i 2 i k12z n − 4i n3
2
Ri
Ri
Ri
105
S = H n1 ( k 2 r Ri ) , T = H n2 ( k 2 r Ri ) , U = −H 1n ( k3 r Ri )
V = −Di k14z −
W=
Di
K
µ i k12z n2 + i µ i k1 z j
2
Ri
Ri
D (1 − µ )
Di
D
K
µi k12z n − i 2 i k12z n + 4i n 3 − 2i n
2
Ri
Ri
Ri
Ri
X =−
Di µi 2 2 2Di ( 1− µ i ) 2 2 Di 4 K i
k1 z n −
k1 z n − 4 n − 2 + ρi hiω 2
2
2
Ri
Ri
Ri
Ri
′
′
′
Y = H n2 ( k1r Re )k 1r , Z = − ρ1ω 2 , A1 = H n1 (k 2 r Re ) k 2 r , B1 = H n2 (k 2 r Re ) k 2 r
′
′
C1 = − ρ 2 ω 2 , D1 = H 1n ( k 2 r Ri ) k 2 r , E1 = H n2 ( k 2 r Ri ) k 2 r , F1 = − ρ 2 ω 2
′
G1 = H 1n (k 3 r Ri ) k 3 r , H1 = − ρ 3ω 2 , I 1 = − p0 ε n ( − j ) n J n (k1 r Re )
′
J 1 = − p 0 ε n ( − j ) n J n ( k1 r Re ) k1 r
0
0
The ten unknown coefficients p1Rn , p 2Tn , p 2Rn , p3Tn , u1n0 , v1n0 , w1n0 , u 2n
, v 2n
and w02n are
obtained in terms of po by solving Equation (4.40). These can be substituted back into
Equations (4.18) to (4.27) to find the displacements of the shell and the acoustic pressures
in series forms.
4-2.1.3. Solution in Terms of the Transmission Loss (TL)
The sound power transmitted to the interior cavity per unit length of the shell is:
2 π T ∂
1
W = Re ∫ p3 ⋅
( w10 ) * Ri dθ
∂t
2
0
T
where r=Ri
(4.41)
where, Re{.} and the superscript * represent the real part and the complex conjugate of
the argument.
106
Substitution of Equations (4.21) and (4.22) for p 3T and w10 into above Equation (4.41)
yields an expression for the components of WnT .
WnT =
=
{
2π
} ∫ cos [nθ ]⋅ R dθ
1
Re p3Tn H1n (k3 r Ri )⋅ ( jω w10n )* ×
2
2
i
where r=Ri
0
π Ri
× Re{ p3Tn H 1n (k 3 r Ri )⋅ ( jω w10n )* }
2ε n
(4.42)
where, ε n =1 for n = 0 and ε n =2 for n=1,2,3,……
TL is defined by
∞
TL = −10log10 ∑ WnT / W I
n= 0
(4.43)
where W I is the incident power per unit length of the shell in the axial direction:
WI =
cos (γ 1 ) p02
× 2Re
ρ1 c1
(4.44)
Thus, an exact expression for TL can be obtained by substituting Equations (4.42) and
(4.44) into (4.43) as follows:
{
}
Re pT3n × H n1 (k3r Ri ) × ( jωw10n ) × ρ1c1π × Ri
TL = − 10log10 ∑
4ε n Re cos(γ 1 )p02
n=0
∞
*
(4.45)
The TL curves of single-walled and double-walled shells calculated in this manner
are shown in Figure 4.2. The geometry of the shells used in the calculation are Re = 0.1
m, Ri = 0.09949 m, he = 0.6 mm, hi= 0.4 mm, hg =0.01 mm, for the double-walled shell,
and 0.1m radius and 1 mm thickness for the single-walled shell.
The material of the
shells is steel, whose Young’s modulus, material density and Poisson’s ratio are E =
1.9×1011 Pa, ρ = 7,750 kg/m3 and µ = 0.3. The temperature of the interior cavity is taken
107
as 103o C, which is a typical operating condition of the automotive muffler. The incident
angle of 45
o
is used for the figures. Parameters used in the calculation to obtain Figure
4.2 are listed in Table 4.1 for the double-shell case, from which parameters for single
shell case can also be obtained by ignoring the parameters corresponding to one of the
shells and the airgap. The condition in Table 4.1 is referred as the muffler condition.
90
80
70
60
TL (dB)
50
40
30
20
10
0
-10
1
2
10
3
10
4
10
10
Frequency (Hz)
Figure 4.2. TLs calculated from 2-D model at muffler condition
-------, single shell;
, double shell
Table 4.1. Parameters to calculate TLs of the double shell at the muffler condition
Temperature
(°c)
Density
(kg/m3 )
Speed of
Sound (m/s)
Outside Air
Outer Shell
Airgap
Inner Shell
Inside Air
20
-
70
-
103
ρ1
1.21
ρe
7750
ρ2
1.03
ρi
7750
ρ3
0.94
c1
343
ce
6100
c2
371
ci
6100
c3
389
108
4-2.2. Sound Transmission Analysis by Plane Wave Model
In this case, the problem is idealized as a one-dimensional wave propagation problem as
shown in Figure 4.3 by assuming that the shells are behaving like a fluid.
The four-pole method can be used conveniently for this type of analysis [46]. The
four-pole equation between the system’s input and output variables are:
pA
=
v A
cos k A l A
j
sin k A l A
rA
jrA sin k Al A cosk B lB
j
cos k A l A sin k Bl B
rB
jrB sin k B lB cos k C lC
j
cos k B l B sin kC lC
rC
jrC sin k C lC
pD
cos kC lC v D
(4.46)
where p and v are the acoustic pressure and particle velocities, k, r, and l indicate the
complex wave number, characteristic impedance, the thickness of each layer respectively.
Subscripts A, B and C refer parameters corresponding to the outer shell, the airgap and
the inner shell, respectively.
Multiplying matrices in Equation (4.46), the system
equation is obtained as:
p A A11
=
v A A21
A12 11
pD
A22 r
D
(4.47)
Since the total particle velocity and pressure v A and pA can be expressed as:
v A = v i − vr =
A
pi − pr
= 21 + A22 p D
rA
rD
(4.48)
Therefore,
A
p i − p r = rA v A = rA 21 + A22 p D
rD
(4.49)
Also,
A
p A = pi + pr = 11 + A12 p D
rD
(4.50)
109
Pi
PAi
PBi
PCi
PD
Cavity
Air
Inner
Shell
Pr
PAr
A
Airgap
Outer
Shell
PBr
PCr
B
C
Exterior
Air
Anechoic
Termination
D
Figure 4.3. Schematic description of the problem: 1-D model
60
50
TL (dB)
40
30
20
10
0
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 4.4. TLs calculated from 1-D model at muffler condition
-------, single shell;
, double shell
110
From Equations (4.49) and (4.50), the incident pressure pi and the reflective pressure
pr can be separated, and the reflection coefficient R can be obtained:
R=
pr
pi
A11
+ A12 −
rD
=
A11
+ A12 +
rD
rA
A22 − rA A22
rD
rA
A +r A
rD 22 A 22
(4.51)
Because the cross-sectional area of the input and output side of the system are the same in
this case, the power transmission coefficient becomes:
Tπ = 1 − R
2
(4.52)
Finally the transmission loss can be estimated from the power transmission coefficient:
TL( dB ) = 10 log 10
1
Tπ
(4.53)
The TL curves calculated by this 1-D model are shown in Figure 4.4 for the same
system at the same condition used to obtain Figure 4.2.
4-2.3. Combined Solutions
The TL curves in Figures 4.2 and 4.4 can be combined to represent the system response
in the entire frequency range. Considering that lower TL means higher transmitted noise,
the rule to combine two TL curves should be picking up the lower TL at each frequency.
Figure 4.5 shows the TL curves of the single shell and the double shell obtained by
combining the curves in Figures 4.2 and 4.4.
These combined TLs are used for
comparisons of the analytical results and the experimental results.
111
90
80
70
60
TL (dB)
50
40
30
20
10
0
-10
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 4.5. Combined TLs from Figures 4.2 and 4.4.
--------, single shell;
, double shell
4-3. Comparison to Experimental Results
4-3.1. Measurement Setup
Figure 4.6 shows the experimental setup to measure TLs of the shells in an anechoic
chamber. Shown is the single shell, which has the same appearance as the double shell.
A pair of microphones shown in the figure is to measure the sound intensity on the
external surface. One more pair of microphones is located at the same position on the
internal surface of the shell to measure the sound intensity on the internal surface.
The geometry of the shells used in the experiment is that, with reference to Figure
4.1, Re = 0.1 m, Ri = 0.098m, he = 1 mm, hi= 1 mm, hg =1 mm, for the double shell, and
0.1m radius and 1 mm thickness for the single-walled shell.
The shells were made of
steel, which has the Young’s modulus E = 1.9 × 1011 Pa, the material density ρ = 7,750
kg/m3 and the Poisson’s ratio µ = 0.3. The temperature was 20o C throughout the entire
system during the measurements. Parameters corresponding to this condition are listed in
112
Table 4.2 for the double shell case, from which parameters for the single shell can be
easily obtained. The condition defined by Table 4.2 is referred as the test condition.
For both shells, the measurement was conducted once using an internal sound source,
and the other time using an external source. The internal source was a 3- inch speaker and
the external source was a B&K (Type 4296) decahedral speaker. A white noise with the
maximum frequency of 6,400 Hz was used to drive the sound sources. The frequency
resolution of the measurement was 16 Hz.
The internal sound source setup represents the theoretical model better because it has
an anechoic condition externally and reverberant condition internally. This setup is a
rough reciprocal of the theoretical model. The external sound source setup has incident
as well as reflected waves both internally and externally.
Figure 4.6. Experimental setup of TL measurement
113
Table 4.2. Parameters to calculate TLs of the double shell at the test condition
Temperature
(°c)
Density
(kg/m3 )
Speed of
Sound (m/s)
Outside Air
Outer Shell
Airgap
Inner Shell
Inside Air
20
-
20
-
20
ρ1
1.21
ρe
7750
ρ2
1.21
ρi
7750
ρ3
1.21
c1
343
ce
6100
c2
343
ci
6100
c3
343
4-3.2. Single Shell Measurement
The analytical solutions of the sound transmission through the single shell can be
obtained ignoring the equations corresponding to one of the shells and the annular space.
Figures 4.7-a and 4.7-b compare the measured and calculated TLs of the single shell.
The calculation was made using the test condition (see Table 4.2 for parameters). The
incident angle of 34o was used in the calculation, which was estimated based on the
relative location between the sound source and the intensity probes. Figure 4.7-a is the
comparison of the calculation and the test with the source inside and Figure 4.7-b is the
comparison of the calculation and the test with the source outside.
The figures show that the calculated TL curves agree reasonably with the measured
TL curves if the differences between the experimental and theoretical models are
considered. It is believed that the measured TL in Figure 4.7-a has weaker frequency
components in the low frequency range as compared to that in Figure 4.7-b because of
the limitation of the performance of the 3-inch speaker used for the former. On the other
hand, the former has stronger frequency components than the latter in the high frequency
range, which is believed to be the effect of the stronger direct field of the internal source
setup used in the former.
114
Calculated TL obviously depends on the choice of the incident angle in the analysis.
This dependency can be removed by averaging TL over all possible incident angles.
According to the Paris formula [41], the average power transmission coefficient τ is
given as:
γ 1max
τ =2
∫
τ (γ 1 )sin γ 1 cos γ 1 dγ 1
(4.54)
0
where, τ (γ 1 ) is the power transmission coefficient calculated for the incident angle γ 1 ,
and γ1max is the maximum incident angle, which is chosen as 80o according to the
suggestion by Mulholland et al. [30]. Then, the average TL is obtained as:
TLavg = 10log
1
τ
(4.55)
The integration in Equation (4.54) is conducted numerically by the Simpson’s rule using
an integration step-size of 2o . Figure 4.7-c compares TLavg to the two measured TLs in a
third octave format, which shows that the TLavg is in reasonably good agreement with the
TL curve measured by using the source outside. Bigger differences between the curves
in the low frequency range may be attributed to the effect of the boundary condition, a
major difference between the experimental and theoretical models, which becomes more
significant in low frequency modes.
4-3.3. Double Shell Measurement
Figures 4.8-a and 4.8-b show comparisons between the calculated and measured TL
curves of the double-walled shell. Similar trends to the single shell case are found in the
double shell case. Figure 4.8-c compares the two measured TL curves and TLavg curve
using a third octave band format for the double-walled shell.
115
70
60
50
TL (dB)
40
30
20
10
0
-10
-20
1
10
2
3
10
10
Frequency (Hz)
Figure 4.7-a. Calculated TL compared with measured TL (a) single shell, source inside
, calculated; ---------, measured
60
50
TL (dB)
40
30
20
10
0
-10
10
1
2
3
10
10
Frequency (Hz)
Figure 4.7-b. Calculated TL compared with measured TL (b) single shell, source outside
, calculated; ---------, measured
116
50
40
TL (dB)
30
20
10
0
-10
1
2
10
3
10
4
10
10
Frequency (Hz)
Figure 4.7-c. Calculated TL compared with measured TL (c) single shell, TL averaged
for random incident angles
, calculated; ∇, measured(source inside); , measured(source outside)
80
70
60
TL (dB)
50
40
30
20
10
0
-10
10
1
2
3
10
10
Frequency (Hz)
Figure 4.8-a. Calculated TL compared with measured TL (a) double shell, source inside
, calculated; ---------, measured
117
80
70
60
50
TL (dB)
40
30
20
10
0
-10
-20
1
10
2
3
10
10
Frequency (Hz)
Figure 4.8-b. Calculated TL compared with measured TL (b) double shell, source outside
, calculated; ---------, measured
60
50
40
TL (dB)
30
20
10
0
-10
-20
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 4.8-c. Calculated TL compared with measured TL (c) double shell, TL averaged
for random incident angles
, calculated; ∇, measured (source inside); , measured (source outside)
118
4-4. Parameter Studies
Effects of design parameters and system parameters are studied analytically in terms of
their effects on the TL. The simulations are made at the muffler condition (see Table
4.1).
4-4.1. Choice of Incident Angle in Analysis
The averaging process in Equation (4.54) is very time consuming, which becomes a
problem in design iterations, in which TLs have to be calculated numerous times.
Therefore, the effect of the choice of the incident angle is studied to see if the calculation
using only one angle can be used for relative comparison purposes. Figure 4.9 compares
the TLs obtained for the incident angles of 30, 45 and 60 degrees. It is known that the
general trend of the TL curves remains the same, although the curves become different as
different incident angles are used. Therefore, the incident angle of 45o is chosen for
parameter studies later in this work.
4-4.2. Effect of the Double-Wall Construction
Figures 4.3 and 4.5 compare the TLs of the single of 1 mm thickness and double shell
composed of 0.6 mm wall and 0.4 mm apart by a very small airgap (0.01mm). Because
the total thickness of the shells is the same, the comparison shows the effect of the double
shell design. From Figures 4.3 and 4.5, it is known that the double-wall structure does
not help to increase TL in the frequency range lower than the coincidence frequency
(about 5,500 Hz in this case).
It will be shown later that the double-wall construction
provides better noise insulation as the airgap size increases, therefore a double-wall
construction with large airgap has an advantage as an acoustic barrier. A large airgap
119
design in mufflers is not available because the double-wall is made by spot welding,
which leaves a very small airgap. Figure 4.10 compares the TLs of the single shell of 1
mm thickness and double shell of two 1 mm walls. In this case the double shell provides
better sound insulation, however at the cost of doubling the weight.
Based on this study, it is believed that the perceived advantage of the double shell
muffler over the single shell counterpart is not from the better sound isolation
characteristics but from the increased damping effect of the double shell.
4-4.3. Effect of Thickness
Figure 4.11 compares the TLs calculated by the bending wave model for the shell
composed of two 1 mm walls and the shell composed of 0.4 mm and 0.6 mm walls. It is
known that the effect of the thickness increase is over a broad range of the frequency.
4-4.4. Effect of the Airgap
Figure 4.12 compares the transmission losses calculated from the bending model for the
double shells of different airgap sizes. Sound transmission through the double shells of
0.6 mm and 0.4mm with three airgap sizes of 0.01 mm, 1mm and 10 mm are considered.
The figure shows that the transmission loss can be increased quite significantly by
enlarging the airgap size, which is not a possible option for mufflers.
The first two dips in the TL curves in Figure 4.12 are noteworthy. It is believed that
the first dip around 100 Hz is caused by the lowest overall system natural frequency. The
second dip observed at around 770 Hz in Figure 4.12 corresponds to the first ring
frequency of the annular airgap. This frequency can be estimated as follows.
120
f cavitiy
ccavity
cos ( γ 1 ) c phase −speed
=
=
2π Ro
2π Ro
(4.55)
where c cavity is the sound speed in the cavity, γ 1 is the incidence angle , Ro is the radius of
the shell and c phase− speed stands for the phase speed in the circumferential direction. The
frequency is calculated to be approximately 770 Hz at the muffler operating condition.
100
90
80
70
TL (dB)
60
50
40
30
20
10
0
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 4.9. Effect of the incidence angle in TLs of the double shell (Re ≅ Ri=0.1m,
hi=0.6mm, he=0.4mm)
--------- , γ1 =30°;
, γ1 =45°;
, γ1 =60°
121
120
100
TL (dB)
80
60
40
20
0
-20
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 4.10. TLs of the single shell (R=0.1m, h=1.0mm) and double shell (Re ≅ Ri=0.1m,
hi=he=1.0mm)
--------, single shell;
, double shell
120
100
TL (dB)
80
60
40
20
0
-20
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 4.11. TLs of the double shell (Re ≅ Ri=0.1m) with respect to thickness
combination
-------, double shell (hi=1.0mm, he=1.0mm);
, double shell
(hi=0.6mm, he=0.4mm)
122
120
100
TL (dB)
80
60
40
20
0
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 4.12. Effect of airgap in TLs of the double shell (Re ≅ Ri=0.1m, hi=0.6mm,
he=0.4mm)
, hg=0.01mm; ------- , hg=1.0mm :
, hg=10mm
4-5. Conclusions
An exact solution procedure is developed to study the sound transmission through a
double-walled cylindrical shell. The solutions obtained from two models, the first of
which describes the sound transmission caused by the bending waves traveling in the
shell and the second describes the sound transmission by the plane waves traveling across
the layers of two shells and the airgap. For the bending wave solution, three acoustic
wave equations and two shell vibration equations are solved simultaneously, which
provides a series solution. This is considered to be the first exact solution to this type of
problem obtained using the full shell vibration equations.
To solve the plane wave
model, the four pole method is used. Both solutions are represented in terms of the
transmission loss (TL), which are then combined into a single TL curve. The idea of
combining the solutions also is used for the first time in this work. The analytical
123
solutions obtained are compared with the measured TLs, which show reasonable
agreement. The effects of important design parameters such as the airgap size and the
thickness ratio are also studied using the analytical solutions.
124
Chapter 5 – Sound Transmission through Double-Panels Lined with
Elastic Porous Material
5-1. Introduction
While flat or curved thin elastic panels lined with porous material are found in many
practical applications, analysis of sound transmission through such structures remains as
a very difficult task. Because the porous material has a solid phase (elastic frame) as well
as a fluid phase (air contained in pores), three wave components are necessary to describe
the acoustic response of the material [47, 48]. Many theories with different degrees of
approximation have been proposed to model the porous material, therefore to solve sound
transmission through structures lined with porous material [48-66]. In some cases it is
assumed that the solid constituent in the porous material is rigid, which allows only a
single longitudinal wave in the material [49-51].
In some models the elasticity of the
frame is considered but its shear deformation is neglected, thus allowing two longitudinal
waves [52-54] in the material but not the oblique incident cases [51, 55].
Bolton et al.
[48] developed an analysis method based on Biot’s theory [47], which considers the
porous material as homogeneous aggregate of the elastic frame and fluid contained in the
pores. The theory allows three waves in the porous material, two dilatational (also called
longitudinal) waves and one rotational (also called shear) wave [47, 48]; therefore, it can
handle oblique waves. One of the dilatational waves is predominantly the elastic wave,
therefore referred to as the frame wave, and the other is predominantly the acoustic wave,
therefore referred to as the airborne wave. The rotational wave is predominantly the
shear wave, or the secondary wave as it is often called. This method will be referred to as
the full method hereafter. The full method [47, 48] was applied to calculate transmission
125
losses (TLs) of elastic double-panel of infinite extent lined with elastic porous liners.
The TLs calculated for random incidences were compared with the measure TLs, which
showed quite good agreement.
Applying the full method to more general cases such as curved panels of finite extent
becomes very difficult because of the complexity of the multi-wave theory in describing
the porous material. The finite element method (FEM) formulation of the multi-wave
theory was conducted by several researchers [56-58]; however, its applications appear to
be still limited to somewhat simple cases.
This work was motivated by a desire to
develop a simplified analysis method as an alternative to the full method so that solutions
to structures of complicated geometry lined with a layer could be obtained with relative
ease and reasonable accuracy.
In this chapter, development of the simplified method is explained. The validity of
the approximate method is checked by comparing the TLs obtained for double-panel
structures from the method and the full method and measurements reported by Bolton et
al. [48]. In Chapter 6, the approximate method is applied to calculate TLs of a doublewalled cylinder sandwiching a porous layer, which serves as an example and
demonstration of applying the approximate method to practical problems. Also it is
considered the first work to calculate TLs through double-walled shells lined with porous
material taking account of the effect of the multi-waves in the porous material.
126
5-2. Development of Simplified Analysis Method
5-2.1. Review of the Full Theory
As a preliminary step to developing the simplified method for analyzing sound
transmission through structures with porous liner, the method developed by Bolton et al.
[48] based on Biot’s theory [47], which is referred to as the full method hereafter, is
reviewed. It is noted that this full method is discussed to explain the simplified method
without claiming any contributions. Readers are encouraged to read references [47, 48,
51, 59, 60] for more detailed discussions on the full method if necessary.
If the porous material is assumed a homogeneous and isotropic aggregate of the
elastic frame and the fluid trapped in pores, its acoustic behavior can be described by the
following: two wave equations [47].
∇ 4 es + A1∇ 2 es + A2 es = 0
(5.1)
∇ 2ϖ + kt2ϖ = 0
(5.2)
Equation (5.1) governs elastic dilatational waves, and Equation (5.2) governs the
rotational wave. In Equations (5.1) and (5.2), es = ∇⋅ u is the solid volumetric strain, u
being the displacement vector of the solid, ϖ = ∇ × u is the rotational strain in the solid
phase.
Also, parameters A1 and A2 are:
A1 =
ω 2 ( ρ11* R − 2 ρ12* Q + ρ22* P)
( PR − Q2 )
(5.3)
A2 =
*
ω 4 [ ρ11
ρ 22* − ( ρ12* ) 2 ]
( PR − Q2 )
(5.4)
*
*
where, ω is the circular frequency, ρ11* , ρ22
, and ρ12
are equivalent masses to be
explained later, R = hE2 , Q = (1 − h) E2 , where h is the porosity and E2 is the bulk
127
modulus of the elasticity of the fluid in the pores, P = A + 2 N , A =
first Lame ′ constant, N =
νE1
is the
(1 + ν )(1 − 2ν )
E1
is the shear modulus of the solid phase of the porous
2(1 + ν )
material, E1 and ν being the in vacuo Young’s modulus and Poisson’s ratio of the bulk
solid phase [47, 61]. Assuming that pores are cylindrical, an expression fo r E2 [51, 52,
60] has been defined as:
{
}
1/2
E2 = E0 1+ [2(γ − 1)/ N 1/2
pr λc − j ]Tc N pr λc − j
−1
(5.5)
where E0 = ρ0 c 2 , γ is the ratio of specific heats, c is the speed of sound in the fluid, j =
−1 , and Npr represents the Prandtl number.
Among equivalent masses, ρ11* can be considered as the effective bulk mass [47] of
*
the elastic part, ρ22
can be considered as the effective bulk mass [47] of the fluid part,
*
and ρ12
, can be considered as the coupling mass [47] between the frame and fluid, which
are obtained as:
*
ρ11* = ρ11 + b / jω , ρ 22
= ρ 22 + b / j ω , ρ12* = ρ12 − b / jω
(5.6)
where, ρ11 = ρ1 + ρa , ρ 22 = ρ 2 + ρa , ρ12 = − ρa , and ρ1 is the bulk density of the solid
phase, ρ 2 = hρ 0 is the bulk density of the fluid phase, ρ0 is the density of interstitial fluid
and ρa is the coupling mass between the solid and fluid phases, which is:
ρ a = ρ 2 (ε '−1)
(5.7)
where, ε' is the geometrical structure factor [48, 53]. b in Equation (5.6) is a viscous
coupling factor [61, 62], and obtained from:
128
b = jωε ' ρ 2 ( ρc* / ρ0 −1)
(5.8)
where, for a cylindrical pore [51, 60]
{
}
ρc* = ρ0 1 − (2/ λc − j )Tc λc − j
−1
(5.9)
λ c in Equation (5.9) is given :
λ2c = 8ωρ 0ε '/ hσ
(5.10)
where, σ is the steady state, macroscopic flow resistivity. Tc in Equation (5.9) is defined
as:
Tc λ c − j = J1 λ c − j / J 0 λ c − j
(5.11)
where, J0 and J1 are Bessel functions of first kind, zero and first order, respectively.
In vacuo Young’s modulus E1 is represented as E1 = Em (1 + j η) , where Em is the
static Young’s modulus and η is the loss factor [67] that reflects the material damping in
the solid phase.
Therefore, ρ1 , Em ,η, υ , σ , ε ', h and ρ o consist a set of macroscopic
parameters that define the vibro-acoustic characteristics of an elastic porous material [51,
52, 60].
Because Equation (5.1), which represents the propagation of dilatational waves in the
solid phase, is a fourth order equation, two dilatational waves exist in the solid. The wave
numbers are:
( A1 + A12 − 4 A2 )
k =
,
2
2
1
( A1 − A12 − 4 A2 )
k =
2
2
2
(5.12)
The wave related to k 1 is referred to as the airborne wave [47, 48], which describes the
wave transmitted predominantly through the interstitial fluid, and the wave related to k 2 is
129
referred to as the frame wave [47, 48], which describes the wave transmitted
predominantly by the elastic structure. From Equation (5.2), the wave number of the
rotational wave is obtained as:
k t2 =
ω2
*
*
× ρ11
− ( ρ12* ) 2 / ρ22
N
(5.13)
Reflected wave
Incident plane wave
θ
x
y
Porous
Layer
C1
C3
C5
C2
C4
C6
Transmitted wave
Figure 5.1. Illustration of wave propagation in the porous layer
If a two-dimensional problem (i.e., an infinite domain problem) as shown in the x-y
plane of Figure 5.1 is considered, the potential of the incident wave can be expressed as:
φ i = e − j( k x + k y y )
(5.14)
where k=ω/c, kx=k×sinθ, k y=k×cosθ, and θ is the angle of incidence (i.e., the angle from
the normal to the surface). Because three types of wave exist in the porous material, six
traveling (3 forward and 3 backward) waves, which all have the same trace wave number,
are induced by an oblique incident wave in a finite depth layer of porous material as
130
shown in Figure 5.1. The x- and y- components of the displacements and stresses of the
solid and fluid phases were derived by Bolton et al. [48] as follows.
