1

The Use of Exact Lamb Waves Modes for Modeling the Power and Energy Transduction of

Structurally-Bonded Piezoelectric Wafer Active Sensors

Bin Lin, Victor Giurgiutiu, Ayman M. Kamal

Department of Mechanical Engineering, University of South Carolina, Columbia, SC, 29208

linbin@cec.sc.edu, victorg@sc.edu, kamal@email.sc.edu

ABSTRACT

This paper presents a theoretical modeling of power and energy transduction of structurally-bonded piezoelectric wafer

active sensors (PWAS) for structural health monitoring (SHM). After a literature review of the state of the art, we

developed a model of power and energy transduction between the PWAS and a structure containing multimodal

ultrasonic guided waves. The use of exact Lamb waves modes for power modeling is an extension of our previously

presented simplified model that considered axial and flexural waves with low frequency approximation. The model

assumptions include: (a) straight-crested multimodal ultrasonic guided wave propagation; (b) ideal bonding (pin-force)

connection between PWAS and structure; (c) ideal excitation source at the transmitter PWAS and fully-resistive

external load at the receiver PWAS. Frequency response functions are developed for voltage, current, complex power,

active power, etc. Multimodal ultrasonic guided wave, normal mode expansion, electromechanical energy

transformation of PWAS and structure were considered. The parametric study of PWAS size and impedance match

gives the PWAS design guideline for PWAS sensing and power harvesting applications

Keywords: multimodal ultrasonic guided waves, normal mode expansion, power, energy, piezoelectric wafer active

sensors

1. INTRODUCTION

1.1 Background

The need for identifying structural damage and monitoring its evolution has stimulated the development of SHM

systems and methodologies in recent years. SHM is a multidisciplinary process that involves several disciplines that

must be closely coordinated. Accurate characterization of structural integrity requires in-situ sensor arrays to reliably

interrogate large areas and detect structural anomalies. The type and efficiency of the SHM sensors play a crucial role in

the SHM system success. Ideally, SHM sensors should be able to actively interrogate the structure and find out its state

of health, its remaining life, and the effective margin of safety. Essential in this determination is to find out the presence

and extend of structural damage. The past two decades have witnessed an extensive development of SHM sensor

technology 1-3

. A wide range of sensors have been developed particularly for generating and receiving acousto-

ultrasonic waves. Piezoelectric wafer active sensors (PWAS) have emerged as one of the major SHM technologies

because, with the same installation of PWAS transducers, one can apply a variety of damage detection methods 1:

propagating ultrasonic guided waves (acousto-ultrasonics), and standing waves (E/M impedance), as well as phased

arrays (Figure 1). PWAS are small, lightweight, unobtrusive, and inexpensive and achieve direct transduction between

electric and elastic wave energies. It is also possible to develop wireless SHM systems which could transmit the data

from the SHM sensors to an observation location outside the danger perimeter. However, the power and energy flow in

active and passive sensing of a SHM system is an import factor and has not systematically addressed.

SPIE Smart Structure and Materials + Nondestructive Evaluation and Health Monitoring 2012,

Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems,

San Diego, CA, 11 - 15 March, 2012, paper # SS12-8345-8

2

Figure 1 The various ways in which PWAS are used for structural sensing includes propagating Lamb waves, standing

Lamb waves and phased arrays. The propagating waves method include: pitch-catch; pulse-echo; thickness mode; and passive detection of impacts and acoustic emission [1].

1.2 State of the Art

The modeling of power and energy transduction has been addressed to a certain extent in classical NDE. Viktorov 4

mentioned the transmissibility function between an NDE transducer and the guided waves in the structure, but did not

give an analytical expression but rather relies on experimentally determination. Auld 5 treated comprehensibly the

power and energy of ultrasonic acoustic fields and developed the complex reciprocity approach to their calculation. He

developed a predictive model for surface acoustic wave (SAW) devices that relied on the simplifying assumption of

single mode nondispersive Rayleigh wave propagation. Rose 9 developed a model for coupling between an angle-beam

ultrasonic transducer and a system of guided Lamb waves in the structure using the normal modes expansion approach.

However, this model was not specific about how the shear transfer takes place through the gel coupling between the

transducer and the structure since the coupling-gel interface did not have clearly predictable behavior.

