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Mechanical equipment often works on variable speed condition, its corresponding vibration signal presents multi-compo- nent, modulation coupling with fast time-varying instantaneous frequency (IF), how to effectively compute IF and realize fasting varying non-stationary signal decoupling separation plays an important role in mechanical system fault diagnosis. In this paper, a sparse representation method called multi-scale chirp sparse representation (MSCSR) is introduced to identify, extract, and trend IF for achieving a highly concentrated time-frequency energy. Simulation demonstrates that the proposed method performs better than traditional IF estimation method. Furthermore, an adaptive time-varying fil- ter is constructed using the extracted instantaneous frequency to decouple non-stationary fast signal. Ultimately, by rapid instantaneous frequency fluctuation experiment, the effectiveness of proposed method for fast strong time-varying signal is validated, it can effectively extract rapid oscillation instantaneous frequency, and the error is less than 10%. Keywords Fast varying signal, instantaneous frequency estimation, adaptive time-varying filter, multi-scale chirp sparse representation, mechanical fault diagnosis Date received: 23 December 2022; accepted: 3 April 2023 Handling Editor: Chenhui Liang scholars have gradually carried out research on the Introduction 3,4 mechanical fault vibration signal separation. In gen- Mechanical equipment under variable speed condition eral, they can be divided into four universal categories, is almost everywhere, such as gearbox under raising speed, its fault vibration signal often reflects multi- Shanghai Dianji University, Shanghai, China component coupling non-stationary property with fast Weiqiao-UCAS Science and Technology Park, Binzhou, Shandong IF. As a result, how to separate fast varying signal Province, China based on IF estimation has become a critical require- Institute of Automation, Chinese Academy of Sciences, Beijing, China The National Joint Engineering Laboratory of Internet Applied ment for vibration-based mechanical equipment condi- Technology of Mines, China University of Mining and Technology, tion monitoring methods. Xuzhou, China In general, the existing signal processing methods are 5 Shandong Energy Zibo Mining Group Information Center, Shandong, majorly applicable to the steady speed working condi- China 1 6 tion. Under variable speed, the fault vibration signal will Liyang Hongda Motor Co., Ltd, Changzhou, China lose the periodic regularity, and present the frequency Corresponding author: modulation, amplitude modulation property, leading to Du Juan, Shanghai Dianji University, Shuihua Road, Shanghai 200240, the invalidity of classical methods. With the develop- China. ment of non-stationary signal processing technology, Email: dujuan@sdju.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (https://creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 2 Advances in Mechanical Engineering 5 6 22 order tracking, order cyclostationary analysis and time- system, in this paper, we introduce it to estimate IF in 7 8 frequency analysis, tine-varying filter. mechanical varying non-stationary signal whose As for order tracking, its main procedure is to matched estimator is constructed according to fast transform non-stationary signal into stationary or varying IF, so as to improve the signal time-frequency cyclically stationary signal by equal interval sampling representation energy aggregation. On the basis, by technology in angle domain, so as to establish a bridge using the extracted IF acting as center frequency of the between signal processing under variable speed condi- filter, we design a adaptive time-varying filter for tion and constant speed condition, so that the classical decomposing mechanical varying non-stationary signal signal processing method can play its role again. ultimately. However, it must be pointed out that the implementa- Finally, the process diagram of proposed methodol- tion of order tracking mostly relies on hardware device ogy is presented in Figure 1. Firstly, the continuous such as encoder or key phase sensor, nevertheless, in wavelet transform is applied to obtain corresponding some critical condition, as sensor cannot be installed wavelet energy spectrogram, and then the number of which brings challenges to order tracking implementa- sub-signal component in vibration signal is accurately tion. Under this