BeatPhenomenonAnalysisofConcreteBeamwith …downloads.hindawi.com/journals/ijdsn/2012/296124.pdf · eﬀect have been applied as actuator and dynamic measure-ment in damage identiﬁcation

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2012, Article ID 296124, 10 pagesdoi:10.1155/2012/296124

Research Article

Beat Phenomenon Analysis of Concrete Beam withPiezoelectric Sensors

Xu Li, Linsheng Huo, and Hongnan Li

Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Linsheng Huo, [email protected]

Received 1 June 2012; Accepted 22 July 2012

Academic Editor: Ting-Hua YI

Copyright © 2012 Xu Li et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The focus of this paper is to give a better understanding of beat phenomenon in the free vibration test of a concrete beam withpiezoelectric ceramic sensors from the view of mathematics. The cause of beat phenomenon from piezoelectric ceramic sensorsembedded in the concrete beam is illustrated and the influence factors of beat phenomenon are discussed. The results show thatthe beat phenomenon from piezoelectric ceramic sensors in the concrete beam is caused by the coupled responses with similarmodel frequencies in different directions. The influence factors of beat phenomenon due to damping effect, impact direction,sensor position and sectional dimension are discussed. As the damping ratios increased, the amplitude of beat signal will die outin an exponential decay. Meanwhile, the damping has a tiny influence on the beat frequency of system response, the amplitudes ofbeat signal both in the time and frequency domain are changed with the variation of impact direction. In addition, the amplitudeof beat signal will be also changed with the position of sensors altered. The beat frequency will get more with the greater differenceof sectional dimension.

1. Introduction

Piezoelectric materials have a broadly applicative prospectin structure health monitoring due to their characters ofelectromechanical coupling, simple structure, low cost, goodreliability, and wide frequency response range. The piezoelec-tric transducers based on positive and negative piezoelectriceffect have been applied as actuator and dynamic measure-ment in damage identification [1–3], impact force location[4, 5], and fatigue crack detection [6, 7] in compositestructures. Besides, the piezoelectric transducers have beensuccessfully utilized in civil engineering structures. Songet al. [8] have developed an overheight collision detectionand evaluation system for concrete bridge girders usingpiezoelectric transducers. Li et al. [9] applied a new typeof cement-based piezoelectric sensor to monitor the trafficflows and concluded that there is a good potential for thecement-based piezoelectric sensor in the engineering appli-cation for monitoring traffic flows in the field of transporta-tion. Gu et al. [10] provided a method of piezoelectric-basedstrength monitoring, in which an innovative experimentalapproach is proposed to conduct the concrete strength

monitoring at early ages. Song et al. [11] exploited smartaggregate (SA), an innovative multifunctional transducer,fabricated by embedding a wired, waterproof piezoelectricceramic patch into a small concrete block. The proposedSA has been successfully utilized in the structural healthmonitoring of a full-size bridge girder [12], reinforcedconcrete shear walls [13], a two-story reinforced concreteframe subjected to progressive collapse [14], circular RCcolumns under cyclic combined loading [15], and circularreinforced concrete columns after seismic excitations [16].

The free vibration of undercritically damped systemwould be decayed exponentially, as shown in Figure 1 [17].However, when conducting the free vibration test with piezo-electric ceramic sensors, instead of an exponential decay,the vibration tends to decrease or increase periodically asshown in Figure 2, which is characterized as the classical beatphenomenon. It will be difficult to evaluate the frequencyor damage of structures from the beat signals. However,there is no definite conclusion about the cause and influencefactors of beat phenomenon in piezoelectric transducerscurrently. In this paper, the cause of beat phenomenon inpiezoelectric ceramic sensors of a simply supported beam is

2 International Journal of Distributed Sensor Networks

t

v(t)

exp(−ξωt)

Figure 1: Free-vibration response of undercritically damped sys-tem.

t

v(t)

Figure 2: The classical beat phenomenon from piezoelectricceramic sensors.

illustrated from the view of mathematics. In addition, theinfluence factors of beat phenomenon due to damping effect,impact direction, sensor position, and sectional dimensionare discussed.

