sensors Article

Influence of the Spatial Dimensions of Ultrasonic Transducers on the Frequency Spectrum of Guided Waves Vykintas Samaitis * and Liudas Mažeika Prof. K.Baršauskas Ultrasound Research Institute, Kaunas University of Technology, K. Baršausko St. 59, LT-51423 Kaunas, Lithuania; [email protected] * Correspondence: [email protected]; Tel.: +370-(37)-351-162 Received: 26 June 2017; Accepted: 4 August 2017; Published: 8 August 2017

Abstract: Ultrasonic guided wave (UGW)-based condition monitoring has shown great promise in detecting, localizing, and characterizing damage in complex systems. However, the application of guided waves for damage detection is challenging due to the existence of multiple modes and dispersion. This results in distorted wave packets with limited resolution and the interference of multiple reflected modes. To develop reliable inspection systems, either the transducers have to be optimized to generate a desired single mode of guided waves with known dispersive properties, or the frequency responses of all modes present in the structure must be known to predict wave interaction. Currently, there is a lack of methods to predict the response spectrum of guided wave modes, especially in cases when multiple modes are being excited simultaneously. Such methods are of vital importance for further understanding wave propagation within the structures as well as wave-damage interaction. In this study, a novel method to predict the response spectrum of guided wave modes was proposed based on Fourier analysis of the particle velocity distribution on the excitation area. The method proposed in this study estimates an excitability function based on the spatial dimensions of the transducer, type of vibration, and dispersive properties of the medium. As a result, the response amplitude as a function of frequency for each guided wave mode present in the structure can be separately obtained. The method was validated with numerical simulations on the aluminum and glass fiber composite samples. The key findings showed that it can be applied to estimate the response spectrum of a guided wave mode on any type of material (either isotropic structures, or multi layered anisotropic composites) and under any type of excitation if the phase velocity dispersion curve and the particle velocity distribution of the wave source was known initially. Thus, the proposed method may be a beneficial tool to explain and predict the response spectrum of guided waves throughout the development of any structural health monitoring system. Keywords: structural health monitoring; ultrasonic guided wave excitation; excitability function; frequency response; MFC transducer

1. Introduction Ultrasonic guided waves (GWs) have shown great potential in the structural health monitoring (SHM) of various ageing engineering components to ensure their safe, reliable, and optimal performance [1,2]. GWs are sensitive to changes in the elastic modulus of the material and possess minor amplitude damping, which enables the inspection of large structures using only a few transducers and the detection of both surface and internal defects in almost real-time [3]. Lamb waves are a kind of guided wave that propagate in plate-like structures in cases where the plate thickness is considerably lower than the wavelength. Lamb waves provide high interrogation volumes and are sensitive to defects of different kinds, hence many studies have been conducted on the Sensors 2017, 17, 1825; doi:10.3390/s17081825

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application of Lamb waves to detect delaminations [4–8], cracks [9–11], notches [12–14], impact damages [15–19], and other structural non-homogeneities. The application of Lamb waves in such a wide variety of areas, all looking for different kinds of defects, shows the huge potential of GW inspection. However, as Lamb waves possess a multi-modal, dispersive, and mode converting nature, complicated signal analysis is a key issue limiting the development of reliable inspection systems. The signals captured from the structure are often distorted and overlapped, while the damage-scattered components are concealed within. To simplify guided wave inspection, various approaches have been used to generate Lamb waves with the desired dispersive properties, sensitivity to defects, excitability, and detectability [20]. Of these, piezoelectric wafers [21–23], interdigital transducers [24–29] and phased-delay excitation [30–32] have attracted the most attention. A proper understanding of the response of various ultrasonic transducers has been of great interest among various researchers. Giurgiutiu et al. [33–35] completed extensive work on analyzing the behavior of piezoelectric wafer active sensors (PWAS) where they used a 2D plane strain model based on the integral transform method and demonstrated that maximal response amplitudes of PWAS were at frequencies, where the width of transducer was equal to odd multiple half wavelengths. In the same fashion, for the even number of half wavelengths response minima can be found. Due to different velocities, the response of A0 and S0 modes of Lamb waves were different, hence the desired mode could either be enhanced or suppressed by selecting the appropriate size of the PWAS. Similarly, Raghavan and Cesnik [36] analyzed the displacement response of a piezo-disc actuator as a function of its length. Grondel et al. [37,38] used a normal mode expansion method to investigate selective A0 mode excitation using surface bonded piezoceramic transducers. Comprehensive research on Lamb wave excitation was presented by Lanza di Scalea et al. [39] where they analyzed the response of surface bonded transducer to Rayleigh and Lamb waves by employing both harmonic and pulse excitation. Zeng et al. [40] analyzed waveform distortions caused by the amplitude dispersion of guided waves and established a method based on the Vold–Kalman filter and Taylor series expansion to remove such effects. Finally, Schubert et al. [41] focused on the impact of amplitude dispersion and sensor size on the response of PWAS attached to composite structures where the authors combined analytical excitability functions, phase velocity, attenuation curves and geometric influences of the transducer to predict the output signal of PWAS. Some research related to the response of piezocomposite transducers, like macro-fiber composites (MFC) [42] can also be found in the literature. MFCs are directional transducers made of uniaxially aligned rectangular cross-section fibers, surrounded by a polymer matrix and comb type electrode pattern [20]. They provide good surface conformability due to the piezocomposite structure, are wideband, non-brittle, thin and low-cost, which makes them a perfect choice for structural health monitoring applications. Matt et al. [43] proposed an approach of MFC rosettes for passive damage localization, which allowed them to extract the direction of the incoming wave. Throughout the development of the rosette approach, the authors estimated the frequency response of a MFC transducer at different wave propagation angles. Furthermore, the authors demonstrated that the frequency response of Lamb waves excited by a MFC transducer showed local minima and maxima, which was related to the length of the transducer and properties of the medium. Birchmeier et al. [44] analyzed the generation of Lamb waves using active fiber composite (AFC) transducers and defined transfer functions for the A0 and S0 modes of Lamb waves and related its oscillations with the length of the element and properties of the wave in the medium. The research mentioned above has demonstrated that the amplitude of guided wave modes is a function of frequency and transducer dimensions, which means that the frequency response is different for every guided wave mode due to different transducer size to wavelength ratio. As the wavelength varies from one material to another, the response also becomes material-dependent. In such cases, the result of wave interference and interaction with damage will vary depending on source type, size, and dispersive properties of the medium. In cases where multiple modes are present in the structure, the signal analysis and interpretation becomes even more complicated. To develop reliable structural health monitoring systems, the source influence on the response spectrum of guided wave modes must always be addressed. The aim of this study was to propose and verify an analytical method to

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help predict the response spectrum of guided wave modes based on initial knowledge on transducer type,Sensors size,2017, and17,propagation medium. First, the scientific problem is explained using the general 1825 3 of 19theory of ultrasonic guided waves, and the proposed excitability function estimation technique is formulated knowledge on transducer size, and propagation medium. First, the scientific problem is based on the Fourier analysis type, of particle velocity distribution on the excitation area. Then, numerical explained using the general theory of ultrasonic guided waves, and the proposed excitability function simulations and different experiments are conducted to validate and investigate the feasibility of the estimation technique is formulated based on the Fourier analysis of particle velocity distribution on proposed technique. Strong experimental evidence is provided about the influence of transducer the excitation area. Then, numerical simulations and different experiments are conducted to validate vibration and size to frequency spectrum guided waves. and investigate thethe feasibility of the proposedoftechnique. Strong experimental evidence is provided about the influence of transducer vibration and size to the frequency spectrum of guided waves.

