Parameter measurement of the cylindrically curved thin layer using low-frequency circumferential...

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Ultrasonics 43 (2005) 357–364

Parameter measurement of the cylindrically curved thin layerusing low-frequency circumferential Lamb waves

Xiao Chen, Mingxi Wan *

Department of Biomedical Engineering, School of Life Science and Technology, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China

Received 10 October 2003; received in revised form 14 June 2004; accepted 18 July 2004

Available online 9 August 2004

Abstract

The characteristic parameters of a cylindrically curved thin layer include its elastic constants, thickness and curved radius. A layer

is considered thin if the echoes from the front and back surfaces of the layer cannot be separated in the time domain, and/or that the

wave arrivals corresponding to longitudinal and shear wave part cannot be identified in the time or space domain. This paper

describes a low-frequency circumferential Lamb wave method to characterize those parameters of a cylindrically curved thin layer.

The technique is based on the measurement of circumferential Lamb wave phase velocity and the unknown parameter is estimated

through minimizing the mean square error obtained by comparing theoretical and experimental phase velocities. The sensitivity and

accuracy of the proposed technique to different parameters are analyzed. Using the present technique, a cylindrically curved thin

layer with thickness down to ten percent of the longitudinal wavelength can be successfully measured with an average relative error

less than two-percent in our experiment.

� 2004 Elsevier B.V. All rights reserved.

PACS: 43.35.Cg; 43.35.YbKeywords: Ultrasonic; Lamb wave; Thin layer; Material characterization

1. Introduction

There are various situations of technological impor-

tance in which one wishes to carry out a non-destructive

evaluation (NDE) of the characteristic parameters (i.e.,

thickness, velocities, etc.) of thin layers, such as protec-

tive surface coatings of pipes, the bonding interface of

composite tubes. A layer can be considered thin when

its thickness is much smaller than the longitudinal wave-

length of the probing ultrasonic wave. It means that theechoes from the front and back surfaces of the plate can

not be separated in the time domain, and/or that the

wave arrivals corresponding to longitudinal and shear

0041-624X/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ultras.2004.07.005

* Corresponding author. Tel.: +86 29 8266 7924; fax: +86 29 8323

7910.

E-mail address: [email protected] (M. Wan).

wave part can not be identified in the time or space do-

main. In this case the conventional characterizationmethods, such as the pulse interference method [1] and

the resonance testing method [2], cannot be applied.

Although using high-frequency ultrasound could broad-

en the limit of the thickness to some extent, it is very

expensive. What is more, high-frequency ultrasound

has a very short penetration depth, which limits its

application to only low-loss materials, and suffers the

interference from micro-structural details of the thinlayer.

Some low-frequency ultrasonic bulk wave methods,

such as the time-domain least square method [3], the fre-

quency-domain transfer function method [4,5], and the

variable trigger and strobe (VTS) method [6] have been

proposed to characterize the thin layer with thickness

down to one percent of the longitudinal wavelength.

mailto:[email protected]

358 X. Chen, M. Wan / Ultrasonics 43 (2005) 357–364

The time-domain least square method and the fre-

quency-domain transfer function method have good

performance for characterizing the thickness of thin lay-

ers, but the computational convergence problem occurs

if the layer thickness is less than one-half of the longitu-

dinal wavelength. Moreover, the aforementioned meth-ods cannot evaluate the shear wave velocity, which is

an indispensable parameter for characterizing the

mechanical properties of the thin layer.

The restrictions of those bulk wave methods have

motivated the development of a new ultrasonic tech-

nique to characterize the thin layer. Compared with bulk

wave, the guided waves are more appropriate for evalu-

ating the characteristic parameters of thin layers in thatthey include both the longitudinal and shear wave infor-

mation [7–10]. One possible method has been developed

by Zhang et al. [10] who used the low-frequency Lamb

F vl; vs; h;R; f ; vð Þ ¼

bIpðx1Þ � Ipþ2ðx1Þ Ipþ2ðy1Þ bNpðx1Þ � Npþ2ðx1Þ Npþ2ðy1ÞbIpðx1Þ � Ip�2ðx1Þ �Ip�2ðy1Þ bNpðx1Þ � Np�2ðx1Þ �Np�2ðy1ÞbIpðx2Þ � Ipþ2ðx2Þ Ipþ2ðy2Þ bNpðx2Þ � Npþ2ðx2Þ Npþ2ðy2ÞbIpðx2Þ � Ip�2ðx2Þ �Ip�2ðy2Þ bNpðx2Þ � Np�2ðx2Þ �Np�2ðy2Þ

��

��¼ 0 ð1Þ

wave phase velocity to determine the material acoustical

parameters of a thin plane layer. The unknown para-

meter is estimated by minimizing the mean square error

obtained by comparing theoretical dispersion curves

with experimental data. However, the previous works

only focus on the thin plane layer.

