An Overview of Vibration Analysis Techniques for the Fault Diagnostics of Rolling Bearings in Machinery

Te perfection of methods and means of nondestructive testing and technical diagnostics is determined by the level of development of science and modern industrial technologies. Te desire to develop technologies determines the extent and degree to which the monitoring of the state of substances, materials, products—and now the state of the natural environment—are becoming increasingly relevant. Te methods and means of condition monitoring and the diagnostics of rolling bearings have been in development for more than 60 years. Despite some successes, however, there is currently no information concerning the veracity of means to completely resolve the bearing diagnostics problem. Tis paper provides a fairly brief overview of methods and means for monitoring the condition and diagnosis of rolling bearings and also describes one of the newest trends in this feld—the analysis of the properties of the characteristic function of vibroacoustic (VA) signals in order to determine the condition of the objects of control and, in particular, rolling bearings. It is shown that the magnitude of the module and the area of the characteristic function of the VA signal are very efective criteria for assessing the technical condition of a rolling bearing.


Introduction
It is known [1][2][3][4][5][6][7][8] that the technical state of industrial mechanical and technological equipment is determined by the state of individual elements in its composition. Terefore, in order to characterize the condition of such equipment, it is necessary to identify the presence or absence of defects in its structural units and parts, such as, pump impeller blades, shafts, elements of sliding and rolling bearings, fasteners, gears, and couplings.
In rotary mechanical systems, rolling bearings are one of the main and also one of the most vulnerable components of the mechanism. Rolling bearings carry out the spatial fxation of the rotor, as most of the loads arising in the mechanism, both static and dynamic, are handled by the bearings. Te malfunction in bearings is highest among other components. Te bearings are the reason for more than 41% of machine breakdowns, the stators are more than 37%, rotors are more than 12%, and others are more than 10% of breakdowns [9,10]. Tus, the fault diagnosis of the bearings of the machinery has great importance [11][12][13]. Terefore, condition monitoring and diagnostics of rolling bearings must be carried out promptly and in a timely manner in order to prevent a sudden failure of the mechanism.

Formulation of the Problem
Te aim of this work is to analyze, from the point of view of the efectiveness of use, the applicability of monitoring methods of nondestructive testing and technical diagnostics, which make it possible to assess the condition of rolling bearings as well as to determine the reasons behind their deterioration (for diagnostic purposes). Te priority had been done for the methods successfully used in the feld. Such methods per ISO 13372: 2012 are usually called diagnostic signs of malfunctions [14].
Tis paper does not consider the methods for intelligent analysis of diagnostic signs and signal parameters used to identify defects and malfunctions. Artifcial intelligence methods, neural networks, fuzzy logic, and others [15][16][17][18][19][20] are a completely out of the current paper subject and a separate task of research in the feld of diagnostics, condition monitoring, and the construction of expert decision-making systems.

