## 2 Dynamic Modeling
The expression of \(V_{\text{L}}\) can be deduced as  
\[V_{\text{L}}=\frac{EI\lambda^{3}}{2}\int_{0}^{1}\Biggl{[}\left(\frac{\partial^ {2}v}{\partial\zeta^{2}}\right)^{2}+\lambda^{2}\biggl{(}\frac{\partial^{2}v}{ \partial\zeta^{2}}\biggr{)}^{2}\biggl{(}\frac{\partial v}{\partial\zeta} \biggr{)}^{2}\Biggr{]}\text{d}\zeta. \tag{5}\]  
The virtual work due to the inertial force and the link foreshortening displacement is obtained as  
\[V_{\text{A}}=-\frac{\dot{\theta}^{2}}{2}\int_{0}^{1}\Biggl{[}m_{\text{P}}(R+L) +\frac{\rho L^{2}\bigl{(}1-\zeta^{2}\bigr{)}}{2}+\rho LR(1-\zeta)\Biggr{]} \Biggl{[}\lambda\biggl{(}\frac{\partial v}{\partial\zeta}\biggr{)}^{2}\Biggr{]} \text{d}\zeta. \tag{6}\]  
Moreover, the potential energy of the torsional spring is  
\[V_{\text{S}}=\frac{k}{2}(\theta_{\text{m}}-\theta)^{2}. \tag{7}\]  
Thus, the entire potential energy of the system is expressed as  
\[V=V_{\text{L}}+V_{\text{A}}+V_{\text{S}}. \tag{8}\]  
In the present paper, the kinetic energy of the slender link is expressed as  
\[T_{\text{L}}=\frac{\rho L}{2}\int_{0}^{1}\biggl{(}\frac{\partial r_{\text{fl} }}{\partial t}\biggr{)}^{\text{T}}\biggl{(}\frac{\partial r_{\text{fl}}}{ \partial t}\biggr{)}\text{d}\zeta. \tag{9}\]  
Considering \(x=\zeta L+R\), the following formulation can be obtained as  
\[T_{\text{L}}=\frac{\rho L}{2}\int_{0}^{1}\Biggl{\{} \begin{pmatrix}\text{d} \theta\\ \text{d}t\end{pmatrix} ^{2}\Bigl{[}\left(\zeta L+R\right)^{2}+u^{2}+2u(\zeta L +R)+v^{2}\Bigr{]}+\\ 2\Bigl{(}\frac{\text{d}\theta}{\text{d}t}\Bigr{)}\Bigl{[}\left(\zeta L+R \right)\Bigl{(}\frac{\partial v}{\partial t}\Bigr{)}+u\Bigl{(}\frac{\partial v }{\partial t}\Bigr{)}-\Bigl{(}\frac{\partial u}{\partial t}\Bigr{)}v\Bigr{]} +\Bigl{(}\frac{\partial v}{\partial t}\Bigr{)}^{2}+\Bigl{(}\frac{\partial u} {\partial t}\Bigr{)}^{2}\Biggr{\}}\text{d}\zeta. \tag{10}\]  
Similarly, the kinetic energy of the tip payload is introduced as  
\[T_{\text{P}}=\frac{m_{\text{P}}}{2}\biggl{(}\left.\frac{\partial r_{\text{fl} }}{\partial t}\right|_{s=L}\biggr{)}^{\text{T}}\biggl{(}\left.\frac{\partial r _{\text{fl}}}{\partial t}\right|_{s=L}\biggr{)}+\frac{J_{\text{P}}}{2}\biggl{(} \left.\frac{\text{d}\theta}{\text{d}t}+\left.\frac{\partial^{2}v}{\partial s \partial t}\right|_{s=L}\biggr{)}^{2}. \tag{11}\]  
Thus, the entire kinetic energy of the flexible system is yielded as  
\[T=\frac{1}{2}J_{\text{m}}\biggl{(}\frac{\text{d}\theta_{\text{m}}}{dt}\biggr{)} ^{2}+\frac{1}{2}J_{\text{h}}\biggl{(}\frac{\text{d}\theta}{\text{d}t}\biggr{)} ^{2}+T_{\text{L}}+T_{\text{P}}. \tag{12}\]  
The virtual work exerted by the rotational actuator torque is \(\delta W=\tau\delta\theta_{\text{m}}\). As mentioned in the introduction, there are three methods to develop the dynamic equations of motion of the flexible link system. In the following deduction, the extended Hamilton's principle is applied as  
\[\int_{t_{1}}^{t_{2}}(\delta T-\delta V+\delta W)\mathrm{d}t=0. \tag{13}\]  
By taking into account that \(\delta\theta_{\mathrm{m}},\delta\theta,\delta v,\ \delta\left(\frac{\partial v}{ \partial\delta}\right)\Big{|}_{\mathrm{s=L}}\) and \(\delta v|_{\mathrm{s=L}}\) are separate variations, the expressions of \(A_{1},A_{2},A_{3},A_{4},\ \mathrm{and}\ \mathrm{and}\ \mathrm{and}\ \mathrm{a}_{5}\) are equal to zero (for more details in Appendix A). Therefore, the dynamical model of a flexible-link, flexible-joint manipulator with tip mass considering the stiffening effect is formulated as  
\[J_{\mathrm{m}}\left(\frac{\mathrm{d}^{2}\theta_{\mathrm{m}}}{\mathrm{d}t^{2}} \right)+\tau-k(\theta_{\mathrm{m}}-\theta)=0, \tag{14}\]  
\[EI\frac{\partial^{4}v}{\partial\delta^{4}}+\left(\frac{\mathrm{d}\theta}{ \mathrm{d}t}\right)^{2}\Bigg{[}m_{\mathrm{p}}L+\frac{\rho\left(L^{2}-s^{2} \right)}{2}\Bigg{]}\Big{(}\frac{\partial^{2}v}{\partial\delta^{2}}\Big{)}+ \rho\Bigg{[}s\left(\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\right)+v \left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^{2}+\left(\frac{\partial^{2 }v}{\partial t^{2}}\right)\Bigg{]}=0, \tag{15}\]  
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