Modeling and Experimentation of Proof-Mass Piezoelectric Wafer Active Sensor Thickness ... · 2015. 6. 19. · Tuncay Kamas1 and Bin Lin2 University of South Carolina, Columbia, SC,

American Institute of Aeronautics and Astronautics

1

Modeling and Experimentation of Proof-Mass Piezoelectric

Wafer Active Sensor Thickness Mode E/M Impedance

Spectroscopy

Tuncay Kamas1 and Bin Lin

2

University of South Carolina, Columbia, SC, 29208

Victor Giurgiutiu3

University of South Carolina, Columbia, SC, 29208, USA

This paper presents theoretical and experimental work on thickness mode electromechanical

impedance spectroscopy (EMIS) of proof-mass piezoelectric wafer active sensor (PM-PWAS) with

proof-mass attached on the sensor. PWAS has recently been extensively employed in many

applications for aero-structural health monitoring and non-destructive evaluations (NDE). EMIS

method is utilized for high frequency local modal sensing to determine the dynamic characteristics

of PWAS bonded on an aero-structural component for in-situ ultrasonics. Analytical analysis

begins with the piezoelectric wafer active sensor (PWAS) under constrained boundary conditions

and it is carried out by using the resonator theory considering the simplified one-dimensional two

layered and three layered models. In two layered model, a PWAS resonator is assumed to be

perfectly bonded on an isotropic elastic plate-like substrate. Global matrix method (GMM) is

employed to carry all the informations from each layer regarding the material properties,

geometric properties as well as the boundary conditions into the eigenvalue problem. The

normalized thickness mode shapes of the PWAS-substrate structure at the resonance frequencies

are obtained for the normal mode expansion (NME) method. The study was followed with the three

layered model to build the basis for the PM-PWAS. PM-PWAS transducer affixed to an isotropic

elastic plate was analytically modeled. Global matrix method (GMM) is also employed to solve the

eigenvalue problem of the three-layered PM-PWAS for the Eigen-vectors and the corresponding

Eigen-frequencies. Eigen-modes are determined for NME method to predict the thickness mode

impedance values of PM-PWAS using the correlation between a proof-mass-piezoelectric

transducer and structural dynamic properties in the substrate structure. Proof masses shift system

resonance towards optimal frequency point. The bonded PWAS and PM-PWAS models are also

numerically generated in a commercial multi-physics finite element analysis (MP-FEA) software,

ANSYS®. The thickness mode EMIS results from analytical, numerical, and experimental analyses

are presented. The analytical constrained PWAS (two-layered) and constrained PM-PWAS (three-

layered) models are verified by MP-FEA computational results and experimental measurement

results. The paper ends with summary and conclusions.

Nomenclature

Ci = amplitude of oscillation

u = particle displacement

γ = wave number

ω = angular frequency

Ea = elastic modulus of material A

ca = phase velocity of material A

dt = time step

σa = normal stress on layer of material A

1 PhD Candidate, Mechanical Engineering, 300 Main Str. Columbia SC 29208.

2 Researcher Scientist, Mechanical Engineering, 300 Main Str. Columbia SC 29208.

3 Professor, Mechanical Engineering, 300 Main Str. Columbia SC 29208.


2

I. Introduction

iezoelectric wafer active sensor (PWAS)1 is light-weighted, inexpensive, unobtrusive, minimally intrusive

sensor requiring low-power. PWAS is made of piezoelectric ceramic with electric field polarization, 3E , across

the electrodes deposited on both surfaces. It has recently been extensively employed in many applications for

structural health monitoring (SHM) and non-destructive evaluations (NDE)2,3

through guided wave generation and

transduction.

Electro-mechanical impedance spectroscopy (EMIS) method has been utilized to determine the local dynamic

characteristics of PWAS bonded on a host structure for in-situ ultrasonics4

such as in the work presented by Sun and

Liang5,6

. They utilized the EMIS method for high frequency local modal sensing.

Many rigorous researches on the thickness (out-of-plane) mode theory have been conducted for piezoelectric

crystal and ceramic resonators. Tiersten7

presented a pioneering work to develop the analytical solution for the

thickness vibration of an anisotropic piezoelectric plate. He used the resonator theory with traction-free 0T

boundary conditions at surfaces of a plate.

