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Loose-part signal properties
Progress in Nuclear Energy, Vol. 28, No. 4, pp. 347-357, 1994 Copyright (c; 1994 Elsevier Science Ltd Primed in Great Britain. All rights reserved 0149-1970/94 $26.00
Pergamon 0149-1970(94)00006-9
LOOSE-PART SIGNAL PROPERTIES C. w. MAYO Departmentof NuclearEngineering,North CarolinaState University, Box 7909Raleigh,NC 27695-7909,U.S.A.
Abstract - Lamb's solution for two-dimensional wave propagation was used to identify plate bending wave characteristics observed in loose part impact signals. The surface displacement wave is shown to have large second derivatives at points that can be used to measure the impact contact time. General features of the initial impact wave shape provide guidance for the use of spectral analysis techniques to identify the contact time and wave arrival time when there is a low signal-to-noise ratio. Improvements in the determination of contact time improve the estimation of loose part mass and energy through the Hertz impact parameters. The shape of the initial impact wave shape can also be used to identify on which side of the plate the impact took place. This information has applications for Loose Part Monitoring System metal impact signal validation and for improved estimation of loose part parameters.
INTRODUCTION Metal impact signal theory for reactor loose parts monitoring has been developed through prior studies (Mayo, 1985, Mayo et al., 1988). The Hertz theory of impact was shown to be applicable for impact response calibration and for the interpretation of impacts against plate type pressure boundary components. This prior work identified the initial impact wave acceleration wave-form experimentally and demonstrated bending wave transmission effects, mechanical resonance response, and multiple path transmission effects in accordance with theory. The results of this work provided a quantitative basis for modeling Loose Part Monitoring System (LPMS) performance, and guidelines for loose part monitoring system specifications, metal impact response calibration, and diagnostic analysis of unknown metal impact signals. The nature of the initial loose part impact wave has not been investigated due in part to the nature of the solution required to include the shear force and rotary inertia for impacts against a structure of finite thickness (Miklowitz, 1953) and the requirement to include the cylindrical nature of the wave propagation for impact against a plate. As an approximation, Lamb's solution for wave propagation on a membrane (Lamb, 1902) provides a two-dimensional, radially symmetric solution for an arbitrary force. The constant wave speed in the membrane solution is equivalent to the effectively constant bending wave velocity associated with the impact of smaller mass loose parts. Lamb's two-dimensional wave demonstrates dominant features of the metal impact wave shape and provides a basis for relating plate bending wave shape features to the initial impact properties. The
347
348
C.W. MAYO
general nature of the impact wave shape can also be used to improve the performance of frequency spectrum analysis techniques for identifying the impact contact time and wave arrival time when there is a low signal-to-noise ratio. These results have application for improved loose part diagnostic analysis.
HERTZ IMPACT THEORY Hertz theory for the impact of a sphere against a plate defines the impact contact time as a function of the impacting object mass, velocity, and contact radius (Harris and Crede, 1961). The impact contact time was associated with the frequency content of the wave that radiates away from the impact point by Raman, who reported that the initial half-period of the plate wave is a factor of 1.6 shorter than the contact time (Raman, 1920). Raman's relationship led to the concept of the primary impact wave frequency content being def'med by the equation
f*-
1.6 2t^
(1)
where fa is the sinusoidal frequency in Hz associated with the initial half-period of the wave t~ is the impact contact time in seconds and the Hertz theory impact contact time can be expressed as
(2)
ts = n kh m'4 V~.2R-.2 where k h is 1.82 x 10.4 ft'81bf'4 for the impact of steel on steel m is the mass of the impacting object in 1bf/32.2 ft/sec 2 Vo is the initial impact velocity R is the contact radius of curvature in ft
These equations have been used to map the general range of loose part signal frequency content as a function of mass and energy as shown in Figure 1. In this example, the frequency associated with the initial half-period is plotted for impact velocities between 1 and 10 ft/sec, for seven different metal spheres having diameters between six inches (32.1 lbs) and 0.5 inches (0.0186 lbs). The Figure 1 values for sphere diameter and mass are given in Table 1. TABLE
- Metal Sphere Diameter and Mass
Diameter
6"
5"
4"
3"
2"
1"
0.5"
Mass (lb)
32.1
18.6
9.5
4.0
1.19
0.15
0.018
The independent parameters of mass and velocity can not be identified uniquely from the single parameter of contact time or the corresponding impact wave frequency. However, as indicated by Figure 1, the initial impact wave frequency can provide useful mass estimates without knowledge of the impact velocity.
Loose-part signal properties
2
349
x 104
1.5 Sphere Diameter - Inches
N
"1i
'-
1"
1
LL 2"
0.5
j
~
i
l o ~'
. . . . . . .
i
. . . . . . . .
10.3
t
. . . . . . . .
10 .2
f
4
i
10"1
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. . . . . . . .
i
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5.6,
.
.
.
.
.
.
.
.
i
101
.
