Sound transmission properties of honeycomb panels and ......Sound transmission properties of honeycomb panels and double-walled structures SATHISH KUMAR Doctoral Thesis Stockholm June

Sound transmission properties of honeycomb panels anddouble-walled structures

SATHISH KUMAR

Doctoral Thesis

Stockholm June 2012

Vinnova Centre of Excellence for ECO2 Vehicle Design

The Marcus Wallenberg Laboratory for Sound and Vibration Research

Department of Aeronautical and Vehicle Engineering

Postal AddressThe Royal Institute of TechnologyMWL/AVESE-100 44 StockholmSweden

Visiting AddressTeknikringen 8Stockholm

ContactTel: +46 8790 9375Email: [email protected]

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stock-holm framläggs till offentlig granskning för avläggande av teknologie doktortexamentorsdagen den 14 June 2012, 13:15 i sal F3 Lindstedtsvägen 26, KTH, Stockholm.

TRITA-AVE-2012:20ISSN-1651-7660ISBN 978-91-7501-334-3

c© Sathish Kumar, June 2012Tryck:US-AB

Abstract

Sandwich panels with aluminium face sheets and honeycomb core material havecertain advantages over panels made of wood. Some of the advantages of these con-structions are low weight, good moisture properties, fire resistance and high stiffness-to-weight ratio etc. As product development is carried out in a fast pace today, there isa strong need for validated prediction tools to assist during early design stages. In thisthesis, tools are developed for predicting the sound transmission through honeycombpanels, typical for inner floors in trains and later through double-walled structurestypical for rail-vehicles, aircrafts and ships.

The sandwich theory for wave propagation and standard orthotropic plate theoryis used to predict the sound transmission loss of honeycomb panels. Honeycomb is ananisotropic material which when used as a core in a sandwich panel, results in a panelwith anisotropic properties. In this thesis, honeycomb panels are treated as being or-thotropic and the wavenumbers are calculated for the two principal directions. Thewavenumbers are then used to calculate the sound transmission using standard or-thotropic theory. These predictions are validated with results from sound transmissionmeasurements. The influence of constrained layer damping treatments on the soundtransmission loss of these panels is investigated. Results show that, after the dampingtreatment, the sound transmission loss of an acoustically bad panel and a normal panelare very similar.

Further, sound transmission through a double-leaf partition based on a honeycombpanel with periodic stiffeners is investigated. The structural response of the periodicstructure due to a harmonic excitation is expressed in terms of a series of space harmon-ics and virtual work theory is applied to calculate the sound transmission. The originalmodel is refined to include sound absorption in the cavity and to account for the or-thotropic property of the honeycomb panels. Since the solution of the space harmonicanalysis is obtained in a series form, a sufficient number of terms has to be includedin the calculation to ensure small errors. Computational accuracy needs to be balancedwith computational cost as calculation times increases with the number of terms. Anew criterion is introduced which reduces the computational time by up to a factorten for the panels studied. For all the double-leaf systems analysed, the sound trans-mission loss predictions from the periodic model with the space harmonic expansionmethod are shown to compare well with laboratory measurements.

vii

Dissertation

The work presented in this Doctoral thesis was carried out within The Centre for ECO2

Vehicle Design at the Department of Aeronautical and Vehicle Engineering, The RoyalInstitute of Technology (KTH) in Stockholm, Sweden.

This thesis consists of two parts. The first part gives an overview of the research witha summary of the performed work. The second part collects the following scientificarticles:

Paper A. Leping Feng and Sathish Kumar, On Application of radiation loss factor inthe prediction of sound transmission loss of a honeycomb panel. International Journal ofAcoustics and Vibration, 17 (2012) 47-51.

Paper B. Sathish Kumar, Leping Feng and Ulf Orrenius, Predicting the sound trans-mission loss of honeycomb panels using the wave propagation approach. Acta Acusticaunited with Acustica, 97 (2011) 869-876.

Paper C. Sathish Kumar, Leping Feng and Ulf Orrenius, Modelling the sound trans-mission through rib-stiffened double-leaf partitions with cavity absorption. Submittedto Journal of Sound and Vibration, January 2012.

Paper D. Sathish Kumar, Leping Feng and Ulf Orrenius, Modelling the sound trans-mission through rib-stiffened sandwich double-leaf partitions using space harmonicanalysis. Submitted to Journal of Sound and Vibration, May 2012.

ix

Contents of this thesis has been presented in the following

conferences

Parts of Paper A has been presented at the following conference:

• The 17th International Congress on Sound and Vibration, Cairo, Egypt, July 2010.

Parts of Paper B has been presented at the following conference:

• The 37th International Congress and Exposition on Noise Control Engineering,Shanghai, China, October 2008.

Parts of Paper C has been presented at the following conference:

• Noise and Vibration: Emerging Methods, NOVEM- 2012, Sorrento, Italy, April2012.

Division of work between the authors

Paper A. Sathish performed the measurements and provided data for computations,Leping performed the computations and wrote the paper.

Paper B. Sathish performed the measurements, computations and wrote the paperwith inputs from Leping and Ulf.

Paper C. Sathish developed the model, performed the computations and wrote thepaper with inputs from Leping and Ulf.

Paper D. Sathish developed the model, performed the measurements, computationsand wrote the paper with inputs from Leping and Ulf.

x

Acknowledgments

The work presented in this thesis has been carried out within the Centre for ECO2 Vehi-cle Design at the Department of Aeronautical and Vehicle Engineering, KTH, Sweden.This thesis is a part of the research project “Coupling shape-vehicle body structure”within the centre. The financial support provided by Vinnova, KTH and the industrialpartners is gratefully acknowledged.

To begin with, I would like to thank my main supervisor, Leping Feng for acceptingme as a PhD student and for his guidance, support and comments since the start of thisthesis. In addition many thanks to my co-supervisor Ulf Orrenius for his support, com-ments on this work and for proofreading the manuscript of the papers and this thesis.I would like to thank Per Wennhage for the useful discussions during the earlier stagesof this thesis. I would also like to thank Gunnar Ziwes from EURO-COMPOSITES R©

S.A. for providing the honeycomb panels investigated in this thesis.The measurements in this thesis wouldn’t be possible without the help of Kent

Lindgren and Danilo Prelevic. Computer support from Urmas Ross is also gratefullyacknowledged.

I also wish to express my gratitude to all my colleagues at the department andwithin the ECO2 centre for providing a great working environment during the last 5years. Thanks, in particular to Ciarán, Dmitry, Hans, Hao, Heiki, Jia, Tomas and Tristan.Thank you guys for the useful and sometimes not-so-useful lunch time discussions.Special thanks to Chenyang for introducing me to LYX and making my life easier whilewriting this thesis.

To my family who has always stood by and supported me for all these years won-dering when I would finish “school”, I want to thank you. I would also like to thankmy friends in Stockholm for making my stay pleasant for all these years.

But first and foremost, I would like to thank my wonderful wife Sharenya whoselove and moral support made this thesis possible. Thank you Kutty!

Sathish Kumar

Stockholm, June 2012

xi

Contents

I Overview and Summary 1

1 Introduction 3

2 Floor constructions in trains 72.1 Floating floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Sandwich constructions and the sandwich effect . . . . . . . . . . . . . . 8

2.3 Honeycomb panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Damping treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Sound transmission through honeycomb panels 113.1 Wave propagation in sandwich structures . . . . . . . . . . . . . . . . . . 11

3.2 Sound transmission through orthotropic plates . . . . . . . . . . . . . . . 13

3.3 Radiation loss factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Sound transmission through double-leaf partitions with periodic stiffeners 174.1 Space harmonic analysis - Isotropic panels . . . . . . . . . . . . . . . . . . 18

4.2 Space harmonic analysis - Sandwich panels . . . . . . . . . . . . . . . . . 22

4.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.3.1 Traditional convergence criterion . . . . . . . . . . . . . . . . . . . 25

4.3.2 New convergence criterion . . . . . . . . . . . . . . . . . . . . . . 26

5 Measurements 295.1 Loss factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Sound reduction index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.3 Vibration velocity level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.4 Bending stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6 Results and discussion 336.1 Honeycomb panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2 Double-leaf partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xiii

7 Conclusions and future work 51

8 Summary of appended papers 538.1 Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.2 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.3 Paper C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548.4 Paper D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

II Appended papers 65

Part I

Overview and Summary

1

Chapter 1

Introduction

A comfortable sound and vibration environment in passenger trains is an importantquality factor. On the other hand, obtaining low noise and vibration levels is oftenexpensive and can add weight to the train. Manufacturers work hard to improve thecomfort standards of trains, while at the same time try to keep the costs down. Thedesign of rail vehicles is driven by a number of functional requirements e.g load carry-ing capacity, acoustic and thermal insulation, good aerodynamics, interior and exterioraesthetics. The components and parts used in trains have been and still being designed,produced and assembled separately, each fulfilling different functions.

An alternate approach to this conventional method is multifunctional design. Theidea of multifunctional design is to design/select a component so that it can have mul-tiple functionalities which can reduce the number of total components. For example, amultifunctional wall or a floor gives more space, higher comfort, lower weight, acous-tic and thermal insulation while at the same time providing mechanical protection.During the design of a multifunctional panel, solutions have to be obtained for severaldesign criteria such as static and dynamic stiffness, thermal insulation, acoustic insu-lation, partition thickness, weight and production costs. In the final design, the vari-ous functional requirements should be met while keeping the material and productioncosts low and avoiding overly complex structures.

One of the main outcomes of a multifunctional panel is the weight reduction. Weightreduction leads to reduced fuel consumption and reduced wear of the equipment.However, an early study showed that it is difficult to comply with the noise and vi-bration requirements in rail vehicles using a light weight structure [3]. It is thereforeimportant to take into account the various aspects of the functions and the constraintswhile designing a multifunctional panel. According to Carlsson [3], the various noisetransmission paths into a passenger compartment for both air-borne and structure-borne noise emissions are as shown in Figure 1.1.

3

Sathish Kumar

Figure 1.1: Noise and vibration transmission paths in a railway compartment [3].

1© Air-borne noise through floors, walls, windows, roofs and auxiliary equipmentslike fans, motors, gears, HVAC units.

2© Structure-borne noise from bogie, diesel engines.

Air-borne sound often originates from external sources and propagates into thevehicle interior through the floor, wall panels and also through the holes in the body-work, door seals, etc. Whereas, structure-borne sound is the result of mechanical vi-brations propagating through the vehicle structure and eventually causing localiseddisplacements in air. The noise from wheel-rail region is often the major source of bothair-borne and structure-borne sound. On vehicles with underfloor diesel engines, noisefrom the engine can be significant and so is the noise from air conditioning systems.Vibrational energy from these sources can be transmitted to the passenger compart-ment. Thus, the floor structure of a passenger compartment is expected to possessgood sound insulation properties. For most passenger trains, floating floors are ap-plied to obtain sufficient noise reduction from the sources under the floor. In addition,lightweight and thin floor designs are desired for increased weight reduction.

