## 3 Application of a Tailored Tether to AWES
### Dynamic Response Model of Simplified AWES Model  
Based on the simplified model shown in Figure 4, the motion of the airborne system (m) can be expressed by Equation (1) [23]. \[ \begin{array}{rl}m\ddot{x}&=F_{1}-P(\delta)=F_{1}-K_{i}\big{[}\delta-(i-1) \big{(}l_{s}-l_{p}\big{)}\big{]}=F_{1}-K_{i}\big{[}u_{1}-u_{2}-(i-1)\big{(}l_{s} -l_{p}\big{)}\big{]}\\ &=F_{1}-K_{i}\big{[}x-X-L-(i-1)\big{(}l_{s}-l_{p}\big{)}\big{]}=F_{1}-K_{i}(x- X-L_{i})\\ \end{array}  \tag{1}\]  
where \(P\) is the internal load, \(\delta\) is the end displacement, \(p\) is the primary element, \(s\) is the secondary element, \(i\) is the failure stage, \(L_{i}\) is the total tether length at the \(i\)th stage, \(L_{p}\) is the tether length of the primary segment, \(L_{s}\) is the tether length of the primary segment, \(K_{i}\) is the equivalent tether stiffness at the \(i\)th stage, \(u_{1}\) is the displacement of the mass m, and \(u_{2}\) is the displacement of the mass M.  
The equation of motion for the station system can also be similarly expressed as  
\[M\ddot{X}=-F_{2}+P(\delta)=-F_{2}+K_{i}(x-X-L_{i}) \tag{2}\]  
At the \(i\)th stage, the global motion of the two-degree-of-freedom system is given by following matrix form. \[\begin{array}{rl}\left[ \begin{array}{c}m0\\ 0M\end{array} \right]\left\{ \begin{array}{c}\ddot{x}\\ \ddot{X}\end{array} \right\}+\left[ \begin{array}{c}K_{i}-K_{i}\\ -K_{i}K_{i}\end{array} \right]\left\{ \begin{array}{c}x\\ X\end{array} \right\}=\left\{\begin{array}  
where, \(q_{mi}(t)\) is the generalized coordinate of the mass m at the \(i\)th stage of the specific time t, \(q_{Mi}(t)\) is the generalized coordinate of the mass M at the \(i\)th stage of the specific time t, and \(\omega_{i}=\pm\sqrt{\frac{K_{i}(m+M)}{mM}}\). At the \(i\)th stage, the initial conditions are  
\[x_{i}\Big{(}\tau_{i}^{0}\Big{)}={x_{i}}^{0},\dot{x}_{i}\Big{(}\tau_{i}^{0} \Big{)}={v_{i}}^{0},X_{i}\Big{(}\tau_{i}^{0}\Big{)}={X_{i}}^{0},\dot{X}_{i} \Big{(}\tau_{i}^{0}\Big{)}={V_{i}}^{0} \tag{5}\]  
Given the initial conditions from Equation (5), the coefficients of Equation (4) are the following. \[A_{i}=\frac{{m{v_{i}}^{0}+M{V_{i}^{0}}}}{{m+M}},B_{i}=\frac{{m{x_{i}^{0}}+M{X_ {i}^{0}}}}{{m+M}},C_{i}=-\frac{{K_{i}L_{i}(m+M)-mM{\omega_{i}}^{2}\big{(}{x_{ i}^{0}-X_{i}^{0}}\big{)}+M{F_{1}}+m{F_{2}}}}{{m(m+M){\omega_{i}}^{2}}},D_{i}= \frac{M\big{(}v_{i}^{0}-V_{i}^{0}\big{)}}{{m(m+M){\omega_{i}}}} \tag{6}\]  
The maximum elongation at the \(i\)th stage is  
\[ \begin{array}{ll}\delta_{i}^{max}(t)&=x_{i}(t)-X_{i}(t)-L_{i}\\ &=\sqrt{\big{(}C_{i}\big{(}1+\frac{m}{M}\big{)}\big{)}^{2}+\big{(}D_{i}\big{(} 1+\frac{m}{M}\big{)}\big{)}^{2}}\sin\big{(}\omega_{i}\big{(}t-\tau_{i}^{0} \big{)}+\alpha_{i}\big{)}+\frac{{F_{1}}+\frac{m}{M}{F_{2}}}{{K_{i}\big{(}1+ \frac{m}{M}\big{)}}}}\end{array}  \tag{7}\]  
where, \(\alpha_{i}=arctan\Big{(}\frac{C_{i}}{D_{i}}\Big{)}\) and \(F_{i}^{max}=\left\{ \begin{array}{ll}P_{p}&1\leq i\leq n\\ P_{\rm s}&i=(n+1)\end{array} \right..\)  
The above equation is valid in the interval \(\tau_{i}^{0}\leq t\leq\tau_{i}^{f}\), where the final time is given by  
\[\tau_{i}^{f}=\frac{1}{\omega_{i}}\left\{ \begin{array}{ll}\frac{{\tau_{i}^{ max}}}{{K_{i}\frac{{F_{1}}+\frac{m}{M}{F_{2}}}}}{{1+\frac{m}{M}}}\\ \sqrt{\big{(}C_{i}\big{(}1+\frac{m}{M}\big{)}\big{)}^{2}+\big{(}D_{i}\big{(}1+ \frac{m}{M}\big{)}\big{)}^{2}}}&-arctan\bigg{(}\frac{C_{i}}{D_{i}}\bigg{)} \end{array} \right\} \tag{8}\]  
Subsequently, if \(\delta_{i}^{u}>\delta_{i+1}^{s}\), the following equations of motion are obtained:  
\[ \begin{array}{ll}m\ddot{x}_{i}=F_{1}\\ M\ddot{X}_{i}=-F_{2}\end{array} \Bigg{\}}\Rightarrow\left\{ \begin{array}{ll} \widetilde{x}_{i}=\frac{F_{1}}{2m}\Big{(}t-\tau_{i}^{f}\Big{)}^{2}+a_{i} \Big{(}t-\tau_{i}^{f}\Big{)}+b_{i}\\ \widetilde{X}_{i}(t)=-\frac{F_{2}}{2M}\Big{(}t-\tau_{i}^{f}\Big{)}^{2}+c_{i} \Big{(}t-\tau_{i}^{f}\Big{)}+d_{i}\end{array} \right.^{\prime}\left\{ \begin{array} []{ll}(x-X)\in\big{(}L+\delta_{i}^{u},L+\delta_{i+1}^{s}\big{)}\\ {applicablefor\tau_{i}^{f}<t\leq\widetilde{\tau}_{i}^{f}}\end{array}