.
.
.
.
.
.
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Impact Energy - Ft-Lbf
Figure 1. Primary Frequency Content for Metal Sphere Impacts The uncertainty in mass estimates based on frequency alone can be characterized by solving equations 1 and 2 for the range in mass that can give a particular primary impact wave frequency over a range of potential impact velocities. These values are shown in Figure 2 for metal spheres with velocities between 1 and 10 ft/sec. The uncertainty in mass is about a factor of four between the minimum and maximum velocity. Still, the range of loose part mass covers about three orders of magnitude, and useful information about the mass of unknown loose parts can be obtained without knowledge of velocity. With the initial mass estimate, velocity and energy can then be estimated through the signal amplitude, and the results refined. Hertz theory has been shown to be remarkably applicable for modeling and interpreting reactor loose part monitoring system signals. The initial assumption of a metal sphere can be relaxed to an object with a finite contact radius and a general shape that behaves as a rigid body during the impact. Both metal spheres and force instrumented metal hammers serve well for metal impact response calibration as a function of known impact parameters. Experimental measurements show that local deformation does not greatly effect the impact parameters over the general range of mass and velocity for reactor loose parts (Mayo et al. 1988).
THE TWO-DIMENSION LAMB WAVE In the early study of impacts against two-dimensional surfaces, Lamb derived a solution for the shape of a two dimensional wave on a membrane as a function of an arbitrary impact force (Lamb, 1902). While this solution does not include shear and rotary inertia effects associated with a plate of finite thickness, it was used in Raman's successful calculation of the relationship between impact contact time and period of the first half-cycle of a plate bending wave using Hertz impact theory (Raman, 1920). This relationship has been demonstrated to be applicable for the loose part impact signals (Mayo et al., 1988).
350
C.W. MAYO
102
10 ~
_1
~
10 °
10"*
Velocity=1 ft/sec
10 .2
~
i 1
01.5
o
Figure 2.
~
11.5
i 2
Frequency - Hz
2.5 x 104
Mass and Initial Wave Frequency Range for Impact Velocities Between 1 10 Ft/sec
Lamb's general solution for an arbitrary impact force function is: u
D(0
2n
e
(3)
o
where: f(t) is an arbitrary force function such that the integral converges r is the distance from the impact point at which the wave is observed c is the constant speed of wave transmission For a force that acts for only a finite time, the limits of integration are effectively restricted to limits for which the integrand is non-zero. Equation (3) can be integrated numerically for the half-sine force function associated with a Hertz impact. The Hertz impact for a metal object weighing a few ounces has a contact time of about 1 x 10.4 seconds and an approximately constant speed of wave transmission due to the shape of the bending wave velocity function for typical reactor pressure boundary thickness (Mayo et al., 1988). Using a half-sine force function with a duration of 1 x 10.4 sec, the corresponding Lamb wave is shown in Figure 3. This wave was calculated with a time step size of 1 x 10-5 seconds. Some of the features of this wave are: The response is zero prior to the time of initial arrival r/c, set here at 1 x 10.4 seconds The amplitude passes through a maximum prior to completion of the impact contact at 2 x 10.4 seconds The wave decays relatively slowly after the completion of the impact contact
Loose-part signal properties
351
0.6 I 0.5 i ~0.4
E J .~0.3 a
0.2
0.1
o
o12
oi,
o'.6
o'.8
;
Time - Seconds
,.2 x 10 "a
Figure 3. Lamb's Surface Wave The derivatives of the half-sine function are discontinuous its beginning and end. The derivatives of a Lamb wave generated by this function are similarly discontinuous at the times that correspond to the beginning and end of contact. These discontinuities are clearly evident in the surface displacement velocity and acceleration functions. The Figure 4 surface acceleration obtained by differentiation of the Figure 3 wave shows these discontinuities at the beginning and end of the impact contact.
0.06
0.04
i
0,02
Q
-0.02
-0.04 0.2
oi,
o16
i
0.8
i
,
Time - Seconds
Figure 4. Lamb Wave Surface Acceleration
,2 x 10 .3
352
C.W. MAYO
The discontinuities in the acceleration function provide distinctive wave shape features that can be used to measure the impact contact time. The fact that the signal is not symmetric shows that it is possible to identify the side of the membrane on which the impact took place by the sign of the derivatives as they approach the discontinuities. The dominant near-sinusoidal signal component is contained in slightly more than one-half cycle that lies between the beginning and end of the impact contact. Defining the half-period as the time that this part of the Figure 4 signal is less than zero gives Raman's relationship between impact contact time and the wave half-period. The short duration of this approximately sinusoidal component and the non-sinusoidal nature of the acceleration signal in general lead to the expectation that Fourier spectrum analysis can give relatively poor resolution of the frequency content associated with the impact contact time.
