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Passive Vibration Attenuation Viscoelastic Damping, Shunt Piezoelectric Patches, and Periodic Structures Mohammad Tawfik

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This book was written to guide researchers through the world of Periodic Structures ... however, it is still work in progress ... tell me what you think about it

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Page 1: Periodic Structures

Passive Vibration Attenuation

Viscoelastic Damping, Shunt Piezoelectric Patches, and Periodic Structures

Mohammad Tawfik

Page 2: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Structures

Passive Vibration Attenuation 2

Contents

1. Periodic Structures: A Passive Vibration Filter .................................................................................... 3

1.1. Periodic Structures ....................................................................................................................... 3

1.2. Literature Survey .......................................................................................................................... 3

1.3. Periodic Analysis .......................................................................................................................... 5

1.4. Periodic Bars ................................................................................................................................ 9

1.4.1. Forward approach for a periodic bar .................................................................................... 9

1.4.2. Reverse approach for a periodic bar ................................................................................... 12

1.4.3. Experimental Work ............................................................................................................. 13

1.5. Periodic Beams ........................................................................................................................... 14

1.5.1. Beams with Periodic Geometry .......................................................................................... 15

1.5.2. Experimental Work ............................................................................................................. 16

1.6. Propagation Surfaces for Periodic Plates ................................................................................... 21

1.6.1. Input-Output Relations ....................................................................................................... 22

1.6.2. Propagation Surfaces .......................................................................................................... 23

1.6.3. Constant angle curves ......................................................................................................... 25

1.6.4. Plates with Periodic Geometry ........................................................................................... 30

1.6.5. Experimental Work ............................................................................................................. 30

1.7. Effect of Shunted Piezoelectric Patches on Propagation Surfaces ............................................ 30

1.7.1. Propagation Surfaces for Coupled System .......................................................................... 30

1.8. Appendices ................................................................................................................................. 34

1.8.1. Appendix A .......................................................................................................................... 34

1.8.2. Nomenclature ..................................................................................................................... 37

1.9. References and Bibliography ..................................................................................................... 38

Page 3: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Structures

Passive Vibration Attenuation 3

1. Periodic Structures: A Passive Vibration Filter

1.1. Periodic Structures

The first question that anyone may ask is: what is a Periodic Structure? The definition of a periodic structure, according to Mead [78], is that it is one that consists fundamentally of a number of identical substructure components that are joined together to form a continuous structure. Periodic structures are seen in many engineering products, examples of periodic structures may include satellite solar panels, railway tracks, aircraft fuselage, multistory buildings, etc …

Following the above definition of periodic structure, there must be a distinction between different substructures that defines the individual unit, that distinction or boundary will introduce a sudden change in the properties of the structure. Two main types of discontinuities may be identifies, namely: geometric discontinuity and material discontinuity. Figure 1.1 shown a sketch of the two different types of discontinuities.

(a) (b)

Figure 1.1. Types of discontinuities (a) Material discontinuity (b) Geometric dicontinuity

Recall what happens to a wave as it travels through a boundary between two different media; part of the light wave refracts inside the water and another part reflects back into the air. Mechanical waves behave in a similar way!

Now, imagine a rod, as example of 1-D structures. As the wave propagates through the rod, it faces a discontinuity in the structure. A part of the wave reflects and another part propagates into the new part. The reflected part of the wave will, definitely, interfere with the incident wave.

Figure 1.2. Sketch of light wave behaviour when

incident on water surface

The interference between the incident and reflected waves will result, in some frequency band, in destructive interference. In the frequency band where destructive interference occurs, there will be reduced vibration level. This band is what we call Stop-Band. Stop bands are the center of interest for the periodic analysis of structures (see section 1.3)

1.2. Literature Survey

In his paper, reviewing the research performed in the area of wave propagation in periodic structures, Mead [78] defined a periodic structure as a structure that consists fundamentally of a number of identical structural components that are joined together to form a continuous structure. Examples of periodic structures can be seen in satellite solar panels, wings and fuselages of aircraft,

Page 4: Periodic Structures

Periodic Structures: A Passive Vibration Filter Literature Survey

Passive Vibration Attenuation 4

petroleum pipe-lines, and many others. An illustration of a simple periodic bar is presented in Figure 1.3.

Figure 1.3. An illustration of a simple periodic bar.

Studies of the characteristics of one-dimensional periodic structures have been extensively reported [79-94]. These structures are easy to analyze because of the simplicity of the geometry as well as the nature of coupling between neighbouring cells. Ungar [79] presented a derivation of an expression that could describe the steady state vibration of an infinite beam uniformly supported on impedances. That formulation, easily allowed for the analysis of the structures with fluid loadings.

Later, Gupta [80] presented an analysis for periodically-supported beams that introduced the concepts of the cell and the associated transfer matrix. He presented the propagation and attenuation parameters’ plots which form the foundation for further studies of one-dimensional periodic structures. Faulkner and Hong [81] presented a study of mono-coupled periodic systems. They analysed the free vibration of spring-mass systems as well as point-supported beams using analytical and finite element methods. Mead and Yaman [82] presented a study for the response of one-dimensional periodic structures subject to periodic loading. Their study involved the generalization of the support condition to involve rotational and displacement springs as well as impedances. The effects of the excitation point as well as the elastic support characteristics on the pass and stop characteristics of the beam are presented.

Other studies have also shown very promising characteristics of periodic structures for wave attenuation [86-94]. Langley [86] investigated the localization of a wave in a damped one-dimensional periodic structure using an energy approach. Later, Cetinkaya [90], by introducing random variation in the periodicity of one-dimensional bi-periodic structure, showed that the vibration can be localized near to the disturbance source. Using the same concept, Ruzzene and Baz [92] used shape memory inserts into a one-dimensional rod, and by activating or deactivating the inserts they introduced aperiodicity which in turn localized the vibration near to the disturbance source. Then, they used a similar concept to actively localize the disturbance waves travelling in a fluid-loaded shell [93]. Thorp et al. [94] applied the same concept to rods provided with shunted periodic piezoelectric patches which again showed very promising results.

