Department of Robotics and Mechatronics, AGH University of Science and Technology, Al. Mickiewicza 30, Krakow, Poland. *corresponding author

Transcription

1 Key Engineering Materials Vol. 588 (214) pp Online available since 213/Oct/11 at (214) Trans Tech Publications, Switzerland doi:1.428/ Comparative Study of Instantaneous Frequency Extraction in Nonlinear Acoustics Used for Structural Damage Detection Dorota Wlodarska a, Andrzej Klepka b, Wieslaw Jerzy Staszewski c, * and Tadeusz Uhl d Department of Robotics and Mechatronics, AGH University of Science and Technology, Al. Mickiewicza 3, 3-59 Krakow, Poland a wlodarsk@agh.edu.pl, b klepka@agh.edu.pl, c w.j.staszewski@agh.edu.pl, d tuhl@agh.edu.pl, *corresponding author Keywords: Structural damage detection, nonlinear acoustics, signal demodulation, instantaneous frequency, Hilbert transform Abstract. Nonlinear acoustics deals with various nonlinear effects that occur in ultrasonic wave propagation. The method is suitable for material characterisation, as it uses different nonlinear phenomena associated with material imperfections. The method has been used for detecting nonlinearities in cracked solids by: measuring distortions of acoustic signals, estimating resonance frequency shifts or assessing nonlinear vibro-acosutic modulations. The latter is the most widely used non-classical approach to probe material nonlinearities. The method involves vibro-acoustic interactions of ultrasonic wave and modal vibration in damaged specimens. Modulation intensity that strongly relates to damage severity - is usually assessed in the frequency domain and often leads to confusing results when large modulations are involved. The paper investigates the time domain analysis of vibro-acoustic modulated signals. Several methods for instantaneous frequency calculation used to assess the intensity of modulation - are compared. Simulated and experimental data are used in these investigations. Introduction Nonlinear acoustic methods have attracted lots of research effort for the last fifteen years. Many different approaches have been used to detect cracks in metals and impact damage in composites. Nonlinear acoustics uses different nonlinear phenomena that can be related to either various material imperfections or non-symmetric thermo-elastic behaviour of interfaces to detect damage. The application of nonlinear acoustics for damage detection is based on different nonlinear effects including: higher-harmonics generation, sub-harmonic generation, shift of resonance frequencies, frequency response mixing, analysis of slow dynamics and signal modulations. All these techniques rely on one common physical phenomenon, i.e. nonlinear crack-wave interaction. It appears that the nonlinear vibro-acoustic wave modulation technique is one of the most widely used and discussed approaches [1-9]. The method involves monitoring various inter-modulation or cross-modulation phenomena of an acoustic wave propagating in damaged structures. This wave is transmitted through a cracked specimen, interacting with low-frequency modal excitation [], as illustrated in Fig. 1. The intact (or undamaged) structure acts as a "linear carrier" for the ultrasonic propagating wave [7] i.e. the response spectrum displays only the main harmonics of frequency components involved. In contrast, when the structure is cracked, the response signal is modulated; modulation sidebands can be observed additionally around the main harmonics of both signals by their co-interaction through the damage interface [3,8]. The number of sidebands and the intensity of modulation can be used for damage detection and its severity assessment [9]. Two modulation mechanism can be observed in acoustical responses from vibro-acoustic interactions. These are frequency and amplitude modulation. These two mechanisms have been briefly explained and presented in the literature (e.g. [2,9]). All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP, (ID: , AGH - Univ. of Science and Technology, Akademia Górniczo-Hutnicza, Kraków, Poland-29/5/14,2:1:21)

