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 nα 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/m2

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 2d

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

2d

P

0w 21 EE

8

0221

3

34 w

SddbhP

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.

2d 0w

1d h

And is the compliance along the direction parallel to the length of the beam. xS

2.direction principalin 1

1,direction principalin 1

222

111

ESS

ESS

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. 1d

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

0w 12G 2112 , νν of the

core, face sheet and entire sandwich structure, by means of the equation:

02

3

34 w

LStPG

⎟⎟⎠

⎞⎜⎜⎝

⎛= , (3)

where

( ) ( ) ( ) ,282 66222

222

1222

112 SnmSnmmnSnmSnmmnSG −+++−−−= (4)

and

,1,,1,1

1266

2

21

1

1212

222

111 G

SEE

SE

SE

S =−=−===νν

and

αcos=m and αsin=n , where α 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

α

α

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

11S 22S

o0=α o45=α 66S 12S

11

.

,

2111

2212

2122

12

νν

ν

SS

SS

=

=− (6)

The shear modulus is also obtained since is known from the

equation

12G 66S

1266

1G

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

0w

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 bD

⎥⎦

⎤⎢⎣

⎡+++=

32

212

32

23l

lclc

lccb tttttEtED , (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).

ct lt

cE lE

.211

,21

,11

2112

1

2112

22

2112

11

ordirectionprincipalforE

Esheetsfacefor

directionprincipalforE

E

directionprincipalforE

Ecorefor

ll

l

l

cc

c

c

cc

c

c

νν

νν

νν

−=

⎪⎪⎩

⎪⎪⎨

⎧

−=

−=

(8)

12

Then the bending stiffness of the entire sandwich beams, in two principal

directions, 1 and 2, can be calculated by Equation (9):

.23

2212

,13

2212

32

232

22

32

231

11

directionprincipalfort

tttt

EtE

D

directionprincipalfort

tttt

EtE

D

llc

lcl

ccs

llc

lcl

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:

( )( )( )( ) .2112

2

,1112

2

2112

32

22

2112

31

11

directionprincipalforttE

D

directionprincipalforttE

D

sslc

ss

sslc

ss

νν

νν

−+

=

−+

=

(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.

o0=α o45=α

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/m3) (kg/m3)

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 o0=α

Plate G Core 6.35 0.5 N/A 163 N/A o45=α

Plate H Entire

sandwich

6.35 0.5 N/A 163 N/A o0=α

Plate I Entire

sandwich

6.35 0.5 N/A 163 N/A o45=α

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).

81.53

4

112

21

3

=cSdd

bhslope ,

6

111 1005.91

×== cc

SEcorefor Pa. (11)

For beam B, the slope of the load-deflection curve and the properties derived are

given by Equations (12).

34.33

4

222

21

3

=cSdd

bhslope ,

6

222 1020.51

×== cc

SEcorefor 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 =

22.53

4

112

21

3

=lSdd

bhslope ,

10

1112 1086.41

×=== lll

SEEsheetfacefor Pa. (13)

For beam D, the slope of the load-deflection curve and the properties derived are

given by Equations (14).

15

03.91853

4

112

21

3

=sSdd

bhslope ,

10

111 100449.11

×== ss

SEsandwichfor Pa. (14)

For beam E, the slope of the load-deflection curve and the properties derived are

given by Equations (15).

15.91843

4

222

21

3

=sSdd

bhslope ,

10

222 100448.11

×== ss

SEsandwichfor 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).

66.109340 2

66

3

==aS

hslope coα ,

7

6612 1003.81

×== cc

SGcorefor 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

,15.33245

2111

2212

22

1221

21222

3

⎪⎪⎩

⎪⎪⎨

⎧

==

=−=

=−

=

cc

cc

c

cc

cc

SS

SS

corefor

aSShslope

νν

ν

α o

(17)

16

For plate H, the slope of the load-deflection curve and the properties derived are

given by Equations (18).

