Periodic Structures - A Passive Vibration Damper

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

Mohammad Tawfik

Periodic Structures: A Passive Vibration Filter Periodic Structures

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
1.6. 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
1.7. 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

Passive Vibration Attenuation 2

Periodic Structures: A Passive Vibration Filter Periodic Structures

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 [‎ 8], is that it is one that consists fundamentally of a number of
7
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 shown a sketch of the two
1
different types of discontinuities.

(a) (b)

Figure ‎ .1. Types of discontinuities (a) Material discontinuity (b) Geometric dicontinuity
1

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 Figure ‎ .2. Sketch of light wave behaviour when
1
the incident wave. 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 ‎ .3)
1

1.2. Literature Survey

In his paper, reviewing the research performed in the area of wave propagation in periodic
structures, Mead [‎ 8] defined a periodic structure as a structure that consists fundamentally of a
7
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,


Periodic Structures: A Passive Vibration Filter Literature Survey

petroleum pipe-lines, and many others. An illustration of a simple periodic bar is presented in
Figure ‎ .3.
1

Figure ‎ .3. An illustration of a simple periodic bar.
1

Studies of the characteristics of one-dimensional periodic structures have been extensively reported
[‎ 9-‎ 4]. These structures are easy to analyze because of the simplicity of the geometry as well as the
7 9
nature of coupling between neighbouring cells. Ungar [‎ 9] presented a derivation of an expression
7
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 [‎ 0] presented an analysis for periodically-supported beams that introduced the
8
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 [‎ 1] presented a study of mono-coupled periodic systems.
8
They analysed the free vibration of spring-mass systems as well as point-supported beams using
analytical and finite element methods. Mead and Yaman [‎ 2] presented a study for the response of
8
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 [‎ 6-‎ 4]. Langley [‎ 6] investigated the localization of a wave in a damped one-
8 9 8
dimensional periodic structure using an energy approach. Later, Cetinkaya [‎ 0], by introducing
9
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
[‎ 2] used shape memory inserts into a one-dimensional rod, and by activating or deactivating the
9
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 [‎ 3]. Thorp et al. [‎ 4] applied the same concept to rods provided with shunted
9 9
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
[‎ 5-‎ 03]. Mead [‎ 5] presented a general theory for the wave propagation in multiply-coupled and
9 1 9
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 [‎ 6] used the energy method [‎ 5] together with characteristic beam modes to describe the
9 9
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. [‎ 7] approached the wave propagation
9


Periodic Structures: A Passive Vibration Filter Periodic Analysis

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 [‎ 8]. In that study, he used the methods developed in his previous three papers to
9
extend the model to predict the structural-acoustic characteristics of a periodically stiffened plate.

Mace [‎ 9] presented an analysis of a periodic plate that is supported on periodically-separated point
9
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 [‎ 00,‎ 01] introduced analytical techniques for predicting the response of two-dimensional
1 1
structures under point loading. The response to harmonic point loading [‎ 00] was studied and
1
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 [‎ 01].
1

The analysis of elastically-supported plates was of great interest to many researchers as it represents
more realistic structures. Warburton and Edney [‎ 02] used the Rayleigh-Ritz method to analyse an
1
elastically-supported periodic plate. Later, Mukherjee and Parathan [‎ 03] used the beam functions
1
of Mead and Parathan [‎ 6] to analyze the behaviour of periodic plates with rotational stiffeners.
9
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


 m11 m12  U1   k11 k12  U1   F1 
m           
 21 m22  U 2  k21 k22  U 2  F2 

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:

 k11   2 m11 k12   2 m12  U1   F1 
     
k21   m21 k22   m22  U 2  F2 
2 2
Figure ‎ .4.
1 General sketch for a
structure

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

 D11 D12  U1   F1 
D   
 21 D22  U 2  F2 


Expanding the two equations, we get:



D11U1  D12U 2  F1
D21U1  D22U 2  F2

Rearranging terms of the equations gives:

 
U 2   D121 D11U1  D121F1
F2  D21U1  D22U 2

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

 
U 2   D121 D11U1  D121 F1
 1
 1
F2  D21  D22 D12 D11 U1  D22 D12 F1

In matrix form:

U 2   
 D121 D11 D121  U1 


  1 1   
 F2   D21  D22 D12 D11 D22 D12   F1 

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

U 2    U1 
 e  
 F2   F1 

Then, we may write:

U1   
 D121 D11 D121  U1 

e   1 1   
 F1   D21  D22 D12 D11  D22 D12   F1 

Giving the input output, transfer, relation as:

T11 T12  U1   U1 
T  F   e  F 
 21 T22   1   1

Where the input output transformation matrix is called the transfer matrix T. From the above
relation, we can clearly see that:

T T 
e   Eigenvalue s  11 12 
T21 T22 

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: Periodic Spring Mass
1



Figure ‎ .5. Sketch of the periodic spring mass system.
1

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

m 0  u1   k
  k  u1   f1 
 0 m     k   
k  u2   f 2 
  u2   

Then, we may get the dynamic stiffness matrix

k   2 m  k  u1   f1 
     
 k k   2 m u2   f 2 

Rearranging terms

  2m 1
 1   u   u 
 k k
 1    2 

k  k   m
2 2
 1
 m   f1   f 2 
2


 k k 

From which we may write the transfer matrix

  2m 1 
 1  
 k k   u1   e   u1 
   

 k  m  k
2 2
  2 m   f1 
 1  f1 

 k k 

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];



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 ‎ .6. Variation of the eigenvalues with the
1 Figure ‎ .7. Variation of the real and imaginary
1
excitation frequency parts of the propagation factor with the excitation
frequency

Figure ‎ .8. Frequency response of a single cell
1 Figure ‎ .9. Frequency response of the six cells
1

From Figure ‎ .6 we may notice that the eigenvalues of the transfer matrix appear as complex
1
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 ‎ .7
1
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


Periodic Structures: A Passive Vibration Filter Periodic Bars

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 ‎ .8 is a plot of the frequency
1
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 ‎ .9 presents the response of the 6-mass spring
1
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 ‎ .3 and its free body diagram shown in Figure ‎ .10, we may define a relation between the
1 1
force f3 and displacement u3 at the right hand side of the cell and f1 and u1 on the left hand side as
follows,

u3    u1 
 e   (1)
 f3   f1 

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,

 D11
1
D12  u1   f1 
1

 1 1     , (2)
 D12 D22  u2   f 2 

 D22
2
D23  u2   f 2 
2
and  2 2    (3)
 D23 D33  u3   f 3 

r
where Dij 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 ‎ .10. A free body diagram for a cell of the periodic bar.
1

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



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,

 D1 1 
  11

1
D12 D12  u1  u2 
1
     (4)
 D1  D11D22 D22   f1   f 2 
1 1 1

 12 1
D12 
1
 D12 

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

 2
D11 1 
  D2 D12  u2  u3 
2
 12   
2 
(5)
 D 2  D11D22 D22   f 2   f 3 
2 2

 12 2
D12 
2
 D12 

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

 D2 1  D1 1 
  11  11
 2 2D12
2
D12  
2
  1 1D12
1
D12  u1 
1
u 
   e  1  (6)
 D11D22  D 2  22   11 1 22  D12
D2 D D
 22   1 
1
1 D f  f1 
 D2 12
D12   D12
2 1 
 12  D12 

which can be rewritten as,

T11 T12  u1   u1 
T  f   e  f  (7)
 21 T22   1   1

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,

 1
  ArcCosh       i (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.



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 4 2 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 ‎ .11. For the frequency band in which the eigenvalues are presented
1
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 ‎ .11. A plot of the variation of the transfer
1 Figure ‎ .12. The variation of the real and imaginary
1
matrix eigenvalues with the excitation frequency. 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 ‎ .12. It should be noted at
1
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

