Contents lists available at ScienceDirect

Journal of Sound and Vibration

Journal of Sound and Vibration 400 (2017) 305–316

http://d0022-46

n CorrE-m

journal homepage: www.elsevier.com/locate/jsvi

Nonlinear spectral correlation for fatigue crack detectionunder noisy environments

Peipei Liu, Hoon Sohn n, Ikgeun JeonDepartment of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, South Korea

a r t i c l e i n f o

Article history:Received 16 January 2017Received in revised form5 April 2017Accepted 12 April 2017Handling Editor: L.G. ThamAvailable online 24 April 2017

Keywords:Nonlinear wave modulationNonlinear spectral correlationFatigue crack detectionNoise reductionUltrasonic wavesWideband inputAir-coupled transducer

x.doi.org/10.1016/j.jsv.2017.04.0210X/& 2017 Elsevier Ltd. All rights reserved.

esponding author.ail address: hoonsohn@kaist.ac.kr (H. Sohn).

a b s t r a c t

When ultrasonic waves at two distinct frequencies are applied to a structure with a fatiguecrack, crack-induced nonlinearity creates nonlinear ultrasonic modulations at the sum anddifference of the two input frequencies. The amplitude of the nonlinear modulationcomponents is typically one or two orders of magnitude smaller than that of the primarylinear components. Therefore, the modulation components can be easily buried undernoise levels and it becomes difficult to extract the nonlinear modulation componentsunder noisy environments using a conventional spectral density function. In this study,nonlinear spectral correlation, which calculates the spectral correlation betweennonlinear modulation components, is proposed to isolate the nonlinear modulationcomponents from noisy environments and used for fatigue crack detection. The proposednonlinear spectral correlation offers the following benefits: (1) Stationary noises havelittle effect on nonlinear spectral correlation; (2) By using a wideband high-frequencyinput and a single low-frequency input, the contrast of nonlinear spectral correlationbetween damage and intact conditions can be enhanced; and (3) The test efficiency can bealso improved via reducing the data collection time. Validation tests are performed onaluminum plates and scaled steel shafts with real fatigue cracks. The experimental resultsdemonstrate that the proposed nonlinear spectral correlation owns a higher sensitivity tofatigue crack than the classical nonlinear coefficient estimated from the spectral densityfunction, and the usage of nonlinear spectral correlation allows the detection of fatiguecrack even using noncontact air-coupled transducers with a low signal-to-noise ratio.

& 2017 Elsevier Ltd. All rights reserved.

1. Introduction

A fatigue crack is a major cause for the failure of metallic structures [1]. Typically, a fatigue crack becomes conspicuousonly after the crack reaches about 80% of the total fatigue life of a structure [2]. Among various non-destructive evaluation(NDE) and structural health monitoring (SHM) techniques such as ultrasonic, acoustic emission, thermography, eddy cur-rent, magnetic particle inspection, X-ray and etc [3–6]., the ultrasonic technique is one of the most promising approaches forfatigue crack detection and has proven its effectiveness in achieving a reasonable compromise between resolution, prac-ticality and detectability. Conventional linear ultrasonic techniques measure variations of the amplitude, phase, and modeconversion of linear ultrasonic waves either transmitted or reflected from a crack [7–10]. However, it is difficult to detect afatigue crack at its early stage using the conventional linear ultrasonic techniques, because the changes of these linear

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316306

features only become prominent when the damage is severe. Recent studies have shown that a fatigue crack and its pre-cursor often serve as a source for generating nonlinear waves, and the sensitivity of the nonlinear ultrasonic techniques to afatigue crack is much higher than what can be achieved by the conventional linear techniques [11–21].

More specifically, nonlinear ultrasonic techniques detect a fatigue crack by investigating accompanying harmonics,modulations of different frequencies, or changing resonance frequencies as the amplitude of the driving input changes. Fornonlinear ultrasonic modulation, normally, a low-frequency input and a high-frequency input are simultaneously applied ona damaged structure to create modulation [12]. For example, a fatigue crack in an aluminum plate was detected by using apiezoelectric stack actuator for generation of a low-frequency input and a surface-mounted piezoelectric transducer forcreation of a high-frequency input [13]. Nonlinear ultrasonic modulation has also been used for detecting fatigue cracks inwelded pipe joints and concrete beams [14,15].

A few technical hurdles still, however, need to be overcome before these nonlinear ultrasonic techniques can maketransitions to real field applications. One issue is that the generation of nonlinear modulation is heavily dependent on thechoice of the input frequencies and can be easily affected by the configuration of the fatigue crack as well as variations of theenvironmental and operational conditions (e.g., temperature and loading) of the target structure [16,17]. Correspondingly,fixed low-frequency and swept high-frequency inputs were used to find an optimal combination of the low-frequency andhigh-frequency inputs that could amplify the amplitude of crack-induced modulation [18]. Fatigue cracks in aircraft fitting-lugmock-up specimens were detected by sweeping both low-frequency and high-frequency inputs [19]. But sweeping inputsignals over wide frequency ranges takes a long data collection time and can be impractical for field applications. In anotherstudy, a pulse laser input was used instead of two individual input frequencies for fatigue crack detection [20]. But since thelinear and nonlinear components overlapped each other in the frequency domain, cracks can only be detected by statisticallycounting the spectral peaks (or energy redistribution) produced by modulations among the broadband input frequencies.

