STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH
COMPOSITE MATERIALS
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 classified information.
____________________________
Zhaohui Yu
Certificate of approval:
____________________________ ____________________________
George T. Flowers Malcolm J. Crocker, Chair
Professor Distinguished University Professor
Mechanical Engineering Mechanical Engineering
____________________________ ____________________________
Hareesh Tippur ZhongYang (Z.-Y.) Cheng
Professor Assistant Professor
Mechanical Engineering Materials Engineering
__________________
Joe F. Pittman
Interim Dean
Graduate School
STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH
COMPOSITE MATERIALS
Zhaohui Yu
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
May 10, 2007
iii
STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH
COMPOSITE MATERIALS
Zhaohui Yu
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon request of individuals or institutions and at their expense.
The author reserves all publication rights.
_____________________________
Signature of Author
_____________________________
Date of Graduation
iv
VITA
Zhaohui Yu, daughter of Shuiyong Yu and Limin Xu, was born on July 25, 1976, in
Qingdao, Shandong Province, China. She graduated from the No. 1 Middle School of
Jimo, Qingdao in 1995. She entered Ocean University of Qingdao, China in September
1995, and graduated with Bachelor of Science degree in Electrical Engineering and a
Minor Diploma in Software Engineering in July 1999. She worked as a signal processing
engineer in HuaYi Building Materials Company of Qingdao from September 1999 to
November 2001. She entered Graduate School, Auburn University, in January 2002. She
married to Yuquan Li on July 23, 2000.
v
DISSERTATION ABSTRACT
STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH
COMPOSITE MATERIALS
Zhaohui Yu
Doctor of Philosophy, May 10, 2007
(B.S., Ocean University of Qingdao, 1999)
132 Typed Pages
Directed by Malcolm J. Crocker
Sandwich composite materials have been widely used in recent years for the
construction of spacecraft, aircraft, and ships, mainly because of their high stiffness-to-
weight ratios and the introduction of a viscoelastic core layer, which has high inherent
damping. One of the main objects of this research is the measurement and estimation of
the bending stiffness and damping of sandwich structures. Knowledge of the elastic
properties of the core and face sheet of the sandwich structures is indispensable for the
analysis and modeling of sandwich strictures. However, traditional methods to determine
the elastic properties are not suitable for the core, which is usually brittle, and the face
sheets, which are usually very thin. A set of special techniques has to be used to estimate
the elastic properties of these materials. The dynamic bending stiffness of such materials
vi
is difficult to measure because it depends on frequency unlike ordinary non-composite
materials. A simple measurement technique for determining the material parameters of
composite beams was used. The damping is an important property used for the analysis
of the acoustical behavior of the sandwich structures, especially for the characterization
of the sound transmission loss. An interesting fact is that damping can be measured as a
byproduct in the procedure of the measurements of dynamic stiffness.
Another main object of the research is to analyze the sound transmission loss of
sandwich structures and to simulate their acoustical behavior using the statistical energy
analysis method (SEA). While solving vibroacoustic problems, FEM and SEA are
commonly used. However, common vibroacoustic problems involve a very large number
of modes over a broad frequency range. At high frequencies these modes become both
expensive to compute and highly sensitive to uncertain physical details of the system.
Many processes involved in noise and vibration are statistical or random in nature. So
SEA is suitable for the high frequency problems such as vibroacoustic problems.
The materials used in the research include sandwich structures with polyurethane foam-
filled honeycomb cores and sandwich structures with closed-cell polyurethane foam
cores. Foam-filled honeycomb cores possess mechanical property advantages over pure
honeycomb and pure foam cores. The honeycomb structure enhances the stiffness of the
entire structure; while the foam improves the damping. Closed-cell polyurethane foam is
CFC-free, rigid, and flame-retardant foam. Both foam-filled honeycomb and closed-cell
foam cores meet the requirements of many aircraft and aerospace manufacturers. Also,
foam-filled honeycomb and closed-cell structures have high strength-to-weight ratios and
vii
great resistance to water absorption, and will not swell, crack, or split on exposure to
water.
viii
ACKNOWLEDGMENTS
I would like to express my sincere appreciation and thanks to my advisor Auburn
University distinguished professor, Dr. Crocker for his guidance and support during my
studies and research. I am grateful for the considerable assistance provided by the other
committee members, Dr. Flowers, Dr. Tippur and Dr. Cheng. I appreciate outside reader,
Dr. Cochran for his time and valuable suggestions to my dissertation. My appreciation
also goes to my friends and colleagues at Auburn University. Last and not least, I want to
express my special thanks to my parents and my husband for their support and love.
ix
Journal used: Journal of Sound and Vibration
Computer software used: Microsoft Word 2002
x
TABLE OF CONTENTS
LIST OF TABLES........................................................................................................... xiii
LIST OF FIGURES ...........................................................................................................xv
CHAPTER 1 BACKGROUND ......................................................................................1
1.1 Introduction..........................................................................................................1
1.2 Objective..............................................................................................................2
1.3 Structure of honeycomb panels............................................................................3
1.4 Organization of Dissertation................................................................................6
CHAPTER 2 STATIC PROPERTIES CHARACTERIZATION...................................8
2.1 Introduction..........................................................................................................8
2.2 Four-point bending method..................................................................................8
2.3 Twisting method ................................................................................................10
2.4 Calculation of static bending stiffness of sandwich beam.................................12
2.5 Experimental results and analysis......................................................................13
2.5.1 Specimen................................................................................................... 13
2.5.2 Four-point bending method....................................................................... 15
2.5.3 Twisting method ....................................................................................... 16
2.5.4 Calculation of static bending stiffness of sandwich beam........................ 17
2.5.5 Summary and analysis .............................................................................. 18
xi
2.6 Finite element method model.............................................................................20
2.6.1 Introduction............................................................................................... 20
2.6.2 FEM in panel problems............................................................................. 20
2.6.3 Element Types .......................................................................................... 23
2.6.4 Finite element modeling of sandwich structures ...................................... 24
2.6.5 Comparison of results from experiments and theoretical analyses........... 30
CHAPTER 3 THEORY FOR SANDWICH BEAMS ..................................................32
3.1 Literature review................................................................................................32
3.2 Theory of sandwich structure.............................................................................35
3.3 Boundary conditions ..........................................................................................43
3.4 Wave numbers ...................................................................................................45
3.5 Least Squares Method........................................................................................50
3.6 Damping measurement methods........................................................................53
CHAPTER 4 DYNAMIC PROPERTIES CHARACTERIZATION ...........................57
4.1 Experiments .......................................................................................................57
4.1.1 Steps.......................................................................................................... 57
4.1.2 Set up ........................................................................................................ 57
4.1.3 Samples..................................................................................................... 60
4.2 Analysis of experimental results........................................................................63
4.2.1 Frequency response functions................................................................... 63
4.2.2 Boundary conditions ................................................................................. 66
4.2.2 Dynamic bending stiffness........................................................................ 69
4.2.3 Damping.................................................................................................... 76
xii
4.3 Conclusions........................................................................................................77
CHAPTER 5 SOUND TRANSMISSION LOSS OF SANDWICH PANELS.............78
5.1 Classical sound transmission analysis ...............................................................78
5.1.1 Mass law sound transmission theory ........................................................ 79
5.1.2 The effect of panel stiffness and damping ................................................ 81
5.1.3 The coincidence effect .............................................................................. 81
5.1.4 Critical frequency...................................................................................... 84
5.1.5 Sound transmission coefficient and transmission loss at coincidence...... 84
5.2 Literature review of the Sound Transmission Loss of Sandwich Panels...........85
5.3 Statistical energy analysis model (SEA)............................................................88
5.3.1 Prediction of Sound Transmission through Sandwich Panels using SEA 88
5.3.2 Simulation using SEA software AutoSEA ............................................... 93
5.4 Review of sound transmission measurement technique: two-room method .....94
5.5 Experiments of TL and simulations using AutoSEA.........................................97
5.6 Summary and conclusions of experiments of TL and simulations using
AutoSEA......................................................................................................................104
CHAPTER 6 SUMMARY AND CONCLUSIONS ...................................................105
REFERENCES ...............................................................................................................109
LIST OF TABLES
Table 1 The dimensions and densities of the specimens tested for their static stiffness ...13
Table 2 Static properties obtained from the four-point bending and twisting methods.....18
Table 3 Static bending stiffness obtained from two ways .................................................19
Table 4 Load-deflection relations for Beam D from the experiments and FEM analysis .31
Table 5 Load-deflection relations for Plate E from the experiments and FEM analyses ..31
Table 6 Boundary conditions for ends of beam.................................................................45
Table 7 Geometrical and material parameters for sandwich beam....................................48
Table 8 Values of
n
? for particular boundary conditions .................................................51
Table 9 Geometry and density of the sandwich beam with foam filled honeycomb core.61
Table 10 Geometry and density of the sandwich beam with foam core............................62
Table 11 Natural frequencies for different measurements on Beam F and their
corresponding bending stiffness. ...............................................................................65
Table 12 Natural frequencies of Beam C for the three different beam boundary
conditions...................................................................................................................67
Table 13 Comparison of static stiffness measured by four-point bending method and two
stiffness limits from dynamic characterization..........................................................70
Table 14 Comparison of shear modulus of the core measured by twisting method and that
from the dynamic stiffness curve for Beam G...........................................................74
xiii
xiv
Table 15 Damping ratio of Beam C...................................................................................76
Table 16 Geometrical parameters of panels under study...................................................97
Table 17 Reverberation times (s) of the receiving room with different panels .................99
xv
LIST OF FIGURES
Figure 1 Sandwich panel with a honeycomb core ...............................................................3
Figure 2 A: Nomex honeycomb core, B: irregular aluminum honeycomb core .................4
Figure 3 Corrugation process used in honeycomb manufacture..........................................5
Figure 4 Geometry and dimensions of the four-point bending test .....................................9
Figure 5 Two principal direction sandwich panel strip for the four-point bending
experiments and its equivalent plate representation ..................................................10
Figure 6 Loading scheme for the pure twisting test...........................................................11
Figure 7 Core, face sheet and entire sandwich structure ...................................................14
Figure 8 Deflection of beam with four-point bending method..........................................25
Figure 9 Out-of-plane shape with un-deformed edge of beam with four-point bending
method........................................................................................................................26
Figure 10 In-plane shear stress contour for beam with four-point bending method..........27
Figure 11 Out-of-plane deflection contour of plate with the twisting method ..................28
Figure 12 Deformed plate with un-deformed edge of plate with the twisting method......29
Figure 13 In-plane shear stress contour for plate with the twisting method......................30
Figure 14 Bending of composite bar or panel by (a) bending and (b) shearing of the core
layer............................................................................................................................35
Figure 15 Excitation of a beam and resulting forces and moments. Dimensions and
material parameters for the laminates and core are indicated....................................37
xvi
Figure 16 Elastic properties and area density of sandwich structure.................................37
Figure 17 Particular boundary conditions..........................................................................44
Figure 18 Wave numbers for beam [16]............................................................................47
Figure 19 Decay rate method used to determine damping ................................................54
Figure 20 Modal bandwidth method to determine damping..............................................55
Figure 21 Power balance method to determine damping...................................................56
Figure 22 Experimental steps to determine some properties of sandwich structures........57
Figure 23 Set Up 1 Using shaker .......................................................................................59
Figure 24 Set Up 2 Using hammer ....................................................................................60
Figure 25 Honeycomb sandwich composite structures; (a) foam-filled honeycomb core,
(b) composite beam....................................................................................................62
Figure 26 Closed-cell foam core........................................................................................63
Figure 27 FRF for the sandwich beam F for free-free boundary condition.......................63
Figure 28 FRF for the sandwich beam G for free-free boundary condition. .....................64
Figure 29 FRF for the sandwich beam E for free-free boundary condition.......................65
Figure 30 Bending stiffness of the sandwich beam C for three different boundary
conditions...................................................................................................................68
Figure 31 Bending stiffness for the sandwich beam C. .....................................................70
Figure 32 Dynamic stiffness for beams C and D...............................................................71
Figure 33 Dynamic stiffness for core in two principal directions .....................................72
Figure 34 Bending stiffness for the sandwich beam G......................................................73
Figure 35 Bending stiffness for the sandwich beam E, F and H........................................75
Figure 36 Damping ratio of sandwich beam C ..................................................................77
xvii
Figure 37 The coincidence effect.......................................................................................83
Figure 38 Idealized plot of transmission loss versus frequency ........................................83
Figure 39 Schematic of the power flow in three-coupled systems using SEA..................89
Figure 40 Sound transmission loss model using the AutoSEA software...........................94
Figure 41 Set up for the two reverberation room sound transmission loss measurement
method........................................................................................................................95
Figure 42 Measurements of TL, mass law and simulation of AutoSEA for panel A......100
Figure 43 Measurements of TL, mass law and simulation of AutoSEA for panel B ......100
Figure 44 Measurements of TL, mass law and simulation of AutoSEA for panel C ......101
Figure 45 Measurements of TL, mass law and simulation of AutoSEA for panel D......101
Figure 46 Measured TL for panels A and B ....................................................................102
Figure 47 Measured TL for panels C and D ....................................................................102
Figure 48 Measurements of TL and simulation by AutoSEA for panel E.......................104
Figure 49 Measured TL for panels A, B, C, D and E ......................................................104
1
CHAPTER 1 BACKGROUND
1.1 Introduction
Applications for sandwich structures are steadily increasing. The term sandwich
panel here refers to a structure consisting of two thin face plates bonded to a thick and
lightweight core. The face plates are typically made of aluminum or some composite
laminate. The core can be a lightweight foam or a honeycomb structure. These types of
sandwich structures having a high strength to weight ratio have been used by the aircraft
industry for over 70 years. However, during the last decade, various types of lightweight
structures have also been introduced in the vehicle industry. This trend is dictated by
demands for higher load capacity for civil and military aircraft, reduced fuel consumption
for passenger cars, increased speed for passenger and navy vessels of catamaran types
and increased acceleration and deceleration for trains to increase their average speeds.
The environmental impact of lightweight vehicles could be considerable in
reducing fuel consumption and increasing load capacity. However, there are also certain
constraints like passenger comfort, safety and costs for new types of vehicles. Passenger
comfort requires low noise and vibration levels in any type of vehicle.
In addition to new materials being introduced, certain types of trains and fast
passenger vessels are being built of aluminum. This means that traditional solutions
developed for steel constructions must be replaced by completely new designs to achieve
the required noise levels. Lightweight structures often have poor acoustical and dynamic
2
properties. In addition, the dynamic properties are often frequency dependent. In order to
avoid noise problems in lightweight vehicles, it is essential that the main structure-borne
sound sources are as weakly coupled as possible to the supporting structure foundation
and to adjoining elements.