2. The Analysis of Beat Phenomenon fromPiezoelectric Ceramic Sensor Embedded inthe Simply Supported Beam

A simply supported beam embedded with piezoelectricceramic sensors is taken as an example to analyze the causeof beat phenomenon. The hypothesis is put up that thecalculative model is the ideal simply supported beam, andthe material of structure is in elastic phase. The size of thebeam is B �W � L, and the piezoelectric ceramic sensor isembedded with the electrode parallel to the beam axis at thelength of z0 to the end of the beam. The detailed illustrationof calculative model is shown in Figure 3. When subject to animpact with initial velocity of v on the middle of the beamfrom θ-direction, the beam would oscillate as free vibration.

2.1. The Stress Condition on the Surface of Sensor Dipole. Theelectrode surface of piezoelectric ceramic sensor has onlyaxial stress when subjected to the curvature movement. Theinitial velocity can be resolved into orthometric components

vx and vy , and the axis stress σ3 is the composition of stresscomponent σ3x and σ3y caused by vx and vy , respectively.Consequently, the stress on the electrode surface is expressedby (1)

σ3 � σ3x � σ3y , (1)

where σ3x and σ3y can be analyzed, respectively. Whensubjected to impact force by vx, the dynamic equation inundercritically damped system is given by [17]

EbIy∂ux

�4��z, t�∂z4

�m∂2ux�z, t�

∂t2

� a1EbIy∂ux

�5��z, t�∂z4∂t

� a0m∂ux�z, t�

∂t� 0,

(2)

where ux�z, t� is the displacement response of the beam in x-direction, Eb is the modulus of elasticity, Iy is the moment ofinertia of the beam in y-direction, m is the mass of the beamin unit length, and a0 and a1 represent the Rayleigh dampingcoefficient related to the mass and stiffness, respectively.Solving (2) with the method of separating variables, thesolution is expressed by Zx�z�Yx�t�, which indicates thatthe free vibration motion is of a specific mode shape Zx�z�having a time-dependent amplitude Yx�t�. Then (2) istransformed by

Zx�4��z�� a4Zx�z� � 0, (3)

Yx�t� � 2ξxωxY�t� � ω2xYx�t� � 0, (4)

where a is a parameter related to the frequency of the beam,in which the natural frequency in x-direction is conveyed as

ωx � a2

�EbIy

m, (5)

where ξx is the damping ratio of the beam in x direction givenby

ξx �a0

2ωx�

a1ωx

2. (6)

The general solution of (3) is

Zx�z��A1 cos�az��A2 sin�az� �A3 cosh�az��A4 sinh�az�.(7)

The boundary conditions of simply supported beam areshown in (8), where M�z� is the bending moment at thelength of z:

Zx�0� � Z�

x�0� �M�0� �M�L� � 0. (8)

Substituting (8) into (7), then the value a is acquired bythe vibration Zx�z�, and natural frequency of each modein x direction can be calculated in each order. The firstmode provides the greatest contribution to the vibration soas the ux�z, t� in 1st mode can be considered as the totaldisplacement response approximately. The value of a in 1st

International Journal of Distributed Sensor Networks 3

Bx

y

z

W

L

z0

vx

vy

v

y0

PZT-1

PZT-1

x0

θ

Figure 3: The calculative model of the simply supported beam.

Poling direction

Electrode

1

2

3

4

5

6

Figure 4: The positive direction of computation model.

Piezoelectric ceramic sensor

KF

CF

RF

Cc

Vo

CP

i

q

Figure 5: The operating principle of charge amplifier.

mode is displayed in (9), and the natural frequency is givenby (10)

a �π

L, (9)

ωx � π2

�EbIy

mL4. (10)

The initial condition 1st mode is calculated as (11) basedon the mode shape orthogonality:

Yx�0� � � L0 mZx�z�v�z, 0�dz� L

0 mZ2x�z�dz cos θ. (11)

By solving (4) Yx�t� is obtained as (12), which is a harmonicsignal decayed exponentially:

Yx�t� � Yx�0�ωDx

sin�ωDxt� exp��ξxωxt�, (12)

where ωDx is the free vibration frequency of the damped

system which is equal to ωx

�1 � ξx

2. The stress on thesurface of sensor dipole caused by vx can be described as

s3x �x0EbIyZ

��

x �z0�Yx�t�Iy

� x0EbZ��

x �z0�Yx�t�,

(13)

where z0 depicts the position of the piezoceramic sensor. x0 isthe distance between x-axis and the centroid of piezoelectricceramic sensor axial section. σ3x is also a harmonic signalwith natural frequency of ωx. Likewise, the stress on thesurface of sensor dipole caused by vy is a harmonic signalwith natural frequency of ωy , which can be conveyed as

σ3y � y0EbZ��

y �z0�Yy�t�, (14)

where y0 is the distance between x-axis and the centroid ofpiezoelectric ceramic sensor axial section. The stress on theelectrode surface is acquired by submitting (13) and (14) into(1).