Another issue is that the amplitude of nonlinear components is at least one or two orders of magnitude smaller than that ofthe linear components, so it is difficult to extract the nonlinear components using a conventional spectral density function(power spectrum) under noisy environments, especially when the noise overlaps the nonlinear components in the frequencydomain. The bispectrum was used to address this issue, which results in a frequency-frequency-amplitude relationshipshowing the coupling between signals at different frequencies [21]. The generation of non-zero bispectrum peaks due todamage-induced harmonics was numerically and experimentally demonstrated in the presence of white noise interference[22], and the bispectrum was also used to detect the modulation created by a fatigue crack in a metal specimen [23]. Fur-thermore, the non-stationary nature of the structural response was considered, and a baseline-free technique based on spectralcorrelation between nonlinear modulation components was developed for fatigue crack detection [24]. Here, the spectralcorrelation method has more advantages than the bispectrum in terms of computation time and flexibility of use [25].

This paper proposes a new nonlinear damage feature named nonlinear spectral correlation, and the combination of awideband high-frequency input and a single low-frequency input is used to enhance the performance of nonlinear spectralcorrelation for fatigue crack detection. The proposed technique offers the following advantages: (1) A new damage featurecoined nonlinear spectral correlation between two nonlinear modulation components is defined; (2) The nonlinear spectralcorrelation is insensitive to noise interference; (3) The contrast between damage and intact conditions is enhanced by usingnonlinear spectral correlation instead of classical nonlinear coefficient; (4) The possibility of satisfying the binding condi-tions, which are necessary for the generation of modulation, is increased using a wideband high-frequency input and asingle low-frequency input; and (5) The employment of a wideband high-frequency input and a single low-frequency inputalso reduces the data collection time significantly.

This paper is organized as follows. Section 2 introduces the proposed nonlinear spectral correlation and its enhancedsensitivity to damage by using a wideband high-frequency input and a single low-frequency input. Section 3 demonstratesthe effectiveness of the nonlinear spectral correlation by detecting real fatigue cracks in aluminum plates. In Section 4, theeffectiveness is further validated by detecting fatigue cracks in scaled steel shafts using noncontact air-coupled transducerswith a low signal-to-noise ratio. Finally, a conclusion is provided in Section 5.

2. Development of nonlinear spectral correlation

2.1. Nonlinear wave modulation

When two inputs with distinct frequencies fa and fb ( )>f fa b are applied to an intact (linear) structure, the structural

response contains the frequency components corresponding only to the input frequencies. Once the structure behavesnonlinearly (e.g., due to fatigue crack), the structural response will contain not only the input frequency components butalso their harmonics (multiples of input frequencies, i.e., f2 a, f2 b, etc.) and modulations (linear combinations of two input

frequencies, i.e., ±f fa b, ±f f2a b, ±f f2 a b, etc.), as illustrated in Fig. 1. This phenomenon is referred to as nonlinear ultrasonic

modulation or nonlinear wave modulation [12,26]. Because this phenomenon occurs only if nonlinear sources exist, it canbe considered a signature of the presence of nonlinearity and thus the existence of a crack, assuming that the inherentmaterial nonlinearity is weak. Considering only the first-order nonlinear modulations at +f fa b and −f fa b, their amplitudes

Fig. 1. Illustration of nonlinear wave modulation.

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316 307

+m and −m are proportional to the amplitudes at fa and fb based on the classical two-fold nonlinear interaction between faand fb [12]:

( )β~ ( )±±m f f ab, 1a b

wherea and b are the amplitudes of the linear responses at fa and fb, and β ( )+ f f,a b and β ( )− f f,a b are the classical nonlinearcoefficients between fa and fb for +m and −m at +f fa b and −f fa b, respectively.

Conventionally, a spectral density function ⎡⎣ ⎤⎦( )( ) ( ) ( )= *P f E X f X fx is used to extract the nonlinear modulation compo-

nents from a structural response ( )x t and estimate β±:

⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

( ) ( )( ) ( ) ( ) ( )

β ( ) =± * ±

* * ( )

± f fE X f f X f f

E X f X f E X f X f,

2a b

a b a b

a a b b

where ( )X f is the Fourier transform of ( )x t , * denotes the complex conjugate, and E is the expectation operation. However,because the amplitude of the nonlinear modulation components is at least one or two orders of magnitude smaller than thatof the linear components, it is difficult to extract the nonlinear modulation using this spectral density function in noisyenvironments, especially when the noise overlaps with the nonlinear modulation components in the frequency domain.Also, for the generation of nonlinear modulation, the following binding conditions should be satisfied in addition to theexistence of a fatigue crack [16,17]: (1) The strain (displacement) at the crack location should be oscillated by both inputs;and (2) The motion induced by one of the two inputs should modulate the other input at the crack location. That is, thegeneration of nonlinear modulation and the value of the nonlinear coefficient β± are heavily dependent on the choice of faand fb and can be easily affected by the configuration of the fatigue crack as well as by variations in the environmental andoperational conditions (e.g., temperature and loading) of the target structure [27].