IMPACT WAVE SHAPE The plate impact bending wave can be observed experimentally. For loose parts monitoring systems, some filtering is usually required in order to remove the shock response of the sensor. A typical broadband loose parts monitoring system channel response to the impact of a 1 inch diameter metal sphere is shown in Figure 5. This mass was chosen since the primary frequency content is in a range where plate bending wave transmission is non dispersive for reactor structures (Mayo et al., 1988) and therefore analogous to the constant speed of propagation for Lamb's two-dimensional wave on a membrane. The sensor mounted resonance frequency is about 15 kHz in this example and can be largely removed by a eight-pole Butterworth low pass filter set at 10 kHz as shown in Figure 6. The initial impact wave-form is shown on an expanded scale in Figure 7. The initial impact wave in Figure 7 has distinct similarities to Lamb's wave solution. The comparison is improved by passing the Lamb wave acceleration from Figure 4 through the same 10 kHz low pass filter used to extract the Figure 7 wave from the Figure 5 signal. This filtered Lamb wave surface acceleration, also inverted in comparison to Figure 4 to match the polarity of the test impact signal, is shown in Figure 8. The experimentally measured bending wave acceleration signals typically show a small precursor not indicated in the Lamb wave and a more complicated post-wave response that incudes more than the filter dynamics as seen in Figure 7. However, the dominant characteristics are highly similar. The detected acceleration signal contains what appears to be three half-cycles, with the time associated with the full width of the central region representing the impact contact time. Direct measurement of both the impact contact time and the associated impact wave shape for metal sphere impacts against a reactor coolant system component (Mayo et al., 1988) verify that the full width of the central region in the initial impact wave acceleration signal provides a direct measure of the impact contact time. While the surface acceleration of Lamb's wave is very close to the surface acceleration of the nondispersive plate bending wave generated by a small mass impact, this comparison of wave shape does not hold true for larger mass. In cases where the plate bending wave velocity is not constant, both waves still show the discontinuities associated with the impact contact time, but the shape of the Lamb wave surface acceleration begins to deviate substantially from the bending wave surface acceleration. Lamb's solution is not suitable for calculating bending wave shape in general, but serves to demonstrate the relationships between impact contact time, discontinuities in the wave surface acceleration, and the concept of a dominant, impact related frequency content based on approximately one-half cycle in the impact wave acceleration signal.
Loose-part signal properties
353
0.8
0.6
0.4
o~ 0.2
|
~
o - 0 . 2
-0.4
-0.6
i
-0.8
0.005
0.01
i
i
l
0.015 0.02 0.025 "lime - Seconds
i
i
0.03
0.035
0.04
Figure 5. Broad Band Impact Signal- One Inch Sphere
0.2
m
0.15
0,1
0.05
i
~
o -0.05
-0.1
-0.15 ~ 0
I
0.005
0.01
0.015 0.02 0.025 Time - Seconds
0.03
0.035
Figure 6. Filtered Impact Signal - One Inch Metal Sphere
0.04
354
C.W. MAYO
3
2
o)
_Q
-1
-2
"30
o12
oi,
o16
o18
;
112
Time - Seconds
1, x 10 .3
Figure 7. Initial Impact Wave from Figure 5
0.04 0.03
0.02
i 0.01
41.01 -0.02
-0.03 ~
012
014
016
018
Time - Seconds
1
112
1.4 x 10 3
Figure 8. Low Pass Filtered Lamb Wave Acceleration
Loose-partsignal properties
355
SPECTRAL ESTIMATION The frequency content of loose part monitoring system metal impact signals is typically estimated with the discrete Fourier transform Power Spectral Density (PSD) M-1
/ ' s / ~ , ) = ~1 I. ~. o xCn) e-~'"t 2
(4)
For this function x(n) is the nth sample of the time series x(t) there are M/2 +1 distinct frequencies including zero the maximum frequency is one-half of the sampling rate R the frequency resolution is if(MAt) where At is the time interval between samples M is typically an integer power of 2 to permit use of the Fast Fourier Transform (FFY) PSD calculations for metal impact signals may use transform lengths of 1-5 x 10z seconds to capture most of the observable impact time response in the transform. The time duration of this total impact response is primarily due to the excitation of lightly damped mechanical resonances associated with the sensor and sensor mounting by the high frequency content of the true impact wave. Background noise, although random, is approximately constant as a function of time. The actual loose part impact wave is much shorter, typically having a duration in the range from about 2 x 104 seconds to 2 x 10.3 seconds. Considering equation 4, increasing the transform length beyond that of the initial impact wave reduces its contribution to the total spectral density relative to the longer time duration of resonance shock response and the background noise. At the same time, reducing the transform length reduces the frequency resolution of the discrete Fourier transform. For estimation of the metal impact contact time via discrete Fourier analysis, the optimum transform length is one that provides the minimum acceptable frequency resolution for mass estimation from the impact wave half-period. These parameters have been investigated for the use of PSD spectrographs for the estimation of loose part impact parameters (Mayo, 1994). A sampling rate of 100 kHz was used to assure accurate peak sampling of the time domain waveform. For this sampling rate, a transform record length of 128 samples was found to provide a good compromise between frequency resolution and the time span of the transform. The resulting resolution of 781.25 Hz is adequate for