Spatial transformation of general sampling-aliasing frequency region for rotating-blade parameter identification with emphasis on single-probe blade tip-timing measurement

Jihui CAO, Zhio YANG,*, Gungrong TENG, Shohu TIAN,Guoyong YE, Xuefeng CHEN

a School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

b Sichuan Gas Turbine Establishment Aero Engine Corporation of China, Mianyang 621000, China

c College of Mechanical and Electrical Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China

KEYWORDS Blade-tip timing (BTT);Extended time–frequency analysis;Image-feature recognition;Parameter extraction;Sampling-aliasing frequency map;Spatial transformation;Vibration analysis

Abstract Blade-health monitoring is intensely required for turbomachinery because of the high failure risk of rotating blades.Blade-Tip Timing(BTT)is considered as the most promising technique for operational blade-vibration monitoring,which obtains the parameters that characterize the blade condition from recorded signals.However,its application is hindered by severe undersampling and stringent probe layouts.An inappropriate probe layout can make most of the existing methods invalid or inaccurate.Additionally, a general conflict arises between the allowed and required layouts because of arrangement restrictions.For the sake of economy and safety, parameter identification based on fewer probes has been preferred by users.In this work, a spatial-transformation-based method for parameter identification is proposed based on a single-probe BTT measurement.To present the general Sampling-Aliasing Frequency (SAFE) map definition, the traditional time–frequency analysis methods are extended to a time-sampling frequency.Then, a SAFE map is projected onto a parameter space using spatial transformation to extract the slope and intercept parameters, which can be physically interpreted as an engine order and a natural frequency using coordinate transformation.Finally,the effectiveness and robustness of the proposed method are verified by simulations and experiments under uniformly and nonuniformly variable speed conditions.?2022 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

Rotating blade is a critical functional component of turbomachinery.However,it is extremely prone to fail because of harsh working conditions such as high speed, extreme temperature,and heavy load.1,2Therefore, blade monitoring is intensely required for safety and integrity of turbomachinery.

Vibration-based condition monitoring is a practical approach because vibration represents the source and manifestation of faults.From the blade vibrations, we can extract specific parameters, e.g., natural frequency and Engine Order(EO), which characterize the blade-health conditions.3,4Blade-Tip Timing (BTT) is a promising vibrationmeasurement method owing to its advantages such as noncontact and lower cost.5BTT measurement obtains all the blades’Time of Arrivals(TOAs)using probes mounted on the casing,which can be used to further derive the tip displacement.

However,severe undersampling and stringent probe layout are the main drawbacks of BTT measurement, which hinders its application.The sampling frequency of BTT measurement is determined by the limited rotation frequency and probe number,which means that the sampling frequency is relatively low due to these limitations.Because of the mismatch in the high target and low sampling frequencies, the BTT signal is severely undersampled.6.

Many multiprobe-based BTT methods have been proposed to overcome undersampling.However, excessive probe number is not feasible because of the high application cost.In the past decades,many advanced methods have been proposed and applied to BTT signal processing.Lin et al.7recovered an undersampled BTT signal and extracted the parameters using sparse reconstruction.Bouchain et al.8investigated a structured sparsity model for a blade-vibration spectrum.Wang et al.9used the Improved Multiple Signal Classification(IMUSIC) to extract the blade frequency.He et al.10proposed an improved BTT method that compensated for the probevibration effect.Li et al.11improved the iterative reweighted least square periodogram to reduce the computation and effect of aliasing.Fan et al.12used the blade-vibration difference instead of the blade vibration to reduce the probe number and eliminate the calculation error of the expected TOAs.

However, probe number still requires three to five probes per stage in existing applications,which leads to another challenge, namely, probe arrangement.Generally, the space inside the turbomachinery is extremely limited for installation13,especially in aero engines.Most multiprobe-based methods have strict requirements for the number and layout of probes.14An inappropriate probe layout will make most of the existing methods invalid or inaccurate.In addition, a conflict exists between the allowed and required layouts because of restrictions in the arrangement.15Furthermore,for the sake of costs and safety,the parameter-identification method based on fewer probes is preferred by users.16,17.

