Sound Transmission Loss of Composite Sandwich Panels
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classi ed information.
Ran Zhou
Certi cate of Approval:
George T. Flowers
Professor
Mechanical Engineering
Malcolm J. Crocker, Chair
Distinguished University Professor
Mechanical Engineering
Winfred A. Foster, Jr.
Professor
Aerospace Engineering
Subhash C. Sinha
Professor
Mechanical Engineering
George T. Flowers
Dean
Graduate School
Sound Transmission Loss of Composite Sandwich Panels
Ran Zhou
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Ful llment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
May 9, 2009
Sound Transmission Loss of Composite Sandwich Panels
Ran Zhou
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Ran Zhou, daughter of Kaiti Zhou and Jiena Shao, was born on October 26, 1977, in
Taiyuan, Shanxi, China. She entered Nanjing University in September, 1995, and graduated
with a GPA of 4.5 (out of 5) with a Bachelor of Science degree in Electronic Science and
Engineering in June, 1999. Then she entered Graduate School, Nanjing University, in
September, 1999, and graduated with a Master of Science degree in Electronic Science and
Engineering in June, 2002. She entered Auburn University in August, 2002.
iv
Dissertation Abstract
Sound Transmission Loss of Composite Sandwich Panels
Ran Zhou
Doctor of Philosophy, May 9, 2009
(M.S., Nanjing University, 2002)
(B.S., Nanjing University, 1999)
205 Typed Pages
Directed by Malcolm J. Crocker
Light composite sandwich panels are increasingly used in automobiles, ships and air-
craft, because of the advantages they o er of high strength-to-weight ratios. However, the
acoustical properties of these light and sti structures can be less desirable than those of
equivalent metal panels. These undesirable properties can lead to high interior noise levels.
A number of researchers have studied the acoustical properties of honeycomb and foam
sandwich panels. Not much work, however, has been carried out on foam- lled honeycomb
sandwich panels.
In this dissertation, governing equations for the forced vibration of asymmetric sand-
wich panels are developed. An analytical expression for modal densities of symmetric sand-
wich panels is derived from a sixth-order governing equation. A boundary element analysis
model for the sound transmission loss of symmetric sandwich panels is proposed. Measure-
ments of the modal density, total loss factor, radiation loss factor, and sound transmission
loss of foam- lled honeycomb sandwich panels with di erent con gurations and thicknesses
v
are presented. Comparisons between the predicted sound transmission loss values obtained
from wave impedance analysis, statistical energy analysis, boundary element analysis, and
experimental values are presented.
The wave impedance analysis model provides accurate predictions of sound transmis-
sion loss for the thin foam- lled honeycomb sandwich panels at frequencies above their rst
resonance frequencies. The predictions from the statistical energy analysis model are in
better agreement with the experimental transmission loss values of the sandwich panels
when the measured radiation loss factor values near coincidence are used instead of the
theoretical values for single-layer panels. The proposed boundary element analysis model
provides more accurate predictions of sound transmission loss for the thick foam- lled hon-
eycomb sandwich panels than either the wave impedance analysis model or the statistical
energy analysis model.
vi
Acknowledgments
I would like to thank Dr. Malcolm J. Crocker for his guidance during this research. I
am also thankful to Dr. George T. Flowers, Dr. Winfred A. Foster, Dr. Subhash C. Sinha,
and Shannon Price for their help. Thanks are also due to my parents for their support and
encouragement.
vii
Style manual or journal used Journal of Approximation Theory (together with the style
known as \aums"). Bibliograpy follows Journal of Sound and Vibration
Computer software used The document preparation package TEX (speci cally LATEX)
together with the departmental style- le aums.sty.
viii
Table of Contents
List of Figures x
1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Literature Review 6
2.1 Wave impedance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Statistical energy analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Boundary element analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Wave Impedance Analysis 18
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Governing equations for forced vibration . . . . . . . . . . . . . . . . . . . . 19
3.3 Sound transmission loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Symmetric sandwich panels . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Other TL models for asymmetric sandwich panels . . . . . . . . . . . . . . 43
3.7 Other governing equations for anti-symmetric motion . . . . . . . . . . . . . 48
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Statistical energy analysis 54
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Assumptions and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Transmission suite model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4 Modal densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5 Internal loss factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.6 Coupling loss factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Sound transmission loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.8 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Boundary element analysis 86
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Finite element analysis models for sandwich structures . . . . . . . . . . . . 87
5.3 Basic concepts of boundary element analysis . . . . . . . . . . . . . . . . . . 91
ix
5.4 Boundary element analysis model for uid-structure- uid systems . . . . . . 95
5.5 Boundary element analysis model for three-layer symmetric sandwich panels 108
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6 Materials and material properties 112
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 Measurement methods for materials . . . . . . . . . . . . . . . . . . . . . . 118
6.4 Experimental resonance frequencies of sandwich beams . . . . . . . . . . . . 121
6.5 Material properties of sandwich panels . . . . . . . . . . . . . . . . . . . . . 128
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7 Dynamic properties of composite sandwich panels 135
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Experimental modal densities . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.3 Experimental total loss factors . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4 Experimental radiation loss factors . . . . . . . . . . . . . . . . . . . . . . . 149
7.5 Experimental internal loss factors . . . . . . . . . . . . . . . . . . . . . . . . 159
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8 Sound transmission loss of composite sandwich panels 161
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2 Experimental sound transmission loss . . . . . . . . . . . . . . . . . . . . . 162
8.3 Sound transmission loss from wave impedance analysis . . . . . . . . . . . . 164
8.4 Sound transmission loss from statistical energy analysis . . . . . . . . . . . 172
8.5 Sound transmission loss from boundary element analysis . . . . . . . . . . . 176
8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9 Conclusions 183
Bibliography 186
A Derivation of stiffness constants of rotated-axis 191
B Partial differential Operators 194
x
List of Figures
3.1 The geometry and loads of a sandwich panel . . . . . . . . . . . . . . . . . 19
3.2 Symmetric and anti-symmetric face sheet displacements . . . . . . . . . . . 21
3.3 The rotated axis system of the orthotropic material . . . . . . . . . . . . . . 23
3.4 Components of pressure elds on a sandwich panel . . . . . . . . . . . . . . 27
3.5 Predicted wave numbers for anti-symmetric waves in panel A . . . . . . . . 35
3.6 Predicted wave speeds for anti-symmetric motion of panel A . . . . . . . . . 37
3.7 Predicted wave speeds for symmetric motion of panel A . . . . . . . . . . . 37
3.8 Wave impedances along two principal directions for panel A . . . . . . . . . 39
3.9 Calculated sound transmission loss values of panel A without damping . . . 39
3.10 Predicted and measured sound transmission loss values of panel A . . . . . 40
3.11 Wave impedances along two principal directions for panel B . . . . . . . . . 42
3.12 Predicted and measured sound transmission loss values of panel B . . . . . 42
3.13 Calculated wave impedances for symmetric panel C . . . . . . . . . . . . . 46
3.14 Predicted sound transmission loss values of panel C from Dym and Lang?s
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.15 Predicted sound transmission loss values of panel C from the present analysis 47
3.16 Predicted sound transmission loss values of panel A made using governing
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1 Block diagram for power ows between the structure and the reverberant eld 57
4.2 Block diagram for power ows between the structure and two reverberation
rooms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
xi
4.3 The transmission suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Constant frequency loci for transverse wave numbers of a simply supported
panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 Wave numbers for free transverse wave in x-axis of panel A . . . . . . . . . 67
4.6 Modal densities of free transverse wave in x-axis of panel A . . . . . . . . . 68
4.7 E ective radiation areas for edge and corner modes . . . . . . . . . . . . . . 73
4.8 Normalized radiation resistances of ba ed simple supported aluminum panels 75
4.9 Sound transmission measurements from the two-room method . . . . . . . . 77
4.10 Estimated transmission loss values of panel D . . . . . . . . . . . . . . . . . 81
4.11 Resonant and non-resonant modes on the sound transmission loss of panel D 82
4.12 Estimated sound transmission loss values of panel A . . . . . . . . . . . . . 83
4.13 The e ects of dimensions of panels and volumes of rooms on sound transmis-
sion loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1 Finite element model for sandwich structures using MSC Nastran . . . . . . 88
5.2 A cantilever sandwich beam with viscoelastic core (beam G) . . . . . . . . . 90
5.3 Half-space V limited by an in nite rigid plane SH and boundary S . . . . . 94
5.4 Sound elds 1 and 2 created by a ba ed planar vibrating structure . . . 97
5.5 Calculated sound transmission loss values of the aluminum panel H for sound
waves at normal incidence using the BEM computer program . . . . . . . . 103
5.6 Calculated sound transmission loss values of the aluminum panel H for sound
waves at oblique incidence using the BEM computer program . . . . . . . . 105
5.7 Predicted sound transmission loss values of the aluminum panel H for sound
waves at normal incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.8 Predicted sound transmission loss values of the aluminum panel H for sound
waves at oblique incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
xii
6.1 Commonly used cell con gurations for honeycomb core materials (a) hexag-
onal (b) square (c) over expanded hexagonal (d) ex . . . . . . . . . . . . . 115
6.2 Manufacture of honeycomb cores - corrugating (top) and expansion (bottom)
processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3 The face sheet (a) and core (b) materials of sandwich structures in this study 118
6.4 The frequency response function of the aluminum beam for the shaker set-up 120
6.5 The frequency response function of the aluminum beam for the impact set-up 120
6.6 The frequency response function of beam Ix . . . . . . . . . . . . . . . . . . 123
6.7 The frequency response function of beam Iy . . . . . . . . . . . . . . . . . . 123
6.8 The frequency response function of beam Jx . . . . . . . . . . . . . . . . . . 123
6.9 The frequency response function of beam Jy . . . . . . . . . . . . . . . . . . 124
6.10 The frequency response function of beam Kx . . . . . . . . . . . . . . . . . 124
6.11 The frequency response function of beam Ky . . . . . . . . . . . . . . . . . 124
6.12 The frequency response function of beam Lx . . . . . . . . . . . . . . . . . 125
6.13 The frequency response function of beam Ly . . . . . . . . . . . . . . . . . . 125
6.14 Loss factors of beams Ix, Iy, Jx and Jy . . . . . . . . . . . . . . . . . . . . . 126
6.15 Loss factors of beams Kx and Ky . . . . . . . . . . . . . . . . . . . . . . . . 127
6.16 The frequency response functions of the aluminum beam . . . . . . . . . . . 127
6.17 Transverse displacement caused by (a) bending and (b) shear . . . . . . . . 129
7.1 Set-up for the modal density and loss factor experiments . . . . . . . . . . . 139
7.2 The inertance of the added mass . . . . . . . . . . . . . . . . . . . . . . . . 140
7.3 The measured point mobility of panel J using the three-channel spectral
analysis (a) real part (b) imaginary part . . . . . . . . . . . . . . . . . . . . 141
7.4 Modal density estimates for panel J without mass correction . . . . . . . . 141
xiii
7.5 Modal density estimates for panel J with mass correction . . . . . . . . . . 142
7.6 Modal density estimates for panel I with mass correction . . . . . . . . . . 143
7.7 Modal density estimates for panel K with mass correction . . . . . . . . . 143
7.8 Modal density estimates for panel L with mass correction . . . . . . . . . . 144
7.9 Loss factor estimates for panel J . . . . . . . . . . . . . . . . . . . . . . . . 147
7.10 Loss factor estimates for panel I . . . . . . . . . . . . . . . . . . . . . . . . 147
7.11 Loss factor estimates for panel K . . . . . . . . . . . . . . . . . . . . . . . . 148
7.12 Loss factor estimates for panel L . . . . . . . . . . . . . . . . . . . . . . . . 148
7.13 Radiation resistance estimates for ba ed clamped panel I . . . . . . . . . . 153
7.14 Radiation resistance estimates for ba ed clamped panel J . . . . . . . . . . 153
7.15 Radiation resistance estimates for ba ed clamped panel K . . . . . . . . . 154
7.16 Radiation resistance estimates for ba ed clamped panel L . . . . . . . . . . 154
7.17 Radiation loss factor estimates for clamped panels I L . . . . . . . . . . . 155
7.18 Radiation resistance estimates for unba ed free-edge panel I . . . . . . . . 156
7.19 Radiation resistance estimates for unba ed free-edge panel J . . . . . . . . 157
7.20 Radiation resistance estimates for unba ed free-edge panel K . . . . . . . . 157
7.21 Radiation resistance estimates for unba ed free-edge panel L . . . . . . . . 158
7.22 Radiation loss factor estimates for unba ed free-edge panels I L . . . . . 158
7.23 Internal loss factor estimates for panels I L . . . . . . . . . . . . . . . . . 159
8.1 Experimental sound transmission loss values of panel I . . . . . . . . . . . . 164
8.2 Experimental sound transmission loss values of panel J . . . . . . . . . . . 165
8.3 Experimental sound transmission loss values of panel K . . . . . . . . . . . 165
xiv
8.4 Experimental sound transmission loss values of panel L . . . . . . . . . . . 166
8.5 Predicted sound transmission loss values of panel I from the wave impedance
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.6 Predicted sound transmission loss values of panel J from the wave impedance
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.7 Predicted sound transmission loss values of panel K from the wave impedance
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.8 Predicted sound transmission loss values of panel L from the wave impedance
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.9 Predicted sound transmission loss values of panel I by using Eq. (??) . . . 170
8.10 Predicted sound transmission loss values of panel J by using Eq. (??) . . . 171
8.11 Predicted sound transmission loss values of panel K by using Eq. (??) . . . 171
8.12 Predicted sound transmission loss values of panel L by using Eq. (??) . . . 172
8.13 Transmission loss estimates for panel I from SEA . . . . . . . . . . . . . . . 174
8.14 Transmission loss estimates for panel I using the measured values of rad . 174
8.15 Transmission loss estimates for panel J using the measured values of rad . 175
8.16 Transmission loss estimates for panel K using the measured values of rad . 175
8.17 Transmission loss estimates for panel L using the measured values of rad . 176
8.18 Predicted sound transmission loss values of panel K from the boundary ele-
ment analysis, for sound waves at oblique incidences . . . . . . . . . . . . . 178
8.19 Predictions of sound transmission loss for panel K made using the boundary
element analysis model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.20 The prediction of sound transmission loss for panel I made using the bound-
ary element analysis model . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.21 The prediction of sound transmission loss for panel J made using the bound-
ary element analysis model . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.22 The prediction of sound transmission loss for panel L made using the bound-
ary element analysis model . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.1 The rotated-axis coordinate system of the orthotropic material . . . . . . . 192
xv
Chapter 1
Introduction
1.1 Background
Sound transmission loss is mostly determined by the mass, and the dynamic sti ness of
structures. A high mass-to-sti ness ratio usually produces a high transmission loss. Because
of the presence of the core, the dynamic sti ness of sandwich structures is strongly depen-
dent on frequency and decreases with increasing frequency. Thus, the sound transmission
loss of sandwich panels can be much di erent from that of single-layer panels. Three ap-
proaches have been used to investigate the sound transmission characteristics of single-layer
panels.
1.2 Approaches
Wave impedance analysis is the most straightforward approach to calculate the sound
transmission loss of panels. The wave impedance of a panel is derived from governing
equations for the forced vibration of the panel. Since the acoustic particle velocity must
match the transverse velocity of the panel at the uid-structure interfaces, the pressures
in the incident, re ected and radiated waves at the interface can be related to the wave
impedance of the panel. Wave impedance analysis assumes that the panel is in nite, so
that only the non-resonant forced motion is considered below the coincidence frequency.
1
Most previous work has used wave impedance analysis to predict the sound transmission
loss of sandwich panels.
Statistical energy analysis (SEA) was developed in the early 1960?s for estimating
the response and radiation properties of structures excited by broadband noise. With
broadband random noise excitation, the statistical properties such as mean square values
and power densities can be used to provide a measure of vibration. SEA works best with
reverberant elds of vibration, and has been used to predict the interaction between resonant
structures and reverberant acoustic elds. SEA is very attractive for use in high frequency
regions where modal densities are high and a deterministic analysis of all the resonant modes
of the vibration of a structure is not practical.
The introduction of computers has permitted increasing use of numerical simulation
analyses, including boundary element analysis and nite element analysis. The boundary
element analysis produces more details of the vibro-acoustic interaction than the wave
impedance analysis or the statistical energy analysis, especially at low frequencies, where the
requirements of SEA may not be met. For nite and boundary element analyses, the mesh of
the structure should provide with at least ve nite element nodes per acoustic wavelength
in the frequency range of interest. At high frequencies, a very re ned discretization is
required, which leads to a large algebraic system. Even with high speed computers, the
computation time for a single frequency is considerable.
The manufacture of high modulus reinforced fabrics increases the application of com-
posite sandwich panels. The relative di erence between the sti nesses of high modulus
2
reinforced face sheets and cores of the sandwich panels in this study is not the same as
that of the traditional sandwich panels for normal constructions [1, 2, 3, 4, 5, 6, 7, 10].
A few experimental sound transmission loss data for sandwich panels with high modulus
reinforced face sheets are available in the literature [8, 59, 60]. In Ref. [8] the sandwich
panels were treated as single-layer panels with an equivalent dynamic bending sti ness in
order to calculate the transmission loss of the sandwich panels. The other two references
only presented comparisons of the experimental results with the mass law values [59, 60].
1.3 Dissertation outline
This dissertation is organized as follows.
Chapter 2 presents a review of previous work on the three analyses, wave impedance
analysis, statistical energy analysis, and boundary element analysis. The section on wave
impedance analysis includes derivations of governing equations for the forced vibration
and the sound transmission loss of sandwich panels. The section on statistical energy
analysis provides SEA applications for the prediction of noise and vibration associated with
structures and acoustic volumes, together with work on the three main parameters, modal
density, internal loss factor, and coupling loss factor. Chapter 2 closes with a brief review
of boundary element analysis on the uid-structure- uid interaction.
Chapter 3 deals with wave impedance analysis. Governing equations for the forced vi-
bration of asymmetric sandwich panels with orthotropic cores are developed, then these are
followed by a sound transmission loss model which makes use of wave impedance analysis for
3
asymmetric sandwich panels. The sound transmission characteristics of two sandwich pan-
els with honeycomb cores are discussed. Comparisons of the governing equations developed
herein and other available governing equations for sandwich panels are provided.
Chapter 4 starts with an introduction of SEA, together with a sound transmission loss
model using SEA. Then theoretical estimation methods for the three main parameters used
in SEA, especially for composite sandwich panels, are discussed. In SEA, the response
of structures is dependent on not only the dimensions of structures, but the dimensions
of acoustic volumes as well. The e ects of these dimensions on the predictions of sound
transmission loss for panels are illustrated.
Chapter 5 rst presents a comparison of di erent nite element models for sandwich
structures. Then concepts of boundary element method in acoustics are introduced and
the boundary element formulations for uid-structure- uid interaction are presented. Com-
parisons of predicted sound transmission loss values of an aluminum panel obtained from
numerical analyses, a BEM computer program in MATLAB language and a transmission
loss model in a commercial software, LMS SYSNOISE, are provided. Finally, a boundary
element analysis model is proposed for three-layer symmetric sandwich panels.
Chapter 6 presents a comprehensive overview of available face sheet and core composite
materials. The experimental methods used to obtain the material properties of the face
sheets and core of sandwich structures are discussed. Then the estimated material properties
of the sandwich panels tested in this study are presented.
4
Chapter 7 concentrates on dynamic properties of composite sandwich panels used in
SEA. The experimental modal densities, radiation loss factors and internal loss factors of
four sandwich panels are provided.
Chapter 8 presents experimental and predicted sound transmission loss values of four
sandwich panels. Both face sheet and core losses are considered in the wave impedance
analysis, and the internal loss factor of the whole structure is used in SEA and boundary
element analysis. Conclusions are presented in Chapter 9.
5
Chapter 2
Literature Review
2.1 Wave impedance analysis
Kurtze and Watters [1], in their classic paper, assumed that the face sheets respond as
elementary plates in bending, and the core acts as a spacer that has mass and only shear
e ects in the core are included. They developed the wave impedance of sandwich panels
from an equivalent electrical circuit analog. Kurtze and Watters added periodic structures,
rigid bridges, in the core to increase the sound insulation. They also illustrated that the loss
tangent of the sandwich panel can be equal to that of the core in the mid-frequency region.
In the analysis, they assumed that the core is soft but incompressible, and the double-wall
resonance frequency is outside the frequency range of interest.
Ford and Walker [2] were the rst to describe the e ects of dilatational modes of
sandwich panels on sound transmission loss. They introduced a dilatational term to describe
the translational motion of the core. Then they developed governing equations for the free
vibration of sandwich panels from energy relationships. Ford and Walker showed that the
dilatational mode of vibration depends primarily on the core thickness and the face sheet
masses, and identi ed the dips in the experimental transmission loss curves as the resonance
frequencies for both exural and dilatational modes.
6
Smolenski and Krokosky [3] corrected some errors in the work of Ford and Walker [2],
and included volumetric and shear terms in the strain energy. They pointed out that in
general exural modes of vibration are insensitive to changes in the Poisson?s ratio and thick-
ness of the core, whereas dilatational modes of vibration respond dramatically to changes
in these core properties.
The rst e ort at calculating the sound transmission loss of sandwich panels by using
wave impedance analysis is attributed to Dym and Lang [4, 5]. They introduced a set of
symmetric and anti-symmetric face sheet displacements as the dependent variables, and
retained the dilatational term. Dym and Lang showed that, for identical face sheets, the
symmetric and anti-symmetric energies are uncoupled naturally. They developed governing
equations for the forced vibration of symmetric sandwich panels by applying Lagrange?s
principle. Then they derived an expression for the sound transmission coe cient in terms
of the anti-symmetric and symmetric wave impedances [5]. They suggested that a high
transmission loss can be achieved by choosing the panel properties in such a way so that
the symmetric and anti-symmetric impedances have similar values.