2.2. The Output Voltage of Piezoelectric Ceramic Sensor. Bymeans of the piezoelectric sensors and signal acquisitionsystem, the changes of stress are transformed into voltagesignal. Following assumptions are made to the computa-tional model of piezoelectric ceramic sensors to facilitatethe analysis according to the actual background and theneed of the analysis. First, the piezoelectric ceramic can beconsidered as the ideal elastic material without free charge. Inaddition, the electrode is isopotential, which means that theelectric field between both electrodes is uniform, and thereis no electric field in any other directions. The piezoelectricequation can be formulated as (15) [18]:

D � ddσ � eσE,

ε � sEσ � dcE,(15)

where D is the electric displacement vector, which means thecharge on unit area. ε denotes the strain vector. σ expresses


0

0

0.1

0.1

0.10.2

0.2

0.3

0.3 0 0.1 0.2 0.3 0 0.1 0.2 0.3

−0.5

−0.1−1

−0.1

0 0

0.5

1

Time (s) Time (s) Time (s)

=×

Acx

exp(ξ xωxt)

cos(ωBt)

sin

V(ωAt)

Figure 6: Illustration of the beat phenomenon with equal Ax and Ay .

the stress vector, and E is the electric intensity. sE is the elasticcompliance matrix, and eσ is the dielectric permittivitymatrix. The superscripts σ and E indicate that the quantityis measured at constant stress and constant electric field,respectively. dd and dc are piezoelectric coefficient matrix,where c and d have been added to differentiate the converseand direct piezoelectric effects. In practice, dd is equal todc. The positive direction of computation model is shownin Figure 4. Equations (16) and (17) are the matrix form ofpiezoelectric equation:

��D1

D2

D3

� �� d31 d32 d33

d15

d24

��

σ1

σ2

σ3

σ4

σ5

σ6

��

��e11

e22

e33

��E1

E2

E3

�,

(16)

��

ε1

ε2

ε3

ε4

ε5

ε6

��

��

s11 s12 s13

s21 s22 s23

s31 s32 s33

s44

s44

s66

�

��

σ1

σ2

σ3

σ4

σ5

σ6

��

��d31

d32

d33

d24

d15

��E1

E2

E3

�.(17)

When the poling of sensor is in 3-direction, the eternalelectric field is zero, and (16) is expressed by

D3 � d31σ1 � d32σ2 � d33σ3. (18)

And the charge Q is

Q �� D3dA3, (19)

where A3 is the area of electrode.Piezoelectric ceramic has very high impedance, in which

the output current is weak. The output signal needs to becollected through charge amplifier which can provide low

V

Y

⇀A

⇀A

y

ωy t

ωt

ωxt

⇀Ax

X

Figure 7: Illustration of V by rotation vector method.

impedance to signal. Figure 5 is the operating principle ofcharge amplifier, where CP is the capacity of piezoelectricceramic sensor, CC is the lead wire capacitance, CF is thefeedback capacitance, and KF is the gain of charge amplifier.The total charge �C� can be written as

C � CC � CP � �1 �KF�CF , (20)

CF is far larger than CC and CP, and the influence of accuracyby CC and CP is less than 0.1% [19]. Therefore, it can beconsidered that the output voltage V only depends on Q andCF , just as

V �

Q

CF. (21)

For typical piezoelectric sheet, the axial thickness is sothin that the shear stress in 1 and 2 directions can beneglected. Substituting (18) and (19) into (21) leads to

V �

d33σ3A3

CF. (22)

The sensitivity of piezoelectric ceramic sensor can bedefined as (23), which represents the relation between outputvoltage and stress on the surface of electrode, and V can bedepicted as (24):

Sq �d33A3

CF, (23)

V � Sqσ3. (24)


Table 1: The parameters of the simply supported beam.