background, scholars propose tacho- judged according to the time-frequency distribution, less order tracking (TLOT) method, it can directly which greatly improves the calculation efficiency. achieve order tracking on non-stationary by using Secondly, MSCSR self-adaptive selects an appropriate time-frequency transformation without hardware, but dynamic time support region according to the IF vary- it still needs to be further improved, mainly in terms of ing law, tracks, and estimates the IF of sub-signal com- accurately tracking and estimating instantaneous fre- ponent which possesses largest energy. Finally, an quency. As instantaneous frequency reflects the adaptive time-varying filter is designed using estimated dynamic state of mechanical equipment, so improving IF as the center frequency, and is applied for decom- the estimation accuracy of IF becomes the core of posing vibration signal into several independent com- TLOT. Furthermore, considering the structure of most ponents. A simulation and mechanical system high mechanical parts has spatial symmetry, its fault vibra- speed fluctuation experiment is to validate the pro- tion signal presents essentially cyclostationary in the posed method, and compared with IF recognition angle domain, by using order tracking and synchro- result and signal decomposition result by using Hilbert- nous averaging, we can convert the non-stationary sig- Huang transform (HHT), continuous wavelet trans- nal in time domain into cyclostationary signal in angle form (CWT), synchrosqueezing wavelet transform 24 25 domain again and establish order cyclostationarity (SWT), variational mode decomposition (VMD), analysis ultimately which is convenient for processing empirical mode decomposition (EMD) respectively, mechanical varying non-stationary signal, such as order the proposed method has a better effectiveness and 11 12 spectrum, order cepstrum, envelope order spec- accuracy. 13 14 15 trum, order bispectrum, and high-order spectrum. As for time-frequency analysis, considering only from An adaptive time-varying filter based on the time domain or frequency domain, we can not instantaneous frequency estimation obtain the instantaneous time-frequency property which is the core of non-stationary signal processing, The key steps of proposed methodology as shown in the time-frequency analysis provides the joint distribu- Figure 1, are discussed in this section. The basic func- tion information in the time domain and frequency tion library of MSCSR for tracking and estimating IF domain which can effectively give attention to both is shown as following. time resolution and frequency resolution, it is very suit- able for estimating instantaneous frequency and separ- Dh ðÞ t ¼ h ðÞ t a;b;I a;b;I ð1Þ ating mechanical varying non-stationary signal. 2 h ðÞ t ¼ K exp½ iaðÞ t + bt 1 ðÞ t a;b;I a;b;I I Finally, as for time-varying filter, its center frequency In the above formulas, D represents base function needs to match the instantaneous frequency and its library, h ðÞ t is the multi-scale chirp base function, parameters also should be set reasonably to achieve a;b;I 2j 2j mechanical varying non-stationary signal separation. I=[kN2 ;(k +1)N2 ] represents dynamic time In conclusion, instantaneous frequency estimation support region, it describes time – varying characteris- plays an important role in non-stationary signal decom- tic of instantaneous frequency, j ¼½ 0; 1; ; log N 1 position. A series of methods have been introduced to symbolizes analysis scale coefficient, N represents sam- achieve this goal, such as reassignment synchrosqueez- pling length, it can reflect the description precision of 18 19 ing transform, multi-scale chirplet path, short-time time-varying characteristics. As for k, it is the sequence Fourier transformation, considering the application number of dynamic analysis section in j scale, and it 21 j of MSCSR on nonlinear FM signal estimation, para- varies from 0 to 221. K is a normalization coeffi- a,b,I meter identification of linear time-varying SDOF cient, a is symbolizes frequency offset coefficient, b is Yan et al. 3 Figure 1. Flow chart of proposed method in this paper. the frequency rake ratio, those parameters can well multi-scale chirp basis function, a specific basis function reflect the varying state of instantaneous frequency. will be obtained that is corresponding to maximum pro- Finally, 1 ðÞ t is a rectangular window function. As for jection coefficient b on each dynamic time support I I the sampling frequency