Vibration control of curved panel using smart damping

Mechanical Systems and Signal Processing 30 (2012) 232–247

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Vibration control of curved panel using smart damping Navin Kumar a,n, S.P. Singh b a b

School of Mechanical, Materials and Energy Engineering, Indian Institute of Technology Ropar, Punjab 140001, India Mechanical Engineering Department, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

a r t i c l e in f o

abstract

Article history: Received 10 September 2009 Received in revised form 4 December 2011 Accepted 6 December 2011 Available online 28 February 2012

Damping designs with active piezoelectric materials and passive viscoelastic materials combine the advantages of both active and passive constrained layer damping (ACLD/ PCLD) treatments. Present study is aimed to examine through experiments, vibration control of curved panel treated with optimally placed active or passive constrained layer damping patches. Placement strategies of constrained layer patches are devised using the modal strain energy method. The optimum location for the application of ACLD/PCLD patches are found for specific modes and the information for different modes is then collated to get the best locations for control of multiple modes. The treatments are then applied to the elements target specific modes of vibrations as well as for control of multiple modes. Extensive experiments are conducted by using number of samples of ACLD and PCLD patches for each configuration to control different modes simultaneously or independently. The results demonstrate the utility of the technique for selecting the locations of the ACLD treatments to achieve desired damping characteristics over a broad frequency range. Experiment results well corroborates with theoretical predictions. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Active constrained layer damping (ACLD) Viscoelastic material Modal strain energy Curved panel

1. Introduction Vibration control of the light weight structures has been extensively researched with the help of piezoelectric actuators and sensors during the past one and half decade [1–5]. One of the drawbacks of this type of treatment is that large control voltage is needed to achieve significant vibration attenuation of the host structure. To reduce the control voltage requirement the active constrained layer damping (ACLD) treatment has attracted significant attention in vibration control of flexible structures in recent years [6,7]. The conventional passive constrained layer damping (PCLD) treatment is used to damp the vibration of structures by attaching a viscoelastic damping layer to the host structure along with a constraining layer. In the active constrained layer damping treatment, the constraining layer is replaced by an active layer made of smart materials such as piezoelectric materials. The transverse shear deformation in the viscoelastic layer can be increased over its passive counterpart and the vibrations of the host structure can be substantially damped out leading to the active or smart damping of the structure. It is easy task to deform the viscoelastic layer than to deform the host structure, thus the piezoelectric materials perform much better to attenuate the vibration of the structures when they are used as active constrained layer treatment than when they are used alone as the distributed actuators. Hence, ACLD treatment has gained tremendous importance for efficient and reliable active control strategies for flexible structures [8–18].

n

Corresponding author. Tel.: þ 91 1881 242170; fax: þ91 1881 223395. E-mail address: [email protected] (N. Kumar).