Comparing with plane layer, the curved thin layer is

difficult to identify in that the guided waves propagat-ing in it are much more complex than the plane layer

[11–13]. Lower studied the guided waves propagation

and their sensitivity to defects [14]. Li formulated a

wave mode expansion solution for transient excitation

of circumferential waves in a thick annulus [15]. Lohr

investigated guided wave ultrasonic method for the pipe

fouling detection potential. The guided wave method

relates energy loss to viscous or semi-solid loadingon a plate or pipe [16]. The results presented in this

paper extend this work to measuring the characteristic

parameters of a cylindrically curved thin layer using cir-

cumferential guided waves. In this paper, the circumfer-

ential wave method is developed to characterize the

acoustical parameters of a cylindrically curved thin

layer. One of the four parameters, thickness, radius,

shear and longitudinal wave velocities of a cylindricallycurved thin elastic layer is deduced when three of

them are known. Sensitivity, accuracy of different

parameters, and dependence of the present method on

the layer thickness are studied experimentally. The

experimental results agree well with the theoretical

calculations.

2. Method

The subject of guided waves in a hollow cylinder has

been studied for many years. The general solution of

harmonic wave propagation in an infinitely long elastic

hollow cylinder has been obtained by Gazis [17] using

elasticity theory. Unlike guided waves in plane layer,

the guided waves in cylindrically curved thin layer are

divided into three kinds of modes, i.e., longitudinal, tor-

sional, and circumferential ones. The characteristic dis-persion equation of the circumferential Lamb waves

propagating in a cylindrically curved isotropic thin layer

is [18]

where vl and vs are the longitudinal and shear wave

velocities respectively. h and R are the thickness and

the curved radius of the thin layer. f is the frequency

of the ultrasonic wave and v is the phase velocity of

the circumferential Lamb waves. p is the angular wave

number and p = 2pf/v. Ip(*) and Np(*) are pth-order Bes-

sel and Neumann functions. The detailed definitions of

x1,2, y1,2 and b can be found in Ref. [18].Using the dispersion curves of the low-order mode of

circumferential Lamb waves, one can measure the

parameters of a cylindrically curved thin layer. One of

the quantities vs, vl, h or R of the cylindrically curved

thin layer can be deduced through a comparison be-

tween the measured and the theoretically predicted dis-

persion curves. The value of vs, vl, h or R of the thin

layer being calculated minimizes the residual error E

defined below [10]

EðqÞ ¼ 1

N

XNi¼1

vi � v�i�� 2 ð2Þ

where ‘‘j j’’ is the absolute value. q represents the valueto be calculated, which may be vs, vl, h or R.

v�i i ¼ 1; . . . ;N are the set of experimental dispersion

data for circumferential Lamb waves in the phase veloc-ity-frequency space. The value of N is directly related to

the effective frequency range from 0.879 to 3.125 MHz,

which corresponds to the bandwidth of 6 dB attenuation

Fig. 2. Schematic of the experiment.

Fig. 1. (a) The waveforms and (b) the amplitude spectrums of the

transmitted pulse.

X. Chen, M. Wan / Ultrasonics 43 (2005) 357–364 359

as shown in Fig. 1. To a fixed angular resolution, the

broader the effective frequency range is, the bigger the

value of N. Besides the bandwidth of the frequency-

thickness product, the angular resolution of the angle

controller can also affect the value of N. Smaller angle

resolution corresponds to the bigger value of N. vii = 1, . . .,N is the theoretical circumferential Lamb wavevelocity calculated from the characteristic dispersion

Eq. (1).

The simplex algorithm [19] has been verified to be the

most appropriate for solving the nonlinear inversion

problem because of its good convergence property

[10]. In this paper, we have also used the simplex algo-

rithm for curve fitting and finding the roots of Eq. (2)in order to obtain the value of a parameter q. Because

it is assumed that one of the four parameters, thickness,

radius, shear and longitudinal wave velocities of cylin-

drically curved thin elastic layers is deduced when three

of them are known in the present paper, one data point

is enough to deduce any of four parameters using Eq. (2)

[10].

