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Comprehensive analysis of aeroengine vibration signals based on wavelet transform method
EURASIP Journal on Advances in Signal Processing volume 2023, Article number: 117 (2023)
Abstract
A single type of signal processing means that it is difficult to analyze vibration signals comprehensively and effectively. By comprehensively using wavelet analysis techniques, a comprehensive and indepth study of aeroengine vibration conditions is realized as a way to carry out health management. By introducing various types of wavelet analysis techniques and using Labview2022 programming, corresponding signal processing tools are developed for the analysis of the collected vibration signals. The comprehensive analysis of aeroengine vibration signals based on the wavelet transform method is realized, and the corresponding products are successfully applied in engineering practice.
1 Introduction
Aeroengine is an extremely precise industrial product, which is difficult and costly to develop [1]. Aeroengine is the core component of the aircraft, whose good condition is related to the safety and reliability of the aircraft. Therefore, the maintenance requirements of the aircraft engine are high [2].
For industrial products, processwise, frequent disassembly, and installation during maintenance is detrimental to the reliability of the equipment and can even lead to fatigue of the mechanical components, thus endangering the safety of the equipment [3, 4]. Therefore, nondestructive testing by radiation detection, electromagnetic testing, and vibration testing has received a great deal of attention in the field of engineering technology. The study of aeroengine working conditions by vibration signal measurement is considered the most intuitive and effective inspection method in the industry [5, 6].
However, for signal analysis, it is often difficult to use a single analysis method to achieve the purpose of comprehensive investigation, and it will show its disadvantage when analyzing the signal of a specific condition. Therefore, here we introduce a variety of analysis methods based on wavelet transform to achieve the goal of comprehensive analysis of the signal. In this study, LabVIEW2022 is used to develop signal processing tools, and a certain type of aeroengine is used as an experimental measurement object for diagnostic analysis.
2 Design and implementation of system functions
2.1 Function designs
By applying the previously designed vibration signal wireless acquisition module based on 2.4G technology development (shown in Fig. 1), the aeroengine vibration signal is acquired and stored in the computer for analysis.
The wavelet transform window changes with the change of frequency and incorporates the idea of Fourier local transform, which can be used to analyze the signal by highlighting the features of the corresponding aspects flexibly through the transform [7, 8]. The specific analysis functions are developed based on wavelet transform, and there are three types: multiresolution analysis based on discrete wavelet transform; multiscale analysis based on wavelet transform; and twodimensional wavelet transform analysis. The functional design of the system is shown in Fig. 2.
2.2 Principle of analysis tools
2.2.1 Principle of multiresolution analysis based on discretetime wavelet transform
Signal flow exists in a discrete signal pattern in a computer. The multiresolution discrete wavelet transform analysis can be represented in the following concrete form [9, 10]. Let \(\left\{ {V_{j} } \right\}\) be a series of closed subspaces in \(L^{2} \left( R \right)\) with \(j \in Z\). To claim that \(\left\{ {V_{j} } \right\}\), \(j \in z\) is a multiresolution analysis, the following six conditions must be satisfied simultaneously [11].

①
There exists \(\left( {j,k} \right) \in Z^{2}\) with \(x\left( t \right) \in V_{j}\), then \(x\left( {t  2^{j} k} \right) \in V_{j}\)

②
There exists \(j \in Z,v_{j} \supset v_{j + 1}\), i.e. \(\cdots V_{0} \supset V_{1} \supset \cdots v_{j} \supset v_{j + 1} \cdots\)

③
There exists \(j \in Z,\) if \(x\left( t \right) \in V_{j}\), then \(x\left( \frac{t}{2} \right) \in v_{j + 1}\)

④
$$V_{j} = \bigcap\limits_{j =  \infty }^{\infty } {V_{j} } = \left\{ 0 \right\}$$(1)

⑤
$$V_{j} = {\text{Closure}}\left( {\bigcup\limits_{j =  \infty }^{\infty } {V_{j} } } \right) = L^{2} \left( R \right)$$(2)