The displacements in the solid phase are:
u x = jk x e − j kx x[
C1 − jk1 y y C2 jk1 y y C3 − jk2 y y C4 jk2 y y
e
+ 2 e
+ 2e
+ 2 e ]
k12
k1
k2
k2
k
− jk y
jk y
− j ty2 e− j kx x [C5 e ty − C6e ty ]
kt
u y = je− j kx x[
k1 y
k
k
k
− jk y
jk y
− jk y
jk y
C1e 1 y − 12y C2 e 1 y + 22y C3 e 2 y − 22y C4e 2 y ]
2
k1
k1
k2
k2
k
− jk y
jk y
+ j x2 e− j kx x[ C5 e ty + C6e ty ]
kt
(5.15)
(5.16)
The displacements in the fluid phase are:
U x = jkx e− j kx x[b1
C1 − jk1 y y
C jk y
C − jk y
C jk y
e
+ b1 22 e 1 y + b2 23 e 2 y + b2 24 e 2 y ]
2
k1
k1
k2
k2
k
− jk y
jk y
− jg ty2 e− j kx x[ C5 e ty − C6e ty ]
kt
U y = je − j kx x[b1
k1 y
k
k
k
− jk y
jk y
− jk y
jk y
C1 e 1 y − b1 12y C2 e 1 y + b2 22y C3 e 2 y − b2 22y C4 e 2 y ]
2
k1
k1
k2
k2
k
− jk y
jk y
+ jg x2 e − j kx x [C5 e ty + C6e ty ]
kt
(5.17)
(5.18)
where in Equation (5.18), b1 and b2 are defined as [48]:
b1 =
*
*
( ρ11* R − ρ12* Q )
( PR − Q 2 )
( ρ11
R − ρ12
Q)
(PR − Q 2 )
2
−
k
,
b
=
−
k22
1
2
*
*
*
*
2
*
*
( ρ22
Q − ρ12* R) ω 2 ( ρ*22Q − ρ12
(
ρ
Q
−
ρ
R
)
R)
ω ( ρ22Q − ρ12 R)
22
12
The stresses in the solid phase are:
131
σy =e
− j kx x
[(2N
k12y
2
1
k
− jk1 y y
+ A + b1Q)C1 e
+ (2 N
k12y
2
1
k
+ A + b1Q)C2 e
jk1 y y
k 22 y
k 22 y
− jk2 y y
jk y
+ (2N 2 + A + b2 Q )C 3e
+ (2N 2 + A + b2 Q)C4e 2 y
k2
k2
kk
+2 N x 2ty (C5 e− jkty y − C6e jkty y )]
kt
τ xy = e − j kx x N [
(5.19)
2k x k1 y
2k k
− jk y
jk y
− jk y
jk y
(C1 e 1 y − C2 e 1 y ) + x 2 2 y (C3 e 2 y − C4 e 2 y )
2
k1
k2
(5.20)
( kx2 − kty2 )
− jk y
jk y
+
( C5 e ty + C6 e ty )]
2
kt
The stresses in the fluid phase are:
− jk
s = e− jk x ( Q + b1 R)C1 e
x
1y y
+ ( Q + b1 R) C2 e
jk1 y y
+ (Q + b2 R) C3 e
− jk2 y y
+ ( Q + b2 R)C4 e
jk2 y y
(5.21)
In above equations, σy can be considered as the force per unit material area acting in the
y-direction on the solid phase, s can be considered as the force per unit material area
acting on the fluid component of the porous material, and τxy being the shear force acting
on the solid phase.
The six constants C1 -C6 in Equations (5.15)-(5.21) have to be determined by applying
boundary conditions (BCs), that are represented in displacements and stresses. Bolton et
al. [48, 52, 65, 66] considered three types of boundary conditions associated with typical
constructions of lined double-panel. A bonded-bonded (B-B) construction occurs when
the porous layer is bonded to both plates as shown in Figure 5.2-a. A bonded-unbonded
(B-U) construction arises when one side of the porous layer is bonded and the other side
is spaced from the plate as shown in Figure 5.2-b. The unbonded–unbonded (U-U)
construction is when both sides of the layer are spaced as shown in Figure 5.2-c.
132
Porous Layer
Panel
Panel
C10
C8
Wp3
C2
Wp11
C13
C6
C9
C7
C1
C5
Wt4
8 BCs:
4 EOM:
1
Wt12
3
3
2
1
2
Total 12 BCs = 8 BCs + 4 EOM
Figure 5.2-a. Illustration of wave propagation in the B-B double-panel
133
Porous Layer
Panel
Panel
Airgap
C10
Wp3
C2
C9
C7
Wt13
C12
C8
C6
C14
C11
C1
C5
Wt4
10 BCs:
3 EOM:
1
3
4
2
1
1
1
Total 13 BCs = 10 BCs + 3 EOM
Figure 5.2-b. Illustration of wave propagation in the B-U double-panel
134
Panel
Airgap
Airgap
Porous Layer
Panel
C11
C2
Wt3
Wt14
C5
C10
C4
C8
C9
C13
C7
C12
C15
C1
C6
12 BCs:
2 EOM:
1
1
4
4
1
1
1
1
Total 14 BCs = 12 BCs + 2 EOM
Figure 5.2-c. Illustration of wave propagation in the U-U double-panel
135
At the open surface of a porous layer (illustrated schematically in Figure 5.3) there
are four BCs to be satisfied. These conditions are:
(i) –h×P = s
(ii) –(1-h)×P = σy
(iii) v y = jω (1− h )u y + jω hU y
(iv) τ xy = 0
where P is the pressure in the exterior acoustic field at the interface, and v y is the normal
component of the acoustic particle velocity of the exterior medium at the interface.
vy
P
x
τ xy
y
jωuy
jωUy
σy
s
Figure 5.3. Detailed cross-sectional view of the open surface of a porous layer [48]
When the elastic porous material is bonded directly to an elastic panel (as illustrated
schematically in Figure 5.4), there exist six BCs. Letting the transverse displacement and
in-plane displacement at the neutral axis be wt ( x , t ) = Wt ( x )e j ωt and wp ( x , t ) = Wp ( x )e j ωt ,
four BCs are from the interface compatibility:
(i) v y = j ωWt (ii) u y = Wt
(iii) U y = Wt
(iv) ux = W p ( − / + )
Two BCs are from equations of motion:
(v) ( + / −)τ yx = ( Dp k x2 − ω 2 ms )W p
(vi) ( + / −) P( − / +) q p − jkx
hp
2
τ xy = ( Dk x4 − ω 2 ms )Wt
136
h p dWt
2 dx
where hp is the panel thickness, D =
Dp =
Eh 3p
is the bending stiffness per unit width,
12(1 − µ 2 )
Ehp
is the membrane stiffness per unit width, ms is the panel mass per unit area, E
1− µ2
is the in vacuo Young’s modulus of the panel, µ is the in vacuo Poisson’s ratio of the
panel, and qp is the normal force per unit panel area exerted on the panel by the elastic
porous material (qp =-σy-s). In BCs (iv)-(vi) [48], the first signs are appropriate when the
porous material is attached to the positive y- facing surface of the panel, and the second
signs when the porous material is attached to the negative y-facing surface.
In an elastic panel that is not in contact with the porous core on its either side, there
are 3 BCs. Two of BCs are:
(i) v1 y = jωWt
(ii) v2 y = j ωWt
The third one is from the equation of motion:
(iii) P1 − P2 = ( Dk x4 − ω 2 ms )Wt
where P1,2 are the acoustic pressures applied on the two sides of the panel and v1, 2 y are
the normal acoustic particle velocities at the two faces of the panels.
vy
Wp
P
x
jωux
jωWt
σy
jωuy
τ xy
s
j ωU y
Figure 5.4. Detailed cross-sectional view of porous layer directly attached to a panel [48]
137
Figures 5.2-a,b,c show the waves propagating in the B-B, B-U and U-U panels and
applicable BCs. For example a total of 13 waves exist in the B-B panel, which are the
incident and reflected waves, four waves in the pla tes (one transverse wave and one shear
wave in each of the two plates), six waves in the porous layer as explained, and one
transmitted wave. Because 2 BCs, which actually are 8 BCs and 4 equations of motion,
are available (Figure 5.2-a), solving the 12 linear equations representing 12 BCs
simultaneously, 12 wave constants can be represented in terms of the input wave strength
C1 . Similarly, there are 14 waves in the B-U panel and 13 BCs are available (Figure 5.2b), and there are 15 waves in the U-U panel and 14 BCs are available (Figure 5.2-c).
The transmission coefficient τ (θ ) is the ratio of the amplitudes of the incident and
transmitted waves; for example, C13 /C1 for the B-B panel. τ (θ ) is obviously a function of
the incidence angle θ.
To consider the random incidences, τ (θ ) can be averaged
according to the Paris formula [41]:
θm
τ = 2 ∫ τ (θ )sin θ cosθ dθ
(5.22)
0
where θm is the maximum incident angle, which is chosen between 70 o and 85o (72o in
this study) as suggested by Mulholland et al. [30]. Integration of Equation (5.22) is
conducted numerically by Simpson’s rule. Finally, the average TL is obtained as:
TLavg = 10log
1
τ
(5.23)
Bolton et al. [48] calculated TLs for three types of double-panel and compared them
with the measured TLs for the structures composed by square aluminum sheets, 1.2 m on
a side, of 1.27 mm and 0.762 mm thickness, and a 27 mm thick, partially reticulated,
138
polyurethane foam slab. The aluminum sheets were assumed to have the following
properties: Young’s modulus, E=7×1010 Pa; Poisson’s ratio, µ=0.33; material density,
ρ=2700 kg/m3 . For the double-panel structures, the 1.27 mm panel was mounted on the
incident side, and the 0.762 mm one was used on the transmission side. In the U-U case,
the airgap on the incident side was assumed to be 2 mm while the airgap on the
transmitted side was assumed to be 6 mm. Finally, in the B-U configuration, the airgap
on the transmitted side was assumed to be 14 mm. The porous material parameters used
in the calculations are: bulk density of solid phase, ρ1 =30 kg/m3; in vacuo bulk Young’s
modulus, Em=8×105 Pa; in vacuo loss factor, υ=0.265; bulk Poisson’s ratio, ν=0.4; flow
resistivity, σ=25×103 MKS Rayls/m; geometrical structure factor, ε’=7.8; porosity,
h=0.9.
The solution process was reconstructed by the author and confirmed to provide the
same results as in the original work to develop the approximate method. The TL from
the full method shown in Figures 5.12-5.15 were calculated by the author’s reconstructed
formulation, while the measured curve was taken from the reference [48].
5-2.2. Development of Approximate Analysis Procedure
Quite a simple idea is used to develop the approximate solution method: only the most
significant component of the three waves in the porous material is kept in the analysis.
Because the most significant wave can be found only after the three waves are calculated
by the full method, the simplified method has to be applied in two steps. Suppose the
problem is to solve sound transmission through a double-walled cylindrical shell with a
porous liner, which will be solved in Chapter 6. At first, a flat double-panel of infinite
139
extent with the same cross sectional construction is solved by using the full method to
find the strongest wave number and corresponding equivalent density. The wave number
and density found as such are used to model the porous layer of the shell. Thus, the
material is essentially modeled as an equivalent fluid.
Figure 5.5 compares the amplitudes of the six displacement components (three waves
in the solid and three waves in the fluid), normalized by the amplitude of the airborne
wave in the fluid phase for the B-B double-panel. These ratios were obtained averaged
over the incident angle to take account of the random incidence. It makes more sense to
compare the waves in terms of their associated energy rather than the displacement. The
energies can be represented as follows.
The airborne wave:
E1 f =
k12y 2
1
h
⋅
(
Q
+
b
R
)
⋅
b
C
1
1
2
k12 1
(5.24)
The frame wave:
E3 f =
k 22 y 2
1
h
⋅
(
Q
+
b
R
)
⋅
b
C
2
2
2
k22 3
where, the subscript f
(5.25)
represents the fluid phase.
The energy associated with the
rotational wave (i.e., the shear wave) is not defined, because it does not contribute to the
fluid stress.
Similarly, the strain energy associated with the displacement in the solid phase can be
defined for each wave component as follows.
The airborne wave:
140
k12y 2
k12y
1
E1 s = (1 − h ) ⋅ 2 N 2 + A + b1Q ⋅ 2 C1
2
k1
k1
(5.26)
The frame wave:
E3 s =
1
(1 − h ) ⋅
2
k22 y 2
k22 y
2
N
+
A
+
b
Q
⋅ 2 C3
2
2
k
2
k2
(5.27)
The rotational wave:
2
k22 y 2
1
E5 s = (1 − h ) ⋅ 2 N 2 C5
2
k2
(5.28)
where, the subscript s represents the solid phase.
Figures 5.6-5.8 show the ratios of the energy carried by the frame wave and the rotational
wave to the airborne wave in the fluid and solid phases: i.e.,
E1 f E3 f E1 s E3 s E5 s
,
,
,
,
E1 f E1 f E1 f E1 f E1 f
for the B-B, B-U and U-U configurations.
From Figure 5.6, it is clear that the frame wave (i.e., k 2 ) is the most significant
component in the entire frequency range in the B-B configuration. For the B-U
configuration, the airborne wave (i.e., k 1 ) is found to be the strongest component in the
frequency range up to 3,000 Hz, while the frame wave (i.e., k 2 ) is the strongest
component in the frequency range above 4,000 Hz. In the U-U configuration, the airborne
wave (i.e., k 1 ) is seen as the dominant over the entire frequency range from Figure 5.8. In
all three cases, the shear wave (rotational wave) is the least contributing component. It is
noted that the strongest wave component may depend on the particular geometry and
construction of the double-panel; therefore, it will be necessary to conduct the analysis by
the full method, which has to be conducted for each new problem.
141
60
Displacement Ratio
50
40
30
20
10
0
0
1000
2000
3000
4000
5000
Frequency (Hz)
Figure 5.5. Frame and shear wave contributions to the fluid and solid displacements in
the y-direction for the B-B double-panel (normalized by the strength of the
airborne wave):
b1 ( k1 y / k1 2 ) C1
, airborne wave in fluid phase to airborne wave in fluid phase (
, frame wave in fluid phase to airborne wave in fluid phase (
∇, shear wave in fluid phase to airborne wave in fluid phase (
b2 ( k2 y / k22 )C3
g ( kx / k t2 )C5
b1 ( k1 y /k 12 )C 1
∗, airborne wave in solid phase to airborne wave in fluid phase (
, shear wave in solid phase to airborne wave in fluid phase (
142
);
b1 ( k1 y /k12 )C1
, frame wave in solid phase to airborne wave in fluid phase (
);
b1 ( k1 y /k12 )C1
);
(k1 y / k12 )C1
b1 ( k1 y /k 12 )C 1
(k 2 y / k 22 )C3
b1 ( k1 y /k12 )C1
(k x /k t2 )C 5
b1 ( k1 y /k 12 )C 1
)
);
);
250
Energy Ratio
200
150
100
50
0
0
1000
2000
3000
4000
5000
Frequency (Hz)
Figure 5.6. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the B-B double-panel (normalized by the energy of the
airborne wave):
E
, airborne wave in fluid phase to airborne wave in fluid phase ( 1 f );
E1 f
E
, frame wave in fluid phase to airborne wave in fluid phase ( 3 f );
E1 f
E
∗, airborne wave in solid phase to airborne wave in fluid phase ( 1 s );
E1 f
E
, frame wave in solid phase to airborne wave in fluid phase ( 3 s );
E1 f
E
, shear wave in solid phase to airborne wave in fluid phase ( 5 s )
E1 f
143
7
6
Energy Ratio
5
4
3
2
1
0
0
1000
2000
3000
Frequency (Hz)
4000
5000
Figure 5.7. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the B-U double-panel (normalized by the energy of the
airborne wave):
, airborne wave in fluid phase to airborne wave in fluid phase (
E1 f
);
E1 f
E3 f
);
E1 f
E
∗, airborne wave in solid phase to airborne wave in fluid phase ( 1 s );
E1 f
E
, frame wave in solid phase to airborne wave in fluid phase ( 3 s );
E1 f
E
, shear wave in solid phase to airborne wave in fluid phase ( 5 s )
E1 f
, frame wave in fluid phase to airborne wave in fluid phase (
144
1
Energy Ratio
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
Frequency (Hz)
Figure 5.8. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the U-U double-panel (normalized by the energy of the
airborne wave):
E
, airborne wave in fluid phase to airborne wave in fluid phase ( 1 f );
E1 f
E
, frame wave in fluid phase to airborne wave in fluid phase ( 3 f );
E1 f
E
∗, airborne wave in solid phase to airborne wave in fluid phase ( 1 s );
E1 f
E
, frame wave in solid phase to airborne wave in fluid phase ( 3 s );
E1 f
E
, shear wave in solid phase to airborne wave in fluid phase ( 5 s )
E1 f
145
Using only the strongest wave, the porous material is now modeled as an equivalent
fluid, whose density is determined from the wave retained. For example, in the B-B
panel of the given dimension, the wave number k 2 in Equation (5.12) and the bulk density
of the solid phase of the porous material (i.e., ρ*11 ) may be used. The wave number k 1 in
Equation (5.12) and the bulk density of the fluid phase (i.e., ρ*22 ) may be used if the
airborne wave is the strongest wave as in the U-U case. The complex speed of sound is
calculated from the wave number and the density as usual.
To explain how this simplified method can be used in a practice, we consider a
curved double-panel lined with a porous core with a B-U construction. The modeling
procedure will be:
1. Build a flat double-panel model of infinite extent with B-U core, keeping the all
other parameters (thickness of the shell, airgap, and the porous layer, elastic
properties of the shell, etc.) the same.
2. Solve the problem by the full method to obtain the ratios of the energy strengths
of the three waves. The porous material is modeled by using the wave number
and equivalent mass density of the strongest wave.
3. Build the double shell model treating the porous material as an equivalent fluid,
which will become a double shell enclosing two layers of fluid, one representing
the porous material and the other representing the airgap between the porous layer
and the shell.
After the simplification as stated above is done, the problem can be solved in a
straightfo rward way by using existing analysis tools: for example, a commercial FEA
software which has a capability to do regular acoustic analysis.
146
The proposed approximate analysis method compromises accuracy for simplicity.
The simplified method is applied to the three cases of sound transmission problems of the
infinite double-panel that were solved by the full method in the work by Bolton et al.
[48], and the results are compared to those from the full method and measurements in the
next sectio n. The comparison helps to elucidate the practicality (or practical limitation)
of the method and the errors introduced by the simplification.
5-3. Comparison of the Solutions from the Simple and Full Analyses
5-3.1. Formulation of the Problem
5-3.1.1. Double-Panel with Bonded-Bonded (B-B) Layer
z
P
pi: Incident
Wave
External
Air
Fluid
Phase
x
θ
c2
T
p
2
pt : Transmitted
Wave
ci: Sound
Speed
External
Air
ρ1a: Density
pR2
pr: Reflected
Wave
hi
hfiber
ct , ρ3a
ht
Figure 5.9. Simplified model of the B-B double-panel
147
Figure 5.9 illustrates the simplified model of a double-panel lined with a porous layer in
the B-B configuration. The mass density and sound speed of the acoustic media in the
incident and transmitted sides are {ρ1a, ci} and {ρ3a, ct }. An oblique wave pi is incident
with an angle θ (measured from the normal axes to the beam) from one side, which
induces the traveling transverse wave in the plate w10 and w20 , the reflected acoustic wave
p r , transmitted wave p t in the external acoustic spaces, and the reflected acoustic wave
T
p 2R , and transmitted wave p 2 in the porous layer.
In the air incident side, the wave equation [38] becomes:
ci ∇ 2 ( pi + p r ) +
∂ 2 ( pi + pr )
=0
∂t 2
(5.29)
where ∇2 the Laplacian operator in the rectangular coordinate and pr is the reflected
wave . The wave equation in the porous material layer becomes:
c 2 ∇ 2 (p2T + p 2R ) +
∂ 2 ( p T2 + p 2R )
=0
∂t 2
(5.30)
Note that c2 is the speed of sound in the equivalent fluid obtained by taking the strongest
wave in the porous material obtained by the full theory as explained in Section 5-2.
In the air in the transmitted side, the wave equation becomes:
c t∇ 2 p t +
∂ 2 pt
=0
∂t 2
(5.31)
Equation of motion of the plate [39] of the incident side is given as:
− Di
∂w10
&&10
+ ( pi + pr ) − ( pT2 + p2R ) = ρ i hi w
∂x 4
(5.32)
Equation of motion the plate on the transmitted side is given as:
148
− Dt
∂w02
&&20
+ ( pT2 + p 2R ) − pt = ρt ht w
4
∂x
(5.33)
where w10 , w20 are the transverse displacements of the plates, Di and Dt are the flexural
rigidity of the plates, ρi and ρt are the material density of the plates, hi and ht are the
thickness of the plates, and the dot notation indicates the derivative with respect to time.
The BC between the air and the plate at the incident side is given as,
∂ ( pi + pr )
∂ 2 w10
= −ρ1 a 2
∂z
∂t
at z = 0
(5.34)
The BCs between the plate and the fluid inside, which represents the porous material,
∂ ( pT2 + p2R )
∂z
= − ρ11*
∂ 2 w10
∂t 2
at z = 0
(5.35)
*
where, ρ11
is the bulk density of the solid phase of the porous material, which is selected
because the dominant wave is the frame wave in this case as discussed. The BC between
the internal fluid and the transmitted side plate is given as,
∂ ( pT2 + p2R )
∂z
= − ρ11*
∂ 2 w02
∂t 2
at z = hfiber
(5.36)
Between the transmitted side plate and the air,
∂pt
∂ 2 w02
= −ρ 3 a 2
∂z
∂t
at z = hfiber
(5.37)
The harmonic incident wave [38, 40] is expressed as:
pi ( x, z , t ) = p0 j (ω t − k1 x x − k1 z z )
(5.38)
where p0 is the effective amplitude of the incident wave; ω angular or rotational
frequency; and
149
k1 =
ω
,
c1
k1 x = k1 sin( θ ) , k1 z = k1 cos(θ )
(5.39)
In the porous layer, the frame wave number obtained in Equation (5.12) is taken as
the equivalent wave number k2 because the frame wave is the dominant carrier.
Therefore,
k 2 x = k1 x ,
k 2 z = k22 − k 22 x
(5.40)
In the transmitted side,
k3 =
ω
,
c3
k 3 x = k1 x ,
k 3 z = k 32 − k 32x
(5.41)
Thus, p r , p2T , p2R , p t , w10 and w02 can be expressed as follows.
pr ( x, z, t ) = picR e j (ω t −k1 x x +k1 zz )
(5.42)
p2R ( x, z , t ) = p2Rc e j ( ωt − k1 x x + k2 zz )
(5.43)
p2T ( x , z , t ) = p2Tce j ( ωt − k1 x x − k2 z z )
(5.44)
pt ( x , z , t ) = ptcT e j ( ωt −k1 x x −k3 zz )
(5.45)
w10 ( x, t) = w10c e j ( ω t − k1 x x)
(5.46)
w20 ( x, t) = w20c e j ( ω t − k1 x x )
(5.47)
By substituting Equations (5.38), (5.42)-(5.47) into Equations (5.32)-(5.33) and
(5.34)-(5.37), which are two equatio ns of motion of the beams and four BCs, six
equations are obtained for 6 unknown complex amplitudes picR , p2Tc , p2Rc , p Ttc , w1c0 and w02c :
w10c ρ i hiω 2 − Di k14x + picR − p2Tc − p2Rc = − p0
(5.48)
150
− jk2 z h fiber
w20c ρ t ht ω 2 − Dt k14x + p2Rc e
+ p2Tc e
jk2 z h fiber
− pTtce
− jk3 z h fiber
=0
(5.49)
picR k1 z j − ρ1 a ω 2 w10c = p 0 k1 z j
(5.50)
jp T2 c k2 z + jp2Rc k2 z − ρ*11ω 2 w10c = 0
(5.51)
jp T2 c k2 z e − jk2 z h fiber + jp2Rc k2 z e jk2 z h fiber − ρ*11ω 2 w20 c = 0
(5.52)
− jptcT k 3 z e
− jk3 z h fiber
− ρ 3 cω 2 w20c = 0
(5.53)
Putting these six equations into a matrix format:
A B
0 F
J 0
0 M
0 P
0 0
C
G
0
N
Q
0
0 D 0 picR E
H 0 I p2Tc 0
0 K 0 p2Rc L
=
0 O 0 ptcT 0
0 0 R w10c 0
S O T w20 c 0
(5.54)
where,
A = 1 , B = −1 , C = −1 , D = ρ i hiω 2 − Di k14x
E = − p0 , F = e − jk2 z h fiber , G = e jk2 z h fiber , H = − e− jk3 zh fiber
I = ρ t ht ω 2 − Dt k14x , J = jk1 z , K = − ρ1a ω 2 , L = jk1 z p0 , M = − jk2 z , N = jk 2 z , O = − ρ*11ω 2
P = − jk 2 z e − jk2 zh fiber , Q = jk 2 z e jk2 z h fiber , R = − ρ*11ω 2 , S = jk3 z e − jk3 z h fiber , T = ρ 3 aω 2
The six unknown coefficients picR , p2Tc , p2Rc , p Ttc , w10c and w20c are obtained in terms of
po by solving Equation (5.54). These can be substituted back into Equations (5.42) to
(5.47) to find the displacements of the shell and the acoustic pressures.
151
5-3.1.2. Bonded-Unbonded (B-U) Configuration
Figure 5.10 shows a double-panel structure lined with a porous layer in the B-U
configuration. The mass density and sound speed of the acoustic media in the incident,
transmitted sides and airgap space are {ρ1a, ci}, {ρ4a, ct} and {ρ3a, c3 }. The mass density
of the fluid phase of the porous layer is defined as ρ *22 if low frequency sound is the
major concern and ρ11* if high frequency sound is the major concern, where, ρ11* and ρ *22
are defined in Equation (5.6).
z
pi: Incident
Wave
x
θ
ci: Sound
Speed
Airgap
Fluid
Phase
c2
pT 2
c3 , ρ3a
pt : Transmitted
Wave
θi2
θt3
θr2
T
p
3
ρ1a: Density
External
Air
pR3
ct , ρ4a
pR2
pr: Reflected
Wave
hi
hfiber
hair
ht
Figure 5.10. Simplified model of the B-U double-panel
152
The wave equations in the incident side, the fluid phase of the porous layer, in the
airgap, and the transmitted side remain the same form as Equations (5.29), (5.30) and
(5.31) only with different variable names. The equations of motion of the plates also
remain the same form as Equations (5.32) and (5.33).
