Russian Journal of Nondestructive Testing, Vol. 41, No. 9, 2005, pp. 561–566. Translated from Defektoskopiya, Vol. 41, No. 9, 2005, pp. 10–18. Original Russian Text Copyright © 2005 by Danilov.
ACOUSTIC METHODS
Effect of the Size of a Flat-Bottom Hole in a Cylindrical Specimen on the Bottom-Signal Level V. N. Danilov Central Scientific and Research Institute for Machine-Building Technology, Sharikopodshipnikovskaya ul. 4, Moscow, 115088 Russia Received May 27, 2005
Abstract—The effect of a change in the size of a flat-bottom hole on the level of a bottom signal measured with a normal probe at an end surface of a cylindrical specimen is considered. It is shown that this effect depends on the specimen’s radius-to-length ratio. It is established that the difference in the levels of a bottom signal and a signal from a hole in the specimen is in good agreement (within 0.5 dB) with the corresponding difference obtained from the calculated DGS diagram.
During ultrasonic inspection of forged pieces, rods, etc., with normal probes aimed at the tuning of the sensitivity of flaw detectors and evaluation of the equivalent area of flaws, a common practice is to use cylindrical calibration blocks with artificial reflectors in the form of flat-bottom holes, the planes of which are oriented perpendicularly to the ultrasonic-beam propagation [1]. Such reflectors are manufactured in the form of holes with flat profiles at different depths, axially symmetric with respect to the specimens. Reliable time selection of signals from flat-bottom reflectors against the background of a bottom signal from the specimen’s surface, which is opposite to the surface of emission, and reception of a longitudinal wave can be achieved at typical distances from the reflectors to the former surface (the depth of the hole) of 15–20 mm. An example of specimens with such reflectors is a KCO-2 kit of calibration blocks (GOST (State Standard) 21397–81) used in ultrasonic testing of intermediate products and articles from aluminum alloys [2]. Analogous specimens are also produced from steel with allowance for the requirements of the above standard [1]. Specimens undergo primary and periodic certification (upon production and during service, respectively) to check the correspondence of their parameters to the required standards. In order to ensure the uniformity of measurements in the determination of the echo-signal amplitudes from the specimen’s flatbottom reflectors, it is necessary to choose a reference signal, the amplitude of which must be compared to the signal amplitudes from the reflectors. For a KCO-2 kit, it was proposed to use an echo signal from a steel ball in an immersion bath as the reference signal, which is used to adjust the bath’s elements and calibrate the flawdetector’s sensitivity [3]. The normalized signal levels obtained are compared to the rated values, the tolerable deviations from which are ±2 dB at most. The necessity of performing rather laborious measurements using a special-purpose setup makes the monitoring of signal levels from the reflectors impracticable not only during operation but also during periodic certification of specimens by metrological agencies. One of the methods for substantially simplifying the determination of the correspondence of signals from flat-bottom reflectors to the rated values is their comparison with the levels of the bottom signals in the same specimens, which are measured at such a position of a normal probe in which the maximum signal from a given reflector is achieved. An advantage of this procedure for choosing the reference signal is that it eliminates the effect of contact conditions between the probe and the specimen’s surface, since both signal levels are measured with the probe staying in the same position. However, it should be noted that the roughness of the surface and the flatness and parallelism of the specimens’ end planes must remain within the limits allowed by the requirements that are imposed on the specimens of the appropriate kit [3]. Figure 1 shows (1) a pulse signal (in arbitrary units) from a flat-bottom reflector with an area of 20.4 mm2 at a distance of hfb = 185 mm and (2) a bottom signal at h = 200 mm (Fig. 2) for a steel specimen of radius 30 mm obtained with use of a è111-2.5-ä12-002 standard normal probe and observed on the screen of the Avgur computer system’s monitor. The amplitude of the pulse from the flat-bottom reflector is 22.2 dB lower than the bottom-signal amplitude (in the positive peaks). Figure 3 shows the spectrum of the bottom signal (also in arbitrary units) computed by the program of the Avgur system. The probe’s operating frequency determined from the spectral peak in Fig. 3 is 2.18 MHz, and from two oscillation periods with the highest positive amplitudes in the pulse (as recommended in [4]), the frequency is somewhat higher— 2.25 MHz. Measuring the signal level from a flat-bottom reflector of 10.2-mm2 area in a specimen with the 1061-8309/05/4109-0561 © 2005 MAIK “Nauka /Interperiodica”
562
DANILOV Echo signal
Pr 2a
b
P
2
1 θ1 hfb
h
R(b) 64
66
68
70
t, µs
72
Fig. 1. Experimental pulses of (1) a signal from a flatbottom hole and (2) a bottom signal in a steel cylindrical specimen detected by a è111-2.5-ä12-002 probe.