⑥
There exists \(\theta \left( t \right)\), such that \(\left\{ {\theta \left( {t  k} \right)} \right\}\) the Riesz basis in V_{0} and \(k \in Z\)
Let V_{0} be a Hilbert space and {\(\theta_{k}\) = \(\theta \left( {t  k} \right)\)},\(k \in Z\) be a set of vectors in V_{0} whose number is the same as the dimension of V_{0}. Any element x in V_{0} can be tabulated as a linear combination of \(\theta_{k}\) [12].
If
is linearly independent and there exists a constant \(0 < A \le B < \infty\) such that
Then {\(\theta_{k}\) = \(\theta \left( {t  k} \right)\)},\(k \in Z\) is the Riesz basis in \(V_{0}\) [11].
Let \(\left\{ {V_{j} } \right\} ,j \in Z\) be a multiresolution analysis and φ(t) be a scale function. If its Fourier transform can be given by the following equation
and make
Then, \(\phi_{j,k} \left( t \right)\) is an orthogonal normalized basis in \(V_{j}\) for all \(j \in Z\).
2.2.2 Principles of wavelet transform multiscale analysis
The multiscale analysis consists of a column of subspaces \(\left\{ {V_{j} } \right\},j \in Z\) of L^{2}(R), where L^{2}(R) denotes the Hilbert space of all square productable functions, and R and Z denote the set of real numbers and the set of integers, respectively [13,14,15]. The multiresolution analysis allows to establish the dual scale equation of the scale function φ(t), i.e., the scale function at one scale can be derived from its own linear combination at the next scale [16,17,18]. The scale function and the wavelet mother function are related to each other, and the wavelet mother function ψ(t) can be obtained from the scale function φ(t).
\(h_{n}\) is the scale function
where \(\sqrt 2\) is the normalization factor and \(g_{n}\) is the coefficient derived from the scale factor \(h_{n}\).
Let \(\psi_{j,k} \left( t \right)\) be the expansion and translation of the wavelet mother function \(\psi \left( t \right)\), and \(\psi_{j,k} \left( t \right)\) form the orthogonal basis of \(f\). \(\phi_{ok} \left( t \right)\) is a telescoping translation of the ruler function \(\phi \left( t \right)\), then at any \(f\left( t \right) \in L^{2} \left( R \right)\), it can be decomposed as:
Among them,
J controls the resolution of the wavelet analysis.
2.2.3 Principle of twodimensional wavelet transform analysis
Since image and computer vision signals are generally two or multidimensional information, it is important to extend the signal analysis to two or more dimensions [19,20,21].

①
Definition
Let \(f\left( {x,y} \right) \in L^{2} \left( {R^{2} } \right)\), then its twodimensional wavelet transform is:
where b_{1}, b_{2} are the translation values of the basic wavelet function in two dimensions, and the twodimensional wavelet function \(\psi \left( {x,y} \right)\) should satisfy the tolerance condition:
It can be shown that the corresponding reconfiguration equation is
Discrete wavelet transform is obtained by discretizing the parameters a, b in Eq. (15): a = 2^{j}, b_{1} = al, b_{2} = am:
In practice, commonly used wavelet functions are binary functions with separable variables.
Twodimensional multiscale analysis is derived from onedimensional multiscale analysis, which in turn leads to twodimensional wavelet space and wavelet function.
Let \(\left( {\left\{ {V_{j}^{1} \left( t \right)} \right\}_{j \in Z} ,\varphi^{1} \left( t \right)} \right),\left( {\left\{ {V_{j}^{2} \left( t \right)} \right\}_{j \in Z} ,\varphi^{2} \left( t \right)} \right)\) be two multiscale analyses of \(L^{2} \left( {R^{2} } \right)\), and \(\psi^{1} \left( t \right),\psi^{2} \left( t \right)\) be the corresponding orthogonal wavelet functions. The tensor product space formed by \(V_{j}^{1}\) and \(V_{j}^{2}\) is defined as
Let \(W_{j}^{1}\) be the orthogonal complementary space of \(V_{j}^{1}\) in \(V_{j  1}^{1}\) and \(W_{j}^{2}\) be the orthogonal complementary space of \(V_{j}^{2}\) in \(V_{j  1}^{2}\), then we have the following expressions [22].