With respect to material and construction costs, sandwich structures can compare
very favorably with other lightweight materials like aluminum. The number of
applications for sandwich panels is steadily increasing. One reason for the growing
interest is that today it is possible to manufacture high quality laminates for many
applications. The material used in the laminates is often glass reinforced plastic (GRP).
The composition of a laminate and thus its material parameters can be considerably
important in the manufacturing process.
Various types of core materials are commercially available. The techniques for
bonding core materials and laminates as well as different plate structure are well
understood, although still under development.
1.2 Objective
As discussed in the previous section, this thesis is mainly concerned with the
static, dynamic and acoustical properties of sandwich structures. In particular, emphasis
is placed on the study of the dynamic response of structural beams to acoustical
excitation. The thesis includes three main parts:
1. Characterization of the static properties of sandwich beams and model the
foam-filled honeycomb sandwich structures by the finite element method (FEM),
2. Study of the vibration response of sandwich structures and
characterization of the dynamic properties of sandwich beams,
3. Analysis of the sound transmission loss of sandwich panels and modeling
of foam-filled honeycomb sandwich structures using the Statistical Energy Analysis
(SEA) method.
1.3 Structure of honeycomb panels
A honeycomb panel is a thin lightweight plate with a honeycomb core with
hexagonal cells. Layered laminates are bonded to both sides of the core as shown in
Figure 1. Each component is by itself relatively weak and flexible. When incorporated
into a sandwich panel the elements form a stiff, strong and lightweight structure. The face
sheets carry the bending loads and the core carries the shear loads. In general, honeycomb
cores are strongly orthotropic.
Figure 1 Sandwich panel with a honeycomb core
The types of core materials in the panels used for the measurements presented
here are either Nomex or aluminum. Nomex is an aramid fiber paper dipped in
phenolicresin with a low shear modulus and low shear strength. A typical thickness of the
3
Nomex honeycomb panels investigated is 10 mm with the thickness of the face sheet or
laminate being between 0.3-0.7 mm. The weight per unit area is of the order of 3 kg/m
2
unit area. Each laminate consists of 3-5 different layers bonded together to give the best
possible strength. The laminates are not necessarily symmetric and are usually
orthotropic. The core acts as a spacer between the two laminates to give the required
bending stiffness for the entire beam. The bending stiffness of the core itself is in general
very low. The bending stiffness of sandwich materials is frequency dependent.
The cells in the core give it an orthotropic structure. The dynamic characteristics
should be expected to be different in each direction. The two main in-plane directions 1
and 2 are defined in Figure 2. The shape of the honeycomb cells of a typical aluminum
core is generally very irregular which makes it impossible to describe its geometry in a
simple way. Nomex cores have very regular shapes as compared to aluminum-cores.
A B
Figure 2 A: Nomex honeycomb core, B: irregular aluminum honeycomb core
4
Honeycomb cores, which were developed starting in the 1940?s primarily for the
aerospace industry, have the greatest shear strength and stiffness-to-weight ratios, but
require special care to ensure adequate bonding of the face sheets to the core since such
cores are hollow. The standard hexagonal honeycomb is the basic and most common
cellular honeycomb configuration, and is currently available in all metallic and
nonmetallic materials. Figure 3 illustrates the manufacturing process, and the L (ribbon
direction) and W (transverse to the ribbon) directions of the hexagonal honeycomb. In
this process, adhesive is applied to the corrugated nodes, the corrugated sheets are
stacked into blocks, the node adhesive cured, and sheets are cut from these blocks to the
required core thickness. The honeycomb cores are suitable for both plane and curved
sandwich applications.
Figure 3 Corrugation process used in honeycomb manufacture
As discussed in the previous chapters, sandwich structures with foam-filled
honeycomb cores have some advantages over pure honeycomb cores. By filling foam in
the honeycomb cells, not only the longitudinal cell walls but also the foam can carry the
5
6
uniaxial load. So the foam is expected to reduce the discontinuities in elastic properties
possessed by with pure honeycomb cores. Also the foam can make the fabrication of
sandwich structures easier than those made with pure honeycomb cores. Another
important advantage of foam-filled honeycomb cores is their improved damping and
shear strength properties.
1.4 Organization of Dissertation
This dissertation contains the results of the present research investigation into the
current objectives. The research was performed in the Sound and Vibration Laboratory of
the Department of Mechanical Engineering at Auburn University. The results reported
are divided into four major parts.
Chapter 2 described the static properties of foam-filled honeycomb sandwich
beams. Orthotropic plate theory is introduced and two methods, the four-point bending
method and the twisting method, are used to measure the shear modulus, Young?s
modulus and other properties of sandwich structures. The finite element method (FEM)
was used to simulate the response of the beams and panels with the four-point bending
method and the twisting method.
A thorough derivation of the beam theory of the sandwich structure is given in
Chapter 3. The theoretical model was derived using Hamilton's principle. The general
dynamic behavior of sandwich structures is discussed. Chapter 3 also describes the
measurements, which include experiments and the analysis of results. In the experiments,
measurements on foam-filled honeycomb sandwich beams with different configurations
were performed and finally the conclusions were drawn from the analysis of the results.
7
Chapter 4 is devoted to the study of sound transmission through sandwich panels.
This chapter starts with a brief introduction of the classical theory of sound transmission
loss. Then the previous research on sound transmission through sandwich panels is
reviewed. The sound transmission loss of several panels with different thickness of core
and face sheet was measured by the two room method. Simulations of the sound
transmission loss were conducted using AutoSEA. Experimental results for sound
transmission loss are presented as well.
The summary and conclusions drawn from this research are given in Chapter 5.
CHAPTER 2 STATIC PROPERTIES CHARACTERIZATION
2.1 Introduction
In this work, an experimental study on the bending stiffness of sandwich beams
was conducted. An experimental procedure to measure the bending stiffness of sandwich
beams was used. The technique includes the standard four-point bending tests of beam
specimens to assist in the evaluation of the in-plane Young?s modulus of the core, face
sheet and entire sandwich structure. In addition, special plate twisting tests have been
used for finding the in-plane shear modulus and Poisson?s ratios of core, face sheet and
entire sandwich structures. Using these elastic properties, the bending stiffness of entire
sandwich structures was calculated in two different ways. The static behavior of the
entire sandwich structures was simulated using the finite element method (FEM).
2.2 Four-point bending method
The four-point bending arrangement used in this work is shown in Figure 4. The
beams were simply supported at the ends of the central span and half of the total load
2
d
P was applied at the ends of the beams, separately. The loading points were located 5
mm from the corners of the plate specimens. The advantage of this bending test is that
normal stresses, but not shear stresses, act over the central span . The central span,
therefore, is in a state of pure bending. Measurement of the load versus central
deflection response enables the calculation of the Young?s moduli of the core,
face sheet and entire sandwich structure, by means of the equation
2
d
P
0
w
21
EE
8
0
2
21
3
3
4
w
Sdd
bh
P
x
?
?
?
?
?
?
?
?
= , (1)
where P is the total load on the ends, is the central span, is the central
deflection, is the outer span and is the thickness as shown in Figure 4.
2
d
0
w
1
d h
And is the compliance along the direction parallel to the length of the beam.
x
S
2.direction principalin
1
1,direction principalin
1
2
22
1
11
E
SS
E
SS
x
x
==
==
(2)
Figure 4 Geometry and dimensions of the four-point bending test
The exact location at which the loads are applied is largely arbitrary, except that
the two outer spans must be equal.
1
d
In the measurement of the deflection, the force required to actuate the deflection
measurement device must be kept small relative to the applied loads to assure a pure
bending state. In this work, a micrometer was used to measure the beam deflections. The
micrometer could be read to the nearest 0.001 mm.
9
Both the principal directions, 1 and 2, of the four-point bend specimens, as shown
in Figures 5 were considered. Since the core pf the sandwich beams used in the
measurements is orthotropic, several extensive reviews of equivalent plate models in the
literature [1-4] were consulted.
Figure 5 Two principal direction sandwich panel strip for the four-point
bending experiments and its equivalent plate representation
2.3 Twisting method
10
Figure 6 Loading scheme for the pure twisting test
The twisting method configuration used in this work is shown in Figure 6. The
plates were simply supported at the three quadrant corners and the load was applied at
the other corner of the plates. Measurement of the load versus central deflection
response enables calculation of shear modulus , Poisson's ratios
P
P
0
w
12
G
2112
, ?? of the
core, face sheet and entire sandwich structure, by means of the equation:
0
2
3
3
4
w
LS
t
P
G
?
?
?
?
?
?
?
?
= , (3)
where
() () ( ) ,282
66
2
22
22
2
12
22
11
2
SnmSnmmnSnmSnmmnS
G
?+++???= (4)
and
,
1
,,
1
,
1
12
66
2
21
1
12
12
2
22
1
11
G
S
EE
S
E
S
E
S =?=?===
??
and
?cos=m and ?sin=n , where ? is the angle between the principal direction 1
and x coordinate.
For various material orientations, Equation (4) can be used to evaluate
(5)
().2,45
,,0
1222
66
SSS
SS
G
G
?=?=
==
o
o
?
?
Compliances and have been obtained from the four-point beam bending
tests. By performing twisting tests, with and , compliances and
can be determined. Combining these results, the equivalent Poisson's ratios are obtained
11
S
22
S
o
0=?
o
45=?
66
S
12
S
11
.
,
21
11
22
12
21
22
12
??
?
S
S
S
S
=
=?
(6)
The shear modulus is also obtained since is known from the
equation
12
G
66
S
12
66
1
G
S = .
The load and supporting points for the twisting tests were located 5 mm from the
corners of the plate specimens. The amount of overhang was not found to be critical. The
method for measuring the displacement of the center of the plate, , was the same as
with the four-point bending tests. The samples tested were
0
w
5.05.0 ? meter square plates.
2.4 Calculation of static bending stiffness of sandwich beam
Once we know the properties of core and face sheet, the bending stiffness per unit
width of the beam is
b
D
?
?
?
?
?
?
+++=
3
2
212
3
2
23
l
lc
lc
l
ccb
t
tt
tt
E
tE
D , (7)
where and are the thicknesses of the core and face sheets (laminate),
respectively. is the effective modulus of the core and is the effective Young?s
modulus of the face sheet (laminate) and they can be calculated using Equation (8).
c
t
l
t
c
E
l
E
.21
1
,2
1
,1
1
2112
1
2112
2
2
2112
1
1
ordirectionprincipalfor
E
Esheetsfacefor
directionprincipalfor
E
E
directionprincipalfor
E
E
corefor
ll
l
l
cc
c
c
cc
c
c
??
??
??
?
=
?
?
?
?
?
?
?
?
=
?
=
(8)
12
Then the bending stiffness of the entire sandwich beams, in two principal
directions, 1 and 2, can be calculated by Equation (9):
.2
3
2
212
,1
3
2
212
3
2
23
2
22
3
2
23
1
11
directionprincipalfor
t
tt
tt
E
tE
D
directionprincipalfor
t
tt
tt
E
tE
D
l
lc
lc
l
ccs
l
lc
lc
l
ccs
?
?
?
?
?
?
?
?
+++=
?
?
?
?
?
?
?
?
+++=
(9)
Alternatively, the bending stiffness per unit width of the entire sandwich beams
can be calculated directly from the modulus of the sandwich beams using the four-point
bending method and the twisting method (see section 2.2 and 2.3) as follows:
()
()
()
()
.2
112
2
,1
112
2
2112
3
2
22
2112
3
1
11
directionprincipalfor
ttE
D
directionprincipalfor
ttE
D
ss
lc
s
s
ss
lc
s
s
??
??
?
+
=
?
+
=
(10)
2.5 Experimental results and analysis
2.5.1 Specimen
The specimens used in the experiments included two beams of honeycomb core
filled with foam in two principal directions, one beam made of a face sheet, one square
plate of core with , one square plate of core with and two entire sandwich
beams cut with the x-coordinate oriented in two the principal directions. All the beams
were uses with the four-point bending method and all the plate were used in twisting
method.
o
0=?
o
45=?
The dimensions and densities of the specimens are listed in Table 1:
Density Specimen
name
content Thickne
ss (mm)
Length
(m)
Width
(mm) Core Face sheet
Direction
13
14
(kg/m
3
) (kg/m
3
)
Beam A Core 6.35 0.61 25.4 163 N/A 1
Beam B Core 6.35 0.61 25.4 163 N/A 2
Beam C Face sheet 0.325 0.61 25.4 N/A 2161 1(2)
Beam D Entire
sandwich
7 0.61 25.4 163 2161 1
Beam E Entire
sandwich
7 0.61 25.4 163 2161 2
Plate F Core 6.35 0.5 N/A 163 N/A
o
0=?
Plate G Core 6.35 0.5 N/A 163 N/A
o
45=?
Plate H Entire
sandwich
6.35 0.5 N/A 163 N/A
o
0=?
Plate I Entire
sandwich
6.35 0.5 N/A 163 N/A
o
45=?
Table 1 The dimensions and densities of the specimens tested for their static stiffness
The pictures of core, face sheet and entire sandwich are shown in Figure 7.
Figure 7 Core, face sheet and entire sandwich structure
2.5.2 Four-point bending method
Beams A, B, C, D and E were measured for their static stiffness using the four-
point bending method.
For beam A, the slope of the load-deflection curve and the properties derived are
given by Equations (11).
81.5
3
4
11
2
21
3
=
c
Sdd
bh
slope ,
6
11
1
1005.9
1
?==
c
c
S
Ecorefor Pa. (11)
For beam B, the slope of the load-deflection curve and the properties derived are
given by Equations (12).
34.3
3
4
22
2
21
3
=
c
Sdd
bh
slope ,
6
22
2
1020.5
1
?==
c
c
S
Ecorefor Pa. (12)
For beam C, the slope of the load-deflection curve and the properties derived are
given by Equations (13). Since the face sheet (laminate) is isotropic in the two in-plane
principal directions .
ll
EE
21
=
22.5
3
4
11
2
21
3
=
l
Sdd
bh
slope ,
10
11
12
1086.4
1
?===
l
ll
S
EEsheetfacefor Pa. (13)
For beam D, the slope of the load-deflection curve and the properties derived are
given by Equations (14).
15
03.9185
3
4
11
2
21
3
=
s
Sdd
bh
slope ,
10
11
1
100449.1
1
?==
s
s
S
Esandwichfor Pa. (14)
For beam E, the slope of the load-deflection curve and the properties derived are
given by Equations (15).
