# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
## 2 Component Modal Synthesis
### Free-Interface Modal Synthesis Techniques  
The presence of rigid body motion is one of the characteristics of free-free or unconstrained systems. For such systems, the stiffness matrix is singular, and hence no flexibility matrix exists, jeopardizing static analysis. In his modal synthesis approach, [PERSON] eliminated the rigid body modes using the inertia relief matrix, facilitating the analysis of the elastic behavior of such unconstrained systems. The free-interface modal synthesis scheme proposed by [PERSON] [61] for free-free systems constitutes attachment modes for unconstrained systems or inertia relief modes, constraint modes, and free-interface normal modes of substructures. For the \(i\)th substructure, the free-interface normal modes are obtained by setting Equation (3) for \(\{p_{B}\}=0\). By substituting \(\{p_{B}\}=0\), Equation (3) reduces to  
\[ \begin{split}&\left[\frac{m_{BB}}{m_{BI}^{\prime}}\vdots\frac{ m_{BI}}{m_{II}}\right]_{i}\left\{\frac{\mu_{B}}{\tilde{u}_{I}}\right\}_{i}\\ &+\left[\frac{k_{BB}}{k_{BI}^{\prime}}\vdots\frac{k_{BI}}{k_{II} }\right]_{i}\left\{\frac{u_{B}}{u_{I}}\right\}_{i}=\left\{\frac{0}{\cdots} \right\}_{i}\end{split}  \tag{12}\]  
Thus,  
\[([k]_{i}-[\omega_{fn}^{2}]_{i}[m]_{i}]\left\{\frac{\phi_{B}}{\phi_{I}}\right\} _{i}=\{0\}\]  
and  
\[\left|[k]_{i}-[\omega_{fn}^{2}]_{i}[m]_{i}\right|=0 \tag{13}\]  
where \([\omega_{fn}^{2}]_{i}=diag(\omega_{fn_{1}}^{2},\ldots,\omega_{fn_{N_{i}}}^{2})\). If the \(i\)th substructure is unrestrained, then rigid body modes are included in \([\omega_{fn}^{2}]_{i}\). The mass normalized eigenvectors corresponding to the free-interface normal modes belonging to the interior and boundary degrees of freedom for the \(i\)th substructure to be retained are gathered in \([\overline{\phi}_{fn}]_{i}\) and \([\overline{\phi}_{Bn}]_{i}\), respectively, thereby retaining the corresponding coordinates. Furthermore, truncation of the free-normal modes renders elimination of the associated degrees of freedom. Let \(W_{i}\) be the set of physical coordinates for the \(i\)th substructure, where \(W_{i}\subset N_{i}\setminus O_{i}\), and where \(O_{i}\) is the statically determinate constraint set sufficient to provide restraint against rigid body motion. Furthermore, let us define a set such that \(L_{i}\subset W_{i}\). The static deflection as a consequence of an applied unit force on \(w_{j}^{i}\in W_{i}\), where \(j\in\{1,\ldots,|W|\}\) while the remaining of the degrees of freedom in the set \(W_{i}\) are devoid of force, is characterized as the attachment mode. Separating the displacement vector for the \(i\)th substructure into pure rigid body displacement \(\{u_{o}\}\) and elastic deformation vector \(\{u_{e}\}\) yields  
\[\{u\}_{i}=\{u_{o}\}_{i}+\{u_{e}\}_{i} \tag{14}\]The displacements in physical coordinates can be expressed in modal coordinates in the following way  
\[\{u\}_{i}=\big{[} \begin{array}{cc}\phi_{0}\end{array} ; \begin{array}{c}\phi_{e }\end{array} \big{]}_{i}\bigg{\{}\begin{array}{c} \begin{array}{c}\eta_{o} \end{array} \\ \begin{array}{c}\cdots\eta_{e}\end{array} \bigg{\}}_{i} \tag{15}\]  
Using the non-homogeneous undamped equation of motion and knowing that \([k]_{i}[\phi_{0}]_{i}=0\), the modal coordinates associated with rigid body motions can be explicitly represented as follows  
\[[m]_{i}[\phi_{0}]_{i}\{\eta_{o}\}_{i}=\{p\}_{i} \tag{16}\]  
Premultiplying Equation (16) by \([\phi_{o}]_{i}^{\intercal}\) yields  
\[\{\dot{\eta}_{o}\}_{i}=[\phi_{o}]_{i}^{\intercal}\{p\}_{i} \tag{17}\]  
where \([\phi_{o}]_{i}^{\intercal}[m]_{i}[\phi_{0}]_{i}=[I]\).