BCs also remain the same form as Equations (5.34)-(5.37) at all interfaces except at
the interface between the equivalent fluid and the internal air, which are:
p2T + p2R = p3T + p3R
uT2 + u2R = u3T + u3R
at z = hfiber
(5.55)
at z = hfiber
(5.56)
where u indicates the particle velocity in the acoustic space.
The harmonic, plane incident wave pi is expressed in Equation (5.38). For the B-U
configuration, the airborne wave or frame wave is the major energy carrier depending on
the frequency of interest. In this particular case, the airborne wave has to be taken for the
frequenc ies lower than 2.7 kHz (see Figure 5.7), and the frame wave if higher than 2.7
kHz. After the wave number k2 and the density of the porous material as explained in
Section 5-2 are chosen, the other wave numbers are described as:
k 2 x = k1 x ,
k 2 z = k 22 − k 22 x
(5.57)
k3 =
ω
,
c3
k 3 x = k1 x ,
k 3 z = k 32 − k 32x
(5.58)
k4 =
ω
,
c3
k 4 x = k1 x ,
k 4 z = k 42 − k 42 x
(5.59)
The traveling waves are expressed in the same way as in the B-B case.
procedure to the B-B case leads to 8 equations as follows:
153
A similar
w10c ρ i hiω 2 − Di k14x + picR − p2Tc − p2Rc = − p0
− jk3 z ( h fiber + hair )
w20c ρ t ht ω 2 − Dt k14x + p3Rc e
(5.60)
+ p3Tc e
jk3 z ( h fiber + hair )
− ptcTe
− jk4 z ( h fiber + hair )
=0
picR k1 z j − ρ1 a ω 2 w10c = p 0 k1 z j
(5.61)
(5.62)
jp T2 k 2 z + jp2R k2 z − ρ*22ω 2 w10c = 0
for low frequency sound
(5.63.a)
jp T k + jp R k −ρ* ω 2 w0 =0
for high frequency sound
(5.63.b)
2
2z
2
2z
11
1c
p2Tc e − jk2 zh fiber + p2Rc e jk2 z h fiber − p3Tc e− jk3 zh fiber − p3Rc e jk3 z h fiber = 0
p2Tc
p2Rc
− jk2 z h fiber
jk h
cos θ i 2 e
− * cos θ r 2 e 2 z fiber
*
ρ 22 c2
ρ 22 c2
pT
pR
− 3c cos θ t 3 e − jk3 z h fiber + 3c cos θ r 3 e jk3 z h fiber = 0
ρ 3 a c3
ρ 3 a c3
pT2 c
p2Rc
− jk2 z h fiber
jk h
cos
θ
e
−
cos θ r 2 e 2 z fiber
i2
*
*
ρ11c2
ρ11c2
pT
pR
− 3c cos θ t 3 e − jk3 z h fiber + 3c cos θ r 3 e jk3 z h fiber = 0
ρ 3 a c3
ρ 3 a c3
(5.64)
for low frequency sound
(5.65.a)
for high frequency sound
(5.65.b)
− jp3T k3 z e− jk3 z ( h fiber + hair ) + jp3R k3 z e jk3 z ( h fiber + hair ) − ρ 3a ω 2 w20 c = 0
− jptcT k 4 z e
− jk4 z (h fiber+ hair )
− ρ 4 cω 2 w02 c = 0
(5.66)
(5.67)
where the incident, reflected, and transmitted waves make the respective angles θi2 , θr2,
and θt3 with the z axis as shown in Figure 5.10. The relationships [38] amongst θi2 , θr2,
and θt3 at the boundary separating two fluids (i.e., z = hfiber) are found as:
cosθ i 2 =
k2 z
sinθ i 2 sinθ t 3
, sinθ i 2 = sinθ r 2 ,
=
, sinθ t 3 = sinθ r 3
k2
c2
c3
These eight equations can be put into a matrix form as follows.
154
(5.68)
A B C
0 0 0
J 0 0
0 M N
0 P Q
0 T U
0 0 0
0 0 0
0 0
F G
0 0
0 0
R S
V W
X Y
0 0
0
H
0
0
0
0
0
A1
D
0
K
O
0
0
0
0
0 picR E
I p2Tc 0
0 p2Rc L
0 p3Tc 0
=
0 p3Rc 0
0 ptcT 0
Z w10c 0
B1 w20 c 0
(5.69)
where, for low frequency sound,
A = 1 , B = −1 , C = −1 , D = ρ i hiω 2 − Di k14x
E = − p0 , F = e− jk3 z ( h fiber + hair ) , G = e jk3 z ( h fiber + hair ) , H = − e− jk4 z ( h fiber + hair )
I = ρt htω 2 − Dt k14x , J = jk1 z , K = − ρ1 a ω 2 , L = jk1 z p0 , M = − jk 2 z , N = jk 2 z , O = − ρ*22ω 2
P = e − jk2 z h fiber , Q = e jk2 zh fiber , R = −e − jk3 z h fiber , S = −e jk3 z h fiber
T=
cos θ i 2 − jk2 zh fiber
cos θ
cos θ t 3 − jk3 z h fiber
cos θ r 3 jk3 z h fiber
e
,U = − * r 2 e jk2 zh fiber ,V = −
e
,W =
e
*
ρ 22 c2
ρ 22 c2
ρ 3 c3
ρ 3c3
X = − jk 3 z e− jk3 z ( h fiber + hair ) , Y = jk3 z e jk3 z ( h fiber + hair ) , Z = − ρ3 a ω 2 , A1 = jk 4 z e − jk4 z ( h fiber + hair ) , B1 = ρ 4 aω 2
For high frequency sound, the terms of O, T and U should be replaced by the following
terms:
*
O = −ρ11
ω2 , T =
cos θ i 2 − jk2 zh fiber
cos θ
e
, U = − * r 2 e jk2 zh fiber
*
ρ11c2
ρ11c2
The eight unkno wn coefficients picR , p2Tc , p2Rc , p3Tc , p3Rc , p Ttc , w10c and w20c are obtained in
terms of po by solving Equation (5.69).
155
5-3.1.3. Unbonded-Unbonded (U-U) Configuration
Figure 5.11 shows a double-panel structure lined with a porous layer in the U-U
configuration. hair1 and hair2 are the depth of the airgaps as shown. The mass density and
sound speed of the acoustic media in the incident and transmitted sides are {ρ1a, ci} and
{ρ5a, ct }. The mass density and sound speed of the airgap spaces in the incident side and
transmitted side are {ρ2a, c2 } and {ρ4a, c4 }. Because the major energy carrier of U-U
panel is the airborne wave, the wave number is taken from Equation (5.12), and the
effective mass density of the fluid phase of the porous layer ρ *22 , which is given in
Equation (5.6).
All equations remain essentially the same as the B-U case except that
there is one more equation because there is one more airgap.
z
Airgap
pi: Incident
Wave
Fluid
Phase
x
External
Air
θ
θi2
c3
pT 3
Airgap
c4 , ρ4a
θi3
θt4
θr3
pT 2
pr: Reflected
Wave
c2
pR3
pR4
ρ2a
hi
hair1
External
Air
pT 4
pR2
ci: Sound
Speed
ρ1a: Density
pt : Transmitted
Wave
ct , ρ5a
hfiber
hair2
ht
Figure 5.11. Simplified model of the U-U double-panel
156
By substituting the assumed solutions for pressure and displacement into two equations
of motion of the beams equations and eight BCs, ten equations are obtained for 10
unknown complex amplitudes picR , p2Tc , p2Rc , p3Tc , p3Rc , p4Tc , p4Rc , p Ttc , w1c0 and w02c . The ten
equations can be put into a matrix equation as follows.
A B
0 0
J 0
0 M
0 P
0 T
0 0
0 0
0 0
0 0
C 0 0 0
0 0 0 F
0 0 0 0
N 0 0 0
Q R S 0
U V W 0
0 X Y Z
0 B1 C1 D1
0 0 0 F1
0 0 0 0
0
G
0
0
0
0
A1
E1
G1
0
0 D 0 p1Rc E
H 0 I p2Tc 0
0 K 0 p2Rc L
0 O 0 p3Tc 0
0 0 0 p3Rc 0
=
0 0 0 p4Tc 0
0 0 0 p4Rc 0
0 0 0 ptcT 0
0 0 H1 w10c 0
I 1 0 J 1 w20 c 0
(5.70)
where,
A = 1 , B = −1 , C = −1 , D = ρ i hiω 2 − Di k14x
E = − p0 , F = e− jk4 z ( hair1 + h fiber + hair 2 ) , G = e jk4 z ( hair 1 +h fiber + hair 2 ) , H = − e− jk5 z ( hair1 + h fiber + hair 2 )
I = ρt htω 2 − Dt k14x , J = jk1 z , K = − ρ1a ω 2 , L = jk1 z p0 , M = − jk2 z , N = jk 2 z , O = −ρ 2 a ω 2
P = e − jk2 z hair1 , Q = e jk2 zh air1 , R = − e− jk3 zh air1 , S = − e jk3 zhair 1
T=
cos θ i 2 − jk2 zh air1r
cos θ r 2 jk2 z hair1
cos θ
cosθ
e
,U = −
e
, V = − * t 3 e− jk3 z hair1 , W = * r 3 e jk3 zh air1
ρ 2 c2
ρ 2 c2
ρ 22 c3
ρ 22c3
− jk3 z ( h air1 + h fiber )
X =e
,Y = e
jk3 z ( hair1 +h fiber )
, Z = −e
− jk4 z ( hair 1 + h fiber )
, A1 = −e
jk4 z ( h air1 + h fiber )
B1 =
cos θ t 3 − jk3 z ( h air1 + h fiber )
cos θ
cos θ t 4 − jk4 z ( h air1 + h fiber )
e
, C1 = − * r 3 e jk3 z ( h air1 + h fiber ) , D1 = −
e
*
ρ 22 c3
ρ 22c3
ρ4 a c4
E1 =
cos θ r 4 jk4 z ( hair 1 +h fiber )
e
, F1 = − jk 4 z e − jk4 z ( hair 1 + h fiber + hair 2 ) , G1 = jk4 z e jk4 z ( h air1 + h fiber +h air 2 )
ρ 4 a c4
157
H1 = −ρ 4 a ω 2 , I1 = jk 5 z e − jk5 z ( h air1 + h fiber + h air2 ) , J1 = ρ5 a ω 2
The relationships amongst θi2 , θr2, and θt3 at the boundary separating two fluids (i.e., z =
hair1 ) are found as [38]:
cosθ i 2 =
k2 z
sinθ i 2 sinθ t 3
, sinθ i 2 = sinθ r 2 ,
=
, sinθ t 3 = sinθ r 3
k2
c2
c3
(5.71)
Similarly, the relationships amongst θt3 , θr3, and θt4 at the boundary separating two fluids
(i.e., z = hair1 +hfiber) are found as:
cosθ t 3 =
k3 z
sinθ t 3 sinθ t 4
, sinθ t 3 = sinθ r 3 ,
=
, sinθ t 4 = sinθ r 4
k3
c3
c4
(5.72)
The ten unknown coefficients picR , p2Tc , p2Rc , p3Tc , p3Rc , p4Tc , p4Rc p Ttc , w10c and w20c are obtained
in terms of po by solving Equation (5.70).
5-3.2. Comparison of the Approximate Solution to the Solution from the
Full Theory
Figure 5.12 shows the comparison of the TL calculated by using the simplified theory
and those from the full method and the measurement (taken from reference [48]) for the
B-B double-panel. Averaging according to the diffusive field theory was taken (see
Equation (5.22)) to calculate the TLs in a random incidence condition, or the measured
condition. As it is shown, the TL curve obtained from the simplified model lies generally
within 5 dB from the other TL curves, and the characteristics of the curves match with
one another very well.
The proposed method can be compared to other approximate methods in the past for
this type of problem. For example, only the structural effect of the porous layer was
considered in some studies, adding half of the mass of the porous layer to each panel and
158
considering continuous spring between the plates [42] to represent the stiffness of the
layer. Figure 5.13 compares the TL obtained from such a model, the TL obtained from
the simplified model and the TL obtained from the full method for the B-B double-panel.
From the comparison in Figure 5.13, it is seen that the simplified model proposed in this
work provides a much better approximation as expected because the model considers the
characteristics of the porous material even though it is in an approximate way.
The TL obtained from the simplified theory is again compared to those from the full
theory and measurement for the B-U configuration in Figure 5.14. In the figure, the TL
obtained by using the airborne wave as the major energy carrier and the TL obtained by
using the frame wave as the major energy carrier are compared with the TL curves
obtained from the full method and experiment. As discussed in Section 5-2.2 and Figure
5.7, the TL calculated by using the airborne wave has to be used in the frequency range
up to 3,000 Hz. The TL based on the airborne wave simplification provides good
agreement in that range. If the interest is in a very high frequency range, above 4,000 Hz,
the TL curve obtained by the frame wave approximation may be used, which shows good
agreement with the TLs from measurement and full theory.
Figure 5.15 shows the TLs obtained for the U-U case, in which the airborne wave is
used in the simplified model. It is seen that the simplified model provides a very good
agreement with the full theory and measured results.
This is expected because the
airborne wave is much stronger than the other two waves in the entire frequency range in
the U-U panel.
159
55
50
45
TL (dB)
40
35
30
25
20
15
10
2
3
10
4
10
Frequency (Hz)
10
Figure 5.12. Comparison of TLs of the B-B double-panel
∇, measured result; - - - , prediction of the simplified
model;
, prediction of the full model
70
60
TL (dB)
50
40
30
20
10
10
2
3
10
Frequency (Hz)
4
10
Figure 5.13. Comparison of calculated TLs of the B-B double-panel
, the full model; , the simplified model; ∇, an approximate
model without considering the multi-wave effect
160
80
70
60
TL (dB)
50
40
30
20
10
0
2
10
3
10
Frequency (Hz)
4
10
Figure 5.14. Comparison of TLs of the B-U double-panel
∇, measured result; - - - , prediction of the simplified model
using the airborne wave ;
, prediction of the simplified
model using the frame wave;
, prediction of the full
model
80
70
60
TL (dB)
50
40
30
20
10
0
10
2
3
10
Frequency (Hz)
4
10
Figure 5.15. Comparison of TLs of the U-U double-panel
∇, measured result; - - - , prediction of the simplified model;
, prediction of the full model
161
5-4. Conclusions
A simplified analysis method was proposed to solve sound transmissions of arbitrary
geometry through structures with porous liners.
The analysis method developed by
Boton et al. [48] based on Biot’s theory [47] is utilized as a pre-processor of the proposed
method. Biot’s theory [47] views the porous material as a homogeneous aggregate of the
elastic frame and fluid pores, therefore describes the material by using three waves: two
dilatational and one rotational waves. The approximate method uses a very simple
concept of using only the strongest among the three waves to model the porous material,
which is essentially modeling the porous layer as an equivalent fluid layer.
The
procedure to apply the method to real structures with arbitrary geometry and boundary
conditions is composed of two steps. In the first step an equivalent flat double-panel of
infinite extent is solved by the full method, keeping the same cross sectional geometry, to
identify the strongest wave by comparing the energies associated with the waves. In the
second step the problem is solved by using the actual geometry but modeling the porous
layer as an equivalent fluid layer. The accuracy (or inaccuracy) of approximate method is
checked by comparing the results with those from the full model and measurements
reported in reference [48].
The comparison shows that the simplified model provides
reasonable agreement with the full model.
Because the proposed method reduces the porous material to an equivalent fluid, the
problem to solve sound transmissions through the structure with a porous liner becomes a
typical acoustic problem composed of multiple fluid layers and structures. Therefore,
most of the existing analysis tools, i.e., FEM software with standard acoustic analysis
capability, can be used without any modifications.
162
The approximation obviously
sacrifices accuracy and theoretical exactness. However, the full method also involves
quite a few idealizations such as homogeneity and isotropy of the porous material. Also
the property of the porous material tends to have wider variations compared to typical
elastic material, depending on its manufacturing and installation procedures. On the
other hand, the potential gain of using the proposed method is significant making
typically unsolvable problems, such as sound transmission through structures lined with
porous layer with realistic geometry, solvable.
163
Chapter 6 – Sound Transmission through Double-Walled Cylinders
Lined with Elastic Porous Core
6-1. Introduction
A double-walled cylindrical shell whose walls sandwich a layer of porous material is
found in practical applications that require a high level of noise insulation such as an
aircraft fuselage. Because of the need to cons ider vibro-acoustic interactions between the
shells, airgaps and porous layer in curvilinear coordinates, applying the full multi-wave
analysis method developed by Bolton et al. [48] based on Biot’s theory [47] (the full
method hereafter) to solve the vibro-acoustic responses of such a system will become
extremely difficult: therefore, the approximate analysis method developed in Chapter 5 is
applied.
There are many analytical studies on the vibration or acoustic responses of cylindrical
structures of various cross sections. In those studies [25-28, 68, 69], it is typical to
assume that the cylinder is infinitely long, the input to the system is a plane wave incident
from outside, and the cavity inside is anechoic. With these assumptions, analytical
solutions can be obtained without ignoring any vibro-acoustic coupling effects. The
solutions from such analyses are often obtained in transmission losses (TLs) and used for
the purpose of comparative designs or qualitative parameter studies.
Koval [25],
Vaicaitis et al. [68, 69], Blaise et al. [26], and Tang et al. [27, 28] investigated sound
transmissions through isotropic and orthotropic shells. Vaicaitis et al. [68, 69] used
Galerkin’s method to calculate the response of a laminated double-walled, fiberreinforced cylindrical shell to random excitations describing the shell motion by the
164
Donnell-Mushtari–Vlasove equations.
In their work, however, structure-acoustic
interactions were not fully considered. Blaise et al. [26] extended Koval’s work [25] to
obtain the transmission coefficient of an orthotropic shell subjected to an oblique plane
sound wave with two independent incident angles.
Tang et al. [27, 28] considered
laminated composite shells of an infinite cylindrical sandwiching honeycomb core when
it is subjected to an oblique plane sound wave. The TL was expressed explicitly in terms
of the modal impedances of the acoustic fluids and the shell. However, the in-plane
displacements of the shell were ignored completely, which is not valid in general. C. L.
Dym et al. [34], Grosveld et al. [70], and C. -Y. Wang et al. [42] studied sound
transmission through sandwich structures, which have lightweight and flexible cores
between relatively stiff skins. Grosveld et al. [70] investigated the problem as a part of
the active control of vibrations and noise transmission / radiation across a double-walled
structure. C. -Y. Wang et al. [42] presented an analytical study on the active control of
vibrations and noise transmissions in simply supported double-walled cylindrical shells
subjected to random inputs. In the work, the shells were modeled by Love’s equation, and
Galerkin’s method was used to obtain the solution.
Lee and Kim [45, 71, 72] studied the sound transmission through single-walled and
double-walled shells using Love’s equation to describe the shell motions and fully
considering structural-acoustic coupling effects. The procedure is generally followed in
this study when the approximate solution method to model is applied, which was
developed in Chapter 5.
165
6-2. Description of the Problem
A double-walled cylindrical shell of infinite length subjected to a plane wave with an
incidence angle γ is shown in Figure 6.1. The radii and thickness of the walls are Ri, Re
and hi, he, in which the subscripts i, e represent the inner and outer shells. A concentric
layer of porous material is installed between the shells in three different ways, which
forms a bonded-bonded (B-B) shell, in which the layer is bonded to both shells, a
bonded-unbonded (B-U) shell, in which the layer is bonded to the external shell but
separated by an airgap from the internal shell, and a unbonded- unbonded (U-U) shell, in
which airgaps exist in both sides of the layer.
pR
Reflected wave
pI
γ
pT
Incident plane wave
Ri
Re
Transmitted wave
y
θ
x
Figure 6.1. Schematic diagram of the double shell with porous layer
166
z
The TLs through the multi- layered walls of the shell is obtained analytically by using
the approximate method developed in Chapter 5.
As the first step, the strongest
component of the three waves in the porous layer is identified by applying the full
method to a double flat panel of infinite extent with the same cross sectional structure
(i.e., the B-B flat panel for the B-B shell). Using the most dominant wave component
enables the porous layer to be represented as an equivalent fluid; therefore, the system is
modeled as a shell with double walls sandwiching multiple layers of fluids.
6-3. Derivation of the System Equation
6-3.1. Double Shell with Bonded-Bonded Porous Material Layer
Figure 6.2 shows the cross-sectional view of the double shell with a porous layer installed
in the B-B configuration of thickness hc. The fluid media in the external space, the porous
core, and the internal space are defined by the density and the speed of sound: {ρ1 , c1 },
{ρ2 , c2 } and {ρ3 , c3 }, respectively.
In the external space, the wave equation [38] is represented as:
(
)
c1∇ 2 p I + p1R +
∂ 2 ( p I + p1R )
=0
∂t 2
(6.1)
where ∇2 is the Laplacian operator in the cylindrical coordinate, p I is the incident wave
and p R1 is the reflected wave. The wave equation in the equivalent fluid representing the
porous core is:
c2 ∇ 2 ( p2T + p2R ) +
∂ 2 ( p2T + p2R )
=0
∂t 2
(6.2)
167
where, p2T and p2R are the pressure of the transmitted and reflected waves, and the speed
of sound c2 has to be calculated from the wave number of the strongest wave component
as explained. In the internal cavity, the wave equation becomes:
∂ 2 p3T
c3 ∇ p + 2 = 0
∂t
2
T
3
(6.3)
where, p3T is the amplitude of the transmitted wave and c3 is the speed of sound in the
cavity inside.
pI
p1R
ρ1 ,c1
ρ2 ,c2
p2T
p2R
Outer shell:
{u,v,w}2
p3T
ρ3 ,c3
hi
hc
Inner shell:
{u,v,w}1
he
Porous core
Figure 6.2. Cross-sectional view of the shell with a B-B porous layer
168
The shell motions are described by Love’s equations [39], fully considering the
displacements in all three directions. Equations of motion in the axial, radial and
circumferential directions of the inner shell are:
L1 {u10 , v10 , w10 } = ρ i hi u&&10
{
(6.4)
}
L2 u10 , v10 , w10 = ρ i hiv&&10
(6.5)
&&10
L3 {u10 , v10 , w10 }+ ( p2T + p 2R ) − p T3 = ρ i hi w
(6.6)
where L1 , L2 and L3 are differential operators which can be found in reference [39], ρi,
hi are in vacuo bulk density and the thickness of the inner shell, {u10 , v 10, w10} represent the
displacements of the inner shell of a point on the neutral surface in the axial,
circumferential, radial directions, and the dot notation indicates the derivative with
respect to time.
Equations of motio n of the outer-shell in the axial, radial and circumferential directions
are:
L1 {u20 , v 20 , w20 } = ρ e he u&&20
{
(6.7)
}
L2 u 02 , v20 , w02 = ρ e he v&&20
(6.8)
&& 20
L3 {u 02 , v20 , w02 }+ ( p I + p1R ) − (p2T + p2R ) = ρ e he w
(6.9)
where, ρe, he
are in vacuo bulk density and the thickness of the outer shell and
{u , v , w } represent the displacements of the outer shell of a point on the neutral surface
0
2
0
2
0
2
in the axial, circumferential, radial directions.
The boundary conditions that have to be satisfied at the two interfaces between the
shells and fluid are:
169
∂ ( p I + p1R )
∂r
∂ ( pT2 + p2R )
∂r
∂ ( pT2 + p2R )
∂r
∂ 2 w02
= − ρ1 2 at r = Re
∂t
(6.10)
= − ρ2
∂ 2 w02
at r = Re
∂t 2
= − ρ2
∂ 2 w10
at r = Ri
∂t 2
(6.12)
at r = Ri
(6.13)
∂p3T
∂ 2 w0
= −ρ 3 21
∂r
∂t
(6.11)
where, in Equations (6.11) and (6.12) the density ρ2 has to be obtained from the effective
bulk density of the medium that corresponds to the strongest wave component as
explained (see Section 5-2.1 of Chapter 5 for this procedure).
Equations (6.1) to (6.9) are solved essentially in the same manner as in the doublewalled shell problem in Chapter 4. The harmonic, plane incident wave [40] pI can be
expressed in cylindrical coordinates as:
∞
p (r , z ,θ , t ) = p0 ∑ ε n (− j ) n J n (k1r r ) cos[nθ ]e j ( ωt −k
I
1 zz)
(6.14)
n =0
where po is the amplitude of the incident wave, n=0,1,2,3,.. indicates the circumferential
mode number, ε n =1 for n = 0 and 2 for n=1,2,,3.., j = −1 , Jn is the Bessel function of
the first kind of order n. Also, wave numbers in Equation (6.14) are defined as:
k1 =
ω
,
c1
k1 z = k1 sin( γ ) , k1 r = k1 cos(γ )
(6.15)
The wave number k2 in the porous core is from the procedure explained in Section 5-2 of
Chapter 5. Wave numbers in the internal cavity are defined as:
170
k3 =
ω
,
c3
k 3 z = k1 z ,
k 3 r = k32 − k32z
(6.16)
Considering the circular cylindrical geometry, the pressures p1R , p 2T , p 2R and p 3T are
expanded as:
∞
p1R (r , z ,θ , t ) = ∑ p1Rn H n2 (k1r r ) cos[nθ ]e j (ωt − k
1z
z)
(6.17)
n =0
∞
p ( r, z,θ , t ) = ∑ p2Tn H n1 (k 2 r r ) cos[nθ ]e j (ω t− k
T
2
1z
z)
(6.18)
n= 0
∞
p2R (r , z ,θ , t ) = ∑ p2Rn H n2 (k 2 r r ) cos[nθ ]e j ( ωt −k
1 zz)
(6.19)
n =0
∞
p3T ( r, z, θ , t ) = ∑ p T3n H n1 (k 3 r r ) cos[nθ ]e j ( ωt −k1z z )
(6.20)
n =0
where H 1n and H n2 are the Hankel functions of the first and second kind of order n.
Notice that the above expressions satisfy the wave equations (Equations (6.1) to (6.3)) as
well as the boundary conditions in the circumferential directions (periodicity)
automatically.
