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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]

0888-32http://d

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Pleasloss

journal homepage: www.elsevier.com/locate/ymssp

Modal analysis of thin cylindrical shells with cardboard linersand estimation of loss factors

Hasan Koruk a,b, Jason T. Dreyer a, Rajendra Singh a,n

a Acoustics and Dynamics Laboratory, Department of Mechanical and Aerospace Engineering, The Ohio State University,Columbus, OH 43210, USAb Istanbul Technical University, Mechanical Engineering Department, 34437 Istanbul, Turkey

a r t i c l e i n f o

Article history:Received 2 August 2013Received in revised form3 October 2013Accepted 31 October 2013

Keywords:Damping mechanismEstimation of material propertiesShell dynamicsAutomotive system applications

70/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.ymssp.2013.10.026

esponding author. Tel.: þ1 614 292 9044.ail addresses: koruk@itu.edu.tr (H. Koruk), d

e cite this article as: H. Koruk, et al.,factors, Mech. Syst. Signal Process. (

a b s t r a c t

Cardboard liners are often installed within automotive drive shafts to reduce radiatednoise over a certain frequency range. However, the precise mechanisms that yield noiseattenuation are not well understood. To overcome this void, a thin shell (under freeboundaries) with different cardboard liner thicknesses is examined using analytical,computational and experimental methods. First, an experimental procedure is introducedto determine the modal behavior of a cylindrical shell with a cardboard liner. Then,acoustic and vibration frequency response functions are measured in acoustic free field,and natural frequencies and the loss factors of structures are determined. The adverseeffects caused by closely spaced modes during the identification of modal loss factors areminimized, and variations in measured natural frequencies and loss factors are explored.Material properties of a cardboard liner are also determined using an elastic plate treatedwith a thin liner. Finally, the natural frequencies and modal loss factors of a cylindricalshell with cardboard liners are estimated using analytical and computational methods,and the sources of damping mechanisms are identified. The proposed procedure can beeffectively used to model a damped cylindrical shell (with a cardboard liner) to predict itsvibro-acoustic response.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Some vehicles include cardboard liners within the drive shafts to control the structureborne and radiated noise over themid-frequency range. Although there are numerous articles on homogenous and undamped cylindrical shells [1–4], no priorstudy has analytically examined the vibro-acoustics of cylindrical shells with cardboard liners. Although a few accelerancefrequency response functions (with and without cardboard liners) have been measured [5–8], the modal parameters of suchstructures have not been quantified in a systematic way. Some attempts have been made to simulate the cardboard liner as adistributed mass element in large scale computational models [9]. Overall, the effect of cardboard thickness has not beenstudied, and their damping mechanisms have not been explored in the literature [6–9]. This paper attempts to fill this voidwith a controlled experimental study and by analytically or computationally examining certain shell vibration modes andassociated damping mechanisms.

All rights reserved.

reyer.24@osu.edu (J.T. Dreyer), singh.3@osu.edu (R. Singh).

Modal analysis of thin cylindrical shells with cardboard liners and estimation of2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

Nomenclature

a acceleration

mnBij modal constant of (m,n) mode for responsepoint i and excitation point j

D bending stiffness of the shelld outer diameter of cylinderE Young’s modulus~E complex Young’s modulusF excitation forceh thickness of the cylinderh1 thickness of base shellh2 thickness of cardboard linerI second moment of inertiaj

ffiffiffiffiffiffiffiffi�1

p

k structural wave numberkθ,kz wave numbers in circumferential and axial

directionsK membrane stiffness of the shellK global stiffness matrix of a cylindrical shell

with a cardboard linerKc stiffness matrix of a composite elementKel stiffness matrix of layer l of a composite

element computed as if the layer is on theneutral plane

Kol stiffness matrix of layer l expressed with

respect to the neutral plane of a compositeelement

l length of cylinderm axial modal indexn circumferential modal indexm,n indices for shell modesM global mass matrix of a cylindrical shell with a

cardboard linerMc mass matrix of a composite elementMe

l mass matrix of layer l of a composite elementcomputed as if the layer is on the neutralplane

Mol mass matrix of layer l expressed with respect

to the neutral plane of a composite elementp sound pressures,q indices for a mode shape of a plate (the

numbers of half-waves in a mode shape alongthe long and short edges)

R radius of the middle surface of a cylindricalshell

T transformation matrix

mnεij residual term of (m,n) mode for response pointi and excitation point j

~Rij compliance function for response point i andexcitation point j

~R′ijðωÞ difference between the actual compliance andthe value of the compliance at the fixedfrequency ϖ

ϖ fixed frequency near natural frequency

Z inverse compliance parameterƞ modal loss factorƞmn modal loss factor for (m,n) modeƞr modal loss factor of rth modeηM loss factor due to material dampingηQ loss factor due to dry frictionηS total (or effective) loss factormƞ average loss factorμ Poisson’s ratioρ mass densitysη standard deviation of the loss factorω excitation frequencyω1,ω2 the frequencies corresponding to half power

pointsωa,ωb the frequencies around the natural frequen-

cies corresponding to angles (φa,φb)ωmn natural frequency of shell mode (m,n)ωr natural frequency of rth modeφa,φb two angles around natural frequency used in

circle-fit method

Subscripts

1 index for base tube2 index for cardboard linerc composite elementi index integerij response and excitation point pairl layer indexmn axial and circumferential modal indicesr mode numberθ circumferential directionz axial direction

Superscripts

e neutral plane of an individual layer of acomposite element

o neutral plane of a composite elementM materialQ frictionalS total (or effective)T transpose of a matrix� complex valued

Operators

j j determinant

Abbreviations

DOFs degrees of freedomFE(M) finite element (method)

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]2

h

d

h

interference fit

l

cardboard

base tube

z=0 z= l

p (r; ω)F (r; ω) a (r; ω)

Fig. 1. Example case: an aluminum cylindrical shell with a cardboard liner; measurements include sound pressure (p) in a free field, structure acceleration(a) and excitation force (F).

