AIAA 2002-1284

American Institute of Aeronautics and Astronautics

1

Active/Passive Vibration Reduction of Periodic 1-D Structures Using Piezoelectric Actuators

Amit Singh

�

Darryll J. Pines

���

Alfred Gessow Rotorcraft Center Department of Aerospace Engineering

University of Maryland College Park, MD-20742-3015

This paper develops an integrated model of periodic 1-D structures with piezoelectric actuators for complete active/passive control. The approach utilizes the property of periodic structural components that create a stop band region in the frequency spectra, predominantly in the higher frequency range. This basic property of periodic structures is enhanced by the application of periodically placed piezoelectric actuators, with piezo forces as a function of displacement. With this control capability, the piezoelectric actuators can introduce the proper force to reduce wave propagation, both in higher and lower frequency range. An analytical model is developed to predict the performance of the periodic rods and beams with piezoelectric actuators acting as controllers. For the purpose of this research, only geometric variations are considered and every cell is assumed to be identical.

Nomenclature

EA axial stiffness EI bending stiffness F1 external point force Gr wave controller of rod Gb wave controller of beam i imaginary unit -1 n number of degree of freedom u axial displacement υ vertical displacement Wle leftward evanescent wave component Wlp leftward propagating wave component Wre rightward evanescent wave component Wrp rightward propagating wave component θ angular displacement ρ A mass per unit length k wave number

_________________________________________________________________ �

Graduate Research Assistant, Smart Structures Laboratory, Alfred Gessow

Rotorcraft Center, Department of Aerospace Engineering; samit@glue.umd.edu ���

Associate Professor, Smart Structures Laboratory, Alfred Gessow Rotorcraft

Center, Department of Aerospace Engineering; djpterp@eng.umd.edu.

Associate Fellow AIAA

Copyright 2002 by Amit Singh and Darryll J. Pines. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

1. Introduction

Periodic structures act as filters for traveling waves, thus serving as a passive mode of vibration reduction. This vibration attenuation capability of the periodic structure is generally not very dominant in the lower frequency range. Piezoelectric actuators placed periodically along the rods and beams, acting as a source of controlled external forces and moments, can greatly enhance the vibration suppression properties of a periodic structure over the complete frequency spectra. Such controlled characteristics are attributed to the unique behavior of the piezoelectric actuators to expand and contract at the application of voltage, thus creating forces on the boundaries constraining it. With such a controllable capability, the piezos can introduce the desired forces and moments.

Theoretical Development A periodic structure consists of an assembly of identical elements connected in a repeating array, which together form a complete structure. Examples of such structures are found in many engineering applications. These include bulkheads, airplane fuselages, and apartment buildings with identical stories. Each such structure has a repeating set of

43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con22-25 April 2002, Denver, Colorado

AIAA 2002-1284

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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stiffeners, placed at regular intervals. The study of periodic structures has a long history. Wave propagation in periodic systems has been investigated for approximately 300 years1. Typically the studies have been related to crystals, optics, and the like. It is only recently that the wave motion in periodic structures has been studied. The salient feature of such structures is the fact that waves can propagate through the structure in some frequency bands (pass bands) and not in others (stop bands )2-7. The present work uses this property of the periodic structure and blends it with the control concepts, to obtain an active/passive periodic structure with enhanced vibration suppression capabilities. This paper is organized into 6 sections. The first section gives a brief introduction of periodic structures. Section 2 presents the fundamentals of wave propagation in a rod. The transfer functions and the propagation parameter of a periodic rod are calculated. In section 3, wave propagation is studied in a periodic beam. Section 4 presents the analysis of an active/passive periodic rod with piezoelectric actuators (acting as controllers), periodically placed along the periodic rod. The control gain is calculated, through which the Dereverberated Transfer Function8 and the propagation parameter are obtained. Section 5 presents the calculation of the control gain and the DTF for a periodic beam. In sections 4 and 5, the DTF results are compared with the RTF results to show the enhanced vibration suppression achieved by using the active/passive periodic structure.

