Novel Investigation of Higher Order Spectral Technologies for Fault Diagnosis of Motor-Based Rotating Machinery

In the last decade, research centered around the fault diagnosis of rotating machinery using non-contact techniques has been significantly on the rise. For the first time worldwide, innovative techniques for the diagnosis of rotating machinery, based on electrical motors, including generic, nonlinear, higher-order cross-correlations of spectral moduli of the third and fourth order (CCSM3 and CCSM4, respectively), have been comprehensively validated by modeling and experiments. The existing higher-order cross-correlations of complex spectra are not sufficiently effective for the fault diagnosis of rotating machinery. The novel technology CCSM3 was comprehensively experimentally validated for induction motor bearing diagnosis via motor current signals. Experimental results, provided by the validated technology, confirmed high overall probabilities of correct diagnosis for bearings at early stages of damage development. The novel diagnosis technologies were compared with existing diagnosis technologies, based on triple and fourth cross-correlations of the complex spectra. The comprehensive validation and comparison of the novel cross-correlation technologies confirmed an important non-traditional novel outcome: the technologies based on cross-correlations of spectral moduli were more effective for damage diagnosis than the technologies based on cross-correlations of the complex spectra. Experimental and simulation validations confirmed a high probability of correct diagnosis via the CCSM at the early stage of fault development. The average total probability of incorrect diagnosis for the CCSM3 for all experimental results of 8 tested bearings, estimated via 6528 diagnostic features, was 1.475%. The effectiveness gains in the total probability of incorrect diagnosis for the CCSM3 in comparison with the CCCS3 were 26.8 for the experimental validation and 18.9 for the simulation validation. The effectiveness gains in the Fisher criterion for the CCSM3 in comparison with the CCCS3 were 50.7 for the simulation validation and 104.7 for the experimental validation.


Introduction
Pumps, gear motors, and cranes are examples of IM applications. Most IMs are required to operate for a long time between maintenance actions. That is why effective condition monitoring is needed in industrial applications of motors [1][2][3][4][5][6][7][8][9], gear motors [10][11][12][13][14][15][16][17], and other structures. Proper fault diagnosis and fault isolation of key components is crucial for continuity of IM operations. Bearings are among the most common parts susceptible to damage in an IM, accounting for 42% of all failures [1]. That is why their efficient diagnosis is highly desired. This need is the reason for constant development of new and more effective diagnostic and condition monitoring techniques. Recent work on this topic more and more frequently highlights higher order spectral signal processing techniques as being very efficient, e.g., ref. [1][2][3][4][5]. Diagnostic methods based on IM stator current are becoming more popular • novel comparison of the CCSM3 and CCCS3 technologies via comprehensive modeling trials • novel comparison of the CCSM4 and CCCS4 technologies via comprehensive modeling trials • novel comparison of the CCSM3 technologies and CCCS3 technologies via comprehensive experimental trials • The objectives of the research are as follows: • perform a novel comprehensive validation of the CCSM3 and CCSM4 via modeling trials for pristine and faulty mechanical components • perform a novel comparison of the CCSM3 with the CCCS3 and the CCSM4 with the CCCS4 via modeling trials for pristine and faulty mechanical components • perform novel comprehensive validation of the CCSM3 via experimental trials for pristine and faulty motor bearings, using motor current signal processing • perform a novel comparison of the CCSM3 with the CCCS3 via experimental trials for pristine and faulty motor bearings, using motor current signal processing

Higher-Order Spectral Cross-Correlations
The first higher-order spectral signal processing technologies examined in this paper are the cross-correlations of spectral moduli (CCSM) proposed in ref. [1]. The CCSM of order n is described by Equation (1) [1]. CCSM where I is the total segment number and mZ j f n is the spectral modulus, related to any complex time-frequency transform or any complex frequency transform of time segment j for frequency f n .
The CCSM of orders three (CCSM3) and four (CCSM4) are expressed by Equations (2) and (3), respectively [1]: In order to estimate diagnostic features, the CCSM3/CCSM4 and CCCS3/CCCS4, the discrete time domain signal has to be divided into time segments of a predefined duration. The following step is to perform the selected frequency transform or time-frequency transform of each time segment. In this paper, the Fourier transform is used. The final step is to estimate the CCSM3/CCSM4 and CCCS3/CCCS4 via Equations (2), (3), (5) and (6), respectively.
To evaluate the diagnostic effectiveness of the technologies, histograms of the considered diagnostic features under various settings are evaluated. Then, each histogram is approximated by the normal (i.e., Gaussian) probability density function. The normal probability density function of random variable X with mean (µ) and standard deviation (σ) is stated as follows: To compare the CCSM with the CCCS, two estimates related to a diagnosis accuracy are used. The first estimate is the total probability of correct diagnosis (TPCD) described by Equation (8) and the second estimate is the Fisher criterion (FC) described by Equation (9).
A threshold-based decision-making rule (an aqua line is used to indicate the diagnostic threshold) is employed via the Bayesian criterion [62] for the decision making for each one-dimensional diagnostic feature.
where rN and prN are the total numbers of correct diagnosis for the defective bearing and new bearing, respectively; rNt and prNt are the total numbers of diagnostic features for the defective bearing and new bearing, respectively.
where bm 1 is the mean value of diagnostic features for the new bearing, bm 2 is the mean value of diagnostic features for the defective bearing, σ 2 1 is the variance of diagnostic features for the new bearing, and σ 2 2 is the variance of diagnostic features for the defective bearing.