Motivated by the above reasons, we propose a Spatial-Transformation-based BTT (ST-BTT) method for parameter identification using a single probe, which is a creative work in the field of BTT because no literature exists on the method for directly identifying the frequency or EO using a single probe under extremely severe undersampling.In the present work, the traditional time–frequency analysis methods are extended to construct general Sampling-Aliasing Frequency(SAFE) maps, which subtly convert the parameter identification into image-feature recognition.Then, a bridge is constructed between the gap of the feature lines in the SAFE map and natural frequency (EO) of the blade.

The remainder of this paper is organized as follows.The BTT principle and the proposed method are described in Section 2.The numerical and experimental validations to demonstrate the effectiveness and robustness of the ST-BTT are presented in Sections 3 and 4, respectively.The conclusions are provided in Section 5.

2.Methodology

2.1.BTT principle

Fig.1 shows the principle of the BTT measurement.A Once-Per-Revolution (OPR) sensor is installed near the shaft to record the rotation frequency.The BTT probes are circumferentially arranged to record the TOAs of the blades.

In the absence of vibrations, the expected TOAs are determined by the position of the probes, rotational speed of the blades, and rotor radius.In the presence of vibrations, the blades arrive at the probes slightly earlier or later than the expected TOAs,and the tip displacement can be derived based on the differences, i.e.,

where y denotes the tip displacement, frdenotes the rotation frequency, t is the measured TOA, and texpis the expected TOA in the absence of vibrations, R denotes the disk radius,i.e., the distance from the blade tip to axis.

2.2.Single-probe BTT measurement and SAFE map

In this paper, we call the devices with independent sampling ability, such as acceleration sensors and strain gauges, as active-sampling sensors.However, BTT probes rely on the blade rotation to sample signals,which we call as passive sampling sensors.In the field of signal analysis, most of the analyzed signals are obtained by active sampling whose sampling frequencies are preset constants.The activesampling capability ensures that undistorted signals can be obtained by choosing an appropriate sampling rate and assisting with the anti-aliasing technique (low-pass filtering).BTT measurements are confronted with undersampling problems due to mismatch between the high natural frequency of the blade and low sampling frequency18.

Most traditional BTT analysis methods always use optimized probe layouts, additional prior, or complex algorithms to match the features from the aliasing signals to overcome undersampling.19However, the number and layout of probes are strictly limited in general, and additional prior is missing.Under a variable speed,the BTT measurement naturally forms a new dimension—sampling frequency—which can be obtained from time t and the speed curve.20Therefore, faand fsform a new plane to display the BTT signal, which also integrate the speed information.In other words, passive sampling is not always an obstacle for signal analysis.The traditional signal analysis comes from the perspective of time t, frequency f, and amplitude A.For the BTT signal analysis, we focus on SAFE.In this study, a SAFE map is constructed using the technique of consistent frequency resolution,which is a derivation of the time–frequency diagram.The connections and differences between the traditional and BTT signals are shown in Fig.2.

Fig.1 Principle of the BTT measurement.

Next, we introduce the single-probe BTT measurement.The signal recorded by the single probe is severely undersampled, and the natural frequency is aliased into [0, fs/2) according to Eq.(2).where fs?[2f,+∞) indicates that the sampling frequency satisfies the Shannon-Nyquist sampling theorem but is beyond the measurement ability of BTT.Thus, interval fs?(0,2f) is investigated.In Eq.(7),the interval(0,2f)consists of subintervals Akand Bkin which G(fs)can be observed as a linear function.The plot of function G(fs)is shown in Fig.3, where M is an arbitrary natural number.

Then,we associate G(fs)with the BTT signal.The sampling frequency of the BTT measurement is related to rotation frequency frand the probe number.In particular, sampling frequency fsis equal to rotation frequency frin the case of a single-probe layout, which means that a map such as G(fs)(Fig.3) can be constructed for a time-variable-speed BTT signal.