Considering the compatibility of the acoustic pressure and the displacement of the
shell in the transverse direction, shell displacements can be expressed as:
∞
w ( z , θ , t ) = ∑ w10n cos[nθ ]e j (ωt − k1z z )
0
1
(6.21)
n =0
∞
u10 ( z ,θ , t ) = ∑ u10n cos[nθ ]e j ( ωt − k
1z
z)
(6.22)
n= 0
∞
v10 ( z ,θ , t ) = ∑ v10n sin [nθ ]e j (ωt − k
1z
z)
(6.23)
n =0
171
∞
w20 ( z ,θ , t ) = ∑ w20n cos[nθ ]e j ( ωt −k
1 zz)
(6.24)
n =0
∞
u 20 ( z ,θ , t ) = ∑ u 20n cos[nθ ]e j (ωt − k
1z
z)
(6.25)
n =0
∞
v 20 ( z ,θ , t ) = ∑ v 02 n sin [nθ ]e j ( ωt −k
1z z )
(6.26)
n =0
Reference [39] explains the choice of the displacement functions for the in-plane
displacements u and v in relation to that for w in the cylindrical shell.
Substitution of the expressions in Equations (6.14) and (6.17)-(6.26) into six shell
equations (Equations (6.4) to (6.9)) and four boundary conditions (Equations (6.10) to
(6.13)) yields 10 equations, which can be decoupled for each mode if the orthogonality
between the trigonometric functions are utilized.
Six of the ten equations are from the
shell vibration equations:
K (1 − µ ) K (1 + µi )
Kµ
u10n ρ i hi ω 2 − K i k12z − i 2 i n 2 − i
nk1 z v0In j − i i k1 z w0In j = 0
2
R
2
R
Ri
i
i
K i (1 − µi ) 2 K i 2 Di (1 − µ i ) 2
k1z − 2 n −
k 1z
−
2
Ri
2 Ri2
Ki
Di
0 K i (1 − µ i )
2
0
u1n
k1z nj +
µ i k 1z nj − 2 µ i k1 z n + v In
2
R
R
R
D
i
2
2
i
i
i
+ R 4 n + ρi hiω
i
(6.27)
(6.28)
K
D (1 − µ )
D
+ w 0In − 2i n − i 2 i k 12z n − 4i n 3 = 0
Ri
Ri
Rii
D
K
D
D (1 − µ )
D
K
u20n − Di k 14z − 2i µ i k12z n 2 + i µ i k1 z j + v 02n 2i µ i k12z n − i 2 i k12z n + 4i n 3 − 2i n
Ri
Ri
Ri
Ri
Ri
Ri
Dµ
2Di (1 − µi ) 2 2 Di 4 K i
+ w20n − i 2 i k12z n 2 −
k1z n − 4 n − 2 + ρi hi ω 2 + p2Tn H n1 (k 2r R i )
2
R
R
R
R
i
i
i
i
+ p 2Rn H n2 (k 2r R i ) − p T3n H n1 (k 3 r Ri ) = 0
172
(6.29)
K (1 − µ ) K (1 + µ e )
K µ
u 02 n ρ e h eω 2 − K ek12z − e 2 e n 2 − e
nk1 z v02 n j − e e k1 z w20n j = 0
2 Re
2 Re
Re
K e (1 − µ e ) 2 Ke 2 De (1 − µ e ) 2
k1z − 2 n −
k1z
−
2
Re
2 Re2
Ke
De
0 K e (1 − µ e )
2
0
u 2n
k1z nj +
µ e k1z nj − 2 µ e k1z n + v 2n
Re
Re
De 2
2
2R e
+ R 4 n + ρ e h eω
e
(6.30)
(6.31)
K
D (1 − µ )
D
+ w20n − e2 n − e 2 e k12z n − 4e n 3 = 0
Re
Re
Re
D (1 − µ )
De
µ e k12z n − e 2 e k12z n
2
R
Re
D
K
u20 n − De k14z − e2 µ e k12z n 2 + e µ e k1 z j + v 02 n e
D
K
R
R
3
e
e
e
e
+ R 4 n − R 2 n
e
e
Dµ
2 De (1 − µ e ) 2 2 De 4 Ke
+ w02 n − e 2 e k12z n 2 −
k1 z n − 4 n − 2 + ρ e heω 2
2
Re
Re
Re
Re
R
2
T
1
R
2
n
+ p1 n H n ( k1r Re ) − p 2 n H n ( k 2 r Re ) − p 2 n H n ( k 2 r Re ) = − p0 ε n ( − j ) J n ( k1 r Re )
(6.32)
In the above equations, each membrane stiffness and bending stiffness K and D of the
inner and outer shells are defined as follows [39]:
Ki =
Ei hi
Eh
, K e = e e2 ,
2
1 − µi
1− µ e
Di =
Ei hi3
Ee he3
and
D
=
e
12(1 − µ i2 )
12(1 − µe2 )
where, Ei and Ee are the Young’s modulus of the inner and outer shells, µi is the Poisson’s
ratio of the inner shell and µe is the Poisson’s ratio of the outer shell.
The rest four equations from the four boundary conditions are:
′
p1Rn H n2 (k1 r Re )k1 r − ρ1 ω 2 w20 n = − p 0εn (− j )n Jn ′ (k1 r Re )k1 r
(6.33)
′
′
p2Tn H 1n ( k2 r R e )k2 r + p2Rn H n2 (k 2 r Re )k2 r − ρ 2ω 2 w20 n = 0
(6.34)
′
′
p2Tn H 1n ( k2 r Ri )k2 r + p2Rn H n2 ( k 2 rR i )k2 r − ρ2ω 2 w10n = 0
(6.35)
′
p3Tn H 1n ( k3 r Re ) k3 r − ρ3ω 2 w10n = 0
(6.36)
173
where, ( ) ' =
These
d
.
dr
equations
can
be
used
to
solve
for
10
unknowns:
0
p1Rn , p2Tn , p 2Rn , p 3T , u1n0 , v1n0 , w1n0 , u20 n , v 2n
and w20 n in terms of the amplitude of the incident
wave po . Equations (6.27) to (6.36) can be put into a form of a matrix equation:
0
0 A B
0 0
0 0
0
0 D E
G H
I
0 J K
0
0 0 0
0 0
0 0
0
0 0 0
0 S T U 0 0
Y 0
0
0 0 0
0 A1 B1 0 0 0
0 D1 E1 0 0 0
0 G1 0 0
0 0
C 0
F 0
L 0
0 M
0 P
0 V
Z 0
C1 0
0 0
0 0
0
0
0
N
Q
W
0
0
0
0
0 p1Rn 0
0 p2Tn 0
0 p2Rn I 1
O p3Tn 0
R u 20n 0
=
X v 02 n 0
0 w20 n J1
0 u10n 0
F1 v10n 0
H1 w10n 0
(6.37)
where,
A = ρe heω 2 − Ke k12z −
D=
K e (1 − µ e ) 2
K (1 + µe )
K µ
n ,B = − e
nk1 z j , C = − e e k1 z j
2
Re
2 Re
2 Re
Ke (1 − µe )
K
D
k1 z nj + e µ e k1 z nj − e2 µe k12z n
2 Re
Re
Re
E=−
K e (1 − µ e ) 2 K e 2 De (1 − µ e ) 2 De 2
k1 z − 2 n −
k1k + 4 n + ρ eheω 2
2
2
Re
2 Re
Re
F =−
D (1 − µ )
Ke
D
n − e 2 e k12z n − e4 n 3 , G = H n2 ( k1 r Re ) , H = −H 1n ( k 2 r Re ) , I = − H n2 ( k 2 r Re ) ,
2
Re
Re
Re
J = −De k14z −
K=
De
K
µe k12z n 2 + e µe k1z j
2
Re
Re
D (1 − µ )
De
D
K
µe k12z n − e 2 e k12z n + e4 n3 − e2 n
2
Re
Re
Re
Re
174
L=−
De µ e 2 2 2 De (1 − µ e ) 2 2 De 4 Ke
k1 z n −
k1 z n − 4 n − 2 + ρe heω 2
2
2
Re
Re
Re
Re
M = ρ i hiω 2 − Ki k12z −
P=
K i (1 − µ i ) 2
K (1 + µ i )
Kµ
n ,N = − i
nk1 z j , O = − i i k1 z j
2
Ri
2 Ri
2 Ri
Ki (1 − µi )
K
D
k1 z nj + i µ i k1z nj − 2i µ i k12z n
2 Ri
Ri
Ri
Q=−
Ki (1 − µ i ) 2 Ki 2 Di (1 − µi ) 2 Di 2
k1 z − 2 n −
k1k + 4 n + ρi hiω 2
2
2
Ri
2 Ri
Ri
R=−
D (1 − µ )
Ki
D
n − i 2 i k12z n − 4i n3 , S = H n1 ( k 2 r Ri ) , T = H n2 ( k 2 r Ri ) , U = −H 1n ( k3 r Ri )
2
Ri
Ri
Ri
V = −Di k14z −
X =−
Di
K
D 1− µ
µ i k12z n2 + i µ i k1 z j , W = Di2 µi k12z n − i ( 2 i ) k12z n + Di4 n 3 − K i2 n
2
Ri
Ri
Ri
Ri
Ri
Ri
Di µi 2 2 2Di ( 1− µ i ) 2 2 Di 4 K i
k1 z n −
k1 z n − 4 n − 2 + ρi hiω 2
Ri2
Ri2
Ri
Ri
′
′
′
Y = H n2 ( k1r Re )k 1r , Z = − ρ1ω 2 , A1 = H n1 (k 2 r Re ) k 2 r , B1 = H n2 (k 2 r Re ) k 2 r , C1 = − ρ2ω 2
′
′
′
D1 = H 1n ( k 2 r Ri ) k 2 r , E1 = H n2 ( k 2 r Ri ) k 2 r , F1 = − ρ2ω 2 , G1 = H 1n (k 3 r Ri ) k 3 r
′
H1 = − ρ3ω 2 , I 1 = − p0 ε n ( − j ) n J n (k1 r Re ) , J 1 = − p 0 ε n ( − j ) n J n ( k1 r Re ) k1 r
0
0
The ten unknown coefficients p1Rn , p 2Tn , p 2Rn , p3Tn , u1n0 , v1n0 , w1n0 , u 2n
, v 2n
and w02n are
obtained in terms of po by solving Equation (6.37) for each mode n, which then can be
substituted back into Equations (6.17) to (6.26) to find the displacements of the shell and
the acoustic pressures in series forms.
175
6-3.2. Double Shell with Bonded-Unbonded Porous Material Layer
Figure 6.3 shows the cross-sectional view of the double shell with a B-U core. The
porous layer is bonded to the outer shell, but is separated from the inner shell by an
airgap. In the figure, hg is the thickness of the airgap, and Rc indicates the radius of the
interface between the porous core and the airgap. The fluid media in the external, the
porous core, the airgap, and the internal space are defined by the density and the speed of
sound: {ρ1 , c1 }, {ρ2 , c2 }, {ρ3 , c3 }, and {ρ4 , c4 }, respectively.
Besides the fact that one more wave equation is necessary because of the airgap, the
wave equations, shell equations, and boundary conditions at the interfaces between the
shell and the fluids remain essentially the same as those of the B-B shell (wave equations
in Equations (6.1) to (6.3), shell equations in Equations (6.4) to (6.9), boundary
conditions in Equations (6.10) to (6.13)) only with different variable names. Therefore,
these equations are not shown. Because of the two additional wave components in the BU shell model compared to the B-B shell model, two additional boundary conditions are
necessary. These are obtained from the interface condition between the equivalent fluid
and the airgap, which at each interface are, with reference to Figure 6.4:
p2Tn H 1n (k 2 r Rc ) + p2Rn Hn2 (k 2 r Rc ) − p3T n H 1n (k 3 r Rc ) − p3Rn H n2 (k 3 r R c ) = 0
pT2 n
p2Rn
1
cos θ i 2 H n ( k 2 r Rc ) −
cos θ r 2 H 2n ( k2 r Rc )
ρ 2 c2
ρ 2 c2
pT
pR
− 3 n cos θ t 3 H 1n ( k3 r Rc ) + 3 n cos θ r 3 H n2 ( k3 r Rc ) = 0
ρ3c 3
ρ3c 3
(6.38)
(6.39)
where, θi2 , θr2, θt3 , and θr3 are the incidence, reflection and transmission angles,
respectively, as shown in Figure 6.4. The relationships among θi2 , θr2, θt3 , and θr3 at the
boundary separating two fluids (i.e., r = hc) are found as [38]:
176
cosθ i 2 =
k2 z
sinθ i 2 sinθ t 3
, sinθ i 2 = sinθ r 2 ,
=
, sinθ t 3 = sinθ r 3
k2
c2
c3
(6.40)
p1R
pI
ρ1 ,c1
ρ2 ,c2
p2T
p2R
ρ3 ,c3
p3R
p3T
p4T
Rc
ρ4 ,c4
Outer shell:
{u,v,w}2
Airgap
hg
Porous core
hc
Inner shell:
{u,v,w}1
Figure 6.3. Cross-sectional view of the shell with a B-U porous layer
177
Incident
Wave
γ
z
Outer
shell
r
Porous
Core
Transmitted
Wave
External
Air
θi2 θ
r2
θt3
Airgap
Reflected Wave
θr3
Inner
shell
Transmitted Wave
Figure 6.4. The incident, reflected, and transmitted waves of the B-U configuration in the
r-z plane
178
Substituting the assumed solutions for the pressures similar to Equations (6.17) (6.20) and of shell displacements identical to Equations (6.21) to (6.26) into six shell
equations and six boundary conditions, and utilizing the orthogonality between the
trigonometric functions yields 12 equations. These equations can be put into a matrix
equation:
0
0
G
0
0
0
Y
0
0
0
0
0
0
0
0
0
0
0
H
I
0
0
0
0
0
0
0
0
0
S
0
0
0
A1 B1 0
0
0 D1
0
0
0
I 1 J1 K 1
M 1 N1 O1
0
0
0
0
0
T
0
0
E1
0
L1
P1
0 A
0 D
0 J
0
0
0
0
U 0
0
0
0 0
0 0
G1 0
0 0
0 0
B
E
K
0
0
0
0
0
0
0
0
0
C 0 0 0 p1Rn 0
F 0 0 0 p2Tn 0
L 0 0 0 p2Rn Q1
0 M N O p3Tn 0
0 P Q R p3Rn 0
0 V W X p4Tn 0
=
Z
0 0 0 u20n R1
C1 0 0 0 v02 n 0
0 0 0 F1 w20 n 0
0 0 0 H1 u10n 0
0 0 0 0 v10n 0
0 0 0 0 w10n 0
where,
A = ρe heω 2 − Ke k12z −
K (1 + µe )
Ke (1 − µ e ) 2
n , B=− e
nk1 z j
2
2Re
2 Re
C=−
K e µe
K (1 − µe )
K
D
k1 z j , D = e
k1 z nj + e µ e k1 z nj − e2 µe k12z n
Re
2 Re
Re
Re
E =−
K e (1 − µ e ) 2 K e 2 De (1 − µe ) 2 De 2
k1 z − 2 n −
k1 z + 4 n + ρ e heω 2
2
2
Re
2Re
Re
F=−
Ke
D (1 − µ )
D
n − e 2 e k12z n − 4e n3
2
Re
Re
Re
G = p1Rn Hn2 ( k1r Re ), H = − p2Tn H1n ( k 1r Re ), I = − p2Rn Hn2 ( k 2 r Re )
179
(6.41)
J = − Dek14z −
L=−
De
K
D
D (1 − µ )
D
K
µ ek12z n 2 + e µek1 z j , K = e2 µe k12z n − e 2 e k12z n + e4 n3 − e2 n
2
Re
Re
Re
Re
Re
Re
De µe 2 2 2 De (1 − µe ) 2 2 De 4 K e
k1 z n −
k1 z n − 4 n − 2 + ρ e heω 2
2
2
Re
Re
Re
Re
M = ρ i hiω 2 − Ki k12z −
P=
Ki (1 − µ i ) 2
K (1 + µ i )
Kµ
n ,N =− i
nk1 z v10n j , O = − i i k1 z j
2
2 Ri
Ri
2Ri
K i (1 − µ i )
K
D
k1 z nj + i µi k1z nj − 2i µi k12z n
2Ri
Ri
Ri
Q=−
Ki (1 − µ i ) 2 Ki 2 Di (1 − µ i ) 2 Di 2
k1 z − 2 n −
k1 z + 4 n + ρi hiω 2
2
2
Ri
2 Ri
Ri
R=−
Ki
D (1 − µ )
D
n − i 2 i k12z n − 4i n3
2
Ri
Ri
Ri
S = pT3 n H1n ( k3 r Ri ), T = p3Rn H n2 ( k3 r Ri ), U = − p 4Tn H 1n (k 4 r R i )
V = − Di k14z −
X =−
Di
K
D
D (1 − µ )
D
K
µi k12z n2 + i µi k1 z j , W = 2i µ i k12z n − i 2 i k12z n + 4i n3 − 2i n
2
Ri
Ri
Ri
Ri
Ri
Ri
Di µi 2 2 2 Di (1 − µi ) 2 2 Di 4 Ki
k1 n −
k1 z n − 4 n − 2 + ρi hiω2
Ri2 z
Ri2
Ri
Ri
Y = p1Rn H n2 ′ ( k1 r Re ) k1 r , Z = − ρ1ω 2
A1 = pT2 n H1n ′ ( k2 r Re ) k2 r , B1 = pR2 n Hn2′ ( k2 r Re ) k2 r , C1 = − ρ 2ω 2
D1 = pT3 n H 1n′ ( k 3 r Ri ) k 3 r , E1 = p3Rn H n2′ ( k 3 r Ri ) k 3 r , F 1 = − ρ3ω 2
G1 = p4Tn Hn1′ ( k4 r Ri ) k4 r , H 1 = − ρ4ω 2
I 1 = p2Tn H 1n ( k2 r Rc ) , J 1 = p2Rn H n2 ( k2 r Rc ) , K1 = − p3Tn H1n ( k3 r Rc ) , L1 = − p3Rn H 2n ( k3 r Rc )
M1=
p2Tn
pR
cosθ i 2 Hn1 ( k 2 r Rc ) , N 1 = − 2 n cos θ r 2 H n2 ( k2 r Rc )
ρ2 c2
ρ2 c2
180
O1 = −
p3Tn
pR
cosθ t 3 H 1n ( k3 r Rc ) , P1 = 3 n cosθ r 3 H n2 ( k3 r Rc )
ρ3 c3
ρ3 c3
Q1 = − p0ε n ( − j )n Jn ( k1 r Re ) , R1 = − p0εn ( − j ) n Jn ′ ( k1r Re ) k1r
0
0
The twelve unknown coefficients p1Rn , p 2Tn , p 2Rn , p3Tn , p3Rn , p4Tn , u1n0 , v1n0 , w1n0 , u 2n
, v 2n
and
w02n are obtained in terms of po by solving Equation (6.41), which are then substituted to
the series expressions for pressures and displacements.
6-3.3. Double Shell with Unbonded-Unbonded Porous Material Layer
ρ1 ,c1
pI
Airgap
ρ2 ,c2
p2T
ρ3 ,c3
p1R
p3T
p2R
Airgap
Rc2
p4T
Rc1
ρ4 ,c4
p5T
p3R
Outer shell:
{u,v,w}2
p4T
Porous
core
ρ5 ,c5
hg1
hc
hg2
Inner shell:
{u,v,w}1
Figure 6.5. Cross-sectional view of the shell with a U-U porous layer
181
Figure 6.5 shows a schematic of the double shell with a porous layer installed in the U-U
configuration. In the figure, hg1 is the thickness of the airgap on the transmitted side, hg2
is the thickness of the airgap on the incident side, Rc1 indicates the radius of the boundary
at interface between the porous core and the airgap on the transmitted side, and Rc2
indicates the radius of the boundary at interface between the porous core and the airgap
on the incident side. The fluid media in the external, the airgap on the incident side, the
porous core, the airgap on the transmitted side, and the internal space are defined by the
density and the speed of sound: {ρ1 , c1 }, {ρ2 , c2 }, {ρ3 , c3 }, {ρ4 , c4}, and {ρ5 , c5},
respectively.
Incident Wave
γ
z
External
Air
Outer
shell r
Transmitted
Wave
Porous
core
Airgap
θi2 θ
r2
θt3
θt4
Reflected Wave
θr3
θr4
Inner
shell
Transmitted Wave
Figure 6.6. The incident, reflected, and transmitted waves of the U-U configuration in the
r-z plane
182
The additional fluid layer in the system compares to the B-U shell, which induces two
additional wave components. Two more boundary conditions can be applied by
considering the interface condition between the external airgap and the porous layer, as
illustrated in Figure 6.6. The derivation procedure is again very similar to the case of the
B-B shell or B-U shell, which leads to a matrix equation of order fourteen as follows.
0
0
G
0
0
0
Y
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A
B
C
0
0
0
H
0
I
0
0
0
0
0
0
0
0
0
0
D
J
E
K
F
L
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
S
0
0
T
0
0
0
0
0
0
U
0 0
0
0
0
M
P
V
N
Q
W
0
A1
0
0
B1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
D1 E1 0
0
0
0
0
0
0
0
0
0
0
I 1 J 1 K 1 L1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 Z
0 C1
0 0
0 G1 0
0 0 0
M 1 N 1 O1 P 1 0 0 0
0
0 Q1 R1 S1 T1 0
0
0 U 1 V 1 W1 X1 0
0
0
0
0
0
0
0 p1Rn 0
0 p2Tn 0
0 p2Rn Y 1
O p T3n 0
R p 3Rn 0
X p4Tn 0
0 p4Rn Z1
=
0 p5Tn 0
F1 u20n 0
H 1 v20n 0
0 w02 n 0
0 u10n 0
0 v10n 0
0 w10n 0
where,
A = ρe heω 2 − Ke k12z −
K (1 + µe )
Ke (1 − µ e ) 2
n , B=− e
nk1 z j
2
2Re
2 Re
C=−
K e µe
K (1 − µe )
K
D
k1 z j , D = e
k1 z nj + e µ e k1 z nj − e2 µe k12z n
Re
2 Re
Re
Re
E =−
K e (1 − µ e ) 2 K e 2 De (1 − µe ) 2 De 2
k1 z − 2 n −
k1 z + 4 n + ρ e heω 2
2
2
Re
2Re
Re
F=−
Ke
D (1 − µ )
D
n − e 2 e k12z n − 4e n3
2
Re
Re
Re
G = p1Rn Hn2 ( k1r Re ), H = − p2Tn H1n ( k 1r Re ), I = − p2Rn Hn2 ( k 2 r Re )
183
(6.42)
J = − Dek14z −
L=−
De
K
D
D (1 − µ )
D
K
µ ek12z n 2 + e µek1 z j , K = e2 µe k12z n − e 2 e k12z n + e4 n3 − e2 n
2
Re
Re
Re
Re
Re
Re
De µe 2 2 2 De (1 − µe ) 2 2 De 4 K e
k1 z n −
k1 z n − 4 n − 2 + ρ e heω 2
2
2
Re
Re
Re
Re
M = ρ i hiω 2 − Ki k12z −
P=
Ki (1 − µ i ) 2
K (1 + µ i )
Kµ
n ,N =− i
nk1 z v10n j , O = − i i k1 z j
2
2 Ri
Ri
2Ri
K i (1 − µ i )
K
D
k1 z nj + i µi k1z nj − 2i µi k12z n
2Ri
Ri
Ri
Q=−
Ki (1 − µ i ) 2 Ki 2 Di (1 − µ i ) 2 Di 2
k1 z − 2 n −
k1 z + 4 n + ρi hiω 2
2
2
Ri
2 Ri
Ri
R=−
Ki
D (1 − µ )
D
n − i 2 i k12z n − 4i n3
2
Ri
Ri
Ri
S = pT4 n Hn1 ( k4 r Ri ), T = p4Rn Hn2 ( k 4 r Ri ), U = − p5Tn H 1n (k 5 r R i )
V = − Di k14z −
X =−
Di
K
D
D (1 − µ )
D
K
µi k12z n2 + i µi k1 z j , W = 2i µ i k12z n − i 2 i k12z n + 4i n3 − 2i n
2
Ri
Ri
Ri
Ri
Ri
Ri
Di µi 2 2 2 Di (1 − µi ) 2 2 Di 4 Ki
k1 n −
k1 z n − 4 n − 2 + ρi hiω2
Ri2 z
Ri2
Ri
Ri
Y = p1Rn H n2 ′ (k1 r Re )k1 r , Z = − ρ1ω 2 , A1 = − p0 ε n ( − j ) n Jn ′ ( k1 r Re ) k1r
A1 = p2Tn H 1n′ ( k2 r Re ) k2 r , B1 = p2Rn Hn2′ (k2 r Re ) k2 r , C1 = − ρ2ω 2
D1 = pT4 n Hn1′ ( k4 r Ri ) k4 r , E1 = pR4 n H n2′ ( k4 r Ri ) k4 r , F1 = −ρ 4ω 2
G1 = p5Tn H 1n ′ ( k5 r Ri )k5 r , H1 = − ρ5ω 2
I 1 = p2Tn H 1n (k 2 r Rc2 ), J1 = p2Rn H n2 ( k2 r Rc 2 ), K1 = − p3Tn H1n (k 3 r Rc2 ), L1 = − p3Rn H n2 ( k3 r Rc2 )
M1=
p2Tn
pR
cosθ i 2 H 1n ( k2 r Rc 2 ) , N 1 = − 2 n cosθ r 2 H n2 ( k2 r Rc 2 )
ρ2 c2
ρ2 c2
184
O1 = −
p3Tn
pR
cosθ t 3 H1n ( k3r Rc 2 ) , P1 = 3 n cos θ r 3 Hn2 ( k 3r Rc 2 )
ρ3 c3
ρ3 c3
Q1 = pT3 n H 1n (k 2 r Rc1 ), R1 = p3Rn Hn2 (k 3 r Rc1 ), S1 = − p 4Tn H n1 (k 4 r Rc1 ), T1 = − p4Rn Hn2 (k 4 r Rc1 )
U1 =
p3Tn
pR
cos θ t 3 H 1n (k 3 r Rc1 ), V1 = − 3 n cos θ r 3 H 2n (k 3 r Rc1 )
ρ3 c3
ρ3 c3
W1 = −
p4Tn
pR
cos θ t 4 Hn1 ( k 4 r Rc 1 ) , X 1 = 4 n cosθ r 4 H n2 ( k4 r Rc1 )
ρ4 c4
ρ4 c 4
Y 1 = − p0ε n ( − j )n J n ( k1 r Re ) , Z 1 = − p0 ε n ( − j ) n J n ′ ( k1 r Re ) k1 r
0
The fourteen unknown coefficients p1Rn , p 2Tn , p 2Rn , p3Tn , p3Rn , p4Tn , p4Rn , p5Tn , u1n0 , v1n0 , w1n0 , u 2n
,
0
v 2n
and w02n can be obtained in terms of po by solving Equation (6.42) for each mode.
These mode solutions can be summed to obtain the pressure and displacement solutions.
