Operating modal analysis from sub-sampled vibration response based on compressed sensing and FastICA

Abstract

In order to identify more high order modal from a small amount of vibration response signal sampling, this paper proposes a method for identifying the OMA of sub-sampled vibration response signals based on compression sensing (CS) and fast independent component analysis (FastICA). Firstly, the method subsamples a large number of vibrational signals to retain the original key information of the signal; Secondly, the subsampled signal is restored by using the compressive sampling matching pursuit (CoSaMP) algorithm of the compression-aware reconstruction algorithm. Finally, the source signal is separated by FastICA to obtain the multi-order natural frequencies and modal mode shapes of the structure. According to Shannon’s theorem, if the analog signal is reproduced without distortion, the sampling frequency should be no less than 2 times the highest frequency in the analog signal spectrum. This method can reduce the amount of original signal acquisition and the requirements for the frequency of the sampled signal, and the recovery signal has a high similarity with the original signal and the natural frequency is not changed. It can effectively identify the higher-order modes of the structure on the basis of breaking through the Nyquist sampling frequency. Experimental results of the subsampled vibration response signal on the uniform steel cantilever beam show that the method can reconstruct the original signal from a small number of subsampled vibration response signals, and identify the high-order nature frequency and modal mode shape that break through the Nyquist sampling frequency.

Keywords

Operational modal analysis sub-sampled compressed sensing FastICA

1. Introduction

Operational modal analysis (OMA) can identify modal parameters of the structure only from the vibration response signals collected by the vibration sensor. In the past decade, the blind source separation method has made great progress in OMA, especially second order blind identification (SOBI) [1] and independent component analysis (ICA) method. Jun Chang et al. [2] proposed a new method of “ICA+IDT” to overcome the problem that ICA could not identify structures with higher damping, which could identify structures with higher damping effectively. Xin Wang et al. [3] proposed the SCA based underdetermined OMA method to identify higher order modes that exceed the number of sensors.

CS [4–7] is a new type of signal compression technology, unlike the traditional sampling theorem, compressed sensing can be much smaller than the Nyquist sampling theorem requires random sampling of signals, and restore the original signal with less error. Nowadays, CS is widely used in areas such as image processing, but it has not been applied to OMA of subsampled vibration response signals.

Aiming at the problems of too large amount of signal data and too high requirement for signal sampling frequency in OMA, this paper proposes a method to compress the signal and identify modal parameters by using compressed sensing and FastICA [8–10] algorithm. The main contributions are as follows: (1) In view of the large amount of vibration signal data and the requirement for sampling frequency, this paper uses subsampling technology to collect data, which effectively reduce the amount of required signal data and the requirement for signal sampling frequency. (2) Compared with traditional OMA method, this paper adopts the method of combining CS and FastICA, which ensures the results of modal parameter identification on the basis of reducing the amount of collected data.

2. Operating modal analysis from sub-sampled vibration response based on compressed sensing and FastICA

2.1. Basic theory of OMA

For a general small damping structure, as long as the natural frequencies of each order of the structure are not equal, there are n modes theoretically, and the vibration displacement response can be expressed in the modal coordinates as: $\begin{eqnarray}\displaystyle \mathbf{X}(t)=\boldsymbol{\it\Phi}\mathbf{Q}(t)=\mathop{\sum }_{i=1}^{n}\vec{{\varphi}}_{i}\vec{q}_{i}(t). & & \displaystyle\end{eqnarray}$ (1)

In Eq. (1). $\mathbf{X}(t)\in \mathbb{R}^{n\times T}$ represents the observation signal matrix, $\mathbf{X}(t)=[\overrightarrow{x_{1}}(t),\overrightarrow{x_{2}}(t),\ldots ,\overrightarrow{x_{i}}(t),\ldots ,\overrightarrow{x_{n}}(t)]^{\mathbf{T}}$ , $\boldsymbol{\it\Phi}\in \mathbb{R}^{n\times n}$ is a modal mode shape matrix composed of $\vec{{\varphi}}_{i}(i=1,2,\ldots ,n)$ of the modal mode shape vector of nth order; $\mathbf{Q}(\text{t})\in \mathbb{R}^{n\times T}$ is a vector matrix consisting of modal responses of each order $\vec{q}_{i}(t)$ .

