Abstract
In order to reduce the time and space complexity of operational modal analysis (OMA) for slow linear timevarying (SLTV) structures based on moving window principal component analysis (MWPCA), this paper proposes a new moving window selfiteration principal component extraction (MWSIPCE) method. Different from getting principal components by singular value decomposition (SVD) or eigenvalue decomposition (EVD) in MWPCA algorithm, MWSIPCE just extracts the firstseveralorders principal components by selfiteration. Comparing with MWPCA, MWSIPCE has lower time and space complexity. What’s more, this paper explains the reason of modal exchange in some data windows in detail, and gives an illustration to how to set window length $L$. The OMA results on nonstationary vibration response simulation signal of timevarying cantilever beam under white noise excitation show that this method can well identify the timevarying transient modal parameters (natural frequencies and modal shapes) for SLTV structures and has less time and space consume, the algorithm is also more precise than MWPCA.
1. Introduction
Modal parameters (Natural frequencies, mode shapes) are essential for dynamics structural damage recognition, health monitoring [1] and independent modal space control [2]. From 1980s, operational modal analysis (OMA) deals with the estimation of modal parameters from vibration data obtained in operational rather than laboratory conditions [3]. OMA differs from experimental modal analysis (EMA) in that it seeks to determine a structure’s dynamic characteristics from responseonly measurements, without precise knowledge of excitation forces. PCA is one of the most classic in multivariate analysis and data mining. The idea was first proposed by the Pearson in 1901, and it was used for the field of biology [4]. Principal components are linear combinations of the original data which help to visualize similarities in an ensemble of signals. Because all principal components are orthogonal and ordered according to the variance contribution of sample, the largest two or three principal components provide an excellent representation of variability within a set of data [5]. Traditional calculation method of PCA is using matrix decomposing, which includes SVD and EVD [6]. However, these ways calculate all principal components in one time, that makes the algorithm always has high timespace complexity. In 1969, NIPALS algorithm [7] is firstly proposed, different from traditional method, the algorithm iteratively calculates principal components. In order to reduce the timespace complexity of traditional PCA, this paper proposes a new selfiteration principal components extraction (SIPCE) algorithm which based on NIPALS algorithm.
With the development of science and technology, high speed, large, complicated and intelligent engineering structures in civil engineering field are applied more frequently. In working condition, these engineering structures usually need to experience a variety of harsh environments, which will lead to many engineering structures parameters (mass, stiffness and damping) changing over time inevitably. In practical engineering structures, the systems are often timevarying and the modal parameters are also changing with time, such as rockets in fight with mass lost [8], the trains crossing bridges [9], etc. YANG summarized the basic theory involved in modal parameter estimation of timevarying structures, including the introduction and discussion about the dynamic theory of linear timevarying systems that presented the dynamic model and the definition and properties of poles of linear timevarying systems. In addition, he established the criteria for slow timevarying systems in the timedomain and frequencydomain, providing the basis and assumptions for modal parameter estimation of timevarying structures respectively. Ramnath [10] pointed out the system whose variation in coefficient is much less than that in solution is called slow timevarying system.
In 1947, structure natural frequency and damping identification method were put forward by Kennedy and Pancu. After 70 years of development and progress, modal parameter identification has experienced from the linear time invariant (LTI) structures to slow linear timevarying (SLTV) structures. At present, OMA method learned from the latest research achievements in the field of control theory, system engineering and signal process [11], etc. Classical OMA method cannot apply to SLTV structures because of the nonstationary response data. The OMA methods of SLTV structures are based mainly on the assumption of short time invariant and “frozen” thought, which promoted modal analysis theory of LTI structures to SLTV structures, and identified the timevarying modal parameters. Another approach is using online or recursive technology to track the timevarying modal parameters [12]. Based on these theories, some popular methods were proposed, including timefrequency analysis based on wavelet transform (WT) [13, 14], HilbertHuang transform (HHT) [15], subspace methods [16], Auto Regressive Moving Average (ARMA) models [17], etc. However, these methods can only identify timevarying transient modal frequency of SLTV structures.
Guan et al. presented the method which combined principal component analysis (PCA) with moving window (MW) for operational modal parameters identification of SLTV structures [18]. That method is based on eigenvalue decomposition of PCA, so the complexity of time and space is very high. This study proposed a timevarying OMA method based on moving window selfiteration principal component analysis (MWSIPCE). MWSIPCE can well identify the timevarying transient modal parameters of SLTV structures and its time and space complexity is lower than MWPCA. Besides that, the reason of modal exchange in some data windows is explained, that has great engineering significance for fault monitoring.
