An Accurate Frequency Estimation Algorithm with

An Accurate Frequency Estimation Algorithm with Application in Modal Analysis Jean Louis NTAKPE Department of Mechanical Engineering, “Eftimie Murgu” University of Resita, P-ta Traian Vuia 1-4, 320085, Resita, Romania, [email protected] Gilbert-Rainer GILLICH Department of Mechanical Engineering, “Eftimie Murgu” University of Resita, P-ta Traian Vuia 1-4, 320085, Resita, Romania, [email protected] Ion Cornel MITULETU Department of Electrical Engineering, “Eftimie Murgu” University of Resita, P-ta Traian Vuia 1-4, 320085, Resita, Romania, [email protected] Zeno-Iosif PRAISACH Department of Mechanical Engineering, “Eftimie Murgu” University of Resita, P-ta Traian Vuia 1-4, 320085, Resita, Romania, [email protected] Nicoleta GILLICH Department of Electrical Engineering, “Eftimie Murgu” University of Resita, P-ta Traian Vuia 1-4, 320085, Resita, Romania, [email protected] Abstract: - In damage detection applications, the precision of evaluating the natural frequencies is crucial, because small structural changes lead to reduced modal parameter changes. Usually, the analyzed timelength is reduced and thus improper frequency resolution is achieved. This makes happen that the true frequency is located between two bins indicated in the spectrum and the modal parameter change is consequently not observed. In the case of beams, which have the natural frequencies of bending modes widely distributed, the frequency estimation can be made in a similar way with that of isolated tones. Algorithms applied for extraction of isolated frequencies are based on interpolation or curve fitting, data involved being the frequency indicated on the spectral bins and the associated amplitudes. Finding the true frequency requires deriving a frequency-amplitude pair located between the displayed bins in the spectrum. Algorithms performing DFT frequency and magnitude interpolation are investigated and compared relative to a proposed method which may offer advantages over actual approaches. Keywords: - spectral analysis, frequency evaluation, interpolation, natural frequency

1. INTRODUCTION Vibration-based detection of damages in structural elements is based on the analysis of modal parameter changes occurred due to alteration of structural integrity [1]-[3]. Most common modal parameters are the natural frequencies, the mode shapes and curvatures, the modal flexibility and the modal damping [4]-[7]. By associating the modal parameter shift with damage size and position predictive model result and damage patterns can be derived [8]-[10]. In many applications, it is necessary to observe these changes as soon as possible, because the damage occurrence may affect the structural safety and harm equipment and persons. RJAV vol XIII issue 2/2016

Mostly used modal parameter is the natural frequency, because it is achieved with simple and robust equipment. The main disadvantage of using this parameter consists of the short-time signal acquired if modal analysis techniques are applied [11]-[12], which restricts the resolution of the estimate provided by the Discrete Fourier Transform (DFT) or by the Fast Fourier Transform (FFT). The time limitation is imposed especially to higher-order frequencies which are subjected to rapid damping [13]. Also, standard frequency estimators as DFT or FFT cannot provide reliable results if computational or other limitations restrict the number of samples which may be processed. It is worth to mention that in all cases the acquired signal is not processed before the spectral analysis. 98

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To overcome this difficulty, several algorithms to increase the frequency readability were proposed in the literature, all of them being resource consuming. This paper introduces a new algorithm to improve the frequency readability, based on the spectral analysis of three signals attained from the originally acquired signal cropped by windowing with rectangular windows. The three picks in the three spectra are finally used for interpolation.

2. ACTUAL INTERPOLATION METHODS The motivation for studying DFT output interpolation or curve fitting is the failure of actual methods for damage detection applications. In Figure 1 an output achieved for standard DFT evaluation for a sinusoidal input is depicted.

The estimation of displacement δ is referred to as the fine frequency estimation, as opposed to the coarse estimation performed by locating DFT maximum [15]. Grandke [16] developed an efficient method which involves the DFT output peak and the largest neighbor, achieved from the time domain signal weighted with a Hann window. In the spectral representation in Figure 1 the maximizer and it's largest neighbor are Ak and Ak 1 . The frequency correction estimate  results from calculating the following ratios: A   k 1 Ak

(1)

2  1  1

(2)



The corrected frequency results from

Ak

fc   k     f

(3)

If the largest maximizer’s neighbor is Ak 1 , the frequency correction estimate results from: A   k 1 Ak

Ak-1 Ak+1 k-1

k

k+1

Figure 1. DFT output for a sinusoid by standard evaluation.

