## Abstract

This paper describes a global non-destructive testing technique for detecting fatigue cracking in engineering components. The technique measures the mixing of two ultrasonic sinusoidal waves which are excited by a small piezoceramic disc bonded to the test structure. This input signal excites very high-order modes of vibration of the test structure within the ultrasonic frequency range. The response of the structure is measured by a second piezoceramic disc and the received waveform is analysed using the bispectrum signal processing technique. Frequency mixing occurs as a result of nonlinearities within the test structure and fatigue cracking is shown to produce a strong mixing effect. The bispectrum is shown to be particularly suitable for this application due to its known insensitivity to noise. Experimental results on steel beams are used to show that fatigue cracks, corresponding to a reduction in the beam section of 8%, can be detected. It is also shown that the bispectrum can be used to quantify the extent of the cracking. A simple nonlinear spring model is used to interpret the results and demonstrate the robustness of the bispectrum for this application.

## 1. Introduction

Fatigue cracks are the most common cause of catastrophic failure of engineering components and structures. In many cases, such as aircraft sub-structures, power generation plant and rail tracks, fatigue cracks cannot be eliminated and so the approach is to rigorously inspect the critical components non-destructively. Typically, this is achieved by inspecting the component point-by-point using a measurement technique such as ultrasound, X-ray or eddy current. As these techniques make local measurements, the inspection of the whole component is often extremely time-consuming. A range of global non-destructive testing (NDT) techniques have been developed such as measurement of the natural frequencies (Ewins 1984), as well as a number of optical thermal techniques (Heller 2001), which, while not truly global, have a large field of view. However, while they have found some application areas, these methods suffer from various limitations. In the case of natural frequency measurement, the sensitivity to small defects is not sufficient for most applications and the approach is inherently sensitive to geometry and boundary condition changes. In the case of optical and thermal techniques, although good sensitivity can be achieved, these techniques are limited to surface breaking or near surface defects and to components with relatively simple geometries. Guided acoustic waves (including Lamb waves) in the low-ultrasonic frequency range provide a potential means of testing large areas of a structure from a single location and systems have been successfully developed for the inspection of simple structures such as pipes and rails (Cawley *et al*. 2003). However, the increase in coverage provided by guided waves is tempered by a reduction in sensitivity to defects and, to date, no reliable means of interpreting signals in more complex structures has been demonstrated in industrial applications.

Recent years have seen the emergence of a number of NDT techniques based on nonlinear vibration and nonlinear ultrasound. These rely on the defect behaving nonlinearly under the applied vibration or ultrasonic illumination. For example, the opening and closing of a crack under forced vibration by sinusoidal excitation forces can result in the generation of harmonics caused by the distortion of the vibration. Wenger *et al*. (2000) and Brotherhood *et al*. (2003) used a local measurement technique, in which a pulse of high-intensity ultrasound was passed through, and reflected from, bonded joints in order to measure their nonlinear response. Regions of poor bonding were shown to cause a distortion in the reflected and transmitted ultrasonic signals. This distortion caused harmonics of the input frequency to be generated and these were used to monitor the severity of the defect. Van Den Abeele *et al*. (2001) used measurements of the change in the low-order structural resonant frequencies with excitation amplitude as a global technique for characterizing distributed micro-cracking in natural slate samples. Other researchers (e.g. Worden *et al*. 1993) have used neural networks to ‘diagnose’ a fault in a component based on analysis of its low-frequency nonlinear dynamic response. Solodov (1998), Kazakov *et al*. (2002) and Morbidini *et al*. (2005) suggested the use of the vibro-acoustic modulation technique, in which the low-frequency resonance of a structure is used to open and close the crack. The crack is then probed by a high-frequency (typically ultrasonic) signal. The low-frequency vibration then modulates the ultrasonic probing pulses producing energy at the sum and difference frequencies. As well as the above experimental studies there has been significant interest in the modelling of the nonlinear dynamics of various defects (e.g. Achenbach & Parikh 1991; Worden *et al*. 1994; Hirsekorn 2001; Pecorari & Poznic 2005).