3. Equipment and experimental procedure

3.1. Setup

The main experimental instrument is Multiscan

Ultrasonics Automated Inspection Systems (MUAIS)

which is made by PANAMETRICS�, Inc. A schematic

diagram of the experimental apparatus for the low-fre-

quency Lamb wave method is shown in Fig. 2. A pair

of accurately matched longitudinal wave transducers,T and R, with a central frequency of 2 MHz is used

for transmitting and receiving waves. The transmitting

and receiving angles are adjusted by the Mechanical

Control Unit to maximize the signal. The optimum inci-

dence angle is adjusted continuously by the mechanical

control unit, whose angle resolution is 0.5�. A computerwith MultiScan� software controls the general experi-

mental procedure. A pulse with a maximum bandwidthof 35 MHz produced by the 5800 Pulser/Receiver is sent

to the excited transducer, which produces the ultrasound

pulse to the specimen. The received signal is received, fil-

tered and amplified by the 5800 Pulser/Receiver, then

digitized at a sampling rate 100 MHz by an 8-bits A/D

converted card in the control computer. To reduce the

random error, each test is repeated and averaged 500

times before it is recorded to the computer for furtheranalysis. The whole experimental procedure is under

the temperature of 20 �C and 0.997 standard atmos-

pheric pressure.

3.2. Experimental specimen

We demonstrate the method using a cylindrically

curved thin aluminum layer as shown in Fig. 2. Theacross section of the specimen is a partial circle with a

central angle of 52�. The length of the specimen is 300mm, which can be considered as infinity. The standard

value of the thickness of the layer is 0.20 mm and the

Transducer

Tank filledwith water

Circumferential Lambwave propagation

Tank filledwith water

T R


standard value of the curved radius is 132.50 mm, which

are measured by 3D high-precision electrical measure-

ment instruments. The measure error is ±0.01 mm.

The density of the specimen is 2700 g/cm3. The density

is measured by using the Archimedes principle. The

standard values of the longitudinal and shear wavevelocities are 6405 and 3081 m/s respectively, which

are measured by the VTS method using the high-fre-

quency longitudinal and shear wave transducers [6].

Specimen

R1

L

2θ

1θ

θ

(a)

(b)

Fig. 3. (a) Experimental setup for circumferential Lamb waves; (b)

determination of the incidence angle.

3.3. Excitation and reception of circumferential Lamb

waves

The circumferential Lamb waves in a cylindricallycurved thin layer are dispersion and multi-modes [14].

It is one of the key elements in parameter characteriza-

tion to select and excite the proper single mode. Gener-

ally, an excitation source can excite all of the modes that

exist within its frequency bandwidth, resulting in a time-

domain signal that is much too complicated to identify.

But it is shown that only the F(1,1) and F(2,1) mode

exit when the product of the frequency and the thicknessis small. To a thin layer, it means that the frequency is in

the low-frequency domain. What is more, the two modes

are not overlapped. The single mode can be excited

through careful selection of the frequency domain and

the incidence angle. So in the experiment, the low-fre-

quency ultrasonic is selected and the low-order modes

are used to characterize the parameters.

The creation of perturbations on the surface of acylindrically curved thin layer to excite Lamb waves

is convenient and simple in realization [10,15,16].

Although the outer and inner surfaces of the specimen

may all be selected as the excited surface, we would

rather use the outer surface since more operation space

can be available. A pair of water tanks was specifically

designed to fit the outer diameter of the specimen, and

the guided waves were sent around the circumferenceof the specimen. The experimental setup for the labora-

tory study is shown in Fig. 3(a).

Fig. 3(b) is geometrical scheme of excite angle. Be-

cause of the curve of the specimen, the angle h1 that isread in the mechanical control unit is not equal to the

incidence angle h. The relation between them is

h ¼ h1 � h2 ð3Þ

h2 ¼ arcsinL2R1

� �ð4Þ

where R1 is the radius of the outer surface of the cylin-

drically curved thin layer, and arcsin (*) denotes the

inverse sine function.

To receive the circumferential Lamb waves of the thin

layer, the received angle should be equal to the incidence

angle [18]. In the experience, the symmetrical axis of the

mechanical control unit is through the center of the

across circle. Therefore, the received tank is symmetri-

cally put with the excited tank [16] as shown in Fig.

3(a). The detailed main designing and obtaining the

dispersion curve are the same as Ref. [10].

3.4. Data analysis

Using the Fourier transform, the received Lamb wave

pulse signals can be used to derive the dispersion curves

in the cylindrically curved elastic layers. Peaks are pre-

sent in the amplitude spectrum indicating the presence

of Lamb wave roots [10]. The phase velocities for theLamb waves are selected by controlling h, which is ad-justed by the Mechanical Control Unit. Because the

phase velocity along the thickness varies linearly with

radius [18], the value of the Lamb wave phase velocity


in Eq. (2) can be calculated through the modified Snell�slaw

v ¼ RR1

vLsin h

ð5Þ

where vL denotes the longitudinal ultrasound velocity inwater, which is taken as 1490 m/s in the experiments.