The true frequency of the evaluated sinusoid is f  5Hz and the signal length is ts  0.92s . Note that the frequency resolution f depends on the signal length, thus for another analysis time the DFT representation differs; by chance, i.e. the signal contains an entire number of cycles, the real frequency will be indicated. One can observe in Figure 1 the three relevant spectral bins. The so-called maximizer, centrally placed, consists in the pick amplitude Ak and its associated spectral bin index k which has two neighbor bins k  1 and k  1 with the output amplitudes Ak 1 and Ak 1 . Some usual DFT output interpolators providing the estimated frequency of a sinusoid are presented in this section. In all cases a real number  which indicates the distance in bin widths from the DFT output maximizer to the true frequency is calculated. The corrected frequency is derived by adjusting the read frequency with a fraction of f . RJAV vol XIII issue 2/2016

(4)

and the corrected frequency is fc   k  1     f

(5)

A similar method was developed by Quinn [17], but it utilizes the DFT output maximizer and both neighbor bins. However, two interpolations are applied, each involving just two bins. The method is simple since it does not require windowing of the time signal before the DFT is applied. Following ratios are calculated for Quinn's estimator: A 1  k 1 Ak

(6)

1 1  1

(7)

A  2  k 1 Ak

(8)

 2 1  2

(9)

1 

and

2 

Corrected frequency f c is derived with Eq. (3), the correction estimate being chosen as following: - if 1  2 and 2  0 then   2 - else   1 . 99

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A similar frequency estimator is proposed by Jain et al. [18]. It is applied in respect to the amplitudes of the maximizer’s neighbor bins. If Ak 1  Ak 1 then Ak Ak 1

(10)

1 1  1

(11)

1  1 

and the corrected frequency is derived as fc   k  1     f

(12)

3. A NEW EVALUATION ALGORITHM The weak point of actual frequency estimators is the distribution of the considered spectral bins k  1 , k , and k  1 . These are located on the main lobe just in the particular case when k fit the true frequency and its neighbors are null. In all other cases, just two of the bins belong to the main lobe, the third being placed on a lateral lobe, as shown in Figure 2. This happens doubtless because, if the true frequency is not located exactly on a spectral bin leakage occurs, and the energy is distributed on more spectral bins.

Else, the correction estimate  results from 1

A  2  k 1 Ak  2 2  1  2

(13)

Spectrogram DFT samples

Ak

0.8

True frequency

(14)

0.6

Ak+1 0.4

and the corrected frequency is derived from Eq. (3).

Ak-1 0.2

The next algorithms make use of three spectral bins for the interpolation at once. Ding [19] proposed a barycentric method, where the correction estimate  results from 

Ak 1  Ak 1 Ak 1  Ak  Ak 1

(15)

Another correction estimate, found by involving the quadratic method, presented by Voglewede in [20], is 

Ak 1  Ak 1 2  2 Ak  Ak 1  Ak 1 

(16)

Instead this quadratic estimator, Jacobsen [20] proposes a new quadratic estimator, which reads:  Jac 

Ak 1  Ak 1 2 Ak  Ak 1  Ak 1

(17)

This estimator was developed heuristically and presented without a proof. Based on Jacobsen's estimator, Candan proposed in [4] the following improvement of  : 

tan( / N )  Jac / N

(18)

For all correction estimates derived from Eq. (13) to (18), the corrected frequency is again calculated using Eq. (3). The analysis of actual estimators has shown that neither leads to enough precise results for damage detection applications. In the following section we propose a new estimator providing improved accuracy. RJAV vol XIII issue 2/2016

0

k-1

k

k+1

Figure 2. Distribution of DFT outputs on the spectral lobes.

The occurrence of a discontinuity in the spectral curve makes difficult finding interpolation methods valid for any position of the three relevant spectral bins. Our idea was to ensure that the three points are on the main lobe. This is possible just if the distance between the selected bins is lower as the frequency resolution. As a consequence, the bins have to belong to different spectra of a signal, for which appropriate frequency resolutions are selected. In fact, the signal is truncated in order to achieve three specific time lengths tS1 , tS 2 , and tS 3 . The lengths are selected to ensure a spectral bin: - at the right of the expected frequency ( 1.05 f  f1  1.15 f ); - at the expected frequency ( f 2  f ); - at the left of the expected frequency ( 0.75 f  f3  0.9 f ), in such way that Ak achieves maxima. Interpolation can be performed using the less resource-consuming method. In essence, the algorithm consists in: 1. defining the three time lengths tS1 , tS 2 , and tS 3 ; 2. applying the DFT for the three time lengths; 3. performing the interpolation.

