## 3 Natural Frequency Analysis of SFF  
The dynamic characteristics of an SFF mechanism consists of the natural frequency of the flexible system in the frequency domain and the dynamic response in time domain. Assuming that the lateral deformation \(v,\)\(\theta,\) and \(\theta_{\mathrm{m}}\) are separable in space and time, which are introduced as  
\[ \begin{array}{c}v(s,t)=\mathbf{\Psi}(s)\sin(\omega t),\ \ \theta=\Theta\sin( \omega t),\ \ \theta_{\mathrm{m}}=\Theta_{\mathrm{m}}\sin(\omega t),\end{array}  \tag{20}\]  
where \(\mathbf{\Psi}(\mathrm{s}),\Theta,\ \mathrm{and}\ \Theta_{\mathrm{m}}\) are parameters in space. By the substitution of Equation (20) into Equations (14) and (15), the expressions are given as below when the external torque \(\tau=0,\)  
\[ \begin{array}{c}k\Theta=\left(k-\omega^{2}J_{\mathrm{m}}\right)\Theta_{ \mathrm{m}},\end{array}  \tag{21}\]  
\[ \begin{array}{c}EI\frac{\partial^{4}\mathbf{\Psi}}{\partial\delta^{4}}\sin( \omega t)+\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^{2}\Bigg{[}m_{ \mathrm{p}}L+\frac{\rho\left(L^{2}-s^{2}\right)}{2}\Bigg{]}\Big{(}\frac{ \partial^{2}\mathbf{\Psi}}{\partial\delta^{2}}\Big{)}\sin(\omega t)+\\ \rho\left[-s\left(\Theta\omega^{2}\right)\sin(\omega t)+\mathbf{\Psi}(s)\sin( \omega t)\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^{2}-\Psi(s)\omega^{2 }\sin(\omega t)\right]=0.\end{array}  \tag{22}\]Regardless of the higher order of \(\frac{\mathrm{d}\Theta}{\mathrm{d}t}\), it is given as  
\[EI\frac{\partial^{4}\Psi(s)}{\partial s^{4}}-\omega^{2}\rho[s\Theta+\Psi(s)]=0. \tag{23}\]  
Let \(\beta^{4}=\omega^{2}\rho/EI\), then it is simplified as  
\[\frac{\partial^{4}\Psi(s)}{\partial s^{4}}-\beta^{4}\Psi(s)-\beta^{4}s\Theta=0, \tag{24}\]  
where  
\[\Psi(s)=A\sin(\beta s)+B\cos(\beta s)+\mathrm{Csinh}(\beta s)+\mathrm{Dcosh}( \beta s)-\Theta\mathrm{s}. \tag{25}\]  
Based on Equation (25) and Equations (17)-(19) can be rewritten as  
\[ \begin{array}{c}\left[-\rho\sin(\beta L)-J_{\mathrm{P}}\beta^{3}\cos(\beta L )\right]A+\left[J_{\mathrm{P}}\beta^{3}\sin(\beta L)-\rho\cos(\beta L)\right] B\\ +\left[\rho\sin\mathrm{h}(\beta L)-J_{\mathrm{P}}\beta^{3}\cos\mathrm{h}( \beta L)\right]C+\left[\rho\cos\mathrm{h}(\beta L)-J_{\mathrm{P}}\beta^{3} \sin\mathrm{h}(\beta L)\right]D=0,\end{array}  \tag{26}\]  
\[ \begin{array}{c}\left[m_{\mathrm{P}}\beta\sin(\beta L)-\rho\cos(\beta L) \right]A+\left[m_{\mathrm{P}}\beta\cos(\beta L)+\rho\sin(\beta L)\right]B\\ +\left[m_{\mathrm{P}}\beta\mathrm{sin}\,\mathrm{h}(\beta L)+\rho\cos\mathrm{ h}(\beta L)\right]\tilde{C}+\left[m_{\mathrm{P}}\beta\mathrm{cos}\, \mathrm{h}(\beta L)+\rho\sin\mathrm{h}(\beta L)\right]D=0,\end{array}  \tag{27}\]  
\[B+D=0, \tag{28}\]  
\[\beta A+\beta C-\Theta=0. \tag{29}\]  
Subsequently, an additional equation is obtained from Equation (16) as  
\[J_{\mathrm{P}}\Theta+J_{\mathrm{P}}\Psi^{\prime}(L)+\frac{\rho L^{3}}{3}\Theta +m_{\mathrm{P}}L^{2}\Theta+\int_{0}^{\mathrm{L}}\rho s\Psi(s)\mathrm{d}s+m_{ \mathrm{P}}L\Psi(L)+J_{\mathrm{m}}\Theta_{\mathrm{m}}=0. \tag{30}\]  
Based on Equations (26)-(30), the coefficients \(A,B,C,D\) and \(\Theta\) need to fulfill  
\[\overbrace{\left[ \begin{array}{cccc}H_{11}&H_{12}&H_{13}&H_{14}&0\\ H_{21}&H_{22}&H_{23}&H_{24}&0\\ H_{31},H_{32},H_{33},H_{34},H_{35}\\ 0&1&0&1&0\\ \beta&0&\beta&0&-1\end{array} \right]}^{X}}_{ \begin{array}{c}A\\ B\\ C\\ D\\ \Theta\end{array} }, \tag{31}\]  
where  
\[