## 2 Materials and Methods
### Identification of Antiresonant Frequency
#### 2.2.1 Theoretical Antiresonant Frequency  
Since the antiresonant frequencies of the lightly damped systems are almost unaffected by damping, the antiresonant frequencies of the undamped multi-degree-of-freedom system represented by the mass matrix \(\mathbf{M}\) and the stiffness matrix \(\mathbf{K}\) are calculated as an example for the convenience of analysis. According to the literature [9; 24; 34], the FRF matrix \(\mathbf{H}(\omega)\) is the inverse of the dynamic stiffness matrix (\(\mathbf{K}-\omega^{2}\mathbf{M}\)), i.e.,  
\[\mathbf{H}(\omega)=\left(\mathbf{K}-\omega^{2}\mathbf{M}\right)^{-1}=\frac{adj\big{(}\bm {K}-\omega^{2}\mathbf{M}\big{)}}{\det(\mathbf{K}-\omega^{2}\mathbf{M})} \tag{2}\]  
where \(adj(\cdot)\) and \(\det(\cdot)\) represent the adjoint matrix and the determinant, respectively. When exciting force is applied to the system at point \(j\), the response is obtained at point \(i\), the frequency response function is  
\[\mathbf{H}_{ij}(\omega)=\frac{adj\big{(}\mathbf{K}_{ij}-\omega^{2}\mathbf{M}_{ij}\big{)}} {\det(\mathbf{K}-\omega^{2}\mathbf{M})}=\frac{(-1)^{i+j}\det\big{(}\mathbf{K}_{ij}-\omega^ {2}\mathbf{M}_{ij}\big{)}}{\det(\mathbf{K}-\omega^{2}\mathbf{M})} \tag{3}\]  
According to Equation (3), the zeros of \(\mathbf{H}_{ij}(\omega)\) are the frequencies \(\omega\) when the numerator is zero. The numerator represents the matrix obtained from the dynamic stiffness matrix of the system after deleting row \(i\) and column \(j\), and its determinant is zero, i.e.,  
\[\det(\mathbf{K}_{ij}-\omega^{2}\mathbf{M}_{ij})=0 \tag{4}\]  
The analytical antiresonant frequencies are obtained from the eigenvalues of Equation (4). In Equation (4), if \(i=j\), the antiresonant frequencies are obtained when the exciting point and the response point are the same (i.e., point FRF); if \(i\
eq j\), the antiresonant frequencies \begin{table}
\begin{tabular}{c c c c} \hline
**Updating Parameter** & **Initial Value** & **Target Value** & **Error (\%)** \\ \hline \(E\) (GPa) & 70 & 77 & \(-9.09\) \\ \(m\) (kg) & 5 & 6.5 & \(-23.08\) \\ \(k_{t}\) (\(10^{7}\) N/m) & 2 & 1 & 100 \\ \(k_{r}\) (\(10^{4}\) N\(\cdot\)m/rad) & 5 & 2 & 150 \\ \hline \end{tabular}
\end{table} 
Table 1: Updating parameters. Figure 1: A cantilever beam with elastic support (adapted from Figure 2 in Ref. [12]). are obtained when the exciting point and the response point are different (i.e., transfer FRF). When the denominator is zero, the resonance frequencies are obtained from Equation (3). It can be known from Equations (3) and (4): (a) the antiresonant frequencies at different exciting points are different, which reflect the local frequency information of the structure. In the FEM updating, the antiresonant frequencies at different measuring points can be increased correspondingly with the increase in the updating parameters. (b) According to the definition of an adjoint matrix, all information related to the exciting point is deleted, so the antiresonant frequencies at the exciting point do not change with the variation of the mass or the stiffness of the point. At the same time, the literature [1; 12; 19] points out that the antiresonant frequencies and the resonance frequencies in the point FRF curves appear alternately, which can provide more frequency information, while the antiresonant frequencies in the transfer FRF have no such law. \[\det(\mathbf{K}-\omega^{2}\mathbf{M})=0 \tag{5}\]  
According to Equation (5), both \(\mathbf{K}\) and \(\mathbf{M}\) are the stiffness and mass matrix of the system, which reflect the global information of the system. If the mass or stiffness at the exciting point is changed, the mass or the stiffness matrix of the system will also be changed, which results in the change of the resonance frequencies.