B �W � L �m3� m �kg/m� Eb �1010N/m2� z0 �m� θ (�) v �m/s� Sq �V/MPa�

0.1 � 0.11 � 1 23.57 2.5 0.25 45 1 0.813

Substitute (1), (13), and (14) into (24), and the coupledresponse from piezoelectric ceramic sensors can be acquiredas (25). Note that for the simply supported beam subject toeccentric impact, the rotating component has no influenceon the output voltage of piezoelectric ceramic sensors,because the piezoelectric coefficient (d36) in the direction ofrotation about PZT axis is zero. Then, the output voltage Vcan be expressed by

V � Ax sin�ωDxt� � Ay sin�ωDyt�, (25)

where

Ax � Sqx0

ωDxEbZ

��

x �z0�Yx�0� exp��ξxωxt�� Acx exp��ξxωxt�,

Ay � Sqy0

ωDyEbZ

��

y �z0�Yy�0� exp��ξyωyt�� Acy exp��ξyωyt�,

(26)

where Acx and Acy are constants unrelated to time. Toillustrate the cause of beat phenomenon, it is assumed theamplitude of voltages from the piezoceramic sensor has thesame value, which means that Ax equals Ay . Then, (25) canbe derived as [20]

V � 2Acx exp��ξxωxt� cos ωDy � ωDx

2t� sin ωDx � ωDy

2t�

� 2Acx exp��ξxωxt� cos�ωBt� sin�ωAt�,(27)

where ωB � �ωDx � ωDy��2 and ωA � �ωDx � ωDy��2.Equation (27) shows the coupled response is an amplitude-modulated harmonic function with the frequency equal toωB and amplitude varying function with the frequency equalto ωA, as illustrated in Figure 6. ωB is defined as the beatfrequency and ωA is the average frequency.

To give a general understanding of beat phenomenon,one can consider the solution of system response by the way

of rotate vector method. As shown in Figure 7,��

A x can beseen as a vector rotating around X-axis with the length of

Ax, where ωDx is the angular velocity of��

A x. Similarly,��

A y

is a vector rotating around X-axis with the length of Ay

and the angular velocity of ωDy .��

A is the summation of��

A x

and��

A y , and the mathematical meaning of V in (25) is the

projective length of��

A in Y -axis and can be expressed by(28), in which ω�t is the phase given by (29), and A is the

length of��

A shown in (30). In this stage, the cause of beatphenomenon can be further examined in Figure 8.Ax and Ay

are illustrated as the amplitudes of system responses causedby vx and vy , respectively; V can be described as a simpleharmonic oscillation with the amplitude of A and the phaseof ω�t. A is the upper envelope of output signal, where theamplitude appears in a way of time-varying with the beatfrequency of ��ωDx �ωDy��2�. Meanwhile, the phase is alteredwith time and Ay�Ax. The beat phenomenon will becomeeasily observable if the beat frequency is in an appropriatescope:

V � A sin�ω�t�, (28)

ω�t �

��

arccos�� 1 � �Ay�Ax� cos��ωDy � ωDx�t��

1 � �Ay�Ax�2� 2�Ay�Ax� cos��ωDy � ωDx�t�

�� ωDxt

if �ωDy � ωDx�t is in 1st and 2nd quadrant

�arccos�� 1 � �Ay�Ax� cos��ωDy � ωDx�t��

1 � �Ay�Ax�2� 2�Ay�Ax� cos��ωDy � ωDx�t�

�� ωDxt

if �ωDy � ωDx�t is in 3 rd and 4th quadrant,

(29)

A �

�Ax

2�Ay

2� 2AxAy cos��ωDy � ωDx�t�. (30)

From previous discussion, the beat frequency ωB isthe major parameter related to beat phenomenon. The

zero value of ωB means that beat phenomenon could notbe observed. The beat phenomenon is characterized as


0 0.1 0.2 0.3

Time

0 0.1 0.2 0.3

Time

0 0.1 0.2 0.3

Time

0.2

0.15

0.1

0.05

0

A ×

0.2

0.1

−0.1

−0.2

0= V

sin

(ω t

)

−0.5

−1

0

0.5

1

Figure 8: Anatomy of the beat phenomenon with unequal Ax and Ay .

Th

e u

pper

env

elop

e of

ou

tpu

t si

gnal

(V

)

ξ = 0

ξ = 0.01

ξ = 0.03

ξ = 0.1

00

0.05 0.1 0.15

Time (s)

0.06

0.05

0.04

0.03

0.02

0.01

Figure 9: The upper envelope of output signal with equal ξx and ξy(ξx � ξy � ξ).

the amplitude modulated periodically. The exponentiallydecayed amplitude implies that there is be phenomenon inthe system response. Hence, the beat frequency ωB and upperenvelope A are researched further in the following.

