Vij D.R. Handbook of Applied Solid State Spectroscopy

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8. Resonance Acoustic Spectroscopy

Figure 8.3 a) The form function for backscattering by a solid aluminum cylinder in water. b, c,

d) Isolated modal resonances of the first, second, and third (n = 1, 2, 3) modes for

backscattering by an aluminum cylinder in water [76].

Figure 8.4a shows the form function of an aluminum cylinder insonified by a

normally incident plane acoustic wave. The backscattered field is composed

of a series of dips, which are due to resonances, superimposed on a smooth

background. Each dip in Figure 8.4a is due to a certain resonance frequency

identified by the integers

),( ln . The first of these two integers defines the

mode number and the second one indicates the eigenfrequency label for a

given mode.

1 l corresponds to a Rayleigh-type wave and ,...3,2 l to a

whispering gallery-type wave. The mode number

n identifies the shape of the

pressure contours around the cylinder at a resonance frequency. For the

breathing mode

)0( n , the pressure contours around the cylinder move

radially in and out. For the dipole mode

)2( n , the pressure contour motion

is rigidly back and forth, and for the quadrupole mode

)2( n , the contours

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8.3.2 Resonance Acoustic Spectroscopy

8.3 Mathematical Models

oscillate between prolate and oblate ellipses. For higher modes, the pressure

contours around the cylinder have rosette patterns with 2n lobes.

The values of the resonance frequencies are dependent on the elastic wave

velocities (or elastic constants) of the solid cylinder. The way these

resonances can be used for material characterization purposes, resonance

acoustic spectroscopy (RAS) [57–59], resembles the way chemical elements

are identified from their optical spectra. The so-called Regge pole trajectories

normalized frequency and n is the mode number. The Regge trajectories for

Figure 8.4a corresponds to a pole on the Regge trajectories plot highlighted by

a “+” in Figure 8.4b. The poles belonging to the same family of surface waves

are connected by a line in Figure 8.4b. Regge poles are obtained by finding

the roots of the denominator of equation (8.43).

Figure 8.4 a) Form function for an aluminum cylinder. b) Regge trajectories for an aluminum

cylinder [76].

Figure 8.5a shows the numerically calculated form function of the same

aluminum cylinder used to generate Figure 8.4 when it is insonified by a

families of resonances have been generated. These resonances are due to

axially guided waves. Resonances associated with axially guided waves are

designated by

²¢ pn, where, similar to resonances associated with Rayleigh

and whispering gallery waves, n is the mode number and p is the frequency

label of the resonance. Regge trajectories of the aluminum cylinder are also

shown in Figure 8.5b.

an aluminum cylinder are shown in Figure 8.4b. Each resonance frequency in

[59] are a plot of resonance frequencies in the (ka, n)-plane where ka is the

plane acoustic wave at an incident angle of D=3 . It can be observed that extra

369

8. Resonance Acoustic Spectroscopy

Figure 8.5 a) Form function for aluminum cylinder. b) Regge trajectories for aluminum

cylinder. Resonances associated with Rayleigh and WG waves are designated by (n, l) and

resonances associated with axially guided waves by n, p [76].

Any component has its own resonances, which depend on the geometry

and material constituents of the component. Presence of any flaws in the

component also affects the resonance frequencies. Each resonance has its own

characteristics and consequently shows sensitivity to certain variations in the

component. One of the objectives of resonance acoustic spectroscopy is to

RAS for characterization of shapes of fluid and solid scatterers. Ripoche and

Maze [59] discussed the use of the method of isolation and identification of

resonances (MIIR) for evaluation of elastic objects. Honarvar and Sinclair

[45] used RAS for monitoring the health condition of clad rods. Ying et al.

[60] implemented RAS in characterizing large-grained cylinders.

areas will be discussed in more detail. However, first we should introduce

some experimental procedures usually used in RAS applications.

In the following sections, applications of RAS in the above-mentioned

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Resonance acoustic spectroscopy has been used by a number of

researchers for characterization of various materials. Brill et al. [57, 58] used

to verify or monitor certain properties of the component.

of the component. After identifying the right resonances, one can use them

identify the resonances and their relative sensitivity to variations in properties

OF RESONANCES (MIIR)

8.4.1 Introduction

RST theoretically shows the possibility of decomposing the backscattered

amplitude spectrum of a penetrable scatterer into a smooth background

component and a resonant component. Measurements of the backscattered

pressure field from immersed elastic targets have been carried out since the

early works of Faran [24] and Hickling [25]. Before 1981, the acoustic

signatures of elastic targets usually were “time signature” and “form

functions” (backscattered pressure versus frequency). The monostatic time

signatures, obtained in the geometry described in Figure 8.1, contain a large

number of backscattered echoes, which are due to the specular echo, Franz

waves, Rayleigh waves, whispering gallery waves, etc. The computed far-

field form functions have been compared to experimental form functions

[25, 68, 69] have been used for the measurement of the form function, it was

not until 1981, when the method of isolation and identification of resonances

(MIIR) was developed, that the real thrust in the experimental procedures

started [36]. Using MIIR, the form function, the resonance spectrum, and the

mode number of individual resonances of a penetrable scatterer can be

experimentally measured using a delicate time-gating procedure. In the

following sections we will discuss two approaches used in MIIR

measurements which are 1) the quasi-harmonic MIIR method, and 2) the

short-pulse MIIR method.