0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.12.012

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

233

Shell structures are popular and widely used in many civil engineering and mechanical structures. Researchers [19–24] have studied the different vibration characteristic in the shell structures. Several researchers [25–32] have actively controlled the vibrations of by using piezoelectric actuator. Passive and active constrained layer damping have been successfully employed on the various types of shell structures. Markus [33] used the unconstrained passive damping treatments to control the vibrations of thin cylindrical shell. Alam and Asnani [34,35] considered vibration and damping analysis of a general multi-layered cylindrical shell having an arbitrary number of orthotropic material layers and viscoelastic layers. Constrained viscoelastic layers have been widely used to reduce excessive vibration and noise in the structures. Vibration of cylindrical shells with constrained damping layer has received attention from many researchers [36,37]. Vibration and damping analyses of isotropic and orthotropic cylindrical shells with full or partial constrained damping layer were carried out by [38,39]. ACLD treatment has been successfully used by Park and Baz [40] as an effective means for dampening the vibration of plates. Shen [41] extended the use of the ACLD treatments to vibration control of shell structure. He has used the deep shell theory for some intuitive results and for energy analysis. Control of axi-symmetric vibrations of cylindrical shells using active constrained layer damping was given by Baz and Chen [42]. They have developed a distributed parametric model for fully covered ACLD shell. Ray et al. [43] developed a finite element model to describe the vibrations of a shell that was partially treated with ACLD. The validity of this model was checked experimentally using simple proportional and derivative controllers applied to a stainless steel cylinder excited with random forces. Ray and Reddy [44] used the ACLD treatment to control the vibration of the laminated composite shell. The attachment of the ACLD patch at the optimum locations is a critical issue. For the plate structure, modal strain energy approach was used by Ro and Baz [45] for placing the ACLD patch at the optimum locations. Most of the studies on ACLD/PCLD treated shell have paid more attention to the closed circular cylindrical shell and they were mainly based on the analytical investigations. Although the effectiveness of the ACLD treatment on the curved panel is analytically demonstrated in previous study [46,47], but it is not possible to control all the modes by attaching ACLD/PCLD at the same place. It is very important to choose the locations where ACLD or PCLD patch should be attached so that higher modes can also be controlled. Moreover to the author’s best knowledge, experimental study on the ACLD/PCLD treated curved panel for control of specific or multiple modes has not yet been addressed. Extensive experimental investigation for ACLD/PCLD treated curved panel system for control of specific and multiple modes has been attempted in this research work. The ACLD/PCLD patches are attached at the locations based on the modal strain energy proposed by Ro and Baz [45]. The ACLD patches are attached on a curved panel, consisting of viscoelastic core, sandwiched between base structure and active constraining layer of piezoelectric material. Separate samples of ACLD and PCLD patches are fabricated and attached at different locations for the each configuration to control different modes simultaneously or independently. Experiments are performed to control the first four modes of vibration of the ACLD treated curved panel. The analytical evaluation of modal strain energy and control effectiveness is done using finite element analysis of the curved panel. 2. Finite element formulation In this study a simple and efficient finite element named serendipity eight-node element is adopted as shown in Fig. 1. This kind of shell element has eight nodes and contains five degrees of freedom at each of the node: u, v, w, a, b where the first three are translations in global directions and the last two are rotations about the local axis. The element geometry can be represented by the natural coordinate system (x–Z–z) where the curvilinear coordinates (x–Z) are in the shell midsurface while z is linear coordinate in the thickness direction. According to the isoparametric formulation, these coordinates (x, Z and z) will vary from  1 to þ1. The coordinates of any point within the element may be expressed as 8 9 8 9 8 9 > < xi > = X zh > < l3i > = = X > y ¼ Ni i m3i ð1Þ N i yi þ > > 2 > :z > ; :n > ; : > ; z i 3i 4 7 ζ

8

ξ 3 η 6 z,w

1

x,u

5

,v 2 Fig. 1. Geometry of eight node serendipity element.

234

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

where Ni are the quadratic serendipity shape functions in (x,Z) plane, hi is the thickness at the nodal points, l3i, m3i and n3i are the direction cosines of normal unit vector v3i at node i. The shape function Ni corresponding to the ith node and shape functions of eight noded serendipity elements are given as follows: 1 1 1 1 2 ð1xÞð1ZÞð1xZÞ, N2 ¼ ð1x Þð1ZÞ, N3 ¼ ð1þ xÞð1ZÞð1þ xZÞ, N4 ¼ ð1 þ xÞð1Z2 Þ 4 2 4 2 1 1 1 1 2 N5 ¼ ð1 þ xÞð1 þ ZÞð1 þ x þ ZÞ, N6 ¼ ð1x Þð1 þ ZÞ, N7 ¼ ð1xÞð1 þ ZÞð1x þ ZÞ, N 8 ¼ ð1xÞð1Z2 Þ 4 2 4 2 N1 ¼

ð2Þ

2.1. Kinematic constraint relationships Complete circular cylindrical shell have only two curved edge boundaries, a curved panel have two curved edge boundaries and two straight edge boundaries. Fig. 2 shows the curved panel, partially treated with active constrained layer damping. The length, thickness, radius and shallowness angle of the panel are denoted by L, tb, r and j, respectively. The top surface of the panel is bonded with the active/passive constrained layer damping treated patches. These patches consist of viscoelastic material layer sandwiched between the base panel and the piezoelectric constraining layer. The thickness of the viscoelastic layer is ts and that of the piezoelectric constraining layer is tc. Middle surface of the base curved panel is considered as the reference surface along which the axial (x), circumferential (y) and radial (z) coordinate system is defined. The longitudinal and circumferential deformations at any point of the constrained layer damping treated curved panel system are given as u and v, respectively. The total displacement at any point in the layer can be generalized as uðx, y,zÞ ¼ u0 ðx, yÞ þ zaðx, yÞ,

uðx, y,zÞ ¼ v0 ðx, yÞ þ zbðx, yÞ,

wðx,y,zÞ ¼ w0 ðx,yÞ

ð3Þ

where u0 and v0 are the displacements at the reference plane. Symbols a and b correspond to the components of the rotational of the normal to the middle surface of the panel in axial and circumferential directions, respectively. Radial displacement (w) assumed to be constant through the thickness of the panel. The generalized displacement variable {d} is the nodal displacement vector of an element and given as   d ¼ ½ u1 v1 w1 a1 b1 u2 v2 :: :: b8  ð4Þ