3. The Influence Factor of Beat Phenomenon

In this section, the influences of beat phenomenon includingdamping effect, impact direction, the position of sensorand anisotropic properties are discussed. Unless mentionedotherwise, the damping ratio of calculative model is 0.01 inboth orthometric directions, and other parameters are listedin Table 1.

3.1. Damping Effect. Figure 9 shows the upper envelope ofoutput signal with equal ξx and ξy . As the damping ratiosincrease, the output signal dies out in an exponential decay.In practice, the damping ratios in different modes are diversemore or less. Figure 10 shows the upper envelope of outputsignal in which ξx is 0.01 and ξy is unequal to ξx. With theincreasing of ξy , the single response with modal frequency ofωy decays rapidly, and the coupled response of sensor tendsto the one with the modal frequency of ωx.

The damping not only leads to the attenuation ofamplitude but also reacts on the beat frequency. Figure 11depicts the change of beat frequency in which ξy/ξx is in the

Th

e u

pper

env

elop

e of

ou

tpu

t si

gnal

(V

)

00

0.05 0.1 0.15

Time (s)

0.05

0.04

0.03

0.02

0.01

ξy = 0.01

ξy = 0.03

ξy = 0.05

ξy = 0.1

Figure 10: The upper envelope of output signal with unequal ξxand ξy .

ξy/ξx

0.9 0.95 1 1.05 1.1

0.1

0.05

0

−0.05

−0.1

ωB/ω

B

Figure 11: The change of beat frequency with different ξy/ξx (ω�

B isthe beat frequency when ξx equals ξy).

range of 0.9 to 1.1, which shows that the difference of dampermakes tiny influence on beat frequency.

3.2. Impact Direction. With the variation of impact direction,the initial conditions in orthorhombic modes would bedifferent, and the amplitude of coupled response of sensorand the magnitudes in each modal frequency are alsochanged. Figure 12 is the time history and frequency domainchart of output signals with the impact direction of 0�, 45�,60�, and 90�. From the figures of time history, the beat


0

0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Time (s)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Time (s)

0

0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Time (s)

0

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Time (s)

0

0

0

Th

e ou

tpu

t si

gnal

(V

)T

he

outp

ut

sign

al (

V)

Th

e ou

tpu

t si

gnal

(V

)

Frequency (Hz)

Frequency (Hz)

Frequency (Hz)

Frequency (Hz)

Mag

nit

ude

(dB

)M

agn

itu

de (

dB)

Mag

nit

ude

(dB

)M

agn

itu

de (

dB)

ξx = ξy = 0ξx = ξy = 0.01

ξx = ξy = 0ξx = ξy = 0.01

10

20

20

20

30

40

40

40

50

60

60

60

70

800.05

0

−0.05

0.05

−0.05

0.04

0.02

−0.02

−0.04

−0.06

Th

e ou

tpu

t si

gnal

(V

)

0.04

0.02

−0.02

−0.04

−0.06

ωxωy

ωxωy

ωxωy

ωxωy

0102030405060

θ = 0◦

θ = 45◦

θ = 60◦

θ = 90◦

Figure 12: The time history and frequency domain chart of output signals.

phenomenon is observed obviously with the eccentric impactforce. If the impact direction is parallel to the symmetricaxis of the section, beat phenomenon would not happen.Also, from the figures of frequency domain, it is difficultto evaluate the structural frequency if beat phenomenonhappened.

Inversely, the maximum and minimum of the upperenvelope expressed as Amax and Amin can be observed frombeat signal. The relation between Amax /Amin and tan θ isshown as (31), where β0 is the Ay/Ax when θ equals 45�.Then the impact direction can be determined, which can be

a practical approach to determine impact direction, such asthe collision monitoring of the ocean platform or bridge in atraffic accident:

Amax

Amin� �1 � β0 tan θ

1 � β0 tan θ�. (31)

3.3. The Sensor Position. The stress on the surface of sensordipole rather than the system frequency will be changedwith the alteration of sensor position. Therefore, the positionof piezoelectric ceramic sensor has no influence on beat


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (s)

Th

e u

pper

env

elop

e of

ou

tpu

t si

gnal

(V

)

z0 = 0.5z0 = 0.4z0 = 0.3

Figure 13: The upper envelope of beat signal with different z0.

