# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
## 5 Conclusions
This work was supported, in part, by Science Foundation Ireland grant \(13/RC/2094\_P2\) and co-funded under the European Regional Development Fund through the Southern & Eastern Regional Operational Programme to Lero--the Science Foundation Ireland Research Centre for Software (www.lero.ie). **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Acknowledgments:** This paper and the research behind it would not have been possible without the support of the IMaR team in the Munster Technological University. **Conflicts of Interest:** The authors declare no conflict of interest. **List of Symbols**  
\([\bar{\phi}_{n}]\)****Matrix of retained fixed-constraint normal modeshapes  
\([\bar{\phi}_{Bn}]\)****Matrix of retained free-interface modeshapes corresponding to boundary degrees of freedom  
\([\bar{\phi}_{n}]\)****Matrix of retained free-interface modeshapes corresponding to interior degrees of freedom  
\([C]\)****Boolean matrix  
\([\bar{k}]\)****Stiffness matrix in generalized coordinates  
\([n]\)****Mass matrix in generalized coordinates  
\([\bar{\rho}]\)****Compatibility matrix  
\([\bar{\rho}^{p}]\)****Boolean matrix  
\([\phi]\)****Matrix of modeshapes for the reduced-order model  
\([\phi^{ss}]\)****Modified modeshape matrix of reduced system  
\([\phi_{a}]\)****Matrix of retained attachment modeshapes  
\([\phi_{c}]\)****Matrix of constraint modeshapes  
\([\phi_{e}]\)****Matrix of elastic modeshapes  
\([\phi_{n}]\)****Matrix of fixed-constraint normal modeshapes  
\([\phi_{0}]\)****Matrix of rigid body modeshapes  
\([\Psi]\)****Inertia-relief matrix  
\([A]\)****State matrix  
\([B]\)****Input matrix  
\([C]\)****Output matrix  
\([D]\)****Feed-forward matrix  
\([k]_{i}\)****Stiffness matrix of the \(i\)th substructure in physical coordinates  
\([K_{r}]\)****Reduced stiffness matrix in physical coordinates  
\([m]_{i}\)****Mass matrix of the \(i\)th substructure in physical coordinates  
\([M_{r}]\)****Reduced mass matrix in physical coordinates  
\([T_{CB}]\)****Craig-Bampton transformation matrix  
\([T_{H}]\)****Hintz's transformation matrix  
\(\epsilon_{i}\)****Percentage error between \(i\)th reduced-order and full-order mode  
\(\omega\)****Natural frequencies of full-order system  
\(\Omega(t_{i}^{d})\)****Reference trajectory at time \(t_{i}^{d}\) in the deceleration regime, and where \(t_{0}^{d}\) indicates the beginning of the braking of the regime  
\(\Omega(t_{i}^{ss})\)****Reference trajectory at time \(t_{i}^{ss}\) in the acceleration and steady state regime, and where  
\(t_{0}^{ss}\)****indicates the vehicle is commencing travel  
\(\omega_{*}\)****Retained modes in reduced system  
\(\omega_{n}\)****Fixed-constraint normal modes  
\(\Omega_{r}\)****Electrical machine rated speed  
\(\omega_{fn}\)****Free-interface normal modes  
\(\tau\)****Total number of substructures or components  
\(\tau_{c}\)****Total number of couplings in the structure  
\(\Theta_{a}\)****A constant and \(\Theta_{a}>0\)  
\(\Theta_{d}\)****A constant and \(\Theta_{d}<0\)  
\(\{\bar{u}\}\)****Global physical displacements  
\(\{a_{B}\}\)****Global physical displacements for uncoupled boundary coordinates  
\(\{\bar{u}\}\)****Local acceleration in physical coordinates  
\(\{\eta\}\)****Modal coordinates