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] 3

2. Problem formulation

Fig. 1 illustrates the example case of a cylindrical shell with a cardboard liner; sensors used for the experimental studyare also shown. The boundary conditions at z¼0 and l are assumed to be free. The following approximate solutions for thenatural frequencies of a homogenous, thin cylindrical shell are first utilized to design the experiment and select the nominaldimensions such as the length (l), thickness (h) and radius (R). The natural frequency, ωmn (rad/s) of a homogeneouscylindrical shell for mode (m,n) is approximated from the following frequency equation that is based on the classical Love'sequation [1] and an infinite shell model [2]:

K k2zmþ 1�μ2 k2θn

� ��ρhω2

mnK 1þμð Þ

2 kzmkθn KμR kzm

K 1þμð Þ2 kzmkθn Kþ D

R2

� �k2mn� 1þμ

2 k2zm� �

�ρhω2mn

KR kθnþ D

R k2mnkθn

� �KμR kzm K

R kθnþ DR k

2mnkθn

� �Dk4mnþ K

R2 �ρhω2mn

�����������

�����������¼ 0 ð1Þ

Here, m and n are axial and circumferential modal indices, respectively; R is the radius of the middle surface and h is thethickness of a cylindrical shell; kθn¼n/R is the wave number in the circumferential direction, kzm is the modal number in the

axial direction and kmn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2zmþk2θn

qis the structural wave number; K ¼ Eh=ð1�μÞ is the membrane stiffness of the shell, and

D¼ Eh3=½12ð1�μ2Þ� is the bending stiffness of the shell where μ is Poisson’s ratio, E is Young’s modulus and ρ is the massdensity. The modal wave number in the axial (z) direction under free–free boundary conditions is given using the beamfunctions kzm ¼ ðm�0:5Þπ=l for mZ2. Note that the beam functions cannot provide a solution for m¼ 0 and m¼ 1.Accordingly, analytical expressions for ω0n and ω1n (in rad/s) for m¼ 0 and m¼ 1 modes are determined using

ω20n ¼

Eρ

h2

12R4

n2ðn2�1Þ2n2þ1

ð2Þ

ω21n ¼ω2

0nðn2þ1Þ n2l2þ24ð1�μÞR2

12R2þn2ðn2þ1Þl2ð3Þ

Both acceleration to force ~AijðωÞ ¼ ~aiðωÞ= ~F jðωÞ and acoustic pressure to force ~GijðωÞ ¼ ~piðωÞ= ~F jðωÞ transfer functions aremeasured on shell samples with different cardboard liners in acoustic free field where “a” is the acceleration, “p” is thesound pressure and “ F ” is the excitation force; i and j are the response and excitation point indices, respectively; ω is theexcitation frequency; and, superscript � implies a complex valued quantity. Preliminary experiments performed on shellswith typical cardboard liners showed that it is difficult to determine their modal parameters (especially the modal lossfactors) when the structure is highly damped. Accordingly, several cardboard samples of increasing thickness (h2) areprepared. Furthermore, results with a different thickness (h2) should provide a better understanding of the modes andsources of damping. Specifically, the shell thickness (h1), outer diameter (d) and length (l) are determined to be 2.1, 152.4 and457.2 mm, respectively, while three cardboard liners are prepared with h2¼0.31, 0.62 and 0.98 mm. The outer diameter ofthe cardboard is equal to the inner diameter of the base (aluminum) cylinder, and thus an interference fit describes theassembled structure. The mass density (ρ) of the aluminum tube is 2704 kg/m3 while the cardboard density is determined tobe 649 kg/m3. Young’s modulus (E) and Poisson’s ratio (μ) of the base aluminum cylinder are 71.0 GPa and 0.33, respectively.The experiments include the same cardboard material and the same base shell (and thus the surface properties do not varyas a function of liner thickness). Therefore, the results are expected to be independent of surface properties.

Overall, the main objectives of this study are as follows: (i) design a controlled vibro-acoustic experiment to measure~AijðωÞ and ~GijðωÞ functions on a cylindrical shell with several cardboard liners; (ii) determine the first ten natural frequenciesand modes of a cylindrical shell without and with the cardboard liners (up to 1950 Hz), and evaluate the accuracy of themeasured natural frequencies; (iii) extract the loss factors of the first five modes of a cylindrical shell without and with thecardboard liners using several frequency response based methods and a number of measured ~AijðωÞ functions, and explorethe uncertainty levels in the measured loss factors; (iv) determine material properties of a cardboard liner using an elasticplate treated with a thin liner; (v) estimate the loss factors of a cylindrical shell with the cardboard liners using an analytical

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

Fig. 2. Simulation of proper locations for suspending the cylinder with a cardboard liner (a), impulse force excitation (b), accelerometer (c) andmicrophone (d).

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]4

approach; (vi) estimate the natural frequencies and modal loss factors of those structures using a computational method;and (vii) identify possible sources of damping.

3. Vibro-acoustic experiments

Before conducting detailed ~AijðωÞ and ~GijðωÞ measurements, the appropriate locations of force excitation, accelerometerand microphone, as well as the shell suspension method [10,11], are identified using a computational (finite element) model[12] of an untreated cylindrical shell. For this, an analysis of the base structure is conducted to determine modal deflectionsfor each mode. It is noted that the degrees of freedom (DOF) with lower displacement response levels (suggesting negligibleinteractions between structure and suspension cord) should provide good suspension locations while the DOFs with highervelocity response levels (which may induce double hit impacts) as well as the nodal points must be avoided whendetermining suitable impulse hammer excitation locations. The acceleration and sound pressure measurements aresimulated via the highest acceleration and surface velocity levels, respectively, over the frequency range of interest.Overall, the best positions are determined by averaging the corresponding parameters for the modes of interest [10,11].Sample results for measurement locations considering the first 12 modes of a cylinder are displayed in Fig. 2. The results foraccelerometer and microphone simulations are very similar, as expected.