Applied Methods of Non-Destructive Testing.
Te assessment of the technical condition of mechanical and technological equipment and its structural elements-in particular, rolling bearings-involves the use of a wide range of methods of nondestructive testing (NDT) and technical diagnostics [9,[21][22][23][24][25].
(i) Visual and Measuring Control (VIC). It is based on obtaining information about the controlled object using visible radiation. Tis is the only type of NDT that can be performed without any equipment using the simplest measuring tools [21]; (ii) Magnetic Method. It is based on the analysis of the interaction of a magnetic feld with a controlled object. Tis method is used to control objects made of ferromagnetic materials [21,24]; (iii) Eddy Current Method. It is based on the registration of changes in the interaction of the electromagnetic feld of an external source (excitation winding of an eddy current transducer) with the electromagnetic feld of eddy currents induced by the transducer in the controlled object [21,24]; (iv) Control By Benetrating Substances (capillary control). It is based on the penetration of test substances into poorly opened external parts and through defects in the solid walls of controlled objects [24]; (v) Ultrasonic Method (UST). It refers to the acoustic type of NDT and is based on the excitation of ultrasonic waves in the controlled object and the further reception of ultrasonic vibrations refected from internal discontinuities (defects), including the analysis of their arrival time, amplitude, shape, and other characteristics [22]; (vi) Acoustic Emission Method (AE). It refers to the acoustic type of allowing one to detect the presence of developing defects by recording and analyzing acoustic waves arising from the process of plastic deformation and crack growth in controlled objects [24,26]; (vii) Electrical Method. It is based on the registration of electric felds and electrical parameters of the controlled object [23]. Te electropotential method is of interest, which is based on recording the potential distribution over the surface of the controlled object. Tis method is used to measure the depth of external cracks in metal, as previously identifed [27]; (viii) Optical Method. It is based on the interaction of optical radiation with an object. To register the parameters of optical radiation, special measuring instruments are used [23]; (ix) Termal Method. It is based on the registration of changes in the thermal or temperature felds of controlled objects. Te main condition for the use of the thermal method is the presence of heat fows in the controlled object [23,28]; It is based on the analysis of changes in the parameters of vibroacoustic processes experienced during the operation of the controlled object [24,[29][30][31][32][33][34][35][36][37][38][39][40]. Te vibroacoustic method allows for the monitoring of the technical condition of the equipment without interfering with its design and decommissioning while maintaining high diagnostic accuracy and reliability [41,42].
Most of the listed NDT methods (VIC, magnetic, eddy current, penetrating substances, ultrasonic inspection, AE, electrical, and optical) assess the properties of the material of the controlled object, i.e., seek to determine its structural parameters. For example, these properties include surface and hidden defects, such as porosity, cracks, fractures, undercuts, notches, scufng marks, and erosion and corrosive wear, and are carried out only if the unit is stopped. As a rule, this is only possible during the renovation period.
In the general case, NDT methods characterize the technical condition of objects by diagnostic parameters, i.e., parameters that indirectly determine the current state of the controlled object, which include thermal, acoustic emission, and vibroacoustic NDT methods. In particular cases, to determine axial displacement or radial beats in centrifugal aggregates, it is possible to use the eddy current method. Te obvious advantage of this method is its application, as in the eddy current method, it is described in the last sentence to assess the technical condition of objects without changing the course of the technological process, i.e., the process of continuous operation.

Assessment of the Technical Condition of Rolling Bearings by the Parameters of Vibroacoustic (VA) Signals.
Condition monitoring and the diagnostics of rolling bearings are based on the analysis of parameters of various characteristics of VA signals. To analyze the state of rolling bearings, the standard deviation is most often used (in vibration diagnostics, it is called the root mean square (RMS) value [8,29,[43][44][45] of vibration parameters). Te kurtosis of the probability density of instantaneous values of the VA signal [8,43,45] is often used less. Tere are known examples of using the parameters of the probability density of the distribution [13] and the entropy of the VA signal for monitoring the state of bearings. To diagnose bearings, analysis of the parameters of the envelope of the vibration signal is also used [8,43,46].

Shock and Vibration
Te main methods for monitoring the technical condition of rolling bearings by the parameters of the VA signal are described in a number of scientifc publications [3,4,8,24,40,45,[47][48][49].

Assessment of the General Level of Vibration Parameters.
Since the inception of the technology for monitoring the state of machines, mechanisms, and their assemblies in terms of vibration parameters, problems have arisen in association with the normalization of the vibration level.
Currently  [37][38][39][50][51][52][53]. It should be noted that most standards provide for the measurement of vibration velocity and (or) vibration displacement [33,[36][37][38][50][51][52][53]. In the standards developed in recent years, in addition to vibration velocity and vibration displacement, norms have been introduced for the level of vibration acceleration [29,30,34,37,39,50]. Obviously, the problem, as indicated in the ISO 10816-3: 1998 standard, is beginning to be solved: "Ideally, the criteria should be presented in the form of constant values of vibration displacement, vibration velocity, and vibration acceleration, depending on the speed range and type of machine. However, at present, the boundaries of states are built only for vibration velocity and vibration displacement [51].
It is worth noting that, in Russia, for the frst time, the standard values for assessing the vibration state of equipment based on joint measurements of displacement, speed, and acceleration on the bodies of machine and mechanism units were defned in a document approved by Gosgortekhnadzor in 1994 [54].
Te assessment of the state of machines and mechanisms by the magnitude of vibration acceleration is a fundamental decision in the feld of VA diagnostics. Tis is due, frst of all, to the fact that a fairly large portion of defects and malfunctions excite VA oscillations at frequencies above 1000 Hz, to which the vibration velocity is measured. First of all, in this range, defects and malfunctions of rolling bearings, the electromagnetic system of AC electric motors, and gearing are manifested.
As for centrifugal machines, measuring the level of highfrequency vibration allows one to detect the occurrence and development of defects that excite these high-frequency vibrations [3,4,8]. However, the identifcation of defects and malfunctions of units of centrifugal machines only by the general level of vibration acceleration, as indicated in the regulatory document [29,34,37,39,50], is very difcult to accomplish. Terefore, to solve this problem, the analysis of the parameters of the probabilistic-statistical and spectral characteristics of VA oscillations is used.
It should be noted that in almost all of these standards, both foreign and domestic, except for [29,30], the values of the normative values are determined on the basis of empirical experience and research that are not available for study and verifcation. Te boundary values of the states in [29,30] were determined based on the analysis of statistical data obtained from the databases of monitoring systems [7,[54][55][56][57][58][59] and the use of probabilistic and statistical decision-making methods [60][61][62][63][64][65][66].