The modeling of structurally integrated active transducers for the design of autonomous SHM systems employing

has been addressed recently. Giurgiutiu et al.1 established a wave tuning model of axial, flexural waves excited by

PWAS. A systematic investigation of power and energy transduction between PWAS and structure during the structural

health monitoring process was recently presented by Lin and Giurgiutiu10-13

. The study used an analytical model to

capture the power and energy flow from the electrical source energizing the transmitter PWAS through various stages of

transduction up to the signal captured by an instrument connected to the receiver PWAS. However, the analytical model

is based on the axial and flexural wave with low frequency approximation.

The Lamb waves modes generated by structurally integrated active transducers has been established. Giurgiutiu et

al. 1 developed the exact Lamb waves modes excited by PWAS. Raghavan and Cesnik extended the analysis to the case

of a circular PWAS14

. Yeum and Sohn further decomposed the Lamb wave mode using concentric ring and circular

piezoelectric transducer15

. Giurgiutiu and Santoni-Bottai 8,17

developed a shear lag solution for the stress and strain

transfer between a structurally attached PWAS and the support structure. The increasing application of ultrasonic

guided wave requires a better understanding and mastering of the power and energy flow for the SHM system.

3

1.3 Motivation

The purpose of the paper is to extend the previous work on power and energy analysis10-13

to the case of multi-mode

Lamb waves used for PWAS-enabled active and passive SHM. The Lamb waves have complicated multi-mode strain

distributions across the thickness. The following issues were addressed: (a) predictive modeling of Lamb wave

propagation; (b) the power and energy transduction under a PWAS excitation; (c) identification of maximum energy

flow.

2. PWAS LAMB WAVE NORMAL MODE EXPANSION This section deals with PWAS Lamb-wave normal mode expansion (NME). For compactness, the following notations

are introduced:

ˆ ( )i tU U U U U Ue U conj Ut x

(1)

2.1 Lamb wave equation

Lamb waves propagate in solid plates. They are elastic waves

whose particle motion lies in the plane that contains the

direction of wave propagation (x axis) and the plate normal

direction (y axis) (Figure 2). The Lamb wave equation for the

straight-crested of motion is

2 22 2 2

2 2 2

2 2 22 2

2 2 2

2

2

y yx x x

y y yx x

u uu u u

x y x yx y t

u u uu u

x y x y x y t

(2)

The solution to Lamb waves equations of motion is found by assuming that the displacement is time-harmonic and by

expressing the displacement in the x and y directions through two scalar potentials. The derivation of the particle

displacement can be found in many Lamb-wave propagation references5-8

. Here we followed the equation and symbol

used in reference 8. The particle displacement solution for symmetric and anti-symmetric waves propagation is

( , , ) cos cos

( , , ) sin sin

Sn

Sn

i x t

x n Sn Sn Sn Sn Sn

n

i x t

y n Sn Sn Sn Sn Sn

n

u x y t B y R y e

u x y t i B y R y e

(Symmetric) (3)

( , , ) sin sin

( , , ) cos cos

An

An

i x t

x n An An An An An

n

i x t

y n An An An An An

n

u x y t A y R y e

u x y t i A y R y e

(Anti-symmetric) (4)

where subscript n denotes the values for each mode; nB , nA are constants be determined later considering individual

mode power normalization; Sn , An are wave numbers evaluated using the relation / c where c is wave speed;

and are functions given by2

2 2

2

pc

and

22 2

2

sc

; ,p sc c are the pressure and shear wave speeds defined

as 2

  pc

and sc

; SnR , AnR are the symmetric and anti-symmetric eigen-coefficients calculated from the

solution of the Rayleigh-Lamb equation for symmetric and antisymmetric modes respectively

2

2 2

2

tan

tan 4

d

d

and

2

22 2

tan 4

tan

d

d

. The stress-free top and bottom surfaces boundary conditions yields

y d

y d Figure 2 Plane wave notations

4

2 2 cos

2 cos

S S S

S

S S S

dR

d

2 2 sin

2 sin

A A A

A

A A A

dR

d

(5)

2.2 Lamb wave normal mode expansion

The NME technique is a physical method to solve a structure with forced loading. It is analogous to the eigenfunction

expansion methods in most mathematics textbooks. The normal modes of the structure serve as the eigenfunctions. The

wave field is expressed in terms of the superposition of guided wave modes. The NME depends on two main

considerations: (a) completeness and (b) orthogonality of the eigenfunctions. The Lamb wave modes of a plate are

assumed to be complete in the space of solutions of the governing equations. To prove the guided wave modes are a set

of orthogonal functions, the reciprocity relations need to be derived first.

