A modal estimation method of rotating blade based on compressed sensing and blade tip timing

Abstract

Structural modal estimation has consistently remained one of the most crucial areas of research in mechanical vibration analysis and signal processing. It is imperative to accurately monitor the vibration signals of rotor blades and their corresponding structural modal parameters. Blade tip timing (BTT) is a promising approach used to measure vibration and monitor the health of blades. However, most existing BTT methods focus on frequency accuracy, neglecting damping, a key physical quantity that represents the strength of the structure. This research focuses on damping modal estimation of blade vibration and proposes a sparse reconstruction algorithm based on a two-dimensional Laplace wavelet family, which addresses the inherent under-sampled problem of BTT technology while enabling estimations of both the structural modal frequency and modal damping of the blades. Firstly, the proposed algorithm is achieved by designing a Laplace wavelet dictionary based on prior information and using a convex optimization objective function with a regularization term. Secondly, Laplace wavelet dictionary matches the signal better than the traditional Fourier dictionary based on the similarity between the damped vibration signal and the Laplace wavelet, then a sparser representation vector can be obtained. Moreover, the research object is not limited to a single blade, but can be easily expanded to multiple stages and multiple rotating blades monitored simultaneously. Finally, the simulation and physical test results indicate that the proposed method exhibits high reconstruction accuracy, reliability, and anti-noise abilities.

Introduction

Rotating machinery blades are key components of aero-engines. During operation, they are subjected to high temperatures, pressures, and loads for long periods of time, which often leads to failures, such as high-cycle fatigue, cracks, and fractures. ¹ There are various reasons for blade failure, such as erosion of high speed airflow, and rotor imbalance. Several methods have been proposed to ensure the stable operation of the blade and extend the fatigue life;^2,3 however, the main reason for blade failure is the high-amplitude and high-order vibration, ⁴ and many articles also focus on the devastating vibration of the blade for fault diagnosis research.^5,6 If the blade dynamic strength is insufficient (especially under the aerodynamic interactions between the elastic blade and the internal flow field), this situation may induce flutter, that is, the blade could generate aero-elastic coupling self-excited vibrations. Once flutter occurs, the blade may break in a very short time because of the severe vibrations, and additional consequences may arise. In terms of ensuring the safety and health of the engine, dynamic monitoring of the blade structure modal parameters has important theoretical research significance and practical engineering value. Blade tip timing (BTT) is a non-contact measurement method for identifying rotor blade vibration by recording the time at which the blade reaches each installed probe. ⁶ In addition to its strong reliability and simple operation, BTT is advantageous in terms of long-term online detection and vibration analysis; according to Ref [7], it has been widely applied for measuring blade vibrations in aero-engines. Refs. [8–11] have demonstrated the feasibility and stability of BTT technology, which was employed to measure and analyze actual blade failures. However, a blade only passes a sensor once during one complete rotation, and therefore, the sampling frequency of the BTT system is determined by the rotation speed and the number of sensors. In general, owing to the limitations of installation conditions and rotation rate, the sampling frequency of the BTT system does not satisfy the Nyquist sampling theorem.^12–14

Researchers have proposed various specialized methods to extract vibration parameters from the under-sampled signal. For the fluttering rotating blade, it is in an unstable motion state. There are several traditional methods to analyze the stability of the blade system by building mathematical models and deriving precise equations of motion.^15–19 A second-order differential multi-harmonic balance equation of the blade vibration was developed in literature. ²⁰ The key determinants of the vibration are the quality, elastic coefficient, and damping coefficient of the blade. Zhao et al. used beam-beam impact modeling and finite element analysis to examine the vibration-shock response of shrouded blades. ²¹ The single degree of freedom (SDOF) approach uses a BTT sensor to measure the rotation speed and the vibration amplitude of the blade when resonance occurs under variable speed conditions. ²² The two-parameter plot (TPP) strategy equally installed four or more probes in casing to collect the data while operating at a constant rotation rate. ²³ The autoregressive method calculates the frequency doubling value of the blade vibration by introducing an autoregressive equation. ²⁴ Joung et al. developed a circular Fourier fit (CFF) method, which relies on the Campbell diagram obtained from finite element analysis to confirm the frequency. ²⁵

In 2006, Donoho et al. proposed a new signal acquisition theory known as compressed sensing (CS) ²⁶ or sparse reconstruction, which breaks through the technical bottleneck of Nyquist sampling theorem. According to this theory, as long as the signal is sparse within a certain transform domain, the original structure of the signal is maintained through linear projection. After solving the optimization problem, the original signal can be accurately reconstructed using only a few sampling points. In recent years, CS theory has been applied to the field of BTT wildly. In 2016, Lin et al. ²⁷ applied this theory to process under-sampled BTT signals and precisely identified the vibration frequency. Pan et al. ²⁸ developed an improved frequency spectrum recovery method that could reach a guaranteed level of sparse solution with biased measurement results, and they studied the influence of measurement uncertainty on the reconstruction. Pan et al. also ²⁹ constructed a dictionary of CS based on a dictionary learning method to make it more adaptive. According to the sparseness of the BTT signal in the order domain, Chen et al. ³⁰ proposed a compressed sensing-based order analysis (CSOA) method, which can accurately obtain the order spectrum under variable rotation speed conditions. This CSOA method also allows for considerable installation angle errors, which makes it more applicable in real-life situations. This method is only valid for synchronous vibration. Recently, compressed sensing method based on deep neural networks^31,32 has shown great advantages. Wu et al. ³³ proposed a deep compressed sensing model, and trained a Generative Adversarial Networks (GAN) to reconstruct the BTT signal, without relying on iteration and prior information. Chen et al. ³⁴ trained a Convolutional Neural Network (CNN) to reconstruct the BTT signal, and applied rectified linear unit (ReLU) layer and batch normalization (BN) layer in the network to improve the training effect of the CNN.

However, most of the existing methods based on compressed sensing focus on the accuracy of the modal frequency, while ignoring the damping, which is a key physical quantity representing the strength of the structure. The damping vibration of blades is the general form of blade vibration, and it is also the initial vibration form of dangerous vibrations such as flutter et al. The modal estimation of damping vibration signals is of great significance to the vibration monitoring and early warning of blade. The modal damping of the static blade can be determined using a half-power bandwidth^35,36 method and/or a hammer method. ³⁷ For rotating blades, damping can be calculated from the stress or pressure measured by installed gauges ³⁸ or miniature pressure sensors. ³⁹ However, such methods are not suitable for comprehensive monitoring of multi-stage blades because they are not sufficiently reliable and require complicated operations.

