## Appendix A  
Based on the assumption of the inextensible flexible link, the following constraint equation is obtained as  
\[\left(1+\frac{\partial u}{\partial s}\right)^{2}+\left(\frac{\partial v}{ \partial s}\right)^{2}=1. \tag{11}\]  
After transforming \(\sqrt{1-\left(\frac{\partial v}{\partial s}\right)^{2}}\) into a power series and retaining the first two items, it is given as  
\[u=-\frac{1}{2}\int_{0}^{\mathrm{L}}\!\left(\frac{\partial v}{\partial s}\right) ^{2}\mathrm{d}s. \tag{12}\]  
By substitution of \(s=\zeta L\), Equation (5) is deduced as  
\[u=-\frac{\lambda}{2}\int_{0}^{1}\!\left(\frac{\partial v}{\partial\zeta} \right)^{2}\mathrm{d}\zeta, \tag{13}\]  
where \(\lambda=1/L\).  
To incorporate the stiffening effect in an SFF system, the potential energy of the system is composed of the potential energy of the flexible link \(V_{\mathrm{L}}\) due to bending, the energy \(V_{\mathrm{A}}\) caused by the axial shortening and the inertial force, and the potential energy of the torsional spring. The expression of \(V_{\mathrm{L}}\) can be deduced as  
\[V_{\mathrm{L}}=\frac{EI\lambda^{3}}{2}\int_{0}^{1}\!\left[\left(\frac{\partial ^{2}v}{\partial\zeta^{2}}\right)^{2}+\lambda^{2}\!\left(\frac{\partial^{2}v}{ \partial\zeta^{2}}\right)^{2}\!\left(\frac{\partial v}{\partial\zeta}\right)^ {2}\right]\!\mathrm{d}\zeta. \tag{14}\]  
Under the assumption that the transversal deflection of the slender link is small, the second term in Equation (14) is neglected. Then, it is simplified as  
\[V_{\mathrm{L}}=\frac{EI\lambda^{3}}{2}\int_{0}^{1}\!\left(\frac{\partial^{2} v}{\partial\zeta^{2}}\right)^{2}\!\mathrm{d}\zeta. \tag{15}\]  
The inertial force on the point of the link due to the rotational velocity is written as  
\[F_{\mathrm{I}}=\dot{\theta}^{2}\!\left[m_{\mathrm{p}}(R+L)+\int_{\zeta}^{1} \!\left(L\zeta+R\right)\rho L\mathrm{d}\zeta\right]\!. \tag{16}\]  
All terms in Equation (A15) are given as  
\[\begin{split} A_{1}&=J_{\text{m}}\Big{(}\frac{d^{2} \theta_{m}}{dt^{2}}\Big{)}+\tau-k(\theta_{m}-\theta),\\ A_{2}&=EI\lambda^{3}\frac{\partial^{4}v}{\partial\xi^{4} }+\Big{(}\frac{\partial\theta}{\text{d}t}\Big{)}^{2}\Big{[}m_{\text{p}}+\frac{ \rho L\big{(}1-\xi^{2}\big{)}}{2}\Big{]}\Big{(}\frac{\partial^{2}v}{\partial \xi^{2}}\Big{)}+\rho L\Big{[}\xi L\Big{(}\frac{d^{2}\theta}{\text{d}t^{2}} \Big{)}+v\Big{(}\frac{\partial\theta}{\text{d}t}\Big{)}^{2}+\Big{(}\frac{ \partial^{2}v}{\partial t^{2}}\Big{)}\Big{]},\\ A_{3}&=J_{p}\Big{(}\frac{d^{2}\theta}{dt^{2}}+ \left.\frac{\partial^{3}v}{\partial s\partial s\partial t^{2}}\right|_{s=L} \Big{)}+\int_{0}^{1}\left\{\rho L\big{(}\xi^{2}L^{2}+v^{2}\big{)}+\left[m_{p}+ \frac{\rho L\big{(}1-\xi^{2}\big{)}}{2}\right]\Big{(}\frac{\partial v}{ \partial\xi}\Big{)}^{2}\right\}\Big{(}\frac{d^{2}\theta}{dt^{2}}\Big{)}d\zeta+ \\ m_{p}\big{(}L^{2}+v^{2}\big{|}_{s=L}\big{)}\Big{(}\frac{d^{2} \theta}{dt^{2}}\Big{)}+\int_{0}^{1}\rho\zeta L^{2}\Big{(}\frac{\partial^{2}v}{ \partial t^{2}}\Big{)}d\zeta+m_{p}L\Big{(}\frac{\partial^{2}v}{\partial t^{2}} \Big{)}\Big{|}_{s=L}+\\ \int_{0}^{1}\rho Lv\Big{(}\frac{\partial v}{\partial t}\Big{)}d \zeta\Big{(}\frac{\partial\theta}{dt}\Big{)}+m_{p}\,v_{|s=L}\Big{(}\frac{ \partial v}{\partial t}\Big{)}\Big{|}_{s=L}\Big{(}\frac{\partial\theta}{dt} \Big{)}+k(\theta_{m}-\theta)\\ A_{4}&=EI\left(\frac{\partial^{2}v}{\partial t^{2}} \right)\Big{|}_{s=L}+J_{p}\Big{(}\frac{d^{2}\theta}{dt^{2}}+\left.\frac{ \partial^{3}v}{\partial s\partial t^{2}}\right|_{s=L}\Big{)},\\ A_{5}&=m_{\text{p}}L\Big{(}\frac{d^{2}\theta}{dt^{2}} \Big{)}+m_{\text{p}}\left(\frac{\partial^{2}v}{\partial t^{2}}\right)\Big{|}_ {s=L}-EI\left(\frac{\partial^{2}v}{\partial s^{2}}\right)\Big{|}_{s=L}.\end{split}\] (A16)