## 3 OMA of Linear Time-Varying Structure Based on MWLPP  
### Problem of OMA of Linear Time-Varying Structure  
The operational modal parameters of time-varying structures vary with time. According to the dynamic theory of the structure, the motion equation of a linear time-varying structure of \(n\) degrees of freedom in the physical coordinate system in \(t\in[T_{BEGIN},T_{END}]\) is,  
\[N:\mathbf{M}(t)\mathbf{\ddot{X}}(t)+\mathbf{C}(t)\mathbf{\dot{X}}(t)+\mathbf{K }(t)\mathbf{X}(t)=\mathbf{F}(t),\,t\in[T_{BEGIN},T_{END}] \tag{11}\]  
Figure 1: Operational modal analysis (OMA) of time-invariant structure based on locality preserving projections (LPP) algorithm. where \(\mathbf{M}(t)\in\mathbb{R}^{n\times n}\), \(\mathbf{C}(t)\in\mathbb{R}^{n\times n}\) and \(\mathbf{K}(t)\in\mathbb{R}^{n\times n}\) are the mass matrix, damping matrix, and stiffness matrix of the time-varying structure. \(\mathbf{X}(t)\in\mathbb{R}^{n\times T}\), \(\dot{\mathbf{X}}(t)\in\mathbb{R}^{n\times T}\) and \(\ddot{\mathbf{X}}(t)\in\mathbb{R}^{n\times T}\) are the time-domain sampling matrix of the displacement response signal, velocity response signal and acceleration response signal of the time-varying structure. \(\mathbf{F}(t)\in\mathbb{R}^{n\times T}\) is the time-domain sampling matrix of the external excitation. According to the \"time-freezing\" theory [40], the mass, damping ratio and stiffness of a linear time-varying structure can be regarded as time-invariant for a short time \(\tau\in[t_{begin},t_{end}]\). Therefore, Equation (11) can be expressed as a set \(N^{\prime}\) consisting of a finite number of linear time-invariant structures \(N^{\prime}(\tau)(\tau\in[t_{begin},t_{end}])\) within the complete time \(t\in[T_{begin},T_{END}]\). \[\left\{ \begin{array}{l}N^{\prime}\triangleq\left\{N^{\prime}(\tau):\mathbf{ M}(\tau)\ddot{\mathbf{X}}(\tau)+\mathbf{M}(\tau)\dot{\mathbf{X}}(\tau)+ \mathbf{M}(\tau)\mathbf{X}(\tau)=\mathbf{F}(\tau),(\tau\in[t_{begin},t_{end} ]\subset(t\in[T_{begin},T_{END}])\right\}\\ \tau=\frac{1}{2}(t_{begin}+t_{end})=\frac{1}{2}(t_{k}+t_{k+1}),k=0,1,\cdots,K \\ t_{0}=T_{begin},t_{K}=T_{END}\end{array} \end{array}\right. \tag{12}\]  
The vibration response data of linear time-varying structures can be divided into a finite number of time-invariant parts by selecting moving window with fixed length \(L\) (\(\tau\in[t_{begin},t_{end}]\)). The displacement response signal \(\mathbf{X}_{L}^{i}(\tau)\in\mathbb{R}^{n\times L}\) in the \(i\)-th (\(\tau\in[t_{begin},t_{end}]\)) window is decomposed into the following Equation (13) in modal coordinates. \[\mathbf{X}_{L}^{i}(\tau)\ \approx\ \mathbf{\Phi}_{L}^{j}\mathbf{Q}_{L}^{i}(\tau)= \sum_{j=1}^{d}\overset{\rightarrow i}{\phi}_{j}^{i}\overset{\rightarrow i}{ \vartheta}_{j}^{i}(\tau) \tag{13}\]  
where \(\mathbf{\Phi}_{L}^{i}=[\overset{\rightarrow i}{\phi}_{1}^{i},\overset{ \rightarrow i}{\phi}_{2}^{i},\cdots,\overset{\rightarrow i}{\phi}_{d}^{i}] \in\mathbb{R}^{n\times d}\) is the modal shape matrix formed by the modal shape vector \(\overset{\rightarrow i}{\phi}_{j}^{i}(j=1,2,\cdots,d)\in\mathbb{R}^{n\times 1}\) of the structure in the \(i\)-th (\(\tau\in\left[t_{begin},t_{end}\right]\)) window. \(\mathbf{Q}_{L}^{i}(t)=[\overset{\rightarrow i}{q}_{1}^{i}(t),\overset{ \rightarrow i}{q}_{2}^{i}(t),\cdots,\overset{\rightarrow i}{q}_{d}^{i}(t)]^{T} \in\mathbb{R}^{d\times L}\) is the modal response matrix formed by the modal response vector \(\overset{\rightarrow i}{q}_{j}^{i}(t)\in\mathbb{R}^{1\times L}(j=1,2,\cdots,d)\) of the structure in \(i\)-th (\(\tau\in\left[t_{begin},t_{end}\right]\)) window. When the order of each natural frequency is different, The main modal shape \(\overset{\rightarrow i}{\phi}_{j}^{i}\) satisfies normalized orthogonality and the modal response \(\overset{\rightarrow i}{q}_{j}^{i}\) of each other is independent. \[(\mathbf{\Phi}_{L}^{i})^{T}\mathbf{\Phi}_{L}^{i}=\mathbf{I}_{d\times d} \tag{14}\]  
\[\mathbf{Q}_{L}^{i}(t)(\mathbf{Q}_{L}^{i}(t))^{T}=\mathbf{\Lambda}_{d\times d}^ {i}=\left[