8.4.2 Quasi-Harmonic MIIR

The classical scattering theory gives the far-field form function of an elastic

cylinder immersed in water and insonified by an infinite plane acoustic wave

(ș) İ ()cos(ș).

fAxn

(8.86)

The computed form function of an aluminum solid cylinder plotted versus

is shown in Figure 8.6a [40].

The cylindrical target, with relatively high length-to-diameter ratio, is

insonified by long incident tone-burst pulses with a carrier frequency of

. In

a monostatic situation, by slowly varying the frequency of the incident signal

between one burst and the next, the quasi-steady state amplitude is recorded

versus normalized frequency. This recording of the pressure amplitude gives

8.4 MIIR

[61--64]. Although both short pulses [62, 65-- 67] and pulses of long duration

371

8.4 METHOD OF ISOLATION AND IDENTIFICATION

obtained by pulse methods by means of the fast Fourier transform (FFT)

as,

8. Resonance Acoustic Spectroscopy

the backscattered signal spectrum (Figure 8.6b) compared to the far-field form

function. The theory agrees with the experiment. The spectrum consists of the

superposition of a smooth background and peaks and valleys corresponding to

resonances. The resonances are located at each minimum , but

Figure 8.6 The form function of the acoustic pressure scattered by a solid elastic cylinder.

8.4.2.1 Resonance Isolation in Quasi-Harmonic MIIR

If the target is insonified by a rectangular wave pulse having a known

frequency when this frequency does not coincide with one of the resonance

frequencies of the target, the received backscattered echo will have a

structure similar to that of the incident pulse with only a change in the

amplitude of the signal (see Figure 8.7a). On the other hand, when

the frequency of the incident wave coincides with one of the resonance

frequencies of the cylinder, the backscattered echo will show a completely

Figure 8.7 a) Incident tone-burst pulse used in quasi-harmonic MIIR. b) Echo scattered from

the cylinder at a resonance frequency.

372

often, it is

difficult to accurately locate them in this spectrum.

computed result and b) measured result [40].

different structure compared to that of the incident pulse, Figure 8.7b.

At a resonance frequency as shown in Figure 8.7b, three distinct parts can

be distinguished in the backscattered echo: (i) a leading transient, (ii) a steady

state vibration during which the amplitude of the signal remains almost

constant (forced vibration part of the signal), and (iii) a second transient part

after the end of the forced vibration during which the amplitude of the signal

starts to decay exponentially (free vibration part of the signal) [70]. If one

measures the amplitude of the signal in the steady state segment while

sweeping the transmitter through a range of frequencies, then a plot of

the measured amplitude versus frequency will produce the form function of

the cylinder as shown in Figure 8.8a. If the same measurements are performed

then the plot of the measured amplitude versus frequency is the resonance

spectrum (Figure 8.8b).

Figure 8.8 a) Measured form function, and b) resonance spectrum of an aluminum cylinder

[40].

8.4.2.2 Resonance Identification in Quasi-Harmonic MIIR

The latter measurement (Figure 8.8b) can also be used to verify the mode

two opposite directions and form standing waves. Therefore, it should be

possible to observe nodal and antinodal points of displacement around the

cylinder.

These nodal and antinodal points can be observed in a bistatic

configuration where one transducer insonifies the cylinder with a long pulse

modulated at a resonance frequency and another transducer (receiver) is

rotated, at a constant radial distance, around the cylinder and measures the

scattered signal. A typical arrangement is shown in Figure 8.9. The receiver

measures the amplitude of the signal at a fixed point during the second

8.4 MIIR

According to RST [71], circumferential waves travel around the cylinder in

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during the free vibration of the cylinder (second transient part of the signal),

shape and, accordingly, the mode number n of a specific resonance.

8. Resonance Acoustic Spectroscopy

transient part of the signal. A plot of amplitude versus angular position will

then reveal the nodal and antinodal points. A typical plot of this kind for n = 4

is shown in Figure 8.10. The mode number n of the resonance is half the

number of nodes or antinodes around the cylinder.

Figure 8.9 Test setup used for normal incidence MIIR measurements and mode shape

measurements [60].

The experimental measurement of the mode number n gives an

“identification of resonances” and allows a distinction of experimental

resonances observed in resonance spectra. Besides, the knowledge of the

mode number n for each resonance gives the experimental phase velocity of

the circumferential waves [22]. Because the response of the target reaches the

steady state, it is called “quasi-harmonic MIIR.”

Figure 8.10 Measured mode shape for n = 4 [76].

Although tone-bursts of long duration were used in early MIIR studies, it

was later shown that it is possible to obtain the same information with

another, much less time-consuming, method. This technique is called short-

pulse MIIR and will be discussed in the next section.