2. Statement of the Problem

2.One Statement of theimportant Problem and versatile features used to describe the modes of guided waves is of the most

frequency response. The frequency of guided wave modesthe depends both the structural One of the most important andresponse versatile features used to describe modes ofon guided waves is frequency response. The frequency response of guided wave modes depends on both the structural properties of the investigated medium and the parameters of the source, such as size and type of properties of thetheoretical investigatedmodels, mediumit and the parameters of the source, such as size and type of excitation. In most is presumed that the conditions ideally possess the uniform excitation. In most theoretical models, it is presumed that the conditions ideally possess the uniform loading and continuous plane wave excitation. However, in reality, the transducers possess limited size loading and continuous plane wave excitation. However, in reality, the transducers possess limited and frequency bandwidth. Hence, the type of excitation determines the mode which is introduced into size and frequency bandwidth. Hence, the type of excitation determines the mode which is the structure, whereas the amplitude of vibrations depends on the size-wavelength ratio of the source. introduced into the structure, whereas the amplitude of vibrations depends on the size-wavelength The of generation ratio the source.mechanism of GW is different compared to the bulk wave case. As both bulk and guided waves may be mechanism generated employing the same principle excitation), whether The generation of GW is different compared to the(i.e., bulkangle wave beam case. As both bulk and a bulk or guided wave is generated mainly depends on the frequency to sample thickness ratio. To guided waves may be generated employing the same principle (i.e., angle beam excitation), whether a bulk or the guided wave is generated depends on the frequency thickness excite GWs, wavelength must bemainly greater than the thickness of to thesample material, whileratio. bulkTowaves excite GWs, the wavelength must medium be greaterwhere than the of theonly material, bulkinfluence waves generally propagate in the infinite thethickness boundaries havewhile a minor on generally propagate infinite medium boundaries only have a minorone influence wave propagation. Dueintothe this reason, in the where case ofthe bulk waves, there are mostly or twoon desired wave propagation. to this reason, in the case of bulk waves, In there are mostly desired non-dispersive wavesDue propagating at constant phase velocity. contrast, the one GWoristwo a superposition non-dispersive waves propagating at constant phase velocity. In contrast, the GW is a superposition of longitudinal and shear waves, which reflect back and forth and convert to other modes. At a of longitudinal and shear waves, which reflect back and forth and convert to other modes. At a given given frequency, the results of such wave interaction may be either constructive, destructive, or frequency, the results of such wave interaction may be either constructive, destructive, or intermediate. AsAs a consequence, GWmodes modesare are produced simultaneously, having its intermediate. a consequence,different different GW produced simultaneously, each each having its ownown frequency-thickness dependent propagation velocity and unique distribution of particle velocity frequency-thickness dependent propagation velocity and unique distribution of particle velocity across the the thickness of of the theultrasonic ultrasonic transducer to diffraction) across thickness thesample. sample.The The response response ofofthe transducer (due(due to diffraction) also also depends on the direction wavegeneration. generation. ItItisiscommon to to calculate the the response function along along depends on the direction ofofwave common calculate response function the transducers axis; however,waves wavescan can also also be inin other directions. For the better the transducers axis; however, begenerated generated other directions. Forsake the of sake of better understanding, GW excitation in a 1D approach was analyzed, and is presented in Figure 1. understanding, GW excitation in a 1D approach was analyzed, and is presented in Figure 1.

Figure1.1.The The analyzed analyzed 1D excitation case.case. Figure 1Dguided guidedwave wave excitation

In this case, the non-dispersive mode is excited using a point type transducer and propagates In thisthe case, thewith non-dispersive mode is excited using a point transducer propagates along plate velocity c. According to general theory of type ultrasonic guidedand waves and the along the plate with wave velocity c. According to general theorydistance of ultrasonic the governing governing equations, the waveform at arbitrary along guided the x axiswaves can beand expressed as wave[45,46]: equations, the waveform at arbitrary distance along the x axis can be expressed as [45,46]:

u(t, x ) = u

(t − x/c ),

u(t,x) = ue xex(t − x/c),

(1)

(1)

where uex(t) is the waveform of particle velocity generated at excitation point (x = 0). If the transducer possesses dimensions,of then Equation (1) can be re-written as: where uex (t) isfinite the waveform particle velocity generated at excitation point (x = 0). If the transducer possesses finite dimensions, then Equationd/2(1) can be re-written as:



u(t, x ) = u ex (t − (x − x ex )/c, x ex )dx ex , d/2 Z −d/2

uex (t − (x − xex )/c, xex )dxex ,

u(t, x) =

−d/2

(2)

(2)

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where d is the length of transducer; xex is the position of excitation point on x axis. Assume that uex (t,xex ) = A(xex ) · uex (t − xex /c), where A(xex ) is the amplitude of the waveform at the excitation point xex . From this assumption, it follows that the excitation waveform is the same at all excitation points; however, it possesses different amplitudes at each node. Next, Equation (2) can be reorganized as follows: d/2 Z

u(t, x) =

A(xex ) · uex (t − (x − xex )/c)dxex .

(3)

−d/2

Assuming that xex = τ·c, Equation (3) can be further reorganized as: u(t, x) = c ·

d/2c Z

A(cø) · uex ((t − x/c) − ø)d(ø).

(4)

−d/2c

Equation (4) is the convolution integral, which can be solved using Fourier transform: u(t, x) = FT−1 {FT[A(cø)] · FT[uex ((t − x/c))]},

(5)

where FT and FT−1 denotes Fourier and inverse Fourier transform respectively. Term FT[A(cτ)] can be called the spatial filter or excitability function as it is only related to the distribution of the excitation amplitudes. This excitability function was investigated in this study. In the case dispersive waves are excited using point type transducer, the waveform at arbitrary distance x can be expressed as convolution of input signal and system’s impulse response [45,47]:   jff!x − u(t, x) = FT−1 FT[uex (t)] · e cp (ω) ,

(6)

where α is attenuation; ω = 2πf; f is the frequency; and cp (ω) is the frequency dependent phased velocity dispersion curve of the mode under analysis. In the case where a transducer possesses finite dimension and the excitation of the corresponding component of particle velocity is not uniform, each point has different waveform uex (t,xex ):

u(t, x) =

d/2 Z

FT

−1

 FT[uex (t, xex )] · e

−

jα!(x−xex ) cp (ω)

 dxex .

(7)

−d/2

If it is assumed that the excitation waveforms in the transmitter area is the same and from point to point differs only in amplitude, it means uex (t,xex ) = A( xex ) · uex (t - xex /c) : u(t, x) =

d/2 Z

  jα!(x−xex ) − dxex , FT−1 FT[A(xex ) · uex (t)] · e cp (ω)

(8)

−d/2

then: u(t, x) =

d/2 Z

A(xex ) · FT

−1

 FT[uex (t)] · e

−

jff!(x−xex ) cp (ω)

 dxex .

(9)

−d/2

It follows from Equation (9) that the waveform of generated waves and their frequency response depends not only on the frequency spectrum of the excitation signal (in most cases assumed as the transducer characteristic), but also on excitation force distribution along the wave propagation direction. If the conventional thickness mode transducer is used to generate the guided waves, which propagates along the sample, the frequency response of the generated waves will be different from conventional frequency response of transducer determined using bulk waves. Additionally, it will be

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different for each mode of the guided waves. Thus, the aim of this study was to develop analytical method tofor help predict response spectrum of guided wave different each modethe of the guided waves. Thus, the aim ofmodes. this study was to develop analytical method to help predict the response spectrum of guided wave modes. 3. Theoretical Model for the Prediction of Frequency Spectrum of Generated Guided Waves 3. Theoretical Model for the Prediction of Frequency Spectrum of Generated Guided Waves The method proposed in this study is based on Fourier analysis of the particle velocity distribution (u) on the excitation surface at theoninitial instant of of excitation. Such particle velocity The method proposed in this study is based Fourier analysis the particle velocity distribution distribution is relatedsurface to the excitation force, whichofisexcitation. required toSuch introduce thevelocity desireddistribution mode into the (u) on the excitation at the initial instant particle is structure. thatforce, the virtual transducer is mounted the surface thethe sample and related to Assuming the excitation whichMFC is required to introduce theon desired mode of into structure. excites the that A0 and 1 modes simultaneously, the initial of excitation, the spatial distribution Assuming the Svirtual MFC transducer is at mounted oninstant the surface of the sample and excites the A0 of the velocity on the excitation area can be Figure 2. The spatial distribution of and S1particle modes simultaneously, at the initial instant ofdescribed excitation,asthe spatial distribution of the particle particle velocity presented in Figure 2 are in 2Dasand it is 2. presumed to be uniform across the width of velocity on the excitation area can be described Figure The spatial distribution of particle velocity the transducer. The 2spatial of particleto velocity was deliberately selected based on the presented in Figure are in distribution 2D and it is presumed be uniform across the width of the transducer. vibration ofdistribution MFC transducer operating at elongation d33 mode. Hence,based it is presumed that theof out-ofThe spatial of particle velocity was deliberately selected on the vibration MFC plane component of the particle velocity at theHence, excitation surface had to the be exclusively transducer operating at elongation d33 mode. it is presumed that out-of-planeconcentrated component at the particle edges ofvelocity the transducer to introduce the had A0 mode 2a). concentrated Similarly, theat saw inof at the excitation surface to be (Figure exclusively thetooth edgeslike of the plane component of thethe particle velocity distribution caused elongation the MFC transducer transducer to introduce A0 mode (Figure 2a). Similarly, theby saw tooth likeof in-plane component of generated S1 mode (Figure 2b). Mathematically distribution presented in Figure be the particlethe velocity distribution caused by elongationthe of the MFC transducer generated the 2S1can mode expressed follows: (Figure 2b).asMathematically the distribution presented in Figure 2 can be expressed as follows: ( 1 z = z or z = z u A 0 (z ) =1  z = z1 1 or z = 2z2, other case uA0 (z) = , 0 0 other case

