Improved minimum variance distortionless response spectrum method for efficient and robust non-uniform undersampled frequency identification in blade tip timing

Abstract

The noncontact blade tip timing (BTT) measurement has been an attractive technology for blade health monitoring (BHM). However, the severe undersampled BTT signal causes a significant challenge for blade vibration parameter identification and fault feature extraction. This study proposes a novel method based on the minimum variance distortionless response (MVDR) of the direction of arrival (DoA) estimation for blade natural frequency estimation from the non-uniformly undersampled BTT signals. First, based on the similarity between the general data acquisition model for BTT and the antenna array model in DoA estimation, the circumferentially arranged probes on the casing can be regarded as a non-uniform linear array. Thus, BTT signal reconstruction is converted into the DoA estimation problem of the non-uniform linear array signal. Second, MVDR is employed to address the severe undersampling issue and recover the BTT undersampled signal. In particular, spatial smoothing is innovatively utilized to enhance the estimation of covariance matrix of the BTT signal to avoid ill-condition or singularity, while improving efficiency and robustness. Lastly, numerical simulation and experimental testing are employed to verify the validity of the proposed method. Monte Carlo simulation results suggest that the proposed method behaves better than conventional methods, especially under a lower signal-to-noise ratio condition. Experimental results indicate that the proposed method can effectively overcome the severe undersampling problem of BTT signal induced by physical limitations, and has a strong potential in the field of BHM.

Graphical abstract

Keywords

blade tip timing (BTT) / frequency identification / minimum variance distortionless response (MVDR) / undersampled / blade health monitoring (BHM)

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Nomenclature

Abbreviations
ADMM	Alternating direction method of multipliers
BHM	Blade health monitoring
BTT	Blade tip timing
DoA	Direction of arrival
ESPRIT	Estimation of signal parameters via rotational invariance techniques
FEM	Finite element modeling
IMVDR	Improved minimum variance distortionless response
IRLS	Iteratively reweighted least squares
MUSIC	Multiple signal classification
MVDR	Minimum variance distortionless response
NUFT	Non-uniform Fourier transform
OMP	Orthogonal matching pursuit
OPR	Once per revolution
RMSE	Root mean square error
SNR	Signal-to-noise ratio
ToA	Time of arrival
Variables
a	Element of steering vector a
a	Steering vector
af	Steering vector with tentative frequency f
A	Array manifold matrix
Ax	Array manifold matrix for signal measurement
ci	Non-zero elements in the ith row of the matrix \sqrt{\mathit{S}/N}
f	Frequency of synthetic signal
fk	kth frequency in frequency grid
{\hat{f}}_i^n	ith estimated frequency in the nth Monte Carlo simulation
fr (t)	Blade’s instantaneous rotation frequency at time t
{\overline{f}}_{\mathrm{r}}\left(N_{\mathrm{r}}\right)	Blade averaged rotation frequency at the Nrth revolution
(f0, f1, ..., fK − 1)	Frequency grid
\left\{f_0^{\prime },f_1^{\prime },\dots ,f_{m-1}^{\prime }\right\}	Frequency set of the blade tip vibration
G	Block matrix
I	Identity matrix
K	Number of frequency grid
L	Length of the snapshot vector
m	Number of the frequencies in the input signal vector x(tn)
M	Length of the input signal vector x(tn)
n	Index of the first value
n(tn)	Zero-mean additive noise vector
N, Nite, Nmax, Nmc, Nr	Numbers of the snapshots, iterations of MVDR, max iterations, Monte Carlo simulation, and the revolution, respectively
Pxx (a)	Power spectral density in steering vector a
Pxx (af)	Power spectral density in steering vector af
{\hat{P}}_{xx}\left(a_{f_k}\right)	Estimated power spectral density in steering vector {\mathit{a}}_{f_k}
Pxx (A)	Diagonal matrix in which diagonal elements represent the power spectral density
Q	Number of probes
R	Radius of the measurement point
Ri	Correlation matrix of the ith snapshot
Rxx	Correlation matrix of the signal x
{\hat{\mathit{R}}}_{xx}	Estimated correlation matrix of the signal x
{\mathit{R}}_{xx}^{\prime }	Estimated correlation matrix of the signal x with spatial smoothing
sij	Element of the matrix \sqrt{\mathit{S}/N} in row i and column j
{s0, s1, ..., sm−1}	Amplitude set of the blade tip vibration
s(tn)	Vector of each frequency value at time tn
S	Covariance matrix of s(tn)
\overline{\mathit{S}}	Spatial smoothed covariance matrix of s(tn)
T_{n_r}	Time interval of adjacent pulses
tact	Actual arrival time of the blade tip
texp	Expected arrival time of the blade tip
tn	Arrival time vector
vi	ith row of the Vandermonde matrix
w	Filter coefficient
w(t)	Filter coefficient at time t
w	Filter coefficient vector
\hat{\mathit{w}}\left(\mathit{a}\right)	Optimal filter coefficient vector
W	Filter coefficient matrix
x	Input signal
x(t)	Vibration displacement of the blade at time t
x(tn)	Input signal vector
xi(tn)	ith snapshots of input signal vector x(tn)
y(tn)	Output signal of the filter at time tn
θq	Installation angle of the qth probe
Δθ	Difference between two adjacent frequencies
ε	Error
{\sigma }_n^2	Variance of Gaussian white noise
\lambda	Regularization parameter of ADMM
\rho	Learning rate of ADMM
{\omega }_0	Center frequency
\left\{{\phi }_0,{\phi }_1,\dots ,{\phi }_{m-1}\right\}	Phase set of the blade tip vibration
\mathbf{\Lambda }	Diagonal matrix

Acknowledgements

We are grateful for the support provided by the National Natural Science Foundation of China (Grant Nos. 52105117 and 51875433), and the Funds for Distinguished Young Talent of Shaanxi Province, China (Grant No. 2019JC-04).

Conflict of Interest

The authors declare that they have no conflict of interest.

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