# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 2 Time Domain Theoretical Response Prediction Model Based on the Modal State Space Model
\tag{7}\]  
Assuming that the system input remains constant within the interval \(t\in[k\Delta t,(k+1)\Delta t]\), i.e., \(u(t)=u(k\Delta t)\), the equation of motion of the \(i\)th mode in the discrete time state space form is as follows:  
\[\mathbf{v}_{i,k+1}=\mathbf{A}_{i}\mathbf{v}_{i,k}+\mathbf{B}_{i}u_{k} \tag{8}\]  
where  
\[\mathbf{A}_{i}=\mathbf{e}^{\hat{\mathbf{A}}_{i}\Delta t},\ \mathbf{B}_{i}= \Big{(}\mathbf{e}^{\hat{\mathbf{A}}_{i}\Delta t}-\mathbf{I}_{2}\Big{)}\Big{(} \ \hat{\mathbf{A}}_{i}\big{)}^{-1}\hat{\mathbf{B}} \tag{9}\]  
where \(\mathbf{I}_{2}\) is the unit matrix of \(2\times 2\). Assuming that the initial condition of the system is \(\mathbf{v}_{i,0}=0\), for the sampling time \(t=0\), \(\Delta t\), \(2\Delta t\), \(\ldots\), \(N_{s}\Delta t\), the input time series of the system is recorded as follows: \(u_{0}\), \(u_{1}\), \(u_{2}\), \(\ldots\), \(u_{N_{s}}\). From Equation (8), the state vector of the \(i\)th mode of the system at the moment \(t=k\Delta t\) can be recursively solved as follows:  
\[\mathbf{v}_{i,k}=\mathbf{A}_{i}^{k-1}\mathbf{B}_{i}u_{0}+\mathbf{A}_{i}^{k-2 }\mathbf{B}_{i}u_{1}+\mathbf{A}_{i}^{k-3}\mathbf{B}_{i}u_{2}+\cdots+\mathbf{B }_{i}u_{k-1},\ k=1,2,3,\ldots,N_{s} \tag{10}\]  
From Equation (10), the modal displacement of the system can be given by the following:  
\[\dddot{q}_{i,k}(f_{i},\zeta_{i})=\mathbf{D}_{1}\Big{(}\mathbf{A}_{i}^{k-1} \mathbf{B}_{i}u_{0}+\mathbf{A}_{i}^{k-2}\mathbf{B}_{i}u_{1}+\mathbf{A}_{i}^{k -3}\mathbf{B}_{i}u_{2}+\cdots+\mathbf{B}_{i}u_{k-1}\Big{)} \tag{11}\]  
where \(\mathbf{D}_{1}= \begin{bmatrix}1&0\end{bmatrix} \). From Equation (11), it can be seen that through the normalization process applied to Equation (1), the modal response is only a function of the dynamical spectral parameters. This normalization effectively disentangles the dynamical spectral parameters from the modal contribution factor in modal space. Since the modal displacement, velocity, acceleration, and inputs of the system satisfy the dynamical equilibrium equations at each moment \(t=t_{k}\), according to Equations (4) and (5), one obtains the following:  
\[\dddot{q}_{i,k}=u_{k}-4\pi^{2}f_{i}^{2}\dddot{q}_{i,k}-4\pi\zeta_{i}f_{i}\dddot {q}_{i,k}=u_{k}-\Big{[}4\pi^{2}f_{i}^{2}&4\pi\zeta_{i}f_{i}\Big{]}\mathbf{v}_ {i,k} \tag{12}\]Substituting Equation (10) into Equation (12), and one then obtains the following:  
\[\overset{\cdot}{\widetilde{q}}_{i,k}(f_{i},\zeta_{i})=u_{k}-\mathbf{D}_{2}\Big{(} \mathbf{A}_{i}^{k-1}\mathbf{B}_{i}u_{0}+\mathbf{A}_{i}^{k-2}\mathbf{B}_{i}u_{1} +\mathbf{A}_{i}^{k-3}\mathbf{B}_{i}u_{2}+\cdots+\mathbf{B}_{i}u_{k-1}\Big{)} \tag{13}\]  
where \(\mathbf{D}_{2}=\Big{[}4\pi^{2}f_{i}^{2}\quad 4\pi\zeta_{i}f_{i}\Big{]}\). Up until now, Equations (11) and (13) have established a prediction model of the theoretical modal response. However, the measured signals are described in physical space, and thus the modal responses obtained from Equations (11) and (13) need to be further transformed into physical space. The modal parameters \(\overline{\theta}=\{f_{i},\zeta_{i},\ \boldsymbol{\varphi}_{i},\gamma_{i}:\ i=1, \ldots,m\}\) are defined. Using \(\eta_{i,k}\) to denote modal displacement \(\tilde{q}_{i,k}\) or modal acceleration \(\tilde{q}_{i,k}\). According to the modal superposition method, the response of the system under the excitation \(u_{k}\) can be given by the following:  
\[\mathbf{y}_{k}(\overline{\theta})=\sum_{i=1}^{m}\boldsymbol{\varphi}_{i}\ \eta_{i,k}=\sum_{i=1}^{m} \boldsymbol{\varphi}_{i}\gamma_{i}\ \widetilde{\eta}_{i,k} \tag{14}\]  
where \(\boldsymbol{\varphi}_{i}\in R^{n}\) is the \(i\)th mode shape consisting of the measured DOFs. \(\psi_{i}=\boldsymbol{\varphi}_{i}\gamma_{i}\), which will be referred to as the spatial parameters (consisting of the mode shape and modal contribution factor) in this paper, is defined.