6-4. Calculation of Transmission Loss (TL)
The sound power transmitted per unit length of the shell can be obtained by integrating
the sound intensity around the circumference:
WT =
2 π
1
Re ∫ p Tj ⋅ ∂ ( w10 ) * Ri dθ
∂t
2
0
where r=Ri
(6.43)
where j=3,4,5 are used, respectively, for the B-B, B-U and U-U configurations, Re{.} and
the superscript * represents the real part and the complex conjugate of the argument.
This, the transmitted power contributed by each mode term WnT is obtained as:
WnT =
=
2π
1
Re pTjn H1n(k jr Ri )⋅ ( jω w10n ) * × ∫ cos2 [ nθ ] ⋅ Ridθ
2
0
{
}
π Ri
× Re{ p Tjn H 1n (k jr Ri ) ⋅ ( jω w10n )* }
2ε n
(6.44)
185
where, ε n =1 for n = 0 and ε n =2 for n=1,2,3,……
The power transmission coefficient is defined by:
∞
τ (γ ) =
∑W
T
n
n= 0
(6.45)
WI
where W I is the incident power per unit length of the shells:
WI =
cos ( γ ) p02
× 2 Re
ρ1c1
(6.46)
Finally, the power transmission coefficient is obtained as a function of the incident angle
γ by substituting Equations (6.44) and (6.46) into (6.45) as follows:
∞
τ (γ ) = ∑
{
}
Re pTjn × Hn1 ( k jr Ri ) × ( jω w10n ) × ρ1c1π × Ri
n =0
*
4ε n Re cos ( γ ) p
(6.47)
2
0
where, j=3,4,5 have to be used for the B-B, B-U and U-U shells, respectively. As was
done in Chapter 5, the transmission coefficient is averaged over all possible angles of
incidence to obtain the average transmission coefficient τ (see Section 5-2.1 and
Equation (5.22) of Chapter 5). Thus, the averaged TL is obtained as:
TLavg = 10log10 (1/ τ )
(6.48)
In the following, the averaged TLs of the three types of the shell are calculated in
terms of the 1/3 octave band for random incidences.
The thickness and radius of the
external shell he and Re are taken as 2.0 mm and 0.2 m, those of the internal shell hi and
Ri are taken as 3.0 mm and 0.15 m, and the shell material is considered aluminum, whose
Young’s modulus, Poisson ratio and density are 7×1010 Pa, 0.33, and 2700 kg/m3 . For
the porous material, the same properties are used as those used in Chapter 5, which are
186
defined by the bulk density of solid phase, ρ1 =30 kg/m3 ; in vacuo bulk Young’s modulus,
Ec=8×105 Pa; in vacuo loss factor [67], η=0.265; bulk Poisson ratio, µc=0.4; flow
resistivity, σ=25×103 MKS Rayls/m; geometrical structure factor, ε’=7.8; porosity,
h=0.9.
6-4.1. B-B Shell
As explained in Chapter 5, a preliminary calculation is conducted by using the full theory
for a flat double-panel of infinite extent with the B-B porous layer. Figure 6.7 compares
the energies associated with three wave components. In this case, the frame wave is
selected to represent the porous material as an equivalent fluid.
Figure 6.8 shows the TL of the B-B shell compared to the TL of the double-walled
shell with the airgap (or without the porous material) between the shells, which shows
that adding the porous material to the double shell in a B-B configuration is not so
effective. No significant increase in the TL is observed except in the frequency range
lower than 250 Hz. It is seen that actually adding the porous layer slightly decreases the
TL in the frequency range between 250Hz and 3000 Hz. However, this may have been
caused by the fact that the simplified model tends to underestimate the TL in the high
frequency range (see Figure 5.13 of Chapter 5).
Considering the approximations
involved, it can be concluded that applying the porous material to the double shell in a BB construction, without leaving any airgap between the layers would only marginally
improve the low frequency range.
187
350
300
Energy Ratio
250
200
150
100
50
0
0
1000
2000
3000
4000
5000
Frequency (Hz)
Figure 6.7. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the B-B double-panel configuration (normalized by the
energy of the airborne wave):
, airborne wave in fluid phase to airborne wave in fluid phase (
E1 f
);
E1 f
E3 f
);
E1 f
E
∗, airborne wave in solid phase to airborne wave in fluid phase ( 1 s );
E1 f
E
, frame wave in solid phase to airborne wave in fluid phase ( 3 s );
E1 f
E
, shear wave in solid phase to airborne wave in fluid phase ( 5 s )
E1 f
, frame wave in fluid phase to airborne wave in fluid phase (
188
160
140
TL (dB)
120
100
80
60
40
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 6.8. Comparison of the calculated TLs of cylindrical double-walled shells in B-B
configuration
, no-foam; , B-B
189
6-4.2. B-U Shell
A double-walled shell with a B-U porous core is considered. The porous layer is bonded
to the external shell, but is apart from the internal shell with an airgap of 17.5 mm.
Therefore Re=0.2 m, Ri=0.15 m, Rc=0.169 m, he=2 mm, hi=3 mm, hg=17.5 mm, and
hc=30 mm. Figure 6.9 shows the ratio of the energies associated with the three wave
components calculated in the preliminary step of the analysis using a flat double-panel of
infinite extent with the B-U construction by the full theory (Bolton-Biot’s method). As
in the flat panel case, the airborne wave is used in the low frequency range, and the frame
wave is used in the high frequency range to represent the porous material as an equivalent
fluid.
Figure 6.10 shows the TLs calculated by using the airborne wave and the frame wave.
According to Figure 6.9, between 1,500 Hz and 3,000 Hz can be considered the transition
frequency range, below which the airborne wave is the main energy carrier, and above
which the frame wave. Therefore, the TL curve obtained by using the airborne wave
model should be used if the interested frequency is relatively low, lower than 1,000 Hz,
and the curve corresponding to the frame wave should be used for relatively high
frequency range, for example above 3,000 Hz. In the middle range, say between 1,000
Hz and 2,500 Hz, the average of the two curves may be used as illustrated in Figure 6.10.
Compared to the B-B or U-U constructions, the B-U shell result is considered to involve
larger error because no single wave is dominant in the entire frequency range. However,
even the lower TL curve is much higher than that of the B-B shell or the double shell
without the porous layer. Therefore, the B-U shell will be much more effective than the
190
B-B shell in terms of noise reduction, which is a conclusion compatible with the flat
panel cases.
30
25
Energy Ratio
20
15
10
5
0
0
1000
2000
3000
4000
5000
Frequency (Hz)
Figure 6.9. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the B-U double-panel (normalized by the energy of the
airborne wave):
E
, airborne wave in fluid phase to airborne wave in fluid phase ( 1 f );
E1 f
E
, frame wave in fluid phase to airborne wave in fluid phase ( 3 f );
E1 f
E
∗, airborne wave in solid phase to airborne wave in fluid phase ( 1 s );
E1 f
E
, frame wave in solid phase to airborne wave in fluid phase ( 3 s );
E1 f
E
, shear wave in solid phase to airborne wave in fluid phase ( 5 s )
E1 f
191
160
Average of
the TL curves
140
Transition
frequencies
TL (dB)
120
100
80
60
40
20
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 6.10. Comparison of the calculated TLs of cylindrical double-walled shells in B-U
configuration
, no-foam; , B-U using the airborne wave; ∇, B-U using the frame wave
192
6-4.3. U-U Shell
In this case, the porous layer is installed with a 7.5 mm gap inside and a 10 mm gap
outside; therefore, Re=0.2 m, Ri=0.15 m, Rc1=0.159 m, Rc2=0.189 m, he=2 mm, hi=3 mm,
hc=30 mm, hg1 =7.5 mm, and hg2 =10 mm. Figure 6.11 shows the ratio of the energies
associated with the three wave components calculated in the preliminary step. In this
case, the airborne wave is selected as the only wave component to represent the porous
material as an equivalent fluid. Figure 6.12 compares the TL calculated the U-U shell
with that of the double-walled shell without any porous layer. As seen in the figure,
adding the porous layer in a U-U configuration substantially increases the TL over the
double-walled shell without the porous layer.
6-4.4. Discussions
Figure 6.13 compares the TLs of the foam- lined cylindrical double-walled shell
calculated for the three configurations, B-B, B-U and U-U.
The B-U and U-U
configurations show much higher TL compared to the B-B shell in the entire frequency
range. Generally, the TL curve of the B-U shell is believed the highest in the frequency
range up to 1,200 Hz, which is also compatible with the study on the flat panel [48]. The
double airgaps in the U-U shell also require additional manufacturing and installation
steps and possibly more space. Therefore, the B-U shell is considered the best design for
noise reduction purposes in curved double shells also, as it was in the case of the flat
double-panel.
193
1
Energy Ratio
0.8
0.6
0.4
0.2
0
0
1000
2000
3000
4000
5000
Frequency (Hz)
Figure 6.11. Frame and shear wave contributions to the fluid and solid strain energies in
the y-direction for the U-U double-panel (normalized by the energy of the
airborne wave):
E
, airborne wave in fluid phase to airborne wave in fluid phase ( 1 f );
E1 f
E
, frame wave in fluid phase to airborne wave in fluid phase ( 3 f );
E1 f
E
∗, airborne wave in solid phase to airborne wave in fluid phase ( 1 s );
E1 f
E
, frame wave in solid phase to airborne wave in fluid phase ( 3 s );
E1 f
E
, shear wave in solid phase to airborne wave in fluid phase ( 5 s )
E1 f
194
160
140
TL (dB)
120
100
80
60
40
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 6.12. Comparison of the calculated TLs of cylindrical double-walled shells in U-U
configuration
, no-foam; , U-U
160
140
TL (dB)
120
100
80
60
40
20
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 6.13. Comparison of the calculated TLs of the three double-walled shells with
three porous layers
, B-B; ∇, B-U; , U-U
195
6-5. Conclusions
Transmission losses (TLs) of double-walled shells sandwiching a layer of porous material
were calculated for three types of layer construction: B-B (the layer bonded to both
shells), B-U (bonded one side and gap in the other side) and U-U (gap in both sides). A
simplified analysis method developed in Chapter 5 was utilized to solve these problems
as a two-step solution procedure. This work is considered the first analytical solution that
calculates the TLs of the structure with double walls sandwiching the porous layer,
considering the acoustic-structural coupling effect as well as the effect of the multi- waves
in the porous layer even if the formulation for the porous material is simplified
substantially. Characteristics of the TL curves obtained for random incidence cases for
B-B, B-U and U-U designs were compared and discussed, which showed similar trends
as in the flat panel cases.
The simplification that uses only one dominant wave to model the porous layer
certainly induces errors in solutions ; however, it makes generally unsolvable problems
solvable. Obviously, the approximate method will have to be used with its intended
purpose in mind: qualitative or comparative design analysis and design parameter studies
of curved shells with a porous material layer. A numerical method will have to be used
for a structure with a porous liner of more complex geometry encountered in practical
problems. However, once the approximate method is applied, sound transmission in
structures can be solved by using widely available standard FEA software with acoustic
solution capabilities.
196
Chapter 7 – Sound Transmission through Stiffened Panels
7-1. Introduction
Thin plates stiffened by parallel, equally spaced line stiffeners are commonly found in
aircraft and marine structures. In the low frequency range in which the wavelength of the
flexural wave in the plate is much longer than the stiffener spacing such a structure can
be modeled as an orthotropic plate. At a relatively high frequency, at which the flexural
wavelength is comparable with stiffener spacing, the structure has to be modeled as a
panel with periodically deployed stiffener.
The structural response of the periodic
structure to harmonic excitations has been obtained by expanding in terms of a series of
space harmonics by Mead et al. [73-75] to investigate structural responses of periodically
stiffened beams and plates. This approach is also adopted in this work to solve the vibroacoustic equation of the stiffened plate subjected to a plane wave input and to calculate
the transmission loss through the structure.
If a panel is stiffened only in one direction, the two-dimensional, periodically
stiffened panel may be modeled as a one-dimensional structure, essentially a periodically
supported beam as illustrated in Figure 7.1.
The stiffener can be represented as a
combination of a lumped mass (M), rotational (K r) and translational (K t ) springs as
shown in Figure 7.1. The system equation is developed by combining the wave equations
in the incident and transmitted side, the beam equation representing the plate and the
effect of the lumped mass and springs representing the stiffener, which are all coupled
with one another.
A unique analysis technique is developed to solve this vibro-
acoustically coupled problem. Utilizing the analysis result, characteristics of the sound
197
transmission through a periodically stiffened plate are studied with special attentions to
the role of the stiffener in the sound transmission. The analysis procedure developed in
this work, which is based on the space harmonic method could be obviously extended to
other types of structures. For example, sound transmission characteristics through a
cylindrical shell with periodic ring stiffeners are being studied by the authors using the
approach in their concurrent work.
The response of stiffened plates excited by random pressure fields can be obtained by
the normal mode expansion method [76-79]. Because the normal modes of the stiffened
plates must be determined, the method is confined to systems with relatively simple
boundary conditions. The structural wave propagation in periodically supported,
undamped beams and grillages was studied by Heckl [80]. It was shown that flexural
waves can propagate freely (without rapidly decaying) only in certain frequency bands.
If the periodic structure is highly damped, a wave approach yields the response much
more readily as demonstrated by Mead and Wilby [81]. A relatively simple formula was
developed for the displacement, curvature or stress at any point in the beam.
Mead and K. K. Pujara [73] developed a space harmonic method, in which the
system response is expanded in terms of the harmonics of the stiffener spacing. This
method is attractive because the sound radiation effects can be very easily incorporated as
the transverse displacement can be expressible as a series of sinusoidal traveling waves.
In their work, the panel is represented as a beam supported at regular intervals on elastic
constraints that oppose both transverse displacements and flexural slope changes. The
response of the beam subjected to a homogeneous random convected pressure field was
solved, in which the coupling between the acoustic system and the structural system is
198
not considered (acoustic pressure acts as structural excitations, but the effect of the
structural response on the acoustic system is not included). Structural responses of twodimensional plates, which are assumed to be simply supported in one direction, with
orthogonal stiffening were investigated by Lin [77, 78], Mercer [79], Ford [82] and
Mercer and Seavey [83].
Sound transmission problems have been studied by some researchers, however there
works are either incomplete [84] or limited to low frequencies due to the related
simplifications [85]. G. P. Mathur et al. [84] proposed a theoretical model based on the
space harmonic approach to calculate the transmission loss through a periodically
stiffened panel and stiffened double-panel structures, however no numerical results were
produced. Schemes to impose the structure-acoustic interactions and convergence of the
solution are developed in this work to make the space harmonic method work for this
type of problems.
W. Desmet et al. [85] presented a method to find the sound
transmission properties of finite double-panel partitions at low frequencies adopting an
experimental approach as well as a theoretical method based on Dowell’s modal coupling
theory [86].
O. K. Bedair [87], and S. Mukherjee [88] investigated the dynamic behavior (e.g., the
fundamental frequency of stiffened plates) of plates stiffened by a system of
interconnected beams or ribs. J. Wei et al. [89, 90] discussed the application of the
Rayleigh- Ritz and extended Rayleigh- Ritz energy methods to finite periodic structures
with sinusoidal displacement functions and also studied the relations between his method
for analyzing finite periodic structures and the theory of infinite periodic structures.
199
Panel
Incident Sound
Y
θ
φ
X
Z
Stiffener
Y
L
X
M
Kr
Kt
Stiffeners have mass, translational and rotational stiffness
Figure 7.1. Schematic representation of a stiffened panel
200
7-2. Formulation of the System Equation
Figure 7.1 illustrates the system to be studied, which is a typical set up that defines the
transmission lo ss. A plane wave is incident to a flat panel with a periodic stiffener, which
induces the reflected wave, the panel motion, and the transmitted wave. The transmitted
side is assumed to be anechoic, therefore there is no reflected wave in the transmitted
side.
Because of the periodic nature of the system, the system response is also expected to
be periodic. Therefore, the transverse motion of the panel in Figure 7.1 can be expressed
as a series of space harmonics [73], i.e.:
+∞
W (x , t ) =
∑
−j
An e
µ + 2 nπ
x
L
e jω t
(7.1)
n =−∞
where, W(x,t) is the panel transverse displacement, coefficients An can be considered as
modal amplitudes of the structure, L is the spacing between stiffeners, and µ is the
characteristic propagation constant which is defined as:
µ
= k x (ω ) − j ψ (ω )
L
(7.2)
where, ω is the angular frequency, ψ is the phase attenuation coefficient and kx is the
component of the wave number along the x axis. Referencing to Figure 7.1, component
wave numbers kx and ky can be obtained as:
kx = k sinθ cos φ
(7.3)
kz = k sinθ sin φ
(7.4)
where k =
ω
is the wave number of the incident plane wave, c is the speed of sound, θ
c
and φ are the incidence angles of the plane wave in the x- y plane and x-z plane. In this
201
study the plane wave is assumed to be incident along X-Y plane, thus i.e., φ=0. Notice
that, in Equation (7.1), the structural wave is expressed as the sum of space harmonics
corresponding to n = 0, ±1, ±2, ±3, …, therefore the forward as well as backward waves,
which represent reflections at the stiffener joint. Each of the space harmonic does not
satisfy the boundary condition, however their sum is forced to satisfy the boundary
condition.
The velocity potential at a point in the incident side half space is composed of the
potentials of the incident and reflected waves. The reflected wave is also expected to be
periodic spatially, the wave velocity potential Φ 1 ( x , y ,t ) is represented:
Φ1 ( x, y, t) = e
− j(
µ
x+ ky 0 y−ω t )
L
+
+∞
∑Be
− j(
µ + 2nπ
x− kyn y −ω t)
L
n
(7.5)
n =−∞
where, the first term represents the potential of the incident wave and the second term in
a series form represents that of the reflected wave. Φ 2 ( x , y, t ) , the velocity potential of
the transmitted wave is also spatially periodic, therefore expressed as:
Φ 2 ( x ,y ,t ) =
+∞
∑ Cne
− j(
µ +2 nπ
x+ kyn y −ω t)
L
(7.6)
n =−∞
In Equations (7.5) and (7.6), kyn is the wave number in the y direction, which can be
obtained from the following relationship.
ω
µ + 2nπ
2
k yn = −
−kz
c
L
2
2
(7.7)
Coefficients Bn and Cn may be considered as modal amplitudes of the reflected and
transmitted waves.
202
The modal amplitudes of the reflected and transmitted waves can be related to those
of the structural wave by considering the boundary conditions of the normal velo cities
[38]. At y=0:
−
∂Φ1
= j ωW
∂y
(7.8)
−
∂Φ 2
= jωW
∂y
(7.9)
Substitution of Equations (7.1), (7.5) and (7.6) into Equations (7.8) and (7.9), we obtain:
ω
∞
∑ Ae
−
µ + 2 nπ
x
L
n
+
n =−∞
ω
∞
∑
An e
∞
∑k
yn
Bn e
−
µ + 2nπ
x
L
−
µ + 2 nπ
x
L
− k y 0e
µ
− x
L
=0
(7.10)
n =−∞
−
µ + 2 nπ
x
L
n =−∞
−
∞
∑
k ynC ne
=0
(7.11)
n =−∞
Because Equations (7.10) and (7.11) should be valid at all values of x, the relationships
between the modal amplitudes are obtained. From Equation (7.10),
Bn = 1 − ω
An
k yn
when n=0
(7.12.a)
= −ω
An
k yn
when n≠0
(7.12.b)
From Equation (7.11),
Cn = ω
An
k yn
(7.13)
Therefore, if the coefficients An , modal amplitudes of the flexural wave in the panel,
are found, all other coefficients are also found. The coefficients An can be found by
solving the system equation, which is derived by applying the principle of virtual work
for one period of the beam as proposed by Mead [73]. The principle states that the sum
203
of the work done by all elements in one period of the system must do no work when the
system moves through any one of the virtual displacements:
δ W = δ Ame
− j(
µ +2 mπ
x− ω t )
L
(7.14)
The virtual work for one period of the panel element is calculated at first. The
equation of motion of the beam representing the unit depth of the panel is [39]:
D
d 4W
− m pω 2W − jωρ 0 ( Φ1 − Φ 2 ) = 0
4
dx
(7.15)
where, mp is the panel mass per unit length and ρ0 is the density of air, and D is the
flexural stiffness of the panel defined as [39]:
Eh 3
D=
12(1 − ν 2 )
(7.16.a)
where, h is the panel thickness, E and ν are the in vacuo Young’s modulus and Poisson’s
ratio of the panel material. If necessary, structural damping of the panel material can be
introduced by taking D as:
D=
Eh 3
(1 + jη )
12(1 − ν 2 )
(7.16.b)
where η is the loss factor of the beam material [67].
The last term in Equation (7.15) represents the acoustic and structural coupling effect. In
Equation (7.15), the equivalent force is applied to a unit length of the beam. Thus, the
virtual work contributed by the panel can be represented as:
d 4W
δΠ p = ∫ D 4 − mpω 2W − jωρ0 ( Φ1 − Φ 2 ) δ W *
dx
x=0
L
204
(7.17)
where, δ W * represents the complex conjugate of the virtual displacement in Equation
(7.14). Therefore, substituting Equations (7.5) and (7.6) to Equation (7.17), the virtual
work done by the panel is obtained:
4
µ + 2 nπ
−j
x
L ∞ µ + 2nπ
δΠ p = δ A ∫ D ∑
An e L e
L
0 n =−∞
*
m
∞
L
−∫
∑mω
L
−j
0 n =−∞
p
− ∫ jωρ0 [e
2
Ane
−j
µ + 2 nπ
x
L
e
µ
x − jk y
y0
L
e
e
j
j
µ + 2 mπ
x
L
µ + 2 mπ
x
L
+
∞
− ∑ Cn e
−j
µ + 2nπ
x
L
e
j
µ + 2 mπ
x
L
e
− jk yn y
n =−∞
µ + 2 mπ
x
L
dx
dx
∞
∑Be
n =−∞
0
j
jkyn y − j
n
e
µ + 2 nπ
x
L
e
j
µ + 2 mπ
x
L
(7.18)
]dx}
The contribution to the virtual work by the translational spring is equal to:
δΠ t = KtW (0) ⋅ δ Am* = δ Am* K t
∞
∑A
(7.19)
n
n =−∞
The contribution to the virtual work by the rotational spring per one period of the system
is equal to:
µ + 2mπ
δΠ r = jK rW ′(0)δ A*m
L
= δ A* K ∞ A µ + 2nπ µ + 2mπ
m r ∑
n
L
L
n =−∞
(7.20)
The contribution of the lumped mass to the virtual work per one period of the system
becomes:
∞
δΠ M = −ω 2 MW (0) ⋅ δ A*m = −ω2 M δ A*m ∑ An
(7.21)
n =−∞
Finally, the virtual work principle requires that
δΠ p + δ Π t + δ Π r + δ Π M = 0
(7.22)
205
Evaluating the integrals involved in δΠ p and noticing that the virtual displacement is
arbitrary, Equation (7.22) results in the following equation.
µ + 2mπ 4
Kt ω2 M ∞
K
2
D
−
m
ω
A
+
An + r
m −
p
∑
L
L n =−∞
L
L
∞
∑A
n
n =−∞
µ + 2nπ µ + 2mπ
L
L
= j ωρ 0 Bm − Cm + 1
when m=0
(7.23.a)
= j ωρ 0 Bm − Cm
when m≠0
(7.23.b)
Substituting the relationships between the modal amplitudes defined in Equations (7.12)
and (7.13), Equation (7.23) becomes:
µ + 2 mπ 4
2 ρ 0ω 2 j
Kt ω 2 M ∞
Kr ∞
µ + 2 nπ µ + 2mπ
2
−
m
ω
+
A
+
−
A
+
An
D
∑
∑
p
m
n
L
k ym
L n =−∞
L n =−∞
L
L
L
= 2ωρ 0 j
for m=0
(7.24.a)
=0
for m =±1, ±2, ±3, …
(7.24.b)
Consideration of the virtual work in any other panel element would have yielded an
identical set of equation.
7-3. Solution Procedure
7-3.1. Solution of the Governing Equation
Equation (7.24) can be solved for unknown coefficients Am’s, from which coefficients
Bm’s and Cm’s can be found using Equations (7.12) and (7.13). The number of terms to
be used in the calculation has to be decided after the convergence of the solution is
checked. For an illustration purpose, we take the terms m = –2, -1, 0, 1, 2 in Equation
(7.24), which results in five equations for the five unknowns A-2 , A-1 , A0 , A+1 , A+2 .
206
µ + 2( −2)π 4
Kt ω 2 M
2ω 2 ρ0 j
2
( A + A−1 + A0 + A1 + A2 )
D
A−2 + −
− m pω + k
L
L −2
y −2
L
2
K r µ + 2( −2)π
µ + 2( −1)π µ + 2( −2)π
+
+ A−1
A−2
L
L
L
L
(7.25)
µ + 2(0)π µ + 2( −2)π
µ + 2(1)π µ + 2( −2)π
+ A0
+ A1
L
L
L
L
µ + 2(2)π µ + 2( −2)π
+ A2
= 0 ( m = − 2)
L
L
µ + 2( −1)π 4
2ω 2 ρ0 j
K t ω 2M
2
D
A−1 + −
− m pω +
( A−2 + A−1 + A0 + A1 + A2 )
L
k y −1
L
L
µ + 2( −2)π µ + 2( −1)π
µ + 2( −1)π
+ A−1
A−2
L
L
L
µ + 2(0)π µ + 2( −1)π
µ + 2(1)π µ + 2( −1)π
+ A0
+ A1
L
L
L
L
µ + 2(2)π µ + 2( −1)π
+ A2
( m = − 1)
= 0
L
L
2
K
+ r
L
µ + 2(0)π 4
Kt ω 2 M
2ω 2 ρ0 j
2
D
−
m
ω
+
A
+
−
( A + A−1 + A0 + A1 + A2 )
p
L
k y 0 0 L
L −2
K µ + 2( −2)π µ + 2(0)π
µ + 2( −1)π µ + 2(0)π
+ r A−2
+ A−1
L
L
L
L
L
µ + 2(0)π
µ + 2(1)π µ + 2(0)π
+ A0
+ A1
L
L
L
µ + 2( 2)π µ + 2(0)π
+ A2
= 2 jωρ0 (m = 0)
L
L
2
207
(7.26)
(7.27)
µ + 2(1)π 4
Kt ω 2 M
2ω 2 ρ0 j
2
( A + A−1+ A0+ A1 + A2 )
D
A1 + −
− m pω + k
L
L −2
y1
L
+
Kr
L
µ + 2( −2)π µ + 2(1)π
A−2
L
L
µ + 2( −1)π µ + 2(1) π
+ A−1
L
L
µ + 2(0)π µ + 2(1)π
µ + 2(1)π
+ A0
+ A1
L
L
L
µ + 2( 2)π µ + 2(1) π
+ A2
( m = 1)
= 0
L
L
(7.28)
2
µ + 2(2)π 4
Kt ω 2 M
2ω 2 ρ0 j
2
( A + A−1 + A0 + A1 + A2 )
D
A2 + −
− mpω + k
L
L −2
y2
L
µ + 2( −1)π µ + 2(2)π
µ + 2( −2)π µ + 2(2)π
+ A−1
A−2
L
L
L
L
µ + 2(0)π µ + 2(2)π
µ + 2(1)π µ + 2(2)π
+ A0
+ A1
L
L
L
L
2
µ + 2(2)π
+ A2
( m= 2 )
=0
L
+
Kr
L
(7.29)
The Equations (7.25) –(7.29) can be put into a matrix equation in Equation (7.30).
