# Estimation of the Frequency Response Function of the Rotational Degree of Freedom
## 3 Vibration Analysis Using FEM
### Rotation/Moment Vibration Analysis Results  
To estimate \(P_{11}\), a mode analysis should be performed. As stated in Section 2.4, mode analysis is performed on the real and imaginary parts using the peak picking method. Similar to the modal analysis of CAE, in the peak picking method, modal analysis of a sufficiently large number of modes must be performed to reduce the influence on higher-order modes. The results of the mode analysis of \(L_{11}\) estimated when \(H_{11}\) and s/1 = 0.01,  
Figure 11: FRF of \(H_{0.5a}\) and \(H_{0.5b}\) at s/1 = 0.5. Figure 9: Error rate magnitude at the first mode. respectively, are illustrated in Figure 12, and the reliability of the mode analysis using modal peak picking is verified. Modal analysis was performed for the first and second modes, and the results were fitted and compared with the FEM results. Although the result are similar to the FRF result of FEM, it cannot be estimated that the initial displacement becomes infinite in the real part. However, because modal analysis is performed at each natural frequency, the infinity of the initial response does not affect the modal parameter estimation. The estimated FRF using Schmitz's method, which is the most widely used method for estimating \(P_{11}\), and the FRF using FEM is illustrated in Figure 13. Residual modes of 12.6 and 41.24 Hz appear between natural frequencies, which are the modes that occur, as the response of \(H_{11}\) approaches zero in Equation (14). A compensation strategy was adopted to eliminate the residual modes. The first compensation strategy is illustrated in Figures 14 and 15 by estimating \(P_{11}\) using the modal parameters of \(H_{11}\) and \(L_{11}\). Figure 14 is illustrated using \(L_{11}\) and \(H_{11}\) by applying the first-order finite difference method according to the distance of the response point, and Figure 15 illustrates the result using the second-order finite difference method. The error increases as the distance between the response points increase, indicating a similar trend to the \(L_{11}(=N_{11})\) result. Figure 12: Modal fitting using modal peak picking; (**a**) is \(H_{11}\) and (**b**) is \(L_{11}\). Figure 13: Comparison of \(P_{11-FEM}\) with [PERSON]’s method. The first compensation strategy using \(L_{11}\), estimated using the first-and second-order finite difference methods when \(s/l=0.01\), is illustrated in Figure 16. Because the error in estimating \(L_{11}\) is negligible, a negligible error occurred in the compensation strategy as well. The accuracy of \(L_{11}\) was confirmed to affect the accuracy of \(P_{11}\). The first compensation strategy has a disadvantage in that it cannot express the free-free boundary condition in which the response becomes infinite at 0 Hz. Because the boundary condition cannot be expressed, the first anti-resonance cannot be expressed as well. It would be more suitable for estimating the FRF of fixed-free or fixed-fixed boundary conditions, where the displacement response does not become infinite near 0 Hz. The second compensation strategy removes residual modes by performing mode analysis after calculating \(P_{11}\) using [PERSON]'s method in Equation (14). As the result of \(P_{11}\) is more accurate and \(L_{11}\) is more accurate through the first compensation strategy, the second compensation strategy is performed for \(s/l=0.01\), which has the highest accuracy of \(L_{11}\). In Figure 17, Compared to the first compensation strategy, the second compensation strategy represents a free-free boundary condition in which the rotational displacement, as well as the anti-resonance, becomes infinite at 0 Hz. The natural frequencies and magnitudes were identical. Over the second natural frequency, it appears different from the FEM result, which will reduce the error if the high-frequency mode number is sufficiently considered. Figure 14: First compensation using the first-order finite difference technique according to \(s/l\). Figure 15: First compensation using the second-order finite difference technique according to \(s/l\).