2.2. Nonlinear spectral correlation

In this study, a new nonlinear damage feature called nonlinear spectral correlation (NSC) is defined by considering thecorrelation between the two nonlinear modulation components at +f fa b and −f fa b induced by inputs at fa and fb as follows:

⎛

⎝⎜⎜⎜

⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

⎞

⎠⎟⎟⎟

⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

( ) ( )( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )

( ) =+ * −

* *

=+ * −

* * ( )

NSC f fE X f f X f f

E X f X f E X f X f

E X f f X f f

E X f X f E X f X f

,

3

a b

a b a b

a a b b

a b a b

a a b b

2

In Eq. (3), ⎡⎣ ⎤⎦) ( )( + * −E X f f X f fa b a b indicates the spectral correlation between the two modulation components at +f fa b

and −f fa b, and NSC is the spectral correlation normalized by the responses at the two input frequencies. Spectral correlation

has been exploited in various fields including diagnosis of gear faults in moving mechanical systems [25,28,29], and channelsensing and spectrum allocation in wireless communication [30–32]. Normally, spectral correlation is used to identifysecond-order cyclostationary stochastic processes whose autocorrelation functions ( )τR t,x vary periodically with time [33]:

⎜ ⎟ ⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠⎤⎦⎥

( )( )

( )

τ τ

τ τ τ

= +

= + * −( )

R t R t T

R t E x t x t

, ,

,2 2 4

x x p

x

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316308

where τ is the time lag, Tp is the cyclic period, and E is the expectation operation with respect to time t . Spectral correlation

is then a double Fourier transform of the ( )τR t,x with respect to t and τ [33]:

∬( ) ( )τ τ= ( )α πα π τ− −S f R t t, e e d d 5x x

t fi2 i2

where α is the cyclic frequency and f is the spectral frequency. Eq. (5) can also be written as:

⎡⎣ ⎤⎦( ) ( )( ) α α= + * − ( )αS f E X f X f/2 /2 6x

Indeed, if the spectral correlation exists for at least one non-zero value of α, the signal is determined to be second-ordercyclostationary, and the corresponding two spectral components α+f /2 and α−f /2 are correlated. It is easy to see that, for

α = 0, ( )S fx0 equals its spectral density function ( ) = ( ) = [ ( ) *( )]S f P f E X f X fx x

0 . Another notable property of spectral correlation is

that stationary noise exhibits no spectral correlation (for α ≠ 0) [34]. So, ⎡⎣ ⎤⎦( ) ( ) ( )= + * −S f E X f f X f fxf

a a b a b2 b ( )α= = ≠f f f, 2 0a b

will be free from stationary noise interferences, and the proposed nonlinear spectral correlation NSC will be less affected bynoise than β±.

For ⎡⎣ ⎤⎦( ) ( )+ * −E X f f X f fa b a b , its corresponding correlation coefficient is defined as [34]:

⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

( ) ( )( ) ( )( )

( ) ( )=

+ * −

+ * + − * − ( )

c f fE X f f X f f

E X f f X f f E X f f X f f,

7a b

a b a b

a b a b a b a b

From Eqs. (2), (3) and (7), the proposed nonlinear spectral correlation NSC can be rewritten as:

β β= ( )+ −NSC c 8

which indicates that NSC includes both the classical nonlinear coefficient β±and the correlation coefficient c.

2.3. Sensitivity enhancement with a wideband high-frequency input

When a structure with a fatigue crack is subjected to inputs at fa and fb with amplitudes ( )A t and ( )B t , the response can be

represented as follows considering only the first-order nonlinear modulations at ±f fa b besides the linear responses at fa and fb:

( ) ( ) ( )= + + ( ) + ( ) ( )π π π π

+( + )

−( − )x t a t b t m t m te e e e 9

f t f t f f t f f ti2 i2 i2 i2a b a b a b

where ( )a t and ( )b t represent the amplitudes of the linear response components at fa and fb, ( )+m t and ( )−m t are the amplitudes

of nonlinear modulation components at +f fa b and −f fa b, respectively. For this response signal ( )x t , ⎡⎣ ⎤⎦( ) ( )+ * −E X f f X f fa b a b

can be derived from Eqs. (5) and (6):

⎜ ⎟ ⎜ ⎟

⎡⎣ ⎤⎦⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠⎤⎦⎥

( ) ( ) ∬∬

( )τ τ

τ τ τ

+ * − =

= + * −( )

π π τ− −

+ −

E X f f X f f R t t

E m t m t t

, e e d d

2 2d d

10

a b a b xf t fi2 2 i2b a

Eq. (10) shows that the spectral correlation ⎡⎣ ⎤⎦( ) ( )+ * −E X f f X f fa b a b between +f fa b and −f fa b depends on the cross-

correlation between ( )+m t and ( )−m t . If two pure sinusoidal signals with constant amplitudes ( )A t and ( )B t at two distinct

frequencies fa and fbare used as inputs, ( ) ( )a t b t, , ( )+m t and ( )−m t become constant and fully correlated. When the effect of

the noise interference is ignored, the nonlinear components at +f fa b and −f fa bare thus fully correlated and the corre-

sponding correlation coefficient c becomes 1, regardless whether the target structure is damaged or not. In summary, whentwo pure sinusoidal signals are used as inputs, (1) c becomes 1 even for the intact case due to the intrinsic materialnonlinearity that exists in an intact structure, and (2) NSC has an identical sensitivity to fatigue crack as β β+ − has

( β β= + −NSC ).In order to increase the sensitivity of the proposed nonlinear spectral correlation NSC to fatigue crack, a wideband high-

frequency signal ( ) πA t e f ti2 a and a single low-frequency signal ( ) πB t e f ti2 b are used as inputs by defining ( )A t and ( )B t as:

( )( )

∑=

= ( )