A1 B 1 C 1
F 1 G1
J1
Symmetric
D1 E1 A−2 0
H 1 I 1 A−1 0
K1 L1 A0 = P1
M 1 N 1 A1 0
O1 A2 0
(7.30)
where, matrix coefficients are:
2ω 2 ρ 0 j Kt ω 2 M K r µ + 2( −2)π
µ + 2( −2)π
2
A1 = D
−
m
ω
+
+
−
+
p
L
ky−2
L
L
L
L
4
Kt ω 2 M K r µ + 2( −1)π µ + 2( −2)π
B1 =
−
+
L
L
L
L
L
C1 =
K t ω 2 M K r µ + 2(0)π µ + 2( −2)π
−
+
L
L
L
L
L
208
2
D1 =
Kt ω 2 M K r µ + 2(1)π µ + 2( −2)π
−
+
L
L
L
L
L
E1 =
Kt ω 2 M K r µ + 2(2)π µ + 2( −2)π
−
+
L
L
L
L
L
2ω 2 ρ 0 j Kt ω 2 M K r µ + 2( −1) π
µ + 2( −1) π
2
F1 = D
+
−
+
− mpω + k
L
L
L
L
L
y −1
4
G1 =
2
Kt ω 2 M K r µ + 2(0)π µ + 2( −1)π
−
+
L
L
L
L
L
K t ω 2 M K r µ + 2(1)π µ + 2( −1)π
H1 =
−
+
L
L
L
L
L
I1 =
Kt ω 2 M K r µ + 2(2)π µ + 2( −1)π
−
+
L
L
L
L
L
2ω 2 ρ 0 j Kt ω 2 M K r µ + 2(0) π
µ + 2(0) π
2
J1 = D
+
−
+
− m pω + k
L
L
L
L
L
y0
4
2
K1 =
Kt ω 2 M K r
−
+
L
L
L
µ + 2(1)π µ + 2(0)π
L
L
L1 =
Kt ω 2 M K r
−
+
L
L
L
µ + 2(2)π µ + 2(0)π
L
L
2ω 2 ρ0 j Kt ω 2 M K r µ + 2(1)π
µ + 2(1)π
2
M1 = D
−
m
ω
+
+
−
+
p
L
ky1
L
L
L
L
4
N1 =
2
Kt ω 2 M K r µ + 2(2)π µ + 2(1) π
−
+
L
L
L
L
L
2 jω 2 ρ 0 K t ω 2 M K r µ + 2(2)π
µ + 2(2)π
2
O1 = D
+
−
+
− m pω + k
L
L
L
L
L
y2
4
P1 = j2ωρ0
209
2
7-3.2. The Transmission Loss (TL) Obtained from the Solution
The power transmission coefficient that is a function of the angle of incidence (θ) can be
obtained as:
τ (θ ) =
It
Ii
(7.31)
where, Ii and It are incident and transmitted normal intensities respectively and are given
by:
Ii =
ωρ 0 k yo
2
(7.32)
and
It =
ωρ0
2
∞
∑C
n
2
Re k yn
(7.33)
n = −∞
The TL is defined as the logarithm of the power transmission coefficient, which will
depend on the incident angle. To estimate the TL for random incidences, the power
transmission coefficient τ (θ ) is averaged according to the Paris formula [41] to obtain the
averaged coefficient τ :
θ lim
τ = 2 ∫ τ (θ )sinθ cos θ dθ
(7.34)
0
where θlim is the limiting angle above which it is assumed that no sound is incident upon
the panel, which is taken as 72o as suggested by Mulholland et al. [30].
Actual
integration of Equation (7.34) was done numerically using the step size of 2o . The
averaged TL is obtained as:
TLavg = 10log10 (1/ τ )
(7.35)
210
The averaged TLs of the unstiffened and stiffened panels are compared in a narrow
band format in Figure 7.2. The simulation conditions used to obtain Figure 7.2 are listed
in Table 7.1. The unstiffened and stiffened panels, on which a plane wave is incident
with an angle 45°, are compared in terms of the TL in a narrow band format as shown in
Figure 7.3.
It is seen that the two TLs in Figures 7.2 and 7.3 show very similar
characteristics. To save related computational work, a single incidence angle 45o is used
for subsequent analyses.
Also, in Figures 7.2 and 7.3, the TLs calculated for the
unstiffened panel are compared with those obtained for the stiffened panel. It is shown
that the effect of the stiffener is pronounced in the low frequency range.
Table 7.1. Dimensions of the panel and simulation conditions
Kt
(N/m)
Kr
(N⋅m/rad)
E
(Pa)
h
(mm)
M
(kg)
3.6×109
60
7.1×1010
ν
ρ
(kg/m3 )
ρ0
(kg/m3 )
0.33
L
(mm)
200
2700
ψ
1°
1.21
c
(m/sec)
343
1.27
θ
0°~72°
φ
0°
0
η
0.1
ω/2π
(Hz)
10 ~ 3,000
211
40
35
30
TL (dB)
25
20
15
10
5
0
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 7.2. Comparison of the predicted averaged TLs between the stiffened and the
unstiffened panels
, W/ stiffener; ---------, W/O stiffener
40
35
30
TL (dB)
25
20
15
10
5
0
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 7.3. Comparison of the predicted TLs between the stiffened and the unstiffened
panels on which a plane wave is incident with an angle 45°
, W/ stiffener; ---------, W/O stiffener
212
7-3.3. Convergence of the Solution
Because the solutions are obtained in series forms, enough terms have to be used in the
calculation to ensure the solutions converge. Once the solution converges at a given
frequency, it can be assumed to converge in all frequencies lower than that.
Therefore,
the necessary number of terms is determined at the highest frequency of interest. A
simple algorithm is used in that the TLs are calculated at the highest frequency of
interest, adding one term to the assumed expansion solution at a time. When the TLs
calculated at two successive calculations are within a pre-set error bound (0.01 dB in this
work), the solution is considered to have converged. The number of coefficients found
this way is used to calculate TL at all other frequencies below this highest frequency of
interest.
37.5
37
36.5
Convergence of TL
TL (dB)
36
35.5
35
34.5
34
33.5
0
5
10
15
20
Mode Number
Figure 7.4. Coefficient convergence diagram for the stiffened panel (t=1.27mm) at 3,000
Hz
213
Figure 7.4 shows how the calculated TL changes as the number of coefficients
increases at the driving frequency of 3,000 Hz. The same data shown in Table 7.1 is used
for the stiffened panel but a single incidence angle of 45° is used.
From the figure, 21
coefficients (n = -10 to 10) are enough to provide a converged solution at 3,000 Hz.
7-4. Parameter Studies
The basic panel dimensions and simulation conditions used in the study are the same as
that is listed in Table 7.1, obviously except the parameter to be studied.
7-4.1. Parameters Related to Modeling
Effect of the Incidence Angle
The TLs calculated for three different incident angles (30o , 45o , 60o ) are plotted in Figure
7.5, which indicates that the transmitted power slightly decreases (TL increases) as the
incidence angle θ decreases. Because the qualitative aspect of the solution does not
change for different angles, the incident angle of 45o is used for all subsequent
calculations, which reduces the related computation time substantially.
Effect of the Phase Attenuation Angle
Figure 7.6 compares the TLs calculated for three different phase attenuation parameters,
0o , 1o and 10o , which are chosen for a very wide range intentionally to see the trend of the
effect. Notice that the choice of the phase attenuation coefficient influences the solution
in a noticeable scale only in the low frequency range, below 100 Hz in this case. The
comparison suggests that using an arbitrarily small value for the attenuation angle will be
acceptable for most purposes. Because the phase attenuation parameter ψ is difficult to
estimate, 1o is used as the angle in all other plots in this study unless it is stated otherwise.
214
45
40
35
TL (dB)
30
25
20
15
10
5
0
1
10
2
3
10
4
10
10
Frequency (Hz)
Figure 7.5. TL curves for the stiffened panel with respect to incidence angle
, θ= 30°; ---------- , θ=45°;
, θ=60°
45
40
35
TL (dB)
30
25
20
15
10
5
0
10
1
2
3
10
4
10
10
Frequency (Hz)
Figure 7.6. TL curves for the stiffened panel with respect to phase attenuation angle
, ψ=0°; ---------- , ψ=1°;
215
, ψ=10°
Loss Factor
The loss factor, which represents the structural damping of the panel, is another
parameter difficult to estimate accurately.
Figure 7.7 compares TLs curves of the
stiffened panel obtained for loss factors of 0, 0.1 and 0.2 compared with the TL curve of
the unstiffened panel. In all other plots in this work, 0.1 is used as the loss factor unless it
is stated otherwise.
45
40
35
30
TL (dB)
25
20
15
10
5
0
-5
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 7.7. TL curves for the stiffened panel with respect to loss factor
, η=0; ---------- , η=0.1;
, η=0.2;
, W/O stiffener (η=0)
7-4.2. Parameter Related to Design
General Effect of Stiffeners
The comparison of the TLs of the stiffened and unstiffened panels in Figure 7.7 reveals
interesting effects of the stiffeners. Let’s use the stiffened shell whose loss factor is 0.1
for the comparison. The figure shows that the positive effect of the stiffeners is restricted
216
largely in the low frequency range, below 110 Hz in this example. In the high frequenc y
range, while the TL curve of the stiffened panel has a slightly higher envelope than that
of the unstiffened panel, it also has several dips. At these frequencies, the stiffener
actually increases the sound transmission. This phenomenon can be explained by the
propagation bands of the periodic structure, which were first described by Heckl [80],
then also by Mead [74]. Free, non-decaying wave propagation in a periodically supported
structure becomes possible only in the frequency bands bounded by the n-th natural
frequency of the hinged single bay of the panel (the lower bounding frequency) and that
of the fully clamped single bay (the upper bounding frequency). Figure 7.7 shows that a
non-propagating band and a propagating band alternate in the stiffened panel. It can be
considered that each dip of the TL curve is caused by the resonance of the single bay of
the panel. In practice, the effect on sound transmission is as though the critical frequency
had been lowered by one or two octaves, the degree of change being dependent upon the
spacing and stiffness (translational and rotational) of the stiffeners [91]. This effect may
be considered when the stiffened plate is designed: for example the spacing may be
determined so that the dip frequencies avoid major excitation frequencies.
Stiffener Mass Effects
TLs calculated when the stiffener has 0%, 10%, 100%, and 200% of the mass of the
panel are plotted in Figure 7.8. The figure indicates that the mass effect caused by the
stiffener has virtually no influence on the TL in the case studied. Figure 7.9 shows the
same comparison, using the TLs calculated for four different stiffener masses but
reducing the trasnslational stiffness of the stiffener drastically, from Kt =3.6×109 N/m to
Kt =1×105 N/m. This indicates that the stiffener mass has to be considered only when the
217
translational spring is very soft, which is not a practical case. Generally, it is considered
that the mass effect of the stiffener will not have to be considered in the analysis.
40
35
30
TL (dB)
25
20
15
10
5
0
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 7.8. TL curves for the stiffened panel with respect to stiffener mass (K t =3.6×109
N/m)
, 0%; ---------- , 10%;
, 100%;
, 200%
40
35
30
TL (dB)
25
20
15
10
5
0
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 7.9. TL curves for the stiffened panel with respect to stiffener mass (K t =1.0×105
N/m)
, 0%; ---------- , 10%;
, 100%;
218
, 200%
Materials
Figure 7.10 shows the effect of the material on the TL of the panel. Materials chosen for
the comparison are steel, aluminum and brass as shown in Table 7.2. The figure shows
that the TL of the brass is comparable with that of the steel in the middle frequencies
ranging from 200 Hz to 2 kHz. Above 2 kHz, the brass will be the most effective material
in the high frequency range. This is as expected because the density of the brass is the
largest, which makes it most effective in the mass controlled high frequency range. The
figure also shows that the aluminum, which has the lowest stiffness, is the least effective
in the low frequency range, which is again as expected because the low frequency range
is controlled by the stiffness. This type of comparison will be useful in practice when the
basic design of a certain structure is to be decided.
Table 7.2. Material properties of the stiffened panel
Density
(ρ: kg/m3 )
Young’s Modulus
(E: Pa)
Poisson’s ratio
(ν)
Steel
Aluminum
Brass
7,750
2,700
8,500
1.9×1011
0.71×1011
1.04×1011
0.3
0.33
0.37
219
50
45
40
35
TL (dB)
30
25
20
15
10
5
0
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 7.10. TL curves for the stiffened panel with respect to plate material
, aluminum; ---------- , steel;
, brass
Panel Thickness
Figure 7.11 shows the effect of the panel thickness on the TL. Changing the thickness
has a broadband effect on TL over the entire range of the frequency. In general, TL
increases more in the low frequency range, or the stiffness controlled region, and less in
the high frequency range, or the mass controlled region. As in the study on the effect of
the stiffener, the structural enhancement is most effective in the low frequency range. In
the high frequency range, other means such as using absorbing materials may be a more
effective solution.
Stiffener Spacing
As shown in Figure 7.12, smaller stiffener spacing substantially increases the TL,
however only in the low frequency range. Decreasing the stiffener spacing increases the
lowest natural frequency of the single bay of panel. Therefore, smaller stiffener spacing
220
may sometimes result in an un-expected effect as shown in the figure in some frequency
ranges. Again, such a situation may be avoided by considering the propagation bands
and bounding frequencies in relation to the excitation frequencies.
60
50
TL (dB)
40
30
20
10
0
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 7.11. TL curves for the stiffened panel with respect to thickness of the panel
, t=0.645 mm; ---------- , t=1.27 mm;
, t=2.54 mm
70
60
TL (dB)
50
40
30
20
10
0
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 7.12. TL curves for the stiffened panel with respect to stiffener spacing
, L=100 mm; ---------- , L=200 mm;
, L=400 mm;
, W/O stiffener
221
Stiffness of the Stiffener
Figure 7.13 shows the effect of the rotational stiffness. The comparison was made for a
very wide range of the rotational stiffness, while the translational stiffness is set to zero.
The figure suggests that the rotational spring almost has no influence on the sound
transmission except in the very low frequency range. This is somewhat expected because
the sound is induced by the transverse motion of the panel.
Figure 7.14 shows the effect of the translational spring stiffness when the stiffener
spacing is L=200 mm. As expected again, the effect is mostly in the low frequency
range. Also the effect of the increase of this parameter becomes saturated after it exceeds
a value that is enough to make the spring virtually a fixed support (7.1×107 N/m in this
case).
35
30
25
TL (dB)
20
15
10
5
0
-5
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 7.13. TL curves for the stiffened panel with respect to rotational stiffness of the
stiffener
,Kr=0 N⋅m/rad;
, Kr =1.2×103 N⋅m/rad; -------, Kr =1.2×104
N⋅m/rad;
, Kr =1.2×105 N⋅m/rad
222
40
35
30
TL (dB)
25
20
15
10
5
0
10
1
10
2
10
3
10
4
Frequency (Hz)
Figure 7.14. TL curves for the stiffened panel with respect to translational stiffness of the
stiffener
,Kt =0 N/m;
, Kt =7.1×105 N/m; --------, Kt =7.1×106 N/m;
7
, Kt =7.1×10 N/m;
, Kt =3.6×109 N/m
7-5. Conclusions
An exact analysis procedure is developed to calculate the sound transmission through the
infinitely long elastic panel stiffened in one direction. The stiffener is modeled as a set of
the lumped mass, rotational and translational springs attached to the panel. The dynamic
equation that describes vibro-acoustic responses of the system is derived using the space
harmonic approach and the virtual energy principle. The interaction between the stiffener
and the panel, and between the panel and the acoustic media are fully considered in the
derivation.
A unique solution procedure is developed by relating the acoustic and
structural space harmonic amplitudes and by considering the boundary conditions
between the plate and the acoustic media. The solution is obtained as a truncated series
of the assumed modes by solving a set of linear equations. A scheme to ensure the
223
convergence of the solution is included in the solution procedure, therefore the series
solution can be considered as an exact solution, which is believed to be the first exact
analytical solution obtained for this type of problem.
Taking the advantage of having an exact solution procedure, the performance of the
stiffened panel as an acoustic barrier is studied in terms of the transmission loss. The
effects of the stiffener and stiffener spacing are discussed in conjunction with the
propagation bands. Parameter studies are conducted for the parameters such as the panel
material, stiffener spacing, size of the stiffener, and thickness of the stiffened panel. The
parameter study demonstrates the value of the analysis developed in this work as a design
tool.
224
Chapter 8 – Sound Transmission through Stiffened Cylindrical Shells
8-1. Introduction
Cylindrical shells stiffened by periodically deployed stiffeners are found in many
practical systems of interest such as in an aircraft fuselage or marine structures. Analysis
of such a system for sound transmission calculation becomes very complicated because
of the presence of stiffeners and the need to model structural, and structure-acoustic
coupling effects. Mead et al. solved free and forced vibration problems of flat panels
with periodic stiffeners using the space harmonic method [73-75].
The analysis
necessary to calculate the sound transmission through a stiffened structure becomes more
complicated because the interactions between the structure and acoustic media have to be
included in the system analysis.
Analytical or numerical studies are found on the
structural responses of periodically stiffened structures [76, 79-81, 84, 85], and
periodically stiffened cylindrical shells [88, 92-103], or sound transmissions through
cylindrical shells [20-29, 45, 71, 72, 103], however no reported work is found that has
obtained analytical solutions for the sound transmission through stiffened shells.
Mead and his collaborators [73-75, 81] investigated the structural responses of
periodically stiffened structures by superposing traveling wave solutions described in
terms of the harmonics of each period of the stiffened structures, which was called the
space harmonics [73].
They studied characteristics of the free and forced vibration
responses of the periodically stiffened structures to the convective acoustic pressure field.
This space harmonics method is adopted in this work to model the structural part of the
system.
225
Structural responses of stiffened cylindrical shells have been studied by some
researchers, although they have focused mostly on numerical methods. Z. Mecitoðlu and
his colleagues [92] studied free vibrations of a stiffened shallow shell using an approach
within the frame of Love’s equation for elastic shells and also by using the finite element
method [93]. He also studied free vibration problems by using a simplified shell theory
within the context of Donnell-Mushtari theory [94].
Egle and Sewall [95] studied the
vibration of othogonally stiffened cylindrical shells with discrete axial stiffeners by using
the Ritz method. Bushnell [96] compared various analytical models to analyze vibration
problems of stiffened shells. Mead and Bardell [97] studied free vibrations of a thin
cylindrical shell with discrete axial and circumferential stiffeners. Mustafa and Ali [98,
99] determined the frequencies of ring stiffened, stringer stiffened and orthogonally
stiffened shell by using super shell finite elements. B. Sivasubramonian et al. [100]
studied free vibration characteristics of longitudinally stiffened curved panels by using
the finite element method. K. Y. Lam and Li Hua [101] investigated the influence of
boundary conditions on the frequency characteristics of a rotating conical shell by using
the Galerkin method. Š. Markuš and D. J. Mead [102, 103] presented the study on
harmonic wave propagation in thick circular orthotropic cylinders and a three- layered
composite thick cylinder, which was carried out within the framework of the complete
three-dimensional theory of elasticity by using Bessel and special Frobenius series.
Sound transmission through various types of cylindrical shells, which however are
not stiffened, has been studied by many investigators [20-29, 45, 71, 72, 104] including
Lee and Kim [45, 71, 72, 104]. Lee and Kim studied exact solutions for the sound
transmissions through cylindrical shells of various cross-sectional structures including a
226
single-walled shell [45], double-walled shell [71], and double-walled shells with a core of
porous layer [104]. In this study, basic modeling and analysis schemes to consider the
structure-acoustic coupling effect are adopted from Lee and Kim’s previous studies [45,
71, 72, 104], while the space harmonic expansion method developed by Mead et al. [73]
is employed to model the effects of periodic stiffeners.
8-2. Formulation of the System Equation
A schematic illustration is shown for a cylindrical shell with periodically deployed
stiffeners in one direction (z-direction in this case) in Figure 8.1. The system is simplified
by three assumptions that have been typically used to calculate the transmission losses
(TLs) of relatively long cylindrical structures [20-29, 45, 71, 72, 104], which are that the
cylinder is infinitely long, the input wave is a plane wave traveling on the plane parallel
to the x-z plane with an incidence angle γ, and the inside cavity is anechoic. This model
approximates the sound transmission problem into an aircraft cabin [20-28] or the
reciprocal of the sound transmission from the hermetic cylindrical machineries running in
the anechoic chamber [72]. As shown in Figure 8.1, the stiffener is modeled by the
combination of the lumped mass M, translational spring Kt and rotational spring Kr. The
shell is characterized by its radius R, wall thickness h, in vacuo bulk mass density ρs, in
vacuo bulk Young’s modulus Es and Poisson’s ratio µs. The acoustic media in the
outside and the inside of the shell are defined by the density and speed of sound: {ρ1 , c1 }
inside and {ρ2 , c2 } outside. Since the structure is spatially periodic, the virtual work done
by only one bay element (including supports) needs to be considered.
227
Incident Sound
Z
γ
R
Y
θ
Stiffening ring
X
L
Z
Shell
M
Kr
X
Kt
Stiffeners have mass, translational and rotational stiffness
Figure 8.1. Schematic representation of a stiffened shell
228
8-2.1. Assumed Solutions
Considering the periodic nature of the structure [73-75] and the assumed mode of the
cylindrical shell [39], the shell displacements can be expressed in terms of a series of
space harmonics as follows.
∞
w1 ( z ,θ , t ) = ∑
+∞
∑w
0
1 nm
−j
cos[ nθ ]e
µ +2 mπ
z
L
e jω t
(8.1)
n =0 m =−∞
∞
u1 ( z ,θ , t ) = ∑
+∞
∑u
0
1 nm
cos[nθ ]e
−j
µ + 2mπ
z
L
e jω t
(8.2)
e jω t
(8.3)
n =0 m =−∞
∞
v1 ( z ,θ, t ) = ∑
+∞
∑
v10nm sin[ nθ ]e
−j
µ + 2mπ
z
L
n = 0 m =−∞
In Equations (8.1) to (8.3), w1 (z,θ,t), u1 (z,θ,t) and v1 (z,θ,t) are the displacements of the
shell in the transverse, longitudinal and circumferential directions, n is the
circumferential mode numbers, m is the space harmonic numbers, L is the spacing of
stiffeners, and µ is the characteristic propagation constant defined as:
µ
= k1z − jψ
L
(8.4)
where, k1 z = k1 sin(γ ) , k 1 =
ω
, and ψ is the phase attenuation coefficient. Expanding the
c1
solutions in terms of space harmonics as in Equations (8.1) to (8.3) is assuming that each
subsection of the shell will have the same type of motion with phase delays [73].
The input harmonic plane wave pI shown in Figure 8.1 can be described in the
cylindrical coordinate system as [40]:
∞
p I ( r , z ,θ, t ) = p0 ∑ ε n ( − j )n J n (k1 r r )cos [nθ ] e − jk1 z ze jω t
n =0
229
(8.5)
where, p0 is the amplitude of the incident wave, j = − 1 , n=0,1,2,3,…, Jn is the Bessel
function of the first kind of order n, ε o =1 for n=0 and ε n =2 for n=1,2,3,….
,
k1 r = k1 cos(γ ) . It is easily seen that k1r = k12 − k12z .
The waves radiated from the shell to the outside p1R and into the cavity p2T will have
the same periodic characteristic s as the structural wave in the shell, therefore can be
represented as:
∞
p1R (r , z ,θ, t ) = ∑
∞
∑
n = 0 m =−∞
∞
p (r , z ,θ, t ) = ∑
T
2
p1RnmH n2 (k1r r)cos [ nθ ]e
∞
∑
p
n =0 m =−∞
T
2 nm
−j
µ+ 2 mπ
z
L
−j
µ+ 2 mπ
z
L
H (k 2r r )cos [nθ ]e
where, k 2 r = k 22 − k 22z , k 2 =
1
n
e j ωt
e j ωt
(8.6)
(8.7)
ω
and H 1n and H n2 are the Hankel functions of the first and
c2
second kind of order n, respectively. The former represents the incoming wave and the
second the outgoing wave.
8-2.2. Boundary Conditions at the Structure-Acoustic Interfaces
The modal amplitudes of the reflected and transmitted acoustic waves can be related to
the modal amplitudes of flexural wave in the shell by applying the boundary conditions
on the internal and external shell surfaces (i.e., r=R). The conditions are from the
continuity of the transverse velocities, which are [38]:
∂ ( p I + p1R )
∂r
= − ρ1
∂p2T
∂2 w
= − ρ2 2 1
∂r
∂t
∂ 2 w1
∂t 2
at r = R
(8.8)
at r = R
(8.9)
230
Substituting the assumed solutions in Equation (8.1) and Equations (8.5) to (8.7) into
Equation (8.8) yields the relationship between the modal amplitudes of the reflected wave
and the modal amplitudes of flexural wave in the shell as:
∞
∞
∑∑
n = 0 m =−∞
p1Rnm H n2 ' (k1 rR )k1 r cos[nθ ]e
−j
µ + 2mπ
z
L
∞
=ρ1ω 2 ∑
n= 0
∞
∑
m =−∞
w10nm cos[nθ ]e
−j
∞
µ + 2 mπ
z
L
− p0 ∑ ε n ( − j ) J ( k1r R ) k1 r cos[nθ ]e
n
'
n
n= 0
where, ( ) ' =
−j
kz
z
L
(8.10)
d
.
dr
Similarly, substituting the equations to Equation (8.9) yields the relationship between the
modal amplitudes of the transmitted wave on the shell and the modal amplitudes of
flexural wave in the shell is identified as:
∞
∞
∑∑
T
2 nm
p
H ( k 2 r R) k 2 r cos[ nθ ]e
1'
n
−j
µ + 2mπ
z
L
=ρ2ω
n = 0 m =−∞
∞
2
∞
∑∑w
0
1nm
cos[ nθ ]e
−j
µ + 2 mπ
z
L
(8.11)
n = 0 m =−∞
At each circumferential mode n, the relationship between the modal amplitudes of the
reflected waves on the shell and the modal amplitudes of flexural wave in the shell is
identified from Equation (8.10) as:
p1Rnm =
ρ1ω 2 w10nm
p0ε n ( − j )n J n' (k 1 r R )k1 r
−
H n2 ' ( k1rR )k 1r
H n2' ( k1rR )k 1 r
ρ1ω 2 w10nm
= 2'
H n ( k1rR )k 1r
for m=0
(8.12.a)
for m ≠ 0
(8.12.b)
Notice that µ / L ≅ k z ( ω ) is assumed to obtain Equation (8.12.a) from Equation (8.10) for
m = 0 case, which will be a very close approximation because the phase attenuation
coefficient ψ is small and the equation is derived for one span of the structure.