2. Periodic Rod A periodic rod cell is shown in Fig. 1 with properties listed in Table 1.

Fig 1. Simple Periodic Rod Cell Geometry

The global stiffness matrix of an element of the cell is given as

+−

−+−

=

−−

−−

− )1(2

2)1(

)1(

),(

2

2

2 ikLikL

ikLikL

ikL ee

ee

e

ikL

L

EA

LkK

(1)

where EA

Ak

ρω= , is the longitudinal wave

number. Assume that the energy is input from the left end with F1 ≠ 0 The global stiffness matrix of the cell is given as

[ ][ ]

+

=

elementB

elementA

cell

KK

LkK

0

0000

0000

0

),(

(2) Using the same procedure, the Global stiffness matrix Krod(k, L) can be formulated for an entire periodic rod structure. Now,

( )

••

=

•• −

∧

∧

∧

∧

0

1

),(

1

11LkK

F

U

F

U

rod

right

left

(3)

E(Pa) q (kg/m3)

EI(Nm2)

ROD 21e10 7850 --- BEAM 21e10

7850 21,721.5(1.5” dia)

314,829(1.5”dia+piezos) 268.2(0.5” dia)

Table 1 Material Properties

The response of ∧

∧

1F

U left and ∧

∧

1F

U right are shown in

Fig. 2 The rod is comprised of 5 cells, with repeating cells of the same dimension and mechanical properties.

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100

102

10-10

10-5

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

Uleft/F1

Fig. 2a

100

102

10-10

10-5

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

Uright/F1

Fig 2b

The forces at the ends of each element are related to the displacements by the relation

=

R

L

RRRL

LRLL

R

L

U

U

KK

KK

F

F

(4) The transfer matrix, T, may be constructed, using the transformation

−+−−= −−

−−

11

11

LRRRLLLRRRRL

LRLLLR

KKKKKK

KKKT

(5) Now, the transfer matrix of the cell shown in Fig. 1 can be computed as

elementAelementBCELL TTT ×= (6)

and for the complete rod

( ) CELLNCELLROD TT = (7)

where, NCELL is the number of cells in the structure. Thus, the eigenvalue problem is formulated as

[ ]RightL

L

LeftL

L

F

U

F

UT

=

λ

(8) From the eigenvalue problem, we can determine the propagation parameter µ which is related to the eigenvalue λ . By definition of the hyperbolic cosine

e µ− + e µ = 2 cosh( µ )

Solving for µ leads to

µ = 2

eacosh

+ −µµ e=

+

2

1

cosh λλ

a

(9) Using the properties listed in Table 1, the real and the imaginary parts of µ can be plotted. Fig. 3 displays these properties as a function of frequency.

0 500 1000 15000

2

4

6

8

10

Frequency(Hz)

Rea

l µ

Fig.3a Real Propagation coefficient

0 500 1000 15000

1

2

3

4

Frequency(Hz)

Imag

inar

y µ

Fig. 3b Imaginary Propagation

Coefficient

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The real part of the propagation coefficient shows the attenuation zone and the imaginary part shows the propagation zone. Fig 3 shows that the attenuation zone lies in the frequency domain from 800 Hz to 1100 Hz.

3. Periodic Beam A periodic beam cell is shown in Fig. 4

Fig 4. Simple Periodic Beam Cell Geometry

The equation of motion for a Bernouli-Euler beam is given by

( ) ( )txft

txA

x

txEI ,

),(,2

2

4

4

=∂

∂+∂

∂ υρυ

(10) Transformed into frequency domain, Eq. (10) becomes

( ) ( ) ( )ωωυωρωυ,,

, 24

4

xfxAx

xEI

∧∧∧

=−∂

∂

(11) The spectral representation for the beam’s flexural vibration are

( ) ( )xLkle

xLiklp

kxre

ikxrp eWeWeWeW −−−−−−

∧+++=υ

( ) ( ) ( ) ( )xLiklp

kxre

ikxrp eikWekWeikW −−−−

∧+−+−=θ

( ) ( )xLkle ekW −−+

(12)

where 4

1

2

= ωρ

EI

Ak is the bending wave

number. Different sign conventions lead to different expressions of nodal displacements and nodal forces. The sign convention used here is shown in Fig. 5 ����������� L ������������������� R

