A new real-time signal processing approach for frequency-varying machinery

Abstract

Development of condition monitoring approaches has played a key role in the stability and safety of frequency-varying machinery operations. Conventional time–frequency analysis methods suffer problems such as analysis results being too complex to realize highly intelligent and automated condition monitoring systems. Blind source separation is an attractive tool due to its excellent performance in separating defect source signals from their mixtures without detailed knowledge of sources and mixing processes; however, it can only be applied under some strict conditions. In this paper, a nonuniform sampling model is built and a new processing algorithm of frequency-varying signal is proposed. The relationship between the power spectral density (PSD) of the vibration signal of frequency-varying machinery and frequencies at different rotational speeds is derived. The proposed method can adaptively eliminate the influence of the varying rotational speed in the revised PSD. Some classical signal analysis methods are implemented to compare with the proposed approach by simulations. An experiment has been conducted by using a JD-1 wheel/rail simulation facility to illustrate the effectiveness of the proposed method.

Keywords

Condition monitoring frequency-varying signal processing nonuniform sampling model wheel/rail simulation experiment

1. Introduction

Condition monitoring of rotating machinery has been a subject of intense academic and practical research for many years (Kankar et al., 2011; Zhang et al., 2015). Many techniques such as acoustics, magnetic fields, eddy currents, and thermal fields have been utilized in the past. Vibration analysis is a common approach for condition monitoring. It is based on the idea that the rotating machinery has a specific vibration signature under standard conditions, and this signature changes with the development of damage (Dong and Wei, 2014; Rohit et al., 2015).

Many solutions proposed in the literature require a constant frequency rotation since the characteristic fault frequencies are directly proportional to the rotating speed. In early studies, the main techniques used for condition monitoring were probability distribution characteristics such as skewness, kurtosis, and Fourier spectrum (Antoni, 2006). These methods have been proven under conditions including steady rotating frequency.

However, general operation of rotating machinery often involves varying motor speed, and in a frequency-varying application the fault characteristic frequencies change as instantaneously as the rotational frequency does (Blodt et al., 2008). Therefore, the Fourier spectrum cannot be used since it simply expands a signal as a linear combination of single frequencies that exist over time. This drawback is the motivation for greater focus on the rapidly developing approaches, time–frequency analysis and blind source separation (BSS), for condition monitoring of rotating machinery.

Each of the time–frequency analysis methods, such as short time Fourier transform (STFT; Djurović and Stanković, 2012), Wigner–Ville distribution (WVD; Stankovic et al., 2014), empirical mode decomposition (EMD; Masoud and Ali, 2012), and wavelet transform (Mckay et al., 2013), has some problems. STFT is only suitable for quasi-stationary signal analysis, and since there is no orthogonal base for STFT, it is difficult to find a fast algorithm to calculate STFT. WVD has good concentration in the time–frequency plane. However, even if support areas of the signal do not overlap each other, interference terms will appear on the time–frequency plane which will mislead the signal analysis. Many improved methods have been proposed to overcome the disadvantages of WVD, such as the Choi–Williams distribution (Wang et al., 2015) and S transform (Jaya et al., 2013), but their algorithms are highly inefficient. EMD is considered an unstable technique, since mode mixing can cause perturbation and result in a set of distorted intrinsic mode functions (IMFs), as reported by Antoniadou et al. (2013). The choice of mother wavelets might influence the results of the wavelet transform. In addition, the complexity of time–frequency analysis results needs human intervention to find the fault features from huge amounts of data. It is difficult to create highly intelligent and automated condition monitoring.

BSS is an attractive tool for condition monitoring due to its excellent performance in separating defective source signals from their mixtures without detailed knowledge of sources and mixing processes (Arnaut and Obiekezie, 2014; Acharjee et al., 2015; Bertrand, 2015; Gu et al., 2015; Zhang et al., 2016). However, BSS is limited to one source with a Gaussian distribution, and sometimes this is incompatible in actual condition monitoring cases. This is due to the central limit theorem which states that the Fourier coefficients quickly tend to Gaussian distribution even if the source signals are non-Gaussian.