,80.27097340 2

66

3

==aS

hslope soα

10

6612 1045.11

×== ss

SGsandwichfor 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

,92.75093245

2111

2212

22

1221

21222

3

ss

ss

c

ss

ss

SS

SS

sandwichfor

aSShslope

νν

ν

α 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

11 1031.9

1×=

−= cc

c

cEEνν

Pa for principal direction 1,

6

2112

22 1035.5

1×=

−= cc

c

cEEνν

Pa for principal direction 2,

for face sheets 10

2112

1 1009.51

×=−

= ll

l

lEEνν

Pa for principal direction 1 or 2. (20)

17

Then using Equation (9), we can calculate the bending stiffness of the entire

sandwich beams:

53.4003

2212

32

231

11 =⎥⎥⎦

⎤

⎢⎢⎣

⎡+++= l

lclc

lccs t

tttt

EtE

D N.m for principal direction 1,

45.4003

2212

32

232

22 =⎥⎥⎦

⎤

⎢⎢⎣

⎡+++= l

lclc

lccs t

tttt

EtE

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:

( )( ) 28.335112

2

2112

31

11 =−+

= sslc

ss ttE

Dνν

N.m for principal direction 1,

( )( ) 25.335112

2

2112

32

22 =−+

= sslc

ss ttE

Dνν

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 1E [ Pa] 2E [ Pa] 12ν 21ν G12 [ Pa]

Core 9.05×106 5.20×106 0.22 0.126 8.03×107

Face sheet 4.86E×1010 4.86×1010 0.213 0.213 1.67×1010

Entire sandwich 1.0449×1010 1.0448×1010 0.3 0.3 8.025×107

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 ,,,, νν

ssss EE 211221 ,,, νν

Bending stiffness per

unit width of the

entire sandwich

beams

From properties

cclcc EEE 2112121 ,,,, νν

[N.m]

From properties

ssss EE 211221 ,,, νν

[N.m]

sD11 335.28 335.25

sD22 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 ( ) o0=α

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 β and the angular displacement

due to shear in the core γ . 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, β and γ is given by

γβ +=∂∂

xw . (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

( )∫∫ =+− 0dxdtATUδ (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 lE 2D lρ and

thickness . The core has effective shear stiffness , its Young's modulus , its

equivalent density

lt eG cE

cρ 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.

ct eG

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 ρρµ += ccc EG ρllE ρ

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

⎥⎦

⎤⎢⎣

⎡+++=+= ∫∫

+

32

21222

32

232/

2/

22/

0

21

llc

lcl

cctt

t l

t

ctttttEtEdyEydyEyD lc

c

c . (24)

In general, . The bending stiffness of one laminate is cl EE >>

12

3

2lltED = . (25)

The mass moment of inertia per unit width is defined as

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+++== ∫

+ 3223

2/

0

2

32

2122 llc

lcl

ccttttttttdyyyI lc ρρρρ , (26)

while the mass per unit area is

llcc tt ρρµ 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, β and γ 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

1U

.21 2

11 ⎟⎠⎞

⎜⎝⎛∂∂

=x

DU β (28)

The potential energy per unit area due to pure bending of the two laminates is 2U

.2

22 ⎟⎠⎞

⎜⎝⎛∂∂

=x

DU γ (29)

The potential energy per unit area due to shear deformation of the core is 3U

38

.21 2

2 γcetGU = (30)

The total potential energy per unit width is thus

( ) .221

0

22

2

2

10 321 ∫∫ ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=++=L

c

LdxGt

xD

xDdxUUUUdx γγβ (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 1T

2

1 21

⎟⎠⎞

⎜⎝⎛∂∂

=twT µ , (32)

and the kinetic energy per unit area due to the rotation of a section of the beam 2T

2

2 21

⎟⎠⎞

⎜⎝⎛∂∂

=t

IT βρ . (33)

This gives the entire kinetic energy per unit width of the beam as

( ) ∫∫ ∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

=+=LL

dxt

ItwdxTTTdx

0

22

0 21 21 βµ ρ . (34)

The total potential energy for the conservative external forces according to Figure

16 is

(35) ( ) ( ) ( ) ( )[ ] ,00 000 1212LLL

MFwpwdxMLMwFLwFpwdxaAdx βββ −+=+−−+=− ∫∫ ∫ [ ]

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 γ , equation (22), and by inserting equations (31), (34) and

(35) into the variational expression (23) the result is

39

[ ] .0

22

0

222

2

2

2

1

=−−−

+⎪⎭

⎪⎬⎫

⎟⎠⎞

⎜⎝⎛∂∂

−⎟⎠⎞

⎜⎝⎛∂∂

−+⎟⎠⎞

⎜⎝⎛∂∂

+⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛∂∂

∫∫ ∫

∫∫

dtMFwpwdxdt

tI

twtG

xD

xDdxdt

L

ce

βδδ

βµγγβδ ρ (36)

The integration over time is from to , and over the length x from 0 to L.