πA t A

B t B

e

11

pf ti2 p

where fp is the frequency within a range of < ≤f F0 p c (the frequency band of the high-frequency input is F2 c), Ap is the

amplitude of the corresponding component at fp in ( )A t , and they satisfy =− −f fp p and = −A Ap p. B is a time-invariant

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316 309

constant. Then, the inputs can be presented as:

∑( ) =

( ) = ( )

π π

π π

( + )A t A

B t B

e e

e e 12

f tp

f f t

f t f t

i2 i2

i2 i2

a a p

b b

where the wideband high-frequency input ( ) πA t e f ti2 a is represented as a summation of pure sinusoidal signals with varyingfrequencies around fa. If the resolution of Fourier transform is ∆f in the frequency domain, we can compute the nonlinearspectral correlation and the correlation coefficient between two nonlinear modulation components induced by inputs

+ ∆f m fa and fb as follows:

⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎤⎦( ) ( ) ( )

( ) ( ) ( ) ( )+ ∆ =+ ∆ + * + ∆ −

+ ∆ * + ∆ [ * ( )NSC f m f f

E X f m f f X f m f f

E X f m f X f m f E X f X f,

13a b

a b a b

a a b b

⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

( ) ( )( ) ( ) ( ) ( ) ( )+ ∆ =

+ ∆ + * + ∆ −

+ ∆ + * + ∆ + + ∆ − * + ∆ − ( )c f m f f

E X f m f f X f m f f

E X f m f f X f m f f E X f m f f X f m f f,

14a b

a b a b

a b a b a b a b

From Eq. (1), the corresponding amplitudes of the nonlinear modulation components induced by + ∆f m fa and fbcan bederived as:

( )

( )

∑

∑

β

β

^ =

^ =( )

π

π

+= ∆ − ∆

∆ + ∆+

−= ∆ − ∆

∆ + ∆−

m t a b

m t a b

e

e15

f m f f

m f f

p pf t

f m f f

m f f

p pf t

0.5

0.5i2

0.5

0.5i2

p

p

p

p

where fp is within the frequency range of − ∆ ≤ ∆f m f f0.5p , ap and b are the amplitudes of the linear responses at +f fa pand

fb, and β±p are the nonlinear coefficients between +f fa p and fb, respectively. Note that, there are no time-invariant terms

( ≠f 0p ) in Eq. (15) and the frequency component fp has different amplitudes of β+a bp p and β−a bp p in ^ ( )+m t and ^ ( )−m t , respec-

tively. When the structure is intact, β±p are weak (close to zero) and the relative variations among β±

p become significant in

comparison to the amplitudes of β±p . For the damage case if + ∆f m fa and fb satisfy the binding conditions, allβ±

p will increase

because +f fa p is within a narrow frequency band of ∆f around + ∆f m fa . In this case, the relative variations among β±p

become negligible in comparison to the increased amplitudes of β±p . Hence, the relative variation between amplitudes of

β+a bp p and β−a bp p is smaller for the damage case than for the intact case. Since the correlation between ^ ( )+m t and ^ ( )−m t is

proportional to the product of β+a bp p and β−a bp p for all fp in the frequency domain [35], consequently, ^ ( )+m t and ^ ( )−m t becomemore correlated and the corresponding correlation coefficient c will increase in return for the damage case.

In summary, by using a wideband high-frequency input and a single low-frequency input as shown in Fig. 2, NSC is notonly robust to noise interferences, but also becomes more sensitive to fatigue crack formation. Moreover, the employment ofa wideband high-frequency input, composing multiple input frequency combinations along with the single low-frequencyinput, can increase the possibility of satisfying the binding conditions for nonlinear modulation generation, and can reducethe data collection time compared to sweeping of pure sinusoidal inputs for the same high-frequency band.

Fig. 2. Nonlinear spectral correlation using a wideband high-frequency input and a single low frequency input.

Fig. 3. Aluminum plate with fatigue crack: (a) Geometrical dimensions and PZT transducer arrangement; (b) Microscopic image of the fatigue crack tip.

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316310

3. Experimental validation on aluminum plates

3.1. Experimental setup

Two 3-mm thick plate specimens made of 6061-T6 aluminum alloy are prepared to validate the performance of theproposed nonlinear spectral correlation. All geometrical information concerning the plate specimen is presented in Fig. 3(a).A cyclic loading test was carried out using a universal testing machine with a 10 Hz cycle rate, a maximum load of 25 kN anda stress ratio of 0.1 so that a fatigue crack can initiate from a notch located in the middle of one edge of a specimen. Thelength of the fatigue crack was around 17 mm, and its overall width was less than 50 μm and less than 10 μm near the cracktip, as shown in Fig. 3(b).

Three identical piezoelectric transducers (APC International) with a diameter of 18 mm and a thickness of 0.508 mmwere attached to each specimen and were labelled as PZT1, PZT2 and PZT3 (Fig. 3(a)). PZT1 and PZT2 were used to generatethe high-frequency and low-frequency inputs, respectively, while PZT3 was used for sensing. All these PZT transducers wereconnected to a data acquisition system [19]. The data acquisition system was comprised of two National Instruments (NI)arbitrary waveform generators (NI-PXI-5421), a high speed digitizer (NI-PXI-5122) and an embedded controller (NI-PXI-8105). The two arbitrary waveform generators were used to generate high-frequency and low-frequency input signals,respectively, and the high speed digitizer to measure the structural response. The embedded controller was for synchro-nizing and controlling the arbitrary waveform generators and the high speed digitizer.