2.2. Strain–displacement relationships Based on an improved shallow shell theory using the modified Donnell’s approximations, with the displacement relations according to the improved shell theory in cylindrical coordinates (x, y, z) as   @u @a 1 @v0 w0 z @b @v0 @b 1 @u0 z @a @w0 1 @w0 v0 þ þz þ , gyz ¼ þb ex ¼ 0 þz , ey ¼ þ þ , gxz ¼  ð5Þ , gxy ¼ r @y r @y r @y r @y r @y @x @x r @x @x @x r

Piezoelectric constraining layer (tc)

x

Viscoelastic layer (ts) Base structure (tb) z

L Piezoelectric sensor layer r





Fig. 2. Attachment of the ACLD patches on the curved panel.

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

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2.3. Stress–strain relationships The stress–strain relations in a layer are listed below. The transverse normal stress sz in the z direction is assumed to be zero:

sx ¼

E ðex þ ney Þ, ð1u2 Þ

sy ¼

E ðe þ nex Þtyz ¼ kGgyz , ð1u2 Þ y

txz ¼ kGgxz txy ¼ Ggxy

ð6Þ

where E is Young’s modulus, G is shear modulus and n is Poisson’s ratio of the material of the layer, k is the shear correction factor and is taken as 5/6. 2.4. Piezoelectric relation The linear constitutive equation for the material of the piezoelectric layer is fsg ¼ ½Cfeg þ½eT fEg

ð7Þ

where {s} represents the stress vector, [C] is the elastic constant matrix, {e} is the strain developed in the piezoelectric layer, [e] is the dielectric permittivity matrix, E is the electric vector. 2.5. Electrical potential function The electrical potential is assumed to be constant over an element of piezoelectric layer and varying linearly through the thickness of the piezoelectric layer. The electric field strength of an element in terms of electrical potential for the actuator layer is expressed as 2 3 0 607 6 7 e e e e ðEc Þ ¼ ½Bc ffc g ¼ 6 7ffc g ð8Þ 415 tc where

½Bec 

e

is the electric gradient matrices of the actuator layer and ffc g is electrical degree of freedom of the piezoelectric layer.

2.6. Kinetic energy integrals The elemental kinetic energy is given as Z T 1 T¼ rfd_ t g fd_ t g dV 2 V

ð9Þ

where r is the mass density. The generalized translational displacement vector in {dt} is given as T

fdt g ¼ fu,v,wg

ð10Þ

where u, v and w are the mid-surface displacements in x, y and z directions, respectively. The generalized translational displacement vector at any point within an element is given as e

fdt g ¼ ½Nt fdt g e

T

ð11Þ T

T

where fdt g ¼ ½fdt1 g fdt2 g . . .fdt8 g  and ½Nt  ¼ ½N t1 N t2 . . .Nt8  Substituting expression in the kinetic energy expression (9) yields the mass matrix [Me] of an element, according to Z e T e 1 T¼ rfd_ t g ½Nt T ½Nt fd_ t gdV ð12Þ 2 V where elemental mass matrix [Me] is defined as Z ½M e  ¼ r½N t T ½N t dV

ð13Þ

V

2.7. Total potential energy Integrals The total potential energy of the shell element is given as T p ¼ UW

ð14Þ

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N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

The elastic strain energy U is given as Z Z 1 1 T fegT fDgfegdV fEe g fDc gdV U¼ 2 V 2 Vc c

ð15Þ

where fegT ¼ fex , ey , gxy , gxz , gyz g and {D} is the stress–strain matrix. The work done by external forces due to applied surface traction and the applied electric charge is given as Z Z Z Z T e T e e W ¼ fd g fse ðx, yÞgdAþ fEec g fqa e ðx, yÞgd ¼ fd g½NT fse gdA þ ffc g½Bec T fqe a ðx, yÞgdA A