231
From Equation (8.11), the relationship between the modal amplitudes of the
transmitted waves on the shell and the modal amplitudes of flexural wave in the shell is
identified as:
p2Tnm =
ρ 2ω 2 w10nm
H 1'n ( k 2 r R) k 2 r
(8.13)
8-2.3. Equations of Motion of the System
The equations of motion of the system can be derived based on the principle of virtual
work following the procedure used by Mead et al. [73-75], which states that the virtual
displacements applied on the system should not do any work.
The Love’s equation [39] is used to describe the equations of motion of the
cylindrical shell in the axial, circumferential and transverse directions. They are:
∞
∞
∑
∑u
n =0 m =−∞
0
1nm
µ +2 mπ
µ + 2mπ 2 K s (1 − µs ) 2
−j
z
2
L
+
n
−
ρ
h
ω
cos[
n
θ
]
e
Ks
s
L
2R2
+j
K s (1 + µ s ) ∞ ∞ 0 µ + 2mπ
∑
∑ v1nmn L
2R
n =0 m =−∞
+j
K s µs ∞ ∞ 0 µ + 2mπ
∑ ∑ w1nm L
R n =0 m =−∞
−j
cos[nθ ]e
−j
cos[nθ ]e
Ds µ + 2 mπ 2
2 µs
n
∞
∞
R
L
0
u1nm
∑
∑
n= 0 m=−∞
K (1 − µs ) µ + 2mπ
− j s
2R
L
µ +2 mπ
z
L
µ +2 mπ
z
L
=0
µ + 2 mπ
sin[nθ ]e − j L z
K s µ + 2mπ
n + R µs
n
L
K s (1 − µ s ) µ + 2mπ 2 K s 2
+
n
2
µ + 2 mπ
∞
∞
2
L
R
sin[nθ ]e − j L z
+ ∑ ∑ v10nm
D (1 − µ ) µ + 2mπ 2 D
n = 0 m =−∞
− s n 2 − ρ hω 2
s
+ s
s
4
2 R2
L
R
∞
+∑
∞
∑w
n = 0 m =−∞
0
1nm
(8.14)
µ + 2m π
Ks
Ds (1 − µ s ) µ + 2 mπ 2 Ds 3
−j
z
L
=0
2 n+
+ 4 n sin[nθ ]e
2
R
L
R
R
232
(8.15)
µ + 2mπ Ds µ + 2mπ 2
K
µ + 2mπ
u10nm Ds
+ 2 µs
n − j s µs
∑
∑
L
L
R
L
R
n= 0 m=−∞
∞
4
∞
2
−j
cos[nθ ]e
µ + 2m π
z
L
Ds µ + 2mπ 2
−
µ
n
s
2
µ + 2mπ
∞
∞
R
L
cos[nθ ]e− j L z
+ ∑ ∑ v10nm
D (1 − µ ) µ + 2mπ 2
n = 0 m =−∞
Ds 3 Ks
+ s 2 s
n
−
n + 2 n
R
L
R4
R
Ds µs µ + 2 mπ 2 2 2Ds (1 − µs ) µ + 2 mπ 2 2
µ + 2 mπ
2
∞
∞
n +
2
n
−j
z
0
R
L
R
L
+ ∑ ∑ w1nm
cos[nθ ]e L
Ds 4 Ks
n = 0 m =−∞
2
+ 4 n + 2 − ρs hω
R
R
− p0
∞
∑ ε (− j ) J
n
n
n (k 1r R)cos[nθ ]e
µ
−j z
L
m=−∞
∞
∞
−∑
∞
∑p
R
1nm
H n2 (k1r R)cos[nθ ]e
−j
(8.16)
µ + 2 mπ
z
L
n= 0 m=−∞
∞
+ ∑ ∑ p2Tnm H 1n (k 2 r R)cos[nθ ]e
−j
µ + 2 mπ
z
L
=0
n = 0 m =−∞
where, Ks and Ds are the membrane and bending stiffness of the shell defined as [39]:
Ks =
Es (1 − jηs )h
1 − µ s2
(8.17)
Ds =
Es (1 − jηs ) h3
12 (1 − µ s2 )
(8.18)
where, ηs is the shell loss factor [67].
Because the inertia terms are represented in the
D’Alembert’s equivalent forces in Equations (8.14) to (8.16), the equations represent the
equivalent forces acting on the shell of unit length and circumferential in three directions.
From the assumed displacements in Equations (8.1)-(8.3), the virtual displacement
can be expressed as any one of the sets of three displacements in the transverse,
circumferential and longitudinal directions described as follows.
−j
δ wpq = δ w cos[ qθ ]e
δ v pq = δ v sin[ qθ ]e
−j
µ +2 pπ
z
L
µ + 2 pπ
z
L
e jω t
(8.19.a)
e jω t
(8.19.b)
233
δ u pq = δ u cos[ qθ ]e
−j
µ + 2 pπ
z
L
e jω t
(8.19.c)
Because the complex algebra is used, the conjugate forms of the virtual displacements are
multiplied to the equivalent forces to calculate the virtual work.
The virtual work done by the cylindrical shell alone (without stiffeners) is obtained
first.
The work done by each virtual displacement is obtained by integrating the
multiplication of the virtual displacement (conjugate form) and the equivalent force in the
corresponding direction. The work done by the longitudinal virtual displacement is
obtained:
µ + 2 mπ 2
Ks
L
2 π L ∞ ∞
K (1 − µs ) 2 − j µ +2 mπ z j µ +2 pπ z
L
δ Π su = δ u * ∫ R ∫ ∑ ∑ u10nm + s
n e
e L cos[nθ ]cos[qθ ]dzdθ
2
2
R
n
=
0
m
=−∞
0
0
− ρ s hω 2
+j
K s (1 + µs )
2R
Kµ
+j s s
R
2π
2π
µ + 2mπ
v10nm n
∑
L
n =0 m =−∞
L ∞
∫0
R∫ ∑
L ∞
∞
0
∞
∑w
∫0 R∫0 ∑
n =0 m =−∞
0
1 nm
µ + 2mπ − j
e
L
−j
e
µ +2 mπ
z
L
µ +2 mπ
z
L
e
e
µ +2 pπ
j
z
L
µ +2 pπ
j
z
L
cos[nθ ]cos[qθ ]dzdθ
cos[nθ ]cos[qθ ]dzdθ
The work done by the circumferential virtual displacement becomes:
234
(8.20)
Ds µ + 2 mπ 2
n
2 µs
L
R
2π L ∞ ∞ 0 K (1 − µ ) µ + 2mπ − j µ+ 2mπ z jµ + 2 pπ z
*
s
s
L
δΠsv = δ v ∫ R ∫ ∑ ∑ u1nm
e L sin[nθ ]sin[ qθ ]dzdθ
n e
2
R
L
− j
0 0 n =0 m =−∞
K s µ + 2 mπ
+ µs
n
L
R
K s (1 − µs ) µ + 2 mπ 2 K s 2
+ R2 n
2
L
2π
L ∞
D (1 − µ ) µ + 2mπ 2
− j µ + 2mπ z jµ + 2 pπ z
∞
e L e L sin[nθ ]sin[ qθ ]dzdθ
+ ∫ R ∫ ∑ ∑ v10nm + s 2 s
2
R
L
n
=
0
m
=−∞
0
0
D 2
− 4s n − ρs hω 2
R
Ks
Ds (1 − µs ) µ + 2 mπ 2
2 n+
− j µ+ 2 mπ z j µ+ 2 pπ z
2
R
L
e L e L sin[nθ ]sin[qθ ]dzdθ
+ ∫ R ∫ ∑ ∑ w10nm R
Ds 3
0
0 n= 0 m=−∞
+
n
4
R
2π
L ∞
∞
(8.21)
The work done by the transverse virtual displacement becomes:
µ + 2 mπ 4
Ds
L
2
2 π L ∞ ∞
D
− j µ +2 mπ z j µ +2 pπ z
µ
+
2
m
π
δΠsw = δ w* ∫ R∫ ∑ ∑ u10nm + 2s µs
n2 e L e L cos[ nθ ]cos[ qθ ]dzdθ
R
L
0 0 n =0 m=−∞
Ks µ + 2mπ
− j µ s
L
R
Ds µ + 2mπ 2
− 2 µ s
µ +2 mπ µ +2pπ
n
2π
L ∞
∞
R
L
e− j L ze j L z cos[nθ ]cos[ qθ ]dzd θ
+ ∫ R∫ ∑ ∑ v10n m
2
D (1 − µ ) µ + 2 mπ
D
K
0
0 n =0 m=−∞
+ s 2 s
n − 4s n 3 + 2s n
R
L
R
R
Ds µs µ + 2mπ 2 2
2
n
R
L
2π
L ∞
∞
2 D (1 − µ ) µ + 2mπ 2 − j µ +2 mπ z jµ +2pπ z
n 2 e L e L cos[nθ ]cos[ qθ ]dzd θ
+ ∫ R∫ ∑ ∑ w10n m + s 2 s
R
L
n
=
0
m
=−∞
0
0
D
K
+ 4s n4 + 2s − ρ shω 2
R
R
µ
∞
−j z
n
L
p0 ∑ ε n ( − j ) Jn ( k1rR )e
m=−∞
2π
L
µ+ 2 mπ µ+ 2 pπ
∞ ∞ R 2
−j
z
j
z
− ∫ R∫ +∑ ∑ p1 nmH n (k 1r R)e L e L cos[ nθ ]cos[qθ ]dzdθ
0
0 n =0 m=−∞
µ + 2 mπ
∞ ∞ T
−j
z
−∑ ∑ p2 nmH n2 (k1rR )e L
n =0 m=−∞
235
(8.22)
The virtual work done by the translational spring is obtained as:
2π
∞
δΠ t = Kt w1 (0,θ , t ) ⋅ δ w* pq = δ w* Kt ∫ R ∑
∞
∑w
n =0 m =−∞
0
0
1nm
cos[nθ ]cos [qθ ]dθ
(8.23)
All three virtual displacements contribute to the work done by the rotational spring,
therefore,
µ + 2 pπ ′
δ Π r = jK r
u1 (0,θ, t )δ u * pq + v 1′(0,θ, t )δ v* pq + w1′(0,θ, t) δ w* pq
L
(
2π
∞
∞
2π
∞
∞
µ + 2 pπ
0 µ + 2 mπ
= δ u * K r
∫ R ∑ ∑ u1 nm
L
L
0 n =0 m =−∞
µ + 2 pπ
+δ v* K r
L
)
cos[nθ ]cos[ qθ ]d θ
(8.24)
µ + 2mπ
sin[nθ ]sin[qθ ]dθ
L
∑ v1nm
∫0 R ∑
n =0 m =−∞
0
2π
∞
∞
µ + 2 pπ
0 µ + 2 mπ
+δ w* K r
R
∫ ∑ ∑ w1 nm
L
L
0 n =0 m =−∞
cos[nθ ]cos[qθ ]d θ
The virtual work done by the lumped mass becomes:
2π
∞
δΠ M = − ω2 Mw1 (0,θ , t )δ w* pq = −ω 2 M δ w* ∫ R∑
0
∞
∑w
n =0 m =−∞
0
1nm
cos[nθ ]cos[ qθ ]dθ
(8.25)
Notice that the coordinate of z=0 and r=R is used to calculate the work done by the
springs and mass because only one set has to be considered.
The virtual work principle requires that
δΠ su + δΠ sv + δΠsw + δΠ t + δΠ r + δΠ M = 0
(8.26)
Because of the othogonality property of the trigonometric functions, integrating
Equations (8.20) to (8.26) with respect to z and θ yields:
236
When p=0:
µ + 2 pπ 2 K s (1 − µ s ) 2
+
n
0 K s
K
(1
+
µ
)
µ
+
2
p
π
s
+ v10np s
n
j
L
2 R2
u1 np
2R
L
*
− ρ s hω 2
δu
∞
µ + 2mπ µ + 2 pπ
0
+ w 0 K s µ s µ + 2 pπ j + K r
u
1n m
1np R
L
L m∑
L
L
=−∞
K s (1 − µ s ) µ + 2 pπ
n
2
2R
L
u 0 Ds µ µ + 2 pπ n − j
1np R 2 s
L
K
µ
+
2
p
π
n
s
+ R µs
L
2
2
K s (1 − µ s ) µ + 2 pπ
D (1 − µ s ) µ + 2 pπ
K
+ 2s n 2 + s
0
2
L
R
2 R2
L
δ v*
+ +v1np
D
2
s 2
− 4 n − ρ shω
R
2
D s (1 − µ s ) µ + 2 pπ D s 3
0 Ks
+ R4 n
2
+w1np R 2 n +
R
L
K r ∞ 0 µ + 2mπ µ + 2 pπ
v1nm
+
∑
L
L
L m=−∞
µ + 2 pπ 4 Ds µ + 2 pπ 2 2
+ 2 µs
n
Ds
L
R
L
u 0
1np K
µ
+
2
p
π
s
µs
− j
R
L
2
2
+v 0 − D s µ µ + 2 pπ n + D s (1 − µ s ) µ + 2 pπ n − Ds n 3 + K s n
1np R 2 s
L
R2
L
R4
R 2
D s µ s µ + 2 pπ 2 2 2 Ds (1 − µ s ) µ + 2 pπ 2 2
n +
n
2
2
L
R
L
R
0 D 4 K
*
+ +w1np + 4s n + 2s − ρ s hω 2
δ w
R
R
H n2 (k1 r R )ρ1ω 2 H 1n (k 2 rR )ρ 2ω 2
− H 2 ' (k R )k + H 1' (k R )k
n
1r
1r
n
2r
2r
2
∞
∞
+ K t − ω M ∑ w10nm + K r ∑ w10nm µ + 2mπ µ + 2 pπ
L
L m =−∞
L m =−∞
L
L
n
2
'
− p ε ( − j )n J ( k R ) − ε n ( − j ) H n (k 2 r R ) J n (k 1 r R ) k1 r
n
1r
2'
0 n
H
(
k
R
)
k
n
1r
1r
=0
237
(8.27)
When p ≠0:
µ + 2 pπ 2 K s (1 − µ s ) 2
+
n
0 K s
K
(1
+
µ
)
µ
+
2
p
π
s
+ v10np s
n
j
L
2R 2
u1 np
2R
L
*
− ρ s hω 2
δ u
∞
µ + 2mπ µ + 2 pπ
0
+ w0 K s µ s µ + 2 pπ j + K r
u1 nm
∑
1 np R
L
L
L
L
m =−∞
Ks (1 − µ s ) µ + 2 pπ
n
2
2R
L
u 0 Ds µ µ + 2 pπ n − j
1 np R 2 s
L
K s µ + 2 pπ
n
+ R µs
L
2
2
K s (1 − µ s ) µ + 2 pπ K s 2 Ds (1 − µ s ) µ + 2 pπ
+ R2 n +
+v 0
2
L
2 R2
L
δ v*
+ 1 np
Ds 2
2
− 4 n − ρs hω
R
2
Ds (1 − µ s ) µ + 2 pπ Ds 3
0 Ks
+ 4n
2
+w1np R2 n +
R
L
R
Kr ∞ 0 µ + 2mπ µ + 2 pπ
v1 nm
+
∑
L
L
L m =−∞
µ + 2 pπ 4 Ds µ + 2 pπ 2 2
+ 2 µs
n
Ds
L
R
L
u 0
1 np K
µ
+
2
p
π
s
−
j
µ
R s
L
2
2
+v 0 − Ds µ µ + 2 pπ n + Ds (1 − µ s ) µ + 2 pπ n − Ds n3 + Ks n
1 np R 2 s
L
R2
L
R4
R2
2
2
δ w*
D µ µ + 2 pπ 2 2 Ds (1 − µ s ) µ + 2 pπ 2
+
n +
n
s2 s
2
R
L
R
L
+w10np + D4s n 4 + K2s − ρ s hω 2
R
R
2
2
1
2
Hn ( k1 r R) ρ1ω
H n (k 2 r R ) ρ 2 ω
−
+
H 2 ' ( k R )k
1'
H
(
k
R
)
k
n
1r
1r
n
2r
2r
K r ∞ 0 µ + 2mπ µ + 2 pπ
Kt ω 2 M ∞ 0
+ L − L ∑ w1 nm + L ∑ w1 nm
L
L
m =−∞
m =−∞
=0
238
(8.28)
Because δ u* ,δ v* and δ w* in Equations (8.27)-(8.28) are virtual displacements, which
are arbitrary, each coefficient of the three virtual displacements should become zero for
the equations to be satisfied. Thus, we obtain three equations of motion for each n, n
=0,1,2,3,…, and for m, p, where p = 0, ± 1, ± 2, ± 3, ± 4, ….:
µ + 2 pπ 2 K s (1 − µs ) 2
n 0 Ks (1 + µs ) µ + 2 pπ
K
+
u10np s
n
L
2R2
+ v1 np
2R
L
2
− ρ hω
s
K µ µ + 2 pπ
K r ∞ 0 µ + 2 mπ µ + 2 pπ
+ w10np s s
j
+
∑ u1nm L
=0
R
L
L m=−∞
L
2
D
µ + 2 pπ
u10np 2s µ s
n−
L
R
K (1 − µ s ) µ + 2 pπ
j s
2R
L
j
(8.29)
n + K s µ µ + 2 pπ n
s
R
L
K (1 − µ s ) µ + 2 pπ 2 K s 2 Ds (1 − µ s ) µ + 2 pπ 2 Ds 2
2
+ v10np s
+ 2n +
− 4 n − ρs hω
2
2
L
R
2R
L
R
+w
0
1np
2
Ks
Ds (1 − µs ) µ + 2 pπ
Ds 3 K r
2 n+
+ 4 n +
2
R
L
R
R
L
∞
∑v
0
1 nm
m=−∞
(8.30)
µ + 2 mπ µ + 2 pπ
= 0
L
L
µ + 2 pπ 4 Ds µ + 2 pπ 2 2
Ds
+ R2 µs
n
L
L
0
u1np
K s µ + 2 pπ
µs
− j
R
L
2
2
D
µ + 2 pπ
Ds (1 − µ s ) µ + 2 pπ
D
K
+v10np − 2s µs
n
+
n − 4s n 3 + 2s n
2
L
R
L
R
R
R
Ds µ s µ + 2 pπ 2 2 2 Ds (1 − µ s ) µ + 2 pπ 2 2
2
n +
n
L
R2
L
R
D
K
+ w10np + 4s n 4 + 2s − ρ shω 2
R
R2
H n ( k1 r R) ρ1ω 2 H 1n (k 2 rR) ρ2 ω2
− H 2 ' (k R )k + H 1' (k R ) k
n
1r
1r
n
2r
2r
K ω 2M ∞ 0
K ∞
µ + 2mπ µ + 2 pπ
+ t −
w1nm + r ∑ w10nm
∑
L m =−∞
L m =−∞
L
L
L
n
2
'
ε ( − j) H n (k 2 r R ) Jn (k 1r R )k1r
= p0 ε n (− j )nJ n (k 1rR ) − n
H n2' (k1rR )k1r
when p=0 (8.31.a)
=0
when p≠0
239
(8.31.b)
0
This constitutes a set of simultaneous equation for u10np , v10np and w1np
. Consideration of
the virtual work in any other shell element would yield an identical set of equation.
Finite terms have to be taken to solve Equations (8.29) - (8.31).
In the actual
calculation of this work, p = -5 to 5, 11 space harmonic terms, were used after the
convergence consideration. Also notice that the series of m in Equations (8.29) – (8.31)
was also summed from m = -5 to m = 5 in this case. For the purpose of easier
explanation, a 3 term space harmonic solution case is explained. In this case p = -1, 0,
and 1, and the space mode is summed for m = -1, 0, 1. This yields nine simultaneous
equations for w10n −1,w01n 0, w10n1, u10n −1, u01n 0, u10n 1 , v10n−1, v10n0 and v10n1 at each n. The 9 equations
for each n is summarized as follows.
Three equations for p = -1:
µ + 2(−1)π 2 K s (1 − µ s ) 2
n 0 K s (1 + µ s ) µ + 2(−1)π
K
+
u10n −1 s
n
L
2 R2
j
+ v1n −1
2
R
L
− ρs hω 2
0 µ + 2(−1)π 2
µ + 2(0)π µ + 2(−1)π
+ u10n0
u1 n−1
K µ µ + 2( −1)π
Kr
L
L
L
+ w10n −1 s s
j
+
R
L
L
0 µ + 2(1) π µ + 2( −1)π
+u1n1
L
L
Ds µ + 2( −1)π 2
u 2 µs
n−
L
R
K (1 − µ s ) µ + 2( −1)π
+ v10n −1 s
2
L
0
1n −1
= 0
K (1 − µ s ) µ + 2( −1)π
K
µ + 2( −1)π
j s
n + s µs
n
2R
L
R
L
2
2
K s 2 Ds (1 − µ s ) µ + 2(−1)π Ds 2
+
n
+
− 4 n − ρs hω 2
R2
2
2R
L
R
2
K
D (1 − µ ) µ + 2(−1)π D s 3
+ w10n −1 2s n + s 2 s
+ R4 n
R
L
R
0 µ + 2(−1)π 2
µ + 2(0)π µ + 2(−1)π
+ v10n0
v1 n−1
Kr
L
L
L
=0
+
L
µ + 2(1)π µ + 2(−1)π
+v10n1
L
L
240
(8.32)
(8.33)
µ + 2(−1)π 4 Ds µ + 2(− 1)π 2 2
Ds
+ R2 µ s
n
L
L
0
u1n −1
K s µ + 2( −1)π
µs
− j
R
L
2
2
D
µ + 2(−1)π
Ds (1 − µ s ) µ + 2(−1)π
D
K
+v10n −1 − 2s µ s
n
+
n − 4s n3 + 2s n
2
L
R
L
R
R
R
Ds µ s µ + 2(−1)π 2 2 2 Ds (1 − µ s ) µ + 2( −1)π 2 2
2
n +
n
L
R2
L
R
D
K
+ w10n− 1 + 4s n 4 + 2s − ρ shω 2
R
R
H n2 ( k1 r R )ρ1ω 2 H 1n (k 2 rR )ρ2ω 2
− H 2 ' (k R )k + H 1' (k R ) k
n
1r
1r
n
2r
2r
K ω 2M 0
0
0
+ t −
w1n −1 + w 1n 0 + w1n 1
L
L
(
)
0 µ + 2( −1)π 2
0 µ + 2(0)π µ + 2( − 1)π
w
+ w1n 0
Kr 1n− 1
L
L
L
+
L 0 µ + 2(1)π µ + 2( −1)π
+ w1n1
L
L
=0
(8.34)
Three equations for p=0:
µ + 2(0) π 2 Ks (1 − µs ) 2
n 0 Ks (1 + µs ) µ + 2(0)π
K
+
u10n0 s
n
L
2R2
j
+ v1 n0
2R
L
2
− ρ s hω
+ w10n 0
0 µ + 2( −1)π µ + 2(0)π 0 µ + 2(0)π 2
u
+ u1n 0
K s µs µ + 2(0)π
K r 1n −1
L
L
L
j
+
=0
R
L
L
0 µ + 2(1)π µ + 2(0) π
+ u1n 1
L
L
241
(8.35)
Ds µ + 2(0)π 2
K (1 − µs ) µ + 2(0)π
K
µ + 2(0)π
u 2 µs
n − j s
n + s µs
n
L
2R
L
R
L
R
K (1 − µ s ) µ + 2(0)π 2 K s 2 Ds (1 − µ s ) µ + 2(0)π 2 D s 2
+ v10n 0 s
+ 2n +
− 4 n − ρ s hω 2
2
2
L
R
2R
L
R
0
1n 0
2
Ks
Ds (1 − µ s ) µ + 2(0)π
D
+w 2 n +
+ 4s n 3
2
R
L
R
R
0 µ + 2(−1)π µ + 2(0)π 0 µ + 2(0)π 2
v
+ v1 n0
K r 1 n−1
L
L
L
= 0
+
L
µ + 2(1)π µ + 2(0)π
+v10n1
L
L
(8.36)
0
1n 0
µ + 2(0)π 4 Ds µ + 2(0)π 2 2
Ds
+ 2 µs
n
L
R
L
0
u1n 0
K s µ + 2(0)π
µs
− j
R
L
+v
0
1n 0
2
Ds µ + 2(0)π 2
Ds (1 − µ s ) µ + 2(0)π
Ds 3 K s
− 2 µ s
n+
n − 4 n + 2 n
2
L
R
L
R
R
R
Ds µs µ + 2(0)π 2 2 2Ds (1 − µs ) µ + 2(0)π 2 2
2
n +
n
L
R2
L
R
Ds 4 K s
0
2
+ w1n 0 + 4 n + 2 − ρs hω
R
R
H n2 (k1r R )ρ1ω 2 H 1n (k 2 rR )ρ 2ω 2
− H 2 ' (k R )k + H 1' (k R )k
n
1r
1r
n
2r
2r
K ω 2M 0
0
0
+ t −
w1n −1 + w 1n 0 + w1n 1
L
L
(
)
2
0 µ + 2( −1)π µ + 2(0)π
0 µ + 2(0)π
+
w
w1n− 1
1n 0
Kr
L
L
L
+
L 0 µ + 2(1)π µ + 2(0)π
+ w1n1
L
L
ε (− j )n H n2 (k 2 rR )Jn' (k 1r R ) k1r
= p0 ε n (− j )n J n ( k1r R ) − n
H n2' (k1rR )k1r
Three equations for p=1:
242
(8.37)
µ + 2(1)π 2 K s ( 1 − µ s ) 2
n 0 K s (1 + µ s ) µ+ 2(1) π
K
+
u10n1 s
L
2R 2
n
j
+ v1n1
2R
L
− ρ shω 2
0 µ + 2(−1)π µ + 2(1)π
u1 n−1
L
L
K µ µ + 2(1)π
K
µ + 2(0)π µ + 2(1)π
+ w10n1 s s
j + r + u10n 0
=0
R
L
L
L
L
2
+u10n1 µ + 2(1)π
L
2
D
µ + 2(1)π
u10n1 2s µ s
n−
L
R
K (1 − µ s ) µ + 2(1)π
j s
2R
L
Ks
n + R µs
µ + 2(1)π
L
(8.38)
n
K (1 − µs ) µ + 2(1)π 2 Ks 2 Ds (1 − µs ) µ + 2(1)π 2 Ds 2
+v10n 1 s
+ 2n +
− 4 n − ρs hω 2
2
2
L
2R
L
R
R
K
D (1 − µ ) µ + 2(1)π 2 Ds 3
+w s2 n + s 2 s
+ R4 n
R
L
R
(8.39)
0
1n1
0 µ + 2(−1)π µ + 2(1)π 0 µ + 2(0)π µ + 2(1)π
+ v1n 0
v1n−1
L
L
L
L
Kr
+
=0
2
L
0 µ + 2(1)π
+v1n1
L
µ + 2(1)π 4 Ds µ + 2(1)π 2 2
Ds
+ R2 µs
n
L
L
0
u1n1
K s µ + 2(1) π
µs
− j
R
L
2
2
D
µ + 2(1)π
Ds (1 − µ s ) µ + 2(1)π
D
K
+v10n1 − 2s µs
n
+
n − 4s n3 + 2s n
2
L
R
L
R
R
R
Ds µs µ + 2(1)π 2 2 2 Ds (1 − µs ) µ + 2 (1)π 2 2
2
n +
n
L
R2
L
R
D
K
+w10n1 + 4s n 4 + 2s − ρs hω 2
R
R2
2
1
2
H n (k1r R) ρ1ω
H n (k 2rR ) ρ2ω
− H 2 ' (k R )k + H 1' (k R )k
n
1r
1r
n
2r
2r
K ω2 M 0
0
0
+ t −
( w1n −1 + w1n 0 + w1n 1 )
L
L
0 µ + 2(−1)π µ + 2(1)π
w1n−1
L
L
Kr
+
2
L 0 µ + 2(1) π
+w1n1
L
0 µ + 2(0)π µ + 2(1)π
+ w1n 0
L
L
243
=0
(8.40)
The Equations (8.32) –(8.40) can be put into a matrix equation in Equation (8.41).
