Blade tip timing signals parameters identification based on Displacement Offset Decreased and Compressed Sensing

Abstract

High-speed rotating blade vibration monitoring is of great significance for the safe operation of turbomachinery. Blade Tip Timing (BTT) is considered a promising technology in the field of blade vibration monitoring. The BTT signal analysis has the following problem, the displacement offset was caused by speed variation which affected the accuracy of the blade vibration parameter identification. Owing to the under-sampled characteristics of the BTT signals and multiple sensor data are required to identify the parameters. The accurate extraction of blade vibration characteristics is challenging. Thus, a vibration analysis method is proposed based on Displacement Offset Decreased and Compressed Sensing (DOD-CS). Firstly, the BTT signal pairwise is subtracted to reduce the impact of the displacement offset, and based on the subtractive results, a basis function is constructed to rebuild the dictionary matrix of the compressed sensing method. Then, we solved the signal sparse representation equation and obtained the spectrum estimation result, that is, the structure vibration parameters. The analysis of simulation data shows that the parameter identification accuracy of the DOD-CS method is higher than that of the OMP-CS (Orthogonal Matching Pursuit and Compressed Sensing) method, and the identification errors of frequency and amplitude are within 1% and 5%, respectively. The experimental results confirm the feasibility of the proposed method in identifying vibration parameters. Compared with existing methods, the innovation of the proposed method is that it eliminates the influence of displacement offset on the accuracy of parameter identification. For under-sampled signals, only two sensor data information are needed to complete spectrum estimation and parameter identification.

Introduction

Rotating blades are critical components in turbomachinery, such as gas turbine engines and aircraft engines, and they achieve energy conversion within the unit during the operation of equipment. They often experience severe vibrations due to various exciting forces, including centrifugal force and aerodynamic forces. ¹ High-cycle fatigue vibration causes blade fatigue failures, such as cracks and even fractures. Therefore, it is necessary to monitor the operating conditions of the blades. Accurately identifying the blade vibration parameters is important for ensuring the safe operation of the equipment.

Blade health monitoring technology is used for condition monitoring of turbine and compressor blades, finding abnormal changes, and failure warnings. The conventional technology for blade vibration measurement mainly involves SG (strain gauges), ² FM (frequency modulation) grids, ³ and SLDV (Scanning Laser Doppler Vibrometer). ⁴ The SG technology extracts alternating strain information arising in key areas of the blade. This technology has the following limitations, including low operating life, and high testing costs, and only a few blades can be monitored online. In addition, strain gauges attached to the blades may alter the structural characteristics of the blades. The FM is a non-contacting and intrusive measurement system, it can only provide vibration information for blades with magnets installed, and using cost is relatively high. The SLDV technology perceives blade vibration by detecting the Doppler frequency of reflected light from the blade surface and is only suitable for monitoring blade vibration at low speeds. In recent years, the BTT (Blade Tip Timing) ⁵ technology has been widely used in this field owing to its non-contact and non-invasive nature. This technology is convenient to use, simple structure, and has lower testing costs. Only a few sensors are required to simultaneously measure the vibration of all blades online. It detects the arrival time of the blade by using sensors mounted on a rig casing. The blade displacement can be calculated according to the arrival time difference to analyze the vibration characteristics. The sampling rate of the BTT depends on the current speed and the number of sensors, and it is lower than the blade vibration frequency. The BTT signal is a typical under-sampled signal.

The common BTT signal processing methods include the single parameter method, ⁶ the two-parameter plots method, ⁷ autoregressive method, ⁸ sine fitting, ⁹ ap-FFT, ¹⁰ and all-blade spectral analysis methods. ¹¹ The single-parameter method uses one sensor to monitor the blade amplitude. Two-parameter plots require two sensors, and the identification results are significantly affected by noise and speed fluctuations. Autoregressive and sine fitting methods require the installation of four sensors at specific angles, with a frequency and amplitude identification error of less than 10%. The above methods have strict limitations in terms of the arrangement and number of sensors. For the optimization and improvement of general methods,^7–9 Mohamed et al. ¹² applied the multi-frequency sine fitting method to identify synchronous vibration parameters, with frequency and amplitude identification errors within 5% and 20%, respectively. Rzadkowski et al. ¹³ proposed the nonlinear least squares method for the analysis of BTT signals and completed parameter identification using information from two sensors. Compared with the least squares method, the amplitude identification accuracy was improved by 10%. Fan et al. ¹⁴ proposed an improved dual parameter method based on amplitude difference, which does not require a speed sensor compared to the original method.