2.2. Compressed sensing fundamentals

The collection method of compressed sensing [11,12] is not to collect data directly, but to perceive the signal through a specific set of waveforms. That is, to project the signal onto a given waveform (measure the correlation with the given waveform), and to perceive a set of compressed data. Finally, the optimal method is used to decrypt the compressed data, and the important information of the original signal is estimated.

Set the observation signal $\mathbf{X}(t)\in \mathbb{R}^{n\times T}$ , $\vec{\mathbf{X}}_{j}(t)$ expressed as a row j (j = 1,2, …, T) observation signal can be expressed as a set of linear combinations of orthogonal basis vectors 𝛹_j: $\begin{eqnarray}\displaystyle x_{j}(t)={\Psi}_{j}\vec{v}_{j}(t). & & \displaystyle\end{eqnarray}$ (2) 𝛹_j ∈ R^T×T is a sparse base (base matrix), $\vec{v}_{j}(t)\in R^{T\times 1}$ is a $\overrightarrow{x_{j}}$ coefficient vector in the 𝛹_j transformation domain, and if only s (s ≪ T) elements in the $\vec{v}_{j}(t)$ are not zero (or much greater than zero, and other elements are close to zero), the $\overrightarrow{v_{j}}(t)$ is said to be s-sparse.

Using an F × T (F < T) sampling matrix 𝜗 to compress the observation signal $\vec{x}_{j}(t)$ to obtain an observation vector $\vec{x}_{i}^{\prime }(\text{t})\in \mathbb{R}^{F\times 1}$ to achieve dimensionality reduction of the observation signal, this process can be expressed as: $\begin{eqnarray}\displaystyle \vec{x}_{j}^{\prime }(t)={\vartheta}{\Psi}_{j}\overrightarrow{v_{j}}(t). & & \displaystyle\end{eqnarray}$ (3)

${\Gamma}_{j}\in \mathbb{R}^{F\times T}$ is the measurement matrix, $\vec{x}_{j}^{\prime }(t)\in \mathbb{R}^{F\times 1}$ is the measurement value. For the specific problem, the 𝜗 and 𝛹_j are known, and therefore the 𝛤_j is also known, so that the coefficient vector can be $\vec{v}_{j}(t)$ can be obtained by measuring the matrix 𝛤_j, and then reconstruct the original signal. Commonly used sampling matrices are stochastic Gaussian matrices, stochastic Bernoulli matrices, and other stochastic matrices.

2.3. OMA of vibration signals based on compression sensing and FastICA

The subsampling $\vec{x}_{i}^{\prime }(\text{t})$ matrix is obtained by sampling the vibration response signal at unequal intervals using the sampling matrix 𝜗 on the sensor. $\begin{eqnarray}\displaystyle \vec{x}_{i}^{\prime }(t)={\vartheta}\vec{x}_{i}(t). & & \displaystyle\end{eqnarray}$ (4)

CS_FastICA method combines compression sensing with FastICA to identify modal parameters with no difference from FastICA’s natural frequencies and small error on the basis of breaking through the Nyquist sampling theorem, and also compresses the amount of data of the signal and breaks through the limit on the frequency of the sampled signal. CS_FastICA steps to go into:

Input: Row i Subsampled signals $\vec{x}_{i}^{\prime }(\text{t})$ , Signal sparsity sp

Output: Mix matrix A , source signal S(t)

The compression matrix $\vec{x}_{i}^{\prime }(t)$ is restored by using the CoSaMP algorithm to obtain the reconstructed signal $\overrightarrow{\hat{x}}_{\mathbf{i}}$ . By calculating the residual r (initial residual $r_{0}=\vec{x}_{i}^{\prime }(t)$ ) of the current iteration and the inner product of each column of the observation matrix, the first 2*sp with the largest result are selected, and then solved by least squares, and finally iteratively obtains $\hat{\mathbf{X}}(t)$ .