15.9184
3
4
22
2
21
3
=
s
Sdd
bh
slope ,
10
22
2
100448.1
1
?==
s
s
S
Esandwichfor Pa. (15)
2.5.3 Twisting method
Plates F, G, H and I were measured under the configuration of twisting method.
For plate F, the slope of the load-deflection curve and the properties derived are
given by Equations (16).
66.109
3
4
0
2
66
3
==
aS
h
slope
c
o
? ,
7
66
12
1003.8
1
?==
c
c
S
Gcorefor Pa. (16)
For plate G, the slope of the load-deflection curve and the properties derived are
given by Equations (17).
()
.
126.0
,22.0
,15.3
3
2
45
21
11
22
12
22
12
21
2
1222
3
?
?
?
?
?
?
?
==
=?=
=
?
=
c
c
c
c
c
c
c
cc
S
S
S
S
corefor
aSS
h
slope
??
?
?
o
(17)
16
For plate H, the slope of the load-deflection curve and the properties derived are
given by Equations (18).
,80.27097
3
4
0
2
66
3
==
aS
h
slope
s
o
?
10
66
12
1045.1
1
?==
s
s
S
Gsandwichfor Pa. (18)
For plate I, the slope of the load-deflection curve and the properties derived are
given by Equations (19).
()
?
?
?
?
?
?
?
==
=?=
=
?
=
.3.0
,3.0
,92.7509
3
2
45
21
11
22
12
22
12
21
2
1222
3
s
s
s
s
c
s
s
ss
S
S
S
S
sandwichfor
aSS
h
slope
??
?
?
o
(19)
2.5.4 Calculation of static bending stiffness of sandwich beam
From Equations (11-15), we know the properties of the core and face sheets.
According to Equation (8), the effective Young?s modulus of the core and face sheets can
be calculated as follows.
for core
6
2112
1
1
1031.9
1
?=
?
=
cc
c
c
E
E
??
Pa for principal direction 1,
6
2112
2
2
1035.5
1
?=
?
=
cc
c
c
E
E
??
Pa for principal direction 2,
for face sheets
10
2112
1
1009.5
1
?=
?
=
ll
l
l
E
E
??
Pa for principal direction 1 or 2. (20)
17
Then using Equation (9), we can calculate the bending stiffness of the entire
sandwich beams:
53.400
3
2
212
3
2
23
1
11
=
?
?
?
?
?
?
?
?
+++=
l
lc
lc
l
ccs
t
tt
tt
E
tE
D N
.
m for principal direction 1,
45.400
3
2
212
3
2
23
2
22
=
?
?
?
?
?
?
?
?
+++=
l
lc
lc
l
ccs
t
tt
tt
E
tE
D N
.
m for principal direction 2. (21)
The bending stiffness per unit width of the entire sandwich beams can be
calculated directly from the modulus of the sandwich beams using four-point bending
method and the twisting method as follows:
()
()
28.335
112
2
2112
3
1
11
=
?
+
=
ss
lc
s
s
ttE
D
??
N
.
m for principal direction 1,
()
()
25.335
112
2
2112
3
2
22
=
?
+
=
ss
lc
s
s
ttE
D
??
N
.
m for principal direction 2. (22)
2.5.5 Summary and analysis
After calculations from the measurements of the four-point bending method and
twisting method, the static properties for the core and entire sandwich structures were
obtained as follows:
Item
1
E
[ Pa]
2
E
[ Pa] 12
?
21
?
G
12
[ Pa]
Core 9.05?10
6
5.20?10
6
0.22 0.126 8.03?10
7
Face sheet 4.86E?10
10
4.86?10
10
0.213 0.213 1.67?10
10
Entire sandwich 1.0449?10
10
1.0448?10
10
0.3 0.3 8.025?10
7
Table 2 Static properties obtained from the four-point bending and twisting
methods
18
The bending stiffness per unit width of the entire sandwich beams were calculated
in two ways, one from the properties of the core and face sheet
and the other directly from the properties of the entire
sandwich beams . All the properties were obtained from the four-
point bending and twisting methods. The comparison of the results from these two ways
(Equation 2 and 3) is given in Table 3:
cclcc
EEE
2112121
,,,, ??
ssss
EE
211221
,,, ??
Bending stiffness per
unit width of the
entire sandwich
beams
From properties
cclcc
EEE
2112121
,,,, ??
[N
.
m]
From properties
ssss
EE
211221
,,, ??
[N
.
m]
s
D
11
335.28 335.25
s
D
22
400.53 400.45
Table 3 Static bending stiffness obtained from two ways
Some conclusions can be drawn from Tables 2 and 3:
1. The shear modulus of the core and that of entire sandwich structure have
very similar values, which means the core bears most of the shear and there is almost no
shear in the face sheet.
2. The results of the measurements are reasonably accurate and the bending
stiffnesses per unit width of the entire sandwich beams obtained from two approaches are
in good agreement.
19
20
3. The measurement methods described in this chapter are useful for the
determination of the static elastic properties of the core, face sheets and entire sandwich
structure.
2.6 Finite element method model
2.6.1 Introduction
Finite element methods have only become of significant practical use with the
introduction of the digital computer in the 1950's. Ideally an elastic structure should be
considered to have an infinite number of connection points or to be made up of an infinite
number of elements. However, it is found that if a structure is represented by a finite,
although normally large number of elements, solutions for the static or dynamic behavior
of the structure may be obtained which are in good agreement with solutions found by
exact methods [5].
The finite element method has been perhaps most widely used for the solution of
structural problems in the aerospace industry, one example being the design of the static
strength of wings. However, FEM is also widely used in other branches of engineering.
In civil engineering it is used for example in the design of dams and shell structures. It
can also be applied to such diverse problems as heat conduction and fluid flow. It should
also be noted that the method is not confined to linear problems but can also be used in
non-linear structural problems where large deformations or creep and plastic deformation
occur [6].
2.6.2 FEM in panel problems
There are several ways of representing a structure in a series of finite elements.
Each method has its advantages and disadvantages. The main difficulty to overcome is
21
the representation of an infinite number of connection points by a finite number. Turner
et al [7] were perhaps the first to advance the concept of finite elements. Their concept
attempts to overcome this difficulty by assuming the real structure to be divided into
elements interconnected only at a finite number of modal points. At these points some
fictitious forces, representative of the distributed stresses actually acting on the elements
boundaries, are supposed to be introduced. This procedure, which at first does not seem
completely convincing, has been given a firm foundation by Zienkiewicz [6].
Many engineering problems in solid mechanics are essentially impossible to solve
using analytical solution techniques. To solve these types of problems, stress analysts
have sought other methods. FEM has become a popular technique which yields
approximate numerical solutions to difficult boundary value problems. This method of
analysis can treat nonlinear problems with irregular boundary shapes and mixed
conditions.
The basic concept of finite element modeling is to replace the solid body to be
analyzed with a network of finite elements [8]. The elements are solid elements whose
properties duplicate the material they replace. These elements are then connected by
nodes. As the size of the finite elements become smaller and smaller, the method yields
results that are more closely related to those obtained from a rigorous mathematical
analysis.
The procedure for dividing a structure into finite elements can be described as
follows [74]:
1) The structure is divided into a finite number of the elements by drawing a
series of imaginary lines.
22
2) The elements are considered to be interconnected at a series of modal
points situated on their boundaries.
3) Functions are then chosen to define the displacement in each element (and
sometimes additional properties such as slope, moment, etc.) in terms of the displacement
at the modal points.
4) The modal displacements are now the unknowns for the system. The
displacement function now defines the state of strain in each element and together with
any initial strains and the elastic constant relates it to the stress at the element boundaries.
5) A system of forces is determined at the nodes which will result in
equilibrium of the boundary stresses with the distributed loading. A stiffness relationship
can now be determined for each element relating the nodal forces to the nodal
displacements.
This method is useful for the analysis of complicated structures which cannot be
approached easily using classical modal analysis. It is most useful for static problems and
for dynamic problems at low frequencies.
The analysis is limited at high frequencies in much the same way as classical
modal analysis. In classical modal analysis, the number of modes which must be included
becomes very large at high frequencies. In the case of finite elements the number of
elements which must be included also becomes very large at high frequencies. It is
desirable to have at least four elements per bending wavelength.
In this research, the FEM technique was used to approximate the bending stiffness
of sandwich panels, and to predict the load-deflection response of sandwich panel
specimens. The finite element software package utilized was ANSYS. Finite element
models for the experimental test method configurations discussed in section 2.6 have
been created using the pre-processor in ANSYS. The basic concepts of these models are
presented in this chapter, while results of the models along with a comparison with
experimental and analysis results is shown in the following section.
Each of the finite element models created for the different test configurations in
this thesis were developed with the computer aided design pre-processor in ANSYS. The
models created in ANSYS were:
1. Entire sandwich beam with the four-point bending model for the principal
direction 1
2. Entire sandwich beam with the four-point bending model for the principal
direction 2
3. Entire sandwich plate with the pure twisting model ( )
o
0=?
The geometries of these models were chosen to be the same as those used in the
actual experiments conducted on the specimens in Table 1.
2.6.3 Element Types
The SHELL99 (linear layered structural shell element) may be used for layered
applications of a structural shell model. The SOLID46 (8-noded, 3-D layered solid
element) is essentially a 3-D version of the layered SHELL99 element type designed to
model thick layered shells or solids. SOLID46 is recommended rather than SHELL99 for
calculating interlaminar stresses, primarily because multiple SOLID46 elements can be
stacked to allow through-the-thickness deformation slope discontinuities. The SHELL99
element was developed assuming that the shear disappears at the top and bottom surfaces
of the element, while the SOLID46 element does not use such an assumption. In the
23
24
SOLID46 formulation, effective thickness-direction properties are calculated using
thickness averaging. The result is that the interlaminar stresses are relatively constant
though the element thickness. To calculate accurate interlaminar stresses, multiple
SOLID46 elements stacked through the thickness are recommended [9]. Solid elements
were used to model the sandwich structure panels to simulate the behavior of load-
deflection and shear stress with the four-point bending and twisting methods. However,
shell elements would likely have sufficed to simulate the load-deflection behavior of the
entire sandwich structures with the four-point bending and twisting methods, because
accurate stresses were not required for the current investigation.
2.6.4 Finite element modeling of sandwich structures
Four-point Bending Method Modeling
The sandwich beam, which is with the four-point bending method, is modeled
using the elements of SOLID46. The geometry of the beam is similar as that of Beam D
in Table 1 and the properties of the three layers of the beam-face sheet, core and face
sheets were obtained from the results of measurements of the four-point bending and
twisting methods as given in section 2.5.
The out-of-plane deflection contour is shown in the following figure:
Figure 8 Deflection of beam with four-point bending method
The out-of-plane deformation shape with an un-deformed edge is shown in the
following figure:
25
Figure 9 Out-of-plane shape with un-deformed edge of beam with four-point
bending method
The in-plane shear stress contour is shown in the following figure:
26
Figure 10 In-plane shear stress contour for beam with four-point bending method
From figure 10, we can see that no shear stresses act over the central span. The
central span, therefore, is in state of pure bending, which is the advantage of using the
four-point bending method instead of the three-point bending method.
Twisting Method Modeling
The sandwich plate, which is with the twisting method, is modeled using the
elements of SOLID46. The geometry of the plate is similar to that of Plate H in Table 1
and the properties of the three layers of the beam-face sheet, core and face sheet were
obtained from the results of measurements made with the four-point bending and twisting
methods as described in section 2.6.
The out-of-plane deflection contour is shown in the following figure:
27
Figure 11 Out-of-plane deflection contour of plate with the twisting method
The out-of-plane deformation shape with un-deformed edge is shown in the
following figure:
28
Figure 12 Deformed plate with un-deformed edge of plate with the twisting method
The in-plane stress contour is shown in the following figure:
29
Figure 13 In-plane shear stress contour for plate with the twisting method
2.6.5 Comparison of results from experiments and theoretical analyses
Table 4 shows the load-deflection relations for Beam D obtained from the experiments
and FEM analysis:
Load (N) Deflection measured in experiments (m) Deflection calculated by FEM (m)
1
0.0001089 0.0001087
2
0.0002177 0.0002175
3
0.0003266 0.0003262
4
0.0004355 0.0004351
30
31
5
0.0005444 0.0005440
6
0.0006532 0.0006526
Table 4 Load-deflection relations for Beam D from the experiments and FEM
analysis
Table 5 shows the load-deflection relations for Plate E from the experiments and FEM
analysis:
Load (N) Deflection measured in experiments (m) Deflection calculated by FEM (m)
1
0.0001089 0.0001088
2
0.0002177 0.0002175
3
0.0003266 0.0003263
4
0.0004355 0.0004351
5
0.0005444 0.000544
6
0.0006532 0.0006526
Table 5 Load-deflection relations for Plate E from the experiments and FEM
analyses
32
CHAPTER 3 THEORY FOR SANDWICH BEAMS
3.1 Literature review
There are a large number of papers and publications on the dynamic properties of
sandwich structures. Already in 1959, Hoff [9] concluded that there was an abundance of
theoretical work in the field. Some of the basic theories are now also summarized in
textbooks. Two of the basic examples are the books by Zenkert [10] and Whitney [11].
The bending of sandwich beams and plates is often described by means of some
simplified models. Often a variational technique is used to derive the basic equations
governing the vibrations of the sandwich structures. Reference is often made to the
Timoshenko [12] and Mindlin [13] models. There are various types of Finite Element
Models. One of the first fundamental works on the bending and buckling of sandwich
plates was published by Hoff [9]. In the paper, Hamilton's principle is used to derive the
differential equations governing the bending and buckling of rectangular sandwich panels
subjected to transverse loads and edgewise compression. Many of basic ideas introduced
by Hoff form the basis for many subsequent papers on bending of sandwich plates.
Another classical paper was published by Kurtze and Waters in 1959 [14]. The
aim of the paper was the development of a simple model for the prediction of sound
transmission through sandwich panels. The thick core is assumed to be isotropic and only
shear effects are included. Using this model, the bending stiffness of the plate is found to
vary between two limits. The high frequency asymptote is determined by the bending
33
stiffness of the laminates. The model introduced by Kurtze and Waters was later
somewhat improved by Dym and Lang [15].
