# 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  
Applying the modal coordinate transformations, a linear time-invariant system with orthogonal damping can be represented by a series of decoupled second-order ordinary differential equations with a single degree of freedom (DOF):  
\[\ddot{q}_{i}+4\pi\zeta_{i}f_{i}\dot{q}_{i}+4\pi^{2}f_{i}^{2}q_{i}=F_{i},\ i=1,2, \cdots,m \tag{1}\]  
where \(f_{i}\) and \(\zeta_{i}\) are the system's \(i\)th natural frequency and damping ratio, respectively, which are collectively referred to as the dynamic spectral parameters in this paper, and \(F_{i}\) is the \(i\)th modal excitation force, which is given by the following:  
\[F_{i}=\gamma_{i}u \tag{2}\]  
where \(u\) is the system input in physical space and \(\gamma_{i}\) is the modal contribution factor. When the system input is a single-point excitation force or the acceleration excitation, \(\gamma_{i}\) can be expressed as follows:  
\[\gamma_{i}=\frac{\[PERSON]_{i}^{\mathrm{T}}\mathbf{B}_{o}}{\Phi_{i}^{\mathrm{T}} \mathbf{M}\Phi_{i}}\ \text{or}\ \gamma_{i}=-\frac{\Phi_{i}^{\mathrm{T}}\mathbf{M}\mathbf{E}}{\Phi_{i}^{\mathrm{ T}}\mathbf{M}\Phi_{i}} \tag{3}\]  
where \(\Phi_{i}\in R^{n_{d}\times 1}\) is the \(i\)th mode shape containing all the DOFs of the structure; \(\mathbf{B}_{o}\in R^{n_{d}\times 1}\) and \(\mathbf{M}\in R^{n_{d}\times n_{d}}\) are the input indication matrix and the mass matrix, respectively; and \(\mathbf{E}\in R^{n_{d}}\) is a vector with all elements being 1. The right term in Equation (1) is related to both \(\gamma_{i}\) and \(u\). Dividing both sides of the equation by \(\gamma_{i}\) results in an expression structurally similar to the equation of motion describing a single DOF in physical space:  
\[\ddot{\widetilde{q}}_{i}+4\pi\zeta_{i}f_{i}\dot{\widetilde{q}}_{i}+4\pi^{2}f_ {i}^{2}\widetilde{q}_{i}=u,\ i=1,2,\cdots,m \tag{4}\]where \(\ddot{\tilde{q}}_{i}=\tilde{q}_{i}/\gamma_{i}\), \(\dot{\tilde{q}}_{i}=\dot{q}_{i}/\gamma_{i}\), and \(\ddot{\tilde{q}}_{i}=q_{i}/\gamma_{i}\) are the modal responses under excitation \(u\). \(\gamma_{i}\) is a time-independent constant, and, by comparing equations of motion (4) with (1), it can be seen that the response of the system under the modal excitation \(F_{i}\) can be obtained by scaling up the \(\gamma_{i}\) times after obtaining the responses \(\ddot{\tilde{q}}_{i}\), \(\dot{\tilde{q}}_{i}\), and \(\ddot{\tilde{q}}_{i}\). The state vector, as follows, can be used to describe the modal displacement and velocity of a structure:  
\[\mathbf{v}_{i}= \begin{cases}\dddot{q}_{i}\\ \dddot{q}_{i}\end{cases}  \tag{5}\]  
Then, Equation (4) can be expressed in the form of state space:  
\[ \begin{split}\dot{\mathbf{v}}_{i}=\hat{\mathbf{A}}_{i}\mathbf{v} _{i}+\hat{\mathbf{B}}u\\ \hat{\mathbf{A}}_{i}=\begin{bmatrix}0&1\\ -4\pi^{2}f_{i}^{2}&-4\pi\xi_{i}f_{i}\end{bmatrix},\ \hat{\mathbf{B}}=\begin{bmatrix}0\\ 1\end{bmatrix}\end{split}  \tag{6}\]  
where \(\hat{\mathbf{A}}_{i}\) is the state transition matrix. For a detailed derivation of Equation (6), please refer to Appendix B. The measured response is a discrete time series in a structural dynamic test assuming a sampling time interval of \(\Delta t\). For a continuous time system represented by Equation (6), the discrete time series are represented by the sampling of continuous time signals:  
\[\mathbf{v}_{i,k}= \begin{cases}\dddot{q}_{i,k}\\ \dddot{q}_{i,k}\end{cases} =\mathbf{v}_{i}(k\Delta t),\ \dddot{q}_{i,k}=\dddot{q}_{i}(k\Delta t),\ u_{k}=u(k \Delta t),\ k=0,1,2,3,\ldots.