2bfb
same dimensions has shown that it is 5.8 ± 0.3 dB lower than that from a reflector of 20.4-mm2 area.
Fig. 2. Schematic of the system for detecting a bottom signal and a signal from a flat-bottom hole in a cylindrical specimen by a normal probe.
When a bottom signal in a cylindrical specimen (CS) with a flat-bottom hole is used as the reference, a question arises as to how the size of the hole affects this signal’s level. Indeed, in the presence of such a hole, the central part of the specimen’s end surface with linear dimensions comparable to the length of a longitudinal wave does not participate in the formation of a bottom-signal pulse (within the first Fresnel zone [5]); this must obviously reduce its amplitude. A simplified estimate of the effect of the size of a flatbottom hole on the bottom-signal level (in a quasi-monochromatic mode) can be obtained using the data from [6] (see formula (14)). The estimate of the normalized pressure on a normal probe of radius a (which produces pressure P directed normally to the center of the CS’s end surface) exerted by longitudinal waves reflected from an axially symmetric surface that is confined by radii bfb of the flat-bottom reflector and b of the CS can be represented in the form b
δP pp
1 = – --- r 1 dr 1 2
∫
b fb
∞
∫
2 (1) J 1 ( ξa )H 0 ( ξr 1 ) exp ( iνh ) dξ
(1)
.
–∞
In (1), δPpp is the normalized pressure of the bottom signal from the CS’s end surface (see Fig. 2) with allowance for a decrease in the reflecting surface in the presence of a flat-bottom hole, ν =
2
2
k l – ξ , kl is
(1)
the wave number of the longitudinal wave, and H 0 is a zero-order Hankel function of the first kind. Performing asymptotic estimates in formula (1) similar to those in [6], we obtain the expression 2 b fb ⎧ h 2 - exp [ i2k l R ( b fb ) ] δP pp ≅ i ⎨ -------------------------- J 1 k l a -------------R ( b fb ) ⎩ k l b fb R ( b fb ) 2 ⎫ h b 2 – --------------------J 1 k l a ------------ exp [ i2k l R ( b ) ] ⎬; 2 R ( b ) klb R(b ) ⎭
R ( b fb ) =
2
2
b fb + h ,
R(b) =
2
(2)
2
b +h .
RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
Vol. 41
No. 9
2005
EFFECT OF THE SIZE OF A FLAT-BOTTOM HOLE
563
2
At bfb
2 ⎫ ⎧ kla h b 2 - exp(i2kl h) – --------------------δPpc ≅ i ⎨ -------J 1 k l a ------------ exp[i2kl R(b)] ⎬ 2 R ( b ) 4h klb R( b ) ⎭ ⎩
h, and δPpp
0, R(bfb)
(see formula (16) in [6]). The first term in curly brackets in (2) describes in a scalar approximation the wave pressure of the bottom signal upon its reflection from an infinite flat boundary with a disk-shaped cut of radius bfb at the center; the second term characterizes the effect of the limited area of reflection from the side surface on the bottomsignal amplitude [6]. Normalizing the modulus of expression (2) to the amplitude of a bottom signal from 2
kla - , and considering the coefficients of reflection of the longitudinal wave an infinite plane [1], |δPbot | = -------4h from an annular edge of the CS’s end surface in an elastic medium, as was done in [6], we obtain the following relationship that allows an estimate of the effect of the size of a flat-bottom hole on the level of the bottom signal from the CS’s end, which also depends on the ratio b/h: 3 b fb 4h 2 δP pp ≅ ⎧ -------------------------------- exp [ i2k l R ( b fb ) ] J 1 k l a -------------⎨ 2 2 2 -----------R ( b fb ) δP bot ⎩ k l b fb a R ( b fb )
(3)
3
4h V ll ( θ 1 )V ll ( π/2 – θ 1 )χ ( θ 1 ) 2 ⎫ b - exp [ i2k l R ( b ) ] ⎬ . – -----------------------------------------------------------------------J 1 k l a ----------2 2 2 R(b) kl b a R( b ) ⎭ In formula (3), θ1 = arcsin[b/R(b)], χ(θ1) is a factor taking into account the longitudinal-to-transversewave conversion upon acoustic emission [1], and Vll is the coefficient of reflectivity for plane longitudinal waves from a free surface [7]. At bfb 0 and R(bfb) h, formula (3) transforms into the expression
2
4 Echo-signal spectrum
6
f, MHz
Fig. 3. Spectrum of an experimental bottom-signal pulse. RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
Vol. 41
No. 9
2005
564
DANILOV
δP pp ----------δP bot
3
4h V ll ( θ 1 )V ll ( π/2 – θ 1 )χ ( θ 1 ) δP pc ≅ exp ( i2k h ) – --------------------------------------------------------------------------------l 2 2 2 δP bot kl b a R( b ) ----- ×
2 J1
(4)
b k l a ------------ exp [ i2k l R ( b ) ] , R(b)
which was obtained earlier in [6] (see formula (17)). δP pc calculated from formula (4) for a Figure 4 shows the modulus of the normalized bottom signal ----------δP bot 0 (in the absence of a flat-bottom hole) as a funcwave reflected from the cylinder’s end surface for bfb tion of the specimen’s radius b (in mm) for a è111-2.5-ä12-002 transducer with parameter kla ≅ 16 (at the rated frequency) and a cylinder of length h = 200 mm. It follows from Fig. 4 that, for a radius of b = 30 mm δP pc ≅ –0.52 dB; i.e., the bottom-signal amplitude (for a quasi-monochromatic of the tested cylinder, ----------δP bot transmission and reception mode) in a CS is slightly lower than that obtained upon reflection from an infinite δP pc plane lying at a distance of h = 200 mm from the probe. For a specimen of radius b = 50 mm, the ----------δP bot value does not exceed 0.01 dB, which is quite low. δP pp calculated from formula (3) for a Figure 5 shows the modulus of the normalized bottom signal ----------δP bot wave reflected from the cylinder’s end with allowance for the influence of a flat-bottom hole as a function of hole radius bfb (from 0.25 to 10 mm) for the same probe. Curves 1 and 2 are plotted for specimens of radii δP pp for the CS with the b = 30 and 50 mm, respectively. At small radii bfb, the normalized bottom signal ----------δP bot smaller radius is lower than that for the CS with b = 50 mm. This follows from the ratio of bottom-signal |δPpc /δPbot|, dB
|δPpp /δPbot|, dB
2.21
–0.24
1.32
–0.48
0.44
–0.72
–0.45
–0.96
–1.34
–1.20
–2.23 10.0
23.3
36.7
b, mm
δP pc Fig. 4. Modulus of normalized bottom signal ------------ as δP bot a function of radius b of a cylindrical article.