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4. NUMERICAL TESTS TO COMPARE THE EFFICIENCY OF THE PROPOSED ALGORITHM AGAINST ACTUAL APPROACHES The test is performed on a harmonic signal with the frequency f  5Hz and time length tS  1.1s . This signal was iteratively shortened with a time step   0.2s and the DFT was applied. Table 1 indicates the signal time lengths, frequency resolution and the

three relevant frequencies and amplitudes in the spectrum. Based on this data, firstly the coefficients  and the frequency correction estimates  are derived for all methods. Afterward, the corrected frequencies are derived and graphically represented in order to ensure a facile comparison. Finally, the proposed algorithm is implemented and the results compared against the actual methods. To provide a clear overview, all achieved results are tabular indicated.

Table 1. The value of the deflection corresponding to the analyzed cases

tSi [s]

fi [Hz]

f k 1 [Hz]

Ak 1 [mm/s2]

fk [Hz]

Ak [mm/s2]

f k 1 [Hz]

Ak 1 [mm/s2]

0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1

1.11111111 1.08695652 1.06382979 1.04166667 1.02040816 1 0.98039216 0.96153846 0.94339623 0.92592593 0.90909091

4.44444444 4.34782609 4.25531915 4.16666667 4.08163265 4 3.92156863 3.84615385 3.77358491 3.7037037 3.63636364

0.185129 0.107339 0.054419 0.026047 0.005814 0 0.00323 0.011886 0.021705 0.029432 0.029897

5.55555556 5.43478261 5.31914894 5.20833333 5.10204082 5 4.90196078 4.80769231 4.71698113 4.62962963 4.54545455

0.221085 0.309599 0.354884 0.444031 0.492248 0.500233 0.475452 0.432661 0.375711 0.30478 0.222326

6.66666667 6.52173913 6.38297872 6.25 6.12244898 6 5.88235294 5.76923077 5.66037736 5.55555556 5.45454545

0.017235 0.017623 0.017674 0.013023 0.004522 0 0.007752 0.029121 0.06615 0.117494 0.185659

5.5

6

5.5

Jacobsen

Grandke

Ding

Jain

Voglewede

frequency [H z]

Frequency [H z]

Quinn

5

4.5

5

4.5 0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1

2

3

4

5

6

7

8

9

10

11

Signal time lenght [s]

Signal time lenght [s]

a)

b)

Figure 3. Deriving the precision of frequency estimation methods based on: (a) the analysis of two bins at once; (b) the analysis based on three bins.

First remark made by reviewing Figures 3 and 4 is the fact that all estimators provide the true frequency value if the signal length contains entire cycles. This happens for tS  1s in our example, the true frequency 5Hz being indicated. A second remark can be made regarding the error range; while the estimators which consider two points for interpolation give errors up to 20%, for the estimators which take three points into consideration the error is limited to 10%. RJAV vol XIII issue 2/2016

A special attention should be given to the estimator proposed by Jain et al., which cannot be used if the signal length is nT ...  n  0.5  T . Here T is a period an n an integer. In contrary, for the analysis time length  n  0.5  T ...nT it gives the best results among all analyzed estimators. Between the frequency evaluator considered in the analysis presented herein, we found out that the most trustful is that base on the barycentric method, introduced by Ding et al. 101

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Table 2. The correction estimates δ and the values of the corrected frequencies for the six considered evaluators

tSi [s] 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1

Grandke δ fC [Hz] [-] -0.783 5.883 -0.838 5.6833 -0.857 5.525 -0.914 5.335 -0.972 5.149 -1 5 -0.979 4.908 -0.919 4.835 -0.836 4.774 -0.735 4.726 -0.644 4.679

Quinn δ fC [-] [Hz] -0.084 5.471 -0.060 5.374 -0.052 5.266 -0.030 5.178 -0.009 5.092 0 5 0.006 4.908 0.028 4.835 0.061 4.778 0.106 4.736 0.155 4.700

