To assure safe and reliable service life but also for an optimized maintenance
strategy, it is necessary to have a precise estimation of lifetime consumption
of critical components. A life monitoring system is usually understood as a
computerized system which takes data from transducers fitted to a piece of plant
and calculates on-line the theoretical creep and fatigue damage experienced
by the plant due to operation and hence, the remanent life of the plant. These
systems have historically been fitted mainly to large boiler plant but are applicable
to other high temperature plant as well (Majle et al.,
1996). It is important to evaluate structural integrity of highly pressurized
piping for power plants. The wall thickness of a pressurized pipe is designed
to satisfy a corrosion margin (Miyazaki, 2002). A study
of fatigue damage is necessarily done since single pipe damage could not be
detected unless the whole piping system must be replaced with the new one. The
signal processing method was used in this study in order to create an estimation
system of fatigue failure detection.
Fatigue is a localized damage process of a component produced by cyclic loading.
It is the cumulative process consisting of crack initiation, propagation and
final fracture of component. During cyclic loading, localized plastic deformation
may occur at the highest stress site. This plastic deformation induces permanent
damage to the component and a crack develops. As the component experiences an
increasing number of loading cycles, the length of the crack also increases.
After a certain number of cycles, the crack will cause the component to fail
(Lee et al., 2005). The fatigue damage starts when
the component goes into service and is subjected to some form of cyclic stress.
As the material is subjected to repeated cyclic loads, additional fatigue damage
occurs. This fatigue damage is cumulative in nature and progresses until a fatigue
crack forms. If the crack is not detected by inspection and the affected equipment
is placed back into service, the crack grows until the component fails by leak,
brittle fracture, or gross plastic deformation due to overload of the remaining
cross sectional area (Gokhale et al., 2007).
Current industrial practice for fatigue life prediction is to use the Palmgren-Miner
(PM) linear damage rule (Abdullah et al., 2005).
For the strain-based fatigue life prediction, this rule is normally applied
with strain-life fatigue damage models. The first strain-life model is the Coffin-Manson
where, E is the material modules of elasticity, εa is a true
strain amplitude, 2Nf is the number of reversals to failure, εf
is a fatigue strength coefficient, b is a fatigue strength exponent, εf
is a fatigue ductility coefficient and c is a fatigue ductility exponent.
In designing for durability,the presence of non zero mean stress can influence fatigue behavior of materials because a tensile or a compressive normal mean stress has been shown to be
responsible for accelerating or decelerating crack initiation and growth (Lee
et al., 2005). In conjunction to the local strain life approach,
many models have been proposed to quantify the effect of mean stresses on fatigue
behavior. Morrow (Morrow, 1968) has proposed the following
relationship when a mean stress is present:
Equation 2 implies the mean normal stress can be taken into
account by modifying the elastic part of the strain-life curve by the mean stress,
σm. Equation 2 has been extensively cited
for steels and used with considerable success in the long-life regime when plastic
strain amplitude is of little significance.
Smith et al. (1970) proposed a method that assumes
that the fatigue damage in a cycle is determined by σmax εa,
where σmax is the maximum tensile stress and εa is
the strain amplitude. Also, the SWT parameter is simply a statement of σaεa
for a fully reversed test is equal to σmax εa for
a mean stress test. The SWT parameter predicts no fatigue damage if the maximum
tensile stress becomes zero and negative. The SWT mean stress correction formula
is expressed as follows:
For loading sequences that are predominantly tensile, the SWT approach is more
conservative and therefore recommended. In a case of the loading being predominantly
compressive, particularly for wholly compressive cycles, the Morrow model provides
more realistic life estimates (Dowling, 1999). Generally,
the selection of mean stress correction approach highly based on the value of
mean stress itself as well as the material. For instance, the unmodified Morrow
approach seem to work reasonably well for steels and in at least some cases
gives better results than the SWT parameter (Dowling, 1999).
However, the SWT model appears to work particularly well for aluminum alloy,
cast iron, hardened carbon steel and microalloy steels (Lee
et al., 2005). According to Lee et al. (2005),
the mean stress correction approach are empirically calculated and it should
be compared to the test data in order to determine the most appropriate model
for a specific material and test condition.
The signal processing methods were utilised in order to analyze the time domain
raw data. The frequency analysis is performed in order to convert a time domain
signal into the frequency domain. The results of a frequency analysis are most
commonly presented by means of graph having frequency on the x-axis and amplitude
on the y-axis. The Fourier analysis is a one of the method to analyze random
data based on the frequency domain analysis. The algorithm that is used to split
the time history into its constituent sinusoidal components is the Fourier transform.
This transform was first defined by the French mathematician and engineer Jean
Baptiste Joseph Fourier who postulated that any periodic function could be expressed
as the summation of sinusoidal waves of varying frequency, amplitude and phase.