f , it must satisfy the following region I. Then, in accordance with maximum projection coefficient b and its corresponding chirp base function, conditions according to Shannon sampling theory. a dynamic linking algorithm called P must be designed a + 2bt\f =2 ð2Þ to connect a series of basis functions, and there is no overlap between the basis functions each other on whole The multi-scale chirp basic function is applied to proj- dynamic time support regions I. ect fault response signal segment by segment, and calcu- lating projection coefficient b and corresponding basic I 2 P 2 iat + bt ½ ðÞ Max K e 1 ðÞ t function h on each dynamic time support region I. If a;b;I I a;b;I I2II ð3Þ the fault signal is more similar to basic function h , a;b;I P ¼fg I ; I ; ::: 2 fg I 1 2 its projection coefficient b will be larger, and the corre- sponding energy of basic function h will be larger. At the same time, the corresponding projection coef- a;b;I As a result, fault response signal must be projected to ficient set and basic function set are shown in following. 4 Advances in Mechanical Engineering Figure 2. The implementation process of instantaneous frequency estimation based on proposed method. b ¼ b , b , ::: region set I ¼fg I ; I ; ::: in chronological order. As a 1 2 I I 1 2 ð4Þ H ¼fg h , h , ::: result, the estimated algorithm of instantaneous a b I a b I 1 1 2 3 frequency can be summarized as shown in Figure 2. The steps of the connection algorithm P in Considering adaptive time-varying filter is based on MSCSSD can be summarized as follows. classical filter, the proposed adaptive time-varying filter based on MSCSR can make its center frequency consis- Step 1: Initialization. i represents the serial number tent with estimated IF according to the mechanical of sub-signal dynamic time support region, E rep- i varying non-stationary signal. As the signal often pre- resents the total energy of sub-signal before dynamic sents obvious amplitude modulation and frequency time support regions I of sequence number i, l rep- i i modulation phenomenon, we can make the estimated resents previous serial number of dynamic time sup- IF that it is corresponding to carrier frequency acting port region attached to i, E indicates energy of as center frequency and side frequency act as cut-off sub-signal corresponding to the projection coeffi- frequency respectively. Therefore, an adaptive filter cient on each dynamic time support region I , initia- H(s,t) design algorithm based on MSCSR is proposed lizing E ¼ 0, l ¼ 0. d i in this paper, it can be obtained by transferring classi- Step 2: For every dynamic time support region I in cal low-pass prototype filter H(s). As its central fre- the set I ¼fg I ; I ; ::: , find all next dynamic time 1 2 quency and filter bandwidth can automatically vary support regions adjacent to it to form a set I ,if according to the non-stationary property, as a result, it is very suitable for mechanical varying non-stationary E + E .E ð5Þ d e d i i j signal separation and the design procedures is shown in Figure 3. There is the following formula, l¼N l l Step 1: As for signal stðÞ;stðÞ ¼ s ðÞ t , l symbo- E ¼ E + E d d e l¼1 j j i ð6Þ lizes the serial number of sub-signal, l ¼ 1; 2; N , l ¼ i N is total amount of sub-signals. As for sub-signal The dynamic linking algorithm called P guarantees with serial number l, its carrier frequency curve and l l extracted sub-signal connected by basic function set H side frequency are w ðÞ t , b ðÞ t ; 0 ł t ł N 1; which z z over the entire dynamic time support region set are obtained by MSCSR, N represents the signal I ¼fg I ; I ; ::: is most similar to original fault response length. 1 2 signal, and instantaneous frequency corresponding basic Step 2: Obtaining the spectrum SfðÞ is correspond- function h can be computed f ðÞ t ¼ a + 2b t; t eI ing to signal stðÞ. a;b;I I m m i i on the dynamic time support region I , the instantaneous Step 3: In order to extract sub-signal s ðÞ t , the adap- frequency f can be tracked and estimated ultimately by tive time-varying filter HtðÞ ; f is designed on connecting the corresponding frequency curve set t ¼ tðÞ 0 ł t ł N 1 as follows. w ðÞ t acts as the 1 1 1 f ¼fg f ; f ; :::: over the whole dynamic time support center frequency, and n harmonic frequency of I I I 1 2 Yan et al. 5 Figure 3. Design flow of adaptive time-varying filter. b ðÞ t is half-bandwidth. Considering fast varying signal often presents modulation characteristic, in order to ensure extracted sub-signal