Adaptive window function and window size based operational modal parameter identification for linear time-varying structure

Abstract

In order to select the window function and window size adaptively before getting the results, we proposed adaptive moving window principle component analysis (AMWPCA) based OMA method to identify modal shapes and modal natural frequencies of slow LTV structures with weekly damped only from non-stationary vibration response signal online. The adaptive is achieved in two ways: change the window function or window size. We develop an adaptive indicator as the basis for window function and window size changes. Our adaptive approach is to make the difference between adjacent eigenvalues not too small. The operational modal parameter identification results in non-stationarity response signal dataset of a three-degree-of-freedom structure with slow time-varying mass show that comparing with fixed size moving window principle component analysis, our AMWPCA method can identify the modal shapes and modal frequencies better.

Keywords

Moving window adaptive window function linear time-varying structure operational modal analysis principle component analysis

1. Introduction

Operational modal analysis (OMA) is used to identify modal parameters (natural frequency, modal shape and modal damping ratio), which can reveal the dynamic nature of the structure [1] In reality, most structures are time-varying. Their parameters (mass, stiffness and damping) will change with time [2] Zhou et al. [3] proposed a vector time-dependent autoregressive model and least squares support vector machine based OMA method to estimate modal parameters of linear time-varying (LTV) structures. Ertveldt et al. [4] developed a frequency domain estimator of LTV structures for analysis and prediction of aeroelastic flutter. Li et al. [5] introduced a data-driven stochastic subspace identification method to extract the modal parameters of LTV structures. Dziedziech et al. [6] presented a new combined non-parametric and parametric method to track time-varying dynamics of mechanical systems. However, these methods are difficult to identify online and in real time.

Moving window method can deal with time-varying and online identification problems. Based on “frozen” technique, the non-stationary vibration response data can be divided into many parts and consider the system as linear time-invariant in each interval [7], which can be represented by data windows. Moving window methods slide the window along the data sample by sample. The newest data is absorbed and the oldest sample is discarded, which efficiently reduces the computation memory and improve the speed of adaptation [8]. Zhi et al. [9] proposed a kernel recursive extended least squares TARMA approach and the moving window is applied to make the algorithm operate online However, window size has a large impact on the recognition results [10] Zhou et al. [11] pointed out that the size of the moving window will directly affect the recognition delay and recognition rate and proposed an algorithm to obtain the optimal window size, but this method must compare the results of all window sizes. Portal et al. [12] proposed a “shape adaptive” moving window to select the number of pixels from a range of results Liu et al. [13] used general relative error and relative error to choose window size, but this method requires input information and does not apply to OMA that only knows the output data. In short, the choice of the window size in current researches is based on experience or recognition results. Furthermore, the window function used in moving window is rectangular window, and the effects of different window functions, such as Hann window and Gaussian window, on the identification process are also worth studying.

Principal component analysis (PCA) is a statistical method. PCA is defined as the orthogonal projection of data onto a low dimensional linear space, such that the variance of the projected data is maximized [14]. PCA is used to identify the modal parameters of linear time invariant structures [15]. Moving window principal component analysis (MWPCA) combines moving window and PCA, and is applied to OMA [16]. In adaptive MWPCA (AMWPCA), we want to choose the window function and window size based on the characteristics of the data adaptively, rather than knowing the recognition results and then adjusting it.

2. Adaptive moving window principal component analysis-based OMA for slow LTV structures

2.1. Problem with moving window based operational modal analysis for slow LTV structure

The moving window technique is used to identify the modal parameters of each window in order, and then connect the results together, which is the modal parameter identification result of the LTV structure.