2.2.1 Reciprocity relation

In acoustics, the reciprocity principle states that a response remains the same when

the source and receiver are interchanged. However, this is just a special case of a

more general reciprocity theorem. In its most general form, acoustic reciprocity

principle establishes a relation between two acoustic states that could occur in one

and the same spatial domain. The complex reciprocity relation is suitable for

waveguide and resonator mode analysis and for velocity and frequency perturbation.

Consider a generic body , and two sources 1F and

2F applied at points 1P and

2P ;

the two force sources produce two fields with velocity and stress 1 1,v T and

2 2,v T .

Using equation of motion and applying the two different sources (1) and (2) and

adding the two fields equations together, we can prove the complex reciprocity form

that relates the velocity responses, tractions and applied sources for harmonic excitation, i.e.,

2 1 1 2 2 1 1 2 v T v T v F v F

(6)

The complex reciprocity relation for time varying harmonic functions takes a simple form, i.e.,

2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 1 2 2 1 1 2

x xx y xy x xx y xy x xy y yy x xy y yy x x x x y y y yv T v T v T v T v T v T v T v T v F v F v F v Fx y

(7)

2.2.2 Orthogonality relation

The concept of orthogonality of a set of functions is an extension of the familiar notion of the orthogonality of vectors in

space, i.e., vectors whose dot products with each other vanish. The orthogonality of wave modes is a straight-forward

method of determining the expansion coefficients. The Lamb wave orthogonality can be proved using the complex

reciprocity relation of Equation (7) with the absence of the body forces (1 2 0F F ). We considered two generic

time-harmonic and space-harmonic separate Lamb modes ( and m n ). Considering the wave number, velocity and stress

are complex ( ,m n , ,n mv v and ,n mT T ), the orthogonality relation can be written as

4 0n m mni P (8)

where

0 if

1Re ( ) ( ) ( ) ( ) if

2

dmn n n n n

x xx y xy

d

m n

Pv y T y v y T y dy m n

(9)

2.2.3 Mode normalization

The Lamb wave stress is given by the equation of elasticity, i.e.,

2 2 2xx xx yy yy xx yy xy xyT S S T S S T S (10)

For both symmetric and anti-symmetric mode, the strain and velocity can be derived from the particle displacement in

Figure 3 Reciprocity relation

F1 F2 v12

v21

1

2

5

equation (3) and (4). After substituting the strain and velocity into equation (9) and performing the integration, we get

2

(symmetric)2

nnn S

BP V

2

(anti-symmetric)2

nnn A

AP V

(11)

where

2 2 2 2 2 2 2 2 2

2 2

sin cos sin cos1 3 4

4 sin cos 2 3 cos sin

S S S S

S S S S S S S S S S S S

S SS

S S S S S S S S S S

d d d dd R R

V

R d d R d d

(12)

2 2 2 2 2 2 2 2 2

2 2

sin cos sin cos1 3 4

4 cos sin 2 3 sin cos

A A A A

A A A A A A A A A A A A

A AA

A A A A A A A A A A

d d d dd R R

V

R d d R d d

(13)

The symmetric mode coefficientnB and the anti-symmetric mode coefficient

nA can be resolved as

2

(symmetric)nn

n

Sn

PB

V

2 (anti-symmetric)nn

n

An

PA

V (14)

For normal modes, we may assume 1nnP ; hence,

2

(symmetric)n

Sn

BV

2

(anti-symmetric)n

An

AV

(15)

2.3 PWAS normal mode expansion

The actuation and sensing between the PWAS and the structure is achieved

through the adhesive layer, which acts predominantly in shear. In a detailed

analysis, the effect of the adhesive layer has to be properly considered1; for a

simplified analysis, the ideal bonding assumption (pin-force model) may be used.

Here, we start with the former and then simplify to the latter.