For the input signals defined in Eq. (12), fa was selected as 183 kHz, fb as 35 kHz, and fp within the range of < ≤f0 5kHzp

( =F 5kHzc ) with 1 Hz interval. The values of fa, fb, and Fc were selected such that the amplitudes of the frequency response

function at the corresponding input and modulation frequencies would become relatively large. Here, the wideband high-frequency input signal can be seen as sum of pure sinusoidal signals with different single frequencies ranging from 178 to188 kHz. The amplitudes of both the high-frequency and low-frequency input signals were set to a peak-to-peak voltage of20 V, and the responses were measured with a 1 MHz sampling rate for 1 s. Each response was measured 10 times andaveraged in the time domain to improve the signal-to-noise ratio. For comparison, the experiments were also conducted bysweeping a sinusoidal high-frequency input from 178 to 188 kHz with 1 kHz increment.

Fig. 4. Correlation coefficient c obtained with (a) a wideband high-frequency input and (b) sweeping high-frequency inputs.

Fig. 5. Sensitivity of c to a fatigue crack when a wideband high-frequency input or sweeping high-frequency inputs were applied.

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316 311

3.2. Test results

In this experiment, the acquired responses were divided into one hundred 0.01 sec segments so that the expectationoperations in the equations listed above can be approximated with one hundred averaging. In this way, the frequencyresolution of Fourier transform becomes 100 Hz (∆ =f 100 Hz), and the nonlinear spectral correlation NSC and the cor-relation coefficient c values for fb and + ∆f m fa can be computed from 178 to 188 kHz with 100 Hz increment. In terms of data

collection time, sweeping of sinusoidal signals for the high-frequency input takes about 100 times longer than using theproposed wideband high-frequency input to achieve the same amount of test results.

Fig. 4(a) plots the correlation coefficient c obtained with the wideband high-frequency input. As discussed previously, cincreases when a fatigue crack presents. Fig. 4(b) shows the c values obtained using sweeping high-frequency inputs. In thiscase, the corresponding c becomes 1 regardless of the existence of fatigue crack. The sensitivity of the correlation coefficientc to a fatigue crack is quantitatively estimated in Fig. 5 based on the following definition of sensitivity:

⎛⎝⎜⎜

⎞⎠⎟⎟∑=

( )=

sN

p

p

1

16n

Nnd

ni

1

2

where pnd and pn

i represent a damage feature obtained from the damage and intact conditions, respectively, and N is the totalnumber of the acquired damage feature values. In the current example, the sensitivity is calculated by defining c as thedamage feature. Fig. 5 compares the sensitivity of the c values, which are obtained under two different input scenarios. Fig. 5shows that, at the presence of a fatigue crack, c increases only when a wideband high-frequency input is employed.

Next, the sensitivities of the proposed nonlinear spectral correlation NSC and the classical nonlinear coefficient β β+ −to afatigue crack are compared. Here, β β+ −is defined as:

⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

( ) ( )( )( ) ( )

( ) ( ) ( ) ( )β β =+ * + − * −

* * ( )+ − f f

E X f f X f f E X f f X f f

E X f X f E X f X f,

17a b

a b a b a b a b

a a b b

Fig. 6 compares the β β+ − and NSC values obtained using the wideband high-frequency input. Though β β+ − and NSC showsimilar patterns, the NSC value is slightly smaller than the β β+ − value, because the c (Fig. 4(a)) value is always less than

Fig. 6. (a) Nonlinear coefficient β β+ −, and (b) nonlinear spectral correlation NSC obtained with a wideband high-frequency input.

Fig. 7. Sensitivities of β β+ − and NSC to a fatigue crack when (a) a wideband high-frequency input and (b) sweeping high-frequency inputs were applied.

Fig. 8. (a) Nonlinear coefficient β β+ −, and (b) nonlinear spectral correlation NSC obtained with sweeping high-frequency inputs.

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316312

1 when a wideband high-frequency input is used. More importantly, NSC is more sensitive to a fatigue crack than β β+ − as

shown in Fig. 7(a). In Fig. 7, Eq. (16) is again used to compare the sensitivities of β β+ − and NSC to a fatigue crack. Here, pnd and

pni represent β β+ − or NSC obtained from the damage and intact conditions.

Fig. 8 plots the β β+ − and NSC values obtained with the sweeping high-frequency inputs. Here, because the c (Fig. 4(b))

value is always around 1, β β+ − in Fig. 8(a) and NSC in Fig. 8(b) are almost identical to each other. For the same token, Fig. 7

(b) shows that the s values for β β+ − and NSC are practically identical when sweeping high-frequency inputs are employed.The robustness of the proposed nonlinear spectral correlation NSC to noise was investigated by adding simulated sta-

tionary white noises with different signal-to-noise ratios (SNRs) to the test signals acquired from the damaged specimen.The SNR is defined as:

= ( )( )

SNRE

E10log dBW

18signal

noise10

where Esignal and Enoise are the energy of the acquired test signal and the added white noise, respectively. SNR varied from 60to 40 dBW with 2 dBW decrements for the test signals obtained using a wideband high-frequency input, and from 60 to 20dBW with 2 dBW decrements for the test signals obtained with sweeping high-frequency inputs, respectively. The c, β β+ −

and NSC values were calculated using these noise-contaminated test signals. The effects of the simulated white noise onthese nonlinear damage features are evaluated by d, which is defined as:

⎛⎝⎜⎜

⎞⎠⎟⎟∑=

^ −×

( )=

dN

q q

q1

100%19n

Nn n

n1

2

where qn represents one of the damage features (c, β β+ − or NSC) calculated from the original test signals, q̂n refers to thecorresponding damage feature calculated from the noise-contaminated test signals with different SNRs. Note that a smallerd value indicates a better robustness with respect to noises.

