Research on tacholess fast-varying instantaneous rotating frequency estimation model based on nonlinear short-time Fourier transform combination with adaptive chirp mode decomposition and its application

Abstract

Instantaneous rotating frequency extraction is one of the key technologies for mechanical health monitoring and fault diagnosis. As the instantaneous rotating frequency presents fast-varying property under high-speed fluctuation, this paper uses a coarse-to-fine strategy to propose a tacholess fast-varying instantaneous rotating frequency estimation model based on nonlinear short-time Fourier transform (NLSTFT) combination with adaptive chirp mode decomposition (ACMD). By self-adaptive matching and decomposing the vibration signal based on its time-frequency distribution, it increases the energy aggregation of time-frequency representation, which not only improves computational efficiency but also avoids the spectral ambiguity problem. As a result, the proposed model is very suitable for extracting instantaneous rotating frequency under severe speed fluctuation; simulation signal and rolling bearing fault vibration signal also validate this conclusion. Furthermore, by integrating with signal decomposition technology, various order components of fault vibration signal can also be self-adaptive extracted.

Keywords

Nonlinear short-time Fourier transform adaptive chirp mode decomposition instantaneous rotating frequency estimation high-speed fluctuation order component extraction fault diagnosis

Introduction

Rotating machinery often works under the variable speed and non-stationary load condition, and its vibration signal presents fast-varying instantaneous rotational frequency and phase function drastically. Besides, non-stationary working condition also leads the vibration signal to produce complex amplitude modulation and frequency modulation phenomenon, which brings difficulties to diagnose fault by using traditional diagnostic methods.^1,2 Considering the instantaneous rotational frequency is not only an important characteristic parameter reflecting the operating state but also a necessary parameter to implement a series of fault diagnosis methods such as order tracking, time-varying filtering, and generalized demodulation analysis, and how to achieve accurate estimation of fast-varying instantaneous rotational frequency is significant for health assessment and early fault diagnosis under unsteady operating condition consequently.^3–5

Traditionally precise measurement methods of instantaneous rotational frequency often employ speedometer combined with displacement sensors. In recent years, optical encoders have been widely used, and above technologies can generally meet current accuracy requirement. However, in many application fields, it is difficult to install these devices, thus failing to obtain accurate instantaneous rotating frequency, which poses challenges for assessing and diagnosing the healthy condition, as a result, it is necessary to precisely estimate instantaneous rotating frequency by solely processing and analyzing vibration signal in the absence of testing device. Robert Gao⁵ combined narrowband phase demodulation and Hilbert transform technology to estimate the instantaneous rotating speed of gearbox equipment. Gryllias K, Gomez JL^6,7 employed short-time scale transformation method to process vibration signal and further extracted fluctuating information of instantaneous rotational frequency. However, these aforementioned methods are limited to the case of finite rotational speed fluctuation. Addressing these shortcomings, Matthieu Kowalski and Gelman Len^8,9 proposed an instantaneous rotating frequency estimation method based on harmonic signal decomposition, transforming the estimation method into an eigenvalue extraction issue. Yet, this method exhibits high sensitivity to background noise.