It is difficult to state all the advantages and disadvantages of these analysis methods because of their different theoretical backgrounds. It is also reasonable to claim that for different situations different methods might work better than the others. The object of this paper is to propose a simple, fast, and accurate method to deal with condition monitoring problems of frequency-varying rotating machinery.

In Section 2, a nonuniform sampling model is built for deriving the relation between the rotational speed of the machinery and the statistical character of the vibration signal. From this a new processing algorithm of a frequency-varying signal is proposed. The validity of formulas is demonstrated by numeric simulations. The conventional time–frequency signal analysis methods, i.e. Choi–Williams time–frequency distribution and EMD, are implemented to compare with the proposed approach.

In Section 3, an experiment is described using a JD-1 wheel/rail simulation facility (Wang et al., 2011) to illustrate the effectiveness of the proposed method. The experiment’s data analysis results show that the proposed method is effective and obtains better condition monitoring results than conventional signal processing methods.

2. A new processing algorithm for a frequency-varying signal

2.1. Nonuniform sampling model

Define the nonuniform sampling times ${t_{n}, n = 0 : N - 1}$ , for all nonnegative integers $k,$ $t_{k} < t_{k + 1}$ . The sampling intervals of the nonuniform sampling times ${ζ_{k} = t_{k + 1} - t_{k}}$ are mutually independent and identically distributed variables. For positive finite valued M, $M \geq ζ_{k} > 0$ . Figure 1 illustrates the nonuniform sampling model. Each pulse width is d, the falling edges of the pulses are the sampling times ${t_{i}}$ , and the sampled values are ${x (t_{i}) / d}$ .

Figure 1.

Nonuniform sampling model.

Define the nonuniform sampled signal model as follows: where the sampling pulse $p (t)$ is defined as

p (t) = {\frac{1}{d}, 0 \leq t_{i} - t \leq d 0, else

(2)

In practical applications the width of sampling pulse $d \to 0$ ; therefore, the nonuniform sampled signal can be rewritten as

x_{s} (t) = \sum_{i} x (t) δ (t - t_{i}) = {lim}_{d \to 0} z (t)

(3)

The length of the n th sampling point after time t is $L_{n} (t)$ , as shown in Figure 2.

Figure 2.

Distance distribution of the nonuniform sampling model.

The distribution function of $L_{n} (t)$ is defined as

G_{n} (a, t) = P [L_{n} (t) \leq a]

where

P [\cdot]

means probability function.

Define β as the average number of the nonuniform sampling points per unit time

β = {lim}_{a \to 0^{+}} \sum_{n = 1}^{\infty} \frac{G_{n} (a, t)}{a}

(4)

Equation (4) means that the average length of n successive sampling intervals is n times the average length of one sampling interval.

In this nonuniform sampling model, the distribution function $G_{n} (a, t)$ possesses the following statistical characteristic

{lim}_{a \to 0^{+}} \frac{G_{2} (a, t)}{a} = 0

which means the probability of finding two or more points in a “small” interval is negligible.

The event $K_{n} (t, τ)$ is defined with t and $t + τ$ as sampling times, and there are n points located in the sampling interval $(t, t + τ)$ , i.e.

K_{n} (t, τ) = {(L_{1} (t) \leq d) \cap (L_{1} (t + τ) \leq d) \cap thereare n pointsbetween (t, t + τ)}

The autocorrelation function of ${x (t_{n})}$ is represented as

r (n) = E [x (t_{m}) x (t_{m + n})]

(5)

where

E [\cdot]

means mathematics expectation.