separating mass effects as shown in Figure 1. One half of this resolution, provided by a 64 point transform, would begin to cause difficulty in resolving frequency differences for larger masses. Spectrographic displays obtained by passing this short-time PSD function through the signal provided good estimates of the frequency content associated with the sinusoidal half period of the initial impact. Valid indications of the impact wave arrival time are observed through broad-band frequency content that is estimated when the transform overlap with the impact wave is too short to estimate a distinct sinusoidal frequency. An example FFT spectrograph of the Figure 7 initial impact wave is shown as equal amplitude surface contours in Figure 9. Much of the visual information about the spectrograph function is lost in blackand-white, and by not examining three dimensional views of the spectrograph surface. However, dominant features can still be observed. The highest contour in Figure 9 is at about 8 kHz, in agreement with Hertz theory. The nature of the finite Fourier transform to estimate very broad spectral content when presented with only fractional cycle of a sinusoid appears as a very short, broad-band transient frequency content when the spectrograph transform first encounters and last overlaps the impact wave signal as seen at about 1.4 and 2.7 milliseconds in Figure 9. The duration of the spectrograph response is simply a measure of the time length used in the individual Fourier transforms, or 1.28 milliseconds in this example. For this
356
C.W. MAYO
measurement, the time length of the transform block is short enough that multiple path wave arrivals corresponding to a distance of about 35 feet around the steam generator can also be observed. Remnants of the low pass filtered sensor resonance at about 15 khz can also be observed with a time delay associated with the bandwidth of the resonance.
2.5
104
1.5 |
==
.-0
0.5
t
0
0.5
1.5
2
i
2.5 3 Time - Seconds
i
i
3.5
4
4.5 x 10 .3
Figure 9. PSD Spectrograph Contours for Impact of One Inch Diameter Sphere
The spectrograph information is degraded by the presence of background noise. However, the ability to observe the dominant frequency content of the impact wave is enhanced by the short time length of the spectral density estimate. Wave arrival time measurements are enhanced by both the ability to find "valleys" of low background spectral density that can be followed to locate the relatively broad impact wave front along the frequency axis, and the fact that the broad band response due to short signal overlap of the Fourier transform readily exceeds the filtered signal bandwidth, generating transient high frequency terms that can be readily observed and used to mark the wave arrival time. Additional examples of how short-time spectrographic analysis indicates the properties of the initial impact wave are being reported (Mayo, 1994).
SUMMARY AND CONCLUSIONS The initial impact wave generated by a loose part impact is very short, in the range from about 104 to 10.3 seconds. The information related to the impact contact time and the peak acceleration is contained a central term which is bounded by large second derivatives in the surface displacement wave at the times associated with the onset and completion of the impact contact. The sinusoidal definition of half-period for this central term is a factor of 0.8 shorter than the contact time. When this characteristic impact wave-form can be observed, the points of discontinuity in the impact force can be used to directly measure the impact contact time from the time domain wave-form. The frequency content of the impact wave associated with the impact contact time can be observed with Fourier analysis. However, considering the short duration of the initial impact wave and non-sinusoidal
Loose-partsignal properties
357
features of its shape, Fourier spectra can have difficulty resolving this information. The spectral density of the initial impact wave is reduced relative to the background noise spectral density when the transform length exceeds the length of the initial impact wave. The shock response of lightly damped mechanical resonances may also dominate the spectral density of the initial impact wave unless sufficiently attenuated by filtering. Short time Power Spectral Density Spectrographs assist in the identification of the initial impact wave primary frequency content. These functions provide an alternate measure of the impact contact time when the initial impact wave acceleration wave-form can not be clearly observed due to background noise. The short time PSD substantially enhances the relative spectral density of the initial impact wave frequency content relative to background and resonance response signals. The short time PSD spectrograph also assists in the identification of the initial wave arrival time through broad-band frequency terms generated when the transform first encounters the initial impact wave on a time scale that is too short to resolve the sinusoidal nature of the wave shape. Time domain identification of the initial impact wave characteristics associated with the beginning and end of the impact contact or spectrographic identification of the initial impact wave arrival time and dominant frequency content provide improved means for loose part impact location and mass estimation. Although not proven experimentally, the asymmetric nature of the initial impact wave should indicate on which side of the structure the impact took place. The information obtained from considering the nature of the initial impact wave and the associated surface acceleration can be used to improve the performance of diagnostic signal analysis for detecting the impact wave arrival time and estimating the associated impact and loose part properties.
