A multi-resolution based method to precise identify the [623056]

A multi-resolution based method to precise identify the
natural frequencies of beams with application in damage
detection
I C Mituletu1*, N Gillich1, J L Ntakpe1, H Furdui1, C Chioncel1
1Department of Mechanical Engineering, “Eft imie Murgu” University of Resita, P-ta
Traian Vuia 1-4, 320085 Resita, Romania
E-mail: [anonimizat]

Abstract. This paper presents an enhanced met hod that accurately identifies natural
frequencies of structures, via an important improvement of spectral lines density. Spectral lines
density increases due to the superposition of many spectra, each spectrum being accomplished
by subtracting certain number of samples from the original signal and using the power spectral
density analysis. The newly achieved overlapp ed-spectrum shows important additions to
spectral line number and makes more accurately identification of frequencies in the peak
position of relevant amplitudes. In this way, modal analysis becomes more reliable and
efficient. Method application assumes smal l frequency changes and the advantage of
establishing more precisely the position and dime nsion of damage. The analysis of all results
explicitly indicates an important assessment in the evaluation of frequency. Considering
relevant aspects, it can be appreciated that the approached method offers a new development
way in the vibration-based damage detection, allowing de possibility to perform more accurate
evaluation of frequencies and introducing new features into the domain.
1. Introduction
In time, whatever physical structure, under the application of a large variety of strength types, gets
damage. By detecting damages in their very early stage, the keeping of structure integrity and
functionality can be ensured via an adequate mainte nance. The impact of early damage detection is
significant for big and complex structures, causing money and work-time saving and also decreasing
the possibility to occur accidents. There is a great area of methods that can perform non-destructive
tests as regard the damage detection. Dynamic met hods offer an import advantage they do not require
access to the damaged place [1].
Within damage detection domain, the vibration- based methods are very popular and also widely
spread. Herein natural frequency shifts are determin ed and afterwards atten tively analyzed [2]-[3].
These methods can be commonly divided into two cat egories those having limitations at the damage
detection or designed to predict detection, location and quantification of damage [4]. Into the second
category, all methods based on the natural frequency shifts are defined as model-based, [5] and [6],
and use finite element models. The readability of frequency shifts determines the accuracy of damage
detection. Also, a great precision in frequency readin g and clear measurements lead to better results in
detection. Small frequency changes caused by early st ructural damage or even small damage cannot be
distinguished from those of the healthy structure, and so that the mathematical model is under the
impossibility to work well. Thus, mostly it is not possible to make a difference between improperly

models and fine changes caused by the occurrence of small or early damage [7] and [8]. And these are
not all the issues regarding the frequency shifts , the temperature action or/and structural loads
application induce more other interferences.
Also, other methods available as well, besides a comprehensive classification and description are presented in [3]. Each method fits a particular app lication, due to its specific advantage. Until now, at
least one method does not meet all the requirement s supposed by the high variety of analyzed
structures and the diversity of imposed conditions. In this paper, a new method to enhance the fre quency evaluation is presented. It is based on an
extensive study regarding the transformation of ac quired time-domain signal in a frequency spectrum.
The achieved spectrum has to provide many possibiliti es to find a better fit for a certain frequency.
Herein, an overlapped-spectrum has been accomp lished via an iterative algorithm, performing a
modulation from the right-side of acquired signal wi ndow. A step-by-step windowing of time signal is
applied to an algorithmic structure implemented in LabView. Each step of iteration consists in
subtracting from the signal right-side of a certain nu mber of time samples. Also, after subtracting the
samples, an analysis applying Power Spectral Density (PSD) feature of LabView is made. The amplitude values of each frequency are stored and at the end of last iteration step, they are displayed
all in one single spectrum, achieving in this wa y a much denser frequency overlapped-spectrum.
2. Description of frequency identification method
Method consists of three steps: si gnal acquisition, reference analysis and algorithm setup. Considering
the spectrogram and Wigner-Vile distribution, as could be seen in figure 1, it is observed that certain
frequencies are damping very fast, providing in this way a very short time length [9]-[10]. As known,
time length gives the resolution of the frequency sp ectrum after Fourier analysis is performed. Thus,
spectrum resulted by applying ordinary Fourier Tr ansform provides poor density and in case of small
damage, determining fine frequency sh ifts, this is impossible to distinguish. So that, even if a faster
acquisition device is taken to achieve much more sa mples in a short time interval, any enhancement of
the frequency spectrum can be accomplished.

a . b .
Figure 1. Wigner-Vile distribution (a) and power spectrum (b) of vibration signal

To significantly enhance the spectrum density , the following argument can be assumed: by
decreasing the length of signal, from the side of low amplitude in the time history, the number of
spectral lines is decreasing as well, but also thei r position changes. From this last point of view,
cropping the signal length become important and can be iteratively applied to achieve a multi-
spectrum. If an iterative step-by-step signal cropping is imagined, which is performed by an algorithm,
and also the obtained spectral lines are afterwards displayed in one single graphical spectrum, we finally can achieve a much denser frequency spectrum.