The inner flooring of a passenger compartment made of sandwich panels with alu-minium face sheets and honeycomb core material have certain advantages over floorpanels made of wood. For this reason, traditional floor panels made of wood is beingreplaced by light weight sandwich structures. Some of the advantages of these con-

4


structions are low weight, good moisture properties, fire resistance and high stiffness-to-weight ratio etc. As mentioned earlier, one main limitation of such light weight struc-ture is the poor sound insulation property. In the transportation industry, automotive,aircraft and rail vehicles are treated with damping materials to reduce structure-bornesound and its effectiveness mainly depends upon parameters such as materials, loca-tion and size of the damping treatment. The basic principle of damping is to convert vi-bratory energy into heat. Traditional damping treatments using viscous damping lay-ers are typically of two types, unconstrained and constrained layer damping [8]. Due tothe shear deformation occurring in the visco-elastic layer, constrained layer dampingtreatments are known to yield significantly larger system damping compared to uncon-strained layer damping, for the same mass of damping material used [19]. The damp-ing treatments are effective in reducing structural vibration and it is often thought thatreduced vibration results in noise reduction. However, when the radiation efficiencyincreases more than the vibration reduction, the total noise radiated from the struc-ture increases [8, 17, 18]. This increased radiation due to the damping treatment doesnot provide the desired effects on sound insulation. The lack of effective predictivemethods for assessing the sound insulation effects due to added damping on complexindustrial structures leads to excessive use of damping materials. Examples are foundin the railway industry where sometimes the damping material applied per carriage ismore than one ton.

As product development is carried out in a fast pace today, there is a strong needfor validated prediction tools to assist during early design stages. There is a need forcomputationally efficient models for calculating the sound insulation properties of thefloors accounting all practical design aspects of typical train structures e.g. orthotropic-ity of the corrugated plates, short circuiting due to boundary conditions and the effectof stiffeners. In this thesis, prediction models for sound transmission through innerfloor panels used in floor structures are developed and validated. The effect of damp-ing treatment on the sound insulation of inner floors is investigated. Also, a soundtransmission model for rib-stiffened double-leaf partitions is developed and extendedto account for complex design aspects such as orthotropicity, sound absorption in thecavity, stud stiffness and placement. The aim of this thesis is to provide simple analyt-ical models of sufficient accuracy and speed to predict the sound insulation of parti-tions. The models should be useful for quick parameter studies at early design stagesand to be used together with optimisation techniques for better sound proofing in en-gineering structures like airplane fuselage and railway floors. A brief introduction tothese structures is presented in the following chapter.

5

Chapter 2

Floor constructions in trains

2.1 Floating floors

Floating floors are used in vehicles to reduce noise and vibration levels. There are sev-eral studies on the successful use of floating floors for structure-borne noise in the shipand aircraft industry [33, 32, 26, 4]. For passenger trains, floating floors are applied toobtain a sufficient noise reduction from sources under the floor. A typical floating floorconstruction consists of an inner floor isolated from the car body with discrete elasticinner layers (studs) resting on supporting beams (stiffeners) as shown in Figure 2.1.

Figure 2.1: Floating floor arrangement.

For thermal and acoustic insulation, the cavity between the inner floor and the carbody framework’s corrugated plate is filled with glass or mineral wool. Traditionallythe inner floor is made of plywood, but in recent years metal and composite sandwichpanels are increasingly used. Among other properties, sandwich structures providesolutions for lightweight and thin floor designs. A properly designed floating floorshould: (i) Provide a vibration isolation for the inner floor for high frequencies. (ii)Reduce the acoustical power radiated by the inner floor. (iii) Reduce transmission ofair-borne sound into the car.

7

Sathish Kumar

2.2 Sandwich constructions and the sandwich effect

Sandwich structures due to their low density in combination with high flexural stiff-ness are extremely popular in aerospace and marine applications where weight is amajor issue e.g. in commercial planes, pleasure boats, space shuttles and satellites. Inground transportation they are increasingly found in cars, buses and trains. A classicalsandwich structure consists of two stiff, strong, thin face sheets bonded to either side ofa relatively thick, weaker, light weight core material. The faces are usually made froma high performance material such as steel, aluminium or fibre composite, whereas thecore is usually a structural solid foam, balsa wood or honeycomb (this can again bemade of aluminium, kraft paper, etc). The structural properties of the face sheets andthe core are less significant as individual panels, but when glued together to form asandwich, they produce a structure of high stiffness and high strength-to-weight ratio,a property which is of great interest to the industry.

The good stiffness properties of a sandwich construction can be illustrated by thefollowing example. A structure made up of a homogeneous material with a givenYoung’s modulus and strength having unit width and thickness ′t′ will have a cer-tain bending stiffness which is normalised as 1. Then the beam is cut into two halvesof thickness ′t/2′ and a core material of thickness ′2t′ is bonded between these twohalves. The corresponding stiffness and strength is ′12′ and ′6′ times more than thehomogeneous beam respectively. This is called the sandwich effect. The core material isassumed to have a surface density much lower than the face sheets and therefore anyaddition in weight to the structure is considered negligible.

t/2

t/2

t

2t

4t

Weight Flexural

Rigidity

Bending

Strength

1

~1

~1

1

12

48

1

6

12

Figure 2.2: The sandwich effect [42].

8


2.3 Honeycomb panels

A honeycomb panel is a lightweight sandwich panel with a honeycomb core. Honey-comb cores can be made of kraft paper, aluminium alloy, glass/phenolic and Nomex R©.In the transportation industry, honeycomb panels with Nomex R© cores are standard inaircraft floor structures, while aluminium cores are increasingly common in railwaycar body structures as well as for certain building elements. For example, in the rail-way car bodies, honeycomb panels are used in door panels, but they are also replacingconventional wooden interior floors due to their resistance to moisture and also to al-low for electrical floor heating systems. Honeycomb cores can be manufactured in avariety of cell shapes, but the most commonly used shape is the hexagonal shape asshown in Figure 2.3. In this thesis, three different honeycomb panels with aluminiumalloy cores are investigated. The panels are differentiated by their core thickness andcore structures as shown Figure 2.3 with dimensions and properties as shown in Table2.1. The panels with 6.4 mm cell were selected to represent a standard design solutionfor railway floor structures and the panel with 19.2 mm was chosen to study the effectof a softer core.

Figure 2.3: Honeycomb panel layout with idealised core.

9

Sathish Kumar

Honeycombcell size (mm)

t f /tc/t f(mm)

Surface densityof HCP (kg m-2)

Panel 1 6.4 1.5/18/1.5 10.4

Panel 2 6.4 1/10/1 6.8

Panel 3 19.2 1/10/1 6.2

Table 2.1: Properties of panels tested.

2.4 Damping treatments

Damping treatments are used to dissipate vibrational energy to control vibrations andnoise. Traditional damping treatments using viscous damping layers are typically oftwo types, unconstrained and constrained layer damping [8]. Constrained layer damp-ing (CLD) treatments have provided an effective means to impart damping to thestructure [2]. Due to the shear deformation occurring in the visco-elastic layer, CLDtreatments are known to yield significantly larger system damping compared to un-constrained layer damping, for the same mass of damping material used [19]. Thoughthese damping materials are effective to attenuate vibration, their effect on the air-borne sound transmission is often limited due to the increased radiation efficiency[8, 17, 18]. In this thesis, the damping treatment applied to the inner floor panels con-sists of a 1 mm thick visco-elastic layer with a constraining layer.

Assuming the core to be weak, the bending stiffness of a sandwich plate is mainlyinfluenced by the face separation distance. Panel 1 having a face separation distance of18 mm has a high initial flexural stiffness. Therefore, a 1 mm thick steel plate (surfacedensity 7.8 kg m-2) was used as a constraining layer for panel 1 and 1 mm thick alu-minium plate (surface density 2.7 kg m-2) was used for panel 2 and panel 3 which hadrelatively low initial bending stiffness. The damping materials can be easily glued tothe honeycomb panel as shown in Figure 2.4.

Figure 2.4: Constrained damping layer attached to a honeycomb panel.

10

Chapter 3

Sound transmission throughhoneycomb panels

3.1 Wave propagation in sandwich structures

Sandwich structures have been the subject of many studies, a large amount of litera-ture has been devoted to the development of theories to study their static and dynamicbehaviours using analytical and numerical methods [20, 15, 9, 10, 31, 29, 25, 34]. Acous-tic analysis of sandwich panels using the classical method was pioneered by Kurtzeand Watters [20] and later by Dym and Lang [9]. Kurtze suggested that using a sand-wich panel can provide better sound insulation between partitions than a homogenouspanel of equivalent mass. Kurtze’s model for sound transmission is based on simplesandwich theory where the effect of core shear is included. Dym [9] improved the accu-racy of Kurtze’s model and later corrected his model [10] by taking into considerationthe acoustic excitation on the lower face sheet of a sandwich panel. Apart from theclassical methods discussed above, statistical energy analysis (SEA) [25] is often used.Commercial softwares like VA One based on SEA and NOVA based on Transfer Ma-trix Method (TMM) can predict the sound transmission loss (STL) of sandwich panelswhen the panel properties are available. A comprehensive assessment of the abovementioned sandwich models has been performed by Wang et al. [40].

A more general approach is given by Nilsson [31] based on the bending of sand-wich beams. The sound insulation property of a sandwich panel is governed by theproperties of the constituent materials, the boundary conditions and the wave mo-tions in the panel. Nilsson’s method is based on classical equations of motion for wavepropagation, as well as theories for excitation and radiation from plates. The generalwave equation is used to describe the displacement in the core and the laminates aretreated as thin Kirchhoff plates while the influence of boundaries are not discussed.

11

Sathish Kumar

The wavenumbers in a sandwich plate is an indicator of the response of the plate to anexternal excitation. In particular, the waves determining the lateral bending of the plateis of importance, since these have the greatest influence on excitation and radiation. Inthe low-frequency limit, the propagation constants are equal to

kB =4

√ω2m

D. (3.1)

Where ω is the angular frequency, m is the surface density of the panel, kB andD are the bending wavenumbers and the apparent bending stiffness of the sandwichpanel respectively. For increasing frequencies, the lateral motion of the structure can nolonger be described by bending alone, the shear and rotation of the core will influencethe deflection of the plate and the apparent bending stiffness of the plate. For thinwalled faces it can be assumed that the displacement of the facesheet is determined bythe flexural and longitudinal waves and the displacement in the thick core is due tobending, rotation, shear and as well as longitudinal deflection.

x

y1E ,

2t

1E , 1t

1t

2G ,2E ,

ip rp

tp

θ

Figure 3.1: Geometry and material parameters of a sandwich panel with an isotropiccore.

