Non-linear Signal Models in Vibroacoustic Machine Diagnostics
A. V. Barkov, N. A. Barkova


Recently, more attention is being paid to monitoring the environmental conditions, including vibroacoustic conditions. The main task of monitoring is to obtain maximum possible information and minimize the loss of information during the analysis of signals under monitoring. Such analysis is possible only when an adequate model of the monitored processes is known. The process of noise and vibration formation which the authors have investigated for many years is not an exception to this rule.

Linear models are most frequently used for the analysis of oscillations in vibroacoustic monitoring. In this case the signal under investigation is considered as a sum of periodic and random components. Each component, in turn, may be a sum of direct and reflected waves that arrived in a measurement point from different sources. As the probability of non-linear signal distortions during sound and vibration propagation is rather low, linear models work quite satisfactorily in most practical problems of signal analysis.

In considering problems of vibroacoustic diagnostics, i.e. interpretation of the object vibroacoustic state changes detected during monitoring, it is necessary to have information about the mechanisms of distortion formation that cause such changes. For example, the process of oscillation forces formation usually is a complex function of many variables and is naturally non-linear. And the non-linear representation of the vibration signals allows the possibility of obtaining additional information about the reasons for vibroacoustic state changes of the object under monitoring. This paper presents a brief analysis of the non-linear models for vibroacoustic signals that helped authors in the practical diagnostics of shipboard machinery during the 1980 and other machines in numerous branches of industry of Russia in 1990s.


The first time a non-linear model of vibroacoustic signals was used by one of the authors was in the mid 1970s for the identification of hydrodynamic noise sources in one of the marine research laboratories. The problem was to distinguish between natural and artificial sources of signals acquired by a passive hydro-locator. Natural sources can be, for example, the noise of the surf, and an artificial source,- a vessel moving by the shore. To solve this problem, a model of random amplitude modulated signal was used:

(1) where:
= measured signal,
= random stationary signal, and
= modulating low frequency signal.

The sea noise usually turned out to be a random stationary process that can be approximated well by a model of, roughly speaking, white noise in the monitored frequency band. The surf noise has close to random low frequency modulation defined by the waves breaking on the shore. The high frequency noise of a vessel is defined by the operation of the propeller, in turn, which has a sharp periodic modulation by the rotation and blade frequencies of the propeller.

Later this model was used by the authors for the diagnostics of friction machine units, in particular, rolling element bearings [1-2] and flow formation units, e.g. working wheels of the pumps and turbines.

This model accurately describes the low frequency fluctuations of the high frequency oscillations power. The function, , is used to express the high frequency oscillations and this should not necessarily be a random process. The main condition is that modulating signal should be the same for all possible components of the signal . In practice, the measured noise and vibration signals contain components, including very powerful ones, that either are not modulated at all or have different rules for power fluctuations. In this case, to get adequate results, it is extremely important to extract from the signal those components, the modulation of which is to be investigated.

Application of this model in practical diagnostics was complicated for many years due to the absence of simple technical means for demodulating a group of signal components extracted from the measured signal by a bandpass filter. A special apparatus assembled from "Bruel & Kjaer" instrumentation was used by authors for this very purpose. Noise and vibration signals were first recorded on tape and then processed by stationary instrumentation. Only during last few years have some European companies, especially Bruel & Kjaer and Diagnostic Instruments started the production of portable instruments capable of power fluctuations analysis of oscillation processes, in particular spectrum analysis of bandpass filtered signal envelope. Advances in models of amplitude modulated signals were closely connected with complications of the modulation function structure, . Thus, in the analysis of rolling element bearings vibration in frames of the model described by equation (1), the authors started to consider its high frequency components excited by the shock pulses. This enabled by an explanation for a number of processes responsible for vibration formation, we derived the quantitative relations between the defect severity and parameters of envelope spectra components including vibration excited by periodic shock pulses. The next step was to further increase the complexity of the modulation function, reflecting, for example, the properties of vibration excited by shock pulses additionally amplitude modulated in turn.

The spectrum analysis of the high frequency vibration (noise) envelope was typically used to assess the modulation function, , parameters. Figure 1 gives examples of high frequency vibration envelope spectra measured on a number of rotating machine units. The first envelope spectrum of random vibration is typical for random processes with no low frequency fluctuations of its power. It can be an envelope spectrum of good rolling or fluid film bearing, or on the case of a pump or a turbine.

The second envelope spectrum illustrates harmonic modulation of high frequency vibration power of a particular machine unit. It can be vibration of bearings, pump or turbine and also a pipeline with liquid or gas flow with an alternative velocity component [4]. The index of random vibration power modulation defines the defect severity. It can be derived from the relations of harmonic and random components of envelope spectrum and also depends on the parameters of measurement instrumentation [2-3].