–1.44 0.25
2
1
3.50
6.75
bfb, mm
δP pp Fig. 5. Modulus of normalized bottom signal ------------ as δP bot a function of radius bfb of a flat-bottom hole (reflector) in a cylindrical article.
RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
Vol. 41
No. 9
2005
EFFECT OF THE SIZE OF A FLAT-BOTTOM HOLE
565
dB DGS diagram δP pc in Fig. 4 for radii of 30 and amplitudes ----------H1 = 25 mm δP bot A1 = 5.3 dB 7 50 mm. When bfb values are sufficiently high H2 = 50 mm A2 = 1.3 dB 1 δP pp value for (above approximately 8 mm), the ----------–5 δP bot Bottom signal the CS with b = 30 mm exceeds the bottom-signal –11 value for the CS with b = 50 mm. Estimates show –17 that, for steel with a longitudinal-wave velocity of 5.92 mm/µs, a specimen of radius b = 50 mm, an –23 operating frequency of 2.5 MHz, and a probe of –29 s = 20.4 mm2 radius a = 6 mm, almost the total value of normalδP pp is determined by the first –35 ized bottom signal ----------–41 δP bot s = 10.2 mm2 term in formula (3) (the contribution of the second –47 0 40 80 120 160 mm term to the result is very low owing to the directivè111-2.5-ä12-002 (no. 96) probe ity of the probe). However, accounting for the Diameter of the piesoplate, 12 mm effect of the wave reflection from the corner Frequency, 2.2 MHz between the side cylindrical surface and the end of Reflector is a flat-bottom hole the specimen with b = 30 mm, which is characterized by the second term in (3), leads to an increased Fig. 6. DGS diagram for flat-bottom reflectors with areas δP pp value. In fact, at b = 10 mm, b = 30 mm, of 20.4 and 10.2 mm2 constructed following the DGS uni-----------fb versal system. δP bot and h = 200 mm, the path difference for edge rays ( b – b fb ) ( b + b fb ) - = 4 mm, which is ~1.7 wavelengths of the longitudinal wave. in (3) is 2(R(b) – R(bfb)) ---------------------------------------h Therefore, the real part of the second term in (3) turns out to be positive, thus ensuring the increase in norδP pp mentioned above. malized bottom signal ----------δP bot
Hence, the effect of a flat-bottom hole on the bottom-signal level is determined not only by the relative size of the hole (i.e., by the hole-to specimen-radius ratio, bfb/b) but also by the relative size of the specimen itself, b/h. It follows from Fig. 5 that the bottom signal decreases owing to the presence of a flat-bottom reflector of 20.4-mm2 area (i.e., of ~2.5-mm radius) by only 0.06 dB at b = 30 mm and by 0.09 dB at b = 50 mm; i.e., this value is small in both cases (for a reflector of 10.2-mm2 area, or 1.8-mm radius, it is even smaller). Therefore, the difference in the levels of the bottom signal and the signal from the aforementioned flat-bottom reflector must actually correspond to that determined from the DGS diagram using the bottom signal as the reference one with allowance for a possible deviation of the signal from the reflector from the rated value (within ±2 dB). The maximum attenuation of the bottom signal resulting from the effect of the flatbottom reflector at bfb = 10 mm and b = 50 mm is 1.44 dB. Figure 6 shows a DGS diagram plotted following the DGS-universal system [8, 9] for flat-bottom reflectors with areas of 20.4 and 10.2 mm2 in steel at depths of up to 210 mm for damping of 0.0004 Np/mm. A value of 0 dB in this diagram corresponds to the bottom-signal level from a depth of 59 mm for a CO-2 calibration block. The diagram was calculated for the parameters of a è111-2.5-ä12-002 normal probe (nos. 96 and 88) used in our experiments (the operating frequency estimated above is 2.2 