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Abstract

The mechanical equipment fault spectrum often contains random noise, and traditional spectrum estimation methods introduce large errors in the fault diagnosis of mechanical equipment including gas blower. Accurate frequency estimation of multi-frequency signals that are contaminated is a common problem. To address the fault spectral characteristics of gas blowers, a novel discrete Fourier transform interpolation algorithm is presented based on traditional practice. The algorithm combines the zero-complement technique and the main lobe fitting technique. The accuracy of the spectrum correction was improved greatly with the new algorithm, and it could effectively promote fault pattern recognition. Application examples and simulation results showed that the new algorithm is very successful in the fault pattern recognition of gas blowers. Under noise conditions, the proposed algorithm has a smaller estimation error than conventional algorithms and could achieve high precision, strong compatibility, and robustness. Theoretically, the algorithm has attained a higher level, and its application in engineering practice is also quite impressive due to its highly expected values.

Keywords

Gas blower interpolation algorithm spectrum correction zero-padding technology

Introduction

Device diagnosis technology is a comprehensive science that involves many disciplines such as machinery, sensing technology, computer technology, and signal processing technology. It relies on advanced sensor technology and online detection technology to collect all kinds of dynamic data. These data are then analyzed and processed to distinguish and confirm its abnormal performance, and its development trend could be further predicted. By identifying the causes of occurrence, location, and severity, specific maintenance measures may be put forward accordingly. With the development of science and technology, the requirements of working parameters for various types of machinery, such as running speed, carrying capacity, and working life, are ever increasing. As people are gaining more in-depth understanding of the hazards, more attention is being paid to diagnostic techniques. Fault pattern recognition of rotating machinery is a hot topic in the field of fault diagnosis. Most machinery and equipment are rotating machines, among which the gas blower is the most typical. Its normal operation is closely associated with smooth production in several industries, such as metallurgy, chemistry, coal, and steel.^1,2 Currently, fan-type rotating machinery is a type of equipment that is widely used in manufacturing. Some of them are the key components in production. Once these devices fail, irreparable economic losses for the enterprise will occur. Therefore, intelligent fault diagnosis technology has been widely used in its fault diagnosis.³ Many methods could be realized for fault pattern recognition, including neural networks, vibration signal processing, cluster analysis, and so on.^4–7 Unfortunately, their failure spectrums are characterized by a periodic length and a large random noise. To address this characteristic, a new spectrum estimation method is proposed in this article that improves on the conventional methods.

For the spectrum analysis, the discrete and continuous modes are both widely used in many areas; the former is the focus of this article. In signal processing, discrete spectrum analysis may achieve the transformation from the time domain to the frequency domain. This feature has been vigorously promoted in several fields including electronics, machinery, radar velocimetry, and laser Doppler. For instance, LH Benedict et al.⁸ proposed an estimation approach, which was based on the turbulent velocity spectrum aiming to analyze laser Doppler data. H-H Kim et al.⁹ applied the discrete Fourier transform (DFT) algorithm to arc fault detection. D Belega et al.¹⁰ introduced parametric estimation to interference suppression.^8–10 Nonetheless, these methods are held back by certain shortcomings, particularly the spectrum leakage problem, which still has not been solved properly. This is a desirable situation that would inevitably cause a greater error of frequency and phase in the calculated signal. Over the past 30 years, the development of the discrete spectrum method has yielded several algorithms, which can be roughly divided into three categories: ratio correction, energy-focus correction, and phase difference.