2. OMA of LTI structures based on SIPCE
2.1. Mathematical model of PCA
The data set of $m$ observation signals$\mathbf{X}\left(t\right)=\left[{\overrightarrow{x}}_{1}\right(t),{\overrightarrow{x}}_{2}(t),\cdots .{\overrightarrow{x}}_{m}(t){]}^{T}$ is composed by $n$ unrelated unknown latent variables$\mathbf{Y}\left(t\right)=\left[{\overrightarrow{y}}_{1}\right(t),{\overrightarrow{y}}_{2}(t),\cdots ,{\overrightarrow{y}}_{n}(t){]}^{T}$ through a linear transformation matrix $\mathbf{W}=[{\overrightarrow{w}}_{1},{\overrightarrow{w}}_{2},\cdots ,{\overrightarrow{w}}_{n}]\in {\mathbb{R}}^{m\times n}$, which is shown in Fq. (1) [19]:
Matrix $\mathbf{W}\in {R}^{m\times n}$ meets condition:
Fig. 1 is the diagram of PCA model. The goal of PCA is to find linear transformation matrix $\mathbf{W}\in {R}^{m\times n}$ and $n$ uncorrelated latent variables $\mathbf{Y}\left(t\right)$ only from observed signal $\mathbf{X}\left(t\right)$. The $n$ uncorrelated latent variables $\mathbf{Y}\left(t\right)$ are called principal components of $m$ observation signal $\mathbf{X}\left(t\right)$.
Fig. 1 is the diagram of PCA model. The goal of PCA is to find linear transformation matrix $\mathbf{W}\in {R}^{m\times n}$ and $n$ uncorrelated latent variables $\mathbf{Y}\left(t\right)$ only from observed signal $\mathbf{X}\left(t\right)$. The $n$ uncorrelated latent variables $\mathbf{Y}\left(t\right)$ are called principal components of $m$ observation signal $\mathbf{X}\left(t\right)$.
Fig. 1Description of PCA problem
2.2. OMA of LTI structure based on PCA model
According to the theory of structural dynamics, for LTI structures with $m$degreeof freedom, the equations of vibration motion in the physical coordinate is [18]:
In the Eq. (3): matrix $\mathbf{M}\in {R}^{m\times m}$ is the mass matrix of the LTI structures, matrix $\mathbf{C}\in {R}^{m\times m}$ is the damping matrix of the LTI structure, matrix $\mathbf{K}\in {R}^{m\times m}$ is the stiffness matrix of the LTI structures, $t$ is the time. Matrix $\mathbf{F}\left(t\right)$ is the time domain random dynamic motivation of the LTI structures, and matrix $\mathbf{X}\left(t\right)$ is the time domain dynamic displacement response of the LTI structures, $\dot{\mathbf{X}}\left(t\right)$ is the time domain dynamic speed response of the LTI structures. And $\ddot{\mathbf{X}}\left(t\right)$ is the time domain dynamic acceleration speed response of the LTI structures.
Viscous damping matrix $\mathbf{C}$ can’t be orthogonally diagonalized by modal shape, so it can’t be decoupled directly. However, in special cases, matrix $\mathbf{C}$ could be orthogonally diagonalized. For example, the viscous damping which is proposed by Rayleigh:
In Eq. (4), $\alpha $, $\beta $ denotes as the outside and inside damping constants of the system, for weak damping vibration system, the above model is effective.
For real modal analysis, in the modal coordinate, the random vibration response signals of mechanical structures with weak damping can be decomposed into the inner product of modal shapes and modal responses:
where $\mathbf{X}\left(t\right)$ is the output displacement, $\mathbf{\Phi}\in {R}^{m\times m}$ is the reversible modal shape matrix which composed by $m$ independent mode shapes ${\overrightarrow{\phi}}_{i}\in {R}^{m\times 1}$, besides that, $\mathbf{\Phi}$ is basisvector matrix in modal coordinate space and $\mathbf{Q}\left(t\right)={\left[{\overrightarrow{q}}_{1}\left(t\right),{\overrightarrow{q}}_{2}\left(t\right),\cdots ,{\overrightarrow{q}}_{i}\left(t\right),\cdots ,{\overrightarrow{q}}_{m}\left(t\right)\right]}^{T}\in {\mathbf{R}}^{m\times T}$ is the modal coordinate matrix which composed by modal response ${\overrightarrow{q}}_{i}\left(t\right)$ of each mode.
Plugging Eq. (5) into Eq. (3), which transports physical coordinate into modal coordinate:
Simplifying Eq. (6):
Then, Eq. (7) is the modal model, where ${\mathbf{M}}_{r}$ is modal mass matrix, ${\mathbf{C}}_{r}$ is modal damping matrix, ${\mathbf{K}}_{r}$ is modal stiffness matrix:
Based on the vibration theory in Eqs. (5)(13), when the structure’s natural frequencies ${f}_{i}$ of each mode are not equal, the modes vectors are orthogonal:
${{\overrightarrow{\mathbf{\phi}}}_{i}}^{T}\mathbf{K}{\overrightarrow{\mathbf{\phi}}}_{{i}^{\text{'}}}=0,i\ne {i}^{\text{'}}.$
And modal response vectors ${\overrightarrow{q}}_{i}\left(t\right)$ are independent from each other. Then $\mathbf{Q}\left(t\right)$ satisfies:
where ${\mathbf{\Lambda}}_{m\times m}^{\text{'}\text{'}}$ is a real positive diagonal matrix.
Natural frequency ${f}_{i}$ and damping ratio ${\xi}_{i}$ can be identified by FFT or single degree of freedom (SDOF) fitting techniques from mode coordinate response vector ${\overrightarrow{q}}_{i}\left(t\right)$. Fig. 2 is the Schematic diagram of OMA based on PCA modal.