REFERENCES Harris, C.M. and Crede, C.E. eds. (1961) Shock and Vibration Handbook Vol. I: Basic Theory and Measurements. New York: McGraw-Hill Book Company, Inc. Lamb, H. (1902) On Wave - Propagation in Two Dimensions. Proceedings of the London Mathematical Society Vol. 35, pp.141-161. Mayo, C.W. (1985) Loose Part Signal Theory. Progress in Nuclear Energy Vol. 15, pp 535 - 543 Mayo, C.W. et al. (1988) Loose Part Monitoring System Improvements. Final Report NP-5743, The Electric Power Research Institute, Palo Alto, CA 94303 Mayo, C.W. (1994) Loose Part Spectrographic Analysis, Proceedings of the Institute for Environmental Sciences, 40th Annual Meeting, the Institute of Environmental Sciences, Mount Prospect, Illinois Milkowitz, J. (1953) Flexural Wave Solutions of Coupled Equations Representing the More Exact Theory of Bending. Journal of Applied Mechanics, 20, pp. 511- 514. Raman, C. (1920) On some applications of Hertz's Theory of Impact. Physical Review Vol.15, pp 277284
Pergamon 0149-1970(94)00006-9
LOOSE-PART SIGNAL PROPERTIES C. w. MAYO Departmentof NuclearEngineering,North CarolinaState University, Box 7909Raleigh,NC 27695-7909,U.S.A.
Abstract - Lamb's solution for two-dimensional wave propagation was used to identify plate bending wave characteristics observed in loose part impact signals. The surface displacement wave is shown to have large second derivatives at points that can be used to measure the impact contact time. General features of the initial impact wave shape provide guidance for the use of spectral analysis techniques to identify the contact time and wave arrival time when there is a low signal-to-noise ratio. Improvements in the determination of contact time improve the estimation of loose part mass and energy through the Hertz impact parameters. The shape of the initial impact wave shape can also be used to identify on which side of the plate the impact took place. This information has applications for Loose Part Monitoring System metal impact signal validation and for improved estimation of loose part parameters.
INTRODUCTION Metal impact signal theory for reactor loose parts monitoring has been developed through prior studies (Mayo, 1985, Mayo et al., 1988). The Hertz theory of impact was shown to be applicable for impact response calibration and for the interpretation of impacts against plate type pressure boundary components. This prior work identified the initial impact wave acceleration wave-form experimentally and demonstrated bending wave transmission effects, mechanical resonance response, and multiple path transmission effects in accordance with theory. The results of this work provided a quantitative basis for modeling Loose Part Monitoring System (LPMS) performance, and guidelines for loose part monitoring system specifications, metal impact response calibration, and diagnostic analysis of unknown metal impact signals. The nature of the initial loose part impact wave has not been investigated due in part to the nature of the solution required to include the shear force and rotary inertia for impacts against a structure of finite thickness (Miklowitz, 1953) and the requirement to include the cylindrical nature of the wave propagation for impact against a plate. As an approximation, Lamb's solution for wave propagation on a membrane (Lamb, 1902) provides a two-dimensional, radially symmetric solution for an arbitrary force. The constant wave speed in the membrane solution is equivalent to the effectively constant bending wave velocity associated with the impact of smaller mass loose parts. Lamb's two-dimensional wave demonstrates dominant features of the metal impact wave shape and provides a basis for relating plate bending wave shape features to the initial impact properties. The
347
348
C.W. MAYO
general nature of the impact wave shape can also be used to improve the performance of frequency spectrum analysis techniques for identifying the impact contact time and wave arrival time when there is a low signal-to-noise ratio. These results have application for improved loose part diagnostic analysis.
HERTZ IMPACT THEORY Hertz theory for the impact of a sphere against a plate defines the impact contact time as a function of the impacting object mass, velocity, and contact radius (Harris and Crede, 1961). The impact contact time was associated with the frequency content of the wave that radiates away from the impact point by Raman, who reported that the initial half-period of the plate wave is a factor of 1.6 shorter than the contact time (Raman, 1920). Raman's relationship led to the concept of the primary impact wave frequency content being def'med by the equation
f*-
1.6 2t^
(1)
where fa is the sinusoidal frequency in Hz associated with the initial half-period of the wave t~ is the impact contact time in seconds and the Hertz theory impact contact time can be expressed as
(2)
ts = n kh m'4 V~.2R-.2 where k h is 1.82 x 10.4 ft'81bf'4 for the impact of steel on steel m is the mass of the impacting object in 1bf/32.2 ft/sec 2 Vo is the initial impact velocity R is the contact radius of curvature in ft
These equations have been used to map the general range of loose part signal frequency content as a function of mass and energy as shown in Figure 1. In this example, the frequency associated with the initial half-period is plotted for impact velocities between 1 and 10 ft/sec, for seven different metal spheres having diameters between six inches (32.1 lbs) and 0.5 inches (0.0186 lbs). The Figure 1 values for sphere diameter and mass are given in Table 1. TABLE
- Metal Sphere Diameter and Mass
Diameter
6"
5"
4"
3"
2"
1"
0.5"
Mass (lb)
32.1
18.6
9.5
4.0
1.19
0.15
0.018
The independent parameters of mass and velocity can not be identified uniquely from the single parameter of contact time or the corresponding impact wave frequency. However, as indicated by Figure 1, the initial impact wave frequency can provide useful mass estimates without knowledge of the impact velocity.