 D11 D12 0  u1   f1 
D    
 12 D22 D23  u2    0 

 0 D23
 D33  u3   f 3 
   

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

 D12u1  D23u3 
u2 
D22

Then substitute into the other equations to obtain



 D11  D 212 / D22  D12 D23 / D22  u1   f1 
     
  D12 D23 / D22 D33  D 2 23 / D22  u3   f 3 

Which may be written as

 D11 D12  u1   f1 
     
 D12 D22  u3   f 3 

If the reduced stiffness matrix is then handled in the same manner as explained in the previous
section, the same results presented in Figure ‎ .11 and Figure ‎ .12 will be obtained.
1 1

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,

 D11 D12 0  u1   f1 
D    
 12 D22 D23  u2    f 2 
 (9)
 0 D23
 D33  u3   f 3 
   

Substituting equation (1) into (9) gives,

 D11 D12 0  u1   f1 
D    
 12 D22 D23e   u2    f 2 
 (10)
 0
 D23e 
D33  u1   f1 
   

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,

 D11  D33 D12  D23e    u1  0
       (11)
 D12  D23e D22  u 2  0

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

  K11  K 33 K12  K 23e    2  M 11  M 33 M 12  M 23e    u1  0
         
 K  K e  
  12 23 K 22  M 12  M 23e M 22  u 2  0

Or

K     2
u  0
M    1     (12)
u2  0

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



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 ‎ .13. Few important notes have to be emphasized at
1
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 ‎ .12) with the
1
independent and dependent variable reversed. Also, the gap existing between both curves of
Figure ‎ .13 corresponds to the frequency band in which the value of the propagation factor has a
1
real part (Figure ‎ .12). The characteristic graphs shown in Figure ‎ .13 are called the propagation
1 1
curves.

Figure ‎ .13. The variation the natural frequency of
1 Figure ‎ .14. the variation of the natural frequency
1
oscillation with the propagation factor. 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 ‎ .14. Similar notes can be taken when
1
comparing the results of Figure ‎ .14 with those of Figure ‎ .12. But it has to be noted that increasing
1 1
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 ‎ .14 fill the gap that exists in Figure ‎ .13. Thus, we
1 1
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 ‎ .15 presents the geometry of one of
1
the experiments conducted by Asiri, and Figure ‎ .16 presents the experimentally obtained frequency
1
response of the bar together with the numerically obtained attenuation curves. The results shown in
Figure ‎ .16 show the degree of accuracy by which the attenuation bands may be predicted by the
1
attenuation curves for the bar.


Periodic Structures: A Passive Vibration Filter Periodic Beams

Figure ‎ .15. Geometry of one of the experiments conducted by Asiri.
1

Figure ‎ .16. Experimental frequency response of the bar with the above mentioned geometry and the corresponding
1
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-



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 ‎ .17.
1

Figure ‎ .17. A sketch of the forces and displacements of a beam cell.
1

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

 D11
1
D12  W1   f1 
1

 1 1    (13)
 D21 D22  W2   f 2 

 D11
2
D12  W2   f 2 
2

 2 2    (14)
 D21 D22  W3   f 3 

w  F 
where Wi   i  , f i   i  , and k ij is the dynamic stiffness matrix term relating the ith
r

w'i  M i 
displacement vector with the jth generalized force vector. The dynamic stiffness matrix can be
assembled for the whole cell

 D11 D12 0  W1   f1 
D    
 12 D22 D23  W2    f 2 

 0
 D23 D33  W3   f 3 
   

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,

 D11 D12  W1   f1 
     
 D21 D22  W3   f 3 

1
where D11  D11  D12 D22 D12 ,



1
D12   D12 D22 D32 ,

1
D21   D32 D22 D12 ,

1
and D22  D33  D32 D22 D32 .