For a sandwich panel with symmetric face sheets and an isotropic core as shown inFigure 3.1, the wavenumbers can be derived as a solution to the following expression[31],

F(k) = (U1 −U2) (V3 −V4)− (U3 + U4) (V1 + V2) . (3.2)

The functions U and V are given in Appendix- Part A. A detailed account aboutderiving these functions can be found in the original Reference [31]. The solutions fork are obtained when F = 0.

Due to the manufacturing of honeycomb cores, the cell wall thickness doubles in thedirection of cell orientation and the actual shape of the honeycomb cell can be irregularmaking the core anisotropic as shown in Figure 3.2.

For simplicity, the honeycomb core can be treated as orthotropic and the whole

12


(a) Regular (b) Irregular

Figure 3.2: Honeycomb core shapes.

panel can be treated as an orthotropic sandwich panel, if the frequency concerned isnot too high. The complex wavenumbers kBx and kBz in the x- and z- directions (alsocalled L- and W- directions) as shown in Figure 3.3 can be solved by using an aver-aged value of the loss factor measured between 100 Hz and 1000 Hz and by applyingdifferent values for the core shear modulus G2x and G2z for the x− and z− directionsrespectively. Along with the geometrical properties, the parameters in Table 3.1 areused to calculate the wavenumbers. The shear moduli in Table 3.1 is supplied by thepanel manufacturer.

Panels G2x(Pa) G2z(Pa) Loss factorHCP

Loss factorHCP + CLD

Poisson’sratio of core

Panel 2 430× 106 220× 106 0.005 0.04 0.33

Panel 3 201× 106 54× 106 0.005 0.04 0.33

Table 3.1: Parameters used for wavenumber calculations.

3.2 Sound transmission through orthotropic plates

For an orthotropic panel as shown in Figure 3.3, the lateral displacement is governedby [17],

Dx∂4W∂x4 + 2Dxz

∂4W∂x2∂z2 + Dz

∂4W∂z4 −ω2mW = jωΦ, (3.3)

where Dx and Dz are the bending stiffness in x− and z− directions respectively andDxz can be approximated by Dxz ≈

√DxDz.

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Sathish Kumar

Figure 3.3: Geometry of a sandwich panel with an anisotropic core.

Dx and Dz can be defined as the bending stiffness of a honeycomb structure havingthe same dynamic properties as a simple homogeneous beam at certain frequencies.According to Heckl [17], for a given angle of incidence θ, the transmission coefficientof an infinite orthotropic plate is given by

τ (θ, φ) =

∣∣∣∣∣∣1 + jωm cos θ

2ρ0c0

1−(

sin2 φ

k2Bx

+cos2 φ

k2Bz

)2

k40 sin4 θ

∣∣∣∣∣∣−2

, (3.4)

where kBx = 4√

ω2m/Dx and kBz = 4√

ω2m/Dz are the wavenumbers in x− andz−directions respectively. For small loss factors, the damping can be included by re-placing Dx and Dz in Equation 3.4 with Dx(1 + jη) and Dz(1 + jη) respectively [17].Assuming the same loss factor for waves in all the directions in the panel, Equation 3.4can be re-arranged as

τ (θ, φ) =

[

1 + ηωm cos θ

2ρ0c0

k40 sin4 θ

k4φ,eqv.

]2

+

[ωm cos θ

2ρ0c0

(1−

k40 sin4 θ

k4φ,eqv.

)]2−1

, (3.5)

where kφ,eqv. is the equivalent wavenumber in direction φ given by

1k2

φ,eqv.=

sin2 φ

k2Bx

+cos2 φ

k2Bz

. (3.6)

The diffuse field sound transmission coefficient over all angles of incidence θ and trans-mission φ is calculated by [36]

14


τ =

´ 2π0´ θlim

0 τ (θ, φ) sin (θ) cos (θ) dθdφ´ 2π0´ θlim

0 sin (θ) cos (θ) dθdφ, (3.7)

where θlim, is the limiting angle above which it is assumed that no sound is incidenton the partition and according to Mulholland et al. [30] θlim can be limited to 78◦ . Thesound transmission loss is then calculated by

STL = 10 log10

(1τ

). (3.8)

3.3 Radiation loss factor

Radiation loss factor is a parameter indicating to what extent the vibration of a struc-ture is damped due to radiation to the surrounding fluid. According to Cremer et al.[8], the radiation loss factor can be defined as

ηrad =ρ0c0σ

ωm, (3.9)

where ηrad is the radiation loss factor and σ is the radiation efficiency which hasbeen investigated by Beranek [2] and Maidanik [27]. According to Leppington et al.[24], a better estimation of the averaged radiation efficiency of a rectangular plate canbe given as

σ̄ ≈ a + bπµkab (µ2 − 1)

{ln(

µ + 1µ− 1

)+

2µ

µ2 − 1

}µ > 1,

σ̄ ≈√

kaH(x) µ = 1, (3.10)

σ̄ ≈ 1/√

1− µ2 µ < 1,

where a and b are the width and length of the plate with a < b, µ = kB/k =√

fc/ fwith fc being the critical frequency of the plate, and H(x) is a function of the panelaspect ratio (a/b) and in our case it can be approximated to 0.41 [24].

For an isotropic panel, there is one critical frequency and the radiation loss factorcan be applied as in Reference 31. But for orthotropic panels, there exists different crit-ical frequencies depending on the transmission angle φ. For anisotropic panels, thereexists no expressions for calculating the wavenumbers and critical frequency as a func-tion of φ. Since the honeycomb panel is treated as being orthotropic , the equivalentcritical frequency in the ribbon (x) and transverse (z) directions can be calculated using

15

Sathish Kumar

the equivalent wavenumber from Equation 3.6 and the related radiation loss factor canbe obtained.

The loss factor is normally measured in 1/3rd octave bands because of the largefluctuations in the measurements. However, Feng and Kumar [13] showed that by us-ing a measured loss factor in 1/3rd octave band, the sound transmission loss will beunderestimated around the critical frequency because of the extremely large radiationloss factor concentrated in a very small frequency band as indicated in Equation 3.10.As a consequence, the radiation loss factor calculated in narrow band has to be addedto the measured loss factor for better sound transmission loss predictions around thecritical frequency [13]. The total loss factor can be expressed as sum of the measuredloss factor and the radiation loss factor.

ηtotal = η0 + 2ηrad. (3.11)

16

Chapter 4

Sound transmission throughdouble-leaf partitions with periodicstiffeners

The transmission loss of regular double-walls is extensively covered in the literature[27, 6, 7, 21, 39, 22, 41]. For simpler structures like single- or double-leaf partitionsmade of homogenous panels, the wave approach by Cremer et al. [8] and Fahy [11]gives a good estimate of the sound transmission loss. The advantage of this method isthat it can be applied to a wide frequency range and takes dominant physical phenom-ena into account, like the wave coincidence and mass law effects by relatively simplecalculations. A limitation of this wave approach is that it cannot be applied to non-homogenous panels and for partitions with stiffeners [35] and hence cannot be appliedto typical engineering structures found in rail vehicles, aircraft and ships. Statisticalenergy models of double-leaf partitions are frequently used due to their simplicity andcomputational efficiency [6, 38, 5]. Novel work by Finnveden and Barbagallo [14] al-lows a physically correct description of the wave transmission at lower frequenciesaccounting for double-wall resonance, but models of rib-stiffened structures need tobe developed further. In particular, detailed effects of parameter variations of studstiffness and placement, stiffener geometry and layout are not readily available withcomputationally efficient models.

In this thesis, space harmonic analysis is used to investigate rib-stiffened double-leaf partitions. The structural response and acoustic pressure of a periodic structuredue to a harmonic excitation has been expressed in terms of a series of space harmon-ics by Mead and Pujara [28]. In their work, the panel was modelled as a beam restingon elastic supports resisting both transverse and rotational displacements. This modelpredicted the structural response due to an acoustic pressure. However, it failed to

17

Sathish Kumar

Figure 4.1: Double-leaf partition with stiffeners and absorbing materials in cavity.

take into account the effect of the structural response on the acoustic system. Lee andKim [21] adopted the method of Mead and Pujara to a 2-D single plate with stiffenersand included the effect of structural response on the acoustic system. The stiffenerswere modelled as periodically distributed lumped masses mounted to the two pan-els, together with a series of uniformly distributed translational and rotational springsand a detailed approach was presented to calculate the sound transmission loss of thestructure. As an extension of Lee’s work [21], Wang et al. [39] studied double-walledrib-stiffened panels and following Lee, Wang disregarded many complicating features(e.g. sound absorbing material in the cavity). In this thesis, Wang’s model has beenextended to include the effects of different fluid properties in the centre cavity whichaccounts for the sound propagation through an absorbing material. Also, the model ismodified so that orthotropic sandwich panels can be included as a part of the double-walled partition. Naturally, the same model can be simplified and used to investigatethe sound transmission properties of a double-leaf partition with standard sandwichpanels.

4.1 Space harmonic analysis - Isotropic panels

A double-walled structure can be modelled with periodic boundary conditions as shownin Figure 4.2a. For simplicity, the system is considered to be infinite in the directionof the stiffeners and the incident of harmonic wave is limited only to the horizontalplane. The panels are considered isotropic with thickness hi, and mass per unit areampi of panel i (where i = 1, 2). The two panels are linked with five vertical “C” shapedaluminum channels. These stiffeners are modelled as periodically spaced translationaland rotational springs and their weights M

′, are equally distributed to the two pan-

els. A stiffener of depth l, web thickness to, Young’s modulus E can be modelled as aspring of translational stiffness K′t = Et0/l per unit length of stiffener [39]. Similarly,the rotational stiffness is estimated by considering the web bending of the stud and is

18


given by K′r = EIz/l where Iz is the second moment of area about the z-axis. For “C”shaped stiffeners studied in this thesis, Iz = t3

o/12.

(a) Periodic representation (b) Notation used in the analysis

Figure 4.2: Cross-section of a double-leaf panel with stiffeners.

