experimental study on vibration and damping of curved panel treated with constrained viscoelastic...

Composite Structures 92 (2010) 233–243

Contents lists available at ScienceDirect

Composite Structures

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

Experimental study on vibration and damping of curved panel treatedwith constrained viscoelastic layer

Navin Kumar a,*, S.P. Singh b

a Design and Manufacturing Institute, Mechanical Engineering Department, Stevens Institute of Technology, Hoboken, NJ 07030, USAb Mechanical Engineering Department, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

a r t i c l e i n f o

Article history:Available online 15 July 2009

Keywords:Constrained layer damping (CLD)Modal strain energy (MSE)Viscoelastic material (VEM)

0263-8223/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.compstruct.2009.07.011

* Corresponding author. Tel.: +1 201 216 8701; faxE-mail address: [email protected] (N. Kum

a b s t r a c t

This paper presents experimental investigation on the damping effects of constrained layer dampingtreatment on a curved panel. Vibration attenuation of the curved panel is achieved by attaching con-straining layer damping patches at the optimal locations. The placement strategies of constrained layerpatches are devised using the modal strain energy (MSE) method. Locations for application of dampingpatches are those, where modal strain energy is maximum for the particular mode. The treatment is thenapplied to the elements that have highest MSE in order to target specific modes of vibrations. Extensiveexperiments are conducted by making number of separate samples of viscoelastic and constrained layerdamping patches for each configuration to damp different modes simultaneously or independently. Theexperimental results demonstrate utility of the modal strain energy technique as an effective tool forselecting the locations of the constrained layer damping treatment to achieve desired damping character-istics over a broad frequency band.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Constrained layer damping (CLD) treatment has been an effec-tive way to suppress vibration and sound radiation from variousstructures. The pioneer work to sandwich beams with viscoelasticcore could be way back to Kerwin [1] for an effective, complex,flexural stiffness of the three-layer beams with damping corelayer. After his work, DiTaranto [2] and Mead and Markus [3]extended Kerwin’s work and developed a sixth order equationof motion governing the axial and bending vibrations of beams.Most of these researchers typically used the closed-form solutionbecause finite element techniques were not readily available forthis class of problems. Nakra [4] and Mead [5] reviewed all thisarea and they have discussed the differences and similarities be-tween the theories. The above theories laid the foundation for theanalysis of sandwich beams with constrained layer dampingtreatments.

Most of these early works dealt with full coverage passive con-strained layer damping [PCLD] treatments that are evidently notpractical in purpose. For partially covered viscoelastically dampedsandwich beams or plates, analytical study was carried out by Lallet al. who solved, by using three different approaches, the eigen-value problem for a beam [6] and for a plate [7] with a singledamping patch. The study of vibration and damping in shells with

ll rights reserved.

: +1 201 216 8963.ar).

added damping treatment has also been of interest to manyresearchers. Unconstrained viscoelastic layers were widely usedas a passive means to damp the vibration of structures. Markus[8] used the unconstrained passive damping treatments to dampthe vibrations of thin cylindrical shell. Pan [9] studied the axisym-metrical vibration of a finite length cylindrical shell with a visco-elastic core. Alam and Asnani [10,11] considered the vibrationand damping analysis of a general multilayered cylindrical shellhaving an arbitrary number of orthotropic material layers and vis-coelastic layers. Constrained viscoelastic layers have been widelyused to reduce excessive vibration and noise in the structures.The high damping capacities of structures with constrained damp-ing layer are mostly due to shear deformation of viscoelastic mate-rials. The vibration of cylindrical shells with constrained dampinglayer has received attention from many researchers [12–16]. Chenand Huang investigated the damping effects of passive constrainedlayer damping (PCLD) treatment of strip type along longitudinal[17] and along circumference [18] directions. Vibration and damp-ing analyses of isotropic and orthotropic cylindrical shells withconstrained damping layer were carried out by Ramesh and Ga-nesh [19]. They adopted the finite element method based on dis-crete layer theory to discuss the cases that the two faces are ofequal thickness. Wang and Chen [20] discussed the damping oforthotropic cylindrical shells with a partially passive constraineddamping layer.