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

T11 T12  W1  W3 
T     
 21 T22   f1   f 3 
1
where T11   D12 D11 ,

1
T12   D12 ,

1
T21  D12  D22 D12 D11 ,
1
and T22  D22 D12 .

We may assume that,

W3    W1 
 e  
 f3   f1 

where  is the propagation factor of the cell.

T 
W1   W1 
 e  
 f1   f1 

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.


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



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 ‎ .18).
1

Figure ‎ .18. The setup of the periodic beam experiment.
1

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 ‎ .19). The beam is then suspended by a thin wire from one
1
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 ‎ .19. A sketch for one cell of the periodic beam.
1

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 ‎ .20).
1



Figure ‎ .20. The excitation piezostack and the output accelerometer.
1

Comparison of Results

The experiment described above was set up and measurements were taken from the two ends of
the beam. Figure ‎ .21 shows the transfer function frequency response of the beam for the plain and
1
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 ‎ .22 presents the frequency response obtained by the finite element
1
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.

40 2.5

20
2

0
Respence Ampl. (dB)

Attenuation Factor

1.5

-20

1

-40 Plain Beam
Periodic Beam
Attenuation Factor
0.5
-60

-80 0
0 1000 2000 3000 4000 5000 6000

Frequency (Hz)

Figure ‎ .21. The frequency response together with the numerical results of the stop bands for the proposed beam.
1



Figure ‎ .22. Frequency response of the beam using finite element model.
1

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 ‎ .23 and Figure ‎ .24. Different cases with varying the lengths L1 and L2 were constructed
1 1
to examine the effect of the geometry on the attenuation characteristics (Figure ‎ .25).
1

Figure ‎ .23. Sketch of the experimental setup for the cantilever beam.
1



Figure ‎ .24. A picture of the experimental setup.
1

Figure ‎ .25. a sketch for the cell geometry of the experiment for the cantilever beam.
1

30 10

9
20

8
10
Transfer Function Amplitude (dB)

7

Attenuatin Factor (rad)
0
6
0 500 1000 1500 2000 2500 3000 3500 4000

-10 5

4
-20

Plain Beam 3
-30 Periodic Beam
Attenuation Factor 2

-40
1

-50 0
Frequency (Hz)

Figure ‎ .26. Experimental results obtained for case #1 compared to plain beam and attenuation curves obtained by
1
numerical model.


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

20 10

9
10
8

0 7
0 500 1000 1500 2000 2500 3000 3500 4000

6
-10

5

-20
4

Plain Beam
-30 3
Periodic Beam
Attenuation Factor
2
-40
1

-50 0
Frequency (Hz)

1
numerical model.

20 10

9
10
8

0 7
0 500 1000 1500 2000 2500 3000 3500 4000

6
-10

5

-20
4

-30 3
Plain Beam
Periodic Beam
2
Attenuation Factor
-40
1

-50 0
Frequency (Hz)

1
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



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 [‎ 5] and Mead et al. [‎ 7]
9 9
introduced relations that could be developed for use with higher order elements.

The input-output relations summarized in Figure ‎ .29 are presented in the following two sets of
1
equations,

w2  e  x w1 , f 2   e  x f1 ,
x  y x  y
w3  e w1 , f3  e f1 ,
 
w4  e y w1 , f 4   e y f1 ,
 
w10  e y w5 , f10  e y f 5 ,
 
w9  e y w6 , f 9  e y f 6 ,
w7  e  x w12 , f 7  e  x f12 ,
and w8  e  x w11 and f 8  e  x f11

where  x  ix and  y  i y with  x and  y denoting the phase factor in the x and y-direction
respectively.