Smith (1999) defined spectral analysis as understanding
a signal by examining the amplitude, frequency and phase of its component sinusoids.
For a periodic time function, x (t), frequency analysis can be performed using
the classical Fourier transform defined by the mathematical definition:
where, X(ω) is the amplitude of Fourier transform in frequency distribution,
ω is the angular frequency and fk.
The frequency analysis data is typically presented in graphical form as Power
Spectral Density (PSD). A PSD is a normalized density plot describing the mean
square amplitude of each sinusoidal wave with respect to its frequency. The
PSD presents the vibrational energy distribution of the signal across the frequency
domain. Each frequency step value of the PSD is characterized by amplitude,
Ak, defined as:
where, S(fk) is the underlying PSD of the signal and fk
is the harmonic frequency.
The PSD can be as an input for generating a time-varying signal by performing
the IDFT or IFFT. However, the time series which was generated from IFFT is
not as accurate as the original signal because the PSD does not contain the
original signal phase information. The assumptions of the signal phase content
can be made in order to regenerate a statistically equivalent time history.
For example, if the time history is taken from an ergodic stationary for Gaussian
and random process, the phase is purely random between -π and +π radians
(Halfpenny, 1999; Li et al.,
The Short-Time Fourier Transform (STFT) is a method of time-frequency analysis which aims to produce frequency information which has a localization in time. The STFT is performed by dividing the signal into small sequential or overlapping
data frames, for which the Fast Fourier Transform (FFT) was applied to each
data frame (Kiymik et al., 2005). It provides
information about when and at what frequencies a signal event occurs. The STFT
approach assumes that if a time-varying signal is divided into several segments,
each can be assumed stationary for analysis purposes. The Fourier transform
is applied to 24 each of the segments using a window function, which is typically
nonzero in the analyzed segment and is set to zero outside. The most important
parameter in the analysis is the window length, which is chosen so as to isolate
the signal in time without any distortions. The STFT was developed from the
Fourier transform and it is mathematically defined as:
where, the Fourier transform of the windowed signal is x(t)e-iωt,
ω is the frequency and τ is the time position of the window (Chui,
The time-frequency resolution depends on the selection of the window length.
The time window length is defined as Δt and the frequency bandwidth is
Δω. Considering the relationship between time and frequency, i.e.:
It can be seen that a good time localization (when Δt is small) or frequency
localization (when Δω is small) can be obtained, but not both simultaneously.
Therefore, the time window length Δt and the frequency bandwidth Δω
MATERIALS AND METHODS
In this study, the strain signal was measured on a laboratory scale water piping system as shown in Fig. 1. Two control valves which each valve located at each pipe were used to open, close and control the flow rate. In this case, both control valves were set up to the similar conditions in order to gain related signals for the particular pipe surfaces. The sections measured were the two horizontal pipes of the same outer diameter but with different internal surface features which were roughened and smooth surfaces at three different flow rates. Two different types of internal surface; smooth and roughened were utilised in order to compare the results for fatigue damage, energy and other features. These two different surfaces were chosen based on the assumption that the results may vary due to the difference of internal surfaces of the pipe. The fatigue data acquisition system was used for the strain signal measurement.
Two millimeter in size strain gauges were used for measuring the strain loading
signals. The strain gauges were mounted on the mirror-polished pipes intentionally
in the same direction of the water flow. For this reason, the strain gauges
can detect any deformation occurs on the pipe surface. The signals were estimated
to be the Variable Amplitude (VA) loading which was then sampled at 50 Hz. The
measured strain signals were shown in Fig. 2.
||Test rig set up for strain data measurement
The original strain signal for smooth (left side) and roughened (right
side) pipe at: (a) normal flow rate, (b) maximum flow rate and (c) variable
For the post
processing stage, the GlyphWorks®software package was used to
calculate the fatigue damage of the pipes and the Matlab® software
was also used to analyze the signal processing approach.
RESULTS AND DISCUSSION
Frequency analysis: Several methods in signal processing have been utilised to process the data obtained during the test. The Power Spectral Density method was used to convert the time domain signal into the frequency domain. It indicated each of frequency existed in the signal. The distribution of vibrational signal energy across the frequency domain can be observed using this method. The plot of PSD for roughened pipe surface can be shown in Fig. 3. As shown in Fig. 3, the frequency for roughened pipe is lower than 0.2 Hz. It means that the fatigue damage in pipe occurs at low frequencies.
Time-frequency analysis: The time domain signal was converted into time-frequency domain using the STFT method. This method allows us to determine the value of frequency at a particular time. High amplitude events in time domain signal was represented by the narrow and wide band power spectrums in the time-frequency representation. It can be seen clearly in Fig. 4 for which the narrow band signals in the original time history occurred as a number of events with narrow bandwidths in the time-frequency mapping added with frequency information. Similarly, wide band signals in time history occurred as wide power spectrum in STFT as in red circle.