These theoretical works and parametric studies on constrainedlayer damping treatments for vibration and noise suppression

http://dx.doi.org/10.1016/j.compstruct.2009.07.011

mailto:[email protected]

http://www.sciencedirect.com/science/journal/02638223

http://www.elsevier.com/locate/compstruct

Constraining layer (tc) x

234 N. Kumar, S.P. Singh / Composite Structures 92 (2010) 233–243

really provide assistance to design decision. However, studiesbased on the optimization are quite less, particularly, in the exper-imental based optimum design of partial constrained layer damp-ing treatment of curved panels, there has been no existingliterature to the authors’ knowledge. Chen and Huang [21] pre-sented a study on optimal placement of PCLD treatment for vibra-tion suppression of plates. In their optimization, the structuraldamping plays the main performance index and the frequencies’shift and PCLD thickness play as penalty functions. Zeng et al.[22] have adopted Genetic algorithm (GA) based penalty functionmethod is employed to find the optimal layout of rectangular pas-sive constrained layer damping patches with aim to minimize thestructural volume displacement of PCLD-treated cylindrical shell.An optimization solution of rectangular PCLD patches locationsand dimensions are obtained under the constraint of total amountof PCLD in terms of percentage added weight to the base structure.Ro and Baz [23] presented optimal damping studies with an activeconstrained viscoelastic damping treatment distributed over re-gions of high strain energy on a plate. Masti and Sansbury [15]have investigated the effectiveness of using a strain energy-basedpartial coating approach for vibration attenuation of cylindricalshells.

These optimal and parametric studies on constrained layerdamping treatments for vibration and noise suppression reallyprovide assistance to design decision. Most of the studies onthe shell treated with constrained layer damping; paid moreattention on the closed circular cylindrical shell and they weremainly based on the analytical investigations. However, thereare few studies based on the constrained layer damping treatedcurved panel, particularly experimental study based on the opti-mally placed constrained layer damping treated curved panels,is not fully explored. Study presented in this paper attempts toarrive at the design of partial constrained layer treatment ofcurved panel by finding an layout of the constrained layer patchesbasis on the modal strain energy. A constrained layer dampingpatch which consists of viscoelastic core, sandwiched betweenbase structure and the constraining layer. The curved panel struc-ture is divided into number of elements and for each element,modal strain energy is calculated. Constrained layer dampingpatches are attached at the locations where modal strain energyis high. Separate samples of viscoelastic layer and constrainedlayer damping patches are fabricated and attached at differentlocations for each configuration to attenuate different modessimultaneously or independently. Experiments are performed todamp first four modes of vibration of the constrained layer damp-ing treated curved panel.

1

2

3

4

5

6

7

8 ξ

η

ζ

θ,v

x,uz,w

Fig. 1. Geometry of eight-node Serendipity element.

2. Finite element formulation

In this study a simple and efficient finite element named Seren-dipity eight-node element is adopted as shown in Fig. 1. This kindof shell element has eight nodes and contains five degrees of free-dom at each of the node:u, v, w, a, b where the first three are trans-lations in global directions and the last two are rotations about thelocal axis. The element geometry can be represented by the naturalcoordinate system (n–g–f) where the curvilinear coordinates (n–g)are in the shell mid-surface while f is linear coordinate in thethickness direction. According to the isoparametric formulation,these coordinates (n, g and f) will vary from �1 to +1.

2.1. Kinematic constraint relationships

Fig. 2 shows the curved panel, partially treated with constrainedlayer damping. The length, thickness, radius and shallowness angleof the panel are denoted by L, tb, r and u, respectively. The top sur-face of the panel is bonded with the constrained layer dampingtreated patches. These patches consist of viscoelastic material layersandwiched between the base panel and the constraining layer.The thickness of the viscoelastic layer is ts and that of the con-straining layer is tc. Middle surface of the base curved panel is con-sidered as the reference surface along which the axial (x),circumferential (h) and radial (z) coordinate system is defined.The longitudinal and circumferential deformations at any pointof the constrained layer damping treated curved panel systemare given as u and v, respectively. The total displacement at anypoint in the layer can be generalized as

uðx; h; zÞ ¼ u0ðx; hÞ þ zaðx; hÞuðx; h; zÞ ¼ v0ðx; hÞ þ zbðx; hÞwðx; y; zÞ ¼ w0ðx; yÞ

ð1Þ

where u0 and v0 are the displacements at the reference plane. Sym-bols a and b correspond to the components of the rotational of thenormal to the middle surface of the panel in axial and circumferen-tial directions, respectively. Radial displacement (w) assumed to beconstant through the thickness of the panel. The generalized dis-placement variable {d} is the nodal displacement vector of an ele-ment and given as

fdg ¼ ½u1 v1 w1 a1 b1 u2 v2 � � � b8� ð2Þ

Base structure (tb)

Viscoelastic layer (ts)

z

θ

L

r

Piezoelectric sensor layer

Fig. 2. Attachment of the constrained layer damping patches on the curved panel.

N. Kumar, S.P. Singh / Composite Structures 92 (2010) 233–243 235

2.2. Strain–displacement relationships

Based on an improved shallow shell theory using the modifiedDonnell’s approximations, with the displacement relations accord-ing to the improved shell theory in cylindrical coordinates (x, h, z)as

ex ¼ @u0@x þ z @a

@x ; eh ¼ 1r@v0@h þ

w0r þ z

r@b@h

� �cxh ¼

@v0@x þ z @b

@x þ 1r@u0@h þ z

r@a@h ; cxz ¼

@w0@x

chz ¼ 1r@w0@h �

v0r þ b

ð3Þ

Table 1Comparison of Eigen values (Hz) of curved panel.