Moore and Lyon [6] were the rst to investigate symmetric sandwich panels with or-
thotropic cores. They used a set of symmetric and anti-symmetric displacements which are
equivalent to those presented by Smolenski and Krokosky [3]. They showed that a high
sound transmission loss can be achieved by using an orthotropic core with a low compres-
sional sti ness and a high shear sti ness, which is quite opposite to the design approach
suggested by Kurtze and Watters [1]. This moves the double-wall resonance frequency to a
7
low frequency, and shifts the coincidence associated with anti-symmetric motion to a high
frequency, then the cancellation of the symmetric and anti-symmetric motions of the face
sheets produces that the transmission loss results are greater than the mass law values in the
mid-frequency region. In their analysis, there is an error in the expression for the sti ness
in the rotated axis system for sandwich panels with orthotropic cores.
Narayanan and Shanbhag [61] derived the acceleration for the forced vibration of sand-
wich panels from the governing equation for sandwich panels presented by Mead and Markus
[23] and applied a transmission loss model that is identical to the transmission loss model
developed by Dym and Lang [5] to examine theoretically the e ects of some core parameters
on the sound transmission loss of sandwich panels.
Dym and Lang [7] extended their model for three-layer symmetric sandwich panels with
isotropic cores to asymmetric sandwich panels with orthotropic cores. Based on the predic-
tions, they found that when the mass is kept constant, panels with asymmetric con guration
have a poorer acoustical performance than those panels with symmetric con guration. The
dependence of sti ness on the angle of rotation of the orthotropic material was not consid-
ered in their model.
Nilsson [8] presented a free vibration dynamic analysis for sandwich panels with glass
reinforced plastic face sheets. The calculation showed that the total loss factor of sand-
wich panels is primarily determined by the loss factors of the face sheets, at low- and
high-frequencies. He treated the sandwich panels as single-layer panels with an equivalent
8
dynamic bending sti ness in order to calculate the sound transmission loss of the sand-
wich panels. He also applied an approximation approach derived from SEA to estimate the
sound transmission loss of an asymmetric sandwich panel. Though Nilsson discussed the
e ects of uid load on sandwich panels and gave an expression for the apparent mass of
the water-loaded panel, he assumed that the e ects of uid load cancel out for symmetric
sandwich panels.
Jones [10] evaluated various full-sized sandwich construction designs experimentally in
a duplex living unit. He pointed out that the measured sound transmission loss values were
higher than the mass law values at low frequencies, because of insu ciently di use sound
elds in the rooms. He found that the sound transmission loss curves of the sandwich
panels with paper honeycomb core have smaller coincidence dips, and those dips do not
return as rapidly towards the mass law curve as do those of the panels with foam cores.
The experimental results also showed that asymmetric sandwich constructions do improve
sound insulation.
Huang and Ng [59] presented experimental sound transmission loss results for honey-
comb sandwich panels with glass reinforced composite face sheets. They showed the e ects
of core thickness on sound transmission loss experimentally. They used an incorrect expres-
sion for the wave impedance of the sandwich panels to predict the sound transmission loss
values. Rajaram et al. [60] conducted experimental studies of the sound transmission loss
of honeycomb sandwich panels with carbon and glass ber composite face sheets.
9
In the analyses of all the papers mentioned above the face sheets of sandwich panels
are assumed to be isotropic that in general is not true for high modulus reinforced ber
materials. The orthotropic face sheets are considered in development of governing equations
for the forced vibration of sandwich panels in this study.
2.2 Statistical energy analysis
Statistical energy analysis (SEA) is a modeling procedure which uses energy ow re-
lationships for the theoretical estimation of the vibration response levels of and the noise
radiation from structures in resonant motion.
Lyon and Maidanik [11] computed the power ow between two random excited, linear
oscillators with small coupling between them. They showed that the power ow is propor-
tional to the di erence in average modal energies of the two oscillators. Then they extended
the model to the coupling between two multimodal systems, and the interaction between
a structure and a reverberant acoustic eld. Lyon and Maidanik also gave a radiation re-
sistance expression for the coupling of a single mode structure to a reverberant acoustic
eld.
Smith [12] calculated the response and sound radiation for one linear resonant mode of
a structure excited by a pure tone. He extended the model to the case of a structure excited
by broadband random noise. Then Smith found that when the modal vibrations are pre-
dominantly damped by sound radiation, the mean square velocity is inversely proportional
to the modal sti ness.
10
Maidanik [13] extended the results presented in the two papers discussed above, from a
single mode formalism to a multimode formalism using two main assumptions. The number
of modes in a combined system is equal to the sum of the numbers of modes of the two
systems; and the modal densities of a combined system also are equal to the sum of the
modal densities of the two systems. Maidanik computed the radiation resistance of a nite
ba ed simply supported single-layer panel for individual modes. He also predicted the
average modal radiation resistance of a ba ed simply supported single-layer panel in a
reverberant acoustic eld and compared the predicted values with experimental results.
Eichler [15] presented a formulation of statistical energy analysis which includes the
relations between the average energies in linear loosely and conservatively coupled systems
in terms of modal densities, internal and coupling loss factors. He showed that the products
of modal density and coupling loss factor are equal within each pair of subsystems. The
noise reduction of a rectangular box was investigated in three frequency regions as presented
by Lyon [14]. It was seen that the sound pressure in the box can exceed that in the incidence
sound eld in both theoretical and experimental cases. Eichler noticed that the predictions
from the classical sound transmission predictions were closer to the measured values near
the critical frequency, because the theory presented in their analysis only considered the
resonant free vibration wave modes.
Crocker and Price [18] presented general power ow relationship equations for a room-
panel-room transmission suite. The power ow between the two rooms was de ned as the
ow between at non-resonant modes, when there are no modes excited in the panel in the
11
frequency band under consideration. Both non-resonant and resonant vibration modes were
taken into consideration. They also provided the experimental determination for the radi-
ation resistance, the coupling factors, the panel response, and the sound transmission loss,
derived using SEA. An aluminum panel was tested in a reverberation room and also clamped
between two reverberation rooms. Comparisons between experimental and predicted sound
transmission loss and radiation resistance values were provided.
Sewell [20] derived an expression for the forced vibration transmission coe cient of a
ba ed single-layer partition in a reverberant acoustic eld using the classical method. The
expression for the forced vibration transmission factor is generally valid when the surface
mass density of the partition is more than 10 kg/m2.
Gomperts [58] provided an expression for the radiation e ciency of a ba ed free-edge
panel and Oppenheimer and Dubowsky [25] studied the radiation e ciency of an unba ed
simply supported panel. Both of these studies were based on the results developed by
Maidanik [13].
The successful prediction of noise and vibration levels of coupled structural elements
and acoustic elds using SEA depends to a large extent on an accurate estimate of three
parameters, 1) the modal density of each subsystem, 2) the internal loss factor of each
subsystem, and 3) the coupling loss factors between the subsystems. Some studies have
been carried out in assessing the parameters experimentally.
12
Clarkson and Pope [21] employed the point mobility technique, developed by Cremer
et al. [42], to estimate modal densities of at plates and cylinders. They found that the
real part of point mobilities of very lightly damped structures can be negative.
Brown [55] showed that modal density estimates can be improved by using a three-
channel spectral analysis which minimizes the erroneous results generated by feedback noise
caused by exciter-structure interaction. Brown and Norton [40] showed that the modal
density measurement for cylindrical pipes can be further improved by using the three-
channel spectral analysis with a mass correction applied to the point mobility measurement.
Keswick and Norton [30] used two mass correction methods, the measured mass method and
the spectral mass method, to obtain the experimental modal densities of a lightly damped
clamped cylindrical pipe. The results showed that the spectral mass method is in better
agreement with theory.
Clarkson and Ranky [22] derived an expression for the modal density of honeycomb
sandwich panels from a reduced form of the governing equation for sandwich structures
presented by Mead and Markus [23] and they evaluated the modal density of honeycomb
plates by using a two-channel spectral analysis without mass correction.
Renji and Nair [26] developed an expression for the modal density of a symmetric sand-
wich panel from a fourth-order equation which was modi ed from the governing equation
of motion for a symmetric laminate by including the shear exibility of the core. In the
work, they considered both real and imaginary parts of the point mobility in the measured
mass correction.
13
The expressions for the modal density of honeycomb sandwich panels given by both
Clarkson and Ranky [22] and Renji and Nair [26] were developed from fourth-order govern-
ing equations, while most governing equations for symmetric honeycomb sandwich panels
are sixth-order [6, 7, 23, 37]. Ferguson and Clarkson [41] presented an expression for the
modal density of honeycomb sandwich panels derived from the sixth-order equation pre-
sented by Mead and Markus [23]. The expression, however, is incorrect.
Clarkson and Pope [21] used a steady state power ow method to estimate the loss
factors of at plates and cylinders. Ranky and Clarkson [57] compared the power ow
method with the enveloped decay method which had been used to obtain internal loss
factors of structures. They found that there is no signi cant di erence between the results
from the two methods when the modes in the chosen band of frequency have similar modal
loss factors. If this is not the case, the decay curve is not a straight line, then the power
ow method provides the result required for SEA calculations.
Renji and Narayan [28] investigated loss factors of honeycomb sandwich panels. They
corrected the e ect of added mass on the driving force by using the measured mass correction
method and assumed that the mass loading of the accelerometer, which was employed to
measure the spatial velocity of the panel is negligible.
Lyon and Maidanik [11] described the experimental determination for the radiation loss
factor of a structure in a reverberant eld. Crocker and Price [18] presented the experimental
determination for the radiation loss factor of a structure clamped between two reverberation
14
rooms. Very little published data exist on the radiation loss factors or radiation resistances
of sandwich panels.
In this study, an expression for model densities of sandwich panels is derived from
a sixth-order governing equation. The experimental results of radiation loss factors for
sandwich panels with di erent boundary conditions are presented.
2.3 Boundary element analysis
The predictions from SEA are more accurate where su cient modes in the frequency
band under consideration. It is impossible to obtain closed-form expressions for the radia-
tion e ciency for structures with arbitrary boundary conditions. Hence, the details of each
mode of nite structures should be considered in the response analysis in order to obtain
better predictions at low frequencies. A lot of studies have been carried out in simulating
uid-structure interactions. Three-domain, uid-structure- uid systems have been modeled
as coupled systems [32, 34] and uncoupled systems [33, 45, 35].
Mariem and Hamdi [32] presented a boundary nite element analysis to compute the
sound transmission loss of a ba ed panel. The elastic potential energy, the kinetic energy
and the work were described in their approach by the displacement of the panel. The
radiated sound pressure eld was associated with the modi ed Green?s function using the
classical formula of Rayleigh. The total load on the panel was given by the pressure step
across the panel. The sound radiation from a ba ed clamped thin circular panel excited by
a normal incident plane wave was computed. The numerical results showed that the radiated
15
energies can be greater than the incident energy near the rst few resonance frequencies.
Except near these frequencies, the numerical results agree well with the experimental values.
However, there are some errors in the expression for the total load on the panel.
Roussos [45] developed an uncoupled analytical model for the sound transmission loss
of a simply supported panel. A Green?s function integral equation was used to link the
plate vibrations to the transmitted far- eld pressure eld.
Barisciano [33] studied the sound transmission loss of honeycomb sandwich panels using
boundary element and nite element models. The nite elements of honeycomb sandwich
panels were constructed using Patran. The computed velocities of the panel excited by uid
forces were imported to a boundary element analysis software as the boundary conditions
of the uid domain. Barisciano treated the uid-structure- uid system as an uncoupled
system and used an incorrect nite element model for the sandwich panels.
Filippi et al. [34] studied the response of a thin elastic rectangular ba ed panel in a
light uid excited by an incidence acoustic eld. The total load on the panel was assumed
to be related to the pressure step across the panel. They only predicted the noise reduction
(di erence in sound pressure levels) across the panel.
Thamburaj and Sun [35] examined the e ects of material and geometrical properties
on the theoretical sound transmission loss of a sandwich beam. The governing equations
for the sandwich beam were derived by applying Lagrange?s principle. They assumed that
the external load on the beams is due to the incident and re ected pressures only.
16
In this work, a boundary element analysis model for the sound transmission loss of
three-layer symmetric sandwich panels is proposed. In the model, the uid-structure- uid
system is treated as a coupled system and the sandwich panel is excited a random incidence
eld.
17
Chapter 3
Wave Impedance Analysis
3.1 Introduction
The wave impedance analysis is used to compute the sound transmission loss of in nite
structures. The pressure loads on the structures are associated with the pressures in the
incident, re ected, radiated waves at the uid-structure- uid interfaces. The pressures in
the radiated and transmitted waves depend on the transverse motions of the structure. The
transverse vibration of the structure is determined by the pressure loads on the structure,
as shown in Fig. 3.1.
Dym and Lang [4, 5] presented a sound transmission loss analysis for symmetric sand-
wich panels with isotropic cores. They later extended their model to asymmetric sandwich
panels with orthotropic cores [7]. They provided the governing di erential equations for
sandwich panels in matrix form. The dependence of sti ness on the angle of rotation of the
orthotropic material was not considered in their model. Moore and Lyon [6] included the
angle of rotation e ects on sti ness of orthotropic material in their sound transmission loss
analysis for symmetric sandwich panels with orthotropic cores. The governing equations
for symmetric and anti-symmetric motions of symmetric sandwich panels were presented in
matrix form, respectively.
18
Figure 3.1: The geometry and loads of a sandwich panel
In this study, the governing equations for the forced vibration of symmetric sandwich
panels with orthotropic cores developed by Moore and Lyon are extended to asymmetric
sandwich panels with orthotropic face sheets and cores. The wave impedance analysis model
for the sound transmission loss of asymmetric sandwich panels is provided. The e ects of the
wave number, wave speed and wave impedance on the prediction of sound transmission loss
for sandwich panels are explained. This sound transmission loss analysis model is compared
with the model given by Dym and Lang [7]. Then a sixth-order governing equation for
anti-symmetric motion of symmetric sandwich panels is derived and compared with the
sixth-order di erential equations for sandwich panels presented by Mead and Markus [23]
and Nilsson and Nilsson [37].
3.2 Governing equations for forced vibration
Both elasticity relationships and energy relationships can be employed to develop gov-
erning equations for three-layer sandwich structures. Since it is extremely di cult to obtain
19
analytical expressions from elasticity relationships, in this study energy relationships were
chosen to derive governing equations for asymmetric sandwich panels. Elastic potential and
kinetic energies are evaluated in terms of the displacements, and the virtual work done by
the pressure loads on the face sheets is also derived in terms of the transverse displace-
ments. Then Lagrange?s equations are applied to obtain governing equations for the forced
vibration of asymmetric sandwich panels.
The basic assumptions made with three-layer sandwich panels are as follows:
1. the face sheets both stretch and bend along the face sheet-core interface;
2. transverse shear and rotatory inertia e ects are neglected in the face sheets;
3. the core is thick compared with the face sheet, and the transverse shear deformation
is included;
4. a \dilatational term" is introduced to allow waves to propagate in-plane in the core.
The displacement functions are assumed as follows and are identical to those given by
Smolenski and Krokosky [3],
u2 =
(us +ua)
z h2
@w
2
@x
cos(kxx); w2 = (ws +wa) sin(kxx); (3.1)
uc =
us + 2zh ua
+ cos
zh
cos(kxx); wc =
2z
h ws +wa
sin(kxx); (3.2)
u1 =
(us ua)
z + h2
@w
1
@x
cos(kxx); w1 = (wa ws) sin(kxx); (3.3)
20
Figure 3.2: Symmetric and anti-symmetric face sheet displacements
where uj and wj are the in-plane and transverse displacements for the face sheet j, as
shown in Fig. 3.1. uc and wc are the in-plane and transverse displacements for the core.
The subscripts s and a denote the displacements caused by symmetric and anti-symmetric
motions, respectively, as shown in Fig. 3.2. cos( z=h) is the dilatational term and kx is
the wave number for the waves in the panel in the x axis direction.
The transverse displacement functions are used to characterize the transverse defor-
mation as either symmetric, with respect to the middle surface z = 0, or anti-symmetric,
with respect to that surface z = (h=2). The in-plane displacements of the face sheets are
obtained by making z = (h=2) in the core displacement, which gives the displacement at
the interface between core and face sheets and adding a term caused by bending, which is
zero at the interface between the core and face sheets.
The strains are obtained from the displacement functions in Eqs. (3.1) (3.3):
"x = @u@x6= 0; for the face sheets; (3.4)
21
"x = @u@x6= 0; "z = @w@z 6= 0; xz = @w@x + @u@z 6= 0; for the core. (3.5)
If all three principal axes for the orthotropic material are aligned with the three axes
of the coordinate system, then the elastic potential energy U can be written as follows with
"y = xy = yz = 0,
U = 12
Z Z
[C11"2x + 2C13"x"z + 2C15"x xz +C33"2z + 2C35"z xz +C55 2xz] dzdx; (3.6)
where Cij is the elastic sti ness constants of the orthotropic material.
Substitution of Eq. (3.4) into Eq. (3.6), yields the elastic potential energy Uj for the
face sheets,
Uj = 12
Z Z "
Ctj11
@u
j
@x
2#
dzdx; j = 1; 2; (3.7)
where Ctj11 is the elastic sti ness constants of the face sheet j.
Similarly, for the orthotropic core, the elastic potential energy Uc becomes,
Uc =
Z Z "C
11
2
@u
c
@x
2
+C13@uc@x @wc@z + C332
@w
c
@z
2
+ C552
@u
c
@z +
@wc
@x
2#
dzdx;
(3.8)
where Cij is the elastic sti ness constants of the core.
If the three axes of the coordinate system are not completely aligned with the three
principal axes of the orthotropic material, as shown in Fig. 3.3, the sti ness in the rotated
22
Figure 3.3: The rotated axis system of the orthotropic material
axis system becomes Cij instead of Cij,
C11 = l4C11 + 2l2m2(C12 + 2C66) +m4C22; (3.9)
C13 = l2C13 +m2C23; (3.10)
C15 = C35 = 0; (3.11)
C33 = C33; (3.12)
C55 = m2C44 +l2C55; (3.13)
where, denotes the angle of rotation, l = cos and m = sin . The details of the derivation
of the sti nesses in the rotated-axis are given in Appendix A.
The kinetic energies are de ned as follows, neglecting the rotational energies,
T = 12
Z Z
!2(u2 +v2 +w2) dzdx: (3.14)
23
The virtual work done by the pressure loads can be expressed as,
W =
Z
[p1w1 p2w2] dx =
Z
[(p1 p2)wa (p1 +p2)ws] dx: (3.15)
Lagrange?s equations are used to obtain the governing equations,
d
dt(
@T
@ _qr)
@T
@qr +
@U
@qr =
@W
@qr; (3.16)
where qr is the generalized displacement which includes us, ws, and for the symmetric
motion, and ua, wa, for the anti-symmetric motion.
The resultant matrix equations are,
2
66
66
66
66
66
66
66
66
4
D11 D12 D13 D14 D15
D12 D22 D23 D24 D25
D13 D23 D33 0 0
D14 D24 0 D44 D45
D15 D25 0 D45 D55
3
77
77
77
77
77
77
77
77
5
8>
>>>
>>>>
>>>>
>>>
><
>>>>
>>>>
>>>
>>>>
>:
ws
us
wa
ua
9>
>>>
>>>>
>>>
>>>>
>=
>>>>
>>>>
>>>
>>>>
>;
=
8
>>>>
>>>>
>>>
>>>>
><
>>>>
>>>>
>>>
>>>>
>:
(p1 +p2)
0
0
(p1 p2)
0
9
>>>>
>>>>
>>>
>>>>
>=
>>>>
>>>
>>>>
>>>>
>;
; (3.17)
with, D11 =
8
>>><
>>>:
(Ct111t31 +Ct211t32)k4x=3 +C55hk2x=3 ( t1skt31 + t2skt32)!2k2x=3
( ch=3)!2 ( t1skt1 + t2skt2)!2 + 4C33=h
9>
>>=
>>>;;
D12 = (Ct111t21 +Ct211t22)k3x=2 2C13kx + ( t1skt21 + t2skt22)!2kx=2;
D22 = (Ct111t1 +Ct211t2)k2x +C11hk2x ( t1skt1 + t2skt2)!2 ch!2;
24
D13 = 4C13kx= 4C55kx= ; D23 = 2C11hk2x= 2 ch!2= ;
D33 = C11hk2x=2 + 2C55=(2h) ch!2=2;
D14 = (Ct111t31 Ct211t32)k4x=3 ( t1skt1 t2skt2)!2 ( t1skt31 t2skt32)!2k2x=3;
D15 = D24 = (Ct111t21 Ct211t22)k3x=2 + ( t1skt21 t2skt22)!2kx=2;
D25 = (Ct111t1 Ct211t2)k2x ( t1skt1 t2skt2)!2;
D44 =
8>
>><
>>>:
(Ct111t31 +Ct211t32)k4x=3 +C55hk2x ( t1skt31 + t2skt32)!2k2x=3
ch!2 ( t1skt1 + t2skt2)!2
9>
>>=
>>>;;
D45 = (Ct111t21 +Ct211t22)k3x=2 + 2C55kx + ( t1skt21 + t2skt22)!2kx=2;
D55 =
8>
>><
>>>:
(Ct111t1 +Ct211t2)k2x +C11hk2x=3 + 4C55=h
( t1skt1 + t2skt2)!2 ch!2=3
9>
>>=
>>>;;
where tjsk, tj denote the mass density and thickness of the face sheet j; and c, h denote
the mass density and thickness of the core.
The solutions for the transverse displacements can be written in terms of the sound
pressure loads,
ws = F11jDj[ (p1 +p2)] + F41jDj[(p1 p2)]; (3.18)
wa = F14jDj[ (p1 +p2)] + F44jDj[(p1 p2)]; (3.19)
where jDj is the determinant of the matrix D, and Fij and is the cofactor of element Dij.
Since the matrix D is a symmetric matrix, then the cofactors must satisfy Fij = Fji.
25
It is convenient to introduce the cofactor ratios,
s1 = F41F
11
; s4 = F14F
44
: (3.20)
After rearrangement, Eqs. (3.18) and (3.19) become,
zs(i!ws) = (p1 +p2) +s1(p1 p2); (3.21)
za(i!wa) = s4(p1 +p2) + (p1 p2); (3.22)
where za, zs are the impedances, za =jDj=(i!F44), and zs =jDj=(i!F11).