Time-frequency analysis methods offer a new approach for estimating instantaneous rotating frequency, such that short-time Fourier transform (STFT) combination with ridge line searching strategy has been widely applied on mechanical equipment monitoring and diagnostics.^10–12 However, the effectiveness of STFT depends on the assumption of local stationary, which is not suitable for rapidly varying instantaneous rotating speed condition. Considering time-frequency reassignment methods involved in reprocessing time-frequency transformation and rearranging the distribution of time-frequency energy to enhance time-frequency localization,¹³ it can be utilized to instantaneous rotating frequency estimation. On the other hand, traditional time-frequency reassignment methods lack reversibility and therefore cannot reconstruct signal in the time-domain. In contrast, synchronous transform methods rearrange signal’s time-frequency representation solely in the frequency direction, providing a reconstructive property.^14,15 However, it neglects the signal’s time-frequency energy distribution in the time direction, thereby limiting its applicability on vibration signal possessing a slowly varying instantaneous rotating frequency.^16–18 Recently, Chen¹⁹ introduced a nonlinear frequency modulation mode decomposition method, by utilizing demodulation technology, this method transforms broadband nonlinear frequency modulated signal into narrowband signal. Combination with an optimization algorithm, a series of order components can be simultaneously extracted. However, there are still some issues that need to be addressed, such that the instantaneous frequency (IF) need to be required in advance for extracting corresponding order component, the ability to self-adaptive update bandwidth coefficients is missing. Therefore, the Adaptive Chirp Mode Decomposition (ACMD) method^20,21 has been developed to address these challenges.

Based on aforementioned studies, in the high-speed fluctuation field, this paper utilizes a combination of nonlinear short-time Fourier transform (NLSTFT) and adaptive chirp mode decomposition (ACMD) to propose an instantaneous rotating frequency (IRF) estimation, and apply it to extract order components. In this proposed model, NSTFT provides a time-frequency spectrum of vibration signal, and ridge detection algorithm based on the spectrum presents an IRF information, thus minimizing human interference. Using the obtained initial IRF, ACMD constructs a matching IRF estimation operator according to the modulation property of vibration signal, given the highest energy attribute in time-frequency spectrum corresponding to IRF, and this approach enables a precise IRF estimation. After K iterations, ACMD decomposes a series of order components and their instantaneous amplitudes (IAs). Simulation and experimental validation demonstrate the effectiveness on estimating rapidly varying IRF, compared with other traditional methods, and it has a superior noise resistance performance. Finally, integrating signal decomposition technology, it further allows to extract a series of order components from fault vibration signal.

Theoretical foundation

Nonlinear short-time Fourier transform (NLSTFT)

NLSTFT serves as a fresh time - frequency analysis technique for handling non - stationary signals, considering mechanical fault vibration signal under large rotating speed fluctuation field presents fast-varying IRF, modulation property, it introduces a time-varying demodulation operator $e^{- i c (t) {(u - t)}^{2} / 2}$ based on linear short-time Fourier transform, $c (t)$ symbolizes the IF’s first-order derivative. If the demodulation operator $e^{- i c (t) {(u - t)}^{2} / 2}$ is consistent with modulation component in fault vibration signal, it can effectively improve time-frequency aggregation. As a result, it is proper to demonstrate the time-frequency property under large rotating speed fluctuation.²² Finally, as for vibration signal $x (t)$ , its NLSTFT $S (t, ω)$ can be expressed as show in formula (1), where $s (t)$ is analytic expression corresponding to $x (t)$ .

S (t, ω) = \int_{- \infty}^{+ \infty} x (u - t) s (u) e^{- i ω u} e^{- \frac{i c (t) {(u - t)}^{2}}{2}} d u

(1)

Adaptive chirp mode decomposition (ACMD)

In general, the mechanical fault vibration signal under variable speed condition can be modeled into a multi amplitude-frequency modulation form. Therefore, the vibration signal $x (t)$ containing n order components can be expressed as formula (2)²³ where N symbolizes the number of order components, $A_{n}$ symbolizes instantaneous amplitude, $f_{n} (s)$ represents IF, and $φ_{n}$ is the initial phase corresponding to the nth order component.

x (t) = \sum_{n = 1}^{N} x_{n} (t) = \sum_{n = 1}^{N} A_{n} \cos [2 π \int_{0}^{t} f_{n} (s) + φ_{n}]

(2)

As for the signal $x (t)$ , by the demodulation process, it can be decomposed into a number of order components which presents narrowband performance, as shown in formula (3) where $α_{n} (t)$ and $β_{n} (t)$ are used to recover the instantaneous amplitude $A_{n} (t)$ corresponding to each order component, and their expressions are shown in formulas (4) and (5), respectively. What’s more, the instantaneous amplitude $A_{n} (t)$ is shown in formula (6).