A

A

ð16Þ

A

where se ðx, yÞ is the surface traction vector, qa e ðx, yÞ the specified surface charge density on the surface of the actuator layer. Strain matrix {e} of Eq. (15) can be represented by e

fegT ¼ ½Bfd g

ð17Þ

Substituting expression (17) in Eq. (15) yield the strain energy of an element in the form U¼

1 e T  e  e  fd g K d 2

ð18Þ

where [Ke] is the stiffness matrix of an element, given as Z Z Z ½K e  ¼ ½BT ½D½BdV, ½K e  ¼ ½K ett ½K ert ½K etr ½K err ½K etc ½K erc , ½K ett  ¼ ½Bt T ½Dtt ½Bt dV, ½K ert  ¼ ½Bt T ½Dtr ½Br dV V ZV Z Z Z V ½K etr  ¼ ½Br T ½Drt ½Bt dV, ½K err  ¼ ½Br T ½Drr ½Br dV, ½K etc  ¼ ½Bt T ½Dtr ½Br dV,, ½K erc  ¼ ½Bt T ½Dtr ½Br dV V

V

Vc

Vc

2.8. Equation of motion To drive the equation for constrained layer damping treated curved panel the Hamilton principle is used: Z t2 ½TU þ Wdt ¼ 0

ð19Þ

t1

where t1 and t2 define the time interval. All variations must vanish at t ¼t1 and t ¼t2. Substituting the potential energy U (Eq. (18)), the work done W (Eq. (16)) and kinetic energy T (Eq. (12)) in Eq. (19) and taking the variation yield the dynamic finite element equations of an element: e e ½Me fd€ t g þ ½K ett fdt g þ½K etr fdr g ¼ fF ect gV þ fF et g

ð20Þ

and e

e

½K err fdr g þ½K etr fdt g ¼ fF ecr gV þ fF er g e

ð21Þ

is the elemental mass matrix, [Kett], [Ketr], [Kert] and [Kerr] are the elemental stiffness matrices, {Fect}, {Fecr} are the coupling vectors and {Fet}, {Fer} are the excitation force vectors corresponding to the translational and

in which [M ] electroelastic rotational co-ordinates, respectively. 2.9. Global equation of motion

In order to obtain global equation of motion, elemental equations of motion are assembled in such a manner to obtain global equations of motion: ½MfX€ t g þ½K tt fX t g þ ½K tr fX r g ¼

a X

fF jct gV j þfF t g

ð22Þ

j¼1

and ½K rt fX t g þ ½K rr fX r g ¼

a X

fF jcr gV j þ fF r g

ð23Þ

j¼1

where [M] and [Ktt], [Ktr], [Krr] are the global mass and stiffness matrices; {Xt}, {Xr} are the global nodal generalized displacement co-ordinates; {Ft}, {Fr} are the global nodal force vectors corresponding to translational and rotational coordinates; a is the number of ACLD patches and for the jth ACLD patch the global nodal electro-elastic coupling vectors are given by X e X e fF ct g, fF jcr g ¼ fF cr g ð24Þ fF jct g ¼ b

b

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

237

where b is the number of elements per ACLD treatment. Involving the boundary conditions, the rotational degree of freedom can be condensed to obtain the global equations of motion in terms of global translational degree of freedom only as follows: ½MfX€ t g þ ½K n fX t g ¼

a X

½fF jct g½K tr ½K rr 1 fF jcr gV j þ fFg

ð25Þ

j¼1

in which ½K n  ¼ ½K tt ½K tr ½K rr 1 ½K rt , and fFg ¼ fF t g½K tr ½K rr 1 fF r g, Actuator layer is supplied with a control voltage proportional to the velocity of the collocated sensor attached at the bottom surface of the curved panel. The control voltage for the jth actuator can be expressed in terms of derivative of the nodal global degree of freedom as V j ¼ K jd ½ej fX_ t g

ð26Þ

where K jd is the controller gain and [ej] is a unit vector with unity as the only non-zero element corresponding to that global degree of freedom, the derivative of which is fed back to the actuator. By substituting Eq. (26) into Eq. (25), the final damped equation of motion is ½MfX€ t g þ ½K n fX t g þ ½C d fX_ t g ¼ fFg

ð27Þ

where ½C d  ¼

a X

K jd ½fF jct g½K tr ½K rr 1 fF jcr g½ej 

j¼1

Eq. (27) can be formulated to compute the frequency response function (FRF).