The analytical in-plane impedance for piezoelectric ceramic transducers such as PWAS has been developed by

Zagrai and Giurgiutiu5.One and two dimensional in-plane E/M impedance models for free PWAS and constrained

PWAS were derived to model the dynamics of PWAS and substrate structure in terms of EMIS. They assumed the

constant electric field, 3E , to derive the in-plane EMIS. However, the analytical study for thickness mode of PWAS-

EMIS has not been fully performed yet. The present work aims to extend the EMIS model of a constrained PWAS at

high frequencies (up to 15MHz). Kamas et-al9 adopted the constant electric displacement assumption used in the

literature10,11

and solved the piezoelectric constitutive equations for the thickness mode. Thickness mode EMIS of

free PWAS was derived and validated using results from coupled field finite element analysis (CF-FEA) and

experimental analysis.

Since the 1980’s, the proof-mass (PM) concept has received considerable attention especially regarding vibration

suppression control 12,13

. PM actuators have been provided for structural vibration control problems with respect to a

broad range of applications. The PM actuator has been modeled as a mass bonded to a structure with dynamic

stiffness and internal damping. The effectiveness of the PM actuator depends strongly on how precisely it is tuned.

In the literature, the models have been developed mostly for flexible structures vibrating in a relatively low

frequency band at a magnitude of 100-1000 Hz 14,15

whereas PWAS needs to work in high frequency range of MHz.

The PM concept is adopted herein to develop a new type of PM transducer for better high frequency local modal

sensing at a desired excitation frequency. Proof masses are affixed on PWAS to tune system resonance towards

optimal frequency point. PM-PWAS concept was first introduced by Kamas et-al16

. In their paper, they developed

the theory of three-bar resonator then carried out a parametric study for the thickness variation effect of proof-mass

on the resonance frequencies.

In the present work, analytical models were implemented for the PWAS and PM-PWAS transducers attached to

an isotropic elastic plate. First, the thickness mode of EMIS of two layered constrained PWAS is analytically

modeled in one dimension. The resonator theory was utilized to obtain the resonance frequencies for the normal

mode expansion (NME) method through global matrix method (GMM). Second, the PM-PWAS transducer was

configured with a PWAS actuator and PM on top. The resonance frequencies are determined for the NME method in

the 1-D simplified analytical thickness mode EMIS model of the three layered PM-PWAS-host structure. The model

was used to build the basis for a PM-PWAS. Coupled-field finite element method (CF-FEM) was used to model and

simulate EMIS of constrained PWAS and constrained PM-PWAS on an isotropic elastic plate. In addition, a set of

experiments was conducted using bonded PWAS on an aluminum plate-like structure. The analytical and numerical

models are validated with the experimental

results. The comparison between theoretical

prediction, simulation, and experimental

data are illustrated and discussed.

II. Theoretical Development on

Thickness Mode E/M Impedance

Modeling of PWAS

A. Constrained PWAS-EMIS

This section introduces a theoretical

framework for a two-layer resonator model

P

PWAS

Plate-like

host structure

, , ,

, , ,

, ,

Figure 1 One dimensional model of a two layer resonator model

including a PWAS bonded on an isotropic elastic bar.


3

including a PWAS and one isotropic elastic bar. The following assumptions were made for the two-bar piezo-

resonator model. First, the geometry and the cross-section area of the two bars are the same although they may have

different materials and different thicknesses. Second, PWAS is assumed to be perfectly bonded to isotropic bar. The

model shown in Figure 1 can be used to develop an analytical solution to obtain the resonance frequencies

essentially to build the basis for the constrained PWAS actuator.