Mode no. FET in [13] ERR in [13] [14] [15] Present

1 870 870 869 870 8692 958 958 957 958 9563 1288 1288 1287 1287 12904 1363 1364 1363 1364 1360

2.3. Stress–strain relationships

The stress strain relations in a layer are listed below. The trans-verse normal stress rz in the z direction is assumed to be zero.

rx ¼ Eð1�t2Þ ðex þ mehÞ; rh ¼ E

ð1�t2Þ ðeh þ mexÞshz ¼ kGchz

sxz ¼ kGcxzsxh ¼ Gcxh

ð4Þ

where E is the Young’s modulus, G the shear modulus, m the Pois-son’s ratio of the material of the layer, k is the shear correction fac-tor and is taken as 5/6.

2.4. Kinetic energy integrals

The elemental kinetic energy is given as

T ¼ 12

ZVqf _dtgTf _dtgdV ð5Þ

where q is the mass density. The generalized translational displace-ment vector in {dt} is given as

fdtgT ¼ fu; v;wg ð6Þ

where u, v and w are the mid-surface displacements in x, h and zdirections.

The generalized translational displacement vector at any pointwithin an element is given as

fdtg ¼ ½Nt �fdetg ð7Þ

where fdetg ¼ ½fdt1gTfdt2gT . . . . . . . . . . . . fdt8gT � and ½Nt� ¼ ½Nt1 Nt2 . . .

. . . . . . . . . . . . Nt8�.Substituting expression in the kinetic energy expression (5)

yields the mass matrix [Me] of an element, according to

T ¼ 12

ZVqf _de

tgT ½Nt�T ½Nt �f _de

tgdV ð8Þ

where elemental mass matrix [Me] defined as

½Me� ¼Z

Vq½Nt �T ½Nt�dV ð9Þ

1 2

3

4

5

6

7

89

11

12

13

14

15

1718

19

20

21

27

2322 16

24

25

26

28

2930

32

33

34

313736

35

38

39

40

41

42

4344

46

45

47

48

49

5051

52

54

55

53

56

5758

59

60

61

62

63

6564

67

66

70

68

69

10

Fig. 3. Curved panel divided in the 7 � 10 number of elements.

2.5. Total potential energy integrals

The total potential energy of the shell element is given as

Tp ¼ U �W; ð10Þ

The elastic strain energy U is given as

U ¼ 12

ZVfegTfDgfegdV ð11Þ

where fegT ¼ fex; eh; cxh; cxz; chzg and {D} is the stress–strain matrix.The work done by external forces due to applied surface trac-

tion is give as

W ¼Z

AfdegTf�reðx; hÞgdA ¼

ZAfdeg½N�Tf�regdA ¼ fFegfdeg ð12Þ

where �reðx; hÞ is the surface traction vector and {Fe}is an elementalforce vector.Strain matrix {e} of Eq. (11) can be represented by

fegT ¼ ½B�fdeg ð13Þ

Substituting expression (13) in Eq. (11), yield the strain energy of anelement in the form

U ¼ 12fdegT ½Ke�fdeg ð14Þ

where [Ke] is the stiffness matrix of an element, given as

½Ke� ¼Z

V½B�T ½D�½B�dV ;

½Ke� ¼ ½Kett � þ ½K

ert� þ ½K

etr� þ ½K

err � þ ½K

etc� þ ½K

erc�

½Kett� ¼

ZV½Bt�T ½Dtt�½Bt �dV

½Kert� ¼

ZV½Bt�T ½Dtr�½Br �dV

½Ketr� ¼

ZV½Br�T ½Drt�½Bt �dV

½Kerr� ¼

ZV½Br�T ½Drr�½Br�dV

2.6. Equation of motion

To drive the equation for constrained layer damping treatedcurved panel the Hamilton principle is used:Z t2

t1

½T � U þW�dt ¼ 0 ð15Þ

t1 and t2 define the time interval. All variations must vanish at t = t1

and t = t2. Substituting the potential energy U (Eq. (14)), the workdone W (Eq. (12)), and kinetic energy T (Eq. (8)) in Eq. (15) and tak-ing the variation, yield the dynamic finite element equations of anelement

Table 2Physical and geometrical specifications of the panel, viscoelastic and constraining layer.