2 34 Smart Diagnostics V Fig.1 Schematic diagram demonstrating the principle of nonlinear vibro-acoustic wave modulation used for damage detection. Previous research shows that when response power spectra are used for damage detection, it is difficult to establish whether the response signal is modulated purely in the frequency or in the amplitude domain or whether both modulation processes are involved []. The time domain analysis needs to be used to distinguish both modulation mechanisms. This requires extraction of instantaneous signal properties. Various approaches can be used to estimate the instantaneous amplitude and frequency. This includes the following methods: [,11-22]: the Hilbert transform (HT), normalised Hilbert transform (NHT), Direct Quadrature (DQ), Teager Energy Operator (TEO), Generalized Zero-Crossing (GZC), cepstrum, Wavelet transform (WT), Empirical Mode Decomposition (EMD), Ensemble Empirical Mode Decomposition (EEMD) and many other approaches. The aim of the recent paper is to explore selected methods for instantaneous amplitude and frequency extraction. Three different signal demodulation methods, namely the Hilbert Transform, Teager Energy Operator and Empirical Mode Decomposition have been used in these investigations. The theoretical background associated with these methods is briefly presented in the second section. Application examples using simulated and experimental data are given in the third section. Finally, conclusions and future work proposal are given in the fourth section of the paper. Instantaneous Signal Characteristics - Theoretical Background Instantaneous frequency. The instantaneous amplitude (or envelope) and instantaneous phase (or frequency) are the two main instantaneous signal characteristics. The envelope function is intuitively a smooth curve joining local signal maxima. Instantaneous frequency can be defined as the derivative of the phase obtained from the complex frequency modulated signal. For any arbitrary signal x(t) these two instantaneous characteristics can be obtained from the analytic signal given as [1-13] z( t) x (1) j t t j xt at e where a (t) is real and positive and (t) is real. Then a (t) is called the instantaneous amplitude (IA) and (t) is called the instantaneous phase (IP) of the signal. The instantaneous frequency (IF) is the time-varying parameter which defines the location of the signal s spectral peak as it varies with time. The IF is defined as f i ( t) ( t) (2) 2

3 Key Engineering Materials Vol This formula is meaningful only for mono-component signals. Multi-component signals have a separate IF for each component. Hilbert transform. The Hilbert transform (HT) can be used for signal demodulation in order to obtain a well defined and physically meaningful IF. The detailed information about the HT can be found in [14, 1-2]. The IF calculated using HT is defined as the derivative of instantaneous phase (IP) of the signal, as described by Eq. (2). The HT concept only acquires a physical sense when applied to narrowband, or quasi-monochromatic signals. For wideband signals band-pass filters are commonly used to obtain the narrowband characteristics, or the Hilbert-Huang transform can be applied, alternatively. Hilbert-Huang transform. The Hilbert Huang transform (HHT) consists of two stages: the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA). The EMD is an intuitive method which has been developed for analysis of non-stationary and nonlinear data by extracting nearly mono-component series of the so-called Intrinsic Mode Functions (IMFs) through sifting processes. The description of the entire procedure can be found in [17-19]. The IMFs represent simple symmetric oscillations with respect to the local mean that satisfies two conditions [2]: 1) the number of zero-crossings and the number of extrema must not be differ by more than one in the whole set of data; 2) the mean value of envelopes defined by the local maxima and envelopes defined by the local minima is zero at any point. Within the above framework, meaningful instantaneous frequencies for all signal components can be obtained using the Hilbert Transform or any other method in the second stage of the procedure [21]. Unlike simple harmonic functions, IMFs can have a variable amplitude and frequency functions. In other words, the entire empirical concept is a generalised Fourier expansion that enables the accommodation of nonlinearity and non-stationarity and the separation of amplitude and frequency modulations [21]. Teager energy operator (TEO). The energy operator is an alternative approach used to compute instantaneous frequency from a signal without involving transforms [14,1]. It is totally based on signal differentiation. As the IF from multi-component signals is a complex function of time, the algorithm for the IF smoothing and approximation - that would capture the most important patterns in data - is needed. The TEO defined as [1] x x 2 xx, (3) represents the total energy of a simple harmonic mass-spring oscillator consisting of mass m suspended by a spring of constant k. From the Newton's law of motion this oscillator can be defined by the energy equation given as [17] 1 E 1 kx 2 mx 2 (4) 2 and oscillations can be described as x t asin( t), (5) where a is the amplitude and k / m is the frequency. The TEO can estimate approximately the squared product of the oscillation amplitude and frequency, when the instantaneous parameters of the signal do not vary too fast with time if compared with the carrier frequency [18]. With this framework the central frequency ω and amplitude a can be defined as