(10) (10)

  2z − l 2z  − l z1 < z < z 2 z1 < z < z2, u S1 (z ) = l uS1 (z) =  l 0 other case , 0 other case

(11) (11)

where zz1 and and zz2 are the coordinates of the front and back edge of the MFC transducer; and l = z2 − z1 where 1 2 are the coordinates of the front and back edge of the MFC transducer; and l = z2 − z1 is the length of transducer. Equations (11) describe the spatial distribution of velocity, particle is the length of transducer. Equations (10)(10) and and (11) describe the spatial distribution of particle velocity, which is graphically illustrated in Figure 2a,b, respectively. Such distribution was selected which is graphically illustrated in Figure 2a,b, respectively. Such distribution was selected based on based onknowledge a-priori knowledge of the vibration a MFCP1) (M-2814 P1) type transducer. a-priori of the vibration of a MFC of (M-2814 type transducer.

(a)

(b)

Figure 2. 2. The MFC transducer transducer on the Figure The presumed presumed spatial spatial distribution distribution of of the the particle particle velocity velocity of of the the MFC on the excitation surface: the vertical or normal component, which mainly generates the A 0 mode (a) and the excitation surface: the vertical or normal component, which mainly generates the A0 mode (a) and the (b). longitudinal or or tangential tangential component, component, which which produces produces the the SS1 mode longitudinal 1 mode (b).

For non-dispersive waves, spatial particle velocity distribution can be transformed to the particle For non-dispersive waves, spatial particle velocity distribution can be transformed to the particle velocity time domain as u(t), using the simple relation t = z/cp, where cp is the phase velocity. For velocity time domain as u(t), using the simple relation t = z/cp , where cp is the phase velocity. For dispersive waves, phase velocity is the function of frequency cp(f), hence this transformation becomes dispersive waves, phase velocity is the function of frequency cp (f), hence this transformation becomes frequency dependent u(z/cp(f)). The excitability function or the response amplitude at the given frequency dependent u(z/cp (f)). The excitability function or the response amplitude at the given discrete frequency fk can be expressed as the magnitude of the Fourier representation of particle

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velocity distribution u(z/cp(fk)). The same procedure must be repeated over the bandwidth of the transducer to collectf the set of magnitude values of excitability function: discrete frequency canwhole be expressed as the magnitude of the Fourier representation of particle k

velocity distribution u(z/cp (fk )). The same procedure mustbe repeated over   the bandwidth of the z ' '   , u A0 transducer to collect the whole magnitude values ofFT excitability function: H A0 (set fk )of ,U =U (12) A0 (f ) A 0 (f ) =  c pA (fk )  f =fk  0  !#  " z 0 0 (12) HA0 (fk ) = UA f , U f = FT u ( ) ( ) , A A0 0 0 f=fk (f ) 0 k   cpA z  , H S1 (fk ) = U 'S1 (f ) , U 'S1 (f ) = FT"u S1  (13) !#  = f f c k    pS1 z(fk )  , US0 1 (f) = FT uS1 HS1 (fk ) = US0 1 (f) (13) , cpS1 (fk ) f=fk where HA0(fk) and HS1(fk) are the analytical excitability functions for the A0 and S1 modes; uA0 and uS1 where HS1 (fkdistributions ) are the analytical excitability functions for the A0 and Sf1k;modes; are theHparticle velocity for A0 and S1 modes at particular frequency and FTudenotes A0 (fk ) and A0 and uthe the particle velocity for Aexcitability at particular frequency fk ; and FT Fourier transform. The distributions above-mentioned function is the transfer function of the 0 and S1 modes S1 are denotes the Fourier transform. The above-mentioned excitability function is the transfer function of ultrasonic transducer, which defines how efficiently it will generate the particular mode at a given the ultrasonic transducer, which defines how efficiently it will generate the particular mode at a given frequency and transducer size. The method proposed in this study was not limited to any particular frequency transducer size. method proposed this study was not limited any particular transducerand or material and canThe be used to predict theinexcitability function on anyto structure, under transducer or material and can be used to predict the excitability function on any structure, under any any type of excitation. To obtain proper results, the method required the phase velocity dispersion type excitation. To obtain proper the method required theinvestigated phase velocity dispersion curve curveofof the analyzed structure andresults, the operation principle of the transducer (particle of the analyzed structure andanalyzed the operation the investigated transducer (particle velocity velocity distribution for the mode)principle as inputof data. distribution for the analyzed mode) as input data. To estimate the excitability function as per Equations (12) and (13), consider a 4 mm thick and To thefiber excitability function per Equations (12)investigated and (13), consider 4 mm and 70 mm estimate wide glass reinforced plasticas (GFRP) plate as an sample,awith thethick material 70 mm wide glass fiber reinforced plastic (GFRP) plate as an investigated sample, with the material properties as follows: Young’s modulus (Ex = 10 GPa, Ez = 35.7 GPa); Poisson’s ratio (υxz = 0.325, υzx = properties follows: GPa, Eρz ==1800 35.7kg/m GPa);3).Poisson’s ratio (υ x = 10 xz = 0.325, 0.091, υyx =as 0.35); ShearYoung’s modulusmodulus (Gxz = 2.8(EGPa; density: The dispersion curves (DC) 3 υofzx the = 0.091, υ = 0.35); Shear modulus (G = 2.8 GPa; density: $ = 1800 kg/m ). The dispersion yx xz analyzed structure can be seen in Figure 3 where the dispersion curves presented were curves (DC) of the analyzed structure can be to seen Figure 3 where the dispersion curves which presented validated experimentally and corresponded thein properties of the real mock-up sample, was were validated experimentally and corresponded to the properties of the real mock-up sample, which used for the experiments in this study. The analytically obtained excitability functions for the A0 and was used for thebeexperiments in this study. analytically obtained excitability functions the A0 S1 modes can seen in Figure 4a,b. TheThe excitability functions presented below were for estimated and S modes can be seen in Figure 4a,b. The excitability functions presented below were estimated 1 considering the same particle velocity distribution as defined by Equations (10) and (11). considering the same particle velocity distribution as defined by Equations (10) and (11).

Figure Figure3. 3. The The phase phase velocity velocity dispersion dispersion curves curves of of 44 mm mm thick thick and and 70 70 mm mm wide wide glass glass fiber fiber reinforced reinforced plastic (GFRP) plate, calculated using the semi analytical finite element method. plastic (GFRP) plate, calculated using the semi analytical finite element method.

The The results results presented presented in in Figure Figure 44 indicated indicated that that in in both both cases cases the the response response amplitude amplitude of of the the excitability excitability function function oscillated oscillated with with an an increase increase of of frequency. frequency. The zero zero values values of of the the excitability excitability function at some frequency components. The periodicity of response functionwere werefound found at some frequency components. The periodicity of amplitude response oscillation amplitude was higher was for the A0 mode, the modes possessing short wavelengths were likely be oscillation higher for themeaning A0 mode,that meaning that the modes possessing short wavelengths were more in comparison to the symmetrical modes. These excitability curves can be used likelydistorted be more distorted in comparison to the symmetrical modes. These excitability curves can to be enhance or suppress the excitation of the desired however, the relative broadband excitation used to enhance or suppress the excitation of themode; desired mode; as however, as the relative broadband excitation is usually used to drive the transducer, multiple modes are being generated anyway. In

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is usually used to drive the transducer, multiple modes are being generated anyway. In such cases, such depending on frequency and of pulse, the of such cases, cases, depending on the theand frequency and the the bandwidth bandwidth of the the excitation excitation pulse, the waveforms waveforms of depending on the frequency the bandwidth of the excitation pulse, the waveforms of at least at least asymmetrical modes will be distorted due to the impact of the excitability function. Thus, the at least asymmetrical modes will be distorted due to the impact of the excitability function. Thus, the asymmetrical modes will be distorted due to the impact of the excitability function. Thus, the influence of function must while influence of the the excitability excitability function must be be considered considered while developing developing methods for SHM. SHM. of influence the excitability function must be considered while developing methodsmethods for SHM.for

((aa))

((bb))

Figure 4. excitability functions on 44 mm GFRP plate for the A and SS11 (b) modes, Figure 4. The The excitability functions mm thick thick plate A00 (a) (a) andmodes, (b) calculated modes, Figure 4. The excitability functions on 4on mm thick GFRPGFRP plate for thefor A0 the (a) and S1 (b) calculated calculated according according to to the the spatial spatial distribution distribution of of the the vertical vertical and and longitudinal longitudinal component component of of particle particle according to the spatial distribution of the vertical and longitudinal component of particle velocity. velocity. velocity.