Length of the panel 260 mm Thickness of the viscoelastic layer 1 mmThickness of the base layer 0.60 mm Density of the viscoelastic layer 1714 kg/m3

Radius of curvature 151 mm Shear modulus of the VEM 5 � 108 (1 + 0.8i) PaIncluded angle 53� Thickness of the constrained layer 0.3 mmDensity of the base 2710 kg/m3 Density of the constraining layer 2710 kg/m3

Thickness of piezo sensor layer 0.1 mm Density of the piezoelectric layer 1780 kg/m3

Electro-mechanical conversion (1 dir.) 23 � 10�12 m/V Electro-mechanical conversion (3 dir.) �23 � 10�12 m/V


½Me�f€dtg þ ½Kett �fd

etg þ ½K

etr�fd

erg ¼ fF

etg ð16Þ

and

½Kerr�fd

erg þ ½K

etr �fd

etg ¼ fF

erg ð17Þ

In which [Me] is the elemental mass matrix, [Kett], [Ke

tr], [Kert] and

[Kerr] are the elemental stiffness matrices, {Fe

t }, {Fer } are the excita-

tion force vectors corresponding to the translational and rotationalcoordinates, respectively.

2.7. Global equation of motion

In order to obtain global equation of motion, elemental equa-tions of motion are assembled in such a manner to obtain globalequations of motion.

½M�f€Xtg þ ½Ktt�fXtg þ ½Ktr �fXrg ¼ fFtg ð18Þ

and

½Krt�fXtg þ ½Krr �fXrg ¼ fFrg ð19Þ

(a)

(c)

Cantilevered P

W0 5

1015

0

5

10

15

Length (Elements)Width (Elements)

05

1015

0

5

10

15

Length (Elements)Width (Elements) W

Fig. 4. Mode shapes of the cantilevered curved panel: (a) first m

The global rotational degree of freedom can be condensed to obtainthe global equations of motion in terms of global translational de-gree of freedom only as follows

½M�f€Xtg þ ½K��fXtg þ ½Cd�f _Xtg ¼ fFg ð20Þ

in which

½K�� ¼ ½Ktt� � ½Ktr �½Krr ��1½Krt �

and

fFg ¼ fFtg � ½Ktr�½Krr��1fFrg

where [M] and [Ktt], [Ktr], [Krr] are the global mass and stiffnessmatrices; {Xt}, {Xr} are the global nodal generalized displacementcoordinates; involving the boundary conditions. Since damping isinherently present in all practical situations, it is included in theanalysis through the matrix [Cd], which can be formed by linearcombination of mass and stiffness matrices as formulated by Wil-son and Penzien [24]. Eq. (20) can be formulated to compute the fre-quency response function (FRF).

(b)

(d)

anel

0 5

1015

0

5

10

15

Length (Elements)idth (Elements)

05

1015

0

5

10

15

Length (Elements)idth (Elements)

ode (b) second mode (c) third mode and (d) fourth mode.

Fig. 6. Complete experimental setup of the constrained layer damping treatedcurved panel.


3. Finite element model validation

To validate the finite element formulation presented in the pre-vious section, present model is compared with those obtained byother researchers [13–15]. They have studied a cylindrical shell pa-nel with clamped edges. We have considered same dimensions,material properties and boundary conditions as taken in papers[13–15], and verified our finite element model. Ref. [13], deter-mined Eigen values by using Triangular finite element (FET) andExtended Rayleigh–Ritz (ERR) method. Table 1 presents the com-parison between the results of Refs. [13–15] and that the presentstudies. The comparison shows good agreement.

4. Materials and mode shapes

Based on the above modeling, finite element program is madefor the vibration analysis of sandwich curved panel. The programis general and can accommodate any number of elements. Butfor the current analysis number of elements are fixed to 70, with245 nodes, as shown in the Fig. 3. For the numerical simulationssame dimensions and properties of the three-layer panel is takenas would be used in experimental studies. The physical and geo-metrical specifications of the curved panel, viscoelastic, piezoelec-tric sensor and constraining layer are given in the Table 2. In thefinite element model of the CLD treated curved panel, after apply-ing the boundary conditions, mode shapes and modal frequenciesare calculated. Mode shapes of the curved panel are presented inthe Fig. 4. The first mode of the cantilevered panel is a twistingmode; the second mode is the bending mode of flapping wing type,in which both corner edges moves up and down simultaneously,the third and the fourth modes are complex bending modes.

0 7 14 21 28 35 42 49 56 63 700

10

20

30

40

50

60

70

80

90

100

Element Number

Mod

al S

trai

n E

nerg

y (%

)

0 7 14 21 28 35 42 49 56 63 700

10

20

30

40

50

60

70

80

90

100

Element Number

Str

ain

Mod

al E

nerg

y (

%)

)a(

)c(

Fig. 5. Modal strain energy plot of the cantilevered curved panel: (a)

5. Modal strain energies: location selection for attachment ofCLD patches

To damp the curved panel vibration effectively, an efficientdamping patch distribution configuration is needed. But the selec-tion of the locations where the CLD patches should be attached todamp the single mode or a set of modes is a critical issue. Strainenergy profile of a particular mode can be an effective mean to at-tach constrained layer damping patches. Procedure to obtain themodal strain energy profile begins by solving the eigenvalue

0 7 14 21 28 35 42 49 56 63 700

10

20

30

40

50

60

70

80

90

100

Element Number

Mod

al S

trai

n E

nerg

y (%

)

0 7 14 21 28 35 42 49 56 63 700

10

20

30

40

50

60

70

80

90

100

Element Number

Mod

al S

trai

n E

nerg

y (%

)

)b(

)d(

first mode (b) second mode (c) third mode and (d) fourth mode.