PCA is a useful tool for linear time-invariant structural modal parameter identification. The result of PCA decomposition corresponds to the modal shape and modal coordinate response matrix [16]. Moving window PCA (MWPCA) combines moving window technology and PCA to have the ability to recognize the modal parameters of LTV structures. The window size is L and the displacement responses $\mathbf{X}_{L}^{({\tau})}\in \mathbb{R}^{N\times L}$ in each window can be decomposed by the modal expansion: $\begin{eqnarray}\displaystyle \mathbf{X}_{L}^{({\tau})}\approx {\Phi}_{L}^{({\tau})}\mathbf{Q}_{L}^{({\tau})},\quad {\tau}=1,∼2,\ldots ,T+1-L & & \displaystyle\end{eqnarray}$ (1) where ${\Phi}_{L}^{({\tau})}\in \mathbb{R}^{N\times N}$ is modal shapes matrix and $\mathbf{Q}_{L}^{({\tau})}\in \mathbb{R}^{N\times L}$ is modal coordinate response matrix. In MWPCA, the window size L is increased by adding new data and reduced by removing old data. Table 1 shows the problem of window size [10]

Table 1

The advantages and disadvantages of different window sizes

	Advantages	Disadvantages
Large window size	Stable and high frequency resolution.	Contain many outdated data and the amount of calculation is large.
Small window size	Adapt to fast time-varying characteristics	Sensitive to noises and disturbances.

The window function of moving window is a rectangular window. There are many kinds of window functions such as Gaussian window, Hanning window and Triangular window. These window functions have their own characteristics. In addition to adjusting the window size, the recognition accuracy can also be improved by adjusting the window function.

2.2. Adaptive indicator of window size and window function

The choice of window size and window function needs to have a quantitative basis, which we call adaptive indicator (AI). This indicator should be related to the vibration response signal.

For the τ-th window, the steps to calculate the indicator are:

Calculate the covariance matrix C_{X
_L
^τ
X
_L
^τT} of the displacement response signal $\mathbf{X}_{L}^{({\tau})}$

Calculate the eigenvalues of C_{X
_L
^τ
X
_L
^τT}

Determine the number of modes m to be extracted. Find m maximum eigenvalues and sort them in descending order, $\vec{E}_{L}^{({\tau})}=[{\lambda}_{1}^{({\tau})},\ldots ,{\lambda}_{j}^{({\tau})},\ldots ,{\lambda}_{m}^{({\tau})}]$ .

Calculate the variance contribution of k-th mode

\begin{eqnarray}\displaystyle {\eta}_{k}^{({\tau})}={\displaystyle \frac{{\lambda}_{k}^{({\tau})}}{\mathop{\sum }_{i=1}^{m}{\lambda}_{i}^{({\tau})}}},\quad k\in \{1,2,\ldots ,m\}. & & \displaystyle\end{eqnarray}

(2)

The AI for window size L and τ-th window is defined as $\begin{eqnarray}\displaystyle AI_{L}^{({\tau})}=\min _{1\leq j\leq m-1}\{({\eta}_{j}^{({\tau})}-{\eta}_{j+1}^{({\tau})})\}. & & \displaystyle\end{eqnarray}$ (3)

Let $AI_{L}^{({\tau})}>{\varepsilon}$ where ϵ is a very small constant. Then, PCA decomposition is performed on $\mathbf{X}_{L}^{({\tau})}$ using a matrix composed of eigenvectors. Figure 1 shows the window function adaptive selection process and Fig. 2 shows the window size adaptive selection process.

Fig. 1.

Window function adaptive selection process of adaptive moving window principal component analysis.

Fig. 2.

Window size adaptive selection process of adaptive moving window principal component analysis.

2.3. Interpretation of adaptive indicator

For non-stationary vibration response signal, the eigenvalues are variable in each window. When the eigenvalues of the adjacent modes are relatively close, a modal switch phenomenon will occur, resulting in a worse recognition effect [17]. Our adaptive indicator is to make the difference between the adjacent eigenvalues not too small, thereby avoiding modal switch as much as possible. The change of eigenvalue is measured by the variance contribution rate. Using window functions or changing the window size can change the variance contribution rate.