4 Amplitude [db] amplitude 3 Smart Diagnostics V x x and x x a. () respectively. The formulas for the instantaneous characteristics - given by Eq. (8) - are very accurate for a simple sinusoidal signal, but their approximate errors for amplitude-modulated (AM), frequency-modulated (FM), and AM-FM signals are relatively small. Therefore these equations can be also used for multi-component signals. Instantaneous Frequency - Application Examples The three methods described in the previous section have been compared using two simulated signals and one experimental signal. The simulated signals were selected intuitively to be similar to nonlinear vibro-acoustic signals that contains frequency and/or amplitude modulations. The experimental signal has been taken from a nonlinear acoustic experiment used for structural damage detection. Band-pass filtering - with the bandwidth equal to 2 khz - has been applied initially to all signals to avoid problems associated with the EMD procedure [22]. Signal processing was performed using MATLAB routines. All results were analysed in the time and frequency domain. Signal 1 - simulated FM signal. The first signal considered was a sum of a sinusoid and FM modulation. This signal was defined as sig t) sin(2 f t) sin(2 f t cos(2 f )), (7) ( 2 1 2t where f1 khz is the carrier frequency and f2 3Hz is the modulation frequency. 1-1 Composite FM signal time [s].1 x Fig. 2 Simulated FM signal and its true IF. Fig. 2 shows the first simulated signal and its true instantaneous frequency. The power spectrum for this signal is presented in Fig. 3. The spectrum has been zoomed around the fundamental carrier frequency to observe a series of modulation sidebands. -1 Power spectrum of a signal X: 5.994e+4 Y: X: e+4 Y: Frequency [Hz] Fig. 3 Zoomed power spectrum for the first simulated signal shown in Fig. 2a.

5 F [Hz] Key Engineering Materials Vol a) IF via HT b) IF via TEO c) Fig. 4 The IFs for the FM signal, calculated via: (a) HT; (b) TEO; (c) EMD-HT. The instantaneous frequencies were calculated using the HT, TEO and EMD-HT procedures. Signal differentiation was used in the HT-based procedure to obtain the IF from the IP. The IF was estimated directly in the TEO approach. The EMD-HT procedure involved two steps. The signal was decomposed to obtain the IMFs and then the HT approach was used - i.e. signal decomposition to obtain IMFs. Then the IMF with the highest concentration of the signal energy has been selected for IF calculation. The results are presented in Fig. 4, where the estimated instantaneous frequencies - given by dashed red lines - are compared with the theoretical characteristics (solid blue lines). The results show that all methods used lead to some discrepancies. The HT approach gives some edge effects. The amplitude of the IF estimated by the TEO procedure is lower than the theoretical amplitude. Undesired strong oscillations appear and in the EMD result. These undesired effects can be fixed by improving the calculation algorithms (for example a change of differentiation technique will eliminate oscillations in the EMD result). Signal 2 - simulated AM-FM signal. In the second example, the simulated signal includes both, i.e. frequency and amplitude, modulations. The second signal was defined as sig t) [1.15sin(2 f t)]sin((2 f t).25cos(2 f )), (8) ( 2 1 2t where f1 7 khz is the carrier frequency, f 2 73 Hz, is the frequency of amplitude and frequency modulation. Fig. 5 gives the original signal with its true instantaneous frequency. As previously, the diagram shows that the true IF oscillates around the carrier frequency of the signal and its oscillation amplitude corresponds to the modulation frequency. The zoomed power spectrum - shown in Fig. - clearly displays a series of modulation sidebands around the carrier component. HT TEO IF of Res. from EMD method using HT EMD time [sec]

6 F [Hz] Amplitude [db] amplitude 38 Smart Diagnostics V 1 AM-FM signal time [s] Fig. 5 The original AM-FM signal and its true IF. Power spectrum of a signal -1-2 X:.993e+4 Y: X: 7e+4 Y: Frequency [Hz] Fig. Power spectrum of analysed signal AM-FM signal zoomed around main harmonic. The estimated IFs - calculated with the three different approaches - are compared with the theoretical IFs in Fig. 7. The results are very similar to the results obtained for the first simulated signals. However, this time the edge effect in the HT procedure is stronger and the IF estimate using the TEO is improved. Again, all these discrepancies can be improved with the appropriate filtering/differentiation procedures used a) IF via HT b) IF via TEO c) Fig. 7 The IFs for the AM-FM signal, calculated via: (a) HT; (b) TEO; (c) EMD-HT. HT TEO IF of Res. from EMD method using HT EMD time [sec]