A more general description of the bending of sandwich beams is given by Nilsson
[16]. In this model, the laminates are again described as thin plates. However, the general
wave equation is used to describe the displacement in the core. The influence of
boundary conditions is not discussed. The model is used for the prediction of the sound
transmission loss of sandwich plates. Some boundary conditions and their influence on
the bending stiffness of a structure were later discussed by Sander [17].
Guyader and Lesueur [18] investigated the sound transmission through
multilayered orthotropic plates. The displacement of each layer is described based on a
model suggested by Sun and Whitney [19]. A considerable computational effort is
required. Renji et al derived a simple differential equation governing the apparent
bending of sandwich panels [20]. The model includes shear effects. Compared with
measured results, this model overestimates the shear effects. In particular, in the high
frequency region, the predicted bending stiffness is too low. A modified Mindlin plate
theory was suggested by Liew [21]. The influence of some boundary conditions is also
discussed in a subsequent paper [22]. Again a considerable computational effort is
required. Maheri and Adams [23], used the Timoshenko beam equations to describe
flexural vibrations of sandwich structures. In particular variations of the shear coefficient
is discussed for obtaining satisfactory results.
Common for many of those references is that the governing differential equations
derived are of the 4th order. Due to the frequency dependence of sandwich, the solutions
34
with four unknown will agree very well for low frequency. With increasing frequency,
the result normally disagrees strongly with measured vibrations.
The main work on sandwich structures has been made on conventional foam-core
structures with various face sheets. Little work has been done on the dynamics of
honeycomb panels. In 1997, Saito et al [24] presented an article on how to identify the
dynamic parameters for aluminum honeycomb panels using orthotropic Timoshenko
beam theory. They used a 4th order differential equation and determined the dynamic
parameters comparing their theories to frequency response measurements. Chao and
Chern [25] proposed a 3-D theory for the calculation of the natural frequencies of
laminated rectangular plates. The paper also includes a long reference list, each reference
being classified according to the method used.
Various finite element methods are often proposed for describing the vibration of
sandwich panels. For example, a finite element vibration analysis of composite beams
based on Hamilton's principle was presented by Shi and Lam [26]. A standard FEM code
was used by Cummingham et al. [27] to determine the natural frequencies of curved
sandwich panels. The agreement between predicted and measured natural frequencies is
found to be very good.
There are certainly a large number of methods available which describe
describing the vibration of sandwich panels. However, the aim of this dissertation is the
formulation of simple but sufficiently accurate differential equations governing the
apparent bending of sandwich beams and plates. Boundary conditions should also be
formulated for the calculation of eigen-frequencies and modes of vibration. The models
should allow simple parameter studies for the optimization of the structures with respect
to their acoustical performance. The aim is also to describe a simple measurement
technique for determining some of the material parameters of composite beams.
3.2 Theory of sandwich structure
The bending of a honeycomb panel cannot be described by means of the basic
Kirchoff thin plate theory. The normal deflection of a honeycomb panel is primarily
caused not only by bending but also by shear and rotation in the core. A honeycomb
panel could be compared to a three-layered panel as shown in Figure 14. This figure
shows the deflection of a beam due to pure bending (a) and due to shear in the core which
is the thickest layer (b).
Figure 14 Bending of composite bar or panel by bending (a), shearing of the core
layer (b) and vibration waves in high frequency range (c).
The total lateral displacement, w, of a sandwich beam is a result of the angular
displacement due to bending of the core as defined by ? and the angular displacement
due to shear in the core ? . Figure 14 (a) and (b) shows the bending of composite bar by
35
bending of the whole sandwich structures and shearing of the core layer. Figure 14 (c)
shows the vibration waves of the whole sandwich structures in high frequency range.
The relationship between w, ? and ? is given by
?? +=
?
?
x
w
. (23)
For a honeycomb beam the lateral displacement w can be found when the
differential equations governing the motion of the structure are determined. The
differential equation can be determined using Hamilton's principle. This basic principle is
for example derived and discussed in references [12, 28-33].
According to Hamilton's principle the time integral of the differential between U
the potential energy per unit length, T the corresponding kinetic energy per unit length
and the potential energy induced per unit length by external and conservative forces is
an extremum. In mathematical terms
A
( )
??
=+? 0dxdtATU? (23)
36
In deriving the equations governing the lateral displacement of the structure
shown in Figure 15, symmetry is assumed. Some properties are shown in Figure 16. The
identical laminates have a Young's modulus , bending stiffness , density
l
E
2
D
l
? and
thickness . The core has effective shear stiffness , its Young's modulus , its
equivalent density
l
t
e
G
c
E
c
? and its thickness . For thick core the parameter is not
necessarily equal to the shear stiffness G as suggested by Timoshenko [34]. The core
itself is assumed to have no stiffness or a very low stiffness in the x-direction. In the y-
direction, the core is assumed to be sufficiently stiff to ensure that the laminates move in
phase within the frequency range of interest.
c
t
e
G
Figure 15 Excitation of a beam and resulting forces and moments. Dimensions and
material parameters for the laminates and core are indicated.
areaunit per Mass
2
llcc
tt ??? +=
ccc
EG ?
ll
E ?
Figure 16 Elastic properties and area density of sandwich structure
According to paper of Nilsson [16], the bending stiffness per unit width of the
beam is
37
?
?
?
?
?
?
+++=+=
??
+
3
2
212
22
3
2
23
2/
2/
2
2/
0
2
1
l
lc
lc
l
cc
tt
t
l
t
c
t
tt
tt
E
tE
dyEydyEyD
lc
c
c
. (24)
In general, . The bending stiffness of one laminate is
cl
EE >>
12
3
2
ll
tE
D = . (25)
The mass moment of inertia per unit width is defined as
()
?
?
?
?
?
?
?
?
+++==
?
+
32
23
2/
0
2
3
2
212
2 l
lc
lc
l
cc
tt
ttt
ttt
dyyyI
lc
?
?
?
?
, (26)
while the mass per unit area is
llcc
tt ??? 2+= . (27)
According to Hamilton's principle, as defined in equation (24), the kinetic and
potential energies of the structure must be given as functions of the displacement of the
beam given by w, ? and ? as in equation (23). The total potential energy of a
honeycomb beam is due to pure bending of the entire beam, bending of both laminates
and shear in the core. The potential energy per unit area due to pure bending of the
entire beam is
1
U
.
2
1
2
11
?
?
?
?
?
?
?
?
=
x
DU
?
(28)
The potential energy per unit area due to pure bending of the two laminates is
2
U
.
2
22
?
?
?
?
?
?
?
?
=
x
DU
?
(29)
The potential energy per unit area due to shear deformation of the core is
3
U
38
.
2
1
2
2
?
ce
tGU = (30)
The total potential energy per unit width is thus
() .2
2
1
0
2
2
2
2
1
0
321
???
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
=++=
L
c
L
dxGt
x
D
x
DdxUUUUdx ?
??
(31)
The kinetic energy of the honeycomb panel consists of two parts, the kinetic
energy per unit area due to vertical motion of the beam
1
T
2
1
2
1
?
?
?
?
?
?
?
?
=
t
w
T ? , (32)
and the kinetic energy per unit area due to the rotation of a section of the beam
2
T
2
2
2
1
?
?
?
?
?
?
?
?
=
t
IT
?
?
. (33)
This gives the entire kinetic energy per unit width of the beam as
()
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
=+=
LL
dx
t
I
t
w
dxTTTdx
0
22
0
21
2
1 ?
?
?
. (34)
The total potential energy for the conservative external forces according to Figure
16 is
(35) () () () ()[],00
0
00
1212
L
LL
MFwpwdxMLMwFLwFpwdxaAdx ??? ?+=+??+=?
???
[]
where F is the force per unit width, M is the moment per unit width and p is the
external dynamic pressure on the beam. The moments and forces are defined in Figure
16. By using the definition of ? , equation (22), and by inserting equations (31), (34) and
(35) into the variational expression (23) the result is
39
[].0
2
2
0
22
2
2
2
2
1
=???
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?? ?
??
dtMFwpwdxdt
t
I
t
w
tG
x
D
x
D
dxdt
L
ce
???
?
??
??
?
?
(36)
The integration over time is from to , and over the length x from 0 to L.
Using Equation (23) and expressing
0
t
1
t
? as function of ? and
x
w
?
?
gives
[].0
2
2
0
222
2
2
2
2
2
1
=???
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?? ?
??
dtMFwpwdxdt
t
I
t
w
x
w
tG
xx
w
D
x
D
dxdt
L
ce
???
?
??
??
?
?
(37)
According to standard procedures we obtain
[].0
2
0
2
2
2
2
21
=??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
??
dtMwFwp
tt
I
t
w
t
w
x
w
x
w
Gt
xx
w
xx
w
D
xx
Ddxdt
L
c
????
????
?
??
?
?
???????
?
(38)
Integrating by parts gives
.0
222
22
0
0
2
2
2
2
2
2
0
2
2
3
3
2
0
2
2
2
3
3
4
4
2
0
2
2
3
3
2
0
2
2
2
2
2
1
0
1
1
0
1
0
=+?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
??
????????
?????
?????
?????
dtMwF
wdxdtpdxdt
t
Idx
t
Idxdt
t
w
wdx
t
w
dxdt
x
w
tGdxdt
xx
w
wtGdt
x
w
wtG
dxdt
xx
w
Ddt
xx
w
Ddxdt
xx
w
wD
dt
xx
w
wDdt
xx
w
x
w
D
x
dxdtDdt
x
D
L
L
t
t
t
t
cece
L
ce
L
LL
L
???
?
?
??
?
???????
???
?
???
?
??
?
??
?
?
?
?
??
??
?
??
?
??
40
(39)
When assuming that the displacement is defined so that 0?y? for
, and for
10
tandtt =
()
0?
?
?
?
?
?
?
?
?
=
?
?
x
y
x
y
?
?
, Equation (39) is reduced to
.02
2
2
2
2
0
2
2
2
0
2
2
21
0
2
2
3
3
2
2
2
2
2
3
3
2
2
2
1
2
2
3
3
4
4
2
2
2
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
dt
xx
w
x
w
D
dtM
xx
w
D
x
D
dtF
xx
w
D
x
w
tGw
dxdt
x
w
tG
t
I
xx
w
D
x
D
dxdtp
t
w
xx
w
D
xx
w
tGw
L
L
L
ce
ce
ce
??
??
??
?
??
?
???
??
?
??
?
?
(40)
Since the expression should be equal to zero for ?? and 0?w? , we obtain five
sets of equations which must be satisfied. The first two brackets in Equation (40) give
two differential equations governing the displacement of the beam as expressed by w
and ? . These are
02
2
2
3
3
4
4
2
2
2
=?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
? p
t
w
xx
w
D
xx
w
tG
ce
?
??
, (41)
and 02
2
2
2
2
3
3
2
2
2
1
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
? ?
???
?
x
w
tG
t
I
xx
w
D
x
D
ce
. (42)
Using these two equations, eliminating ? by using the simplification
and , the equation governing w is given by:
(xkti
x
Aew
?
=
? ) )(xkti
x
Be
?
=
?
?
41
()
() .2
222
2
2
2
2
21
4
4
2
2
4
4
1
22
4
21
24
6
2
6
6
12
t
w
IptG
x
p
DD
t
w
I
t
w
x
w
DtG
tx
w
tGIDD
tx
w
ID
x
w
DD
ce
cece
?
?
??
?
?
+=
?
?
+
?
?
?
?
?
?
?
?
?
?
+
?
?
+
??
?
++?
??
?
+
?
?
?
??
??
?
???
(43)
Eliminating instead gives the corresponding equation for w ? as
()
.2
222
3
3
2
4
4
2
2
4
4
1
22
4
21
24
6
2
6
6
12
x
p
tG
x
p
D
t
I
tx
DtG
tx
tGIDD
tx
ID
x
DD
ce
cece
?
?
+
?
?
?=
?
?
+
?
?
?
?
?
?
?
?
?
?
+
?
?
+
??
?
++?
??
?
+
?
?
?
?
?
?
?
??
??
??
?
??
(45)
The shear angle ? can be shown to satisfy the same differential equation as ? in
Equation (45). They are the same results that obtained by Nilsson in [16].
Each one of the last three integrals in Equation (40) must also be equal to zero.
For these conditions to be satisfied, it follows that at the boundaries of the beam the
following conditions must be satisfied
,0,0
,0,2
,0,2
2
2
2
2
21
2
2
3
3
2
=
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
=?=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
==
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
x
w
or
xx
w
orM
xx
w
D
x
D
worF
xx
w
D
x
w
tG
ce
?
?
??
?
?
(46)
for x=0 and x=L. These equations satisfy the boundary conditions for a beam.
For free vibration, there are no external forces and moments. Then the boundary
conditions for two ends can be expressed from Equation (46) as
42
00
,02
,02
2
2
2
2
21
2
2
3
3
2
=
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
x
w
or
xx
w
or
xx
w
D
x
D
wor
xx
w
D
x
w
tG
ce
?
?
??
?
?
(47)
Nilsson obtained the same boundary condition equations in [16].
Using the wave Equations (43) and (44), together with the six boundary
conditions, the displacements and w ? can be determined. For free vibrations the
external pressure p is equal to zero and for honeycomb panels the moment of inertia can
be assumed to be very small. Using these assumptions, the resulting equations for w
Equation (43) are reduced to
,0
2
2
2
2
1
4
4
22
4
1
21
6
6
2
=
?
?
?
?
?
?
?
?
?
?
+
?
?
?
??
?
?
?
?
?
?
?
?
? +
+
?
?
t
w
Dx
w
tG
tx
w
D
DD
x
w
D
ce
?
? (48)
which is the wave equation for the bending of beams neglecting the moment of
inertia. The corresponding equation for ? , Equation (43) is reduced to
.0
2
2
2
2
1
4
4
22
4
1
21
6
6
2
=
?
?
?
?
?
?
?
?
?
?
+
?
?
?
??
?
?
?
?
?
?
?
?
? +
+
?
?
tDx
tG
txD
DD
x
D
ce
????
?
?
(49)
As seen from the equations the expressions ? and satisfy the same differential
equation for zero external exciting force.
w
3.3 Boundary conditions
The beam must satisfy certain boundary conditions at each end. Boundary
conditions such as simply supported, clamped and free can be defined by means of the
Equations (43) through (47). For each boundary condition, certain requirements for the
43
displacement and the angular displacement w
x
w
?
?
as well as forces and bending
moments must be considered.
Figure 17 Particular boundary conditions
The three boundary conditions for a beam shown in Figure 17 are summarized in Table 6.
Clamped ends
Simply
supported ends
0=
?
?
x
w
0=?
0=w
0
2
2
=
?
?
x
w
0=
?