The plane wave pi, incident on the periodic panel with stiffeners induces a reflectedwave pr, a transmitted wave pt and the panel motion. Since the structure is periodic,the system response is also expected to be periodic [28], the motion of each panels canbe expressed in terms of space harmonic expansions. The plane wave is assumed toincident along the x-y plane at an angle θ as shown in Figure 4.2b and the incidentwavenumber components kx and ky are given as

kx = k sin θ , ky = k cos θ , (4.1)

where k = ω/c0 and ω is the angular frequency (2π f ) and c0 is the speed of sound inair. Similarly, the velocity potentials in the incident, cavity and the transmitted spacecan be expressed in terms of space harmonic series [28]. The wavenumber in the y-direction in, kyn is given by the expression

kyn =

√(k)2 −

(kx +

2nπ

L

)2

. (4.2)

Since the cavity is filled with an absorbing material, the wavenumber in the y-directionin the cavity, kync is given by the expression

19

Sathish Kumar

kync =

√(kc)

2 −(

kx +2nπ

L

)2

. (4.3)

where kc is the complex wavenumber in the absorbing material and is related to thedynamic density ρc(ω) and the dynamic bulk modulus Kc(ω) of the absorbing materialas

kc = ω

√ρc(ω)

Kc(ω). (4.4)

The dynamic density and the dynamic bulk modulus are obtained through Allard-Champoux model for rigid frame materials [1]

ρc(ω) = ρ0

[1 +

1j2π

(σ

ρ0 f

)G1

(ρo fσ

)], (4.5)

Kc(ω) = γP0

(γ− γ− 1

1 +(1/j8πNpr

)(ρo f /σ)−1 G2(ρ0 f /σ)

)−1

. (4.6)

The functions G1(ρ0 f /σ) and G2(ρ0 f /σ) are given by G1(ρ0 f /σ) =√[1 + jπ (ρ0 f /σ)]

and G2(ρ0 f /σ) = G1[(ρ0 f /σ) 4Npr

], where ρ0 is the density of air, σ is the static air

flow resistivity, γ is the specific heat ratio, P0 is the air equilibrium pressure and Npr

is the Prandtl number. A real part of the wavenumber greater than unity indicates alower speed of sound through the absorbing material and a negative imaginary partindicates the dissipation of energy. Applying the boundary conditions at the fluid-panel interface, the modal amplitudes can be found in terms of the amplitudes of thespace harmonics. The amplitudes of the space harmonics can be found by applying theprinciple of virtual work as explained in Appendix-Part B. Making use of the couplingrelations between the modal amplitudes of waves in air and the flexural motion ofpanel, the following relations are obtained.

20


D1

(kx +

2mπ

L

)4

−mp1ω2 +jω2ρ0

kyn−

jω2ρc

(1 + e2jkync H

)kync

(1− e2jkync H

) α1,m

+(

K′t −ω2M

′)( +∞

∑n=−∞

α1,n

)+ K

′r

[+∞

∑n=−∞

α1,n

(kx +

2nπ

L

)](kx +

2mπ

L

)

−K′t

(+∞

∑n=−∞

α2,n

)− K

′r

[+∞

∑n=−∞

α2,n

(kx +

2nπ

L

)](kx +

2mπ

L

)

+2jω2ρcejkync H

kync

(1− e2jkync H

)α2,m =

2jωρ0 I, f or m = 0,

0, m 6= 0

(4.7)

D2

(kx +

2mπ

L

)4

−mp2ω2 +jω2ρ0ejkyn H

kyn− 2jω2ρcejkync H

kync

(1− e2jkync H

)− α2,m

+(

K′t −ω2M

′)( +∞

∑n=−∞

α2,n

)+ K

′r

[+∞

∑n=−∞

α2,n

(kx +

2nπ

L

)](kx +

2mπ

L

)

−K′t

(+∞

∑n=−∞

α1,n

)− K

′r

[+∞

∑n=−∞

α1,n

(kx +

2nπ

L

)](kx +

2mπ

L

)

+jω2ρc

(1 + e2jkync H

)kync

(1− e2jkync H

) α1,m = 0.

(4.8)

These equations are solved for αi,m which can be used to find the modal amplitudes.The power transmission coefficient is then calculated by

τ (θ) =∑n=+∞

n=−∞ |ξn|2Re(kyn)

|I|2ky. (4.9)

The diffuse field sound transmission coefficient over all angles of incidence θ withinthe incident plane is calculated by [36]

τ =

´ θlim0 τ (θ) sin (θ) cos (θ) dθ´ θlim

0 sin (θ) cos (θ) dθ. (4.10)

21

Sathish Kumar

4.2 Space harmonic analysis - Sandwich panels

The double-leaf partition with isotropic panels discussed in the previous section couldbe ideal for walls in building constructions, but not as floors e.g. in trains because oftheir poor load carrying capacity. By replacing the homogenous panels with sandwichmaterials, the load carrying capacity of such partitions can be increased significantlywithout adding additional weight. The model presented in the previous section hasbeen extended to replace the isotropic panels with sandwich panels. Specifically, sand-wich panels with different properties in the orthotropic directions are studied in thissection.

Following the methodology for isotropic plates in the previous section, the double-leaf partition as shown in Figure 4.3a with orthotropic sandwich panels connected bystiffeners can be modelled with periodic boundary conditions . For simplicity, the sys-tem is considered to be infinite in the direction of the stiffeners and the angle of inci-dence of harmonic wave is limited only to the horizontal plane. The panels are consid-ered orthotropic with thickness hi, mass per unit area mpi and Young’s moduli Exi, Ezi

of panel i (where i = 1, 2 and x and z are the orthotropic directions). The stiffeners aremodelled as periodically spaced translational and rotational springs with their massesM′, equally distributed on the two panels. The translational and rotational stiffness of

the beam shaped stiffeners can be estimated using the formulas in the previous section.

(a) Periodic representation (b) Notation used in the analysis

Figure 4.3: Cross-section of a double-leaf panel with stiffeners.

The plane wave pi, incident on the periodic panel with stiffeners induces a reflectedwave pr, a transmitted wave pt and the panel motion. Since the structure is periodic,

22


the system response is also expected to be periodic [28], the motion of each panels canbe expressed in terms of space harmonic expansions. The plane wave is assumed toincident along the x-y plane at an angle θ as shown in Figure 4.3b and the incidentwavenumber components kx , ky and kz are given as

kx = k sin θ cos φ , ky = k cos θ , kz = k sin θ sin φ , (4.11)

where k = ω/c0 and ω is the angular frequency (2π f ) and c0 is the speed of sound inair. The wavenumbers in the y-direction, kyn and kync are given by the expressions

kyn =

√(k)2 −

(kx +

2nπ

L

)2

− (kz)2 and (4.12)

kync =

√(kc)

2 −(

kx +2nπ

L

)2

− (kz)2. (4.13)

Applying the boundary conditions at the fluid-panel interface, the relationship be-tween modal amplitudes and the amplitudes of the space harmonics are obtained. Ap-plying the principle of virtual work, making use of the coupling relations between themodal amplitudes of waves in the fluid and the flexural motion of panel, the followingrelations are obtained.

[Dx1

(kx +

2mπ

L

)4

+ Dxz1

(kx +

2mπ

L

)2

(kz)2 + Dz1 (kz)

4 −mp1ω2

+jω2ρ0

kyn−

jω2ρc

(1 + e2jkync H

)kync

(1− e2jkync H

) α1,m +

(K′t −ω2M

′)( +∞

∑n=−∞

α1,n

)

+K′r

[+∞

∑n=−∞

α1,n

(kx +

2nπ

L

)](kx +

2mπ

L

)− K

′t

(+∞

∑n=−∞

α2,n

)

−K′r

[+∞

∑n=−∞

α2,n

(kx +

2nπ

L

)](kx +

2mπ

L

)

+2jω2ρcejkync H

kync

(1− e2jkync H

)α2,m =

2jωρ0 I, f or m = 0,

0, m 6= 0

(4.14)

23

Sathish Kumar

[Dx2

(kx +

2mπ

L

)4

+ Dxz2

(kx +

2mπ

L

)2

(kz)2 + Dz2 (kz)

4 −mp2ω2

+jω2ρ0ejkyn H

kyn− 2jω2ρcejkync H

kync

(1− e2jkync H

)− α2,m +

(K′t −ω2M

′)( +∞

∑n=−∞

α2,n

)

+K′r

[+∞

∑n=−∞

α2,n

(kx +

2nπ

L

)](kx +

2mπ

L

)− K

′t

(+∞

∑n=−∞

α1,n

)

−K′r

[+∞

∑n=−∞

α1,n

(kx +

2nπ

L

)](kx +

2mπ

L

)

+jω2ρc

(1 + e2jkync H

)kync

(1− e2jkync H

) α1,m = 0.

(4.15)

Where, Dxi and Dzi are the bending stiffness of the sandwich panels in two orthotropicdirections and Dxzi can be approximated by Dxzi ≈

√DxiDzi. For a honeycomb panel,

Dxi and Dzi can be obtained using the method described in Chapter 3.

These equations are solved for αi,m which can be used to find the modal amplitudes.The power transmission coefficient is then calculated by

τ (θ, φ) =∑n=+∞

n=−∞ |ξn|2Re(kyn)

|I|2ky. (4.16)

The diffuse field sound transmission coefficient over all angles of incidence θ and trans-mission φ is calculated by [36]

τ =

´ 2π0´ θlim

0 τ (θ, φ) sin (θ) cos (θ) dθdφ´ 2π0´ θlim

0 sin (θ) cos (θ) dθdφ. (4.17)

Where θlim, the angle above which it is assumed that no sound is received is limited to78◦ [30]. The sound transmission coefficients from Equations 4.10 & 4.17 can be usedto calculate the sound transmission loss by

STL = 10 log10

(1τ

). (4.18)

The model in this thesis is developed for a double-leaf partition made of two or-thotropic sandwich panels but naturally, isotropic material properties can also be ap-plied.

24


4.3 Convergence

Since the solution to this space harmonic analysis is obtained in a series form, a conver-gence check is performed to establish the number of terms required for accurate soundtransmission loss predictions.

4.3.1 Traditional convergence criterion

The sound transmission loss is calculated using a simple algorithm adding one termat a time to the assumed expansion solution for selected frequencies (e.g. f = 200 Hz,1000 Hz, 5000 Hz, & 10000 Hz). The solution is considered to have converged when theSTL difference between two successive calculations is less than a preset error of 0.01dB. Figure 4.4 shows an example of how the calculated STL changes with increasingthe number of terms and the convergence associated with the selected frequencies. Atlow frequencies few terms are enough to ensure convergence, but the number of termsrequired increases with frequency. Following Lee and Kim [21], it can be concluded thatwhen a solution converges at a given frequency, it converges for all lower frequencies.

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

Number of Terms

ST

L (d

B)

200 Hz1000 Hz5000 Hz10000 Hz

Convergence line

Figure 4.4: Convergence check for different frequencies.

The standard procedure is to find the number of terms required at the highest fre-quency of interest and thereafter calculate the sound transmission for all other frequen-cies with the same number of terms. The cost of using a fixed number of terms for theseries expansion is the high computation time. To reduce computation time, a new cri-terion is developed here for selecting the space harmonic terms for quick convergence.