The theory of ratio correction (also known as interpolation) was first proposed in 1975 by JC Burgess¹¹ and has been successfully used in solving accurate measurement problems in the discrete higher harmonic signal parameters in the electricity field. For over 20 years, it has been repeatedly validated and widely used in engineering practice. In 1990, C Offelli and D Petri¹² proposed the energy-focus correction method, which was improved by Ding Kang and Xie Ming who developed a means for the three-point convolution amplitude correction, significantly improving the accuracy of the discrete spectrum analysis.^13,14 To solve the general problem of windowing functions, Santamria and Pantaleon employed the algorithm of phase difference to complete the single-frequency signal estimation. This is an algorithm that is applicable for adding any symmetrical window function, and notably has noise immunity.¹⁵ Despite the above features, different disadvantages of the above algorithms cannot be avoided, especially when random noise is present in the signal spectrum of mechanical failure, resulting in the low accuracy of their estimates.^16–18 The noise will also seriously affect the accuracy of the spectrum estimation. The frequency estimation error is larger than that without random noise under the condition of low signal-to-noise ratio (SNR). Therefore, a new DFT interpolation algorithm is proposed, which could be applied to gas blowers or other rotating machinery’s fault spectrum. Specific types of faults can finally be judged by the proposed method. In the new algorithm, the interpolation algorithm and zero-padding technique are used to solve the problem in which the real frequency of the signal falls between the two lines in the main lobe of the discrete spectrum to generate a larger frequency error, thus a more accurate spectrum is obtained. Moreover, for signals with random noise, the method can preprocess the input signal via the use of multiple autocorrelation operations and detect a weak useful signal submerged in the noise to improve the SNR. Even under noisy conditions, this methodology could still obtain accurate frequency. These advantages enable it to play an important role in the fields related to mechanical equipment fault diagnosis. The validity of the method is proven by means of computer simulation and numerical comparison, and then a comparative study with some traditional methods is conducted. Computer simulation results show that when working on the mechanical vibration frequency spectrum with random noise, the method presents an impressive anti-noise performance and high estimation accuracy. Its further application in other engineering fields may be highly expected. In this work, a new interpolation DFT algorithm is first proposed and then applied to obtain accurate estimates of fault signal in the scenario of random noise. The remaining sections are organized as follows. In section “Theoretical background,” a brief description of the traditional interpolation DFT algorithms is given, and then several symbols and formulas are defined for use in subsequent chapters. In section “Proposed algorithm,” the whole process of the new algorithm is deduced and extended to other classical window functions. In section “Experiment,” the proposed algorithm is applied to estimate the fault spectrum of a gas blower, followed by careful analysis on the estimation results. In section “Study of noise influence,” a comparative study of this proposed algorithm and the conventional interpolation DFT algorithms with respect to estimating the discrete spectrum is carried out. Finally, in section “Conclusion,” some of the main conclusions are summarized.

Theoretical background

To derive the algorithm theory on the discrete spectrum estimation, we assume a single-frequency cosine signal with Gaussian white noise, which is given in the form

$y (t) = A \cos (2 π f_{0} t + φ_{0}) + z (t)$ (1)

where t is a continuous time variable, $0 \leq t \leq T$ . The parameters A, f₀, and $φ_{0}$ , respectively, denote the amplitude, frequency, and the initial phase of the signal; z(t) is the white noise. After sampling at frequency f_s over the observation interval $N Δ t (Δ t = 1 / f_{s})$ , the following discrete cosine signal of N samples

$y (n) = A \cos (2 π \frac{f_{0}}{f_{s}} n + φ_{0}) + z (n), n = 0, 1, \dots, N - 1$ (2)

becomes available. To satisfy the Nyquist sampling theorem, it is assumed that f_s surpasses 2f₀. The frequency resolution is obtained by $Δ f = f_{s} / N$ . If the signal is asynchronously sampled, the normalized frequency $ψ_{0}$ lies between the two largest spectral lines. Therefore, $ψ_{0}$ can further be written as

$ψ_{0} = β_{w} + μ_{w}$ (3)

where $β_{w}$ and $μ_{w}$ ( $- 0.5 < μ_{w} \leq 0.5$ ) are, respectively, the integer part and the fractional part. If the SNR needs to be greater than a set threshold value, the value of $β_{w}$ can be determined using the window spectral peak searching algorithm. Additionally, $ψ_{0}$ represents the number of samples included in the periodic sinusoidal signal; its relationship with the sampling frequency could be defined as