Preliminary ~AijðωÞ and ~GijðωÞ measurements are conducted to select the best signal processing parameters. For instance,~AijðωÞ measurements are obtained with a frequency resolution of 0.078, 0.125 and 0.250 Hz, while ~GijðωÞ measurements areperformed with 0.063 and 0.125 Hz resolution. Measured data with a finer frequency resolution is used to identify thedamping at the lower modes. The autospectra of the force signal are examined to ensure that the structure is properlyexcited over the frequency range of interest. Each power spectrum is found using five averages.

Since the mass loading by an accelerometer would shift the frequencies, the natural frequencies of the shell structure aredetermined by the acoustic functions ~GijðωÞ. Nevertheless, the modal loss factors are estimated using structural accelerances~AijðωÞ; half-power, line-fit and circle-fit methods [10,13,14] are utilized. Also, a very fine frequency resolution is employedfor better accuracy and for a clear recognition of closely spaced modes. The measurement period is selected to be sufficientlylong to ensure that the signal approaches zero at the end of its minimal period. Consequently, there is no need to apply aparticular windowing function; hence, there should be minimal uncertainty in the identified modal loss factors. Also,experiments are performed only under free–free boundary conditions to eliminate any uncertainty due to boundarydamping and stiffness. Overall, the first ten shell modes of structures are successfully identified.

The assembled structure consists of the main shell and a cardboard liner. A nonlinear interface with a dry frictionelement is expected between the base tube and cardboard liner. Therefore, the variations in the measured ~AijðωÞ and ~GijðωÞshould be identified before determining the modal parameters from the measured ~AijðωÞ and ~GijðωÞ database. For thispurpose, some precautions are taken. First, in order to verify the repeatability of the measurement system, including thebehavior of the assembly, three ~AijðωÞ functions were measured on the treated cylindrical shell with h2¼0.62 mm for thesame response and excitation points; these are overlaid in Fig. 3. Here, the response and excitation locations are on the sameline on the outer surface of the cylinder, i.e., i is given by r¼d/2, θ¼0, z¼200 mm and j is given by r¼d/2, θ¼0, z¼150 mm.It is seen that the ~AijðωÞ functions are nearly the same even at the anti-resonances, though there are some deviations beyond

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

200 400 600 800 1000 1200 1400 1600 1800 2000-70

-60

-50

-40

-30

-20

-10

0

10

20

30

Frequency (Hz)

Aij,

dB re

1.0

m/s

2 -N

1st 2nd 3rd

Fig. 3. Three structural accelerance Aij magnitudes measured on a treated cylindrical shell with a cardboard liner (with h2¼0.62 mm) for repeatabilitycheck. Key: 1st, 2nd and _____ 3rd.

200 400 600 800 1000 1200 1400 1600 1800 2000-70

-60

-50

-40

-30

-20

-10

0

10

20

30

Frequency (Hz)

Aij (

dB),

ref.

1 m

/s2 /N

Aij Aji

Fig. 4. Two structural accelerance Aij magnitudes measured on a treated cylindrical shell with a cardboard liner (with h2¼0.62 mm) for reciprocity check.Key: Aij and _____ Aji.

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] 5

1950 Hz. Also, the reciprocity experiment is performed, i.e., the measurements are repeated by interchanging the excitation(i) and response (j) coordinates as given above. The corresponding ~AijðωÞ functions measured on the treated cylindrical shell(with h2¼0.62 mm) are compared in Fig. 4. Observe that the ~AijðωÞ functions are nearly the same below 1200 Hz thoughthere are negligible amplitude deviations between 1200 Hz and 1950 Hz. However, higher deviations are seen beyond1950 Hz. Such variations at higher frequencies as determined via repeatability and reciprocity experiments are attributed tothe nonlinear behavior of the assembly of the base tube and cardboard liner due to friction, the inability to properly excitethe structure, and the effect of damping. It is noted that higher increased damping is expected at higher frequencies due tothe enhanced effects of friction and radiation damping. Therefore, the lower modes are more important from the vehiclevibration and sound perspective. Accordingly, in this paper, the natural frequencies of the first ten modes (below 1950 Hz)and the damping levels of only the first five modes are identified. Overall, it is seen that ~AijðωÞ and ~GijðωÞ measurements arevery repeatable, and the coherence function is at least 0.999 at and around the natural frequencies. It is noted that thefrequency domain based methods utilize data at and around natural frequencies to identify modal parameters.

Typical ~AijðωÞ and ~GijðωÞ for the same excitation, accelerometer and microphone points measured on the untreated andtreated shells with h2¼0.31, 0.62 and 0.98 mm are depicted in Figs. 5 and 6, respectively. It is seen that the resonance peaksmove to the left as h2 increases. The closely spaced peaks (degenerate modes) can be identified for the untreated shell andfor the treated shells with h2¼0.31 and 0.62 mm, but the degenerate modes disappear when h2¼0.98 mm due to the higherdamping. Also, Fig. 7 shows the effect of cardboard thickness (h2). Although the degenerate modes are not completelysymmetrical due to imperfection especially for a treated shell, they may be assumed to be almost symmetrical.

4. Extraction of modal parameters

Since it is virtually impossible to have a perfectly symmetrical shell, closely spaced peaks are found at each mode inmeasured accelerances ~AijðωÞ as previously mentioned. Also, some resonant peaks may contain more than one mode.

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

200 400 600 800 1000 1200 1400 1600 1800 2000-60

-40

-20

0

20

40

60

Frequency (Hz)

Aij,

dB re

1 m

/s2 -N

(0,2)

(1,2)

(3,4)

(1,3)

(1,4)

(2,4)(2,3)

(0,3) (2,2)

(0,4)

h2 = 0, h2 = 0.31 mm, h2 = 0.62 mm h2 = 0.98 mm

Fig. 5. Effect of cardboard liner thickness (h2) on measured accelerances. Key: h2¼0, h2¼0.31 mm, h2¼0.62 mm andh2¼0.98 mm.