ToTo illustrate the on the thefrequency frequencyspectrum spectrumof guided wave illustrate the influence of excitability function guided wave To illustrate theinfluence influenceof ofexcitability excitability function function on on the frequency spectrum ofof guided wave modes, we thetransducer transducer was driven a tone a Gaussian envelope modes, we assumed that was driven by burst with envelope of modes, weassumed assumedthat that the the transducer was driven by aby a tone tone burstburst with awith a Gaussian Gaussian envelope of three three of three periods and central frequency of 80 kHz. The length of the transducer was 28 mm along periods periods and and central central frequency frequency of of 80 80 kHz. kHz. The The length length of of the the transducer transducer was was 28 28 mm mm along along the the wave wavethe wave propagation direction a particle velocity distribution as shown in Figure 2. Next, propagation direction and had aa particle velocity distribution as in 2. the propagation direction andand hadhad particle velocity distribution as shown shown in Figure Figure 2. Next, Next, thethe magnitude spectrum looked likethat thatrepresented representedby bythe the solid line in Figure magnitude spectrum the excitation pulse solid line in 5, magnitude spectrumofof ofthe theexcitation excitationpulse pulse looked looked like like that represented by the solid line in Figure Figure 5, 5, which also shows the excitability function and its product with the magnitude spectrum of excitation which also shows the excitability function and product with the magnitude spectrum of excitation which also shows the excitability function and its with the magnitude spectrum of excitation pulse dash-dot and dashed It observed that under such and pulse asas dash-dot lines,respectively. respectively. It was observed under such excitation and pulse as dash-dotand anddashed dashed lines, lines, respectively. It was was observed thatthat under such excitation excitation and size size of the source, some significant filtering was present in the spectrum of A mode (Figure 5a). ofof the source, some significant filtering was present in in thethe spectrum of the thethe A00A mode (Figure 5a). size the source, some significant filtering was present spectrum of mode (Figure 5a). 0 Meanwhile, 11 mode was not filtered due the significantly larger wavelength compared to the Meanwhile, the mode was not filtered due to the significantlylarger largerwavelength wavelengthcompared comparedto tothe theA0 Meanwhile, thethe S1SSmode was not filtered due toto the significantly A mm 13.1 mm To interpret the wave propagation, A00 mode mode (45.5versus mm versus versus 13.1@ mm @ 80 80 kHz). kHz). To correctly correctly interpret the guided guided wave propagation, mode (45.5 (45.5 mm 13.1 mm 80 @ kHz). To correctly interpret the guided wave propagation, mode mode interference, and interaction with defects, the likely filtering effects must be mode interference, and interaction withthe defects, likelyeffects filtering effects must be considered, considered, interference, and interaction with defects, likely the filtering must be considered, although it’s although usually still neglected in although it’s usually in still neglected in most most research. research. usually stillit’s neglected most research.

((aa))

((bb))

Figure The influence of to the frequency response of waves for Figure 5.The Theinfluence influence of of the the excitability excitability function toto thethe frequency response of the the guided waveswaves for Figure 5. 5. the excitabilityfunction function frequency response ofguided the guided 0 (a) and symmetric S 1 (b) modes (solid line: spectrum of the excitation pulse, dashthe asymmetric A 0 (a) and symmetric S1 (b) modes (solid line: spectrum of the excitation pulse, dashthe asymmetric A for the asymmetric A0 (a) and symmetric S1 (b) modes (solid line: spectrum of the excitation pulse, dot excitability function, dashed line: of pulse function). dot line: line:line: excitability function, dashed line: product product of excitation excitation pulse and and excitability excitability function). dash-dot excitability function, dashed line: product of excitation pulse and excitability function).

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4. Validation of the Proposed Method

The goal of this section was to validate the excitability function estimation technique presented 4. Validation of the Proposed Method previously and to verify that the magnitude spectrum of each guided wave mode present in the The can goalbe of expressed this sectionaswas to validate thefrequency excitability functionof estimation technique structure a product of the spectrum the excitation pulsepresented and the previously and to verify that the magnitude spectrum of each guided wave mode present in the excitability function. For this purpose, the 3D finite element (FE) model of the transient wave structure can bethe expressed as a investigation product of the spectrum of the excitation pulse and propagation and experimental wasfrequency performed on the GFRP sample. the excitability function. For this purpose, the 3D finite element (FE) model of the transient wave propagation experimental wasElement performed the GFRP sample. 4.1. Validation and of thethe Proposed Methodinvestigation Employing Finite (FE)on Model 3D finite element (FE)Method modelEmploying of the transient propagation 4.1. The Validation of the Proposed Finite wave Element (FE) Modelwas employed for a complex shaped GFRP plate with dimensions of 800 mm × 70 mm × 4 mm. Next, the frequency spectrum of The 3D finite (FE) model the transient propagation employed for a complex the fundamental A0element and S1 modes wereofextracted fromwave the FE simulationwas data and compared to the shaped GFRP plate with dimensions of 800 mm × 70 mm × 4 mm. Next, the frequency spectrum analytically predicted excitability functions. In the upcoming chapters, the numerical model of theof the fundamental and S1 described, modes were extracted the FE simulation data and to A the GFRP sample will A be0 briefly followed by from the procedure used to extract thecompared spectrum of 0 analytically predicted excitability functions. In the upcoming chapters, the numerical model of the and S1 modes. The chapter will be concluded with the comparison between the spectra of A0, S1 modes GFRP sampleexcitability will be briefly described, followed by the procedure used to extract the spectrum of and analytical functions. A0 and S1 modes. The chapter will be concluded with the comparison between the spectra of A0 , S1 modes and analytical 4.1.1. Description of theexcitability Glass Fiberfunctions. Reinforced Plastic (GFRP) FE Model The graphic representation of the model used in FE simulations can be seen in Figure 6a. The 4.1.1. Description of the Glass Fiber Reinforced Plastic (GFRP) FE Model geometrical shape and dimensions of the simulated sample were deliberately selected to correspond The graphic representation of the model used in FE simulations can be seen in Figure 6a. The to the real mock-up specimen, which was used later for experimental verification. In the numerical geometrical shape and dimensions of the simulated sample were deliberately selected to correspond to model, the operation of the MFC (M-2814 P1) actuator, with an active area of 28 mm × 14 mm working the real mock-up specimen, which was used later for experimental verification. In the numerical model, in the elongation d33 mode was simulated. The distance along the z axis between the center of the the operation of the MFC (M-2814 P1) actuator, with an active area of 28 mm × 14 mm working in the virtual transducer and the nearest end of the sample was set to 165 mm. The excitation area was elongation d mode was simulated. The distance along the z axis between the center of the virtual aligned to the 33 center of the sample with respect to the x axis. The thin layer of wax (Young’s modulus: transducer and the nearest end of the sample was set to 165 mm. The excitation area was aligned to the E = 1.81 GPa, Poisson’s ratio: υ = 0.49; density: ρ = 951 kg/m3) was used as a coupler between the center of the sample with respect to the x axis. The thin layer of wax (Young’s modulus: E = 1.81 GPa, specimen and the transducer. To simulate the operation of the MFC transducer in the elongation d33 Poisson’s ratio: υ = 0.49; density: $ = 951 kg/m3 ) was used as a coupler between the specimen and the mode, the active surface of the actuator was divided into two separate areas referred to as Zone A transducer. To simulate the operation of the MFC transducer in the elongation d33 mode, the active and Zone B, each with dimensions of 14 mm × 14 mm (see Figure 6b). Then the excitation of guided surface of the actuator was divided into two separate areas referred to as Zone A and Zone B, each with waves was simulated by applying the monotonically increasing excitation force of the opposite phase dimensions of 14 mm × 14 mm (see Figure 6b). Then the excitation of guided waves was simulated to the nodal points in “zone A” and “zone B” as illustrated in Figure 6b. In this way, the waves by applying the monotonically increasing excitation force of the opposite phase to the nodal points generated in Zone A propagated along the positive direction of the z axis while the waves introduced in “zone A” and “zone B” as illustrated in Figure 6b. In this way, the waves generated in Zone A in Zone B propagated backwards. Under this type of excitation, typically both fundamental propagated along the positive direction of the z axis while the waves introduced in Zone B propagated asymmetrical and symmetrical modes are generated simultaneously. The waveform of the excitation backwards. Under this type of excitation, typically both fundamental asymmetrical and symmetrical force was a burst with a Gaussian envelop of three periods, a central frequency of 80 kHz and a modes are generated simultaneously. The waveform of the excitation force was a burst with a Gaussian bandwidth of 42.9 kHz at −6 dB. envelop of three periods, a central frequency of 80 kHz and a bandwidth of 42.9 kHz at −6 dB.