$\frac{f_{0}}{f_{s}} = \frac{ψ_{0}}{N}$ (4)

The integer part $β_{w}$ can be readily and correctly determined by means of a maximum search routine, as long as the SNR is above the threshold. The weighted samples can be acquired when signal samples y(n) are multiplied by data window values w(n)

$y_{w} (n) = A \cos (2 π ψ_{0} n + φ_{0}) w (n)$ (5)

Then the signal added window undergoes an implement DFT, combined with equation (5); thus, equation (5) can further be expressed as

$Y_{w} (k) = \frac{A}{2} e^{j φ_{0}} W_{N} (k - ψ_{0}) + \frac{A}{2} e^{- j φ_{0}} W_{N} (k + ψ_{0})$ (6)

In equation (6), W_N(k) represents the window w(n) from the discrete time Fourier transform (DTFT). Putting equation (3) into equation (6), the largest magnitude would be acquired through

$| Y_{w} (β_{w}) | = \frac{A}{2} | e^{j φ_{0}} W_{N} (- μ_{w}) + e^{j φ_{0}} W_{N} (2 β_{w} + μ_{w}) |$ (7)

In equation (7), the second sign on the right side denotes leakage offered by the imaginary part of the spectrum. The conditions are postulated that $ψ_{0}$ meets the requirements of $ψ_{0} > 5$ and $ψ_{0} < N / 2 - 5$ to ensure that leakage caused by the spectrum’s negative part is the smallest and could be neglected.

Similarly, the second and the third largest spectral lines can also be determined. With proper combination of two or more spectral lines, a ratio $χ$ can be obtained which only depends on the selected window W_N(k) and the frequency deviation $μ_{w}$ . In other words, if the data window is known, $μ_{w}$ can be solely determined based on $χ$ . It could be shown as

$μ_{w} = h (χ)$ (8)

For maximum side-lobe decay windows,^{2,3,5,7,8,13,14} $μ_{w}$ can be written as a function of $χ$ in a simple and explicit form. For others, the polynomial approximation solution is suggested. In the next part, we will introduce an interpolation algorithm that is simple and can be applied to almost all classic windows.

Proposed algorithm

Description of proposed algorithm

This section aims to elaborate the advantages of zero-padding technology and the interpolation algorithm. A new means will be proposed so that the integral and fractional parts of the spectrum can be quickly determined to hit the target of correction. A reasonable choice of spectral estimation parameters (such as sample frequency f_s and sampling points N) may yield the desired frequency resolution of spectral estimation. The zero-padding concept is rooted in adding zero to the finite-length sequence of the original N points, so that the total length $L (L > N)$ is available for the spectrum correction. The discrete signal samples with zero added are expressed as

$y_{az} (n) = {\begin{matrix} y_{w} (n), 0 \leq n \leq N \\ 0, N \leq n \leq L \end{matrix}$ (9)

where L is the total signal length. If $λ = ((L - N) / (N))$ , then the discrete signal has a total of $L - N$ zeros that are added to obtain $λ$ spectral lines. Zero padding may help yield a finer structure of the spectrum to better estimate the effect of the spectrum.¹⁹ However, padding the data with zeros and computing a longer fast Fourier transform (FFT) does yield more frequency domain points; it does not improve the fundamental limit, nor does it alter the effects of aliasing error. The resolution limits are established by the observation interval and the sampling rate. No amount of zero padding can improve these basic limits, and the spectrum parameters, such as SNR and the spectrum leakage level, remain unchanged. The DFT of equation (9) could be written as

$Y (k) = \sum_{n = 0}^{N - 1} y_{az} (n) e^{- j \frac{2 π}{N} \frac{k}{λ} n}$ (10)

Compared with equation (4), the following formula is obtained

$Y (k) = Y_{w} (\frac{k}{λ})$ (11)

Similarly, equation (7) can be reformulated as

$| Y (β) | = \frac{A}{2} | e^{j φ_{0}} W_{N} (- μ) + e^{- j φ_{0}} W_{N} (\frac{2 β}{λ} + μ) |$ (12)

where $β$ and $μ$ are, respectively, the integer part and the fractional part of $λ ψ_{0}$ . Similarly, $β$ is returned by the maximum search routine of Y(k). At this stage, the three largest lines of Y(k) are given by $| Y (β) |$ , $| Y (β + 1) |$ , and $| Y (β - 1) |$ .