To comprehensively estimate the structural vibration modal parameters of rotating blades, this article proposes a signal reconstruction method based on the compressed sensing theory and blade tip timing technology, named Laplace Wavelet-based Basis Pursuit (LWBP). We utilize the two-dimensional Laplacian wavelet family to construct a sparse representation dictionary, which can precisely assess the similarity between these functions and the structural response signal. LWBP can fully mine the multiple structure modals information and completely restore the structural response signal of the blade, which provides algorithmic support for the performance monitoring of rotating blades. Its anti-noise performance is also excellent, and it can be employed in low signal-to-noise ratio (SNR) situations.

Blade tip timing principle

Typically, the BTT sensor is installed along the circumference of the casing, which can be uneven. A pulse is generated at the instant the blade passes the BTT sensor. Meanwhile, a reference sensor is installed on the rotating shaft, and a reflective mark is placed on the shaft. Each time the impeller completes one revolution, a rotation speed signal can be obtained. Figure 1 shows the various components of the BTT system. Theoretically, the time recorded by the sensor when the blade does not vibrate can be calculated using equation (1), ⁴⁰

${{\overline{t}}_{i.n}}^k=\frac{1}{2\pi f_r}\left(2\pi n+{\alpha }_i-{\theta }_k\right)$

(1)

where $f_r$ is the rotation frequency, ${\alpha }_i$ is the installation angle of the BTT sensor, $n$ denotes the number of turns, and ${\theta }_k$ represents the angle between the kth blade and the reflective mark on the shaft. Because of the vibration generated during operation, when the kth blade passes the ith sensor, the actual arrival time can be expressed as shown in equation (2), ⁴⁰

${t_{i.n}}^k=\frac{1}{2\pi f_r}\left(2\pi n+{\alpha }_i-{\theta }_k-\frac{y}{r}\right)$

(2)

where $y$ represents the displacement of the blade, and $r$ is the radius of rotation of the blade. Then, the displacement of the blade can be obtained using equation (3):

$y=2\pi f_r\times r\times \left({t_{i.n}}^k-{{\overline{t}}_{i.n}}^k\right)$

(3)

Figure 1.

Blade tip timing system.

A blade can only be measured once by a sensor in a single rotation. Therefore, the sampling rate of the BTT measurement is determined by the number of sensors and the rotation frequency, according to the relationship in equation (4),

$f_{BTT}=I{f}_r$

(4)

where $I$ is the number of sensors.

LWBP method

Compressed sensing model of BTT sampling

In this section, compressed sensing model, which is the theoretical basis of LWBP, will be introduced. The sampling frequency of the BTT system is often less than the vibration frequency of the blade because of the limitations of the installation conditions and rotation speed. Therefore, the signal obtained by BTT sampling does not satisfy the Shannon sampling theorem $\left(I{f}_r=2f_{\max }\right)$ , where $f_{\max }$ is the maximum frequency of the signal. Fortunately, recent researches suggest that the blade vibration signal can be reconstructed from the under-sampled BTT signal through compressive sensing theory.^26–34

According to the compressed sensing theory, the signal is sparse if it can be represented by several coefficients in the sparse domain, which is composed of basic functions. A collection of these basis functions is known as a dictionary, wherein each basis function is called an atom. Based on the sparsity, data compression can be completed while sampling, and the original signal can be restored using fewer measured values. For the kth blade, $N$ data points can be collected over a given period of time.

Suppose that the displacement signal containing all information for an arbitrary blade can be recorded by a total of $L$ BTT probes installed uniformly around the casing $\left(L{f}_r\ge 2f_{\max }\right)$ . For the kth blade, the displacement signal $X_{N\times 1}$ sampled by $L$ sensors can be expressed as shown in equation (5),

$X_{N\times 1}={\displaystyle \sum }_{i=1}^{\stackrel{\sim }{N}}{\theta }_i{\psi }_i={\mathrm{\Psi }}_{N\times \stackrel{\sim }{N}}{\theta }_{\stackrel{\sim }{N}\times 1}$

(5)

where ${\mathrm{\Psi }}_{N\times \stackrel{\sim }{N}}={\left\{{\psi }_i\right\}}_{i=1}^{\stackrel{\sim }{N}}$ is a set of basic functions, and ${\theta }_{\stackrel{\sim }{N}\times 1}$ is a sparse vector containing only few non-zero elements. Only $I$ sensors are installed among $L$ positions. Thus, the sampling stream for an arbitrary blade displacement signal can be regarded as discarding all but $I$ samples in every rotation of $I$ samples periodically. A set $C=\left\{c_i\right\}\left(0<c_i<I\right)$ describes the positions of $I$ samples. Therefore, the measurement result $Y_{M\times 1}$ is generated by extracting $M$ data points from $X_{N\times 1}$ , which can be expressed as shown in equation (6),

$Y_{M\times 1}={\mathrm{\Phi }}_{M\times N}{X}_{N\times 1}$

(6)

where ${\mathrm{\Phi }}_{M\times N}=\left[\begin{array}{cccc}1& 0& \mathbf{L}& \begin{array}{cc}\mathbf{L}& \begin{array}{cc}\mathbf{L}& 0\end{array}\end{array}\\ 0& \mathbf{L}& 1& \begin{array}{cc}0& \begin{array}{cc}\mathbf{L}& 0\end{array}\end{array}\\ \mathbf{M}& \mathbf{L}& \mathbf{L}& \begin{array}{cc}\mathbf{L}& \begin{array}{cc}\mathbf{L}& \mathbf{M}\end{array}\end{array}\\ 0& \mathbf{L}& \mathbf{L}& \begin{array}{cc}1& \begin{array}{cc}\mathbf{L}& 0\end{array}\end{array}\end{array}\right]$ is the measurement matrix, which is formed by extracting specific rows from an $N\times N$ unit matrix. According to the BTT sampling principle, the column k where element 1 is located satisfy $k=c_i+nL\left(i=\mathrm{1,2},KI,n=\mathrm{1,2},KN/L\right)$ . Based on a certain dictionary, if the sparse representation of the blade displacement signal can be obtained from the under-sampling signal measured by the BTT system, it is possible to reconstruct the non-under-sampled signal of the blade. This can be achieved by solving an $l_0-norm$ optimization problem with the linear constraints shown in equation (7),

$\begin{array}{l}\underset{\theta }{\mathrm{argmin}}{\Vert \theta \Vert }_0\\ s.t.Y_{M\times 1}={\mathrm{\Phi }}_{M\times N}{\mathrm{\Psi }}_{N\times \stackrel{\sim }{N}}{\theta }_{\stackrel{\sim }{N}\times 1}\end{array}$