x (t) = \sum_{n = 1}^{N} {α_{n} (t) \cos [2 π \int_{0}^{t} {\tilde{f}}_{n} (s) d s] + β_{n} (t) \sin [2 π \int_{0}^{t} {\tilde{f}}_{n} (s) d s]}

(3)

α_{n} (t) = A_{n} \cos {2 π \int_{0}^{t} [f_{n} (s) - {\tilde{f}}_{n} (s)] d s + φ_{n}}

(4)

β_{n} (t) = - A_{n} \sin {2 π \int_{0}^{t} [f_{n} (s) - {\tilde{f}}_{n} (s)] d s + φ_{n}}

(5)

A_{n} (t) = \sqrt{α_{n}^{2} (t) + β_{n}^{2} (t)}

(6)

By analyzing formulas (4) and (5), when $f_{n} (s)$ is equal ${\tilde{f}}_{n} (s)$ , $α_{n} (t)$ and $β_{n} (t)$ will evolve into two pure amplitude modulated narrowest bandwidth signal. Therefore, by minimizing the bandwidths corresponding to $α_{n} (t)$ and $β_{n} (t)$ , ACMD can extract a series of order components and estimate its corresponding instantaneous frequencies. The procedure is shown in formula (7). In this formula, $x_{i} (t)$ denotes the extracted order component, $∥ \cdot ∥_{2}$ denotes the L2 norm, $∥ x (t) - x_{i} (t) ∥_{2}^{2}$ is the residual energy, and $α$ is the weight factor.

{\begin{cases} \min_{{p_{i} (t)}, {q_{i} (t)}, {7 (t)}} {{‖ p_{i}^{″} (t) ‖}_{2}^{2} + {‖ q_{i}^{″} (t) ‖}_{2}^{2} + α {‖ x (t) - x_{i} (t) ‖}_{2}^{2}} \\ s . t . x_{i} (t) = p_{i} (t) \cos (2 π \int_{0}^{t} f_{i} (τ) d τ) + q_{i} (t) \sin (2 π \int_{0}^{t} f_{i} (τ) d τ) \end{cases}

(7)

In order to facilitate processing, the vibration signal is discretely sampled at time

t = t_{0}, \dots, t_{N - 1}

, and N is the number. Therefore, formula (7) is rewritten as shown in formula (8).

\min_{{v_{i}}, {f_{i}}} {{‖ Θ v_{i} ‖}_{2}^{2} + α {‖ x - G_{i} v_{i} ‖}_{2}^{2}}

(8)

where

Θ = [\begin{array}{l} Ω & 0 \\ 0 & Ω \end{array}]

Ω

is the second-order difference matrix,

v_{i} = {[p_{i}^{T}, q_{i}^{T}]}^{T}

x = {[x (t_{0}), \dots, x (t_{N - 1})]}^{T}

G_{i} = [C_{i}, S_{i}]

. Where

C_{i} = d i a g [\cos (φ_{i} (t_{0})), \dots, \cos (φ_{i} (t_{N - 1}))]

S_{i} = d i a g [\sin (φ_{i} (t_{0})), \dots, \sin (φ_{i} (t_{N - 1}))]

φ_{i} (t) = 2 π \int_{0}^{t} f_{i} (τ) d τ_{0}

In order to compute formula (8), this paper introduces an iterative algorithm to alternately update each intermediate variable. After a series of iterations, the IF corresponding to ith order component is updated as shown in formula (9), where $f_{i}^{j} = {[{\tilde{f}}_{i}^{j} (t_{0}), \dots, {\tilde{f}}_{i}^{j} (t_{N - 1})]}^{T}$ , $Δ {\tilde{f}}_{i}^{j}$ symbolizes frequency increment matrix, I represents unit matrix, and $β$ is the penalty coefficient.