The third envelope spectrum of random vibration was measured on a bearing shield with a wear of rolling element bearing outer (stationary) race, its loaded sector. The modulation of vibration power is also very smooth. It is defined by periodic increases of the rolling friction coefficient while each roller passes the worn sector of the outer race.

The fourth envelope spectrum was measured on a rolling element bearing with a cracked outer race producing periodic, short duration, shock pulses. In the envelope spectrum, in this case, you will find a great many components with nearly equal amplitudes with the frequencies equal to multiples of shock pulse repetition frequency. Similar spectrums can be observed on a fluid film bearing with so called "hydrodynamic shock pulses".

The fifth envelope spectrum of rolling element bearing random vibrations illustrates a more complicated signal. Here the modulation function is also amplitude modulated. This situation occurs, for example, in rolling element bearings with a cracked inner (rotating) race. In this case, the amplitude of shock pulse excited by the interaction of each roller and crack on rotating race depends on the load on this particular roller. The load, in turn depends on the rotation angle of the shaft. Similar complicated cases can be found with the defects of other rotating machine units, for example, gear transmissions [5].

The sixth envelope spectrum of random vibration has major differences from the previous ones. This illustrates the case when the modulation process, , is random itself, though low frequency one. In this case, spectrum analysis of the vibration envelope does not provide complete information about the parameters of the modulation function and sometimes special types of envelope analysis are used. This spectrum presents the information about the condition of such machine units as fluid film bearings, pump (turbine) impellers and some others.

During investigation of bladed machines’ noise and vibration, the authors started to apply a number of different methods for the analysis of oscillation process power fluctuations. Besides random vibration, envelope spectrum analysis, the shape of envelope was analyzed. This showed additional information that helps to interpret the processes of oscillation force formation. Other methods are correlation and coherent analysis of envelopes of vibration signal components selected by different bandpass filters. These and other complex methods of envelope analysis were used mainly, not for mass condition diagnostics of machines and equipment, but for the investigations of samples of new machinery types aimed at operation optimization and further noise and vibration reduction. These types of investigations led to new technical solutions, including the invention of new designs of machine units that significantly decrease vibration levels and prolong machine service life.

There is one more method of high frequency random vibration envelope analysis. The authors widely applied it, not for condition diagnostics, but for the balancing of bladed machines. This is a method of amplitude-phase analysis of harmonic components, e.g. with the rotation frequency of an impeller, in the vibration envelope. The first time the problem of the consideration of liquid flow properties by the authors was during pumps impeller balancing in the 1980s [7]. Pump vibration at the rotation speed actually depends on mechanic unbalances of the pump, hydrodynamic unbalances of impeller, and pulsations of the hydrodynamic resistance of the flow. This last factor can occur due to the impeller wobbling. This pulsation can be detected by the envelope spectrum of the pipeline high frequency vibration [4] and can significantly change as balancing weights are mounted. For this very reason in the balancing of pumps, the authors started to consider, not only amplitude-phase characteristics of pump vibration at rotation frequency, but also similar characteristics of the high frequency vibration envelope of pump and pipelines as well.

Figure 1a, 1b, 1c, 1d, 1e, 1f(below). Examples of rotating machine units high frequency vibration envelope spectra.

Fig. 1a.(above). Good machine unit (rolling element bearing, fluid film bearing, pump impeller, etc.)

Fig. 1b(above). Machine unit with defects on initial stage of their development. Such defects do not lead to the shock pulses appearance (good bearing when shaft wobbling occurs, etc.)

Fig. 1c(Above). Rolling element bearing with significant wear of outer race friction surface.

Fig. 1d(Above). Rolling element bearing with well developed cavity (spall) of outer race.

Fig. 1e(Above). Rolling element bearing with deep cavity on inner race friction surface.

Fig. 1f(Above). Defective machine units (fluid film bearing, pump impeller) when the high frequency vibration is modulated by a random low frequency process.

II. Frequency Modulated Signals

In the early 1980s while developing methods for electric machines vibration diagnostics, the authors faced the necessity of using an additional non-linear model of a signal, a periodic signal with fluctuations of the period. In this case, the processes responsible for this type of fluctuation may contain a great many of both periodic and random components. A model of such a signal can be represented by the following equation:

(2) where

= a single signal component,
= average period of the signal, and
= modulating process.