In view of Eq. (1) and Eq. (5), for random vibrations of weakly damped systems, there is a onetoone mapping between the orthogonal mode matrix $\mathbf{\Phi}$ and the linear transformational matrix $\mathbf{W}$ in Eq. (2). Besides that, there is a onetoone mapping between modal response matrix $\mathbf{Q}\left(t\right)$ in and the PCs $\mathbf{Y}\left(t\right)$. Therefore, the existence, uniqueness and deterministic of OMA based on PCA algorithm can be proved by PCA decomposition.
2.3. Selfiteration and principal component extraction algorithm
SIPCE algorithm is based on NIPALS algorithm. Firstly, choosing a column from the observation signal matrix $\mathbf{X}\left(t\right)\in {R}^{m\times T}t=\mathrm{1,2},\cdots ,T$. Then calculating the first principal component vectors ${\overrightarrow{y}}_{1}\left(t\right)\in {R}^{1\times T}$ and the first transformation matrix vector ${\overrightarrow{w}}_{1}\in {R}^{m\times 1}$ iteratively.
As Eq. (16) shows, updating error matrix to calculate the second principal component vectors ${\overrightarrow{y}}_{2}\left(t\right)\in {R}^{1\times T}$ and the second transformation matrix vector ${\overrightarrow{w}}_{2}\in {R}^{m\times 1}$ and so on:
Fig. 2Schematic diagram of OMA based on PCA modal
(1) Choosing the arbitrary column data vector, and assume that the chosen column is the $l$th column. So, vector ${\overrightarrow{x}}_{l}\left(t\right)\in {R}^{1\times T}$ is got from signal matrix $\mathbf{X}\left(t\right)\in {R}^{m\times T}$. $i$ denotes the $i$th principal moment, and $j$ is the $j$th loop which calculates the $i$th principal moment, $i=$ 1, $j=$ 1, and ${\overrightarrow{y}}_{i}^{\left(j\right)}\left(t\right)\leftarrow {\overrightarrow{x}}_{l}\left(t\right)$; Set the maximal number of iterations is ${J}_{\mathrm{m}\mathrm{a}\mathrm{x}}$.
(2) Calculate ${\overrightarrow{w}}_{i}^{\left(j\right)}\in {R}^{m\times 1}$: ${\overrightarrow{w}}_{i}^{\left(j\right)}\leftarrow \left(\mathbf{X}\right(t\left){{\overrightarrow{y}}_{i}^{\left(j\right)}}^{T}\right(t\left)\right)/\left({\overrightarrow{y}}_{i}^{\left(j\right)}\right(t\left){{\overrightarrow{y}}_{i}^{\left(j\right)}}^{T}\right(t\left)\right)$;
(3) Normalize the vector ${\overrightarrow{w}}_{i}^{\left(j\right)}$: ${\overrightarrow{w}}_{i}^{\left(j\right)}\leftarrow {\overrightarrow{w}}_{i}^{\left(j\right)}/\Vert {\overrightarrow{w}}_{i}^{\left(j\right)}\Vert $;
(4) Update ${\overrightarrow{y}}_{i}^{(j+1)}\left(t\right)$: ${\overrightarrow{y}}_{i}^{(j+1)}\left(t\right)\leftarrow {{\overrightarrow{w}}_{i}^{\left(j\right)}}^{T}\mathbf{X}\left(t\right)/\left({{\overrightarrow{w}}_{i}^{\left(j\right)}}^{T}{\overrightarrow{w}}_{i}^{\left(j\right)}\right)$;
(5) Compare the ${\overrightarrow{y}}_{i}^{(j+1)}$in the step (4) with the ${\overrightarrow{y}}_{i}^{\left(j\right)}$ in the step (2). Define variable ${\beta}_{i}=\Vert {\overrightarrow{y}}_{i}^{(j+1)}{\overrightarrow{y}}_{i}^{\left(j\right)}\Vert /\Vert {\overrightarrow{y}}_{i}^{\left(j\right)}\Vert $, and $\alpha $ denotes accuracy requirement. If ${\beta}_{i}\le \alpha $, their difference is in the prescribed scope (the algorithm is convergent). Then return to the step (6), or $j\leftarrow j+1$ and return to the step (2).
(6) Calculate the error matrix:
(7) Calculate ${\lambda}_{i}$: ${\lambda}_{i}\leftarrow \Vert \frac{1}{T}\mathbf{X}\left(t\right){\mathbf{X}}^{T}\left(t\right){\overrightarrow{w}}_{i}\Vert /\Vert {\overrightarrow{w}}_{i}\Vert $.
(8) Define variable ${\epsilon}_{i}={\lambda}_{i}/\sum _{s=1}^{i}{\lambda}_{s}$, $\eta $ is truncation error, if ${\epsilon}_{i}>\eta $, $i\leftarrow i+1$ and return to the step (1), calculate ${\overrightarrow{y}}_{2}\left(t\right),{\overrightarrow{y}}_{3}\left(t\right),\cdots ,{\overrightarrow{y}}_{i}\left(t\right)$ and ${\lambda}_{2},{\lambda}_{3},\cdots ,{\lambda}_{i}$.