The BTT signal measurement, data processing, and analysis involve uncertainties. The blade displacement offset is one of the uncertainties arising from the influence of aerodynamic loads that vary with the rotational speed. ¹⁵ The windward side of the blade bears more flow loads, causing the blade to bend and displacement offset. Before using the BTT algorithm for parameter identification, the displacement offset must be removed from the BTT signal, which may weaken the resonance peak amplitude and affect the accuracy of the frequency and amplitude identification. ¹⁶ The methods for removing the displacement offset include cubic spline fitting and the Savitzky Golay filter. ¹⁷

For the parameter estimation of under-sampled signals, Pan et al. and Spada et al. proposed a new signal analysis theory called Compressed Sensing (CS).^18,19 This method is not limited by the Nyquist sampling frequency and expands the signal analysis bandwidth. It can reconstruct the original signal with limited sensor information. Kharyton and Zachariah ²⁰ constructed a measurement matrix using a single sine and cosine function as a dictionary atom, and used sparse iterative covariance estimation to solve the sparse representation model of the BTT signal. Bouchain et al. ²¹ considered the multi-mode vibration of blades selected the superposition of multiple sine and cosine functions as dictionary atoms, and used the block orthogonal matching pursuit (OMP) method for BTT signal model solving. Lin et al. applied compressive sensing to blade vibration monitoring and crack identification, ²² focusing on the blind reconstruction of under-sampled multi-frequency blade vibration signals.

In addition to the above methods, scholars use frequency estimation methods for the BTT signal analysis. Kharyton et al. ²³ compared the parameter identification capabilities of several analytical methods, which consist of NUDFT, ²⁴ Lomb-Scargle Periodogram, ²⁵ Minimum variance spectrum estimation (MVSE), ²⁶ I-Channel Interpolation, ²⁷ Iterative variable threshold, ²⁸ and Quasi cross-spectrum estimate. The NUDFT and MVSE methods can find the modal frequencies in the medium and high bands and have certain recognition capabilities for small amplitude asynchronous vibrations. These two methods have advantages in dealing with non-uniform signal recognition. Carassale et al. ²⁹ separates and identifies multi-frequency signals under harmonic interference by ICA. Salhi et al. ³⁰ studied a subspace algorithm to reconstruct under-sampled signals, which solved the parameter identification under variable speed and speed fluctuation.

There are two problems with the parameter identification of BTT signals. First, the displacement offset that occurs with the speed variation in the signal affects the accuracy of parameter identification. Secondly, owing to the under-sampled characteristics of the BTT signals, multiple sensors (at least four sensors) must be arranged at specific angles to accurately identify vibration parameters using the BTT algorithm. To address these problems, this study proposes a parameter identification method based on Displacement Offset Decreased and Compressed Sensing (DOD-CS). The DOD-CS method first subtracts the signals of two sensors to remove the displacement offset; and second, based on compressive sensing theory, constructs basis functions and dictionary matrices by subtractive signals. The sparse representation equation of signals is solved to obtain the spectrum estimation results. The innovation of this method is that it can directly process the original signal, and not require the step of removing displacement offset from the original signal. It can eliminate the influence of displacement offset on signal parameter recognition. For BTT under-sampled signals, only two sensor information can be used to complete signal spectrum recovery and parameter identification.

The principles of the BTT and the DOD-CS method are explained from the perspective of signal reconstruction and model solution in section “Method.” The feasibility of DOD-CS was verified through numerical simulations and experiments, as described in sections “Numerical simulation and verification” and “Experimental data analysis and verification.” A conclusion is presented in section “Conclusion.”

Method

Principle of BTT measurement

The basic principle of the BTT technology is to measure the blade arrival time using BTT sensors mounted on the rotor casing and a rotating speed sensor near the shaft. When the impeller rotates, the BTT sensor detects the passage of blades and generates pulse signals. The principle of the BTT measurement is shown in Figure 1.