②
Reconstruction formula for twodimensional wavelets
From the definition of 1D multiscale analysis there are
Thus, the sequence of subspaces \(\left\{ {\tilde{V}_{j} } \right\}\)_{j∈Z} satisfies the definition of multiscale analysis, and thus we obtain a multiscale analysis of the twodimensional space \((\left\{ {\tilde{V}_{j} } \right\}\)_{j∈Z},\(\varphi \left( {x,y} \right) )\).
of which
Therefore, for any \(f\left( {x,y} \right) \in L^{2} \left( {R^{2} } \right)\) there is a unique decomposition.
where \(g_{j} \left( {x,y} \right) \in \tilde{W}_{j}\), so the following expressions are available.
From orthogonality it is known that
From the definition of twodimensional wavelet transform, we know that \(a_{k,m}^{j}\), \({\upbeta }_{k,m}^{j}\), \(\gamma_{k,m}^{j}\) are twodimensional discrete wavelet transforms of \(f\left( {x,y} \right)\); \(\psi^{1} \left( {x,y} \right)\), \(\psi^{2} \left( {x,y} \right)\), \(\psi^{3} \left( {x,y} \right)\) are three different wavelet functions, all of them are functions of variable separation type consisting of a onedimensional wavelet function and a scale function [23,24,25].
is the scale function corresponding to the three different wavelet functions.
2.3 Programming design
The project team used LabVIEW2022 to develop and implement functions based on discrete wavelet transform multiresolution analysis, wavelet transformbased multiscale analysis, and twodimensional wavelet transform analysis.
The system enables a comprehensive study of the signals, allowing engineers to have a deeper grasp of the vibration of the aeroengine under test. In this way, it can determine the working status of the aeroengine, realize the fault diagnosis, and play the role of health management. The program panel is shown in Fig. 3.
3 Experiments and discussion
3.1 Experiment I
A model of a small aeroengine (teaching prototype) was used as the experiment object. By starting the engine on the ground bench, the project team collected its vibration signal in normal operation (economic cruise state) as the normal reference signal. By setting the engine stuck fault, its vibration signal in that operation state was collected and compared with the former for illustration.
As shown in Fig. 4, the upper panel is the result screen of running a normal reference signal; the lower panel is the result screen of running a stuck fault vibration signal. The system interface presents a combination of four display frames: the original waveform map of the signal, the discrete wavelet transformbased multiresolution analysis map, the wavelet transformbased multiscale analysis map, and the twodimensional wavelet transform analysis map. In general, each analysis method can effectively deduce the characteristics of the signal. When this type of aeroengine is in cruise, the signal shows smoothness, indicating that the unit is running steadily; when it is in a stuck state, the plot boxes of each method demonstrate the corresponding state characteristics.
As shown in Fig. 5, the upper figure is the original waveform of the normal reference signal, and the lower figure is the original waveform of the vibration signal when the engine is in a stuck fault condition. The two can be clearly distinguished from each other by the smoothness of the signal.
As shown in Fig. 6, the left and right figures show the results based on the discrete wavelet transform multiresolution analysis of the above signals: the left figure presents the results for the normal reference signal, and the right figure presents the results when it is in a stuck fault condition. There is a clear distinction between the two features.
As shown in Fig. 7, two signals are presented based on the multiscale analysis of the wavelet transform: the upper panel shows the results of the analysis of the normal reference signal; the lower panel shows the results of the analysis of the signal when the engine is in a stuck fault condition. The interface has 3 display windows, and the differences between the two are very obvious when examined in terms of the periodicity and smoothness of the signals.
As shown in Fig. 8, what is presented is a plot of the signal after analysis by 2D wavelet transform: the upper plot is the result of analyzing the normal baseline signal; the lower plot is the result of analyzing the signal when the engine is in a stuck fault condition. Twodimensional mapping has the property of showing the energy condition of the signal, so it is practical to examine the signal condition from the smoothness of the signal and the regularity of the energy distribution. The normal reference signal has good smoothness, and the energy distribution shows obvious regularity; while in the stuck fault state, the signal smoothness is poor and the energy distribution varies widely. The differences between the above two signals are obvious.
3.2 Experiment II