From equations (3) and (5), the autocorrelation function of the nonuniform sampled sequence is given by

R_{s} (τ) = E [x_{s} (t + τ) x_{s} (t)] = {lim}_{d \to 0} E [\frac{x (t_{m + n})}{d} \frac{x (t_{m})}{d}] P [K_{n} (t, τ)] = {lim}_{d \to 0} \frac{1}{d^{2}} \sum_{n = 0}^{\infty} r (n) P [K_{n} (t, τ)]

(6)

Now we discuss the autocorrelation function according to the value of τ.

While $0 \leq τ \leq d$

As far as $0 \leq τ \leq d$ , there is $n = 0$ . The event $K_{0} (t, τ)$ denotes that t and $t + τ$ are sampling times, and there are no sampling points in the interval $(t, t + τ) .$ That is to say, the time t and the time $t + τ$ lie in the one sampling plus. Referring to Figure 2, the probability function of event $K_{0} (t, τ)$ is given by

P [K_{0} (t, τ)] = P [τ \leq L_{1} (t) \leq d] = G_{1} (d, t) - G_{1} (τ, t)

Equation (6) can now be written as

{lim}_{d \to 0} \frac{1}{d^{2}} r (0) P [K_{0} (t, τ)] = {lim}_{d \to 0} r (0) \frac{G_{1} (d, t) - G_{1} (τ, t)}{d^{2}}

(7)

Let $n = 1$ , a equal d and τ in equation (4). Therefore,

{lim}_{d \to 0} G_{1} (d, t) = β d, {lim}_{τ \to 0} G_{1} (τ, t) = β τ

(8)

From equations (7) and (8), we obtain

{lim}_{d \to 0} \frac{G_{1} (d, t) - G_{1} (τ, t)}{d^{2}} = {lim}_{d \to 0} \frac{β (d - τ)}{d^{2}} |_{0 \leq τ \leq d}

which is one of the expressions for the delta function

δ (τ)

; therefore equation (7) can be written as

{lim}_{d \to 0} \frac{1}{d^{2}} r (0) P [K_{0} (t, τ)] = {lim}_{d \to 0} r (0) \frac{β (d - τ)}{d^{2}} |_{0 \leq τ \leq d} = r (0) β δ (τ)

(9)

While $τ > d$

As far as $τ > d$ , there is $n \geq 1$ . The probability function of event $K_{n} (t, τ)$ is as follows

P [K_{n} (t, τ)] = P [(0 \leq L_{1} (t) \leq d) \cap τ \leq L_{n + 1} (t) \leq τ + d] = P [0 \leq L_{1} (t) \leq d] \cdot P [τ - d \leq L_{n + 1} (t) - L_{1} (t) \leq τ + d]

(10)

Therefore, we have

\frac{1}{d^{2}} {lim}_{d \to 0} P [K_{n} (t, τ)] = {lim}_{d \to 0} \frac{1}{d^{2}} G_{1} (d, t) P [τ - d \leq L_{n + 1} (t) - L_{1} (t) \leq τ + d] = β {lim}_{d \to 0} \frac{1}{d} P [τ - d \leq L_{n + 1} (t) - L_{1} (t) \leq τ + d]

(11)

Defining $y_{n} (t) = L_{n + 1} (t) - L_{1} (t) = \sum_{i = 1}^{n} ζ_{i}$ as the length of the n successive sampling intervals and $f_{y}^{n} (t)$ as the probability density function of $y_{n} (t)$ , equation (11) can be rewritten as

β {lim}_{d \to 0} \frac{1}{d} \int_{τ - d}^{τ + d} f_{y}^{n} (t) d t = 2 β f_{y}^{n} (τ)

(12)

Combining equations (9) and (12) with equation (6), the autocorrelation function of the nonuniform sampled signal is given by

R_{s} (τ) = β [r (0) δ (τ) + 2 \sum_{n = 1}^{\infty} r (n) f_{y}^{n} (τ)]

(13)