7 Amplitude Amplitude Key Engineering Materials Vol To quantify the comparison of IF extraction methods for both simulated signals the root mean squared error (RMSE) has been additionally calculated. Table 1 summarises the results of calculations. Table 1. The RMSE of IF calculations using different methods. RMSE of simulated FM signal Method used for IF calculation RMSE of simulated AM-FM signal HT 1.2e e+2 TEO 5.38e e+2 EMD e e+2 It can be noticed from the first look over above indexes, that the HT appears to be the most effective method of IF extraction for signals for which the real IF is known. It gives the smallest error in comparison with other methods. The principal reason of falsely high values of the RMSE index in EMD method disclosed above are the edge effects and the oscillations of IF amplitude around it real value that can be observed in Fig 4 and 7 for both simulated signals. Signal 3 - experimental signal. The third signal considered was taken from the nonlinear vibro-acoustic experiment. The composite plate impacted at the centre with an 3,9 J low frequency velocity impact with average area of damage equal to 32mm 2 was used in the experiment. The specimen was obtained from unidirectional carbon/epoxy (Seal HS1/REM) prepreg layers of stacking sequence of the laminate - [3/93]s and an average laminate thickness equal to 2 mm. Nonlinear acoustics test was performed as follows. A high frequency ( khz) acoustic sine wave (peak-to-peak amplitude = V) was transmitted through the structure undergoing simultaneously a low frequency vibration (459 Hz - 7 th vibration mode of the composite plate) with amplitude equal to 12 V pp. The test was performed using the surface-bonded piezoceramic transducers: the lowprofile piezoceramic transducer for the ultrasonic wave and the piezoceramic stack actuator the for low frequency vibration. The PSV Polytec scanning laser vibrometer measuring out-of-plane vibration was used to acquire response signal. x a) Experimental signal b) 2 x 1-3 Filtered signal Fig. 8 Nonlinear vibro-acoustic response: (a) original experimental data b) filtered data. Fig. 8 shows the original vibro-acoustic response and the filter response. Filtering was used to eliminate the strong, low-frequency component resulting from modal excitation. The zoomed power spectrum in Fig. 9 shows the fundamental khz component and two pairs of modulation sidebands embedded in the background noise.

8 F [Hz] F [Hz] Amplitude 4 Smart Diagnostics V Power spectrum of a signal X: 5.999e+4 Y: Frequency Fig. 9 Zoomed the power spectrum for the nonlinear vibro-acoustic experimental response..1 a) IF via HT b) IF via TEO c) Instantaneous Frequency from EMD method time [sec] Fig. 1 The IFs for the AM-FM signal, calculated via: (a) HT; (b) TEO; (c) EMD-HT. The IFs obtained by applying the HT, TEO and EMD-HT methods are shown in Fig. 1. The zoomed time traces of these IFs are compared in Fig Instantaneous Frequency comparison EMD TEO HT time [sec] Fig. 11 The close-up view of the comparison of calculated IFs. The estimated IFs are very similar for all methods employed. All methods capture the main oscillations. However, the TEO approach gives the largest amplitude from all of the methods used. x 1-3

9 Key Engineering Materials Vol Conclusions and further work suggestions A number of different approaches have been developed for instantaneous frequency estimations. This paper briefly describes three different procedures based on the Hilbert transform (strict mathematical approach), the Hilbert-Huang transform (empirical procedure) and the Teager energy operator (procedure that combines signal processing with physics). The methods have been compared with respect to possible applications to damage detection based on nonlinear vibro-acoustic modulations. Two simulated and one experimental signals were used in the presented analysis. The results presented in the paper are not conclusive. All methods used exhibit some discrepancies in the estimated instantaneous frequencies. These discrepancies include the following effects: the edge effect, undesired oscillations and reduced amplitude. It is anticipated that some of the problems encountered can be addressed using simple signal processing procedures (e.g. improved differentiation or filtering). Nevertheless future work should examine the effectiveness of the method for the time-domain estimation of the modulation intensity in damage detection experimental tests based on nonlinear vibro-acoustic modulations. Acknowledgements The work presented in this paper was supported by The Foundation for Polish Science under the research WELCOME project no. 21-3/2. References [1] K. E.-A. Van Den Abeele, P. A. Johnson, A. Sutin, Non-Linear Elastic Wave Spectroscopy (NEWS) Techniques to Discern Material Damage, Part I: Non-Linear Wave Modulation Spectroscopy (NWMS), Res. Nondestr. Eval.. 12 (2) 17-3 [2] K.-Y. Jhang, Nonlinear Ultrasonic Techniques for Nondestructive Assessment of Micro Damage in Material: A Review, Int. J. Prec. Eng. Manuf. 1 (29) [3] O. Buck, W. L. Morris, J. M. Richardson, Acoustic Harmonic Generation at Unbonded Interfaces and Fatigue Cracks, Appl. Phys. Lett. 33 (1978) [4] P. Duffor, M. Morbidini, P. Cawley, A study of the vibro-acoustic modulation technique for the detection of cracks in metals, J. Acoust. Soc. Am., 119 (2) [5] A. M. Sutin, V. E. Nazarov, Nonlinear Acoustic Methods of Crack Diagnostics, Radiographics and Quantum Electronics, 38 (1995) [] H. F. Hu, W. J. Staszewski, N. Q. Hu, R. B. Jenal, G. J. Qin, Crack detection using nonlinear acoustic and piezoceramic transducers instantaneous amplitude and frequency analysis, Smart Mater. Struct. 19 (21) 1-1 [7] F. Aymerich, W. J. Staszewski, Experimental study of impact-damage detection in composite laminates using a cross-modulation vibro-acoustic technique, Struct. Health Monit. 9 (21) [8] H.-C. Wu and K. Waenemuende, Mechanism Aspects of Non-linear Acoustic Signal Modulation due to Crack Damage, in T. Kundu, Advanced Ultrasonic Methods for Material and Structure Inspection, ISTE, London, 21, pp , doi: 1.12/ ch8 [9] K. Haller, Nonlinear acoustics applied to nondestructive testing, Printfabriken, Karlskrona, 27 [1] R. Carmona, W.-L. Hwang, B. Torresani, Practical Time-Frequency Analysis Gabor and Wavelet Transforms with an Implementation in S, Academic Press, New York, 1998 [11] L. Cohen, Time-Frequency Analysis, Prentice Hall, Jew Jersey, 1995 [12] B. Boashash, Time Frequency Signal Analysis and Processing A Comprehensive Reference, first ed., Elsevier Ltd., 23