Jain δ fC [-] [Hz] 0.544 4.988 0.742 5.090 0.867 5.122 0.944 5.111 0.988 5.069 0 5 -0.016 4.885 -0.072 4.735 -0.213 4.503 -0.627 4.002 -5.063 -0.517

Regarding the evaluator proposed by Candan, it takes really effect just if the numbers of samples in the signal are less than 150, which is unusual for vibration measurement. Else, it indicates the same values as the Jacobsen evaluator. The evaluation algorithm proposed in this paper involves three points for interpolation, all of them certainly situated on the main lobe. This is ensured by choosing the proper signal’s time length tS1 , tS 2 respectively tS 3 for the spectral analysis. As an example, let us consider a beam having in the initial, intact state, the frequency fU  5.2 Hz . In the actual state the beam could be damaged and consequently have a diminished frequency, let’s say it is f  5Hz , similar with the signal tested before. Because the damage is not observable by visual control, the presumption is that the expected signal has 5.2Hz. Thus, following signals time lengths for the DFT analysis are chosen from Table 1: - The time length tS 2  0.96s , because the time 0.961538 seconds correspond to 5 entire cycles for the expected frequency. - The time length tS1  0.92s , because the time 0.91575 seconds correspond to 5 entire cycles for an increased frequency with 5%. - The time length tS 3  1.1s , because the time 1.09265 seconds correspond to 5 entire cycles for an decreased frequency with 12%. From the DFT of the three signals sections represented in Figure 4, achieved from the same original signal by truncation, the three pick values and the corresponding frequencies are selected. The values are: for tS1  0.92s : f k1  5.43478Hz Ak1  0.3096 for tS 2  0.96s : f k 2  5.20833Hz Ak 2  0.444031 for tS 3  1.1s : f k 3  5.43478Hz Ak 3  4.5454 RJAV vol XIII issue 2/2016

Jacobsen δ fC [-] [Hz] -0.70 4.855 -0.181 5.253 -0.057 5.261 -0.015 5.192 -0.001 5.100 0 5 0.004 4.906 0.021 4.828 0.067 4.783 0.190 4.819 0.679 5.225

Ding δ fC [-] [Hz] -0.396 5.159 -0.206 5.228 -0.086 5.233 -0.026 5.181 -0.002 5.099 0 5 0.009 4.911 0.036 4.844 0.095 4.812 0.194 4.824 0.355 4.901

Voglewede δ fC [-] [Hz] -0.350 5.205 -0.090 5.344 -0.0288 5.290 -0.007 5.200 -0.001 5.101 0 5 0.002 4.904 0.010 4.818 0.033 4.750 0.095 4.724 0.339 4.885

a)

b)

c) Figure 4. DFT representation of truncated signals: (a) for tS1  0.92s ; (b) for tS 2  0.96s ; (c) for tS 3  1.1s 102

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0.5 0.45 0.4 0.35 0.3 0.25 0.2

2

y = -1.0436x + 10.514x - 26.005 2 R =1

0.15 0.1 0.05 0 4.5

4.6

4.7

4.8

4.9

5

5.1

5.2

5.3

5.4

5.5

Figure 5. DFT representation of truncated signals

The graphical representation of the three points is given in Figure 5; here the function for the trend line is also indicated. By derivation of the trend line function and equaling the result with zero, result the equality 2.0872 x  10.514  0

(19)

thus the maximum is achieved for x  5.039 which is the evaluated frequency. One observes that the precision is considerably improved, the error being less then 1%. This error limit is valid also if other time lengths are chosen for the analysis.

5. CONCLUSION The study presented in this paper revealed that frequency estimators basing on three-point interpolation methods are more precise then these based on just two points, the error limit being 10% for the first class of interpolators in comparison with 20% achieved with the second class. It was also found that the reason for such high errors is the location of the points involved in the interpolation, not all three being located on the main lobe. The algorithm proposed herein avoids this disadvantage and essentially increase the frequency readability, for all simulations the errors being les then 1%.