Constrained layer damping patch

Sensor amplifier System

Piezoelectric Sensor

Base curved panel

Dynamic Signal Analyzer

Impact hammer

Data acquisition system

Point of impact

Fig. 7. Schematic diagram of complete experimental setup.

Fig. 8. Cantilevered bare curved panel with piezoelectric sensor attached at theback side.

Table 3Eigen values (Hz) from FEM and experimental studies.

Mode no. FEM study Experimental study

1 60 602 120 1193 193 1904 243 238

Fig. 9. (a) Curved panel with constrained layer damping (CLD) patches attached


problem given by Eq. (20) with {F} = 0, to determine the eigenvec-tors of the untreated curved panel. The modal strain energy (Uij) ofany curved panel element i at any mode of vibration j, is deter-mined as:

Uij ¼ ½wij�T ½Ki�½wij� ð21Þ

where i = 1, . . . , N and j = 1, . . . M.With N and M denoting the number of elements and number of

modes considered, respectively. [Ki], denotes the stiffness matrix ofelement i and [wij] defines the eigenvectors of element i at mode j.The strain energy plots for first four modes of the curved panel areplotted as shown in the Fig. 5. It is observed that for first mode themodal strain energy is maximum at both the side edges near thefixed end. It is due to the fact that first mode a twisting modeand strain will be maximum at the corners of the fixed end. Forthe second mode, modal strain energy is maximum near the fixedend and its magnitude decreases from fixed end to free end. Simi-larly modal strain energy is plotted for the third and the fourthmode. As these modes are complex bending modes, modal strainenergy is maximum near the cantilevered end of the curved panel.

The attachment of the CLD patches is based on the modal strainenergy. The optimum location for the application of CLD patches,are those, where modal strain energy is maximum for particularmode. These locations will also give maximum passive dampingaddition in case of failure of active damping. Based on the modelstrain energy, CLD patches have been attached to damp the firstfour modes independently or simultaneously.

on both the edges. (b) Elements numbers where CLD patches are attached.


6. Experiments on the CLD treated curved panel

From the finite element model of the curved panel, modal fre-quencies, mode shape and modal strain energy are obtained.

0 50 100 150 200 250 3000

20

40

60

Frequency (Hz)

Mag

nitu

de (

db)

Fig. 10. Vibration attenuation of the first mode by attaching two VEM and CLDpatches on the both edges: (________), bare; (. . .. . .. . .. . ..), VEM; (___ _ ____ _ ____)CLD.

0.5 1.0-3

-2

-1

0

1

2

3

Time (s)

Sens

or O

utpu

t (V

)

Fig. 11. Response of the first mode for: (- - -), bare; and (——————), constrainedlayer damping; treated curved panel.

Fig. 12. (a) Curved panel with constrained layer damping patch attached in the mid

Experiments are conducted to damp the first four modes of thepartially constrained layer damping treated curved panel.

dle. (b) Elements numbers where constrained layer damping patch is attached.

0 50 100 150 200 250 3000

20

40

60

Frequency (Hz)

Mag

nitu

de (

db)

Fig. 13. Vibration attenuation of the second mode by attaching VEM and CLD patchesin the middle of curved panel: (_______), bare; (. . .. . .. . .), VEM; (___ _ ___ _ ___), CLD.

0 0.2 0.4 0.6-4

-3

-2

-1

0

1

2

3

4

Time (s)

Sens

or O

utpu

t (V

)

Fig. 14. Experimental time response of the second mode for: (- - - - -), bare curvedpanel ; (______) CLD; treated curved panel.

Fig. 15. (a) Curved panel with constrained layer damping (CLD) patches attached on both the edges as well as in the middle. (b) Elements numbers where CLD patches areattached.

0 50 100 150 200 250 3000

20

40

60

Frequency (Hz)

Mag

nitu

de (

db)

Fig. 16. Vibration attenuation of first four modes by attaching two VEM and CLDpatches on the both edges and one VEM and CLD patch in the middle: (_______),bare; (__ __ __ __ __), VEM; (___ _ ___ _ ___ _ ___) CLD.


