Quasi least square (LS) technique was used to identify first- and second-order hidden periodicities in vibration signal. It was shown, that the low-frequency vibration component (< 2 kHz) is adequately described by the periodically non-stationary random processes (PNRP) model. The amplitude spectra of the deterministic oscillations and the time changes in the power of the stochastic part are analyzed. The value of an indicator based on the mean function harmonic power is decisive for detecting failures and monitoring the motor condition. The structures of the PNRP moment functions and the high values of the condition indicator enable us to deduce that motor is in a critical state. electric motor vibration; periodically non-stationary random process; mean and covariance function; basic frequency estimators; condition indicators.
To calculate the mean, covariance function and their Fourier coefficients using experimental data, we need to apply methods of statistical analysis of PNRP, such as coherent (synchronous) averaging [1–10], or the component [11] or the least squares [12] methods. The suitability of a given technique depends on the specifics of the experimental data, the purpose of the analysis, and the required accuracy, but all these methods can be used in cases where the non-stationarity period (basic frequency) is known. In vibration analysis, this period can be in many cases calculated based on technical parameters of investigated mechanism. However, the period values obtained in this way are not sufficiently accurate, and may also vary under real-world conditions. Hence, to ensure that the PNRP analysis is effective, we need to determine the period based on the selected vibration realizations.
To determine the non-stationarity period, special functionals can be used [9–16] that are similar to coherent or component statistics, except that a test period is inserted to replace the true period. These functionals have extreme values at points that are asymptotically unbiased and consistent period estimators. The biases of these estimators are on the order O (T −2), and the variances are on the order O (T −3), where T is the realization length. To improve the efficiency of estimating the
0000-0001-5546-453X (R. Yuzefovych); 0000-0003-0243-6652 (I. Javorskyj); 0000-0001-5559-1969 (O. Lychak); 0000-0003-4121-6011 (P. Semenov); 0009-0003-1855-6226 (R. Khmil)
© 2025 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). basic frequency, the LS method was proposed in [12]. As the realization length increases, the LS functional for determining the basic frequency of the mean function quickly tends to where
T and 2 T is the realization length, h= is the sampling step, and L1 is the number of harmonics.
k The maximum point f^0 of (1) is asymptotically unbiased and consistent estimator of the mean basic frequency [12]. The functional at the maximum point f =f^0 is close to the sum of the time-averaged power of the chosen harmonics: The quantities m^ ks ( f^ ) and m^ck ( f^0) are asymptotically unbiased and consistent estimators of the 0 Fourier coefficients for the mean function [17-19]. Hence, based on the statistical expression The expression in (3) is the interpolation formula for the mean function for all t ∈[ 0 ; f −01]if the condition
, is satisfied [11]. The function in (2) describes the deterministic oscillations. Their amplitudes and phase spectra are defined by the expressions: we can form the mean function estimator
As a summary of the foregoing discussion, we present step-by-step procedures for determining the deterministic part of the vibration as follows:
Using the experimental time series ξ (nh), we form the statistics in (2); By substituting into (1) the formulae in (2) we can form the functional for the numerical analysis. The number L1 is chosen to be close to the ratio f m / f r, where f m is the signal spectrum high boundary frequency and f r is the rotation frequency; (1) (2) (3) (4)
We then calculate the functional in (1) for the frequencies belonging to the interval[ f 1 , f 2] , which contain the rotation frequency f r. If we want to find the basic frequency to an accuracy of 10-3 Hz, we choose a calculation step of Δ f =10−3 Hz; The maximum points of (10) are taken as the estimator of the mean basic frequency f^0, and the value F ( f^0) is accepted as the estimator of power for deterministic oscillations; By substituting f =f^0 into (11), we calculate the mean Fourier coefficients, and the amplitude and phase spectra; Using the interpolation formula in (3), we calculate the mean values for all t ∈[ 0 ; P^ ], where P^=1/ f^ . Assuming that m^ (t , f^0)=m (t + P^ , f^0), we can separate the stochastic part ξ˚ (nh)=ξ (nh)− m^ (nh , f^o)
.
The LS functional for estimating the variance in the basic frequency for large in the form: can be represented where and L2 is the number of harmonics. The maximum point of the expression in (5) is an asymptotically unbiased and consistent estimator of the variance in the basic frequency. By substituting f =f^0 into (6), we obtain statistics for calculating the Fourier coefficients of the variance. Hence, the quantity We can obtain the variance estimator for t ∈[ 0 ; f −01]all on the basis of R^ck (0 , f^0), R^ks (0 , f^0) and the statistical expression using the interpolation formula Aliasing errors are absent if the inequality (5) (6)
It follows from the above that the procedures used for estimating the vibration variance are similar to those for estimating the mean. In the present case, only the squared centered realization is processed, and the functional in (5) is calculated. Since the variance in the time changes is the result of correlations of the harmonic belonging to the signal spectrum and shifted bykf 0 , where k is number of the variance harmonic, then the integer number L2 is limited by the ratio of the signal bandwidth to the rotation frequency.
The mean and variance spectra are used to define the health status of the mechanism. The condition indicators can also be formed on the basis of a harmonic composition. The first indicator is determined by the ratio of the sum of the time-averaged powers of the harmonics for deterministic oscillations to the time-averaged power of the stochastic component The second indicator is a measure of the signal non-stationarity of the second order. It is defined as the ratio of the sum of amplitudes of the variance harmonics to its timeaveraged value, .