REFERENCES [1] Gillich G.-R., Birdeanu E.D., Gillich N., Amariei D., Iancu V., Jurcau C.S, Detection of damages in simple elements, Annals of DAAAM and Proceedings of the International DAAAM Symposium, 2009, pp. 623-624. [2] Bratu P., Vibratiile sistemelor elastice, Ed. Intact, Bucuresti, 2012. [3] Gillich G.-R., Minda P.F., Praisach Z.-I., Minda A.A., Natural frequencies of damaged beams - a new approach, Romanian Journal of Acoustics and Vibration, 9(2), 2012, pp. 101-108. [4] Gillich G.-R., Praisach Z.-I., Detection and quantitative assessment of damages in beam structures using frequency and stiffness changes, Key Engineering Materials, 569-570, 2013, pp. 1013-1020.

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[5] Yang Z.B., Radzienski M., Kudela P., Ostachowicz W., Scalewavenumber domain filtering method for curvature modal damage detection, Composite Structures, 154, 2016, pp. 396– 409. [6] Praisach Z.-I., Minda P.F., Gillich G.-R., Minda A.A., Relative frequency shift curves fitting using FEM modal analyses, Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements, Paris, 2011, pp. 82-87. [7] Song Y.Z., Bowen C.R., Kim A.H., Nassehi A., Padget J., Gathercole N., Virtual visual sensors and their application in structural health monitoring, Structural Health Monitoring, 13(3), 2014, pp. 251-264. [8] Sinha J.K., Friswell M.I., Edwards S., Simplified models for the location of cracks in beam structures using measured vibration data. Journal of Sound and Vibration, 251(1), 2002, pp. 13-38. [9] Gillich G.-R., Maia N.M.M., Mituletu I.-C., Tufoi M., Iancu, V., Korka, Z., A new approach for severity estimation of transversal cracks in multi-layered beams, Latin American Journal of Solids and Structures, 13(8), 2016, pp. 1526-1544. DOI: 10.1590/1679-78252541 [10] Wang L., Lie S.T., Zhang Y., Damage detection using frequency shift path, Mechanical Systems and Signal Processing, 66-67, 2016, pp. 298–313. [11] Gillich G.R., Praisach Z.I., Abdel Wahab M., Vasile O., Localization of transversal cracks in sandwich beams and evaluation of their severity, Shock and Vibration, 2014, 607125, DOI: 10.1155/2014/607125 [12] Minda P.F., Praisach Z.-I., Gillich N., Minda, A.A., Gillich, G.-R., On the efficiency of different dissimilarity estimators used in damage detection, Romanian Journal of Acoustics and Vibration, 10(1), 2013, pp. 15-18. [13] Gillich G.-R., Frunzaverde D., Gillich N., Amariei D., The use of virtual instruments in engineering education, Procedia - Social and Behavioral Sciences, 2 (2), 2010, pp. 3806-3810. DOI: 10.1016/j.sbspro.2010.03.594 [14] Chioncel, C.P., Gillich, N., Tirian, G.O., Ntakpe, J.L., Limits of the discrete fourier transform in exact identifying of the vibrations frequency, Romanian Journal of Acoustics and Vibration, 12(1), 2015, pp. 16-19. [15] Djukanović S., Popović T., Mitrović A., Precise sinusoid frequency estimation based on parabolic interpolation, 24th Telecommunications Forum TELFOR, Belgrade, 2016, pp. 1-4. [16] Grandke T., Interpolation Algorithms for Discrete Fourier Transforms of Weighted Signals, IEEE Trans. on Instrumentation and Measurement, Vol. IM-32, 1983, pp 350-355. [17] Quinn B.G., Estimating Frequency by Interpolation Using Fourier Coefficients, IEEE Trans. Signal Processing, 42, 1994, pp. 1264-1268. [18] Jain V.K., Collins W.L., Davis D.C., High-Accuracy Analog Measurements via Interpolated FFT, IEEE Trans. Instrumentation and Measurement, Vol. IM-28, 1979, pp 113-122. [19] Ding K, Zheng C., Yang Z., Frequency Estimation Accuracy Analysis and Improvement of Energy Barycenter Correction Method for Discrete Spectrum, Journal of Mechanical Engineering, 46(05), 2010. [20] Voglewede P., Parabola approximation for peak determination, Global DSP Mag., 3(5), 2004, pp. 13-17. [21] Jacobsen E., Kootsookos P., Fast, accurate frequency estimators, IEEE Signal Processing Magazine, 24(3), 2007, pp. 123-125. [22] Candan C., A method for fine resolution frequency estimation from three DFT samples, IEEE Signal Processing Letters, 18(6), 2011, pp. 351-354.

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