0 0.1 0.2 0.3 0.4 0.51

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Time (s)

Ay/A

x

z0 = 0.5z0 = 0.4z0 = 0.3

Figure 14: The Ay�Ax with different z0.

frequency but has influence on the amplitude of sensorresponse. Figure 13 represents the upper envelope of beatsignal with various z0. Figure 14 shows the ratio of singleamplitudes (Ay/Ax) in orthometric directions with variousz0. Note that the change of z0 has no influence on theproportion of response in each single mode. It can beconcluded that the beat phenomenon can be observed moreeasily if the piezoelectric ceramic sensors are placed near tothe middle of the beam.

Figure 15 is the upper envelope of beat signal withdifferent y0, and the Ay/Ax with different y0 is charted inFigure 16. For the system with similar orthometric modes,the dipole surface stress caused by each single mode isdifferent with the variation of position in the same section,so the amplitude in single mode will be also changed, whichcan cause the alteration of the amplitude of coupled responseand the proportion of response in each single mode.

3.4. Sectional Dimension. Sectional dimension can beregarded as a significant factor accounting for the beat

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

y0 = 0.02y0 = 0.025

y0 = 0.03

Th

e u

pper

env

elop

e of

ou

tpu

t si

gnal

(V

)

Figure 15: The upper envelope of beat signal with different y0 (x0 �

0.025 m).

0 0.1 0.2 0.3 0.4 0.50.8

1

1.2

1.4

1.6

1.8

2

2.2

Time (s)

Ay/A

x

y0 = 0.02y0 = 0.025y0 = 0.03

Figure 16: The Ay/Ax with different y0 (x0 � 0.025 m).

phenomenon, for the natural frequencies in orthogonaldirections of the beam are related to material properties.Figure 17 is the beat frequency in which the B�W is from0.9 to 1.1, and Figure 18 is the upper envelope of outputsignal with different B�W . It can be concluded that thebeat frequency will get more with the greater difference ofsectional dimension.

4. Conclusion

The beat phenomenon in the free vibration test of a simplysupported beam is analyzed from the view of mathematics.The results show that the beat phenomenon of piezoelectricceramic sensors is caused by coupled response with similarfrequency in different directions. As the damping ratiosincrease, the amplitude of beat phenomenon will die outin an exponential decay. However, the damping effect hasa tiny influence on beat frequency. With the variation ofimpact direction, the proportion of response in each single


0.9 0.95 1 1.05 1.10

10

20

30

40

50

Bea

t fr

equ

ency

(H

z)

B/W

Figure 17: The beat frequency with different B�W .

0 0.1 0.2 0.3 0.4 0.50

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

Th

e u

pper

env

elop

e of

ou

tpu

t si

gnal

(V

)

B/W = 1B/W = 1.05B/W = 1.1

Figure 18: The upper envelope of output signal with differentsectional dimension.

mode will be changed. Hence, the amplitude of coupledresponse of sensor and the magnitude in frequency domainare changed. The amplitude of system response can alsobe changed with the variation of sensor position in axialdirection, which has no influence on the proportion ofresponse in single mode. The location shift of sensors in thesame section will change not only the coupled response butalso the proportion of amplitude in both single modes. Thesectional dimension is a vital factor for the beat frequency,which will get more with the greater difference of sectionaldimension.

Acknowledgments

The authors are grateful for the support from Science Fundfor Creative Research Groups of the National Natural ScienceFoundation of China (Grant no. 51121005).

References

[1] H. S. Tzou and C. I. Tseng, “Distributed piezoelectricsensor/actuator design for dynamic measurement/control ofdistributed parameter systems: a piezoelectric finite elementapproach,” Journal of Sound and Vibration, vol. 138, no. 1, pp.17–34, 1990.

[2] R. Seydel and F. K. Chang, “Impact identification of stiffenedcomposite panels: I. System development,” Smart Materialsand Structures, vol. 10, no. 2, pp. 354–369, 2001.

[3] R. Seydel and F. K. Chang, “Impact identification of stiffenedcomposite panels: II. Implementation studies,” Smart Materi-als and Structures, vol. 10, no. 2, pp. 370–379, 2001.