Technology Validation via Modeling
To compare the proposed CCSM and CCCS techniques, healthy and damaged mechanical components were modeled. A generic bilinear oscillator was used to model these components. A generic dynamic bi-linear oscillator used for modeling is widely employed for modeling various electromechanical components of industrial electromechanical systems and complex structures, e.g., sub-harmonic resonances of offshore structures [63], loading towers [64], components with clearances [65], and the following components with local damage: rolling element bearings with local damage in outer races, inner races, cages, and rolling elements [66,67], gearboxes with local damage in gear teeth [68], and rotating blades and shafts of turbomachinery with fatigue damage [69][70][71][72][73][74][75]. The model describes "in-process damage" as local damage such as fatigue cracks, pitting, local scratches, etc. Thus, this model is suitable for the investigation proposed in the paper.
The equation for the bilinear oscillator is shown below [73]: d 2 y dt 2 + 2h dy dt + ω 2 s x(t) = Acos(ωt + ϕ), x ≥ 0 (10) d 2 y dt 2 + 2h dy dt + ω 2 c x(t) = Acos(ωt + ϕ), x < 0 (11) where X(t) is the displacement, x(t) is the normalized displacement, m is the mass, h is damping, c is the damping coefficient, k s is the stiffness of positive displacement, k c is the stiffness of negative displacement, A 1 is the constant amplitude of an input sinusoidal signal (i.e., the excitation), A is the normalized amplitude of the excitation, ω is the constant angular frequency of the excitation, and ϕ is the random initial phase of the excitation. ω s and ω c are the resonance frequencies at positive and negative displacements. The random initial phase is uniformly distributed in the range [0; 2π]. Equations (10)-(12) are used to describe a mechanical component with and without damage. The stiffness of positive displacement k s and the stiffness of negative displacement k c are equal for an undamaged component, whereas for a component with damage, the stiffness of positive displacement k s and the stiffness of negative displacement k c are different. A relative damage severity is characterized by the stiffness ratio k * = k c −k s k c . The resonance frequency of a component with damage is calculated via Equation (13) below [73]: A total of 300 time domain realizations were simulated via Equations (10)-(13) to investigate the cross-correlation techniques, including: • for healthy components • for damaged components with relative damage severity of 5% • for damaged components with relative damage severity of 10% These signals were simulated using resonance frequencies ω c of 14 and 30 Hz one set being undamaged and another set having damage levels of 5% and 10%. The signals with a resonance frequency ω c of 14 Hz were sampled by a sampling frequency of 340 Hz, while those with a resonance frequency ω c of 30 Hz were sampled with a sampling frequency of 720 Hz. All signals for healthy and damaged components were simulated with a damping of 2.45 and the time duration of each simulated signal was 80 s.
The following parameters were used to evaluate the cross-correlation techniques: the internal window was 2 s, the frequency resolution was 0.5 Hz, the time window was the Hanning window, and the overlapping of internal windows was 0%. Gaussian white noise was added to each simulated signal to simulate an early stage of damage.
The modeling flow chart for the CCSM3/CCCS3 and CCSM4/CCCS4 is given below in Figure 1.
an undamaged component, whereas for a component with damage, the stiffness of positive displacement k and the stiffness of negative displacement k are different. A relative damage severity is characterized by the stiffness ratio k * = . The resonance frequency of a component with damage is calculated via Equation (13) below [73]: A total of 300 time domain realizations were simulated via Equations (10)-(13) to investigate the cross-correlation techniques, including: • for healthy components • for damaged components with relative damage severity of 5% • for damaged components with relative damage severity of 10% These signals were simulated using resonance frequencies ω of 14 and 30 Hz one set being undamaged and another set having damage levels of 5% and 10%. The signals with a resonance frequency ω of 14 Hz were sampled by a sampling frequency of 340 Hz, while those with a resonance frequency ω of 30 Hz were sampled with a sampling frequency of 720 Hz. All signals for healthy and damaged components were simulated with a damping of 2.45 and the time duration of each simulated signal was 80 s.
The following parameters were used to evaluate the cross-correlation techniques: the internal window was 2 s, the frequency resolution was 0.5 Hz, the time window was the Hanning window, and the overlapping of internal windows was 0%. Gaussian white noise was added to each simulated signal to simulate an early stage of damage.
The modeling flow chart for the CCSM3/CCCS3 and CCSM4/CCCS4 is given below in Figure 1.

Validation of Third Order Cross-Correlations via Modeling
The CCSM3 and CCCS3 are functions of three independent frequencies. To estimate the diagnostic effectiveness of the CCSM3 and CCCS3, four sets of harmonics were investigated using the simulated signals from components with resonance frequencies ω of 14 and 30 Hz. The investigated sets of harmonics of component resonance oscillations were as follows: (1,1,3), (1,3,5), (1,2,4), and (1,2,5). The triple cross-correlations for harmonic set (1,1,3) were estimated for the three resonance harmonics: the fundamental resonance harmonic that was employed twice and the third resonance harmonic. The triple