The analysis of periodic plates is of a specific importance as it relates to many practical structures [95-103]. Mead [95] presented a general theory for the wave propagation in multiply-coupled and two-dimensional periodic structures by reducing the number of degrees of freedom of the system based on the propagation relation existing between the two ends of the structure. Mead and Parathan [96] used the energy method [95] together with characteristic beam modes to describe the behaviour of plates. In that paper, they introduced the concept of “Propagation Surfaces” that reflects the change of the dynamical behaviour of the periodic plate with the change in the direction and phase of propagating waves. Finally, Mead et al. [97] approached the wave propagation

Page 5: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Analysis

Passive Vibration Attenuation 5

problem of a periodically stiffened plate using the finite element approach which utilized hierarchical polynomials. The investigation of the acoustic characteristics of a periodic plate was also studied by Mead [98]. In that study, he used the methods developed in his previous three papers to extend the model to predict the structural-acoustic characteristics of a periodically stiffened plate.

Mace [99] presented an analysis of a periodic plate that is supported on periodically-separated point supports. The solution procedure involved the use of the Fourier transform of the equation of motion and the support conditions. The analysis also extended to the prediction of the acoustic loading and radiation from the vibrating surface of the plate.

Langley [100,101] introduced analytical techniques for predicting the response of two-dimensional structures under point loading. The response to harmonic point loading [100] was studied and conclusions were drawn that showed the potential of using periodic two-dimensional structures as filters. Similar results were obtained when analyzing the response of a periodic plate to point impulsive loading [101].

The analysis of elastically-supported plates was of great interest to many researchers as it represents more realistic structures. Warburton and Edney [102] used the Rayleigh-Ritz method to analyse an elastically-supported periodic plate. Later, Mukherjee and Parathan [103] used the beam functions of Mead and Parathan [96] to analyze the behaviour of periodic plates with rotational stiffeners. They concluded that their proposed method is computationally efficient compared to finite element method.

1.3. Periodic Analysis

Periodic structures can be modeled like any ordinary structure, but in a periodic structure, the study of the behavior of one cell is enough to determine the stop and pass bands of the complete structure independent of the number of cells.

Recall the equations of motion for a general body

2

1

2

1

2221

1211

2

1

2221

1211

F

F

U

U

kk

kk

U

U

mm

mm

Where U is a vector presenting the displacements at a certain

point in the structure, F is a general force vector; m and k are

general mass and stiffness terms depending on the modelling

method. For harmonic excitation, we may write:

2

1

2

1

22

2

2221

2

21

12

2

1211

2

11

F

F

U

U

mkmk

mkmk

Figure 1.4. General sketch for a

structure

From which, the dynamic stiffness matrix may be written as follows:

2

1

2

1

2221

1211

F

F

U

U

DD

DD

Expanding the two equations, we get:

Page 6: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Analysis

Passive Vibration Attenuation 6

2222121

1212111

FUDUD

FUDUD

Rearranging terms of the equations gives:

2221212

1

1

12111

1

122

UDUDF

FDUDDU

Collecting right hand displacements and forces on the right hand side of the equations gives:

1

1

1222111

1

1222212

1

1

12111

1

122

FDDUDDDDF

FDUDDU

In matrix form:

1

1

1

122211

1

122221

1

1211

1

12

2

2

F

U

DDDDDD

DDD

F

U

Now, assume the input output relation for the given cell are in the form:

1

1

2

2

F

Ue

F

U

Then, we may write:

1

1

1

122211

1

122221

1

1211

1

12

1

1

F

U

DDDDDD

DDD

F

Ue

Giving the input output, transfer, relation as:

1

1

1

1

2221

1211

F

Ue

F

U

TT

TT

Where the input output transformation matrix is called the transfer matrix T. From the above

relation, we can clearly see that:

2221

1211

TT

TTsEigenvaluee

Note that the transfer matrix is dependent on the excitation frequency, hence, the propagation factor is dependent on the frequency. Also, it can be proven that the eigenvalues of the transfer

matrix will appear in reciprocal pairs ().

Example 1.1: Periodic Spring Mass

Page 7: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Analysis

Passive Vibration Attenuation 7

Figure 1.5. Sketch of the periodic spring mass system.

Write down the equations of motion for the cell given by 2 half masses and one spring

2

1

2

1

2

1

0

0

f

f

u

u

kk

kk

u

u

m

m

Then, we may get the dynamic stiffness matrix

2

1

2

1

2

2

f

f

u

u

mkk

kmk

Rearranging terms

2

2

1

1

222

2

1

11

f

u

f

u

k

m

k

mkk

kk

m

From which we may write the transfer matrix

1

1

1

1

222

2

1

11

f

ue

f

u

k

mk

k

mk

kk

m

Below, is the MATLAB code used to generate the results of this example.

m=1; k=1; mc=[m,0;0,m]; kc=[k,-k;-k,k]; mg=[m 0 0 0 0 0 0 2*m 0 0 0 0 0 0 2*m 0 0 0 0 0 0 2*m 0 0 0 0 0 0 2*m 0 0 0 0 0 0 m]; kg=[k -k 0 0 0 0 -k 2*k -k 0 0 0 0 -k 2*k -k 0 0 0 0 -k 2*k -k 0 0 0 0 -k 2*k -k 0 0 0 0 -k k];

Page 8: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Analysis

Passive Vibration Attenuation 8

for ii=1:1001 freq(ii)=(ii-1)*0.002; KD=kc-freq(ii)*freq(ii)*mc; TT=[-KD(1,1)/KD(1,2) 1/KD(1,2) KD(2,2)*KD(1,1)/KD(1,2)-KD(2,1) -KD(2,2)/KD(1,2)]; Lamda(:,ii)=sort(eig(TT)); Mew(ii)=acosh(0.5*(Lamda(1,ii)+Lamda(2,ii))); Resp=inv(KD)*[1;0]; xx(ii)=20*log(abs(Resp(2))); KG=kg-freq(ii)*freq(ii)*mg; Resp=inv(KG)*[1;0;0;0;0;0]; yy(ii)=20*log(abs(Resp(6))); end subplot(4,1,1); plot(freq,Lamda(1,:),freq,Lamda(2,:)); grid subplot(4,1,2); plot(freq,real(Mew),freq,imag(Mew)); grid subplot(4,1,3); plot(freq,xx); grid subplot(4,1,4); plot(freq,yy); grid

Figure 1.6. Variation of the eigenvalues with the

excitation frequency

Figure 1.7. Variation of the real and imaginary

parts of the propagation factor with the excitation

frequency

Figure 1.8. Frequency response of a single cell

Figure 1.9. Frequency response of the six cells

From Figure 1.6 we may notice that the eigenvalues of the transfer matrix appear as complex conjugate for all frequencies below the cut-off frequency of the cell (Only real part is plotted). Fro frequencies above the cut-off frequency, the eigenvalues appear in real reciprocal pairs. Figure 1.7 presents plot for the variation of the real and imaginary parts of the propagation factor μ. Note here that the real part of the propagation factor is equal to zero for all frequency values below the cut-off

Page 9: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Bars

Passive Vibration Attenuation 9

frequency. Further, we may notice that the imaginary part varies from 0 to π then it stays constant for the frequency values at which the real part is non-zero. Figure 1.8 is a plot of the frequency response of the cell. In this plot we may also note that the response of the cell becomes less than unity (0 dB) for higher frequencies. Finally, Figure 1.9 presents the response of the 6-mass spring system in which we may notice that the response also becomes less than unity for the higher frequencies similar to that of a single cell.