25

Sathish Kumar

4.3.2 New convergence criterion

The basis of the new convergence criterion is that the pressure field at a given fre-quency can be expressed with a limited number of propagating and evanescent waves.At low frequencies this number of propagating waves is low, but it increases with fre-quency and a larger number of terms is needed for convergence. In this new methodonly the space harmonics corresponding to the propagating and a few evanescentwaves are included in the solution for each frequency. Neglecting cavity absorption,propagating and evanescent waves can be separated by checking the wavenumbersin the y-direction. For double-leaf partitions with isotropic panels, the wavenumberin the y-direction is given by Equation 4.2. When ω/c0 >| kx + 2nπ/L | the wave ispropagating and when ω/c0 <| kx + 2nπ/L | the wave is evanescent (wavenumberhas only an imaginary part). This is illustrated in Figure 4.5 where the wavenumberkyn is plotted against the space harmonics at selected frequencies. Since the evanescentwaves do not possess a real part, their amplitude is shown as 0.

−40 −30 −20 −10 0 10 20 30 40

0

20

40

60

80

100

120

140

160

180

200

n

real

(k

yn)

1000 Hz5000 Hz10000 Hz

Space Harmonicscorresponding to Propagating waves

Space Harmonicscorresponding toEvanescent waves

Figure 4.5: Real kyn plotted against the space harmonic terms, θ = 0◦ (Old criterion).

For a given frequency f , angle of incidence θ and stiffener separation distance L,the propagating waves are limited to the region nmin ≤ n ≤ nmax where

nmin = −L fc0

(1 + sinθ) , (4.19)

nmax = +L fc0

(1− sinθ) . (4.20)

26


By evaluating α1,n and α2,n in Equations 4.7 & 4.8 with n ranging from nmin tonmax, includes only the propagating waves in the solution. Since the panels are closelyspaced, some evanescent waves do not decay sufficiently and have considerable am-plitudes when reaching the second panel. The space harmonic terms correspondingto these evanescent waves cannot be ignored and therefore, a number of evanescentwaves (∆n) should be included in the solution (n = nmin − ∆n to nmax + ∆n). A nu-merical investigation revealed that evanescent waves with amplitude decays of morethan 60 dB when reaching the second panel, do not significantly affect the sound trans-mission loss and therefore need not be included in the calculation. The old and the newcriterion for convergence can be illustrated in Figures 4.6 & 4.7.

−20−10

010

20

1002000

40006000

800010000

0

50

100

150

200

Space Harmonic termsFrequency (Hz)

real

(k yn

)

Propagating waves

Evanescent waves

(a) Old criterion

−20−10

010

20

1002000

40006000

800010000

0

50

100

150

200

Space Harmonic termsFrequency (Hz)

real

(k yn

)

Propagating waves

Evanescent waves

(b) New criterion

Figure 4.6: Real kyn plotted against the space harmonics and frequency, θ = 0◦.

−20−10

010

20

010

2030

4050

6070

8090

0

20

40

60

80

100

Space Harmonic termsAngle of Incidence (θ)

real

(k yn

)

Propagating waves

Evanescent waves

(a) Old criterion

−20−10

010

20

020

4060

80

0

20

40

60

80

100

Space Harmonic termsAngle of Incidence (θ)

real

(k yn

)

Propagating waves

Evanescent waves

(b) New criterion

Figure 4.7: Real kyn plotted against the space harmonics and angle of incidence, f =5000 Hz.

In Figure 4.6, the wavenumbers are shown as a function of frequency with normalincidence, wave heading (θ = 0◦), whereas in Figure 4.7, the wavenumbers are plottedat 5000 Hz as a function of wave heading angle. For the new criterion, only the spaceharmonics which are included in the solution are plotted against the real part of the

27

Sathish Kumar

wavenumber kyn for different frequencies. The non-zero wavenumbers in the plot cor-respond to propagating waves and the others corresponds to evanescent waves. Thenumber of evanescent waves included in the solution, ∆n = 5.

And for orthotropic plates, from Equation 4.12, it can be determined that the wavesare propagating when ω/c0 >| (kx + 2nπ/L) + kz | and evanescent when ω/c0 <|(kx + 2nπ/L) + kz |. Hence, for a given frequency f , angle of incidence θ, and angle ofpropagation in the plate φ and stiffener separation distance L, the propagating wavesare limited to the region nmin ≤ n ≤ nmax where

nmin = −L fc0

(√1− (sinθsinφ)2 + sinθcosφ

), (4.21)

nmax = +L fc0

(√1− (sinθsinφ)2 − sinθcosφ

). (4.22)

28

Chapter 5

Measurements

5.1 Loss factor

Loss factor is a good measure of the structural damping present in a system. From theseveral methods available, a simple and robust decay method was used to measurethe loss factor of the test panels in 1/3rd octave bands. The test panel was suspendedin springs to achieve free-free boundary conditions minimizing boundary losses apartfrom radiation. For the same reason, impulse excitation was used to avoid unwantedexternal damping due to a shaker mounting. The decay method uses the fact that freevibrations decay with time and by measuring the reverberation time of the structure,the loss factor can be calculated by [8]

η =2.2fnT

, (5.1)

where fn is the frequency and T is the reverberation time in seconds. Here, randomaccelerometer positions and excitation points were used to get a spatial average of themeasured vibration response and to reduce the influence from individual modes.

After the damping layers were attached, the reverberation time decreased signifi-cantly making decay measurements difficult. Therefore, power injection method wasused to measure the loss factor of the structures with damping treatment. This methodis not ideal for complex built-up structures, but works very well for freely hangingsimple structures as in our case. For a point excited system, the input power is relatedto the spatial average of the vibration velocity as P = Sm′ωηv2

∆/2 [8]. This equationcan be re-arranged to calculate the loss factor, as

η =2P

Sm′ωv2∆

, (5.2)

29

Sathish Kumar

where P is the input power and S, m′, v2∆ the area, surface density and the mean

square vibration velocity of the test structure respectively.In practice, using harmonic excitation, the loss factor is determined according to

Feng [12], as

η =−Img(G f a)

Sm′Gaa, (5.3)

with G f a being the cross-spectrum of the excitation force and response accelerationand Gaa being the power spectrum of acceleration.

5.2 Sound reduction index

The sound reduction index (SRI) is a measure of the sound insulation of a partition andwas measured using the sound intensity technique according to ISO 15186− 1 : 2000.The test partition was mounted between a reverberation room and an anechoic roomas shown in Figure 5.1. Four loudspeakers were used to generate white noise in thereverberation room in the frequency range 100 Hz to 5000 Hz to create a diffuse soundfield. A condenser microphone attached to a rotating boom was used to measure theaverage sound pressure level inside the reverberation room. The sound intensity wasmeasured on a plane parallel to the panel surface facing the anechoic room.

Anechoic Room Reverberation Room

Test Panel

Rotating Boom Loud SpeakerIntensity probe

Anechoic Room Reverberation Room

Test Panel

Intensity probe

PC &

Signal Generator

Amplifier

Rotating

Boom

Loud-

speaker

Figure 5.1: Setup for sound reduction index measurement.

The panel was mounted in such a way that flanking transmission was minimised

30


thus ensuring that the sound transmission was only through the test panel. Then theair-borne sound reduction index was calculated as given below

RI = LP − 6− (LI + 10 log (Sm/S)) . (5.4)

Where RI is the sound reduction index, LP is the sound pressure level in the rever-beration room, LI is the sound intensity measured in the anechoic room, Sm is the mea-sured surface area and S is the area of the test specimen. Weighted (apparent) soundreduction index R′W , a single numbered quantity used to describe the sound insulationof partitions was calculated according to ISO 717− 1 : 2006.

5.3 Vibration velocity level

Vibrations in a structure can be quantified by measuring the average vibration veloc-ity levels. The influence of damping treatments on the vibration of a structure canbe seen by comparing the vibration velocity levels of the structure before and afterthe damping treatment. To measure the vibration velocity, the test panel was fixedbetween the reverberation room and anechoic room as in the sound reduction indexsetup. Eight piezoelectric accelerometers were placed on the panel surface facing theanechoic room. Vibration velocity was measured on the panel surface for an acousticexcitation from the reverberation room. The measurement was repeated for differentaccelerometer positions. By taking a spatial average of the vibration velocity from allaccelerometer positions, the vibration velocity level was calculated using the expres-sion below

Lv = 20log10

(< v >

vre f

), (5.5)

where Lv is the vibration velocity level, < v > is the average panel normal velocityand vre f = 10−9 m s-1 is the reference value for vibration velocity.

5.4 Bending stiffness

For homogeneous panels the bending stiffness is frequency independent, but for mul-tilayer panels, like honeycomb panels, the bending stiffness becomes frequency depen-dent. Simplified modal tests of freely hanging honeycomb beams were made to mea-sure their natural frequencies and to calculate their apparent bending stiffness. Sections(2050 mm x 70 mm) of honeycomb panel 2 and panel 3 were cut out in W-direction (Fig-

31

Sathish Kumar

ure 2.3) and suspended using rubber strings to achieve free-free boundary conditions.The measurement setup is shown in Figure 5.2.

BeamPC & DAQ

Rubber StringsCharge Amplifier

Accelerometer

Impact Hammer

Figure 5.2: Setup for bending stiffness measurement.

An impact hammer was used to excite the beam and an accelerometer was usedto measure the response. The inertial effects due to accelerometer mass was neglectedsince the mass of the accelerometer was too small (2.4 grams) compared to the testbeams (975 grams and 889 grams). Different accelerometer positions were selected andmeasured to make sure to find all relevant modes. From the measured natural frequen-cies the bending stiffness was calculated by [42]

Dx =ωnm′L4

xα4

n, Dz =

ωnm′L4z

α4n

, (5.6)

where ωn is the natural angular frequency, L the length of the beam, m′ the beamsurface density, n is the order of the natural frequency and α1 = 4.73, α2 = 7.85, α3 =

11.00 , α4 = 14.14 , αn = nπ + π/2 for n > 4 . From the measured bending stiffnesses,the flexural wavenumbers of the beams were calculated from the relations

kx = 4

√ω2

nm′

Dx, kz =

4

√ω2

nm′

Dz. (5.7)

32

Chapter 6

Results and discussion

6.1 Honeycomb panels

The loss factor of the honeycomb panels were measured using the decay method asexplained in Section 5.1. The measured loss factors for all three honeycomb panelswithout and with the damping treatment are shown in Figure 6.1.

100 125 160 200 250 315 400 500 630 800 10000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency (Hz)

Loss

Fac

tor

Panel 1Panel 1 + CLD

(a) Panel 1

100 125 160 200 250 315 400 500 630 800 10000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency (Hz)

Loss

Fac

tor


(b) Panel 2

100 125 160 200 250 315 400 500 630 800 10000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Frequency (Hz)

Loss

Fac

tor


(c) Panel 3

Figure 6.1: Measured loss factor of the honeycomb panels without and with dampingtreatment.