without distor- tion, as Chebyshev II filter has fluctuation in the stop-band and a steep transition band, which is suit- able for signal separation, this paper adopts it acting as prototype low-pass filter. l l l l 1; w ðÞ t nb ðÞ t ł f ł w ðÞ t + nb ðÞ t 1 1 1 1 z z z z HtðÞ ; f ¼ 0; else ð7Þ Figure 4. The wavelet scalogram of simulated signal. Step 4: The adaptive time-varying filter HtðÞ ; f is applied to obtain corresponding filtered sub-signal S ðÞ f in frequency domain at the time t ¼ t (SNR) is defined as in formula (10), A and A signal noise ðÞ 0 ł t ł N 1 . symbolize the RMS value of simulation signal and noise component respectively. S ðÞ f ¼ 23SfðÞ3HtðÞ ; f ð8Þ xtðÞ ¼ x ðÞ t + x ðÞ t + sðÞ t 1 2 x ðÞ t ¼ 9sinðÞ 40pt ð9Þ Step 5: The IFFT transformation is used to obtain x ðÞ t ¼ 15sinðÞ 128pt corresponding sub-signal s ðÞ t in time domain at the time t ¼ tðÞ 0 ł t ł N 1 , repeating above process 1 1 IRFðÞ t ¼ d[½ x ðÞ t =dt ¼ 80pt=2p ¼ 40t at any time to obtain filtered sub-signal s ðÞ t . ð10Þ signal signal : SNR ¼ 10lg ¼ 20lg ðÞ dB Step 6: Repeating above procedures from step 1 to A A noise noise step 5, and achieving signal separation stðÞ completely. Firstly, we use the wavelet spectrum by using contin- uous wavelet transform to observe the number of sub- signals as shown in Figure 4. It can be significantly seen Simulation analysis that the simulation signal contains two sub-signals and In order to analyze the effectiveness of proposed their varying IFs conform to linear varying trend, and method on varying non-stationary signal decomposi- mainly concentrate in the range of [0 Hz, 40 Hz] and tion, a simulation signal x(t) containing three sub- [0 Hz, 128 Hz]. signals is presented in formula (9). Considering actual Secondly, we apply the MSCSR to estimate IFs. We noise effect, adding 20% SNR level Gaussian white set the frequency variation range on MSCSR is from 10 noise sðÞ t to simulation signal, the sampling frequency to 400 Hz, f =30Hz, f = 300 Hz, search frequency min max is 512 Hz and sampling time is 1 s. Signal-to-noise ratio slope is ranging from 0 to 200 Hz, search resolution is 6 Advances in Mechanical Engineering Figure 5. Estimated instantaneous frequency value by using MSCSR: (a) time-frequency representation of signal component x ðÞ t , (b) instantaneous frequency estimation of signal component x ðÞ t by using MSCSR, (c) time-frequency representation of signal component x ðÞ t by using MSCSR, and (d) instantaneous frequency estimation of signal component x ðÞ t by using MSCSR. 2 2 (a) (b) Figure 6. The instantaneous frequency extracted by CWT, HHT, and SWT: (a) instantaneous frequency estimation of signal component x ðÞ t and (b) instantaneous frequency estimation of signal component x ðÞ t . 1 2 1 Hz/s, and the extracted IFs f (t), f (t)are extract IF, as shown in Figure 6. Compared Figure 5 extract1 extract2 shown in Figure 5(b) and (d) respectively. with Figure 6. It can be clearly seen that the tracking Time-frequency representation in Figure 5(a) and (c) effect by using CWT in the beginning and end stages present the varying law of instantaneous frequency in has certain deviation, and estimated value has a large x ðÞ t and x ðÞ t , it can be seen that the IFs possess linear fluctuation phenomenon. On the other hand, although 1 2 varying property. Compared with Figure 5(a) and (c), SWT can achieve the tracking in general, the accuracy the estimated IF by MSCSR as shown in Figure 5(b) is not high. As for HHT, as IF presents a linear varying and (d) is consistent with the theoretical value, this result trend, it can not effectively achieve tracking, this phe- presents the estimated value by using the proposed nomenon implies that the existing methods can not be method has a perfect precision. effectively used to estimate IF under large fluctuation For comparison, we also uses Hilbert-Huang background, while the estimated IF based on MSCSR Transform (HHT), Continuous Wavelet Transform is more consistent with theoretical value, which (CWT), synchrosqueezing wavelet transform (SWT) to indicates the proposed method in this paper adopts Yan et al. 7 Table 1. IA index identification effect of instantaneous value by using proposed method is less than compared frequency on simulation signal xtðÞ. methods which is all less than 5%, indicating that the proposed method has high accuracy and good noise Method MSCSR CWT SWT HHT resistance. Through above simulation signal, the advantage of IA1 6.7 33.4 76.4 80.3 proposed method in IF tracking