f_{i}^{j + 1} = f_{i}^{j} + {(\frac{1}{β} Ω^{T} Ω + I)}^{- 1} Δ {\tilde{f}}_{i}^{j}

(9)

By extracting the ith order component ${\tilde{x}}_{i} (t)$ , the residual signal $R_{i} = x (t) - {\tilde{x}}_{i} (t)$ is acted as the original signal, repeating the above procedure, and the vibration signal $x (t)$ is decomposed into a series of order components ultimately in formula (10).

x (t) = \sum_{i = 1}^{K} {\tilde{x}}_{i} (t) + X_{K} (t)

(10)

In conclusion, as ACMD only requires the initial IF of a specific order component, this approach enhances flexibility and adaptability.

Ridge line detection

Unlike the original ACMD which uses the Hilbert transform to estimate the initial IF, in this paper, we will use the ridge detection algorithm combination with NLSTFT time-frequency distribution to implement the initial IF estimation of order components. The ridge detection algorithm can estimate the IF more accurately, which facilitates the order components extraction using ACMD. Scholars have conducted a lot of researches on ridge line detection method based on time-frequency distribution. Literature²⁴ estimates the IF by detecting the maximum value in time-frequency representation, which is susceptible to interference. Literature²⁵ can extract all the order components simultaneously by calculating the local maxima of a particular function, but the number of order components needs to be determined in advance. In real vibration signal, the number of order components is extremely difficult to determine due to the interference. In addition, ACMD does not need to provide the initial IFs of all order components, but only the IF information of a certain order component to be extracted. Therefore, in this paper, the ridge line detection method from literature²⁶ will be used, and it can estimate the IF information of order components that has the highest energy, which is coincident with ACMD.

Considering that the order component corresponding to IRF possesses the highest energy in NLSTFT time-frequency distribution; it can be well used for the coarse estimation of IRF, and the specific steps are shown as follows.

1. Detecting the location of maximum energy in the time-frequency distribution $(t_{0}, f_{0}) .$

2. Assuming $t_{r} = t_{0} + 1 / f s, t_{l} = t_{0} - 1 / f s, f_{l} = f_{0}, f_{r} = f_{0},$ where $f s$ is the sampling frequency.

3. Calculating the IF at $t_{l}$ and $t_{r}$ , $I F (t_{l}) = {\tilde{f}}_{1}, I F (t_{r}) = {\tilde{f}}_{0},$ where ${\tilde{f}}_{1}$ and ${\tilde{f}}_{0}$ are the frequencies corresponding to maximum energy at $(t_{l}, f_{l} - Δ f) \sim (t_{l}, f_{l} + Δ f)$ and $(t_{r}, f_{r} - Δ f) \sim (t_{r}, f_{r} + Δ f)$ , respectively, and $Δ f$ is the maximum allowable frequency transformation.

4. Assuming the values are all zero at $(t_{l}, f_{l} - Δ f) \sim (t_{l}, f_{l} + Δ f)$ and $(t_{r}, f_{r} - Δ f) \sim (t_{r}, f_{r} + Δ f)$ .

5. Assuming $t_{r} = t_{r} + 1 / f_{s}, t_{l} = t_{l} - 1 / f_{s}$ .

Repeating above step 3∼step5 until a series of order component are found.

From above analysis, it can be seen that the proposed method is very simple to implement, as it only requires the time-frequency representation of vibration signal and the maximum varying allowable frequency range. As a result, we use NLSTFT combination with ridge line detection method to achieve IF coarse estimation.

The proposed method

This paper adopts the “from coarse-to-fine” strategy to propose a tacholess fast-varying instantaneous frequency estimation method based on NLSTFT combination with ACMD and applies it to extract order components, the specific procedures are shown in Figure 1.

Step one: Conducting the vibration signal’s time-frequency distribution using NLSTFT.