2.3.1 Mode participation factor under shear excitation

We considered traction forces and velocity given in Equation (6). Coefficients

( )na x are the x-varying modal participation factors that depend on the mode under

consideration and the excitation used to generate the field. The modal participation

factors are the same for the all the acoustic fields (velocity, traction force, etc.).

The mode participation factor can be solved from a first order ODE in the form of

1 1ˆ4 ( )

d

nn n n n nd

P i a xx

v T v T y (16)

Recall that the orthogonality relation is obtaining by requiring that the normal modes of the plate satisfy the traction free

condition, i.e., 0n

yz dT

. For a shear excitation ( )yz zd

T t x at the upper surface, we get the shear component to be

zero on the lower surface. The mode participation factor of the forward wave solution outside the excitation region can

be derived as

( ) ( )4

n n

n a

x i x i x

n x

nn a

v da x e t x dx e

P

(17)

The total particle velocity using NME can be written as

( , ) ( ) ( )n n

n

x y a x yv v (18)

where ( )n yv is the velocity modeshape of the thn mode, i.e., ( )

( )( )

n

x

n n

y

v yy

v y

v . ( )n yv can be derived using the

combination of the symmetric displacement in equation (3) with symmetric normalization coefficient and anti-

symmetric particle displacement equation (4) and anti-symmetric normalization coefficient equation.

Figure 4 PWAS pin-force relation

- -

-

-

-

-

6

2.3.2 Normal mode expansion for ideal bonding (pin-force model)

Considering the ideal bonding assumption, the load transfer takes place over an infinitesimal region at the PWAS end.

Assuming a PWAS with center at 0 0x and length 2al a , the shear stress can be written as

0 0ˆ(x)= [ ( ) ( )] [ ( ) ( )]xt a x a x a F x a x a (19)

Here 0F̂ is the pin-end force per unit width.

Substitution of equation (19) into equation (17), we got the mode participation factor under PWAS excitation is

0 0ˆ ˆ( ) [ ]

4n n n n

n

x i a i a i x i xPWAS

n n

nn

v da x F e e e g F e

P

(20)

where ng is the coefficient

[ ] sin

4 2n n

n n

x xi a i a

n n

nn nn

v d v dg e e a

P P

The Lamb wave NME of the particle displacement under PWAS excitation is

0

1 ˆ( , ) ( )ni x

n n

n

u x y g F e yi

v (21)

The displacement in x-direction at the PWAS end ( , x a y d ) is

2

0

1 ˆ( , ) sin2

n

n

x i a

x n

n nn

v du a d a F e

i P

(22)

2.3.3 PWAS excitation voltage and pin-force relation

Consider a PWAS of length 2l a , width b and thickness PWASh ; the relation between PWAS pin-force and

transmitter input voltage are now analyzed. When the PWAS transmitter is excited by an oscillatory voltage, its volume

expands in phase with the voltage in accordance with the piezoelectric effect. Expansion of the PWAS mounted on the

surface of the plate induces a surface reaction form the plate in the form of the force distributed around the PWAS end.

The PWAS end displacement is constrained by the plate and is equal to the plate displacement at x a . The reaction

force along the PWAS edge, .0F b , depends on the PWAS displacement, PWASu t ,and on the frequency-dependent

dynamic stiffness, strk ,presented by the structure to the PWAS.

0 ,str xF b k u a d (23)

Under harmonic excitation, the dynamic stiffness, strk is obtained by dividing the force by the displacement given

by Equation (21), i.e.,

1

0( )

ˆ ( , )ni a

str n n

nx

F bk i b g e d

u a d

v (24)

The static stiffness PWASk of a free PWAS is

11

PWAS

PWAS E

h bk

s l (25)

The dynamic stiffness ratio is defined as the ratio between strk and PWASk , i.e.,

str

PWAS

kr

k

(26)

The relation between pin-force force per unit width and the static stiffness of the PWAS is

0 ,

1PWAS ISA PWAS ISA

rF b k u a d u k u

r

(27)

where ISAu is the induced strain, i.e., 31

ˆ

ISA

PWAS

Vu ld

h

7

3. PWAS POWER ANALYSIS

The power and energy transduction flow chart for a PWAS transmitter on a structure is shown in Figure 5. The

electrical energy of the input voltage applied at the PWAS terminals is converted through piezoelectric transduction into

mechanical energy that activates the expansion-contraction motion of the PWAS transducer. This motion is transmitted

to the underlying structure through the shear stress in the adhesive layer at the PWAS-structure interface. As a result,

ultrasonic guided waves are excited into the underlying structure. The mechanical power at the interface becomes the

acoustic wave power and the generated Lamb waves propagate in the structure.