Pulsating electromagnetic torque is a typical reason for the appearance of frequency modulation in the vibration of electric machines. As a rule, in defect free electric machines, there are no such torques and they only appear with defects of the electromagnetic system of the machine. Given the appearance of pulsating torques at twice the slip frequency and frequency modulation of the main vibration components, it is very easy to detect the defects of the rotor of an induction motor [6].

A number of investigations were launched to investigate the means of frequency modulation detection of low frequency signals. The authors paid special attention to the simultaneous appearance of frequency and amplitude modulation. These two modulation types have rather different features and can be detected by the analysis of, for example, narrow band spectrum [2].

Further, the detection of modulating function in the frequency modulated signal was done directly by the wave fronts in the periodic signal A(t). Primary attention was paid to the analysis of shock pulse frequency modulation which can be easily be extracted from the background of machine stationary vibration.

So, simultaneous with the envelope analysis of high frequency random vibration caused by shock pulses in rolling element bearings, the analysis of the processes modulating the intervals between shock pulses was carried out. These processes can contain independent and reliable information about the degree of cage wear, but only in case when one of the bearing races has a cavity or spall as a source of shock pulses.

The model of frequency modulated vibration was used for the diagnostics of cylinder-piston groups in reciprocating machines. Taking into account that the shock pulses produced in the moments of fuel combustion and piston movement direction changes can be easily detected in the background of vibrations of other origin, the modulating function extraction can be easily done, for example, by the analysis of leading edge of shock pulses. Further spectrum analysis of this function and random vibration envelope, if necessary, allows the diagnostics of both combustion systems and the degree of piston-cylinder group wear.

Figure 2 contains an example of high frequency vibration waveform of a rolling element bearing excited by shock pulses. Consider the intervals between shock pulse fluctuations in case of good and severely worn cage (curves 1 and 2). A time waveform of low frequency vibration of internal combustion engine excited by shock pulses and the intervals between fronts of these pulses for the engine with combustion system defects and worn cylinders surface is also presented.

Fig.2a, 2b, 2c, 2d (below). Shock pulses in machines’ units: time wave forms of vibration and the distribution of shock and their periods.

Fig. 2a(above). Time wave form of a rolling element bearing with a spall on the outer race.

Fig. 2b(above). Distribution of time intervals between shock pulses in a rolling element bearing with a spall on outer race during the initial stage of defect development.

Fig. 2c(above). Distribution of time intervals between shock pulses in a rolling element bearing with a spall on outer race in presence of severe wear of the bearing cage.

Fig. 2d(Above). Time wave form of an automobile engine. The shock pulses appear due to the change of piston motion direction and fuel combustion.

Fig. 2e(above). Distribution of time intervals between shock pulses in automobile engine in the case of combustion system defects.

III. Other non-linear models of signals.

Until recently, only the simplest of non-linear models had found their application in the practical vibration and noise diagnostics of the machines. These simplest types are a periodic signal amplitude modulated by another periodic process, random signals modulated by low frequency periodic processes, and a periodic signal frequency modulated by a low frequency process with both periodic and random components. One more simple non-linear model is a signal limited by amplitude or duration.

The models of signals limited by amplitude are used by authors mainly in two practical problems, detection and identification of magnetic saturation of core in electric machines and apparatuses, and local liquid flow cavitation in the flow formation and distribution mechanisms. Such a limitation can be detected either by the time waveform analysis or spectrum analysis of the pressure pulsation or the resultant vibration.

The model of signal limited by duration is, first of all, the model of short shock pulses frequently found in machine vibration diagnostics. The volume of information that can be extracted from shock pulses analysis increases with the increase in knowledge about the nature of these pulses and methods of their analysis.

Early on, the method of short shock pulses was closely connected with the knowledge about the attenuated self oscillations of those units and elements that produce shock pulses in their interaction. Mainly, these are self oscillations of rolling elements and the stationary race of rolling bearings [8]. But the diagnostic information about the defective machine condition is contained first of all in the exciting forces parameters, i.e. in the parameters of the shock pulse, but not in the reactions of the oscillations produced by the shock pulse, especially the resonant system. For this very reason, the diagnostic information obtained by the analysis of the most strong and easily measured vibration on the resonance frequencies of the machine units never was a complete one.

Later, specialists began measuring vibration excited by shock pulses on frequencies different from the resonant frequencies of the machine units [9]. Still, the problem of shock pulse analysis was complicated because of the use of resonant transducers having low damping factors which caused distortion of the information about the modulating processes contained in the broader frequency bands away from the transducer resonances.

