Some Aspects of Ultrasonic Attenuation in Common Metals

S. TANAKA2

Synopsis

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As one of the means to measure the attenuation of ultrasonic wave in some common metals, a resonance method is discussed. The results obtained are

com-pared with that by the pulse method on which the writer reported previously.

It is shown that the resonance method is useful for determining the attenuation

constant, by theoretically deriving the relation between the quality factor of resonance

and the attenuation constant in wave transmission, on the basis of the

experi-mental results.

1. Introduction

In the previous study, the writer C1j measured the ultrasonic attenuation in

some common metals by the use of pulse techniques, in the frequency range from

500 Kc/s to 6 Mc/s. The attenuation constant a was measured from the

exponen-tial decay of the multiple-reflection patterns. Although this method has been

widely applied in such measurement, many factors should be taken into account

for the arrangements of the apparatus and the corrections of the measuring

values. And also it is often experienced that the observed patterns change consid-erably in figure depending on the condition of the measurement. Sufficient care,

therefore, should be taken for the accuracy of the measured values, when the

attenuation of a given specimen is to be measured by means of commonly used

ultrasonic flaw detector especially of one-probe type.

Now that the attenuation constants of metals stand in close relation with the

physical conditions of the materials, and its frequency characteristics are necessary data for the practical use of the ultrasonic flaw detector and the thickness detector,

it is desirable that an easy and practical method for measurement of such values

Rep. No. 90 (in the European language) of the Institute of High Speed Mechanics, Thoku University. Read at the 32nd General Meeting of the Japan Society of

Me-chanical Engineers, on April 8, 1955.

Professor of the Institute of High Speed Mechanics, Thoku University, Sendai, Japan.

150 Rep. Inst. High Sp. Mech., Japan, Vol. 9 (1958), No. 90

be devised. In prospect of the resonance method being of service in estimating the attenuation with comparative facility, the measurement by this method was subjected to further scrutiny. This resonance method is not for directly measuring

the attenuation constant, but it was presumed that the measured value is related with the quality factor (sharpness in resonance curve) of the specimen.

In this report, therefore, the writer tried to derive the correlation of the

loga-rithmic decrement a obtained by the pulse method to the presumable quality

factor Q acquired by application of the resonance method, and to establish the

correspondence of these two methods on the basis of the measured results. Though

the attenuation values obtained by the resonance method are relative values, it is possible to determine the values applicable to various materials from the above mentioned correspondence by taking the value of a suitable material obtained by the pulse method as a standard.

2. Results of Measurements by Pulse and Resonance Methods

Fig. i shows the block diagram of the apparatus used in the previous experi-ments on the pulse method. An ultrasonic pulse is sent out from one end of the

specimen by the Xcut quartz crystal resonating with carrier frequency. The

radiated pulse is subjected to repeated reflections at both ends of the specimen

Fig. 1. Block Diagram of the Equipment on the Pulse Method

and is picked up by the receiving crystal, which is separated from the sending

one. Induced voltage is shown as the standing pattern on the cathode-ray tube screen after having been amplified and detected. Fig. 2 shows an example of such

multiple-reflection pattern obtained with mild steel*. The measured values were

* Referring to the picture in Fig. 2, the uniform high part on the left-hand side is due to a saturation of the amplification caused by the intensity of the received signal, and the measurement of the ratio of amplitude has been effected within the scope of the right-hand half showing the logarithmic decrement in linear characteristics, as have been

previously verified. P.R.F. D1FF. RINGING qJ Osc AND AMP. TEST PIECE Ose. AMP.

SWEEP VIDEO VIDEO

DET AMP.

ER.

Fig. 2. An Example of Multiple-Echo Pattern Obtained with Mild Steel corrected by experimentally determining the errors due to the spherical divergence of the wave train and the loss of energy on account of reflection at the boundaries

of the specimen. Then the values shown in Fig. 3 were obtained as representing the attenuation characteristics versus frequency with the most probable precision. The results show that, generally speaking, the attenuation constant tends to in-crease in linear proportion with the rise of the frequency. The steep increase in

2

-J

O L

O I 2 3 4 5 6

FREQUENCY N MEGACYCLES PER SEC.

Fig. 3. Relation between Observed Attenuation Constant with Frequency in Various Metals

152 Rep. Inst. High Sp. Mech., Japan, Vol. 9 (1958), No. 90

a part of the curves given by copper and cast-iron is explained, according to

W. P. Mason and the others 2j, as being due to the added influence of the loss caused by scattering at the grain boundaries.