?
x
?
0=w
44
Free ends
2
2
2
2
1
t
I
x
D
?
?
=
?
? ??
?
0
2
2
=
?
?
x
w
0=
?
?
x
?
Table 6 Boundary conditions for ends of beam
The natural frequencies for a clamped and a free beam are identical, according to
the Euler theory.
However, when shear is considered, as in Table 6, there is a difference between
the natural frequencies for these two conditions. The natural frequencies for the clamped
boundaries are the lowest. This is due to the fact that shear in the beam is induced by the
clamped boundaries to a clamped boundaries to a larger extend as compared to the case
with free boundaries. The apparent bending stiffness for the clamped beam is therefore
somewhat lower than for the beam with free ends.
3.4 Wave numbers
In Sections 3.2 and 3.3, the wave equation and the boundary conditions for
honeycomb beams were determined. The equation governing the free flexural vibrations
of a honeycomb beam is in Section 3.2 given by Equation (43) for ( ) 0=p
()
.0
222
4
4
2
2
4
4
1
22
4
21
24
6
2
6
6
12
=
?
?
+
?
?
?
?
?
?
?
?
?
?
+
?
?
+
??
?
++?
??
?
+
?
?
?
t
w
I
t
w
x
w
DtG
tx
w
tGIDD
tx
w
ID
x
w
DD
cece
?
???
?
??
(50)
Neglecting losses in the structure, the bending and shear stiffnesses , and
are real quantities. By assuming a solution
1
D
2
D
e
G
()
.~
xkti
x
Aew
??
45
the wave number must satisfy
x
k
.0
22
2
4
1
2
1
422
11
22
4
1
2
6
2
=+
?
?
?
?
?
?
?
?
?+
?
?
?
?
?
?
?
?
++??
?
?
?
?
????
?
?
?
D
I
D
ktGk
D
tGI
D
D
kI
D
D
kD
xcex
ce
xx
(51)
This equation has the six solutions
321
,, ??? iik
x
???= where the ? variables
are real quantities if the losses are neglected. If the stiffness is defined as ( )?iDD += 1
0
and ()?iGG += 1
0
, losses can be included. The absolute values of the wave numbers are
shown in Figure 18.
46
Figure 18 Wave numbers for beam [16].
Solid line ? real root of the wave number
and Dashed and dotted lines ? purely imaginary roots
Dash-dotted line ? real root
Upper asymptote ? corresponds to the bending of one laminate
Lower asymptote ? corresponds to the bending of the entire beam
The material and geometrical parameters describing the beam are given in Table
7. The lower of the two parallel lines in Figure 19 represents the wave number
corresponding to pure bending of the entire beam. The upper line represents the wave
47
number for pure bending of one of the identical laminates. The parallel lines define the
low- and high-frequency limits for the wave number
1
? for the first propagating mode. In
the mid-frequency region, shear and rotation become important. As these effects increase,
the wave number deviates from the lower asymptote and shifts towards the upper one.
For high frequencies, the wave numbers of the structure adjust to the asymptote for the
wave number for the bending waves propagating in one of the identical laminates.
L
(m)
b
(m)
l
t
(mm)
c
t
(mm)
l
?
(kg/m
3
)
c
?
(kg/m
3
)
?
(kg/m
2
)
(MPa)
e
G (GPa)
l
E
1.2 0.1 0.5 10 1264 137.6 2.64 80 55
Table 7 Geometrical and material parameters for sandwich beam
The wave number for a propagating wave is defined as
x
k
x
x
D
k
2
4
??
= , (52)
where is the apparent bending stiffness of the beam. The apparent bending
stiffness is defined as the bending stiffness of an equivalent orthotropic beam of mass
x
D
x
D
? and with wave number , for the first mode of propagating bending waves.
Considering this definition, the agreement with the asymptotes confirms that the bending
stiffness for low frequencies is determined as the bending stiffness of the entire beam
given in Equation (24). For high frequencies the bending stiffness is determined by the
bending of the laminates only, as given by Equation (25). The bending stiffness given in
Equations (24) and (25) will give the limiting values for the calculated bending stiffness.
x
k
48
The transition of the bending stiffness between the two asymptotes is determined by
shear and rotation in the core.
The dotted and the dashed lines in Figure 18 represent the purely imaginary roots
given by
2
? and
3
? and correspond to the near field solutions or the evanescent waves
for the in-phase motion of the laminates. The constant value for low frequencies, the
dotted line, is determined by the thickness of the core. For increasing frequencies, this
curve approaches the limit determined by the wave number for evanescent waves in one
of the identical laminates. Decreasing the thickness of the core will increase the constant
value in the lowest frequency range. The other near field solution closely follows the
asymptote for the bending of the entire beam for low frequencies.
In summary the limiting values for the wave numbers are
()
,
2
lim;lim
4/1
21
21
2
1
4/1
1
2
0
1 ?
?
?
?
?
? +
=
?
?
?
?
?
?
=
???
DD
DD
D
ff
??
?
??
? (53)
,lim;lim
2/1
1
2
2
4/1
1
2
0
2
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
=
???
D
I
D
ff
?
?
??
?
?
(54)
,
2
lim;
2
lim
4/1
2
2
3
2/1
2
0
3 ?
?
?
?
?
?
=
?
?
?
?
?
?
=
???
DD
GH
ff
??
?? (55)
or
2
? we see a minimum at a certain frequency , where
p
f
2
? shifts from being
entirely complex to real and describes a propagating wave.
As the frequency increases,
2
? approaches zero for
p
ff = where
?
? I
tG
f
ce
p
2
1
= . (56)
49
Below this frequency ,
p
f
2
? is imaginary, and describes evanescent waves. For
higher frequencies
2
? is real and describe propagating waves.
3.5 Least Squares Method
In the previous section the wave equation governing the displacement of a
honeycomb beam was derived. Based on this differential equation, wave numbers, natural
frequencies and modes of vibration can be determined for different boundary conditions.
For the response of a beam to be calculated, all the material parameters of the beam must
be known. The dynamic properties of a composite beam are not always well defined. This
is due to the fact that the elements of the assembled structure perform differently.
However, the main dynamic properties of a composite beam can be determined from
measurements of the first few natural frequencies when the structure is freely suspended.
The wave number for the first propagating bending mode is
x
k
x
x
D
k
2
4
??
= , (57)
where is the apparent bending stiffness of the structure. Consequently is
defined as the bending stiffness of a simple homogeneous beam, which at a certain
frequency has the same dynamic properties as the honeycomb structure. By inserting the
definition for in the wave equation (51) the result becomes
x
D
x
D
x
k
02
2
2/1
1
2/3
2/1
=?+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
DDD
D
DtG
xx
xce
??
, (58)
50
if the moment of inertia is neglected. In the low frequency range, or as 0?? ,
the first part of the equation dominates why . The bending stiffness is
consequently determined by pure bending of the beam.
1
DD
x
?
In the high frequency range, when ??? . For high frequencies, the
laminates are assumed to move in phase. In this frequency range, the bending stiffness for
the entire beam is equal to the sum of the bending stiffnesses of the laminates. This result
agrees with the results discussed in the previous sections.
2
2DD
x
?
For a composite beam, the wave number for flexural waves is defined by
Equation (57). For a beam with boundary conditions well defined, the bending stiffness
can be determined by means of simple measurements. The natural frequencies- for a
beam of length-L are given by the expression
x
k
n
f
;,
4
4/1
22
n
xn
n
n
L
D
f
L ?
??
? =
?
?
?
?
?
?
= n=1, 2, 3? ,(59)where
n
? for a
beam with various boundaries is given in Table 8.
Boundary conditions n 1 2 3 4 5 n>5
Free-free
Clamped-clamped
n
?
4.73 7.85 11.00 14.14 17.28 2/?? +n
Free-clamped
n
?
1.88 4.69 7.85 11.00 14.14 2/?? ?n
Simply supported
n
?
3.14 6.28 9.42 12.56 15.7 ?n
Table 8 Values of for particular boundary conditions
n
?
51
Measurements reveal the fundamental natural frequency of the beam. By
arranging Equation (59), the apparent bending stiffness for mode n having the
natural frequency is for a beam of length ? and mass per unit area
xn
D
n
f L ? , given by
...3,2,1
4
4
422
== nfor
Lf
D
n
n
xn
?
??
. (60)
The bending stiffness of a composite beam is strongly frequency dependent as
given by equation (60). Since most of the parameters are unknown, only the frequency
dependent parameters are preserved and the equation is rewritten as
,0
2/12/3
=?+? CDD
f
B
D
f
A
xxx
(61)
where
.2;
2
;
2
2
2/1
1
2/1
DC
tG
B
D
tG
A
cece
===
????
(62)
For non-metallic materials, Young's modulus can exhibit frequency dependence
as discussed in, for example, Reference [12] and demonstrated in References [32] and
[33]. However, within the frequency range of interest, here up to 4 kHz, the
parameters , and in Equation (62) are assumed to be constant for the structures
investigated. Using the measured data the constants A, B and C can be determined by
means of the least squares method. The quantity Q is defined by
1
D
2
D G
,
2
2/12/3
?
?
?
?
?
?
?
?
?
?+?=
i
xixi
i
xi
i
CDD
f
B
D
f
A
Q (63)
where is the measured bending stiffness at the specific frequency . The
constants A, B and C are chosen to give the minimum of Q. The shear modulus and
xi
D
i
f
c
G
52
the bending stiffness and can be determined, once the parameters A, B and C are
predicted. With known constants A, B and C the dynamic parameters can be determined
1
D
2
D
./2
,2/
2/1
2
ce
tBG
CD
??=
=
(64)
The parameters are predicted from the MATLAB program.
3.6 Damping measurement methods
Basically there are four measures of damping, the loss factor ?, the quality factor
Q, the damping ratio ? , and the imaginary part of the complex modulus. However, they
are related to each other. The loss factor or damping ratio is used in measurements:
??
?
? tan
2
2
1
2
=
?
??
=====
E
E
C
C
QW
D
c
. (65)
Here D and W are the dissipated and total powers in one cycle of vibration, C and
C
c
are the damping coefficient and the critical damping, 'E and "E are the real and
imaginary parts of the complex modulus.
Many references present reviews of damping measurements [35-39]. Generally,
there are three sorts of experimental methods.
Decay rate method
This method can be used to measure the damping of a single resonance mode or
the average of a group of modes in a frequency band. The structure is given an excitation
by a force in a given frequency band, the excitation is cut off, the output of the transducer
is passed through a band pass filter and then the envelope of the decay is observed. The
damping ratio can be calculated from the slope of the envelope of the log magnitude-time
53
plot, as shown in Figure 19. One of the disadvantages of this method is that the effect of
54
noise is considerable.
Figure 19 Decay rate method used to determine damping
mn
n
A
= ln
1
? , (66)
Am
+
()
22
2 ??
?
?
+
= , (67)
where ? is the decay rate.
se function (log magnitude-time plot or Nyquist
diagram
Modal bandwidth method
With the frequency respon
), the modal bandwidth method, also called as half-power point method is the
most common form used to determine the damping (shown in Figure 20). This method
applies only to the determination of the damping of a single mode. The shortcoming of
this method is that the repeatability is, in general, rather poor. The loss factor or the
damping can therefore only be estimated, based on averages of several measurements.
For sandwich structures, the loss factor is frequency dependent. In this work, the same
experimental set up was used to determine the dynamic stiffness and to estimate the
damping for the sandwich structures.
Figure 20 Modal bandwidth method to determine damping
,
2
12
ff ?
=? . (68)
n
f
where and are the closest frequencies at which the power is dropped 3 dB from
od
d on the relationship between the input power and the
dissipa
1
f
2
f
that at the mode frequency.
Power balance meth
The SEA method is base
ted power. So the loss factor can be determined by measuring the input power and
the total energy of a modal subsystem (shown in Figure 21). Assuming a stationary
55
power input at a fixed location, the input power must be equal to the dissipated power
under steady state conditions. The disadvantage of this method is that it is relatively hard
to determine the input power and the total energy of a modal subsystem.
Figure 21 Power balance method to determine damping
tot
in
W
=? ,
W2
(69)
where is the input power and is the total energy of a modal subsystem.
in
W
tot
W
56
CHAPTER 4 DYNAMIC PROPERTIES CHARACTERIZATION
4.1 Experiments
4.1.1 Steps
The steps used to determine the dynamic stiffness and static properties of
entire beams and face plates are as follows:
57
Figure 22 Experimental steps to determine some properties of sandwich
structures
These steps provide an interesting approach to determine some parameters of
sandwich structures.
4.1.2 Set up
Since honeycomb plates are typically anisotropic, measurements are
performed on beams representing the two main in-plane directions of the plate. For
Calculate dynamic bending stiffness, , for mode n
n
D
Determine parameters A, B and C
Draw dynamic stiffness curve
x
D Calculate dynamic propertiesG
21
,, DD
e
Measure natural frequencies
n
f
Calculate damping for natural
frequencies
n
f
materials tested in two directions, the structure and the results are given a subscript of
1 or 2 to indicate the orientations of the beam. The tests have been made for different
boundary conditions.
Due to the low mass of the material ~ kg/m5.2
2
an accelerometer would have
a certain influence on the vibrations of the beam. The vibration measurements were
therefore made with a laser vibrometer to avoid transducer contact. The frequency
response function, FRF, was determined to give the natural frequencies for the beam.
Based on the frequency response function, the loss factor or damping was also
determined by the Modal Bandwidth Method.
The experiment set up 1 is shown in Figure 23. The set up includes: a B&K
Pulse System (Sound & Vibration Multi-analyzer), laser vibrometer, sample, shaker
and power amplifier.
The sample shown in the figure has simply supported ends. The other
boundaries such as free-free and clamped-clamped were tested as well. The B&K
Pulse system gives the output of the frequency response function of the beam when it
is excited by white noise. The advantage of set up 1 is that high frequencies can be
excited. But one of the disadvantages is that mass is added to the lightweight structure
by the probe or the needle of the shaker. Since the sample is very light weight, a little
weight added can influence the result to some degree. The other disadvantage is that
some resonances may be missed since the probe of the shaker effects the
configuration of the beam in some cases.
Set up 2 is shown in Figure 24, which includes a B&K pulse system, laser
vibrometer, modal hammer and sample. The modal hammer gives an impulse to the
58
sample. The dynamic signal analyzer provides the output of the frequency response
function of the beam when it is excited by the impulse given by modal hammer. The
advantage of the set up 2 is that no mass is added to the lightweight structure so there
is no mass loading problem. The disadvantage is that it is hard to high frequency
vibration with the modal hammer.