Step two: Roughly estimating the IF whose order component has the highest energy in time-frequency representation by using ridge line detection, which is generally the IRF.

Step three: Using ACMD to process the roughly IRF for estimating a precise IRF and its corresponding first-order component as well as instantaneous amplitude.

Step four: Subtracting the extracted first-order component from original signal and obtaining the residual component, repeating step one-three to process the residual component for obtaining a series of order components and their instantaneous frequencies and amplitudes.

Figure 1.

A tacholess fast-varying instantaneous rotating frequency estimation flowchart based on NLSTFT combination with ACMD.

Simulation analysis

A rapidly varying simulation signal under speed fluctuation condition

Considering that mechanical equipment usually operates in a variable speed state, we have generated a fault vibration signal with rapid instantaneous speed changes, which includes linear variable speed state, speed fluctuation state, and large fluctuation speed fast changing turntable. This symbolizes that mechanical equipment operates in an extreme condition, and the simulation signal whose instantaneous angular frequency $ω (t)$ is shown in formula (11), and its corresponding IRF is $f (t)$ = $ω (t) / 2 π$ with a duration of 30s.

W (t) = {\begin{cases} 3 π t + 30 π & 0 < t < 10 \\ 60 π & 10 < t < 12 \\ 60 π + 40 \sin (0.6 π t) & 12 < t < 17 \\ 60 π + 80 \sin (0.5 π t) & 17 < t < 19 \\ 60 π + 40 \sin (0.6 π t) & 19 < t < 24 \\ 60 π & 24 < t < 26 \\ 60 π - 7.5 π t & 26 < t < 30 \end{cases}

(11)

Furthermore, we take this IRF $f (t)$ to act as the base frequency, and a vibration simulation signal x(t) is constructed as shown in formula (12) which contains two double IRF and three double IRF where η(t) symbolizes Gaussian white noise, which is set to −5 dB in signal x(t). Finally, its time-domain waveform is shown in Figure 2.

x (t) = \sin (\int_{0}^{t} ω τ d τ) + 0.5 \sin (2 \int_{0}^{t} ω τ d τ) + 0.3 \sin (3 \int_{0}^{t} ω τ d τ) + η (t)

(12)

Figure 2.

Time-domain of simulation signal (SNR = −5 dB).

As shown above, firstly we obtain the signal’s time-frequency spectrum by using NLSTFT as shown in Figure 3(a). Here, the parameter in formula (1) $c (t)$ is 0, sampling frequency is 400, and the window function length is 220. Next, since the order component corresponding to IRF has the highest energy in the simulated signal, we use ridge line searching method to estimate instantaneous angular frequency initially and use the least squares method to smooth it as shown in Figure 3(b). Analyzing Figure 3(b), the estimation method using “NLSTFT + ridge line searching” may experience fluctuations and cannot accurately capture rapidly changing mid frequencies, resulting in significant errors. Furthermore, based on the proposed method’s procedure, we use ACMD to process the initial angular frequency for obtaining a more precise angular frequency as shown in Figure 3(c).

Figure 3.

Signal time-frequency diagram and comparison of rapidly varying instantaneous angular frequency estimation.

Figure 3(c) displays the result by applying the proposed model in this paper, and the estimation result closely aligns with the true frequency. This indicates that both models can accurately extract IRF when the mechanical equipment is operating at steady-state or linearly varying speeds, but the proposed model is significantly superior to classical methods when operating under severe conditions. Table 1 also quantitatively validates the aforementioned conclusion. Here, we use the average relative error

erro r_{avg}

and the maximum relative error

erro r_{\max}

, to evaluate the effect which is shown in formulas (13) and (14).

e r r o r_{a v g} = \frac{1}{N} \sum_{i = 1}^{N} | \frac{f_{r, i} - f_{e, i}}{f_{r, i}} |

(13)

e r r o r_{\max} = \max (| \frac{I F_{e s t i m t e} - I F_{t r u e}}{I F_{t r u e}} \times 100 % |)

(14)

where N is the total number of simulation signal,

f_{r, i}

is the ith sample point of true frequency, and

f_{e, i}

is the ith sample point of estimated frequency.