The step (8) is to calculate the oversized contribution rates which correspond to the principal components. So, ${\lambda}_{i}$ is the approximate contribution rate, which corresponds to the $i$th principal component. Variable ${\epsilon}_{i}$ denotes cumulative variance proportion of the current principal component of calculated principal components.
Setting truncation error as $\eta $. If ${\epsilon}_{i}<\eta $, the current principal component is not important enough, the algorithm stops iteration, and the process of principal components extraction is end.
In the steps of the SIPCE algorithm, ${\overrightarrow{y}}_{i}\left(t\right)$ is the $i$th principal component of the residual error matrix. The comparison of the ${\overrightarrow{y}}_{i}\left(t\right)$ in the step (5) is to judge whether ${\overrightarrow{y}}_{i}\left(t\right)$ meets the demand. If it does, the algorithm updates the error matrix and get the first principal component of the new error matrix $({\overrightarrow{y}}_{2}\left(t\right),{\overrightarrow{y}}_{3}\left(t\right),\cdots ,{\overrightarrow{y}}_{i}\left(t\right),\cdots ,{\overrightarrow{y}}_{m}\left(t\right))$. Eq. (17) and Eq. (18) show that the contribution of ${\overrightarrow{y}}_{i}\left(t\right)$ is larger than ${\overrightarrow{y}}_{i+1}\left(t\right)$. In conclusion, ${\overrightarrow{y}}_{1}\left(t\right)$ is the first principal component of observation signal matrix, and ${\overrightarrow{y}}_{2}\left(t\right)$ is the second principal component and so on.
2.4. Approach of selfiteration and principal component extraction algorithm to OMA
Fig. 3 shows the flow of OMA based on SIPCE. Variable ${\lambda}_{i}(i=1,2,\dots ,m)$ represents the contribution of its corresponding principal components. ${\epsilon}_{i}$ denotes the importance measure of next principal components, and $\eta $ is a setting threshold value that we could use to control the iterations. Therefore, it realizes the principal components extraction.
2.5. The comparison between SIPCE and PCA
The traditional batch PCA gets linear transformation matrix and principal components by singular value decomposition (SVD) or eigenvalue decomposition (EVD) [20]. However, there are singular value and illposed problems in SVD and EVD [21, 22]. Because of these defects, the traditional batch PCA algorithm may not be able to accurately identify the modal vibration mode and natural frequency of the structures. SIPCE algorithm directly gets principal components by recursion, and the accuracy of selfiteration and principal component extraction results only depends on the threshold value. Therefore, selfiteration and principal component extraction based on OMA could avoid singular value decomposition of matrix and the illposed problems of eigenvalue decomposition effectively.
SIPCE calculates principal components one by one, and its goal is to identify several order models by user requirements. However, traditional batch principal component analysis gets all principal components by matrix decomposition. That not only increases the computation time, but also spends lots of memory to store large matrices.
3. OMA of SLTV structures based on MWSIPCE
3.1. OMA of SLTV structures based on “Timefreezing” theory
Based on the assumption of “timeinvariant” in a minor interval, the countless SLTV structures sets ${S}^{\text{'}}$ could be gotten by mass, stiffness and damping matrix of each instantaneous moment $\tau \in [{t}_{begin},{t}_{end}]$ in “freezing” timevarying. The set ${S}^{\text{'}}$ is expressed by Eq. (19):
Equation ${S}^{\text{'}}$ is the LTI structures at $\tau $ moment, and the set of ${S}^{\text{'}}$ is “timefreezing” represent of SLTV structure $S$. Because the SLTV structures and LTI structures have the same mass, stiffness and damping matrixes at every moment, the varying characteristics of them are equipotent [23, 24].
For small damping structures, the collected data of the structure response can be divided into several parts. And in the $\tau $th part (data window), the modal coordinates can decompose the response of the linear structure:
Fig. 3The flowchart of OMA based on selfiteration principal component extraction algorithm
The variables $\mathbf{\Phi}\left(\tau \right)$ and $\mathbf{Q}(\tau ,t)$ are modal shape matrix and modal response vector of the $\tau $th data window. We normalize main orthogonal modal shape vector ${\overrightarrow{\phi}}_{i}\left(\tau \right)$ to meet each modal response that is independent when each structure natural frequency ${\overrightarrow{q}}_{i}(\tau ,t)$ are unequal. Then the natural frequency can be identified from modal respond as well as OMA of LTI structure are based on PCA:
$=\left[\begin{array}{ccccc}{\overrightarrow{q}}_{1}\left(\tau ,t\right){\overrightarrow{q}}_{1}^{T}\left(\tau ,t\right)& \cdots & 0& \cdots & 0\\ \vdots & \ddots & \vdots & \ddots & \vdots \\ 0& \cdots & {\overrightarrow{q}}_{i}\left(\tau ,t\right){\overrightarrow{q}}_{i}^{T}\left(\tau ,t\right)& \cdots & 0\\ \vdots & \ddots & \vdots & \ddots & \vdots \\ 0& \cdots & 0& \cdots & {\overrightarrow{q}}_{m}\left(\tau ,t\right){\overrightarrow{q}}_{m}^{T}\left(\tau ,t\right)\end{array}\right].$
3.2. OMA of SLTV structure based on MWSIPCE
3.2.1. The model of MWSIPCE
In order to identify timevarying transient modal parameters, we divide a time quantum into many minimum transient and then identify the modal parameters of every transient by the SIPCE. The process of segmentation can be mathematically modeled as Fig. 4 shows, and rectangular window is selected to add new data and discard part of old data. Variate $i$ is to express the $i$th data window $k=1,\cdots ,NL+1$.