33

Sathish Kumar

From the figures it can be concluded that there is a significant increase in the struc-tural loss factor with damping treatment for all the three panels. The decay measure-ments were limited to a frequency of 1000 Hz above which, the reverberation time wastoo short to get a useful result. Due to the previously mentioned problem of obtainingloss factor results using the decay method with the damping treatment, power injectionmethod was used later. The principal use of damping treatments is to reduce vibrationsin the structure. To see the effect of damping treatments on vibration attenuation, thevibration velocity level of the panels were compared without and with the dampingtreatment. Figures 6.2a, 6.3a & 6.4a show the averaged vibration velocity level of thepanels to an acoustic excitation without and with the damping treatment. A diffusesound field was created in the reverberation room and it was ensured that the acousticexcitation levels were same for all panels and measurements. As expected, the damp-ing treatments reduce the vibration levels for all three panels. Figures 6.2b, 6.3b & 6.4bshows the vibration level difference of the panels before and after the treatment.

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ratio

n V

eloc

ity L

evel

(dB

)


(a) Lv without and with damping

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Frequency (Hz)

∆ L v (

dB)

(b) ∆Lv

Figure 6.2: Measured vibration velocity level (Lv) of panel 1.

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34


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(b) ∆Lv


The sound reduction index of the panels was measured using the sound intensitytechnique as explained in Section 5.2. The procedure was repeated after the dampingtreatments were applied. The measured sound reduction index of panel 1 without andwith the damping treatment is shown in Figure 6.5. Just as observed in the loss factorresults, the damping treatment increased the sound reduction index of the panel. How-ever, this increase is not entirely due to the damping treatment. A part of this increaseis due to the mass added to the panel due to the damping treatment (masslaw).

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45

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Sou

nd R

educ

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Inde

x (d

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Figure 6.5: Measured sound reduction index of panel 1 without and with damping.

For this reason, mass normalisation was done to remove any effect of added masson the results. This was done by subtracting the mass contribution ∆ = 20 log(m′/m′0)from the measured sound reduction indices in each frequency bands, where m′ and m′0are the mass densities with and without damping layers attached. The measured sound

35

Sathish Kumar

reduction index of panel 1 compared to the mass normalised sound reduction indexwith damping treatment is shown in Figure 6.6a. The result shows that the dampingtreatment has very little influence on the sound reduction index of the panel. Similarly,the sound reduction index of panel 2 and panel 3 are mass normalised and the resultsare shown in Figure 6.6b and Figure 6.6c respectively. As observed for panel 1, thedamping treatment on panel 2 has very little influence. On the other hand, for panel 3a significant increase in sound transmission loss can be seen above 500 Hz.

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(a) Panel 1

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Panel 2Panel 2+ CLD

(b) Panel 2

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(c) Panel 3

Figure 6.6: Measured sound reduction index of the honeycomb panels without andwith damping treatment (mass normalised).

To investigate and quantify this influence, weighted sound reduction index R′W , asingle number quantity characterising the sound insulation of a partition was calcu-lated according to ISO 717− 1 : 2006 and is presented in Table 6.1. For panel 1, thedamping treatment increases the sound insulation by 5 dB whereas the influence fromadded mass is 4.9 dB which shows that the increased losses does not increase soundinsulation. On the other hand for panel 3, the sound insulation increases by 8 dB outof which only 3.9 dB is due to the added mass, clearly showing that damping treat-ment increases sound insulation. From Table 6.1, it can also be inferred that withoutthe damping treatment, panel 2 has a weighted sound reduction index 3 dB more thanpanel 3. But, once the damping material is attached, both panels have the same sound

36


insulation rating. It should be noted that the only structural difference between panel2 and 3 is the honeycomb cell size affecting the shear stiffness of the core.

Panels HCPR′w(dB)

HCP + CLDR′w(dB)

∆(dB)

Panel 1 25 30 4.9

Panel 2 22 27 3.6

Panel 3 19 27 3.9

Table 6.1: Weighted sound reduction index of the honeycomb panels without and withdamping treatment.

The lowest frequency at which the wavelength of bending waves in plate is equal tothe wavelength in air is known as the critical frequency. For a single-leaf isotropic panel,there exists one critical frequency fc whereas, for a honeycomb panel there exists arange of critical frequencies between fcL and fcW (where the subscripts L and W arethe two principal orthotropic directions). This critical frequencies in the honeycombpanels can be better illustrated using dispersion curves, where the wavenumbers ofthe panels are plotted together with the wavenumbers in air.

Using the sandwich theory, the wavenumbers of the honeycomb panels were cal-culated in the two orthotropic directions (L and W) and plotted in Figures 6.7a and6.7b. In these figures, critical frequency is observed where the plate wavenumbers(kLand kW) intersect the wavenumbers in air (kair). In Figure 6.7, the parallel lines arethe asymptotes to the wavenumbers. The lower asymptote (klower) corresponds to thewavenumber for pure bending of the entire construction whereas, the upper asymp-tote (kupper) corresponds to the wavenumber of flexural waves propagating in one ofthe laminate. It can be seen in Figure 6.7b that the wavenumbers in W-direction (kW)matches with the wavenumbers in air (kair) over a large frequency range (Figure 6.8).This implies that the wavelength of the bending wave in the plate is close to the wave-length of sound in air which leads to coincidence over the entire frequency range.

37

Sathish Kumar

125 250 500 1000 2000 4000 8000

101

102

Frequency (Hz)

Wav

e N

umbe

r (1

/m)

kair

klower

kupper

kL

kw

(a) Panel 2

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102

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Wav

e N

umbe

r (1

/m)

kair

klower

kupper

kL

kw

(b) Panel 3

Figure 6.7: Wavenumbers in the panels (predicted) and air.

2000 4000 6000

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102

Frequency (Hz)

Wav

e N

umbe

r (1

/m)

kair

klower

kupper

kL

kw

Figure 6.8: Extended critical frequency range, Panel 3.

Later, the panels were cut in the W-direction into beams (2050 mm x 70 mm) andtheir bending wavenumbers were measured. The predicted wavenumbers are com-pared with the measurements in Figure 6.9 and a good agreement is achieved. Theexact values of the measured and predicted critical frequencies for the two panels areshown in Table. 6.2. The measured and predicted critical frequencies in W-directionwere found to be within a range of 2%.

38


Panels fcL Hzprediction

fcW Hzprediction

fcW Hzmeasurement

Panel 2 792 888 904

Panel 3 851 3632 3556

Table 6.2: Critical frequencies of panel 2 and panel 3 in L- and W-directions.

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e N

umbe

r (1

/m)

AirPanel 2−MeasuredPanel 3−MeasuredPanel 2−PredictedPanel 3−Predicted

Figure 6.9: Measured and predicted bending wavenumbers.

Further, the vibration velocity level difference of the panel is compared with thesound reduction index difference before and after the application of damping. A typi-cal effect of damping treatments can be seen in Figure 6.10a. For panel 1, even thoughsignificant vibration reduction is achieved with damping treatment, the sound radiatedfrom the structure is not reduced to the same extent. For panel 2, the increased soundreduction index is approximately equal to the vibration reduction as seen in Figure6.10b. On the other hand for panel 3 in Figure 6.10c, an interesting phenomenon is ob-served. For panel 3, the increase in sound reduction index is more than the vibrationreduction at certain frequencies. The reason for this effect is the extended coincidencerange for panel 3 as observed before. It is well know that damping is more effective inand around the critical frequency. Naturally, when damping treatments are applied toa panel with a wide critical frequency range, their effect is greater than for a standardpanel.

39

Sathish Kumar

125 250 500 1000 2000 40000

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Frequency (Hz)

∆ (d

B)

∆ L

v

∆ SRI

(a) Panel 1

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∆ (d

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∆ L

v

∆ SRI

(b) Panel 2

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Frequency (Hz)

∆ (d

B)

∆ L

v

∆ SRI

(c) Panel 3

Figure 6.10: Comparison of the difference in vibration velocity level and sound reduc-tion index without and with the damping treatment.

The sound reduction index of the honeycomb panel was predicted using the modeldescribed in Section 3. Figure 6.11 shows the measured SRI of panel 2 compared withthe predictions made using the measured structural loss factor averaged over fre-quency. It can be seen that for panel 2, the predictions with the average loss factor holdsgood for frequencies below critical frequency ( f < fc) but at ( f = fc) and above thecritical frequency ( f > fc), the sound reduction index is under predicted. According toHeckl [17], the radiation loss factor can have a significant contribution for structuressuch as composite structures that are very lightly damped similar to those investigatedin this thesis. As a consequence, the radiation loss factor calculated in narrow bandfrequency as in Equation 3.10 is added to the measured loss factor for better sound re-duction index predictions around the critical frequency as shown by Feng and Kumar[13].

40


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Panel 2 − MeasuredPanel 2 − Prediction

Figure 6.11: Measured and predicted sound reduction index of panel 2 .

Figure 6.12a shows the predicted sound reduction index calculated with an addedradiation loss factor. A good agreement can be seen between the measured curve andthe predictions when the theoretical radiation loss factor is added. The same compari-son for panel 3 is shown in Figure 6.12b.

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Panel 2 − MeasuredPanel 2 − with Radiation Lossfactor

(a) Panel 2

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Panel 3 − MeasuredPanel 3 − with Radiation Lossfactor

(b) Panel 3

Figure 6.12: Measured and predicted sound reduction index with added radiationlosses.

41

Sathish Kumar

6.2 Double-leaf partitions

The periodic model derived in Section 4.1 is used to examine the vibro-acoustic perfor-mance of a double-leaf partition with homogenous aluminium panels and aluminiumstiffeners.The predicted transmission loss is then compared to the experimental datafrom Legault and Atalla [23]. The parameters listed in Table 6.3 are used to describe thepartition and the sound absorbing material in the cavity. Assuming the air in the cavityto be at standard temperature and pressure we have ρ0 = 1.21 kg m-3, Npr = 0.702,and P0 = 101325 Pa and the wavenumbers in the cavity are obtained using Equation4.4.

h1(m) 1× 10−3 η1 0.01 l(m) 50.8× 10−3

h2(m) 2× 10−3 η2 0.01 to(m) 3.175× 10−3

ρ1(kg m-3) 2742 ν1 0.33 L(m) 508× 10−3

ρ2(kg m-3) 2742 ν2 0.33 H(m) 50.8× 10−3

E1(Pa) 70× 109 co(m s-1) 342 K′t(Pa) 4.37× 109

E2(Pa) 70× 109 ρ0(kgm-3)

1.21 K′r(Pa) 3.67× 103

M(kg) 0.885 Porosity 91% Flowresistivity(N s m-4)

17700

Table 6.3: Dimensions and material data of panels and fibrous material [23].