Fig. 9(a) and (b) plot the d values of c, β β+ − and NSC obtained using a wideband high-frequency input and sweeping high-

frequency inputs, respectively. NSC is much more robust with respect to the increased noises than c or β β+ −. This superior

Fig. 9. Robustness of c , β β+ − and NSC with respect to the increased noise levels when (a) a wideband high-frequency input and (b) sweeping high-frequency inputs were applied.

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316 313

robustness of NSC is attributed to the facts that (1) Stationary noise has no influence on spectral correlation⎡⎣ ⎤⎦( ) ( )+ * −E X f f X f fa b a b ; and (2) The noise effect on ⎡⎣ ⎤⎦( ) ( )*E X f X fa a and ⎡⎣ ⎤⎦( ) ( )*E X f X fb b is much smaller than its effect on

⎡⎣ ⎤⎦) )( + *( +E X f f X f fa b a b and ⎡⎣ ⎤⎦) ( )( − * −E X f f X f fa b a b because the amplitude of nonlinear modulations is at least one or two

orders of magnitude smaller than that of the linear responses.

4. Fatigue crack detection on steel shafts

4.1. Experimental setup

The proposed nonlinear spectral correlation was also applied to detect fatigue cracks in steel shaft specimens. The testspecimens are scale-down version of real shafts used in automobile assembly lines. Shafts are commonly used as parts ofrotating components, and they are susceptible to fatigue crack as they are subjected to repeated sudden acceleration anddeceleration. The dimensions of the shaft specimens are presented in Fig. 10(a). Fatigue cracks were introduced to two ofthree specimens, and the fatigue cracks were initiated from the point where the diameter of the shaft suddenly changed asindicated in Fig. 10. The specimens were subjected to a torsional cyclic loading of 5 kN∙m for about 15,000 cycles (Fig. 11).The existence of the crack was confirmed with a separate penetrant test as shown in Fig. 10(b).

Note that, because shafts in real applications need to rotate, it is difficult to use contact-type transducers like the pre-vious PZT transducers. Instead, three noncontact piezoelectric air-coupled transducers were adopted in this experiment,labelled ACT1, ACT2 and ACT3 in Fig. 10(a). For generating high-frequency and low-frequency inputs, NCG200-S38 (TheUltran Group) and NCG50-S38 (The Ultran Group) with a central frequency of 200 kHz and 50 kHz were used, respectively.BAT-1 (MicroAcoustic Instruments) was used to measure the response with a frequency band of 40 kHz to 2.25 MHz. Thedistance from the air-coupled transducers to the shaft specimen was around 20 mm and their incidence angle was set to bearound °2 based on Snell’s law [36]. This experiment adopted the same data acquisition system from the previous ex-periment. The only difference is that a linear amplifier (Piezo Systems, EPA-104) was adopted to increase the peak-to-peakvoltage of the wideband high-frequency input signal to 100 V so that ACT1 can effectively generate ultrasonic waves on theshaft specimen. For the input signals in this experiment (Eq. (12)), fa was selected as 189 kHz, fb as 38 kHz, and fp within the

range of < ≤f0 5kHzp ( =F 5kHzc ) with 1 Hz interval. Again, the wideband high-frequency input signal can be viewed as the

Fig. 10. Steel shaft with fatigue crack: (a) Dimensions and ACT transducer arrangement; (b) A close-up of the fatigue crack.

Fig. 11. Fatigue test on steel shafts.

Fig. 12. Correlation coefficient c obtained from (a) damage I and (b) damage II specimens.

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316314

summation of pure sinusoidal signals with varying frequency values ranging from 184 to 194 kHz. The responses weremeasured with a 1 MHz sampling rate for 1 s. Each response was measured 10 times and averaged in the time domain toimprove the signal-to-noise ratio (SNR). Because the efficiency of energy transmission between the transducers and thetarget specimen via air is quite low, the SNR in this experiment will be much lower than the SNR achieved in the previousexperiment using contact-type transducers. This poor SNR provides a good opportunity to demonstrate the effectiveness ofNSC in detecting a fatigue damage under noisy environments. Indeed, with only a few exceptions [37,38], little work hasbeen done in terms of detecting damage-induced nonlinear modulation using air-coupled transducers.

4.2. Test results

Fig. 12 plots the correlation coefficient c values obtained from three shaft specimens, labeled intact, damage I and damage II,respectively. Note that the c values in Fig. 12 are much lower than the c values from the previous experiment (Fig. 4(a)). Thisdifference is mainly attributed to the low SNR of the signals acquired using the ACTs. Eq. (16) is used to evaluate the sensitivity of c to

fatigue cracks in damage I and damage II specimens, respectively. Here, pni represents c obtained from the intact specimen, pn

d

represents c obtained from damage I or damage II specimen. Fig. 13 shows the s values obtained for damages I and II specimens. Itshows that the c value increased for both damage I and II specimens relative to its value obtained from the intact specimen when awideband high-frequency input was applied.

Fig. 13. Sensitivity of c to fatigue cracks in damage I and II specimens.