�������� L R

FL FR

ML MR

Fig. 5 Single Cell Element Nodal displacements are given by

−−

−−=

−−

−−

−−

−−

le

lp

re

rp

kLikL

kLikl

kLikL

kLikL

R

R

L

L

W

W

W

W

kikkeike

ee

keikekik

ee

11

11

θυθυ

(13) The nodal forces

0

3

3

=∂∂=

x

L xEIF

υ,

Lx

R xEIF

=∂∂−=

3

3υ

0

2

2

=∂∂−=

x

L xEIM

υ,

Lx

R xEIM

=∂∂=

2

2υ

(14) are motion dependent and are given by

−−−−

−−−−

=

−−

−−

−−

−−

le

lp

re

rp

kLikL

kLikL

kLikL

kLikL

R

R

L

L

W

W

W

W

ee

kikkeike

ee

keikekik

EIk

M

F

M

F

11

112

(15) From Eq. 13 and Eq. 15, we can get

[ ]

=

R

R

L

L

R

R

L

L

Kelement

M

F

M

F

θυθυ

(16)

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The global stiffness matrix of the cell is given as [ ] [ ]elementBelementAcell KKLkK +=),(

(17) Using the same procedure as before, it is not difficult to assemble the Global stiffness matrix for the complete beam Kbeam(k, L). Now, assuming the energy input from the left, with F1 �����01 ����

( )

••••••••

=

••

••

••

••

−

∧∧∧∧

∧∧∧∧

∧∧∧∧

∧∧∧∧

00

00

10

01

),(

//

//

//

//

1

11

11

11

11

LkK

MF

MF

MF

MF

beam

RR

RR

LL

LL

θθ

υυ

θθ

υυ

(18)

Using the properties of the beam listed in Table 1,

the response of ∧∧

1/ FLυ , ∧∧

1/ MLυ , ……., ∧∧

1/ FRθ

and ∧∧

1/ MRθ are shown in Fig. 5

The beam comprises of 5 cells

100

102

104

10-10

10-5

100

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

Fig. 5a ∧∧

1/ FLυ

100

102

104

10-10

10-5

100

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

Fig. 5b ∧∧

1/ FRυ

100

102

104

10-8

10-6

10-4

10-2

100

Frequency(Hz)

Dis

plac

emen

t(m

)/M

omen

t(N

-m)

Fig. 5c ∧∧

1/ MLθ

100

102

104

10-8

10-6

10-4

10-2

100

Frequency(Hz)

Dis

plac

emen

t(m

)/M

omen

t(N

-m)

Fig. 5d ∧∧

1/ MRθ

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100

102

104

10-8

10-6

10-4

10-2

100

Frequency(Hz)

Dis

plac

emen

t(m

)/M

omen

t(N

-m)

Fig. 5e ∧∧

1/ MLυ

100

102

104

10-10

10-5

100

Frequency(Hz)

Dis

plac

emen

t(m

)/M

omen

t(N

-m)

Fig. 5f ∧∧

1/ MRυ

100

102

104

10-8

10-6

10-4

10-2

100

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

Fig. 5g ∧∧

1/ FLθ

Notice in all the transfer functions, there is a distinct high frequency that is absent of resonant dynamics.

100

102

104

10-10

10-5

100

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

Fig. 5h ∧∧

1/ FRθ

Following the same procedure as before, the real and imaginary Propagation Coefficient for the beam can be plotted.

0 2000 4000 6000 8000 100000

1

2

3

4

Frequency(Hz)

Rea

l µ

Fig. 5i Real Propagation Coefficient

0 2000 4000 6000 8000 100000

1

2

3

4

Frequency(Hz)

Imag

inar

y µ

Fig. 5j Imaginary Propagation Coefficient

Fig 5i plots the real part of the bending propagation parameter. This figure illustrates the frequency range of attenuation. The results presented in Fig 5 for a passive periodic beam will be compared in the section below for a

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combined Active/Passive beam. Piezoelectric Actuators mounted on both, beam and rod will be used to extend the real part of µ over a broader frequency range.

4. Active/Passive Periodic Rod In this section the knowledge gained from the analysis of the passive periodic rod will be used. A picture of the integrated structure is shown in Fig. 6. In this case we have considered 4 piezos, separated by 900 each.