1.4. Periodic Bars

One-dimensional periodic structures will be our key-way towards better understanding of the phenomena associated with general periodic structures. Consider a unit cell of the periodic structure of Figure 1.3 and its free body diagram shown in Figure 1.10, we may define a relation between the force f3 and displacement u3 at the right hand side of the cell and f1 and u1 on the left hand side as follows,

1

1

3

3

f

ue

f

u (1)

where is the propagation factor.

On the other hand, the force-displacement relations of each of the parts of the cell could be written in terms of the dynamic stiffness matrix as follows,

2

1

2

1

1

22

1

12

1

12

1

11

f

f

u

u

DD

DD, (2)

and

3

2

3

2

2

33

2

23

2

23

2

22

f

f

u

u

DD

DD (3)

where r

ijD is the dynamic stiffness coefficient relating the i’th force to the j’th displacement of the

r’th element that can be determined using any technique such as finite element. Remember that the dynamic stiffness matrix of an element is a function of the excitation frequency.

Figure 1.10. A free body diagram for a cell of the periodic bar.

1.4.1. Forward approach for a periodic bar

The approach presented in this section for the analysis of the periodic characteristic of a bar is going to be named the “forward approach”, in contrast with the “reverse approach” that will be presented

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Periodic Structures: A Passive Vibration Filter Periodic Bars

Passive Vibration Attenuation 10

later. The forward approach starts with a physical input (excitation frequency) and advances to determine the periodic characteristics of the bar, mainly presented in the propagation factor.

For the first element, we may rearrange the equation (2-a) to be in the form,

2

2

1

1

1

12

1

22

1

12

1

22

1

111

12

1

12

1

12

1

11 1

f

u

f

u

D

D

D

DDD

DD

D

(4)

Similarly, for the second element, equation (3) can take the following form,

3

3

2

2

2

12

2

22

2

12

2

22

2

112

12

2

12

2

12

2

11 1

f

u

f

u

D

D

D

DDD

DD

D

(5)

Combining equations (1), (3) and (4) gives,

1

1

1

1

1

12

1

221

121

12

1

22

1

11

1

12

1

12

1

11

2

12

2

222

122

12

2

22

2

11

2

12

2

12

2

11 11

f

ue

f

u

D

DD

D

DD

DD

D

D

DD

D

DD

DD

D

(6)

which can be rewritten as,

1

1

1

1

2221

1211

f

ue

f

u

TT

TT (7)

where, [T] is called the transfer matrix of the cell. The above equation is an Eigenvalue problem, similar to that obtained previously for the periodic mass spring system, in [T] which can be solved directly yielding the required Eigenvalues. Recall that the transfer matrix was derived from the dynamic stiffness matrix which is a function of the excitation frequency. It may be shown that the

eigenvalues (’s) of the transfer matrix [T] appear in pairs such that one is the reciprocal of the other

(i.e. /1& ). Suggesting that these eigenvalues are e and e , which we can use to write as

follows,

iArcCosh

1 (8)

In general, the value obtained for the propagation factor from equation (8) is a complex value

whose imaginary part defines the phase difference between the input and the output vibration waves, while the real part denotes the attenuation in the vibration amplitude between the input and the output.

Page 11: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Bars

Passive Vibration Attenuation 11

To demonstrate the previous concepts, a test case was considered in which the modulus of elasticity

(E) for both parts of the bar is 71 GPa, density () 2700 Kg/m3, smaller diameter 4 cm, larger

diameter 24 cm, and length of each part 1 m.

The variation of the eigenvalues of the transfer matrix function of a unit cell with the excitation frequency is plotted in Figure 1.11. For the frequency band in which the eigenvalues are presented by one branch, they appear as a complex conjugate pair. While, for the frequency band in which they have two distinct branches, the eigenvalues are real.

Figure 1.11. A plot of the variation of the transfer

matrix eigenvalues with the excitation frequency.

Figure 1.12. The variation of the real and imaginary

parts of the propagation factor with the excitation

frequency.

The variation of the propagation parameter can thus be determined through equation (8). The real and imaginary parts of the propagation parameter are plotted in Figure 1.12. It should be noted at this point that the real and imaginary parts of the propagation parameter are varying with

frequency. The frequency band in which the real part is zero, the imaginary part varies from 0 to

and from to 0. While, through the frequency bands in which the real part is positive, the imaginary

part is constant at the values of or 0. This note is going to help us understanding the behaviour of the propagation surfaces of two-dimensional plates later.

Another way for obtaining the propagation factor is through dynamic condensation of the dynamic stiffness matrix after assembling the cell global matrix. The condensation is obtain through the following procedure; assemble the dynamic stiffness matrix to obtain

3

1

3

2

1

3323

232212

1211

0

0

0

f

f

u

u

u

DD

DDD

DD

Then evaluate the internal degrees of freedom in terms of the boundary degrees of freedom using the second equation

22

3231122

D

uDuDu

Then substitute into the other equations to obtain

Page 12: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Bars

Passive Vibration Attenuation 12

3

1

3

1

22232

33222312

22231222122

11

//

//

f

f

u

u

DDDDDD

DDDDDD

Which may be written as

3

1

3

1

2212

1211

f

f

u

u

DD

DD

If the reduced stiffness matrix is then handled in the same manner as explained in the previous section, the same results presented in Figure 1.11 and Figure 1.12 will be obtained.