3. Simulation identificatio

3.1. Simulation setting

We design a simulation of linear time-varying three-degree-of-freedom (DOF) spring oscillator system with slowly varying mass. The simulation details can be seen in [10]. The modal assurance criterion (MAC) is used to describe the accuracy of modal shape recognition which is defined as $\begin{eqnarray}\displaystyle \text{MAC}={\displaystyle \frac{(\vec{{\varphi}}_{k}^{T}\vec{{\phi}}_{k})^{2}}{(\vec{{\varphi}}_{k}^{T}\vec{{\varphi}}_{k})(\vec{{\phi}}_{k}^{T}\vec{{\phi}}_{k})}} & & \displaystyle\end{eqnarray}$ (4) where $\vec{{\varphi}}_{k}$ is the kth order theoretical modal shape and $\vec{{\phi}}_{k}$ is the kth order identified modal shape. The relative error (RE) is used to evaluate the accuracy of the frequency identification. $\begin{eqnarray}\displaystyle RE={\displaystyle \frac{|f_{k}^{identified}-f_{k}^{theoretical}|}{f_{k}^{theoretical}}} & & \displaystyle\end{eqnarray}$ (5) where $f_{k}^{theoretical}$ is the kth order theoretical modal natural frequency and $f_{k}^{identified}$ is the kth order identified natural frequency.

3.2. Window function results at a certain point and analysis

In this simulation, to study window function in AMWPCA, the window size is initialized to 1024. We randomly selected point 526.4s for research.

The window function contains 11 types in MATLAB. Figures 3(a) and 3(b) show the frequency and modal shape identification results with different window function at 526.4s. Figures 3(c) shows that the window function can change the difference of the second and third order variance contributions.

Fig. 3.

The identified results with 11 different window functions at 526.4s.

From Fig. 3, we can see that some window functions can make the identification results better. The window functions can change the variance contributions. So, our AI can be used as the basis for window function selection.

3.3. Window size results at a certain point and analysis

The same points 526.4s is used to study the effectiveness of our AI in window size. The window function is initialized to a rectangular window and the range of window changes is 64 to 4096. Figures 4(a)–4(b) show the changes in frequency and modal shape identification results at 526.4s. Figure 4(c) shows the variance contribution of each order at different points.

Fig. 4.

The identified results when the size of the window changes at 526.4s.

In Figure 4(a) the RE increases as the window size increases. In Figs. 4(b)–4(c), when the MAC recognition result is not good, the second-order and third-order variance contributions are close.

From Fig. 4, we can know that the size of data window has great influence on identification results. The frequency results are filtered such that no frequency exchange occurs at the point where the MAC results are poor. In Fig. 4(b) and Fig. 4(c), when the value AI is small, the recognition result tends to be bad. Therefore, our AI can be used to adjust the window size and improve the identification accuracy.

3.4. Adaptive method results and analysis

The results of our adaptive method for frequency and MAC are similar. Therefore, we mainly study MAC changes The parameters setting is: ϵ = 0.05, l = 1024, L₀ = 64 and r _max = 4. Figure 5(a) shows the recognition results of the MAC when the window size changes according to our AI and Fig. 5(b) shows the adaptive results using different window functions.

Fig. 5.

The adaptive method results.

In Fig. 5(a), all the identification results are the same as or better than the original result. In Fig. 5(b), the MAC value of most points has improved, especially at higher orders. From Fig. 5 we can know that our adaptive methods have ability to improve the modal parameters recognition accuracy The window size improvement effect is better than the window function.

4. Conclusion

This paper proposes that window function and window size will affect the modal parameter identification results in MWPCA. The effect of window size is more pronounced. Then, an adaptive indicator is designed to adjust the window size and window function The simulation results show that our adaptive methods can improve recognition accuracy of modal parameters. The future work is to research better methods to make the recognition results more accurate and apply our method to complex LTV structures In addition, the other adaptive indicator of moving windows is also a further research direction.

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