Fig. 4The rectangular data window with moving window length L
Setting the raw data matrix $\mathbf{X}\left(t\right)\in {R}^{m\times N}$ with $m$ variables and $N$ samples when the rectangular data window with memory length $L$ is introduced, and then the raw data matrix will be changed into ${\mathbf{X}}_{L}^{\left(k\right)}\in {R}^{m\times L}$. With $L$ sampling length, ${\mathbf{X}}_{L}^{\left(k\right)}$ is used to estimate the statistical average modal value of middle time. When the transient time is changed, the rectangular window moves. In addition, the data window matrix will be updated as Fig. 4 shows. Therefore, the transient mathematical model of timevarying structure is built, and then combines every transient to identify SLTV structure modal parameter.
Based on the assumption of short time invariant and “frozen” thought, at every transient time, the structure is regarded as a LTI structure. Then, identifying the modal parameters in every rectangular data window by SIPCE. The OMA of SLTV structures is turned into the OMA based on SIPCE for LTI structures. In the end, we combine the modal of every transient by curvefitting technique, then measure and monitor the modal parameters of SLTV structures realtimely.
Fig. 5 shows the flow of MWSIPCE. For the raw matrix $\mathbf{X}\left(t\right)$, the algorithm gets the transient data window ${\mathbf{X}}_{L}^{\left(k\right)}$, and then calculate the principal components and the corresponding change vector. Updating the data window until all transient modal parameters are calculated.
Fig. 5The flowchart of OMA based on MWSIPCE for SLTV
3.2.2. The selection of window length $\mathit{L}$
As an important parameter for MWSIPCE based Operational Modal Identification under a slow SLTV Structure, the window length $L$ should not be too large or too small. If the window length $L$ is too long, the nonstationary random response signals of the slow SLTV structure cannot approximately be seen as the stationary random response of LTI structure. The $i$th order statistical average mode nature frequency ${f}_{L}^{\left(k\right)}\left(i\right)$ and statistical average modal shapes ${\overrightarrow{\mathbf{\phi}}}_{L}^{\left(k\right)}\left(i\right)$ during $L$ periods are not regarded as an approximate evaluations of transient mode nature frequency ${f}_{}^{(k+(L1)/2)}\left(i\right)$, and transient mode shapes ${\overrightarrow{\mathbf{\phi}}}^{(k+(L1)/2)}\left(i\right)$ at time$(k+(L1)/2)$. If the window length is too short, identification error will be larger owing to inadequate sample and large frequency resolution. Moreover, the precision of identified natural frequencies got by FFT depends on frequency resolution $\mathrm{\Delta}f$. It is more precise when the length $L$ of FFT is larger. And $\mathrm{\Delta}f$ is inversely proportional to sampling frequency $f$. Frequency resolution $\mathrm{\Delta}f$ of FFT is defined [25]:
The factors effect the choice of window length $L$ is showed below:
1) The changing speed of SLTV structure and nonstationary degree of vibration response signal. If the SLTV structure is fast changed, there is a high nonstationary degree and we should reduce the window length $L$. Define average frequency variation of the $i$th mode in a data window as $\mathrm{\Delta}{f}_{L}^{}\left(i\right)$:
where variables ${f}^{end}\left(i\right)$, ${f}^{begin}\left(i\right)$, ${t}_{end}$, ${t}_{begin}$ are endfrequency of the $i$th mode, begin frequency of the $i$th mode, endtime, begintime of the whole data.
2) Sample frequency $f$ of vibration response signal and frequency resolution $\mathrm{\Delta}f$. Window length $L$ is proportional to sample frequency $f$ and frequency resolution $\mathrm{\Delta}f$.
3) The power of storage and computing for embedded structure. The memory and computation of MWPCA and MWSIPCE mode are proportional to moving window length $L$ [26]. Furthermore, computation of FFT is depending on window length $L$.
3.3. The comparison between MWSIPCE and MWPCA
In order to ensure the precision of timevarying transient modal parameter identification based on assumption of short time invariant, the response time should be divided into minimum period, so that we could produce many response data.
According to user requirements, comparing with MWPCA algorithm calculates all components in each rectangular data window, MWSIPCE algorithm just extracts several principal components in each rectangular data window and does not need to calculate other components. Therefor MWSIPCE algorithm mostly reduces time complexity. Besides that, MWSIPCE iteratively gets principal parameters, so, it doesn’t need to store many large matrixes and avoids the illposed or singular value problems.
For raw data $\mathbf{X}\left(t\right)\in {R}_{m\times N}$, different algorithms are compared in various aspects. The size of space complexity depends on whether the program stores past matrixes. And the results are expressed in Table 1.