Algorithm to analysis and iteratively crop the si gnal is implemented in LabView under a logical
loop structure. Loop considers a start point and a stop one, within these points performing a number of
iteration. From the beginning have to be establishe d some functional parameters, as initial length of
time signal, entire cropping portion of signal and number of samples cropped at the iteration step.
2.1. Signal acquisition and pre-processing
In order to prove the method performance, make a right comparison of results and have a better
flexibility for input signals, we provide as time signa ls those generated by LabView. Of course, they
have been generated to meet few important require ments: short length, are mixed and have different
amplitudes. Figure 2 shows an auto-generated ti me signal, different cropped and mixed from two
signals having the following characteristics: entire time length t, total number of samples N, frequency
f and amplitude A. First signal S
1 has f 1 = 4.2 Hz and A 1 = 1 mm/s2, the second one S2 is at f 2 = 23.7 Hz
and A 2 = 0.5 mm/s2.

Figure 2. Mixed sine signal with two va riable amplitude and cropped.

Left image of figure 2 presents the entire signal t = 1 s, middle image shows the signal cropped
with a period T 1 = 0.0(6) s and the right-side one has a cropped period T 2 = 0.037. The reason of
cropping is assumed by algorithm setup and will be later detailed.
2.2. The reason of applying reference analysis
Actually, reference analysis is performed to acquire an initial spectrum and establish who the
frequencies of interests in a certain case are. At th is time, the cropping period is chosen taking into
account the lowest frequency considered as expected from experience reasons. It is very important to
establish the right frequencies of interest, due to th e fact that in correlation with them the algorithm
must be set. In the presented case, this second step is redundant, the freque ncies are well known being
generated by ourselves. In real cases, the result of th is test can be assumed as being similar to that
shown in figure 1 b.
2.3. Explaining the algorithm implementation
Algorithm is a logic structure implemented in a virt ual environment employed to ensure an explicitly
work and satisfy certain requirements. This algorith m is formulated to get the signal, compute some
parameters, iteratively crop and analyze the signa l, and display all the results in one graphical
frequency spectrum. First consideration is about the i nput signal, which can be auto-generated or real
acquisitioned and stored as a file with a given ex tension. In LabView, the "Block Diagram" ( BD)
window ensures the placement and interconnectivity of functional blocks. Auto-generated signal can
be applied by using the functi onal block "Simulate Signal" ( SS), where the waveform, amplitude,
frequency, length and number of samples were set in this case. Main part of algorithm block diagram
(ABD ) is the logic structure placed into the limite d loop, where "Extract Portion of Signal" ( EPS) and
"Spectral Measurements" ( SM) blocks perform signal cropping and spectral analysis. At the end of
loop execution, all data achieved from the ite rative analysis is sending to the "Graph" ( G) block. In a
different window "Front Panel" ( FP), data is show as graphical spectrum. The parameters chosen to be
set or visualize are also shown in FP.

Figure 3 presents the BD image that is meant to comprehens ively clear up all the aspects associated
with the algorithm implementation.

Figure 3. Software implementation of fre quency identification algorithm.
In table 1 parameter abbreviations included in the figure 3 are presented and detailed as well. Also,
for each of them the sense of consideration (as inpu t or output) is presented, regarding the displayed
values meaning that can be divided in two categories, set by hand or resulted after computing.
Table 1. Explanation of parameters involved in the identification algorithm.
Parameter Sense Detail
A1,2 Input Signal amplitude
F1,2 Input Signal frequency
Ph3 Input Signal phase – not used in these tests
Ref.F. Input Reference frequency – exp ected to reach by analysis
Ref.P. Output Reference period – corresponds to Ref.F.
T.S.N. Input Total samples number – all signal samples considered in test, equal to N
S.I. Input Sampling interval – whole signal length in time, equal to t
S.T. Output Sampling time
B.S. Input Begin sample – first sample from the signal left-side
A.S.N. Output All samples number – remaining samples after last iteration is done
Start.It. Input Start iteration – sample whence algorithm begins the iterative cropping
It.S. Input Iteration step – number of samples cropped at iteration
Stop.It. Output Stop iteration – number of samples cropped at the iteration end
In.It.N. Output Initial iteration number – resulted by computing
Real.It.N. Output Real iteration number – resulted by applying It.S. to In.It.N. and rounding
O.S. Output Original signal – graphically displayed
P.S. Output Portioned signal – graphically displayed when the iteration ends
D.R.S. Output Displaying of resulted data – data displayed as overlapped-spectrum

