# Dynamic Reduction-Based Virtual Models for Digital Twins--A Comparative Study
## 4 Results  
This section presents a comprehensive assessment of the efficacy of dynamic model reduction approaches based on the evaluation methodology outlined in Section 3. The structure utilized to compare the dynamic reduction approaches is an existing commercial off-road vehicle. The structure was designed, modeled, and analyzed using the Ansys 2020R2 platform. Due to confidentiality considerations, the structure shown in Figure 1 is only a partial line diagram. The overall mass of the structure is 359 tonnes, and translational movement is restricted along the \(X\)- and \(Z\)-axis, while travel along the \(Y\)-axis is unrestricted. The built-in libraries of the dynamic reduction methods outlined in Section 2 are available in Ansys 2020R2, and [79; 80] provides a detailed description of the procedure to develop reduced-order models from full-order finite element models using these libraries. Figure 1: Line Diagram with Boundary Conditions and Couplings for the Structure. Table 1 shows a comparison of the modes up to 30 Hz between full-order and reduced-order models obtained by dynamic reduction techniques. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode No. **} & **FULL-ORDER** & \ \begin{tabular}{c} **FREE-INTERFACE** \\ **CMS** \\ \end{tabular}  & \begin{tabular}{c} **FIXED-INTERFACE** \\ **CMS** \\ \end{tabular}  \\ \cline{2-4}  & **Freq (Hz)** & **Freq (Hz)** & **Freq (Hz)** \\ \hline
1 & 0.0000 & 0.0000 & 0.0000 \\ \hline
2 & 0.1857 & 0.1857 & 0.1857 \\ \hline
3 & 0.4302 & 0.4302 & 0.4302 \\ \hline
4 &  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode No. **} & \multirow{2}{*}{**FULL-ORDER**} & \multicolumn{2}{c}{**FREE-INTERFACE**} & \multicolumn{2}{c}{**FIXED-INTERFACE**} \\  & & **CMS** & **CMS** \\ \cline{2-3}  & **Freq (Hz)** & **Freq (Hz)** & **Freq (Hz)** \\ \hline
36 & 8.5915 & 8.5831 & 8.5831 \\ \hline
37 & 8.6763 & 8.6649 & 8.6649 \\ \hline
38 & 8.8309 & 8.8182  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode No. **} & **FULL-ORDER** & \ \begin{tabular}{c} **FREE-INTERFACE** \\ **CMS** \\ \end{tabular}  & \begin{tabular}{c} **FIXED-INTERFACE** \\ **CMS** \\ \end{tabular}  \\ \cline{2-4}  & **Freq (Hz)** & **Freq (Hz)** & **Freq (Hz)** \\ \hline
74 & 19.5310 & 19.5261 & 19.5267 \\ \hline
75 & 19.7960 & 19.7869 & 19.7893 \\ \hline
76 & 19.9100 & 19  
Figure 2 depicts that the frequency deviation of the reduced-order model from the full-order finite element model for both free-interface and fixed-interface modal synthesis is less than 0.8% over a frequency range of 0-30 Hz. Figure 3 shows the modal vector orthogonality verification of the component modal synthesis-based reduced-order model for modes up to 30 Hz. The diagonal terms are unity, but all off-diagonal terms are near to zero for the reduced-order models obtained from both fixed-interface and free-interface component modal synthesis, as shown in the figure, which is consistent with the orthogonality relations specified in Equation (42). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode No.**} & **FULL-ORDER** & \
\begin{tabular}{c} **FREE-INTERFACE** \\ **CMS** \\ \end{tabular} &
\begin{tabular}{c} **FIXED-INTERFACE** \\ **CMS** \\ \end{tabular} \\ \cline{2-4}  & **Freq (Hz)** & **Freq (Hz)** & **Freq (Hz)** \\ \hline
112 & 28.2750 & 28.2502 & 28.2612 \\ \hline
113 & 28.4250 & 28.4160 & 28.4172 \\ \hline
114 & 28.7100 & 28.7304 & 28.7353 \\ \hline
115 & 28.9110 & 28.9312 & 28.9342 \\ \hline
116 & 29.0080 & 29.0032 & 29.0053 \\ \hline
117 & 29.3980 & 29.4209 & 29.4278 \\ \hline
118 & 29.6380 & 29.6299 & 29.6304 \\ \hline
119 & 29.6620 & 29.6699 & 29.6710 \\ \hline
120 & 29.9190 & 29.8785 & 29.8789 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 2: Percentage deviations of modes. Figure 4 illustrates the digital model of the structure created utilizing either of the modal synthesis methods and integrating the corresponding driveline, electrical, and powertrain control modules in the MATLAB Simulink environment.