Figure 1.

The principle of BTT measurement.

During rotation, when the blade vibrates, it generates a forward or backward vibration displacement. The blade arrival time through the BTT sensor is either ahead or behind compared to the absence of vibration. The solid signal in Figure 1 represents the pulse signal collected by the sensor when the blade is not vibrating and t_bs0 represents the theoretical arrival time of the blade. The dashed line represents the pulse signal collected by the sensor during blade vibration, t_nbs represents the actual arrival time of the blade, and T_n denotes the rotational period. The blade vibration displacement is calculated as follows:

$$y_{\mathrm{nbs}}=\frac{2\pi r(t_{\mathrm{nbs}}-t_{\mathrm{bs}0})}{T_n}$$

(1)

Where y_nbs is the blade vibration displacement vector in which the b-th blade passes through the s-th sensor during the n-th circle, and r is the impeller radius.

DOD-CS method

Blade vibration can be expressed through simple harmonic motion, and the vibration equation is written as

$$y=\sum _{i=1}^Q[A_i\sin (2\pi f_{i}t)+B_i\cos (2\pi f_{i}t)]+C$$

(2)

Where y is the blade vibration displacement, Q is the number of vibration modes, t is the time, f_i is the natural frequency of the blade corresponding to the i-th mode, A_i and B_i are the amplitude coefficients, and C is the DC component.

With one revolution, the blade vibration displacements are y_i and y_j through the i-th and j-th sensors, respectively, corresponding to the arrival times of t_i and t_j. Where f denotes blade vibration frequency. Assuming that the blade amplitudes are the same, the blade vibration displacement is represented as

$$y_i=A\sin (2\pi f{t}_i)+B\cos (2\pi f{t}_i)+C$$

(3)

$$y_j=A\sin (2\pi f{t}_j)+B\cos (2\pi f{t}_j)+C$$

(4)

The Sum-to-Product Identities are shown as

$$\sin (\alpha )-\sin (\beta )=2\cos \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$$

(5)

$$\cos (\alpha )-\cos (\beta )=-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$$

(6)

According to equations (5) and (6), the blade vibration displacement difference measured by the two sensors can be expressed as

$$\begin{array}{c}{y}_j-y_i=2\sin \pi f(t_j-t_i)*\hfill \\ (A\cos \pi f(t_j+t_i)-B\sin \pi f(t_j+t_i))\hfill \end{array}$$

(7)

Assuming that λ_ji = 2sin(πf(t_j−t_i)) and φ_ji = πf(t_j + t_i), the blade vibration displacement difference can be expressed as

$$y_j-y_i={\lambda }_{\mathrm{ji}}(A\cos {\phi }_{\mathrm{ji}}-B\sin {\phi }_{\mathrm{ji}})$$

(8)

The y_j−y_i can be written as the product of vectors as follows

$$y_j-y_i=\left[\begin{array}{cc}{\lambda }_{\mathrm{ji}}\cos {\phi }_{\mathrm{ji}}\hfill & {\lambda }_{\mathrm{ji}}\sin {\phi }_{\mathrm{ji}}\hfill \end{array}\right]·\left[\begin{array}{c}A\hfill \\ \multicolumn{1}{c}{-B}\end{array}\right]$$

(9)

According to equation (9), λ_jicosφ_ji and λ_ji sin φ_ji are selected as basis functions to construct the CS matrix Φ. The frequency range f = [f₁, f₂, …, f_n] and matrix Φ are expressed as