Using Equation (23) and expressing

0t 1t

γ as function of β and xw∂∂ gives

[ ] .0

22

0

2222

2

2

2

2

1

=−−−

+⎪⎭

⎪⎬⎫

⎟⎠⎞

⎜⎝⎛∂∂

−⎟⎠⎞

⎜⎝⎛∂∂

−⎟⎠⎞

⎜⎝⎛ −∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛∂∂

∫∫ ∫

∫∫

dtMFwpwdxdt

tI

tw

xwtG

xxwD

xDdxdt

L

ce

βδδ

βµβββδ ρ

(37)

According to standard procedures we obtain

[ ] .0

2

0

2

2

2

2

21

=−−⎭⎬⎫

−⎟⎠⎞

⎜⎝⎛∂∂

⎟⎠⎞

⎜⎝⎛∂∂

−⎟⎠⎞

⎜⎝⎛∂∂

⎟⎠⎞

⎜⎝⎛∂∂

−

⎟⎠⎞

⎜⎝⎛ −∂∂

⎟⎠⎞

⎜⎝⎛ −∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+⎩⎨⎧

∂∂

∂∂

∫

∫∫

dtMwFwptt

Itw

tw

xw

xwGt

xxw

xxwD

xxDdxdt

L

c

δβδδδββδµ

δβδβδβδβδββ

ρ

(38)

Integrating by parts gives

.0

222

22

00

2

2

2

2

2

2

0

2

2

3

3

2

02

2

23

3

4

4

2

02

2

3

3

2

02

2

22

2

10

1

1

0

1

0

=+−

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎠⎞

⎜⎝⎛∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−

⎟⎠⎞

⎜⎝⎛ −∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎠⎞

⎜⎝⎛ −∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

∂∂

+∂∂

−∂∂

∫∫

∫∫∫∫∫∫ ∫∫

∫∫∫∫∫

∫∫∫∫∫

∫∫∫∫∫

dtMwF

wdxdtpdxdtt

Idxt

Idxdttwwdx

tw

dxdtxwtGdxdt

xxwwtGdt

xwwtG

dxdtxx

wDdtxx

wDdxdtxx

wwD

dtxx

wwDdtxx

wxwD

xdxdtDdt

xD

LL

t

t

t

t

cece

L

ce

L

LLL

δβδ

δβδββδβδµδηµ

βδββδβδ

βδββδββδ

βδβδδββδββ

ρρ

40

(39)

When assuming that the displacement is defined so that 0≡yδ for

, and for 10 tandtt = ( ) 0≡⎟⎠⎞

⎜⎝⎛∂∂

=∂

∂xy

xy δδ , Equation (39) is reduced to

.02

2

2

2

2

02

2

2

02

2

21

02

2

3

3

2

2

2

2

2

3

3

22

2

1

2

2

3

3

4

4

22

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

∂∂

+

⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂∂

+

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎠⎞

⎜⎝⎛ −∂∂

+

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+∂∂

−+

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−

∫

∫

∫

∫∫

∫∫

dtxx

wxwD

dtMxx

wDx

D

dtFxx

wDxwtGw

dxdtxwtG

tI

xxwD

xD

dxdtptw

xxwD

xxwtGw

L

L

L

ce

ce

ce

βδ

ββδβ

ββδ

ββββδβ

µββδ

ρ

(40)

Since the expression should be equal to zero for δβ and 0≠wδ , 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 β . These are

02 2

2

3

3

4

4

22

2

=−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

− ptw

xxwD

xxwtG ce µββ , (41)

and 02 2

2

2

2

3

3

22

2

1 =⎟⎠⎞

⎜⎝⎛ −∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+∂∂

− ββββρ x

wtGt

Ixx

wDx

D ce . (42)

Using these two equations, eliminating β by using the simplification

and , the equation governing w is given by: ( xkti xAew −= ω ) )( xkti xBe −= ωβ

41

( )