Loose-part signal properties
2
349
x 104
1.5 Sphere Diameter - Inches
N
"1i
'-
1"
1
LL 2"
0.5
j
~
i
l o ~'
. . . . . . .
i
. . . . . . . .
10.3
t
. . . . . . . .
10 .2
f
4
i
10"1
"
. . . . . . . .
i
10 0
5.6,
.
.
.
.
.
.
.
.
i
101
.
.
.
.
.
.
.
10 2
Impact Energy - Ft-Lbf
Figure 1. Primary Frequency Content for Metal Sphere Impacts The uncertainty in mass estimates based on frequency alone can be characterized by solving equations 1 and 2 for the range in mass that can give a particular primary impact wave frequency over a range of potential impact velocities. These values are shown in Figure 2 for metal spheres with velocities between 1 and 10 ft/sec. The uncertainty in mass is about a factor of four between the minimum and maximum velocity. Still, the range of loose part mass covers about three orders of magnitude, and useful information about the mass of unknown loose parts can be obtained without knowledge of velocity. With the initial mass estimate, velocity and energy can then be estimated through the signal amplitude, and the results refined. Hertz theory has been shown to be remarkably applicable for modeling and interpreting reactor loose part monitoring system signals. The initial assumption of a metal sphere can be relaxed to an object with a finite contact radius and a general shape that behaves as a rigid body during the impact. Both metal spheres and force instrumented metal hammers serve well for metal impact response calibration as a function of known impact parameters. Experimental measurements show that local deformation does not greatly effect the impact parameters over the general range of mass and velocity for reactor loose parts (Mayo et al. 1988).
THE TWO-DIMENSION LAMB WAVE In the early study of impacts against two-dimensional surfaces, Lamb derived a solution for the shape of a two dimensional wave on a membrane as a function of an arbitrary impact force (Lamb, 1902). While this solution does not include shear and rotary inertia effects associated with a plate of finite thickness, it was used in Raman's successful calculation of the relationship between impact contact time and period of the first half-cycle of a plate bending wave using Hertz impact theory (Raman, 1920). This relationship has been demonstrated to be applicable for the loose part impact signals (Mayo et al., 1988).
350
C.W. MAYO
102
10 ~
_1
~
10 °
10"*
Velocity=1 ft/sec
10 .2
~
i 1
01.5
o
Figure 2.
~
11.5
i 2
Frequency - Hz
2.5 x 104
Mass and Initial Wave Frequency Range for Impact Velocities Between 1 10 Ft/sec
Lamb's general solution for an arbitrary impact force function is: u
D(0
2n
e
(3)
o
where: f(t) is an arbitrary force function such that the integral converges r is the distance from the impact point at which the wave is observed c is the constant speed of wave transmission For a force that acts for only a finite time, the limits of integration are effectively restricted to limits for which the integrand is non-zero. Equation (3) can be integrated numerically for the half-sine force function associated with a Hertz impact. The Hertz impact for a metal object weighing a few ounces has a contact time of about 1 x 10.4 seconds and an approximately constant speed of wave transmission due to the shape of the bending wave velocity function for typical reactor pressure boundary thickness (Mayo et al., 1988). Using a half-sine force function with a duration of 1 x 10.4 sec, the corresponding Lamb wave is shown in Figure 3. This wave was calculated with a time step size of 1 x 10-5 seconds. Some of the features of this wave are: The response is zero prior to the time of initial arrival r/c, set here at 1 x 10.4 seconds The amplitude passes through a maximum prior to completion of the impact contact at 2 x 10.4 seconds The wave decays relatively slowly after the completion of the impact contact
Loose-part signal properties
351
0.6 I 0.5 i ~0.4
E J .~0.3 a
0.2
0.1
o
o12
oi,
o'.6
o'.8
;
Time - Seconds
,.2 x 10 "a
Figure 3. Lamb's Surface Wave The derivatives of the half-sine function are discontinuous its beginning and end. The derivatives of a Lamb wave generated by this function are similarly discontinuous at the times that correspond to the beginning and end of contact. These discontinuities are clearly evident in the surface displacement velocity and acceleration functions. The Figure 4 surface acceleration obtained by differentiation of the Figure 3 wave shows these discontinuities at the beginning and end of the impact contact.