Damping is incorporated by allowing the sti ness constants in the material to become
complex,
C ij = Cij(1 +i ); (3.23)
where is the energy loss factor of the material.
For an isotropic material, a special case of an orthotropic material, the sti ness con-
stants are described in terms of the Lame constant and the shear modulus ,
C11 = C22 = C33 = + ; C12 = C13 = C23 = ; C44 = C55 = C66 = : (3.24)
26
Figure 3.4: Components of pressure elds on a sandwich panel
3.3 Sound transmission loss
Consider a sandwich panel of in nite extent, separating two semi-in nite air spaces as
shown in Fig. 3.4. All of the sound waves shown are assumed to be plane waves. Let a
pressure wave be incident on the face sheet 1 at an angle .
The incident re ected, radiated, and transmitted sound pressures can be expressed as,
pinc(x;z) = Pinc exp[i(!t kxsin kzcos )]; for z< 0; (3.25)
pref(x;z) = Pref exp[i(!t kxsin +kzcos )]; for z< 0; (3.26)
prad(x;z) = Prad exp[i(!t kxsin +kzcos )]; for z< 0; (3.27)
ptra(x;z) = Ptra exp[i(!t kxsin kzcos )]; for z> 0; (3.28)
where Pinc, Pref, Prad and Ptra are the amplitudes of the incident, re ected, radiated,
and transmitted sound pressures, respectively. k is the wave number of sound in air. The
27
pressure in the re ected wave is assumed to be equal in magnitude to the pressure in the
incident wave.
As a result of the matching of the face sheet velocity and the acoustic eld velocity
at the interface, the radiated pressure can be determined from the acoustic momentum
equation,
@prad
@z
z=0
= i air! _w1; (3.29)
where air is the mass density of air, and _w1 is the transverse velocity of the face sheet 1.
Integrating Eq. (3.29) with respect to z, we have
prad = zair _w1cos with zair = aircair; (3.30)
where zair is the acoustic impedance of air and cair is the speed of sound in air.
Similarly, the transmitted pressure produced by the transverse motion of the face sheet
2 is,
ptra = zair _w2cos ; (3.31)
where _w2 is the transverse velocity of the face sheet 2.
The pressure load on the face sheet 1 is,
p1 = (pinc +pref +prad)jz=0: (3.32)
28
On the transmission side, the acoustic eld pressure is simply the transmitted pressure.
The pressure load on the face sheet 2 is,
p2 = ptrajz=0: (3.33)
Substituting Eq. (3.30) into Eq. (3.32), and Eq. (3.31) into Eq. (3.33), yields the
incident and transmitted pressures,
2pinc = p1 + zaircos ( _wa _ws); (3.34)
ptra = p2 = zaircos ( _wa + _ws): (3.35)
Eliminating p1 and p2 from Eqs. (3.34) and (3.35), gives the equations of the transverse
velocities in terms of the incident pressure,
2
66
64
zs + 2zair=cos 2s1zair=cos
2s4zair=cos za + 2zair=cos
3
77
75
8>
>><
>>>
:
_ws
_wa
9>
>>=
>>>
;
=
8>
>><
>>>
:
2pinc(s1 1)
2pinc(1 s4)
9>
>>=
>>>;: (3.36)
The impedances and cofactor ratios are evaluated by replacing wave-number kx by ksin in
Eq. (3.17).
29
The sound transmission coe cient can be evaluated in terms of the impedances and
the cofactor ratios,
( ; ) =
ptra
pinc
2
=
(1 s4)zs=(2z0air) (1 s1)za=(2z0air)h
1 +zs=(2z0air)
ih
1 +za=(2z0air)
i
s4s1
2
with z0air = zaircos ; (3.37)
( ; ) =
ptra
pinc
2
=
zs=(2z0air) za=(2z0air)h
1 +zs=(2z0air)
ih
1 +za=(2z0air)
i
s4s1
2
for s4zs = s1za: (3.38)
In practice, sound waves are usually incident upon a structure from many angles si-
multaneously. Therefore an idealized random incidence model is usually assumed, in which
plane waves of equal amplitude are incident from all directions with equal probability and
which have random phases. The random incidence transmission coe cient, , is obtained
by averaging ( ; ) over all angles of incidence and rotation as follows,
=
R2
0
R lim
0 ( ; ) sin cos d d
R2
0
R lim
0 sin cos d d
: (3.39)
Based on eld and laboratory measurements [38], the limiting angle lim is usually assumed
to be 78o.
For sandwich panels with isotropic face sheets and cores, the sti ness constants are
independent of the angle of rotation. Then the random incidence transmission coe cient
becomes,
=
R lim
0 ( ; ) sin cos d R
lim
0 sin cos d
: (3.40)
30
Finally, the sound transmission loss is de ned by
TL = 10log10(1 ) dB. (3.41)
3.4 Symmetric sandwich panels
For sandwich panels with identical face sheets, one has
Ct111 = Ct211 = Ct; t1sk = t2sk = t; t1 = t2 = t: (3.42)
Then the symmetric and anti-symmetric motions are uncoupled naturally,
D14 = D15 = D24 = D25 = 0; s1 = s4 = 0: (3.43)
The governing matrix equations for symmetric and anti-symmetric motions can be
written as follows.
[Ms]
8>
>>>
>>><
>>>
>>>>
:
ws
us
9>
>>>
>>>=
>>>
>>>>
;
=
2
66
66
66
64
D11 D12 D13
D12 D22 D23
D13 D23 D33
3
77
77
77
75
8
>>>>
>>><
>>>
>>>>
:
ws
us
9
>>>>
>>>=
>>>
>>>>
;
=
8
>>>>
>>><
>>>
>>>>
:
(p1 +p2)
0
0
9
>>>>
>>>=
>>>
>>>>
;
; (3.44)
D11 = 2Ctt3k4x=3 + 4C33=h+C55hk2x=3 2mt2!2k2x=3 ch!2=3 2m!2;
D12 = Ctt2k3x 2C13kx +mt!2kx; D13 = 4C13kx= 4C55kx= ;
31
D22 = 2Cttk2x +C11hk2x 2m!2 ch!2; D23 = 2C11hk2x= 2 ch!2= ;
D33 = C11hk2x=2 + 2C55=2h ch!2=2; and
[Ma]
8>
>><
>>>
:
wa
ua
9>
>>=
>>>
;
=
2
66
64
D44 D45
D45 D55
3
77
75
8>
>><
>>>
:
wa
ua
9>
>>=
>>>
;
=
8>
>><
>>>
:
p1 p2
0
9>
>>=
>>>
;
; (3.45)
D44 = 2Ctt3k4x=3 +C55hk2x 2mt2!2k2x=3 ch!2 2m!2;
D45 = 2C55kx Ctt2k3x +mt!2kx;
D55 = 2Ctk2xt+C11hk2x=3 + 4C55=h 2(m+ ch=6)!2;
where the surface mass density of a single face sheet is m = tt.
The symmetric and anti-symmetric wave impedances for transverse motion are,
zs = (p1 +p2)i!w
s
= jMsj
i!
D22 D23
D23 D33
; (3.46)
za = (p1 p2)i!w
a
= jMaji!D
55
= 1i!
D44 D
245
D55
!
: (3.47)
The contributions of anti-symmetric and symmetric motions to the sound transmission
coe cients can be evaluated separately,
a( ; ) =
1
1 +za=(2z0air)
2
; s( ; ) =
1
1 +zs=(2z0air)
2
: (3.48)
32
The sound transmission coe cient due to anti-symmetric and symmetric motions is,
( ; ) =
1
1 +zs=(2z0air)
1
1 +za=(2z0air)
2
: (3.49)
If, under particular conditions, zs za, in the frequency range of interest, the sound
transmission coe cient can be approximated by,
( ; ) a( ; ): (3.50)
If the two wave impedances are nearly equal in sign and magnitude in certain frequency
bands, high transmission loss values are expected in those bands.
The expressions for the governing equations, Eqs. (3.44) and (3.45), the sound trans-
mission coe cient, Eq. (3.49) for symmetric sandwich panels are equivalent to those given
by Moore and Lyon [6]. Their expression for the di use eld transmission coe cient is
equivalent to Eq. (3.39). However, they neglected the contribution of C66 on the rotated
axis sti ness C11 (see Eq. (3.9)).
3.5 Numerical results
The wave speed and wave impedance for panels provide a way to predict sound trans-
mission characteristics of the panels. The wave speed in a panel is de ned by the wave
number in the panel and the circular frequency, c = !=kp. In the absence of damping, zeros
occur in the wave impedance, where the trace wave speed matches the wave speed for freely
33
propagating waves in the panel, ctrace = c. The trace wave speed is de ned by the trace
wave number in the panel and the circular frequency, and depends on the angle of incidence
of the acoustic plane wave,
ctrace = c0sin ; ktrace = ksin ; ctrace = !k
trace
: (3.51)
The trace wave speed is always greater than c0, the speed of sound in air. It is equal to c0
at grazing incidence, = 90o, and becomes in nite at = 0o. Thus, coincidence may occur
when the wave speed in the panel is greater than or equal to the speed of sound in air.
The calculated wave numbers for anti-symmetric waves in a symmetric sandwich panel
with a honeycomb core are shown in Fig. 3.5. The wave numbers are evaluated from the
zeros in the wave impedances assuming zero damping in both the face sheets and core. The
face sheets of panel A were assumed to be isotropic and the honeycomb core was assumed
to be orthotropic. The proprieties of panel A are given in Table 3.1. The wave numbers
for the orthotropic core are dependent on the propagation direction of the incident acoustic
wave relative to the principal axes in the honeycomb core. The four solid curves denote the
wave numbers along the directions where the angles of rotation are equal to 0o, 30o, 60o
and 90o, from top to bottom in the gure.
Two parallel dotted lines are indicated in Fig. 3.5. The lower dotted line corresponds
to the wave number kl for pure bending of the entire panel. The upper dotted line represents
the wave number ku for exural waves propagating in a single face sheet loaded with half
34
Figure 3.5: Predicted wave numbers for anti-symmetric waves in panel A
the mass of the core.
kl =
"
(2 tt+ ch)!2
Ct(h2t=2 +ht2 + 2t3=3)
#1=4
; ku =
"
( tt+ ch=2)!2
Ctt3=12
#1=4
: (3.52)
It is seen that the wave number of the rst propagating mode for anti-symmetric motion
of the panel asymptotically approaches the lower line for decreasing frequencies. In the high
frequency region the upper line is the asymptote. The di erences between the four solid
curves become smaller when the curves approach either the lower or upper lines. Thus, the
anti-symmetric motion of the panel is mainly determined by the pure bending sti ness of
the entire panel in the low frequency region, and by the bending sti ness of one face sheet
loaded with half the mass of the core in the high frequency region.
35
The wave numbers for symmetric waves in panel A along the four directions were also
calculated. It was found that the wave numbers for symmetric waves are complex below
6000 Hz.
The predicted wave speeds for anti-symmetric and symmetric waves in panel A are
shown in Figs. 3.6 and 3.7. The dotted horizontal lines indicate the speed of sound in air
in both gures. It was found that anti-symmetric waves of panel A travel fastest along
= 90o and slowest along = 0o. The anti-symmetric wave speeds exceed the speed of
sound in air near to 200 Hz.
No freely propagating symmetric waves in the four directions exist below 6000 Hz.
The near vertical lines indicate the freely propagating symmetric waves in the panel, which
travel at high speeds and which depend considerable on frequency. This behavior is similar
to a double wall resonance.
fw = 12
s
C33
h(2 tt+ ch=3): (3.53)
For panel A, the double wall resonance frequency is about 5200 Hz. At double wall
resonance, the face sheet motions are uniform, which corresponds to a trace wave number of
zero and to an in nite wave speed. An incident wave at normal incidence excites symmetric
panel motion at frequencies near the double wall resonance. The wave speeds for symmetric
waves do not monotonically increase with increasing frequency as those for anti-symmetric
36
Figure 3.6: Predicted wave speeds for anti-symmetric motion of panel A
Figure 3.7: Predicted wave speeds for symmetric motion of panel A
37
waves do. The speeds of freely symmetric waves are greater than the speed of sound in air
in the frequency range of interest.
In the absence of damping, the wave impedances are imaginary. The wave impedances
were calculated for the two waves in = 0o, 90o directions at oblique incidence = 76:5o,
as shown in Fig. 3.8. It is seen that anti-symmetric coincidences occur near to 250 Hz
and no symmetric coincidence occurs in the frequency range of interest. The symmetric
impedance is insensitive to the propagation direction in panel A. A dip in the symmetric
wave impedance occurs near to 4000 Hz, below the double wall resonance frequency. The
symmetric wave impedances are much higher than the anti-symmetric wave impedances
below 2000 Hz for this oblique angle of incidence. The sound transmission loss values
caused by anti-symmetric or symmetric wave motions are compared with the predictions of
sound transmission loss generated by both motions for panel A, as shown in Fig. 3.9.
Since most measured sound transmission loss results for sandwich panels are presented
in one-third octave bands, the predicted sound transmission loss values were frequency-
averaged for comparison purpose. The transmission loss of panel A is dominated by the
anti-symmetric motion in the frequency range of interest. The contribution of the symmetric
motion is negligible up to 5000 Hz. The coincidence dip near to 250 Hz is associated
with anti-symmetric motion of panel A, which is consistent with the wave speed curves.
The sound transmission loss values of panel A were calculated and are compared with the
measured transmission loss results given in Ref. [6], as shown in Fig. 3.10. The predictions
were generated for two di erent values of energy loss factor of the core.
38
Figure 3.8: Wave impedances along two principal directions for panel A
Figure 3.9: Calculated sound transmission loss values of panel A without damping
39
Figure 3.10: Predicted and measured sound transmission loss values of panel A
The e ects of core damping are noticeable at frequencies above the onset of coincidence,
where the core sti nesses a ect the sound transmission loss of the panel. The transmission
loss curves do not return to the mass law curve at frequencies above coincidence, and are
separated by more than 10 dB.
Theoretically, when the wave impedances for anti-symmetric and symmetric motions
are equal in sign and magnitude, there is no transmitted sound pressure. So the transmission
loss can exceed mass law values when the two wave impedances have similar values. This
behavior has been demonstrated analytically and experimentally for a honeycomb sandwich
panel in Ref. [6]. This acoustical behavior is a result of the cancellation of symmetric and
anti-symmetric motions of the face sheets. The honeycomb core was uncommonly orientated
so that the cells lay in the plane of the panel. The wave impedances of two motions along
40
Table 3.1: Properties of sandwich panels A and B
Property Panel A Panel B
Face Surface mass density m (kg/m2) 4.17 5.7
Sheet Sti ness Ct11 (GPa) 7.0 7.0
Thickness t (mm) 6.35 9.53
Density c (kg/m3) 28 21
Thickness h (mm) 76.2 84.1
C11 (MPa) 4.0 0.4
C22 (MPa) 4.0 95
C33 (MPa) 370 0.4
Core C44 (MPa) 50 7.6
C55 (MPa) 23 0.17
C66 (MPa) 0.2 4.2
C12 = C13 = C23 (MPa) 0.5 0.04
Energy loss factor c 0.03, 0.05 0.03
= 0o, 90o at oblique incidence = 76:5o for panel B are shown in Fig. 3.11. The rst
symmetric coincidence occurs near to 200 Hz, and the second symmetric coincidence and
the rst anti-symmetric coincidence are around 2000 Hz. It is seen that, in the frequency
region f > 400Hz, the wave impedances are nearly equal in sign and magnitude. Thus high
transmission loss values are expected above 400 Hz.
Sound transmission loss values of panel B were calculated and are compared with the
measured results given in Ref. [6], as shown in Fig. 3.12. The predictions were generated
for three di erent values of energy loss factor of the core. It is seen that damping does not
have a noticeable e ect on the sound transmission loss of panel B in the region 200 Hz <
f < 2000 Hz, between two symmetric coincidences.
41
Figure 3.11: Wave impedances along two principal directions for panel B
Figure 3.12: Predicted and measured sound transmission loss values of panel B
42
3.6 Other TL models for asymmetric sandwich panels
Dym and Lang [7] have presented a sound transmission loss analysis for asymmetric
sandwich panels with orthotropic cores. In their model, the displacement functions of the
face sheets and the core are written as,
u2 =
u h2 @ w@x
+
~u h2 @ ~w@x
z h2
@w
2
@x ;w2 = ~w + w; (3.54)
uc =
u h2 @ w@x
+ 2zh
~u h2 @ ~w@x
+gcos
z
h
;wc = 2zh w + ~w; (3.55)
u1 =
u h2 @ w@x
~u h2 @ ~w@x
z + h2
@w
1
@x ;w1 = ~w w; (3.56)
where w, u and g are for the symmetric motion, and ~u, ~w are for the anti-symmetric motion,
respectively. They assumed that all displacements are in phase,
u; w; g; ~u; ~w exp[i(!t kxsin )]; (3.57)
while the transverse displacements and the in-plane displacements are assumed to be out
of phase in other works [2, 3, 6].
It is noted that the displacement functions assumed in section 3.2 are equal to those
in Eqs. (3.54) (3.56), if
us = u h2 @ w@x; g = ; ws = w; and ua = ~u h2 @ ~w@x; wa = ~w: (3.58)
43
Dym and Lang applied equations the same to Eqs. (3.7), (3.8), (3.14) and (3.15) to
evaluate the elastic potentials, the kinetic energies of the sandwich panel, and the potential
for surface pressures, respectively. But they introduced a shear correction factor in the
transverse shear strain of the core, xz. Applying Hamilton?s principle, they derived the
governing di erential equations for the forced motion,
2
66
66
66
66
66
66
66
66
4
B11 B12 B13 B14 B15
B12 B22 B23 B24 B25
B13 B23 B33 0 0
B14 B24 0 B44 B45
B15 B25 0 B45 B55
3
77
77
77
77
77
77
77
77
5
8>
>>>>
>>>
>>>>
>>>
><
>>>
>>>>
>>>
>>>>
>>:
w
u
g
~w
~u
9>
>>>
>>>>
>>>>
>>>
>=
>>>
>>>>
>>>
>>>>
>>;
=
8>
>>>
>>>>
>>>>
>>>
><
>>>>
>>>>
>>>
>>>>
>:
(p1 +p2)
0
0
(p1 p2)
0
9>
>>>
>>>>
>>>
>>>>
>=
>>>>
>>>>
>>>
>>>>
>;
: (3.59)
The elements of the matrix B are given in Appendix B.
Since the in-plane displacement functions in their model are not the same as those used
in our model, Eq. (3.1), and the factor is in the elastic potential of the core, the matrix
B is not equivalent to the matrix D in Eq. (3.17). Thus the wave impedances computed
from this model are not the same as those from our model.
Dym and Lang developed the sound transmission coe cient of asymmetric sandwich
panels using the same approach described in Sec. 3.3. However, they neglected the depen-
dence of the sti ness of orthotropic materials on the angle of rotation between the axes
of the coordinate system and the principal axes of the orthotropic materials. Then they
44
Table 3.2: Properties of sandwich panel C
Property Panel C
Density 1 = 2 (kg/m3) 985.0
Face Young?s modulus E1 = E2 (GPa) 4.71
Sheet Thickness t1 = t2 (mm) 3.68
Poisson?s ration 1 = 2 0.3
Loss factor 1 = 2 0.01
Density c (kg/m3) 30
Thickness h (mm) 50.7
Core Lame constant (MPa) 21.57
Shear modulus (MPa) 2.14
Energy loss factor c 0.03
did not consider the dependence of the angle of rotation in the random incidence transmis-
sion coe cient calculation. The random incidence transmission coe cient is de ned Eq.
(3.40), so it is only valid for sandwich panels with isotropic materials.
No experimental data for sandwich panels with asymmetric con gurations are available
in the literature. Dym and Lang [7] have studied the e ects of the face sheet thicknesses of
panel C. The properties of panel C are given in Table 3.2.
The predicted wave impedances for two wave motions at oblique incidence = 76:5o
from the present model and Dym and Lang?s model are shown in Fig. 3.13. The predictions
are almost the same except near coincidence. The symmetric coincidence of panel C is near
to 1600 Hz, while the anti-symmetric coincidence is above 8000 Hz for this oblique incidence.
It was found that in the frequency range 6300 Hz >f > 3150 Hz, the two wave impedances
are nearly equal in sign and magnitude, at this oblique coincidence. Thus high transmission
loss values are expected in this region for panel C.
45
Figure 3.13: Calculated wave impedances for symmetric panel C
The transmission loss values of panel C with asymmetric con gurations were calculated
and are shown in Figs. 3.14 and 3.15. To avoid the e ects introduced by adding mass, the
sum of the face sheet thicknesses, t1+t2, was xed, and the ratio of the face sheet thicknesses
t1=t2 was varied for panel C. Since the face sheets and core of panel C were assumed to
be isotropic, the discrepancy between the two models is caused by the di erences in the
displacement functions and the elastic potential of the core. The two models produce similar
transmission loss values. At low frequencies, below where symmetric coincidence occurs for
the panels the predicted sound transmission loss results follow the eld incidence mass law
transmission loss curve.
The predicted sound transmission loss values exceed mass law values in some regions,
especially for the symmetric con guration, t1 = t2. This behavior is the result of the
46
Figure 3.14: Predicted sound transmission loss values of panel C from Dym and Lang?s
model
Figure 3.15: Predicted sound transmission loss values of panel C from the present analysis
47
cancellation of the symmetric and anti-symmetric motions of the face sheets. The symmetric
con guration enhances the cancellation. Asymmetric con gurations of the sandwich panel,
t1 6= t2, exhibit some improvement in sound isolation near coincidence.