10 42 Smart Diagnostics V [13] F. Hlawatsh, F. Auger, Time-Frequency Analysis: Concepts and methods, ISTE Ltd., 28 [14] N. E. Huang, W. Zhaohua, S. R. Long, K. C. Arnold, X. Chen, K. Blank, On Instantaneous Frequency, Adv. Adapt. Data Anal. 1 (29) [15] E. Kvedalen, Signal processing using the Teager Energy Operator and other nonlinear operators, Cand. Scient Thesis, University of Oslo, 23 ( [1] M. F. Ghazadi, et al., Comparative study of instantaneous frequency based methods for leak detection in pipeline networks, Mech. Syst. Signal Process. (211), doi: 1.11/j.ymssp [17] M. Fieldman, Hilbert transform applications in mechanical vibration, first ed., John Wiley & Sons, 211 [18] G. Rilling, P. Flandrin, P. Goncalves, On Empirical Mode Decomposition and its algoritms, in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing/NSIP-3, Grado, Italy (23) [19] N. E. Huang, An adaptive data analysis method for nonlinear and nonstationary time series: the Empirical Mode Decomposition and Hilbert Spectral Analysis, in T. Qian, M. I. Vai, Y. Xu (Eds.), Wavelet analysis and applications, Birkhäuser Basel, Berlin, 27, pp [2] D. Donnelly, The Fast Fourier and Hilbert-Huang Transforms: A Comparison, Int. J. of Computers, Communications & Control 4 (2) [21] N. E. Huang, Introduction to Hilbert-Huang transform and its related mathematical problems, in Ed. N. E. Huang, S. S. P. Shen (Eds.), The Hilbert-Huang transform and its applications, World Scientific Publishing Company, Singapore, 25, pp. 1-2 [22] Y. Chen, M. Q. Feng, A technique to improve the empirical mode decomposition in the Hilbert-Huang transform, Earthq. Eng. Eng. Vib. 2 (23) 75-85

11 Smart Diagnostics V 1.428/ Comparative Study of Instantaneous Frequency Extraction in Nonlinear Acoustics Used for Structural Damage Detection 1.428/ DOI References [1] K. E. -A. Van Den Abeele, P. A. Johnson, A. Sutin, Non-Linear Elastic Wave Spectroscopy (NEWS) Techniques to Discern Material Damage, Part I: Non-Linear Wave Modulation Spectroscopy (NWMS), Res. Nondestr. Eval. 12 (2) [3] O. Buck, W. L. Morris, J. M. Richardson, Acoustic Harmonic Generation at Unbonded Interfaces and Fatigue Cracks, Appl. Phys. Lett. 33 (1978) [7] F. Aymerich, W. J. Staszewski, Experimental study of impact-damage detection in composite laminates using a cross-modulation vibro-acoustic technique, Struct. Health Monit. 9 (21) [14] N. E. Huang, W. Zhaohua, S. R. Long, K. C. Arnold, X. Chen, K. Blank, On Instantaneous Frequency, Adv. Adapt. Data Anal. 1 (29) [22] Y. Chen, M. Q. Feng, A technique to improve the empirical mode decomposition in the Hilbert-Huang transform, Earthq. Eng. Eng. Vib. 2 (23)