Using the traditional convergence criterion, the number of terms required at thehighest frequency of interest is determined and the sound transmission is calculatedfor all other frequencies with the same number of terms. From the convergence curve inFigure 6.13a, it is found that at least 65 terms are needed for convergence at 10 kHz. Thesound transmission loss of the partition predicted using the traditional convergencecriterion is compared to the measured data in Figure 6.13b. The calculations are madein narrow bands with a frequency step of 10 Hz and later averaged over 1/3rd octavebands. From the figure it can be concluded that an overall agreement is achieved.

42


0 10 20 30 40 50 60 70 80 90 1000

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Number of Terms

ST

L (d

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200 Hz1000 Hz5000 Hz10000 Hz

Convergence line

(a) Convergence analysis

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MeasuredPredictedn= −32 to +32

(b) Measured and predicted (traditional conver-gence).

Figure 6.13: Convergence analysis and sound reduction index of the partition.

To reduce computation time, a new frequency dependent convergence scheme is in-troduced as explained in Section 4.3.2. In this new scheme, the space harmonics corre-sponding to all propagating waves and some evanescent waves which are consideredto be influential for sound transmission are included in the calculation. The propagat-ing waves are limited within a region given by Equations 4.19 & 4.20. However, thenumber of evanescent terms (∆n) included in the solution has to be estimated. Fig-ure 6.14 shows the calculated sound transmission loss of the partition using the newconvergence criterion with different number of evanescent terms (∆n).

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∆n = 2∆n = 5∆n = 10

Figure 6.14: Predicted sound reduction index using the new convergence scheme withdifferent ∆n.

It can be observed that the sound transmission results obtained with ∆n = 5 is verysimilar to that obtained with ∆n = 10 but with a smaller computational expense. A

43

Sathish Kumar

numerical investigation revealed that when ∆n > 5, the evanescent waves decayed by60 dB when reaching the second panel. Therefore, for the partition investigated here itcan be concluded that ∆n = 5 is enough to achieve numerical convergence. Figure 6.15shows the measured sound transmission loss compared to the predictions obtained bythe traditional fixed term method and the new adaptive method. For this partition,the new method takes one tenth of the computation time of the traditional techniquewithout compromising the accuracy of the result.

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MeasuredTraditional Convergence (n = −32 to +32)New Convergence with ∆n = 5

Figure 6.15: Measured and predicted sound reduction index with different conver-gence schemes.

Double-leaf partitions were constructed in the lab using aluminium honeycombsandwich panels and 2 mm thick aluminum panel. The two panels were linked withfive vertical “C” shaped aluminum channels, each spaced by a distance L of 505 mmand the cavity was filled with mineral wool as shown in Figure 6.16b. The thickness ofthe stiffeners was 3 mm, while their web length H measured 50 mm and their flangelength measured 30 mm. The stiffeners and the aluminum panels have a Young’s mod-ulus of 70× 109 Pa, Poisson’s ratio of 0.33, and a density of 2742 kg m-3. The honey-comb panels discussed in the previous section (panel 2 and panel 3) were used to buildtwo different double-leaf partitions. The stiffeners were connected to the honeycombpanels using glue as shown in Figure 6.16a and the aluminium panels were connectedto the stiffeners using evenly spaced screws (100 mm) mounted at the middle of theflange. The properties of the aluminium panels, stiffeners and the mineral wool com-mon for both the partitions is given in Table. 6.4 and the properties of the honeycombpanels are as given in Table. 2.1 and Table. 3.1.

44


(a) Stiffener glued to the honeycombpanel

(b) Partition with mineral wool but without aluminium panel

Figure 6.16: Assembly of double-leaf partition with honeycomb panel.

h2(m) 2× 10−3 co(m s-1) 342 L(m) 505× 10−3

ρ2(kg m-3) 2742 ρ0(kg m-3) 1.21 H(m) 50× 10−3

E2(Pa) 70× 109 l(m) 44× 10−3 K′t(Pa) 4.77× 109

η2 0.01 to(m) 3× 10−3 K′r(Pa) 3.57× 103

ν2 0.33 Porosity 94% M(kg) 0.89

Flowresistivity(N s m-4)

35000

Table 6.4: Dimensions and material data of panels and mineral wool.

The sound transmission loss of the partitions were measured using the sound in-tensity technique between 100 Hz and 5000 Hz as described in Section 5.2. Figure 6.17shows the measured sound transmission loss of the two partitions. A difference in thesound transmission loss can be seen in the higher frequencies. The lower transmis-sion loss of the partition with honeycomb panel 3 is due to its softer core. The soundtransmission loss of the two partitions were calculated using the periodic model forsandwich panels as described in Section 4.2 and the results are shown in Figures 6.18& 6.19. The bending stiffness of the honeycomb panels used in the calculation wereobtained analytically using the wave propagation theory as described in Reference 31.

45

Sathish Kumar

125 250 500 1000 2000 40000

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Partition with HC panel 2 (6.4 mm core)Partition with HC panel 3 (19.2 mm core)

Figure 6.17: Measured sound reduction index of the two partitions.

Using the traditional convergence criterion, it was determined that at least 37 spaceharmonic terms were necessary for the solution to converge at 5000 Hz . Whereas, us-ing the new convergence criterion with ∆n = 5 , the computation time for the partitionswas reduced by a factor of 10. Figures 6.18 & 6.19 shows that the predictions using theperiodic model agree well with the measurements.

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MeasuredPrediction

Figure 6.18: Measured and predicted sound reduction index of the partition with hon-eycomb panel 2.

46


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MeasuredPrediction

Figure 6.19: Measured and predicted sound reduction index of the partition with hon-eycomb panel 3.

Modelling the stud stiffness

In this section, estimation of stiffness for beam-like studs are discussed. Such studsare common in both aerospace and rail vehicle structures. The stiffness of the studis a critical parameter in sound transmission predictions. For a “C” shaped stud, it isassumed that the flange, and particularly the web, possess a bending resilient characterwhich deforms under lateral force and hence acting as a spring between the two panels.For the partitions described in the previous section, the translational stiffness per unitlength of the stud with thickness 3 mm is K′t = 4.37× 109 Pa when calculated usingK′t = Et0/l . If the stiffener is much thinner for example, with thickness 0.3 mm, thisexpression leads to K′t = 0.43× 109 Pa which by the order of magnitude is much stifferthan reality. The reason is that the stiffness expression (K′t = Et0/l ) assumes thatthe deformation is mainly due to axial deformation in the stud web. This assumptionholds true when the stud thickness is high (in our case 3 mm), but when the studthickness is small, stud deformation mainly results from web bending and the aboveexpression grossly over estimates the stiffness. The translational stiffness of a flexibleconnector accounting for web bending has been discussed by Legault and Atalla [23]and is calculated by K′tb = Et3

0/3l L2F where E is the Young’s modulus, to is the web

thickness, l is the depth, and LF is the length of the flange of the stiffener. For a stiffenerof thickness 0.3 mm, this expression leads to a stud stiffness of K′tb = 2.27× 104 Pa.

47

Sathish Kumar

(a) t0 = 3 mm (b) t0 = 0.3 mm

Figure 6.20: Influence of thickness on the translational stiffness of an aluminium stiff-ener.

A double-leaf partition consisting of two identical 13 mm thick gypsum boardsseparated by 65 mm steel studs of thickness 0.5 mm is investigated. The cavity be-tween the boards is filled with mineral fibres. This partition corresponds to wall #12in Reference 16. The parameters listed in Table 6.5 is used to describe the wall and themineral fibre. For a stiffener with thickness 0.5 mm, the translational stiffness becomesK′tb = 2.15× 105 Pa.

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70

Frequency (Hz)

Sou

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educ

tion

Inde

x (d

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MeasuredPeriodic model

Figure 6.21: Comparison of the measured and predicted sound reduction index.

Figure 6.21 shows a comparison of the predictions, using the new convergence cri-terion, to the test data from Reference 16 and it can be concluded that an overall agree-ment is achieved. The space harmonics corresponding to all propagating waves anda few evanescent (∆n = 3) waves were included in the calculation. For intermediatestud thicknesses, the stiffness is rather complex but can be obtained using experimentsor through detailed stiffener vibration models [37].

48


h1(m) 13× 10−3 η1 0.02 l(m) 65× 10−3

h2(m) 13× 10−3 η2 0.02 to(m) 0.5× 10−3

ρ1(kg m-3) 630 ν1 0.28 L(m) 406× 10−3

ρ2(kg m-3) 630 ν2 0.28 H(m) 65× 10−3

E1(Pa) 2× 109 co(m s-1) 342 K′t(Pa) 2.15× 105

E2(Pa) 2× 109 ρ0(kg m-3) 1.21 K′r(Pa) 33.65

M(kg) 0.5 Porosity 93% Flowresistivity(N s m-4)

11400

Table 6.5: Dimensions and material data of panels and mineral fibre [16].

49

Chapter 7

Conclusions and future work

The measurements carried out during this thesis illustrates that constrained layer damp-ing treatments increase the loss factors for all the honeycomb panels tested. As ex-pected, the vibration velocity levels also decreased after damping treatment. The effecton the sound transmission properties is more complex. For panels 1 and panel 2 thedamping treatment does not affect the sound transmission loss apart from the effectsof the added mass. However, for panel 3, the same damping treatment increases thesound insulation due to the panel’s extended coincidence range as demonstrated bythe dispersion curves. It is known that damping has its most influence around the crit-ical frequency where the transmission is resonant. Naturally, panel 3 with its extendedcoincidence range had its sound insulation improved when the damping treatmentwas applied. The weighted sound reduction index, R′w of panel 2 was 3 dB higherthan that of panel 3 but with the damping treatment applied both panels have thesame weighted sound reduction index (R′w). In-short, the visco-elastic damping treat-ment increases the sound insulation of an acoustically bad panel (in our case panel 3).Whereas, the same treatment on a normal panel (panel 1 and 2) does not improve thesound insulation apart from the effects of added mass.

Radiation losses can be a significant contributor to the total losses for lightly dampedcomposite structures. Since the measured loss factor is averaged over 1/3rd octavebands, the abrupt increase of loss factor around the critical frequency is not captured inthe measurements. Therefore, the measured structural loss factor alone is not enough tocharacterise the total losses of a light weight panel around the critical frequency. Apartfrom the measured structural loss factor, the radiation loss factor calculated in narrowbands must also be included in the model to accurately predict the sound transmis-sion loss of lightweight panels like those discussed in this thesis. The sound reductionindex calculated using the wave propagation approach, combined with orthotropictheory, agrees very well with the measurements.