According to the same conditions as the previous experiment, a model of a small aeroengine (teaching prototype) was used as the experimental object. Fixed on the ground bench to start, the project team collected its vibration signal in normal operation (economic cruise state) as the normal reference signal. By setting up an engine sticky fault, the project team collected its vibration signal in that operating condition and compared it with the former for illustrative purposes.
As shown in Fig. 9, the upper panel is the result screen of running a normal reference signal; the lower panel is the result screen of running a sticky fault vibration signal. When this type of aeroengine is in a sticky state, the signal shows smoothness and a certain regularity in the energy distribution from the operation results of individual analysis methods. However, operation in a sticky state is also a fault.
As shown in Fig. 10, the upper figure is the original waveform of the normal reference signal, and the lower figure is the original waveform of the vibration signal when the engine is in sticking condition. From the characteristics of the waveform, the waveform graph of the vibration signal in the sticky state already has a certain smoothness. The differentiation of the signal characteristics is not as obvious as the previous stuck signal waveform. Therefore, it will affect the accuracy of fault determination using this tool.
As shown in Fig. 11, the left and right plots show the results based on the discrete wavelet transform multiresolution analysis of the above signal. The left figure presents the results for the normal reference signal, while the right figure presents the results when it is in the sticky state. The features between the two are clearly distinguished.
As shown in Fig. 12, two signals are presented based on the multiscale analysis of wavelet transform. The top panel is the result of analyzing the normal reference signal; the bottom panel is the result of analyzing the signal when the engine is in a sticky state. The interface has 3 display windows, and the differences between both are very obvious when examined in terms of the periodicity and smoothness of the signals, but the signal in the sticky state shows good periodicity and smoothness. The distinguishability appears weaker than the previous card resistance signal. Therefore, it can affect the accuracy of the judgment using this tool.
As shown in Fig. 13, the plot is presented after the signal is analyzed by twodimensional wavelet transform. The upper figure is the result of analyzing the normal reference signal; the lower figure is the result of analyzing the signal when the engine is in a sticky state. Although the plots of signals in the sticky state after processing by this method are somewhat distinguishable from those formed by the constant reference signal, they are more similar to the signals generated in the stuck state (as shown in Fig. 8) and are not very easy to distinguish. Therefore, it is not reliable to use this method as a single criterion for fault type. By combining multiple analysis methods, we can analyze the signal comprehensively to achieve fault diagnosis and effective identification.
4 Conclusion
This study uses LabVIEW2022 to successfully implement a system for comprehensive analysis of aeroengine vibration signals based on the wavelet transform method, which can quickly carry out fault diagnosis based on aeroengine, and the design has achieved the expected goal.
The practical application proves that all subfunctions of the system can be operated effectively. For different processing requirements, the system can also flexibly set each parameter for detailed analysis. The tool has successfully served the frontline work and has been unanimously recognized and welcomed by engineers and technicians in related fields.
By using wireless communication technology and a wavelet analysis tool developed based on LABVIEW, the research team successfully realized wireless monitoring and troubleshooting of an aeroengine. In terms of design and application, the system, on the one hand, is able to monitor the operation of the aeroengine and, on the other hand, is able to carry out auxiliary diagnostics of the type of faults based on the visualization and analysis of the signals.
Availability of data and materials
The original data have been stored in” https://figshare.com/articles/figure/PIC3/21724298”.
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Acknowledgements
We would like to express our deep gratitude for the strong support provided by Nanjing Univ Aeronaut & Astronaut, Coll Energy & Power Engn, Jiangsu Prov Key Lab Aerosp Power Syst.
Funding
This research was supported by the ‘Innovation Research Fund for Doctoral Students of Nanjing University of Aeronautics and Astronautics,’‘National Natural Science Foundation of China,’ and 'Funding of Jiangsu Innovation Program for Graduate Education.’
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Xin, M., Ye, Z., Zhao, Y. et al. Comprehensive analysis of aeroengine vibration signals based on wavelet transform method. EURASIP J. Adv. Signal Process. 2023, 117 (2023). https://doi.org/10.1186/s1363402301079y
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DOI: https://doi.org/10.1186/s1363402301079y