L1 L2 L1 L2 FC Piezoelectric Actuator

FC Fig. 6 Periodic Structure with Piezoelectric Actuators Figure 6 shows a two cell integrated periodic rod. Two cells are shown in the figure to make it clear to the reader that there will be no piezo forces acting on the extreme left node and the last two nodes. The system displayed here presents one strategy for active-passive control. The first step now, is to calculate the stiffness matrix of the second element of a cell, which includes piezoelectric actuators. This is achieved by solving the energy equation: u(x) u1 u2 Fig. 7

Here, 2),(1),(),( 21 uxguxgxu ωωω +=

Substituting the values of g1 and g2 for both the rod and the piezos, we get

( )[ ][ ] +

−−= −

−−−

11

),(1

11

2

2

ue

eexu

Lik

xLikxik

rod ω

( ) ( )[ ][ ] 21 1

11

2u

e

eeLik

xLikxLik

−

−−+−

−+−

( )[ ][ ] +

−−= −

−−−

11

),(1

22

2

2

ue

eexu

Lik

xLikxik

piezo ω

( ) ( )[ ][ ] 21 2

22

2u

e

eeLik

xLikxLik

−

−−+−

−+−

(19) where, k1 and k2 are the longitudinal wave numbers of the rod and the piezo, respectively. Now,

dxt

uAdx

t

uA

dxx

uAEdx

x

uAE

Lpiezo

piezopiezo

Lrod

rodrod

Lpiezo

piezopiezo

Lrod

rodrod

2

0

2

0

2

0

2

0

22

1

22

1

∫∫

∫∫

∂

∂−

∂∂

−

∂

∂+

∂∂

ρρ

= [ ]

2

121

u

u

KK

KKuu

RRRL

LRLL (20)

where,

RRRL

LRLL

KK

KK, is the stiffness matrix for

the second element. Proceeding in the same way as for a periodic rod in section 2, RTF for this integrated system is formulated. Now, assume that energy is input from the left end with F1 ≠ 0. As the incident energy arrives at the end of the element, a control force Fc = Gru is designed to satisfy two objectives:1) Ensure equilibrium of the forces at those ends and 2) prevent energy reflection from those ends. This control force is to be provided by piezoelectric actuators. However, before the

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expression for control gain is determined, the values of equivalent Young Modulus, Area, density and wave number for the second element have to be calculated. The equivalent values are calculated below: Aequivalent(B) = Arod + 4Apiezo

Eequivalent(B) = (ErodArod + 4EpiezoApiezo)/At

Mequivalent(B) = ρ rodArodLrod +

4 ρ piezoApiezoLpiezo

Vequivalent(B) = ArodLrod + 4ApiezoLpiezo

ρ equivalent(B) = Mequivalent(B)/Vequivalent(B)

kequivalent(B) = )()(

)()(

BequivalentBequivalent

BequivalentBequivalent

AE

Aρω

(21) where, A, E, M, V, ρ , k are the area, young modulus, mass, volume, density and wave number respectively. The force equilibrium and displacement compatibility conditions at the interface between Element A and B are given by:

0)( =++ leftBequivalent

rightelementAC FFF

leftBequivalent

rightelementA uu )(=

and Wlp(ω ) = 0 (22) where,

elementALx

elementAelementAright

elementA

x

uAEF

=

∧

∧

∂

∂=

0

)()()(

=

∧

∧

∂

∂=x

BequivalentBequivalentleft

Bequivalent

x

uAEF

(23) are the motion dependent forces. After some algebra, the expression for the control gain is determined to be

)()()(1 BequivalentBequivalentBequivalentAAA AEikAEikG −= (24) Similarly, applying the force equilibrium and the displacement compatibility conditions at the interface between Element B and A leads to:

0)( =++ rightBequivalent

leftelementAC FFF

rightBequivalent

leftelementA uu )(=

and Wlp(ω ) = 0 (25) where,

0=

∧

∧

∂

∂=x

elementAelementAleft

elementA

x

uAEF

BLx

BequivalentBequivalentright

Bequivalent

x

uAEF

=

∧

∧

∂

∂= )()()(

(26) are the motion dependent forces. After some algebra, the expression for the control gain is determined to be

AAABequivalentBequivalentBequivalent AEikAEikG −= )()()(2

(27) It should be noted here, that the gains G1 and G2 just differ in sign, because the force generated by the piezos on both sides is of same magnitude, but opposite in sign. RTF matrix of this integrated system is calculated using the same procedure as in the section of simple periodic rod. Now, with the RTF for a cell available, the