Fig. 14. (a) Nonlinear coefficient β β+ −, and (b) nonlinear spectral correlation NSCobtained from three steel shaft specimens.

Fig. 15. Sensitivities of β β+ − and NSC to a fatigue crack for (a) damage I and (b) damage II specimens.

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316 315

The nonlinear coefficient β β+ − and the proposed nonlinear spectral correlation NSC values obtained from three shaftspecimens are plotted in Fig. 14(a) and (b), respectively. Due to the low SNR, the NSC values become much smaller than theβ β+ − values, which also indicates that the proposed nonlinear spectral correlation NSC is much less affected by noise in-terferences. The sensitivities of β β+ − and NSC to a fatigue crack are also calculated using Eq. (16) and shown in Fig. 15. Here,

pni represents β β+ − or NSC obtained from the intact specimen, pn

d represents β β+ − or NSC obtained from damage I or damage II

specimen. Fig. 15 proves again that, compared with the classical nonlinear coefficient β β+ −, the proposed nonlinear spectralcorrelation NSC is more sensitive to a fatigue crack for both damage I and damage II specimens. Moreover, the employmentof a wideband high-frequency input can reduce data collection time significantly when compared to sweeping sinusoidalinputs for the identical high-frequency band.

5. Conclusions

This study proposes a new nonlinear damage feature named nonlinear spectral correlation by considering the spectral cor-relation between two nonlinear modulation components at sum and difference of the two input frequencies. A wideband high-frequency signal and a single low-frequency signal are also used as inputs to enhance the contrast of the proposed nonlinearspectral correlation between damage and intact conditions, and to reduce the data collection time. The major advantages of theproposed nonlinear spectral correlation over classical nonlinear coefficient are: (1) The nonlinear spectral correlation is much lessaffected by noise interferences; and (2) The nonlinear spectral correlation has a higher sensitivity to fatigue crack.

The experimental tests performed on aluminum plates indicate that, the sensitivity of the nonlinear spectral correlationwas two times better than the classical nonlinear coefficient when a wideband high-frequency input was used instead ofsweeping sinusoidal inputs for the same frequency band. When SNR decreased from 60 to 20 dBW, the variation in thenonlinear spectral correlation was less than 0.5% while the correlation coefficient and the classical nonlinear coefficientshowed significant changes of more than 40%, which might lead to serious false alarms in real field applications. The testresults performed on steel shafts with air-coupled transducers show that the proposed nonlinear spectral correlation ismuch less affected by noise, and maintains its higher sensitivity to fatigue crack than the classical nonlinear coefficient evenunder noisy environments.

P. Liu et al. / Journal of Sound and Vibration 400 (2017) 305–316316

Acknowledgement

This work was supported by the Eurostars-2 Joint Programme (E!9679/N0001699) with co-funding from the EuropeanUnion Horizon 2020 Research and Innovation Programme and Korea Institute for Advancement of Technology (KIAT), and agrant (13SCIPA01) from Smart Civil Infrastructure Research Program funded by Ministry of Land, Infrastructure andTransport (MOLIT) of Korea government and Korea Agency for Infrastructure Technology Advancement (KAIA).

References

[1] P.G. Forrest, Fatigue of Metals, Pergamon Press, Oxford, 1962.[2] J.Y. Kim, L.J. Jacobs, J. Qu, Nonlinear ultrasonic techniques for nondestructive damage assessment in metallic materials, in: Proceedings of the 8th

International Workshop on Structural Health Monitoring, September, Stanford, CA, 2011.[3] V. Zilberstein, D. Schlicker, K. Walrath, V. Weiss, N. Goldfine, MWM eddy current sensors for monitoring of crack initiation and growth during fatigue

tests and in sevice, Int. J. Fatigue 23 (2001) 477–485.[4] Y.K. An, J.M. Kim, H. Sohn, Laser lock-in thermography for detection of surface-breaking fatigue cracks on uncoated steel structures, NDTE Int. 65

(2014) 54–63.[5] J.J. Williams, K.E. Yazzie, E. Padilla, N. Chawla, X. Xiao, F. De Carlo, Understanding fatigue crack growth in aluminum alloys by in situ X-ray synchrotron

tomography, Int. J. Fatigue 57 (2013) 79–85.[6] S.K. Ho, R.M. White, J. Lucas, A vision system for automated crack detection in welds, Meas. Sci. Technol. 1 (1990) 287–294.[7] J. Tian, Z. Li, X. Su, Crack detection in beams by wavelet analysis of transient flexural waves, J. Sound Vib. 4 (2003) 715–727.[8] D.A. Cook, Y.H. Berthelot, Detection of small surface-breaking fatigue cracks in steel using scattering of Rayleigh waves, NDT & E Int. 34 (2001)

483–492.[9] S. Park, C. Lee, H. Sohn, Reference-free crack detection using transfer impedances waves, J. Sound Vib. 12 (2010) 2337–2348.

[10] L. Qiu, S. Yuan, Q. Bao, H. Mei, Y. Ren, Crack propagation monitoring in a full-scale aircraft fatigue test based on guided wave-Gaussian mixture model,Smart Mater. Struct. 25 (2016) 055048.

[11] N. Pugno, C. Surace, R. Ruotolo, Evaluation of the non-linear dynamic response to harmonic excitation of a beamwith several breathing cracks, J. SoundVib. 235 (2000) 749–762.