Signal separation. The reconstructed signal $\hat{\mathbf{X}}(t)$ blind source separation using FastICA algorithm to obtain the mix matrix A and the source signal S (t). The mixed matrix A is found to be convergent when it satisfies the inequality | A ^T ∗ A _old − 1| < ϵ. A _old denotes the mixing matrix obtained in the previous stage.

The modal mode shape Φ is obtained from the mixture matrix A , and the natural frequency of the reconstructed signal $\overrightarrow{\hat{x}}_{\boldsymbol{i}}(t)$ is determined by using the Fourier transform.

The algorithm mainly compresses the $\vec{x}_{i}$ of each observation signal to less than the amount of data required by Nyquist’s sampling theorem to obtain a compressed signal $x_{i}^{\prime }$ , and then uses the CoSaMP [13–15] algorithm to recover the recovery signal $\overrightarrow{\hat{x}}_{\boldsymbol{i}}(t)$ . Finally the FastICA algorithm is used to separate the recovery signal $\overrightarrow{\hat{x}}_{\boldsymbol{i}}(t)$ blindly to obtain the mixed matrix. The algorithm flow is shown in Fig 1.

3. Experimental validation

3.1. Introduction to experimental objects and datasets

Five displacement sensors are mounted on the beam using a uniform steel cantilever beam of 0.9 × 0.05 × 0.008 m³ to pick up the displacement response excited by the impact hammer. At a sampling frequency of 1600 Hz and a cutoff frequency of 800 Hz for all five channels, the output signal is digitally sampled by the DASP [16]. Figure 2 shows some of the experimental setups.

Fig. 1.

Subsampling OMA flowchart based on CS and FastICA.

Fig. 2.

Experimental setup—One cantilever beam and five displacement sensors.

Due to the lack of subsampled data sets, subsampled signals are generated using experimental data sets and sampling matrix simulations. The relationship between sampling time, data volume and frequency: $\begin{eqnarray}\displaystyle f=L/t & & \displaystyle\end{eqnarray}$ (5) where t is the sampling time, f denotes the sampling frequency, and L denotes the amount of data sampled.

The 2.25 s of experimental data was collected by the experimental equipment, with a total data length of 3600 and a sampling frequency of 1600 Hz. According to Nyquist’s sampling theorem to identify a frequency of 300 Hz, the sampling frequency must be 600 Hz or more, so the number of sampling points required is at least 1350 for the same time (2.25 seconds). In the following experimental procedure, the experimental data is compressed to 1260 samples in 2.25 seconds.

3.2. Evaluation indicators and methods

Use Cosine Similarity to determine how similar the recovered signal is to the original signal: $\begin{eqnarray}\displaystyle \cos ({\theta}_{i})=\frac{\mathop{\sum }_{i=1}^{n}x_{i}\times \hat{x}_{i}(t)}{\sqrt{\mathop{\sum }_{i=1}^{n}(x_{i}(t))^{2}}\times \sqrt{\mathop{\sum }_{i=1}^{n}(\hat{x}_{i}(t))^{2}}} & & \displaystyle\end{eqnarray}$ (6) where $x_{i}(t),\hat{x}_{i}(t)$ represent the ith value of the two vectors X(t) and $\hat{\text{X}}(t)$ , respectively. If the value of cos(θ_i) is closer to 1, its means that the similarity between the two vectors is higher.

In this paper, the modal assurance criterion (MAC) evaluation method is used to verify the experiment. When comparing the correlation of two vectors in a high-dimensional space, a common metric is MAC. As shown in Eq. (7). $\begin{eqnarray}\displaystyle \text{MAC}(\vec{\hat{{\phi}}_{i}},\vec{{\phi}}_{i})=\frac{(\vec{{\phi}}_{i}^{T}\vec{\hat{{\phi}}}_{i})^{2}}{(\vec{\hat{{\phi}}}_{i}^{T}\vec{\hat{{\phi}}}_{i})\cdot (\vec{{\phi}}_{i}^{T}\vec{{\phi}}_{i})}. & & \displaystyle\end{eqnarray}$ (7)

$\vec{{\phi}}_{i}$ and $\vec{{\phi}}_{i}^{\hat{∼}}$ are the modal mode modes of order i theory and identified. MAC values range from 0 to 1, where 0 means no correlation and 1 means full correlation.