In this work,the map that reflects the relationship between the aliasing and sampling frequencies is called a SAFE map.In the SAFE map, the zero points, i.e., intersections of the polyline and horizontal axis, can represent the synchronous resonance centers because natural frequency f0is an integral multiple of fr.For the BTT signal, the absolute value of the polyline slope is physically interpreted as EO.Therefore, EO identification is equivalent to slope calculation in the SAFE map, which means that the probe number required to identify EO is reduced to one.In this work,parameter identification is premised on the construction of a SAFE map.In contrast to the traditional signal with a constant sampling frequency, the sampling frequency of the BTT signal for analysis is variable.The traditional time–frequency analysis method, e.g., discrete Short-Time Fourier Transform(STFT),analyzes the separated data using a fixed-length window function.

Fig.2 Connections and differences between traditional and BTT signals.

where ΔR represents the frequency resolution and NLdenotes the window length of h(n).

Under a fixed-length window,ΔR is inconsistent because of variable fs(fr), which prevents the result of Fourier transform in each interval from being spliced along the time direction to form a complete time–frequency map, as shown in Fig.4(a).Therefore, an Adaptive Window-length STFT (AW-STFT) is proposed to ensure consistent frequency resolution, as expressed in Eq.(10).

where variable ΔRfdenotes the frequency resolution, which is artificially set.

Fig.3 Plot of G(fs).

Fig.4 Single-probe BTT signal analysis.

The window length is inversely proportional to λs.And λsdecreases with the increase of fs,this is,the window length varies linearly with fs, which ensures consistent frequency resolution, i.e., fs/ NL= ΔRf, where NLdenotes the length of data intercepted in the window.

Further,the change in the rotation frequency is usually limited at small intervals.Thus, the average rotation frequency can be approximately regarded as the sampling frequency at a small-time interval.Furthermore, the time–frequency distribution obtained by AW-STFT（Fig.4(b)）can be converted into a SAFE map (Fig.4(c)) according to the time-varying speed information.

The flow of the BTT signal with a variable speed analyzed using AW-STFT is listed in Table 1.

fris divided into a series of subintervals to determine the center of the window,and the resolutions of fsand faare fixed,as shown in Fig.4(c).It needs to note that AW-STFT can be applied to both uniformly and nonuniformly variable speeds because of the known rotation frequency in the BTT measurements.In fact,the proposed method can also be further developed using BTT signals to generate a rotational speed to work without using an OPR sensor.Furthermore,the idea of consistent resolution can also be extended to the Winger-Ville Distribution (WVD) and Wavelet Transform (WT) by scaling the window function and wavelet basis, respectively, to construct a general SAFE map.

2.3.General SAFE and extended time-frequency analysis

In the process of constructing a SAFE map of a BTT signal,we find that many traditional time–frequency analysis methods can be extended to a time-sampling frequency (t, fs, fa).Incidentally, analysis from the sampling-frequency perspective is only useful for undersampled signals in which the frequency is aliased.Thus, the frequency is called aliasing frequency for time-sampling frequency and sampling-frequency analysis to emphasize its particularity.For the traditional signal obtained by active sampling, fsis considered constant.Thus, the timesampling-frequency analyses can be simplified into a time–frequency analysis.However, for a BTT signal, fsis a variable with rotational frequency, which defines a new dimension—sampling frequency of signal analysis.

In Section 2.2, we have described in detail how to extend STFT to AW-STFT based on the principle of constant resolution.Similarly, other time–frequency analysis methods canalso be generalized by introducing new dimensions—sampling frequency.In this study, we consider WVD and WT as additional examples to emphasize the general SAFE analysis.

Table 1 AW-STFT algorithm flow.