1.4.2. Reverse approach for a periodic bar

In this section, the reverse approach will be introduced in order to illustrate the concept of propagation lines which will be extended to the propagation surfaces for plates. Using the finite element model presented earlier, we may assemble the global dynamic stiffness matrix of the cell as follows,

3

2

1

3

2

1

3323

232212

1211

0

0

f

f

f

u

u

u

DD

DDD

DD

(9)

Substituting equation (1) into (9) gives,

1

2

1

1

2

1

3323

232212

1211

0

0

f

f

f

u

u

u

DeD

eDDD

DD

(10)

Since the resultant force f2 at point two is zero, we may add the first and last equations of the above system and simplify the result to get,

0

0

2

1

222312

23123311

u

u

DeDD

eDDDD

(11)

Separating the mass and stiffness terms in the above equation, we get

0

0

2

1

222312

231233112

222312

23123311

u

u

MeMM

eMMMM

KeKK

eKKKK

Or

0

0

2

12

u

uMK (12)

Equation (12) presents an eigenvalue problem of the vibration as a function of the propagation

parameter . We will call this approach the “reverse approach” as the independent variable of the

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Periodic Structures: A Passive Vibration Filter Periodic Bars

Passive Vibration Attenuation 13

problem, , is a quantity that we have no direct access to, in contrast with the “forward approach”

in which the independent variable is the excitation frequency which is a quantity we can physically control and measure.

To demonstrate the relationship between both approaches, the values of the propagation factor is

constrained to be imaginary values varying from 0 to . The resulting values of the natural frequencies of oscillation are shown in Figure 1.13. Few important notes have to be emphasized at this point. The curves presenting the variation of the excitation frequency are identical to those presenting the variation of the imaginary part of the propagation factor (Figure 1.12) with the independent and dependent variable reversed. Also, the gap existing between both curves of Figure 1.13 corresponds to the frequency band in which the value of the propagation factor has a real part (Figure 1.12). The characteristic graphs shown in Figure 1.13 are called the propagation curves.

Figure 1.13. The variation the natural frequency of

oscillation with the propagation factor.

Figure 1.14. the variation of the natural frequency

of oscillation with the real part of the propagation factor

(imaginary part =)

Now, varying the values of the real part of the propagation factor, for a constant value of the imaginary part, results in the characteristics shown in Figure 1.14. Similar notes can be taken when comparing the results of Figure 1.14 with those of Figure 1.12. But it has to be noted that increasing the value of the real part above the maximum obtained by the “forward approach” results in obtaining complex pairs for the excitation frequencies indicating going beyond the physical boundaries. Nevertheless, the graphs of Figure 1.14 fill the gap that exists in Figure 1.13. Thus, we may call that gap “the attenuation band”, or “stop band”, and the curves, “the attenuation curves”.

1.4.3. Experimental Work

In an extended research of the characteristics of periodic bars, Asiri conducted different experiments on bars with periodic configurations. His results were assessed by numerical results for the pass and stop bands obtained from a spectral finite element model. His results emphasized the effectiveness of the periodic configurations in attenuating the vibration response in the stop bands indicated by the numerical model for the different configurations. Figure 1.15 presents the geometry of one of the experiments conducted by Asiri, and Figure 1.16 presents the experimentally obtained frequency response of the bar together with the numerically obtained attenuation curves. The results shown in Figure 1.16 show the degree of accuracy by which the attenuation bands may be predicted by the attenuation curves for the bar.

Page 14: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Beams

Passive Vibration Attenuation 14

Figure 1.15. Geometry of one of the experiments conducted by Asiri.

Figure 1.16. Experimental frequency response of the bar with the above mentioned geometry and the corresponding attenuation curves.

1.5. Periodic Beams

Periodic beams have been of special interest to researchers in the past decades due to their relation to railroad structures. The fact that the railway is supported at equal distances presents an almost-

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Periodic Structures: A Passive Vibration Filter Periodic Beams

Passive Vibration Attenuation 15

ideal case for the study of infinite simply supported beams, further, the effect of the foundation elasticity, presenting the ground elasticity, was widely introduced to the studies.

1.5.1. Beams with Periodic Geometry

Numerical Model

The spectral finite element model presented earlier for the plate case was simplified to be suitable for the beam case. The degrees of freedom and generalized forces of the beam cell at the three nodes are shown in Figure 1.17.

Figure 1.17. A sketch of the forces and displacements of a beam cell.

The equations of motion of the beam elements could be written as follows,

2

1

2

1

1

22

1

21

1

12

1

11

f

f

W

W

DD

DD (13)

3

2

3

2

2

22

2

21

2

12

2

11

f

f

W

W

DD

DD (14)

where

i

i

iw

wW

',

i

i

iM

Ff , and r

ijk is the dynamic stiffness matrix term relating the ith

displacement vector with the jth generalized force vector. The dynamic stiffness matrix can be assembled for the whole cell

3

2

1

3

2

1

3323

232212

1211

0

0

f

f

f

W

W

W

DD

DDD

DD

Condensing the above system to remove the internal displacement vector (W2) and assuming no internal forces on the cell, i.e. f2 is zero, we get,

3

1

3

1

2221

1211

f

f

W

W

DD

DD

where 12

1

22121111 DDDDD ,

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Periodic Structures: A Passive Vibration Filter Periodic Beams

Passive Vibration Attenuation 16

32

1

221212 DDDD ,

12

1

223221 DDDD ,

and 32

1

22323322 DDDDD .

Rearranging the equations to put them in an input output relation, we get,

3

3

1

1

2221

1211

f

W

f

W

TT

TT

where 11

1

1211 DDT

,

1

1212

DT ,

11

1

12221221 DDDDT

,

and 1

122222

DDT .

We may assume that,

1

1

3

3

f

We

f

W

where is the propagation factor of the cell.

1

1

1

1

f

We

f

WT

The above equation can be solved as an eigenvalue problem for the eigenvalues e. It can be proven that the eigenvalues of this problem will appear in pairs each if which is the reciprocal of the other.

1.5.2. Experimental Work

Due to the lack of experimental studies that emphasize the periodic characteristics of structures, it was decided to study the characteristics of the periodic beam to give a broader and more in-depth understanding of the behaviour of the periodic structures. In the forthcoming sections, the understanding of the periodic beam and plate structures will be emphasized through the experimental and numerical results obtained.

At this point, differentiations between two techniques of analysis have to be outlined; the periodic analysis and the finite element analysis. When periodic analysis is mentioned, it is to point towards the process of investigating the pass and stop bands through the study of the propagation curves and surfaces and related characteristics. On the other hand, the “finite element analysis” term will be used to point towards the use of ordinary finite element techniques that would apply to any

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Periodic Structures: A Passive Vibration Filter Periodic Beams

Passive Vibration Attenuation 17

structure’s geometry rather than to periodic structures in specific. This distinction had to be made as most of the periodic analysis will be derived from a finite element model.