MWSIPCE algorithm is more precise than MWPCA algorithm. Moreover, it enhances the efficiency of operating modal parameter identification.
Table 1The comparison of algorithm performance
Algorithm  Time complexity  The least space complexity  Algorithm robustness 
MWPCA based on EVD  $o\left({m}^{2}\right)\times N$ High  ${R}^{m\times m}\times 2+{R}^{L\times L}+{R}^{m\times L}$  Illposed problems 
MWPCA based on SVD  $o\left({m}^{2}\right)\times N$ High  ${R}^{m\times m}\times 2+{R}^{m\times L}\times 2$  Singular value problem 
MWSIPCE  $o\left(r{J}_{max}\right)\times N$ Low  ${R}^{m\times L}$  Avoids the illposed and singular value problems 
3.4. The reason of modal exchange in MWSIPCE
Based on MWSIPCE algorithm, the frequencies of SLTV structure are ordered by the contribution rather than the frequency value. At one transient, the contributions of each identified modal can be represented as ${\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{i}$, and they meet the following relationship:
For SLTV structure, engineering structure parameters (mass, stiffness and damping) change over time, so the modal contribution rate of each order is timevarying. Therefore, there is a special phenomenon called modal exchange. Therefore, contribution order is not equal to the frequency order, the correspondence of two ranking order maybe change over time. For example, at transient time ${t}_{1}$, there is a onetoone match between the firstfourorder identified modal and the firstfourorder real modal, but at transient time ${t}_{2}$, the third order identified modal is corresponded to the forthorder real modal and the forthorder identified modal is corresponded to the thirdorder real modal.
For SLTV structure, MWSIPCE algorithm can find modal exchange based on changing frequency, and monitor running state of SLTV structure. CPV $\eta $ denotes project require, at ${t}_{a}$ moments, $\eta $ is the variance contribution of the first 3 order PCs. With the new data is added, the contribution of the third modal is decreasing and the forth order’s contribution is increasing. Maybe at a moment ${t}_{b}$, when the CPV $\eta $ meets project requirement, the modals involved in CPV are the first order, the second modal and the forth modal.
3.5. Limitations and applications of timevarying OMA based on movingwindow MWSIPCE
Timevarying operational modal analysis with MWSIPCE method has the following limitations and application scores:
1) Each order of modal shapes gotten by PCA is normalized orthogonal, which loses amplitude information. This is a characteristic of PCA or ICA based OMA and a characteristic of MWSIPCE based timevarying OMA.
2) It can just apply to slow linear timevarying structures. Only for slow linear timevarying structures, the theory of “timefreezing” is established, and the nonstationary random response signals of the slow LTV structures can approximately be seen as the stationary random response time series of LTI structures in short time interval.
3) The length of moving window $L$ is selected with prior knowledge of timevarying speed of LTV structure and fixed. However, the length of moving window $L$. should be fully selfadaptive selected and changed only by nonstationary vibration response signals when there is no prior knowledge of timevarying speed then.
4) PCA method for operational modal analysis can identify mode shapes, modal frequencies and mode damping ratios for LTI structure only from stationary vibration response signals. MWSIPCE method for operational modal analysis can also identify timevarying transient mode shapes, modal frequencies and mode damping ratios only from nonstationary vibration response signals. However, theoretical analysis and numerical simulation indicate that mass decreasing or moving will generates an additional damping [27, 28] for LTV structure. So, the timevarying transient mode damping ratio identified by LMRPCA method cannot be compared with mode damping ratio calculate by FEM directly.
5) The modes which calculated by SIPCE are ordered by the contribution of vibration response rather than frequency. For timevarying structure, the contribution of each mode is varying. MWSIPCE is based on SIPCE, there might be modal exchange in the calculation.
4. Simulation verification of timevarying transient operational modal identification
In order to validate the accuracy and efficiency of MWSIPCE method, we set up a dynamic model of a timevarying cantilever beam, and analysis the influence of mass change for timevarying structure. A simulation with cantilever beam is designed, which is a timevarying structure whose mass is changing over time. For experimental verification, additional damping caused by moving mass and the limitation of experimental condition make it quite difficult [29], so simulation verification is a frequently used method. In the generation of simulation data, we gave full consideration to boundary conditions, and the Gaussian noise is added into the data to simulate real scene. The algorithm can be well validated by the simulation of timevarying cantilever beam [25].
4.1. Evaluation criteria
Theoretical values and FEA values are set as standard, which are calculated under the condition of undamped real modal shape and undamped real modal nature frequency, and MWPCA algorithm is based on EVD method.
In order to quantitatively compare the modal vibration mode, modal assurance criterion (MAC) is investigated [30], which is an important standard when considering the effectiveness of modal shape identification [31]. Eq. (25) is the detailed equation of calculated MAC value:
where ${\overrightarrow{\phi}}_{i}$ is the identified shape, ${\overrightarrow{\varphi}}_{i}$ represents the real shape, and (${\overrightarrow{\phi}}_{i}^{T}{\overrightarrow{\varphi}}_{i}$) represents the inner product of the two vectors. MAC values oscillate between 0 (no coincidence) and 1 (complete coincidence), The more MAC value is close to 1, the more accurate estimated shape is. However, MAC just can reveal the information of direction and shape, the size of amplitudemode is not contained.