Figure ‎ .29. A sketch representing the relations between the input and output displacements.
1



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,

 w1 
w 
 5
 w6 
  

 m11  m19   k11  k19   w11 
2     
             w12   0 (15)
 
 m91  m99  k91
    k99  w13 
  

w14 
 
w15 
w16 
 

1.6.2. Propagation Surfaces

The concept of propagation surfaces was introduced by Mead and Parathan [‎ 6] as a graphical
9
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 x ,  y respectively, and the average wave numbers in the x and y-directions could be given by

x y
kx  & ky  (16)
a b
where a and b are the plate-period length in the x and y-directions respectively.

Mead and Parathan [‎ 6] used displacement functions to describe the vibration of beams then
9
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,

t 2 a 4
 (
D p 4

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:



  
 2 m  k  0

The propagation surfaces resulting from the solution of the above eigenvalue problem are shown in
Figure ‎ .30.
1

In Figure ‎ .31, which is the same as Figure ‎ .30 but from a different viewing point, we can clearly see
1 1
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”.



y

Figure ‎ .30. Propagation surfaces
1 resulting from the solution of the eigenvalue problem of the finite element model.
x





x y

Figure ‎ .31. The plot of the propagation surfaces from a planar point of view.
1

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 ‎ .32 shows the
1
resulting curves drawn with the independent variable (x) on the vertical axis.



o
Figure ‎ .32. The curves of the propagation surfaces at angle 45 .
1

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 ‎ .33 presents the attenuation curves for a wave propagating at 45o and with the
1
imaginary part of the propagation factor set equal to zero. While Figure ‎ .34 presents the
1
o
attenuation curves with a wave propagating at 45 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.



o
Figure ‎ .33. The “attenuation surface” at angel 45 with the imaginary part set to zero.
1

o
Figure ‎ .34. The “attenuation surface” at angel 45 with the imaginary part set to .
1





y

x

Figure ‎ .35. The attenuation surfaces for the plate with the imaginary part set to zero.
1





y

x

Figure ‎ .36. The attenuation surfaces for the first two attenuation bands of the plate with the imaginary part set to .
1


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

1.6.4. Plates with Periodic Geometry


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 ‎ .37).
1

Figure ‎ .37. Frequency response for a vibration absorber.
1

When an inductance is added to the system, a similar result is obtained for the propagation surfaces
as shown in Figure ‎ .38. It is obvious that the shape of the propagation surfaces exhibits shape
1
similar to that of the frequency response of a two-degree of freedom spring-mass system.





y

x

Figure ‎ .38. The two surfaces resulting from adding the inductance compared to the original surface.
1

Drawing the curves with a wave propagating at 45o (Figure ‎ .39), we can visualize the gap introduced
1
in the band that was originally covered by the first propagation surface. That gap now presents a
stop band.



Figure ‎ .39. The propagation curves resulting from introducing the shunted inductance.
1

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 ‎ .40 and Figure ‎ .41).
1 1

Figure ‎ .40. The attenuation curves resulting from introducing the shunted inductance at phase angle .
1



Figure ‎ .41. The attenuation curves resulting from the introduction of the shunted inductance at phase angle zero.
1


Periodic Structures: A Passive Vibration Filter Appendices

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,

 w1 
w 
 5
 w6 
  

 m11  m19   k11  k19   wb 
 
   
              wc   0
 
 m91  m99  k 91
    k 99   wd 
  

 we 
w 
 f
 wg 
 

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),

x  y y y x  y
k11  k11  k12e  x  k13e  k14e  k 21e   x  k 22  k 23e  k 24e
 x  y  y  y  x  y
k 31e  k 32e  k 33  k 34e   x  k 41e  k 42e  k 43e  x  k 44
y  x   y  x  y  y
k12  k15  k1a e  k 25e   x  k 2 a e  k 35e  k 3a e   x  k 45e  k 4a
y  x  y  x  y  y
k13  k16  k19e  k 26e   x  k 29e  k 36e  k 39e   x  k 46e  k 49
 y  x  y  y  y
k14  k18e  x  k1b  k 28  k 2b e   x  k 38e  k 3b e  k 48e  k 4b e
 y  x  y  x  y  y
k15  k17e  x  k1c  k 27  k 2c e   x  k 37e  k 3c e  k 47e  k 4c e
 x  y  y
k16  k1d  k 2 d e   x  k 3d e  k4d e
 x  y  y
k17  k1e  k 2e e   x  k 3e e  k 4e e
 x  y  y
k18  k1 f  k 2 f e   x  k 3 f e  k4 f e
 x  y  y
k19  k1g  k 2 g e   x  k 3 g e  k4g e



