[american institute of aeronautics and astronautics 43rd aiaa/asme/asce/ahs/asc structures,...

AIAA-2002-1547

A Phased Sensor/Actuator Array for Detecting Damage in 2-D Structures

Ashish Purekar∗ Darryll J. Pines†

Alfred Gessow Rotorcraft CenterDepartment of Aerospace Engineering

University of MarylandCollege Park, MD 20742

Abstract

A phased array concept is introduced for qualitativemonitoring of vibrations in plate structures. Used asa sensor array, the phased array listens for waves com-ing from different directions on the plate. Piezoelec-tric materials are used to both sense structural vibra-tions as well as introduce a source of disturbance inthe structure. The array is used to send out a signalinto the plate and then monitor the vibrations to lookfor reflections from discontinuities. Incipient waves ap-proach damage, in the form of discontinuities, produc-ing reflections which are monitored qualitatively.

Introduction

Research for damage detection has increased in orderto develop technologies which can reliably and accu-rately diagnose the presence of damage in structures.Much work has been done in developing ways of up-dating conventional dynamic models of structures.1–4

Mass and stiffness matrices are updated to includedamage effects. Local methods, such as ultrasonics,approach the damage detection problem by probingcritical points on the structure in hopes of seeing achange caused by damage.

Wave models are used instead of conventionalmass and stiffness models to determine the responseof a structure as a combination of reverberant waves.Extension of wave models to damage detection involvebacking out the size and location of damage by lookingat incident, reflected, and transmitted waves and theirproperties.5–7 The work done using wave methods arereadily applied toward 1-D structures such as beams or

∗Graduate Research Assistant, Member: AIAA. Tel: (301)405-1142 Fax: (301) 314-9001, email: [email protected]

†Associate Professor, Member: AIAA. Tel:(301) 405-0263Fax:(301) 314-9001, email: [email protected]

Copyright c© 2002 by Ashish Purekar and Darryll J. Pines. Pub-lished by the American Institute for Aeronautics and Astronau-tics, Inc., with permission.

rods, however, the application of wave methods towardmore complicated structures such as plates becomesmore difficult. Reflections do not occur at specific lo-cations along the structure, namely the end points orjunctions in a beam, but occur at distributed locationsalong the structure. Nevertheless, boundaries and dis-continuities of a plate produce reflections in much thesame way as the 1-D counterpart, the beam. An ac-tive damage detection system could look for reflectionsfrom discontinuities which would point out the locationand extent of damage.

Arrays in acoustic or electromagnetic applicationsare normally physically rotated toward a source in or-der to get the largest response. For static arrays ar-tificial phasing is used to tune into a source. Phasedarrays in acoustics and antenna theory uses the oscil-latory nature of acoustic and electromagnetic waves tosteer and focus in on a signal coming from specific di-rections. Dynamics of structures can also be expressedin a wavelike nature and so the phased array conceptcan also be applied toward sensing structural dynam-ics. High frequency beam dynamics has been expressedusing two directional propagating waves, therefore aphased array on a beam would look in one of two di-rections. In a plate, structural waves which are presentpropagate in all directions and a phased array shouldbe able to distinguish these waves from each other. Auseful application of phased arrays is towards damagedetection in plate structures.

Active materials, such as piezoceramics, providealternate ways of exciting and sensing vibrations thannormal methods. Piezo patch actuators on a beamprovide moment excitation and shaped piezo sensorsare able to measure the slope. Furthermore, activematerials enable passive as well active elimination ofvibration in the structure. For damage detection ap-plications, piezoceramics can be used simultaneouslyto send out an interrogating signal and sense the cor-responding structural vibrations.

1American Institute of Aeronautics and Astronautics

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

AIAA 2002-1547

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

Plate Dynamics

As with models of typical structures, the structuraldynamics of a plate can be decomposed into modalcoordinates. An alternate view, and the one whichis taken in this paper, is to view the structure as amedium through which energy propagates in a wave-like nature. The two mechanisms of energy transfer,lateral and transverse, are governed by the materialand cross sectional properties of the structure. Thework presented here deals with transverse dynamicsbut the same sort of methodology can be used withlateral, or in-plane, dynamics also.