The measurements of the Lamb roots are repeated for

a set of angles hj, i.e., for a set of phase velocity values.The mechanical structure of the transducer holder per-

mits the incident angle to be adjusted ranging from

16� to 54� with an angle resolution of 30 0, which corre-sponds to a range of 1800–5406 m/s for the phase veloc-

ities of Lamb waves to be detected. Fig. 1(a) shows thetime domain reference signal with 80 mm apart between

the transmitted and received transducers. The frequency

spectrum analysis of the reference signal in Fig. 1(b)

shows that the central frequency of the reference signal

is 1.998 MHz and its 6 dB attenuation points are

0.879 and 3.125 MHz.

Fig. 4. The ev � eq curve of (a) F(2,1) mode and (b) F(1,1) mode.

Solid line: vl; dashed line: vs; dotted line: R; dot-dashed line: h.

4. Sensitivity analysis

It has been mentioned in the previous section that the

dispersion of the cylindrically curved thin layer depends

on both the frequency and the thickness of the layer.

Therefore, careful attention must be given to the sensi-

tivity problem before considering the inverse problem.

The relative sensitivity of the phase velocity of Lambwaves is defined in Ref. [10]. From the point of view

of designing the experiment for the purpose of solving

the inverse problem, the relative sensitivity is the most

important parameter. If the relative sensitivity is large,

a small change in q will result in a large change in v,

and vice versa.

Shown in the Fig. 4(a) are the theoretical curves of

the relative error of the circumferential Lamb wavevelocity ev versus the relative error of the parameter

eq when the specimen is characterized by F(2,1) mode

of the circumferential Lamb wave velocity using a pair

of broadband transducers with 2 MHz center fre-

quency on the condition of N = 1. It can be seen from

Fig. 4(a) that the relative error of vs is slightly smaller

than that of vl with the same measurement error of

phase velocity. But the F(2,1) mode has very poor sen-sitivity for measuring thickness of the same curved

aluminum thin layer. In other words, the error of h

is much larger than those of vs, vl and R at the same

level of ev.

The fact that F(2,1) mode has very poor sensitivity in

measurement of the thickness can be explained from the

dispersion curve of the thin layer. It is shown that the

dispersion property of the F(2,1) mode has very littlechange when the product of the frequency and the thick-

ness changes. When the thickness increases or decreases

at a fixed frequency, the Lamb wave phase velocity has

almost no variety in the low-frequency domain.

Shown in Fig. 4(b) are the theoretical curves of the

relative error of the circumferential Lamb wave velocity

ev versus the relative error of the parameter eq when the

specimen is characterized by F(1,1) mode of the circum-

ferential Lamb wave velocity using a pair of broadbandtransducers with 2 MHz center frequency on the condi-

tion of N = 1. It can be seen from Fig. 4(b) that the rel-

ative error of vl is slightly smaller than that of vs with the

same measurement error of phase velocity. On the other

hand, the sensitivity of the F(1,1) mode to h is much lar-

ger than that of the F(2,1) mode. The sensitivities of the

two modes to R are almost the same.

From the sensitivity analysis above, one concludesthat vs, vl and R of the cylindrically curved thin layer

can be evaluated by the two modes, while the F(1,1)

mode is more suitable to characterize h.

Fig. 6. The dispersion of circumferential F(1,1) and F(2,1) mode

Lamb waves, and the experimental data of a cylindrically curved thin

aluminum layer.


5. Experimental results and discussion

Fig. 5(a) shows the received pulses at different inci-

dence angle in time domain which are plotted by Scan-

View� software. Fig. 5(b) shows the corresponding

amplitude spectrum at different incidence angle in fre-quency domain which are analyzed by ScanView� soft-

ware. The maximum value locations of the peak of the

spectrum amplitude are also indicated in the same fig-

ure. Notice how they change when the incidence angle

varies. For F(1,1) mode, the maximum value locations

of the peak decreased with the increase of the incidence

angle.

It is noticed that only one point is measured in Fig. 6for F(2,1) mode. Because the F(2,1) mode are almost

non-dispersive in the low-frequency domain. The phase

velocity cannot be adjusted so small because of the limit

of the angle resolution. So the measured frequency is se-

lected in the dispersive domain if possible in the charac-

terization of parameters in order that the parameters

can be deduced efficiently using Eq. (2). This is different

to that in the defect inspection. Because in that condi-tion, the non-dispersive waves are more suitable for

inspection and the measured frequency is selected on

Fig. 5. (a) The waveforms and (b) the ampl

the non-dispersive domain in order that the wave can

propagate as long as it can [14,16].

itude spectrums of the received pulses.