6.1. Experimental setup

Experiments are conducted on bare and partially CLD treatedcurved panel with different configurations. A suitable supportmatching the curvature of the curved panel is fabricated and alu-minum curved panel is mounted on a fixed support to provide can-tilevered boundary condition. Figs. 6 and 7 show photograph of theexperimental setup and schematic drawing of partially constrainedlayer damping treated cantilevered curved panel, respectively. Pie-zoelectric sensor (Measurement specialties, Inc.) layers are bondedon the other side of the curved panel (Fig. 8). The cantilever panel

Table 4Reduction in the vibration level of first four modes.

Conditions

Targeting the twisting mode (VEM/CLD patches attached at both the edges)

Targeting the bending modes (VEM/CLD patches attached in the middle only)

Targeting of both twisting as well as bending modes (VEM/CLD patches attached at boththe middle)

is disturbed by hitting it at an element number 26 near the root byan impact hammer. Impact hammer has piezoelectric sensor at-tached at its impact tip. As a result of the impact, the curved panelgets disturbed. Both the impact hammer response and the piezo-electric sensor response are stored both in Fourier transformationspectrum analyzer (FFT analyzer) and in data acquisition unitand then processed to get the frequency response function. Inthe second step, separate viscoelastic layer patches of exact sizebased on modal strain energy of particular mode are cut fromthe viscoelastic material sheet (EAR ISODAMP C-2003). The dimen-sion of single element in x and h directions are 26 mm and 20 mm,respectively. Total 29 numbers of viscoelastic material patches aremade. Viscoelastic patches have inbuilt adhesive on one side, it candirectly bonded with the base panel. The time response and fre-quency response functions are obtained for each configuration todamp first four modes independently or simultaneously. The mag-nitude of the frequency response function (FRF) is in dB relative to1 V/N. In the last step, viscoelastic layer patches are removed fromthose places and different samples of constrained layer dampingpatches are made, aluminum constraining layer is used andbonded with the viscoelastic layer. Again, total 29 numbers con-strained layer damping patches are made. Constrained layer damp-ing patches are attached on the locations where modal strainenergy is high for the particular mode, frequency response functionare plotted to damp the first four modes.

6.2. Modal damping by CLD treatment

In the present study, experiments are conducted to attenuatethe first four modes of the curved panel, first mode is the twistingmode and other three modes are complex bending modes. Firsttwo modes are tried to attenuate independently, and then all fourmodes are attempted to damp simultaneously. Comparisons be-tween Eigen frequencies obtained from finite element study and

Reduction in the vibration level (dB)

Firstmode

Secondmode

Thirdmode

Fourthmode

VEM 12 3 3 5CLD 24 6 7 9

VEM 3 10 7 10CLD 7 19 20 23

the edges as well as in VEM 13 12 8 13CLD 26 19 20 23


obtained from experimental study are presented in Table 3. Thereis good agreement between analytical and experimental studies.

6.2.1. Vibration attenuation of the first mode by attaching CLD patchFrom the modal strain energy plot of the first mode (twisting

mode), it is evident that it is maximum near the fixed end at both

Table 5Modal loss factors of the first four modes when first mode is targeted.

Configurations of curved panel First mode loss factor Second mod

0.0181 0.0117

0.0301 0.0134

0.0380 0.0146

0.0420 0.0151

Table 6Modal loss factors of the first four modes when second mode is targeted.


0.015 0.011

0.020 0.032

0.023 0.041

the edges of the curved panel. Therefore to damp the twistingmode vibration, a viscoelastic layer patch is bonded at elements(1-8-15-22) and other viscoelastic patch is attached at elements(7-14-21-28) of the base curved panel as shown in Fig. 9a and b.Dimensions of the viscoelastic patches in x, h and z directions are104 mm, 20 mm and 1 mm, respectively. Piezoelectric sensors of

e loss factor Third mode loss factor Fourth mode loss factor

0.0155 0.0121

0.0176 0.0148

0.0181 0.0150

0.0190 0.0162


0.018 0.014

0.034 0.024

0.051 0.028


same dimension are attached at the collocated positions at theother face of the curved panel. The time response of the VEM trea-ted panel is obtained from the oscilloscope and transfer functionsis obtained from the impact hammer input to the sensor output.By attaching viscoelastic layer patches on the base curved panel,a reduction of 12 dB is achieved in the vibration level of the firstmode, 3 dB for the second mode, 3 dB and 5 dB for the third andthe fourth modes, respectively as shown in Fig. 10. After measuringthe transfer function, viscoelastic patches are removed from thoselocations, constrained layer damping patches are attached at thesame locations. The sizes of constrained layer damping patchesare 104 mm, 20 mm and 1.3 mm in x, h and z directions, respec-tively. There is a 24 dB reduction in the vibration level for the firstmode, for the second mode it is 6 dB, 7 dB and 9 dB for the thirdand the fourth modes, respectively. Hence there is a considerablereduction in the vibration level of the first mode by attaching vis-coelastic and constrained layer damping patches at the locationsbased on the modal strain energy approach. Fig. 11 presents the

Table 7Modal loss factors of the first four modes when all the mode are targeted.