( ) .2

222

2

2

2

2

214

4

2

2

4

4

122

4

2124

6

26

6

12

twIptG

xpDD

twI

tw

xwDtG

txwtGIDD

txwID

xwDD

ce

cece

∂∂

−−∂∂

+=∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

∂++−

∂∂∂

+∂∂

−

ρρ

ρρ

µ

µµµ

(43)

Eliminating instead gives the corresponding equation for w β as

( )

.2

222

3

3

24

4

2

2

4

4

122

4

2124

6

26

6

12

xptG

xpD

tI

txDtG

txtGIDD

txID

xDD

ce

cece

∂∂

+∂∂

−=∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

∂++−

∂∂∂

+∂∂

−

βµ

βµββµµββ

ρ

ρρ

(45)

The shear angle γ can be shown to satisfy the same differential equation as β 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

=∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂∂

==⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−⎟⎠⎞

⎜⎝⎛ −∂∂

xwor

xxw

orMxx

wDx

D

worFxx

wDxwtG ce

β

βββ

ββ

(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

=∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=⎟⎠⎞

⎜⎝⎛ −∂∂

xwor

xxw

orxx

wDx

D

worxx

wDxwtG ce

β

βββ

ββ

(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 β 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

,02

2 2

2

14

4

22

4

1

216

6

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−∂∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

∂∂

tw

DxwtG

txw

DDD

xwD ce

µµ (48)

which is the wave equation for the bending of beams neglecting the moment of

inertia. The corresponding equation for β , Equation (43) is reduced to

.02

2 2

2

14

4

22

4

1

216

6

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

−∂∂

∂⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

∂∂

tDxtG

txDDD

xD ce

βµββµβ (49)

As seen from the equations the expressions β 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 wxw∂∂ 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=∂∂

xw0=β0=w

02

2

=∂∂

xw0=

∂∂

xβ

0=w

44

Free ends 2

2

2

2

1 tI

xD

∂∂

=∂∂ ββ

ρ02

2

=∂∂

xw 0=

∂∂

xβ

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

122

4

2124

6

26

6

12

=∂∂

+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

∂++−

∂∂∂

+∂∂

−

twI

tw

xwDtG

txwtGIDD

txwID

xwDD cece

µ

µµµ

ρ

ρρ

(50)

Neglecting losses in the structure, the bending and shear stiffnesses , and

are real quantities. By assuming a solution

1D 2D

eG

( ).~ xkti xAew −ω

45

the wave number must satisfy xk

.0

222

4

1

2

1

422

11

224

1

262

=+

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛++−−

ωµ

ωµωµµω

ρ

ρρ

DI

DktGk

DtGI

DDkI

DDkD xcex

cexx

(51)

This equation has the six solutions 321 ,, κκκ iikx ±±±= where the κ variables

are real quantities if the losses are neglected. If the stiffness is defined as ( )ηiDD += 10

and ( )ηiGG += 10 , 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 1κ 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)

lt

(mm)

ct

(mm)

lρ

(kg/m3)

cρ

(kg/m3)

µ

(kg/m2) (MPa) eG (GPa)lE

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 xk

x

x Dk

24 µω= , (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

xD

xD

µ 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.

xk

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 2κ and 3κ 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;lim4/1

21

212

1

4/1

1

2

01 ⎥

⎦

⎤⎢⎣

⎡ +=⎥

⎦

⎤⎢⎣

⎡=

∞→→ DDDD

D ff

µωκµωκ (53)

,lim;lim2/1

1

2

2

4/1

1

2

02

⎥⎥⎦

⎤

⎢⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

∞→→ DI

D ff

ωκµωκ ρ (54)

,2

lim;2

lim4/1

2

2

3

2/1

203 ⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

∞→→ DDGH

ff

µωκκ (55)

or 2κ we see a minimum at a certain frequency , where pf 2κ shifts from being

entirely complex to real and describes a propagating wave.

As the frequency increases, 2κ approaches zero for pff = where

ρπ ItGf ce

p 21

= . (56)

49

Below this frequency , pf 2κ is imaginary, and describes evanescent waves. For

higher frequencies 2κ 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 xk

x

x Dk

24 µω= , (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

xD xD

xk

02 22/1

1

2/3

2/1 =−+⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛ DDDD

DtGxx

xce

ωµ, (58)

50

if the moment of inertia is neglected. In the low frequency range, or as 0→ω ,

the first part of the equation dominates why . The bending stiffness is

consequently determined by pure bending of the beam.