Above parameters have not been considered only for simple signal cropping and iteration, but also
to create many possibilities of data and results ma nagement from whatever section of the algorithmic
structure. Therefore, considering important the way in which identification algorithm works and can
be set, in order to get suitable results more details are provided in the next chapter.
3. Easy frequency evaluation by adequate algorithm setup
From the beginning has to be mentioned the importance of following few steps to properly setup the
identification algorithm and easy achieve clear results:

• Chose the lowest expected frequency value, decr ease its value to at least 10% and use it as
Ref.F.
• Perform reference analysis and make the c hoice regarding the frequencies of interest.
• For each of these frequencies a new algorithm setup has to be applied and also a new analysis
with the identification algorithm must be taken.

Let us consider figure 4 as example for how al gorithm parameters have to be managed and what
for. Firstly, image from figure 4 has been accomplished by writing the following values for
parameters: T.S.N. = 50000, S.I. = 1 s, Ref.F. = 3.5 and It.S. = 50. Ref.F. was deduced by assuming a
lowest expected frequency in about of 4 Hz, subt racting 10% and rounding to 3.5 Hz. Afterwards,
overlapped-spectrum achieved is zoomed in around of 4 Hz, so as the image offers a good vision.

Figure 4. Overlapped-spectrum achieved at Ref.F. = 4 Hz.

Overlapped-spectrum is realized from 286 graphical lines, being resulted from the same number of
iteration performed by the algorithm. Image can tric k us, due to its apparent complexity, and make
enough difficult the evaluation of frequency because of the presence of some superposed small lobes.
To clear it up, values Ref.F. as well as Start.It. are recommended to be modified. By increasing the
value of Ref.F., smaller lobes disappear and right-side of the main lobe is going to clear as well. Also,
increasing the value of Start.It., left-side of the main lobe become more definite.
But in the overlapped-spectrum two values are di stinguished, as well as in the auto-generated time
signal provided to the input of identification algorit hm. Magnifying area in around of 23 Hz, the image
of the other main lobe is depicted in figure 5.

Figure 5. Overlapped-spectrum achieved at Ref.F. = 23 Hz.

As easily can be seen, more superposed lobes are affecting the image of the main lobe at about of
23 Hz. To refine its area, values Ref.F. and Start.It. have to be managed differe ntly than before. First
modification is addressed to Ref.F., which must be significantly increased at least up to the value of
expected frequency or even more, if the superposed lobes do not disappear. As for lobe's left-side,
Start.It. value will be set much lower, otherwise lobe will entirely vanish.
4. Results analysis and discussion
Taking into account the previously presented tech nique for managing the parameters of frequency
identification algorithm, we offer the values of those strictly involved in clearing lobe areas. In this
way, making easy reading frequencies within th e overlapped-spectrum resulted after iteration
completes.

Figure 6. Frequency identification at 4.2 Hz, after performing an adequate algorithm setup.
Figure 6 shows how easy is the identification of frequency after perform ing adequate setting of
algorithm parameters, frequency matching area being ma gnified and placed at the up-right side in the
same image. Due to the clearness of the lobe area, simply identify the peak of lobe, and there must be
the best frequency matching. Being in LabView, th e cursor numerically shows values for frequency
and amplitude in the pointed location.

The values of input parameters that suitable charact erize the first frequency identification at 4.2 Hz
are the following: Ref.F. = 15, Start.It. = 1300 and It.S. = 50.
For the other frequency value of 23.7 Hz that is also provided into the mixed signal, a new
parameterization and analysis have been performed, and the identification was made using a graphical
evolution as presented in figure 7.

Figure 7. Second frequency is clearly identified at 23. 7 Hz, after performing recommended setup.