For the simple Euler-Bernoulli beam, the solutionto the equation of motion, EIwiv+ρAw = 0, is a com-bination of propagating and near field terms. The nearfield solutions die out spatially whereas the propagat-ing solutions are wavelike and move throughout thewhole beam. In the simple plate case, the governingequation for transverse motion is:

D∇4w +mw = 0 (1)

whereD is the rigidity of the plate, Et3/12(1−ν2), andm = ρt, and the operator ∇4 =

(∂2/∂x2 + ∂2/∂y2

)2.In polar coordinates, the differential equation can bereduced to an ordinary differential equation if thereis no variation in the angular coordinate, θ, and themotion is considered to be harmonic in time.

D

(∂4w

∂r4+ 2

1r

∂3w

∂r3+1r2

∂2w

∂r2

)− ω2mw = 0 (2)

The closed form solution uses Bessel functions whoseproperties have been widely studied.

w = c1H(1)0 (βr) + c2H

(2)0 (βr) + c3I0 (βr) + c4K0 (βr)

(3)where β = 4

√ω2m/D and r is the radial position. The

solutions are analogous to the beam case where I0 andK0 are the near field solutions and H

(1)0 and H

(2)0 are

the propagating solutions. The wavenumber-frequencyrelationship, β, describes the properties of the struc-tural waves.

Of the solutions of the equation governing the mo-tion of a plate, the H(1)

0 (βr) andH(2)0 (βr) terms are of

interest because they describes the motion of a wavesmoving away from and toward the origin. For a largeargument,

H(1)0 (βr) ≈

√2

πβrei(βr−π/4) (4)

H(2)0 (βr) ≈

√2

πβre−i(βr−π/4) (5)

An important aspect of the wave approach tostructural dynamics is their interactions with disconti-nuities such as boundary conditions or damage. InEuler-Bernoulli beam theory, the states of the sys-tems are displacement, slope, moment, and shear inthe beam. These quantities can be expressed in termsof the near field and propagating wave solutions anda scattering matrix relationship is developed which re-lates the incoming near field or propagating wave tothe outgoing near field or propagating wave. Eachboundary condition has a unique scattering matrixwhich relates incoming dynamics to outgoing dynam-ics. A similar situation holds when considering dam-age in the form of a crack. There is an effective changein cross-sectional properties and a complicated scat-tering relationship can be formed. An observation ofthe interaction of structural waves with damage is thattransmitted and reflected waves are produced when anincident wave approaches a damaged region.8,9 Thesame analogy holds when considering plate dynamics.An incident wave to a damage, such as a hole, willproduce an outgoing, or reflected wave.

Phased Arrays

Normally, arrays are physically pointed at the sourceto be listened to, as in Figure 1, but static arrays arenot able to be moved physically. Phased arrays areable to selectively listen to a source from a particulardirection or transmit information in a particular direc-tion using a static array.10 This is done by weightingthe individual responses of a group of sensors in orderto filter out signals coming from undesired directions.The general equation for the response from a continu-ous sensing array, as shown in Figure 2 is given by:

ψ(�r) =∫

A0

X (�r0)D (�r0) g (|�r − �r0|) dA0 (6)

where X is a user specified gain applied to the sensors,D is a function which describes the distribution of thearray and g is the response at �r0 from a disturbanceat �r. The transmitting case is very similar and the ψdescribes the response at �r from signals leaving �r0.

In acoustics, the g term is the Green’s functionwhich describes the dynamics of a source at �r to asensor at �r0 in an unbounded space with no reflections.Typically, the Green’s function involves a magnitudeterm which is inversely proportional to the distancefrom the source and a complex term which describesthe change of phase over that distance.

g (|�r − �r0|) = 14π|�r − �r0|e

−iβ|�r−�r0| (7)


originalorientation

steeredorientation

steering

array source

Figure 1: Steering of an Array toward a Source

continuous arrayx

y

z r&

0r&

Figure 2: Diagram of array.

The β term is a function of the properties of themedium and is inversely proportional to the wave-length at a given frequency, β = 2π/λ. Simplificationof the array equation, given in Equation 6, involvesreplacing the g (|�r − �r0|) term by an approximate, butmore manageable, expression. By assuming that �r ismuch greater in magnitude than �r0 and using only thefirst two terms of a binomial expansion of the complexexponential, the g (|�r − �r0|) term can be reduced to

g (|�r − �r0|) ≈ 14πr

e−iβreiβr·�r0 (8)

In this expression, r is the unit vector to the source at�r and is composed of direction cosines, ai + bj + ck.Combining βr into one expression gives the wavenum-ber vector, �β, which describes both the properties andthe direction of a wave coming from �r.