I F_{true}

is the true angular frequency and

I F_{e s t i m t e}

is the estimated angular frequency.

Table 1.

Comparison of error analysis.

Estimation method	Average relative error $erro r_{a v g} (%)$	Maximum relative error $e r r o r_{\max} (%)$
NLSTFT combination with ridge line searching	3.07	20.90
NLSTFT combination with ACMD	0.54	12.80

When selecting parameters for NLSTFT and ACMD, careful consideration is essential to ensure the model’s performance and accuracy. In ACMD, alpha0 serves as the penalty parameter controlling the filter bandwidth. This parameter is typically selected through experimentation to achieve a balance where the bandwidth is sufficient for effective filtering without excessively smoothing the signal. Beta controls the penalty parameter for the smoothness of the IF increment during iterations, and it is also determined experimentally to ensure smoothness without over-smoothing. Tol represents the convergence criterion tolerance, commonly set at values such as 1 × 10⁻⁷, 1 × 10⁻⁸, or 1 × 10⁻⁹, with its impact on convergence speed and accuracy validated through experiments. For NLSTFT, hlength denotes the window function length, which is usually chosen to be between 1/4 and 1/10 of the signal length, with its impact on time-frequency representation tested experimentally. The parameter c represents the first derivative of the signal’s instantaneous frequency, chosen experimentally to optimize the modulation effect of the nonlinear short-time Fourier transform. Fs is the sampling frequency, selected based on the signal’s sampling rate to balance time and frequency resolution.

To ensure extraction effectiveness, the selection of Alpha0, Beta, and Tol parameters should be as small as possible. It is recommended to refer to the parameter settings in this study when processing other signals: Alpha0 = 1 × 10⁻⁶, Beta = 1 × 10⁻⁹, and Tol = 1 × 10⁻⁸. Initially, c is set to 0 and is updated iteratively, so it does not need to be set separately. For the selection of the hlength parameter, the method for selecting the window function length in short-time Fourier transform (STFT) can be referenced. The optimal value of the window function length can be between 1/4 and 1/10 of the signal length. Based on experience, 220 was chosen as the hlength setting in this paper. Through the careful selection of these parameters, the model’s effectiveness and accuracy can be ensured.

In summary, when the mechanical engineer works under a steady-state or linearly varying rotating speed, traditional model based on time-frequency analysis combined with ridge line detection can accurately extract the IRF. However, in extremely fluctuating rotating speed or rapidly varying rotating speed, the proposed approach in this paper, which employs a “coarse-to-fine” estimation strategy, yields more precise estimation result.

Experimental validation

Semi-physical simulation

In this section, we use the simulation signal generated by the dynamic signal processing system built in the laboratory for semi-physical verification. The experimental system consists of a dual-channel signal generator of Puyuan DG1032, a dynamic vibration signal acquisition card, and an industrial computer, as shown in Figure 4(a). A signal generator is used to generate a linear swept component $X_{LSF 1}$ , which has an amplitude of 5 m/s² and its IF varies from 0 Hz to 50 Hz conforming to linearly increasing law, with a sampling frequency of 1000 Hz, while another signal generator is used to simulate the noise component $X_{Noise}$ . Finally, we use Matlab to generate another linear swept component $X_{LSF 2}$ which has an amplitude of 5 m/s², and its IF varies from 0 Hz to 100 Hz conforming to linearly increasing law. The above three components are superimposed to obtain the validation signal $X_{Simulation}$ . Further, the validation signal $X_{S i m u l a t i o n}$ is captured by using the dynamic signal acquisition card and displays on the interface by using Labview as shown in Figure 4(b).

Figure 4.