Note that the relationship between the indicators І1 and І2 may differ depending on the type of mechanism analyzed, the type of fault, and its stage of development. is fulfilled [11]. The function in (7) describes the shape of the periodic time changes in the power of the stochastic part. To characterize these changes, it is advisable to calculate the amplitude and phase spectra as follows: (8)
A NELCON Port crane with a FLENDER D 46393 Bocholt gearbox driven by two identical Siemens 1LL8 317-4 PC-Z motors [20] was the subject of our investigation (Figure 1). The main parameters of a single motor were as follows: power – 400 kW, nominal rotational speed 1495 rpm, cosφ 0.88. Vibration signals were acquired simultaneously by two piezoceramic accelerometers fixed onto the cast iron body with powerful magnets, close to the cage of the bearing, on top of the motor. The sensitive axis of the sensors was set to the vertical direction to coincide with direction of gravity, to provide a maximum range for the vibration signal (in accordance with [21–25]). Data acquisition system settings were tuned to a filter cutoff frequency of 12.5 kHz with a sampling frequency of 25 kHz. The signals were stored on the hard disk of a notebook, and were processed offline with PNRP methods.
We consider the frequency band [0 Hz, 2 kHz]. The segment of the realization of signal is shown in Figure 1. To ascertain its covariance properties and spectral composition, we calculate the covariance function and spectral density for the stationary approximation, using the formulae: (10) τ where L= m is some natural number,
h Hamming window:
is the of the correlogram cut-off and is k (nh)the Graphs of the estimators for the covariance function (9) and spectral density (10) are shown in Figure 2. The time-averaged power of the signal in the low-frequency band is R (0)=1.23 (m / s2)2 . The covariance function estimator has an undamped tail (Figure 5a) which can be explained by the presence of deterministic oscillations in the signal, with power of 0,3 (m / s2), equal to 0.25 of the signal power in this frequency range. The deterministic oscillations cause sharp peaks in the spectral density estimator (Figure 5b).
We separate the deterministic oscillations and compute their amplitude spectrum. To do this, we first use the functional in (1) to determine their basic frequency. The dependence of this functional on the test frequency is shown in Figure 3.
As we can see, the maximum value of this quantity is reached at f =28.56 Hz, which we accept as the basic frequency estimator f^0. By inserting this value into (2) f =f^0, we can calculate the Fourier coefficients of the mean function and hence the amplitude spectrum (4). A diagram of the latter is shown in Figure 4b, and the numerical values of the harmonic amplitudes are listed in Table 1. As expected, the first harmonic has the largest amplitude, exceeding the second harmonic by more than nine times.
Table 1. Amplitudes of harmonics for deterministic oscillations The graph of the time dependence of the deterministic component obtained using the interpolation formula therefore has the form of a basic harmonic, on which the low-power oscillations of the harmonic with higher frequency are superposed (Figure 4a).
By separating the stochastic component ξ˚ (nh)=ξ (nh)− m^ (nh , f^o), we can identify the hidden periodicity of the second order. The dependence of the quadratic functional in (5) on the test frequency is shown in Figure 5. The clear, strong peak in the graph shows that the power of the stochastic part changes periodically over time. The maximum point of the functional in (5) is considered here as the estimator of the variance basic frequency f^0=28.43 Hz. The difference between this value and mean basic frequency obtained above is only 0.13 Hz, which can be considered the statistical error in the estimation. It is evident that the variance amplitude spectrum is narrow, and the dominant part of the power for variance time changes belongs to the first five harmonics. The sum of the amplitudes for 20 harmonics is equal 4.014 (m / s2)2to while the time-averaged value of the variance is equal to R0I(0)=2.689 (m / s2)2 . Thus, for the indicator of the periodical non-stationarity of the second order, we have I 2I= 2Σ0 V^ I (k f^0)/ R0I=1.43. A graph of the time dependence of the variance, which k=1 was calculated on the basis of the interpolation formula in (7), is presented in Figure 6a. In the same way as for the mean time changes, the oscillations at the basic frequency are most noticeable in this graph.
We have shown that the low-frequency component of the motor vibrations (< 2 kHz) is described by the PNRP model, where the basic frequency is determined by the shaft rotation frequency. The first harmonics of the amplitude spectra of the mean function and variance time changes are dominant, allowing us to deduce that these vibrations result from the rotor imbalance. This implies that the defect affects the properties of both the deterministic and stochastic parts of the vibration. The condition of the motor can therefore be characterized by the indicators of the first (I1) and the second (I2) orders.(1 is defined as the ratio of the total power of the harmonics of the deterministic oscillations to the time-averaged power of the stochastic part. I2 is equal to the ratio of the sum of the amplitudes of the variance harmonics to its time-averaged value. It was found that in the case considered here, the numerical value of I1 considerably exceeded the value of I2. The values of the parameters describing the PNRP structure for the vibrations indicate that the motor’s condition is unsatisfactory. Further research involves analyzing the high-frequency component of the vibration signal spectrum.
The author(s) have not employed any Generative AI tools.
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