## 4 Stiffening Effect  
In the previous section, the dynamic model of the SFF manipulator system is developed on a basis of the small deformation assumption in free vibration. However, the stiffening effect is highly related to its rotational velocity in Equation (15), which is neglected in Section 3. By reviewing Equation (15), the stiffening effect of the SFF system becomes more prominent when a high angular velocity is considered. To reduce the complexity of the issue, the angular velocity remains constant so that the term \(\left(\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\right)\) in Equation (15) is equal to 0. Thus, Equation (15) with a constant angular velocity is simplified as  
\[EI\frac{\partial^{4}v}{\partial s^{4}}+\left(\frac{\mathrm{d}\theta}{\mathrm{ d}t}\right)^{2}\left[m_{\mathrm{P}}L+\frac{\rho\left(L^{2}-s^{2}\right)}{2} \right]\left(\frac{\partial^{2}v}{\partial s^{2}}\right)+\rho\left[v\left( \frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^{2}+\left(\frac{\partial^{2}v}{ \partial t^{2}}\right)\right]=0. \tag{34}\]  
Figure 8: Modal shapes of the SFF system. After several mathematical combination manipulations, the governing equation of motion is obtained as  
\[EI\frac{\partial^{4}v}{\partial s^{4}}+\rho\Big{(}\frac{\partial^{2}v}{\partial t ^{2}}\Big{)}+\varnothing\Big{(}\Gamma\frac{\partial^{2}v}{\partial s^{2}}+\rho v \Big{)}=0, \tag{35}\]  
where \(\varnothing=\Big{(}\frac{\partial\theta}{\partial t}\Big{)}^{2}\) and \(\Gamma=m_{\rm P}L+\frac{\rho\big{(}L^{2}-s^{2}\big{)}}{2}\). Since Equation (35) is not able to be expressed in an explicit form, the Finite Element Method (FEM) is used in this section to analyze the natural frequency of the SFF system, considering the stiffening effect. By employing the approach mentioned in [22], the lateral deformation of the \(i\)th element is denoted as  
\[v_{i}(x,t)=\Psi_{e}(x)Q_{e}^{i}(t), \tag{36}\]  
where \(\Psi_{e}(x)\) represents a shape function of the link in terms of length \(x\), and \(Q_{e}(t)\) is referred to as the nodal displacement with respect to time. By following the shape function defined in [12; 23] with Hermite interpolation, it is written as  
\[\Psi_{e}(x)=[\varphi_{1}(x),\ \varphi_{2}(x),\ \varphi_{3}(x),\ \varphi_{4}(x)], \tag{37}\]  
where \(\varphi_{1}(x)=1-\frac{3x^{2}}{l^{2}}+\frac{2x^{3}}{l^{3}}\), \(\varphi_{2}(x)=x-\  
where \(E_{x}=\varnothing\bigg{[}\frac{m_{p}L}{\rho}+\frac{L^{2}-(il-l+x)^{2}}{2}\bigg{]}\), \(x\in\{0,l\}\). Then, Equation (43) can be simplified as  
\[ \begin{array}{c}\ddot{v}_{i-1}(t)+\varnothing\Big{[}1-\frac{6}{2}\Big{(}\frac{ m_{p}L}{\rho}+\frac{L^{2}-s_{0}^{2}}{2}\Big{)}\Big{]}v_{i-1}(t)+\varnothing \bigg{[}-\Big{(}\frac{m_{p}L}{\rho}+\frac{L^{2}-s_{0}^{2}}{2}\Big{)}\frac{4}{l }\Big{]}\theta_{i-1}(t)\\ +\varnothing\Big{[}\Big{(}\frac{m_{p}L}{\rho}+\frac{L^{2}-s_{0}^{2}}{2}\Big{)} \frac{6}{l}\Big{]}v_{i}(t)+\varnothing\Big{[}-\Big{(}\frac{m_{p}L}{\rho}+\frac {L^{2}-s_{0}^{2}}{2}\Big{)}\frac{7}{l}\Big{]}\theta_{i}(t)=0,\text{ if }x=0,s_{0}=(i-1)l,\end{array}  \tag{44}\]  
\[ \begin{array}{c}\ddot{v}_{i}(t)+\varnothing\Big{[}\Big{(}\frac{m_{p}L}{\rho}+ \frac{L^{2}-s_{0}^{2}}{2}\Big{)}\frac{6}{l}\Big{]}v_{i-1}(t)+\varnothing\Big{[} \Big{(}\frac{m_{p}L}{\rho}+\frac{L^{2}-s_{0}^{2}}{2}\Big{)}\frac{7}{l}\Big{]} \theta_{i-1}(t)\\ +\varnothing\Big{[}1-\Big{(}\frac{m_{p}L}{\rho}+\frac{L^{2}-s_{0}^{2}}{2}\Big{)} \frac{6}{l}\Big{]}v_{i}(t)+\varnothing\Big{[}\Big{(}\frac{m_{p}L}{\rho}+\frac {L^{2}-s_{0}^{2}}{2}\Big{)}\frac{4}{l}\Big{]}\theta_{i}(t)=0,\text{ if }x=l,s_{l}=il.\end{array}  \tag{45}\]  
Since \(v_{i}(t)\) is only related to time at the curvilinear point of \(s=il\), it is expressed as \(v_{i}(t)=\Psi(il)\sin(\omega t)\) with \(\omega\) of the natural frequency. An additional equation is obtained  
\[\ddot{v}_{i}(t)=-\omega^{2}\Psi(il)\sin(\omega t)=-\omega^{2}v_{i}(t). \tag{46}\]  
By combining Equation (44) to Equation (46), the transfer formula is yielded as  
\[