200 400 600 800 1000 1200 1400 1600 1800 2000

-60

-50

-40

-30

-20

-10

0

10

20

30

Frequency (Hz)

Gij,

dB re

1.0

Pa/

N

(0,4)(2,4)

(1,3)

(1,4)

(0,2)(1,2)

(0,3)(2,3) (2,2)

(3,4)

h2 = 0 mm, h2 = 0.31 mm, h2 = 0.62 mm h2 = 0.98 mm

Fig. 6. Effect of cardboard liner thickness (h2) on measured acoustic frequency response functions. Key: h2¼0 mm, h2¼0.31 mm,h2¼0.62 mm and h2¼0.98 mm.

230 235 240 245 250 255 260 265-20

-10

0

10

20

30

40

Frequency (Hz)

Aij,

dB re

1.0

m/s

2 -N (0,2)(0,2)

(0,2)

(0,2)(1,2)(1,2)

(1,2)(1,2)

h2 = 0 mm, h2 = 0.31 mm, h2 = 0.62 mm h2 = 0.98 mm

Fig. 7. Effect of cardboard liner thickness (h2) on measured accelerances showing (0,2) and (1,2) type mode pairs. Key: h2¼0 mm,h2¼0.31 mm, h2¼0.62 mm and h2¼0.98 mm.

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]6

The adverse effects of such closely spaced modes encountered during the identification process are minimized via the circle-fit [13] and line-fit [14] type methods that take into account both accelerance ~AijðωÞ amplitude and phase data when distinctmodes are well excited; these methods are used in addition to the conventional half-power method [10].

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] 7

In the half-power method [10], the loss factor ðηmnÞ for mode (m,n) is determined by the following:

ηmn ¼ 2ζmn ¼ω2mn;2�ω2

mn;1

2ω2mn

ð4Þ

where ζmn is the corresponding viscous damping ratio and ðωmn;1;ωmn;2Þ are the frequencies corresponding to half powerpoints around ωmn: In the circle-fit method [13], the modal loss factor is determined by the following:

ηmn ¼ω2mn;b�ω2

mn;a

ω2mnð tan ðφmn;a=2Þþ tan ðφmn;b=2ÞÞ

ð5Þ

where two frequencies ðωmn;a; ωmn;bÞ correspond to the angles ðφmn;a; φmn;bÞ around ωmn when the dynamic compliancefunction is plotted using the Nyquist diagram [10]. Note that the expression in Eq. (5) is equal to the expression in Eq. (4)when the angles ðφmn;a;φmn;bÞ are selected as π/2. The line-fit method [14] is shown to be quite effective for thedetermination of modal loss factors [15]. In this method, the compliance function ~Rij for response point i and excitationpoint j can be written as follows:

~RijðωÞωCωmnffi mnBij

ω2mn�ω2þ jηmnω2

mnþmnεij ð6Þ

where mnBij is the modal constant and mnε ij is the residual term for the mode (m,n) corresponding to i and j. A new form ofcompliance function, ~R′ijðωÞ, which is the difference between the actual compliance function and the value of thecompliance function at one fixed frequency, ϖ, in the range of interest is defined as follows:

~R′ijðωÞ ¼ ~RijðωÞ� ~RijðϖÞ ð7Þ

to cancel the residual term in the compliance function. After that, an inverse parameter is defined using ~R′ijðωÞ for modalanalysis [10]; it is given in Eq. (8). The expression in Eq. (8) can also be expressed as in Eq. (9):

ZðωÞ ¼ ðω2�ϖ2Þ~R′ijðωÞ

ð8Þ

ZðωÞ ¼ ðω2mn�ω2þ jηmnω

2mnÞðω2

mn�ϖ2þ jηmnω2mnÞ

mnBijð9Þ

The inverse parameter in Eq. (9) is separated into real and imaginary parts which are supposed to fit onto a line whenplotted against ω2. The modal parameters are then estimated by determining the best lines that provide the best fits for themeasured data. See Refs. [10,13,15] for more details.

5. Comparison of natural frequencies

The natural frequencies of the untreated shell are first calculated using the approximate analytical formulas given byEqs. (1)–(3). Then, the homogeneous shell elements (such as S4R in [16] and 3SHL04 in [17]) are utilized for computational(finite element) modal analysis. The mesh convergence is checked and the finite element models of the cylindrical shell aredeveloped using 2500 (3SHL04) and 8554 (S4R) elements.

Before presenting the natural frequencies of cylinders, variations in natural frequencies need to be addressed. When themeasured acoustic ~GijðωÞ functions are analyzed for each test sample, small variations in natural frequencies are found,though such variations increase as the cardboard thickness h2 increases. For example, the natural frequencies for thedegenerate mode pairs of the first mode of the treated sample with h2¼0.31 mm are 243.16, 243.22, 243.12, 243.05, 243.14and 243.25 Hz for the first peak and 244.15, 244.25, 244.48, 244.50, 244.54 and 244.52 Hz for the second peak when sixmeasured ~GijðωÞ spectra for various excitation and response points are analyzed. Similarly, the natural frequencies arebetween 238.52 and 238.74 Hz for the first peak and from 239.75 to 240.12 Hz for the second peak for the degenerate modepairs of the first mode of the treated shell with h2¼0.62 mm. Overall, when all ~GijðωÞ functions are analyzed for each modeof each test sample, it is seen that the variations in the natural frequencies are less than 70.03%, 70.05%, 70.10% and70.25% with h2¼0, 0.31, 0.62 and 0.98 mm, respectively. Such variations are mainly due to finite frequency resolution, theeffect of excitation position, the nonlinear effect of the cardboard liner and dry friction. However, they are deemednegligible.

Table 1 compares the experimental, analytical and computational natural frequencies of a homogeneous cylindrical shell.The degenerate mode pairs are also identified for both untreated and treated samples with h2¼0.31 and 0.62 mm liners,though the degenerate modes disappear when a thicker (h2¼0.98 mm) liner is employed due to higher damping. Observethat the error is less than 1% for the computational method, but higher errors (say up to 6%) are seen for the analyticalmethod. The averaged errors are about 0.4 and 4.4% for the computational and approximate analytical methods,respectively. Error in the analytical method is due to the simplifying assumptions made in the shell vibration formulation[2], especially for the (0,n) and (1,n) modes.

