INTRODUCTION

Vibrations are an intrinsic effect in engineering systems (civil structures, mechanical systems such as airplanes, vehicles, boats, mobile machinery and others); especially in machines, which work in a changing environment or under dynamic load. The vibrations can be: (i) harmful, (ii) useful, or (iii) a source of information ^{1}. The classical analysis of vibrations focuses on the mechanical behavior after overcoming a level of the threshold by the amplitude or frequency domain, which for different types of machines has been defined by national standards (e.g. Polish Standard -PN-) or international standards (e.g. ISO) ^{2}. But, to determine the mechanical condition of a machine, the natural vibration process during the lifetime can be used as source of information ^{3}.

The state of mechanical condition applied to engineering systems can be described through a set of parameter values in a given moment of a dynamic time, *t*^{4}^{,}^{5}. The time series of the condition mechanical state could be considered as the time of machine operating time, Θ. Using mathematical methods of assessment (forecastting, correlation functions, etc.) allows the qualification of the state of mechanical condition of machines in a particular period of time ^{6}^{,}^{7}.

The state of art includes studies ^{8}^{,}^{9} that have been focused on the mechanical modelling of the dynamic variation, considering the change of the loads to assess the mechanical condition. The dynamic assessment of the mechanical condition requires the association of their functional parameters (state features) through measurement dataset _{ Y(t ,Θ) = {y1, y2,…yn}, } and an analysis of the associated physical processes * Γ(X,S), * to get the symptom dataset

*S(T*=

*,Θ)*_{ {s1, s2,…sn} }(see Figure 1).

Previous studies ^{(9, 10)} has been exposed methodologies of technical assessment in order to get a mechanical condition assessment through theoretical or analytical models, without a implementation of an applied case.

Other studies ^{11}^{-}^{13} have used the Singular Value Decomposition (SVD) method in order to obtain different modes of failure. In the other hand, researchers ^{14}^{-}^{16} have demonstrated that the SVD method is an appropriate tool for analyzing a mapping from one space vector into another, possibly with a different dimensional vectoring. Others studies ^{(12, 17)} applied SVD to get an appropriate set of symptoms in the observation matrix, S*(t,*Θ*).*

A set of researchers ^{(15, 17, 18)} presented a validation of mechanical condition assessment from the SVD method, the researches obtained signals, which was generated by vibration of a vehicle engine, and the influence of the engine in the variation of the vibro-acoustic data was analyzed, and the influence of the engine in the variation of the vibro-acoustic data was analyzed; then, using the SVD method was possible to identify which symptoms in the observation matrix has better information to assess the state of mechanical condition of combustion engines; nevertheless, the tests were developed on laboratory conditions (using a bed test), then, an analysis in operation real condition not was feasible.

The reviewed literature has a limited scope, due to the theoretical SVD workframe has been applied to engineering systems with controlled conditions. This paper proposes an implementation of the SVD method via recording of dynamic variables, using vibro-estimators technique; then, the study aims to develop an experimental test on industrial machines in a manufacturing process to assess the its state of mechanical condition.

Below, the abbreviations, acronyms, coefficients and constants are presented which will be used throughout the paper:

FFT Fast Fourier Transform.

PDF Probability Density Function.

MD Multivariable Diagnostic.

SIBI Computer System of Identification Investigations.

SVD Singular Value Decomposition.

*A* Probability of unnecessary reparations.

*G* Reliability of the system on service.

_{ Fi ith } function of failure progress.

_{ Pi } Measurement points

*S*Symptom matrix observation.

_{ SDi ith } discriminant of failure.

*t*Dynamic time domain.

_{ Upp } , _{ Vrr } Left and right orthogonal singular matrix, respectively.

_{ vi } , _{ ui } _{ ith } eigenvector components.

_{ wF } Failure condition hypothesis.

_{ wG } Good condition hypothesis.

_{ wu } Bayes decision rule from hypothesis.

_{ σ i } _{ ith } eigenvalue.

_{ Φi } System variation.

_{ Σpr } Diagonal matrix of the singular values.

Θ Machine operation time, lifetime.

THEORETICAL ANALYSIS

Given a set of symptoms from vibro-acoustic signals which describe the mechanical condition, i.e. it contains relevant information about the state of the technical behavior ^{15}. The general procedure is shown in Figure 2.