The power spectral density (PSD) is a Fourier transform of the autocorrelation. Performing the Fourier transform of equation (13) and letting $F_{y}^{n} (ω) = \int_{- \infty}^{+ \infty} f_{y}^{n} (τ) e^{- j ω τ} d τ$ , the PSD of the nonuniform sampled signal $x_{s} (t)$ is

S_{s} (ω) = β \cdot r (0) + β \sum_{| n | = 1}^{\infty} r (n) F_{y}^{n} (ω)

(14)

Noticing that $F_{y}^{n} (ω) = E [e^{- j ω y_{n} (t)}]$ , we have

F_{y}^{n} (ω) = E [e^{- j ω y_{n} (t)}] = E [exp (- j ω \sum_{i = 1}^{n} ζ_{i})] = E [Π_{i = 1}^{n} exp (- j ω ζ_{i})]

(15)

By letting the nonuniform sampling interval ${ζ_{i}}$ be replaced by a random variable ζ, equation (15) can be rewritten as

F_{y}^{n} (ω) = E {[e^{- j ω ζ}]^{n}} = E [e^{- j n ω ζ}]

(16)

Combining equation (14) with equation (16), the PSD of the nonuniform sampling sequence is as follows

S_{s} (ω) = β \cdot r (0) + β \sum_{n \neq 0} r (n) E [e^{- j n ω ζ}] = β \sum_{n = - \infty}^{+ \infty} r (n) E [e^{- j n ω ζ}]

(17)

The PSD of the conventional uniform sampling sequence is

S_{s} (ω) = \sum_{n = - \infty}^{+ \infty} r (n) e^{- j n ω}

(18)

A high degree of similarity can be seen by comparing the proposed PSD (equation (17)) of the nonuniform sampling process with the PSD (equation (18)) of the uniform sampling case. In equation (17), if we let ${ζ_{i} = T}$ and therefore $β = 1 / T$ , where T is the sampling interval, then equation (17) changes into

S_{s} (ω) = β \sum_{n = - \infty}^{+ \infty} r (n) E [e^{- j n ω ζ}] = \frac{1}{T} \sum_{n = - \infty}^{+ \infty} r (n) e^{- j n ω T}

(19)

When the sampling interval T equals one second, equation (19) = equation (18), and it is obvious that the proposed PSD of the nonuniform sampling process covers the special case of uniform sampling. Therefore, the accuracy of the proposed PSD is valid.

2.2. Frequency-varying signal processing

In this subsection we propose a new processing algorithm for the frequency-varying signal based on the nonuniform sampling model.

Let $f_{ζ} (ζ)$ denote the probability density function of the sampling interval ζ. In equation (17) there is

E [e^{- j n ω ζ}] = \int_{- \infty}^{+ \infty} e^{- j n ω ζ} f_{ζ} (ζ) d ζ

(20)

In order to eliminate the spectrum distortion caused by variable frequency of the rotating machine, the sampling interval ζ should decrease with the increment of rotating speed v. That is to say, ζ is inversed to v, $ζ = p / v$ (where $p$ is a positive constant), and both $ζ (v)$ and its inverse function $v (ζ)$ are single-valued throughout the useful range of ζ.

Let $f_{v} (v)$ denote the probability density function of the rotating speed v. Notice the relationship between the probability density function $f_{ζ} (ζ)$ and the probability density function $f_{v} (v)$

f_{ζ} (ζ) = f_{v} (v) | \frac{d v}{d ζ} |

Therefore, equation (20) can be rewritten as follows

E [e^{- j n ω ζ}] = \int_{v (- \infty)}^{v (+ \infty)} e^{- j n ω p / v} f_{v} (v) d v

(21)

Then calculate the probability density function $f_{v} (v)$ according to the records of the instantaneous rotating speed v_i. The useful range of rotating speed $[v_{l}, v_{h}]$ is uniformly divided into K shares, where K is a positive constant. Assuming the total number of sample points is N, and there are N_i points of instantaneous rotating speed lying in the i th speed section. Therefore, the discrete estimate of PSD $f_{v} (v)$ of i th speed section is

p_{i} = \frac{N_{i} K}{N (v_{h} - v_{l})}

(22)

The estimate is related to the number of speed sections divided by K. Our studies show that the optimal estimation results can be obtained if the value of K makes each section possess instantaneous rotating speed data.