The above experimental approaches have shown significant promise as NDT techniques. There is a body of evidence showing that a range of common defects, such as fatigue cracking, distributed micro-cracking and some adhesive defects exhibit dynamic nonlinear responses over a range of frequencies. However, apart from the local ultrasonic techniques, none of the experimental techniques developed to date have been shown to be sufficiently reliable for widespread engineering application. This paper presents a novel nonlinear measurement technique, which enables the global detection of fatigue cracks. In particular, some of the key limitations of existing techniques such as amplifier noise, random noise and geometry variations are overcome or reduced. The test structure is excited by a single piezoceramic transducer, which emits two summed sinusoids at different ultrasonic frequencies. The higher frequency is chosen as a non-integer multiple of the lower frequency so as to generate mixing frequencies unrelated to the harmonics of the lower frequency. A second piezoceramic sensor is used to record the response of the test structure. The result is processed using a signal processing technique known as bispectral analysis (Nikias & Raghuveer 1987), which has been developed by workers in the nonlinear dynamics field and allows nonlinear signals to be characterized with very low sensitivity to noise. Experiments and simulations are used to assess the sensitivity of the measurement technique to the detection of fatigue cracking and its insensitivity to amplifier distortions and other noise sources. In this work, amplifier distortion is taken to mean both harmonic distortion (the generation of harmonics when a single frequency signal is amplified) and intermodulation distortion (frequency mixing when two or more signals of different frequencies are amplified). The effects of both types of distortion are considered in later sections.

The paper is divided as follows: §2 describes the theoretical background to the bispectral processing method; §3 presents a simulation study to demonstrate the use of the bispectrum and its insensitivity to noise; and §4 presents the experimental procedure and describes results obtained from the application of this technique to steel specimens containing fatigue cracks.

## 2. Bispectral analysis

### (a) Background

Bispectral analysis belongs to the group of higher-order spectral analysis techniques, or polyspectra. It is well documented in the literature that bispectral analysis can provide information on the nonlinear properties of a system that is not available using traditional signal processing techniques such as power spectral analysis.

Detailed discussions on the theory behind and the interpretation of bispectral analysis may be found, for example, in Nikias & Raghuveer (1987), Fackrell *et al*. (1995*a*) and Collis *et al*. (1998). As well as providing information regarding system nonlinearities, bispectral analysis is known to provide increased sensitivity compared to power spectral analysis due to its inherent property of Gaussian noise suppression (Raghuveer & Nikias 1985). It is therefore an attractive signal analysis tool for detecting nonlinearities due to damage. For example, Howard (1997) demonstrated that the bispectrum could detect phase-coupling present in amplitude and phase modulated signals, and used higher-order signal processing techniques to show that vibration signals from a gearbox with a bearing defect were nonlinear. Stack *et al*. (2004) successfully extended this approach to detect defects in bearings, which resulted in amplitude modulated vibration signals. Fackrell *et al*. (1995*b*) investigated the use of bicoherance (a normalized version of the bispectrum) analysis for detecting loose bolted connections through white noise excitation. Rivola & White (1998) used bicoherence analysis to detect fatigue cracks in beams excited with Gaussian noise and it was concluded that, although the information contained in the bicoherence representation of a signal was difficult to interpret, there was scope for using bicoherence analysis to detect damage. The bispectrum has also been used in conjunction with neural networks (Xiang & Tso 2002; Chen *et al*. 2002; Yang *et al*. 2002) in order to characterize different types of structural defects.

In this section, we consider the key features of bispectral analysis that may be used to detect fatigue crack defects. Through simple analytical examples, we demonstrate the motivation for exciting the structure at two independent frequencies to detect fatigue cracks.