3.7 Other governing equations for anti-symmetric motion
When the transverse vibration is dominated by anti-symmetric transverse vibration,
the governing equation for transverse vibration of symmetric sandwich panels becomes,
q =
2
66
64
2Ctt3k4x=3 +C55hk2x 2mt2!2k2x=3 !2
(2C55kx Ctk3xt2+m!2kxt)22Cttk2
x+C11hk2x=3+4C55=h 2(m+ ch=6)!2
3
77
75w; (3.60)
where the pressure step across the panel is, q = p1 p2, and the surface mass density is
= ch+ 2m.
The governing equation can be written in the alternative form,
h
Ak6x +Bk4x +Ck2x +D
i
w =
1 + C11h6C
tt
k2x + 2C55hC
tt
1C
tt
m+ ch6
!2
q; (3.61)
where; A = Ctt
3
6
1 + 2C11h3C
tt
; D = !
2
2Ctt
!2
2m+ ch3
4C55h
;
B = C55
"
(t+h)2
h +
t2
3h +
C11h2
6Ctt
#
!2
"
t2
3
m+ ch3
mthC119C
t
#
;
C = !2
(
mt2!2
6tCt
2
ch
3 +m
1 + hC116tC
t
C55tC
t
"
ch2
6 +m
(t+h)2
h +
2t2
3h
!#)
:
48
When the sti nesses of the face sheets are much greater than the sti nesses of the core,
and high order !2 terms are negligible, the governing equation becomes,
h
Ak6x +Bk4x +Ck2x +D
i
w =
k2x + 2C55hC
tt
q; (3.62)
where, A = Ctt
3
6 ; B = C55
"
(t+h)2
h +
t2
3h
#
; C = !2f g; D = !
2
2Ctt
4C55h
:
Mead and Markus [23] presented an equation of motion in terms of the transverse
displacement for a three-layer damped sandwich beam with a viscoelastic core. In their
analysis, rotatory inertia was ignored and elasticity relationships were applied to derive the
sixth-order di erential equation:
@6w
@x6 g(1 +Y)
@4w
@x4 +
Dt
@4w
@x2@t2 g
@2w
@t2
!
= 1D
t
@2q
@x2 gq
!
; (3.63)
where q is the total load, Dt is the bending rigidity of a single face sheet, g is the shear
parameter, Y is the geometric parameter and is the surface mass density of the panel.
For symmetric sandwich panels, the parameters can be evaluated as,
Dt = Ett
3
6(1 2t ); g =
2C55
ht
1 2t
Et ; Y = (h+t)
2 Ett
2(1 2t )Dt; = ch+ 2m: (3.64)
49
The governing Eq. (3.63) can be written as follows,
h
A0k6x +B0k4x +C0k2x +D0
i
w =
"
k2x + 2C55ht 1
2t
Et
#
q; (3.65)
where, A0 = Ett
3
6(1 2t ); B
0 = C55t2
3h +
C55(t+h)2
h ; C
0 = !2; D0 = 2C55 !2(1 2t )
Ctth :
The equation above is identical to Eq. (3.62), which is an approximate expression for Eq.
(3.61) under certain conditions.
The impedance is given by,
z = qi!w = 1i!
(
!2 + Dt[k
6 +g(1 +Y)k4]
k2 +g
)
: (3.66)
Nilsson and Nilsson [37] have presented an equation of motion for a three-layer sym-
metric sandwich beam. In their analysis, rotatory inertia e ects were considered and energy
relationships were applied to derive the sixth-order di erential equation.
2D2@
6w
@x6 +
2D2
D1 I
@6w
@x4@t2
+ 2D2D
1
+ I GehD
1
@4w
@x2@t2 +Geh
@4w
@x4 +
D1
@2w
@t2
!
+ I D
1
@
4w
@t4 =
1 + 2D2D
1
@2q
@x2 +
Geh
D1 q +
I
D1
@2q
@t2; (3.67)
where D1 is the bending rigidity of the entire beam, D2 is the bending rigidity of a single
face sheet, I is the mass moment of inertia, Ge is the equivalent shear sti ness, and is
50
the surface mass density of the beam.
D1 = E1h
3
12(1 2c) +
Et1
(1 2t )
h2t
2 +ht
2 + 2t3
3
!
; D2 = E
t1t3
12(1 2t );
I = ch
3
12 + t
h2t
2 +ht
2 + 2t3
3
!
; Ge = C55(t+h)
2
h2 ; = ch+ 2m:
The governing equation (3.67) can be written as ,
"
2D2k6 +
Geh 2D2D
1
I !2
k4
+ 2D2D
1
+ I GehD
1
k2!2
Geh I !2
!2
D1
#
w
=
G
eh
D1 +
1 + 2D2D
1
k2 I D
1
!2
q: (3.68)
The impedance is,
z = qi!w = 1i!
8<
: !
2 + 2D2k
6 +
Geh 2D2D1 I !2
k4 I GehD1 k2!2
Geh
D1 +
1 + 2D2D1
k2 I D1!2
9=
;: (3.69)
The sound transmission coe cient can be evaluated by Eq. (3.48). The predicted sound
transmission loss values of panel A obtained by using the wave impedances computed from
the three governing equations are compared with the measured results, as shown in Fig.
3.16. The energy loss factor in the core was assumed to be 0.03 for all three cases. The
three governing equations generate similar transmission loss predictions for panel A in the
region between 250 Hz and 4000 Hz, where the anti-symmetric transverse displacement is
dominant.
51
Figure 3.16: Predicted sound transmission loss values of panel A made using governing
equations
3.8 Conclusions
The wave impedance analysis provides the sound transmission loss of in nite sand-
wich panels. If damping is included in the theoretical calculations, the e ect is noticeable
above coincidence, where the sti nesses of the face sheets and core a ect the sound trans-
mission loss of the panels. For sandwich panels with sti cores, typical honeycomb cores,
anti-symmetric coincidence occurs at low frequencies, and symmetric coincidence is at high
frequencies. Thus, in the frequency range of interest, the anti-symmetric wave motion is
dominant. The sound transmission loss caused by anti-symmetric wave motion provides
an accurate approximation of the sound transmission loss produced by anti-symmetric and
symmetric motions. For sandwich panels with soft cores, the sound transmission loss values
52
can exceed the mass law values because of the cancellation of symmetric and anti-symmetric
motions of the face sheets. This acoustical behavior occurs in the frequency region be-
tween symmetric and anti-symmetric coincidence. Asymmetric con gurations lessen the
cancellation of symmetric and anti-symmetric motions of the face sheets, and produce some
improvement in sound isolation near coincidence.
53
Chapter 4
Statistical energy analysis
4.1 Introduction
The modal vibration of a nite panel consists of standing waves. Each standing wave
can be considered to be composed of two wave types: forced traveling waves set up by exter-
nal loads and free (re ected) waves that are generated by secondary and necessary to satisfy
at the boundaries. There is little hope to obtain a detailed classical solution for a nite
panel because of the existence of many modes of vibration. For panels excited by broadband
noise, the detailed response characteristics may be neglected and statistical properties such
as mean square values and power spectra can be used to provide an approximate measure
of vibration.
Statistical energy analysis (SEA) was developed from studies of the power ow of two
randomly excited linearly coupled oscillators in the 1960?s. It was found that for conservative
coupling the power ow is proportional to the average modal energy di erence between two
oscillators. A thermal argument concludes that the products of modal density and coupling
loss factor are equal in the two coupled oscillators [11]. The modal averaged radiation
resistance of a ba ed simply supported single-layer panel excited by a reverberant eld was
derived by Maidanik [13]. Later this analysis was extended from systems consisting of two
subsystems to multiple subsystems [39]. Crocker and Price [18] presented a three-subsystem
54
SEA model to estimate the sound transmission loss of a single-layer panel with both resonant
and non-resonant modes under consideration. Price and Crocker [19] presented a ve-
subsystem SEA model to estimate sound transmission through a double wall.
In this chapter, the assumptions and concepts of SEA are brie y reviewed. A three-
subsystem SEA model for a transmission suite is introduced. An analytical expression for
modal densities of sandwich panels is derived from a sixth-order governing equation. The
expression is compared with other analytical expressions for modal densities of sandwich
panels [22, 41]. The sound transmission loss of the structure between two reverberation
rooms is developed. Then the sound transmission loss estimates for a single-layer panel and
a honeycomb sandwich panel are compared with those experimental values given in Refs.
[6, 18]. The e ects of test area of the panels and volume of the rooms? acoustic spaces on
sound transmission loss are also studied.
4.2 Assumptions and concepts
The fundamental assumptions made in SEA are:
1. the couplings between the di erent subsystems are small, linear and conservative;
2. the power ows are between the subsystems having resonance frequencies in the band
of interest;
3. the subsystems are excited by broadband random excitation;
4. the total motion is regarded as a sum of independent motions in individual modes;
55
5. equipartition of energy exists between all modes at resonance within a given frequency
band in a given subsystem.
An individual oscillator driven in steady state conditions at a single frequency has
potential and kinetic energy stored within it. The power dissipated via the damping is
related to the energy stored in the oscillator.
diss = cv _x2 = 2 !nm_x2 = 2 !nE = !n E; (4.1)
where cv is the viscous damping coe cient, is the damping ratio, !n is the radian natural
frequency, m is the oscillator mass, E is the stored energy, and is the energy loss factor.
The power dissipation concepts for a single oscillator can be extended to a collection
of oscillators in a speci ed frequency band,
diss = !n E; (4.2)
where ! is the geometric mean center frequency of the band, and is the mean energy loss
factor of all the modes in the band.
SEA is closely related to room acoustics and thermodynamics. It is assumed that
energy ows from oscillators of higher energy to those of lower energy. It was shown that
the power ow is proportional to the average modal energy di erence of two loosely coupled
56
Figure 4.1: Block diagram for power ows between the structure and the reverberant eld
and randomly excited oscillators [11] and can be expressed in power dissipation terms,
h 12i= 12n1!fhE1i=n1 hE2i=n2g; (4.3)
where n1 and n2 are the modal densities of the two groups of oscillators. nij is the coupling
loss factor, describing the ow of energy from subsystem i to subsystem j, and hEji=nj is
the modal energy of group j.
Now consider a two-subsystem model with numerous modes in each subsystem where
one subsystem is driven by external forces and the other subsystem is driven through the
coupling. The typical application is the response of a structure excited in a reverberant
eld. The model is illustrated in Fig 4.1. The steady state power ow balance equations
for the two groups of oscillators are,
in1 = diss1 + 12; in1 = ! 1E1 +! 12n1(E1=n1 E2=n2); (4.4)
in2 = diss2 12; 0 = ! 2E2 ! 12n1(E1=n1 E2=n2); (4.5)
57
where in1 is the power input to subsystem 1, the structure; in2 = 0, is the power input
to subsystem 2, the acoustic eld. n1 and n2 are the modal densities. 1 and 2 are the
internal loss factors. 12 and 21 are the coupling loss factors associated with energy ow
from subsystem 1 to 2 and subsystem 2 to 1, respectively. E1 and E2 are the vibration
energies associated with subsystems 1 and 2. All uctuating terms are assumed to be both
time- and space-averaged, and the brackets have been removed for convenience. Since the
products of the modal density and coupling loss factor are equal in each pair of subsystems
[15], ni ij = nj ji, the Eqs. (4.4) and (4.5) can be written in matrix form,
!
2
66
64
n1 1 +n1 12 n1 12
n2 21 n2 2 +n2 21
3
77
75
8>
>><
>>>:
E1=n1
E2=n2
9>
>>=
>>>; =
8>
>><
>>>:
in1
0
9>
>>=
>>>;: (4.6)
By rearranging the bottom equation of the matrix Eqs. (4.6), the steady state modal
energy ratio between the two subsystems is,
E2=n2
E1=n1 =
21
2 + 21: (4.7)
The modal energy of subsystem 2 is always less than that of subsystem 1. If 2 21,
the ratio approaches 1, which indicates that the additional damping provided to subsystem
2, the reverberant eld, will be ine ective unless 2 is about the same as 21.
The two-subsystem model can be extended to a multiple-subsystem. In the general
case, N groups of oscillators yield N simultaneous power ow balance equations which can
58
be written in matrix form. The steady state power ow balance matrix is,
!A
8
>>>>
>>>
>>>>
><
>>>>
>>>>
>>>
>:
E1=n1
E2=n2
EN=nN
9
>>>>
>>>
>>>>
>=
>>>>
>>>>
>>>
>;
=
8
>>>>
>>>
>>>>
><
>>>>
>>>
>>>>
>:
in1
in2
inN
9>
>>>>
>>>
>>>>
=
>>>>
>>>
>>>>
>;
; (4.8)
with A =
2
66
66
66
66
66
66
4
( 1 +PNj6=1 1j)n1 12n1 1Nn1
21n2 ( 2 +PNj6=2 2j)n2 2Nn2
N1nN ( N +PNj6=N Nj)nN
3
77
77
77
77
77
77
5
:
4.3 Transmission suite model
A transmission suite is considered to consist of three coupled systems as illustrated in
Fig. 4.2. The power ow balance equations for the three subsystems are written as,
in1 = diss1 + 12 + 13; in1 = !
2
66
64
1E1 + 12n1(E1=n1 E2=n2)
+ 13n1(E1=n1 E3=n3)
3
77
75; (4.9)
in2 = diss2 12 + 23; in2 = !
2
66
64
2E2 12n1(E1=n1 E2=n2)
+ 23n2(E2=n2 E3=n3)
3
77
75; (4.10)
59
Figure 4.2: Block diagram for power ows between the structure and two reverberation
rooms
in3 = diss3 13 23; in3 = !
2
66
64
3E3 13n1(E1=n1 E3=n3)
23n2(E2=n2 E3=n3)
3
77
75; (4.11)
where 13 is the power ow from subsystem 1 to subsystem 3 when there are no modes
excited in subsystem 2 in the frequency band under consideration. The non-resonant power
ow modes 13 is due to modes which are resonant outside of the frequency band under
consideration.
For transmission loss measurements, a panel is clamped between two reverberation
rooms and excited by noise in the source room, in2 = in3 = 0, as shown in Fig. 4.3.
Equations (4.9) (4.11) can be written in matrix form,
!A
8>
>>>>
>><
>>>>
>>>
:
E1=n1
E2=n2
E3=n3
9>
>>>>
>>=
>>>>
>>>
;
= !
2
66
66
66
64
a11 a12 a13
a21 a22 a23
a31 a32 a33
3
77
77
77
75
8>
>>>
>>><
>>>>
>>>
:
E1=n1
E2=n2
E3=n3
9>
>>>
>>>=
>>>>
>>>
;
=
8>
>>>
>>><
>>>
>>>>
:
in1
0
0
9>
>>>
>>>=
>>>
>>>>
;
; (4.12)
60
Figure 4.3: The transmission suite
with A =
2
66
66
66
64
( 1 + 12 + 13)n1 12n1 13n1
21n2 ( 2 + 21 + 23)n2 23n2
13n1 23n2 3n3 + 13n1 + 23n2
3
77
77
77
75
:
Then, we have the modal energies,
E1
n1 =
F11
jAj in1 =
a22 a23
a32 a33
jAj in1;
E3
n3 =
F13
jAj in1 =
a21 a22
a31 a32
jAj in1; (4.13)
where jAj is the determinant of the matrix A and Fij is the cofactor of element aij.
Hence the modal energy ratio is,
E1=n1
E3=n3 =
a22a33 a23a32
a21a32 a22a31 = 1 +
2 radn2 3n3 + ( 3n3 + radn2) 2n2
2radn22 + 13(2 rad + 2)n1n2 ; (4.14)
with 21 = 23 = rad.
61
The successful prediction of the modal energy ratio using SEA depends to a large
extent on knowledge of the modal densities, internal loss factors and coupling loss factors
associated with the subsystems.
4.4 Modal densities
The modal density of the volume of an acoustic space depends on whether the volume
is one-dimensional, two-dimensional, or three-dimensional. The modal density of a volume,
a three-dimensional enclosure, is,
n(f) = 4 f
2V
c3 +
fS
2c2 +
P
8c; (4.15)
where V is the volume of the enclosure, A is the total surface area, and P is the total length
of the edges.
The modal density of structures depends on their boundary conditions and the gov-
erning equation of motion. For simply supported panels, the wave number for a freely
propagating wave is,
kmn =
q
k2x +k2y =
q
(m =lx)2 + (n =ly)2; (4.16)
where m and n are the mode numbers, lx and ly are the dimensions of the panel.
62
Figure 4.4: Constant frequency loci for transverse wave numbers of a simply supported
panel
Then the modal density is associated with the constant frequency loci of the wave
number, as illustrated in Fig. 4.4.
n(!) = =4 k
2
( =lx)( =ly) ! =
Ap
4
dk2
d!; n(f) = 2 n(!) =
Ap
2
dk2
d!: (4.17)
The governing equation for free motion of a single-layer panel is,
D@
4w
@x4 +m
@2w
@t2 = 0; with D =
Et3
12(1 2); m = t; (4.18)
where D is the bending rigidity, m is the surface density, E is the Young?s modulus, and
is the Poisson?s ratio.
Then the wave number k must satisfy,
Dk4 m!2 = 0 or k2 = !
q
m=D: (4.19)
63
Hence, for a simply supported single-layer panel, the modal density is constant, inde-
pendent of frequency.
n(f) = Ap2
s
12(1 2)
Et2 : (4.20)
For boundary conditions other than simply supported, analytical expressions for the
wave number of free motion are not available. High order modes of free motion are less
sensitive to boundary conditions than low order modes. Thus, except for the rst several
modes, the modal density for simply supported panels provides an approximation for that
of panels with other boundary conditions.
The modal densities of sandwich panels are more complicated because not only are
they frequency dependent, but this frequency dependence is not a linear function. Clarkson
and Ranky [22] derived the square wave number, k2, from the sixth-order equation given by
Mead and Markus [23], by assuming the bending rigidity of the face sheets Dt is negligible,
k2 !
2 +!p( !)2 + 4 g2Dt(1 +Y)
2gDt(1 +Y) ; (4.21)
with Y = [h+ (t1 +t3)=2]
2E1t1E3t3
Dt(E1t1 +E3t3) ; g =
Gc
h
1
E1t1 +
1
E3t3
; Dt = E1t
31 +E3t33
12 ;
where Ej is the Young?s modulus of the face sheet j and Gc is the shear modulus of the
core; tj and h are the thickness of the face sheet j and the core, respectively; and is the
64
surface density of the sandwich panel.
Hence, n(f) = ApfgD
t(1 +Y)
1 + !
2 + 2g2Dt(1 +Y)
p( !2)2 + 4 (g!)2D
t(1 +Y)
!
: (4.22)
Ferguson and Clarkson [41] presented the modal density derived from the same sixth-
order equation,
n(f) = Ap9
P 2=3 dPd! cos
3
P1=3 dPd!sin
3
; with (4.23)
P = 3p3
(
!2
Dt +g
2 (1 +Y)2
2
)3=2
; cos( ) = 272P
"
2
27g
3(1 +Y)3 + g!2
Dt
1 +Y
3 1
#
:
It was found that the modal density presented above produced a considerable di erence
from that derived by Clarkson and Ranky [22], as shown later in this section.
For sandwich panels with sti cores, such as honeycomb cores, the anti-symmetric
motion is dominant in the frequency range of interest. The governing equation for anti-
symmetric motion of sandwich panels can be written as a cubic equation with respect to
k2,
k6 +a2k4 +a1k2 +a0 = 0: (4.24)
The standard solutions are,
fk2g1 = 13a2 + (S +T); (4.25)
65
fk2g2 = 13a2 (S +T)2 +i
p3
2 (S T); (4.26)
fk2g3 = 13a2 (S +T)2 i
p3
2 (S T); (4.27)
where S = 3
q
R+pD; T = 3
q
R pD; D = Q3 +R2;
Q = (3a1 a22)=9; R = (9a2a1 27a0 2a32)=54:
In the absence of damping, the wave number of free anti-symmetric transverse motion
is always real. Then the freely propagating wave number must satisfy the equation,
k2 = (a2=3) + (S +T): (4.28)
Hence the modal density can be obtained from,
dk2
d! =
13 da2d! + (dSd! + dTd!)
; (4.29)
with dSd! = 13S 2
dR
d! +
1
2pD
dD
d!
; dTd! = 13T 2
dR
d!
1
2pD
dD
d!
;
dD
d! = 3Q
2 dQ
d! + 2R
dR
d!;
dR
d! =
a1
6
da2
d! +
a2
6
da1
d!
1
2
da0
d!
a22
9
da2
d!;
dQ
d! =
1
3
da1
d!
2a2
9
da2
d!:
66
Figure 4.5: Wave numbers for free transverse wave in x-axis of panel A
Equation (4.24) is equivalent to the sixth-order governing equation for free motion of
sandwich panels presented by Mead and Markus [23], if
a2 = g(1 +Y); a1 = !2=Dt; a0 = g!2=Dt: (4.30)
The wave numbers and modal densities of sandwich panel A, were computed, shown
in Figs. 4.5 and 4.6. The properties of panel A are given in Table 3.1, and the dimensions
are 1.22 m 2.44 m. It is shown that the e ect of bending rigidity of the face sheets, Dt,
on wave numbers is noticeable above 2000 Hz. While the e ect of bending rigidity of the
face sheets on modal densities is apparent above 2000 Hz. The modal density presented by
Fergusan and Clarkson, Eq. (4.23), generates quite di erent data from the data obtained
67
Figure 4.6: Modal densities of free transverse wave in x-axis of panel A
from the other two equations. Some factors might be missing. Two other sixth-order anti-
symmetric equations, Nilsson and Nilsson?s, Eq. (3.67) and the one developed in previous
chapter, Eq. (3.60), also were employed to compute the wave numbers and the modal
densities. It was found that the results from these two more complicated equations are the
same as those from the sixth-order equation given by Mead and Markus, Eq. (4.30).
4.5 Internal loss factors
Internal loss factors of the volumes of acoustic spaces can be obtained from the rever-
beration time T60, the reverberation time being the time that the energy level in the volume
takes to decay to 10 6 of its original value. The internal loss factor of an acoustic volume
68
is given by,
exp( !T60) = 10 6: (4.31)
Hence,
= 2:2fT
60
: (4.32)
Analytical expressions are not generally available for internal loss factors of structural
components. Very little consistent information is readily available about the internal loss
factors of structural elements. The internal loss factor often varies from mode to mode, and
it is widely recognized that it is the major source of uncertainty in the estimation of the
dynamic response of a system.