(a)

(b)

Figure Figure6.6.The Theschematic schematicview viewofofthe theGFRP GFRPplate plateused usedininFE FEsimulations simulations(a), (a),the theprinciple principleofofguided guided 33 mode of MFC transducer (b). wave generation in the FE model simulating the d wave generation in the FE model simulating the d33 mode of MFC transducer (b).

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For transient simulation, simulation,the theNewmark Newmark time integration scheme applied. integration For transient time integration scheme waswas applied. The The integration step step in time domain was 0.625 µs, which is 1/20 of the period at 80 kHz central frequency. The sample in time domain was 0.625 µs, which is 1/20 of the period at 80 kHz central frequency. The sample was was meshed using SOLID64 elements, which had eight nodes each with three degrees of freedom. meshed using SOLID64 elements, which had eight nodes each with three degrees of freedom. The The average spatial of element the element equal 1 mm possessing 14 nodes wavelength for average spatial size size of the was was equal to 1 to mm possessing 14 nodes per per wavelength for the the slowest A 0 mode at 80 kHz frequency. The finite elements of the regular areas were created using slowest A0 mode at 80 kHz frequency. The finite elements of the regular areas were created using the the mapped mapped mesh, mesh, whereas whereas for for the the regions regions close close to to cut-outs, cut-outs, the the free free sweep sweep mesh mesh was was implemented. implemented. Material properties for the GFRP were selected as the same to those presented in the Material properties for the GFRP were selected as the same to those presented in theprevious previoussection. section. The variables monitored in this study was a vertical (y) and longitudinal (z) component of particle The variables monitored in this study was a vertical (y) and longitudinal (z) component of particle velocity atthe thenodal nodalpoints points situated along centerline the sample. The B-scan image of the velocity at situated along the the centerline of theofsample. The B-scan image of the vertical vertical and longitudinal component of particle velocity can be seen in Figure 7. and longitudinal component of particle velocity can be seen in Figure 7.

(a)

(b)

Figure 7. The and longitudinal longitudinal (b) component of guided Figure 7. The simulated simulated B-scan B-scan images images of of the the vertical vertical (a) (a) and (b) component of guided waves along the the centerline centerline of of the the GFRP GFRP sample. sample. waves along

At the end of the simulation routine, the frequency spectrum of the fundamental A0 and S1 modes At the end of the simulation routine, the frequency spectrum of the fundamental A0 and S1 modes present in the structure was extracted from the B-scan data of the vertical and longitudinal present in the structure was extracted from the B-scan data of the vertical and longitudinal component of component of particle velocity. For this purpose, the B-scan data u(t,x) was transferred to the particle velocity. For this purpose, the B-scan data u(t,x) was transferred to the frequency-wavenumber frequency-wavenumber domain u(f,ξ) using the 2DFFT approach [48]. The frequency-wavenumber domain u(f,ξ) using the 2DFFT approach [48]. The frequency-wavenumber representation of the B-scan representation of the B-scan showed the wavenumber DC for all modes available in the structure. As showed the wavenumber DC for all modes available in the structure. As the fundamental A and the fundamental A0 and S1 waves were the modes of interest, the appropriate DC were filtered0from S1 waves were the modes of interest, the appropriate DC were filtered from u(f,ξ) by employing the u(f,ξ) by employing the Gaussian filter, which mathematically can be described by the following Gaussian filter, which mathematically can be described by the following equation: equation: (ξ(ξ−−ξ ) , ξ00((ff))) F(f,Fξ(f, ) ξ=) =e−eA−A , 22

(14)

where A is the coefficient that determines the width of the Gaussian filter along wavenumber wavenumber axis; axis; ; ak ;k and bk are of the equation; k = 1 …K; total number a k a⋅kf ·+fb+k b ξ00((ff))== ak and bk the are coefficients the coefficients oflinear the linear equation; k = 1K. .is. the K; K is the total number linear points; fk , fk+1 defines the frequency bandwidth investigation. fk];, fand k+1 defines the frequency bandwidth underunder investigation. The of linearofpoints; f ∈ [fk,ffk+1∈];[fand k ,fk+1 The Gaussian filter was adjusted to fit the DC of each mode of interest, meaning that each mode was Gaussian filter was adjusted to fit the DC of each mode of interest, meaning that each mode linear segments segments in in the the frequency-wavenumber frequency-wavenumber domain which were selected approximated by a set of linear modes can can be described by the equations: manually. As the consequence, the filtered filtered A A00 and S1 modes ξ)),, A 0ξ()f, ξ UA0U(f, =) = u(uf,(f, ξ)ξ )· ⋅FFAA00((f,f, ξ

(15) (15)

US1U(f,S1ξ(f, ) ξ=) =u(uf,(f,ξ)ξ )· ⋅FFSS11((f,f,ξ)),

(16)

where u(f,ξ) are the wavenumber dispersion curves obtained from the B-scan data using the 2D FFT approach; and F(f,ξ) are the Gaussian filters for A0 and S1 modes. As a result of filtering, the

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wavenumber dispersion curves were obtained separately for the A0 and S1 modes. Next, the frequency spectra of these modes were calculated using the following expressions: where u(f,ξ) are the wavenumber dispersion curves obtained from the B-scan data using the 2D FFT approach; and F(f,ξ) are the Gaussian filters for AU and the 0A U A max (f ) = max (f, ξS)1 ,modes. As a result of filtering, (17) 0 0 ξ wavenumber dispersion curves were obtained separately for the A0 and S1 modes. Next, the frequency spectra of these modes were calculated using the following expressions: U A max (f ) = max U0 (f, ξ)
, , (18) A0 N((ff)) = UAU0 max (17) A ξ U 0 max A max (f )

(

f

(

)

0

)

(

)

UA max (f) UA0U ) =(f ) = max0 U S (f, ξ )
,, NS(fmax 1 (f) 1 max ξUA0 max

(18) (19)

f


US1 max (f) = max US1 (f, ξ) , (19) ξ U S max (f ) 1 U S N (f ) = . (20) 1 US1U max (f) (f ) max S1max
. US1 N (f) = (20) f max US1 max (f) f As the goal of this study was to compare the simulated spectrum of the A0 and S1 modes (UA0N(f), As the goal of this study was to compare the simulated spectrum of the A0 and S1 modes (UA0 N (f), US N(f)) with the analytically predicted excitability functions (HA (fk), HS1(fk)), their estimation will be US11 N (f)) with the analytically predicted excitability functions (H0A0 (fk ), H S1 (fk )), their estimation will briefly described in the next section. be briefly described in the next section.

(

)

4.1.2. Estimation of the Excitability Functions for the Considered Problem Problem To estimate the excitability functions using the technique described in Section 3, the spatial particle velocity velocity for for the the A A00 and S11 modes distribution of the particle modes had to first be defined. In this case, it was presumed that that the theparticle particlevelocity velocityofofthe thevertical verticalcomponent component(y) (y)ononthe the active area transducer presumed active area of of thethe transducer at 0 mode (see Figure 8a). Similarly, at the first instant of the modeling represented the excitation of the A the first instant of the modeling represented the excitation of the A0 mode (see Figure 8a). Similarly, the the particle velocity of longitudinal component (z)first at the firstwas instant used tothe describe the particle velocity of longitudinal component (z) at the instant usedwas to describe generation mode (see generation of the S1 Figure of the S1 mode (see 8b). Figure 8b).