The second term on the right in equation (12) represents the contribution from the imaginary part in the spectrum. If $ψ_{0}$ satisfies that $5 < ψ_{0} < N / 2 - 5$ , the interference from the second term will be very small and can be neglected. With this assumption, equation (12) reduces to

$| Y (β) | ≅ \frac{A}{2} | W_{N} (- μ) |$ (13)

Similarly, we have $| Y (β + 1) | ≅ (A / 2) | W_{N} (σ - μ) |$ and $| Y (β - 1) | ≅ (A / 2) | W_{N} (σ + μ) |$ , where $σ = 1 / λ$ . Particularly, for the Hanning window, we have $| W_{N} (k) | ≅ \sin (π k) / h (k), h (k) = 2 π k$

$| Y (β) | ≅ \frac{A}{2} \frac{\sin (π μ)}{h (μ)}$ (14a)

$| Y (β + 1) | ≅ \frac{A}{2} \frac{\sin π (σ - μ)}{h (σ - μ)}$ (14b)

$| Y (β - 1) | ≅ \frac{A}{2} \frac{\sin π (σ + μ)}{h (σ + μ)}$ (14c)

Expanding the sine terms in equations (14b) and (14c) and combining them, we achieve that

$h (σ + μ) | Y (β - 1) | - h (σ - μ) | Y (β + 1) | ≅ A \sin (π μ) \cos (π σ)$ (15)

Combining equations (14a) and (15) yields

$h (σ + μ) | Y (β - 1) | + h (σ - μ) | Y (β + 1) | ≅ 2 | Y (β) | h (μ) \cos (π σ)$ (16)

We now introduce two variables $ε_{1}$ and $ε_{2}$ , defined as

$ε_{1} = \frac{| Y (β - 1) |}{| Y (β) |}$ (17a)

and

$ε_{2} = \frac{| Y (β + 1) |}{| Y (β) |}$ (17b)

For simplicity and conciseness, in the following parts, we replace “=” in equation (16) with “≅,” but remember that the approximation relationship still remains. Now, equation (16) can be rewritten as

$h (μ + σ) ε_{1} + h (μ - σ) ε_{2} ≅ 2 h (μ) \cos (π σ)$ (18)

Expanding the terms $h (σ + μ)$ , $h (μ - σ)$ , and $h (μ)$ , the value of $μ$ can finally be obtained as

$μ = \frac{σ (ε_{1} - ε_{2})}{\cos (π σ) - ε_{1} - ε_{2}}$ (19)

Because the value belongs to the interval $(- 0.5, 0.5]$ , the decimal part of the line frequency correction can eventually be obtained as follows

$\hat{μ} = \frac{σ (ε_{1} - ε_{2})}{\cos (π σ) - ε_{1} - ε_{2}}$ (20)

Extension for other classic windows with main lobe fitting

In the section above, we have deduced the interpolation for the Hanning window with zero padding. In this section, it will be extended for other classic windows with the main lobe fitting technique. Set

$X (k) = {[W_{O} (k)]}^{p} + W_{H} (k)$ (21)

where $W_{O} (k)$ denotes the normalized spectrum of any classic window, $W_{H} (k)$ denotes the normalized spectrum of the Hanning window, and $p = X L_{Han} / X L_{w (n)}$ . XL is the scalloping loss (SL) of the Hanning window and the corresponding window selected in equation (21). SL is defined as the ratio of coherent gain for a frequency component located halfway between DFT bins to the coherent gain for a frequency component located exactly at a DFT bin, indicated as