The phase portion of the approximations thatvary with r, e±iβr, is equivalent to the phase term inEquation 7. Because of this similarity, the structuralwaves present in the plate can be thought of in muchthe same way as the acoustic or electromagnetic waveswhen dealing with phased array theory.

For a finite array which is distributed along a line,the response of the array reduces from an area integralto an integral along a line. With the approximationfor g (|�r − �r0|) described above, the array response in

Equation 6 becomes simpler to handle and looks like

ψ (�r) =14πr

e−iβr

∫ L/2

L/2

X (x0) eiβxx0dx0 (9)

where βx = aβ and a is the direction cosine of r tothe array axis, as shown in Figure 3. When there areno weights applied to the sensor array, the integral inEquation 9 can be found to be a sinc function whichis dependent on the direction cosine, a.

ψ(a) = Lsin

(πL

λa)

πLλa

(10)

where the leading constant terms in Equation 9 are

array

y

x

r

θ

a2/L− 2/L

Figure 3: Diagram of line array in 2D space.

disregarded. The array response is a function of a, thedirection cosine of the incoming wave, and is greatestwhen a = 0 which corresponds to a wave traveling per-pendicular to the array distribution. The array can bethought of as a filter which has different gains for wavestraveling from different directions. For the unsteeredcase, the phased array would apply the highest gainto a wave whose front is parallel to the array distri-bution while applying lower gains to waves from otherdirections.

Steering is accomplished by adding the appropri-ate weights along the array,X(x0), such that the arrayresponse is greatest when a = a′. This is done by set-ting the appropriate β′

x in X(x0) = exp(−iβ′xx0) so

that the array response becomes

ψ(a) = Lsin

(πL

λ (a− a′))

πLλ (a− a′)

(11)

where a′ is the direction to be steered toward. Figure4 shows unsteered and steered responses of an array.The array selectively listens to the direction that issteered to. This filter like behavior of the phased ar-ray is particularly advantageous when coupled with thewavelike nature of structural dynamics.


−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a L / λ

|ψ /

L|a = 0 a’ = −λ / L a’ = λ / L a’ = 2 λ / L

Figure 4: Analytical Unsteered and Steered Responses.

A measure of the width of the main lobe of theunsteered array response is found by determining whenthe array response reaches

√2/2 of its maximum value.

The beamwidth is found by solving the equation

sin(πL

λa

)−

√22

(πL

λa

)= 0 (12)

for a. This relationship is satisfied when a = ±.443λ/Lso the beamwidth of the lobe when the array is steeredtoward 90◦ is ∆a = .886λ/L. The quantity ∆a givesa measure of the sensitivity of the array to the di-rection of the incoming wave. A wave which ap-proaches the array outside of the ∆a region would notbe seen as clearly as a wave within the ∆a region. Thebeamwidth, ∆a, is preserved when the array is steeredsince steering just produces a lateral shift of the un-steered response in Figure 4.

While the plots in Figure 4 are useful in showingthe effects that steering has on the response of the ar-ray from signals coming from different directions, it ismore useful to look at array response in terms of polarcoordinates. Figure 5 shows the array being steeredtoward 60◦ and 45◦.

Note that the main lobe corresponding to the ar-ray response in Figure 5 which is steered toward 45◦ iswider than the unsteered lobe. The width of the mainlobe of the array response grows as the array is steeredaway from 90◦ (broadside) toward either 180◦ or 0◦

(endfire) though this is not readily apparent when con-sidering the direction cosine space as shown in Figure4. When shifting from the direction cosine variable,a, to the angular variable, θ, an inverse cosine needsto be used. A relatively small ∆a around a ≈ ±1yields a large ∆θ causing the lobe to grow wider. Awave which approachs from the endfire locations wouldbe more difficult to resolve than a wave coming from

0.2

0.6

1

30

60

90

120

150

180 0

unsteered

45°

60°

Figure 5: Analytical Unsteered and Steered Normal-ized Responses. (L = 1, λ = .3)

broadside.

Discrete Array

Whereas continuous arrays provide nice characteris-tics and are easy to model, typically arrays composedof discrete elements are used in common applications,shown in Figure 6. An acoustic array would be com-posed of a set of microphones set in a pattern andon a structure such as a plate, individual point mea-surements, such as accelerometers or strain gages, areused as elements of an array. The expression whichdescribes the response of the array changes from anintegral, as in Equation 9 where the constant termsare removed, to a summation over the elements in thearray scaled by the number of elements.

