# Estimation of the Frequency Response Function of the Rotational Degree of Freedom
## 3 Vibration Analysis Using FEM
### Rotation/Force(=Displacement/Moment) Vibration Analysis Results  
The estimated FRF, \(L_{11}(=N_{11})\), using the first-order finite difference method according to the response point distance, is illustrated in Figure 7. The natural frequency coincided with the FEM results regardless of the distance between the response points, but the magnitude of the natural frequency tended to decrease as the distance between the response points increased. The anti-resonance also exhibited a tendency to move as the distance of the response point increased.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Length** & **Diameter** & **Density** & **Elastic Modulus** & **Damping Ratio** \\ \hline
1 m & 0.005 m & 7850 kg/m\({}^{3}\) & 210 GPa & 0.01 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Properties of the beam model.  
Figure 5: Response point of the beam model.  
Figure 6: Verification of FEM and theory.  
The estimated FRF, \(L_{11}(=N_{11})\), using the second-order finite difference method according to the response point distance, is illustrated in Figure 8. Similar to the first-order method, the natural frequency coincided with the FEM result, and an error occurred in the natural frequency magnitude and anti-resonance. As the distance between the response points increased, the error increased.  
The error rate for the first natural frequency magnitude of the FEM and the estimated FRF of the first-and second-order finite difference methods are illustrated in Figure 9. In both the first and second order, the error rate increased as the distance of the response point increased. When s/1 was 0.1, the error rate of the first order was 0.4%, and the second order was 2.1%. When s/1 was 0.2, the error rate of the first order was 2.9%, and the second order was 12.3%. The magnitude error increased as the higher-order mode goes. The estimation of \(L_{11}(=N_{11})\) using the finite difference method yielded more accurate results in the first order, and the shorter the distance between the response points, the more accurate the results. The ratio of the response point distance to the total length of the structure during the experiment should not exceed 0.05. Even considering the structure and sensor size, it should not exceed 0.1 at most.  
Figure 8: Estimation for \(L_{11}(=N_{11})\) for the second-order finite difference method.  
Figure 7: Estimation for \(L_{11}(=N_{11})\) for the first-order finite difference method.  
The responses of \(H_{0.01a}\) and \(H_{0.01b}\) when s/1 = 0.01 are illustrated in Figure 10. The two response points clearly indicate the resonance and anti-resonance, and these points were well-estimated in the rotational FRF using these points. However, as illustrated in Figure 11, both response points do not exhibit resonance and anti-resonance. This is related to the shape of the mode, and because it does not indicate resonance and anti-resonance, it causes a significant error in the rotational FRF estimation using this. Therefore, when selecting a response point, it is necessary to select a point where the resonance and anti-resonance modes are expressed.