The wave equations for each layer of the two bar resonator can be obtained from Newton’s equation of motion as

follows;

Equation Chapter 1 Section 1 2 2

2

2 2

u uc

y t

(1)

The general wave equation solution for the first layer can be determined as

1 2

1 2ˆa ai y y i y y i t i t

a au C e C e e u e

(2)

where au is the displacement for the isotropic elastic material on the left hand side, Similarly, the displacement, pu

, for PWAS can be determined. The capital coefficients denote the axial wave amplitudes as forward and backward

directions respectively in x axis and / , /a a p pc c denote the wave numbers for each material of the

divisions and related to the wave speed in each material. Linear Hooke’s law applied to determine the stress-strain

constitutive equations using the strain-displacement relation, /u x , for each layer in 1-D two-bar resonator

problem as follows;

1 2

1 2a ai y y i y y i t

a a a a aE iE C e C e e

(3)

The stress equation p was determined in similar manner. Two unknown coefficients in each stress equation exist

and four unknown coefficients in total therefore four boundary conditions should be implied to the general wave

solutions to obtain the solutions of the wave equations in terms of the displacement mode shapes and the frequency

responses at structural resonances for different modes.

1. Eigenvalue problem

The stress and displacement boundary conditions to be imposed are; stress-free boundary condition at free ends,

stress continuity condition and displacement continuity condition at the interface. The linearly dependent equation

system was determined by implying the boundary conditions in a matrix form to provide a solution of the Eigen-

frequencies and to eventually obtain the four Eigen-vectors (the displacement amplitudes). Therefore, this problem

turned out to be an eigenvalue problem which requires a matrix as functions of frequencyr and the matrix to be

multiplied by a tensor that contains the displacement amplitudes and the product of the two matrices was equal to

zero as shown in Eq.(4).

1 2

2 31 2

2 31 2

2 3

( )

( )( ) 1

2

3( )( )

4

( )

1 0 0

1

0

1 1

0 0 1

a

pa

pa

p

i y yc

i y yi y ycp pc a a

a p a p

i y yi y ycc

i y yc

e

CE Ec ce e

CE c E c

C

e e C

e

(4)

The determinant of the global matrix 4 4xA must equal to zero to have non-trivial solution for the displacement

amplitudes. Then one can say that the global matrix is apparently a singular matrix and it does not have a unique

solution. Therefore, we can find the basis of the eigenvector by assuming one of the unknowns is one i.e. 4 1C .

This assumption helps us obtain 3 3xA non-singular matrix simply excluding 4

th row that no longer represents an

independent equation. The 4th

column of the global matrix is moved to the right hand side of the equation to

eventually obtain a non-trivial basis of the solution for the unknown constants.


4

2. Normalized mode shapes

The next step was the verification of orthogonality of the mode shapes The mass-weighted integral was used to

verify the orthogonality of the mode shapes and to find the modal participation factor, im of each mode to scale the

mode shape amplitudes. For the two bar resonator problem, it takes the following form. The surface area, A , is

omitted since it is equal for each layer in this problem;

32

1 2

0 , if

, if

yy

a a p p

a i j p i j

iy y

i jU U dy U U dy

m i j

(5)

Then, the normalization of the mode shapes was carried out to find normal modes after the modal participation

factors (modal mass) were obtained which are the values on the diagonal of the orthogonality matrix. The values can

be used to normalize the mode shapes by the following relation; /new

iU U m

3. Calculation of Frequency Response Function Through Normal Mode Expansion Method (NME)

The normalized mode shapes are substituted into the frequency response function (FRF) equation (6) that was

derived through the NME method

2 21

1( , ) ( )

a a p i t

n

n n

U y U y tu y t U y e

A

(6)

to obtain the FRF to the single input single output excitation applied by the PWAS. The axial FRF in thickness

mode is expressed as

2(3) (3)

2 2

1ˆ

2

u u

u u u

n a p n a

PWAS

n n n

U y t U yu

A i

(7)

The SISO FRF is the same as the dynamic structural compliance seen by the PWAS modal sensor placed on the

structure. The dynamic structural stiffness is the reciprocal of the compliance

12

(3) (3)

2 2

1( )

ˆ 2

u u

u u u

n a p n a

str

nPWAS n n

U y t U yk A

u i

(8)

is defined in Eq.(8) to calculate the frequency dependent stiffness ratio, ( ) ( ) /str PWASr k k where ( )strk is

function of frequency and 33 /D

PWASk Ac t , dependent on the area of the surface whose normal is on y axis and the

stiffness of PWAS in thickness direction and the thickness of the PWAS.