[4] Zhou Wanlin, Wang Xinwei, and Hu zili, “On load identifica-tion for piezoelectric smart structures,” Acta Mechanica Sinica,vol. 36, no. 4, pp. 491–495, 2004.

[5] X. Zhao, H. Gao, G. Zhang et al., “Active health monitoring ofan aircraft wing with embedded piezoelectric sensor/actuatornetwork: I. Defect detection, localization and growth moni-toring,” Smart Materials and Structures, vol. 16, no. 4, articleno. 032, pp. 1208–1217, 2007.

[6] J.-B. Ihn and F. K. Chang, “Detection and monitoring ofhidden fatigue crack growth using a built-in piezoelectricsensor/actuator network: I. Diagnostics,” Smart Materials andStructures, vol. 13, no. 3, pp. 609–620, 2004.

[7] J. B. Ihn and F. K. Chang, “Detection and monitoring ofhidden fatigue crack growth using a built-in piezoelectricsensor/actuator network: II. Validation using riveted jointsand repair patches,” Smart Materials and Structures, vol. 13,no. 3, pp. 621–630, 2004.

[8] G. Song, C. Olmi, and H. Gu, “An overheight vehicle-bridgecollision monitoring system using piezoelectric transducers,”Smart Materials and Structures, vol. 16, no. 2, article no. 026,pp. 462–468, 2007.

[9] Z. X. Li, X. M. Yang, and Z. Li, “Application of cement-basedPiezoelectric sensors for monitoring traffic flows,” Journal ofTransportation Engineering, vol. 132, no. 7, pp. 565–573, 2006.

[10] H. Gu, G. Song, H. Dhonde, Y. L. Mo, and S. Yan, “Concreteearly-age strength monitoring using embedded piezoelectrictransducers,” Smart Materials and Structures, vol. 15, no. 6,article no. 038, pp. 1837–1845, 2006.

[11] G. Song, H. Gu, and Y. L. Mo, “Smart aggregates: multi-functional sensors for concrete structures—a tutorial and areview,” Smart Materials and Structures, vol. 17, no. 3, ArticleID 033001, pp. 1–17, 2008.

[12] G. Song, H. Gu, Y. L. Mo, T. T. C. Hsu, and H. Dhonde,“Concrete structural health monitoring using embeddedpiezoceramic transducers,” Smart Materials and Structures,vol. 16, no. 4, article no. 003, pp. 959959–968968, 2007.

[13] S. Yan, W. Sun, G. Song et al., “Health monitoring ofreinforced concrete shear walls using smart aggregates,” SmartMaterials and Structures, vol. 18, no. 4, Article ID 047001, pp.1–7, 2009.

[14] A. Laskar, H. Gu, Y. L. Mo, and G. Song, “Progressive collapseof a two-story reinforced concrete frame with embedded smartaggregates,” Smart Materials and Structures, vol. 18, no. 7,Article ID 075001, 2009.

[15] Y. Moslehy, H. Gu, A. Belarbi, Y. L. Mo, and G. Song,“Smart aggregate based damage detection of circular RCcolumns under cyclic combined loading,” Smart Materials andStructures, vol. 19, no. 6, Article ID 065021, 2010.

[16] H. Gu, Y. Moslehy, D. Sanders, G. Song, and Y. L. Mo,“Multi-functional smart aggregate-based structural healthmonitoring of circular reinforced concrete columns subjected


to seismic excitations,” Smart Materials and Structures, vol. 19,no. 6, Article ID 065026, 2010.

[17] R. Clough and J. Penzien, Dynamics of Structures, ASMETransactions, USA, 1975.

[18] J. Sirohi and I. Chopra, “Fundamental understanding ofpiezoelectric strain sensors,” Journal of Intelligent MaterialSystems and Structures, vol. 11, no. 4, pp. 246–257, 2000.

[19] W. Changa-Ming, K. De-Ren, and H. Yun-Feng, Sensing andTesting Technology, Beihang University Press, Beijing, China,2005.

[20] L.-S. Huo and H.-N. Li, “The beat phenomenon in struc-tural vibration control using circular tuned liquid columndampers,” Chinese Journal of Computational Mechanics, vol.27, no. 3, pp. 322–326, 2010.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Documents

BeatPhenomenonAnalysisofConcreteBeamwith …downloads.hindawi.com/journals/ijdsn/2012/296124.pdf · eﬀect have been applied as actuator and dynamic measure-ment in damage identiﬁcation