x  y y y  x  y
k 21  k 51  k 52e  x  k 53e  k 54e  k a1e  ka2e  k a 3e  x  k a 4
y  y
k 22  k 55  k 5 a e  ka5e  k aa
y  y
k 23  k 56  k 59e  ka6e  k a9
x  x  y  y
k 24  k 58e  k 5b  k a 8 e  k abe
 x  y  y
k 25  k 57e  x  k 5c  k a 7 e  k ace
 y
k 26  k 5 d  k ad e
 y
k 27  k 5e  k aee
 y
k 28  k 5 f  k af e
 y
k 29  k 5 g  k ag e

x  y y y  x  y
k 31  k 61  k 62e  x  k 63e  k 64e  k 91e  k 92e  k 93e  x  k 94
y  y
k 32  k 65  k 6 a e  k 95e  k9a
y  y
k 33  k 66  k 69e  k 96e  k 99
 x  y  y
k 34  k 68e  x  k 6b  k 98e  k 9b e
 x  y  y
k 35  k 67e  x  k 6c  k 97e  k9c e
 y
k 36  k 6 d  k 9 d e
 y
k 37  k 6e  k 9 e e
 y
k 38  k 6 f  k 9 f e
 y
k 39  k 6 g  k 9 g e

y x  y x  y y
k 41  k 81e   x  k 82  k 83e  k 84e  k b1  k b 2 e  x  k b 3 e  kb4 e
 x  y y
k 42  k 85e   x  k 8 a e  k b 5  k ba e
 x  y y
k 43  k 86e   x  k 89e  k b 6  k b9 e
k 44  k 88  k 8b e   x  k b8 e  x  k bb
k 45  k 87  k 8c e   x  k b 7 e  x  k bc
k 46  k 8 d e   x  k bd
k 47  k 8e e   x  k be
k 48  k 8 f e   x  k bf
k 49  k 8 g e   x  k bg



y x  y x  y y
k 51  k 71e   x  k 72  k 73e  k 74e  k c1  k c 2 e  x  k c 3e  kc 4e
 x   y y
k 52  k 75e   x  k 7 a e  k c 5  k ca e
 x  y y
k 53  k 76e   x  k 79e  kc6  kc9e
k 54  k 78  k 7 b e   x  k c8 e  x  k cb
k 55  k 77  k 7 c e   x  k c 7 e  x  k cc
k 56  k 7 d e   x  k cd
k 57  k 7 e e   x  k ce
k 58  k 7 f e   x  k cf
k 59  k 7 g e   x  k cg

x  y y
k 61  k d 1  k d 2 e  x  k d 3 e  kd 4e
y
k 62  k d 5  k dae
y
k 63  k d 6  k d 9 e
k 64  k d 8 e  x  k db
k 65  k d 7 e  x  k dc
k 66  k dd
k 67  k de
k 68  k df
k 69  k dg

x  y y
k 71  k e1  k e 2 e  x  k e3 e  ke4 e
y
k 72  k e5  k ea e
y
k 73  k e 6  k e9 e
k 74  k e8 e  x  k eb
k 75  k e 7 e  x  k ec
k 76  k ed
k 77  k ee
k 78  k ef
k 79  k eg



x  y y
k81  k f 1  k f 2 e  x  k f 3 e  k f 4e
y
k82  k f 5  k fa e
y
k83  k f 6  k f 9 e
k84  k f 8 e  x  k fb
k85  k f 7 e  x  k fc
k86  k fd
k87  k fe
k88  k ff
k89  k fg

x  y y
k91  k g1  k g 2 e  x  k g 3e  k g 4e
y
k92  k g 5  k gae
y
k93  k g 6  k g 9 e
k94  k g 8e  x  k gb
k95  k g 7 e  x  k gc
k96  k gd
k97  k ge
k98  k gf
k99  k gg