The impedance model shown in Eq. (9) for constrained PWAS in thickness mode is derived in Appendix A. by

using the resonator theory under constrained boundary conditions and constant electrical displacement, 3D ,

assumption. The coefficients defined in the procedure can be adjusted by the GMM method that conveys the new

boundary conditions into the FRF through NME method and then substituted into the thicknes mode EMIS equation

for constrained PWAS by the stiffness ratio.

2

33

0

1 11

cott t

VZ

I i C r

(9)

B. Constrained PM-PWAS EMIS

This section introduces a theoretical framework for a three-bar resonator model including a PWAS and two

isotropic elastic bars bonded on both surfaces of the PWAS. The following assumptions were made for the three-bar

piezo-resonator model. First, the geometry and the cross-section area of all three layers are the same although they

may have different materials and different thicknesses. Second, the two isotropic bars are assumed to be perfectly


5

bonded to the PWAS. The model shown

in Figure 2 can be used to develop an

analytical solution to obtain the

resonance frequencies essentially to

build the basis for the proof-mass PWAS

(PM-PWAS) E/M impedance

spectroscopy (EMIS).

A similar procedure can be pursued

for the constrained PM-PWAS EMIS.

Only difference is the global matrix as

seen in Eq.(10) defined for the three-bar

resonator model additionally including

proof-mass bonded on PWAS. The

elements of the matrix discretized in

each box correspond to one of the three

layers. The elements in the left box are from the substrate material, in the middle box are from the PWAS layer, and

in the right are from the proof-mass layer. The matrix elements are currently functions of the wave-number. They

can be converted to functions of frequency by the wave number frequency relation,

/ , / , / a a p p b bc c c . An eigenvalue problem with respect to the new three-bar resonator problem is

solved in the similar manner described in the previous Eigenvalue problem subsection in order to turn the singular

matrix into non-singular matrix. Therefore, the basis of the solution for the six eigenvectors can be found at each

corresponding Eigenmodes. Then, the modal participation factor is obtained through the verification of the

orthogonality of the mode shapes by the mass weighted integral method. The normalized mode shapes are

determined by the calculated participation factor for each modes.

1

2

3

4

5

6

1 0 0 0 0

0 0

1 1 0 00

0 0 1 1

0 0

0 0 0 0 1

a a

p pa a

p pa a

p p b b

p p b b

b b

i H

i Hi H

a a a a p p p p

i Hi H

i H i H

i H i H

p p p p b b b b

i H

e C

E e E E E e C

Ce e

Ce e

CE e E E E e

Ce

(10)

III. Methodology

A. CF-FEA Numerical PM-PWAS EMIS Simulation

ANSYS multi-physics software with the implicit solver is used to obtain EMIS computation in frequency

domain. To perform the coupled field analysis of PWAS transducers, Coupled field piezoelectric elements which

could represent both mechanical and electrical fields can be used so that the stress field and electric field are coupled

and change in one field will induce change in the other field of the piezoelectric materials. The coupled field finite

elements used in our analysis are the 2-D coupled field solid elements (PLANE 13) that have 4 nodes with up to 7

degrees of freedom (DOF) including electric voltage as another DOF when used for piezoelectric analysis; the

electric voltage can be added in addition to the displacement DOF. A PWAS of dimension 37 0.2 mm was

modeled. The APC-850 material properties were assigned to the PWAS as follows. pC , the stiffness matrix,

p , the dielectric matrix, and pe , the piezoelectric matrix are determined as in the previous numerical

simulation by Kamas9. The density of the PWAS material is assumed to be 37700 kg/m

Figure 2 One dimensional model of a three layer resonator model

including a PWAS constrained by two isotropic elastic bars.

, , ,

, , ,

, ,

, , ,

PWAS

Proof-mass

Plate-like

host

structure


6

The aluminum beam dimensions are 100mm 0.8 mm and the

PWAS 7mm 0.2 mm . For a plane strain analysis, only a

longitudinal section of the specimen and PWAS were analyzed;

hence 2D meshed CF_FEA model was generated which reduced

considerably the computational time. The 2D plane element

PLANE42 is used for the aluminum beam; this element has 4

nodes and 2 DOF at each node. The 2D plane element PLANE13

is used to model the PWAS using the coupled field formulation.

Then, the impedance spectrum up to 15 MHz was calculated.