WVD is the most important and simplest Cohen class bilinear time–frequency distribution.An adaptive length window function related to fsis introduced into the traditional WVD to ensure consistent frequency resolution.The extended WVD, called adaptive window length WVD (AW-WVD) is defined as follows:

where λvis the window-length scaling factor of the WVD,which is defined as.

where fcdenotes the center frequency of the selected mother wavelet.

WT provides a new perspective, i.e., time scale, to analyze nonstationary signals.Because of finite support, the local signals can be analyzed by translation and scaling of the wavelet base.To obtain a consistent frequency resolution, original scale a is adjusted according to sampling frequency fs.Only single variable fsexists in the scaling factors (λs, λv,λw).Thus,the scaling factor can be expressed as.

where α is the generalized scaling factor and κ is a transformation constant related to the specific time–frequency analysis method.

SAFE map is actually a projection of the time-samplingfrequency distributions to the SAFE plane.Furthermore, fsis related to t according to the time-varying rotational frequency, which is obtained by an OPR sensor or BTT probe.Therefore, the time–frequency distribution can also be converted into a SAFE distribution, i.e., SAFE map.

where SPEC(t,f) denotes the time–frequency spectrogram,SAFE(fs,fa) denotes the SAFE spectrogram, which is called SAFE map in this study, and g(t) is a function of fswith respect to t.g-1(fs) is the inverse function of g(t), e.g., for BTT measurements, in the presence of an equispaced layout(including a single-probe layout), g(t) is expressed as.

where npdenotes the probe number and fr(t) represents the rotational frequency, which is an artificially set function of time.For the BTT measurement, fr(t) can be easily obtained from an OPR or BTT signal.

By analyzing the undersampled signal obtained by passive sampling,the importance of the sampling-frequency dimension for signal analysis is revealed,which is always ignored in traditional signals due to the constant sampling frequency and absence of undersampling.Similarly, in actual conditions,many objects, phenomena, and laws are observed due to imperfections, deficiencies, and abnormalities.

2.4.Spatial transformation and parameter identification

In this section, we present the identification of the parameters in the SAFE map.The feature line in the SAFE map is compressed to a parameter point using spatial transformation to identify the parameters(EO and natural frequency).In existing spatial-transformation methods, Hough Transform (HT) and Radon Transform (RT) are two commonly used methods to identify lines, which are discussed hereafter.

Both HT and RT transform the Cartesian- or polarcoordinate space into a parameter space that represents the characteristics of lines.HT is implemented based on a voting mechanism, and RT depends on the projection integral.Furthermore, binarization of the original image is necessary in HT.

A line can be represented by slope k and intercept b in the Cartesian-coordinate system.

(β,ρ)or(k,b)can form a parameter space in which a line is represented by a point, as shown in Fig.5.The point on the line in Cartesian-coordinate space can be expressed as a curve in the parameter space, for instance, ponit A (x1,y1) in Fig.5(a) is equal to the blue curve in Fig.5(b).The curves in Fig.5(b) corresponding to different points on the line in Fig.5(a)intersects at the intersection point (β0,ρ0) which is exactly the parameters of the line in Fig.5(b).Therefore, we can extract the parameters of polylines in the SAFE map by finding intersection points in the parameter space.In this study,HT and RT are used to construct parameter space and extract line parameters, respectively.

The HT and RT principles in a SAFE map are shown in Fig.6, which can be expressed by the following formula:

where R2denotes the image plane and f(x,y ) is the value at(x,y) of the image.Generally, for HT, f(x,y ) is binary.For RT, f(x,y ) is a gray value.δ(?) denotes the Dirac function.

After obtaining characteristic parameters (β,ρ) of a line through space transformation, we need to convert them into Cartesian-coordinate (X1O1Y1) parameters with the frequency value (fs= 0, fa= 0) as the origin, which is called the frequency origin (O1), to distinguish it from the image origin(O2).In this converted coordinate system, the slope and intercept of the line can be physically interpreted as EO and natural frequency fn,as shown in Fig.7.The slope and intercept of the line in(X1O1Y1)are calculated according to(β,ρ)in the polar coordinate with O2as the origin.The physical meanings of X1and Y1axes are (aliasing) frequency f and sampling frequency fs.