Table 1

The calculated parameters for cylindrically curved thin aluminum layer using the F(1,1) and F(2,1) mode

Parameter Value Estimated using F(1,1) mode Estimated using F(2,1) mode

Value Error (%) Value Error (%)

h (mm) 0.20 0.1982 �0.90 – –

R (mm) 132.50 131.24 �0.95 131.73 �0.58vs (m/s) 3081 3054.8 �0.85 3055.4 �0.83vl (m/s) 6405 6361.4 �0.68 6299.3 �1.65

Fig. 7. Possible measurement errors of the Lamb wave velocity at

different incidence angles with the angle resolution of 0.5�.


Denoted by the asterisk, the experimental Lamb wave

dispersion data of F(1,1) and F(2,1) mode for an cylin-

drically curved thin aluminum layer with the thickness

of 0.2 mm is shown in Fig. 6, which is obtained usingthe method described in Section 3.

The main objective of this work is to develop a tech-

nique suitable for characterizing the shear wave velocity,

the longitudinal wave velocity, the thickness and the

radius of cylindrically curved elastic layers whose thick-

ness is thin comparing with the body longitudinal wave-

length. A specimen described in the Section 3 has been

measured using the developed low-frequency Lambwave method. The ratio of the thickness to the longitu-

dinal central wavelength is less than 0.1. The measured

results of the thin layer using the low-frequency F(1,1)

and F(2,1) mode Lamb wave method are presented in

Table 1, in which the blanks represent the evaluation er-

rors that are larger than 10% and the corresponding

measured value is omitted. The standard parameter

values are also listed in the same table. It is revealedin Table 1 that the agreement between the standard

value and estimated one is good. In most of the situa-

tion, the error is less than two percent. The F(2,1) mode

fails to characterize the thickness of thin layer. The in-

verse error of F(2,1) mode to vl is much more than that

of the F(1,1) mode. The two modes have almost the

same error to vs.

There are two main factors that affect the precision ofthe estimating result in characterizing the acoustical

parameters. One is the measurement error of the Lamb

wave velocity, and the other is the inverse arithmetic.

One cause of the evaluation errors is attributed to the

measurement error of the cylindrically Lamb wave

velocity, which has a direct relation with the angle accu-

racy of the transmitter and receiver. On the condition

that the angle resolution is 0.5�, the possible measure-ment errors of the cylindrically curved Lamb wave

velocity at different incidence angles are shown in Fig.

7. It can be noticed that the measurement error of the

Lamb wave velocity decreases as the incidence angle in-

creases (corresponding to the decrease of Lamb wave

velocity). As shown in Fig. 6, F(2,1) mode Lamb wave

phase velocity decreases with the layer thickness, the

possible measurement error of the Lamb wave velocityalso decreases, which can improve the accuracy of vsand vl besides the increased sensitivities. Moreover, it

can be deduced that the F(2,1) mode is more appropri-

ate to measure the specimens with lower acoustic wave

velocity. F(1,1) mode Lamb wave phase velocity in-

creases with the layer thickness, the possible measure-

ment error of the Lamb wave velocity also increases.

Therefore the F(1,1) mode is more appropriate to meas-ure thinner specimens.

The other main cause of the evaluation errors is

attributed to the LMS arithmetic that is used to find

the value of the parameters from the experimental data.

Since the minimum value of the root-mean-square resid-

ual error function shifts from the true value. Even in the

point of the true value, the function does not reach zero

due to the effect of the measurement error. The detaildiscussion can be found in Ref. [5]. It is well known that

the random noise is ineluctable in the measurement. In

the experiment, the received signals are averaged coher-

ent in order to improve the signal-to-noise ratio.

6. Summary

A low-frequency circumferential Lamb wave method

is developed to measure anyone of the four acoustic

parameters, thickness, radius, shear and longitudinal

wave velocities, of cylindrically curved thin elastic layers


given the other three parameters. The method has been

proved to be accurate with thickness of the specimen

down to ten percent of the longitudinal wavelength in

our experiment. The average relative error is less than

two percent. Further work should be done to study

the limitation of the method, no matter theoreticallyor experimentally. In order to resolve the relative poor

sensitivity of the F(2,1) mode to the evaluation of the

thickness, one can use the F(1,1) mode. While each

mode has its advantages and disadvantages, use of sev-

eral modes simultaneously can increase the prediction

accuracy.

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