Experimental Setup

In order to develop more understanding of the of the behaviour of the periodic beams as well as developing a numerical model to study its characteristics, an experiment was set for a periodic beam with free-free boundary conditions (Figure 1.18).

Figure 1.18. The setup of the periodic beam experiment.

The beam is aluminium beam which is 40 cm long and 5 cm wide with 1 mm thickness. The periodicity was introduced onto the beam by bonding 5 cm by 5 cm pieces of the same material on both surfaces separated by 5 cm (Figure 1.19). The beam is then suspended by a thin wire from one of its end to simulate free-free boundary conditions. Thus, the beam is set up with four identical cells each of which has free-free boundary conditions.

Figure 1.19. A sketch for one cell of the periodic beam.

The beam is then excited by a piezostack (model AE0505D16 NEC Tokin, Union City, CA, 94587) at one end and the measurement was taken by an accelerometer from the other end (Figure 1.20).

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Periodic Structures: A Passive Vibration Filter Periodic Beams

Passive Vibration Attenuation 18

Figure 1.20. The excitation piezostack and the output accelerometer.

Comparison of Results

The experiment described above was set up and measurements were taken from the two ends of the beam. Figure 1.21 shows the transfer function frequency response of the beam for the plain and periodic beams. The attenuation factor of the beam, as calculated by the real part of the propagation factor of the periodic model, is plotted below the frequency response for the sake of comparison. The results shown emphasize the accuracy of the periodic model used to predict the behaviour of the beam. Figure 1.22 presents the frequency response obtained by the finite element model of the described beam. Comparing the results of both figures, we can note clearly the consistency of results obtained by the three models, experimental, periodic and finite element.

Figure 1.21. The frequency response together with the numerical results of the stop bands for the proposed beam.

-80

-60

-40

-20

0

20

40

0 1000 2000 3000 4000 5000 6000

Frequency (Hz)

Respence A

mpl. (

dB

)

0

0.5

1

1.5

2

2.5

Attenuation F

acto

r

Plain Beam

Periodic Beam

Attenuation Factor

Page 19: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Beams

Passive Vibration Attenuation 19

Figure 1.22. Frequency response of the beam using finite element model.

Another experiment was set up for a set of beams with cantilever boundary conditions. The experiments was set up with two accelerometers and excited by a piezoelectric actutator as shown in Figure 1.23 and Figure 1.24. Different cases with varying the lengths L1 and L2 were constructed to examine the effect of the geometry on the attenuation characteristics (Figure 1.25).

Figure 1.23. Sketch of the experimental setup for the cantilever beam.

Page 20: Periodic Structures

Periodic Structures: A Passive Vibration Filter Periodic Beams

Passive Vibration Attenuation 20

Figure 1.24. A picture of the experimental setup.

Figure 1.25. a sketch for the cell geometry of the experiment for the cantilever beam.

Figure 1.26. Experimental results obtained for case #1 compared to plain beam and attenuation curves obtained by numerical model.

-50

-40

-30

-20

-10

0

10

20

30

0 500 1000 1500 2000 2500 3000 3500 4000

Frequency (Hz)

Tra

ns

fer

Fu

nc

tio

n A

mp

litu

de

(d

B)

0

1

2

3

4

5

6

7

8

9

10

Att

en

ua

tin

Fa

cto

r (r

ad

)

Plain Beam

Periodic Beam

Attenuation Factor

Page 21: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 21

Figure 1.27. Experimental results obtained for case #2 compared to plain beam and attenuation curves obtained by numerical model.

Figure 1.28. Experimental results obtained for case #3 compared to plain beam and attenuation curves obtained by numerical model.

1.6. Propagation Surfaces for Periodic Plates

It is naturally understood that the beam is a special case of the plate structure. The thin beam and plate structures have similar approximate theories that describe their behaviour. From dynamics

-50

-40

-30

-20

-10

0

10

20

0 500 1000 1500 2000 2500 3000 3500 4000

Frequency (Hz)

Tra

ns

fer

Fu

nc

tio

n A

mp

litu

de

(dB

)

0

1

2

3

4

5

6

7

8

9

10

Att

en

uat

in F

act

or

(ra

d)

Plain Beam

Periodic Beam

Attenuation Factor

-50

-40

-30

-20

-10

0

10

20

0 500 1000 1500 2000 2500 3000 3500 4000

Frequency (Hz)

Tra

ns

fer

Fu

nc

tio

n A

mp

litu

de

(d

B)

0

1

2

3

4

5

6

7

8

9

10

Att

en

ua

tin

Fa

cto

r (r

ad

)

Plain Beam

Periodic Beam

Attenuation Factor

Page 22: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 22

point of view, the beam would be characterized by having the bending waves travelling in one dimension, along the direction of the beam axis. Due to the characteristic of the beam being of short width relative to the length, its modes of vibration in the shorter direction are associated with very high frequencies.

On the other hand, the plate is the general case in which the length and width dimensions are of the same order giving way for bending waves to travel in both directions with similar characteristics. That specific nature of the plate introduces a lot of complexities to the study. A basic problem that arises from the 2-dimensional effect is the fact that the source of vibration at a certain point on the plate can not be pointed out due to the fact that reflections from the tips of the structure are interfering together with the fact that in a periodic structure we are introducing more reflections that would travel in all directions increasing the degree of complexity.

1.6.1. Input-Output Relations

To establish a system of equations that can be used for the “reverse approach” study of the periodic behaviour of the plate, relations between the displacements of the different nodes are developed and implemented similar to those introduced in equation(1). Mead [95] and Mead et al. [97] introduced relations that could be developed for use with higher order elements.

The input-output relations summarized in Figure 1.29 are presented in the following two sets of equations,

118

127

69

510

14

13

12

,

,

,

,

,

,

wewand

wew

wew

wew

wew

wew

wew

x

x

y

y

y

yx

x

118

127

69

510

14

13

12

,

,

,

,

,

,

fefand

fef

fef

fef

fef

fef

fef

x

x

y

y

y

yx

x

where xx i and yy i with x and y denoting the phase factor in the x and y-direction

respectively.

Figure 1.29. A sketch representing the relations between the input and output displacements.