Fig.5 Spatial transformation.

Fig.6 Schematic of RT in a SAFE map.

The calculation process of physical slope and intercept is as follows:

where tan-1(?) is an inverse tangent function.

Fig.7 Four cases of line detection in a SAFE map.

We need to note that the image origin and positive direction defined by HT and RT are unfixed in different versions.For example, the origin of HT is at the upper left corner of the image in MATLAB.Therefore, Eq.(27) needs to be adjusted according to the image origin in the specific algorithm.

Furthermore, errors from measurement and method can affect the accuracy of the parameter identification, especially slope k.Fortunately, we can obtain additional information that k is ideally an integer.Therefore, the identified error of the slopes can be eliminated by rounding k, which can further correct the error in b.Finally, the estimation of EO and natural frequency are equal to the absolute values of k and b respectively.

3.Numerical validation

Numerical validations at uniformly and nonuniformly variable speeds were performed.The simulation validations were performed according to the following steps.

Step 1.A multiprobe layout, which satisfied the Nyquist-Shannon sampling theorem, was first employed for the BTT measurement of the blade vibrations.The nonundersampled signal was analyzed using Discrete Fourier Transform (DFT)to obtain the natural frequency,which served as an additional reference solution of the ST-BTT method.

Step 2.The signal recorded by one of the probes in the multiprobe system was used as the signal of ST-BTT.Then,SAFE maps were constructed using AW-STFT, AW-WVD, and ASWT.The natural frequency and EO were identified using spatial transformation, which were compared with the reference solutions to demonstrate the accuracy of the proposed method.

Step 3.Signals from the other single probes were individually analyzed to verify the robustness of ST-BTT by comparing the consistency of the results.

3.1.Simulation description

Verification was performed on a simulated signal based on the assumption that the response of the blade represented the superposition of several simple sinusoidal motions:

where x(t) is the displacement of the blade tip.f0, A0, and φ0represent the amplitude, frequency, and phase of the naturalfrequency component, respectively.fi,Ai, and φi(i ≥1)represent the amplitude, frequency, and phase of the ith harmonic components of the rotation frequency, respectively.N(t)denotes the noise term.

The specific parameters of the simulation signal are listed in Table 2.

3.2.Uniformly variable speed

The blade vibrations obtained by 12 equispaced probes at a uniformly variable speed are shown in Fig.8.Theoretically,the sampling frequency satisfies the Shannon-Nyquist sampling theorem when the rotation frequency exceeds 140 Hz,according to Eq.(18).Thus, we selected the data near 180 Hz for the spectrum analysis.In the spectrum obtained by DFT, we could observe the rotation frequency and its harmonics,namely,179.6,359.3,and 718.5 Hz,as well as the natural frequency of 875.7 Hz.

Table 2 Parameters of the sinusoidal signal superposition model.

Fig.8 Simulation signal recorded by 12 equispaced probes.

However, installing numerous probes on the rotor at each stage is impractical.An extreme case with the displacement recorded by a single probe and its spectrum was provided(Fig.9).Excessive undersampling led to serious frequency aliasing where accurately identifying the frequency became challenging.

3.2.1.Construction of SAFE map

To overcome the challenge caused by severe undersampling,SAFE maps were constructed using AW-STFT, AW-WVD,and AS-WT to identify EO and the frequency from extremely undersampled BTT signals,as shown in Fig.10.The trend was removed in the raw data recorded by a single probe, which eliminated the bright band at the bottom of SAFE map and highlighted the slanted polylines.The same operation was performed during the construction of SAFE maps in the following.

To prove that the ST-BTT method does not depend on the resonance, the SAFE map was cut to remove the resonance center.Furthermore, the SAFE map was locally enlarged to improve the accuracy of the parameter identification.The adjusted SAFE map was converted into a gray image, which represented the input of the subsequent spatial transformation(Fig.11).