1DDx →

In the high frequency range, when ∞→ω . 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.

22DDx →

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

xk

nf

;,4

4/122

nxn

nn L

Df

L απµ

κ =⎥⎦

⎤⎢⎣

⎡= n=1, 2, 3… ,(59)where nα 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 nα 4.73 7.85 11.00 14.14 17.28 2/ππ +n

Free-clamped nα 1.88 4.69 7.85 11.00 14.14 2/ππ −n

Simply supported nα 3.14 6.28 9.42 12.56 15.7 πn

Table 8 Values of for particular boundary conditions nα

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

xnD

nf L µ , given by

...3,2,144

422

== nforLfDn

nxn α

µπ . (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

,02/12/3 =−+− CDDfBD

fA

xxx (61)

where

.2;2

;2 22/1

12/1 DC

tGB

DtG

A cece ===πµπµ

(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

1D 2D G

,2

2/12/3∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛−+−=

ixixi

ixi

i

CDDfBD

fAQ (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

xiD if

cG

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

1D 2D

./2

,2/2/1

2

ce tBGCD

µπ=

= (64)

The parameters are predicted from the MATLAB program.

3.6 Damping measurement methods

Basically there are four measures of damping, the loss factor η, the quality factor

Q, the damping ratio ζ , 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:

φζπ

η tan2212

=′′′

=====EE

CC

QWD

c

. (65)

Here D and W are the dissipated and total powers in one cycle of vibration, C and

Cc 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

nA= ln1δ , (66)

Am +

( ) 222 δπ

δζ+

= , (67)

where δ 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 −=ζ . (68)

nf

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

1f 2f

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

inW=ζ ,

W2 (69)

where is the input power and is the total energy of a modal subsystem. inW totW

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 nD

Determine parameters A, B and C

Draw dynamic stiffness curve xD Calculate dynamic propertiesG 21 ,, DDe

Measure natural frequencies nf

Calculate damping for natural frequencies nf

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/m3) 162.73 162.73 362.331 362.331

Density (kg/m2) 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/m3) 343.148 292.3406 334.5601 268.3411

Density (kg/m2) 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

-700 500 1000 1500 2000

Frequency [Hz]

Rel

ativ

e Le

vel [

dB]

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

ativ

e Le

vel [

dB]

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]

Rel

ativ

e Le

vel [

dB]

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

rf [Hz] D [Nm] rf [Hz] D [Nm] rf [Hz] D [Nm] rf [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.

100

101

102

103

104

0

50

100

150

200

250

300

350

400

450bending stiffness for beam

frequency [Hz]

bend

ing

stiff

ness

[Nm

]

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.

100 101 102 103 1040

50

100

150

200

250

300

350

400bending stiffness for beam

frequency [Hz]

bend

ing

stiff

ness

[Nm

]

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

1D

2D

0→ω , 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

1D

∞→ω , 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:

22D

Static stiffness by four-point method Two stiffness limits from dynamic

characterization

1D [Nm] 2D [Nm] 0→ω [Nm] ∞→ω

[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

100

101

102

103

104

0

50

100

150

200

250

300

350

400

450bending stiffness for beam

frequency [Hz]

bend

ing

stiff

ness

[Nm

]

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.

100 101 102 103 1041

2

3

4

5

6

7bending stiffness for beam

frequency [Hz]

bend

ing

stiff

ness

[Nm

]

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.

100 101 102 103 1041000

2000

3000

4000

5000

6000

7000

8000bending stiffness for beam

frequency [Hz]

bend

ing

stiff

ness

[Nm

]

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.

cG by twisting method cG from dynamic characterization

5.189×107 Pa 5.226×107 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

100 101 102 103 1040

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000bending stiffness for beam

frequency [Hz]

bend

ing

stiff

ness

[Nm

]

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 3500Frequency (Hz)

Dam

ping

rat

io (%

)

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

WW

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.

IW TW

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

WW

=τ , (71)

78

where τ is the sound transmission coefficient. Comparing the definitions of

transmission loss and sound transmission coefficient, we have

⎟⎠⎞

⎜⎝⎛=τ1lg10TL , (72)

or

. (73) 10/TL10−=τ

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

cos1lg101lg10TL2

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟

⎠⎞

⎜⎝⎛=

cmρ

θωτ

(74)

where )Hz(2)rad/s( fπω = =sound frequency

m = panel mass density per unit surface area , 2kg/m

θ = angle of incidence,

ρ = air density,

c = speed of sound in air.