Page 23: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 23

Note that in the above relations, wi stands for the vector of degrees of freedom if the ith node; i.e. {w,wx,wy,wxy,D}. Implementing those relations in the element equations of motion, and assuming harmonic vibration, we may obtain the following relation,

0

16

15

14

13

12

11

6

5

1

9991

1911

9991

1911

2

w

w

w

w

w

w

w

w

w

kk

kk

mm

mm

(15)

1.6.2. Propagation Surfaces

The concept of propagation surfaces was introduced by Mead and Parathan [96] as a graphical presentation of the change in the dynamic characteristics of the periodic plate with the change in the wave direction. For a planar wave travelling in a periodically supported plate at an inclination

angle from the x-axis, the phase difference between two adjacent periods in the x and y-directions

are yx , respectively, and the average wave numbers in the x and y-directions could be given by

b

ka

ky

yx

x

& (16)

where a and b are the plate-period length in the x and y-directions respectively.

Mead and Parathan [96] used displacement functions to describe the vibration of beams then extended the model to two dimensions by multiplying two polynomials (the x-polynomial and the y-polynomial). Then, the stiffness and mass matrices were constructed and the natural frequencies

were calculated. A plot of the non-dimensional frequency with the phase difference (i.e. the propagation surfaces) for a simply-supported periodic plate was then presented. The non-

dimensional frequency is defined as,

4

42

pD

at (

17)

where t and Dp are the plate thickness and flexural rigidity respectively.

Using the developed 16-node element, the set of 16 matrix equations can be reduced to a set of 9

matrix equations, and can be solved as an eigenvalue problem for the non-dimensional frequency Such that:

Page 24: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 24

02 km

The propagation surfaces resulting from the solution of the above eigenvalue problem are shown in Figure 1.30.

In Figure 1.31, which is the same as Figure 1.30 but from a different viewing point, we can clearly see the bands over which the propagation surfaces reside. These frequency bands are the bands in which the vibration would propagate from the input to the output nodes in an analogous manner to the propagation bands identified earlier for the periodic bar. Gaps that exist between the surfaces over bands of frequencies can also be identified as “attenuation or stop bands”.

Figure 1.30. Propagation surfaces resulting from the solution of the eigenvalue problem of the finite element model.

x

y

Page 25: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 25

Figure 1.31. The plot of the propagation surfaces from a planar point of view.

1.6.3. Constant angle curves

To simplify the graphical representation of the “reverse approach”, we are going to examine the propagation surfaces at constant angle. A wave propagation angle of 45o is considered. By varying

the imaginary part of the propagation factor from 0 to , setting the real part to 0 and taking y to

be equal to x, we can obtain the propagation curves for the different bands. Figure 1.32 shows the

resulting curves drawn with the independent variable (x) on the vertical axis.

x y

Page 26: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 26

Figure 1.32. The curves of the propagation surfaces at angle 45o.

Approaching the problem from the perspective of the attenuation factor (the real part of the propagation factor), we can draw the “Attenuation Surfaces” or the “Attenuation Curves”. Setting the imaginary part of the propagation factor to zero, we can obtain the attenuation curves (or the stop bands). Figure 1.33 presents the attenuation curves for a wave propagating at 45o and with the imaginary part of the propagation factor set equal to zero. While Figure 1.34 presents the attenuation curves with a wave propagating at 45o and with the imaginary part of the propagation

factor set to .

An interesting feature appears in these graphs, namely, the overlapping of the propagation and attenuation bands. This property of the bands comes from the fact that the wave is now propagating in a square plate in contrast with the one-dimensional structures considered earlier. In a simply-supported square plate, the 2nd and the 3rd vibration modes coincide (namely the (1,2) and (2,1) modes). Nevertheless, the energy flow in both directions is distinct and occurs between two different set of nodes.

Getting back to the three dimensional surfaces, we can now obtain the “Attenuation Surfaces” for

the plate by setting the values of the imaginary part of the propagation factor to 0 or . The resulting surfaces present the attenuation bands associated with the periodic plate of interest. It has to be noted, again at this point, that the overlapping of the surfaces does not contradict the fact that the bands are distinct. In other words, within the stop bands, the vibration of certain propagation mode while the other modes that undergo propagation phases are still propagating vibration.

Page 27: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 27

Figure 1.33. The “attenuation surface” at angel 45o with the imaginary part set to zero.

Figure 1.34. The “attenuation surface” at angel 45o with the imaginary part set to .

Page 28: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 28

Figure 1.35. The attenuation surfaces for the plate with the imaginary part set to zero.

x

y

Page 29: Periodic Structures

Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates

Passive Vibration Attenuation 29

Figure 1.36. The attenuation surfaces for the first two attenuation bands of the plate with the imaginary part set to .

x

y

Page 30: Periodic Structures

Periodic Structures: A Passive Vibration Filter Effect of Shunted Piezoelectric Patches on Propagation Surfaces

Passive Vibration Attenuation 30

1.6.4. Plates with Periodic Geometry

1.6.5. Experimental Work

1.7. Effect of Shunted Piezoelectric Patches on Propagation Surfaces

1.7.1. Propagation Surfaces for Coupled System

It is interesting to visualize the result of adding an inductor to the shunt circuit. It simply splits the mode that is targeted into two modes with one surface above and another below the original propagation surface; just like the case of adding a secondary mass-spring system to a primary system as takes place in the classical vibration absorber problem (Figure 1.37).

Figure 1.37. Frequency response for a vibration absorber.

When an inductance is added to the system, a similar result is obtained for the propagation surfaces as shown in Figure 1.38. It is obvious that the shape of the propagation surfaces exhibits shape similar to that of the frequency response of a two-degree of freedom spring-mass system.

Page 31: Periodic Structures

Periodic Structures: A Passive Vibration Filter Effect of Shunted Piezoelectric Patches on Propagation Surfaces

Passive Vibration Attenuation 31

Figure 1.38. The two surfaces resulting from adding the inductance compared to the original surface.

Drawing the curves with a wave propagating at 45o (Figure 1.39), we can visualize the gap introduced in the band that was originally covered by the first propagation surface. That gap now presents a stop band.

x

y

Page 32: Periodic Structures

Periodic Structures: A Passive Vibration Filter Effect of Shunted Piezoelectric Patches on Propagation Surfaces

Passive Vibration Attenuation 32

Figure 1.39. The propagation curves resulting from introducing the shunted inductance.

Plotting the attenuation curves for this case, we can notice the introduction of an attenuation factor in that band indicating that the propagating wave is expected to decay. (Figure 1.40 and Figure 1.41).

Figure 1.40. The attenuation curves resulting from introducing the shunted inductance at phase angle .