Validation of Third Order Cross-Correlations via Modeling
The CCSM3 and CCCS3 are functions of three independent frequencies. To estimate the diagnostic effectiveness of the CCSM3 and CCCS3, four sets of harmonics were investigated using the simulated signals from components with resonance frequencies ω c of 14 and 30 Hz. The investigated sets of harmonics of component resonance oscillations were as follows: (1,1,3), (1,3,5), (1,2,4), and (1,2,5). The triple cross-correlations for harmonic set (1,1,3) were estimated for the three resonance harmonics: the fundamental resonance harmonic that was employed twice and the third resonance harmonic. The triple crosscorrelations for harmonic set (1,3,5) were estimated for the three resonance harmonics: the fundamental resonance harmonic, the third resonance harmonic, and the fifth resonance harmonic. The triple cross-correlations for harmonic set (1,2,4) were estimated for the three resonance harmonics: the fundamental resonance harmonic, the second resonance harmonic, and the fourth resonance harmonic of the component resonance oscillations. Finally, the triple cross-correlations for harmonic set (1,2,5) were estimated for the three resonance harmonics: the fundamental resonance harmonic, the second resonance harmonic, and the fifth resonance harmonic.
To estimate the CCSM3 and CCCS3, SNR of 22 dB was used for 5% relative damage severity and SNR of 18 dB was used for 10% relative damage severity. Figure 2a,b show histograms for the selected diagnostic features. The histograms for additional frequency sets, taken into consideration for the CCSM3 and CCCS3, are presented in Figures A36-A39 in Appendix A. The superiority of the CCSM3 over the CCCS3 was also observed on these histograms. cross-correlations for harmonic set (1,3,5) were estimated for the three resonance harmonics: the fundamental resonance harmonic, the third resonance harmonic, and the fifth resonance harmonic. The triple cross-correlations for harmonic set (1,2,4) were estimated for the three resonance harmonics: the fundamental resonance harmonic, the second resonance harmonic, and the fourth resonance harmonic of the component resonance oscillations. Finally, the triple cross-correlations for harmonic set (1,2,5) were estimated for the three resonance harmonics: the fundamental resonance harmonic, the second resonance harmonic, and the fifth resonance harmonic.
To estimate the CCSM3 and CCCS3, SNR of 22 dB was used for 5% relative damage severity and SNR of 18 dB was used for 10% relative damage severity. Figure 2a,b show histograms for the selected diagnostic features. The histograms for additional frequency sets, taken into consideration for the CCSM3 and CCCS3, are presented in Figures A36-A39 in Appendix A. The superiority of the CCSM3 over the CCCS3 was also observed on these histograms. The main parameter that affects an accurate histogram evaluation is the bin number. The bin number is selected to accurately extract the shape of the histograms, using the following methodology. For each pair of the obtained modeling histograms in Sections 3.1 and 3.2, the total number of diagnostic features for the damaged and undamaged conditions were both 300. The standard "Rule of Thumb" is to divide a feature range into k equal bins, where k is the square root of a feature number. So, for each histogram (i.e., for the damaged and undamaged conditions), 18 bins were needed. Taking into account a distance between histograms for the undamaged and damaged conditions, a "safety factor" of 1.25 was applied and the obtained total number of bins was 45. The bin range for each pair of histograms was defined based on a minimum diagnostic feature value for the undamaged conditions and a maximum diagnostic feature value for the damaged conditions. Once the bin range was defined, the bin size was automatically defined based on the defined bin range and selected number of bins. If we choose a bin number that is too big for a given feature number, the bar height at each bin will suffer significant statistical fluctuation due to the paucity of samples in each bin. If we choose a bin number that is too small, the histogram cannot represent the shape of the underlying distribution because the resolution is insufficient. Figure 3a,b show the TPCDs and Fisher criteria for the CCSM3 and CCCS3 for all investigated harmonic sets using resonance frequencies ω of 14 and 30 Hz, without fault and with relative fault severities of 5% and 10% introduced for the defective cases. The main parameter that affects an accurate histogram evaluation is the bin number. The bin number is selected to accurately extract the shape of the histograms, using the following methodology. For each pair of the obtained modeling histograms in Section 3.1 and 3.2, the total number of diagnostic features for the damaged and undamaged conditions were both 300. The standard "Rule of Thumb" is to divide a feature range into k equal bins, where k is the square root of a feature number. So, for each histogram (i.e., for the damaged and undamaged conditions), 18 bins were needed. Taking into account a distance between histograms for the undamaged and damaged conditions, a "safety factor" of 1.25 was applied and the obtained total number of bins was 45. The bin range for each pair of histograms was defined based on a minimum diagnostic feature value for the undamaged conditions and a maximum diagnostic feature value for the damaged conditions. Once the bin range was defined, the bin size was automatically defined based on the defined bin range and selected number of bins. If we choose a bin number that is too big for a given feature number, the bar height at each bin will suffer significant statistical fluctuation due to the paucity of samples in each bin. If we choose a bin number that is too small, the histogram cannot represent the shape of the underlying distribution because the resolution is insufficient. Figure 3a,b show the TPCDs and Fisher criteria for the CCSM3 and CCCS3 for all investigated harmonic sets using resonance frequencies ω c of 14 and 30 Hz, without fault and with relative fault severities of 5% and 10% introduced for the defective cases.
The superiority of the CCSM3 over the CCCS3 can be seen in Figure 3a, with the CCSM3 having the total probability of correct diagnosis as high as 98% and as low as 92% and with the CCCS3 having 52% as the highest value of the total probability of correct diagnosis.
The Fisher criteria also validated the higher performance of the CCSM3 over the CCCS3 for all investigated cases. The CCSM3 s Fisher criterion result range was 3-8, as shown in Figure 3b, while that of CCCS3 was less than 1 for all investigated cases. The superiority of the CCSM3 over the CCCS3 can be seen in Figure 3a, with CCSM3 having the total probability of correct diagnosis as high as 98% and as low as and with the CCCS3 having 52% as the highest value of the total probability of co diagnosis.
The Fisher criteria also validated the higher performance of the CCSM3 over CCCS3 for all investigated cases. The CCSM3′s Fisher criterion result range was 3shown in Figure 3b, while that of CCCS3 was less than 1 for all investigated cases.
The simulation results generated from processing the modeled bilinear signals resonance frequencies of 14 and 30 Hz, with and without damage at relative severiti 5% and 10%, and with the addition of white Gaussian noise demonstrated that the CC technique was highly effective in detecting nonlinearities resulting from compo faults. The performance of the CCSM3 technology was compared to the performanc the CCCS3 technology using the TPCDs and Fisher criteria, which confirmed the sup ority of the CCSM3 in diagnosing early-stage faults. Therefore, the CCSM3 techno was an efficient technique for identifying incipient faults.

Validation of Fourth Order Cross-Correlations via Modeling
The CCSM4 and CCCS4 are functions of four independent frequencies. The CC and CCCS4 were estimated using the following harmonic sets: (1,1,2,3), (1,2,4,5), (1,2 and (1,3,4,5). The fourth cross-correlations for harmonic set (1,1,2,3) were estimated the four resonance harmonics: the fundamental resonance harmonic employed twice The simulation results generated from processing the modeled bilinear signals with resonance frequencies of 14 and 30 Hz, with and without damage at relative severities of 5% and 10%, and with the addition of white Gaussian noise demonstrated that the CCSM3 technique was highly effective in detecting nonlinearities resulting from component faults. The performance of the CCSM3 technology was compared to the performance of the CCCS3 technology using the TPCDs and Fisher criteria, which confirmed the superiority of the CCSM3 in diagnosing early-stage faults. Therefore, the CCSM3 technology was an efficient technique for identifying incipient faults.