### (b) Theory of bispectral analysis

#### (i) Definition

We begin by considering a zero-mean random stationary process *x*(*t*) with Fourier transform *X*(*f*), where *t* and *f* are time and frequency, respectively. *X*(*f*) has the expected value *E*{*X*(*f*)}, which is equal to a weighted average of the possible outcomes of *X*(*f*) with the weights based upon the probabilities of those outcomes. *E*{*X*(*f*)} is alternatively known as the mean or first moment of *X*(*f*).

Accordingly, the power spectrum of *x*(*t*) is the second moment of *X*(*f*) and is given by(2.1)where ^{*} denotes complex conjugation.

The power spectrum contains no phase information and so can provide no direct evidence of phase coupling between frequency components of *X*(*f*), which in turn provides evidence of nonlinearity in the process *x*(*t*) (Nikias & Raghuveer 1987).

The bispectrum is the third-order equivalent, providing information on the skewness of the signal when decomposed over frequency. The time-domain representation of the bispectrum may be found in Raghuveer & Nikias (1985). It is more intuitive, however, to use the frequency-domain representation, which is given by(2.2)where *f*_{1} and *f*_{2} are two independent frequencies and *X*(*f*_{1}) is the component of the Fourier transform of *x*(*t*) at the frequency *f*_{1}.

The bispectrum takes the form of a surface with peaks at frequency pairs between two of the three spectral components {*f*_{1}, *f*_{2}, *f*_{1}+*f*_{2}}. These peaks indicate quadratic phase coupling (QPC), which is the coupling of frequency components within a signal resulting from their interaction with some non-symmetric, nonlinear mechanism within the measured system (Hinich 1982). Furthermore, the amplitude of these peaks provides a measure of the level of coupling present and hence the severity of the nonlinearity. A linear system would not exhibit this phase coupling and would have a zero bispectrum. Significantly, uncorrelated Gaussian noise also has a zero bispectrum (Raghuveer & Nikias 1985). This property is useful when analysing experimental data, which is often contaminated with Gaussian transducer noise (Rivola & White 1998).

In practice, the bispectrum must be estimated. This may be achieved by taking *N* separate datasets, but often is achieved by dividing a single dataset into *N* segments. A windowing function is applied to each segment and the Fourier transforms of all segments are averaged. Thus, the estimated bispectrum is given by(2.3)

A number of detailed studies discussing the mathematical properties of the bispectrum are presented in the literature (e.g. Nikias & Petropula 1993). In §2*b*(ii), we consider some of the features associated with the bispectrum that are key to the application described in this paper. It is worth noting that for broadband signals the bicoherence, a normalized version of the bispectrum, is often used (Rivola & White 1998); however, as the signals in this work have discrete frequency components it is not used here.

#### (ii) Properties of the bispectrum

In order to demonstrate the properties of the bispectrum, we proceed with a simple numerical example. It is common in the related literature to consider a signal composed of three frequency components when explaining the properties of the bispectrum (e.g. Nikias & Raghuveer 1987). Hence, we consider the continuous-time sinusoidal signal,(2.4)where *F*_{2}>*F*_{1} and *F*_{3}=*F*_{1}+*F*_{2}. This could represent the output from a nonlinear system excited by input signals at frequencies *F*_{1} and *F*_{2} (i.e. the signal component at the frequency *F*_{3} is generated as a result of the nonlinear system response, as discussed later in the section). The Fourier transform of this signal is given by(2.5)where *δ*(*f*−*F*) is the Dirac-delta function centred at frequency *F*. It follows that, *based upon this single set of data*, the bispectrum (equation (2.2)) is(2.6)where *δ*(*f*_{1}−*F*_{a}, *f*_{2}−*F*_{b}) is a two-dimensional Dirac-delta function defined in the (*f*_{1}, *f*_{2}) plane and may be written as the product of two one-dimensional Dirac-delta functions . The bispectrum is represented graphically in figure 1, for a window length of 2048 and frequencies {*F*_{1}=0.14*f*_{s}, *F*_{2}=0.2*f*_{s}}, where *f*_{s} is the sampling frequency.