3.3. Experimental parameter settings

In the CoSaMP algorithm, the sparsity sp is set to 50.

In the FastICA algorithm, the maximum number of iterations is 5000 and the convergence conditions is ϵ = e⁻¹⁰.

3.4. Experimental verification

3.4.1. Comparison of experimental results of observation signals and reconstructed signals

The observation signal sampling matrix is restored by CoSaMP to obtain the reconstructed signal. Figure 3 is a comparison between the reconstructed signal and the observation signal. Table 1 shows the similarity table of observed signals to reconstructed signals.

Fig. 3.

Comparison diagram of observed and reconstructed signals.

Fig. 4.

Modal shapes identification results by CS_FastICA.

Table 1

Similarities between observed and reconstructed signals

Observation signal	1	2	3	4	5
Similarity	0.9962	0.9914	0.9990	0.9980	0.9992

3.4.2. Comparison of experimental results of observation signals and reconstructed signals

The reconstructed signal obtains a mix matrix (modal mode shape) by FastICA. Figure 4 shows a comparison chart of the standard modal mode shape, the modal mode shape identified by the CS_FastICA, and the modal mode shape identified by FastICA. Table 2 shows results of natural frequencies identified by FastICA, natural frequencies identified by CS_ICA and MAC results.

Table 2
Identified results for natural frequencies and MAC results

Modal order Frequencies (Hz) MAC

Nature Frequencies FastICA CS_FastICA FastICA CS_FastICA

1 8.9 9.7778 9.7778 1.0000 1.0000

2 55.78 58.6667 58.6667 0.9995 0.9994

3 156.20 163.1111 163.1111 0.9999 0.9995

4 306.09 319.1111 319.1111 0.7626 0.9606

5 505.84 — — — —

Modal order	Frequencies (Hz)	MAC
1	8.9	9.7778	9.7778	1.0000	1.0000
2	55.78	58.6667	58.6667	0.9995	0.9994
3	156.20	163.1111	163.1111	0.9999	0.9995
4	306.09	319.1111	319.1111	0.7626	0.9606
5	505.84	—	—	—	—

3.5. Analysis of results

As can be seen from Fig. 3 and Table 1, the observation signal and the reconstructed signal are nearly coincident in time domain.

Figure 4 shows that compared to the standard results CS_FastICA methods have a high degree of similarity in the 1st to 4th order modes.

In Table 2 it can be concluded that FastICA is consistent with the natural frequencies identified by the CS_FastICA. And the MAC results show that the FastICA experimental results are similar to the experimental results of CS_FastICA. However, its fifth-order mode cannot be identified, and there is a phenomenon of modal absence.

4. Conclusions

This paper uses a combination of compression sensing and FastICA algorithms to identify modal mode shape of vibration signals. CS_FastICA method is a good way to identify higher-order nature frequencies and modal mode shape from a small number of subsampled vibration response signals.

However, we didn’t obtain subsampled signals directly from the vibration sensor, and the dataset will be obtained through related experiments in the future. There are still some deficiencies, and the problem of absence of the fifth-order mode will continue to be studied and discussed in the future. In the OMA method of sub-sampled vibration signal based on compression sensing and FastICA, it is found that the difference in sampling matrix has a greater impact on the experimental results, and the research on sampling matrix and different reconstruction algorithms still needs further exploration.

Footnotes

Acknowledgements

This research was supported by National Natural Science Foundation of China (Grant No. 51305142,51305143), the Guiding Project of Fujian Science and Technology Plan (Grant No. 2021H0019) and the Scientific Research Funds of Huaqiao University (16BS304).

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Modal order	Frequencies (Hz)			MAC
	Nature Frequencies	FastICA	CS_FastICA	FastICA	CS_FastICA
1	8.9	9.7778	9.7778	1.0000	1.0000
2	55.78	58.6667	58.6667	0.9995	0.9994
3	156.20	163.1111	163.1111	0.9999	0.9995
4	306.09	319.1111	319.1111	0.7626	0.9606
5	505.84	—	—	—	—