In measurements made, these two set ups were combined to obtain the natural
frequencies in the frequency band of interest.
Sound & Vibration Multi-
analyzer Pulse system
V(t)
Laser vibrometer
Sample
Force transducer F(t)
Shaker
White noise
Power amplifier
Figure 23 Set Up 1 Using shaker
59
Sound & Vibration
Multi-analyzer Pulse
system
F(t)
V(t)
Laser Vibrometer
Modal Hammer
Sample
(b)
Figure 24 Set Up 2 Using hammer
4.1.3 Samples
The composite sandwich structures with honeycomb cores and sandwich
structures with closed-cell foam cores, used in the experiments, are shown in Figures
25 and 26. The face sheets are made with woven cloth impregnated with a resin, and
the core is a lightweight foam-filled honeycomb structure and closed-cell foam.
60
61
The geometry and density of the sample beams used in the experiments are
shown in the Tables 9 and 10.
Beam No. Beam A Beam B Beam C Beam D
Content Honeycomb
core filled
with foam
Honeycomb
core filled
with foam
Entire
sandwich
beam with
foam filled
honeycomb
core
Entire
sandwich
beam with
foam filled
honeycomb
core
Length (m) 0.6 0.6 0.6 0.6
Thickness(t) (m) 0.00635 0.00635 0.00705 0.00705
Width (m) 0.0254 0.0254 0.0254 0.0254
Density (kg/m
3
) 162.73 162.73 362.331 362.331
Density (kg/m
2
) 1.03334 1.03334 2.554 2.554
Direction 1 2 1 2
Face sheet N/A N/A Single sheet Single sheet
Table 9 Geometry and density of the sandwich beam with foam filled honeycomb
core.
Beam No. Beam E Beam F Beam G Beam H
Content
Entire
sandwich
Entire
sandwich
Entire
sandwich
Entire
sandwich
beam with
foam core
beam with
foam core
beam with
foam core
beam with
foam core
Length (m) 0.457 0.612 0.854 0.608
Thickness(t) (m) 0.0256 0.0256 0.0256 0.0256
Width (m) 0.029 0.029 0.028 0.030
Density (kg/m
3
) 343.148 292.3406 334.5601 268.3411
Density (kg/m
2
) 8.9218 7.6009 8.6986 6.9769
Table 10 Geometry and density of the sandwich beam with foam core.
Figure 25 Honeycomb sandwich composite structures; (a) foam-filled
honeycomb core, (b) composite beam
62
Figure 26 Closed-cell foam core
4.2 Analysis of experimental results
4.2.1 Frequency response functions
Some typical measured frequency response functions for beams C, E, F and G
with free edges are shown in Figures 27, 28, 29 and 30. For low frequencies, the
peaks are easily resolved. For increasing frequencies, the peaks are not easily
separated. This is due to the hammer excitation. It is difficult to excite the higher
modes using an impact hammer. The peaks representing the natural frequencies for
the beams are marked in the figures. The unmarked peaks result from the mounting of
the beams and twisting modes.
FRF for Beam F
-150
-140
-130
-120
-110
-100
-90
-80
-70
0 500 1000 1500 2000
Frequency [Hz]
Relat
i
ve
L
e
ve
l [d
B]
Figure 27 FRF for the sandwich beam F for free-free boundary condition.
63
FRF for Beam G
-7.00E+01
-6.00E+01
-5.00E+01
-4.00E+01
-3.00E+01
-2.00E+01
-1.00E+01
0.00E+00
1.00E+01
2.00E+01
0.00E+00 5.00E+02 1.00E+03 1.50E+03 2.00E+03 2.50E+03 3.00E+03
Frequency [Hz]
Rel
a
t
i
ve
L
e
ve
l [d
B]
Figure 28 FRF for the sandwich beam G for free-free boundary condition.
64
FRF for Beam E
-7.00E+01
-6.00E+01
-5.00E+01
-4.00E+01
-3.00E+01
-2.00E+01
-1.00E+01
0.00E+00
1.00E+01
0.00E+00 5.00E+02 1.00E+03 1.50E+03 2.00E+03 2.50E+03 3.00E+03
Frequency [Hz]
Relat
i
ve L
e
v
e
l [d
B]
Figure 29 FRF for the sandwich beam E for free-free boundary condition.
The natural frequency values vary slightly for different measurements,
although the repeatability is rather good. Four measurements on the same beam are
compared in Table 11. The set-ups were disassembled between measurements.
Measurement 1 Measurement 2 Measurement 3 Measurement 4
r
f
[Hz] D [Nm]
r
f
[Hz] D [Nm]
r
f
[Hz] D [Nm]
r
f
[Hz] D [Nm]
294 6771.4 296 6863.8 295 6817.5 296 6863.8
705 5046.2 707 5074.9 706 5060.5 706 5060.5
1194 3767.8 1197 3786.7 1195 3774.1 1196 3780.4
1683 2739.5 1686 2749.2 1684 2742.7 1685 2746.0
Table 11 Natural frequencies for different measurements on Beam F and their
corresponding bending stiffness.
65
The relative error, , in the bending stiffness is equal to where
is the uncertainty in the recorded eigenfrequency of the beam. Consequently the
error is fairly large in the low frequency region. For high frequencies, the result is
less sensitive to this type of variation. The table reveals some variations in the
bending stiffness in the low frequency range. For increasing frequencies the
differences are negligible.
DD /? ff /2?
f?
D?
4.2.2 Boundary conditions
In the measurements, the boundary conditions were found to have some
effect. For a clamped beam, shear is induced at the boundaries, thus rendering the
beam more flexible as compared to a beam with free ends. The natural frequencies of
a clamped beam are consequently lower than the corresponding natural frequencies
for the same beam with free ends. Hence, the frequency-dependent stiffness is
affected by the boundaries. The natural frequencies for the three special boundary
conditions are shown in the Table 12. This is due to the fact that shear in the beam is
induced by the clamped boundaries to a larger extend as compared to the case with
free boundaries. The apparent bending stiffness for the clamped beam is therefore
somewhat lower than that for the beam with free ends as shown in Figure 30.
Boundary
conditions
Free-free Simply supported-
simply supported
Clamped-clamped
Mode Frequencies (Hz)
1 119 53.7 118
2 317.5 206.5 308.5
66
67
3 583 432.5 562
4 883 711 864
5 1206.5 1024 1177.5
6 1539 1335 1498
7 1890 1676 1835
8 2246 2011 2175
9 2806.1 2542.2 2789
10 3247.3 2963.8 3228
Table 12 Natural frequencies of Beam C for the three different beam boundary
conditions.
10
0
10
1
10
2
10
3
10
4
0
50
100
150
200
250
300
350
400
450
bending stiffness for beam
frequency [Hz]
bend
i
n
g
s
t
i
f
f
nes
s
[
N
m
]
Figure 30 Bending stiffness of the sandwich beam C for three different boundary
conditions.
Measurement points *, and : bending stiffness for the natural frequencies of
the beam with free, simply supported and clamped ends, respectively.
+
o
Solid, dashed and dotted curves: calculated static bending stiffness of the beam
with free, simply supported and clamped ends, respectively.
68
4.2.2 Dynamic bending stiffness
The dynamic bending stiffness for the sandwich Beam C is shown in Figure
31. The bending stiffness is frequency dependent and decreases with increasing
frequency. The bending stiffness at low frequency is dominated by the bending
stiffness of the entire beam. At high frequency, the bending stiffness is dominated by
the bending of the face plates only.
10
0
10
1
10
2
10
3
10
4
0
50
100
150
200
250
300
350
400
bending stiffness for beam
frequency [Hz]
bend
i
n
g
s
t
i
f
f
nes
s
[
N
m
]
69
Figure 31 Bending stiffness for the sandwich beam C.
Measurement points *: bending stiffness for the natural frequencies.
Curves -: calculated static bending stiffness, based on material parameters for
face plates and core.
The static stiffness of the entire beam and the static stiffness of the
faceplate are determined by use of the four-point bending method. In the low
frequency range, when
1
D
2
D
0?? , the apparent dynamic stiffness is close to . The
bending stiffness is consequently determined by pure bending of the beam. In the
high frequency range, when
1
D
??? , and then the apparent dynamic stiffness is close
to . The face plates are assumed to move in phase. In this frequency range, the
bending stiffness for the entire beam is equal to the sum of the bending stiffness of
the two face plates. The comparison is as shown in Table 13:
2
2D
Static stiffness by four-point method
Two stiffness limits from dynamic
characterization
1
D [Nm]
2
D [Nm] 0?? [Nm]
???
[Nm]
344 24 345.2 48.6
Table 13 Comparison of static stiffness measured by four-point bending method
and two stiffness limits from dynamic characterization.
The dynamic stiffness plots in the two principal directions are shown in Figure
32:
70
10
0
10
1
10
2
10
3
10
4
0
50
100
150
200
250
300
350
400
450
bending stiffness for beam
frequency [Hz]
be
nd
i
n
g s
t
i
f
f
n
es
s
[
N
m
]
Figure 32 Dynamic stiffness for beams C and D
From Figure 32, we can see that the stiffness in the two principal directions is
in very good agreement. The reason is that the stiffness of entire sandwich structures
is dominated by the face sheet while the stiffness of the face sheets for the sandwich
structures, usually, is much higher than that of the core.
71
The dynamic bending stiffness for the core in two directions is shown in
Figure 33.
10
0
10
1
10
2
10
3
10
4
1
2
3
4
5
6
7
bending stiffness for beam
frequency [Hz]
bendi
n
g
s
t
i
f
f
nes
s
[
N
m
]
Figure 33 Dynamic stiffness for core in two principal directions
72
Since the core is orthotropic, the stiffness is different in the two principal
directions. But the effect of orthotropy on the stiffness of the whole sandwich
structures is very small.
10
0
10
1
10
2
10
3
10
4
1000
2000
3000
4000
5000
6000
7000
8000
bending stiffness for beam
frequency [Hz]
ben
di
ng
s
t
i
f
f
n
es
s
[
N
m
]
Figure 34 Bending stiffness for the sandwich beam G.
The dynamic bending stiffness for Beam G is shown in Figure 34. Since Beam
G is the longest beam among the beam samples E, F, G and H, eight natural
frequencies could be measured using the hammer method but only four or five could
be measured for the others. So this implies that the longer is the beam, the more
73
natural frequencies and the lower is the first natural frequency and the more accurate
stiffness curve that can be obtained.
c
G by twisting method
c
G from dynamic characterization
5.189?10
7
Pa 5.226?10
7
Pa
Table 14 Comparison of shear modulus of the core measured by twisting method
and that from the dynamic stiffness curve for Beam G.
From Table 14, it can be seen that the shear modulus of the core measured by
twisting method and that from dynamic characterization are in good agreement.
74
10
0
10
1
10
2
10
3
10
4
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
bending stiffness for beam
frequency [Hz]
be
ndi
ng s
t
i
f
f
nes
s
[
N
m
]
Figure 35 Bending stiffness for the sandwich beam E, F and H.
Red line: Beam E; Green line: Beam F; Blue line: Beam H
The dynamic bending stiffness curves for beams E, F, and G are shown in
Figure 35. Note that the order of the area density for beams E, F and G is from the
smallest to the largest is E>F>G. From this plot, such conclusions can be drawn that
the stiffness increases if the area density increases.
75
76
4.2.3 Damping
The damping ratio of beam C is given in this table:
Natural frequency (Hz) Damping ratio (%)
53.7 0.315
206.5 0.367
432.5 0.471
711 0.492
1024 0.479
1335 0.466
1676 0.5
2011 0.557
2542.2 0.689
2963.8 0.81
Table 15 Damping ratio of Beam C
The damping for beam C is shown in this figure.
damping of sandwich Beam C with 1/4 in thickness and single face sheet
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 500 1000 1500 2000 2500 3000 3500
Frequency (Hz)
D
a
m
p
i
ng ra
t
i
o
(
%
)
Figure 36 Damping ratio of sandwich beam C
This figure proves that the damping for the sandwich structures under
investigation is very high compared to the traditional materials such as aluminum.
4.3 Conclusions
Several theoretical models for the determination of the dynamic bending
stiffness of sandwich beams were reviewed. Nilsson?s sixth-order differential
equation model was discussed. A simple measurement technique for determining the
material parameters of composite beams was used. The experimental results show
that this technique can be used to determine the dynamic stiffness of composite
sandwich beams.
77
CHAPTER 5 SOUND TRANSMISSION LOSS OF SANDWICH PANELS
The sound transmission loss (TL), or sound reduction index, is a measure of the
sound insulation provided by a wall or other structural element. Sound transmission loss
(TL) of a panel is given by
TL
?
?
?
?
?
?
?
?
=
T
I
W
W
lg10 , (70)
where is sound power incident on the panel and is sound power
transmitted through the panel. Since the sound transmission loss is dependent on the
frequency of the sound, it is usually reported for each octave band or each one-third
octave band. This number indicates the noise insulation capability of the panel. The
sound transmission loss is often an important consideration in the analysis and design of
partitions or panels separating adjoining spaces in industry, housing and various types of
vehicles.
I
W
T
W
5.1 Classical sound transmission analysis
The classical sound transmission analysis theories are described by Wilson in the
book [80].
The sound transmission coefficient is the fraction of the sound power transmitted
through a wall or barrier. Thus:
I
T
W
W
=? , (71)
78
where ? is the sound transmission coefficient. Comparing the definitions of
transmission loss and sound transmission coefficient, we have
?
?
?
?
?
?
=
?
1
lg10TL , (72)
or
. (73)
10/TL
10
?
=?
5.1.1 Mass law sound transmission theory
If the panel is thin, that is, the panel thickness is small compared with a
wavelength of sound in air, and if the panel stiffness and damping may be neglected, then
the sound transmission is mass-controlled.
The mass law transmission loss equation based on theoretical considerations is:
,
2
cos
1lg10
1
lg10TL
2
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+=
?
?
?
?
?
?
=
c
m
?
??
?
(74)
where )Hz(2)rad/s( f?? = =sound frequency
m = panel mass density per unit surface area ,
2
kg/m
? = angle of incidence,
? = air density,
c = speed of sound in air.
If the angle of incidence is equal to zero, the above equation becomes:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+=
2
0
2
1lg10TL
c
m
?
?