The SVD method to dimension the symptoms observation matrix, is expressed as follows ^{(13, 15)}

where _{ Σpr } is the main diagonal matrix of the singular values, then _{ Σpr=diag } ({σ_{1}, …, σ_{ i}) } becomes the eigenvalues with a physical interpretation, σ_{ i } represents the fraction of damage (fault contribution) in the engineering systems. The SVD has the following mathematical property

where _{ vi, } and _{ ui } are the singular vectors, and Φ_{pr} describes the behavior variation of the system due operational condition, i.e. its wear or its failure modes.

The function of failure progress in the machine lifetime can be described through a discriminant, _{ SD i(Θk). } It is therefore possible to affirm that _{ SD i(Θk) } ~ _{ F i(Θk) }^{11}.

_{ SDi } is a sensible symptom of a specific failure and therefore, each _{ SDi } is used to identify a failure between different symptoms that evolve in the system. _{ SDi } contains independent information about the failure mode and also on the intensity of this failure ^{(2, 12)}.

The SVD result is analyzed by a joint of _{ SDi } and the decision theory concept. Then, two hypotheses in which the symptom will be classified are considered ^{(11, 19)}: good condition, w_{G}; and failure condition, w_{F}. The *posteriori* probability of each condition is with , the decision rule using Bayes theory is defined as wG if and only if

otherwise is wF, where: is the Probability Density Function (PDF) according to a good symptom condition S; and is the PDF of the failure state of *S.*

The *posteriori* conditional PDF is got with the *a-priori* probability *P*(_{ wu } ). Then, the decision rule can be formulated as the probability ratio _{ wG } in the case that

And _{ wF } in the contrary case. Generally, the bounder of the decision is defined through the equality of the pondered conditional PDFs ^{11}.

Neyman-Pearson lemma may be used to get particular test-statistics. Neyman-Pearson allows identifying the error of wrongly classifying a good condition as the beginning of a failure over a probability level of *A*^{21}.

Where *A* is the probability of unnecessary reparations, and *G* is the reliability of the system under service. It must be recalled that the preceding integral is nothing else but the reliability of the previously defined symptom ^{22}. The diagnostic of the probability of a wrong classification for a good condition is usually fixed to a low level (e.g. *A* = 0.05).

The final stage of the analysis is the correlation symptoms procedure with the identified failure, which allows ranking the symptoms due to the correlation coefficient value.

CASE OF STUDY

This study is applied to industrial machines for lighting elements manufacturing in series (see Figure 3). The machines were made by the Philips Co. in the Netherlands, into a production line model HOOR-6002 and the year of production was 1980. The maximum output capacity of the line is 6900 lighting elements per hour (tube lights). The machines are subject to preventive maintenance once a week, and also it is carried out a detailed inspection quarterly basis to review all systems.

The research has been conducted on two identical machines. One machine is having a change in the dynamic state that generates an increased amount of damaged production elements (lighting elements), which will be called fault machine, M_{F}. The second machine does not have production errors, representing the nominal behavior of the machine dynamic state, i.e. the second machine shows the reference values, _{ MR }^{20}.

The physical vibration performance has been recorded in a DAQ system by means of a set of single-axial piezoelectric accelerometers, PCB-352C68-ICP model. The signals in time domain recorded from DAQ system were processed to a real-time hardware package; it has been setted as following:

(i) a four-channel module VIBdaq (24 VDC; 2,4 mA) + Anti-aliasing filtering data individually adjusted (gain error < 0.5%), and a sampling frequency of 100 kilo-sampling per second with 16 bit resolution; and

(ii) a digital processing signal software called SIGVIEW, which has an optimized FFT algorithm with windowing and zero-padding.