$$\begin{array}{c}\mathrm{\Phi }=\hfill \\ \left[\begin{array}{ccccc}{\lambda }_{111}\cos ({\phi }_{111})\hfill & {\lambda }_{111}\sin ({\phi }_{111})\hfill & \cdots \hfill & {\lambda }_{11n}\cos ({\phi }_{11n})\hfill & {\lambda }_{11n}\sin ({\phi }_{11n})\hfill \\ \multicolumn{1}{c}{{\lambda }_{121}\cos ({\phi }_{121})}& {\lambda }_{121}\sin ({\phi }_{121})\hfill & \cdots \hfill & {\lambda }_{12n}\cos ({\phi }_{12n})\hfill & {\lambda }_{12n}\sin ({\phi }_{12n})\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & \hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{{\lambda }_{\mathrm{ij}1}\cos ({\phi }_{\mathrm{ij}1})}& {\lambda }_{\mathrm{ij}1}\sin ({\phi }_{\mathrm{ij}1})\hfill & \cdots \hfill & {\lambda }_{\mathrm{ijn}}\cos ({\phi }_{\mathrm{ijn}})\hfill & {\lambda }_{\mathrm{ijn}}\sin ({\phi }_{\mathrm{ijn}})\hfill \end{array}\right]\hfill \end{array}$$

(10)

Where λ_jin = 2sin(πf_n*(t_j−t_i)), φ_jin = πf_n*(t_j + t_i).

Next, we built a compressed sensing model for signals based on the known input signal and CS matrix Φ. Assuming that virtual BTT sensors are installed in the casing and the number is I, the sampling frequency of the BTT system can satisfy the Nyquist sampling theorem. N points were used to represent the blade vibration signal Y_N measured by I virtual sensors over a period of time. If the representation of the blade vibration signal Y_N under dictionary D is sparse, it is represented as

$$Y_N=D·s$$

(11)

Where s is the coefficient vector, containing a finite number of non-zero elements.

The actual sensor arrangement of the BTT measurement system can be regarded as selecting J positions from I virtual sensors, J << I. The BTT measurement result can be regarded as the extraction of M data points from the ideal signal Y_N. The signal Y_M is expressed as

$$Y_M=H·Y_N$$

(12)

Where H denotes the sensor position matrix. According to equations (11) and (12), the under-sampled BTT signal Y_M can be represented as the product of the CS matrix Φ and the coefficient vector s. It is represented as

$$Y_M=H·D·s=\mathrm{\Phi }·s$$

(13)

The number of unknowns in the above equation is larger than the number of measurements and is a typical underdetermined system. The solution of the equation can be transformed into an L−1 norm minimization problem, which is expressed as follows:

$$\arg min\frac{1}{2}\Vert Y_M-\mathrm{\Phi }·s{\Vert }_2^2+\mu \Vert s{\Vert }_1$$

(14)

Where μ is the regularization parameter.

The sparse representation of the blade vibration signal is shown in equation (13), where the CS matrix Φ is constructed in equation (10), and the corresponding measurement signal Y_M is the signal difference between the two sensors, Y_M = y_j−y_i. Signal reconstruction algorithms can be used to solve the equation (14). To balance the solution efficiency and accuracy, the Compressive Sampling Matching Pursuit method was adopted in the greedy algorithm. Based on the compressed sensing model presented in this section, solving the equation yields the coefficient vector s, which is the recovered signal spectrum.

There are the following explanations about the issue of DOD features embedded in the CS model. The CS model includes the construction of a dictionary matrix and equation solving. According to the blade vibration, it can be expressed in a harmonic motion as shown in equation (2), the dictionary atoms are generally selected as sine and cosine functions. Considering the blade vibration displacement offset, multiple sine and cosine functions were selected to construct a dictionary matrix and solve the equation, resulting in incorrect frequency results. Therefore, a new method for constructing dictionary matrices has been proposed. The two sensor signals are subtracted to remove displacement offset, and the subtractive results are used as a dictionary atom, which is the DOD feature. As shown in equation (9), the subtracted signal can be expressed as the product of DOD features and corresponding coefficient vectors. The vibration frequency parameters can be obtained by solving the equation in equation (14).