MHz, and the radius is 6 mm). The levels of the bottom signal (for a 200-mm depth) and signals from flat-bottom reflectors (for a 185-mm depth) are marked with open circles on the corresponding curves. It follows from the DGS diagram in Fig. 6 that the level of the bottom signal from a 200-mm depth exceeds the signal level from the flat-bottom reflector of 20.4-mm2 area by ~22.5 dB (22.2 dB in the experiment); the signal from the reflector with an area of 10.2 mm2 is ~5.5 dB lower than that from the flat-bottom reflector with an area of 20.4 mm2 (the value obtained experimentally for two specimens is 5.8 ± 0.3 dB). As was mentioned above, a deviation of signal values from those produced by nominal flat-bottom reflectors of at most ±2 dB is in principle allowable. Therefore, the agreement between the theoretical and experimental data is good (the discrepancy RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
Vol. 41
No. 9
2005
566
DANILOV
is no more than 0.5 dB) and confirms that, for the CSs with flat-bottom reflectors (holes) used in the experiments, the effect of their size on the bottom-signal level is insignificant and can be ignored. CONCLUSIONS (1) The effect of a flat-bottom hole on the bottom-signal level in a cylindrical specimen is determined by both the hole-to-specimen-radius ratio and the specimen radius-to-length ratio. In the case of a flat-bottom reflector with an area of 20.4 mm2, a decrease in the bottom signal is insignificant and can be ignored. (2) The reflectivity of a flat-bottom hole (reflector) in a cylindrical specimen can be evaluated by comparing the difference in the levels of the bottom signal and the signal from the hole with the corresponding difference from the calculated DGS diagram constructed for the parameters of the probe used in the measurements. REFERENCES 1. Ermolov, I.N. and Lange, Yu.V., Nerazrushayshchii kontrol’. Spravochnik v 7-mi tomakh (Nondestructive Testing. Handbook in 7 Volumes.), Klyuev, V.V., Ed., vol. 3: Ultrasonic Testing, Moscow: Mashinostroenie, 2004. 2. GOST (State Standard) 21397-81: Nondestructive Testing. Set of Calibration Blocks for Ultrasonic Testing of Semi-Finished and Aluminum Alloy Products. Specifications, 1982. 3. Kruglov, L.D. and Grishina, V.M., A Set of Standard Specimens KSO-2 and its Measurement Assurance, Defektoskopiya, 1984, no. 5, pp. 23–27. 4. Danilov, V.N., Ermolov, I.N, and Shcherbakov, A.A., Determination of the Vibration Frequency in Nondestructive Testing, Defektoskopiya, 2003, no. 3, pp. 3–11 [Rus. J. of Nondestructive Testing (Engl. Transl.), 2003, vol. 39, no. 3, p. 171]. 5. Yavorskii, B.M. and Detlaf, A.A., Spravochnik po fizike dlya inzhenerov i studentov vuzov (Handbook on Physics for Engineers and Students of Colleges), Moscow: Nauka, 1968. 6. Danilov, V.N., Estimating the Parameters of Signals Observed during Ultrasonic Testing of a Cylindrical Article with a Normal Probe on the End Surface, Defektoskopiya, 2005, no. 2, pp. 55–71 [Rus. J. of Nondestructive Testing (Engl. Transl.), 2005, vol. 41, no. 2, p. 102]. 7. Brekhovskikh, L.M., Volny v sloistykh sredakh (Waves in Layered Media), Moscow: Nauka, 1973. 8. Danilov, V.N. and Voronkov, V.A., On the Calculation of DGS Diagrams, V Mire Nerazrush. Kontrol., 2001, no. 2, pp. 20–22. 9. Voronkov, V.A. and Danilov, V.N., Sensitivity Calibration of Ultrasonic Detectors Based on DGS Diagrams, Defektoskopiya, 2001, no. 1, pp. 56–60 [Rus. J. of Nondestructive Testing (Engl. Transl.), 2001, vol. 37, no. 1, p. 44].
RUSSIAN JOURNAL OF NONDESTRUCTIVE TESTING
Vol. 41
No. 9
2005