Figure 5 PWAS transmitter power flow

Questions that need to be answered through predictive modeling are:

(1) How much of the applied electrical energy is converted in the wave energy?

(2) How much energy is lost through the shear transfer at the PWAS-structure interface?

(3) How much of the applied electrical energy gets rejected back into the electrical source?

(4) What are the optimal combinations of PWAS geometry, excitation frequency, and wave mode for transmitting the

maximum energy as ultrasonic waves into the structure?

To answer the above questions, we developed the electrical power, mechanical power and Lamb wave power in the

structure.

3.1 Electrical power

To calculate the transmitter electrical power and energy, we need to calculate the input electrical power by using input

admittance of the PWAS when attached to the structure. Because of the electromechanical coupling, the impedance is

strongly influenced by the dynamic behavior of the substructure and is substantially different from the free-PWAS

impedance.

Under harmonic excitation, the time-averaged power is the average amount of energy converted per unit time with

continuous harmonic excitation. The time-averaged product of the two harmonic variables is one half the product of one

variable times the conjugate of the other. When a harmonic voltage is applied to the transmitter PWAS, the current is

ˆI YV (28)

The constrained PWAS admittance can be expressed1 using the frequency dependent stiffness ratio of Equation (26),

i.e.,

2

0 31

11 1

( )cot ( )Y i C k

r

(29)

where ( ) a . A simplified form of Equation (29) can be obtained under the quasi-static assumption in which the

PWAS dynamics are assumed to happen at much higher frequencies than the Lamb-wave propagation

( ( ) 0 , ( )cot ( ) 1 ), i.e.,

2

0 3111

rY i C k

r

(30)

Piezoelectric

transduction:

Elec.→ Mech.

PWAS-structure

interaction

Shear-stress

excitation

of structure

Transmitter

Input AV

Transmitter PWAS

(Wave Exciter)

V

B2

Lamb waves

0x a 1x a

Ultrasonic guided waves

from transmitter PWAS

8

The power rating, time-averaged active power and reactive power are

2 2 21 ˆ

2rating active reactiveP Y V P P

21 ˆ2

active RP Y V 21 ˆ

2reactive IP Y V (31)

where RY is the real part of admittance and

IY is the imaginary part of admittance.

The active power is the power that converts to the mechanical power at the interface. The reactive power is the

imaginary part of complex power that is not consumed and is recirculated to the power supply. The power rating is the

power requirement of the power supply without distortion. In induced-strain transmitter applications, the reactive power

is the dominant factor, since the transmitter impedance is dominated by its capacitive behavior. Managing high reactive

power requirements is one of the challenges of using induced-strain transmitters.

3.2 Mechanical power

When the acoustic energy emanating from the transmitter PWAS, the input electrical energy applied to the PWAS is

transduced into mechano-acoustic energy. This mechanical power can be calculate by the pin-force and particle velocity

at the PWAS end.

The frequency response function (FRF) of pin-force 0F̂ at the end of PWAS in response of applied voltage V̂ is

defined as

0ˆ ˆ/FVFRF F V (32)

The Lamb wave NME of the particle velocity under PWAS excitation is derived from equation as

0ˆ( , ) ( )ni a

n n

n

v a d g F e d

v (33)

The FRF of the particle velocity in response of applied voltage is

0ˆ( , )

( ) ( ) ( )ˆ ˆ

n ni a i a

vV n n FV n n

n n

Fv a dFRF g e d FRF g e d

V V

v v (34)

The time-averaged power is defined as

0

1 T

P P t dtT

(35)

The time-averaged product of two harmonic variables is one half the product of one variable times the conjugate of the

other. The time-averaged mechanical power at PWAS-structure interface is

0 0 0

1 1ˆ ˆ2 2

FV vVP F v FRF V FRF V (36)

3.3 Lamb Wave Power and Energy

The acoustic energy is distributed into the several guided wave modes concomitantly existent in the substructure. Not

all the modes are excited equally and mode-tuning principles applies. Here we calculate the total Lamb wave kinetic

energy and potential energy at cross section of the PWAS transmitter edge ( ,x a y d ) across whole thickness. The

kinetic energy for Lamb wave is defined as

2 21( , )

2e x y

A

k x t v v dA (37)

The time-averaged kinetic energy can be calculated by the velocity times the conjugate of itself over two.