Combining equation (17) with equation (21), we propose a novel, simple, and fast algorithm for the revised PSD to deal with real-time vibration analysis of frequency-varying machinery

S (ω) = β \sum_{n = 1}^{N} \sum_{i = 1}^{K} r (n) e^{- j n ω p / v_{i}} \frac{N_{i} K}{N (v_{h} - v_{l})}

(23)

As far as the vibration condition monitoring system, the energy centrobaric spectrum is used as the indicator to reflect the machinery condition, which is defined as follows

FC = \frac{\sum_{i = 1}^{N} ω_{i} S_{i}}{\sum_{i = 1}^{N} S_{i}}

(24)

where

ω_{i}

is the frequency value of i th revised PSD component and S_i is the value of i th revised PSD component.

Corresponding to different wear conditions, rotating machinery has different vibration signatures that largely affect the spectrum centrobaric. Therefore, the energy centrobaric spectrum can be calculated by the proposed method to supply a rapid alert in condition monitoring of rotating machinery.

2.3. Simulation

In this subsection, the conventional signal analysis methods and the proposed algorithm are used to analyze the frequency-varying signal as follows

x (t) = f (t) + ε (t) = \sin (13 π t^{0.5}) + \sin (29 π t^{0.5}) + \sin (37 π t^{0.5}) + ε (t)

where

f (t)

is a combined signal of three frequency-varying sine signals, and

ε (t)

is a Gaussian random signal.

$f (t)$ , $x (t)$ , and its conventional PSD, $X (ω)$ , are shown in Figure 3. Obviously the conventional PSD fails to reveal the essential characteristic hidden in the frequency-varying signal $x (t)$ .

Figure 3.

Frequency-varying signal $f (t)$ , combined signal $x (t)$ of $f (t)$ and a Gaussian random signal, and conventional PSD of $x (t)$ .

Then, the EMD of the frequency-varying signal $x (t)$ is performed, and the result is shown in Figure 4. Affected by frequency variance, there are nine distorted periodic components separated from the signal $x (t)$ . This again illustrates the limitation of the conventional signal analysis method.

Figure 4.

Processing result of signal $x (t)$ by EMD.

Next, the performance results of the Choi–Williams time–frequency analysis for the frequency-varying signal $x (t)$ are shown in Figure 5. The analysis took a lot of time, and can only vaguely identify the frequency-varying phenomenon from the analysis result. In practical applications, the analyzed signal is unknown so it is very difficult to reveal the essential characteristic from Figure 5.

Figure 5.

Choi–Williams time–frequency distribution of signal $x (t)$ .

Finally, the proposed algorithm is used to analyze the frequency-varying signal $x (t)$ . The result is shown in Figure 6. The three main frequency compositions can be recognized distinctly from Figure 6. The influence of the distortion spectrum from frequency variance has been eliminated, and the essential characteristic can be observed clearly from the processing result.

Figure 6.

Processing result of signal $x (t)$ by proposed method.

3. Experiments

All experiments for condition monitoring of rotating machinery with varying frequency are conducted on a JD-1 wheel/rail simulation facility (Wang et al., 2011), as shown in Figure 7.

Figure 7.

JD-1 wheel/rail simulation facility.

The tester is composed of a small wheel roller and a large wheel roller which are driven by DC motors A and B, respectively. Motor speed imposed on the rollers can be controlled accurately.

The experimental purpose is vibration monitoring and signal analysis when the small wheel rotates in uniform and varying speed, respectively. An acceleration sensor is used to measure longitudinal vibration signals. The small wheel is made as shown in Figure 8, and a cavity is machined on this wheel, as shown in Figure 9.