4.2. Parameters of timevarying cantilever beam
The dimension of cantilever beam model is 1 m×0.02 m×0.002 m (length×height×width) while cross sectional area $A=W\times H=$ 4$\times $10^{4} m^{2}, density ${\rho}_{0}=$ 7860 kg/m^{3}, moment of inertial $I=W{H}^{3}/12$, and $E=$ 3×10^{11} N/m^{2 }is Young’s modulus. Timevarying mass property is simulated by changing the density of beam, given as Eq. (26):
The simulation time is set as 4 s, and the first ten order timevarying natural frequency values at the start and finish time are compared in Table 2.
Table 2The change curve of natural frequencies (04 s)
t/s  Order  
1  2  3  4  5  6  7  8  9  10  
0 s  16.9997  104.6552  293.038  574.2391  949.2649  1418.0561  1980.635  2637.0355  3384.3056  4231.5200 
4 s  $\downarrow $ 19.6807  123.3374  345.3485  $\downarrow $ 676.7473  1118.71  1671.1952  2334.2016  3107.7761  $\downarrow $ 3991.9779  4986.8942 
In Finite element method (FEM), the continuum cantilever beam is generally dispersed with finite multidegreeoffreedom structures and the mass, stiffness and damping matrices can be expressed as follows:
where ${\mathbf{M}}^{e}\left(t\right)$, ${\mathbf{K}}^{e}\left(t\right)$ and ${\mathbf{C}}^{e}\left(t\right)$ are timevarying element mass, stiffness and damping matrices respectively. ${\mathbf{C}}^{e}\left(t\right)$ denotes proportional damping matrix, and $L$ is the length of each element. And ${\beta}_{M}\text{,}$${\beta}_{K}$ are the proportional coefficients with responding to mass and stiffness matrices, respectively.
In FEM, the continuums beam is divided into 40 elements, each finite element is an output. Only one translational freedom and one rotational freedom are considered for each node without axial freedom, which is depicted in Fig. 6.
The forced response signals are acquired by Newmark – $\beta $. The proportional coefficients are set as ${\beta}_{M}=$ 4×10^{4}, and ${\beta}_{K}=$ 1×10^{7}. The initial velocities and displacement are zero while the white noise is imposed as excitation. From Table 2, it is found that the frequencies are increasing over time. As frequency is 4986.8942 Hz at the finish time, the sampling frequency is set as 10000 Hz based on sampling theorem. The simulation time is set as 4 s while integration step is 1/10000 s for Newmark$\beta $, and the number of sample points $N=$ 40000.
Level of 2.0 % Gauss measurement noise is added into response signals in time domain. The calculation formula of noise is as follows:
In Eq. (30), $e$ denotes a normal random vector, its mean value is zero and standard deviation is one. $r$ is the noise level. The measurement noise level above is 2.0 %.
Fig. 6FEM Model of cantilever beam
4.3. Parameters of timevarying OMA based on MWSIPCE
1) Limited memory length $L$ is set as 4096 data points [25].
2) Set precision threshold $\alpha =$ 0.000001.
3) Set maximum number of iteration ${J}_{\mathrm{m}\mathrm{a}\mathrm{x}}=$ 100.
4) Set truncation error $\eta =$ 0.05.
5) Computer configuration
Operating system: 64bit Windows 10.
Manufacturer: HP.
System Type: HP Z240 SFF Workstation.
Processor: Intel(R) Core(TM) i56500 CPU @ 3.20GHz (4 CPUs).
Memory: 8192MB RAM.
Programing language and environment: MATLAB 2014a.
4.4. Timevarying transient modal parameters identification results
In the FFT figure, the FFT peak frequency (horizontal axis) is corresponding to the nature frequency. Fig. 7(ad) shows the first three order nature frequencies that identified by MWSIPCE at random moments (0.9954 second, 1.4954 second, 2.4954 second, 3.4954 second).
Fig. 8(ad) shows the first three order shapes that identified by MWSIPCE at random moments, and the modal shapes are normalized (The moments are same as Fig. 7).
Connecting the nature frequency at every moment Comparison results of the first three orders frequency between theoretical calculation and MWSIPCE are shown in Fig. 9. The frequency identification changes from 0.5 second to 3.7048 second.
Fig. 10 is the records of the MAC values at every moment, and we have corrected the correspondence when there is modal exchange in the third order. The frequency identification changes from 0.7048 s to 3.7048 s, and we just identify the absolute timevarying transients to ensure recognition accuracy. The calculated equation is showed in section 4.1.
For MWSIPCE algorithm, its accuracy depends on threshold $\alpha $. Table 3 is the precise MAC comparison of the third order between MWPCA and MWSIPCE at some moments. Table 4 is the selfMAC of some moments to check the modal vector orthogonality.
Table 5 is the comparisons of timespace consumptions. MWSIPCE extracts the first 4 order modes, which meet the requirement.