Validation of Fourth Order Cross-Correlations via Modeling
The CCSM4 and CCCS4 are functions of four independent frequencies. The CCSM4 and CCCS4 were estimated using the following harmonic sets: (1,1,2,3), (1,2,4,5), (1,2,3,4) and (1,3,4,5). The fourth cross-correlations for harmonic set (1,1,2,3) were estimated for the four resonance harmonics: the fundamental resonance harmonic employed twice, the second resonance harmonic, and the third resonance harmonic. The triple cross-correlations for harmonic set (1,2,4,5) were estimated for the four resonance harmonics: the fundamental resonance harmonic, the second resonance harmonic, the fourth resonance harmonic, and the fifth resonance harmonic. The fourth cross-correlations for harmonic set (1,2,3,4) were estimated for the four resonance harmonics: the fundamental resonance harmonic, the second resonance harmonic, the third resonance harmonic, and the fourth resonance harmonic. Finally, the fourth cross-correlations for harmonic set (1,3,4,5) were estimated for the four resonance harmonics: the fundamental resonance harmonic, the third resonance harmonic, the fourth resonance harmonic, and the fifth resonance harmonic. These sets of resonance harmonics were investigated using resonance frequencies ω c of 14 and 30 Hz for components without damage and with relative damage severities of 5% and 10% introduced for the damaged cases.
To estimate the CCSM4 and CCCS4, SNR of 24 dB was used for 5% relative damage severity and SNR of 19 dB was used for 10% relative damage severity for resonance frequencies of 14 and 30 Hz Figure 4a,b show histograms for the selected diagnostic features. The histograms for additional frequency sets, taken into consideration for the CCSM4 and CCCS4, are presented in Figures A40-A43 in Appendix A. The superiority of the CCSM4 over the CCCS4 was also observed on these histograms. damental resonance harmonic, the second resonance harmonic, the fourth resonance harmonic, and the fifth resonance harmonic. The fourth cross-correlations for harmonic set (1,2,3,4) were estimated for the four resonance harmonics: the fundamental resonance harmonic, the second resonance harmonic, the third resonance harmonic, and the fourth resonance harmonic. Finally, the fourth cross-correlations for harmonic set (1,3,4,5) were estimated for the four resonance harmonics: the fundamental resonance harmonic, the third resonance harmonic, the fourth resonance harmonic, and the fifth resonance harmonic. These sets of resonance harmonics were investigated using resonance frequencies ω of 14 and 30 Hz for components without damage and with relative damage severities of 5% and 10% introduced for the damaged cases.
To estimate the CCSM4 and CCCS4, SNR of 24 dB was used for 5% relative damage severity and SNR of 19 dB was used for 10% relative damage severity for resonance frequencies of 14 and 30 Hz Figure 4a,b show histograms for the selected diagnostic features. The histograms for additional frequency sets, taken into consideration for the CCSM4 and CCCS4, are presented in Figures A40-A43 in Appendix A. The superiority of the CCSM4 over the CCCS4 was also observed on these histograms.     (1,2,3,4) were estimated for the four resonance harmonics: the fundamental resonan monic, the second resonance harmonic, the third resonance harmonic, and the fou onance harmonic. Finally, the fourth cross-correlations for harmonic set (1,3,4,5) w timated for the four resonance harmonics: the fundamental resonance harmonic, th resonance harmonic, the fourth resonance harmonic, and the fifth resonance har These sets of resonance harmonics were investigated using resonance frequencies 14 and 30 Hz for components without damage and with relative damage severitie and 10% introduced for the damaged cases.
To estimate the CCSM4 and CCCS4, SNR of 24 dB was used for 5% relative d severity and SNR of 19 dB was used for 10% relative damage severity for resonan quencies of 14 and 30 Hz Figure 4a,b show histograms for the selected diagnostic fe The histograms for additional frequency sets, taken into consideration for the CCSM CCCS4, are presented in Figures A40-A43 in Appendix A. The superiority of the C over the CCCS4 was also observed on these histograms.   The superiority of the CCSM4 over the CCCS4 is shown in Figure 5a, with the C having a TPCD as high as 98% and as low as 92%, while the CCCS4 had 56% as the h value of the TPCD. The Fisher criteria also validated the higher performance CCSM4 over the CCCS4 for all investigated cases. The CCSM4′s Fisher criteria ran 3.0-6.4, as shown in Figure 5b, while for the CCCS4, it was less than 1 for all inves cases.
The simulation results acquired by processing the modeled bilinear componen resonance frequencies of 14 and 30 Hz, at fault severity ratios of 5% and 10%, and presence of white Gaussian noise suggested that the CCSM4 technology was profic detecting nonlinearities resulting from component faults. The TPCDs and Fisher were utilized to compare the performance of the CCSM4 to the CCCS4, revealing perior diagnostic capability of the CCSM4 in identifying early-stage faults. Therefo CCSM4 technology was an efficient technique for diagnosing incipient faults.

Experimental Technology Validation
The laboratory test rig used in this research is described in detail in ref. [1,2]. tograph of the test rig is presented in Figure 6. The IM used in this experiment was a phase 1.1 kW motor with two pole pairs. The tested IM was supplied directly fr electrical power grid. The superiority of the CCSM4 over the CCCS4 is shown in Figure 5a, with the CCSM4 having a TPCD as high as 98% and as low as 92%, while the CCCS4 had 56% as the highest value of the TPCD. The Fisher criteria also validated the higher performance of the CCSM4 over the CCCS4 for all investigated cases. The CCSM4 s Fisher criteria range was 3.0-6.4, as shown in Figure 5b, while for the CCCS4, it was less than 1 for all investigated cases.
The simulation results acquired by processing the modeled bilinear components with resonance frequencies of 14 and 30 Hz, at fault severity ratios of 5% and 10%, and in the presence of white Gaussian noise suggested that the CCSM4 technology was proficient in detecting nonlinearities resulting from component faults. The TPCDs and Fisher criteria were utilized to compare the performance of the CCSM4 to the CCCS4, revealing the superior diagnostic capability of the CCSM4 in identifying early-stage faults. Therefore, the CCSM4 technology was an efficient technique for diagnosing incipient faults.