(a)

(b)

Figure 8. 8. The velocity at the excitation excitation surface surface Figure The vertical vertical (a) (a) and and longitudinal longitudinal (b) (b) component component of of particle particle velocity at the at the first time instant of the modeling. at the first time instant of the modeling.

The spatial particle velocity distributions presented in Figure 8 were used to calculate the The spatial particle velocity distributions presented in Figure 8 were used to calculate the excitability functions. The same material properties were also used to define the DC for a 4 mm GFRP excitability functions. The same material properties were also used to define the DC for a 4 mm plate. The excitability functions in this case were estimated in a frequency band up to 200 kHz. Once GFRP plate. The excitability functions in this case were estimated in a frequency band up to 200 kHz. the calculations were completed, the excitability functions HA0(fkH ) and HS1(fk)Hwere compared to the Once the calculations were completed, the excitability functions A0 (fk ) and S1 (fk ) were compared magnitude spectra of the appropriate mode U A0N(f) and US1N(f), estimated by Equations (12) and (13). to the magnitude spectra of the appropriate mode UA0 N (f) and US1 N (f), estimated by Equations (12) The comparison of the analytical and numerical results canresults be seencan in Figure they show good and (13). The comparison of the analytical and numerical be seen9 where in Figure 9 where they

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agreement between the numerical calculations and analytical estimation. From the results, it could show good agreement between the numerical calculations and analytical estimation. From the results, be concluded that the magnitude spectrum of each guided wave modewave was actually theactually productthe of it could be concluded that the magnitude spectrum of each guided mode was ref(f) and the excitability function H, which itself depends on the the excitation spectrum product of thepulse excitation pulse U spectrum Uref (f) and the excitability function H, which itself depends properties of the material, type of type excitation, and sizeand of the If at least of the on the properties of the material, of excitation, sizesource. of the source. If one at least oneaboveof the mentioned parameters changes, the spectrum of the of mode present in thein structure will also above-mentioned parameters changes, the spectrum the mode present the structure will vary. also vary.

(a)

(b)

Figure Figure9.9.The Thenumerically numericallyestimated estimatedfrequency frequency spectra spectra of of the the pure pure A A00 (a) (a)and andSS11(b) (b)mode modeon onaa44mm mm GFRP GFRPsample sample(solid (solidline) line)along alongwith with analytically analytically predicted predicted excitability excitability functions functions (dashed (dashed line). line).

4.2. 4.2.Experimental ExperimentalVerification Verification of of the the Method Method on on the the Anisotropic Anisotropic GFRP GFRP Plate Plate To To further further investigate investigate the the feasibility feasibilityof ofthe theproposed proposedexcitability excitabilityfunction functionestimation estimationtechnique, technique, experiments were conducted on GFRP samples possessing the same geometry, mechanical experiments were conducted on GFRP samples possessing the same geometry, mechanicalproperties properties and sensor allocation, as described in Sections 4.1 and 4.1.1. The goal of this study was to and sensor allocation, as described in Sections 4.1 and 4.1.1. The goal of this study was to compare compare the theexperimentally experimentallyobtained obtainedfrequency frequencyresponse responseof of the the asymmetrical asymmetrical A A00mode modewith withthe theanalytically analytically predicted predicted excitability excitability function. function. A MFC actuator (M-2814 P1,Smart SmartMaterial MaterialCorp. Corp.Dresden, Dresden,Germany) Germany) was mounted of A MFC actuator (M-2814 P1, was mounted onon toptop of the the GFRP sample using a thin layer of wax for coupling as shown in Figure 10. The actuator was GFRP sample using a thin layer of wax for coupling as shown in Figure 10. The actuator was driven driven a 150cycle, 50 Vpulse square pulse with afrequency central frequency kHz tomultiple generatemodes multiple with a 1with cycle, V square with a central of 80 kHz of to 80 generate in a modes in a wide frequencyTobandwidth. To collect the experimental data,low a frequency custom made low wide frequency bandwidth. collect the experimental data, a custom made thickness frequency thickness mode frequency transducer240 (central 240 bandwidthatof−the transducer −6 mode transducer (central kHz;frequency bandwidth of kHz; the transducer 6 dB level, 340 at kHz) dB 340 kHz) was attached the surface of the sample andwave scanned the waslevel, attached perpendicularly to perpendicularly the surface of theto sample and scanned along the path along of guided wave path of guided waves. The receiver moved away from the transmitter up to 380 mm with a step waves. The receiver moved away from the transmitter up to 380 mm with a step increment of 1 mm. increment 1 mm. The initialthe spacing between the actuator and sensor wasTo equal to 5reliable mm. Toacoustic obtain The initialofspacing between actuator and sensor was equal to 5 mm. obtain reliable acoustic the contact between sensor and thetextolite specimen, a glasswith textolite protector contact between sensor and thethe specimen, a glass protector a contact area ofwith 3 mma2 2 was used. The waveforms were recorded using a 25 MHz sampling frequency. contact area of 3 mm was used. The waveforms were recorded using a 25 MHz sampling frequency. The response signals The signals eighttotimes and averaged to to ensure to noise ratio. wereresponse measured eight were timesmeasured and averaged ensure better a signal noisebetter ratio. aInsignal this way, the B-scan In this way, the B-scan dataset of the out was collected at the The centerline of the dataset of the out of-plane component wasof-plane collectedcomponent at the centerline of the sample. experimental sample. The experimental set-up is graphically illustrated in Figure 10. set-up is graphically illustrated in Figure 10.

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Figure 10. The experimental set-up investigating source influence on guided wave generation. Figure10. 10.The Theexperimental experimentalset-up set-upinvestigating investigatingsource sourceinfluence influence on on guided guided wave wave generation. generation. Figure

The experimentally obtained raw B-scan u(t,x) image of the out-of-plane component of guided waves is experimentally presented in Figure 11a. raw Analogically, the image frequency wavenumber representation the BThe experimentally obtained raw B-scan B-scan u(t,x) u(t,x) image of of the the out-of-plane component of ofofguided guided The obtained out-of-plane component scan data u(f,ξ) showing the11a. modes available the inthe the structure can be seenrepresentation inrepresentation Figure 11b.ofThe waves in Analogically, frequency wavenumber the B-scan waves is presented presented inFigure Figure 11a. Analogically, frequency wavenumber of results the B0 was in Figure 11a,b shows modes were mainly present in structure, and that the A data u(f,ξ)u(f,ξ) showing thethat modes available in the structure cancan be the seen in Figure 11b. The results in scan data showing thetwo modes available in the structure be seen in Figure 11b. The results dominant. Figure 11a,b shows thatthat two modes were were mainlymainly presentpresent in the structure, and that and the Athat dominant. in Figure 11a,b shows two modes in the structure, the A0 was 0 was dominant.

(a) (b) ( a ) ( b) the 4 mm thick GFRP Figure 11. The experimental B-scan of the out-of-plane component of GW along

Figure 11. The experimental B-scan of the out-of-plane component of GW along the 4 mm thick GFRP sample11. (a),The andexperimental the frequency-wavenumber frequency-wavenumber representation of the the B-scan B-scan data (b). Figure B-scan of the out-of-plane component of GWdata along(b). the 4 mm thick GFRP sample (a), and the representation of sample (a), and the frequency-wavenumber representation of the B-scan data (b).