B. Experimental PM-PWAS EMIS Measurement

The E/M impedance SHM method is direct and convenient to

implement, the only required equipment being an electrical

impedance analyzer, such as HP 4194A impedance analyzer. An

example of performing PWAS E/M impedance spectroscopy is

presented in Figure 3. A HP 4194A impedance analyzer was used

for the experimental analysis. The impedance analyzer reads the

E/M impedance of PWAS itself as well as the in-situ E/M

impedance of PWAS attached to a specimen. It is applied by

scanning a predetermined frequency range in high frequency band

(up to 15MHz) and recording the complex impedance spectrum. A

LabView data acquisition program was used to control the

impedance analyzer and sweep the frequency range in steps that

was predefined and to attain the data in a format that assists to data analysis. During the visualization of the

frequency sweep, the real part of the E/M impedance, Re Z , follows up and down variation as the structural

impedance goes through the peaks and valleys of the structural resonances and anti-resonances.

IV. Results and Discussion

We considered a PM-PWAS under constrained boundary conditions to conduct a one-dimensional theoretical

E/M impedance analysis. The results in terms of impedance are compared with CF-FEA as well as experimental

analysis. During the visualization of the frequency sweep, the peaks appearing in the frequency range up to 2500

kHz are associated with the anti-resonances of the in-plane modes. These peaks are progressively smaller, with the

fundamental resonance being the strongest. This is consistent with the fact that higher modes need more energy to

get excited. Under constant energy excitation, higher modes would have

lower amplitudes. At around 11 MHz, a new solitary strong peak

appears. This is associated with the fundamental resonance of the

thickness model. The first and second in-plane impedance spectra in the

experimental results were found to be significantly different from the

theoretical predictions. The significant differences are indicative of the

2-D stiffening effect, typical of in-plane vibrations of low-aspect ratio

piezo-plates. These 2-D stiffening effects could not be captured by the

1-D theory at higher modes. However, the 2-D stiffening effect

diminishes, and the agreement between theory and experiment improves

at the thickness mode spectra because the thickness mode is hardly

affected by the in-plane shape of the PWAS.

For constrained PWAS, a circular PWAS (Table 1) in diameter of

7mm and in thickness of 0.2mm is bonded in very center point of a

circular aluminum plate in thickness of 0.8mm to use the axial

symmetry so that avoid the instensity of in-plane mode in the first

thickness mode impedance region and the confusion between the in-

plane modes and the first thickness mode impedance

a

b

Figure 3 Experimental setup for measuring

the impedance characteristics of a) circular

PZT active sensor bonded on a circular

aluminum plate with b) HP 4194A

Impedance Phase-Gain Analyzer.

Property APC 850

7700

400

-175

26

-12.4

15.3

17.3

1750

0.63

0.72

0.36

Poisson’s ratio 0.35

3( / )kg m12

33 (10 / )d m V

12

31(10 / )d m V

3

33 (10 / )g Vm N

3

31(10 / )g Vm N

12 2

11(10 / )Es m N

12 2

33 (10 / )Es m N

33 0/T

p

33

31

Table 1 Properties of APC 850

piezoelectric ceramic

(www.americanpiezo.com)


7

A. Experimental and CF-FEA Validation of

constrained PWAS model

Spectra from anaytical, numerical simulation and

experiment of the constrained-PWAS are obtained for

out-of-plane EMIS in high frequency range. Constrained

PWAS-EMIS measurement results is seen in Figure 5

for 0.2mm thick PWAS on 0.8mm aluminum plate.

Globally good matching is observed as compared to

verify the analytical constrained PWAS-EMIS model.

Also 2-D CF-FEA model is used to validate the

corresponding analytical model however some

discrepancies between the analytical PWAS-EMIS and

CF-FEA PWAS-EMIS are visible for the thickness

mode peak as can be seen in Figure 4. Small differences

at high frequencies are expected between the analytical

and the numerical responses due to the simplifying

assumptions made in the one-dimensional analytical

analysis. The first impedance peak is predicted by using

the thickness mode analytical and numerical models and

found to be at 2.31 MHz that has good agreement with

the predictions. 2-D CF-FEA model results seem to

match better in comparison with the 1-D analytical

two-bar resonator model results. It is also noticable

that in higher thickness modes, the impedance results

agree reasonably well in comparing the analytical

thickness mode EMIS model of PWAS with the

corresponding CF-FEA and experimental results.