3. Finite element analysis Based on the above formulation a finite element program is made for the vibration analysis of sandwich curved panel in MATLAB. The program is general and can accommodate any number of elements. For the current analysis, number of elements is fixed to 70, with 245 nodes, as shown in the Fig. 3. For present finite element model, same dimensions and properties of the three layers panel is taken as would be used in experimental studies. From the finite element model of the bare panel, mode shapes and the modal frequencies are obtained. The flow chart for curved panel finite element solution strategies is given in the Fig. 4. Mode shapes of the curved panel are presented in the Fig. 5. First mode is a twisting mode; second mode is a bending mode of flapping wing type, in which both corner edges move up simultaneously, third and fourth modes are also complex bending modes.

1

64

65

45 52

59

40

61 68

55 62

69 63

33 27 34

48

41 35

12

19

26

54

67

5

25

39

4

11

18

32

47

3

10 17

31

46

53

60

16 24

38

2

9

23

30

37

44

51

58 66

36

43

50 57

22

29

8

15

28

20

13

6 7

14 21

42 49

56

70

Fig. 3. Curved panel divided in the 7X10 number of elements.

238

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

Define the initial parameters like thicknesses, elasticity constants, density of all three layer, radius of curvature of the panel, length of the curve panel, included angle, rotational and translational degree of freedoms

Define the shape functions, rigidity matrix, elemental mass matrix, elemental stiffness matrix, Jacobian, Strain displacement matrix, Boundary conditions.

Fined global mass, stiffness and damping matrix.

Determine the modal frequencies and mode Shape

Determine the modal strain energy Ui=1,……n, j=1,…….m

Determine the elements have higher modal strain energies

Apply the ACLD patch at the elements have highest strain energy at mode p

Apply the ACLD patch at the next elements have higher strain energy at mode p Fig. 4. The flow chart of finite element solution strategies for curved panel.

3.1. Finite element model validation To validate the finite element formulation presented in the previous section, present model is compared with those obtained by other researchers [48–50]. They have studied a cylindrical shell panel with clamped edges. We have considered same dimensions, material properties and boundary conditions as taken in papers [48–50], and verified our finite element model. Petyt [48] determined eigen values by using triangular finite element (FET) and extended Rayleigh– Ritz (ERR) method. Table 1 presents the comparison between the results of Refs [48–50] and that the present studies. The comparison shows good agreement. 3.2. Location selection for attachment of ACLD/PCLD patches The selection of the locations where the ACLD/PCLD patches should be attached to control the single mode or a set of modes is a critical issue. Strain energy profile of a particular mode [45] can be one of the effective means to attach the constrained layer damping patches. The procedure to obtain the modal strain energy profile begins by solving the eigen problem given by Eq. (27) with {F}¼0, to determine the eigenvectors of the untreated curved panel. The modal strain energy (Uij) of any curved panel element i at any mode of vibration j is determined as U ij ¼ ½cij T ½K i ½cij 

ð28Þ

where i¼1,y,N and j ¼1,y,M. With N and M denoting the number of elements and number of modes considered, respectively. Also, [Ki] denotes the stiffness matrix of element i and [cij] defines the eigenvectors of element i for mode j. The strain energy plots for first four

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

15 Wi

dth

15

10 (El

5

em

ent

s)

5 0

0

th

Leng

239

15

10 ents) (Elem

15

10

Wid

th (

10

5

Ele

men

ts)

5 0

0

ts)

emen

h (El Lengt

Cantilevered Panel

15

15 15

10

Wi

dth

5

(El

em

ent

s)

5 0

0

nts)

eme h (El

t

Leng

15

10

Wi

dth

10

10

5

(El

em e

nts

)

5 0 0

th Leng

ents)

(Elem

Fig. 5. Mode shapes of the cantilevered curved panel: (a) first mode, (b) second mode, (c) third mode and (d) fourth mode.

Table 1 Comparison of eigen values (Hz) of curved panel. Mode no.