3.2.2.Spatial transformation and parameter identification

First, the adjusted SAFE maps (Fig.11) were projected from Cartesian spaces to parameter spaces using RT, as shown in Fig.12 and Fig.13.The lines in the SAFE maps represented the peak points in the parameter spaces.Furthermore, we could observe that the peaks in the parameter spaces of RT in the SAFE map obtained by AW-WVD were more significant due to the remarkable local aggregation of WVD.

Then, (β,ρ) marked with white in parameters spaces could be determined by searching the peaks in the parameter spaces,which were further converted into(k,b)in the Cartesian coordinate with a frequency origin to bridge the gap between the feature lines in the SAFE map and natural frequency (EO)of the blade.The parameter identification results of the simulation signal are listed in Table 3.Hough transformation is another important spatial transformation method, which is also used to obtain the characteristic parameters, as shown in Fig.14.

Fig.9 Simulation signal recorded by a single probe.

Fig.10 Simulated SAFE maps obtained by different methods.

Fig.11 Inputs of the spatial transformation in the simulation.

Fig.12 Parameter spaces of RT of the AW-STFT simulated SAFE map.

Estimated slopes k were not integers because of the limited resolutions of the frequency and image.This error in k could be corrected by rounding.The absolute value of adjusted k was physically interpreted as EO.Further, natural frequency fnwas calculated using Eq.(27).We can see that the natural frequencies obtained by ST-BTT were close to the reference solution of 875 Hz, and the maximum error was less than 0.5 %, as shown in Fig.15.

The SAFE maps constructed using the different extended time–frequency methods (AW-STFT, AW-WVD, and AWST) could be used as input to ST-BTT.In addition, each line in the SAFE map could be calculated to obtain EO and fn.As the line number increased,i.e.,EO decreased,the identification error gradually decreased, which could be attributed to the high sampling-frequency resolution and significant energy at high rotation frequency.

Fig.13 Parameter spaces of RT of the simulated SAFE map obtained by AW-WVD and AS-WT.

Table 3 Parameter identification results of simulation signal in a uniformly variable speed.

Fig.14 Parameter spaces of HT of the AW-STFT simulated SAFE map.

Fig.15 Natural frequency identification errors of the simulation signal at a uniformly variable speed.

Furthermore, the robustness of the proposed method was first verified by analyzing different signals from other single probes.The results of parameter-identification (EO and fn)from 12 individual probes were stacked in Fig.16, which are marked with circles of different colors.Some circles overlapped because of the same parameters.The error bar denoted the mean and variance of 12 parameters, which was used to show the accuracy and consistency of results from different probes.In the stack plots,the identification results from different single probes were closely packed,which demonstrated the robustness of the SP-BTT method.Therefore, ST-BTT could be expected to extract EO and the natural frequency using a single sensor at any position without any additional prior information.

3.3.Nonuniformly variable speed

Fig.16 Stack plots of the parameter identification results from different single probes at a uniformly variable speed.

Fig.17 Performances of ST-BTT versus speed fluctuation and SNR.

3.3.1.Construction of the SAFE map

Because an S-shaped curve is popular in the operation of mechanical equipment owing to its smoothness, it was used in this simulation.The displacement recorded by a single probe and its SAFE map are shown in Fig.18.

3.3.2.Spatial transformation and parameter identification

Similarly,the SAFE maps were projected to a parameter space using HT and RT,and EO and the natural frequency were calculated according to the parameters listed in Table 4.

Slopes k of the lines are non-integers,which approached the corresponding EOs.The limited slope errors (<0.5) ensured accuracy of EO obtained by rounding.Furthermore,the natural frequencies were calculated from the SAFE maps constructed using AW-STFT, AW-WVD, and AW-ST.The maximum error was 0.43 % (3.8 Hz) compared with that of the reference solution, which was perfectly acceptable, as shown in Fig.19.