On the other hand, the resonance method proposed hereunder has been

devel-oped for the past several years by N. G. Branson, and also further studied by the

writer 3j for its use in measuring the thickness of a metal plate. In this method,

connecting the basic circuit as shown in Fig. 4, an X-cut quartz crystal is placed

upon the surface of a metal plate, and excited by a variable frequency

self-oscillator, and when the frequency comes to coincide with that of the thickness

resonance

of the metal plate, the

z plate current of the oscillator tube

is abruptly increased. On the

as-sumption that the degree of increase

As experimental materials, discs

of 5 cm in diameter and 3 cm in

thickness were prepared of eight

kinds of different metals, namely,

aluminium, two kinds of steel with unequal carbon contents, copper, duralumin,

cast-iron, brass and gun-metal, and the surface finished to the same degree of

precision with the view to eliminating the possible influence of the difference in the shape of the specimens upon the obtained values of Q. The quartz crystal used

had a natural frequency of 3 Mc and the variable range of driving frequency was selected from 1.5 to 2.5 Mc/s. The change of the plate current at the frequency of thickness resonance was rendered audible by frequency modulation, and the output voltage obtained through the amplifier was measured by a valve-voltmeter connected in parallel with the receiver. Fig. 5 illustrates an example of the measured values thus obtained, in which the output voltages at the resonance

frequency for the specimens of various materials are connected with straight lines.

Besides, fur the purpose of ascertaining how these results are related with the

attenuation constant a in wave transmission described above, the attenuation

constants for frequency of 2.1 Mc/s in the same materials were measured by means

of the pulse method. The results are shown in 'fable 1 in correlation with the

values of the output voltage V as measured by the resonance method. It is easy

to see that the two sets of values obtained by the two methods stand in some

TEST

PIECE

QUARTZ

Fig. 4. Basic Circuit Diagram of the

w w -j o > a-o u-o w Q z

I

FREQUENCY IN MEGACYCLES PER SEC.

Fig. 5. Change of Output Voltage at the Resonance Frequency in Various Metals

correlation. In brass and gun-metal, the attenuation was too great to admit detection by the resonance method, and reflection from the bottom plane was

observable when the pulse method was applied.

Table 1. Correlation of Observed Attenuation Values obtained at 2.1 Mc/s by the Pulse and the Resonance Methods

3. The Relation between the Attenuation Constant and the Quality Factor

In general, the equation of the longitudinal vibration of a slender rod with

internal damping may be obtained using the complex elastic modulus E, as follows

C4J:

Material AttenuationConstant

a(db/cm) Resonance Indication V (volt) Quality Factor Q Remarks

Aluminum 0.08 1.70 1,150 Cold worked

Steel (hard) 0.12 1.15 820 Tool steels

Copper 0.17 0.70 730 nW011( fine

Steel (soft) 0.20 0.40 490 C = 0.27%

Duralumin 0.45 0.20 _{210 Cast}

Cast iron 0.55 0.20 220 Gray iron

Brass -.. Cast

154 Rep. Inst. High Sp. Mech., Japan, Vol. 9 (1958), No. 90

c=,/--at-

ax-E= Ei (cose + j

E=E1+jE2, jTTj,

(1)

where E1 shows the ordinary elastic modulus, and E3 may be tentatively assumed

to be a constant, which is to be subjected to experimental verification. Now, let

O stand for the argument of E, and we have

E1 i

E1tanO

When a sinusoidal force Eit is applied at one end of the rod, the transmission

of the wave in x - direction may be expressed as a solution of equation (1 ) as

follows: e A ---7--X f O\ I

11

w --- sin-- )x

'

AE* )tWX

)

(2)

Then the attenuation constant a is obtained as

wO

a =

_c2

c

it is shown that it rises in proportion with the increaseof frequency. The results

are as shown in Fig. 3, and indicate that when the loss due to scattering is left out of consideration, the value of E2 may be taken to express the material constant. Now, as stated above, AC-pulse modulated by carrier frequency were used in the

experiments, but the solution in respect of a stationary state obtained above may

be assumed to be approximately accurate.

Next, we will proceed to calculate the equivalent lumped constant of a metal

specimen in a state of thickness resonance. In general, it is usual to separateeach

constant from the solution for forced vibration, but since in this case the boundary conditions of the longitudinal vibration of the rod is so simple, that one end is driven and the other free, the solution may be immediately obtained in the fol-lowing way. The driving point impedance for unit area of a specimen with the length i is given as follows,

= pòtanh (jal),

Frum this, in the well known impedance formula

f s

Iwm--w

the equivalent lumped constant ni and . satisfied in the vicinity of the frequency

of n-th resonance may be caluculated in approximation. As the n-th complex

resonance frequency j is in the relation of j?=nc/2l, taking w in the vicinity of

th,, in th =-n(-7 ), assuming E1 and substituting w = th(1±E), we get

/ . w th,l . th1

'

thl

tanh( i-t-1f = jtan {____(1+E)} =j tan(nir ± E

Since, however,

i

= w-

W=-(

2\

Then, the lumped constant for unit area is given by

i 7t2p

n',r-m = --pl, s -= 'z

21

- 21 (Ej±jE).