DTF’s ∧

∧

1F

U left and ∧

∧

1F

U right can be obtained using the

following expression:

+=

−

−

∧

∧

∧

∧

∧

∧

00

1

1

1

1 1

2

1

212

G

G

G

RTF

F

U

F

U

F

U

right

node

left

(28) In a real system, forces at the leftmost node and the rightmost two nodes cannot be applied, as discussed earlier. This has been taken care of in the system when gain was added to the system. The method to

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calculate the propagation parameter is the same as discussed in the section on the simple passive periodic rod. Because the controllers are designed to eliminate nodal reflections, the DTF is obtained for Active/Passive rod. System, with 5 cells is analyzed and compared with the simple periodic rod system with the same geometry but no controllers attached. The results are shown in Fig.8

100

102

10-10

10-5

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

Uleft/F1DTFRTF

Fig. 8a

100

102

10-10

10-5

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

Uright/F1DTFRTF

Fig. 8b

0 500 1000 15000

5

10

15

Frequency(Hz)

Rea

l µ

DTFRTF

Fig. 8c Real Propagation Coefficient

0 500 1000 15000

1

2

3

4

Frequency(Hz)

Imag

inar

y µ

DTFRTF

Fig. 8d Imaginary Propagation Coefficient

Notice from Fig 8c that the effect of the controller is to extend the real part of the propagation parameter µ over a broader frequency range. This has the effect of damping the vibration modes at low frequency while maintaining the performance at high frequency by the periodic properties of the rod.

5. Active/Passive Periodic Beam This section uses the knowledge gained from the analysis of the passive periodic beam. A picture of the integrated structure is shown in Fig. 9. In this case 2 piezos separated by 1800, are considered.

L1 L2 L1 L2 FC

FC

Fig. 9 Fig 9 shows a two cell integrated periodic beam. The system provides an active/passive control. In this section, same procedure to calculate the stiffness matrix is used as in the previous section for the active/passive periodic rod. From the stiffness matrix, RTF for the integrated system is obtained. However, one difference from the active/passive periodic rod case is that the two piezos apply force in the opposite directions, creating a moment. Now, assume that energy is input from the left end with F1 ≠ 0, M1 ≠ 0. As the incident energy arrives

at the element ends, a control moment Mc = Gb

θυ

is

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designed to satisfy two objectives: 1) Ensure equilibrium of the moments at the ends and 2) prevent energy reflection from those ends. It should be noted that the controllers provide no force to balance the shear forces at the nodes. The control force, to be provided by these controllers (piezoelectric actuators) can be developed by supplying voltage to the actuators. Relationship can be obtained between voltage and the actuator force. Before the expression for control gain is obtained, values of equivalent Young Modulus, Area density and wave number for the second element have to be determined. The equivalent values are calculated as Aequivalent(B) = Arod + 2Apiezo

Eequivalent(B) = (ErodArod + 2EpiezoApiezo)/At

Mequivalent(B) = ρ rodArodLrod +

2 ρ piezoApiezoLpiezo

Vequivalent(B) = ArodLrod + 2ApiezoLpiezo

ρ equivalent(B) = Mequivalent(B)/Vequivalent(B)

kequivalent(B) = 4

1

2

)()(

)()(

ω

ρ

BequivalentBequivalent

BequivalentBequivalent

IE

A

(29) where, A,E,M,V, ρ ,k are area, young modulus, mass, volume, density and wave number respectively. The moment equilibrium compatibility condition at the interface between Element A and B is given by:

0)( =++ leftBequivalent

rightelementAC MMM

(30)

=

0

0

)(

)(

ωω

le

lp

W

W

(31) where,

Mc = Gb

elementA

elementA

θυ

(32) After the complex algebra, the expression for control gain is determined to be

+−−+−

−=

QQQAAAQQQAAA kIEkIEikIEkIEi

G

)(1()(

0022

1

(33) where,

BQelementAA

BequivalentQelementAABequivalentQ

elementAABequivalentQelementAA

LLLL

kkkkII

IIEEEE

==

===

===

,

,,,

,,,

)()(

)(

(34) Similarly, applying the moment equilibrium compatibility condition at the interface between Element B and A leads to:

0)( =++ rightBequivalent

leftelementAC MMM

(35)

=

0

0

)(

)(

ωω

le

lp

W

W

(36) where,

Mc = Gb

)(

)(

Bequivalent

Bequivalent

θυ

(37) From here we get G2 in the same way as G1

RTF matrix of this integrated system is calculated using the same procedure as in the section for the simple periodic beam. Now, when this matrix control law is used, the DTF’s can be obtained. For a single cell, with energy input F1 and M1 from the left, the DTF is given as:

∧∧∧∧

∧∧∧∧

∧∧∧∧

∧∧∧∧

∧∧∧∧

∧∧∧∧

11

11

1212

1212

11

11

//

//

//

//

//

//

MF

MF

MF

MF

MF

MF

RR

RR

nodenode

nodenode

LL

LL

θθ

υυ

θθ

υυ

θθ

υυ

=

×

+

−

×

−

×

×

×

00

00

00

00

10

011

662

1

21

22

22

22

G

G

G

RTF

(38) A 5-cell active/passive periodic beam is analyzed, and the transfer function responses are plotted. Precautions, like no control forces on the extreme left node and the rightmost two nodes, have been taken

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care of, in this analysis too. The results are plotted in Fig. 10 and compared with the results of passive periodic rod with same geometry

100

102

104

10-10

10-5

100

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N) DTF

RTF

Fig. 10a ∧∧

1/ FLυ

100

102

104

10-10

10-5

100

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N) DTF

RTF

Fig. 10b ∧∧

1/ FRυ

100

102

104

10-8

10-6

10-4

10-2

100

Frequency(Hz)

Dis

plac

emen

t(m

)/M

omen

t(N

-m)

DTFRTF

Fig. 10c ∧∧

1/ MLθ

100

102

104

10-8

10-6

10-4

10-2

100

Frequency(Hz)

Dis

plac

emen

t(m

)/M

omen

t(N

-m) DTF

RTF

Fig. 10d∧∧

1/ MRθ

100

102

104

10-8

10-6

10-4

10-2

100

Frequency(Hz)

Dis

plac

emen

t(m

)/M

omen

t(N

-m)

DTFRTF

Fig. 10e ∧∧

1/ MLυ

100

102

104

10-10

10-5

100

Frequency(Hz)

Dis

plac

emen

t(m

)/M

omen

t(N

-m)

DTFRTF

Fig. 10f ∧∧

1/ MRυ

American Institute of Aeronautics and Astronautics

12

100

102

104

10-8

10-6

10-4

10-2

100

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

DTFRTF

Fig. 10g ∧∧

1/ FLθ

100

102

104

10-10

10-5

100

Frequency(Hz)

Dis

plac

emen

t(m

)/F

orce

(N)

DTFRTF

Fig. 10h ∧∧

1/ FRθ

The method to calculate the propagation parameter is the same as discussed in the section for simple periodic beam, but here instead of the DTF, the RTF is used.

0 2000 4000 6000 8000 100000

1

2

3

4

Frequency(Hz)

Rea

l µ

DTFRTF

Fig. 10i Real Propagation coeffient

0 2000 4000 6000 8000 100000

1

2

3

4

Frequency(Hz)

Imag

inar

y µ

DTFRTF

Fig. 10j Imaginary Propagation Coefficient Again as in the case of the rod, Fig 10i and Fig 10j clearly illustrate the effect of the controller on the real part of the propagation coefficient. Notice that the real part extend over a broader frequency range than in the case of purely passive periodic beam. This permits damping of low frequency vibration modes. In the case of the beam, the passive periodicity with the controller is not as well preserved at high frequency as in the case of the rod.

6. Summary and Conclusions This paper has presented the first combined Active/Passive periodic properties for a rod and beam. Results from simulation experiments suggest that Active control and the concept of Dereverberation can be used to cancel reflected waves at low frequencies while preserving high frequency properties of the passive periodic structures.

Acknowledgements

This work is supported by the National Rotorcraft Technology under Grant NGT252273, with Dr. Yung Yu serving as Program Monitor.

American Institute of Aeronautics and Astronautics

13

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