Then, the stack plots of the parameter identification from the different single probes are shown in Fig.20 to demonstrate the robustness of ST-BTT.The estimated results from the different probes that were collected together approached those of the references.

4.Experimental validation

Experimental validations were performed according to the following steps.

Step 1.A multiprobe layout was first employed for the BTT measurement of the vibrations of a bladed disk.Then, IMUSIC was used to calculate the natural frequency of each blade by analyzing the multiprobe signals, which served as a reference for the ST-BTT method.

Step 2.The signal recorded by one of the probes in the multiprobe system was used as the ST-BTT signal.Then,EOs and the natural frequencies were calculated from the SAFE maps using spatial transformation, which were compared with the reference solutions to demonstrate the accuracy of the STBTT method.

Fig.18 Simulation signal recorded by a single probe at a nonuniformly variable speed.

Table 4 Parameter identification results of the simulation signal at a nonuniformly variable speed.

Step 3.The signals from the other single probes were individually analyzed to verify the robustness of ST-BTT by comparing the consistency of the results.

4.1.Experiment description

This section presents the experimental validations at both uniformly and nonuniformly variable speeds, which were conducted to verify the effectiveness and robustness of STBTT.The probe layout and bladed disk used in the experiment are shown in Fig.21.Blades 3 and 7 of the bladed disk were preset with damages.

4.2.Uniformly variable speed

The experimental signal at a uniformly variable speed is shown in Fig.22 where the acceleration of frwas set to 1 Hz/s.In addition, the multiprobe signals at a constant speed (183 Hz)were input to IMUSIC to calculate the reference value.

4.2.1.Construction of the SAFE map

SAFE maps were constructed using AW-STF,AW-WVD,and AS-WT to identify EO and the frequency from the extremely undersampled BTT signals, as shown in Fig.23.

Fig.24 is the grayed image of the partial SAFE map.Using Fig.24 as the input of spatial transformation can avoid or reduce the influences of highlighted part at the bottom and line length on identification accuracy.

4.2.2.Spatial transformation and parameter identification

First, the adjusted SAFE maps (Fig.24) were converted into parameter spaces (Fig.25 and Fig.26) using Radon transformation.Another parameter space obtained by HT is shown in Fig.27.Then, (β,ρ) were obtained by searching the peaks in the parameter spaces.

Fig.19 Natural frequency identification errors of the simulation signal at a nonuniformly variable speed.

Fig.20 Stack plots of the parameter-identification results from different single probes at a nonuniformly variable speed.

Then, line features (β,ρ) in the parameter space were converted into (k,b) in the Cartesian coordinate with a frequency origin to bridge the gap between the feature lines in the SAFE map and natural frequency(EO)of the blade.Table 5 lists the parameters calculated by ST-BTT where natural frequencies fnapproached those of the reference.The maximum error was - 1.44 % (-12.7 Hz), as shown in Fig.28.Furthermore,the calculated natural frequencies were slightly lower than those of the reference obtained by IMUSIC,and the difference generally decreased with the increase in EO, which could be attributed to the sampling-frequency resolution, rotation effect, and significant energy at a high rotation frequency.

Furthermore,the stack plots of the results from blade 1 are shown in Fig.29 to illustrate the robustness of SP-BTT.The estimated parameters (EO and fn) from the different single probes were approximate and indicated that ST-BTT is effective and robust.

Finally, fnof all blades identified by ST-BTT are shown in Fig.30.The results from each single probe were close to the reference obtained by IMUSIC with a similar tendency.The maximum error was-1.28%,which came from the damaged blade (Blade 7), which is perfectly acceptable.

4.3.Nonuniformly variable speed

In the nonuniformly variable speed experiment,frwas set as an S-curve.The displacement and rotation frequency are shown in Fig.31.

4.3.1.Construction of the SAFE map

The proposed methods (AW-STFT, AW-WVD, and AS-WT)for constructing a SAFE map are suitable for both uniformly and nonuniformly variable speeds.Therefore,the SAFE maps from the signals in an S-shaped speed are shown in Fig.32.