51

Sathish Kumar

An analytical model is developed for double-leaf partitions with periodic stiffenersand sound absorption material in the cavity. The stiffeners are modelled as uniformlydistributed lumped masses attached to the two panels with a set of translational androtational springs. The sound absorption in the cavity is characterised by an equiv-alent fluid model. The vibro-acoustic response is derived by using a series of spaceharmonics and applying the principle of virtual work. The sound transmission lossesof isotropic double-leaf partitions are calculated using the periodic model and the re-sults are compared with published test data. The periodic model is later extended toinclude orthotropic sandwich panels. The frequency dependent bending stiffness ofthe orthotropic panels are obtained using sandwich theory. Two different double-leafpartitions are built using an aluminium panel and two different honeycomb panelsand their sound transmission losses are measured and compared to predictions fromthe periodic model. The results show that the periodic model using the space harmonicexpansion method provides a good estimate of the sound transmission loss for all thesystems analysed.

Since the solution to this method is obtained in a series form, computational accu-racy needs to be balanced with computation cost as calculation times increases withthe number of terms. For improved computational efficiency, an adaptive criterion isproposed to select the appropriate space harmonic terms which are considered to beinfluential for sound transmission through the partition. This criterion results in quickconvergence of the solution. For the partitions studied in this thesis, the computationtime was reduced by a factor of at least 10. This decrease in computation time enablesto use this model for quick parametric studies in early design stages.

The stud stiffness plays an important role in the sound transmission of double-leafpartitions. A partition was chosen to illustrate the importance of selecting the right ex-pression for calculating the stud stiffness based on stiffener thickness. Two such modelsfor calculating the equivalent stud stiffness have been discussed. However, more workis needed on developing simplified models to calculate the stud stiffness at higher fre-quencies and studs with complex geometry. It is common to use visco-elastic mountsin a floating floor assembly. The effects of visco-elastic mounts on the sound trans-mission should be investigated. The periodic model developed in this thesis could beused in conjunction with structural optimisation techniques for better sound proofingin engineering structures like airplane fuselages and railway floors in the future.

52

Chapter 8

Summary of appended papers

8.1 Paper A

On Application of radiation loss factor in the prediction of sound transmission lossof a honeycomb panelLeping Feng and Sathish Kumar.

This article shows the importance of radiation loss factor in predicting the soundtransmission property of lightly damped panels, especially around the critical fre-quency region. When the sound transmission properties of a panel is investigated,the radiation loss factor is often neglected for panels unless the structural dampingis very less or when the sound transmission around the critical frequency of the panelis of interest. This article shows that by including the theoretical radiation loss factorin narrow bands to the measured structural loss factor, the sound reduction index ofhoneycomb sandwich panels can be predicted with good accuracy around the criti-cal frequency. The wave propagation constants are calculated as given by Nilsson [31]and the sound reduction index of the panels are predicted using the orthotropic paneltheory. The predictions are validated through laboratory measurements.

8.2 Paper B

Predicting the sound transmission loss of honeycomb panels using the wave propa-gation approachSathish Kumar, Leping Feng and Ulf Orrenius.

The wave propagation approach can be used to predict the sound transmissionproperties of sandwich panels having an isotropic core. In this article, the wave propa-gation approach is used to predict the sound transmission loss of honeycomb panels. Ahoneycomb panel is a lightweight sandwich panel with a honeycomb core of hexagon

53

Sathish Kumar

cell. Due to the manufacturing process of honeycomb cores, the cell wall thicknessdoubles in the direction of cell orientation and the actual shape of the honeycomb cellcan be irregular making the core anisotropic. As an approximation, the honeycombcore can be treated as orthotropic and the whole panel can be treated as a orthotropicsandwich panel. The wavenumbers are calculated for the two principal orthotropic di-rections. The orthotropic panel theory is used to predict the sound transmission lossof panels with two different core structures. Visco-elastic damping with constraininglayers (CLD) are attached to these panel and the sound transmission loss is calculated.The predicted wavenumbers and the sound transmission loss of the panels agree wellwith laboratory measurements.

8.3 Paper C

Modelling the sound transmission through rib-stiffened double-leaf partitions withcavity absorptionSathish Kumar, Leping Feng and Ulf Orrenius.

The sound transmission loss of infinite, periodically rib-stiffened double-leaf parti-tions with different fluids in the cavity is investigated. The sound transmission modelfor double-leaf partitions with periodic stiffeners has been extended to include soundabsorbing materials in the cavity. The vibro-acoustic response of the panels is expressedas a series of space harmonics and virtual work theory has been successfully appliedfor calculating the sound transmission loss of the partition. The sound propagationin the cavity is defined by assuming an equivalent fluid model. The number of spaceharmonic terms required for convergence of the sound transmission loss is studied. Toreduce the computation time, a new criterion is introduced to select the appropriatespace harmonics based on frequency and angle of incidence. This new criterion en-sures quick convergence and reduced computation time. For the partitions analysed,the computation time was reduced by a factor of ten. The model is verified throughcomparison with experimental data available in the literature.

8.4 Paper D

Modelling the sound transmission through rib-stiffened sandwich double-leaf par-titions using space harmonic analysisSathish Kumar, Leping Feng and Ulf Orrenius.

An analytical model is developed for the sound insulation of infinite, periodicallyrib-stiffened double-leaf partitions consisting of sandwich panels. The two panels are

54


separated by “C” shaped stiffeners and the cavity between the panels is filled withmineral wool. The sandwich panel investigated in this paper has a honeycomb corewhich makes the panels orthotropic. The apparent bending stiffness of the sandwichpanels in the two orthotropic directions are derived using sandwich theory. The cavitybetween the sandwich panels is filled with absorbing material and the sound propaga-tion in the cavity is defined by assuming an equivalent fluid model. The vibro-acousticresponse of the panels is expressed as a series of space harmonics and virtual worktheory has been applied for calculating the sound transmission loss of the partition. Anew criterion is employed for quick convergence which reduces the computation time.For the partitions analysed, the computation time was reduced by a factor of ten. Thepredicted transmission loss is validated through comparison with laboratory measure-ments.

55

Appendix

Part A

Functions used to solve for the wavenumber ′k′,

U1 = (1− eα)L1

[ζ2

L − k2ν2/ (1− ν2)]

; U2 = (1 + eα)ζL

[k4 − k4

1

]

U3 = (1− e−β)ζTkL1 [(1− 2ν2) / (1− ν2)] ; U4 = (1 + e−β)k[k4 − k4

1

]

V1 = (1− eα)k[k2

L1 − k2]

; V2 = (1 + eα)2ζLM1k

V3 = (1− e−β)ζT

[k2

L1 − k2]

; V4 = (1 + e−β)2M1

[k2 − k2

L1/ (1− ν2)]

L1 =12E2 (1− ν2)

(1− ν2

1)

E1t31 (1 + ν2) (1− 2ν2)

; M1 =E2(1− ν2

1)

2E1t1 (1 + ν2)

α = ζLλL; β = ζLλT

ζL =

√k2 − ω2ρ2 (1 + ν2) (1− 2ν2)

E2 (1− ν2); ζT =

√k2 − 2ω2ρ2 (1 + ν2)

E2

λL =

√ω2ρ2

E′2 − k2 ; λT =

√ω2ρ2

G2 − k2

56


Part B

Principle of Virtual work

The principle of virtual work states that the sum of the works done by all the elementsin one period of the system must do no virtual work when moved through any one ofthe virtual displacements [28]

δWi = δαi,me−j(kx+2mπ/L)x. (8.1)

The total virtual work is the sum of virtual works of the two panels elements, thevirtual work of the translational and rotational springs and from the lumped mass ofthe studs. The principle of virtual work requires that

δΠp1 + δΠp2 + δΠt1 + δΠt2 + δΠr1 + δΠr2 + δΠm1 + δΠm2 = 0. (8.2)

Virtual work of panel elements: Isotropic

The virtual work contributed by the two panel elements in one period of a panel ele-ment can expressed as

δΠp1 =

Lˆ

x=0

(D1

∂4W1

∂x4 + mp1∂2W1

∂t2 − jωρoΦ1 + jωρcΦ2

)δW∗1 dx at y = 0, (8.3)

δΠp2 =

Lˆ

x=0

(D2

∂4W2

∂x4 + mp2∂2W2

∂t2 − jωρcΦ2 + jωρoΦ3

)δW∗2 dx at y = H. (8.4)

where δW∗i is the complex conjugate of the virtual displacement.

Virtual work of panel elements: Orthotropic

The virtual work contributed by the two panel elements in one period of a panel ele-ment can expressed as

57

Sathish Kumar

δΠp1 =

Lˆ

x=0

(Dx1

∂4W1

∂x4 + 2Dxz1∂4W1

∂x2∂z2 + Dz1∂4W1

∂z4 + mp1∂2W1

∂t2 − jωρoΦ1 + jωρcΦ2

)δW∗1 dx at y = 0,

(8.5)

δΠp2 =

Lˆ

x=0

(Dx2

∂4W2

∂x4 + 2Dxz2∂4W2

∂x2∂z2 + Dz2∂4W2

∂z4 + mp2∂2W2

∂t2 − jωρcΦ2 + jωρoΦ3

)δW∗2 dx at y = H.

(8.6)

where δW∗i is the complex conjugate of the virtual displacement.

Virtual work of translational springs

The contribution to the virtual work by the translational spring is equal to

δΠt1 = Kt (W1 (0)−W2 (0)) δα∗1,m = Kt

[+∞

∑n=−∞

α1,n −+∞

∑n=−∞

α2,n

]δα∗1,m, (8.7)

δΠt2 = Kt (W2 (0)−W1 (0)) δα∗2,m = Kt

[+∞

∑n=−∞

α2,n −+∞

∑n=−∞

α1,n

]δα∗2,m. (8.8)

Virtual work of rotational springs

The contribution to the virtual work by the rotational spring is equal to

δΠr1 = Kr

(W′1 (0)−W

′2 (0)

)j(

kx +2mπ

L

)δα∗1,m

= Kr

[+∞

∑n=−∞

α1,n

(kx +

2nπ

L

)−

+∞

∑n=−∞

α2,n

(kx +

2nπ

L

)](kx +

2mπ

L

)δα∗1,m,

(8.9)

58


δΠr2 = Kr

(W′2 (0)−W

′1 (0)

)j(

kx +2mπ

L

)δα∗2,m

= Kr

[+∞

∑n=−∞

α2,n

(kx +

2nπ

L

)−

+∞

∑n=−∞

α1,n

(kx +

2nπ

L

)](kx +

2mπ

L

)δα∗2,m.

(8.10)

Virtual work of lumped masses

The contribution to the virtual work by the lumped mass is equal to

δΠm1 = −ω2MW1 (0) δα∗1,m = −ω2M

[+∞

∑n=−∞

α1,n

]δα∗1,m, (8.11)

δΠm2 = −ω2MW2 (0) δα∗2,m = −ω2M

[+∞

∑n=−∞

α2,n

]δα∗2,m. (8.12)

59

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