A useful property of the bispectrum is its capability to detect QPC. First we consider the situation where we have multiple independent datasets containing signals of the form given in equation (2.4) and *ϕ*_{1}, *ϕ*_{2} and *ϕ*_{3} are independent random variables (i.e. the three frequency components are not coupled). In this case, since the bispectrum uses the expectation over all signals, due to the random variation in the phase of the bispectrum peaks (equation (2.6)), the averaged bispectrum over the multiple datasets reduces to zero as *N*→∞ (Nikias & Raghuveer 1987). However, for the case where *ϕ*_{3}=*ϕ*_{1}+*ϕ*_{2} (i.e. the three frequency components within the signal are quadratically phase coupled), we find that the resulting bispectrum is real-valued and non-zero.

As with the power spectrum, due to symmetry, the bispectrum need not be calculated over the whole frequency range. It can be shown that the region bounded by *f*_{1}=*f*_{2} and *f*_{1}=0 for positive *f*_{1} and *f*_{2} contains the complete description of the bispectrum. This region is further reduced when considering discrete-time signals (Raghuveer & Nikias 1985; Hinich & Wolinsky 1988). Considering only this region, equation (2.6) can be rewritten as the reduced bispectrum, *B*′(*f*_{1}, *f*_{2}) and is given by(2.7)

#### (iii) Quadratic systems

As previously stated, QPC can result from the modification of a signal by a quadratic nonlinearity. In order to demonstrate this we now consider a quadratic system described by(2.8)where *r* is the input to the system and *y* is the output. If the input is at a single frequency, i.e. *r*=sin(2*πFt*), then the bispectrum of the output, noting that the mean of *y* is removed before the bispectrum is performed, is given by(2.9)For a linear system, *a* is zero and, hence, so is the bispectrum of *y*. Thus, a single input frequency may be used to demonstrate quadratic nonlinearity. With a single excitation frequency, however, it is possible in practice to detect power amplifier nonlinearities which are manifest by the presence of harmonics of *F* within the input signal (Brotherhood *et al*. 2003). If this were the case, then these harmonics could mask the effects of system nonlinearity. The effects of amplifier nonlinearity are discussed in §4.

This problem may be overcome by exciting the system at two frequencies as discussed in §4. Hence, we now have the function(2.10)where *F*_{2}>*F*_{1} and *F*_{2} is not a harmonic of *F*_{1}. If the system is taken to be quadratic, using equation (2.8), then the output (after removing the mean) is given by(2.11)The resulting bispectrum is given by(2.12)where *α*=1 when 2*F*_{1}<*F*_{2} and *α*=0 when 2*F*_{1}>*F*_{2}. The similar condition on *β* is that *β*=1 when 3*F*_{1}<*F*_{2} and *β*=0 when 3*F*_{1}>*F*_{2}. In both cases, we obtain peaks in the bispectrum at points corresponding to the interaction between the difference frequency (*F*_{2}−*F*_{1}) and one or other of the input frequencies. Again it can be seen that, for a linear system with *a*=0, the resulting bispectrum will be zero.

## 3. Simulation of the dynamics of a cracked beam

In this section, as a first approximation, we use a model of a bilinear spring to represent a cracked beam due to the change in stiffness between compressive and tensile states. This is represented graphically in figure 2. For a given force input, *F*, the displacement *u* is measured. When in compression the crack is assumed to be fully closed and the beam effectively behaves as if undamaged with a stiffness *K*. When in tension the crack is assumed to be fully open and the beam behaves as a linear system with a reduced stiffness, *γK*. It is the transition between these two states that results in the nonlinear behaviour of the system.

The bispectrum method using two input frequencies is applied and used to provide a measure of the level of damage present in the measured system. Figure 3 shows the bilinear stiffness curve corresponding to 10% change in stiffness (*γ*=0.9) for positive displacement compared to negative displacement.