(7)

where ${\Vert \theta \Vert }_0$ represents the number of the non-zero values in $\theta$ . However, equation (7) is an NP (non-deterministic polynomial)-hard problem, which is difficult to solve in polynomial time. In some cases, greedy algorithms, such as orthogonal matching pursuit (OMP) can be used to solve equation (7). However, the OMP algorithm approximates the $l_0-norm$ sparsest solution through iteration; thus, the result may be inaccurate. Therefore, some researchers suggest that $l_0-norm$ optimization problem can be transformed into $l_1-norm$ optimization problem which is not Np-hard.^41,42 Candes et al. ⁴³ confirmed that if the solution is sufficiently sparse, equation (7) can be transformed into a $l_1-norm$ regularization model:

$\begin{array}{l}\underset{\theta }{\mathrm{argmin}}{\Vert \theta \Vert }_1\\ s.t.Y_{M\times 1}={\mathrm{\Phi }}_{M\times N}{\mathrm{\Psi }}_{N\times \stackrel{\sim }{N}}{\theta }_{\stackrel{\sim }{N}\times 1}\end{array}$

(8)

In general, there are measurement uncertainties in the BTT sampling process, resulting in uncertain noise in BTT signal. ⁴⁴ Therefore, the equality constraints in equation (8) can be relaxed into the following inequality constraints:

$\begin{array}{l}\underset{\theta }{\mathrm{argmin}}{\Vert \theta \Vert }_1\\ s.t.{\Vert Y_{M\times 1}-{\mathrm{\Phi }}_{M\times N}{\mathrm{\Psi }}_{N\times \stackrel{\sim }{N}}{\theta }_{\stackrel{\sim }{N}\times 1}\Vert }_2^2\le \epsilon \end{array}$

(9)

where $\epsilon$ is a threshold determined by experience. In fact, an unconstrained optimization problem is easier to solve than a constrained one.^45,46 Therefore, it is necessary to transform the constraint term into a penalty term. For a better expression and more options of solving method, the constraint term in equation (9) can be written as a penalty term:

$\underset{\theta }{\mathrm{argmin}}{\Vert Y-\mathrm{\Phi }\mathrm{\Psi }\theta \Vert }_2^2+\lambda {\Vert \theta \Vert }_1$

(10)

where $\lambda$ is the regularization parameter and value of $\lambda$ determines the sparsity of $\theta$ . For noisy signals

$X_{L,noise}=X_L+\sigma e$

(11)

where e is Gaussian white noise, σ is noise’s variance. For a signal of length N, the value of $\lambda$ can be determined as

$\lambda =\sigma \sqrt{2\log \left(N\right)}$

(12)

The compressed sensing model provides a theoretical basis for under-sampled signal reconstruction, and the LWBP method is also implemented based on this model.

Designing a dictionary with a Laplace wavelet family

The traditional BTT signal reconstruction methods based on the compress sensing model are mainly aimed at the blade vibration signals with good sparsity in the frequency domain. These signals are non-damped linear combinations of one or more harmonics. Therefore, Fourier dictionary is applied in sparse reconstruction as the dictionary that best matches the frequency characteristic of non-damped signals. However, it is difficult for damping signals to strictly satisfy the requirement of sparsity in frequency domain, resulting in low accuracy of traditional reconstruction methods applying Fourier dictionary (See Table 1). In order to solve the deficiencies of traditional reconstruction methods in damping signals, this research constructs a Laplace wavelet dictionary to obtain a better reconstruction performance.

Table 1.

Modal parameters calculated by the matrix pencil method.

Dictionary		${\omega }_1\left(Hz\right)$	${\zeta }_1$	${\omega }_2\left(Hz\right)$	${\zeta }_2$
	Original signal	130	0.0045	210	0.0095
Fourier	Reconstructed signal	130.1226	0.0037	179.7918	0.0014
	Error (%)	0.094	17.778	14.385	85.263
Laplace wavelet	Reconstructed signal	130	0.0045	210	0.0095
	Error (%)	0	0	0	0

Accurate signal recovery depends on the sparsity of the signal in the transform domain, which requires high similarity between the atoms and the original signal. In general, the structural response signal with a single-modal is described by the mathematical model presented in equation (13),

$f\left(t\right)=e^{-\omega \zeta t}\cos \left(\omega t-t_0\right)\hspace{0.17em}{t}_0\ge 0$

(13)

where $\omega$ is the frequency of the modal, $\zeta$ is the damping which is negative when $f\left(t\right)$ diverges and positive when $f\left(t\right)$ converges, $t_0$ is the time delay. Considering the characteristics of the single-modal signal, a set of real-part and imaginary-part functions of a Laplace wavelet are selected as the atoms to construct the dictionary $\mathrm{\Psi }$ . The definition of a Laplace wavelet and its real-part and imaginary-part function are shown in equations (14)–(16), respectively,

$W_{\omega }\left(t\right)=\{\begin{array}{cc}{Ae}^{-\zeta /\sqrt{1-{\zeta }^2\omega \left(t-\tau \right)}}{e}^{-j\omega \left(t-\tau \right)}& ,tϵ\left[\tau ,+\infty \right]\\ 0& ,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\end{array}$

(14)

${\psi }_{c,\omega ,\zeta }\left(t\right)=\{\begin{array}{cc}{Ae}^{-\zeta /\sqrt{1-{\zeta }^2\omega \left(t-\tau \right)}}{\mathit{cos}}^{\left(\omega \left(t-\tau \right)\right)}& ,tϵ\left[\tau ,+\infty \right]\\ 0& ,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\end{array}$

(15)

${\psi }_{s,\omega ,\zeta }\left(t\right)=\{\begin{array}{cc}{Ae}^{-\zeta /\sqrt{1-{\zeta }^2\omega \left(t-\tau \right)}}{\mathit{sin}}^{\left(\omega \left(t-\tau \right)\right)}& ,tϵ\left[\tau ,+\infty \right]\\ 0& ,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\end{array}$

(16)

where $A$ is the normalization coefficient and is set to 1 in this paper, $\tau$ is the time parameter which is set to zero in the dictionary design. Equations (13)–(16) reveal the similar vibration characteristics of $f\left(t\right)$ and $\psi \left(t\right)$ . Hence, different Laplace wavelet functions can be generated by adjusting parameters $\omega$ and $\zeta$ to construct a dictionary and realize the sparse representation of structural response signal. For $m$ modals, assume that the ranges of parameters ω and ζ in the mth dictionary are $\left[{\omega }_{m,\mathit{min}},{\omega }_{m,\mathit{max}}\right]$ and $\left[{\zeta }_{m,\mathit{min}},{\zeta }_{m,\mathit{max}}\right]$ , respectively. We can discretize these two intervals by dividing them into equal intervals of Δω and Δζ. This will give us discrete parameter sets $\left\{{\omega }_{m1},K,{\omega }_{mn}\right\}$ and $\left\{{\zeta }_{m1},K,{\zeta }_{mk}\right\}$ ,where n and k is the length of two sets, respectively. Arbitrarily combine the elements pairwise from $\left\{{\omega }_{m1},K,{\omega }_{mn}\right\}$ and $\left\{{\zeta }_{m1},K,{\zeta }_{mk}\right\}$ , resulting in a pair of atoms:

$d_{mc,i,j}=e^{-\frac{{\zeta }_{mj}}{\sqrt{1-{\zeta }_{{mj}^2}}}{\omega }_{ki}t}\cos \left({\omega }_{mi}t\right)$

(17)

$d_{ms,i,j}=e^{-\frac{{\zeta }_{mj}}{\sqrt{1-{\zeta }_{{mj}^2}}}{\omega }_{ki}t}\sin \left({\omega }_{mi}t\right)$

(18)

where $i=\mathrm{1,2},K,n,j=\mathrm{1,2},K,k$ Normalize these atoms

$d^{\prime }=\frac{d}{{\Vert d\Vert }_2}$

(19)

By combining all the results, a total of $2kn$ atoms can be obtained, which can be organized into an atomic library format:

$D_{mc}=\left\{\begin{array}{c}\begin{array}{c}{d}_{mc,i=1,j=1}^{\prime }\\ \mathrm{M}\\ d_{mc,i=1,j=k}^{\prime }\end{array}\}i=1\\ \mathrm{M}\\ \begin{array}{c}{d}_{mc,i=n,j=1}^{\prime }\\ \mathrm{M}\\ d_{mc,i=n,j=k}^{\prime }\end{array}\}i=n\end{array}\right\},D_{ks}=\left\{\begin{array}{c}\begin{array}{c}{d}_{ms,i=1,j=1}^{\prime }\\ \mathrm{M}\\ d_{ms,i=1,j=k}^{\prime }\end{array}\}i=1\\ \mathrm{M}\\ \begin{array}{c}{d}_{ms,i=n,j=1}^{\prime }\\ \mathrm{M}\\ d_{ms,i=n,j=k}^{\prime }\end{array}\}i=n\end{array}\right\}$

(20)

Then, the new dictionary composed of multiple sub-dictionaries can be expressed as shown in equation (21):

$\mathrm{\Psi }=\left[D_{1c},D_{1s},\dots ,D_{mc},D_{ms}\right]$

(21)

Each column in the dictionary represents an atom and the number of atoms is determined by $m,n,k$ , that is, $2mnk=\stackrel{\sim }{N}$ . If the number of atoms in the dictionary is equal to the length of the original signal $\left(\stackrel{\sim }{N}=N\right)$ , the dictionary expressed in equation (21) is called a complete dictionary. For fixed-length original signals, adding more atoms to the dictionary can make the representation sparser, which leads to an over-complete dictionary $\left(\stackrel{\sim }{N}>N\right)$ . In general, the matrix dimension should not be too large considering the calculation time.

A Laplace wavelet dictionary $\mathrm{\Psi }$ is constructed, then equation (13) can be expressed as

$f\left(t\right)=\mathrm{\Psi }\theta$

(22)

where $\theta$ is sparse indicator vector. Solving equation (20) is a parametric searching procedure. The atoms of $\mathrm{\Psi }$ that best matches the structural response signal $f\left(t\right)$ will be selected. Their column positions correspond to the positions of the non-zero elements in the indicator vector $\theta$ , while the zero elements in $\theta$ serve to remove other irrelevant atoms. According to equation (6) and equation (22), BTT signal $y\left(t\right)$ can be expressed as shown in equation (23):

$y\left(t\right)=\mathrm{\Phi }\mathrm{\Psi }\theta$

(23)

According to the compressed sensing model of BTT signal in Section 3.1, equation (21) can be transformed into a sparse reconstruction problem. The problem of equation (23) can be expressed as a $l_1-norm$ regularization model with a penalty term:

$\underset{\theta }{\mathrm{argmin}}{\Vert y\left(t\right)-\mathrm{\Phi }\mathrm{\Psi }\theta \Vert }_2^2+\lambda {\Vert \theta \Vert }_1$

(24)

For the model in equation (24), the BP algorithm ⁴⁷ is applied to achieve the sparsest solution of $\theta$ , which can accurately obtain the solution of $l_1-norm$ model through linear programming. Then, the sparse indicator vector $\theta$ is obtained and the original signal $f\left(t\right)$ can be successfully reconstruct through equation (22).

In summary, this article proposes a signal reconstruction method based on the compressed sensing theory and blade tip timing technology. On the basis of the characteristics of the structural response signal with small damping, LWBP method is proposed, in which Laplace wavelet dictionary is constructed to search for the damping modal of blade vibration signal, and BP algorithm is applied to realize the accurate reconstruction of damping structural response signal.

Simulations

Verification of two vibration modals

Simulations were performed to verify the feasibility of recovering a structural response signal using LWBP. To quantitatively evaluate the effect of the reconstruction, we define the mean square error (MSE) as shown in equation (25):

$MSE={\Vert x-x^{\prime }\Vert }_2^2$

(25)

where $x$ is the original signal, and $x^{\prime }$ is the reconstructed signal. The simulation signal containing two modals is shown in equation (26):

$\begin{array}{l}h\left(t\right)={\displaystyle \sum }_{i=1}^2{e}^{-2\pi {\omega }_i{\zeta }_{i}t}\cos \left(2\pi {\omega }_i\left(t-{\varphi }_i\right)\right)\\ {\omega }_i=\left\{\mathrm{130,210}\right\},{\zeta }_i=\left\{\mathrm{0.004,0.009}\right\},{\varphi }_i=\left\{\mathrm{0,0}\right\}\end{array}$

(26)

The time and frequency domain diagrams of the simulated signal are shown in Figure 2. Assuming that the blades are evenly installed in the circumferential direction of the casing, and only small damping vibration occurs, the blade disk rotating speed is stable. Setting $L=32$ and the rotation speed $f_r=50Hz$ , the ideal sampling rate can be computed as $f_{ideal}=32\times 50=1600H\mathrm{z}$ . It is clear that the sampling rate of the BTT system is not enough to collect the complete set of information regarding $h\left(t\right)$ . Setting $I=4$ and placement of probes $C=\left\{\mathrm{1,9,17,25}\right\}$ , the actual sampling rate is $f_{BTT}=4\times 50=200H\mathrm{z}$ . The ideal vibration signal sampled by $L$ probes and the BTT signal sampled by $I$ sensors are shown in Figure 3. The data collected by $I$ sensors (Figure 3(b)) does not include the complete information of the vibration signal characteristics, whereas the massive amount of data sampled by $L$ sensors (Figure 3(a)) can completely recover the original signal. However, the data in Figure 3(a) cannot be obtained because there is not enough space to install $L$ sensors. Therefore, a compressed sensing algorithm is used to recover the signal collected by $L$ sensors from the data collected by $I$ sensors.

Figure 2.

Time domain and frequency domain diagrams of the original signal.