4.6 Coupling loss factors
The coupling loss factors for the SEA transmission suite model have two classes of
factor, the structure-acoustic volume coupling loss factor and the acoustic-acoustic volume
coupling loss factor. The structure-acoustic volume coupling loss factor can be associated
with the radiation resistance of the structure.
The power radiated by a structure is given by,
= Rradhv2pi: (4.33)
69
The power dissipated due to radiation is,
= rad!Mphv2pi: (4.34)
Hence, the coupling loss factor due to radiation damping is,
rad = Rrad!M
p
: (4.35)
When a structure is excited in a reverberant acoustic eld, the radiation resistance can
be expressed as [11, 13],
Rrad = (16= ) ck2
Z Z
d~r1d~r2 (~r1;~r2) (~r1;~r2); (4.36)
where k is the wave number in the air; is the cross correlation of the vibrational eld and
is the cross correlation of the pressure eld; and and c are the mass density and the
speed of sound in air, respectively.
For free transverse waves on an in nite structure, yields
Rrad
Ap =
8
>>><
>>>:
0; kp >k
c(1 k2p=k2) 1=2; kp
>><
>>>
:
Q2
i=1[sin(xikpx) sin(yikpy)]; 0 xi >lx and 0 >yi >ly
; (4.38)
where the wave number in the panel is, kp =
q
k2px +k2py. The cross correlation of the
pressure eld is given by,
(~r1;~r2) = sinkj~r1 ~r2jkj~r
1 ~r2j
: (4.39)
Then the radiation resistance can be written as,
Rrad = (64 ck2= 2)
Z 1
0
IxIy d ; with (4.40)
Ix =
k2
px
k4
!
0
BB
B@
cos2
sin2
1
CC
CA
[ 2 (k2px=k2)]2
klx
2 ; Ix =
k2
py
k4
!Z =2
0
0
BB
B@
cos2
sin2
1
CC
CA
[(k2py=k2) ( sin )2]2
kly
2 sin d ;
=
q
f=fc; =
p
1 2;
where fc is the critical frequency of the single-layer panel. The cos2 and sin2 are to be chosen
according to whether the mode, either in x or y direction, is odd or even, respectively.
71
The resonant modes of a panel can be divided into three classes. Modes which have
wave phase speeds in both edge directions less than the speed of sound are termed \corner"
modes. Modes having wave phase speeds in one edge direction greater than the speed of
sound and wave phase speeds in the other edge direction less than the speed of sound are
termed \edge" modes. Modes having wave phase speeds greater than the speed of sound in
air are termed \surface" modes.
The approximated radiation resistance of the modes is,
kp k, kpy >k, kpx k, kpx >k, corner mode,
Rrad = 8 ck
2
k2pxk2py:
It is shown theoretically that surface modes have high radiation e ciencies. Corner
modes have lower radiation e ciencies than edge modes. The theoretical results for the
radiation and classi cation of modes can also be given a simple physical explanation. Figure
4.7 shows a typical modal pattern in a ba ed simply-supported panel. The dotted lines
represent panel nodes. For corner modes, the uid will produce pressure waves which will
travel faster in the uid than the panel transverse waves and the acoustic pressures created
by the quarter wave cells will cancelled everywhere except at the corners as shown. For edge
modes, cancellation can only occur in one edge direction and the quarter wave cells shown
will cancel everywhere except at x-edges. For surface modes, the uid cannot produce
73
pressure waves which will move fast enough to cause any cancellation. The modes radiate
from the whole surface area of a panel.
The results for the single mode can be extended to the reverberant vibrational eld
radiation resistance. The modal averaged radiation resistance of a ba ed simply supported
single-layer panel in an acoustic reverberant eld given by [13] is, as corrected in Ref. [18],
R2 rad = Ap c
8>
>>>
>>><
>>>
>>>>
:
corner + edge; f fc
with (4.42)
corner =
8
>>><
>>>:
( c a=Ap) 2(8= 4)[(1 2 2)= =p1 2]; f fc=2
;
edge = 14 2 P cA
p
(1 2) ln [(1 + )=(1 )] + 2
(1 2)3=2 ;
a = cf; c = cf
c
; =
s
f
fc:
Maidanik [13] also noted that well below the critical frequency, the radiation resistance of
a clamped panel is twice that of a simply supported panel. Later Nikiforov [44] and Berry
et al. [24] showed that this conclusion is restricted to large structures or high order modes.
Typical frequency-averaged radiation resistances of ba ed simply supported single-
layer panels are illustrated in Fig. 4.8. The radiation resistances of two di erent dimensions
of aluminum panels, 1.22 m 2.44 m and 0.42 m 0.84 m, were calculated. The thicknesses
74
Figure 4.8: Normalized radiation resistances of ba ed simple supported aluminum panels
of both panels are 6.35 mm. It is shown that the small panel has higher values of radiation
e ciency than the large panel below the critical frequency. This is because more e ective
radiation areas per unit area exist in the small panel.
The acoustic-acoustic volume coupling loss factors describe the power ow between
two reverberation rooms when there are no modes excited in the structure in the frequency
band of interest.
During steady state conditions, the sound power owing from the source room to the
receiving room due to sound transmission is,
tr = IpAp; with Ip = hp
2i
4 c; (4.43)
75
where Ip is the incident sound intensity on the structure in the source room and is the
transmission coe cient of the structure. The power ow can be written in dissipation terms,
tr = tr!E = tr!hp
2i
c2 V: (4.44)
Hence, the source-receiving coupling loss factor is associated with the non-resonant trans-
mission coe cient, nr,
13 = nr = cAp4V
1!
nr; (4.45)
where V1 is the volume of the source room.
Since mass law transmission is derived by assuming zero sti ness and damping in the
in nite structure and o resonance, then 13 can be obtained from the eld incidence mass
law transmission coe cient,
10log10
1
nr
= 20log10
!
2 c
5 dB; (4.46)
where is the surface mass density of the structure.
Sewell [20] derived the transmission coe cient due to forced vibration,
10log10
1
nr
20log10
"
!
2 c
1 !
2
!2c
!#
10log10
"
ln(k
q
Ap) + 14 k2A
p
#
5 dB; (4.47)
where !c is the critical frequency of the structure. This formula is not for lightweight panels,
and it requires, > 10 kg/m2.
76
Figure 4.9: Sound transmission measurements from the two-room method
4.7 Sound transmission loss
The sound transmission loss of partition can be obtained experimentally by the two-
room method. The sound power 12 owing from the source room to the receiving room
must equal the sound power 21 owing back into the source room from the receiving room
plus the sound power , which is absorbed within the receiving room, as illustrated in
Fig. 4.9.
12 = 21 + : (4.48)
The sound power, 1, incident upon the surface of the partition is,
1 = hp
21i
4 cAp; (4.49)
where Ap is the surface area of the partition between two rooms, and hp21i is the mean
square sound pressure in the source room. Likewise, the sound power incident upon the
77
receiving room side of the partition is,
2 = hp
22i
4 cAp; (4.50)
wherehp22iis the mean square sound pressure in the receiving room. The sound power which
is transmitted from the source room to the receiving room and the sound power transmitted
from the receiving room back to the source room are,
12 = 1 ; 21 = 2 ; (4.51)
where is the transmission coe cient of the partition. The sound power absorbed by the
receiving room is,
= hp
22i
4 cApS2 2; (4.52)
where S2 is the total surface area of the receiving room and 2 is the average absorption
coe cient in the receiving room.
Substituting Eqs. (4.52) and (4.51) into Eq. (4.48), yields
1
=
Ap
S2 2
hp21i
hp22i 1
!
= hp
21i
hp22i
Ap
S2 2 + Ap: (4.53)
The energy density ratio of the transmission suite SEA model can be evaluated by Eq.
(4.14). Since the mean square sound pressure ratio is equivalent to the sound energy density
78
ratio between the two reverberation rooms,
E1=V1
E3=V3 =
hp21i
hp22i: (4.54)
The transmission loss of the structure in a transmission suite SEA model can be esti-
mated from,
TL = 10log10
1
= 10log10
A
p
S3 3
E
1=V1
E3=V3 1
; (4.55)
where S3 is the total surface area of the receiving room. 3 is the average absorption
coe cient in the receiving room. S3 3 is associated with the reverberation time of the
receiving room T3, which is related to the internal loss factor of the receiving room, 3.
The modal density of large acoustic volumes is generally approximated by the rst
term of Eq. (4.15). Hence, the transmission loss can be expressed as,
TL = 10log10
A
pT3
0161V3
E
1=V1
E3=V3 1
= 10log10
4 13:7fApn
3 3c3
E
1=V1
E3=V3 1
; (4.56)
with, E1=n1E
3=n3
1 = 2 radn2 3n3 + ( 3n3 + radn2) 2n2 2
radn2n3 + 13(2 rad + 2)n1n2
:
The transmission coe cient due to non-resonant modes is,
1
=
4 f
c3
13:7Ap
13n1 =
1
nr; with rad = 0: (4.57)
79
The transmission coe cient due to resonant modes is,
1
= 4
13:7fAp
c3
2 rad + [1 + ( radn2)=(n3 3)] 2
2radn2 ; with 13 = 0: (4.58)
In the absence of structural damping, the transmission coe cient becomes,
1
= 4
13:7fAp
c3
2
radn2 + 2 13n1: (4.59)
4.8 Numerical results
Since most transmission loss measurements are conducted in reverberation rooms, the
reverberation times were assumed in this numerical study instead of the internal loss factors
of the two rooms. The reverberation times of two rooms were assumed to be 1.4 seconds at
all frequencies.
Two non-resonant transmission coe cients, Eqs. (4.46) and (4.47), were used in cal-
culating the sound transmission loss of an aluminum panel, panel D, as shown in Fig. 4.10.
The estimates were compared with the measured values in Ref. [18]. The material prop-
erties of the aluminum panels were assumed as, Young?s modulus E = 70 GPa, Poisson?s
ratio = 0.3, mass density = 2700 kg/m3, energy loss factor int = 0.001. The thickness
and the dimensions are given in Table 4.1. The volumes of the two reverberation rooms
were, 127.4 m3 [18]. The estimates using the eld incidence mass law as the non-resonant
80
Figure 4.10: Estimated transmission loss values of panel D
Table 4.1: Properties of aluminum panels D, E and F
Property Panel D Panel E Panel F
Thickness (mm) 3.175 6.35 6.35
Dimension (m m) 1:97 1:55 1:22 2:44 0:42 0:84
transmission coe cient are much closer to the measured values than those using Sewell?s
formula, Eq. (4.47), below the critical frequency.
The sound transmission loss values due to resonant and non-resonant modes were also
computed, as shown in Fig. 4.11. The non-resonant coupling loss factors were obtained
from eld incidence mass law, Eq. (4.46). Below the critical frequency, non-resonant modes
are dominant in transmission, and above that frequency, resonant modes are substantial.
Two sets of modal densities were employed in the calculation of sound transmission
loss for honeycomb sandwich panel A. The predictions of sound transmission loss are shown
81
Figure 4.11: Resonant and non-resonant modes on the sound transmission loss of panel D
and compared with the measured values [6]. The properties of panel A are given in Table
3.1. The internal loss factor was assumed to be int = 0.03. The volumes of the two rooms
were taken as 100 m3.
The theoretical radiation resistances for sandwich panels are not readily available.
From the previous chapter, the critical frequency of panel A is near to 200 Hz, and the
radiation resistance is independent of the wave number for structures above the critical
frequency, then the radiation resistances of sandwich panels are only associated with the
critical frequencies above that frequency. The predictions using SEA were computed from
250 Hz, above the critical frequency. Both predictions using SEA produced similar values of
sound transmission loss, the di erence between them is noticeable above 2000 Hz, as shown
in Fig. 4.12. Since the approximate modal density function overestimates modal densities
82
Figure 4.12: Estimated sound transmission loss values of panel A
at high frequencies, as shown in Fig. 4.6, the estimated sound transmission loss values
are smaller than those using the modal densities derived from the sixth-order governing
equation.
The e ects of dimensions of panels and volumes of rooms on sound transmission loss
were studied, as illustrated in Fig. 4.13. The volumes of the large rooms were 127.4 m3, and
52 m3 for the small rooms. Both panels are made of aluminum, and have the same material
properties as those of panel D. The thicknesses and the dimensions are given in Table 4.1.
Two panels with the identical thickness, then yields the same surface mass density and
critical frequency. The sound transmission loss predictions from SEA are insensitive to the
size of the panels and the volumes of the rooms above the critical frequency. The sound
83
Figure 4.13: The e ects of dimensions of panels and volumes of rooms on sound transmission
loss
transmission loss predictions for the panels reduces with decreasing the size of the panels
or increasing the volumes of the rooms below the critical frequency.
4.9 Conclusions
It is shown that for lightweight panels, the SEA model using non-resonant coupling
loss factor associated with the eld incidence mass law generates better results than the
model using Sewell?s formula. For sandwich panels with sti cores, anti-symmetric motions
dominant in the frequency range of interest, SEA produces reliable sound transmission
loss estimates above the critical frequency. The derived modal density of a traditional
honeycomb sandwich panel was found to be one half of the approximate modal density
84
that was obtained from a fourth-order governing equation, which yields a 2 dB di erence
in the sound transmission loss. SEA produces similar sound transmission loss estimates for
single-layer panels made of the identical material and with the same thickness above the
critical frequency. Below that critical frequency, the estimates depend on the dimension of
the panels and the volumes of the rooms.
85
Chapter 5
Boundary element analysis
5.1 Introduction
For a ba ed planar structure, boundary element analysis has the advantage over nite
element analysis of avoiding the discretization of the uid domain. Thus it is e cient
for solving exterior interactions, where the uid occupies an unbounded domain. Three-
domain, uid-structure- uid systems have been modeled as coupled systems [32, 34, 46]
and uncoupled systems [33, 35, 45]. The uncoupled approach assumes that the structure is
rigid, and that the external load of the structure is only associated with the sound pressure
in the incident and re ected waves at the interface. The coupled approach considers the
e ect of the sound pressure in the radiated waves at the interfaces.
In this chapter, nite element models of sandwich structures are reviewed and some
simulation results from MSC Nastran are compared with the exact solutions from the classic
analysis. The concepts of boundary element analysis are introduced. The coupled boundary
element analysis model for uid-structure- uid systems is discussed. A computer program in
MATLAB language is developed to compute the sound transmission loss of a ba ed simply
supported aluminum panel. The estimates of sound transmission loss for the aluminum
panel from a sound transmission loss model in a commercial boundary element analysis
software, LMS SYSNOISE, are provided and compared with the results from the computer
86
program. Finally, a coupled boundary element analysis model for the sound transmission
loss of three-layer symmetric sandwich panels is presented.
5.2 Finite element analysis models for sandwich structures
The displacement compatibility over the entire interfaces between the core and the face
sheets is required for modeling of sandwich structures. Some authors have developed nite
element programs for sandwich beams [35, 49], while others presented modeling methods
using commercially available nite element analysis softwares [48]. Finite element programs
for sandwich beams follow the same procedure for the development of the governing equation
for transverse motion of three-layer sandwich panels described in Sec. 3.2. The face sheets
are treated as elementary bent plates. The mid-plane displacements of the face sheets and
the displacements of the core are assumed to satisfy the displacement compatibility over
the face sheet-core interfaces.
Earlier nite element methods implemented with MSC Nastran required four layers of
nodes and extensive constraint equations to achieve the proper bending-shearing behavior of
a three-layer sandwich structure [47]. Johnson and Kienholz [48] proposed a nite element
model for sandwich structures with viscoelastic cores using only two layers of nodes, as
illustrated in Fig 5.1. The face sheets are modeled with plate elements, such as CQUAD
and CTRIA , with two rotational and three translational degrees of freedom per node. The
viscoelastic core is modeled with solid elements, such as HEXA and PENTA, with three
translational degrees of freedom per node. The plate elements are o set to one surface of
87
Figure 5.1: Finite element model for sandwich structures using MSC Nastran
the plate, coincident with the nodes of the adjoining solid elements. The plate elements are
able to account for the stretching and bending of the face sheets. The modal loss factor is
de ned as,
(r) = v[V (r)v =V (r)]; (5.1)
where v is the energy loss factor of the viscoelastic core evaluated at the rth calculated
resonance frequency and V (r)v =V (r) is the fraction of elastic strain energy attributable to
the core when the structure deforms in the rth mode shape.
The sixth-order di erential equation of motion formulated in terms of the transverse
displacement w, for a three-layer sandwich beam with a viscoelastic core is [23],
@6w
@x6 g(1 +Y)
@4w
@x4 +
Dt
@4w
@x2@t2 g
@2w
@t2
!
= 1D
t
@2q
@x2 gq
!
; (5.2)
with Dt = E1t
31 +E3t33
12 ; g =
Gc
h
1
E1t1 +
1
E3t3
; Y = [h+ (t1 +t3)=2]
2E1t1E3t3
Dt(E1t1 +E3t3) ;
88
Table 5.1: Classic boundary conditions
Clamped w = 0 rw = 0 r5w gYr3w = 0
Free r2w = 0 r4w ( =Dt)!2w = 0 r5w g(1 +Y)r3w ( =Dt)!2rw = 0
where Gc is the shear modulus of the core, q is the external load, and is the surface mass
density.
The exact solution of Eq. (5.2), is of the form,
w = (A1e ik1x +A2e ik2x +A3e ik3x +A4e ik4x +A5e ik5x +A6e ik6x)ei!t: (5.3)
The amplitude Aj can be determined from the boundary conditions [23, 50], given in Table
5.1.
For a cantilever sandwich beam, as shown in Fig. 5.2, the equations for amplitudes Aj
of free transverse motion can be written in matrix form as,
2
66
66
66
66
66
66
66
66
66
66
64
b11 b12 b13 b14 b15 b16
b21 b22 b23 b24 b25 b26
b31 b32 b33 b34 b35 b36
b41 b42 b43 b44 b45 b46
b51 b52 b53 b54 b55 b56
b61 b62 b63 b64 b65 b66
3
77
77
77
77
77
77
77
77
77
77
75
8
>>>>
>>>>
>>>
>>>>
>>>>
>><
>>>>
>>>
>>>>
>>>>
>>>
>>>:
A1
A2
A3
A4
A5
A6
9
>>>>
>>>>
>>>
>>>>
>>>
>>>=
>>>>
>>>
>>>>
>>>>
>>>
>>>;
=
8
>>>>
>>>
>>>>
>>>>
>>>
>>><
>>>
>>>>
>>>>
>>>
>>>>
>>>:
0
0
0
0
0
0
9
>>>>
>>>
>>>>
>>>>
>>>
>>>=
>>>
>>>>
>>>>
>>>
>>>>
>>>
;
; (5.4)
with, b1j = 1; b2j = ikj;b3j = i(k5j +gYk3j); b4j = k2je ikjL;
89
Figure 5.2: A cantilever sandwich beam with viscoelastic core (beam G)
b5j =
k4j D
t
!2
e ikjL; b6j =
ik5j ik3jg(1 +Y) +ikj D
t
!2
e ikjL:
The modal loss factor is determined by the perturbation method [51],
= Im(!
2)
Re(!2); where !
2 = Dt
k6 +g(1 +Y)k4
k2 +g : (5.5)
The cantilever beam, beam G, has identical aluminum face sheets and a viscoelastic
core, the properties are given in Table 5.2. The results for beam G, as obtained from the
sixth-order equation and two nite element models using MSC Nastran, are presented in
Table 5.3. One nite element model had CQUAD4 elements o set from the solid nodes by
a half of the thickness of the face sheets, as shown in Fig. 5.1, and the other was without
reference to the surface o set. Both nite element models had 50 CQUAD4 plate elements
and 50 HEXA solid elements in the lengthwise direction, x-axis.
90
Table 5.2: Properties of beam G
Property Face sheet Core
Young?s modulus (Pa) 69 109 2:1 106
Shear modulus (Pa) 26:5 109 6:23 105
Poisson?s ratio 0:3 0:685
Thickness (mm) 1:524 0:127
Mass density (kg/m3) 2800 970
Table 5.3: Comparisons of natural frequencies for beam G
Sixth-order Eq. Finite element model Finite element model
Mode CQUAD4 with o set CQUAD4 without o set
Natural frequency(Hz) Natural frequency(Hz) V =V Natural frequency(Hz)
1 63 63 0.29 39
2 291 291 0.23 241
3 735 734 0.14 675
4 1383 1381 0.08 1325
5 2249 2243 0.05 2193
The natural frequencies predicted from Johnson and Kienholz?s model and the sixth-
order equation are almost the same for the rst ve modes, while the nite element model
without o set predicts large di erences for low order modes.
5.3 Basic concepts of boundary element analysis
A time-harmonic sound pressure eld is represented by,
p(x;y;z;t) = p(x;y;z)ei!t: (5.6)
The Helmholtz equation is,
r2p+k2p = 0: (5.7)
91
The acoustic particle velocity is related to the normal derivative of the sound pressure as,
v = i !@p@n: (5.8)
The fundamental solution to the Helmholtz equation, Eq. (5.7) in three dimensions is,
G(x;X0) = e
ikr
4 r ; (5.9)
where r is the distance between the eld point x and the source point X0. The above
fundamental solution satis es,
r2G(x;X0) +k2G(x;X0) = (x;X0); (5.10)
where is the Dirac delta function.