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

Table 1Measured and calculated natural frequencies of a homogeneous thin cylinder with free boundaries.

Mode Experimental Approximate analytical solution Finite element model (FEM)

FEM-1 [16] FEM-2 [17]

Index (m,n) ωmn (Hz) ωave (Hz) ωmn (Hz) Error (%) ωmn (Hz) Error (%) ωmn (Hz) Error (%)

1 (0,2) 247.0, 248.4 247.7 234.9 �5.2 248.7 0.4 248.9 0.52 (1,2) 258.6, 259.9 259.3 245.4 �5.3 258.9 �0.1 259.1 �0.13 (0,3) 699.4, 701.9 700.6 664.5 �5.2 704.6 0.6 704.2 0.54 (1,3) 714.3, 716.0 715.1 679.1 �5.0 717.8 0.4 717.5 0.35 (2,3) 1023.4, 1025.8 1024.6 988.2 �3.5 1017.8 �0.7 1016.8 �0.86 (2,2) 1339.3, 1343.8 1341.5 1284.7 �4.2 1348.2 0.5 1347.2 0.47 (0,4) 1353.9, 1358.3 1356.1 1274.0 �6.1 1354.1 �0.1 1350.9 �0.48 (1,4) 1365.6, 1372.1 1368.8 1290.5 �5.7 1368.0 �0.1 1364.7 �0.39 (2,4) 1472.4, 1476.0 1474.2 1451.0 �1.6 1481.9 0.5 1477.9 0.3

10 (3,4) 1834.6, 1841.8 1838.2 1791.6 �2.5 1840.8 0.1 1835.4 �0.2

Table 2Effect of cardboard linear thickness (h2) on measured frequencies. See Table 1 for frequencies with h2¼0.

Mode h2¼0.31 mm h2¼0.62 mm h2¼0.98 mm

Index (m,n) ωmn (Hz) ωave (Hz) ωmn (Hz) ωave (Hz) ωmn (Hz)

1 (0,2) 243.2, 244.5 243.8 238.6, 240.0 239.3 236.32 (1,2) 254.3, 255.7 255.0 249.5, 250.8 250.2 246.83 (0,3) 688.2, 691.0 689.6 675.3, 678.5 676.9 672.84 (1,3) 702.7, 704.6 703.6 689.7, 692.2 690.9 686.55 (2,3) 1007.5, 1010.4 1008.9 990.8, 995.8 993.3 983.86 (2,2) 1317.6, 1322.3 1319.9 1297.1, 1300.2 1298.6 1286.87 (0,4) 1332.1, 1336.3 1334.2 1312.0, 1316.0 1314.0 1301.58 (1,4) 1345.4, 1352.1 1348.7 1326.0, 1336.0 1331.0 1314.09 (2,4) 1450.3, 1453.1 1451.7 1419.0, 1427.0 1423.0 1414.0

10 (3,4) 1808.0, 1812.7 1810.4 1776.0, 1783.5 1779.8 1762.4

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]8

The measured natural frequencies of the treated shells (with different h2 values) are given in Table 2. Also, selected modeshapes of the cylindrical shells are displayed in Fig. 8. It is noted that the mode shapes of the cylindrical shell with acardboard liner are expected to be the same as that of an untreated structure. Observe that the natural frequencies of atreated shell decrease at all modes as h2 is increased. The overall reductions are about 1.6%, 3.2% and 4.2% for the treatedcylinders with h2¼0.31, 0.62 and 0.98 mm.

6. Extraction of modal loss factors

Although most sources of uncertainties are minimized as much as possible using well-established experimentalmethodology, damping levels still contain some uncertainty mainly due to the nonlinear nature of damping, the degeneratemode pairs of cylindrical shells and the dry friction between the main shell and a cardboard liner. Therefore, the issue aboutuncertainty in the loss factor should be addressed first. For this purpose, a number of ~AijðωÞ functions are measured, and themodal loss factors are determined for well-excited modes. For example, the loss factors for the first mode of the treatedsample with h2¼0.62 mm are determined to be 0.00654, 0.00768, 0.00607, 0.00600 and 0.00626 using the circle-fit,0.00586, 0.00593, 0.00569, 0.00579 and 0.00565 using the line-fit and 0.00632, 0.00659, 0.00622, 0.00619 and 0.00644using the half-power when five accelerance ~AijðωÞ functions are analyzed. Similarly, the loss factors for the second mode ofthis structure are determined to be 0.00696, 0.00740, 0.00714, 0.00697 and 0.00673 using circle-fit, 0.00634, 0.00664,0.00661, 0.00613 and 0.00599 using the line-fit, and 0.00737, 0.00795, 0.00712, 0.00730 and 0.00653 using the half-power.Typical loss factor ranges, averages ðμηÞ and standard deviations ðsηÞ of the treated cylindrical shell with h2¼0.62 mm arelisted in Table 3. It is seen that three methods predict similar loss factors and their standard deviations are about 10% or less.This value is quite reasonable considering the nature of damping and closely spaced modes. Also, the averaging processreduces random errors, and thus the results are believed to be quite accurate.

The averaged modal loss factors ðηmnÞ of untreated and treated shells are compared in Table 4. It is seen that the averageloss factor of the aluminum cylinder is about 0.0009; it is comparable to values found in the literature. The modal lossfactors of the treated cylindrical shells increase as cardboard thickness (h2) increases as expected. The loss factors are about

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

Table 3Loss factor range, averages (mƞ) and standard deviations (sη) for a treated cylindrical shell with h2¼0.62 mm.