B. CF-FEA Validation of PM-PWAS model

EMIS results are obtained from CF-FEA for 2-D

PWAS perfectly bonded in the center of an aluminum

plate with 100mm length and 1.0mm thickness to

verify the corresponding analytical model in this

example. The same four configurations are employed

in the FEA model as in the analytical models. We

came up with the impedance peaks harmonically

2 3 4 5 6 7 8 9 10 11

20

25

30

35

40

45

50

55

60

X: 4.53

Y: 33.07

Experimental Impedance Circular PWAS on Circular Al Plate

Frequency (MHz)

Re(Z

)

X: 8.05

Y: 26.96

X: 10.67

Y: 60.88

X: 2.31

Y: 54.52

Figure 4 EMIS measurement result for a circular

PWAS in diameter of 7mm and in thickness of

0.2mm is bonded on a circular aluminum plate in

diameter of 100mm and in thickness of 0.8mm

a b 2 3 4 5 6 7 8 9 10 110

0.1

0.2

0.3

0.4

0.5

0.6

0.7 X: 10.86

Y: 0.7422

FEA Impedance of Bonded PWAS on Aluminum Plate

Frequency (kHz)

Re(Z

)

X: 8.296

Y: 0.3886

X: 5.181

Y: 0.2383X: 2.262

Y: 0.2002

2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

X: 9.752

Y: 2.886

1-D Analytical Bonded PWAS Impedance, ReZ

X: 2.331

Y: 1.702

X: 5.085

Y: 6.155

X: 7.404

Y: 6.847

Table 2 E/M impedance frequencies in the first four

thickness modes as the proof-masses incline from no

mass up to 0.15mm PM height

Figure 5 EMIS results from a) 2-D CF-FEA b) 1-D analytical models for PWAS perfectly bonded in the

center of an aluminum plate with 100mm length and 0.8mm thickness


8

increasing as overtones and the overtones appearing at nearly the same anti-resonance frequencies. The downward

shift phenomena were also clearly seen at the first frequency as well as the overtone anti-frequencies in the E/M

impedance plots (Fig. 7) as the proof-masses incline from no mass up to 0.15mm PM height. The numerical values

of the anti-resonance frequencies can also be seen in Table 2. Each overtone frequencies are analyzed one by one

and each obviously confirmed frequency shift as shown for the first two thickness impedance modes in Fig. 6.

The stiffer the material is the further down shift in resonance frequency occurs. Thus, the analytical first

E/M impedance peak obviously shifts down as much.

1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

16

18

Frequency, MHz

ReZ

, O

hm

s

Analytical PM-PWAS Impedance

0.15mmPM

0.10mmPM

0.05mmPM

No PM

a b 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

1.2

1.4

FEA PM-PWAS Impedance

Frequency (MHz)

Re(Z

) (O

hm

s)

0.15mm PM

0.10mm PM

0.05mm PM

No PM

Figure 6 Analytical and CF-FEA results regarding the frequency shifts of anti-resonance frequencies by

adding a proof-mass on PWAS and increasing the thickness of PM-PWAS at a) the first thickness mode and

b) the second thickness mode

a b

Figure 7 EMIS results from a) 2-D CF-FEA and b) 1-D analytical models for PM-PWAS perfectly bonded on

an aluminum plate-like substrate in 1.0mm thickness. Proof-mass thickness varies by 0.05mm to show the

phenomenalogical agreement in the frequency downshift at thickness mode E/M impedance


9

Appendix

A. ONE-DIMENSIONAL ANALYSIS OF A

THICKNESS MODE OF A CONSTRAINED PWAS

ON BOTH SURFACES

Consider a PWAS of length al , thickness

at , and

width ab , undergoing thickness expansion,

3u , induced by

the thickness polarization electric field, 3E . The electric

field is produced by the application of a harmonic voltage

ˆ i tV t Ve between the top and bottom surface

electrodes. The resulting electric field in the thickness

mode, /E V t , is assumed non-uniform with respect to

3 3/ 0x E x as opposed to the longitudinal mode; however, the electric displacement, D , is assumed uniform

with respect to 3 3/ 0x D x . The length, width, and thickness are assumed to have widely separated values

a a at b l such that the length, width, and thickness motions are practically uncoupled.