A1
F1
K1
P1
0
0
E 2
0
0
B1 C1 D1 0
0
G1 H 1 0
I1
0
L1 M 1 0
0
N1
0
0 Q1 R1 S 1
U1 0 V 1 W1 X 1
0
Z1 A 2 B 2 C 2
0
0 F2 0
0
J2 0
0 K2 0
0 P2 0
0 Q2
0 u1 n −1 0
0 u1n 0 0
O1 u1 n1 0
0 v1n − 1 0
0 v1n 0 = 0
D2 v1 n1 0
I 2 w1 n −1 0
N2 w1 n 0 O2
T 2 w1 n1 0
E1 0
0
J1
0
0
T1
0
0
Y1
0
0
G2 H 2
L2 M2
R2 S 2
where,
2
2
µ + 2( −1)π Ks (1 − µ s ) 2
K r µ + 2( −1)π
2
A1 = Ks
+
n
−
ρ
h
ω
+
s
L
2R2
L
L
B1 =
K r µ + 2(0)π µ + 2(−1)π
L
L
L
Kr µ + 2(1)π µ + 2(−1)π
,
C1 = L
L
L
D1 =
K s (1 + µ s ) µ + 2(− 1)π
K µ µ + 2( −1)π
n
j , E1 = s s
j
2R
L
R
L
F1 =
K r µ + 2( −1)π µ + 2(0)π
L
L
L
K µ + 2(0)π
µ + 2(0)π K s (1 − µs ) 2
G1 = K s
+
n − ρ shω 2 + r
2
L
2R
L
L
2
H1 =
K r µ + 2(1)π µ + 2(0) π
L
L
L
Ks (1 + µs ) µ + 2(0)π
,
n
I1 =
2R
L
J1 =
K µ + 2( −1)π µ + 2(1)π
Ks µs µ + 2(0)π
j , K1 = r
R
L
L
L
L
L1 =
K r µ + 2(0)π µ + 2(1) π
L
L
L
j
K µ + 2(1)π
µ + 2(1)π Ks (1 − µs ) 2
M1 = K s
+
n − ρ s hω 2 + r
2
L
2R
L
L
2
244
2
2
(8.41)
N1 =
Ks (1 + µs ) µ+ 2(1) π
Kµ
n
j , O1 = s s
2R
L
R
P1 =
Ds µ + 2(−1)π
µ
n−
R 2 s
L
µ + 2(1)π
j
L
K (1 − µ s ) µ + 2(− 1)π
j s
2R
L
2
n + K s µ µ + 2(−1)π
R s
L
K (1 − µ s ) µ + 2(−1)π
K s 2 Ds (1 − µ s ) µ + 2(− 1)π
Q1 = s
+ 2n +
2
L
R
2 R2
L
2
n
2
Ds 2
K µ + 2( −1)π
n − ρs hω2 + r
4
R
L
L
2
−
R1 =
K r µ + 2(0)π µ + 2(−1)π
L
L
L
T1 =
Ds (1 − µ s ) µ + 2( −1)π 2 Ds 3
Ks
n
+
+ R4 n
R2
R2
L
U1 =
Ds µ + 2(0)π
µ
n−
R 2 s
L
V1 =
Kr µ + 2( −1)π µ + 2(0)π
L
L
L
2
Kr µ + 2(1)π µ + 2( −1)π
,
S1 = L
L
L
K (1 − µ s ) µ + 2(0)π
K
µ + 2(0)π
j s
n + s µ s
2R
L
R
L
n
K (1 − µs ) µ + 2(0)π K s 2 Ds (1 − µs ) µ + 2(0)π
W1 = s
+ 2n +
2
L
2R 2
L
R
2
−
Ds 2
K
n − ρ s hω 2 + r
4
R
L
µ + 2(0)π
L
2
X1 =
Kr µ + 2(1)π µ + 2(0)π
L
L
L
Z1 =
Ds µ + 2(1)π
µ
n−
R 2 s
L
2
A2 =
2
Ds (1 − µ s ) µ + 2(0)π Ds 3
Ks
,
Y 1 = R2 n +
+ R4 n
R2
L
2
K (1 − µ s ) µ + 2(1)π
K
µ + 2(1)π
j s
n + s µ s
2R
L
R
L
K r µ + 2( −1)π µ + 2(1)π
L
L
L
,
K µ + 2(0)π µ + 2(1)π
B2 = Lr
L
L
245
n
K (1 − µ s ) µ + 2(1)π K s 2 Ds (1 − µ s ) µ + 2(1)π
C2 = s
+ 2n +
2
L
2R 2
L
R
2
−
2
Ds 2
K µ + 2(1) π
n − ρs hω2 + r
4
R
L
L
2
Ds (1 − µ s ) µ + 2(1) π 2 Ds 3
Ks
D2 = 2 n +
+ R4 n
R
R2
L
µ + 2(−1)π Ds µ + 2(−1)π 2
K s µ + 2(−1)π
E 2 = Ds
+ R2 µ s
n − j R µs
L
L
L
4
2
D
µ + 2( −1)π
D (1 − µ ) µ + 2( −1)π
D
K
F 2 = − s2 µ s
n + s 2 s
n − s4 n3 + s2 n
R
L
R
L
R
R
2
2
D µ µ + 2( −1)π 2 2Ds (1 − µs ) µ + 2( −1)π 2 Ds 4 K s
G2 = s 2 s
n +
n + 4 n + 2 − ρs hω 2
2
R
L
R
L
R
R
2
−
2
H n2 (k1r R )ρ1ω 2 H n1 (k 2 r R )ρ2ω 2 K t ω 2M
+ 1'
+ −
H n2 ' (k1rR )k1r
H n (k 2 r R )k 2r
L
L
K ω 2M
H2 = t −
L
L
K r µ + 2( −1)π
+ L
L
2
K r µ + 2(0)π µ + 2( −1)π
+ L
L
L
K ω 2 M K r µ + 2(1)π µ + 2( −1)π
I2 = t −
+
L L
L
L
L
µ + 2(0)π Ds µ + 2(0)π 2
K
µ + 2(0)π
J 2 = Ds
+ 2 µs
n − j s µs
L
R
L
R
L
4
2
Ds µ + 2(0)π
D (1 − µ ) µ + 2(0)π
D
K
µs
n + s 2 s
n − 4s n3 + 2s n
2
R
L
R
L
R
R
2
K2 = −
L2 =
+
2
2
2
Ds µs µ + 2(0)π 2 2 Ds (1 − µs ) µ + 2(0)π 2
n
+
n
R 2
L
R2
L
Ds 4 Ks
Hn2 (k 1r R) ρ1ω 2 Hn1 (k 2r R) ρ2ω2
2
n
+
−
ρ
h
ω
−
+
s
R4
R2
H n2 ' (k1 rR )k1 r
H 1'n (k 2 r R ) k2 r
K ω 2M K r µ + 2( −1)π µ + 2(0)π
+ t −
+
L L
L
L
L
K ω 2M K r µ + 2(0)π ,
K ω 2 M K r µ + 2(1)π µ + 2(0)π
M2 = t −
+
N2 = t −
+
L L
L
L L
L
L
L
L
2
246
ε (− j )n H n2 (k 2 rR )Jn' (k 1r R ) k1r
O 2 = p0 ε n ( − j )nJ n (k 1rR ) − n
H n2' (k1 rR )k1 r
µ + 2(1)π Ds µ + 2(1)π 2
K
µ + 2(1)π
P 2 = Ds
+ 2 µs
n − j s µ s
L
L
R
L
R
4
2
D
µ + 2(1)π
D (1 − µ ) µ + 2(1)π
D
K
Q 2 = − 2s µs
n + s 2 s
n − 4s n 3 + 2s n
R
L
R
L
R
R
2
2
2
2
Ds µs µ + 2(1)π 2 2Ds (1 − µs ) µ + 2(1)π 2
R2 = 2
n +
n
R
L
R2
L
2
2
1
D
K
H ( k R )ρ1ω
H (k R )ρ2ω 2
+ s4 n 4 + s2 − ρs hω 2 − n 2 ' 1r
+ n 1' 2 r
R
R
H n (k1rR )k1r
H n ( k2 rR )k 2r
+
Kt ω 2 M Kr µ + 2( −1)π µ + 2(1)π
−
+
L
L
L
L
L
S2 =
Kt ω 2 M Kr µ + 2(0)π µ + 2(1)π
−
+
L
L
L
L
L
K ω 2 M K r µ + 2(1)π
T2 = t −
+
L
L
L
L
2
As shown in Equations (8.1)-(8.3) and (8.5)-(8.7), the solutions are obtained in a
double series of the circumferential modes and space harmonic modes. The maximum
space harmonics of 5 (i.e., m=-5 to m=5) provided converged results for each n in this
case, therefore adopted for all studies in this work. Then, the maximum number of the
circumferential mode n was decided after checking the convergence trend of the solution
as the maximum n changes.
8-3. Calculation of Transmission Loss
The transmitted sound power per unit length of the shell in the interior cavity can be
defined as:
WT =
2π
1
Re ∫ p2T ⋅ ∂ ( w1 ) * Rdθ
∂t
2
0
where r=R
247
(8.42)
Substitution of appropriate equations for p2T and w1 into Equation (8.42) yields an
expression for the components of WnT
WnT =
1
∞
2π
Re ∑ p2Tnm H 1n (k 2 r R) ⋅ ( jω w10nm )* × ∫ cos 2 [ nθ ] ⋅ Rdθ where r=R
2
m =−∞
0
(
)
(8.43)
where Re{.} and the superscript * in Equations (8.42) and (8.43) represent real part and
the complex conjugate of the argument, respectively. And ε n =1 for n = 0 and ε n =2 for
n=1,2,3,…… The transmitted sound power is defined as:
WnT =
*
πR
∞
× Re ∑ p2Tnm × H 1n (k 2 r R) × ( jω w10nm )
2ε n
m =−∞
(8.44)
The power transmission coefficient is defined by
∞
τ (γ ) =
∑W
n =0
T
n
(8.45)
WI
where W I is the incident power with an angle γ per unit length of the shells in the axial
direction
cos ( γ ) p02
W =
× 2R
ρ1c1
I
(8.46)
Then, a closed form for the power transmission coefficient that is a function of the angle
of incidence (γ) can be obtained by substituting Equations (8.44) and (8.46) into (8.45) as
follows:
*
∞ T
1
0
Re
∑ p2 nm × H n ( k2 r R ) × ( jω w1 nm ) × ρ 1c1π
∞
τ ( γ ) = ∑ m =−∞
2
4ε n cos ( γ ) p0
n =0
248
(8.47)
To estimate the random incidence TL, the TL is computed at a particular angle of
incidence. Then the TL averaged over all possible angles of incidence, τ , is found
according to the Paris formula [41] as:
γ lim
τ = 2 ∫ τ (γ )sin γ cos γ dγ
(8.48)
0
where γlim is the limiting angle above which it is assumed that no sound is incident upon
the shell. Calculated TL obviously depends on the choice of the angle of incidence in the
analysis. This dependency can be removed by averaging TL over all possible incident
angles as earlier mentioned. In the calculations to be presented in this paper, the power
transmission coefficient has been calculated in steps of 2° from 0° to 80°, which was
suggested by Mulholland et al. [30], and then Equation (8.48) has been evaluated
numerically. Finally, the averaged (or random incidence) TL is obtained as:
TLavg = 10log10 (1/ τ )
(8.49)
Figure 8.2 shows the random incidence TLs of the unstiffened and stiffened shells
compared in a narrow band format when the mass effect of the stiffener is neglected. The
simulation conditions used to obtain Figure 8.2 are listed in Table 8.1. The unstiffened
and stiffened shells for the single incident angle case of 45° are compared in terms of the
TL in a narrow band format as shown in Figure 8.3 when the mass effect of the stiffener
is also neglected. Figure 8.3 shows comparison of the TLs of the stiffened shell and
unstiffened shell. It is interesting to see that the influence of stiffeners is only in the very
low frequency and that stiffeners actually reduce the TL in some frequencies. It can be
explained by considering the conditions that allows free waves to propagate.
Qualitatively, it is related to the resonance frequency of the single bay of the shell
249
between the stiffeners. Mead explained such a phenomenon by the propagating and nonpropagating bands and bounding frequencies for flat stiffened panels [74]. In practice,
the effect on sound transmission is as though the critical frequency had been lowered by
one or two octaves, the degree of change being dependent upon the phase attenuation
coefficient, spacing and stiffness (translational and rotational) of the stiffeners [91].
Table 8.1. Dimensions of the cylindrical shell and simulation conditions
Kt
(N/m)
Kr
(N⋅m/rad)
Es
(Pa)
µs
h
(mm)
M
(kg)
1.0×105
1.0×103
1.9×1011
0.3
1.0
0.0
ρs
(kg/m3 )
ρ1
(kg/m3 )
ρ2
(kg/m3 )
c1
(m/sec)
c2
(m/sec)
ω/2π
(Hz)
250
7750
γ
0°~80°
1.21
L
(mm)
100
1.21
ψ
1°
343
R
(m)
0.1
343
ηs
0.1
10 ~ 3,000
-
-
100
90
TL (dB)
80
70
60
50
40
30
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 8.2. Comparison of the predicted averaged TLs between the stiffened and the
unstiffened shells
, W/ stiffener; ---------, W/O stiffener
100
90
80
TL (dB)
70
60
50
40
30
20
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 8.3. Comparison of the predicted TLs between the stiffened and the unstiffened
shells on which a plane wave is incident with an angle 45°
, W/ stiffener; ---------, W/O stiffener
251
8-4. Convergence of the Solution
As one can see in Equations (8.1)-(8.3) and (8.6)-(8.7), the solutions are obtained in
series forms, which requires enough terms be used in the calculation to ensure the
solutions converge. Once the solution converges at a given frequency, it can be assumed
to converge in all frequencies lower than that, because more terms are necessary to be
used in the calculation for a higher frequency.
Therefore, the necessary number of
circumferential modes has to be determined at the highest frequency of interest.
A
simple algorithm can be used to ensure the convergence of the solution in that the TL is
calculated at the highest frequency of interest, adding one term at a time. When the TLs
calculated at two successive calculations are within a pre-set error bound (0.01 dB in this
work), the solution is considered to have converged. The number of modes found this
way is used to calculate TL at all other frequencies below this highest frequency of
interest.
Figure 8.4 shows how the calculated TL as the number of circumferential modes
included (n) increases, while the number of the space harmonics is fixed as 11 (p = -5 to
5) at the driving frequency of 3,000 Hz. The same data shown in Table 8.1 is used for the
stiffened shell but the incidence angle θ is taken as 45°. From the figure, it is known that
at least 6 circumferential modes (n = 0 to 5) have to be used to obtain a converged
solution at 3,000 Hz. This makes a 66 term solution because 11 space harmonics and 6
circumferential modes are used.
The necessary number of terms will have to be
determined considering the highest frequency of interest and the structural modes.
252
80
75
70
65
TL (dB)
60
55
Convergence of TL
50
45
40
35
30
0
10
20
30
40
50
Mode Number
Figure 8.4. TL convergence diagram for the stiffened cylindrical shell (R=0.1m, t=1.0
mm) at 3,000 Hz
8-5. Parameter Studies
Studies on important design parameters are conducted for the stiffened shell with the
same specification listed in Table 8.1. TLs calculated for three different incident angles
(30o , 45o , 60o ) are shown in Figure 8.5, which indicates that the transmitted power
slightly decreases (TL increases) as the incidence angle γ increases.
Because the
qualitative aspect of the solution does not change for different angles, the incident angle
of 45o is used for all subsequent calculations, which reduces the related computation time
substantially compared to the random incidence case.
253
110
100
90
TL (dB)
80
70
60
50
40
30
20
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 8.5. TL curves for the stiffened shell with respect to incidence angle
, γ=30°; ---------- , γ=45°;
, γ=60°
8-5.1. Parameters Related to Modeling
Phase Attenuation Coefficient Angle and Shell Loss Factor
The phase attenuation coefficient angle is difficult to estimate especially in the modeling
stage. Figure 8.6 shows the effect of different phase attenuation coefficients used in the
simulation. As shown in Figure 8.6, even for such a wide range chosen intentionally for
a comparison purpose, any appreciable effect of the coefficient is limited to the low
frequency range: the TL increases if the phase attenuation coefficient is increased.
The loss factor represents the internal damping characteristics of the structure, which
is also difficult to estimate accurately. The TLs obtained for the loss factors of 0, 0.1 and
0.2 are compared in Figure 8.7. Again, a very wide range of the parameter is used for a
comparison purpose. The comparison shows that the effect of the loss factor has small
effects on the estimated TL in the frequencies around the dips of the TL curves, however
254
this has a relatively small effect. From this, it is concluded that a damping treatment such
as a coating will not increase the TL of a stiffened shell in a significant extent, which is
different from the stiffened panel [91].
Both the attenuation coefficient and loss factor are typically small, therefore relatively
arbitrary small values are chosen somewhat arbitrarily in this study. In all calculations,
ψ=1° and ηs=0.1 are chosen as the phase attenuation coefficient angle and the loss factor
unless it is stated otherwise.
100
90
80
TL (dB)
70
60
50
40
30
20
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 8.6. TL curves for the stiffened shell with respect to phase attenuation angle
, ψ=0°; ---------- , ψ=1°;
, ψ=10°;
, W/O stiffener
255
100
90
80
70
TL (dB)
60
50
40
30
20
10
0
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 8.7. TL curves for the stiffened shell with respect to loss factor
, ηs=0; ---------- , ηs=0.1;
, ηs=0.2;
,
W/O stiffener(ηs=0)
8-5.2. Study of Design Parameters
Materials
Figure 8.8 shows the TL curves obtained for systems of different material.
Materials
chosen for the comparison are steel, aluminum and brass as shown in Table 8.2. The
figure shows that the steel is the most effective above 100 Hz, as expected because the
stiffness of the steel is the largest. The figure also shows that the aluminum, which has
the lowest stiffness and density, is the least effective in the low frequency range, which is
again as expected. Such a comparison may be used in practical design situations for
basic design of the system.
256
100
90
80
70
60
50
40
30
20
10
10
1
2
3
10
4
10
10
Figure 8.8. TL curves for the stiffened shell with respect to shell material
, aluminum; ---------- , steel;
, brass
Table 8.2. Material properties of the stiffened shell
Density
(ρ: kg/m3 )
Young’s Modulus
(E: Pa)
Poisson’s ratio
(ν)
Steel
Aluminum
Brass
7,750
2,700
8,500
1.9×1011
0.71×1011
1.04×1011
0.3
0.33
0.37
Shell Thickness
As it is seen in Figure 8.9, changing the thickness has a broadband effect on TL over the
entire range of the frequency. In general, TL increases 6 dB as the thickness doubles
except in the very low frequency range, which is anticipated. In the low frequency range,
the reduction is lower than 6 dB because it is controlled more by the membrane stiffness.
In a practical situation when the shell has to be designed only as thick as necessary
257
because of the weight constraint, this type of analys is will be very useful. For example,
if a target TL is known from the consideration of the noise level, a proper thickness of the
shell may be calculated correspondingly.
Stiffener Spacing
As shown in Figure 8.10, smaller stiffener spacing has much higher effect in the low
frequency range.
As the stiffener spacing increases, the TL of the stiffened shell
decreases, and eventually become close to that of the unstiffened shell as shown in Figure
8.10.
100
90
80
TL (dB)
70
60
50
40
30
20
10
1
2
3
10
10
4
10
Frequency (Hz)
Figure 8.9. TL curves for the stiffened shell with respect to shell thickness
, t=1.0 mm; ---------- , t=2.0 mm;
258
, t=3.0 mm
120
110
100
90
TL (dB)
80
70
60
50
40
30
20
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 8.10. TL curves for the stiffened shell with respect to stiffener spacing
, L=10 mm; ---------- , L=50 mm;
, L=100 mm;
, W/O stiffener
Stiffness of the Stiffener
The TLs calculated for four different stiffness values are compared in Figure 8.11. The
effect of increasing the stiffness is also more beneficial in the low frequency range as
shown in the figure. It was found that increasing of the stiffness beyond a certain value
(K t = 2 × 107 N/m in this case) did not contribute at all to change the TL. This may be
explained because the stiffener becomes virtually a rigid support at the value.
259
120
110
100
TL (dB)
90
80
70
60
50
40
30
1
10
2
3
10
10
4
10
Frequency (Hz)
Figure 8.11. TL curves for the stiffened shell with respect to trans lational stiffness of the
stiffener
,Kt =0 N/m;
, Kt =2.0×106 N/m; --------, Kt =1.0×107 N/m;
7
Kt =2.0×10 N/m
8-6. Conclusions
Solutions of the vibro-acoustic responses of a periodically stiffened cylindrical shell of
infinite length are obtained analytically in this work. The system equation is derived by
applying the virtual energy method fully considering coupling effects between the
structures and between the structural and acoustic systems. The Love’s equation is used
to describe the shell motion, therefore all three components of the shell motion are
considered. The motion of the shell, the reflected and transmitted waves induced by an
incident plane wave are expanded in terms of the space harmonics in the solution
procedure, which was developed by Mead et al. [73-75]. The solution is obtained as a
truncated series, therefore a convergence-checking scheme is built in the solution
procedure to ensure converged solutions are obtained. The solution obtained in this work
260
,
is believed to be the first exact solution for this type of the problem: the sound
transmission through stiffened cylindrical shell. Using the solutions represented in the
transmission losses, characteristics of the stiffened shell are studied, which helps to
understand the effects of important modeling and design parameters.
261
Chapter 9 – Conclusions
9-1. Summary
The major accomplishments of this dissertation are (1) development of a new
experimental method to identify damping matrices (2) improvement and development of
vibro-acoustic analysis techniques to study sound transmission through a cylindrical
single-walled shell and a cylindrical double-walled shell.
A new algorithm was proposed for experimental identification of the damping
matrices, which identifies separately the viscous and structural damping matrices of the
equation of motion of a dynamic system. The new algorithm is very simple, therefore
provides more accurate and robust results compared to the method previously used [14].
An analytical model was developed to calculate the noise transmission through the
cylindrical single-walled shell and the cylindrical double-walled shell by considering all
three motion directions of the shell and all the vibro-acoustic coupling effects. This is
considered to be the first exact solution to this type of problem obtained using the full
shell vibration equations. The analytical solution was verified by comparing the measured
TLs of a finite shell.
A simplified analysis method was developed to solve sound transmissions of arbitrary
geometry through structures with porous liners. The approximate method uses a very
simple concept of using only the strongest among the three waves to model the porous
material, which is essentially modeling the porous layer as an equivalent fluid layer. For
262
the first time, sound transmission through the double-walled shells for three types of
porous liner were calculated by the simplified analysis.
Exact analysis procedures are developed to calculate the sound transmission through
the infinitely long elastic panel and the cylindrical shell that are stiffened in one direction.
The stiffener is modeled as a set of the lumped mass, rotational and translational springs
attached to the structures. The dynamic equation that describes vibro-acoustic responses
of the system is derived using the space harmonic approach and the virtual energy
principle. The solution is obtained as a truncated series of the assumed modes by solving
a set of linear equations. A scheme to ensure the convergence of the solution is included
in the solution procedure, therefore the series solution can be considered as an exact
solution, which is believed to be the first exact analytical solution obtained for this type
of problems. Parameter studies are conducted for the design parameters of the stiffened
panel and the stiffened shell. The parameter study demonstrates the value of the analysis
developed in this work as a design tool.
9-2. Contributions
Primary contributions of this research to the state of the art are found in the vibroacoustic analysis technique as well as in the damping identification technique.
1. A new experimental method to identify damping matrices of a dynamic system
using measured frequency response functions (FRFs) is developed and verified in
this work. The method provides a much more robust and accurate results, and
also can be more easily extended to new applications.
validation of the method is also considered to be the first.
263
The experimental
2. An exact analytical procedure to calculate vibro-acoustic responses of thin
cylindrical shells of various cross-sectional structures was developed for the first
time. The full shell equation, the Love’s equation, was used and the convergence
of the solution is ensured in the solution procedure.
3. A simplified analysis method to solve sound transmission through structures with
porous liners of arbitrary geometry is developed and verified analytically and
experimentally. Analysis of sound transmission through a cylindrical doublewalled shell lined with elastic porous core is also conducted by the simplified
analysis for the first time considering the acoustic-structural coupling effect as
well as the effect of the multi-waves in the porous layer.
4. Vibro-acous tic models for stiffened panels and stiffened cylindrical shells are
developed for the first time by employing space harmonic expansion method and
the virtual work principle.
9-3. Recommendations for Further Research
The following are possible research areas that may be extended from this study:
1. Development of damping identification techniques can be extended to rotating
systems by modifying the procedure and refining the measurement techniques.
2. Development of damping identification procedure using the strain frequency
response function (SFRF) instead of displacement frequency response function
(DFRF or FRF), which will improve the accuracy of the identification results
from the FRFs measured at a point that is near the boundaries, thus moving very
little.
264
3. More general numerical formulation of the implementation of the vibro-acoustic
analysis work, especially the porous material modeling and analysis, in
conjunction with the finite element method or boundary element method.
265
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