The use of regularization parameters in equation (14) ensures the accurate identification of the vibration frequency, and the reconstruction accuracy of the amplitude decreases. The least-squares method was used to calculate the amplitude. Taking the blade single-mode vibration, as an example, it is given by equation (15). The blade vibration measurement displacement y₀ and the coefficient matrix D are represented as

$$y=A\sin (2\pi \mathrm{ft})+B\cos (2\pi \mathrm{ft})+C$$

(15)

$$y_0=\left[\begin{array}{c}y\left[1\right]\hfill \\ \multicolumn{1}{c}{y\left[2\right]}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{y\left[M\right]}\end{array}\right]$$

(16)

$$D=\left[\begin{array}{ccc}\sin (2\pi f_1{t}_1)\hfill & \cos (2\pi f_1{t}_1)\hfill & 1\hfill \\ \multicolumn{1}{c}{\sin (2\pi f_1{t}_2)}& \cos (2\pi f_1{t}_2)\hfill & 1\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{\sin (2\pi f_1{t}_M)}& \cos (2\pi f_1{t}_M)\hfill & 1\hfill \end{array}\right]$$

(17)

Where my[1], y[2], …, y[M] are the blade vibration displacement sequences sampled at times t₁, t₂, …, t_M. The coefficient matrix x is solved as follows:

$$x=\left[\begin{array}{c}A\hfill \\ \multicolumn{1}{c}{B}\\ \multicolumn{1}{c}{C}\end{array}\right]$$

(18)

The coefficient matrix x can be solved using the least squares method and is calculated as

$$x={\left(D^{T}D\right)}^{-1}\left(D^T{y}_0\right)$$

(19)

The blade vibration frequency f is known, and the corresponding amplitude and phase can be calculated using x as follows:

$$P=\sqrt{A^2+B^2},\phi =-\arctan (B/A)$$

(20)

Through this process, the blade vibration frequency f, amplitude P, phase j, and DC component C are obtained. The steps for parameter identification based on compressed sensing are as follows:

• (1) Collecting blade vibration displacement sequence y by BTT measurement system.

• (2) Calculate the blade vibration displacement difference y_j−y_i from two sensor signals and construct a matrix Φ.

• (3) The compressive sampling matching pursuit algorithm is used to solve the signal sparse representation in equation (6) and complete the signal spectrum estimation and the identification of the blade vibration frequency.

• (4) Using the least squares method to calculate blade amplitude.

Numerical simulation and verification

To verify the performance of the proposed method, in terms of precision of parameter identification and anti-noise ability, the blade disk simulation models proposed in Dimitriadis et al. ³¹ were used to output the simulation signal.

Consider the following two situations to compare the proposed method: (1) Comparison of identification results with the OMP-CS method. ²¹ (2) Parameter identification accuracy under different noise levels.

There are two sets of simulation data. The first set is mistuned blade vibration data which contains two synchronous vibrations. In the simulation model, each blade has the same mass but different stiffness, and the coupling stiffness between blades is considered to simulate mistuned. The second group is blade asynchronous vibration data. The simulation parameters are listed in Table 1. The time-domain diagrams of the two sets of simulation data are shown in Figure 2(a) and (b). The sweep rate of the rotating speed is 0.025 Hz/s.

Table 1.

Signal simulation parameter configuration.

Data set	1	2
Frequency (Hz)	180, 182	320
Engine order	2	3.2
Amplitude (mm)	1	1
Sensor angle (°)	68.5, 120.1, 198.5	114, 167.2
Rotating speed range (Hz)	[85, 95]	[95, 105]
Sweep frequency time (s)	400
Adding zero drift (mm)	0.5

Figure 2.

Time domain of simulation data: (a) synchronous vibration and (b) asynchronous vibration.

Comparison of results with OMP-CS method

Using the first set of data as an example to illustrate the parameter identification process. First of all, two sensor signals were subtracted as the inputs for the signal sparse representation equation. The signal to be analyzed is centered around the maximum resonance amplitude, with a signal length of 20 cycles. Secondly, the CS matrix Φ is calculated based on the blade arrival time information and frequency range, which should cover the blade’s natural frequency. The frequency range of the first set of data is [0, 400 Hz]. The Compressive Sampling Matching Pursuit method is used to solve equation (14), the number of iterations is 50. The sparse vector s obtained is the spectrum estimation result. The analysis results of the proposed method and OMP-CS method are shown in Figures 3 and 4. The dashed line represents the theoretical value and the solid line represents the estimated value.

Figure 3.

The spectrum estimation results by DOD-CS method: (a) peak1 and (b) peak2.

Figure 4.

The spectrum estimation results by OMP-CS method: (a) peak1 and (b) peak2.