In this analytical model, PWAS is assumed to be constrained by structural stiffness on top and bottom surfaces

as seen in Figure . The analytical analysis starts with the general piezoelectric constitutive equations expressing the

linear relation between stress-strain and stress-electric displacement in thickness mode are

3 33 3 33 3

3 33 3 33 3

a-)

b-)

D

S

T c S h D

E h S D

(11)

The relations of the four piezoelectric constants to each other are in thickness mode17

. IEEE Standard on

Piezoelectricity11

provides other relations to alternate the forms of the constitutive equations. In our model, the

overall stiffness applied to the PWAS has been split into two equal components applied to the PWAS surfaces

1 1 1

; 2 2

total str

total str str

k kk k k

(12)

The boundary conditions applied at the PWAS ends balance the stress resultants, 3T bl

3 3 322 2

str

t tT x bl k u

(13)

Mechanical response

The resonance theory begins with the wave equation (1). Introducing the wave speed in direction of3x axis,

3 33 /Dc c , and the wave number in thickness mode, 3/t c ; the particle displacement

3u is given by

3 3 1 3 2 3

ˆ ( ) sin cost tu x C x C x (14)

1C and2C are to be determined from the boundary conditions. Note that

3 3 3 3/S u x u and substitute Eq(14) into

Eq. (13). Impose the boundary conditions on the PWAS surfaces that balance the stress resultant with the spring

reaction force 32 strk u . Introducing the quasi-static PWAS stiffness,

33 /D

PWASk Ac t and the stiffness ratio

/str PWASr k k . We can rearrange the equation using the ratio and it yields the following linear system in 1C and

2C

by substitution of the general solution. Rearrange using 0.5t t

Figure 8 PWAS constrained by structural stiffness,

kstr


10

33 3

1 2

33

cos sin sin cos2

t t t t t t D

h Dtr C r C

c (15)

Upon subtraction, we obtain 2 0C . Now add the two equations to obtain

33 3

1

33

1

2 cos sinD

t t t

h DtC

rc

(16)

Recall the strain 3 3

ˆS u

33 3

3 3 3

33

ˆ coscos sin

t tD

t t t

h DS u x

c r

(17)

Electrical response

/ 2

3 3

/2

3 3

a-)

b-)

t

t

A

V E dx

dI D dA i D bl

dt

(18)

Recall the second constitutive equation and substitute Eq. (17) into the equation

2

33 3

3 3 33 3

33

coscos sin

s

t tD

t t t

h DE x D

c r

(19)

Recalling the piezoelectric constant relations, one can derive these relations 2 2 2

33 33 33/ sh e ; 33 331/s s ; and

33 33 33/ De d s ; finally one can come up with the expression, 2

2

33 33 33 33/s Dh d s , and plug it into Eq(19) noting that

33 331/D Dc s and introduce the electro-mechanical coupling coefficient ,33 , defined as 2 2

33 33 33 33/ D Se c 11

2 3

3 33 3 33

cos1

cos sin

S t t

t t t

xE D

r

(20)

Upon substitution of Eq. (20) into Eq. (18)a, we obtain after rearrangement

2

33 3 33

sin1

cos sin

S t

t t t

V D tr

(21)

Recalling /I Q Q t and Eq. (18)b, we obtain 3I i D bl . Substitute Eq. (21) and electrical current equation

into the impedance /Z V I and recall the capacitance of the material 0 33/ SC bl t

2

33

0

1 11

cott t

VZ

I i C r

(22)


11

Acknowledgments

Support from National Science Foundation Grant # CMS-0925466; Air Force Office of Scientific Research

#FA9550-11-1-0133, Dr. David Stargel, Program Manager; are thankfully acknowledged.

References

1 Giurgiutiu, V., Bao, J., and Zhao, W., “Active Sensor Wave Propagation Health Monitoring of Beam and Plate

Structures,” Proc of SPIE’s 8th International Symposium on Smart Structures and Materials, Newport Beach, CA: 2001.

