# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
## 2 Component Modal Synthesis
### Fixed-Interface Modal Synthesis Technique
The interface coordinates being entirely constrained in fixed-constraint normal mode analysis renders the upper partition of Equation (3) redundant. It is worth mentioning that the constraint modes represent the static characteristics of the substructure, while the fixed constraint normal modes, as determined by vibration analysis, demonstrate the dynamic behavior of the substructure. Finally, the transformation from physical coordinates to generalized coordinates for the \(i\)th substructure is then  
\[ \begin{split}\{u\}_{i}&=\left[\begin{smallmatrix}I& \vdots&0\\ \overbrace{\phi_{c}}^{\ast}&\vdots&\overbrace{\phi_{n}}^{\ast} \end{smallmatrix}\right]_{i}\left\{\begin{smallmatrix}\underline{u_{B}}\\ \overbrace{\eta_{n}}^{\ast}\end{smallmatrix}\right\}_{i}\\ &=\left[T_{{}_{C\!B}}\right]_{i}\left\{\begin{smallmatrix}\underline{u_{B}} \\ \overbrace{\eta_{n}}^{\ast}\end{smallmatrix}\right\}_{i}\end{split}  \tag{7}\]  
It is worth noting that truncating the constraint modes is not feasible since they characterize the entire motion of the system. The reduced mass and stiffness matrix of the \(i\)th substructure in generalized coordinates is then expressed as  
\[ \begin{split}[\bar{m}]_{i}&=\left[T_{{}_{C\!B}}\right] _{i}^{\intercal}[m]_{i}[T_{{}_{C\!B}}]_{i}\\ &[\bar{k}]_{i}=\left[T_{{}_{C\!B}}\right]_{i}^{\intercal}[k]_{i}[ T_{{}_{C\!B}}]_{i}\end{split}  \tag{8}\]  
The equations of motion for the overall structure in the unconnected form are eventually obtained by grouping together the equations of motion for all \(\tau\in\mathbb{N}^{+}\) substructures. A compatibility matrix \(([\beta])\) renders constraints on the interconnected substructures, ensuring compatible displacements at substructure boundaries. The boundary coordinates for any two adjoining substructures \(i\) and \(j\) are expressed in their respective local coordinate frames of reference, \(\{u\}_{i}\) and \(\{u\}_{j}\), and consequently, it is imperative to express their frame of reference in a shared global reference frame, \(\{\bar{u}\}\), prior to establishing compatible displacement connections. Let the boundary coordinates for the \(i\)th and \(j\)th substructures in the global frame of reference be \(\{\bar{u}_{B}\}_{i}^{s}\) and \(\{\bar{u}_{B}\}_{j}^{s}\), respectively, at a common connection point s. Then, the following necessary and sufficient conditions must be satisfied at the boundary to ensure that a substructure's displacements on its boundary correspond to those of its adjoining substructures in the global frame of reference:  
\[\{\bar{u}_{B}\}_{i}^{s}=\{\bar{u}_{B}\}_{j}^{s} \tag{9}\]  
For the sake of brevity, superscript (s) in Equation (9) will be omitted. In terms of interconnected substructures, there are no compatibility constraints for the modal coordinates. Furthermore, any arbitrary substructure \(i\) with multiple couplings to adjoining substructures may be expressed using a matrix notation \([\beta]_{i}\). Finally, the general form of the transformation of coordinates to represent the overall structure is as follows:  
\[ \begin{split}\begin{pmatrix}u_{B_{1}}\\ \eta_{n_{1}}\\ \vdots\\ u_{B_{\tau}}\end{pmatrix}&=\begin{bmatrix}\beta_{1}\\ \vdots\\ \beta_{\tau}\end{bmatrix}\begin{pmatrix}\bar{u}_{B_{1}}\\ \vdots\\ \bar{u}_{B_{\tau}}\\ \eta_{n_{1}}\\ \vdots\\ \eta_{n_{\tau}}\end{pmatrix}&=[\beta]\{\mu\}\end{split}  \tag{10}\]  
The uncoupled set of boundary coordinates mentioned in Equation (10) is where external excitations are applied. Finally, the equation of motion for free vibration of the complete structure may be expressed as  
\[[M_{\mu}]\{\bar{\mu}\}+[K_{\mu}]\{\mu\}=0 \tag{11}\]where \([M_{\mu}]=[\beta]^{\intercal}[\bar{M}_{\mu}][\beta]\) and \([K_{\mu}]=[\beta]^{\intercal}[\bar{K}_{\mu}][\beta]\), and where  
\[\bar{M}_{\mu}= \begin{bmatrix}[\tilde{m}]_{1}&&\\ &\ddots&\\ &&[\tilde{m}]_{\pi}\end{bmatrix} ,\quad\bar{K}_{\mu}= \begin{bmatrix}[\bar{k}]_{1 }&&\\ &\ddots&\\ &&[\bar{k}]_{\pi}\end{bmatrix}