Page 33: Periodic Structures

Periodic Structures: A Passive Vibration Filter Effect of Shunted Piezoelectric Patches on Propagation Surfaces

Passive Vibration Attenuation 33

Figure 1.41. The attenuation curves resulting from the introduction of the shunted inductance at phase angle zero.

Page 34: Periodic Structures

Periodic Structures: A Passive Vibration Filter Appendices

Passive Vibration Attenuation 34

1.8. Appendices

1.8.1. Appendix A

When the input output relations of the different nodes are implemented into the equations of motion of the plate elements, then terms get collected, the 16 equations reduce to 9 equations given by,

0

6

5

1

9991

1911

9991

1911

g

f

e

d

c

b

w

w

w

w

w

w

w

w

w

kk

kk

mm

mm

Where the terms of the above equation are 4x4 matrix each given by the following set of relations (not that the letters a to g are used instead of the number 10 to 16 for the sake of clarity),

yyxx

yyxx

yyxx

yyxx

yyxyxyxx

yyyxyxx

yxyxyxxy

yxyxyxxy

xyxyxyyx

yxyxyyxx

ekekekkk

ekekekkk

ekekekkk

ekekekkk

ekekekekekkkekk

ekekekekekkkekk

kekekekekekekkk

kekekekekekekkk

kekekekekkekek

ekekkekekekekkk

gggg

ffff

eeee

dddd

cccc

bbbb

aaaa

432119

432118

432117

432116

44733722711715

44833822811814

494639362926191613

44533522511512

4443424134333231

242322211413121111

Page 35: Periodic Structures

Periodic Structures: A Passive Vibration Filter Appendices

Passive Vibration Attenuation 35

y

y

y

y

yyxx

yyxx

yy

yy

xyxyyyxx

ekkk

ekkk

ekkk

ekkk

ekekkekk

ekekkekk

kekekkk

kekekkk

kekekekekekekkk

agg

aff

aee

add

acac

abab

aa

aaaa

aaaa

529

528

527

526

755725

855824

96595623

555522

43215453525121

y

y

y

y

yyxx

yyxx

yy

yy

xyxyyyxx

ekkk

ekkk

ekkk

ekkk

ekekkekk

ekekkekk

kekekkk

kekekkk

kekekekekekekkk

gg

ff

ee

dd

cc

bb

aa

9639

9638

9637

9636

99766735

99866834

9996696633

99566532

949392916463626131

bgg

bff

bee

bdd

bcbc

bbbb

bb

baba

bbbb

kekk

kekk

kekk

kekk

kekekkk

kekekkk

ekkekekk

ekkekekk

ekekekkekekkekk

x

x

x

x

xx

xx

yyxx

yyxx

yyxxyxyx

849

848

847

846

788745

888844

96898643

588542

43218483828141

Page 36: Periodic Structures

Periodic Structures: A Passive Vibration Filter Appendices

Passive Vibration Attenuation 36

cgg

cff

cee

cdd

cccc

cbcb

cc

caca

cccc

kekk

kekk

kekk

kekk

kekekkk

kekekkk

ekkekekk

ekkekekk

ekekekkekekkekk

x

x

x

x

xx

xx

yyxx

yyxx

yyxxyxyx

759

758

757

756

777755

877854

96797653

577552

43217473727151

dg

df

de

dd

dcd

dbd

dd

dad

dddd

kk

kk

kk

kk

kekk

kekk

ekkk

ekkk

ekekekkk

x

x

y

y

yyxx

69

68

67

66

765

864

9663

562

432161

eg

ef

ee

ed

ece

ebe

ee

eae

eeee

kk

kk

kk

kk

kekk

kekk

ekkk

ekkk

ekekekkk

x

x

y

y

yyxx

79

78

77

76

775

874

9673

572

432171

Page 37: Periodic Structures

Periodic Structures: A Passive Vibration Filter Appendices

Passive Vibration Attenuation 37

fg

ff

fe

fd

fcf

fbf

ff

faf

ffff

kk

kk

kk

kk

kekk

kekk

ekkk

ekkk

ekekekkk

x

x

y

y

yyxx

89

88

87

86

785

884

9683

582

432181

gg

gf

ge

gd

gcg

gbg

gg

gag

gggg

kk

kk

kk

kk

kekk

kekk

ekkk

ekkk

ekekekkk

x

x

y

y

yyxx

99

98

97

96

795

894

9693

592

432191

1.8.2. Nomenclature

A Area ai Undetermined coefficients of the transverse displacement shape function bi Undetermined coefficients of the electric displacement shape function D Electric displacement DP Plate flexural rigidity d Piezoelectric coefficient di Nodal electric displacement E Young’s modulus of elasticity

Electric field e Piezoelectric material constant relating stress to electric field Hw,HD Transverse displacement and electric displacement interpolation functions

respectively k,kx,ky Wave number, component of wave number in x and y-directions respectively kb,kD,kbD Element bending, electric, and displacement-electric coupling stiffness matrices

respectively mb,mD Element bending and electric mass matrices respectively Nw,ND Lateral displacement and electric displacement shape functions respectively Q Plane stress plane strain constitutive relation T Kinetic energy

E

Page 38: Periodic Structures

Periodic Structures: A Passive Vibration Filter References and Bibliography

Passive Vibration Attenuation 38

U Potential energy V Volume W External work w Transverse displacement wb,wD Nodal transverse and electric displacements respectively

(.) First variation

Strain

xy Shear strain

Curvature

The propagation factor

Mass density

Stress

Wave propagation angle

Poisson’s ratio

Frequency Dielectric constant

Phase angle Subscripts D Related to electric degrees of freedom w Related to transverse deflection b Related to bending degrees of freedom x In the x-direction ,x Derivative in the x-direction y In the y-direction ,y Derivative in the y-direction Superscript D At constant electric displacement

At constant electric field T Matrix transpose

1.9. References and Bibliography 1. Wada, B. K., Fanson, J. L., and Crawley, E. F., “Adaptive Structures,” Journal of

Intelligent Material Systems and Structures, Vol. 1, No. 2, 1990, pp. 157-174.

2. Crawley, E. F., “Intelligent Structures for Aerospace: A Technology Overview and

Assessment,” AIAA Journal, Vol. 32, No. 8, 1994, pp. 1689-1699.

3. Rao, S. S., and Sunar, M., “Piezoelectricity and Its Use in Disturbance Sensing and

Control of Flexible Structures: A Survey,” Applied Mechanics Review, Vol. 47, No. 4,

1994, pp. 113-123.