FET in [48]

ERR in [48]

[49]

[50]

Present

1 2 3 4

870 958 1288 1363

870 958 1288 1364

869 957 1287 1363

870 958 1287 1364

869 956 1290 1360

modes of the curved panel are as shown in the Fig. 6. It is observed that for first mode the modal strain energy is maximum at both the side edges near the fixed end. It is due to the fact that first mode is a twisting mode and strain will be maximum at the corners of the fixed end. For the second mode, modal strain energy is maximum near the fixed end and its magnitude decreases from fixed end to free end. Similarly modal strain energy plots are drawn for the third and the fourth mode. The optimum locations for the application of ACLD/PCLD patches are those, where modal strain energy is maximum for particular mode. These locations will also give maximum passive damping addition in case of failure of active damping. With the help of finite element model, the modal strain energies associated with each mode are obtained. Based on the model strain energy, ACLD/PCLD patches have been attached to control first four modes independently or simultaneously. In case of simultaneous control of multiple modes actuators at the collated locations for all the desired modes are activated.

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

100

100

80

80

Modal Strain Energy (%)

Modal Strain Energy (%)

240

60 40 20

60 40 20 0

0 1 (N

o.

2

3

4 W of idth 5 6 ele 7 me nts )

9 8 7 6 5 ) gth 4 Len ements 3 2 el f o 1 . (No

10

1 (N

o.

2

3

4 W 5 of idth 6 ele 7 me nts )

2

1

10

6 gth ) Len ements el f o No. 5

4

3

9

8

7

(

100

100

80

80

Modal Strain Energy (%)

Modal Strain Energy (%)

Cantilevered Panel

60 40 20 0

60 40 20 0

1

2

3 4 o. Wid 5 of t 6 ele h 7 me nts )

(N

1

2

3

4

5

6

7

8

9

gth Len ments) le e f o.o

(N

10

1 (N

o.

2

3

7

4

W 5 of idth 6 ele 7 me nts )

1

2

3

4

8

9

10

6 gth s) Len ement e f l o . o

5 (N

Fig. 6. Modal strain energy plot of the cantilevered curved panel: (a) first mode, (b) second mode, (d) third mode and (d) fourth mode.

4. Experiments on the ACLD treated curved panel From the analytical model of the curved panel, modal frequencies as well as mode shape are obtained. Experiments are conducted for the partially covered curved panel. Physical and geometrical properties of the test panel, viscoelastic material (EAR ISODAMP C-2003) and the piezoelectric layer (measurement specialties, Inc.) are given in the Table 2. Photograph of complete experimental setup of ACLD treated curved panel is shown in the Fig. 7. The cantilever panel is disturbed with the instrumented hammer. Instrumented hammer has piezoelectric sensor attached at its impact tip. As a result of the impact, the curved panel gets disturbed. The transverse response is measured with the help of the piezoelectric collocated sensors, which are bonded on the other side of the curved panel. Negative velocity feedback controller is design in the Labview program. Two inputs and two outputs variable labview program is made to control the first mode. Input channel 1 is used for sensors located at one edge of the curved panel whereas input channel 2 is used for sensors located at other edge of the curved panel. Similarly output channels are used for collocated actuators. Hence actuator patches at both the edges will have same magnitude of actuation but of reverse polarity. Active gain to two individual signals from input channels 1&2 can be applied independently. Sensor responses are acquired through a data acquisition card (PXI-6040E of National Instruments) on to a PXI based real time (RT) system (PXI-8170). RT system does control calculations on the sensor signal as per the Labview computer program which is downloaded on it from the host PC via an ethernet cable. Control output of the PXI-RT system is amplified by a voltage amplifier and fed to the piezoelectric actuator. Both instrumented hammer and piezoelectric sensor signal are also analyzed in the FFT analyzer. Fig. 8 presents the schematic drawing of partially active constrained layer damping treated cantilevered curved panel. First, only the aluminum curved panel (without PCLD or ACLD layer) is mounted on the fixed support as shown in the Fig. 9. Piezoelectric sensors layers are attached on the other side of the curved panel. Response of the bare curved panel is measured from the FFT analyzer. Comparisons between eigen frequencies obtained from finite element study and obtained from experimental study are presented in Table 3. There is good agreement between analytical and experimental studies. In the second step, separate constrained layer patches consists of piezoelectric layer bonded on the viscoelastic material

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

241

Table 2 Physical and geometrical specifications of the panel, viscoelastic and piezoelectric layer. Length of the panel Thickness of the base layer Radius of curvature Included angle Density of the base

260 mm 0.56 mm 151 mm 521 2710 kg/m3

Thickness of the viscoelastic layer Density of the viscoelastic layer Shear modulus of the VEM Thickness of the piezo layer Density of the piezolayer Layer

0.4 mm 1714 kg/m3 5  108(1þ 0.8i) Pa 0.1 mm 1780 kg/m3

Fig. 7. Complete experimental set up for the ACLD treated curved panel.