Experimental Technology Validation
The laboratory test rig used in this research is described in detail in ref. [1,2]. A photograph of the test rig is presented in Figure 6. The IM used in this experiment was a three-phase 1.1 kW motor with two pole pairs. The tested IM was supplied directly from the electrical power grid.
All measurements used in this experiment were made under the conditions of a full motor load. All experimental tests were performed with 6204C3 Koyo bearings. One of the new tested bearings is shown in Figure 7. The IM current recordings were performed for 4 new bearings and 4 defective bearings with intentionally made damage of different types and severity. Close-up photographs of the introduced damage are presented in Figure 8. The description of each damage is presented in Table 1. All introduced defects were local faults to reproduce conditions of early-stage bearing defect development. The new bearings used in this experiment are described as p1, p2, p3, and p4.
The acquired data for each diagnostic case included three digital standard current signals of three phases of the induction motor. These signals were not normalized. The length of each signal was 65 s acquired with a sampling frequency of 65 kHz. The current signals were captured for stationary motor operation under full load conditions. The main component of each signal was the 50 Hz supply current component. Graphical illustrations of fragments of the recorded signals for the damaged and pristine bearings are presented in Figures 9 and 10, respectively.  Table 1. All introduced defects were local faults to reproduce conditions of early-stage bearing defect development. The new bearings used in this experiment are described as p1, p2, p3, and p4.  . Experimental test rig: arrow A directs to the motor, the B arrows direct to the laser sensors, arrow C directs to the magnetic coupling, arrow D directs to the electromagnetic brake, arrow E directs to the cooling hoses, and the F arrows directs to the vibration isolation pads. Figure 6. Experimental test rig: arrow A directs to the motor, the B arrows direct to the laser sensors, arrow C directs to the magnetic coupling, arrow D directs to the electromagnetic brake, arrow E directs to the cooling hoses, and the F arrows directs to the vibration isolation pads.
All measurements used in this experiment were made under the conditions of a full motor load. All experimental tests were performed with 6204C3 Koyo bearings. One of the new tested bearings is shown in Figure 7. The IM current recordings were performed for 4 new bearings and 4 defective bearings with intentionally made damage of different types and severity. Close-up photographs of the introduced damage are presented in Figure 8. The description of each damage is presented in Table 1. All introduced defects were local faults to reproduce conditions of early-stage bearing defect development. The new bearings used in this experiment are described as p1, p2, p3, and p4.  The captured signals were split into 13 non-overlapping time segments of 5 s. The Blackman time window was used for each segment of the signal. The frequency of the IM shaft rotation f r and the frequency of the power supply f p were estimated from the IM current spectrum for each time segment separately. It is crucial in the examined technologies to track changes in the aforementioned frequencies in each time segment in order to track the bearing defect frequencies. The values of the IM shaft rotation frequency f r were estimated using the detection of rotor slot harmonics (RSH) frequency and the parameters of the used IM constructions [46,76]. After detection of the RSH frequency, Equation (14) was used for each f r estimation [46,76]: where f RSH is the frequency of RSH, f p is the frequency of the power supply, f r is the IM shaft rotation frequency, k is the number of rotor bar harmonics, and N R is the number of rotor bars.   Figure  in1 inner race pit damage with diameter of 1 mm and depth of 0.5 mm 1.20% Figure 8A in2 inner race scratch damage across the bearing rolling direction with length of 3 mm, width of 1 mm, and depth of 0.7 mm 1.20% Figure 8B out1 outer race scratch damage along the bearing rolling direction with length of 3 mm, width of 1 mm, and depth of 0.5 mm 2.23% Figure 8C out2 outer race scratch damage across the bearing rolling direction with length of 3 mm, width of 1 mm, and depth of 0.5 mm 0.74% Figure 8D The acquired data for each diagnostic case included three digital standard current signals of three phases of the induction motor. These signals were not normalized. The length of each signal was 65 s acquired with a sampling frequency of 65 kHz. The current signals were captured for stationary motor operation under full load conditions. The main component of each signal was the 50 Hz supply current component. Graphical illustrations of fragments of the recorded signals for the damaged and pristine bearings are presented in Figures 9 and 10, respectively.   Figure   in1 inner race pit damage with diameter of 1 mm and depth of 0.5 mm 1.20% Figure 8A in2 inner race scratch damage across the bearing rolling direction with length of 3 mm, width of 1 mm, and depth of 0.7 mm 1.20% Figure 8B out1 outer race scratch damage along the bearing rolling direction with length of 3 mm, width of 1 mm, and depth of 0.5 mm 2.23% Figure 8C out2 outer race scratch damage across the bearing rolling direction with length of 3 mm, width of 1 mm, and depth of 0.5 mm 0.74% Figure 8D The values of these frequencies are needed to estimate the characteristic frequencies related to bearing damage. Next, the complex spectral amplitude for the CCCS3 and the modulus of the complex spectral amplitude for the CCSM3 from the current spectra were obtained for three bearing defect-related characteristic frequencies for each segment. The following step is to use Equation (2) for the CCSM3 estimation and Equation (5)   The captured signals were split into 13 non-overlapping time segments of 5 s. The Blackman time window was used for each segment of the signal. The frequency of the IM shaft rotation and the frequency of the power supply were estimated from the IM current spectrum for each time segment separately. It is crucial in the examined technologies to track changes in the aforementioned frequencies in each time segment in order to track the bearing defect frequencies. The values of the IM shaft rotation frequency were estimated using the detection of rotor slot harmonics (RSH) frequency and the parameters of the used IM constructions [46,76]. After detection of the RSH frequency, Equation (14) was used for each estimation [46,76]:  The captured signals were split into 13 non-overlapping time segments of 5 s. The Blackman time window was used for each segment of the signal. The frequency of the IM shaft rotation and the frequency of the power supply were estimated from the IM current spectrum for each time segment separately. It is crucial in the examined technologies to track changes in the aforementioned frequencies in each time segment in order to track the bearing defect frequencies. The values of the IM shaft rotation frequency were estimated using the detection of rotor slot harmonics (RSH) frequency and the parameters of the used IM constructions [46,76]. After detection of the RSH frequency, Equation (14) was used for each estimation [46,76]: The bearing defect frequencies are described by the following equations [1,2,19]: where f out is the defect frequency for the outer ring, f in is the defect frequency for the inner ring, f r is the shaft rotation frequency, N b is the number of balls or rollers, B d is the diameter of the balls or rollers, P d is the pitch diameter of a bearing, and α is the angle of thrust. FCC o is the fault characteristic coefficient for the outer race, while FCC i is the fault characteristic coefficient for the inner race. Coefficients FCC i and FCC o are provided by SKF.
To compare the CCSM3 with the CCCS3, 51 diagnostic features for each bearing were calculated for each examined technology. Time segments overlapping from 0% to 50% with 1% steps were used for these calculations. The values of 102 diagnostic features were used for estimations of the TPCD and FC. IM current signals were recorded for four new bearings and four defective bearings. The four pairs (new-defective) of bearings were split into two cases: two bearings with outer race defects and two bearings with inner race defects. Each pair of bearings was examined separately and the obtained results were averaged. The recorded signals were processed via the examined CCSM3 and CCCS3 technologies in order to make a comparison.
In the current stage, the acquired data were processed on a separate computer via Matlab R2021a. The computational/measurement complexity are discussed below. All data in the experimental part were processed on PC with Intel Core i7-2600 CPU and 8 GB of RAM (by Intel Corporation, Santa Clara, CA, USA). The computation time on this hardware acquired with the Matlab cpu time command was 1.9812 s for 0% overlapping and 3.3696 s for 50% overlapping. It took 123.5 s to calculate 51 diagnostic features. If the CCSM3 algorithm software was organized, using the calculation of the CCSM3 during the sample intervals, then the CCSM3 feature could be estimated in "real time." The signals were the three phases of the motor current, the length was 65 s, the sampling frequency was 65,536 Hz, and 24 bits were used for the sampling.
The total number of operations acquired with Matlab socFunctionAnalyzer (by Math-Works, Natick, MA, USA) was 310,778,788 for the original 65,536 Hz sampling frequency, 65 s of signal duration, and 5 s duration of time segments with no overlapping. The number of operations could be significantly decreased by decreasing the sampling frequency used in this experiment. After downsampling the signal by 2 (with the same all other parameters) the total number of operations was decreased to 155,458,468. The total number of operations for further downsampling is presented in the Table 2. Based on the theoretical basis, the number of operations can be estimated as follows.
Considering a conventional method of CCSM3 estimation, the main method for estimation of the CCSM3 is the direct method. Extension of this method to the CCSM4 or to any higher CCSM order can be made easily. Considering the direct method of estimation of the numerator of the CCSM3 for one current phase (since estimation of the denominator of the CCSM3 is a computationally easy task) for one diagnostic feature, the following steps should be employed: a. segment current data into K non-overlapped segments of N samples each b.
estimate DFT coefficients for each segment c.
calculate CCSM3 estimates for all segments via multiplications of moduli of the selected DFT coefficients and average these estimates over K segments The number of multiplication operations to estimate the DFT for each segment is Nlog 2 N, and the number of addition operations to estimate the DFT for each segment is 2Nlog 2 N.
Finally, ignoring a small number of addition operations for the averaging of the CCSM3 over K segments and a small number of multiplication operations for obtaining the CCSM3 for each segment, the total number of multiplication operations for K segments is K × Nlog 2 N and total number of addition operations for K segments is K × Nlog 2 N.
Using the above methodology to estimate the total number of operations for calculation of the numerator of the CCSM3, the numbers of multiplication and addition operations were 78,040,269 and 156,080,538, respectively, for the processing parameters employed in the experimental trials, i.e., the segment size was 5 s, the number of segments was 13, and the sampling frequency was 65.536 kHz with 0% segment overlapping. Therefore, the total number of operations (i.e., additions and multiplications) for calculation of the numerator of the CCSM3 was 234,120,807.