The boundary integral equation can be found from Green?s second identity,
Z
S
p@G@n G@p@n
dS =
Z
V
pr2G Gr2p
dV: (5.11)
Substituting the Laplacian r2p on the left-hand side of Eq. (5.7), yields
Z
S
p@G@n G@p@n
dS =
Z
V
r2G+k2G
p dV: (5.12)
92
Equations (5.10) and (5.12) give the boundary integral representation of the sound pressure
eld,
p(x) =
Z
S
@p(X0)
@n G(x;X
0)dS
Z
S
p(X0)@G(x;X
0)
@n dS: (5.13)
The normal derivative of the fundamental solution is,
@G
@n =
e ikr
4 r
ik + 1r
@r
@n: (5.14)
For in nite regions, a far- eld boundary condition is necessary to guarantee that the solution
of the mathematical problem will be a sound wave propagating from the source to in nity,
and not vice versa. This condition is called the Sommerfeld radiation condition at in nity,
limr!1r
@
@r +ik
= 0: (5.15)
For larger values of r,
@r
@n!1;
1
r !0; (5.16)
then Eq. (5.14) reduces to,
@G
@n +ikG = 0: (5.17)
Consider the domain V limited by an in nite rigid plane boundary, SH, ba e, and
another boundary S, as shown in Fig. 5.3. The plane is rigid, so that total re ection occurs
93
Figure 5.3: Half-space V limited by an in nite rigid plane SH and boundary S
for waves at any angle of incidence at SH,
v = 0)@p=@n = 0: (5.18)
Then the boundary integral equation for the sound pressure eld becomes,
p(x) =
Z
S
@p(X0)
@n G(x;X
0)dS
Z
S+SH
p(X0)@G(x;X
0)
@n dS: (5.19)
The sound pressure p is not zero at SH. To avoid the discretization of the in nite
boundary, the fundamental solution G has to be modi ed to be satis ed over SH,
@G(x;X0)
@n = 0: (5.20)
94
Using the method of images, the half-space Green?s function can be of the form,
G(x;X0) = e
ikr1
4 r1 +
e ikr2
4 r2 ; (5.21)
where r1 denotes the distance from x to X0 and r2 is the distance from x to X00 (the image
of X0 with respect to SH). The second term of Eq. (5.21) represents the re ected waves
due to the presence of the in nite plane SH.
The normal derivative of the half-space Green?s function is,
@G(x;X0)
@n =
1
4
1r2 ikr
1
e ikr1@r1@n +
1r2 ikr
2
e ikr2@r2@n0
; (5.22)
where n0 is the image of n with respect to the plane SH. For any point x along SH, we have
r2 = r1; while @r2=@n = @r1=@n: (5.23)
Then the integral Eq. (5.19) reduces to
p(x) =
Z
S
@p(X0)
@n G(x;X
0) dS
Z
S
p(X0)@G(x;X
0)
@n dS: (5.24)
5.4 Boundary element analysis model for uid-structure- uid systems
For the conventional multi-domain systems, the acoustic domains are rst divided into
several subdomains, and the Helmholtz integral equation is applied to each subdomain. Two
95
interface conditions, the continuity of normal particle velocity and the continuity of sound
pressure, are then enforced at the interface between two neighboring subdomains. For the
uid-structure- uid system, the structure involved is elastic, and the sound pressure has a
step across the interface. The uid-structure- uid interaction requires a slight modi cation
of the interface conditions. The continuity of normal particle velocity is the same as for
uid- uid systems, and the second condition becomes the continuity of the normal stress,
which relates the normal displacement and the sti ness and mass matrices of the structure
to the pressure step across the structure. The sti ness and mass matrices can be obtained
from the nite element analysis.
Consider a uid-structure- uid system, as illustrated in Fig. 5.4. An elastic panel,
occupies the domain on the plane z = 0 in a three-dimensional space. The ba e occupies
the region 0 and is perfectly rigid. The two half spaces 1 (z< 0) and 2 (z> 0) contain
a uid. The system is excited by a simple harmonic sound source, O(Q)ei!t, located in 1
at point Q, (x; y; z).
Let w(U) = w(x; y) be the panel normal displacement of node U, located at (x; y; 0);
p2(Q) = p2(x; y; z) and p1(Q) = p1(x; y; z) denote the sound pressure elds, in 2 and
1 elds, respectively. The sound pressure step q(Q) = q(x; y) across the panel, is de ned
by,
q(x;y) = lim
l!0
jp1(x; y; l) p2(x; y; l)j; l> 0: (5.25)
96
Figure 5.4: Sound elds 1 and 2 created by a ba ed planar vibrating structure
Then the sound pressure functions p2(Q) and p1(Q) satisfy the Helmholtz equation,
(r2 +k2)p2(Q) = 0; Q2 2 and (r2 +k2)p1(Q) = O(Q); Q2 1: (5.26)
At the interface, the continuity of normal stress produces,
(K M!2)w(U) = q(U); U2 ; (5.27)
where K and M are the sti ness and the mass matrix of the panel, respectively.
The continuity of normal velocity gives,
@p2(U)
@n =
@p1(U)
@n = f!
2w(U); U2 : (5.28)
97
where f is the mass density of the uid, and n is the unit normal vector to the surface ,
outward to the sound pressure eld 2.
On the ba e plane, we have,
@p2(U)
@n =
@p1(U)
@n = 0 and w(U) = 0; U2
0: (5.29)
The half-space Green?s function is expressed as,
G(Q;Q0) = e
ikr(Q;Q0)
4 r(Q;Q0) +
e ikr(Q;Q0 )
4 r(Q;Q0 ); (5.30)
where the coordinates of the points Q0 and Q0 are (x0;y0;z0) and (x0;y0; z0), respectively.
The structure and the ba e are coplanar. Thus, the normal derivative of the modi ed
Green?s function is zero along . The sound pressure elds can be written as,
p2(Q) = !2 f
Z
w(U0)G(Q;U0)dS(U0); Q2 2; (5.31)
p1(Q) = p0(Q) !2 f
Z
w(U0)G(Q;U0)dS(U0); Q2 1; (5.32)
where p0(Q) is the sound pressure generated by the source and its image, in the presence
of the ba e.
Hence, we have the equation for the panel displacement:
(K M!2)w(U) + 2 f!2
Z
w(U0)G(U;U0)dS(U0) = p0(U); U2 : (5.33)
98
The discretization by nite elements of the panel yields the matrix equation,
[K (M + 2B)!2]w = F; (5.34)
where B is the complex symmetric matrix associated with the uid, which depends on
the wave number in the z direction, kz, and F is the loading matrix acting on the plane
structure. The equation above is similar to the equation derived by Mariem and Hamdi
[32], although the factor of B is 4 in their work.
The sound pressure in the incident plane wave can be written as,
pinc = Pincexpfi!t ik(xsin cos +ysin sin +zcos )g; (5.35)
where Pinc is the amplitude of the source, is the angle of incidence, and is the angle of
rotation. Then the wave number k in Eq. (5.30) is evaluated as kcos .
The sound transmission coe cient is de ned as,
= WrW
i
; (5.36)
where Wi denotes the virtual sound power ow, which would pass through the surface of
the panel, if the panel were removed.
Wi = jPincj
2Scos
2 fcf ; (5.37)
99
where cf is the sound speed in the uid. The sound power radiated by the panel in the
semi-in nite uid domain, Wr, is given by,
Wr = 12Re
Z
S
v n(U)pr(U)dS(U)
= 12Re
Z
S
i!w (U)pr(U)dS(U)
; U2 : (5.38)
with pr(U) = !2 fR w(U0)G(U;U0)dS(U0).
Theoretically, modal superposition method is numerically equivalent to the direct re-
sponse method if the modal basis consists of all modes of the structure. The modal super-
position method can be used to evaluate the frequency response of the normal displacement
in the form of a linear combination of modal eigenvectors,
w = X
j
ajf jg= [ ]fag; (5.39)
wherefagcontains the modal participation factors, and [ ] is a matrix, whose columns are
the modal eigenvectors. Compared to the direct response method, the modal superposition
method has the advantage of allowing faster calculations once the modes are determined.
Then Eq. (5.34) can be expressed as,
([ ]T[K (M + 2B)!2][ ])fag= [ ]TF: (5.40)
100
The structural damping can be introduced by adding a fraction matrix ,
([ ]T[K(1 +i ) (M + 2B)!2][ ])fag= [ ]TF: (5.41)
When the modal eigenvectors are generalized with respect to mass, the following rela-
tions are obtained,
f igTKf jg=
8
>>><
>>>:
i = !2i i = j
0 i6= j
; and f igTMf jg=
8
>>><
>>>:
1 i = j
0 i6= j
: (5.42)
Hence Eqs. (5.40) and (5.41) are only associated with the element mesh, the eigenvalues ,
and the generalized modal eigenvectors, .
The natural frequency fmn of a simply supported single-layer panel can be evaluated
by,
Dk4 = m!2; with k2mn = (m =lx)2 + (n =ly)2; ! = 2 f; (5.43)
where lx and ly are the dimensions of the panel. The modal eigenvectors of normal dis-
placements can be expressed as,
mn = Vmn sin(m x=lx) sin(m y=ly); (5.44)
where the amplitude Vmn can be obtained from nite element analysis.
101
Table 5.4: Natural frequencies and generalized modal amplitudes of panel H
From Eq.(5.43) MSC Nastran
Mode (m;n) Frequency (Hz) Frequency (Hz) Vmn
1 (1,1) 52.9 52.9 1.1414531
2 (2,1) 85.5 85.5 1.1432148
3 (3,1) 140.0 140.2 1.1461568
4 (1,2) 178.7 179.5 1.1463532
5 (2,2) 211.4 211.9 1.1481224
6 (4,1) 216.2 217.2 1.1502883
7 (3,2) 265.8 266.1 1.1510770
8 (5,1) 314.2 316.9 1.1556219
9 (4,2) 342.1 342.5 1.1552263
10 (1,3) 388.4 392.8 1.1545663
11 (2,3) 421.1 424.9 1.1563482
12 (6,1) 434.0 440.0 1.1621734
13 (5,2) 440.1 441.6 1.1605831
14 (3,3) 475.6 478.5 1.1593239
15 (4,3) 551.8 554.0 1.1635029
16 (6,2) 559.9 563.8 1.1671624
The generalized modal eigenvectors of a 3.175 mm thick aluminum panel, panel H,
were computed by MSC Nastran. The 0.84 m 0.428 m panel was divided into 40 24
CQUAD4 elements. The generalized modal amplitudes Vmn with respect to the mass are
given in Table 5.4.
A boundary element method (BEM) computer program in MATLAB language was
developed to calculate the sound transmission loss of the aluminum panel H. The modal
supposition method was employed and the rst 16 modes were included. The structural
modal damping was assumed to be constant over all frequencies. Two di erent structural
loss factors were examined, = 0:001, 0.005. The frequency increment used was 1 Hz.
The calculated sound transmission loss values of the aluminum panel H for sound waves at
normal incidence are shown in Fig. 5.5.
102
Figure 5.5: Calculated sound transmission loss values of the aluminum panel H for sound
waves at normal incidence using the BEM computer program
It was found that the presence of the uid modi es the resonance characteristics of
the panel, slightly shifting modal frequencies to lower frequencies. The e ect of damping
on sound transmission loss is noticeable around resonance frequencies. It is seen that the
radiated power is greater than the virtual power ow at the resonance frequencies of the
rst several modes.
The speci c acoustic impedance of the half-space Green?s function, Eq. (5.30), is
identical to that of a spherical wave,
Z = i fcfkr1 +ikr = fcf
k2r2
1 +k2r2 +i
kr
1 +k2r2
!
: (5.45)
103
Unlike the speci c acoustic impedance of a plane sound wave, the speci c acoustic impedance
of a spherical sound wave has both resistive and reactive components. When kr>> 1, the
speci c acoustic impedance approaches fcf, the speci c acoustic impedance of a plane
wave. For boundary element analysis, both the resistive and reactive components are re-
quired. The virtual sound power ow, Eq. (5.37), is de ned by using plane wave concepts,
and the radiated power, Eq. (5.38), is associated with the speci c acoustic impedance of
spherical waves. That may explain why the radiated sound power is greater than the virtual
sound power ow at low order resonance frequencies.
When a plane wave is normally incident on the panel, the sound pressure generated by
the source and its image on the panel, p0(U), is uniform (see Eq. (5.35)). Only odd-odd
(volume displacing) modes are excited, so only odd-odd modes radiate power. It is seen
that the transmission loss curve generated by the rst mode, (1,1), is much higher than the
mass law curve above its resonance frequency, and it approaches the mass law curve with
increasing frequency.
The sound pressure in a plane wave obliquely incident on the panel, depends on the
angle of incidence, , and the angle of rotation, . Thus it should be expected that the
transmission loss of the aluminum panel will also depend on the angle of rotation, . A
comparison of the sound transmission loss values of the aluminum panel H for sound waves
at oblique incidences predicted using the computer program is presented in Fig. 5.6.
When a plane wave is obliquely incident on the panel, along the x axis direction, ( =
0o), the sound pressure generated by the source and its image on the panel, p0(U), is constant
104
Figure 5.6: Calculated sound transmission loss values of the aluminum panel H for sound
waves at oblique incidence using the BEM computer program
along the y axis direction. Then the radiated power for odd-even or even-even modes of
the panel is quite small. Hence the resonance dips in the predicted sound transmission loss
along the x-axis direction ( = 0o), occur at the resonance frequencies whose y-axis mode
numbers are odd. Likewise, the resonance dips in the predicted transmission loss along the
y-axis direction ( = 90o), occur at the resonance frequencies whose x-axis mode numbers
are odd (see Table 5.4). Since the sound pressure generated by the source and its image on
the panel, p0(U), is less sensitive to the angles of incidence and rotation at low frequencies,
the e ect of angle of rotation on sound transmission loss is negligible in that region.
A commercial boundary element analysis software, LMS SYSNOISE, was also used to
calculate the sound transmission loss of the aluminum panel H. A nite element model
105
database and a boundary element indirect ba ed model database were de ned in the soft-
ware. The element mesh, 40 24 CQUAD4 elements, was imported to the software as
the structural meshes for both databases and to occupy on the plane z = 0. The com-
puted eigenvectors of the rst 16 modes obtained from MSC Nastran were imported to
the software as the structural modes in the nite element model database. A plane wave
source was de ned in the boundary element model database at 5 m below the center of the
panel. The two databases were linked to solve the displacement of the aluminum panel.
LMS SYSNOISE does not directly calculate the radiated power of planar structures, but
it does provide an alternative method to estimate the radiated sound intensity from planar
structures, by integration of the sound power through a eld point mesh, which covers the
receiver side of the structure. A hemisphere with 1 m radius was used for the eld point
mesh.
Comparisons of sound transmission losses of the aluminum panel H for plane sound
waves at normal and oblique incidences calculated using the BEM computer program and
the transmission loss model in the boundary element analysis software are given in Figs. 5.7
and 5.8. In both cases, the structural damping was assumed to be constant for all modes
and the structural loss factor was assumed to be = 0:001; the frequency increment used
was 1 Hz.
The predictions from the BEM computer program and the TL model in the boundary
element analysis software are quite similar, although there are some di erences. In the
software, the velocity eld is obtained by di erentiation of the pressure eld at the Gauss
106
Figure 5.7: Predicted sound transmission loss values of the aluminum panel H for sound
waves at normal incidence
Figure 5.8: Predicted sound transmission loss values of the aluminum panel H for sound
waves at oblique incidence
107
points of the elements of the eld point mesh and then extrapolation and averaging are
carried out at the nodes. Hence the estimated sound power radiated depends on the eld
point mesh and the Gauss points. The estimates of radiated sound power can become more
accurate if a ner eld point mesh or more Gauss points are used.
5.5 Boundary element analysis model for three-layer symmetric sandwich pan-
els
For a three-layer sandwich panel, the generalized modal eigenvectors of the normal
displacements of the face sheets can be written as,
mn =
8
>>><
>>>
:
Vmn;2func(m;n;x;y;z); z> 0
Vmn;1func(m;n;x;y;z); z< 0
: (5.46)
These generalized eigenvectors can be obtained from the nite element model for sandwich
structures given by Johnson and Kienholz [48]. In their model, for symmetric sandwich
panels, the normal displacements of the face sheets are the same. Hence the three-layer
symmetric sandwich panel can be treated as an equivalent single-layer panel whose gener-
alized modal eigenvectors satisfy,
0mn = V0mnfunc(m;n;x;y) with V0mn = Vmn;2 = Vmn;1: (5.47)
108
In the previous section, it was shown that Eqs. (5.40) and (5.41) are only associated
with the element mesh, the eigenvalues , and the generalized modal eigenvectors of the
structure. Then only the generalized eigenvectors for one face sheet of the symmetric sand-
wich panel are required in this boundary element analysis model. The sound transmission
coe cient can be computed from Eqs. (5.36) (5.38) and the sound transmission loss can
be determined with Eq. (3.41).
To verify the proposed model, the aluminum panel H was modeled as a single-layer
panel and a three-layer panel using the nite element model presented by Johnson and
Kienholz [48], t1 = t2 = 0:24 mm, h = 2:195 mm. The nite element mesh of the three-
layer panel was generated in MSC Nastran, and consisted of 40 24 CQUAD4 elements
for each face sheet, and 40 24 HEXA elements for the core. The nite element mesh
of the single-layer panel consisted of 40 24 CQUAD4 elements. The calculated natural
frequencies and generalized amplitudes Vmn, with respect to the mass, for the clamped
aluminum panel H, are given in Table 5.5. It is seen that the two nite element models
predict almost the same natural frequencies and generalized amplitudes.
The sound transmission characteristics of the clamped aluminum panel H in the single-
layer model case and the three-layer model case were computed by using LMS SYSNOISE.
In the single-layer model case, the sound transmission loss values were obtained by using
the TL model described in Sec. 5.4. In the three-layer model case, the structural mesh used
in the boundary element analysis is not the entire element mesh that consists of CQUAD4
and HEXA elements. Only the element mesh of one face sheet, 40 24 CQUAD4 elements,
109
Table 5.5: Natural frequencies and generalized amplitudes of the clamped aluminum panel
H
Single-layer model Three-layer model
Mode (m,n) Frequency (Hz) Amplitude Vmn Frequency (Hz) Amplitude Vmn
1 (1,1) 104.9 1.3873931 104.5 1.3862565
2 (2,1) 137.0 1.3394921 136.4 1.3382779
3 (3,1) 194.6 1.3268757 193.5 1.3257434
4 (1,2) 273.6 1.2870189 271.5 1.2858151
5 (4,1) 277.6 1.3145773 275.6 1.3140515
6 (2,2) 304.4 1.2745483 302.1 1.2627527
7 (3,2) 357.7 1.2642025 355.1 1.3862565
8 (5,1) 385.4 1.3166149 381.7 1.3161944
was imported to the software as the structural meshes for both databases and to occupy
on the plane z = 0. Then the computed eigenvectors of the face sheet were imported to
the software as the structural modes in the nite element model database. The estimates
of the radiated sound powers were found to be almost the same for both cases.
5.6 Conclusions
The presence of uid modi es the resonance characteristics of the structure. The
radiated sound power of a planar structure can be higher than the virtual sound power ow
near the lower order resonance frequencies. The damping increases the sound transmission
loss near the resonance frequencies. For nite single-layer isotropic rectangular panels,
the sound transmission loss also depends on the angle of rotation, . The contribution of
other than odd-odd modes on the sound transmission loss for single-layer isotropic panels
is negligible for plane sound waves at normal incidence, while those can be substantial for
plane sound waves at oblique incidence. The sound transmission loss values are less sensitive
110
to the damping and the angle of incidence at low frequencies. Good agreement was found
between the predications obtained from the BEM computer program and the TL model of
the boundary element analysis software.
A boundary element analysis model is presented to obtain the sound transmission
loss of three-layer symmetric sandwich panels. Only the eigenvectors of one face sheet are
required in the analysis. Hence, this incurs only a minimum increase in computation e ort
relative to single-layer structures.
111
Chapter 6
Materials and material properties
6.1 Introduction
A sandwich structure consists of three or more constituents, the face sheets, the core
and the adhesive joints. The introduction of ber composites allows the choice of a great
number of face sheet materials. The number of available cores also has increased because
of the introduction of cellular plastics. Since the material properties of composites are
very dependent on the manufacturing process, there is usually a large number of material
data for composites, especially data for ber composites. In a few papers on sound trans-
mission through sandwich composite structures, parameters of material properties used in
predictions were determined experimentally, based on resonance frequency measurements
[6, 8]. Moore and Lyon [6] determined the core material parameters experimentally, based
on resonance frequency measurements with a layer of core material sandwiched between
rigid metal discs. Nilsson and Nilsson [8] separated the face sheets from the foam core and
simulated a free-free beam boundary condition to determine the E-moduli of the face sheets
and foam core.
In this chapter, the materials commonly used in sandwich structures are reviewed,
especially the composite materials for the sandwich structures investigated in this study.
The experimental methods used to obtain material properties of the face sheets and core
112
of sandwich structures are discussed. The estimated material properties of the sandwich
structures used in this study are also presented.
6.2 Materials
\Almost any structural material which is available in the form of a thin sheet may be
used to form the faces of a sandwich panel" [52], gives a good view of material selection
for the face sheets of sandwich structures. The properties of primary interest for the face
sheets of sandwich structures are sti ness, strength, impact resistance, surface nish, en-
vironmental resistance and wear resistance. Common face sheet materials can be divided
into two main groups: metallic and non-metallic materials.
The advantages of using metal for the face sheets include high sti ness and strength,
low cost, good surface nish, and high impact resistance. The drawbacks include high mass
density and di culty in manufacturing sandwich structures.
Most non-metallic composites o er strength properties similar to those of metals. The
manufacturing of non-metallic composite sandwich structures is much easier than the man-
ufacturing of metal face sheet sandwich structures. The most important non-metallic mate-
rials are ber reinforced composites. Glass bers have good mechanical and environmental
resistance properties. Their main drawbacks are that their elastic moduli are fairly low and
their mass densities are higher than those of other reinforcements.
Aramid bers made from aromatic polymid, have low mass densities, and high sti ness
and high strength properties. It is di cult to machine aramid bers, however, because
113
of their extremely wear resistance. Graphite bers are built-up from long carbon-carbon
molecular chain yielding very strong bers. Graphite bers are among the strongest and
sti est composite materials when they are combined with matrix systems to produce high-
performance structures. Graphite bers have a low coe cient of thermal expansion, and
good friction properties. Their main drawbacks are their high cost and relatively brittleness.
Common composite face sheet material forms can be divided into two main groups,
laminate and textile structures. Textile structures include unitapes and 2-D woven fabrics.