Using the Signal Crest Factor.
Te "crest factor" conventionally combines several methods for assessing the condition of rolling bearings. Tese methods have various practical modifcations, but they are based on the same physical meaning. Te essence of the methods is to compare the overall level of the VA signal with the value of the peaks (surges) in the VA signal. Te criterion for making a decision is the following hypothesis: the more the peak value exceeds the value of the general level, the more the defect is developed in the bearing.
Depending on the method of comparing the levels of the peak values and the level of the VA signal, the following are distinguished: (v) PeakVue is a method for analyzing high-frequency components in a VA signal [8].
Te method of assessing the state in terms of the ratio of the peaks of amplitudes and RMS in the VA signal (PEAK/ RMS) refers to the classical method of assessing the state by the "peak factor." Te method is not complicated, and with the correct implementation of the technical means, it is quite sensitive. To use this method, it is sufcient to have a vibrometer that allows the measurement of the RMS and the amplitude of vibration peaks.
Te high-frequency detection (HFD) method of detecting a high-frequency signal is based on measuring the numerical values of high-frequency vibrations created by incipient defects that excite oscillations in the frequency range from 5 kHz to 60 kHz. Typically, the measurement is carried out at the resonant frequency of the transducer to amplify the low-level signal. Since HFD is measured using an accelerometer, the resulting value is displayed in terms of acceleration due to gravity {g}. Tis method provides an early warning of bearing problems [68][69][70].
Te spike energy (SE) method was originally developed to detect signals generated by defective rolling bearings. Te term "Spike Energy" means very short pulses or bursts of a VA signal generated by the action of rolling elements on microscopic cracks and chips. SE is a measure of the intensity of the energy generated by such repetitive mechanical shocks. Impacting energy intensity is a function of pulse amplitude and repetition rate. Te measured signal value is expressed in {gSE} (SE acceleration units). SE measurement reveals early signs of rolling bearing failure [68,70,71].

Shock and Vibration 3
Shock pulse measurement (SPM) is a method based on the registration and analysis of shock impulses of rolling bearings. Shock impulses-elastic waves or low-energy vibrations-are excited as a result of the collision of elements and changes in pressure in the rolling zone of rolling bearings. Tis method was developed in 1969 [72]. Te measurement of shock pulses by the SPM method is carried out by a specially developed piezoelectric transducer, which perceives and amplifes incoming shock pulses at their resonant frequency in the range of 32 . . . 37 kHz. At the output of the measuring device, there is a sequence of pulses, each of which, as a rule, has a deviation from a certain average value. Te technical condition of rolling bearings is estimated by the magnitude of the peaks of the values of individual impulses and expressed in decibels [69,72]. Te SPM method makes it possible to detect the deterioration of lubrication conditions and the appearance of defects in rolling bearings at an early stage [69,70]. Signal spectrum analysis reveals the cause of bearing condition changes.
Te PeakVue method is based on vibration analysis and is used to detect microshocks. Te main idea of the method can be reduced to the synchronous detection of high-frequency harmonics of the VA signal, while the low-frequency components, which are below 3-5 harmonics of the defect frequency, are fltered out. It should be borne in mind that when using this method, it is necessary to select the synchronous detection frequencies for each defect [67,68]. Te efciency of the method is close to that of the SPM method using spectral analysis and the envelope method.
An overview of the methods for assessing the technical condition of rolling bearings according to the "peak factor" made it possible to formulate their main advantages and disadvantages. Te obvious advantages include the following: (i) Monitoring of bearing operating conditions (e.g., lubrication); (ii) High sensitivity; (iii) Early detection of defects; (iv) Te possibility of use by specialists who do not have sufciently high qualifcations.
Disadvantages include the following: (i) Te need to select the parameters of measuring instruments (for example, the order and range of the flter, the reference frequency) and the values of the state criteria individually for each control object in order to obtain sufciently reliable results; (ii) Te complexity of determining the type of defect, the degree of its development, and, therefore, the difculty in predicting the residual resource; (iii) Sensitivity decreases with increasing RMS VA signals.