(a) Semi-physical validation platform. (b) Time-domain of validation signal $X_{S i m u l a t i o n}$ .

Finally, we use the proposed model to extract the IF corresponding to signal components $X_{LSF 1}$ and $X_{LSF 2}$ , respectively, as shown in Figure 5(a) and (b). It can be observed that the extracted IF closely align with the actual frequency, indicating that the estimation method has high accuracy.

Figure 5.

Instantaneous frequency estimation and order component extraction corresponding to validation signal $X_{Simulation}$ .

Using the two estimation IFs, the corresponding order components $X_{E x t r a c t e d - L S F 1}$ and $X_{E x t r a c t e d - L S F 2}$ can also been extracted as shown in Figure 5(c) and (d); it also presents the comparison between extracted order components and real order components $X_{L S F 1}$ and $X_{L S F 2}$ that they are generally consistent.

When dealing with order components, as they often present numerous zero crossing phenomenona, directly calculating the average relative error may not accurately reflect the extraction performance. Therefore, we choose to use the mean squared error ( $M S E$ ) and the Pearson product-moment correlation coefficient $r_{x y}$ as evaluation indexes. Their formulas are given by formulas (15) and (16).

M S E = \frac{1}{N} \sum_{i = 1}^{N} {(x_{i} - {\hat{x}}_{i})}^{2}

(15)

Here,

x_{i}

represents the true value,

{\hat{x}}_{i}

represents the extracted value, and

N

is the length. A smaller MSE value indicates better extraction performance.

r_{x_{i} {\hat{x}}_{i}} = \frac{\sum_{i = 1}^{n} (x_{i} - \bar{x}) ({\hat{x}}_{i} - \bar{\hat{x}})}{\sqrt{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} \sum_{i = 1}^{n} {({\hat{x}}_{i} - \bar{\hat{x}})}^{2}}}

(16)

In general, the extraction performance is positive correlation with the Pearson product-moment correlation coefficient,

\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} x_{i}

\bar{\hat{x}} = \frac{1}{n} \sum_{i = 1}^{n} {\hat{x}}_{i}

Furthermore, we quantitatively analyze the similarity between the extracted order components and original order components, and here, we also select two indicators which are mean squared error, correlation coefficient, and the result is shown in Table 2. In general, the correlation coefficients are relatively all close to 1, indicating that the extraction results are accurate. Therefore, the proposed method is suitable for signal decomposition under varying speed condition.

Table 2.

Comparison of accuracy analysis.

Order component	Mean squared error $M S E$	Correlation coefficient $r_{x y}$
X_LSF1	1.33	0.96
X_LSF2	0.87	0.97

Rolling bearing fault vibration signal validation under varying speed condition

We use the rolling bearing vibration test signal under variable speed condition from Xi’an Jiao tong University, and the experiment uses the SQ (Spectra Quest) company’s mechanical failure comprehensive simulation test bench to simulate the motor bearing outer ring and inner ring failure. The structure of the test bench is shown in Figure 6; the test bench consists of motor, rotor, and load three major components. The experiment uses piezoelectric acceleration sensors to collect motor bearing vibration signal. The data acquisition instrument used is CoCo80, and the sampling frequency is 25.6 kHz. The style of motor bearing is NSK6203.

Figure 6.

Rolling bearing test bench under varying speed condition.

The “NC-REC3642-ch2.” is designated as experimental validation signal whose speed conforms to linear varying law, as original signal has a high sampling frequency and large number of sampling points, and it is a challenge for direct time-frequency analysis that will lead to a series of issues such as insufficient memory and a low computation efficiency. In order to address this problem, it is essential to down-sample experimental signal initially, with a down-sampling factor of 100. Finally, the corresponding rolling bearing fault vibration signal is depicted in Figure 7.

Figure 7.

Time-domain of motor rolling bearing vibration signal.