has been preliminarily IA2 4.3 29.7 57.8 76.2 validated, and it has potential application value in non- stationary signal separation based on filter technology. Considering that the actual mechanical equipment multi-scale chirp basic function whose parameters are often run in large speed fluctuation condition, its IF adjustable, can dynamically and self-adaptive track and often presents curvilinear varying trend. As a result, we estimate fast varying IF. simulate a mechanical varying non-stationary signal as Furthermore, in order to quantify IF identification shown in formula (12). It contains two sub-signals s (t) accuracy, we adopt root mean square (RMS) to act as and s (t), they reflect modulation property and their precision index (Index of accuracy, IA). f ðÞ n represents instantaneous frequencies are overlapping as shown in IF identification value, f ðÞ n symbolizes IF theoretical formula (13) and (14), sðÞ t represents gaussian white value, N is number of points. Generally, if IA index has noise, sampling frequency is 1024 Hz, the correspond- a smaller value, it indicates that the recognition value is ing time waveform is shown in Figure 8. connected with theoretical value excellently, that is, the identification accuracy is more higher. stðÞ ¼ s ðÞ t + s ðÞ t + sðÞ t ð12Þ 1 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P s ðÞ t ¼ðÞ 1 + cosðÞ 2pðÞ 20t 10cosðÞ pt N 2 ð13Þ ðÞ f ðÞ n f ðÞ n i t i¼1 cosðÞ 2pðÞ 350t + 14:29sinðÞ 1:4pt qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j ¼ ð11Þ N 2 ðÞ f ðÞ n i¼1 s ðÞ t ¼ðÞ 1 + cosðÞ 2pðÞ 20t 10cosðÞ pt ð14Þ cosðÞ 2pðÞ 200t + 14:29sinðÞ 1:4pt Table 1 shows corresponding IA values by using four methods, MSCSR, SWT, CWT, and HHT, IA1 and It can be seen that the simulation signal stðÞ has IA2 represent the recognition accuracy of sub-signals frequency-modulation and amplitude-modulation prop- x ðÞ t , x ðÞ t respectively. As for simulation signal xtðÞ,it 1 2 erty, and the time-varying carrier frequencies for s (t) can be seen from Table 1 that the proposed method is and s (t) are shown in formula (15) and (16). Besides, as obviously superior to HHT, CWT, and SWT. for the IF corresponding to amplitude modulation com- In order to further verify the anti-noise ability and ponent in s (t)and s (t) is shown in formula (17). As a 1 2 accuracy of proposed method on instantaneous fre- result, the simulation signal stðÞ contains six instanta- quency estimation under strong noise, compared with neous frequencies, f (t)2f (t), f (t), f (t)+ f (t), 1 b 1 1 b SWT, CWT, and HHT, we add different SNR level f (t)2f (t), f (t), f (t)+ f (t). 1 b 2 2 b Gaussian white noise sðÞ t into simulation signal x(t) which is ranging from 210 to 20 dB, the IA index under f ðÞ t ¼ 350 + 20pcosðÞ 1:4pt ð15Þ different SNR is calculated and shown in Figure 7. As f ðÞ t ¼ 200 + 20pcosðÞ 1:4pt ð16Þ can be seen from the Figure 7, the corresponding IA (a) (b) Figure 7. IA value of instantaneous frequency estimation under different SNR value: (a) IA value of instantaneous frequency estimation of signal component x ðÞ t under different SNR value and (b) IA value of instantaneous frequency estimation of signal component x ðÞ t under different SNR value. 2 8 Advances in Mechanical Engineering Figure 8. Time domain waveform of simulation signal. Figure 10. Amplitude-frequency response of simulation signal. Figure 11. Time-frequency ridge line of signal components by Figure 9. STFT spectrum of simulation signal. MSCSR. f ðÞ t ¼ 20 + 10psinðÞ pt ð17Þ Considering the simulation signal stðÞ represents remarkable non-stationary property, we firstly use short-time Fourier transform (STFT) spectrum to determine the number of sub-signal as shown in Figure 9, six time-varying instantaneous frequencies, especially two time-varying carrier frequencies can be highly legi- ble, and the sub-signal’s side-bands have similar varying trend, which sufficiently reflects the coupling appear- ance in simulation signal. Furthermore, from Figure 10, compared with side frequency, the energy of carrier fre- Figure 12. Amplitude-frequency response of time-varying quency is more powerful, we will extract two carrier fre- adaptive filter H ðÞ s; t . quencies primarily. Considering above issues, we apply the proposed method on simulation signal to identify instantaneous central frequency and estimated side-band frequency is carrier frequency, the estimated frequency f (t), extract1 corresponding cutoff frequency, and the filter’s time- f (t) and correspondingly theoretical