N element array

y

x

r

θ

a0dx

Figure 6: Diagram of discrete array in 2D space.

ψ (�r) =1N

∑N

X (xn) eiβxxn (13)

If there are an odd number of equally spaced elementsin the array and there is no weighting, X(xn), of theelements then the summation becomes:

ψ(a) =1N

sin(Nπ dx0

λ a)

sin(π dx0

λ a) (14)


where dx0 is the element spacing. Steering of the dis-crete array is done in much the same way as the con-tinuous array where each element of the array has aspecific weighting, β′

x based on its position, X(xn) =exp(−iβ′

xxn). The steered array response then be-comes:

ψ(a) =1N

sin(Nπ dx0

λ (a− a′))

sin(π dx0

λ (a− a′)) (15)

The continuous array response in Equation 12has a maximum value at a single position, a = 0.The discrete array equivalent, Equation 13, reachesa maximum value periodically with a because of thesinusoidal nature of the numerator and denominator.A situation can occur where multiple main lobes arepresent within 0 ≤ θ ≤ π, as shown in Figure 7. One ofthose lobes is the main lobe while the others are grat-ings lobes and are undesirable because they corruptthe array response. If the grating lobes are present,the array is as sensitive to a wave coming from thegrating lobe direction as it is from the steered direc-tion. In order to remove this source of error from thearray response, a restriction needs to be placed on thearray. Grating lobes can be avoided if the Nyquist cri-terion, dx0 < λ/2, is satisfied. The wavelength λ is thesmallest wavelength under consideration.

0.2

0.6

1

30

60

90

120

150

180 0

main lobe

grating lobe grating lobe

Figure 7: Example of Grating Lobes in Array Response(λ = .3, dx0 = .4, N = 5)

Damage Detection

A damage detection methodology using structuralwaves could look for reflected waves from damage muchthe same way as radar would work in an electromag-netic medium. In an infinite plate, a pulse introducedinto the structure would produce a reflection only ifa structural discontinuity existed, shown in Figure 8.The case is different for a finite plate where boundaryeffects take a significant role. Reflections off of bound-aries are present and need to be taken into account.

incidentpulse

damagelocation

reflectedwave

Figure 8: Incident Pulse and Corresponding Reflectionfrom Damage.

An array on the plate needs to be able to produce anincident wave in the plate and sense any reflectionsthat follow.

Self Sensing Actuators

The constitutive equations which describe the behaviorof piezoceramics are:

D = d T + εTE (Direct Effect)S = sET + d E (Converse Effect) (16)

where d, εT , and sE and properties of the piezoelectricmaterial. The direct effect is used in sensing applica-tions as the stress in the material, T , causes an elec-tric displacement, D, to be developed. Similarly, theconverse effect is used in actuator applications as anelectric field, E, causes a strain, S, in the material.

For a strain sensor in one direction, the secondequation in (16) is rearranged to be T = (1/sE)S −(d/sE)E. Substituting this expression into the DirectEffect equation and holding the electric field, E, con-stant becomes:

∆D =(

d

sE

)∆S (17)

A change in electric displacement on the element is afunction of change in strain. The electric displacement,D, is related to the charge by q =

∫ADdA and for small

elements, q = DA. The charge on the element, q, isthe integral of the current, expressed with Laplace’soperator as q = i/s. A current integrator is then usedin order to determine the charge on the element whichcorresponds to the strain of the material.

Finding the mechanical strain using a piezoce-ramic self sensing actuator involves separating thecharge on the piezo into a charge due to actuationand a charge based on mechanical deformation. Dosch,et. al.11 used a bridge circuit to cancel out the actua-tion voltage for purposes of control. The bridge circuit


actV

strainV

piezo

2V 1VpC

rC

bC bC

Figure 9: Bridge circuit for self sensing piezo.

shown in Figure 9 is used to send an actuation volt-age, Vact, to a piezo element on the plate. The piezoelement in the figure is comprised of a voltage source,Vstrain, and a capacitance, Cp. The voltage, Vstrain,is caused by mechanical deformation and is a measureof the strain at the element location on the plate.

Vstrain =qmech

Cp=

Ad

sECpS (18)

where Cp = ε33A/t is the capacitance of the piezoelement at constant strain. If the bridge is prop-erly balanced, Cp = Cr , then the voltage differenceVout = V1 − V2, is proportional to the voltage due tomechanical deformation.