, (75)
79
the normal incidence mass law TL for approximating the transmission loss of
panels with incident sound in the mass-controlled frequency.
o
0
When studying transmission loss between two rooms, the sound source may
produce a reverberant field on the source side. Thus, the incident sound may strike the
wall at any angle between 0 and . For field incidence, it is assumed that the angle of
incidence lies between 0 and . This results in a field incidence transmission loss of
about
o
90
o
72
dB5TLTL
0
?= . (76)
The following additional changes are made in the mass law equation. Converting
the frequency of the sound from radians per second to hertz, where
),Hz(2)ad/s( fr ?? = (77)
using typical values for the speed of sound in air and air density
( )elyapproximatc skg/m400
2
?=? , (78)
and assuming that
1
2
>>
c
m
?
?
, (79)
We obtain the field incidence mass law equation
( ) dB.47lg20TL ?= fm (80)
Based on the above equation for the mass-controlled frequency region, the
transmission loss of a panel increases by 6 dB per octave. In addition, a doubling of panel
thickness or doubling of panel mass per unit area should produce a 6 dB increase in
transmission loss at a given frequency. While the above equations are useful for
prediction of material behavior, the transmission loss of actual structural elements should
be confirmed by laboratory or field testing whenever possible.
80
5.1.2 The effect of panel stiffness and damping
Low-frequency sound transmission is governed largely by panel bending stiffness.
Transmission of somewhat higher frequency sound is governed by panel resonances. The
sound transmission loss of the panel may be considered mass controlled when responding
to frequencies above twice the lowest resonance frequency, but below the critical
frequency.
The sound transmission coefficient equation for panels with significant bending
stiffness and damping is given by Ver and Holmer (1971) as follows:
,
sin
1
2
cossin
2
cos
1
1
2
4
42
2
4
42
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+=
mc
B
c
m
mc
B
c
m ??
?
????
?
??
?? (81)
where it is assumed that the thickness of the panel is small compared with the
wavelength of the incident sound, and
B = panel bending stiffness ( mN? ),
= panel surface density ( ), m
2
kg/m
? = composite loss factor (dimensionless).
5.1.3 The coincidence effect
Sound in panels and other structural elements can be converted into vibration and
propagated as bending waves, longitudinal waves, and transverse waves. Bending waves
are of particular importance because of the coincidence effect. Figure 37 shows a panel
with an airborne sound wave of wavelength ? and incident at angle? . Assume bending
wave of wavelength
B
? is excited in the panel. The propagation velocity of bending
81
waves is frequency dependent, with higher frequencies of vibration corresponding to
higher propagation velocities. The coincidence effect occurs when
,arcsin
*
?
?
?
?
?
?
?
?
==
B
?
?
?? (82)
where the asterisk (*) is used to indicate coincidence and is called the
coincidence angle. When wave coincidence occurs, the sound pressure on the surface of
the panel is in phase with the bending displacement. The result is high-efficiency energy
transfer from airborne sound waves in the source space to bending waves in the panel,
and thence to airborne sound waves in the receiving space. Efficient sound energy
transfer from air space to air space is obviously undesirable from a noise control
standpoint. Figure 38 is an idealized plot of transmission loss versus frequency, showing
the stiffness-controlled, resonance-controlled, mass-controlled, and coincidence-
controlled regions for a panel. Note that the transmission loss curve dips substantially at
frequencies beyond the critical frequency due to the coincidence effect.
*
?
82
Figure 37 The coincidence effect
83
Figure 38 Idealized plot of transmission loss versus frequency
f (Octave scale)
Stiffness
controlled
Resonance
controlled
Mass
controlled
Coincidence
controlled
6 dB/octave
slope
T
L
(
d
B
)
Coincidence
dip
5.1.4 Critical frequency
It can be seen from Equation (82) that the coincidence effect cannot occur if the
wavelength of airborne sound ? is greater than
B
? , the bending wavelength in the panel.
The lowest coincidence frequency, called the critical frequency, occurs at the critical
airborne sound wavelength
, (84)
B
??? ==
*
where critical frequency
*
*
?
c
f = , (85)
which corresponds to grazing incidence ( )
o
90=? . For any frequency above the
critical frequency, there is a critical angle at which coincidence will occur.
*
?
Unfortunately, the critical frequency often falls within the range of speech
frequencies, thereby limiting the effectiveness of some partitions intended to provided
privacy and prevent speech interference.
5.1.5 Sound transmission coefficient and transmission loss at coincidence
The coincidence effect depends on the characteristics of the plate or panel and on
the airborne sound wave. Coincidence occurs when
1
sin
4
42
=
mc
B ??
. (86)
Substituting the above condition in the equation for the sound transmission
coefficient of a plate results in the coincidence condition sound transmission coefficient,
2
*
*
2
cos
1
1
?
?
?
?
?
?
+
==
c
m
?
???
?? , (87)
84
where ? is the loss factor of the panel.
and the coincidence condition transmission loss [72]
.
cos
1lg20
2
cos
1lg10
1
lg10
,
2
cos
1
1
*
2
*
*
*
2
*
*
?
?
?
?
?
?
?
?
+=
?
?
?
?
?
?
?
?
+=
?
?
?
?
?
?
==
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
==
c
fm
c
m
TLTL
c
m
?
???
?
???
?
?
??
?
??
(88)
Based on the above equations, we should predict
and (89) 1
*
=? ,0
*
=TL
for undamped panels (i.e., if loss factor 0=? . There is some damping, however,
in all construction materials. Note that the above transmission loss equation is based on
theoretical behavior of an infinite plate. The actual constraints of windows and walls may
produce a different response to sound waves. It is recommended that the behavior of
actual materials be verified by laboratory or field tests.
5.2 Literature review of the Sound Transmission Loss of Sandwich Panels
The purpose of research on the sound transmission loss of sandwich structures is
to improve the sound transmission loss without compromising the stiffness to density
ratio so that the critical frequency of the panel can be raised out of the audio frequency
range. As early as 1959, Kurtze and Watters designed some kinds of sandwich plates and
analyzed their acoustical behavior [40]. They inserted a soft core between two thin face
sheets to introduce shear waves in the middle frequency range. If the shear wave speed is
less than the speed of sound in air, the critical frequency is then shifted to higher
frequencies, which avoids locating the critical frequency in the range of interest. But one
of the shortcomings in their model is that they assumed that the core material is
85
86
incompressible, which is impossible in reality for soft core. Although their experimental
results agree with the theoretical estimation of the anti-symmetric motion, their model
cannot predict symmetric motion.
In the work of Ford, Lord and Walker, they assumed the polyurethane foam core
to be a compressible material, and they studied both the anti-symmetric and symmetric
modes of vibration [41]. Smolenski and Krokosky corrected the energy expression used
in [42] and investigated the influence of the core material properties on the critical
frequency due to the symmetric mode [43].
Dym and Lang derived five equations of motion, representing the symmetric and
anti-symmetric vibration modes, for sandwich panels with identical face sheets in [44].
By using their five equations, TL can be calculated for both the symmetric and anti-
symmetric cases. Lang and Dym presented optimal TL properties for sandwich panels in
[44] and they found that by increasing in the stiffness of the core the coincidence effect
caused by the symmetric vibration mode could be eliminated, but the anti-symmetric
coincidence effect would still occur at a low frequency.
Since in some cases, the core of sandwich structures is orthotropic and the face
sheets are not identical, Dym and Lang expanded their theoretical model to include these
cases in [46]. One of the improvements is assumption of damping in both the face sheets
and the core [46]. Ordubadi and Lyon studied the effect of orthotropy on the sound
transmission through plywood panels [48]. They presented an analytical expression for
the TL of such panels.
The most important contribution of the study of Narayanan and Shanbhag is that
they found the transmission loss is more sensitive to the variation of the core shear
87
parameter than to the change of other parameters. However, in their models, they did not
consider symmetric modes. And also they did not integrate the TL results calculated by
their model at some particular angles of incidence to obtain the field incidence
representation.
Moore and Lyon developed an analytical model for sandwich panels with
isotropic and orthotropic cores [50]. They considered both symmetric and anti-symmetric
modes. They further developed a design approach which lowers the double wall
resonance frequency to below the frequency band of interest, and shifts the critical
frequency to higher frequencies.
Wang, Sokolinsky, Rajaram and Nutt derived expressions to predict the TL in
infinitely wide sandwich panels using two models, (1) the consistent high-order
approach, and (2) the two-parameter foundation model [52]. In both the models, the TL is
calculated using a decoupled equation which represents the symmetric and anti-
symmetric motions. They compared their numerical prediction with experimental results.
The consistent high-order approach is more accurate, while the two-parameter
fundamental model is more convenient.
The TL of multi-layer panels has also been analyzed by some other researchers.
Guyander and Lesueur studied the equation of motion, the modal density and the TL of
viscoelastic and orthotropic multi-layer plates [53-55]. They used both plane wave and
reverberant sound excitations to study the TL. Panneton and Atalla developed a three-
dimensional finite element model to predict the TL through a multi-layer system made of
elastic, acoustic and porous-elastic media [56]. The three-dimensional Biot theory was
used to model the porous-elastic medium. However, at low frequencies (lower than 100
Hz, and sometimes even 200 Hz), the predicted behavior is completely incorrect. For
higher frequencies, the model is only useful for unbonded plates. Kurra and Arditi used
the ASTM and ISO standards to measure the sound transmission loss of multi-layered
plates [57, 58]. Uris and Estelles studied the sound transmission of multi-layered
sandwich plates using different configurations of polyurethane and polystyrene layers
[59]. They found that multi-layered sandwich plates possess better sound transmission
loss, and the coincidence effect is not as obvious as with three-layer sandwich plates.
These observations result from the fact that the multi-layered plates are much thicker and
the increased surface densities help to increase the sound transmission loss.
5.3 Statistical energy analysis model (SEA)
5.3.1 Prediction of Sound Transmission through Sandwich Panels using
SEA
SEA method was first developed by Lyon and others in the 1960?s and later Lyon
gave the detailed description of theory and application about SEA in [66]. Crocker and
Price used SEA to predict the sound transmission loss of isotropic single-layered panels
[81]. Here we used the same theoretical model to predict the sound transmission loss of
sandwich panels.
The subsystems and energy flow relationship are illustrated schematically in
Figure 40. The source room and the receiving room are the first and third subsystems,
and the panel under study is the second subsystem. Here the two rooms are assumed to be
reverberant. This means that the sound pressure level measured in each room is the same
at any position in that particular case. is the power input from loudspeakers in the
in
1
?
88
source room, is the power dissipated in the i-th subsystem, and is the power
flow from the i-th to the j-th subsystem.
diss
i
?
ij
?
Figure 39 Schematic of the power flow in three-coupled systems using SEA
If only the source room is excited using loudspeakers and there is no other power
input to the other subsystems, the power balance equations can be expressed as:
, (90)
131211
?+?+?=?
dissin
, (91)
23212
?+?=?
diss
. (92)
13233
?+?=?
diss
The power dissipated in a system in a specified frequency band is related to the
energy stored in the system, , through the internal loss factor
i
E
i
? , namely,
, (93)
ii
diss
i
E ??=?
where ? is the center frequency of the frequency band. The power flow between
subsystems and is i j
?
?
?
?
?
?
?
?
?=?
j
j
i
i
iijij
n
E
n
E
n?? , (94)
89
where n
i
and n
j
are the modal densities of subsystems i and j; ?
ij
is the coupling
loss factor from subsystem i to subsystem j. The equation (95) must be satisfied.
i
j
ji
ij
n
n
=
?
?
, (95)
33
2
2
air
i
i
c
V
n
?
?
= and similar to , (96)
j
n
where is the room volume.
i
V
Applying Equations (94) to (90)-(92) gives the power balance of the partition in a
frequency band with the center frequency ?:
23222
2
2
1
1
22112
?????? EE
n
E
n
E
n +=
?
?
?
?
?
?
?
?
?=? , (97)
33
1
1
33113
???? E
n
E
n ==? . (98)
Note that generally the sound pressure level in the source room is significantly greater
than that in the receiving room,
33 1
/En En
1
/<< , so the term can be neglected in
Equation (95). The subsystems are coupled and the coupling loss factors ?
3
/En
3
23
and ?
23
are
related to the sound radiation efficiency ?
rad
:
??
??
???
?
s
rad
rad
c
2
2321
=== , (99)
where
rad
? is the acoustic radiation loss factor of the panel and the subscript 2?
means that
2
rad
?
? represents the one-sided radiation efficiency.
Thus Equations (91) and (92) can be reduced:
90
,
2
21
2
2
2
?
?
?
?
?
?
?
?
+
=
rad
rad
n
E
n
E
??
?
(99)
.
32313
232131
3
???
??
++
+
=
EE
E (100)
Substituting Equations (99) into (100), gives the ratio of energy between two
rooms,
.
32313
232131
3
???
??
++
+
=
EE
E (101)
Assuming the two rooms are large enough, , and ignoring high order
of loss terms,
231
, nnn >>
.
13
3
3
1
3
1
?
?
+=
n
n
E
E
(102)
The coupling loss factor
13
? due to non-resonant transmission is obtained from
[66],
?
?
?
?
?
?
+?=
?
?
1
101310
4
log10TLlog10
V
cA
airp
, (103)
where is the effective area of the panel.
p
A
Then the TL can be expressed,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?=
3
1
3
1
31
10
4
log10TL
n
n
E
E
V
cA
airp
??
. (104)
Since the source room sound field is assumed to be reverberant, the energy stored in it is
expressed by the pressure p
1
, and volume V
1
:
2
1
2
11
/ cVpE ?>=< . (105)
91
The mechanical energy stored in the panel is expressed by its velocity and mass:
ps
AvE ?>=<
2
2
, (106)
where ?
s
and A
p
are the surface density and area of the panel.
Combining the results above, the averaged squared velocity of the panel is obtained
2 2
2 rad
21
1rad2
2
sp
nV
v
nA
?
?
?
??? ?
1
p
c
< >
<>=? ? ?
+
. (107)
Thus the power radiated by the resonant modes into the receiving room is
22
23
vcA
radp
?
??=? . (108)
Similarly,
13 13 1
WE??= . The coupling loss factor ?
13
due to non-resonant random
incidence mass-law transmission is obtained from [66]
10 13 RI 10
1
10log TL 10log
4
p
Ac
V
?
?
??
=? +
?
??
?
2
a
, (109)
2
RI 10 10
TL 10log 10log ln(1 )a? ?=? +
? ?
, (110)
where / 2
s
ac???= , and TL
RI
is called the random incidence transmission loss.
Substituting (110) in (109), ?
13
can be derived as
23
22
13 32 22
1
ln 1
4
p
s
s
Ac
Vc
?
? ?
?
?? ?
? ?
=+
? ?
? ?
. (111)
?