Mode Circle-fit Line-fit Half-power mƞ (%) sη (%)

Index (m,n) ƞmn (%)

1 (0,2) 0.600–0.768 0.565–0.743 0.619–0.762 0.651 0.0662 (1,2) 0.673–0.740 0.599–0.710 0.653–0.795 0.693 0.0493 (0,3) 0.647–0.705 0.630–0.705 0.647–0.717 0.668 0.0274 (1,3) 0.651–0.789 0.624–0.783 0.687–0.759 0.716 0.0655 (2,3) 0.577–0.656 0.554–0.711 0.568–0.711 0.638 0.061

m = 0 m = 1 m = 2

n = 2

n = 3

n = 4

Fig. 8. First nine mode shapes of a thin cylindrical shell with free boundaries.

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] 9

0.004, 0.007 and 0.012 with h2¼0.31, 0.62 and 0.98 mm, respectively. Note that the degenerate modes at a higher frequencywith h2¼0.98 mm could not be separated, and thus no results are reported for such modes.

7. Determination of material properties of cardboard liners

The material properties of the cardboard liner used in this paper are determined using a lightly damped elastic platetreated with a cardboard liner. The dimensions of the base plate are selected so that this plate would have no symmetricalmodes, and the modes would also not be closely spaced. Two steel plates (300 mm � 201 mm � 1 mm) are treated withcardboard liners with h2¼0.31 and 0.62 mm. The density of the base steel plate is 7664 kg/m3 while Poisson’s ratio is 0.3.The cardboard is glued to the base plate using an adhesive whose elastic modulus is much lower than the liner. Furthermore,the thickness of the adhesive is negligible as the thickness of the treated plate is nearly equal to the thickness of the baseplate plus the thickness of cardboard. Therefore, the effect of the adhesive is assumed to be negligible. Also, each liner withdifferent thickness used here can be considered as a single layer which is nearly isotropic although cardboard materials areexpected to be anisotropic [18].

A preliminary test on the plates showed that the accelerometer provided additional damping. Therefore, modalparameters of the plates are again determined using acoustic ~GijðωÞ functions measured in the acoustic free field and half-power method. Both untreated and treated plates are modeled using homogeneous and composite finite element (FE)formulations, which have already been found to predict the modal parameters of structures with high accuracy [17]; this FEformulation will be explained in Section 8.2. Overall, the plates are modeled using a quite fine mesh (2400 elements).

First, Young’s modulus and loss factor of the untreated plate (steel) are determined by comparing experimentaland theoretical modal parameters. Then, the material properties of the cardboard liner are extracted by comparingexperimental and theoretical modal parameters of the treated plates and by the material properties of the base steel plate

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

Table 4Effect of cardboard liner thickness on measured loss factors. See Tables 1 and 2 for natural frequencies.

Mode ƞmn (%)

Index (m,n) h2¼0 mm h2¼0.31 mm h2¼0.62 mm h2¼0.98 mm

1 (0,2) 0.130 0.514 0.651 1.1922 (1,2) 0.114 0.434 0.693 1.0933 (0,3) 0.068 0.364 0.668 1.2524 (1,3) 0.055 0.392 0.716 a

5 (2,3) 0.073 0.357 0.638 a

ηð%Þ ¼∑3 or 5k ¼ 1 ηmn;k=5� 100 0.088 0.412 0.673 1.179

a Could not be identified.

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]10

determined before. Overall, Young’s moduli (E) and material loss factors (η) of steel and cardboard liner used in this studyare determined to be as follows: E¼205.80 and 4.72 GPa, and η¼0.001 and 0.034, respectively, with h2¼0.31 and 0.62. Themeasured and predicted natural frequencies and modal loss factors of the plates using the identified material properties arecompared in Table 5, where s and q are the number of half-waves in a mode shape along the long and short edges,respectively. It is seen that the natural frequencies match well as the average differences between the experimental andtheoretical natural frequencies are 0.53, 0.55 and 0.88% for plates with h2¼0, 0.31 and 0.62 mm, respectively. The averageexperimental loss factors are ηr¼0.0010, 0.0021 and 0.0043 with h2¼0, 0.31 and 0.62 mm, respectively, while the predictedloss factors of the plates are ηr¼0.0021 and 0.0042 with h2¼0.31 and 0.62 mm, respectively, for all modes. It seems both theuntreated and treated plates are proportionally damped structures.

8. Estimation of damping

Two approaches are introduced below to model a cylindrical shell with a cardboard liner. Each proposed approachattempts to predict the damping introduced by the cardboard liner.

8.1. Analytical approach

The Ross–Kerwin–Ungar (RKU) equations [19] are used to predict the equivalent elastic and damping properties of acomposite structure assuming that the cardboard liner acts as a free-layer damping treatment. The flexural rigidity, EcIc, ofthe two-layer system can be written as follows:

EcIcE1I1

¼ 1þEh3þ3ð1þhÞ2 Eh

1þEh

!ð10Þ

where I is the second moment of area, E¼ E2=E1 and h¼ h2=h1; and subscripts 1, 2 and c refer to the base structure, coated(cardboard) layer and composite structure, respectively. It should be stated that as the damping of the base structure couldbe significant, and consequently it should be taken into account, the base structure damping is usually ignored in theliterature [19]. When the loss factors of the base structure and the cardboard layer are η1 and η2, respectively, rewriteEq. (10) as follows:

EcIc 1þ jηc� �

E1I1 1þ jη1� � ¼ 1þEh

3ð1þ jη2Þþ3ð1þhÞ2 Ehð1þ jη2Þ1þEhð1þ jη2Þ

" #ð11Þ

where ηc is the loss factor of a two-layer structure and j¼ffiffiffiffiffiffiffiffi�1

p. Here, it is assumed that the cardboard liner follows the

elastic deformation of the base tube and the interfacial damping mechanism is negligible. As a result, the averaged lossfactors of the treated cylindrical shells with h2¼0.31, 0.62 and 0.98 mm are found to be 0.0023, 0.0041 and 0.0068,respectively. These values are, however, smaller than the measured results of Section 6.