0.06
0.04
i
0,02
Q
-0.02
-0.04 0.2
oi,
o16
i
0.8
i
,
Time - Seconds
Figure 4. Lamb Wave Surface Acceleration
,2 x 10 .3
352
C.W. MAYO
The discontinuities in the acceleration function provide distinctive wave shape features that can be used to measure the impact contact time. The fact that the signal is not symmetric shows that it is possible to identify the side of the membrane on which the impact took place by the sign of the derivatives as they approach the discontinuities. The dominant near-sinusoidal signal component is contained in slightly more than one-half cycle that lies between the beginning and end of the impact contact. Defining the half-period as the time that this part of the Figure 4 signal is less than zero gives Raman's relationship between impact contact time and the wave half-period. The short duration of this approximately sinusoidal component and the non-sinusoidal nature of the acceleration signal in general lead to the expectation that Fourier spectrum analysis can give relatively poor resolution of the frequency content associated with the impact contact time.
IMPACT WAVE SHAPE The plate impact bending wave can be observed experimentally. For loose parts monitoring systems, some filtering is usually required in order to remove the shock response of the sensor. A typical broadband loose parts monitoring system channel response to the impact of a 1 inch diameter metal sphere is shown in Figure 5. This mass was chosen since the primary frequency content is in a range where plate bending wave transmission is non dispersive for reactor structures (Mayo et al., 1988) and therefore analogous to the constant speed of propagation for Lamb's two-dimensional wave on a membrane. The sensor mounted resonance frequency is about 15 kHz in this example and can be largely removed by a eight-pole Butterworth low pass filter set at 10 kHz as shown in Figure 6. The initial impact wave-form is shown on an expanded scale in Figure 7. The initial impact wave in Figure 7 has distinct similarities to Lamb's wave solution. The comparison is improved by passing the Lamb wave acceleration from Figure 4 through the same 10 kHz low pass filter used to extract the Figure 7 wave from the Figure 5 signal. This filtered Lamb wave surface acceleration, also inverted in comparison to Figure 4 to match the polarity of the test impact signal, is shown in Figure 8. The experimentally measured bending wave acceleration signals typically show a small precursor not indicated in the Lamb wave and a more complicated post-wave response that incudes more than the filter dynamics as seen in Figure 7. However, the dominant characteristics are highly similar. The detected acceleration signal contains what appears to be three half-cycles, with the time associated with the full width of the central region representing the impact contact time. Direct measurement of both the impact contact time and the associated impact wave shape for metal sphere impacts against a reactor coolant system component (Mayo et al., 1988) verify that the full width of the central region in the initial impact wave acceleration signal provides a direct measure of the impact contact time. While the surface acceleration of Lamb's wave is very close to the surface acceleration of the nondispersive plate bending wave generated by a small mass impact, this comparison of wave shape does not hold true for larger mass. In cases where the plate bending wave velocity is not constant, both waves still show the discontinuities associated with the impact contact time, but the shape of the Lamb wave surface acceleration begins to deviate substantially from the bending wave surface acceleration. Lamb's solution is not suitable for calculating bending wave shape in general, but serves to demonstrate the relationships between impact contact time, discontinuities in the wave surface acceleration, and the concept of a dominant, impact related frequency content based on approximately one-half cycle in the impact wave acceleration signal.
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SPECTRAL ESTIMATION The frequency content of loose part monitoring system metal impact signals is typically estimated with the discrete Fourier transform Power Spectral Density (PSD) M-1
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For this function x(n) is the nth sample of the time series x(t) there are M/2 +1 distinct frequencies including zero the maximum frequency is one-half of the sampling rate R the frequency resolution is if(MAt) where At is the time interval between samples M is typically an integer power of 2 to permit use of the Fast Fourier Transform (FFY) PSD calculations for metal impact signals may use transform lengths of 1-5 x 10z seconds to capture most of the observable impact time response in the transform. The time duration of this total impact response is primarily due to the excitation of lightly damped mechanical resonances associated with the sensor and sensor mounting by the high frequency content of the true impact wave. Background noise, although random, is approximately constant as a function of time. The actual loose part impact wave is much shorter, typically having a duration in the range from about 2 x 104 seconds to 2 x 10.3 seconds. Considering equation 4, increasing the transform length beyond that of the initial impact wave reduces its contribution to the total spectral density relative to the longer time duration of resonance shock response and the background noise. At the same time, reducing the transform length reduces the frequency resolution of the discrete Fourier transform. For estimation of the metal impact contact time via discrete Fourier analysis, the optimum transform length is one that provides the minimum acceptable frequency resolution for mass estimation from the impact wave half-period. These parameters have been investigated for the use of PSD spectrographs for the estimation of loose part impact parameters (Mayo, 1994). A sampling rate of 100 kHz was used to assure accurate peak sampling of the time domain waveform. For this sampling rate, a transform record length of 128 samples was found to provide a good compromise between frequency resolution and the time span of the transform. The resulting resolution of 781.25 Hz is adequate for separating mass effects as shown in Figure 1. One half of this resolution, provided by a 64 point transform, would begin to cause difficulty in resolving frequency differences for larger masses. Spectrographic displays obtained by passing this short-time PSD function through the signal provided good estimates of the frequency content associated with the sinusoidal half period of the initial impact. Valid indications of the impact wave arrival time are observed through broad-band frequency content that is estimated when the transform overlap with the impact wave is too short to estimate a distinct sinusoidal frequency. An example FFT spectrograph of the Figure 7 initial impact wave is shown as equal amplitude surface contours in Figure 9. Much of the visual information about the spectrograph function is lost in blackand-white, and by not examining three dimensional views of the spectrograph surface. However, dominant features can still be observed. The highest contour in Figure 9 is at about 8 kHz, in agreement with Hertz theory. The nature of the finite Fourier transform to estimate very broad spectral content when presented with only fractional cycle of a sinusoid appears as a very short, broad-band transient frequency content when the spectrograph transform first encounters and last overlaps the impact wave signal as seen at about 1.4 and 2.7 milliseconds in Figure 9. The duration of the spectrograph response is simply a measure of the time length used in the individual Fourier transforms, or 1.28 milliseconds in this example. For this
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measurement, the time length of the transform block is short enough that multiple path wave arrivals corresponding to a distance of about 35 feet around the steam generator can also be observed. Remnants of the low pass filtered sensor resonance at about 15 khz can also be observed with a time delay associated with the bandwidth of the resonance.