1

. .4

e x x y y

A

k v v v v dA (38)

Actual strains for Lamb waves are evaluated from the displacement in equation (21) after introducing normalization and

modal participation factors. For a beam structure, wave solution is considered invariant in width direction. The area

integral is reduced to integration over thickness. The kinetic energy contains both symmetric and anti-symmetric wave

energy and the time-averaged kinetic energy takes the form

n ne eS eA

n n

k k k (39)

9

where neSk and

neAk is the time-average kinetic energy for symmetric mode nS and anti-symmetric mode

nA , i.e.,

2 2 22 ( ) cos cos sin sin

4n

n

d

S

eS n n Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn

d

bk g B F y R y y R y dy

(40)

2 2 22 ( ) sin sin cos cos

4n

n

d

A

eA n n An An An An An An An An An An

d

bk g A F y R y y R y dy

(41)

where the terms nS

ng and nA

ng represent the coefficients in equation (20) for symmetric and anti-symmetric modes.

The Lamb Wave potential energy is defined as

1

( , )2

e xx xx xy xy yy yy

A

v x t T S T S T S dA (42)

Then time-averaged potential energy is

1

2 2 2 24

e xx xx yy xx yy yy xy xy

A

v S S S S S S S S dA (43)

Similar to the kinetic energy, the time-averaged potential energy is summation of the potential energy of all modes, i.e.

n ne eS eA

n n

v v v (44)

whereneSv and

neAv are the time-average potential energy for symmetric mode nS and anti-symmetric mode

nA ,

i.e.,

22

2 2 2 2 2 2 2 2 2

2

02

2

2 cos cos

2 cos cos ( )cos cos

( )4

2 cos cos

2 sin2

n

n

d

Sn Sn Sn Sn Sn Sn

d

d

Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn

dS

eS n n d

Sn Sn Sn Sn Sn Sn

d

Sn Sn Sn

y R y dy

y R y R y y dyb

v g B F

y R y dy

y

2

2 2 sin

d

Sn Sn Sn Sn

d

R y dy

(45)

22

2 2 2 2 2 2 2 2 2

2

02

2

2 sin sin

2 sin sin ( )sin sin

( )4

2 sin sin

2 cos2

n

n

d

An An An An An An

d

d

An An An An An An Sn An An An An An An An

dA

eA n n d

An An An An An An

d

An An An

y R y dy

y R y R y y dyb

v g A F

y R y dy

y

2

2 2 cos

d

An An An An

d

R y dy

(46)

The total energy for exact Lamb waves per unit length is summation of the kinetic and potential energy. The total time-

averaged Lamb wave energy at the cross section of PWAS end is

e e ee k v (47)

4. SIMULATION RESULTS We considered a PWAS transmitter attached to the center of an infinite aluminum plate. Straight crested Lamb-waves

are assumed hence the process is z-invariant. Numerical simulation was performed with the parameters given in Table

1. Constant 10-V excitation voltage from an ideal electrical source was assumed at the transmitter PWAS. In our

simulation, the PWAS size was varied from 5 to 25 mm, whereas the frequency was spanned from 1 to 2000 kHz .

10

Table 1 Simulation Parameters

Beam structure

(2024 Al alloy)

Transmitter PWAS

(PZT850)

Length ∞ 5-25 mm

Height 1 mm 0.2 mm

Width 7mm 7mm

Young’s Modulus (E) 70e9 63e9

Density 2700 7600

Constant Voltage Input 10 V

Frequency Frequency sweep 1 - 2000 kHz

It was found (Figure 6) that the reactive electrical power required for PWAS excitation is orders of magnitude

larger than the active electrical power (compare Figure 6a with Figure 6b). Hence, the power rating of the PWAS

transmitter is dominated by the reactive power, i.e., by the capacitive behavior of the PWAS. We note that the

transmitter reactive power is directly proportional to the transmitter admittance ( Y i C ), whereas the transmitter

active power is the power converted into the ultrasonic acoustic waves generated into the structure from the transmitter.