8.2. Computational approach

A recently developed finite element formulation for composite structures [17] is used to model a cylindrical shell with acardboard liner. This formulation is based on a complex eigenvalue method, and it represents multilayer compositestructures with an equivalent layer. In this formulation, the damping capability is added to the element by defining thecomplex Young’s modulus of the material as ~E ¼ Eð1þ jηÞ where E and η are the storage Young’s modulus and the loss factor,respectively. The stiffness and mass matrices for individual layers of the composite element are expressed with respect to

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

Table 5Measured and predicted modal parameters of the untreated and treated elastic plates.

Mode h2¼0 h2¼0.31 mm h2¼0.62 mm

Measured Predicted(ηr¼0.10%for all modes)

Measured Predicted(ηr¼0.21%for all modes)

Measured Predicted(ηr¼0.42%for all modes)

Index, r (s,q) ωr (Hz) ƞr (%) ωr (Hz) ωr (Hz) ƞr (%) ωr (Hz) ωr (Hz) ƞr (%) ωr (Hz)

1 (2,2) 57.12 0.10 57.01 57.25 0.19 57.28 58.20 0.41 58.572 (3,1) 61.14 0.10 61.15 61.62 0.22 61.45 63.24 0.42 62.843 (3,2) 132.38 0.10 131.61 132.76 a 132.23 134.88 0.40 135.214 (1,3) 141.76 0.12 141.17 141.75 0.22 141.85 144.12 0.43 145.065 (4,1) 164.38 0.10 164.34 165.38 0.22 165.13 168.94 0.42 168.866 (2,3) 188.26 0.11 190.27 189.01 0.20 191.18 193.89 0.40 195.507 (4,2) 244.50 a 244.15 245.90 a 245.31 251.54 0.45 250.838 (3,3) 280.06 0.10 279.99 280.66 0.23 281.32 285.54 a 287.649 (5,1) 341.12 0.09 342.81 344.51 0.22 344.45 355.46 0.49 352.22

10 (1,4) 377.16 0.11 382.63 376.77 a 384.45 382.00 0.45 393.1211 (5,2) 410.92 0.09 415.35 414.52 0.21 417.33 421.08 0.40 426.7212 (4,3) 415.59 a 419.38 416.14 a 421.35 425.05 a 430.81

ηð%Þ ¼∑12r ¼ 1ηr=12� 100 0.10 0.21 0.43

a Could not be identified.

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] 11

the neutral plane of the composite element are defined as follows:

Kol ¼

T 0 0 0T 0 0

T 0T

26664

37775T

Kel

T 0 0 0T 0 0

T 0T

26664

37775; l¼ 1;2 ð12Þ

Mol ¼

T 0 0 0T 0 0

T 0T

26664

37775T

Mel

T 0 0 0T 0 0

T 0T

26664

37775; l¼ 1;2 ð13Þ

where Kel and Me

l are the stiffness and mass matrices of each layer of the composite element which are computed as if eachlayer is on the neutral plane, T is a transformation matrix of dimension 6 � 6, 0 is a 6 � 6 null matrix, and subscript l refersto the individual layer of a composite element [17]. Although any shell element can be used to develop the compositeelement, the specific shell element utilized in this formulation includes the “physical drilling degrees of freedom” in thenormal direction to the element [17]. The stiffness and mass matrices for the composite element are obtained by adding thecontributions of individual layers via a summation process as follows:

Kc ¼ ∑2

l ¼ 1Kol ; ð14Þ

Mc ¼ ∑2

l ¼ 1Mo

l ; ð15Þ

where the subscripts c refers to a composite shell. It should be noted that the summation above represents the matrixbuilding process rather than a simple matrix summation. After the elemental stiffness and mass matrices are obtained withrespect to the neutral plane of the composite element, these matrices are transformed to a common global coordinatesystem and the natural frequencies and modal damping values for a damped system can be obtained by solving theconventional eigenvalue problem:

ðK�λ2MÞΨ¼ 0 ð16Þwhere K and M are the global stiffness and mass matrices of a cylindrical shell with the cardboard liners, respectively, λ2 isthe eigenvalue and Ψ is the eigenvector. The solution of the eigenvalue problem above results in complex eigenvalues ðλ2r Þand complex eigenvectors ðΨrÞ. The natural frequencies ðωrÞ and loss factors ðηrÞ are determined by defining λ2r ¼ω2

r ð1þ jηrÞand by the following:

ω2r ¼ Reðλ2r Þ ð17Þ

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H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]12

ηr ¼ Imðλ2r Þ=Reðλ2r Þ ð18Þ

As a result, the loss factors of the treated cylindrical shells are determined using the FE formulation presented here, andthe results are listed in Table 6. This method yields averaged loss factors to be about 0.0021, 0.0037 and 0.0059% for h2¼0.31,0.62 mm and 0.98 mm, respectively. Observe that these values are close to the results predicted via the analytical method.

The predicted natural frequencies and corresponding errors are given in Table 7a. It is seen that the predicted naturalfrequencies are higher than the measured ones. This is expected because the layers are assumed to be fully joined in thecomputational method. In Section 6, it was shown that the natural frequencies decrease as h2 increases. It is clear that thecardboard liner has mostly the mass effect. Therefore, the cardboard liner is modeled as a distributed mass and its stiffnesseffect is ignored. The natural frequencies of the treated cylinders for this case are given in Table 7b. It is seen that thepredicted natural frequencies are very close to the measured ones, i.e., the error is less than 1% even for h2¼0.98 mm.Although the strain field within the cardboard liner would yield some damping, experimental results suggest that the elasticeffect of the liner can be ignored when calculating the natural frequencies of a cylinder with a liner. The effects of surfaceproperties of the test samples, frictional sliding between the base shell and cardboard liner, etc. is the subject of afuture work.

9. Identification of damping mechanisms

The averaged loss factors (for the first 5 modes) of a cylindrical shell with h2¼0.31, 0.62 and 0.98 mm are determined tobe ηr¼0.0023, 0.0041 and 0.0068 via the analytical approach and ηr¼0.0021, 0.0037 and 0.0059 via the computational

Table 6Predicted modal loss factors of the treated cylinders using a computational method.