Unitapes have maximum structural properties in the ber direction, while they are much
poorer in the direction transverse that in the ber direction. 2-D woven fabrics are more
expensive than unitapes, however the lay-up lab requirements are reduced in manufacturing
operations. Common 2-D woven fabrics include unidirectional fabrics, plain weave fabrics
and satin weave fabrics. Among them, plain weave is the most stable construction and has
minimum slippage. The strength is uniform in both directions. A laminate is a stack of
lamina comprised of a layer of bers in a matrix. The rule-of-mixtures is used to estimate
the properties of a lamina [62]. Classical lamination theory is applied to calculate the
properties of a laminate.
Core materials are of the same importance as the face sheet materials and the least
knowledge exists about their properties. Cores in sandwich structures can be divided into
two main groups, honeycombs and foams. The properties of primary interest of the core
include mass density, shear modulus, shear strength, sti ness perpendicular to the face
sheets, thermal insulation and acoustic insulation.
114
Figure 6.1: Commonly used cell con gurations for honeycomb core materials (a) hexagonal
(b) square (c) over expanded hexagonal (d) ex
Figure 6.2: Manufacture of honeycomb cores - corrugating (top) and expansion (bottom)
processes
Honeycomb core materials have been developed and used primarily in aerospace ap-
plications. Honeycomb materials can be manufactured with a variety of cell shapes. The
commonly used cell con gurations for honeycombs are shown in Fig. 6.1.
The manufacture of honeycombs is conducted in two di erent ways, as illustrated in
Fig. 6.2. The corrugating process requires that pre-corrugated metal sheets are stacked
into blocks and bonded together. When the adhesive has cured, blocks with the required
thickness can be cut from the stack. The corrugating process is usually used for high density
metal honeycombs.
115
The expansion process begins with the stacking of thin plane sheets of the web material
on which adhesive node lines have been printed. When the adhesive has cured, the honey-
comb block may be expanded by pulling in the W direction until a desired cell shape has
been achieved (see Fig. 6.1). Metal honeycombs are cut into the desired thickness prior to
expansion, and when expanded they retain their shapes since the material yields plastically.
Non-metallic materials, such as impregnated ber mats or paper, are heat treated after
expansion to retain their shapes. Then the materials are dipped in resin, which is cured in
an oven. After this process is completed, the core is sliced.
Due to the manufacturing methods involved, most honeycombs have not only di erent
out-of-plane properties but also di erent in-plane properties from each other. Honeycombs
have excellent mechanical properties. These include very high sti ness perpendicular to
the plane, and the highest shear sti ness- and strength-to-weight ratios of all available core
materials. Their main drawbacks are high cost and di cult handling during lay up of
sandwich element.
Gibson and Ashby [53] derived an expression for the out-of-plane shear moduli of regular
hexagonal honeycomb cores,
GLT = GWT = 1:15Gst=s; (6.1)
where t is the cell wall thickness, s is the diameter of a circle inscribed in the hexagonal
cell, and Gs is the shear modulus of the cell wall material. In practice, regular hexagonal
116
honeycomb cores have double cell walls in the L-direction due to their manufacture (see
Fig. 6.1). Then the shear modulus estimation is modi ed to [54],
GLT = 4tGs=(3s); GWT = 16tGs=(30s): (6.2)
For square cells the shear moduli are [54],
GLT = GWT = tGs=s: (6.3)
Foams do not o er the same high sti ness- and strength-to-weight ratios as honey-
combs. Foams can be manufactured from a variety of synetheticpolyers and are generally
less expensive. Foams o er high thermal insulation and acoustical damping. The foam
surface is easy for bonding. The manufacturing operation of sandwich elements with foams
is much easier than that with honeycombs.
Polyurethane (PUR) foams have low thermal conductivity, very good insulation prop-
erties and poor mechanical properties. PUR foams are probably the least expensive of all
available core materials. That PUR foams can also be foamed in-situ gives an integrating
manufacturing in conjunction with the manufacturing of the sandwich elements.
Polystyrene (PS) foams have good mechanical and thermal insulation properties, and
they are low cost. Their main drawback is their sensitivity to solvent. Polyvinylchloride
(PVC) foams are the most frequently used foams and they have quite good mechanical
117
Figure 6.3: The face sheet (a) and core (b) materials of sandwich structures in this study
properties. Polymethacrylimide (PMI) foams have the best mechanical properties and are
more expensive than other foams.
All sandwich structures tested in this study have plain weave fabric-reinforced graphite
composites as the face sheet materials, and PUR foam- lled paper honeycombs as the core
materials. The combination of PUR foam and honeycomb materials gives the core the
advantage of possessing both foam and honeycomb properties, a high shear modulus, and
a large bonding area.
6.3 Measurement methods for materials
Moore and Lyon [6] estimated the honeycomb sti nesses from resonance frequency
measurements on a test sample consisting of a thin layer of honeycomb sandwiched between
rigid metal disks. They found that such tests gave values for the main diagonal axial
sti nesses C11, C22, C44 and C55. The o -diagonal sti nesses C13, C23, and C12 were
arbitrarily assumed to be equal to 0.1 times the softer of the axial sti nesses.
118
Nilsson [8] simulated a free-free beam boundary condition by suspending a beam by
strings. The beam was excited by an impact in a direction perpendicular to the plane of the
beam. The loss factors were derived from half-power bandwidth measurements for various
resonances.
In this study, a free-free beam boundary condition was simulated in two ways. In one
case the beam was excited with white noise by a shaker mounted at its center. Since the
force is applied at the center of the beam, then the center can be considered approximately
as a node of the standing waves. It is expected that only odd modes of the beams are
excited successfully. In the other case the beam was suspended by strings at the two ends.
The beam was excited by an impact in a direction perpendicular to the plane of the beam.
In both cases, the response functions were measured by an accelerometer mounted on the
beam. The modal loss factors were derived from half-power bandwidth method at various
resonances determined from the response function measurements.
In order to verify the hypotheses concerning free-free boundary conditions, a 61 cm
long, 2.54 mm wide, 6.35 mm thick, aluminum beam was excited in both ways. The
admittance-frequency response functions of the aluminum beam for two cases are shown in
Figs 6.4 and 6.5.
It is seen that the even modes are suppressed when the beam is excited by a shaker
mounted at the center. The resonance frequencies of odd modes are slightly lower than
those obtained from the impact set-up. There is more noise in the impact system than in
119
Figure 6.4: The frequency response function of the aluminum beam for the shaker set-up
Figure 6.5: The frequency response function of the aluminum beam for the impact set-up
the shaker system. The measured resonance frequencies and loss factors of the aluminum
beam are given in Table 6.1.
The natural frequencies of a free-free single-layer beam are,
f = 8L2
s
Et2
12 ; = 3:011
2; 52; 72; ; (6.4)
The exact natural frequencies were computed by assuming E = 70 GPa, = 2700kg/m3.
120
Table 6.1: Measured resonance frequencies and loss factors of the aluminum beam
Mode 1 2 3 4 5 6 7 8
Shaker f (Hz) 83 442 1102 2060
(%) 1.8 0.21 0.1 0.04
Impact f (Hz) 88 244 476 789 1174 1642 2178 2798
(%) 1.15 0.41 0.22 0.12 0.1 0.09 0.06 0.05
Exact f (Hz) 89 246 483 790 1192 1665 2216 2847
The resonance frequencies obtained from the impact set-up are all within 2% of the
predicted values, when a free-free boundary condition is assumed. The resonance frequencies
obtained from the shaker set-up are within 9% of the predicted values for the same condition.
The two set-ups give the similar modal loss factor values, of the order 0.1%, except in the
low frequency region. The experimental loss factor includes both the internal loss in the
structure and the loss at the boundaries of the structure element in the low frequency region.
This fact may explain the high values of the rst few modal loss factors.
6.4 Experimental resonance frequencies of sandwich beams
Honeycomb materials have di erent sti nesses in planes perpendicular and parallel to
the direction of the cells. It was assumed that the cells of the honeycomb are aligned
perpendicular to the plane of the sandwich structures in this study. Most foam cores are
only moderately anisotropic and have fairly similar in-plane properties. Thus the foam- lled
honeycomb cores were assumed to be orthotropic.
Measurements were performed on beams representing the two main in-plane directions
of the panels. For materials tested in two directions of the structure, the results are assigned
subscripts x or y to indicate the orientations of the beam. Test samples in the form of beams
121
Table 6.2: Properties of the sandwich beams
Face sheets Core
Beam Length Width Thickness Density Thickness Density
(cm) (cm) (mm) (kg/m3) (cm) (kg/m3)
Ix 114.3 3.0 0.33 1900 0.635 160
Iy 53.3 3.2 0.33 1900 0.635 160
Jx 103.8 3.2 0.50 1600 0.635 160
Jy 54.5 3.2 0.50 1600 0.635 160
Kx 101.6 4.4 0.50 1600 1.27 120
Ky 53.0 4.1 0.50 1600 1.27 120
Lx 104.8 3.2 0.33 1900 2.54 70
Ly 54.9 3.5 0.33 1900 2.54 70
were cut out from four symmetric composite sandwich panels. The con gurations are given
in Table 6.2. The estimated densities of the face sheets include the density of the plain
weave fabrics and the adhesive. The sandwich structures were assumed to be homogenous.
Beams Ix, Iy, Jx and Jy have the same foam- lled honeycomb core. Beams Jx, Jy,
Kx and Ky with two layers of plain weave fabrics for each face sheet were treated as three-
layer sandwich structures. All sandwich beams are lightweight, 3 kg/m2. In order to
minimize e ects of mass loading, an Endevco model 22 piezoelectric accelerometer, mass
of 0.14 g, was employed to measure the responses of the beams. For each test specimen,
the measurements were repeated several times. The resonance frequencies of all sandwich
beams were obtained by using the impact method discussed in the previous section. The
admittance-frequency functions of all eight beams are shown in Figs. 6.6 6.13.
For sandwich beams Ix, Jx and Kx, at least the rst 10 modes can be identi ed, and
for sandwich beams Iy, Jy and Ky, the rst six modes can be identi ed, corresponding to
122
Figure 6.6: The frequency response function of beam Ix
Figure 6.7: The frequency response function of beam Iy
Figure 6.8: The frequency response function of beam Jx
123
Figure 6.9: The frequency response function of beam Jy
Figure 6.10: The frequency response function of beam Kx
Figure 6.11: The frequency response function of beam Ky
124
Figure 6.12: The frequency response function of beam Lx
Figure 6.13: The frequency response function of beam Ly
frequencies up to 2 kHz. While for beams Lx and Ly, only a few resonant modes can be
registered. The modal loss factors of beams Ix, Jx, Iy and Jy are similar, of the order 1%,
as shown in Fig. 6.14. The modal loss factors of beams Kx and Ky are about 2% around
1200 Hz, as shown in Fig. 6.15.
The e ects of the mass loading of the accelerometer on the frequency response function
were also investigated. Both a laser Doppler vibrometer and an accelerometer were employed
to measure the response of a light single-layer beam, 1.9 kg/m2. The beam was excited
with white noise by a shaker mounted at its center. The length, width and thickness of the
125
Figure 6.14: Loss factors of beams Ix, Iy, Jx and Jy
beam are 61 cm, 2.54 cm, and 6.35 mm, respectively. The frequency response functions of
the beam are shown in Fig. 6.16.
The frequency response functions of the two transducers are seen to be similar. The
resonance frequencies measured by the accelerometer are slightly lower than those measured
by the laser vibrometer. The laser vibrometer and the accelerometer all require some mass
to be attached the beam. The mass of the accelerometer is 0.14 g, while the mass of the
metal piece needed for the laser vibrometer measurement is much smaller. There is about
5 dB di erence between the measured amplitudes of the two frequency response functions.
The modal loss factors were determined by the half-power point method. The resonance
frequencies and modal loss factors are given in Table 6.3.
126
Figure 6.15: Loss factors of beams Kx and Ky
Figure 6.16: The frequency response functions of the aluminum beam
127
Table 6.3: Resonance frequencies and modal loss factors of the beam
Laser f(Hz) 67 182 352 580 859 1193 1582 2486 3020
vibrometer (%) 4.2 3.2 2.6 2.7 2.2 1.8 1.9 2.0 2.0
Accelerometer f(Hz) 67 180 350 580 856 1188 1577 2482 3002
(%) 3.0 2.8 2.6 2.7 2.6 3.5 2.8 3.9 3.1
The e ect of mass loading of the accelerometer on the resonance frequencies of the
beam was found to be negligible. The resonance frequencies measured by the accelerometer
system are all within 1% of those obtained by the laser system. The laser system has more
noise than the accelerometer system, and it cannot provide precise resonance frequencies
and modal loss factors except at low frequencies.
6.5 Material properties of sandwich panels
The well known sixth-order governing di erential equation of transverse displacement
for a sandwich beam was originally developed by Mead and Markus for sandwich beams
with viscoelastic cores [23]. In general these beams have thin cores, which are di erent
from the sandwich structures examined in this study. Nilsson and Nilsson [37] developed a
sixth-order governing equation for symmetric sandwich beams with thick honeycomb and
foam cores, which includes rotatory inertia e ects of the face sheets and core. They assumed
that the total transverse displacement w of a honeycomb sandwich is primarily caused by
bending, shear and rotation in the core, as shown in Fig. 6.17.
@!=@x = + : (6.5)
128
Figure 6.17: Transverse displacement caused by (a) bending and (b) shear
The di erential equation governingw, and is determined using Hamilton?s principle.
The total potential energy of a honeycomb sandwich beam is assumed to be caused by pure
bending of the entire beam, bending of both face sheets and shear in the core. The kinetic
energy of the honeycomb sandwich beam is due to the transverse motion of the beam and
the rotation of a section of the beam.
Then the equation governing w can be written as,
2D2@
6w
@x6 +
2D2
D1 I
@6w
@x4@t2
+ 2D2D
1
+ I GehD
1
@4w
@x2@t2 +Geh
@4w
@x4 +
D1
@2w
@t2
!
+ I D
1
@
4w
@t4 =
1 + 2D2D
1
@2q
@x2 +
Geh
D1 q +
I
D1
@2q
@t2; (6.6)
with D1 = E2h
3
12 +E1
h2t
2 +ht
2 + 2t3
3
!
; I = ch
3
12 + t
h2t
2 +ht
2 + 2t3
3
!
D2 = E1t
3
12 ; = 2 tt+ ch; Ge = G2
1 + th
2
;
where h and t are the thickness of the face sheets and core, respectively. The mass densities
of the face sheets and the core are t and c, respectively.
129
Assuming w = PjAjei!te ikjx, and allowing the external load q to equal zero, the
wave number kj must satisfy the following expression,
2D2k6+
Geh 2D2D
1
I !2
k4
+ 2D2D
1
+ I GehD
1
k2!2 GehD
1
!2+ I D
1
!4 = 0: (6.7)
Nilsson and Nilsson presented the boundary conditions, given in Table 6.4, in terms of
w, and . It was shown that can be expressed in a similar way to w, = PjBjei!te ikjx,
and it must satisfy the following equation,
D1@
2
@x2 + 2D2
@3w
@x3
@2
@x2
!
+I @
2
@t2 Geh
@w
@x
= 0: (6.8)
Then the amplitude Bj can be determined as a function of Aj,
Bj = Aj 2D2k
3j +Gehkj
(D1 + 2D2)k2j +Geh I !2 = AjXj: (6.9)
Since both composite face sheet and core materials have losses, the damping is intro-
duced by a complex E-modulus,
E = ER(1 +i ); where ER is real: (6.10)
130
Table 6.4: Basic boundary conditions
Clamped w = 0 = 0 rw = 0
Free r2w = 0 r = 0 D1@2 @x2 = I @2 @t2
For a free-free boundary sandwich beam, the six boundary conditions in combination
with Eq. (6.9), yield the equations for Aj, which written in matrix form are,
2
66
66
66
66
66
66
66
66
66
66
64
b11 b12 b13 b14 b15 b16
b21 b22 b23 b24 b25 b26
b31 b32 b33 b34 b35 b36
b41 b42 b43 b44 b45 b46
b51 b52 b53 b54 b55 b56
b61 b62 b63 b64 b65 b66
3
77
77
77
77
77
77
77
77
77
77
75
8
>>>>
>>>
>>>>
>>>>
>>>
>>><
>>>>
>>>
>>>>
>>>
>>>>
>>>:
A1
A2
A3
A4
A5
A6
9
>>>>
>>>
>>>>
>>>>
>>>
>>>=
>>>
>>>>
>>>>
>>>
>>>>
>>>;
=
8>
>>>>
>>>
>>>>
>>>>
>>>
>><
>>>
>>>>
>>>
>>>>
>>>>
>>>
:
0
0
0
0
0
0
9>
>>>>
>>>
>>>>
>>>
>>>>
>>=
>>>
>>>>
>>>
>>>>
>>>>
>>>
;
; (6.11)
with b1j = k2j; b2j = ikjXj; b3j = D1Xjk2j I Xj!2; b4j = k2je ikjL;
b5j = ikjXje ikjL; b6j = (D1Xjk2j I Xj!2)e ikjL:
The material properties of the face sheets and cores of the sandwich structures tested
in this study were estimated using Eqs. (6.7) and (6.11), based on the experimental data
for free-free sandwich beams. It is well known that the motion of sandwich structures is
primarily determined by the face sheets at low frequencies. Thus the E-moduli of the face
sheets can be estimated from the rst several resonance frequencies. Then the G-moduli of
the cores can be estimated from the higher modes. Since only the rst few modes can be
131
Table 6.5: The estimated moduli of sandwich panels I, J, K and L
Sandwich panel I J K L
Face sheets Ex (GPa) 32 49 39 32
Ey (GPa) 34 49 39 24
Cores Gxz (MPa) 100 90 100
Gyz (MPa) 150 140 60
identi ed for beams Lx and Ly, the shear moduli, Gxz and Gyz, of the core of panel L can?t
be determined from measurements. The material property parameters were assumed to
be constant for all frequencies. Then the estimated main material properties for sandwich
panels I, J, K and L are given in Table 6.5.
The resonance frequencies of sandwich beams are insensitive to the core E-modulus
perpendicular to the plane, which was assumed to be 2.3 times the sti er out-of-plane shear
modulus of the core. Based on the estimated material properties of the face sheets and the
cores, the predicted natural frequencies obtained from the models of Nilsson and Nilsson
[37], Mead and Markus [23] (MM), Johnson and Kienholz [48] (FEM) are given in Tables 6.6
6.11. The FEM model had 60 CQUAD4 plate elements and 60 HEXA solid elements in
the lengthwise direction, x-axis. Poisson?s ratios were assumed to be 0.15. The o -diagonal
sti ness constants C13, C23, and C12 were arbitrarily assumed to be equal to 0.1 times the
softer of the axial sti ness constants.
The predicted resonance frequencies are seen to agree well with the measured resonance
frequencies. The modal frequencies predicted from governing equations, the models of
Nilsson and Nilsson, and Mead and Markus, are similar, then the e ects of the rotatory
inertia of the face sheets and core on modal frequencies are negligible for the sandwich
132
Table 6.6: The modal frequencies of beam Ix
Mode 1 2 3 4 5 6 7 8 9 10
Nilsson (Hz) 28 77 149 245 361 498 652 823 1009 1208
MM (Hz) 28 76 148 243 359 494 648 819 1004 1203
Johnson (Hz) 28 76 149 243 360 495 650 821 1007 1207
Measured (Hz) 27 74 143 238 353 486 639 808 992 1189
Mode 11 12 13 14 15
Nilsson (Hz) 1418 1638 1867 2103 2345
MM (Hz) 1413 1633 1862 2098 2340
Johnson (Hz) 1419 1642 1875 2116 2366
Measured (Hz) 1400 1622 1852 2085 2331
Table 6.7: The modal frequencies of beam Iy
Mode 1 2 3 4 5 6 7
Nilsson (Hz) 133 360 690 1109 1601 2152 2748
MM (Hz) 131 356 684 1100 1591 2141 2738
Johnson (Hz) 131 357 685 1103 1596 2150 2750
Measured (Hz) 126 353 683 1096 1586 2132 2720
Table 6.8: The modal frequencies of beam Jx
Mode 1 2 3 4 5 6 7 8 9 10
Nilsson(Hz) 48 132 253 407 589 794 1017 1254 1501 1755
MM(Hz) 48 130 250 403 584 788 1010 1246 1492 1746
Johnson(Hz) 48 131 252 406 588 793 1016 1254 1504 1761
Measured(Hz) 47 131 251 404 583 781 999 1224 1460 1702
Mode 11 12
Nilsson (Hz) 2015 2278
MM (Hz) 2005 2268
Johnson (Hz) 2026 2296
Measured (Hz) 1950 2195
Table 6.9: The modal frequencies of beam Jy
Mode 1 2 3 4 5 6
Nilsson (Hz) 177 474 888 1389 1949 2548
MM (Hz) 176 472 886 1387 1949 2549
Johnson (Hz) 177 472 885 1386 1947 2549
Measured (Hz) 177 467 877 1372 1929 2477
133
Table 6.10: The modal frequencies of beam Kx
Mode 1 2 3 4 5 6 7 8 9 10
Nilsson (Hz) 81 219 416 661 943 1253 1582 1924 2275 2630
MM (Hz) 80 216 411 655 936 1245 1574 1917 2268 2624
Johnson (Hz) 80 217 413 657 941 1254 1589 1939 2300 2671
Measured (Hz) 79 216 413 651 919 1216 1536 1987 2230 2592
Table 6.11: The modal frequencies of beam Ky
Mode 1 2 3 4 5 6
Nilsson (Hz) 285 707 1218 1759 2300 2845
MM (Hz) 283 704 1216 1758 2305 2847
Johnson (Hz) 284 708 1227 1780 2343 2904
Measured (Hz) 281 702 1227 1757 2277 2727
structures tested in this study. The di erence of the predictions between the FEM model
and the other two models is larger for thick sandwich beams than for thin sandwich beams.