Furthermore, Figure 8(a) and (b) presents a comparison between two tacholess rotating frequency’s estimation results and estimation result based on phase-locked pulse signal. Obviously, it is evident that the estimation result by the proposed model successfully captures a complete acceleration and deceleration process. Specifically, it starts from a stationary state, gradually accelerates, maintains stability, and eventually decelerates. The original data indicates that the rotating speed remains stable state after accelerating to 3000 rpm, which corresponds to 50 Hz (3000/60 = 50 Hz). From Figure 8(b), it is apparent that during the time interval from 4 s to 12 s, the estimated rotating frequency aligns with the true rotating frequency and effectively reflects the actual rotating speed’s varying law. Considering the vibration signal undoubtedly contains noise, the estimation result presents some fluctuations in the beginning and end stages.

Figure 8.

Instantaneous frequency extraction results and a series of extracted order components and residual component.

Finally, we sequentially estimated the remaining instantaneous frequencies and extracted corresponding order components, residual component, as shown in Figure 8(c) (d).

In conclusion, when we employ the proposed model to estimate IRF and extract a series of order components in the context of multi-component coupling fault diagnosis application, it not only yields an accurate estimation result but also facilitates the rapid and efficient identification. Consequently, the proposed method can be utilized where it is not possible to place an encoder to achieve rotating speed measurement.

Conclusion

(1) A tacholess fast instantaneous rotating frequency estimation model is proposed based on NSTFT combined with ACMD, adopting a “coarse-to-fine” strategy. Firstly, NSTFT constructs a fault vibration signal’s time-frequency spectrum, followed by ridge line detection to coarsely estimate the IRF with the highest energy. Furthermore, ACMD precisely estimates the IRF. This strategy effectively solves spectral ambiguity and provides accurate IRF estimation, confirmed by simulations and experiments, showing significant advantages over traditional methods for estimating fast-varying IRF.

(2) Based on the estimated IRF combination with cyclic extraction strategy, a series of order components in vibration signal can be obtained sequentially. It solves the unavoidable problems of traditional methods caused by speed fluctuation, noise, amplitude and frequency modulation, and other irrelevant components, and can present an ideal time-frequency representation with high resolution, which has strong application value. However, when this model directly is applied on the low SNR condition, the extracted order components have amplitude distortion. Considering the Vold-Kalman filter has been widely employed to extract order components, and the proposed method has a high estimation accuracy on IRF, we suggest the proposed model combined with the Vold-Kalman filter can be used to improve the extraction accuracy of order components, effectively reducing the interference of noise on the signal and extracting more precise order components to address amplitude distortion issues.

(3) In future, combining the proposed model with traditional demodulation models, we can provide a novel tacholess phase-amplitude demodulation technique, which can be used to diagnose fault under variable working condition. In addition, considering the use of NLSTFT to process big data, there is still a high degree of spectral ambiguity in the time-frequency representation, and in order to improve the energy concentration of time-frequency representation, higher precision time-frequency distribution such as Synchronous Compression Transform (SST) can be attempted. Finally, a novel order component extraction model can also be constructed by combining key phase instantaneous rotating frequency estimation with ACMD.

(4) In mechanical systems, certain critical components cannot accommodate traditional speed sensors due to space limitations or other reasons. MEMS (Micro-Electro-Mechanical Systems) sensors, such as accelerometers and gyroscopes, can fill this gap. In the future, the method proposed in this paper can be combined with MEMS technology to further enhance the fault diagnosis capabilities of mechanical systems. By integrating MEMS sensors into mechanical systems, we can achieve continuous condition monitoring and obtain real-time vibration and displacement data of critical components, thereby improving the accuracy of fault diagnosis.

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 paper is supported by National Natural Science Foundation of China (Number: 62303300) and Shang Hai Professional Technical Service Platform Project (Number: 23DZ22905000).

ORCID iDs

Lu Yan

Qin Xiao Chen

Jin Wei Bie

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