Fig. 7FFT of the first three orders principal components at random moments
a) 0.9954 second
b) 1.4954 second
c) 2.4954 second
d) 3.4954 second
Table 3Comparison of MAC the third order between MWPCA and MWSIPCE at some moments
Moments (s)  MWPCA  MWSIPCE 
1.3153  0.7580  0.8607 
1.3336  0.6878  0.8125 
1.3763  0.5679  0.7068 
1.4612  0.6740  0.8212 
Fig. 8Modal shape comparison between MWSIPCE and theoretical value
a) 0.9954 second
b) 1.4954 second
c) 2.4954 second
d) 3.4954 second
Fig. 9The frequency comparison of cantilever beam
a) The first order
b) The second order
c) The third order
Fig. 10MAC value of every moment
Table 4The selfMAC of some moments
1.0201 s  2.0201 s  3.0201 s  
Order  1  2  3  Order  1  2  3  Order  1  2  3 
1  1.000  0  0  1  1.000  0  0  1  1.000  0  0 
2  0  1.000  0  2  0  1.000  0  2  0  1.000  0 
3  0  0  1.000  3  0  0  1.000  3  0  0  1.000 
Table 5Timespace consumptions based on MWSIPCA
Method  Absolute time consumption (s)  Memory consumption (MB) 
MWPCA  374.135  7196 
MWSIPCE  200.397  2397 
4.5. The analysis of simulation results
When the time is in the interval of 0.5002 s0.972 s and 3.6520 s3.7820 s, there is a big deviation between the thirdorder frequency identified by MWSIPCE and the thirdorder theoretical value. However, the thirdorder frequency identified by MWSIPCE is similar to the forthorder theoretical value. In addition, it is because in these transient times, the contribution rates of the third modal and the forth modal are changed, so the third modal identified by MWSIPCE is changed. As we can see in Table 2, the thirdorder modal frequency identified by MWSIPCE is corresponding to the forthorder nature frequency. Because of the modal exchange, Fig. 8(c) shows that the thirdorder real shape is corresponding to the fourthorder modal shape, which is identified by MWSIPCE.
Fig. 7, Fig. 8, Fig. 9 and Fig. 10 approve that MWSIPCE algorithm can identify the modal shape and nature frequency of SLTV structure accurately. Table 3 shows that MWSIPCE is more precise than MWPCA at some moments. The reason is there are singular value decomposition of matrix or the illposed problems for MWPCA. Table 4 shows that the different modal shapes identified by MWSIPCE are complete orthogonal.
When the limited memory length is 8192, $\mathrm{\Delta}f=$ 1.22 Hz is less than $\mathrm{\Delta}{f}_{L}\left(3\right)=$ 12.2436 Hz and the identified effect is good while it needs high time and space complexity. When the limited memory length is 2048, its $\mathrm{\Delta}f=$ 4.88 Hz is more than $\mathrm{\Delta}{f}_{L}\left(3\right)=$ 3.0609 Hz. In addition, it needs low time and space complexity, but the identified effect is the worst, even in some time, the modal parameters are not identified for the data points are not enough. Therefore, the limited memory length is chosen to be 4096 according to compromise, because its $\mathrm{\Delta}f=$ 2.44 Hz is less than $\mathrm{\Delta}{f}_{L}\left(3\right)=$ 6.1218 Hz and its time and space complexity is not too high.
MWSIPCE algorithm extracts several principal components from every rectangular data window. The number of identified order depends on the requirements of the user. However, MWPCA must calculate all components. Table 5 shows that the MWSIPCE algorithm has less time and space consumption.
5. Conclusions
The simulation results of cantilever beam show that the MWSIPCE method can track the timevarying transient modal frequency and modal shape of SLTV structure well. Simulation results show that MWSIPCE needs less time and space consumption than MWPCA, and it realtimely measures and monitor SLTV structure. Furthermore, this paper gives detail explanations to the modal exchange in the simulation, which has great significance for malfunction monitoring of SLTV structure. Besides, the article illustrates the way of setting length $L$ and the limitation of MWSIPCE method.
However, this study has some limitations. Each order of modal shapes got by PCA is normalized orthogonal, which loses amplitude information. It may be inappropriate to use average values to estimate the intermediate time for a period of time in the FFT of PCs, which would cause more burrs in the results of FFT of PCs. In the simulation verification, level of 2.0 % Gauss measurement noise is added into response signals to simulate the noise in the real scene, the design of real experiments and experimental verification of SIPCE are the future work. The next research is to identify the modal parameters in realtime using online and recursive techniques and integrate the algorithms into embedded device for mechanical fault diagnosis. How to select $L$ scientifically and adaptively is also a future study.
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About this article
This work was financially supported by National Natural Science Foundation of China (Grant No. 51305142, 51305143), China Postdoctoral Science Foundation Project (No. 2014M552429), Postgraduate Scientific Research Innovation Ability Training Plan Funding Projects of Huaqiao University (No. 1511314011).
Tianshu Zhang implemented the method and completed the experimental verification. He also writhed the paper. Cheng Wang proposed the ideas, and gave guides in writing the paper. Jianying Wang provided with the data, and helped accomplish the experiment. Yewang Chen gave help in algorithm optimization. Yiwen Zhang helped modify English mistakes.