Inner Race Defect Analysis
Components with the frequency related to bearing inner race defects, described by Equation (16), did not directly appear in the IM current spectrum. The frequencies of the components of the IM current related to bearing inner race defects can be described by Equation (17) [28,36]: where f in is the inner race defect frequency, f p is the frequency of the power supply, f r is the frequency of the IM shaft rotation, and i, j, k are integer coefficients related to each frequency. The bearings used in this part of experiment were the pairs of bearings in1, p1 and in2, p2. The results obtained in this part were estimated with the use of 84 components, which were used to calculate the CCSM3 and CCCS3 for 28 different combinations of three components for each bearing. The 84 components were obtained with the i, j, k coefficients [1]: i = 3, 6, 9; j = −3, −1, 1, 3; k = −3, −2, −1, 0, 1, 2, 3.
For each i, different sets of j and k coefficients were used. The values of each diagnostic feature based on the CCSM3 and CCCS3 were estimated for three individual combinations of i, j, k coefficients. The values of the i, j, k coefficients corresponding with each number of the CCSM3 and CCCS3 correlation are listed in Table A2 in Appendix A. TPCDs, estimated via Equation (7) for the CCSM3 and CCCS3 technologies, are presented in Figure 11 The average TPCD, calculated for all 28 correlations, was 99% for the CCSM3 and 61% for the CCCS3. For the same data, the Fisher criteria were estimated. The values of the Fisher criteria for each correlation number (Table A1)