Figure 3.

Blade displacement sampled by (a) $L$ probes or (b) $I$ probes.

Setting the simulation time to 0.64 s yields a total of 1024 data points. Setting $m=1,n=32,k=32$ , ${\omega }_{\psi 1}=0\times 2\pi$ , $\mathrm{\Delta }\omega =10\times 2\pi$ , and $\mathrm{\Delta }\zeta =0.001$ , the frequency range of atoms in the dictionary $\mathrm{\Psi }$ is determined to be $0\sim 310Hz$ , and the damping range is $0.001\sim 0.032$ . It is clear that the modal of $h\left(t\right)$ exactly matches the 839^th and 1361^th atoms in the designed dictionary, and its sparsity is equal to 2 in the transform domain. After optimizing equation (8), the two atoms mentioned above in the dictionary can be identified accurately, and the original signal is perfectly recovered via the compressed sensing algorithm, with $MSE=1.2848e^{-4}$ . Figure 4(a) shows the time domains of the original and reconstructed signals, as well as the transform domain of the reconstructed signal, which directly displays the positions of the selected atoms in the dictionary. Figure 4(b) shows the frequency domain of the original and reconstructed signals. These results confirm that when the modal(s) of the original signal exist in the dictionary, the compressed sensing algorithm can accurately find them and reconstruct the signal using the selected atom(s). In contrast, the reconstruction result using a dictionary constructed by a Fourier basis function is shown in Figure 4(c) and (d). The modal parameters in the reconstructed signal calculated by a matrix pencil method are shown in Table 1. It can be seen that both frequency and damping of the signal reconstructed by Fourier dictionary have unacceptable errors. Considering the attenuation characteristics of the original signal, its energy is not concentrated in the frequency domain, which causes a loss in sparsity. Thus, the compressed sensing algorithm fails to restore the original signal from the under-sampled signal. The modal parameters in the reconstructed signal calculated by a matrix pencil method are shown in Table 1.

Figure 4.

Reconstructed signal: (a) Time domain and (b) frequency domain of the signal reconstructed by the Laplace wavelet dictionary (c) time domain and (d) frequency domain of the signal reconstructed by a Fourier dictionary.

In fact, Figure 4(a) and (b) presents the ideal situation. In most cases, the modals of the original signal cannot be found in the designed dictionary. Some prior information is needed to focus the frequency and damping of the atoms within a certain range. Therefore, we generate an attenuated signal including two modals, as shown in equation (27):

$\begin{array}{l}h\left(t\right)={\displaystyle \sum }_{i=1}^2{e}^{-2\pi {\omega }_i{\zeta }_{i}t}\cos \left(2\pi {\omega }_i\left(t-{\varphi }_i\right)\right)\\ {\omega }_i=\left\{\mathrm{130.5,210.5}\right\},{\zeta }_i=\left\{\mathrm{0.0045,0.0095}\right\},{\varphi }_i=\left\{\frac{\pi }{3},\frac{\pi }{6}\right\}\end{array}$

(27)

Assuming that the number of modals and approximate range of parameters have been obtained from the prior information, we set $\mathrm{\Delta }\omega =2\pi$ , $\mathrm{\Delta }\zeta =0.001$ , ${\omega }_{\psi 1}=114\times 2\pi$ , ${\omega }_{\psi 2}=194\times 2\pi$ , and $m=2,n=32,k=16$ , to obtain the frequency range of the atoms in the dictionary ( $114\sim 145Hz$ and $194\sim 225Hz$ ) and the damping range ( $0.001\sim 0.016$ ). The time domains of the original and reconstructed signals, as well as the transform domain of the reconstructed signal are shown in Figure 5(a). The frequency domain of the original and reconstructed signal are shown in Figure 5(b). Although the modal of the original signal could not be found in the dictionary, it is still approximately represented by a series of atoms with similar vibrational characteristics. Despite the weakened sparsity, the results still reach an acceptable level. In this case, $MSE=0.238$ , which confirms the high reconstruction accuracy of the LWBP. The frequency and damping of the modals in the reconstructed signal calculated by a matrix pencil method are shown in Table 2.

Figure 5.

(a) Time domain and (b) frequency domain of signal reconstructed using the designed dictionary.

Table 2.

Modal parameters calculated by the matrix pencil method.

	${\omega }_1\left(Hz\right)$	${\zeta }_1$	${\omega }_2\left(Hz\right)$	${\zeta }_2$
Original signal	130.5	0.0045	210.5	0.0095
Reconstructed signal	130.5015	0.0046	210.5034	0.0094
Error (%)	0.001	2.222	0.002	1.053

Table 2 clearly indicates that the errors in the modal parameters between the reconstructed signal and the original signal are very small. When the sampling time is longer, the atoms can be set more densely within a certain range, and the reconstruction effect improves. For signals with multiple modals, setting enough atoms around each modal enables enhanced reconstruction of the multi-modal structure response signal.

Anti-noise analysis

Simulation experiments were designed to test the anti-noise performance of LWBP and obtain the optimal regularization parameter $\lambda$ under different noise conditions. The simulated signal is regarded as two modal signals corrupted by measurement uncertainties, as expressed in equation (28):

$h\left(t\right)={\displaystyle \sum }_{i=1}^2{e}^{-2\pi {\omega }_i{\zeta }_{i}t}\cos \left(2\pi {\omega }_i\left(t-{\phi }_i\right)\right)+v\left(t\right)$

(28)

Among them, $v\left(t\right)$ can be considered as White Gaussian Noise with a zero-mean normal distribution. To evaluate the influence of measurement uncertainty on the accuracy of the restored modal parameters, the SNR and error are introduced, as defined in equations (29) and (30), respectively,

$SNR=10\log {\left(\frac{h\left(t\right)-v\left(t\right)}{v\left(t\right)}\right)}^2$

(29)

$error=\left({\alpha }_{real}-{\alpha }_{rec}\right)\times 100\%$

(30)

where ${\alpha }_{real}$ represents the modal parameters of the simulated signal, and ${\alpha }_{rec}$ represents the modal parameter calculated by the matrix pencil method from the reconstructed signal. First, it is necessary to verify the influence of the regularization parameters $\lambda$ on the calculated modal parameters under different signal-noise ratios. The parameters of the simulation signal were set to ${\omega }_i=\left\{\mathrm{130.5,210.5}\right\},{\zeta }_i=\left\{\mathrm{0.0045,0.0095}\right\},{\varphi }_i=\left\{\pi /3,\pi /6\right\}$ . After setting the rotation speed $f_r=50Hz$ , the designed dictionary and sensor layout are the same as those described in Section 4.1, and the frequency and damping can be calculated to determine the errors following the signal reconstruction. The simulated results are shown in Table 3, and the changes in the modal parameter errors as a function of the regularization parameter $\lambda$ are presented in Figure 6.