Point of impact

Piezoelectric constraining layer Base curved panel Impact hammer

Feedback algorithm Piezoelectric sensors Viscoelastic layer Data acquisition system

Sensor amplifier System

Actuation amplifier Dynamic Signal Analyzer

Fig. 8. Schematic diagram of complete experimental setup.

are fabricated. Viscoelastic patches have inbuilt adhesive on one side, it can be directly bonded with the base panel. The time response and receptance are obtained for each configuration to damp first four modes independently or simultaneously. In the last step, negative velocity feedback controlled is designed in the Labview program and the input signals obtained from the sensor layers got amplified and feedback to the piezoelectric constraining layers after applying the negative velocity feedback control law.

4.1. Modal control by ACLD treatment Experiments are conducted to control the first four modes of the curved panel. First two vibration modes are attempted to control independently, and then first four modes are targeted to control simultaneously.

242

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

Piezoelectric Sensors

Fig. 9. Bare curved panel with piezoelectric sensors attached at the back side.

Table 3 Eigen values (Hz) from FEM and experimental studies. Mode no.

FEM study

Experimental study

1 2 3 4

60 120 193 243

60 119 190 238

ACLD Patches 29 36

22

15

8

1

21

14

7

28 42

35

Piezoelectric sensors Fig. 10. (a) Curved panel with active constrained layer patches attached on both the edges, (b) elements number where ACLD patches are attached.

4.1.1. Control of the first mode by attaching ACLD patch From the modal strain energy plot of the first mode, the modal strain energy is maximum near the fixed end at both the edges of the curved panel. Therefore to control the twisting mode, an ACLD patch is attached at elements (1-8-15-22-2936) and other ACLD patch is attached at elements (7-14-21-28-38-42) of the base curved panel as shown in the Fig. 10. Piezoelectric sensors are fixed on the collocated positions at the other face of the curved panel. In case of twisting mode one edge of the curved panel moves up while the other edge moves down. The voltage applied from the ACLD patches are opposite in polarity so that it applied the force in up direction if that edge is moving down and applied the force in down direction on the other edge if it is moving up. The transfer function (receptance) obtained from the hammer input to the sensor output of the PCLD and ACLD treated panel are plotted in Fig. 11. There is a nearly 57% reduction in the vibration level of first mode, for second mode it is only 6%, and 13% and 9% for third and fourth mode, respectively. Hence there is a considerable reduction in the vibration level of the first mode. It is due to the reason that being a twisting mode, it is possible to apply an opposite torque related motion by using two ACLD patches.

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

243

12

Receptance (mm/N)

10

8

6

4

0 0

50

100

150 200 250 Frequency (Hz)

300

350

400

Fig. 11. Control of the first mode by attaching two ACLD patches on the both edges: (??), PCLD; (——), ACLD.

1.5

Displacement (mm)

1 0.5 0 -0.5 -1 -1.5 0.2

0.4 Time (s)

0.6

Fig. 12. Response of the first mode for: (- - - - ), PCLD; and (——) ACLD; treated curved panel, when both side patches are active.

Fig. 12 presents the time responses of the first mode of the PCLD and ACLD treated curved panel. Damping ratios for these two configurations are calculated from the logarithmic decrement method. For PCLD treated panel, damping ratio is 0.0161 and for ACLD treated panel it is 0.0366. Hence by attaching the ACLD patches at the position where modal strain energy is high, the first mode the damping ratio increases by about 56% in comparison with the PCLD treated panels. 4.1.2. Control of the second mode by attaching ACLD patch To control the bending modes (second mode) an ACLD patch is mounted in the middle and near the fixed end of the curved panel at the elements numbers as shown in Fig. 13. The piezoelectric sensors are attached at the collocated position on the other side of the panel. The transfer function (receptances) of the ACLD and PCLD treated curved panel are presented in the Fig. 14. There is only 7% reduction of vibration level for the first mode, 52%, 45% and 43% reduction in vibration level of second mode, third and fourth mode, respectively. Thus the bending modes get improved but the torsional mode is not much affected. Fig. 15 present second mode time responses of the PCLD and ACLD treated curved panel. The damping ratios for both the configurations are calculated from the logarithmic decrement method. The damping ratio of the second mode (bending mode) is 0.00808, for PCLD treated panel and for ACLD treated panel it is 0.0155. Hence there is an increase of about 48% in the damping ratio of the second mode from PCLD to ACLD treated curved panel.