[12] K.E.A. Van Den Abeele, P.A. Johnson, A. Sutin, Nonlinear elastic wave spectroscopy (NEWS) techniques to discern material damage, Part I: nonlinearwave modulation spectroscopy (NWMS), Res. Nondestruct. Eval. 12 (2000) 17–30.

[13] Z. Parsons, W.J. Staszewski, Nonlinear acoustics with low-profile piezoceramic excitation for crack detection in metallic structures, Smart Mater. Struct.15 (2006) 1110–1118.

[14] D.M. Donskoy, A.M. Sutin, Vibro-acoustic modulation nondestructive evaluation technique, J. Intell. Mater. Syst. Struct. 9 (1998) 765–771.[15] I.N. Didenkulov, A.M. Sutin, A.E. Ekmov, V.V. Kazakov, Interaction of sound and vibrations in concrete with cracks, AIP Conference Proceedings,

vol. 524, 2000, pp. 279–282.[16] H.J. Lim, H. Sohn, P. Liu, Binding conditions for nonlinear ultrasonic generation unifying wave propagation and vibration, Appl. Phys. Lett. 104 (2014)

1–5.[17] V.Y. Zaitsev, L.A. Matveev, A.L. Matveyev, On the ultimate sensitivity of nonlinear modulation method of crack detection, NDT & E Int. 42 (2009)

622–629.[18] N.C. Yoder, D.E. Adams, Vibro-acoustic modulation using a swept probing signal for robust crack detection, Struct. Health Monit. 9 (2010) 257–267.[19] H. Sohn, H.J. Lim, M.P. DeSimio, K. Brown, M. Derisso, Nonlinear ultrasonic wave modulation for fatigue crack detection, J. Sound Vib. 333 (2013)

1473–1484.[20] P. Liu, H. Sohn, T. Kundu, S. Yang, Noncontact detection of fatigue cracks by laser nonlinear wave modulation spectroscopy (LNWMS), NDT & E Int. 66

(2014) 106–116.[21] C.R.P. Courtney, S.A. Neild, P.D. Wilcox, B.W. Drinkwater, Application of the bispectrum for detection of small nonlinearities excited sinusoidally, J.

Sound Vib. 329 (2010) 4279–4293.[22] A. Rivola, P.R. White, Bispectral analysis of the bilinear oscillator with application to the detection of fatigue cracks, J. Sound Vib. 216 (1998) 889–910.[23] A.J. Hillis, S.A. Neild, B.W. Drinkwater, P.D. Wilcox, Global crack detection using bispectral analysis, Proc. R. Soc. A 462 (2006) 1515–1530.[24] P. Liu, H. Sohn, S. Yang, H.J. Lim, Baseline-free fatigue crack detection based on spectral correlation and nonlinear wave modulation, Smart Mater.

Struct. 25 (2016) 125034.[25] L. Bouillaut, M. Sidahmed, Cyclostationary approach and bilinear approach: comparison, applications to early diagnosis for helicopter gearbox and

classification method based on HOCS, Mech. Syst. Signal Process. 15 (2001) 923–943.[26] W.J.N. de Lima, M.F. Hamilton, Finite-amplitude waves in isotropic elastic plates, J. Sound Vib. 265 (2003) 819–839.[27] H.J. Lim, H. Sohn, M.P. DeSimio, K. Brown, Reference-free fatigue crack detection using nonlinear ultrasonic modulation under changing temperature

and loading conditions, Mech. Syst. Signal Process. 45 (2014) 468–478.[28] G. Dalpiaz, A. Rivola, R. Rubini, Effectiveness and sensitivity of vibration processing techniques for local fault detection in gears, Mech. Syst. Signal

Process. 14 (2000) 387–412.[29] J. Antoni, Cyclostationarity by examples, Mech. Syst. Signal Process. 23 (2009) 987–1036.[30] N. Han, S.H. Shon, J.H. Chung, J.M. Kim, Spectral correlation based signal detection method for spectrum sensing in IEEE 802.22 WRAN systems, IEEE

in: Proceedings of the 8th International Conference Advanced Communication Technology, vol. 3, 2006, pp. 1765–1770.[31] A. Fehske, J. Gaeddert, J. Reed, A new approach to signal classification using spectral correlation and neural networks, in: IEEE International Sym-

posium on New Frontiers in Dynamic Spectrum Access Networks, 2005, pp. 144–150.[32] D.S. Yoo, J. Lim, M.H. Kang, ATSC digital television signal detection with spectral correlation density, J. Commun. Netw. 16 (2014) 600–612.[33] W.A. Gardner, Cyclostationarity: half a century of research, Signal Process. 86 (2006) 639–697.[34] W.A. Gardner, Measurement of spectral correlation, IEEE Trans. Acoust. Speech Signal Process. 34 (1986) 1111–1123.[35] E. Kreyszig, Advanced Engineering Mathematics, ninth ed, Wiley, 2006.[36] M. Castaings, P. Cawley, The generation, propagation, and detection of Lamb waves in plates using air‐coupled ultrasonic transducers, J. Acoust. Soc.

Am. 100 (1996) 3070–3077.[37] H.J. Lim, B. Song, B. Park, H. Sohn, Noncontact fatigue crack visualization using nonlinear ultrasonic modulation, NDT & E Int. 73 (2015) 8–14.[38] E.M. Ballad, S.Y. Vezirov, K. Pfleiderer, I.Y. Solodov, G. Busse, Nonlinear modulation technique for NDE with air-coupled ultrasound, Ultrasonics 42

(2004) 1031–1036.