Estimation of the Spectral Components of the VA Signal.
Currently, to assess the condition of rolling bearings, spectral characteristics of VA signals are used, such as the following: (i) Te VA signal envelope spectrum [8,45,73]; (ii) Te direct spectrum of vibration acceleration, vibration velocity, and vibration displacement [8, 61-63, 65, 66, 74].
Te method for determining the technical condition of bearings by the parameters of the spectrum of the envelope of the VA signal is based on the analysis of high-frequency vibration; while when using a band-pass flter, a narrow frequency range in a band up to 10 . . . 15 kHz is selected from the entire signal. Te received signal, most often vibration acceleration, is detected by an amplitude detector (or using the Hilbert transform), after which the noise fltering algorithm and the extraction of useful components are applied [3]. A description of this method was initially provided in the works of Mori et al. [75,76] in the mid-1960s and later by Veshkurtsev in [19].
Te choice of the frequency band of the flter, with which the high-frequency component of vibration is isolated, is an urgent problem, since the selected frequency range afects the types of identifed defects and the value of the amplitude modulation coefcient, by the value of which the technical condition of the bearing is judged.
One of the approaches to the selection of the flter bandwidth is the "resonant method" or the high-frequency resonance technique (HFRT) [70,77]. Te physical basis of the method is as follows: Whenever a defect comes into contact with a moving element of the bearing, a short pulse is generated, which periodically excites resonant oscillations at a characteristic frequency associated with the location of the defect and the parameters of the acoustic environment. Tus, the resonant frequencies are amplitude-modulated by the frequency of the characteristic defect. By demodulating one of these resonances, it is possible to recover a signal indicative of the type of defect. Tis approach works well in the absence of other defects whose frequencies fall within the envelope selection band.
Also, the center frequency and bandwidth of the flter can be determined using the spectral kurtosis method or spectral kurtosis (SK) [47,48,70,[78][79][80], which has proven to be very efective in detecting impulse components excited by defects. Its main idea is to consider kurtosis or kurtosis in the frequency domain as a measure of impulse component detection, which will allow the selection of the optimal frequency band and improve the signal-to-noise ratio in the spectrum of the VA signal envelope.
Te authors of [11,78] classifed the studied VA process as conditionally unsteady (conditionally non-stationary, CNS) non-Gaussian processes; depending on the duration of the implementation, the process can be taken as stationary, but a separately taken implementation at an arbitrary moment in time is a nonstationary process. In particular, the problem was formulated by the authors as a problem of detecting transients in strong additive noise, with the possibility of separating non-Gaussian components. Frequency-domain kurtosis or, in other words, spectral kurtosis, was used to perform this separation. Ideally, spectral kurtosis takes on zero values at those frequencies where only stationary Gaussian noise is present and high positive values at those frequencies where transients occur. 4 Shock and Vibration It should be noted that kurtosis-based methods can lead to inaccurate results in the presence of relatively strong non-Gaussian noise containing peaks with a high amplitude or a relatively high pulse repetition rate, i.e., individual impulses from faults must be separated in the temporal implementation [80]. Another disadvantage is the use of shorttime Fourier transformation (STFT) flters and fnite impulse response (FIR) flters, both of which have inherent disadvantages. For example, STFT requires a tradeof between time and frequency resolution due to window length limitations, and the FIR flter parameters cannot match every signal that represents a fault [63].
To expand the capabilities of the spectral kurtosis method, various algorithms have been developed based on the use of autoregressive (AR) models [81], complex Morlet wavelets [82], and minimum entropy deconvolution (MED) [83].
Te development of technologies for measuring VA signals makes it possible to expand the frequency ranges for measuring vibration acceleration with an allowable error of up to 20 kHz without using complex methods of mounting vibration accelerometers [45]. In this case, it becomes possible to analyze the absolute vibration in this range, as well as the use of such methods as SPM (using the sensor's own resonance) and envelope extraction [49].
After choosing the optimal flter bandwidth and executing the algorithm for obtaining the spectrum of the VA signal envelope [8,24,48], the level of the components at the defect frequencies is estimated. Te calculation of the characteristic frequencies of bearing defects is given in [8,24].
Te obvious advantages of the method for determining the technical state of a rolling bearing from the spectrum of the VA signal envelope are as follows: high sensitivity, noise immunity, and the ability to identify the type and location of the defect. Te disadvantages include difculty in choosing the flter bandwidth that arises when the envelope-fnding algorithm is performed. An incorrectly selected bandpass flter frequency range can result in missing components associated with bearing failure and, as a result, missing a defect.
Te assessment of the technical condition by the level of components in the spectrum of vibration velocity is now more often carried out in cases where the cost of equipment diagnostics is quite small and commensurate with the losses from its breakdown, since this method allows detecting bearing defects only at the last stages of development. Te main advantages are minimal technical costs and no requirement for special qualifcations on the part of personnel. Te disadvantages of this method include rather a low detection rate of defects since, and in the vibration velocity, spectral causes appear with signifcant damage to the bearing and with other defects in the mechanism.
3.6. Wavelet Transforms. Another way to detect defects is wavelet analysis. Te wavelet transform makes it possible to obtain a time-frequency distribution with a variable resolution, from which the periodic pulses generated when the rolling elements pass over the defect can be separated. Kumenko [84] in 1996 applied discrete wavelet transform to VA signals in order to detect the occurrence of chipping in rolling bearings. Te values of the wavelet coefcients during impulse responses increase as a fracture approaches. Te assumption about the possible occurrence of a fracture in the bearing is based on the analysis of trends in the maximum values of the wavelet coefcients. Te authors of [85] showed in laboratory conditions that the discrete wavelet transform can be used as a tool for detecting single and multiple faults in ball bearings.
Te advantages of the wavelet transform method include the early detection of a defect in comparison with the method of detecting defects by the spectrum of the VA signal envelope [79].
Te main obstacle to the widespread use of wavelet transform in VA diagnostics is the efect of noise (interference): Even small interferences can cause a signifcant distortion of the results. In addition, when trying to automate signal analysis, it becomes necessary to use signifcant computational tools [78]. Tere is practically no publicly available information on the successful use of wavelet transform in automatic or automated processing of real VA signals nor is information available on the reliability of determining defects in real signals.