carrier fre- extract2 frequency response is shown in Figure 12. It can be seen quency f (t), f (t) are shown in Figure 11. It can be seen 1 2 intuitively that the amplitude-frequency property of that the proposed method in this paper also has good time-varying filter is very consistent with signal stðÞ. tracking accuracy for fast varying instantaneous fre- Considering amplitude-frequency property of simu- quency, as a result, other four side-band frequencies lation signal stðÞ in Figure 10, adopting time-varying f (t)2f (t), f (t)+ f (t), f (t)2f (t), f (t)+ f (t) are filter H ðÞ s; t decomposes this signal, it can well obtain 1 b 1 b 1 b 2 b 2 also estimated respectively. a signal s ðÞ t in Figure 13 corresponding to sub-signal p2 Additionally, a time-varying filter HðÞ s; t is designed s ðÞ t according to the following formula (18). 2 2 based on extracted frequency for decomposing simula- Compared with the theoretical simulation sub-signal tion signal stðÞ. According to formula (7) and (8), the s ðÞ t , whose time-frequency spectrum presents in estimated instantaneous carrier frequency is taken as Figure 9, the spectrum of signal s ðÞ t has some p2 Yan et al. 9 paper, we also use several different methods to analyze the signal and estimate its instantaneous frequency, the result is shown in Figure 17. It can be obviously seen from Figure 17 that except HHT, the estimated values corresponding to other methods can well match with the theoretical result. Moreover, the proposed method performs best in esti- mation accuracy, this conclusion can also be proved by the IA index as shown in Table 2. On the other hand, we analyze another fault vibra- tion signal as shown in Figure 18 under a large speed Figure 13. Filtered result by time-varying adaptive filter fluctuation condition which presents sinusoidal varying H ðÞ s; t . instantaneous frequency, and the estimated IF is shown in Figure 19. Although there is a certain deviation between the distortion, as the adaptive time-varying filter exists instantaneous frequency value identified by proposed amplitude loss, while the loss is not very serious from method and theoretical value, it is basically consistent Figure 13. with theoretical value on the trend, and the recognition effect is obviously better than SWT and HHT. s ðÞ t ¼ HðÞ s; t s ðÞ t ð18Þ p2 2 2 Although SWT can also identify the corresponding trend, the estimated instantaneous frequency curve still Considering the adaptive decomposition method contains a lot of burrs. Besides, the IA value in Table 3 owns unique advantage without presetting basis func- also can prove above conclusion. Furthermore, in order tion, we select empirical mode decomposition (EMD), to evaluate the efficiency of proposed method in experi- variational mode decomposition (VMD) for compari- ment, we set another evaluation index, which is the cal- son as shown in Figure 14. It can directly see that the culation time t, Table 4 shows the consumed time by number of sub-signals obtained by adaptive decompo- each methods. We can see that the proposed method in sition method is not equal to real number, and each this paper has certain advantage in computational effi- sub-component presents modulation attribute, which ciency, which indicates that the MSCSR can be applied indicates that the decomposition results reflect aliased to the field of rotating machinery online monitoring. phenomenon. In conclusion, we can know that when Considering that the rotating machinery works the signal has a fast varying IF, adopting signal decom- under the large speed fluctuation condition, its instan- position based on presetting a series of basis functions taneous rotating frequency often reflect significantly will be more realistic. non-linear property, in order to solve this problem, we can divide the instantaneous frequency into a series of segments for converting the global non-linearity into Experimental application local linearity, or we can use polynomial frequency fac- In this section, we use the collected rolling bearing fault tor instead of linear frequency factor in MSCSR whose vibration signal under variable speed condition as formula isftðÞ ¼ a + 2bt. shown in Figure 15 to validate the accuracy of the pro- posed method on instantaneous rotating frequency esti- mation. The rolling bearing fault test platform is Conclusion composed of AC motor, motor speed controller, rotat- An adaptive time-varying filter design method based ing shaft, loading system and experimental rolling bear- on instantaneous