# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
## 2 Component Modal Synthesis
### Fixed-Interface Modal Synthesis Technique  
[PERSON] pioneered the notion of fixed-interface component modal synthesis [56], but the method was incompatible with automation. [PERSON] [57] modified [PERSON]'s approach, removing the limitations that facilitated the implementation of the modal synthesis process on computing platforms. [PERSON]'s method analyzes components by the superimposition of constraint and fixed-constraint normal modes. Constraint modes are the displacements of interior coordinates in a substructure induced by sequential unit displacements of interface constraints, with the remaining boundary constraints being constrained. The boundary constraints are usually imposed on the coupled points between interconnected substructures. These coupled points are subjected to unit displacement in the direction of each degree of freedom, considered one at a time, resulting in the deflection of the substructure. The deflected configurations of the substructure are the constraint modes for that individual substructure. For the \(i\)th substructure, the linear relationship between the externally applied force and stiffness for static analysis is given by  
\[\left\{ \begin{array}{c}\underline{p_{B}}\\ \underline{p_{I}}\end{array} \right\}_{i}=\left[ \begin{array}{cc}\underline{ k_{BB}}&\vdots&\underline{k_{BI}}\\ \underline{k_{BI}}&\vdots&\underline{k_{II}}\end{array} \right]_{i}\left\{ \begin{array}{c}\underline{u_{B}}\\ \underline{u_{I}}\end{array} \right\}_{i} \tag{1}\]  
Because there is no external force acting on the unconstrained coordinates in constraint mode analysis, taking the bottom partition of Equation (1) yields  
\[ \begin{array}{c}[k_{BI}]_{i}^{\intercal}\{u_{B}\}_{i}+[k_{II}]_{i}\{u_{I}\}_ {i}=\{0\}\\ \{u_{I}\}_{i}=-[k_{II}]_{i}^{-1}[k_{BI}]_{i}^{\intercal}\{u_{B}\}_{i}=[\phi_{ c}]_{i}\{u_{B}\}_{i}\end{array}  \tag{2}\]  
Fixed-constraint normal modes are defined by displacements of interior points in the component relative to the interface constraints. These are the normal modes of vibration with all interface constraints fixed, typically computed by eigenvalue analysis of the substructure. The zero-input response and eigenanalysis of a system are analogous, and therefore, in the absence of external excitation in the internal degrees of freedom, the undamped equation of motion for the \(i\)th substructure can be expressed as,  
\[ \begin{array}{c}\left[\begin{array}{cc}\underline{m_{BB}}&\vdots&\underline {m_{BI}}\\ \underline{m_{BI}^{\intercal}}&\vdots&\underline{m_{II}}\end{array}\right]_{i} \left\{\begin{array}{c}\underline{\bar{u}_{B}}\\ \underline{\bar{u}_{I}}\end{array}\right\}_{i}\\ +\left[\begin{array}{cc}\underline{k_{BB}}&\vdots&\underline{k_{BI}}\\ \underline{k_{BI}^{\intercal}}&\vdots&\underline{k_{II}}\end{array}\right]_{i} \left\{\begin{array}{c}\underline{u_{B}}\\ \underline{u_{I}}\end{array}\right\}_{i}=\left\{\begin{array}{c}\underline{ \underline{p_{B}}}\\ \underline{0}\end{array}\right\}_{i}\end{array}  \tag{3}\]  
In fixed-interface normal mode analysis, the constraints on the interface coordinates for a substructure are constrained, resulting in  
\[\{u_{B}\}=0\]  
and  
\[\{\bar{u}_{B}\}=0 \tag{4}\]  
Substituting Equation (4) into Equation (3) and using the lower partition yields  
\[[m_{II}]_{i}\{\bar{u}_{I}\}_{i}+[k_{II}]_{i}\{u_{I}\}_{i}=\{0\} \tag{5}\]  
The free vibration solution for Equation (5) is of the form \(\{u_{I}\}_{i}=[\phi_{n}]_{i}\{\eta_{n}\}_{i}\). Therefore, Equation (5) reduces to the eigenvalue problem  
\[([k_{II}]_{i}-[\omega_{n}^{2}]_{i}[m_{II}]_{i})\{\phi_{I}\}_{i}=0\]  
and  
\[\left|[k_{II}]_{i}-[\omega_{n}^{2}]_{i}[m_{II}]_{i}\right|=0 \tag{6}\]  
where \([\omega_{n}^{2}]_{i}=diag(\omega_{n_{1}}^{2},\ldots,\omega_{n_{N_{i}}^{2}}^{2})\). The mass normalized eigenvectors form the respective columns of \([\phi_{n}]_{i}\) of the \(i\)th constrained substructure. The elements of \([\phi_{n}]_{i}\) to be retained are then collected in \([\bar{\phi}_{n}]_{i}\) and thereby retain the associated interior degrees of freedom of the substructure.