2 Giurgiutiu, V., and Zagrai, A., “Damage Detection in Simulated Aging-Aircraft Panels Using The Electro-Mechanical

Impedance Technique,” Adaptive Structures and Material Systems Symposium, ASME Winter Annual Meeting, Orlando, FL:

2000, pp. 1–10.

3 Giurgiutiu, V., Zagrai, a., and Jing Bao, J., “Piezoelectric Wafer Embedded Active Sensors for Aging Aircraft Structural

Health Monitoring,” Structural Health Monitoring, vol. 1, Jul. 2002, pp. 41–61.

4 Zagrai, A. N., and Giurgiutiu, V., “Electro-Mechanical Impedance Method for Damage Identification in Circular Plates,”

vol. 40, 2001.

5 Liang, C., Sun, F. P., and Rogers, C. a., “Coupled Electro-Mechanical Analysis of Adaptive Material Systems --

Determination of the Actuator Power Consumption and System Energy Transfer,” Journal of Intelligent Material Systems and

Structures, vol. 5, Jan. 1994, pp. 12–20.

6 Sun, F. P., Liang, C., and Rogers, C. A., “Structural modal analysis using collocated piezoelectric actuator/sensors: an

electromechanical approach,” Proc. SPIE 2190, Smart Structures and Materials 1994: Smart Structures and Intelligent Systems,

238, Orlando, FL: 1994.

7 Tiersten, H. F., “Thickness Vibrations of Piezoelectric Plates,” J. Acoustic Society of America, vol. 35, 1963, pp. 53–58.

8 Zagrai, A., and Giurgiutiu, V., “Electro-Mechanical Impedance Method for Crack Detection in Thin Plates,” Journal of

Intelligent Material Systems and Structures, vol. 12, 2001.

9 Kamas, T., Lin, B., and Giurgiutiu, V., “Analytical modeling of PWAS in-plane and out-of-plane electromechanical

impedance spectroscopy (EMIS),” SPIE Smart Structure and Materials + Nondestructive Evaluation and Health Monitoring

2013, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems, J.P. Lynch, C.-B. Yun, and K.-

W. Wang, eds., 2013, p. 869227.

10 Meeker, T. R., “Thickness mode piezoelectric transducers,” Ultrasonics, vol. 10, 1972, pp. 26–36.

11 IEEE Ultrasonics, “IEEE Standard on Piezoelectricity,” 1987.

12 Zimmerman, D. C., and Inman, D. J., “On the nature of the interaction between structures and proof-mass actuators,”

Journal of guidance, control, and dynamics, vol. 13, 1990, pp. 82–88.

13 Griffin, S., Lane, S. A., Hansen, C., and Cazzolato, B., “Active Structural Acoustic Control of a Rocket Fairing Using

Proof Mass Actuators,” Journal of Spacecraft and Rockets, vol. 38, 2013, pp. 219–225.

14 Huyanan, S., and Sims, N. D., “Vibration Control Strategies for Proof-mass Actuators,” Journal of Vibration and

Control, vol. 13, Dec. 2007, pp. 1785–1806.

15 Díaz, C. G., Paulitsch, C., and Gardonio, P., “Active damping control unit using a small scale proof mass electrodynamic

actuator,” Journal of the Acoustical Society of America, vol. 124, 2008, pp. 886–897.

16 Kamas, T., Giurgiutiu, V., and Lin, B., “Analytical Modeling of Proof-Mass Piezoelectric Wafer Active Sensor for

Symmetric Lamb Waves Tuning,” International workshop of Structural Health Monitoring, IWSHM, 2013.

17 Berlincourt, D. A., Curran, D. R., and Jaffe, H., “Piezoelectric and Piezomagnetic Materials and Their Function in

Transducers,” Physical Acoustics and The Properties of Solids, W.P. Mason, ed., Van Nostrand: Van Nostrand Company Inc.,

1958, pp. 169–188.


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Modeling and Experimentation of Proof-Mass Piezoelectric Wafer Active Sensor Thickness ... · 2015. 6. 19. · Tuncay Kamas1 and Bin Lin2 University of South Carolina, Columbia, SC,