A remarkable variation of active power with frequency is shown in Figure 6b: we notice that the active power (i.e., the

power converted into the ultrasonic waves) is not monotonic with frequency, but manifests peaks and valleys. As a

result, the ratio between the reactive and active powers is not constant, but presents the peaks and valleys. The increase

and decrease of active power with frequency corresponds to the PWAS tuning in and out of various ultrasonic waves

traveling into the structure. The maximum active power seems to be ~ 4 Wm .

(a)

0 500 1000 1500 20000

0.5

1

1.5

2

frequency [kHz]

Pow

er

[Watt

]

Electrical Reactive Power

(b) 0 500 1000 1500 2000

0

1

2

3

4

5

frequency [kHz]

Pow

er

[mW

att

]

Electrical Active Power

Figure 6 Electrical power required at the PWAS terminals: (a) reactive power; (b) active power

The acoustic energy is distributed into the several guided wave modes concomitantly existent in the substructure. In

the simulation, only S0 and A0 modes exist. The kinetic and potential energy at the cross section is shown in Figure 7a.

The total wave energy is shown in Figure 7b.

Figure 8 presents the results of a parameter study of various PWAS sizes and frequencies. The resulting parameter

plots are presented as 3D mesh plots. Figure 8a shows a 3D mesh plot of the electric active power vs. frequency and

transmitter size. It indicates the active power that PWAS generates into the structure contains the tuning effect of

transmitter size and excitation frequency. A larger PWAS does not necessarily produce more wave power at a given

frequency. The maximum wave power output in this simulation is ~ 5 Wm . This study provides guidelines for the

design of transmitter size and excitation frequency in order to obtain maximum wave power into the SHM structure.

Similarly, the total Lamb wave energy shows the same tuning trend in relation of PWAS sizes and excitation frequency

(Figure 8b).

11

0 500 1000 1500 20000

2

4

6x 10

-7

Energ

y,

[Joule

s]

Kinetic and Potential Wave Energies

K.E

P.E

0 500 1000 1500 20000

0.5

1x 10

-6

frequency [kHz]

Energ

y,

[Joule

s]

Total Wave Energy

Total

Figure 7 The acoustic energy at the cross section area of the PWAS transmitter edge (a) Lamb wave kinetic energy and

potential energy, (b) total wav energy.

(a)

0200

400600

8001000 5

10

15

20

25

0

1

2

3

4

5

Transmitter size [mm]

Constant voltage (10V) Input

frequency [kHz]

Active P

ow

er

[mW

att

]

(b)

0200

400600

8001000 5

10

15

20

25

0

0.5

1

x 10-6

Transmitter size [mm]

Constant voltage (10V) Input

frequency [kHz]

Wave T

ota

l E

nerg

y [

J]

Figure 8 PWAS transmitter under constant voltage excitation (a) active power; (b) total wave energy

5. CONCLUSION AND FURTHER WORK An analytical investigation of power and energy transduction in PWAS attached on structure was developed using the

Lamb wave normal mode expansion method. The predictive modeling of the multi-physics power and energy

transduction between structurally guided waves and ideal bonded PWAS were presented and discussed. It was shown

that a judicious combination of PWAS size, structural thickness, and excitation frequency can ensure optimal energy

transduction and coupling with the ultrasonic guided waves traveling in the structure. The analytical solution was based

on the use of all the Lamb wave modes. Simulation on S0 and A0 was performed to show the power and energy trend

for the PWAS transmitter. Future work should also focus on power and energy analysis with assumptions in circular

crested (2-D) multi-mode Lamb wave propagation.

12

6. ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under Grant # CMS- 0925466, Shih-

Chi Liu, Program Director and by the US Department of Commerce, National Institute of Standards and Technology,

Technology Innovation Program, Cooperative Agreement Number 70NANB9H9007. Any opinions, findings, and

conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the

views of the National Science Foundation or the US Department of Commerce, National Institute of Standards and

Technology.

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