0.022 0.020

0.031 0.0230

0.036 0.028

0.039 0.031

0.043 0.045

0.045 0.058

time responses of the first mode of the bare and CLD treated curvedpanel. Damping ratios for these two configurations are calculatedfrom the logarithmic decrement method. For curved panel withoutany layer the damping ratio is 0.0021 and with CLD treatment it is0.0201. Hence by attaching the CLD patches at the position wheremodal strain energy is high, the first mode damping ratio is in-creased by about 90% in comparison with the bare panel.

6.2.2. Vibration attenuation of the second mode by attaching CLDpatch

To reduce the bending mode (second mode) vibrations, separatepatches of viscoelastic and constrained layer damping are mountedin the middle and near the fixed end of the curved panel at ele-ments (2-3-4-5-6-9-10-11-12-13) and piezoelectric sensor are at-tached at the collocated position on the other side of the panelas shown in the Fig. 12. The sizes of the viscoelastic patches in x,h and z directions are 52 mm, 100 mm and 1 mm, respectively,

whereas the sizes of constrained layer damping treated patches


0.0195 0.0160

0.021 0.017

0.025 0.0190

0.029 0.021

0.041 0.038

0.056 0.042


in x, h and z directions are 52 mm, 100 mm and 1.3 mm, respec-tively. Transfer function for each case of bare curved panel, paneltreated with viscoelastic layer and constrained layer damping arepresented in the Fig. 13. With only the viscoelastic patch on thecurved panel, there is only 3 dB reduction of vibration level ofthe first mode, 10 dB, 8 dB and 10 dB reductions in vibration levelof the second, third and the fourth mode, respectively. With con-strained layer damping patches, there is 7 dB reduction of vibrationlevel of the first mode, 19 dB, 20 dB and 23 dB reduction in vibra-tion level of second mode, third and fourth mode, respectively.Fig. 14 presents second mode time responses of the bare and con-strained layer damping treated curved panel. The damping ratiosfor both the configurations are calculated from the logarithmicdecrement method. The damping ratio of the second mode (bend-ing mode) of the bare curved panel is 0.00305, and for CLD treatedpanel it is 0.01601. Hence there is an increase of about 81% in thedamping ratio of the second mode from bare panel to CLD treatedcurved panel.

6.2.3. Vibration attenuation of both twisting and bending modesTo reduce both bending and twisting modes vibrations, sepa-

rate patches of viscoelastic layer and constrained layer dampingare mounted at the elements (2-3-4-5-6-9-10-11-12-13) in themiddle and two patches are attached each at one of the edge atelements (1-7-8-14-15-16-21-22-28) as shown in Fig. 15. The sizeof viscoelastic patch and constrained layer damping treated patchbonded at both the edges and in the middle are same as the sizesof the patch used to attenuate the first (twisting) and the second(bending) mode, respectively. Piezoelectric sensors are attachedat the collocated position on the other side of the panel. Transferfunction for each case of bare curved panel, panel treated withviscoelastic layer and curved panel treated with constrained layerdamping are presented in the Fig. 16. With only the viscoelasticlayer there is 13 dB reductions in the vibration level of the firstmode (twisting mode), second mode is reduced by 12 dB, thirdmode by 8 dB and fourth mode amplitude is reduced by 13 dB.With constrained layer damping, there is about 26 dB reductionin the vibration level of the first mode (twisting mode), secondmode level reduced by 19 dB, third mode by 20 dB and fourthmode amplitude is reduced by 23 dB. Thus by placing the visco-elastic layer and constrained layer damping patches based onthe modal strain energy of first four modes, considerable levelof vibration reduction has been achieved. A summary of thevibration level reduction of the different modes is presented inthe Table 4.

7. Placement of CLD patches on the curved panel for differenttargeting modes

By increasing the coverage of constrained layer dampingpatches obtained by modal strain energy approach, largest damp-ing for the particular mode or for number of modes can beachieved. Layouts with different number of patches to damp differ-ent modes are presented in Tables 5–7. Effect of treatment size onthe modal damping factors of the first four modes when first modeis targeted is presented in Table 5. The locations of the CLD patchesare determined by the MSE distribution of the first mode. Thisclearly shows that the modal damping factor of the first mode isthe highest of all the considered modes. The modal loss factor ofthe first mode increased from 0.018 to 0.042 when the treatmentsize is about 12% of the curved panel. Similar effects can be seenin Table 6, when we target the second mode, loss factor increasesfrom 0.015 to 0.023 for the treatment size is about 22% of thecurved panel. When we are targeting all the four modes simulta-neously (Table 7), the first mode loss factor increases from 0.022

to 0.045, second mode loss factor increases from 0.020 to 0.058,for third and forth mode it is from 0.0195 to 0.056 and 0.016 to0.042, respectively for treatment size of about 33% of the curvedpanel.