6.6 Conclusions
The main material moduli of composite sandwich panels have been determined exper-
imentally from data for free-free sandwich beams. The predicted natural frequencies were
found to agree well with the measured resonance frequencies. Due to the small mass, 0.14
g, the e ect of mass loading of the accelerometer on the measured resonance frequencies
of the sandwich beams was found to be negligible. The modal loss factors of the sandwich
beams are much higher than that of the aluminum beam, about 10 times.
134
Chapter 7
Dynamic properties of composite sandwich panels
7.1 Introduction
The parameters that are used in statistical energy analysis to describe the dynamic
behavior of a structure are the modal density, internal loss and coupling loss parameters
of the structure. The accuracy prediction of response using SEA greatly depends on an
accurate estimate of the three parameters.
In principle the modal density can be obtained by exciting the structure with a sinu-
soidal force of varying frequency and counting the number of modes that are excited in each
frequency band. However, the mode count method is not suitable for structures that have a
high modal density and a high modal overlap or those with heavily damped modes present.
Because of these reasons, the point mobility technique, described by Cremer et al. [42] is
a more suitable method for measuring modal densities. The accuracy of this technique is
critically dependent on the reliable measurement of force and velocity.
Two common direct experimental techniques for obtaining internal loss factors are 1)
the half-power bandwidth method and 2) the envelope decay method. Only the internal
loss factors of the non-overlapping modes can be obtained from the half-power bandwidth
method. For SEA applications, the primary property of interest is the band-averaged loss
factor not the modal loss factor. The envelope decay method is based on the logarithmic
135
decrement of the transient structural response, which is obtained from measurements of the
decay of the vibration after the excitation is cut o . The steady state power ow method
is an indirect approach to experimentally obtain the band-averaged loss factor.
Most experiments used to measure the loss factor of a structure have been conducted
in air. In such cases, the loss factor reported is the total loss factor, which includes the
radiation loss factor.
In this study, modal densities, internal loss factors and radiation loss factors of four
composite sandwich panels were estimated experimentally. The dimensions of all four panels
are the same, 1.12 m 0.62 m. A three-channel spectral analysis was employed to obtain
the point mobilities of the sandwich panels. The modal densities of the composite sandwich
panels were experimentally determined with the spectral mass correction method. The total
loss factors of the panels were evaluated by using the power ow method. The experimental
radiation loss factors of the unba ed and ba ed composite sandwich panels were compared
with the theoretical estimates.
7.2 Experimental modal densities
The modal density of a structure can be obtained from the measurement of the spatially
averaged point mobility frequency response function [42],
n(f) = 4MpRe[Y(f)]; (7.1)
136
where Y(f) = V(f)=F(f) is the driving point mobility of the structure at frequency f, and
Mp is the mass of the structure. The band-averaged modal density is given by,
n(f) = 14f
Z
4MpRe[Y(f)] df: (7.2)
In the conventional two-channel spectral analysis, the point mobility is determined by
the cross-spectrum of the force, and velocity and the auto-spectrum of the input force,
Y(f) = Gfv(f)G
ff(f)
: (7.3)
For lightly damped systems, the driving force at resonance is very small, Gff(f)!0. Any
feedback due to exciter-structure interaction can produce bias error which can sometimes
result in negative peaks [55].
In the three-channel spectral analysis, the point mobility is determined by using the
relation,
Y(f) = Gsv(f)G
sf(f)
: (7.4)
where Gsv(f) and Gsf(f) are the cross-spectra between the original input and the measured
velocity, and the original input and the measured force.
Mass corrections must be considered when making any frequency response measurement
on a lightweight structure. In the case of point mobility measurements, there will always be
some added mass between the force gauge of the impedance head and the structure. The
137
added mass will corrupt the force measurement because some portion of the force measured
is used to drive against the inertial resistance of the added mass.
The point mobility measurement can be corrected for the mass loading e ect as follows,
Yc = VmF
c
= VmF
m MAm
= Vm=Fm1 i!MV
m=Fm
= Ym1 i!MY
m
; (7.5)
where Am and Fm are the acceleration and force measured by the impedance head, Vm and
Ym are the measured velocity and point mobility, Fc and Yc are the corrected force and point
mobility, respectively. M is the added mass between the force gauge and the structure.
The added mass M can be evaluated by adding the manufacturer?s speci cations for
the mass below the force gauge to the mass of the attachment components or by measuring
the point mobility of the added mass attached to the impedance head when it is separated
from the structure. The rst correction method is termed as the measured mass method.
The second correction method is termed as the spectral mass method.
Hence, the real and imaginary parts of the corrected point mobility are,
Re(Yc) = Re(Ym)[1 +!MIm(Y
m)]2 + [!MRe(Ym)]2
; (7.6)
Im(Yc) = !Mf[Im(Ym)]
2 + [Re(Ym)]2g+ Im(Ym)
[1 +!MIm(Ym)]2 + [!MRe(Ym)]2 : (7.7)
In this study, the modal density of a sandwich panel was obtained by averaging the
modal densities measured at four randomly chosen points on the panel. The point mobility
138
Figure 7.1: Set-up for the modal density and loss factor experiments
was measured at each position with a B&K impedance head type 8000 that was attached
to a B&K vibration exciter type 4809 by a stud. The impedance head was attached to the
panel with wax. The sandwich panel was suspended by strings and excited by a conventional
electrodynamic shaker with a broadband random force, as shown in Fig. 7.1. The measured
inertance of the added mass between the force gauge and the panel is between 860 (m/s2)/N
and 960 (m/s2)/N in the frequency range of 200 Hz 5600 Hz, as shown in Fig. 7.2. Then
the e ective dynamic mass of the added mass is between 1.06 g and 1.14 g, which is slightly
smaller than 1.2 g, the mass below the force gauge of the impedance head speci ed by the
manufacturer.
The frequency analysis bandwidths chosen were one-third octave and a constant band-
width of 400 Hz. The rst is consistent with most previous work and the second is chosen
to have at least ve resonance frequencies in each analysis band. The frequency analysis
139
Figure 7.2: The inertance of the added mass
resolution was chosen to be 1 Hz. Four composite sandwich panels with foam- lled honey-
comb cores, panels I L, were investigated. The dimensions of the four panels are given
in Table 6.2.
The measured point mobility of panel J at one location using the three-channel spectral
analysis is illustrated in Fig. 7.3. It was found that only slight di erences between the
measured point mobilities of all four panels obtained by the two-channel and three-channel
spectral analyses exist at very low frequencies.
The modal density estimates without mass correction, as shown in Fig. 7.4, only
provide a reasonable approximation at low frequencies, where the e ect of the added mass
is negligible. The theoretical modal density predictions were derived for simply supported
panels. The theoretical predictions were obtained from 1) Mead and Markus?s sixth-order
governing equation by following the procedure that is described in Sec. 4.4, and 2) a reduced
140
Figure 7.3: The measured point mobility of panel J using the three-channel spectral analysis
(a) real part (b) imaginary part
Figure 7.4: Modal density estimates for panel J without mass correction
141
Figure 7.5: Modal density estimates for panel J with mass correction
fourth-order governing equation, by use of Eq. (4.22). It was found that both equations
produce the same modal density values for all four panels at frequencies below 6000 Hz.
The modal density estimates for panel J with mass correction, agree with the theoret-
ical predictions as shown in Fig. 7.5. The modal density estimates for the other panels, I,
K and L are shown in Figs. 7.6 7.8.
Since Mead and Markus?s sixth-order governing equation was developed for sandwich
beams or isotropic sandwich panels, and panels J and K have similar sti ness constants
along the two main in-plane directions, as shown in Table 6.5, then the modal density
estimates for these two panels are closer to the theoretical predictions than those for panel
I are. The core shear modulus of panel L could not be determined from the resonance
measurements. So the theoretical modal density predictions were computed by assuming
142
Figure 7.6: Modal density estimates for panel I with mass correction
Figure 7.7: Modal density estimates for panel K with mass correction
143
Figure 7.8: Modal density estimates for panel L with mass correction
that the equivalent shear moduli are 30 MPa and 40 MPa, respectively. The theoretical
values imply that the modal density of panel L is insensitive to the core shear moduli at
frequencies below 1000 Hz.
7.3 Experimental total loss factors
Unlike modal densities, theoretical expressions for loss factors of structures are not
available. The loss factor of a structure can be obtained from the measurement of the force
supplied to the structure and the spatially averaged square velocity produced. In steady
state conditions, the average power input is equal to the average power dissipated, and then
144
the average loss factor is,
= F
2(t)Re(Y)
Mphv2(t)i2 f: (7.8)
Thus the band-averaged loss factor is,
(f) = 14f
Z F2(t)Re(Y)
Mphv2(t)i2 f df: (7.9)
As discussed in the previous section, the force measurement can be mass corrected as
follows,
Fc = Fm MAm = Fm i!MYcFc)Fc = Fm1 +i!MY
c
; (7.10)
then, F2c = F
2m
[1 !MIm(Yc)]2 + [!MRe(Yc)]2: (7.11)
In the case of the measurement of high frequency vibration of lightly damped structures,
considerable care should be taken when using an accelerometer because of the mass loading
e ect. Well below its resonance frequency, the accelerometer can be assumed to act as a
pure mass. The velocity of the structure Vc can be assumed to be reduced to Va by the
presence of the accelerometer [56],
Va
Vc =
Z
Z +i!ma; (7.12)
where Z is the mechanical impedance of the test element, and ma is the accelerometer mass.
145
In this study, the mass loading of the accelerometer was assumed to be,
V 2a
V 2c =
1
1 + [!maRe(Yc)]2; (7.13)
where Yc is the corrected point mobility. The corrected point mobility in the loss factor
analysis is the estimate obtained from the spectral mass correction method.
The frequency bandwidth of the loss factor analysis chosen was one-third octave. The
frequency analysis resolution was 1 Hz. The velocities of the panels were determined by
measuring the panel responses with an Endevco model 2226c piezoelectric accelerometer at
ve randomly chosen positions. The loss factor estimates for panel J are shown in Fig. 7.9.
The mass loading e ect becomes apparent at frequencies above 3150 Hz for panel J. It
was found that the corrected loss factor of panel J is less than 3%. The modal loss factors
of panel J, obtained from the half-power point method are given in Table 7.1. The loss
factor estimates are in good agreement with the modal loss factors.
Table 7.1: The modal loss factors of panel J
Frequency (Hz) 161 222 238 320 336 366 382 412 454 490 554 601
Loss factor (%) 0.7 0.8 0.7 1.6 0.6 0.5 0.6 0.6 0.7 0.7 0.6 0.9
The loss factor estimates for panels I, K and L are shown in Figs. 7.10 7.12. It was
found that the loss factor of panel L is much higher than those of the three other panels.
The e ect of the mass loading of the accelerometer is small at frequencies below 2000 Hz
for all four panels.
146
Figure 7.9: Loss factor estimates for panel J
Figure 7.10: Loss factor estimates for panel I
147
Figure 7.11: Loss factor estimates for panel K
Figure 7.12: Loss factor estimates for panel L
148
7.4 Experimental radiation loss factors
The radiation resistance of a structure in a reverberant eld can be experimentally
obtained by studying the energy ow relations between the structure and the reverberation
room [13]. Consider a panel that is excited by a shaker in a reverberation room. The steady
state power ow balance equations are,
in1 = diss1 + 12; 0 = diss2 12: (7.14)
The total power supplied to the panel by the shaker is,
in1 = Rtothv2i= Rtot E1M
p
= (Rint +Rrad)hv2i; (7.15)
where Rtot, Rrad and Rint are the total, radiation and mechanical resistances of the panel,
respectively. Mp is the mass of the panel and hv2i is the mean square velocity of the
structure. The power dissipated by the structure is,
diss1 = Rint E1M
p
= Rinthv2i: (7.16)
The rate of internal energy dissipation by the reverberation room is,
diss2 = E2 room! = room!hp
2i
c2 Vroom; (7.17)
149
where room andVroom are the internal loss factor and the volume of the room, respectively.
hp2i is the mean square pressure of the room.
Substitution of Eqs. (7.15) (7.17) into (7.14), yields
Rradhv2i= room!hp
2i
c2 V2: (7.18)
The equation above can be rewritten as,
Rrad = room! SpS
v c2
Vroom = 13:8SpTroomS
v c2
Vroom; (7.19)
where Sp is the pressure spectral density function of the room, Sv is the velocity spectral
density function of the structure, and Troom is the reverberation time of the room.
Similarly, the radiation resistance of structures excited by a shaker between two rever-
beration rooms can be estimated experimentally [18]. The steady state power ow balance
equations are,
0 = diss1 + 12 + 13; in2 = diss2 12 + 23; 0 = diss3 13 23: (7.20)
Then, we have
in2 diss2 = diss1 + diss3 = Rradhv2i= ! c2 ( 1hp21iV1 + 3hp23iV3): (7.21)
150
Likewise, the radiation resistance of the structure between two reverberation rooms is,
Rrad = 13:8S
v c2
S
p1
T1 V1 +
Sp3
T3 V3
: (7.22)
It is noted that in both radiation resistance determinations, Eqs. (7.19) and (7.22), the
radiation resistance is also termed as R4 rad, because that the radiating area of the panel is
twice of the area of the panel.
Gomperts [58] showed that the radiation e ciency of a ba ed free-edge panel at fre-
quencies well below the critical frequency is,
baf; free = (2=5)(f=fc)2 baf; ss = (2=5)(f=fc)2( corner + edge); (7.23)
where fc is the critical frequency of the panel. corner and edge are the radiation e -
ciencies of ba ed simply supported panel for corner and edge modes, which were derived
by Maidanik [13] (see Eq. (4.42)).
When the panel is unba ed, uid ow around the panel edges reduces the sound
radiation. Oppenheimer and Dubowsky [25] have provided an expression for the radiation
e ciency for unba ed simply supported panels,
unbaf; ss = Fplate(Fcorner corner +Fedge edge); f
>>>
>>><
>>>
>>>>
:
x
y
z
yz
xz
xy
9>
>>>
>>>=
>>>
>>>>
;
=
2
66
66
66
64
C11 C12 C13 0 0 0
C12 C22 C23 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 0 0 C66
3
77
77
77
75
8>
>>>
>>><
>>>
>>>>
:
x
y
z
yz
xz
xy
9>
>>>
>>>=
>>>
>>>>
;
where
C11 = 1 23 32 E1; C12 = 12 + 13 32 E2 = 21 + 23 31 E1;
C13 = 13 + 12 23 E3 = 31 + 21 32 E1; C22 = 1 13 31 E2;
C23 = 32 + 12 31 E2 = 23 + 21 13 E3; C33 = 1 12 21 E3;
= 1 12 21 23 32 13 31 2 12 23 31;
C44 = G23; C55 = G13; C66 = G12:
If the Young?s modulus in one principal axis of orthotropic material is much sti er than
those in the others, E3 >> E1; E2; 31; 32 >> 13; 23, the sti ness constants can be
approximated,
C11 = E11
12 21
; C12 = 121
12 21
E2 = 211
12 21
E1;
C13 = 131
12 21
E3 = 311
12 21
E1; C22 = E21
12 21
;
C23 = 321
12 21
E2 = 231
12 21
E3; C33 = E31
12 21
:
191
Figure A.1: The rotated-axis coordinate system of the orthotropic material
According to the tensor transformation rule, the stresses and strains along the principal
axes can be written with the stresses and strains in x, y, z axes,
2
64 1 12 13
12 2 23
13 23 3
3
75 = A
2
64 x xy xz
xy y yz
xz yz z
3
75AT;
2
64 "1 12=2 13=2
12=2 "2 23=2
13=2 23=2 "3
3
75 = A
2
64 "x xy=2 xz=2
xy=2 "y yz=2
xz=2 yz=2 "z
3
75AT;
where A =
2
64 cos sin 0 sin cos 0
0 0 1
3
75 =
2
64 l m 0 m l 0
0 0 1
3
75:
Expanding the equations above, produces
8>
>>>
>>><
>>>
>>>>
:
1
2
3
23
13
12
9>
>>>
>>>=
>>>
>>>>
;
= T
8
>>>>
>>><
>>>
>>>>
:
x
y
z
yz
xz
xy
9
>>>>
>>>=
>>>
>>>>
;
=
2
66
66
66
64
l2 m2 0 0 0 2lm
m2 l2 0 0 0 2lm
0 0 1 0 0 0
0 0 0 l m 0
0 0 0 m l 0
lm lm 0 0 0 l2 m2
3
77
77
77
75
8
>>>>
>>><
>>>
>>>>
:
x
y
z
yz
xz
xy
9
>>>>
>>>=
>>>
>>>>
;
; and
8
>>>>
>>>
<
>>>>
>>>:
"1
"2
"3
23=2
13=2
12=2
9
>>>>
>>>
=
>>>>
>>>;
= T
8
>>>>
>>>
<
>>>>
>>>:
"x
"y
"z
yz=2
xz=2
xy=2
9>
>>>>
>>=
>>>>
>>>;
:
192
The relationships between engineering strains and tensorial strains can be expressed
as,
8>
>>>>
>><
>>>>
>>>:
"x
"y
"z
yz
xz
xy
9>
>>>>
>>=
>>>>
>>>
;
= R
8>
>>>>
>><
>>>>
>>>
:
"x
"y
"z
yz=2
xz=2
xy=2
9>
>>>>
>>=
>>>>
>>>
;
=
2
66
66
66
64
1
1
1
2
2
2
3
77
77
77
75
8>
>>>
>>><
>>>>
>>>
:
"x
"y
"z
yz=2
xz=2
xy=2
9>
>>>
>>>=
>>>>
>>>
;
:
Applying the stress-strain relations in the three principal axes and introducing the
developed transformations above, we obtain
8
>>>>
>>>
<
>>>>
>>>:
1
2
3
23
13
12
9
>>>>
>>>
=
>>>>
>>>;
= T
8>
>>>>
>><
>>>>
>>>
:
x
y
z
yz
xz
xy
9>
>>>>
>>=
>>>>
>>>
;
= C
8>
>>>
>>><
>>>>
>>>
:
"1
"2
"3
23
13
12
9>
>>>
>>>=
>>>>
>>>
;
= CR
8>
>>>
>>><
>>>
>>>>
:
"1
"2
"3
23=2
13=2
12=2
9>
>>>
>>>=
>>>
>>>>
;
= CRT
8>
>>>
>>><
>>>
>>>>
:
"x
"y
"z
yz=2
xz=2
xy=2
9>
>>>
>>>=
>>>
>>>>
;
;
then,
8
>>>>
>>><
>>>
>>>>
:
x
y
z
yz
xz
xy
9
>>>>
>>>=
>>>
>>>>
;
= C
8
>>>>
>>>
<
>>>
>>>>
:
"x
"y
"z
yz
xz
xy
9
>>>>
>>>
=
>>>
>>>>
;
; where C = T 1CRTR 1;
C11 = l4C11 + 2l2m2(C12 + 2C66) +m4C22;
C12 = l2m2C11 + (l4 +m4)C12 +l2m2C22 4l2m2C66;
C13 = l2C13 +m2C23;
C16 = l3mC11 + (lm3 l3m)C12 lm3C22 + 2(lm3 l3m)C66;
C22 = m4C11 + 2l2m2(C12 + 2C66) +l4C22;
C26 = lm3C11 + (l3m lm3)C12 l3mC22 + 2(l3m lm3)C66;
C33 = C33; C44 = m2C55 +l2C44; C55 = m2C44 +l2C55;
C66 = l2m2C11 2l2m2C12 +l2m2C22 + (l2 m2)C66;
C34 = C35 = C36 = C14 = C15 = C24 = C25 = 0:
193
Appendix B
Partial differential Operators
The elements of the matrix B of Eq. (3.59) are di erential operators which are listed
below:
B11 = mh
2
4
@4
@x2@t2 +m
@2
@t2 +D
@4
@x4 h
2C13 + C553
+ 4C13h ;
B12 = mh2 @
3
@x@t2 F
@3
@x3 + 2C13
@
@x; B13 =
h2
@3
@x@t2
C11h2
@3
@x3 +
4
(C13 +C55)
@
@x;
B22 = m@
2
@t2 +C
@2
@x2; B23 =
2C11h
@2
@x2
2 h
@2
@t2;
B33 = h2 @
2
@t2 +
C11h
2
@2
@x2 +
2C55
2h ; B34 = B35 = 0;
B14 = h
2( 1t1 2t2)
4
@4
@x2@t2 ( 1t1 2t2)
@2
@t2 (D1 D2)
@4
@x4;
B24 = B15 = h( 1t1 2t2)2 @
3
@x@t2 + (F1 F2)
@3
@x3;
B25 = ( 1t1 2t2) @
2
@t2 (C1 C2)
@2
@x2;
B44 = m
h2
4
@4
@x2@t2 +m
@2
@t2 +
eD @4
@x4;
B45 = m
h
2
@3
@x@t2
eF @3
@x3; B55 = m
@2
@t2 +
eC @2
@x2
4C55
h ;
with, m = 1t1 + 2t2 + h; m = 1t1 + 2t2 + h3 ;
C = C1 +C2 +C11h; F = F1 +F2 + C11h
2
2 ;
eC = C1 +C2 + C11h
3 ;
eF = F1 +F2 + C11h2
6 ;
D = D1 +D2 + C11h
3
4 ; Di =
Ei
3(1 v2i )
"
h
2 +ti
3
+
h
2
3#
;
Ci = Eiti(1 v2
i )
; Fi = Ei2(1 v2
i )
"
h
2 +ti
2
+
h
2
2#
;
where j and denote mass densities of the face sheet j and the core; tj and h stand for
thicknesses of the face sheet j and the core, respectively. Ej is the Young?s modulus of the
194
face sheet j. Cij is the sti ness constants of the core, and the directions denote as follows,
8
>>>>
>>><
>>>
>>>>
:
x
y
z
yz
xz
xy
9
>>>>
>>>=
>>>
>>>>
;
=
2
66
66
66
64
C11 C12 C13 0 0 0
C12 C22 C23 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 0 0 C66
3
77
77
77
75
8
>>>>
>>><
>>>
>>>>
:
x
y
z
yz
xz
xy
9
>>>>
>>>=
>>>
>>>>
;
:
195