Simulations were conducted using the bilinear system with a range of different stiffness changes (again using frequencies *F*_{1}=0.14*F*_{s} and *F*_{2}=0.2*F*_{s}). Figure 4 demonstrates the advantage of using the bispectrum over the power spectrum.

A bilinear system was simulated with a small change in stiffness (2%) and Gaussian signal noise was added at a level 60 dB down from the input signal power. No difference is seen between the power spectrum plots for the linear and nonlinear systems (we would expect to see signal components at frequencies *F*_{2}±*F*_{1}, but these peaks are lost in the noise). Inspection of the corresponding bispectra for the two systems, however, reveals substantial differences. The linear system is seen to produce peaks in the bispectrum of its output, which is apparently at odds with the theory presented in §2. This is due to signal noise and represents the signal-to-noise ratio of the bispectrum method. The bispectrum for the nonlinear system, despite the small nonlinear effect, shows a large increase in the amplitude of the peaks. In particular, considering the peak at (*F*_{1}, *F*_{2}−*F*_{1}), we see an increase in amplitude by a factor of greater than 2. Hence, it is shown that the bispectrum can potentially provide a very sensitive method for detecting small system nonlinearities.

Figure 5 shows the amplitude of the bispectrum peak at (*F*_{1}, *F*_{2}−*F*_{1}) for the bilinear system as the percentage change in stiffness increases.

It is seen that the amplitude of the peak increases in a predictable fashion, which implies that this measure may be used to track the progression of damage.

In summary, we have described a method whereby the system to be examined is excited using two frequencies. This enables the detection of system nonlinearity, which cannot be confused with other effects such as amplifier harmonic distortion in an experimental setup. When combined with the bispectrum method we create a sensitive method for detecting and quantifying system nonlinearity. In §4, this method is applied to real test specimens, where the nonlinearity is a result of fatigue damage.

## 4. Experimental results

The technique described in §3 was applied to four steel specimens with dimensions 60 mm×60 mm×400 mm. One was undamaged and the remaining three contained fatigue cracks across their full widths and with depths of 5, 15 and 25 mm, respectively. These cracks were produced using four-point loading. The experimental setup was as shown in figure 6.

### (a) Amplifier nonlinearity in experimental setups

It is well known that amplifiers can introduce nonlinearity into a system. In order to demonstrate the effects of amplifier nonlinearity in the experimental setup, a simple test was conducted. The setup shown in figure 6 was used with an undamaged specimen excited at a single frequency (280 kHz) through a single amplifier. Two different amplifiers were used, one of which was known to produce greater levels of harmonic distortion than the other. The resulting bispectra for the two tests are shown in figure 7.

The plots for both amplifiers show peaks at (280,280)kHz, which corresponds to an interaction between the driving frequency and its first harmonic, which is generated by the amplifier nonlinearity. It is seen that the peak is greater in amplitude by a factor of approximately 10 for the more nonlinear amplifier. This effect is likely to be far greater than that resulting from damage in the specimen, which would consequently be masked. It is also possible for amplifiers to display intermodulation distortion when amplifying two or more separate frequencies (e.g. Meyer *et al*. 1972). This would result in mixing of the two driving frequencies in the technique discussed in §3, and, consequently, this effect would mask the nonlinear effect of the material damage. Significant intermodulation distortion was not found with the experimental setup used here, as demonstrated in figure 8*a*, which shows no significant bispectral peaks for the undamaged beam. If such a problem were to occur, it may be overcome by using two amplifiers and two transducers to excite the specimen at the two separate frequencies. In this case, significant mixing of the two frequencies will only occur as a result of material damage.