Figure 5 and 6 show the examples of time-frequency
representation of the smooth and roughened pipe surfaces at normal, maximum
and variable flow rates. The characteristics of fatigue damage events varied
for each flow rate. It was because, though every flow rate had a number of fatigue
damage events, but the time and frequency localizations of the events were different
for each situation as plotted in Fig. 5 and 6.
For the smooth surface, high power spectrums occurred at the normal flow rate
where the red stripes exist for a couple of times as shown in Fig.
5. By contrary, the high power spectrums occurred at every flow rate for
roughened pipe surface though it happened at different times as shown in Fig.
6. It might be caused by the ununiformity of the surfaces in the roughened
pipe. In addition, high power spectrum events occurred more often in the roughened
pipe and not depending on the type of flow rates. Obviously in Fig.
5 and 6, the distribution of the high power spectrum happened
more frequently across time for roughened pipe compared to the smooth pipe surface.
|| Signals for roughened surfaced pipe for a normal flow rate: (a) time
history and (b) PSD plot
|| Representation of smooth pipe signal at the maximum flow rate: (a) time
history and (b) STFT
The total of energy was calculated using the power spectrum method which represents
the power distribution in the signal as shown in Table 1 and
2. For both of the smooth and roughened pipe surfaces, the
highest total of energy occurs at normal flow rate which are 1.060x107
and 0.440x107 με2 Hz-1, respectively.
To our conscious, it appears that the lowest total of energy differs for each
pipe surface where it happened at the maximum flow rate for smooth pipe and
variable flow rate for the roughened pipe surface.
Fatigue damage analysis: The strain-life module in GlyphWorks®
was utilised to calculate the fatigue damage for smooth and roughened pipe surfaces.
||Total of energy for smooth pipe
|| Total of energy for roughened pipe
The fatigue damage calculation was based on the Smith-Watson-Topper (SWT) relationship
in Eq. 3. Thus, the fatigue damage (D) was calculated using
||STFT localization for smooth pipe at different flow rate, (a) normal,
(b) maximum and (c) variable
The fatigue damage histogram for smooth pipe surface was shown in Fig.
7, while Fig. 8 showed the fatigue damage histogram for
roughened surface pipe at three different flow rates. The red color plot represents
the highest value of fatigue damage. Each color of the plot presents different
levels of damage which were dark blue is for the lowest, followed by blue, green,
yellow, orange and red. There is only low fatigue damage events exist in the
smooth pipe surface as shown in Fig. 7. At normal flow rate,
there are a number of dark blue plots which represents the low fatigue damage
amplitude occur. Gradually, the fatigue damage amplitude increases at maximum
and variable flow rates where the light blue plots are more than the dark blue
|| Fatigue damage ratio for smooth pipe
But the amplitude of fatigue damage show an increment in the roughened pipe
surface as can be shown in Fig. 8. It is visible to us that
the number of fatigue damage events and its amplitude do not exhibit much difference
even though the flow rates are different. It showed that fatigue damage amplitude
is higher for roughened pipe surface than the smooth pipe. The fatigue damage ratios for both signals are shown in Table
3 and 4. The ratio was calculated with reference to the
normal water flow rate. For the smooth pipe surface, the fatigue damage ratio
of the maximum and the variable flow rates with respect to normal flow rate
was 75% and 189%.
||STFT localisation for roughened pipe at different flow rate: (a) normal,
(b) maximum and (c) variable
||Fatigue damage histograms for roughened pipe at (a) normal, (b) maximum
and (c) variable flow rate
||Fatigue damage histograms for roughened pipe at (a) normal, (b) maximum,
(c) variable flow rate
|| Fatigue damage ratio for roughened pipe
It means that the fatigue damage for the maximum flow rate was lower than
the normal flow rate. In other words, the fatigue damage value for the variable
flow rate is larger than the normal flow rate.
Similar observation can be seen for roughened surface pipe where the fatigue damage ratio of maximum and variable flow rates is 68% of the normal flow rate and 114%, respectively for the normal flow rate.
The fatigue damage value obtained for both types of pipe surface indicated that the maximum flow rate contributed to the lowest fatigue damage in the piping system. In the meantime, the fatigue damage ratios at variable flow rate are higher for the smooth and roughened pipe surfaces. From the results, the signal processing approach used in this study was capable in verifying the fatigue damage events in piping system. This study demonstrated the combination approach of signal analysis and fatigue life assessment of piping system in the context of signal patterns and fatigue damage value. By applying the Short Time Fourier Transform (STFT), the fatigue damage events in piping were clarified from the features obtained. This combination could be a very useful tool for the reliable and quick analysis of the structural integrity of piping system.
The authors would like to express their gratitude to Universiti Kebangsaan Malaysia and Ministry of Higher Education, Malaysia through the research fundings of UKM-KK-02-FRGS0016-2006 and UKM-KK-02-FRGS0012-2006, for supporting this research.