## 3 Natural Frequency Analysis of SFF
\begin{array}{c}H_{11}=-\rho\sin(\beta L)-J_{\mathrm{P}}\beta^{3}\cos(\beta L ),H_{12}=J_{\mathrm{P}}\beta^{3}\sin(\beta L)-\rho\cos(\beta L),\\ H_{13}=\rho\mathrm{sinh}\,h(\beta L)-J_{\mathrm{P}}\beta^{3}\cos\mathrm{h}( \beta L),H_{14}=\rho\mathrm{cos}\,h(\beta L)-J_{\mathrm{P}}\beta^{3}\sin \mathrm{h}(\beta L),\\ H_{21}=\mathrm{m}_{\mathrm{P}}\beta\sin(\beta L)-\rho\cos(\beta L),\,H_{22}= \mathrm{m}_{\mathrm{P}}\beta\cos(\beta L)+\rho\sin(\beta L),\\ H_{23}=m_{\mathrm{P}}\beta\mathrm{sinh}\,h(\beta L)+\rho\mathrm{cos}\,h( \beta L),H_{24}=m_{\mathrm{P}}\beta\mathrm{cos}\,h(\beta L)+\rho\sin\mathrm{ h}(\beta L),\\ H_{31}=m_{\mathrm{P}}L\sin(\beta L)+J_{\mathrm{P}}\beta\cos(\beta L)+\rho\int_{0 }^{L}\sin(\beta s)s\mathrm{d}s,\\ H_{32}=m_{\mathrm{P}}L\cos(\beta L)-J_{\mathrm{P}}\beta\sin(\beta L)+\rho\int_{ 0}^{L}\cos(\beta s)s\mathrm{d}s,\\ H_{33}=m_{\mathrm{P}}L\mathrm{sinh}\,h(\beta L)+J_{\mathrm{P}}\beta\mathrm{cos }\,h(\beta L)+\rho\int_{0}^{L}\sin\mathrm{h}(\beta s)s\mathrm{d}s,\\ H_{34}=m_{\mathrm{P}}L\mathrm{cos}\,h(\beta L)+J_{\mathrm{P}}\beta\mathrm{sin }\,h(\beta L)\rho\int_{0}^{L}\cos\mathrm{h}(\beta s)s\mathrm{d}s,\\ H_{35}=\frac{\rho}{3}L^{3}+m_{\mathrm{P}}L^{2}+J_{\mathrm{P}}+\frac{\rho k_{ \mathrm{m}}}{\rho k-\beta^{4}EI_{\mathrm{m}}}.\end{array} \]  
In Equation (31), only the item \(H_{35}\) is related to the parameters of the flexible joint and the motor. If \(k\) is extremely large, the SFF system is transformed into a Single Flexible-link Rigid-joint (SFR) system. Under this circumstance of an SFR system, it is equivalent to a cantilever beam with tip mass. To achieve the non-trivial solution of vector \(\mathbf{x}\) in Equation (31), the determinant of matrix \(\mathbf{\Pi}\) must be equal to zero. Once finding the parameter \(\beta\), the natural frequency \(\omega\) is able to be calculated according to \(\beta^{4}=\omega^{2}\rho/EI\). To compare the results, the values of the physical parameters of the SFF manipulator are chosen with those in [12], as below in Table 1.  
Based on the assumption of the small displacement, the natural frequencies are listed and compared in Table 2. When the order of the system model is higher, the natural frequency is closer to that with the Global Mode Method (GMM) in [12]. The natural frequencies in this paper show good agreement with the results in [12]. As mentioned in [23], the fundamental frequency of a cantilever beam of mass \(m_{\text{L}}\) with tip mass of \(m_{\text{P}}\) as  
\[\omega_{1}=\sqrt{\frac{3 EI}{\left(m_{\text{P}}+\frac{33}{140}m_{\text{L}} \right)L^{3}}}. \tag{32}\]  
By substituting of the values in Table 1, the fundamental natural frequency of the SFF system from Equation (32) is \(\omega_{1}=5.4581\text{ rad}/\text{s}\). The value of the fundamental frequency obtained from the proposed method is 5.1486 rad/s with an error of 5.67%. As shown in Figure 5, the obtained fundamental frequencies from Equations (31) and (32) are well matched. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Item** & **Parameter** & **Value** & **Unit** \\ \hline Mass density & \(\rho\) & 0.41 & kg/m \\ Link bending stiffness & \(EI\) & 20.82 & N\(\cdot\)m\({}^{2}\) \\ Link length & \(L\) & 1 & m \\ Motor moment of inertia & \(J_{\text{m}}\) & 173 & kg\(\cdot\)m\({}^{2}\) \\ Stiffness of the flexible joint & \(k\) & 624.6 & N\(\cdot\)m/rad \\ Payload moment of inertia & \(J_{\text{P}}\) & 0.005 & kg\(\cdot\)m\({}^{2}\) \\ Payload mass & \(m_{\text{P}}\) & 2 & kg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical and Material Parameters of an SFF manipulator. Figure 5: Fundamental natural frequencies of the SFF system and the cantilever beam. To verify the analytical results, two simulations are implemented with commercial software ANSYS. The first simulation is performed as a free vibration of a cantilever flexible beam with tip mass. The model is developed with tip mass but without the flexible joint. A flexible joint (i.e., a torsional joint) is added to the model in the second simulation.