The field test procedure requires a selection of a measurement points set. The criteria to select the measurement points are: (i) structural design of the machine, and (ii) physical access to the particular point in the machine, as well as an appropriate surface to attach the transducer, see Appendix A. The measures are focused on the main bearings due to the characteristic of the type industrial machine applied as case of study (rotating machine). A total of nine (9) measurement points _{ Pt = {p1, …, p9} } have been identified on each machine, _{ MF } and _{ MR. } The set of points _{ {p1, p2, p3, p9} } has been located on the side of the machine actuator, see Figure 7, and the set of points _{ {p4, p5, p4, p7, p8} } on the side of the feed tape, see Figure 8. The set of points _{ {p1 p2, p3, p4, p4, p5, p9} } has axial direction related to the main bearings, and the set of points _{ {p7, p8} } has radial direction. All datasets recorded are available from the authors of this paper, as an example, the data recorded from _{ MF } and _{ MR } are shown in Appendix B; a minimum 50kHz sample frequency has been established, including an anti-aliasing filter setting at a twice sampling frequency. The signals recorded as dataset were transformed to the frequency domain by Fast Fourier Transform (FFT) algorithm. FFT was used to ensure proper excitation in frequency range of interest which from 0 to 8 kHz.

ANALYSIS OF RESULTS

The dataset have been analyzed with an algorithm called Computer System of Identification Investigations (SIBI). SIBI algorithm was used to validate the research results ^{(2, 16)} and allowed generating a set of symptoms, *,* which represent: (i) _{ xave } - Average; (ii) _{ ARMS } - Root Mean Square amplitude; (iii) σ_{ j } - Standard deviation; (iv) *I* - Shape factor; (v) *C* - Crest factor; and (vi) *I* - Impulse factor. The selected symptoms _{ Sj } are the most common measures used in analyzing the experimental test results for this type of machines; also, these measures have a high sensitivity to the mechanical variation, which allows the identification of the mechanical condition.

S is analyzed by using SVD for the quantitative identification of symptoms related to an individual damage participation _{ SDi }^{16}. The results of the failures contribution percentage _{ SDi } (see Figure 4) show that _{ MR } has one representative fault indicator (SD_{1}), while _{ MF } presents an increase in the quantity of fault items _{ (SD1, SD2). SD1 } values indicate a low dynamical level in the machine (verified *posteriori),* and larger _{ SD2 } values mean a damage development. Figure 4 shows _{ SD1 } values decrease from _{ MR } to _{ MF } (10% aprox.), and _{ SD2 } values increase from _{ MR } to _{ MF } (10% aprox.); then, the development of damage is observed which causes the machine faulty behavior.

The histogram of _{ SD1 } (see Figure 5) allows the failure assessment of the machine. The standard deviation of _{ SD1 } decreases from _{ MR } to _{ MF } (8,5% aprox.), then, the principal fault converges to a central value, the SVD method allows identify a set of suitable symptoms.

Analyzing in detail the principal fault (see Figure 6), it is possible to break down _{ SD1 } into the eigenvector components to identify the symptoms S_{j}, which have the highest level of influence, i.e. the most sensible symptoms regarding the fault.

The values of the symptoms correlation, independent to the dynamic state, show an equal ranking of their participation in quantitative and qualitative assessment of the dynamic state of the machine, it means the results are consistent and they provide a stable index.

The results show the development of damage causes the machine works on fault condition. According to the results from the object to study (rotating machine), the principal fault has been produced in the main bearing on the side of the equipment. The results also show the influence of an electric motor bearing used to drive the machine.

CONCLUSIONS AND FUTURE WORK

The vibrations signals allow describing the dynamic condition of machines by a set of indexes (eigenvector components of SDj) from different symptoms _{ S={s1,s2, sr}, } which they are related to the fault.

The assessment of mechanical condition based on a multidimensional analysis (vibroacoustics signal) gives the opportunity to identify the relationship between symptoms S, whose values have changed as a result of damage.

The proposed method allows ranking the symptoms to describe the machine technical condition and, to identify the damage causes. Besides, this method allows generating a new set of independent symptoms that may be useful for assessment of the mechanical condition; and the method can be used to reject symptoms that do not provide important information about the mechanical condition.

The Neyman-Pearson analysis applied to the main fault, _{ SD1, } give the probability of not exceeding a limit value, which allows the assessment of importance of damage.

To generate a dynamic map of the mechanical condition, it is necessary to make a series of tests approaching the lifetime of the machine, Θ. This approach will enable monitoring the system to get the mechanical behavior of the machine, thereby increasing its reliability in terms of its implementation in production processes.

The next step of the quantitative and qualitative analysis description of participation symptoms in the assessment of mechanical condition is modelling the cause-effect relationships using multiple regressions analysis.