Mode h2¼0.31 mm h2¼0.62 mm h2¼0.98 mm

Index (m,n) ƞmn (%)

1 (0,2) 0.219 0.385 0.6262 (1,2) 0.219 0.385 0.6263 (0,3) 0.218 0.383 0.6224 (1,3) 0.218 0.383 0.6225 (2,3) 0.183 0.294 0.459

Table 7Predicted natural frequencies of the treated cylinders determined via a computational method where the elastic effect of the liner is included in (a) andignored in (b).

Mode h2¼0.31 mm h2¼0.62 mm h2¼0.98 mm

Index (m,n) ωmn (Hz) Error (%) ωmn (Hz) Error (%) ωmn (Hz) Error (%)

(a) Elastic effect of the liner is included1 (0,2) 249.3 2.24 252.1 5.36 258.7 9.482 (1,2) 259.5 1.77 262.5 4.91 269.3 9.103 (0,3) 705.3 2.28 713.3 5.38 731.8 8.764 (1,3) 718.6 2.13 726.7 5.17 745.4 8.585 (2,3) 1012.5 0.35 1014.2 2.11 1024.7 4.156 (2,2) 1331.6 0.88 1317.5 1.45 1303.1 1.277 (0,4) 1352.8 1.40 1367.8 4.09 1402.6 7.778 (1,4) 1366.7 1.33 1381.7 3.85 1416.9 7.839 (2,4) 1478.6 1.86 1492.6 4.89 1526.9 7.9910 (3,4) 1830.3 1.10 1834.6 3.08 1827.4 3.69Average absolute error (%) 1.53 4.03 6.86

(b) Elastic effect of the liner is ignored1 (0,2) 244.7 0.34 240.7 0.56 236.3 0.002 (1,2) 254.7 �0.12 250.5 0.14 246.0 �0.343 (0,3) 692.3 0.39 681.1 0.62 668.8 �0.604 (1,3) 705.4 0.25 693.9 0.43 681.3 �0.755 (2,3) 999.6 �0.93 983.3 �1.01 965.4 �1.876 (2,2) 1324.2 0.32 1302.4 0.29 1278.4 �0.657 (0,4) 1328.1 �0.46 1306.5 �0.57 1282.9 �1.438 (1,4) 1341.7 �0.52 1319.9 �0.80 1296.0 �1.379 (2,4) 1453.0 0.09 1429.3 0.44 1403.4 �0.7510 (3,4) 1804.3 �0.33 1774.9 �0.27 1742.5 �1.13Average absolute error (%) 0.37 0.51 0.89

Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i

H. Koruk et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] 13

method, respectively. The corresponding measured loss factors are 0.0041, 0.0067 and 0.0118 for the cylindrical shell withh2¼0.31, 0.62 and 0.98, respectively. Since both analytical and computational methods consider only the material damping,the difference between measured (real) and computational or analytical results (based on simplifying formulation) could beattributed to the dry friction damping. The loss factors due to friction should increase as the cardboard thickness (and thusthe normal load) is increased. Indeed, the values of frictional (the difference between experimental and computational) lossfactors are 0.0020, 0.0031 and 0.0059 for the treated cylinders with h2¼0.31, 0.62 and 0.98 mm, respectively.

The total (equivalent) loss factor ðηSÞ of a composite cylindrical shell with a cardboard liner can be approximated byηS ¼ ηMþηQ where ηM and ηQ are the loss factors due to material damping and friction. The ηM can be predicted usinganalytical or computational methods once the material properties of a cardboard liner are known. Interestingly, the resultsof this paper show that there is a relationship between frictional and material (or total) loss factors. The ratios between theηQ and ηS are 0.49, 0.46 and 0.50 for the treated cylinders with h2¼0.31, 0.62 and 0.98 mm, respectively. In fact, thefollowing relationship ηQ ¼ cηM is seen where cffi0:85�1:00 for the cases presented here. Therefore, the effective dampingof a cylindrical shell with a cardboard liner can be simply approximated by ηS ¼ c1ηM where c1ffi1:85�2:00 for the casespresented here.

10. Conclusion

This paper investigates the effect of cardboard liners on the resonant vibro-acoustic responses of cylindrical shells.Specific contributions of this paper include the following. A controlled experiment is first designed to measure acoustic andvibration frequency response functions with different cardboard thicknesses in acoustic free field. By analyzing theexperimental data, the difficulties in the identification of modal parameters of cylindrical shells with cardboard liners areexplored, the natural frequencies and loss factors of structures are determined using several frequency domain methods andmeasured data, and the variations of natural frequencies and uncertainty levels in loss factors are explored. The materialproperties of a cardboard liner are also determined by utilizing an elastic plate treated with a cardboard liner. The modalloss factors of a cylindrical shell with cardboard liners are then estimated using analytical and computational approacheswhile the natural frequencies of cylinders are predicted using a computational method. Finally, potential sources of dampingmechanisms are identified by comparing experimental, analytical and computational damping levels. This suggests that thedry friction contributes to about half of the overall damping.

The results show that an equivalent loss factor for a cylindrical shell with a cardboard liner can be calculated. Once theequivalent values of loss factor and flexural rigidity are available, analytical methods for a homogeneous shell can be used topredict the modal behavior of a cylindrical shell with a cardboard liner. Similarly, computational methods like the one givenin Section 8.2 can be used to predict the vibro-acoustic behavior of a cylindrical shell with a cardboard liner. Subsequently,one could calculate the modal radiation efficiency of a cylindrical shell using the methodology of this paper. The effect ofsurface properties and the role of sliding friction between base shell and cardboard liner should be pursued in future.

Acknowledgment

We acknowledge the Smart Vehicle Concepts Center (www.SmartVehicleCenter.org) and the National Science Founda-tion Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) for supporting thisfundamental study.

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Please cite this article as: H. Koruk, et al., Modal analysis of thin cylindrical shells with cardboard liners and estimation ofloss factors, Mech. Syst. Signal Process. (2013), http://dx.doi.org/10.1016/j.ymssp.2013.10.026i