To fulfil the scope of the experiment, the spectrum of the A0 mode was extracted from the To fulfil the scope of the experiment, the spectrum of the A0 mode was extracted from frequency-wavenumber data (Figurethe11b) by implementing the procedure described in To fulfil the scope ofu(f,ξ) the experiment, spectrum of the A0 mode was extracted from the the frequency-wavenumber u(f,ξ) data (Figure 11b) by implementing the procedure described in Section 4.1.1. To calculate the excitability function, presumed the that procedure the loadingdescribed distribution frequency-wavenumber u(f,ξ) data (Figure 11b) byit was implementing in Section 4.1.1. To calculate the excitability function, it was presumed that the loading distribution required4.1.1. to introduce the Athe 0 mode was thefunction, same as that presented in Figure 8a. loading Next, thedistribution excitability Section To calculate excitability it was presumed that the required to introduce the A0 mode was the same as that presented in Figure 8a. Next, the excitability was calculated assame per as Equation (12) in ainfrequency band up to 200 kHz, function to ofintroduce the A0 mode required the A 0 mode was the that presented Figure 8a. Next, the excitability function of the A0 mode was calculated as per Equation (12) in a frequency band up to 200 kHz, assumingofthat material properties of the investigated GFRP were the same as200 listed in was calculated Equation (12) in asample frequency up toas kHz, function the the A0 mode assuming that the material properties ofas theper investigated GFRP sample wereband the same listed in Section 3. The comparison between the experimentally obtained spectrum of the A 0 mode U A0expin (f) assuming that the material properties of the investigated GFRP sample were same as listed Section 3. The comparison between the experimentally obtained spectrum of the A0 mode UA0 exp (f) Section The comparison between theH obtained spectrum of the A0 mode UA0exp(f) and the the3.analytical analytical excitability function Hexperimentally A0(f in 12. and excitability function (fk))can canbe beseen seen inFigure Figure 12. A 0 k and the analytical excitability function HA0(fk) can be seen in Figure 12.

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Figure Figure 12. 12. The The comparison comparison between between the the experimental experimental spectrum spectrum of of the the A A00 mode mode and and the the analytical analytical excitability excitability function. function.

The results showed good agreement between the analytical predictions and experiments. Note The results showed good agreement between the analytical predictions and experiments. Note that in this case, the transducer was driven by a square pulse of 1 cycle, which means that there might that in this case, the transducer was driven by a square pulse of 1 cycle, which means that there might be additional zero harmonics in the frequency spectrum of the A0 mode due to the spectrum of input be additional zero harmonics in the frequency spectrum of the A0 mode due to the spectrum of input signal. This can be seen in Figure 12 where a local minimum was present at the frequency slightly signal. This can be seen in Figure 12 where a local minimum was present at the frequency slightly above 100 kHz. above 100 kHz. 5. Aperture 5. Demonstration Demonstration of of the the Method Method Using Using Linear Linear Phased Phased Array Array with with Variable Variable Aperture

In In previous previous sections, sections, the theexcitability excitabilityfunction functionestimation estimationtechnique techniquewas wasintroduced introducedand andverified. verified. Furthermore, it was demonstrated with numerical simulations and validated with experiments Furthermore, it was demonstrated with numerical simulations and validated with experiments that that the excitation type influences the spectrum of each GW mode. In this section, the influence of source the excitation type influences the spectrum of each GW mode. In this section, the influence of source size size on on the the forced forced guided guided wave wave excitation excitation was was demonstrated demonstrated and and validated. validated. For For this this purpose, purpose, the the experiments were carried out on 0.5 mm thick aluminum plate with dimensions of 1250 mm 700 experiments were carried out on 0.5 mm thick aluminum plate with dimensions of 1250 mm × 700× mm. mm. A material with well-known properties was deliberately thistostudy to simplify the A material with well-known properties was deliberately selectedselected for this for study simplify the analysis analysis of GW signals. It was estimated that for this type of material, only fundamental modes (A of GW signals. It was estimated that for this type of material, only fundamental modes (A0 and S00) in the frequency up to 3ToMHz. of the study, a 2.25 128 and S0) existed existed in the frequency band upband to 3 MHz. fulfil To thefulfil scopethe of scope the study, a 2.25 MHz, 128MHz, elements elements phased array (2.25L128 96Px12 I3 P 2.5HY, Olympus NDT, PA, USA) was used an actuator phased array (2.25L128 96 x12 I3 2.5HY, Olympus NDT, PA, USA) was used as anasactuator and and attached to the surface of the plate using oil for acoustic coupling. The array was positioned attached to the surface of the plate using oil for acoustic coupling. The array was positioned along along the the centerline centerline of of the the sample sample at at aa distance distance of of 425 425 mm mm to to the the closest closest edge edge of of the the plate. plate. In In total, total, four four independent experiments were carried out (referred as Experiment #1, Experiment #2, etc.) independent experiments were carried out (referred as Experiment #1, Experiment #2, etc.) to to obtain obtain the response from fromthe thestructure structure at different actuator theexperiment, first experiment, only aelement single the response at different actuator sizes.sizes. In theInfirst only a single element of an array was In excited. each of the subsequent experiments, active of aperture an of an array was excited. each ofInthe subsequent experiments, the activethe aperture an arrayofwas array was incremented by adding one neighboring array element to an active aperture, which meant incremented by adding one neighboring array element to an active aperture, which meant that in the that in the second experiment, two arraywere elements were excited soof on. each of the second experiment, two array elements excited at once, andatsoonce, on. Inand each theInexperiments, experiments, arrayby was driven byof a tone burst 1 cycle and 200 V with a central frequency of the array wasthe driven a tone burst 1 cycle andof200 V with a central frequency of 2.25 MHz. To 2.25 MHz. To collect the experimental data, a thickness mode transducer with a point type protector collect the experimental data, a thickness mode transducer with a point type protector was attached was attached perpendicularly tothe theplate surface the plate andthe scanned along the sample. wave path the perpendicularly to the surface of andof scanned along wave path of the Theof initial sample. The initial distance between the array and the receiver was equal to 100 mm, while in each distance between the array and the receiver was equal to 100 mm, while in each case the receiver was case the receiver wasthe scanned away from to increment 300 mm with a step of 0.1 scanned away from transmitter up to the 300transmitter mm with aup step of 0.1 mm.increment All waveforms mm. waveforms recorded using the 100 MHz sampling frequency averaged eight times. wereAll recorded usingwere the 100 MHz sampling frequency and averaged eightand times. The experimental The experimental set-up and aperture configurations are presented in Figure 13. set-up and aperture configurations are presented in Figure 13.

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(a)

(b) Figure 13. 13. The The experimental experimental set-up set-up for for investigation investigation of of source source size size influence influence on on guided guided wave wave mode mode Figure spectrum (a) (a) and and the the array array aperture aperture configurations configurations used used throughout throughoutexperiments experiments(b). (b). spectrum

According to the datasheet of the array, the width of each element was equal to 0.5 mm with an According to the datasheet of the array, the width of each element was equal to 0.5 mm with an inter-element distance of 0.25 mm (pitch 0.75 mm). In this way, the active aperture in Experiment #1 inter-element distance of 0.25 mm (pitch 0.75 mm). In this way, the active aperture in Experiment #1 was equal to 0.5 mm, while in each of the subsequent experiments the aperture was equal to—1.25 was equal to 0.5 mm, while in each of the subsequent experiments the aperture was equal to—1.25 mm, mm, 2 mm, and 2.75 mm respectively. At the end of the experiments, a total of four B-scan datasets 2 mm, and 2.75 mm respectively. At the end of the experiments, a total of four B-scan datasets were were created, each at a different active aperture of the actuator. In this study, the S0 mode was selected created, each at a different active aperture of the actuator. In this study, the S0 mode was selected as as the mode of interest. Thus, the frequency spectrum was estimated from each of the B-scans (namely the mode of interest. Thus, the frequency spectrum was estimated from each of the B-scans (namely US exp#1(f), US exp#2(f), US exp#3(f), US0exp#4(f)) by employing the procedure described in Section 4.1.1. To US00 exp #1 (f), U0 S0 exp #2 (f),0US0 exp #3 (f), US0 exp #4 (f)) by employing the procedure described in Section 4.1.1. estimate thethe excitability functions, it it was the To estimate excitability functions, waspresumed presumedthat thatthe theparticle particlevelocity velocitywas was concentrated concentrated at at the edges of of each each element. element. The The example example of of the the spatial spatial particle particle velocity velocity distribution distribution where where four four elements elements edges were excited at once (Experiment #4) is presented in Figure 14. The particle velocity distributions for were excited at once (Experiment #4) is presented in Figure 14. The particle velocity distributions the other experiments were defined in a similar fashion, depending on how many elements were for the other experiments were defined in a similar fashion, depending on how many elements were fired at at the the same same time. time. To To describe describe the the phase phase velocities velocities of of the the guided guided waves, waves, the the following following material material fired properties of an aluminum sample were defined: Young’s modulus (72 GPa), and Poisson’s ratio properties of an aluminum sample were defined: Young’s modulus (72 GPa), and Poisson’s ratio (0.35, 3 ). In total, four excitability S0#1(fk), HS0#2(fk), (0.35, the density: 27803 ).kg/m the density: 2780 kg/m In total, four excitability functions functions (referred as(referred HS0 #1 (fk ),asHH (fk ), HS0 #3 (fk ), S0 #2 (f k),))H S0#4(fkestimated )) were estimated for each experiment. HSS0#3 (f were for each experiment. #4 k 0

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Figure 14. 14. The particle velocity distribution which was required required to to excite excite the the SS00 mode in case the four Figure array elements were fired fired at at the the same same time. time.