If the angle of incidence is equal to zero, the above equation becomes:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

2

0 21lg10TL

cmρω , (75)

79

the normal incidence mass law TL for approximating the transmission loss of

panels with incident sound in the mass-controlled frequency. o0

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

o90

o72

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 πω = (77)

using typical values for the speed of sound in air and air density

( )elyapproximatc skg/m400 2 ⋅=ρ , (78)

and assuming that

12

>>c

mρω , (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:

,sin12

cossin2

cos1

12

4

422

4

42−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=

mcB

cm

mcB

cm θω

ρθωθω

ρθωητ (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 2kg/m

η = 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 λ and incident at angleθ . Assume bending

wave of wavelength Bλ 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*⎟⎟⎠

⎞⎜⎜⎝

⎛==

Bλλθθ (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.

*θ

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 TL (dB

)

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 λ is greater than Bλ , the bending wavelength in the panel.

The lowest coincidence frequency, called the critical frequency, occurs at the critical

airborne sound wavelength

, (84) Bλλλ == *

where critical frequency

**

λcf = , (85)

which corresponds to grazing incidence ( )o90=θ . For any frequency above the

critical frequency, there is a critical angle at which coincidence will occur. *θ

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

1sin4

42

=mc

B θω . (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*

*

2cos1

1

⎥⎦

⎤⎢⎣

⎡+

==

cmρ

θηωττ , (87)

84

where η is the loss factor of the panel.

and the coincidence condition transmission loss [72]

.cos1lg202

cos1lg101lg10

,

2cos1

1

*2*

**

2*

*

⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟⎟

⎠

⎞⎜⎜⎝

⎛+=⎟

⎠⎞

⎜⎝⎛==

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

==

cfm

cmTLTL

cm

ρθηπ

ρθηω

τ

ρθωη

ττ

(88)

Based on the above equations, we should predict

and (89) 1* =τ ,0* =TL

for undamped panels (i.e., if loss factor 0=η . 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 in1Π

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.

dissiΠ ijΠ

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 Π+Π+Π=Π dissin

, (91) 23212 Π+Π=Π diss

. (92) 13233 Π+Π=Π 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 iE iη , namely,

, (93) iidissi E ωη=Π

where ω is the center frequency of the frequency band. The power flow between

subsystems and is i j

⎟⎟⎠

⎞⎜⎜⎝

⎛−=Π

j

j

i

iiijij n

EnEnωη , (94)

89

where ni and nj are the modal densities of subsystems i and j; ηij is the coupling

loss factor from subsystem i to subsystem j. The equation (95) must be satisfied.

i

j

ji

ij

nn

=ηη

, (95)

33

2

2 air

ii c

Vnπω

= and similar to , (96) jn

where is the room volume. iV

Applying Equations (94) to (90)-(92) gives the power balance of the partition in a

frequency band with the center frequency ω:

232222

2

1

122112 ωηωηωη EE

nE

nE

n +=⎟⎟⎠

⎞⎜⎜⎝

⎛−=Π , (97)

331

133113 ωηωη E

nEn ==Π . (98)

Note that generally the sound pressure level in the source room is significantly greater

than that in the receiving room, 3 3 1/E n E n1/<< , so the term can be neglected in

Equation (95). The subsystems are coupled and the coupling loss factors η

3 /E n3

23 and η23 are

related to the sound radiation efficiency σrad:

ωρσρ

ηηηπ

s

radrad

c 2

2321 === , (99)

where radη is the acoustic radiation loss factor of the panel and the subscript 2π

means that 2radπσ represents the one-sided radiation efficiency.