4. Park, C. H., and Baz, A., “Vibration Damping and Control Using Active Constrained

Layer Damping: A Survey,” The Shock and Vibration Digest, Vol. 31, No. 5, 1999, pp.

355-364.

5. Benjeddou, A., “Recent Advances in Hybrid Active-Passive Vibration Control,” Journal

of Vibration and Control, Accepted for Publishing.

6. Chee, C. Y. K., Tong, L., and Steven, G. P., “A Review on The Modeling of

Piezoelectric Sensors and Actuators Incorporated in Intelligent Structures,” Journal of

Intelligent Material Systems and Structures, Vol. 9, No. 1, 1998, pp. 3-19.

1

E

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Passive Vibration Attenuation 39

7. Crawley E. F. and de Luis J., “Use of Piezoelectric Actuators as Elements of Intelligent

Structures,” AIAA Journal, Vol. 25, No. 10, 1987, pp. 1373-1385.

8. Hagood, N. W., Chung, W. H., and von Flotow, A., “Modeling of Piezoelectric Actuator

Dynamics for Active Structureal Control,” AIAA paper, AIAA-90-1087-CP, 1990.

9. Koshigoe, S. and Murdock, J. W., “A Unified Analysis of Both Active and Passive

Damping for a Plate with Piezoelectric Transducers,” Journal of the Acoustic Society of

America, Vol. 93, No. 1, 1993, pp. 346-355.

10. Vel, S. S. and Batra, R. C., “Cylendrical Bending of Laminated Plates with Distributed

and Segmented Piezoelectric Actuators/Sensors,” AIAA Journal, Vol. 38, No. 5, 2000,

pp. 857-867.

11. Dosch, J. J., Inman, D. J., and Garcia, E., "A Self-Sensing Piezoelectric Actutator for

Collocated Control," Journal of Intelligent Material Systems and Structures, Vol. 3, No.

1, 1992, pp. 166-185.

12. Anderson, E. H., Hagood, N. W., and Goodliffe, J. M., "Self-Sensing Piezoelectric

Actuation: analysis and Application to Controlled Structures," Proceeedings of the

AIAA/ASME/ASCE/AHS/ASC 33rd

Structures, Structural Dynamics, and Materials

Conference (Dallas, TX), AIAA, Washington, DC, 1992, pp. 2141-2155.

13. Vipperman, J. S., and Clark, R. L., "Implementation of An Adaptive Piezoelectic

Sensoriactuator," AIAA Journal, Vol. 34, No. 10, 1996, pp. 2102-2109.

14. Dongi, F., Dinkler, D., and Kroplin, B., "Active Panel Suppression Using Self-Sensing

Piezoactuators," AIAA Journal, Vol. 34, No. 6, 1996, pp. 1224-1230.

15. Cady, W. G., “The Piezo-Electric Resonator,” Proceedings of the Institute od Radio

Engineering, Vol. 10, 1922, pp. 83-114.

16. Lesieutre, G. A., "Vibration Damping and Control Using Shunted Piezoelectric

Materials," The Shock and Vibration Digest, Vol. 30, No. 3, May 1998, pp. 187-195.

17. Hagood, N. W., and von Flotow, A, "Damping of Structural Vibration with Piezoelctric

Materials and Passive Electrical Networks," Journal of Sound and Vibration, Vol. 146,

1991, No. 2, pp. 243-264.

18. Hollkamp, J. J. and Starchville, T. F. Jr., “A Self-Tuning Piezoelectric Vibration

Absorber,” Journal of Intelligent Material Systems and Structures, Vol. 5, No. 4, 1994,

pp. 559-566.

19. Wu, S., “Piezoelectric Shunts with Parallel R-L Circuit of Structural Damping and

Vibration Control,” Proceedings of SPIE, Vol. 2720, 1996, pp. 259-265.

20. Park, C. H., Kabeya, K., and Inman D. J., “Enhanced Piezoelectric Shunt design,”

Proceedings ASME Adaptive Structures and Materials Systems, Vol. 83, 1998, pp. 149-

155.

21. Law, H. H., Rossiter, P. L., Simon, G. P., and Koss, L. L., “Characterization of

Mechanical Vibration Damping by Piezoelectric Material,” Journal of Sound and

Vibration, Vol. 197, No. 4, 1996, pp. 489-513.

22. Tsai, M. S., and Wang, K. W., "Some Insight on active-Passive Hybrid Piezoelectric

Networks for Structural Controls," Proceedings of SPIE's 5th

Annual Symposium on

smart Structures and Materials, Vol. 3048, March 1997, pp. 82-93.

23. Tsai, M. S., and Wang, K. W., "On The Structural Damping Characteristics of Active

Piezoelectric Actuators with Passive Shunt," Journal of Sound and Vibration, Vol 221,

No. 1, 1999, pp. 1-22.

24. Hollkamp, J. J., "Multimodal Passive Vibration Suppression with Piezoelectric Materials

and Resonant Shunts," Journal of Intelligent Material Systems and Structures, Vol. 5,

No. 1, 1994, pp. 49-57.

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Passive Vibration Attenuation 40

25. Wu, S. Y., "Method for Multiple Mode Shunt Damping of Structural Vibration Using

Single PZT Transducer," Proceedings of SPIE's 6th

Annual Symposium on smart

Structures and Materials, Vol. 3327, March 1998, pp. 159-168.

26. Wu, S., "Broadband Piezoelectric Shunts for Passive Structural Vibration Control,"

Proceedings of SPIE 2001, Vol. 4331, March 2001, pp. 251-261.

27. Behrens, S., Fleming, A. J., and Moheimani, S. O. R., “New Method for Multiple-Mode

Shunt Damping of Structural Vibration Using Single Piezoelectric Transducer,”

Proceedings of SPIE 2001, Vol. 4331, pp. 239-250.

28. Park, C. H. and Baz, A., “Modeling of A Negative Capacitance Shunt Damper with IDE

Piezoceramics,” Submitted for publication Journal of Vibration and Control.

29. Forward, R. L., “Electromechanical Transducer-Coupled Mechanical Structure with

Negative Capacitance Compensation Circuit,” US Patent Number 4,158,787, 19th

of

June 1979.

30. Browning, D. R. and Wynn, W. D., “Vibration Damping System Using Active Negative

Capacitance Shunt Reaction Mass Actuator,” US Patent Number 5,558,477, 24th

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