From the spectrum estimation results in Figures 3 to 4, it can be observed that the frequency identification results of the DOD-CS method are consistent with the theoretical value. In contrast, there are interference components with large amplitudes in the results obtained by OMP-CS. Owing to the presence of the displacement offset, the relationship between blade vibration displacement and frequency parameters has been changed. There is an error in solving the sparse vector s based on the inner product of displacement input and measurement matrix, resulting in errors in spectrum estimation results.

Based on the spectral estimation results, the blade vibration frequency parameters can be obtained, and then the least squares fitting method is used to identify the blade amplitude parameters. The parameter identification results and corresponding identification errors of simulation data are shown in Tables 2 and 3. The relative identification error of the frequency is within 1%. The relative identification error of the amplitude is within 5%.

Table 2.

Parameters identification results.

Data set	Frequency (Hz)	Engine order	Amplitude (mm)
1	180	2	1.167
	183	2	1.059
2	320	3.2	1.480

Table 3.

Parameters identification error.

Data	Frequency (Hz)			Amplitude (mm)
	Data1		Data2	Data1		Data2
Theoretical value	180	182	320	1.116	1.017	1.490
Identified value	180	183	320	1.167	1.059	1.480
Relative errors (%)	0	0.55	0	4.6	4.1	0.67

The reconstruction of blade vibration displacement is based on the amplitude identification results. The reconstruction results of displacement curves measured by each sensor are shown in Figure 5. The displacement reconstruction process is as follows. Firstly, the entire section of data is segmented with a block length of 20 cycles. Then, the DOD-CS method is used to obtain the spectrum estimation result and the displacement reconstruction value is calculated according to equation (13). Finally, connect the reconstructed displacements of each block. In Figure 5, near the resonance peak, there is an error between the reconstructed value and the original value. Overall, the reconstructed results of the blade vibration displacement are consistent with the theoretical value.

Figure 5.

Reconstructing results of blade vibration displacement of different sensors: (a) Sensor1, (b) Sensor2, and (c) Sensor3.

Parameter identification results under different noise levels

To investigate the influences of noise on blade vibration parameter identification, noises with different signal-to-noise ratios are added to the simulation data. Comparing the accuracy of parameter identification under different noise levels based on DOD-CS methods. The applied noise is Gaussian white noise, with a sound intensity range of [3, 20 dB]. Under different noise levels, the frequency identification results based on the DOD-CS method are consistent with the theoretical values. It indicates that the proposed method has a better noise resistance performance. When the noise intensity is 3.5 dB, the relative recognition errors of the amplitude parameters of the two sets of simulation data are 7.62% and 3.26%, respectively (Figure 6).

Figure 6.

The amplitude identification relative errors of the DOD-CS method at different noise levels: (a) first set of data and (b) second set of data.

Experimental data analysis and verification

The rotating blade test system consists of a rotating blade test rig, data acquisition, and signal analysis software. The rotating blade test rig includes a frequency converter, motor, rotor, sensor, and casing as shown in Figure 7.

Figure 7.

The rotating blade test rig.

The motor speed was controlled using a frequency converter with a speed range of 0–1000 rpm, the sweep time was 60 s. Twelve blades (blade thickness is 0.5 mm) were uniformly distributed on the blade disk. The sampling rate of the BTT measurement system depends on the current speed and the number of sensors.

According to the finite element analysis results, the Campbell diagram of the blade is shown in Figure 8. The first-order modal parameters of the blade can be obtained from the Campbell diagram as shown in Table 4.

Figure 8.

Campbell diagram of the blade.

Table 4.

Blade first-order modal vibration parameters.

Engine order	3	4	5
Speed (rpm)	680	480	370
Frequency (Hz)	34	32	30.83

The vibration test data of the fifth blade are presented in Figure 8. The black curve represents the speed, which rises to 1000 rpm and then decreases. The other four curves with different colors represent the vibration signals measured by the four optical fiber sensors with the corresponding installation angles of 30°, 75°, 148°, and 222°. From the time-domain diagram, it can be observed that, during the increase in rotational speed, the blade vibration displacement drifts downwards.