Thus Equations (91) and (92) can be reduced:

90

,221

2

2

2⎟⎟⎠

⎞⎜⎜⎝

⎛+

=rad

rad

nE

nE

ηηη (99)

.32313

2321313 ηηη

ηη++

+=

EEE (100)

Substituting Equations (99) into (100), gives the ratio of energy between two

rooms,

.32313

2321313 ηηη

ηη++

+=

EEE (101)

Assuming the two rooms are large enough, , and ignoring high order

of loss terms,

231, nnn >>

.13

3

3

1

3

1

ηη

+=nn

EE (102)

The coupling loss factor 13η due to non-resonant transmission is obtained from

[66],

⎥⎦

⎤⎢⎣

⎡+−=

ωη

1101310 4

log10TLlog10VcA airp , (103)

where is the effective area of the panel. pA

Then the TL can be expressed,

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−=

3

1

3

1

3110 4

log10TLnn

EE

VcA airp

ωη. (104)

Since the source room sound field is assumed to be reverberant, the energy stored in it is

expressed by the pressure p1, and volume V1:

21

211 / cVpE ρ>=< . (105)

91

The mechanical energy stored in the panel is expressed by its velocity and mass:

ps AvE ρ>=< 22 , (106)

where ρs and Ap are the surface density and area of the panel.

Combining the results above, the averaged squared velocity of the panel is obtained

2 22 rad2 1

21 rad 22 s p

n Vvn A

π

π

ηη η ρ ρ

12

pc

< >< >= ⋅ ⋅ ⋅

+. (107)

Thus the power radiated by the resonant modes into the receiving room is

2223 vcA radp

πσρ=Π . (108)

Similarly, 13 13 1W Eωη= . The coupling loss factor η13 due to non-resonant random

incidence mass-law transmission is obtained from [66]

10 13 RI 101

10 log TL 10log4

pA cV

ηω

⎛ ⎞= − + ⎜

⎝ ⎠⎟

2a

, (109)

2RI 10 10TL 10log 10log ln(1 )a ⎡ ⎤= − +⎣ ⎦ , (110)

where / 2sa cωρ ρ= , and TLRI is called the random incidence transmission loss.

Substituting (110) in (109), η13 can be derived as

2 3 2 2

13 3 2 2 21

ln 14

p s

s

A cV c

ρ ω ρηω ρ ρ

⎡ ⎤= +⎢ ⎥

⎣ ⎦. (111)

⎥⎦

⎤⎢⎣

⎡+=Π 22

22

2

21

13 41ln

)( cl

cAps

s

p

ρρω

ωρ

ρ. (112)

In the source room, the sound power incident on the dividing partition of area is: pA

pinc Ac

pρ4

21 ><

=Π . (113)

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

⎥⎦

⎤⎢⎣

⎡++⋅⋅

+⋅

=Π+Π

Π=

22

22

221

22

2

1

21323

41ln

)(2

4/1

ccA

cV

nn

cA

s

s

prad

srad

rad

pinc

ρρω

ωρρ

σρηη

η

ρτ π

π

π .

(114)

Then, the sound transmission loss TL can be calculated from

TL 10log(1/ )τ= . (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 m3.

The area of the panels under test is 0.36 m2. 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

pRs A

ALL lg10TL

, (131)

where = space averaged sound pressure level in the source room sL

= space averages sound pressure level in the receiving room RL

= area of the panel under test pA

= equivalent absorption area of the receiving room RA

and

2

2 10ln24

RR cT

VA = , (132)

where and V2RT 2 are the reverberation time and volume of the receiving room

[74].

So transmission loss can be expressed as,

96

10ln24

log1010ln24

log102

2

2

2

VcTA

NRV

cTALLTL 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/m3)

Surface

density

(kg/m2)

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).

SL

RL

SL RL

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 10000Frequency (Hz)

TL (d

B)

MeasuredAutoSEA with 2% dampingAutoSEA with 5% dampingMass LawAutoSEA 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 w ith 2% damping

AutoSEA w ith 5% damping

Mass Law

Figure 43 Measurements of TL, mass law and simulation of AutoSEA for panel B

100

05

1015202530354045

100 1000 10000

Frequency (Hz)

TL (d

B)

MeasuredAutoSEA with 2% dampingAutoSEA with 5% dampingMass Law

Figure 44 Measurements of TL, mass law and simulation of AutoSEA for panel C

05

1015202530354045

100 1000 10000

Frequency (Hz)

TL (d

B)

MeasuredAutoSEA with 2% dampingAutoSEA with 5% dampingMass 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 APanel 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 CPanel 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

05

10

1520

25

30

35

100 1000 10000

Frequency (Hz)

TL (d

B)

measuredautosea 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 (d

B)

panel Apanel Bpanel Cpanel Daluminum 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 ,, κκκ iikx ±±±= . The real solution 1κ 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 2κi 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 3κi 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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