V1 =Cp

Cb + Cp(Vact + Vstrain) (19)

V2 =Cr

Cb + CrVact (20)

In an unbalanced bridge, however, the input voltageVact enters into the expression for Vout and could po-tentially drown out Vstrain. This phenomenon is re-ferred to as electrical feedthrough of the excitationvoltage into the output voltage.

Piezoelectric patches on a plate serve as elementsof the phased array as well as a source of an inter-rogating signal. The center piezoelectric patch of thearray imparts a disturbance into the structure by us-ing the self-sensing actuator circuit. The center patchproduces a pulse onto the plate and causes structuralwaves to propagate radially outward. In the pres-ence of damage, reflected waves should return fromthe damage location and the relative location of thedamaged region is then determined by the phased ar-ray, which uses information from all of the piezoelectricelements. A diagram of the conceptual damage detec-tion strategy is shown in Figure 10.

array

θ

damage

incident

reflected

piezo element

actVstrainV

plate

Figure 10: Incident Pulse and Corresponding Reflec-tion from Damage.

Experimental Setup

A 5 mm thick, .6 m × .9 m plate was instrumentedwith 11 PZT-5H piezoelectric patches along one edgeto form the array. Each element was 3.2 mm × 3.2mm and the elements of the array were separated by adistance of 1.25 cm. The spacing of the array limitedthe smallest wavelength to be about 2.5 cm. In thefrequency domain, this translated to a cutoff frequencyof about 7600 Hz. Two opposite edges of the platewere clamped and the other two were free with thearray placed along one of the free edges.

Damage, in the form of a hole, crack, or delamina-tion, can be considered as a relative impedance changefrom the normal structure. In this sense the structuraldynamics change as a result of the change in impedanceof a region of the structure. A wave passing throughthe structure would see a relative impedance changeresulting in transmitted and reflected waves. Anothermethod of altering the local impedance of the structureis by the addition of a mass which would also cause areflection of incident energy. A 130 gram mass with a3.8 cm × 7.6 cm footprint was attached to the plateat various angles relative to the array at a radius of38 cm from the center of the array. A diagram of theexperimental setup is shown in Figure 11.

Normally, the boundary conditions would be asource of concern when dealing with structural dynam-ics since natural frequencies and mode shapes are de-pendent on the type and precision of the boundary. Apinned or clamped boundary condition which is notclose to perfect could alter the natural frequency andmode shapes dramatically. In this situation, however,the boundary conditions are of secondary importancebecause the reflected wave from a damage located onthe interior of the plate should arrive earlier than wavesarriving from the boundary. The reflections from theboundary can be windowed out of the gathered data


array

θ

mass

incident

reflected

plate

"15

"2/1

"10

'3

'2

Figure 11: Experimental Setup.

thereby eliminating them from corrupting the sensorresponse.

The signal into the middle element of the array,used to excite the structure, was a chirp with a band-width between 1000 Hz and 8000 Hz and a durationof .001 sec, shown in Figure 12. The chirp signal pro-vided the incident signal which is to be reflected backto the array from the damage location.

0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 12: Input signal.

Results

The input chirp as well as the resulting reflections fromboundaries and simulated damage was recorded foreach piezoelectric sensor. An undamaged case, whereno mass was placed, was tested first, and the signals

from the sensors was recorded. A source of error inthe tests are the reflections from the boundaries. Eachsensor signal gathered from the undamaged case weresubtracted from the corresponding sensor signal fromthe damaged case.

∆si = sdami − sundam

i for i = 1, · · · , N (21)

This was done in order to remove the reflections fromthe boundaries allowing the reflections from the dam-aged region to become more prominent in the signals.Furthermore, the signals were also windowed to removeany secondary reflections. The response of the arraywas then found using these differential signals, ∆si.

ψ(θ, ω) = F (θ,∆s1, · · · ,∆sN ) (22)

where F is the operator to construct the array responseand θ is the angle to steer toward.

The square of the magnitude of the array responsewas used as a measure of the energy reflected backtoward the array from the simulated damage.

ψ (θ) =∫ ω2

ω1

|ψ (θ, ω) |2dω (23)

The response of the array was found for steering anglesbetween 0◦ and 180◦. The results are shown in Figure13 where the simulated damage was placed at variouslocations along the 38 cm radius from the center of thearray.