Figure 8.

Small wheel.

Figure 9.

Wheel cavity.

First, normal and faulty wheels rotate at a uniform speed of 35 r/min. The vibration signals $x_{0} (t)$ of the normal wheel and $x_{1} (t)$ of the faulty wheel are shown in Figure 10. $X_{0} (ω)$ and $X_{1} (ω)$ are PSDs of the normal and the faulty wheels, respectively, as shown in Figure 11.

Figure 10.

Vibration signal $x_{0} (t)$ of the normal wheel and $x_{1} (t)$ of the faulty wheel in uniform speed.

Figure 11.

PSD $X_{0} (ω)$ of the normal wheel and PSD $X_{1} (ω)$ of the faulty wheel in uniform speed.

$X_{1} (ω)$ denotes the actual spectrum characteristic of the faulty wheel which contains the dominant rotational frequency component, the fault feature frequency component, and the high frequency random component due to surface irregularities.

Then the faculty wheel rotates with varying speeds of 35 r/min, 56 r/min, and 98 r/min, sequentially. The vibration signal $x_{2} (t)$ is shown in Figure 12. The PSD $X_{2} (ω)$ of the varying speed signal $x_{2} (t)$ is shown in Figure 13. It is obvious that the varying speed leads to distortion of the signal spectrum, which doesn’t reflect the actual condition of the small wheel.

Figure 12.

Vibration signal $x_{2} (t)$ in varying speed.

Figure 13.

PSD $X_{2} (ω)$ of vibration signal $x_{2} (t)$ .

Finally, the proposed algorithm is used to analyze the frequency-varying vibration signal $x_{2} (t)$ . The revised PSD $X_{3} (ω)$ by the proposed algorithm is shown in Figure 14. Under varying rotation speed, the vibration signal is nonstationary and its statistic characteristics vary with time, making it difficult to detect any defect frequencies. For such a signal, conventional PSD doesn’t provide accurate information for the spectrum of the signal due to the non-constant speed, as shown in Figure 13. Comparing the revised PSD $X_{3} (ω)$ of the frequency-varying signals in Figure 14 with the uniform frequency case in Figure 11, the proposed approach can effectively correct the changing trends of frequency components as well as their contributions, and the revised PSD determined by the proposed method consistent well with the real condition characteristic. The influence of the distortion spectrum relating to and caused by the varying rotational speed has been adaptively eliminated. Therefore, the defect features of the small wheel can be extracted clearly from the revised PSD. The experimental data analysis result verifies the performance of the proposed method in dealing with nonstationary problems.

Figure 14.

Revised PSD $X_{3} (ω)$ by proposed method.

4. Conclusion

Vibration monitoring and real-time calculation of machinery condition are beneficial to the stable operation of machinery and personal safety of operators. As far as frequency-varying machinery is concerned, the conventional signal analysis methods experience problems. At a constant rotational speed, the PSD can reflect the true defect characteristic of machinery. When the rotational speed is varying, the nonstationary signal in time domain generates adequate frequency components in frequency domain.

This paper presents a nonuniform sampling model and derives a revised PSD algorithm. The relationship between the PSD of vibration signal of frequency-varying machinery and frequencies at different rotational speed is derived. The proposed revised PSD expression is essentially a type of adaptive filter algorithm, which can adaptively eliminate the influence of the varying rotational speed in revised PSD. Simulations illustrate the applicability of our approach.

Vibration monitoring experiments of varying frequencies are conducted using a JD-1 wheel/rail simulation facility. The condition monitoring performance of the proposed method is successfully tested. The experimental data analysis results illustrate the effectiveness of the proposed method. The proposed method can simply and quickly determine if a fault exists, and accurately reflect the true defect characteristic.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant numbers 51205323, 51275426, and 51475393) and the State Scholarship Fund of China.

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