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Several studies were developed independently by three laboratories (NRL,

Washington; GPS. ENS, Paris; LEAH. US, Le Havre) to obtain resonance

spectra by means of short-pulse methods. The pulse method of Dragonette,

Numrich, and Frank [61, 62] offers some possibilities to obtain the total form

function of targets immersed in water. This function is given after processing

the incident signal is a short pulse. In 1984, Numrich et al. [72] used this

method, however only a selected portion of the time record of the echo was

transformed by the FFT algorithm. The frequency signature, which was

obtained for cylindrical bodies with hemispherical end caps was the elastic

response of the target. A method was developed independently by de Billy

[73] for the study of a cylinder (2a = 0.7mm) where a portion of the time

signature is isolated by a gate, then transformed into a frequency signature by

a spectrum analyzer.

In Le Havre, Maze et al. [74] could obtain the form function and the

resonance spectrum of the scatterer by calculating the FFT of the back-

scattered echo of short-pulse MIIR with one single measurement.

8.4.3.1 Resonance Isolation in Short Pulse MIIR

In short-pulse MIIR, the scatterer is insonified by pulses having a duration

that is short compared to the target's diameter divided by sonic velocity. The

first arrival at the receiver of the backscattered echo of the short pulse

corresponds to the specular reflection of the incident pulse from the vertex of

the scatterer. The subsequent echoes following this first echo (see Figure

8.11) are due to the resonances generated by the surface waves encircling the

body of the scatterer and producing standing waves.

Figure 8.11 Backscattered echo from an aluminum cylinder [75].

8.4 MIIR 375

8.4.3 Short-Pulse MIIR

the time signatures by means of a fast Fourier transform (FFT) algorithm when

8. Resonance Acoustic Spectroscopy

By calculating the FFT of the backscattered echo, one can obtain the form

function of the elastic scatterer for a broad range of frequencies with one

single measurement. Moreover, if the specular echo is removed from the

backscattered signal by proper time-gating, the FFT of the remaining part of

the signal will be the resonance spectrum of the scatterer (see Figure 8.12

[75]).

Figure 8.12 Measured form function and resonance spectrum of an aluminum cylinder at

incident angle of

Į 0

[76].

8.4.3.2 Resonance Identification in Short-Pulse MIIR

The mode number of the resonances contained within the frequency range of

the incident pulse can also be measured using short-pulse MIIR [77, 78]. The

bistatic measurements are similar to those of the quasi-harmonic MIIR. For

example, in the case of a long circular cylinder, returned echoes (with the

specular echo removed) are measured around the cylinder (e.g., at 5º

increments) and are stored in the computer memory. By finding the FFT of

each echo, the mode shape of each resonance can be plotted in a polar

diagram as in Figure 8.13 [76].

Figure 8.13 Identification patterns of two resonances, n = 2 and 3 [76].

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Nowadays, short-pulse MIIR is more often used for the measurement of the

scattered field, because it is much less time-consuming than quasi-harmonic

MIIR.

8.5 EXPERIMENTAL AND NUMERICAL RESULTS

8.5.1 Introduction

Echoes of acoustic waves reflected from elastic targets carry within them

certain resonance features caused by the excitation of the eigenvibrations of

the target. By means of suitable background subtraction (either

mathematically or experimentally), it is possible to obtain the resonance

spectrum of the target. This resonance spectrum characterizes the target just

as an optical spectrum characterizes the chemical element or compound that

emits it. Extracting the resonance information from the echo provides the

possibility of identifying the target as to its size, shape, composition, and

integrity. In the following sections, first, the primary applications of RAS in

characterizing the target shape will be presented. Then, various examples of

engineering applications of RAS for nondestructive evaluation (NDE) and on-

line monitoring of a number of cylindrical components including simple

elastic rods and wires, elastic, viscoelastic and explosively welded clad rods,

and fiber-reinforced composite (transversely isotropic) rods will be discussed.

8.5.2 Characterization of Target Shape by RAS

In this section, some early applications of RAS for identification of the shape

of a target are discussed. The dependence of the resonance spectra on target

shape for a fluid object in vacuo and elastic targets in a fluid were studied by

Brill et al. [57, 58]. As these two cases have similar mathematical

formulations, they will be considered together. Their eigenfrequency spectra

have been illustrated in the form of “level schemes” similar to that given by

atomic spectroscopy. The eigenfrequency spectrum of an object was tracked

through various geometrical shape changes, starting with a spherical shape

where the eigenfrequencies show azimuthal degeneracy. By removing the

azimuthal degeneracy, as the sphere is deformed into a slightly eccentric, and

finally, into a strongly eccentric prolate spheroid, it is observed that the eigen-

frequencies begin to split. The spheroid spectrum is compared to that of a

finite circular cylinder having the same ratio of length of axes. It is

demonstrated that the two spectra approach each other as the spheroid and

cylinder become longer, until they finally coalesce into the axially degenerate

resonances of the infinite cylinder.

8.5 Experimental and Numerical Results

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