Outer Race
Components with the frequency related to bearing outer race defects described by Equation (14) did not directly appear in the IM current spectrum. The frequencies of the components of the IM current related to bearing outer race defects can be described by Equation (17) [5,28,36]: where f out is the outer race defect frequency, f p is the frequency of the power supply, and i, j are the integer coefficients related to each frequency. The bearings used in this part of the experiment were the pairs of bearings out1, p3 and out2, p4. The results obtained in this part were estimated via the use of 12 components to calculate the CCSM3 and CCCS3 for 4 different combinations of three components for each bearing. The 12 components were obtained following the use of i, j coefficients in Figure 11 The average TPCD, calculated for all 28 correlations, was 99% for the CCSM3 and 61% for the CCCS3. For the same data, the Fisher criteria were estimated. The values of the Fisher criteria for each correlation number (Table A1) are presented in Figure 12. The average Fisher criterion values were 19.5 for the CCSM3 and 0.2 for the CCCS3. Histograms of the diagnostic features values based on the CCSM3 are presented in Figures  A1-A28 in Appendix A. Each pair of presented histograms displayed full separation between the cases of new and defective bearings. Figure 11. Graph of the probability of proper inner race diagnosis with the CCSM3 (blue) and CCCS3 (red) for each correlation number (Table A1).  (Table A1). Figure 11. Graph of the probability of proper inner race diagnosis with the CCSM3 (blue) and CCCS3 (red) for each correlation number (Table A1).
in Figure 11 The average TPCD, calculated for all 28 correlations, was 99% for the CCSM3 and 61% for the CCCS3. For the same data, the Fisher criteria were estimated. The values of the Fisher criteria for each correlation number (Table A1) are presented in Figure 12. The average Fisher criterion values were 19.5 for the CCSM3 and 0.2 for the CCCS3. Histograms of the diagnostic features values based on the CCSM3 are presented in Figures  A1-A28 in Appendix A. Each pair of presented histograms displayed full separation between the cases of new and defective bearings. Figure 11. Graph of the probability of proper inner race diagnosis with the CCSM3 (blue) and CCCS3 (red) for each correlation number (Table A1).  (Table A1).  (Table A1).
For each coefficient i, different sets of j coefficients were used. The values of each diagnostic feature based on the CCSM3 and CCCS3 were estimated for three individual combinations of the i, j coefficients. The values of the i, j coefficients corresponding with each number of the CCSM3 and CCCS3 correlations are listed in Table A2 in Appendix A. The TPCDs, estimated via Equation (7) for the CCSM3 and CCCS3 technologies are presented in the Figure 13. The average TPCD calculated for all 4 correlations was 98% for the CCSM3 and 55% for the CCCS3. For the same data, the Fisher criteria were estimated. The values of the Fisher criterion for each correlation number (Table A2) are presented in Figure 14. The average Fisher criterion values were 13.2 for the CCSM3 and 0.03 for the CCCS3. Histograms of the diagnostic features values based on the CCSM3 are presented in Figures A29-A35 in Appendix A. Each pair of presented histograms displayed full separation between the cases of new and defective bearings.
combinations of the , coefficients. The values of the , coefficients corresponding with each number of the CCSM3 and CCCS3 correlations are listed in Table A2 in Appendix A. The TPCDs, estimated via Equation (7) for the CCSM3 and CCCS3 technologies are presented in the Figure 13. The average TPCD calculated for all 4 correlations was 98% for the CCSM3 and 55% for the CCCS3. For the same data, the Fisher criteria were estimated. The values of the Fisher criterion for each correlation number (Table A2) are presented in Figure 14. The average Fisher criterion values were 13.2 for the CCSM3 and 0.03 for the CCCS3. Histograms of the diagnostic features values based on the CCSM3 are presented in Figures A29-A35 in Appendix A. Each pair of presented histograms displayed full separation between the cases of new and defective bearings. Figure 13. The complete probability of correct outer race diagnosis with the CCSM3 (blue) and CCCS3 (red) for each correlation number (Table A2). Figure 13. The complete probability of correct outer race diagnosis with the CCSM3 (blue) and CCCS3 (red) for each correlation number (Table A2).  (Table A2).

Technology Comparisons
The simulations and experimental results clearly showed that the CCSM3 was a more efficient signal processing technique than the CCCS3 for bearing fault diagnosis. The simulations were carried out for signal-to-noise ratios varying from 18 to 22 dB and fault severity ratios of 5% and 10%. The simulations included 4 different diagnosis features for two different resonance frequencies of 14 and 30 Hz for undamaged components. The experimental validation was carried out on 8 bearings: two with inner race damage, two with outer race damage, and 4 pristine bearings. For diagnostic purposes, the recorded motor current signals were processed. In the inner race experiments, 28 different diagnosis features were examined. In the outer race experiments, 4 different diagnosis features  (Table A2).