Hilbert-Huang Transform.
Recently, in many branches of science and technology devoted to solving various problems, the Hilbert−Huang transform (HHT) has been used, which refers to alternative methods of time-frequency analysis of nonstationary processes [86]. Tere is a method for determining the technical condition of rolling bearings based on the use of empirical mode decomposition (EMD), a component of HHT [87,88]. As a result of calculations using the EMD algorithm, empirical modes or internal oscillations (Intrinsic Mode Functions, IMF) have been found. A new method of splitting the IMF into three combined mode functions (CMF) can then be applied and, fnally, the vibration signal can be divided into three parts, namely, the noise component, the useful signal part, and the trend part. Spectral analysis of empirically determined local amplitude is used to further extract related failure symptoms from the resulting signals. According to the results of the study, the authors concluded that the proposed method for diagnostics of rolling bearings makes it possible to identify bearing faults at an early stage.
However, it is known [68] that in the study of real physical processes, the efect of mode mixing begins to manifest itself when, during empirical mode decomposition of the signal, segments of other mode functions appear at some time intervals, which reduces the efciency of the method.

Statistical
Estimates of the VA Signal. Statistical assessments of the VA signal are also used as criteria for assessing the technical condition of rolling bearings as given as follows: mean square value (RMS), peak level, crest factor, skewness, kurtosis, variance, standard deviation, clearance Shock and Vibration factor, impulse factor, shape factor, correlation function, and others [8,55,[61][62][63][64][65][66]. Te assessment of the technical condition of rolling bearings in accordance with the requirements of regulatory documents [29] should be carried out according to the value of the RMS VA signal.
However, the RMS value is of limited use since it is not sensitive to defects at an early stage, contributes little energy to the oscillatory process, refects only the energy of the original signal, and does not display information about the type of defect. Te RMS value also does not permit the determination of the presence of short-term surges in the signal, which can subsequently become critical and culminate in the destruction of the bearing element.
Te kurtosis coefcient is a measure that determines the acuity of the peak in the distribution of the VA signal. In other words, it determines the presence of peak values in the time signal and estimates their magnitude since the outliers in the signal when a shock disturbance appears and distorts the shape of the probability density curve, which afects the magnitude of the kurtosis [3,8,74].
Te sufciently high sensitivity of the kurtosis coefcient to a change in the technical condition of rolling bearings made it possible to develop a method for assessing the technical condition [89]. Te method is based on obtaining and evaluating the kurtosis coefcient in four vibration frequency ranges as follows: from 3 kHz to 5 kHz, from 5 kHz to 10 kHz, from 10 kHz to 15 kHz, and from 15 kHz to 20 kHz. Via the deviation of the coefcient of kurtosis, the degree of development of the defect can be judged. Te threshold values of the kurtosis coefcient are empirically determined: If the kurtosis coefcient is less than 3, the bearing is in good condition; if the kurtosis coefcient is more than 3, operation with restrictions is permissible; if the kurtosis coefcient is more than 5, the operation of the bearing assembly is unacceptable. Unfortunately, it was not possible to fnd a theoretical or empirical substantiation of the given boundary values in the available sources.
Despite the fairly widespread occurrence of kurtosis as a criterion for assessing technical condition, mainly of rolling bearings, there are no publicly available data on the reliability of assessing the condition when using kurtosis.

Probabilistic-Statistical Estimates of VA Signals Using the Characteristic Function.
In the theory of probability, in order to obtain analytical information about a random process, in addition to the generally accepted functions (distribution function, probability density, and correlation function), it is possible to use other characteristics that fully describe and refect all of the properties of the process under study. Such an alternative way of representing random variables is the characteristic function (CF) [90][91][92][93][94].