The frequency spectrum (US exp#1(f)) and the excitability function (HS #1(fk)) of the S0 mode when The frequency spectrum (U0S0 exp #1 (f)) and the excitability function 0(HS0 #1 (fk )) of the S0 mode one single was fired (Experiment #1) can be The results showed there when one element single element was fired (Experiment #1)seen can in beFigure seen in15a. Figure 15a. The resultsthat showed were no zero harmonics in the S 0 mode spectrum, which was caused by the size of the source if that there were no zero harmonics in the S0 mode spectrum, which was caused by the size ofonly the one array element was fired. Thewas results forThe Experiment can be seen#2incan Figure 15b in where in 15b the source if only one array element fired. results for#2Experiment be seen Figure (US0spectrum exp#2(f)) when elements were latter picture, the solid line the represents therepresents S0 mode spectrum where in the latter picture, solid line the S0 mode (US0two exp #2 (f)) when two elements were fired simultaneously. excitability function )) the wasdash-dot plotted in the while dash-dot k)) was (H plotted line, the fired simultaneously. The excitabilityThe function (HS0#2(f S0 #2 (fkin line, while the round-dot line was the product of U (f) and H (f ). The results for Experiments S0 #2 k S0Sexp #1 round-dot line was the product of US0exp#1(f) and H 0#2(fk). The results for Experiments #3 and #4 are #3 and #4 are presented in a similar fashion, where the round-dot line represents US0 exp #1 (f)·HS0 #3 (fk ) presented in a similar fashion, where the round-dot line represents US exp#1(f)·HS0#3(fk) and and US0 exp #1 (f)·HS0 #4 (fk ), respectively. The results in Figure 15b–d demonstrate0the influence of the US0exp#1(f)·H S0#4(fk), respectively. The results in Figure 15b–d demonstrate the influence of the source source size to the frequency response of the S0 mode and observed that the amount of low amplitude size to thecomponents frequency response of the S0 function mode and amount ofBy low amplitude frequency in the excitability wasobserved related tothat the the source width. knowing the frequencyspectrum components inathe excitability function was related to the source width.atBydifferent knowing the response when single array element is fired, each subsequent spectrum array response spectrum when a single array element is fired, each subsequent spectrum at different array apertures can be predicted by calculating the product of US0 exp #1 (f) and the appropriate excitability exp#1 (f) and15, theit appropriate excitability aperturesHcan the product of US0in function (fk )predicted (i = 2, 3, by 4). calculating From the results presented Figure was observed that the S0 # ibe magnitude of the experimental spectrum U (f) (solid line) and product of U (f) · H function HS0#i(fk) (i = 2, 3, 4). From the results in Figure 15, it was Sobserved that the S0 exp #presented i S0 # i (f k) 0 exp #1 (round-dot matched at only some frequencies. calculation theproduct product U ·HS0S##ii (f exp #1 (f) k) magnitude line) of the experimental spectrum US0exp#iThe (f) (solid line) of and ofS0U S0exp#1 (f)·H 0 (fk) in all cases was based on the spectrum obtained from Experiment #1 US0 exp # i (f) and the appropriate (round-dot line) matched at only some frequencies. The calculation of the product US exp#1(f)·HS #i(fk) excitability function, which depended on array aperture. Thus, in Figure 15b, U0S0 exp #2 (f) 0was in all casesto was on·Hthe spectrum obtained from Experiment #1 US exp#i(f) and the appropriate compared USbased (f) S0 #2 (fk ). Consequently, in Figure 15c, the US00 exp #3 (f) was compared to 0 exp #1 U HS0 #3 (fk ). which Hence,depended the magnitude of energy distribution was different due(f)towas a comparison excitability on array aperture. Thus, in Figure 15b, US0exp#2 compared S0 exp #1 (f)·function, of at different array apertures. Furthermore, in each case, the spectrum was normalized with to spectra US0exp#1(f)·H S0#2(fk). Consequently, in Figure 15c, the US0exp#3(f) was compared to US0exp#1(f)·HS0#3(fk). respect to its own maximum value. As the goal of this comparison was to verify the appearance of Hence, the magnitude of energy distribution was different due to a comparison of spectra at different low amplitude frequency components at different frequencies due to source size, the differences in array apertures. Furthermore, in each case, the spectrum was normalized with respect to its own magnitude were not further investigated. maximum value. As the goal of this comparison was to verify the appearance of low amplitude frequency components at different frequencies due to source size, the differences in magnitude were not further investigated.

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(a)

(b)

(c)

(d)

Figure spectrum of of the the SS0 mode on the 0.5 mm Figure 15. 15. The The influence influence of of the the source source size size to to the the frequency frequency spectrum 0 mode on the 0.5 mm aluminum sample when the active aperture was 0.5 mm (Experiment #1) (a); 1.25 mm (Experiment aluminum sample when the active aperture was 0.5 mm (Experiment #1) (a); 1.25 mm (Experiment #2) (b); #2) (b); 2 mm (Experiment #3) (c); and 2.75 mm (Experiment #4) (d). 2 mm (Experiment #3) (c); and 2.75 mm (Experiment #4) (d).

6. Discussion and Conclusions 6. Discussion and Conclusions In this study, the influence of the source on the frequency response of guided waves was In this study, the influence of the source on the frequency response of guided waves was demonstrated and explained. It was shown through numerical simulations and experiments that the demonstrated and explained. It was shown through numerical simulations and experiments that the frequency response of each guided wave mode was a product of the spectrum of excitation pulse and frequency response of each guided wave mode was a product of the spectrum of excitation pulse the excitability function, which itself depended on the type of excitation, material properties, and size and the excitability function, which itself depended on the type of excitation, material properties, of the source. Therefore, the interaction between the guided waves and structural defects depended and size of the source. Therefore, the interaction between the guided waves and structural defects on the initial properties of each guided wave mode and it became mandatory to predict the response depended on the initial properties of each guided wave mode and it became mandatory to predict the spectrum of each mode to further develop reliable methods for damage detection. In this research, response spectrum of each mode to further develop reliable methods for damage detection. In this the novel excitability function estimation technique based on Fourier analysis of particle velocity research, the novel excitability function estimation technique based on Fourier analysis of particle distribution on the excitation area was proposed, which enabled the response amplitude as a function velocity distribution on the excitation area was proposed, which enabled the response amplitude as a of frequency separately for each GW mode to be estimated. The performance of the proposed function of frequency separately for each GW mode to be estimated. The performance of the proposed technique was demonstrated only for the fundamental asymmetrical and 1st order symmetrical technique was demonstrated only for the fundamental asymmetrical and 1st order symmetrical modes; modes; however, the method itself could also be used to predict the excitability functions of other however, the method itself could also be used to predict the excitability functions of other modes. The modes. The proposed excitability function estimation technique was validated with numerical simulations and experiments on the GFRP and aluminum samples. The numerical and experimental

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proposed excitability function estimation technique was validated with numerical simulations and experiments on the GFRP and aluminum samples. The numerical and experimental results showed good agreement with the theoretical predictions. Furthermore, it was demonstrated that the excitability function was related to the wavelength-size ratio of the transducer, therefore the asymmetrical modes (which possess short wavelengths) always had more frequencies with amplitudes close to zero at the same band compared to the symmetrical modes. The technique developed in this study can be further used as a support for existing guided wave signal processing methods to improve their performance, or as one of the predictive modeling tools during the design, implementation, and optimization of structural health monitoring systems. Author Contributions: Vykintas Samaitis contributed to the formulation of the method, designed and performed numerical simulations and experiments, analyzed the data and wrote the manuscript. Liudas Mažeika proposed the method to predict the frequency spectrum of guided waves, supervised the research and contributed to the writing of the final revision of the paper. Conflicts of Interest: The authors declare no conflict of interest.

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