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The spectrograph information is degraded by the presence of background noise. However, the ability to observe the dominant frequency content of the impact wave is enhanced by the short time length of the spectral density estimate. Wave arrival time measurements are enhanced by both the ability to find "valleys" of low background spectral density that can be followed to locate the relatively broad impact wave front along the frequency axis, and the fact that the broad band response due to short signal overlap of the Fourier transform readily exceeds the filtered signal bandwidth, generating transient high frequency terms that can be readily observed and used to mark the wave arrival time. Additional examples of how short-time spectrographic analysis indicates the properties of the initial impact wave are being reported (Mayo, 1994).
SUMMARY AND CONCLUSIONS The initial impact wave generated by a loose part impact is very short, in the range from about 104 to 10.3 seconds. The information related to the impact contact time and the peak acceleration is contained a central term which is bounded by large second derivatives in the surface displacement wave at the times associated with the onset and completion of the impact contact. The sinusoidal definition of half-period for this central term is a factor of 0.8 shorter than the contact time. When this characteristic impact wave-form can be observed, the points of discontinuity in the impact force can be used to directly measure the impact contact time from the time domain wave-form. The frequency content of the impact wave associated with the impact contact time can be observed with Fourier analysis. However, considering the short duration of the initial impact wave and non-sinusoidal
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features of its shape, Fourier spectra can have difficulty resolving this information. The spectral density of the initial impact wave is reduced relative to the background noise spectral density when the transform length exceeds the length of the initial impact wave. The shock response of lightly damped mechanical resonances may also dominate the spectral density of the initial impact wave unless sufficiently attenuated by filtering. Short time Power Spectral Density Spectrographs assist in the identification of the initial impact wave primary frequency content. These functions provide an alternate measure of the impact contact time when the initial impact wave acceleration wave-form can not be clearly observed due to background noise. The short time PSD substantially enhances the relative spectral density of the initial impact wave frequency content relative to background and resonance response signals. The short time PSD spectrograph also assists in the identification of the initial wave arrival time through broad-band frequency terms generated when the transform first encounters the initial impact wave on a time scale that is too short to resolve the sinusoidal nature of the wave shape. Time domain identification of the initial impact wave characteristics associated with the beginning and end of the impact contact or spectrographic identification of the initial impact wave arrival time and dominant frequency content provide improved means for loose part impact location and mass estimation. Although not proven experimentally, the asymmetric nature of the initial impact wave should indicate on which side of the structure the impact took place. The information obtained from considering the nature of the initial impact wave and the associated surface acceleration can be used to improve the performance of diagnostic signal analysis for detecting the impact wave arrival time and estimating the associated impact and loose part properties.
REFERENCES Harris, C.M. and Crede, C.E. eds. (1961) Shock and Vibration Handbook Vol. I: Basic Theory and Measurements. New York: McGraw-Hill Book Company, Inc. Lamb, H. (1902) On Wave - Propagation in Two Dimensions. Proceedings of the London Mathematical Society Vol. 35, pp.141-161. Mayo, C.W. (1985) Loose Part Signal Theory. Progress in Nuclear Energy Vol. 15, pp 535 - 543 Mayo, C.W. et al. (1988) Loose Part Monitoring System Improvements. Final Report NP-5743, The Electric Power Research Institute, Palo Alto, CA 94303 Mayo, C.W. (1994) Loose Part Spectrographic Analysis, Proceedings of the Institute for Environmental Sciences, 40th Annual Meeting, the Institute of Environmental Sciences, Mount Prospect, Illinois Milkowitz, J. (1953) Flexural Wave Solutions of Coupled Equations Representing the More Exact Theory of Bending. Journal of Applied Mechanics, 20, pp. 511- 514. Raman, C. (1920) On some applications of Hertz's Theory of Impact. Physical Review Vol.15, pp 277284