Te characteristic function (CF) was frst proposed in 1902 by the outstanding Russian mathematician A.M. Lyapunov to prove the central limit theorem of probability theory. Later, CF was used in applied science. For example, the use of CF in the feld of detection, demodulation, and fltering of signals in various devices made it possible to improve the metrological characteristics of known devices by an order of magnitude or more [92,93,95]. It is also known that CF can be used as a tool for evaluating various models and quantities in econometrics [96].
Characteristic functions are a very convenient way for solving a fairly wide range of problems. Te use of the CF method opens up opportunities for obtaining new results, including in the feld of VA diagnostics [41,57,97].

Experimental Results
In order to test the hypothesis about the possibility of using the CF parameters of VA signals to assess the state of rolling bearings, studies were carried out using a special rotor kit and certifed measuring equipment. Te rotor kit is used for quality control of rolling bearings (see Figures 1 and 2).
Te rotor kit consists of mechanical units that allow the simple installation of the bearing on the spindle. Te rotation of the spindle and the inner ring of the bearing is provided by electric motor. Outer ring of the bearing is fxed. Load of the bearing in the axial and radial directions allows receiving a vibration signal in the radial direction at the place of greatest loading (see Figure 3). Te special software allows somebody to promptly analyze the signal, spectrum, vibration acceleration envelope, and some other parameters of the vibration signal, as well as record the vibration signals in the database.
Te research was conducted on rolling bearings with previously confrmed defects (Table 1), and the condition of the bearings was checked for compliance with the requirements of GOST 32106 [29] (Table 1). Bearing 7316 (according to ISO-30316) had an operational defect in the form of a chipped roller (BSF), and two bearings, 46416 (1) and 46416 (2) (according to ISO-7416) had artifcially created defects of the inner (BPFI) and outer (BPFO) rings. Instantaneous VA values for each bearing were obtained using a test bench [42], after which the RMS was determined and the condition was assessed in accordance with established norms [29]: bearing 317 (ISO 6317)-GOOD, 46416 (1)-PERMISSIBLE, 46416 (2)-REQUIRED ACTION, 7316-NOT PERMISSIBLE (Table 1) [42].
In Table 1, the following conventions are adopted: d BPFO -diameter of the outer ring; d BPFI -inner ring diameter; d BSF is the diameter of the rolling elements; z-number of rolling bodies; α-contact angle.
Under the same loading conditions and the rotation frequency of the inner ring of the bearings, VA signals were measured, empirical characteristic functions (ECF) were calculated, and their graphs were plotted (Figure 4).
In order to test the hypothesis about the dependence of the CF parameters on the state of the bearings according to the formula for the CF of the normal law [90], a theoretical CF (TCF) for VA signals was calculated and constructed ( Figure 4). In this case, the values of the mean square deviation (RMS) in the calculations corresponded to the experimental values (a r.m.s. ) of the RMS (Table 1). To assess the deviation of ECF from TCP, the multiple coefcient of determination R 2 or Linder's measure was used [98].
For the CF for VA signals of the bearings under study, the coefcient of determination was the following: R 2 317 � 0,9996, 6 Shock and Vibration R 2 46416(1) � 0,9994, R 2 46416(2) � 0,9994, and � 0,9928, which indicates the nearly maximum possible degree of coincidence of the ECF and TCF curves for various states of the bearing, which corresponds to the normal distribution law.
Te following parameters of the CF, which are the criteria for the state of the bearings, were used [7,56]: (1) Te magnitude of the CF of the VA signal for a given argument. Using the CF modulus value for a given argument is the simplest in terms of calculations, since it is enough to calculate only one CF value for a given argument value. In this case, the CF value lies on the interval from zero to unity.
Te value of the argument at a given value of the CF module can be used for narrowband signals, when the CF curve is a "wide" bell. In this case, the CF argument will have a signifcant range of changes -from zero to several units of the reciprocal of the process parameter. However, the calculation of the argument requires the calculation of several CF values.
Te use of these two abovementioned parameters is advisable if the signal is a multifrequency process that