Table 3.

Errors in the modal parameters with different SNR.

SNR (dB)		Error in modal parameters (%)
		$\lambda =0.001$	$\lambda =0.005$	$\lambda =0.01$	$\lambda =0.05$	$\lambda =0.1$	$\lambda =0.5$
SNR = 5	${\omega }_1$	4.978	5.256	0.413	0.347	0.407	0.353
	${\zeta }_1$	19.243	19.212	10.736	10.179	20.445	7.876
	${\omega }_2$	2.271	1.100	0.858	0.325	0.186	0.234
	${\zeta }_2$	57.7	37.258	27.095	25.822	18.541	32.918
SNR = 10	${\omega }_1$	10.195	0.387	0.394	0.405	0.386	0.382
	${\zeta }_1$	17.281	10.248	9.346	13.047	18.506	13.582
	${\omega }_2$	3.186	1.133	0.536	0.255	0.251	0.19
	${\zeta }_2$	58.615	22.509	22.823	7.437	8.845	17.831
SNR = 15	${\omega }_1$	0.378	0.38	0.386	0.378	0.381	0.384
	${\zeta }_1$	4.102	5.812	8.44	11.389	15.943	12.98
	${\omega }_2$	1.596	0.718	0.218	0.225	0.222	0.187
	${\zeta }_2$	30.567	9.125	5.962	6.931	6.677	14.674
SNR = 20	${\omega }_1$	0.386	0.385	0.383	0.383	0.389	0.39
	${\zeta }_1$	3.11	5.11	6.16	12.25	16.16	12.28
	${\omega }_2$	0.851	0.21	0.21	0.206	0.212	0.23
	${\zeta }_2$	8.79	4.94	4.67	5.12	6.37	12.8

Figure 6.

Errors in the modal parameters of the signals reconstructed with different $\lambda$ when the SNR is equal to (a) 5 dB, (b) 10 dB, (c) 15 dB, or (d) 20 dB.

The error values listed in Table 3 are the averages of the results from 20 experiments performed under the same conditions, and the smallest error in each set is bolded. Under different SNR conditions, the error in the frequency between the reconstructed signal and the original signal is very small, but the damping is significantly affected by $\lambda$ . Figure 6 shows the errors as a function of $\lambda$ under various signal-to-noise ratios. As the noise component increases, $\lambda$ must be reduced to minimize the damping error, thereby sacrificing a certain degree of sparsity and increasing the proportion of ${\Vert Y-\mathrm{\Phi }\mathrm{\Psi }\theta \Vert }_2^2$ (equation (8)). In addition, since the vibration signal gradually decays to zero over time, the noise accounts for a relatively large proportion of the collected data, and therefore, it has an appreciable impact on the accuracy of the reconstructed signal. Therefore, as the SNR increases, the estimation errors associated with the modal parameters inevitably increase.

Simulation of a three-modal signal with noise

The simulation described above confirmed that the compressed sensing algorithm can use the dictionary comprising two-dimensional Laplace wavelet family real-part functions to (i) accurately reconstruct the BTT signal and (ii) accurately estimate the modal parameters of the original signal through the matrix pencil method. Moreover, this method demonstrates good anti-noise performance. Therefore, experiments were conducted using a simulation signal containing three modals with noise to verify the effectiveness of LWBP. First, we suppose that a blade has three vibration modals when operating, as expressed in equation (31):

$\begin{array}{l}h\left(t\right)={\displaystyle \sum }_{i=1}^3{e}^{-2\pi {\omega }_i{\zeta }_{i}t}\cos \left(2\pi {\omega }_i\left(t-{\varphi }_i\right)\right)+v\left(t\right)\\ {\omega }_i=\left\{\mathrm{130,210,833}\right\},{\zeta }_i=\left\{\mathrm{0.03,0.07,0.05}\right\},{\varphi }_i=\left\{\frac{\pi }{3},\frac{\pi }{6},\frac{\pi }{4}\right\}\end{array}$

(31)

Considering a sensor layout identical to that described in Section 4.1, the actual sampling rate $f_{BTT}=1200Hz$ , the ideal sampling rate $f_{ideal}=9600Hz$ , the simulation time is equal to $0.107s$ , and the ideal sampling point number is 1024. Assuming that we have obtained the approximate ranges of the three vibration modal parameters in advance (based on finite element analysis), three sub-dictionaries are designed to fit the three modals. We set $m=3$ , $k=32$ , $n=64$ , ${\omega }_{\mathrm{\Psi }1}=114\times 2\pi$ , ${\omega }_{\mathrm{\Psi }2}=194\times 2\pi$ , ${\omega }_{\mathrm{\Psi }3}=817\times 2\pi$ , $\mathrm{\Delta }\omega =0.5\times 2\pi$ , and $\mathrm{\Delta }\zeta =0.005$ . This designed dictionary is an over-complete dictionary. The advantage is that additional atoms could be introduced when the number of sampling points is fixed to maintain the accuracy of the reconstructed signal. Setting the $SNR=15$ and applying the analysis presented in Section 4.2 reveals that the reconstruction effect is enhanced when the canonical parameter $\lambda =0.005$ . After substituting the optimal solution of equation (10) into equation (5), allows the reconstructed signal (Figure 7) to be obtained and its modal parameters to be calculated. The results are shown in Table 4.

Figure 7.

(a) Time domain and (b) frequency domain of reconstructed three-modal signal when SNR = 15 db.

Table 4.

Errors in the modal parameters of the three-modal signal.

	${\omega }_1\left(Hz\right)$	${\zeta }_1$	${\omega }_2\left(Hz\right)$	${\zeta }_2$	${\omega }_3\left(Hz\right)$	${\zeta }_3$
Original signal	130	0.03	210	0.07	833	0.05
Reconstructed signal	130.55	0.029	211.38	0.067	834.41	0.052
Error (%)	0.423	3.333	0.657	4.286	0.169	4

Figure 7(a) shows the reconstructed signal and highlights the atoms that were used to fit the original signal. The front portion of the attenuated signal reflects the vibrational characteristics of the signal. In Figure 7(b), the fitting effect of the reconstructed signal in frequency domain is also exact. The modal parameters of the original signal and the reconstructed signal are listed in Table 4. The frequency errors of the three modals are all <1%, and the damping errors are all <5%, thus confirming that the LWBP can accurately reconstruct the original signal with noise.

Simulation without prior information

In the previous simulation experiments, it is necessary to obtain priori information to determine the parameter range of atoms in the dictionary. The advantage is to reduce the dimension of the dictionary matrix and the computational complexity. In this section, we decide to verify how well our algorithm performs without prior information.