Qualitatively, the results show that the array doessee a reflection coming back from the mass placed onthe plate. As expected, the main lobe of the arrayenergy, ψ(θ), points in the direction of the mass onthe plate. For most cases, the error associated withsteering was about 2◦ to 4◦ off of the true angle. In thethe 40◦ case, an additional lobe appears at 0◦. Thiscould be due to a reflection leaking into the sensorsignals. Additionally, the array does have considerabletrouble for the 160◦ case. In this case a lobe appearsat 90◦. As with the 40◦ case, this could be due to awave leaking into the sensors despite the windowing.The main lobe seems to point at 180◦. For moderatesteering angles between 30◦ and 150◦, the array candistinguish the relative direction of the mass howeverthe array has trouble outside of this region.

Conclusions

The results provide validity to using phased array the-ory in conjunction with wave models of plate struc-tures to determine the source direction of structuralwaves. Approximations were made to the solutions ofthe differential equation of a plate. The approxima-tions behaved much like acoustic or electromagnetic


waves in their phase variations with space. This sim-ilarity is crucial in order to apply the correct phasingnecessary for the use of phased arrays.

The experimental results suggest that the pres-ence of a damaged region could be inferred based onreflected waves from a damaged region. A damagedregion is seen as an impedance change in the structureand partially reflects incident energy. A mass on theplate, simulating damage, was seen to reflect incidentenergy back toward the phased array. The incidentenergy was produced by an element in the array toprovide an interrogating signal. The element was partof a self-sensing circuit which simultaneously excitedthe structure as well as provided an indication of thedeflection at that point.

References

1 C. Farhat and F. Hemez, “Updating finite elementdynamic models using an element-by-element sensi-tivity methodology,” AIAA Journal 31, pp. 1702–1711, 1993.

2 T. Kashangaki, S. W. Smith, and T. W.Lim, “Underlying modal data issues for detect-ing damage in truss structures,” in Proceedingsof the 33rd AIAA/ASME/AHS/ASC Structures,Structural Dynamics, and Materials Conference,pp. 1437–1446, 1992.

3 H. Luo and S. Hanagud, “An integral equation forchanges in the structural dynamics characteristicsof damaged structures,” International Journal ofSolids and Structures 34, pp. 4557–4579, 1997.

4 D. C. Zimmerman and M. Kaouk, “Structural dam-age detection using a subspace rotation algorithm,”in Proceedings of the 33rd AIAA/ASME/AHS/ASCStructures, Structural Dynamics, and MaterialsConference, pp. 2341–2350, 1992.

5 J. F. Doyle, “Determining the size and location oftransverse cracks in beams,” Journal of Experimen-tal Mechanics 35, pp. 272–280, 1995.

6 K. A. Lakshmanan and D. J. Pines, “Local dam-age detection in structures using wave models,”Proceedings from the AIAA/ASME/AHS AdaptiveStructures Forum 1996, pp. 381–386, 1996.

7 J. Ma and D. J. Pines, “Dereverberation and its ap-plications to damage detection in one-dimensionalstructures,” AIAA Journal 39, pp. 902–918, 2001.

8 J. F. Doyle and S. Kamle, “An experimental studyof the reflections and transmission of flexural waves

at discontinuities,” Journal of Applied Mechanics52, pp. 669–673, 1985.

9 A. S. Purekar and D. J. Pines, “Near field ef-fects of damage,” in Proceedings of SPIE, vol. 4327,pp. 732–742, 2001.

10 L. J. Ziomek, Fundamentals of Acoustic Field The-ory and Space-Time Signal Processing, CRC Press,1995.

11 J. J. Dosch, D. J. Inman, and E. Garcia, “A self-sensing piezoelectric actuator for collocated con-trol,” Journal of Intelligent Material Systems andStructures 3, pp. 166–185, 1992.


0.2

0.4

0.6

0.8

1

30

60

90

120

150

180 0

(a) Mass located at 40◦

0.2

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0.8

1

30

60

90

120

150

180 0

(b) Mass located at 70◦

0.2

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0.8

1

30

60

90

120

150

180 0

(c) Mass located at 100◦

0.2

0.4

0.6

0.8

1

30

60

90

120

150

180 0

(d) Mass located at 130◦

0.2

0.4

0.6

0.8

1

30

60

90

120

150

180 0

(e) Mass located at 160◦

Figure 13: Normalized ψ(θ) Using Experimental Data.


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