From this, the equivalent stiffness s, and the equivalent resistance r,, are given as follows:

nr-

s = ----E,,

jco = 21 (J)E =

The equivalent resistance r,, is given as a function of the frequency, but if the value ro at the resonance frequency of the n-th order is considered approximately

instead of r,,, the sharpness of resonance Q becomes co

(4)

156 Rep. inst. High Sp. Mech., Japan, Vol. 9 (1958), No. 90

It is a material constant independent of the frequency. This tendency is shown

in the example of the measured values -obtained by the resonance method illus-trated in Fig. 5. That is to say, the out-put resonance voltage V is in approxi-mately constant value with the order of resonance. The slight rise observed in the

range of higher frequency is due to increase sensitivity of detection, as the variable capacitance in the resonance circuit becomes reduced.

From ( 3 ) and ( 7), the relation

Wi

Wi

a

_c2Q

_{c 20}

is deduced. This correlation of a and Q will he verified by measured results in

- the following.

4. Examination of Measured Results

'rable i shows the attenuation constant a obtained from transmission of pulse

train, and the out-put voltage V obtained by the resonance method, of the common

metals used in the experiments. Besides the equivalent values of Q converted

from a dy the equation (9) above are also given in correlation, for illustrating

the mutual correspondence of the values.

Table 2. Correspondence of Attenuation Constant a

with Quality Factor Q

(9)

Material Velocity_(cm/s) Frequency_(Mc/s) Att. Coust._-r(db/cm) Quality Factor_Q

Aluminum 6220 3.1 0.06 2,300 4.0 0.09 2,000 4.9 0.12 1,800 Mild steel 5870 2.0 0.08 1,200 (C=O.35%) 3.0 0.12 1,200 3.8 0.13 1,300 5.3 0.20 1,200 Mild steel 5870 3.1 0.34 420 (C=0.12%) 3.9 0.45 400 5.3 0.60 410 Copper 4620 0.85 0.14 350 1.0 0.18 330 2.0 0.50 240 Cast iron 4810 0.60 0.16 210 1.0 0.24 230 Q= E1 i

(8)

For the sake of reference, the attenuation constant a obtained in some common

metals in Fig. 3 are converted into their equivalent values of Q in Table 2. For

copper and cast-iron, the calculation was undertaken only for the lower frequency

region where no scattering loss is occasioned, and yet, in this range also, the values of Q were about only one-tenth of that of aluminium in which the atten-uation is far lower.

5. Conclusions

The purport of this paper has been to propose a reference materialon the use of the ultrasonic flaw detector or the thickness detector, and the range of frequency from 0.5 to 6 Mc/s used in experiment is most generally used for these instruments. The writer could ascertain that, within this frequency range, the attenuation constant

a of mild steel and aluminium increases approximately in linear relation with the

frequency, and the writer has tried to provided a theoretical foundation to the

process of determining the penetrating degree into such materials of the ultrasonic

waves utilizing this property. From this point of view, if we admit that, within

this range of frequency, a is linear with the frequency, the argument Q of the

complex Young's modulus E is represented by a material constnat independent of

the frequency, and from this, the Q-value of the material within this range of

frequency is given as the reciprocal of 9. In general, of course O is a function

the frequency, but for using an ultrasonic flaw detector, it is imperative to be

informed of the attenuation degree of the tested material, and it is also doubtless that the material has a certain Q-value.

Next, it has been known for a long time that standing wave techniques may be used for Q-measurernent, but the writer points out that it is difficult and im-practical to make measurement by the standing wave techniques in current use when the materials are of very high Q-values, such as aluminium or mild steel. In consideration of this defect, the writer has adopted the method of measuring the relative Q-value using a detection method by means of frequency modulation

developed as an ultrasonic thickness detector. The writer has set forth a proposition

that the Q-values above may be easily and practically estimated by this method.

Acknowledgment

The writer wishes to express his appreciation to Prof. F. Numachi, Director of the Research Institute, for his helpful criticism and suggestions.

158 Rep. Inst. High Sp. Mech., Japan, Vol. 9 (1958), No. 90

Bibliography

S. Tanaka and T. Anzai, Measurement of Ultrasonic Atteunation in

Metals by the Pulse Method, Sci. Rep. Res. Inst. Tôholcu Univ. A-4

(1952), p. 643.

W. P. Mason and H J. McSkirnin, Energy Losses of Sound Waves in Metals Due to Scattering and Diffusion, Jour. App. Phys. 19 (1948), p. 940. N G. Branson, Portable Ultrasonic Thickness Gage, Electronics, (1948),

p. 88.

S. Tanaka, Measurement of the Thickness of Metal Plate by Ultrasonic

Harmonic Method, I and II, Sci. Rep. Res. Inst. Tohoku Univ. A-2 (1950),

p. 917 and A-3 (1951), p. 201.

(4 J N O. Mykiestad, The Concept of Complex Daniping, Jour. App. Mech.,