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ACLD Patch 2 3 9 4 16 10 23 17 11 5 24 18 12 6 25 19 13 26 20 27

Piezoelectric sensors

Fig. 13. (a) Curved panel with active constrained layer patches attached in the middle, (b) elements numbers where ACLD patches are attached.

10

Receptance (mm/N)

8

6

4

2

0 0

100

200 Frequency (Hz)

300

400

Fig. 14. Control of the bending mode by attaching an ACLD patch in the middle of the curved panel near the fixed end: (- - - - ), PCLD; (——) ACLD.

Displacement (mm)

2

0

-2

0.2

0.4

0.6

0.8

1.0

Time (s) Fig. 15. Experimental time response of the 2nd mode for: (- - - - ), PCLD; (——) ACLD; treated curved panel.

4.1.3. Control of both twisting and bending modes To control both bending and twisting modes, one ACLD patch is attached in the middle and two ACLD patches are attached each at one of the edge at element numbers as shown in Fig. 16. Piezoelectric sensors are attached at both the edges and the middle at the same elements on the other side of the curved panel.

N. Kumar, S.P. Singh / Mechanical Systems and Signal Processing 30 (2012) 232–247

245

ACLD patches 1 2 8 3 15 9 10 4 22 16 17 11 5 23 29 24 18 12 6 36 25 19 13 7 26 20 14 27 21 28 35 42

Piezoelectric sensors

Fig. 16. (a) Curved panel with active constrained layer patches attached on both the edges as well as in the middle, (b) elements numbers where ACLD patches are attached.

Receptance (mm/N)

9

6

3

0 0

100

200 Frequency (Hz)

300

400

Fig. 17. Control of both bending and twisting mode by attaching an ACLD patch in the middle and other two patches on both the edges of the curved panel: (??), PCLD; (——) ACLD.

Table 4 Percentage reduction in the vibration level of different modes and peak control voltage. Conditions

Percentage reduction in the vibration level First mode

Control of the twisting mode (ACLD/PCLD patches are attached at both the edges) 57.2 Control of bending modes (ACLD/PCLD patches are attached in middle only) 7.5 Control of both twisting as well as bending modes (ACLD/PCLD patches are attached at both 52.4 the edges as well as in the middle)

Second mode

Third mode

Fourth mode

6.0 52.3 40.0

13.3 45.0 46.9

9.3 43.2 34.6

Peak voltage (V)

165 165 165

Both side patches and middle patch are made active to control both twisting as well as bending modes. Sensor signals from the middle and from both the side edges are amplified and feed back to corresponding actuators by applying the negative feedback control law. Both the side edges are given opposite voltages to control the twisting mode. Fig. 17 presents the receptance of this case, there is about 36% reduction in the vibration level of the first mode (twisting mode), second mode is controlled by 32%, third mode by 34.5% and fourth mode amplitude is reduced by 39%. Thus by making both side and middle patch active, an overall reduction in the vibration level of the first four modes are achieved. A summary of the vibration level reduction and peak voltage applied to control the different modes is presented in the Table 4. Same peak control voltage is applied to control the different modes independently or simultaneously. It can be observed that for each case considerable amount of vibration level reduction has been achieved.

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5. Conclusions In the present study, extensive experiments are conducted on a partially covered active constrained layer treated curved panel. The ACLD treated curved panel is modeled by finite element method. Mode shapes as well as modal strain energies are plotted for the each mode of the curved panel. ACLD treated patches are attached at the elements where the modal strain energy is high. The twisting mode (first mode) is controlled by attaching ACLD patches at edges elements near the cantilevered end of the curved panel and bending modes are controlled by attaching ACLD treated patch at the middle elements near the fixed end of the curved panel. ACLD patches are attached at both the side edges as well as in the middle of the curved panel to control first four modes. Hence it is possible to control independently and simultaneously the twisting as well as bending modes of vibrations by attaching ACLD patches at the different locations of the curved panel. It is worth to point out that the work presented in this paper is limited to the case where the curved panel is fixed at its one curved ends. It is, therefore, meaningful to carry out further study to examine the influence of boundary conditions on the ACLD layout for minimizing its vibration response. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

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