As the rotational speed increases, the displacement offset also increases. During the decrease in rotational speed, the displacement offset decreases. When the rotational speed was zero, the blade vibration displacement returned to the zero line.

The parameter identification algorithms include the sine fitting method, the OMP-CS method proposed in Bouchain et al., ²¹ and the DOD-CS method. The first and second methods require data from four sensors, and a step to remove displacement offset is required before data analysis. The DOD-CS method uses data from two sensors with corresponding installation angles of 30° and 75°.

The resonance regions of the blade are shown in Figure 9 as 1, 2, 3, and 4. Refer to Table 4 for the blade resonance frequency, set the frequency range to [26, 40 Hz]. The spectrum estimation results based on the DOD-CS method are shown in Figure 10, where (a), (b), (c), and (d) represent the data analysis results at the four marked positions in Figure 9, respectively. The parameter identification results are listed in Table 5.

Figure 9.

Time domain diagram of blade vibration test data.

Figure 10.

The spectrum estimation results of DOD-CS method: (a) position 1, (b) position 2, (c) position 3, and (d) position 4.

Table 5.

Parameters identification results (DOD-CS method).

Position	Frequency (Hz)	Engine order	Amplitude (mm)	Speed (rpm)
1	31.2	4	0.201	458
2	34.7	3	0.132	668
3	34.3	3	0.770	694
4	30	4	0.546	451

The parameter identification results of the sine fitting method and OMP-CS method are shown in Tables 6 and 7. The blade resonance frequency shown in the Campbell plot in Table 4 is set as a theoretical value, and the relative errors of the frequency identification for the three methods are listed in Table 8. According to the parameter identification results in Table 8, the frequency identification error based on the DOD-CS method is within 7%, which is lower than that of the other two methods. The DOD-CS method only requires information from two sensors to complete the blade vibration parameter identification.

Table 6.

Parameters identification results (Sine Fitting method).

Position	Frequency (Hz)	Engine order	Amplitude (mm)	Speed (rpm)
1	31.5	4	0.33	472
2	33.7	3	0.42	674
3	34.2	3	0.48	684
4	29.2	4	0.39	438

Table 7.

Parameters identification results (OMP-CS method).

Position	Frequency (Hz)	Engine order	Amplitude (mm)	Speed (rpm)
1	31.2	4	0.24	458
2	33.3	3	0.39	668
3	33.8	3	0.38	694
4	29.9	4	0.684	451

Table 8.

Frequency identification error (three methods).

Method	Data’s position	Frequency (Hz)	Relative error (%)
DOD-CS	1	31.2	2.5
	2	34.7	2.06
	3	34.3	0.88
	4	30	6.25
OMP-CS	1	31.2	2.5
	2	33.3	2.05
	3	33.8	0.58
	4	29.9	6.56
Sine fitting	1	31.5	1.56
	2	33.7	0.88
	3	34.2	0.58
	4	29.2	8.75

Conclusion

For the problem of low precision of BTT signal parameter identification with displacement offset, and the under-sampled characteristics of the signal requiring multiple sensor data to identify parameters. This paper proposes a parameter identification method based on Displacement Offset Decreased and Compressed Sensing (DOD-CS). The following results were obtained through the analysis of simulation and experimental data.

The analysis of simulation data includes synchronous vibration with mistuned and single-mode asynchronous vibration under variable speed. In the absence of noise, the frequency and amplitude identification errors of synchronous vibration are 0.55% and 4.6%, and the frequency and amplitude identification errors of asynchronous vibration are 0% and 0.67%. When a noise level of 3 dB is applied, the amplitude recognition errors of synchronous and asynchronous vibrations are 7.62% and 3.26%. Directing on the analysis of experimental data, the frequency recognition errors of DOD-CS, sine fitting, and OMP-CS methods are 6.25%, 6.56%, and 8.75%, respectively. Compared with previous methods, the advantage of the DOD-CS method is that it eliminates the influence of displacement offset components on parameter identification, which can effectively improve the parameter identification accuracy. Only two sensors are required to accurately identify the parameters for under-sampled signals. In addition, the proposed method is only applicable to the analysis of the vibration characteristics of the blisk and blade, and the parameter identification of signals under linear variable speed conditions.