Suitable values for parameters setting, ensuring a clear area of the lobe, in this case, are Ref.F. =
27, Start.It. = 1200 and It.S. = 30. By these settings , a number of 22 spectra have been accomplished,
which formed the overlapped-spectrum.
Method limits can be defined as belonging to th ree groups: signal structure, acquisition device
capability and computing power. First group refers to number of signal samples, in case of short signal
length, at low-frequency value and for small frequency shifts; we have to achieve as many samples as
possible. Also, signal length has to stay bigger than one period of expected lowest frequency. Second
group considers the characteristics of acquisition devi ce. As for the sensitivity, a higher number of bits
for analog-to-digital conversion are requested, and for s hort signals faster devices have to be provided.
Third group treats computing power, regarding th e number of iteration. Once frequency shift value
decreases, much iteration must be performed, and gr eater computing power to diminish the analyzing
time is recommended.
In the end, it has to be specified, these frequenc y values actually were chosen as being nearby to
those from real cases [11]-[14], in order to give significance and weighty to the results of performed
tests. So that, results confirm the important improvement introduced by this method, and also
highlights the great advantage in using the describ ed new technique and algorithm, in the frequency
identification field.
5. Conclusion
The paper presents an enhance method to accurately identify frequencies within a multi-resolution
spectrum, here introduced as overlapped-spectrum. Th e method consists of a software algorithm, for
iteratively cropping of the signal and generating the overlapped-spectrum, and a technique to adequate
setup the algorithm, which clears the spectrum area in around of desired frequency.
Performing analyses on signals similar to real once, method confirms its effectiveness in improving
the readability of frequencies, especially in the case of short time signals. The clarity and easiness of frequency identification, strongly suggests this method for performing
in cases as rapid vibration damping or early damage occurrence.

The maximum precision in frequency reading depe nds only on the accuracy of the acquisition
signal and the number of samples, when short time signals are analysed.

Acknowledgments
The work has been funded by the Sectoral Op erational Programme Human Resources Development
2007-2013 of the Romanian Ministry for Europ ean Funds through the Financial Agreement
POSDRU/159/1.5/S/132395.
References
[1] Balageas D, Fritzen C P and Güemes A 2006 St ructural health monitoring, ISTE Ltd, London
[2] Cawley P and Adams R D 1979 The location of defects in structures from measurements of
natural frequencies Journal of Strain Analysis 14 2 pp 49-57
[3] Doebling S W, Farrar C R, Prime M B and Sh evitz D W 1996 Damage identification and health
monitoring of structural and mechanical systems from changes in their vibration
characteristics: a literature review Rep. No. LA 13070-MS Los Alamos National
Laboratory (Los Alamos, NM)
[4] Salawu O S 1997 Detection of structural da mage through changes in frequency: a review
Engineering Structures 19 9 pp 718-723.
[5] Rytter A 1993 Vibration based insp ection of civil engineering structures Ph.D. Thesis (Aalborg
University, Denmark)
[6] Morassi A and Vestroni F 2008 Dynamic Methods for Damage Detection in Structures
(CISM Courses and Lectures vol 499) (Springer Wien New York)
[7] Friswell M I 2007 Damage identification using inverse methods Phil. Trans. R. Soc. A 365 pp
393-410
[8] Ewins D J 2000 Model Testing: Theory, Practice and Applications (2nd Edition) Research
Studies (Press Ltd., Hertfordshire, England)
[9] Gillich G R and Praisach Z I 2014 Modal iden tification and damage detection in beam-like
structures using the power spectrum and time–frequency analysis Signal Process. 96(Part A)
29–44
[10] Onchis-Moaca D, Gillich G R and Frunza R 2012 Gradually improving the readability of the
time-frequency spectra for natural freque ncy identification in cantilever beams Proceedings of
the 20th European Signal Processing Conference – EUSIPCO 2012 (Bucharest)
[11] Gillich G R, Praisach Z I and Negru I 2012 Damages influence on dynamic behaviour of
composite structures reinforced with continuous fibers Mater. Plast. 49(3) 186-191
[12] Gillich G R, Praisach Z I, Wahab M A and Vas ile O 2014 Localization of transversal cracks in
sandwich beams and evaluation of their severity Shock Vib. 9(4-5) 607125
[13] Praisach Z I, Gillich G R and Birdeanu D E 2010 Considerations on na tural frequency changes
in damaged cantilever beams using FEM Latest trends on engineering mechanics, structures,
engineering geology 214-9
[14] Gillich G R and Praisach Z I 2013 Detection a nd Quantitative Assessment of Damages in Beam
Structures Using Frequency and Stiffness Changes Key Eng. Mat. 569 1013-20