We note that the resonance traces in the middle part of the SAFE maps was divergent due to rapidly changing fr.Therefore, the acceleration must be < 3 Hz/s for our blades whose frequency was close to 875 Hz.A 0.1–1.5 Hz/s acceleration was then recommended.

4.3.2.Spatial transformation and parameter identification

Similarly, the SAFE maps were projected onto the parameter space using HT and RT, and EO and the natural frequency were calculated using Eq.(27).The calculated results of fnapproached those of the reference.The maximum error was - 1.65 % (-14.62 Hz), which was slightly higher than the error(-1.44%)at the uniformly variable speed.However,the error at the nonuniformly variable speed (Fig.33)approached that of the uniformly variable speed(Fig.28)with a similar trend.

Fig.21 Experimental preparation.

Fig.22 Experimental signal at a uniformly variable speed.

Fig.23 Experimental SAFE maps at a uniformly variable speed obtained by different methods.

Fig.24 Inputs of the spatial transformation in the experimental.

The stack plots of the parameter identification from the different single probes are shown in Fig.34 to demonstrate the robustness of ST-BTT.The maximum difference between the frequency from different individual probes are 2.3 Hz(0.68 %), which indicates the consistency of results between different probes.

Finally,the fnvalues of all blades identified by ST-BTT are shown in Fig.35, which were close to those of the references.The maximum error was - 1.46 %, which is perfectly acceptable.

Fig.25 Parameter spaces of RT of the AW-STFT experimental SAFE map.

Fig.27 Parameter spaces of HT of the AW-STFT experimental SAFE map.

Table 5 Parameter-identification results of the experimental signal at a uniformly variable speed.

Fig.28 Natural frequency identification errors of the experimental signal at a uniformly variable speed(the values are reversed for better observation).

Fig.29 Stack plots of experimental Blade 1 from different single probes at a uniformly variable speed.

Fig.30 Results extracted by ST-BTT at a uniformly variable speed.

Fig.31 Experimental signal at a nonuniformly variable speed.

Fig.32 Experimental SAFE maps at a nonuniformly variable speed obtained by different methods.

Fig.33 Natural frequency identification errors of the experimental signal at a nonuniformly variable speed (the values are reversed for better observation).

Fig.34 Stack plots of experimental Blade 1 from the different single probes at a nonuniformly variable speed.

Fig.35 Results extracted by ST-BTT at a nonuniformly variable speed.

5.Conclusions

This paper has proposed a spatial-transformation-based method for parameter identification using single-probe BTT.SAFE map is the intermediate product of ST-BTT,which subtly converts the parameter identification into image-feature recognition.

First, by analyzing the undersampled BTT signal obtained by passive sampling,the importance of the sampling-frequency dimension for signal analysis is revealed, which is always ignored in the traditional signals due to a constant sampling frequency and absence of undersampling.Therefore,the traditional time–frequency analysis methods (STFT, WVD, and WT) are extended to the time-sampling frequency, which serves as the basis for constructing the SAFE maps.

Then, the SAFE map is projected onto a parameter space using spatial transformation where the lines are compressed into points.Parameters (β,ρ) of the line can be determined by searching the peaks in the parameter spaces,which are further converted into (k,b) in the Cartesian coordinate with a frequency origin to bridge the gap between the feature lines in the SAFE map and natural frequencies (EOs) of the blade.We point out that the slope and intercept of the line can be physically interpreted as EO and natural frequency fnin this converted coordinate system.

Finally,the effectiveness and robustness of ST-BTT are verified by simulations and experiments at uniformly and nonuniformly variable speeds.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Key Research and Development Program of China (No.2020YFB2010800), the National Natural Science Foundation of China (Nos.51875433 and 92060302), the Natural Science Foundation of Shaanxi Province,China (No.2019KJXX-043, 2021JC-04),the Fundamental Research Funds for the Central Universities and the Foundation of Beilin District ,China(No.GX2029).