8. Conclusions

In the present study, extensive experiments are conducted ona partially covered constrained layer damping treated curved pa-nel. The CLD treated curved panel is modeled by using finite ele-ment method. Mode shapes and modal strain energies areplotted for the each mode of the curved panel. Extensive exper-iment are conducted to demonstrate the utility of the modalstrain energy approach as an effective mean for selecting thelocations of the CLD treatment to achieve desired damping char-acteristics by targeting individual mode or multiple modes overa broad frequency range. It is experimentally verified that byincreasing the coverage of constrained layer damping patch basison modal strain energy approach, largest damping for the partic-ular mode or for number of modes can be achieved. It is mean-ingful to carry out further study to examine the influence ofboundary conditions on the CLD layout for minimizing its vibra-tion response.

References

[1] Kerwin Jr EM. Damping of flexural waves by a constrained viscoelastic layer. JAcoust Soc Am 1959;31:952–62.

[2] DiTaranto RA. Theory of vibratory bending for elastic and viscoelastic layeredfinite-length beams. ASME J Appl Mech 1965:881–6.

[3] Mead DJ, Markus S. The forced vibration of a three-layer, damped sandwichbeam with arbitrary boundary conditions. J Sound Vib 1969;10:163–75.

[4] Nakra BC. Vibration control with viscoelastic materials. Shock Vib Dig1976;8:3–12.

[5] Mead DJ. A comparison of some equations for flexural vibration of dampedsandwich beams. J Sound Vib 1982;83(3):363–77.

[6] Lall AK, Asnani NT, Nakra BC. Damping analysis of partially covered sandwichbeams. J Sound Vib 1988;123:247–59.

[7] Lall AK, Asnani NT, Nakra BC. Vibration and damping analysis of rectangularplate with partially covered constrained viscoelastic layer. Am Soc Mech Eng JVib Acoust Stress Reliab Des 1987;109:241–7.

[8] Markus S. Damping properties of layered cylindrical shells, vibration in axiallysymmetric modes. J Sound Vib 1976;48(4):511–24.

[9] Pan HH. Axisymmetrical vibrations of a circular sandwich shell with aviscoelastic core layer. J Sound Vib 1969;9:338–48.

[10] Alam N, Asnani NT. Vibration and damping analysis of a multilayeredcylindrical shell, part-1: theoretical analysis. AIAA 1984;22(6):803–10.

[11] Alam N, Asnani NT. Vibration and damping analysis of a multilayeredcylindrical shell, part-2: numerical results. AIAA 1984;22(6):975–81.

[12] Leissa AW, Iyer KM. Modal response of circular cylindrical shells withstructural damping. J Sound Vib 1981;77(1):1–10.

[13] Petyt M. Vibration of curved plates. J Sound Vib 1971;15(3):381–95.[14] Au FTK, Cheung YK. Free vibration and stability analysis of shells by the

isoparametric spline finite strip methods. Thin Wall Struct 1996;24:53–82.[15] Masti RS, Sainsbury MG. Vibration damping of cylindrical shells partially

coated with a constrained viscoelastic treatment having a standoff layer. ThinWall Struct 2005;43:1355–79.

[16] Ray MC. Smart damping of laminated thin cylindrical panels usingpiezoelectric fiber reinforced composites. Int J Solids Struct 2007;44:587–602.

[17] Chen LH, Huang SC. Vibrations of a cylindrical shell with partially constrainedlayer damping (CLD) treatment. Int J Mech Sci 1999;41:1485–98.

[18] Chen LH, Huang SC. Vibration attenuation of a cylindrical shell with partiallyconstrained layer damping strips treatment. Comput Struct 2001;79:1355–62.

[19] Ramesh TC, Ganesh N. Orthotropic cylindrical shells with viscoelastic core: avibration and damping analysis. J Sound Vib 1994;175:535–55.

[20] Wang HJ, Chen LW. Finite element analysis of orthotropic cylindrical shell witha constrained damping layer. Finite Elem Anal Des 2004;40:737–55.

[21] Chen YC, Huang SC. An optimal placement of CLD treatment for vibrationsuppression of plates. Int J Mech Sci 2002;44:1801–21.

[22] Zheng H, Cai C, Pau GSH, Liu GR. Minimizing vibration response of cylindricalshells through layout optimization of passive constrained layer dampingtreatments. J Sound Vib 2005;279:739–56.

[23] Ro J, Baz A. Optimum placement and control of active constrained layerdamping using modal strain energy approach. J Vib Control 2002;8:861–76.

[24] Wilson EL, Penzien J. Evaluation of orthogonal damping matrices. Int J NumerMethods Eng 1972;4:5–10.

experimental study on vibration and damping of curved panel treated with constrained viscoelastic...

Documents