## 5 Limitations  
While the GRCFBS method provides a reliable approximation of system vibrational characteristics, several limitations affect its accuracy at higher frequencies.  
* Euler-Bernoulli beam theory (EBT) assumptions: the Euler-Bernoulli beam theory used in the current analysis assumes negligible shear deformation and rotational inertia. At higher frequencies, these assumptions become less valid, which leads to discrepancies in the predicted magnitudes of the resonance peaks. This limitation is particularly relevant for systems operating at higher frequencies (above 2000 Hz), where rotational inertia and shear effects are more significant;
* Absence of material damping: the model does not include material damping, which plays a critical role in attenuating vibrations at high frequencies;
* Alternative beam theories: to improve accuracy at higher frequencies, future work could incorporate alternative beam theories, such as the Timoshenko beam theory, which accounts for shear deformation and rotational inertia. Additionally, including material damping in future models would help more accurately reflect the behavior of real-world systems at high frequencies;
* High-frequency behavior and inertia effects: both the FEA and GRCFBS methods exhibit a noticeable frequency shift beyond 2000 Hz, suggesting that higher-order effects are more pronounced in the FEA results. However, both methods encounter limitations when modeling high-frequency dynamics, where inertia effects (such as shear deformation and rotational inertia) become dominant. These effects are not fully captured by the assumptions in the EBT or the beam models used in FEA, which could lead to misrepresentations of the system's true behavior at higher frequencies;  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case Study** & **Mode** &
\begin{tabular}{c} **Receptance** \\ **Component** \\ \end{tabular} & **GRCFBS** & **FEA** & **Error \%** \\ \hline \multirow{8}{*}{1 (AB)} & Resonance & & 383.1 & 368.1 & 3.8 \\  & & 1025.7 & 1020.9 & 0.4 \\ \cline{2-6}  & & \(H_{y_{1}y_{1}}^{AB}\) & 688.8 & 673.3 & 2.3 \\ \cline{2-6}  & & \(H_{y_{2}y_{2}}^{AB}\) & 756 & 754.2 & 0.24 \\ \cline{2-6}  & & \(H_{y_{1}y_{2}}^{AB}\) & 1219.7 & 1186.7 & 2.78 \\ \cline{2-6}  & & \(H_{y_{1}y_{2}}^{AB}\) & 438.3 & 421.2 & 4.06 \\ \cline{2-6}  & resonance & \(H_{y_{1}y_{2}}^{AB}\) & 1448.4 & 1466.3 & 1.22 \\ \cline{2-6}  & & \(H_{y_{2}y_{2}}^{AB}\) & 1733.9 & 1792.7 & 3.28 \\ \cline{2-6}  & & \(H_{y_{2}y_{2}}^{AB}\) & 682.8 & 672 & 1.61 \\ \cline{2-6}  & & \(H_{y_{1}y_{2}}^{AB}\) & 1042.4 & 1056.3 & 1.33 \\ \hline \multirow{2}{*}{2 (AC)} & Resonance & & 483 & 468 & 3 \\ \cline{2-6}  & & \(H_{y_{1}y_{2}}^{AB}\) & 839.5 & 807.1 & 4.01 \\ \cline{2-6}  & resonance & \(H_{y_{2}y_{2}}^{AB}\) & 1733.9 & 1792.7 & 3.28 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of dominant modes of vibration for GRCFBS and FEA methods below 2000 Hz.  
* Rotational acceptance accuracy beyond 2000 Hz: despite the observed discrepancies in magnitude, the rotational acceptance (both direct and cross components) shows good agreement for frequencies beyond 2000 Hz. This indicates that while the methods may not fully capture the magnitude of the system's vibrational response at high frequencies, they remain consistent in predicting the resonance frequencies where rotational effects are more pronounced.