### (b) Application of the technique to specimens with fatigue cracks

Two arbitrary function generators were used to generate continuous sinusoidal excitation signals with frequencies of 280 and 462 kHz, respectively. The signals were amplified and then used to excite the specimens via 2 mm thick, 5 mm diameter piezo-ceramic elements (type PZ27). The input transducers were bonded to one end of the specimens and identical elements were used to measure their responses at the opposite ends. The ultrasonic excitation frequencies were chosen arbitrarily, the only constraints being that both driving frequencies and the sum and difference frequencies lay within the linear operating region of the transducers, and that *F*_{2} was not an integer multiple of *F*_{1}. The amplitudes of each of the input signals were adjusted, such that the amplitudes of the components at *F*_{1} and *F*_{2} in the measured response signal were identical in each test (8Vpp). This was to compensate for differences between the transducers on each specimen and the excitability of each frequency. Figure 8 shows the resulting bispectra for the four specimens.

It is observed that, as for the simulated results in §3, the amplitude of the bispectral peaks increases as the level of damage increases. Furthermore, if we consider the peak at (*F*_{1}, *F*_{2}−*F*_{1})=(280, 182 kHz), we see from figure 9 (the line corresponding to excitation at 462 and 280 kHz) that a similar trend is observed for this data as for the simulated data in figure 5. Hence, it appears that the experimental technique combined with bispectral analysis provides a highly sensitive method for detecting, quantifying and tracking material damage. Also shown in figure 9 are similar trends obtained from different combinations of excitation frequencies. Similar trends are observed in all cases, demonstrating the insensitivity of the technique to frequency selection. The amplitudes of the peaks increase as the excitation frequencies increase. This is assumed to result from the different energy levels and mode shapes associated with the different excitations and is a subject of ongoing research. It is worth noting that the noise level measured for the undamaged sample is low relative to the peak detected for the beam with a 5 mm crack, suggesting that the technique has potential for detecting far smaller cracks.

In order to demonstrate the insensitivity of the technique to geometry variations, a fresh undamaged beam was inspected using the technique described in §3. The geometry of this beam was then progressively altered by drilling a series of 13 mm diameter holes to a depth of half of the beam section. In total, three of these holes were drilled, and after each new hole the beam was inspected. Figure 10 shows the bispectrum results from the series of inspections.

It is observed that in each case the bispectral peaks are of very similar amplitude (and small compared to the damaged cases in figure 8), demonstrating that the technique is insensitive to small geometry variations.

## 5. Conclusions

This paper has described a nonlinear NDT technique, in which the test structure is excited by a waveform consisting of two sinusoidal waves of different frequencies. The response of a linear system consists of the same two frequencies, whereas the response of a nonlinear system contains mixing frequencies at the sum and difference of the input frequencies. The mixing frequencies are often too small to be readily identifiable in the frequency response function. It was shown that bispectral signal processing can be used to enhance the signal-to-noise ratio of the nonlinear effects, principally by reducing the noise in the signal. Experimental results using simple piezo-ceramic discs as exciters and receivers show that the technique can detect small fatigue cracks in steel bars. The measurement can also be used to provide an accurate, reliable method for tracking damage progression. Together, this combination of a cheap and simple sensor system and a robust analysis procedure offer the prospect of a global NDT technique, which can be applied reliably to engineering structures. The nonlinear system was modelled using a simple nonlinear spring, and was shown to produce similar results to those observed experimentally. The model was also used to demonstrate that the bispectrum changes in a predictable way as the extent of the nonlinearity is increased. It was shown both experimentally and in the model that, for a bilinear system, the amplitude of the mixing frequency peak (*F*_{2}−*F*_{1}, *F*_{1}) increases in line with the stiffness change. Experimentally, a 5 mm crack was detected with sufficient signal-to-noise ratio to suggest that the technique has potential for detecting far smaller cracks. It was also demonstrated experimentally that the technique is insensitive to small geometry variations.

## Acknowledgments

This work was funded under the EC project AERONEWS (FP6-502927) and through Dr Drinkwater's EPSRC fellowship.

## Footnotes

- Received August 11, 2005.
- Accepted November 22, 2005.

- © 2006 The Royal Society