The simulated signal is regarded as two modal signals corrupted by measurement uncertainties, as expressed in equation (32):

$\begin{array}{l}h\left(t\right)={\displaystyle \sum }_{i=1}^2{e}^{-2\pi {\omega }_i{\zeta }_{i}t}\cos \left(2\pi {\omega }_i\left(t-{\varphi }_i\right)\right)+v\left(t\right)\\ {\omega }_i=\left\{\mathrm{108,437}\right\},{\zeta }_i=\left\{\mathrm{0.027,0.053}\right\},{\varphi }_i=\left\{\frac{\pi }{3},\frac{\pi }{6}\right\}\end{array}$

(32)

The sensor layout is the same as in Section 4.1, set the rotational speed $f_r=80Hz$ , the ideal sampling rate $f_{ideal}=2560Hz$ , and the actual sampling rate $f_{BTT}=320Hz$ . Since there is no prior information, the parameter range of atoms in the dictionary can only be expanded, which means increasing the dimension of the dictionary to ensure that the main modals in the original signal are covered in the dictionary. The main thing is that the frequency range of atoms in the dictionary needs to be greatly expanded. For that, we set $m=1$ , $n=1024$ , $k=32$ , ${\omega }_{\mathrm{\Psi }1}=0\times 2\pi$ , $\mathrm{\Delta }\omega =0.5Hz$ , $\mathrm{\Delta }\zeta =0.005$ . An over-complete dictionary with 65536 atoms is designed, where the atoms have a frequency range of $0\sim 511.5H\mathrm{z}$ , and a damping range of $0\sim 0.16$ . The length of sampled time series is 1024 while the simulation time is set to 0.4s. Under the condition of SNR=10db, setting the canonical parameter $\lambda =0.001$ . After substituting the optimal solution of equation (10) into equation (5), the obtained original signal and reconstructed signal are shown in Figure 8 and its modal parameters are calculated. The results are shown in Table 5.

Figure 8.

(a) Time domain and (b) frequency domain of original and reconstructed signal without prior information when SNR = 10 db.

Table 5.

Errors in the modal parameters of the signal.

	${\omega }_1\left(Hz\right)$	${\zeta }_1$	${\omega }_2\left(Hz\right)$	${\zeta }_2$
Original signal	108	0.027	437	0.053
Reconstructed signal	108.055	0.030	436.716	0.055
Error (%)	0.051	11.111	0.065	3.774

It can be seen directly from Figure 8 that the reconstructed signal tracks the original signal well in both the time domain and the frequency domain. Although some spurious modals are present (Figure 8(b)), the dominant modal is precisely identified. The modal parameter errors (Table 5) are all in a very small order of magnitude, which confirms that the algorithm is still effective even in the absence of prior information. However, as the dimension of the dictionary increases, the computing time will inevitably increase.

Experiments

We use the data obtained from a certain type of aero-engine in the external bypass surge test to further verify the reconstruction ability of LWBP under the condition of constant speed. The rotating speed of engine is 6900 r/min, by adjusting the engine bleed valve, the blade flutter occurs and disappears after a period of time. Totally 18 blades are uniformly installed on the impeller, and a BTT fiber optic sensor is installed on the casing to record the arrival time of each blade. Figure 9 shows the vibration displacement signal of a single blade. Since all blades vibrate in a same mode when flutter occurs, it could be equivalent to measuring the vibration displacement of one blade with 18 uniformly distributed BTT sensors, which means $L=18$ , and the sampling rate is $115\times 18=2070Hz$ . The complete vibration displacement signal pieced from 18 blades is shown in Figure 10. When flutter occurs, the blade vibration displacement gradually diverges, then the blades vibrate intensely in 6.5-8.2 s, which may lead to the risk of scraping the casing. After 8.2 s, the vibration displacement gradually attenuates and the flutter disappears.

Figure 9.

Vibration displacement of a single blade.

Figure 10.

Vibration displacement integrated from 18 blades.

The original signal which measured by 18 BTT sensors with a length of 900 (4.35 s to 4.78 s in Figure 10) is intercepted from the divergence process, and its frequency spectrum is shown in Figure 11. It has two dominant modals of $420.9Hz$ and $535.9Hz$ , and the damping are, respectively, $-4.16e^{-4}$ and $-2.62e^{-4}$ . Extract two positions from 18 sensors, setting $I=2$ and the sensors distribution $C=\left\{\mathrm{1,10}\right\}$ , then the under-sampled signal with sampling rate of $115\times 2=230Hz$ could be obtained and its frequency spectrum is shown in Figure 12. By setting $m=2,n=40,k=32$ , ${\omega }_{\psi 1}=410\times 2\pi$ , ${\omega }_{\psi 2}=530\times 2\pi$ , $\mathrm{\Delta }\omega =0.5\times 2\pi$ , and $\mathrm{\Delta }\zeta =-0.00002$ , an over-complete dictionary $\mathrm{\Psi }$ was designed. After substituting the optimal solution of equation (10) into equation (5), the obtained original signal and reconstructed signal are shown in Figure 13. Likewise, a signal of length 900 (8.26 s to 8.69 s in Figure 10) was intercepted from the convergence process. After down-sampling and reconstructing, the result is shown in Figure 14. Then, the modal parameters of the reconstructed signal were obtained using the matrix pencil method and the results are shown in Table 6.

Figure 11.

Frequency spectrum of original signal in divergence process.

Figure 12.

Frequency spectrum of under-sampled signal in divergence process.

Figure 13.

(a) Time domain and (b) frequency domain of original and reconstructed signal in divergence process.

Figure 14.

(a) Time domain and (b) frequency domain of original and reconstructed signal in convergence process.

Table 6.

Comparison of experiment results.

		${\omega }_1\left(Hz\right)$	${\zeta }_1$	${\omega }_2\left(Hz\right)$	${\zeta }_2$
Divergence	Original signal	420.87	−4.16e-4	535.88	−2.62e-4
	Reconstructed signal	420.87	−4.30e-4	535.94	−2.42e-4
	Error (%)	0	3.37	0.01	7.63
Convergence	Original signal	421.81	1.51e-3	537.40	1.20e-3
	Reconstructed signal	421.89	1.52e-3	537.51	1.20e-3
	Error (%)	0.02	0.67	0.02	0

Since the non-dominant modals are not considered when designing the dictionary, the non-dominant modals are filtered out as noise during the reconstruction process. Regardless of the divergence stage or the convergence stage, the reconstruction effect of the vibration displacement signal of the blade has achieved good results, and the dominant modals are accurately reconstructed. The errors of the two parameters we care about (frequency and damping) are relatively small, which verifies LWBP and shows that the occurrence and retreat of blade flutter could be identified in a very short time based on the LWBP.