Shock and Vibration
(1) Te properties of the CF of the VA signal-in particular, the modulus, the CF argument, and the area under the CF curve-change their value in accordance with changes in the state of the object and correspond to the current state at least as much as this state of the RMS VA signal, particularly the RMS vibration acceleration.
(2) Te multiple determination coefcient R 2 (or Linder's measure) of the CF, constructed from experimental data and calculated using the formulas of the classical CF of the normal distribution law [90], demonstrates an almost perfect coincidence-the diference is tenths of a percent, which confrms the adequacy of the theoretical provisions of the proposed hypothesis of the dependence parameters of CF VA signals from changes in the state of rolling bearings.
(3) In several works [47,57,97,[100][101][102][103][104][105], probabilistic and statistical methods were used to assess the reliability of the condition assessment, which includes the probability of missing a dangerous bearing condition and the probability of a false alarm.

Conclusions
A review of the methods and techniques used to assess the technical condition of rolling bearings made it possible to draw the following conclusions: (1) Te most informative and efective method for assessing the technical condition of bearings is the VA diagnostics method, which includes various methods for analyzing the VA processes accompanying the operation of a rolling bearing; (2) All of the abovementioned methods can be divided into methods for assessing the technical condition (peak factor, statistical estimates of the VA signal) and methods for diagnosing a specifc type of rolling bearing defect (Fourier transform, wavelet analysis, Hilbert−Huang transform); (3) Based on the identifed advantages and disadvantages of methods for assessing and diagnosing the technical condition of rolling bearings, it can be argued that, at present, there is not a single tool for analyzing VA processes that allows one to reliably determine the presence of incipient rolling bearing defects, as well as their condition, since for any method or technique, except for CF, the reliability of the assessment of the state has not been determined, which is confrmed by the lack of information in scientifc publications.
(4) None of the used classical NDT methods allows one to estimate the change in the VA signal when the state of the diagnosed object changes, taking into account the probabilistic and statistical properties of the VA signal. To assess the states, particular statistical parameters of the VA signal are used.
(5) Te characteristic function of a random process, the VA signal, is one of the probabilistic and statistical characteristics of a random process that makes it possible to estimate the statistical parameters of a random process using, in some cases, simpler signal processing [41,57,97]. Te use of the properties of CF of VA signals for assessing the state of objects under control makes it possible to detect changes in the characteristics of VA signals associated with a change in the state of objects under control, which allows us to speak of its higher efciency and reliability for assessing the state of objects, including rolling bearings. (6) Currently, research is being conducted on the parameters of the CF of VA signals of rolling bearings on rotary kit (see Figures 5 and 6) using National Instruments measuring equipment and a portable device for measuring vibration UNISCOPE. Te stand allows to simulate several bearing defects as damage to the outer and inner rings and damage to the rolling elements and the separator, as well as imbalance, misalignment, and some others. Te UNISCOPE device implements the function of measuring the CF of the VA signal in real time, which allows promptly assess the change in the structure of the VA signal and detect various defects and malfunctions. Te VA signals are stored in a database for further detail analysis.

Data Availability
In study and references. Te copies of the data mentioned in the study can be obtained free of charge.

Conflicts of Interest
Te authors declare that they have no conficts of interest.