frequency estimation is proposed and ing, NI CompactDAQ. It can carry out rolling bearing applied for achieving mechanical varying non- fault experiment under different rotating speeds. The stationary signal separation. The proposed method is test rolling bearing is 52732QT and we adopt electric also validated by simulation signal, compared with discharge machining to cause outer ring damage. Here, other algorithms, the accuracy of proposed method is we generate two non-stationary signals, whose instanta- proved. Finally, an instantaneous rotating frequency neous rotating frequencies are in accordance with fast estimation experiment based on rolling bearing fault linear varying trend and sinusoidal varying trend vibration signal verifies application reliability. The respectively. main conclusion of this article is as follows. Firstly, we collect the fault vibration signal with line- arly varying frequency as shown in Figure 16 and the rotating speed increases from 900 to 1080 rad/min in (1) As the mechanical fault vibration signal pre- 6 s. Furthermore, except the proposed method in this sents significantly fast varying instantaneous 10 Advances in Mechanical Engineering Figure 14. (continued) Yan et al. 11 Figure 14. Decomposition result by VMD and EMD: (a) VMD decomposition, (b) amplitude-frequency spectrum corresponding to VMD decomposition, (c) EMD decomposition and (d) amplitude-frequency spectrum corresponding to EMD decomposition. Figure 15. Experimental bench for instantaneous frequency. Figure 17. The identified instantaneous frequency by using Figure 16. Fault vibration signal with linearly varying different methods. instantaneous frequency. 12 Advances in Mechanical Engineering Table 2. Identification effect of linearly varying instantaneous performance computing, high accuracy, and frequency by using IA index. strong noise resistance, the MSCSR using this basis function can effectively extract the instan- Method MSCSR SWT HHT taneous frequency that presents curvilinear varying law. Furthermore, an adaptive filter IA 1.1 2.3 83.5 designed with the extracted instantaneous fre- quency has high calculation efficiency which solves the problem that the classical filter needs to select filter’s parameters. Finally, the result proves that the proposed method can extract the modulation component and decomposing mechanical varying non-stationary signal. (2) Considering the instantaneous rotating fre- quency under large speed fluctuation condition shows fast varying nonlinear property, in order to use MSCSR to better estimate it, we propose two solutions which utilizes transformation idea and alternative idea respectively. As for the former, we can divide the globally nonlinear instantaneous frequency into several locally lin- Figure 18. Fault vibration signal with sinusoidal varying instantaneous frequency. ear instantaneous frequencies, and then use MSCSR to achieve accurate estimation. As for the latter, we can replace the linear frequency factor ftðÞ ¼ a + 2bt in the existing MSCSR with the polynomial frequency factor, which can better conform to the nonlinear varying frequency. 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 Figure 19. The identified instantaneous frequency presenting sinusoidal varying feature. The author(s) disclosed receipt of the following financial sup- port for the research, authorship, and/or publication of this article: This study is supported by the Lin-gang’s integration between industry and education public service project of Table 3. IA index identification effect of sinusoidal varying Shanghai Dianji University (22B0203) and The Ministry of instantaneous frequency on fault vibration signal. Education’s Cooperative Education Project (BINTECH- Method MSCSR SWT HHT KJZX-20220831-51). IA 8.4 15.2 108.7 ORCID iD Lu Yan https://orcid.org/0000-0003-3650-1504 Table 4. Computing time of different methods on References instantaneous frequency estimation. 1. Shi H, Gan C, Zhang X, et al. A fault diagnosis method for rolling bearings based on RDDAN under multivari- Method MSCSR SWT HHT able working conditions. Meas Sci Technol 2023; 34: t(s) 10 15 7 025003. 2. Choudhury MD, Blincoe K and Dhupia JS. 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Advances in Mechanical Engineering – SAGE
Published: May 1, 2023
Keywords: Fast varying signal; instantaneous frequency estimation; adaptive time-varying filter; multi-scale chirp sparse representation; mechanical fault diagnosis
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