?
?
?
?
?
+=?
22
22
2
2
1
13
4
1ln
)( c
l
cAp
s
s
p
?
??
??
?
. (112)
In the source room, the sound power incident on the dividing partition of area is:
p
A
pinc
A
c
p
?4
2
1
><
=? . (13)
92
Below the critical frequency, the sound transmission loss of a finite panel is more
controlled by the contribution of those modes that have their resonance frequencies
outside the frequency band of the excitation signal than by those with resonance
frequencies within that band. So taking into account both the forced and resonant panel
motions, the transmission coefficient can be approximated by
?
?
?
?
?
?
++??
+
?
=
?+?
?
=
22
22
2
21
2
2
2
1
2
1323
4
1ln
)(2
4/
1
c
cA
c
V
n
n
cA
s
s
p
rad
srad
rad
p
inc
?
??
??
?
?
???
?
?
?
?
?
? .
(114)
Then, the sound transmission loss TL can be calculated from
TL 10log(1/ )?= . (130)
5.3.2 Simulation using SEA software AutoSEA
AutoSEA is an interactive vibro-acoustics simulation tool based on the SEA
method. In order to calculate the sound transmission loss and the radiation efficiency of a
panel, two virtual rooms and a panel must be created in AutoSEA, as shown in Figure 41.
The two virtual rooms were assumed each to have a diffuse sound field, and to have
identical volumes as do the two real rooms in the Sound and Vibration Laboratory. A
random sound source was used to generate the white noise generation in the source room.
The panel under investigation has clamped boundary conditions. To create a sandwich
panel in AutoSEA, the material properties of the face sheets and the core are required
separately. The face sheets of a sandwich panel must only be isotropic while the core can
be orthotropic. The material properties assumed for the face sheets and the core are listed
in Appendix B.
93
Figure 40 Sound transmission loss model using the AutoSEA software.
5.4 Review of sound transmission measurement technique: two-room
method
There are several methods which have been devised for the measurement of
transmission loss. The most widely used method consists of the use a transmission suite.
94
Figure 41 Set up for the two reverberation room sound transmission loss
measurement method
The two-room method was used to determine the sound transmission loss of panels in the
two room suit in the Sound and Vibration Laboratory at Auburn University. The two-
room suite consists of two reverberation rooms with the panels under investigation
mounted in a window in the walls between the two rooms, as shown in Figure 42. The
four edges of the panels under investigation were clamed on a window between two
rooms. The volumes of the source room and receiving room are both equal to 51.15 m
3
.
The area of the panels under test is 0.36 m
2
. These two reverberation chambers are
95
vibration-isolated from each other and the ground to reduce environmental noise. Each
room has two walls made of wood with fiberglass filled in between them. The two rooms
are also separated from each other using fiberglass. Springs are installed under the rooms
to reduce the mechanical noise between the rooms and the ground. It should be noted that
the transmission suite method cannot be used exactly ?in the field? since the sound fields
are rarely sufficiently diffuse. So the sound absorption in each room should be taken into
account. By assuming that the rooms are designed to minimize sound transmission paths
other than through the test specimen and the panel under test is the only path that the
sound travels through, the sound transmission loss measured using the two-room method
is given by
?
?
?
?
?
?
?
?
+?=
R
p
Rs
A
A
LL lg10TL
, (131)
where = space averaged sound pressure level in the source room
s
L
= space averages sound pressure level in the receiving room
R
L
= area of the panel under test
p
A
= equivalent absorption area of the receiving room
R
A
and
2
2
10ln24
R
R
cT
V
A = , (132)
where and V
2R
T
2
are the reverberation time and volume of the receiving room
[74].
So transmission loss can be expressed as,
96
10ln24
log10
10ln24
log10
2
2
2
2
V
cTA
NR
V
cTA
LLTL
RpRp
RS
+=+?= , (133)
where is the noise reduction between these two rooms. NR
5.5 Experiments of TL and simulations using AutoSEA
Table 15 lists the properties of the panel whose TL was measured by two room
method.
Sandwich panels under study
Panel
Core
thickness
(mm)
Face sheet
thickness
(mm)
Density
(kg/m
3
)
Surface
density
(kg/m
2
)
A 6.35 0.35 362.33 2.55
B 6.35 0.7 525.88 4.07
C 12.7 0.7 345.19 4.86
D 25.4 0.7 248.28 6.65
Aluminum panels under study
Panel Thickness (mm) Surface density (kg/m2)
E 6.35 17.15
Table 16 Geometrical parameters of panels under study
97
98
First of all, the reverberation times in the receiving room were measured for each
panel using a B&K PULSE system. Table 16 shows the reverberation times measured for
one-third octave frequency bands from 80 Hz to 8 kHz for Panel A ~ D.
Center
frequency (Hz)
Panel A Panel B Panel C Panel D
80 0.380 0.375 0.421 0.377
100 0.632 0.544 0.557 0.506
125 0.493 0.395 0.352 0.360
160 0.636 0.659 0.699 0.634
200 0.578 0.740 0.736 0.698
250 0.903 0.818 0.856 0.860
315 1.048 1.086 1.034 1.043
400 1.240 1.162 1.155 1.111
500 1.264 1.254 1.309 1.345
630 1.316 1.407 1.324 1.344
800 1.411 1.449 1.473 1.430
1000 1.326 1.364 1.368 1.338
1250 1.238 1.248 1.224 1.247
1600 1.132 1.157 1.121 1.146
2000 1.028 1.036 1.016 1.016
2500 0.947 0.916 0.947 0.937
3150 0.846 0.809 0.865 0.830
4000 0.778 0.759 0.780 0.747
5000 0.698 0.688 0.690 0.713
6300 0.633 0.639 0.629 0.652
8000 0.556 0.545 0.555 0.538
Table 17 Reverberation times (s) of the receiving room with different panels
Then a steady white noise from the generator in the Pulse system was provided to
two loudspeakers in the source room. An air jet nozzle was used to increase the noise
level in the high frequency region since the noise from the two loudspeakers is not
enough in the high frequencies. With the loudspeakers and air jet, we can assume that the
white noise is generated by these sound sources. The sound pressure levels in the two
rooms were measured using two B&K microphones, type 4188, whose optimized
frequency response range is from 8 Hz to 12.5 kHz. For the panels listed in Table 16, the
measurements of the sound pressure levels in both the source and receiving rooms (
and ) were repeated eight times by putting the two microphones at eight randomly
selected positions. The spatial averages were calculated to obtain the noise reduction
(NR). Note that the background noise in the source and receiving rooms was also
measured and subtracted from and . The sound transmission loss was then calculated
using Equation (133).
S
L
R
L
S
L
R
L
Figures 42, 43, 44 and 45 compare the measured transmission loss of panels A, B,
C and D, with the results simulated using AutoSEA software in which the panel damping
was set to either 2% or 5%. By comparing the two simulated cases with the measured
99
result, it can be seen that the simulated result with damping of 5% is closer to the
measured result. That means the damping of these sandwich panels is close to 5% or loss
factor is close to 10%, or at least damping is greater than 2%.
0
5
10
15
20
25
30
35
40
100 1000 10000
Frequency (Hz)
TL (
d
B
)
Measured
AutoSEA with 2% damping
AutoSEA with 5% damping
Mass Law
AutoSEA with measured damping values
Figure 42 Measurements of TL, mass law and simulation of AutoSEA for panel A
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
100 1000 10000
Frequency (Hz)
TL
(
d
B
)
Measured
AutoSEA with 2% damping
AutoSEA with 5% damping
Mas s Law
Figure 43 Measurements of TL, mass law and simulation of AutoSEA for panel B
100
0
5
10
15
20
25
30
35
40
45
100 1000 10000
Frequency (Hz)
TL (
d
B
)
Measured
AutoSEA with 2% damping
AutoSEA with 5% damping
Mass Law
Figure 44 Measurements of TL, mass law and simulation of AutoSEA for panel C
0
5
10
15
20
25
30
35
40
45
100 1000 10000
Frequency (Hz)
TL (
d
B
)
Measured
AutoSEA with 2% damping
AutoSEA with 5% damping
Mass Law
Figure 45 Measurements of TL, mass law and simulation of AutoSEA for panel D
101
0
5
10
15
20
25
30
100 1000 10000
Frequency (Hz)
TL (
d
B
)
Panel A
Panel B
Figure 46 Measured TL for panels A and B
0
5
10
15
20
25
30
100 1000 10000
Frequency (Hz)
TL
(
d
B
)
Panel C
Panel D
Figure 47 Measured TL for panels C and D
Figure 46 shows the TL for panels A and B. The TL increases when the thickness
of the face sheet increases.
Figure 47 shows the TL for panels C and D. The TL increases when the thickness
of the core increases.
102
One aluminum panel E was simulated using AutoSEA shown in Figure 48. Panel
E has the same thickness with panel A. From Figure 49, it can be seen that the critical
frequency is above 6 kHz, which is an advantage for such sandwich panels since the
human hearing tends to be more sensitive to sound in the range of approximately 2-6
kHz. Figure 49 shows the comparison of the TL of sandwich panels A, B, C and D and
aluminum panel E. It is seen that the sound transmission loss values of the sandwich
panels are smaller than that of the aluminum panel. This implies that the surface mass
density still dominates the overall transmission loss behavior of these panels. The foam-
filled honeycomb sandwich design does not demonstrate any advantage of sound
transmission over heavier metal counterpart, although the sandwich structures have
higher damping. That means such a foam-filled honeycomb sandwich design must be
modified if it is to obtain higher sound transmission loss. But considering the light weight
materials, the sandwich structures can be used as a good substitute for heavier metal
panels when the weight and static bending strength are important factors for design.
TL Aluminum 0.25 inch
0
5
10
15
20
25
30
35
100 1000 10000
Frequency (Hz)
TL (
d
B
)
measured
autosea 1%
autodea 0.01%
103
Figure 48 Measurements of TL and simulation by AutoSEA for panel E
0
5
10
15
20
25
30
35
100 1000 10000
frequency (Hz)
TL (dB
)
panel A
panel B
panel C
panel D
aluminum 0.25 inch
Figure 49 Measured TL for panels A, B, C, D and E
5.6 Summary and conclusions of experiments of TL and simulations using
AutoSEA
The four sandwich panels with the same original honeycomb have similar
acoustical performances in the frequency range 125~8000Hz. Increasing the thickness
either of the face sheets or core increases the first resonance frequency. Increasing the
thickness in the core improves the damping. The estimated transmission losses from SEA
agree well with the experimental values. The foam filled honeycomb core sandwich panel
has high damping, greater than 2% and close to 5%.
104
105
CHAPTER 6 SUMMARY AND CONCLUSIONS
The materials we used include the sandwich structures with honeycomb core
filled with foam and the sandwich structures with close-cell foam core. We purchased the
foam filled honeycomb core panels, close-cell foam core panels and face sheets
according to our needs. The professor, Dr. Vaidya, at Birmingham made the sandwich
panels with the materials we provided. The author improved the existing methods and
used them to study the sandwich materials. The sandwich materials we designed were
proved to have good stiffness and damping properties and fine sound transmission
behavior.
From the measurements using the four-point bending method and twisting
method, the static properties for the core and entire sandwich structures were calculated.
The finite element method was used to model the static response of the sandwich beams
with four-point bending method and twisting method.
Dynamic properties of thin sandwich structures with honeycomb cores were
discussed in this dissertation. The structure was considered anisotropic in the theoretical
models. The dynamic parameters of the laminates and the core are usually not known.
For a bonded honeycomb structure though, the mass per unit area, the dimensions of the
entire structure and the laminates are easily determined.
Hamilton's principle was used to derive a six order wave equation governing the
bending stiffness of a sandwich structure. Measurements gave the first few natural
frequencies for the beams and from these results their corresponding equivalent bending
stiffnesses at the natural frequencies could be determined. Using the least squares method
on the bending stiffnesses at the natural frequencies, the bending stiffness is determined
in the entire frequency range.
The results for the frequency dependent bending stiffness curve allow the
determination of the Young's modulus of the bonded laminates and the shear modulus of
the core. The value of bending stiffness in the low frequency region give the Young's
modulus for the bonded laminates and the slope of the decreasing stiffness reveals the
shear modulus of the core.
For the structures the necessary dynamics parameters can be now estimated. The
wave number for the honeycomb structure can be determined based on the six order wave
equation.
The wave number equation has the six solutions of the
form
321
,, ??? iik
x
???= . The real solution
1
? represents a propagating wave. In the
low frequency region, the characteristics of the honeycomb structure are determined by
the characteristics of the entire bonded structure. The core serves as a spacer between the
laminates, and in the mid frequency region it provides shear. In the high frequency region
the wave number plot confirms that the vibration of a structure is determined by the
laminates only. The imaginary solution
2
?i represents an evanescent wave which
becomes real and propagating above a certain frequency. This change denotes the
frequency for which the moment of inertia becomes of importance for the calculations.
The third solution
3
?i is determined by the core thickness and is constant in the low
106
107
frequency region and for increasing frequencies adjusts to the asymptote for the wave
number of the laminates.
The boundary conditions for the honeycomb structures are shown to influence on
the dynamic properties with decreased effect of shear when tightening the boundary
conditions.
The theory presented provides a good estimate for the bending stiffness of
traditional sandwich elements with foam cores.
The analysis of the sound transmission loss of foam-filled honeycomb sandwich
panels was carried out. A theoretical model was used for the statistical energy analysis
(SEA) method. A parameter study was made for the sound transmission loss. It can be
concluded that an increase in the core thickness will increase the critical frequency.
Measurements on five different panels, including one isotropic aluminum panel and four
sandwich panels, were carried out using the two-room method. Predictions of sound
transmission loss were conducted using AutoSEA software. The measured results agree
quite well with the results predicted using AutoSEA. The surface densities of the
sandwich panels are much smaller than those of the aluminum panels. The measured
results show that the overall sound transmission loss values of sandwich panels are
smaller than the aluminum panel. This implies that the surface density still dominates the
overall transmission loss behavior of these panels. The foam-filled honeycomb sandwich
design does not show any sound reduction advantage over their heavier metal
counterparts, although the sandwich structures have higher damping capacity. That means
that such a foam-filled honeycomb sandwich design must be modified to obtain higher
sound transmission loss properties. The measured transmission loss curves also show that
108
the critical frequency increases as the core thickness is increased, if other parameters do
not change, which agrees with the theoretical prediction. The results from the
measurements and the simulation by AutoSEA imply that the foam-filled honeycomb
core sandwich panel has high damping, greater than 2% and close to 5%.
109
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