$SL = XL = \frac{W (0.5)}{W (0)}$ (22)

where $W (•)$ is the DTFT of the relevant window. It can be found that $| S (k) |$ is very small for most classic windows if k is in the range $(- 0.5, 0.5]$ . For example, the maximum value of $| S (k) |$ is less than $10^{- 5}$ for the Hanning window and less than $10^{- 4}$ for the Blackman window. This means that $[W_{O} (k {)]}^{p}$ and $W_{H} (k)$ fit well with each other in the middle of their main lobes. It is implied that the above interpolation algorithm can be extended for the classic window as long as k is limited in the range of $[- 0.5, 0.5]$ . With the zero-padding technique ( $λ \geq 3$ ), it is easy for the three largest spectral lines, which are used in the interpolation algorithm, to satisfy the requirement. Given the amount of computation and the radix-2 FFT algorithm, we often choose $λ = 4$ in practice measurement.

Experiment

The proposed algorithm is highly capable of spectrum estimation. In this section, the gas blower of a steel plant was selected as the research object to validate the speed and accuracy in mechanical fault diagnosis. Gas blowers may encounter different types of faults; one of the most common faults is shaft misalignment. Statistics show that 60% of the rotating mechanical faults are related to shaft misalignment. In this article, fault identification was performed in the case of shaft misalignment using the proposed method to determine the effectiveness of the new algorithm in practical application. To truly evaluate the ability to estimate the frequency deviation, the new algorithm was validated using the programming tools MATLAB 7.0 and VC 6.0. The normalized frequency deviation $μ$ was limited in the range of $(- 0.5, 0.5)$ , with a scanning step of 0.05 Hz. For the convenience of research, the sampling frequency f_s was 1024 Hz and sampling points N equaled 1024. A gas blower was taken as an example to verify the validity of the new algorithm. In the experiment, a D450-type gas blower was used. First, some acceleration sensors and speed sensors were placed to measure the vibration signals of the gas blower; then the HY-106 patrol instrument was used to collect the vibration data of the gas blower. Because the vibration between the gas blower and the reducer is considerable, a failure is likely to occur; thus, the sensor is mounted on the casing near the reducer side. Figure 1 shows a diagram of the measuring points of an acceleration sensor in collecting gas blower vibration.

Figure 1.

Measurement site.

After a period of time, the fault spectrum on bush no. 3 of the gas blower obtained from the acceleration sensor is shown in Figure 2.

Figure 2.

Time-domain waveform of bearing bush no. 3.

The time-domain waveform of the fault was estimated by the new algorithm; the frequency domain waveform is shown in Figure 3.

Figure 3.

Estimated frequency spectrum of bearing bush no. 3.

Figure 3 shows that the fault characteristic frequency is 184.5 Hz, exactly three times the blower speed frequency (f = 61.5 Hz), and the shaft misalignment fault frequency is exactly 185 Hz. This analysis presented a preliminary result that a shaft misalignment fault occurred in the gas blower. Then workers were arranged to open the gas blower. The on-site inspection revealed that no. 2 and no. 3 shafts were indeed in misalignment. After the correction measure was taken, the fault phenomenon disappeared. It is now believed that the new algorithm has higher precision in terms of the spectral estimation of mechanical failure.

Study of noise influence

In the frequency correction, estimation accuracy under noisy conditions is an important indicator of the superiority of an algorithm. When noise is present and kept at a higher level, even if the SNR is set high enough to be much larger than the threshold value, the traditional methodologies for frequency correction may still estimate the correct spectral line in the wrong position. This occurrence in spectrum correction is often referred to as an incorrect polarity estimate (IPE). For example, the near-coherent sampling in two-point algorithms and nearly half of the cycle in three-point algorithms are prone to the IPE phenomenon. Regarding the occurrence of the IPE phenomenon, not only the signal inversion results but also the value $\hat{μ}$ will be greatly affected.