Technology Comparisons
The simulations and experimental results clearly showed that the CCSM3 was a more efficient signal processing technique than the CCCS3 for bearing fault diagnosis. The simulations were carried out for signal-to-noise ratios varying from 18 to 22 dB and fault severity ratios of 5% and 10%. The simulations included 4 different diagnosis features for two different resonance frequencies of 14 and 30 Hz for undamaged components. The experimental validation was carried out on 8 bearings: two with inner race damage, two with outer race damage, and 4 pristine bearings. For diagnostic purposes, the recorded motor current signals were processed. In the inner race experiments, 28 different diagnosis features were examined. In the outer race experiments, 4 different diagnosis features were examined.
A comparison of the achieved results is presented in Table 3. The gain of effectiveness for FC was calculated as the ratio of FC for the CCSM3 and for the CCCS3. The effectiveness gains for FC were 50.7 for the simulation results, 97.5 for the inner race experiment results, and 440 for the outer race experiment results. The gain in probability of incorrect diagnosis was calculated as the ratio of estimated probability of incorrect diagnosis for the CCCS3 and for the CCSM3. The effectiveness gains for the probability of incorrect diagnosis were 11.4 for the simulation results, 27.9 for the inner race experiment results, and 22.5 for the outer race experiment results. Both the simulation and experimental results confirmed that the CCSM3 method was a more efficient diagnosis method than the CCCS3 method. The presented results clearly showed that the CCSM3 was an efficient method to detect mechanical damage in electromechanical systems. The simulations revealed that this new signal processing technique can perform well even with signals with low SNR. This feature is highly desirable especially for damage diagnosis with the use of electrical signals, such as a motor current where component amplitudes are comparable with noise levels. The conducted research also showed that the results for the CCCS3 were far below reasonable for any diagnostic system.
From the definitions of bicoherence and tricoherence, it is clear that the analyzed spectral components must exhibit statistical dependencies in order to obtain non-zero results, which are associated with bearing faults. However, bicoherence and tricoherence could be used in specific cases in which fault-related spectral components follow "the bicoherence and the tricoherence frequency rules" [77][78][79]. These higher-order spectra frequency requirements may not be met for all spectral components, which carry valuable information for a fault diagnosis. Therefore, these higher-order spectra frequency requirements limit the usage of spectral components generated by faults, which could be used for a fault diagnosis via motor current signature analysis and thus limited fault types, which could be effectively diagnosed by higher-order spectra.
The proposed cross-correlations do not need to follow "the bicoherence and the tricoherence frequency rules." Contrary to the bicoherence, tricoherence, and other higherorder spectra, the CCSM technologies allow to take into account statistical dependencies between any needed combinations of multiple spectral components appearing due to a fault, without limitations, i.e., regardless of dependencies between their central frequencies.
Compared to existing technologies, the CCSM technologies also have the advantage that they can be applied by acquiring motor current signals, without the need to install sensors outside the motor electrical box. Indeed, motors are widely used in harsh environments, which makes installation of sensors on a motor very difficult or even impossible. In addition, unlike other diagnostic technologies, the collection of motor stator current data will offer the possibility of diagnosing various possible motor faults, not only bearing faults. This is possible using various MCSA-based technologies. The minor disadvantage of the proposed technologies is a relatively high computational complexity related to the diagnostic feature estimation, which could be easily overcome by powerful IT tools.
Data diversity should be addressed by proper training of the proposed technologies, taking into account multiple faults and possible combinations of the speed and load of various rotating machinery. One needs to obtain proper training data related to these combinations of speed and load in order to reliably diagnose various faults.
If a failure appears due to an unknown fault, the key step in generalizing/investigating the unknown fault is to perform the nine key steps of a classical failure analysis. These steps should be sufficient to define an unknown fault and a failure mode of that fault. In addition, in order to generalize possible known and unknown faults, one needs to create/use a generalized, comprehensive theoretical model of faults.
Finally, the comprehensive validation and comparison of the novel cross-correlation technologies confirmed an important, non-traditional, novel outcome of this research: the technologies based on cross-correlations of spectral moduli were more effective for damage diagnosis than the technologies based on cross-correlations of complex spectra. This non-traditional outcome will allow wide industrial implementation of the novel crosscorrelations of spectral moduli for predictive maintenance of various rotating machinery.

Conclusions
This study comprehensively investigated and validated for the first time by modeling and experiments new, online, nonlinear, higher-order CCSM technologies for the fault diagnosis of rotating machinery. The main contributions of this paper are the novel modeling validation of the technologies, novel experimental validation of the technologies, and novel comparisons with advanced higher-order CCCS technologies.
Experimental trials were undertaken using a test rig dedicated for IM testing. Motor current signals were captured for pristine and damaged bearings. Experimental validation was performed for four pristine tested bearings, two bearings with inner race damage at a 1.2% relative fault severity ratio, and two bearings with outer race damage at relative fault severity ratios of 2.23% and 0.74%.
Experimental and simulation validation confirmed the high probability of correct diagnosis via the CCSM at early-stage fault development. The average total probability of incorrect diagnosis for the CCSM3 technology for all experimental results of 8 tested bearings, estimated via 6528 diagnostic features, was 1.5%. The average total probability of incorrect diagnosis for the CCSM3 technology, estimated via 2400 simulated signals, was 4.3%. The average total probability of incorrect diagnosis for the CCSM4 technology, estimated via 2400 simulated signals, was 5%.
The effectiveness gains in the total probability of incorrect diagnosis for the CCSM3 in comparison with the CCCS3 were 26.8 for the experimental validation and 18.9 for the simulation validation. The effectiveness gains in the Fisher criterion for the CCSM3 in comparison with the CCCS3 were 50.7 for the simulation validation and 104.7 for the experimental validation.
Finally, the comprehensive validation and comparison of the novel cross-correlation technologies confirmed an important non-traditional novel outcome: the technologies based on cross-correlations of spectral moduli were more effective for damage diagnosis than the technologies based on cross-correlations of complex spectra.
The presented higher-order spectral technologies are very powerful technologies for performing damage diagnosis in various electromechanical systems using motor current signature analysis (MCSA). The proposed technologies effectively work even for signals with low signal-to-noise ratio. For the presented research, the main limitations are as follows: • the quality of data capture equipment. A low noise current transducer (i.e., self-noise was −120 dB in the whole frequency range) and 24-bit DAQ are minimal requirements for current data capture • the computational complexity of the estimations of the proposed technologies. This limitation is easy to overcome via powerful IT tools.
The proposed technologies are novel, as these technologies do not use a phase spectrum. This non-traditional outcome will open the door for wide industrial implementation of the novel cross-correlations of spectral moduli for predictive maintenance of various rotating machinery. Therefore, the future scope of the research is to prove its effectiveness for other MCSA and vibration applications.
The results of this study are significant for damage diagnosis. The proposed crosscorrelations of spectral moduli present a novel conceptualization and will make a considerable impact on damage diagnosis in electrical and mechanical engineering via motor current signature analysis, vibration analysis, ultrasound analysis, etc.

Conflicts of Interest:
The authors declare no conflict of interest.