The investigations of shock pulses and diagnostic information contained in them carried out by the authors showed that special methods should be used for the reconstruction of the shock pulse parameters from vibration signals measured at points of considerable distance from the source point of shock pulses. It was found that the main diagnostic information is contained, not in the shock pulse itself, but in the processes reflecting the fluctuations of amplitude, shape, duration and intervals between pulses in the measured series.

Many years experience in the investigation of the vibration of artificial objects caused the authors to come to the conviction that the diagnostic information contained, in the vibration signal of rotating machinery is very small compared to the information that can be obtained by the analysis of oscillations of natural objects, especially in the core of the earth and in human being.

In 1993, the authors joined their efforts with the group of medical experts in the development of methods to extract information from the oscillation processes connected to the human heart operation. These investigations together with physiologists resulted in the development of the first systems for psycho-physiological human monitoring. Such a system can be used by any person himself with the purpose of objective monitor one’s fatigue, the influence of physical or psychological loads, drugs, etc. on the organism. The means of signals measurement and software for data processing and analysis can be as simple as the ones for machinery condition diagnostics.

One more non-linear model is used by the authors for the diagnostics of some machinery units. This model, which is analogous to pseudo-noise in acoustics, can be considered as pseudo-vibration, signals that are the results of transducer effects including measurement system physical aperture, transducer error signals from non-vibration sources, and other very localized phenomena such as pulsating pressure induced accelerations caused by turbulence. It can also be the result of rolling or slide friction surfaces that contact each other at many points of friction. The estimation of pseudo-vibration inputs in the overall random vibration level in the measurement points enables a measurement system user to find the points with maximum levels of pseudo-vibration and define their level, which can allow other parameters to be estimated from the level of the pseudo-vibration signals. This method can be used, for example, to find the place and degree of inner surface wear of a pump or a pipeline by the vibration of its outer surface [10].


  1. The analysis of noise and vibration signals with non-linear models, from which are based many of modern machinery condition diagnostic and prediction methods, enables the following conclusions:
  2. The improvement of vibroacoustic methods for machinery condition diagnostics depends mainly on the complexity of signal models used and the corresponding methods of their analysis, and only in very rare cases does improvement come by increasing the volume of diagnostic measurements.
  3. The application of non-linear models and optimization of signal analysis methods significantly increases the volume of information about the oscillation forces that excite noise and vibration in defective machines and units. These are the properties of oscillation forces that most completely reflect the peculiarities of defects and significantly change during the defect development.
  4. Among all non-linear models of noise and vibration signals used in the condition diagnostics, the most reliable information is provided from the models of simplest signals modulated in amplitude, frequency or shape with low frequency processes with several components of different nature.

The volume of diagnostic information extracted from the noise and vibration signals, taking into account the degree of their non-linearity, is usually is enough to provide condition diagnosis and forecast for many types of machinery with no need for other types of measurements, such as temperature, etc.


  1. Soviet patent N868416 "Method for rolling element bearings condition assessment" A. V. Barkov and others. Priority date 1979.
  2. Alexandrov, A.V. Barkov, N. A. Barkova, V. A. Shafransky, Vibration and Vibrodiagnostics of Electrical Equipment in Ships, -Sudostroenie (Shipbuilding), Leningrad, 1986.
  3. Barkov A. V., Barkova N. A., Mitchell J. S., "Condition Assessment and Life Prediction of Rolling Element Bearings", Sound & Vibration, 1995, June pp.10-17, September, pp.27-31.
  4. N. V. Barkova, M. A. Barkova "A Method of Detection the Low Frequency Fluctuations of Liquid Flow", Proceedings of the 4th International Congress on Sound and Vibration, St. Petersburg, Russia, June 24-27, 1996, Volume 3, pp. 1591-1594
  5. A. V. Barkov, N. A. Barkova, "Diagnostics of Gearings and Geared Couplings Using Envelope Spectrum Methods", Proceedings of the 20th Annual Meeting of the Vibration Institute, Saint Louis, Missouri, USA, 1996, pp. 75-83
  6. Soviet patent N1279850 "Method for detection of short circuited rotor windings defects in induction motor" A.V. Barkov and others. Priority date 1981.
  7. Soviet patent N1453199 "A device for rotor balancing" A. A. Alexandrov, A.V. Barkov, N. A. Barkova. Priority date 1987.
  8. US patent N3554012 "Method and arrangement for Determining the mechanical state of Machines" E.V. Shoel, Tumba, Sweden 1971
  9. US patent N3842663 "Demodulated resonance analysis system" Darrell R. Harting, John W. Taylor, 1974
  10. Soviet patent N1516831 "Method vibration diagnostics of mechanisms" A. V. Barkov, N. A. Barkova and others. Priority date 1987.

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