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
E 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


Periodic Structures: A Passive Vibration Filter References and Bibliography

U Potential energy
V Volume
W External work
w Transverse displacement
wb,wD Nodal transverse and electric displacements respectively
1


(.) 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
E 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.



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.



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 of
September 1996.
31. Wu, S. Y., “Broadband Piezoelectric Shunts for Structural Vibration Control,” US Patent
Number 6,075,309, 13th of June 2000.
32. McGowan, A. R., "An Examination of Applying Shunted Piezoelectrics to Reduce
Aeroelastic Response," CEAS/AIAA/ICASE/NASA Langley International Forum on
Aeroelasticity and Structural Dynamics 1999, Williamsburg, Virginia, June 22-25, 1999.
33. Cross, C. J. and Fleeter, S., “Shunted Piezoelectrics for Passive Control of
Turbomachine Blading Flow-Induced Vibration,” Smart Materials and Structures, Vol.
11, No. 2, 2002, pp 239-248.
34. Zhang, J. M., Chang, W., Varadan, V. K., and Varadan, V. V., “Passive Underwater
Acoustic Damping Using Shunted Piezoelectric Coatings,” Smart Materials and
Structures, Vol. 10, No. 2, pp. 414-420.
35. Warkentin, D. J., and Hagood, N. W., "Nonlinear Shunting for Structural Damping,"
Proceedings of SPIE's 5th Annual Symposium on smart Structures and Materials, Vol.
3041, March 1997, pp. 747-757.
36. Davis, C. L. and Lesieutre, G. A., “A Modal Strain Energy Approach To The Prediction
of Resistively Shunted Piezoelectric Damping,” Journal of Sound and Vibration, Vol.
184, No. 1, 1995, pp. 129-139.
37. Saravanos, D. A., "Damped Vibration of Composite Plates with Passive Piezoelctric-
Resistor elements," Journal of Sound and Vibration, Vol. 221, No. 5, 1999, pp. 867-885.
38. Saravanos, D. A. and Christoforou, A. P., “Impact Response of Adaptive Piezoelectric
Laminated Plates,” AIAA-2000-1498, 41st AIAA/ASME/ASCE/AHS/ASC Structures,
Structural Dynamics, and Materials Conference and Exhibit, Atlanta, GA, 3-6 April
2000.
39. Park, C. H. and Inman D. J., “A Uniform Model for Series R-L and Parallel R-L Shunt
Circuits and Power Consumption,” SPIE Conference Proceedings on Smart Structure
and Integrated Systems, Newport Beach, CA, March 1999, Vol. 3668, pp. 797-804.
40. Caruso, G., “A Critical Analysis of Electric Shunt Circuits Employed in Piezoelectric
Passive Vibration Damping,” Smart Material and Structures, Vol. 10, No. 5, 2001, pp.
1059-1068.
41. Benjeddou, A., “Advances in Piezoelectric Finite Element Modeling of Adaptive
Structural Elements: A Survey,” Computers and Structures, Vol. 76, 2000, pp. 347-363.
42. Tzou, H. S. and Tseng, C. I., “Distributed Piezoelectric Sensor/Actuator Design For
Dynamic Measurement/Control of Distributed Parameter Systems: A Piezoelectric Finite
Element Approach,” Journal of Sound and Vibration, Vol. 138, No. 1, 1990, pp. 17-34.


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