# Estimation of the Frequency Response Function of the Rotational Degree of Freedom
## 2 Theoretical Background
### Rotation/Force and Displacement/Moment FRF  
The rotation/force (N) and displacement/moment (L) FRFs can be estimated using two or three translational FRFs measured at a constant distance, as illustrated in Figure 2. Rotational FRF can be estimated using the measured translational FRF transformation matrix, as expressed in Equation (5):  
\[[H_{estimation}]=[T][H_{Measurement}][T]^{T} \tag{5}\]  
where the measured translational FRF is shown as Equation (6):  
\[[H_{Measurement}]=\left[\begin{array}{cc}H_{1b1b}&H_{1b1a}&H_{1b1}\\ H_{1a1b}&\left[\begin{array}{cc}H_{1a1a}&H_{1a1}\\ H_{11a}&H_{11}\end{array}\right]\\ H_{11b}&H_{11}\end{array}\right] \tag{6}\]  
The rotation transformation matrix using the finite difference method is obtained using Lagrange interpolation. When the domain for Lagrange interpolation in the experimental boundary is \((a,\ b)\) and the measurement point \(x_{i}\) satisfies \(a\leq x_{i}\leq b\ (i=0,1,2\cdots n-1)\), the two-dimensional mode shape is as follows by Lagrange polynomials [7; 18; 19; 20]:  
Figure 2: Impact and response point for rotational D.O.F.  
\[y(x)=\sum\limits_{i=0}^{n-1}y_{i}a_{i}(x) \tag{7}\]  
where, \(a_{i}(x)=\prod\limits_{k\
eq i}^{n-1}\sum\limits_{k=0}^{x-x_{k}}\frac{x-x_{k}}{ x_{i}-x_{k}}\), \(i=0,1,2,\cdots,n-1\).  
The rotation matrix differentiates Equation (7) and is as follows:  
\[\delta_{br}(x)=y^{\prime}(x)=\sum\limits_{i=0}^{n-1}y_{i}a_{i}^{\prime}(x) \tag{8}\]  
Table 1 shows the results of the 2 nd and 3 rd interpolation. Using this interpolation formula, it is possible to obtain a transformation matrix (T) that calculates the rotational freedom of the coupling region from the translational degrees of freedom in each direction at the nodes near the coupling region.  
When the third order or higher is reached, the FRF of the translational degrees of freedom to be measured increases, which may increase the experimental error. For this reason, only the first and second interpolation methods are used in this study. The transformation matrix (\(T\)) using the finite difference method has first-order and second-order approximations, and the first-order requires two measurement points. Depending on the position of the rotational FRF to be estimated, forward and backward difference transformation matrices are adopted, as expressed in Equations (9) and (10), respectively.  
\[\left[T_{1f}\right]=\left[\begin{array}{cc}0&1\\ 1/s&-1/s\end{array}\right] \tag{9}\]  
\[\left[T_{1b}\right]=\left[\begin{array}{cc}0&1\\ -1/s&1/s\end{array}\right] \tag{10}\]  
The second order requires three measurement points, and they include the forward, central, and backward difference transformation matrices. Equations (11)-(13) are similar.  
\[\left[T_{2f}\right]=\frac{1}{2s}\left[\begin{array}{cc}0&0&2s\\ -1&4&-3\end{array}\right] \tag{11}\]  
\[\left[T_{2c}\right]=\frac{1}{2s}\left[\begin{array}{cc}0&2s&0\\ -1&0&1\end{array}\right] \tag{12}\]  
\[\left[T_{2b}\right]=\frac{1}{2s}\left[\begin{array}{cc}0&0&2s\\ 1&-4&3\end{array}\right] \tag{13}\]  
In each transformation matrix, the forward, central, and backward parameters are related to the position to obtain the rotational FRF [11; 12; 13; 14; 15; 16].