Fault signals of machinery and equipment often contain much noise. To verify the performance of the new algorithm against additive noise, this section will study the new algorithm in comparison with the conventional method. Before the comparative study was performed, it was assumed that there was a theoretical signal with additive noise, where the SNR was set as −5 dB, so that the correct polarity estimates may occur (IPE phenomenon). We set the step length as 0.025 Hz, the scanning frequency ranged from 255.5 to 256.5 Hz, and the random phase was uniformly distributed in a range of $[- π, π]$ . In practical measurement, the theoretical signal is usually interfered with by the noise caused by various uncontrollable factors. The estimation accuracy may be greatly reduced and the frequency estimation becomes a random variable. To compare the sensitivity of the new method to white noise, a signal with additive zero mean Gaussian noise was generated, and the noise level was adjusted to obtain the test signal. For each frequency, 50,000 separate instances were generated accordingly. Figure 4 shows the standard deviation error and the mean absolute error of both the proposed algorithm and the conventional algorithm in handling the frequency deviation function in the Hanning window.

Figure 4.

Average absolute errors of various algorithms at the Hanning window (SNR = −5 dB).

Figure 4 exhibits three conventional algorithms with high estimation accuracy under additive noise conditions. As $| μ |$ increased, the frequency correction errors of these three kinds of algorithms (including the standard deviation error and the average absolute error) increased. When the value $μ$ ranged in $(- 0.5, 0.5)$ , the Ding Kang algorithm presented an estimation error that topped all algorithms under noise conditions. The error curve first increased to the highest point and then continued to fall, finally staying at a higher level. Remarkably, the newly proposed algorithm, when $μ$ was valued in the scope of $(- 0.5, 0.5)$ , had estimation error values that were almost equal, except for a few fluctuations, and ranked the least among all estimating results. To more intuitively show the above algorithm’s performance, the extreme values of the frequency error for each algorithm at the Hanning window are shown in Table 1.

Table 1.

The extreme values of the frequency error for each algorithm at the Hanning window.

Various methods	Minimum frequency error	Maximum frequency error
Ding Kang	0.0182	0.0363
Xie Ming	0.0173	0.0271
Belega	0.0184	0.0218
New algorithm	0.0115	0.0183

Figures 5 and 6 show the estimated standard error of Hanning window and Blackman window, respectively. As seen from Figures 5 and 6, including the proposed algorithms, the results of each method at the Hanning window are basically the same as the simulation results in Figure 4.

Figure 5.

Frequency error correction at Hanning window.

Figure 6.

Frequency error correction of each method at Blackman window.

In summary, the simulation study shows that the conventional algorithms worked with low estimation errors under noisy conditions. For the standard deviation of the estimation error, the new algorithm is ranked the lowest of all algorithms, demonstrating its excellent anti-IPE performance. Additionally, the new algorithm may include other advantages. For conventional algorithms, the line spacing is defined as the frequency resolution; thus, the noise correlation may be significantly weakened. Moreover, the spectrum of this algorithm can be more powerful, where its spectral interval is only $1 / λ$ of the frequency resolution. Therefore, this may contain similar noises in adjacent lines, making it able to eliminate most of the noises.

Conclusion

Aiming at the fault spectral characteristics of a gas blower with random noise, based on zero-padding technology and the main lobe fitting techniques, a new frequency correction methodology was proposed based on interpolation DFT. Being simple, fast, and practical, the methodology is compatible with many other classic window functions. It does not need to know the spectral data window function or make any prior calculations. The cost is an increase in computation of FFT. Through computer simulation, the effectiveness of the new algorithm and conventional algorithms was compared and validated for a variety of window functions. Analysis of application instances showed that the new algorithm could be effectively used for the accurate correction of the fault spectrums of a gas blower and similar rotating machinery. Finally, the issue of spectrum correction error under noise conditions for the new algorithm and the conventional algorithms was also discussed. A comparative study and simulation results showed that the new algorithm presents less estimation error in the presence of noise and has a greater robustness against IPE. In summary, the new algorithm may be more powerful in resisting additive noise.

Footnotes

Academic Editor: Chi-man Vong

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) received no financial support for the research,authorship,and/or publication of this article.

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