It's interesting to consider whether the LWBP can fully recreate all of the vibration modals of the original signal if these non-dominant modals are also included in the construction of the sub-dictionary. Figure 11 shows that the original signal also contains 11 low energy non-dominant modals in addition to the two dominant modals. Four of these modals are chosen because they have a high energy level, additionally, 4 Laplace wavelet sub-dictionaries are designed, and 2 sensors and 4 sensors are used, respectively, to reconstruct the original signal after compression sampling. The parameters chosen for reconstructing the signal in Figure 13 are consistent with the frequency parameters of the atoms in the sub-dictionary. Then, an over-complete dictionary $\mathrm{\Psi }$ was designed. After substituting the optimal solution of equation (10) into equation (5), the obtained original signal and reconstructed signal are shown in Figure 15. Then, the modal parameters of the reconstructed signal were obtained using the matrix pencil method and the results are shown in Table 7.

Figure 15.

(a) Time domain and (b) frequency domain of original and reconstructed signal with 2 sensors in divergence process. (c) Time domain and (d) frequency domain of original and reconstructed signal with 4 sensors in divergence process.

Table 7.

Modal parameter identification results in divergence process using 4 sensors.

		Original	Reconstructed	Error (%)
Dominant modals	${\omega }_1\left(Hz\right)$	420.87	420.27	0.14
	${\zeta }_1$	−5.04e-4	−4.71e-4	6.55
	${\omega }_2\left(Hz\right)$	535.89	535.74	0.03
	${\zeta }_2$	−2.62e-4	2.78e-4	6.11
Non-dominant modals	${\omega }_3\left(Hz\right)$	690.09	689.92	0.02
	${\zeta }_3$	−5.38e-5	−5.56e-4	3.35
	${\omega }_4\left(Hz\right)$	766.23	766.27	0.01
	${\zeta }_4$	−4.74e-4	−4.60e-4	2.95
	${\omega }_5\left(Hz\right)$	804.97	804.64	0.04
	${\zeta }_5$	−9.54e-5	−1.31e-4	37.3
	${\omega }_6\left(Hz\right)$	995.60	995.05	0.06
	${\zeta }_6$	1.03e-4	−1.11e-4	7.77

Figure 15 illustrates that when we focus on six modals, just two sensors cannot be used to accurately reconstruct the signal, and there are too many atoms to prevent the major modals from being reconstructed. The total power of the reconstructed signal is virtually the same as that of the original signal, according to the time-domain figure. LWBP only successfully reconstructs the amplitude-frequency characteristics of one major modal from the frequency domain diagram, and it performs poorly when trying to do the same for the other five modals. Four actual sensors increase the amount of information in the compressed sample signal, which doubles. Currently, just one non-dominant modal is present, and the reconstruction findings of the two main modals and three non-dominant modals are fairly precise. Although the modal's amplitude-frequency curve reconstruction effect is poor, the technique nonetheless accurately locates the modal component in the frequency domain and detects the presence of the modal in the compressed sampling signal.

Combining Figure 15(d) and Table 7, it can be shown that the reconstructed signal's spectral peaks for modal 1 and 2 are marginally greater than those of the original signal. The reconstructed signal's modal 3 and modal 4 spectral lines essentially match those of the original signal. The spectral peak of modal 5 in the reconstructed signal is closer to the original signal, but it produces two spurious modals near 804Hz, which makes the estimated value of ${\zeta }_4$ far from the original signal. Modal 6's spectral line deviates slightly from the original signal. With the exception of ${\zeta }_4$ , all six modals' frequency parameter reconstruction errors are below 0.2%, and the damping parameter reconstruction errors are all below 10%, demonstrating the accuracy of the reconstruction using LWBP. The estimation of the modal parameters of structural signals is typically more precise and can not only identify the dominant modal but also the non-dominant modal.

Conclusions

This study aimed to comprehensively estimate the structure modal parameters of rotating machinery blades. Based on compressed sensing theory and blade tip timing technology, a new reconstruction method LWBP for impulse response signals was developed to fully evaluate the blade structure modal information contained in the BTT signal. Simulations and experiments were performed to demonstrate the advantages and validity of the LWBP. It was confirmed that this approach provides algorithmic support for (i) estimation of rotating blade multi-modal parameters under the action of aerodynamic forces and (ii) comprehensive monitoring of the dynamic strength performance of the blade. The key procedures and results are summarized as follows:

1) In the field of BTT, the research on damping blade vibration is insufficient. This research studies on damping blade vibration signal, and proposes a BTT sampled damping signal reconstruction method LWBP based on compressive sensing theory. The LWBP method successfully reconstructs the damping signal, fills the gap in the field of BTT signal processing.

2) This research proposes the LWBP method, which constructs a new dictionary comprising a Laplace wavelet family. For an unknown signal, prior information is needed to increase the atom density within a certain range around the modals. The signal can be reconstructed based on the designed dictionary and a sparse solution, which is obtained by optimizing a convex unconstrained regularization model with penalty. For the similarity of structure response signal and Laplace wavelet function, the atoms of the Laplace wavelet dictionary match the damping blade vibration signal more than the atoms of the Fourier dictionary, resulting in a sparser characteristic of reconstructed signal.

3) When the modals of the structure response signal match the atoms in the dictionary, the signal has the greatest sparseness in the transform domain, and the LWBP’s reconstruction effect is excellent. When the modals of the structure response signal do not match the atoms, the sparsity in the transform domain is weaker, but the structural response signal can still be recovered well with a specially designed dictionary.

4) LWBP demonstrates certain anti-noise performance, it can work normally when the SNR is 5dB, and the variation in the modal parameter error with the regularization parameter was obtained.

5) LWBP can accurately identify the major mode and take into account the non-dominant mode under the presumption of proper parameter selection. It can also detect and rebuild the vibration signal of multi-mode tiny damping vibration.

6) In the conducted experiment, the complete signal can be recovered from the under-sampled BTT signal, then the modal parameter could be accurately estimated by the matrix pencil method.

7) The LWBP method still has some room for improvement (a) When there is no prior knowledge, it is still possible to rebuild the principal modals by creating the Laplace wavelet dictionary in a wider frequency range, but the computing complexity is significantly increased. (b) When the number of modals rises, there is a greater chance that some of the reconstructed signals will contain spurious modals. This will reduce the reconstruction accuracy and have a significant impact on the estimation of the modal parameters. (c)The setting principles for the frequency of atoms in the dictionary and the interval of damping parameters are also very important to consider. A more sensible technique for setting parameters can be derived using formula derivation in the subsequent work.

Supplemental Material

Supplemental Material - A modal estimation method of rotating blade based on compressed sensing and blade tip timing

Supplemental Material for A modal estimation method of rotating blade based on compressed sensing and blade tip timing by Hua Zheng, Guanyu Fang, Zhenglong Wu, Shiqiang Duan, and Jiangtao Zhou in Journal of Low Frequency Noise, Vibration and Active Control