## 3 Application of a Tailored Tether to AWES
### Approximate Response Modeling of Simplified AWES
\begin{array}{cc}\ddot{x}_{2}\\ \ddot{X}_{2}\end{array} \right\}-\left[ \begin{array}{cc}K_{d}&-K_{d}\\ -K_{d}&K_{d}\end{array} \right]\left\{ \begin{array}{c}x_{2}\\ X_{2}\end{array} \right\}=\left\{ \begin{array}{c}-F_{1}-K_{d}(L+\delta_{1}) \\ F_{1}+K_{d}(L+\delta_{1})\end{array} \right\} \tag{21}\]  
Next, the forced vibration response can be obtained as follows:  
\[ \begin{array}{ll}x_{2}(t)=q_{m}(t)+q_{M}(t)=A_{2}\big{(}t-\tau_{1}\big{)}+B _{2}+C_{2}Exp\big{(}\omega_{2}\big{(}t-\tau_{1}\big{)}\big{)}+D_{2}Exp\big{(} -\omega_{2}\big{(}t-\tau_{1}\big{)}\big{)}\frac{1}{\overline{n}}\frac{1}{ \omega_{2}^{2}}[F_{1}+K_{d}(\delta_{1}+L)]\\ X_{2}(t)=q_{m}(t)-\frac{m}{M}q_{M}(t)=A_{2}\big{(}t-\tau_{1}\big{)}+B_{2}- \frac{m}{M}C_{2}Exp\big{(}\omega_{2}\big{(}t-\tau_{1}\big{)}\big{)}-\frac{m}{ M}D_{2}Exp\big{(}-\omega_{2}\big{(}t-\tau_{1}\big{)}\big{)}-\frac{m}{M}\frac{1}{ \overline{n}}\frac{1}{\omega_{2}^{2}}[F_{1}+K_{d}(\delta_{1}+L)]\\ \end{array}  \tag{22}\]  
where \({\omega_{2}}^{2}=\frac{K_{d}}{m}\big{(}1+\frac{m}{M}\big{)}\) (14). Figure 6: Force and displacement plot for the first and second stages. By applying the initial conditions \(x_{2}\Big{(}\tau_{f1}\Big{)}=x_{f1}\), \(\dot{x}_{2}\Big{(}\tau_{f1}\Big{)}=v_{f1}\), \(\;\;X_{2}\Big{(}\tau_{f1}\Big{)}=X_{f1}\), \(\;\dot{X}_{2}\Big{(}\tau_{f1}\Big{)}=V_{f1}\) to Equation (22), we can obtain the coefficients:  
\[ \begin{array}{l}A_{2}=\frac{mv_{f1}+MV_{f1}}{m+M},\;\;\;B_{2}=\frac{mv_{f1}+ MK_{f1}}{m+M},\\ C_{2}=-\left[\begin{array}{c}(m+M)F_{1}+mM\omega_{2}\Big{[}-\omega_{2}\Big{(} x_{f1}-X_{f1}\Big{)}-\Big{(}v_{f1}-V_{f1}\Big{)}\Big{]}\\ +(m+M)K_{d}(L+\delta_{1})\end{array}\right]/\big{[}m(m+M)\omega_{1}^{2}\big{]} \\ D_{2}=-\left[\begin{array}{c}(m+M)F_{1}+mM\omega_{2}\Big{[}-\omega_{2}\Big{(} x_{f1}-X_{f1}\Big{)}+\Big{(}v_{f1}-V_{f1}\Big{)}\Big{]}\\ +(m+M)K_{d}(L+\delta_{1})\end{array}\right]\end{array} \right]/\big{[}m(m+M) \omega_{1}^{2}]\end{array} \tag{23}\]  
_Tether status check at stage II_  
The tether failure can be checked according to the work-energy theorem. \[ \begin{array}{ll}-W&=(Total\;\;Area)=A_{126}+A_{2346}-A_{345}\\ &=\frac{1}{2}F_{1}\delta_{1}+\int_{\delta_{1}}^{\delta^{*}}[F_{1}-K_{d}( \delta-\delta_{1})]d\delta-\int_{d}^{\delta^{*}}[F^{*}+K_{1}(\delta-\delta^{* })]d\delta\end{array}  \tag{24}\]  
where \(\delta^{*}=x_{2}(t^{*})-X_{2}(t^{*})-L,F^{*}=F_{1}-K_{d}(\delta^{*}-\delta_{1} ),andd=\frac{F^{*}}{K_{1}}+\delta^{*}\)  
The kinetic energy variation can be expressed as:  
\[\Delta T=T^{*}-T_{0}=\frac{1}{2}m\Big{(}{v_{*}}^{2}+{V_{*}}^{2}\Big{)}-\frac{ 1}{2}m\Big{(}{v_{0}}^{2}+{V_{0}}^{2}\Big{)} \tag{25}\]  
In cases in which the kinetic energy variation of the moving flying object (m) is higher than the amount of work produced by the external force and elongation (\(\delta_{2}\)), all the main elements will fail. _Response at the second stage (II) when the failure of primary elements occurs_  
The corresponding response at the second stage when some of the primary elements fail is as follows:  
\[ \begin{array}{ll}x_{3}(t)=q_{m}(t)+&q_{M}(t)\\ &=A_{3}\Big{(}t-\tau_{f2}\Big{)}+B_{3}+C_{3}cos\omega_{3}\Big{(}t-\tau_{f2} \Big{)}+D_{3}sin\omega_{3}\Big{(}t-\tau_{f2}\Big{)}\\ &+\frac{1}{mv_{3}^{2}}[-F^{*}+K_{1}(\delta^{*}+L)]\\ X_{3}(t)=q_{m}(t)-&\frac{m}{M}q_{M}(t)\\ &=A_{3}\Big{(}t-\tau_{f2}\Big{)}+B_{3}-\frac{m}{M}C_{3}cos\omega_{3}\Big{(}t- \tau_{f2}\Big{)}-\frac{m}{M}D_{3}sin\omega_{3}(t\\ &-\tau_{f2}\Big{)}-\frac{m}{M}\frac{1}{mv_{3}^{2}}[-F^{*}+K_{1}(\delta^{*}+L)] \end{array}  \tag{26}\]  
where \({\omega_{3}}^{2}=\frac{(m+M)K_{1}}{mM}\). By applying the initial conditions \(x_{3}\Big{(}\tau_{f2}\Big{)}=x_{f2},\dot{x}_{3}\Big{(}\tau_{f2}\Big{)}=v_{f2}, X_{3}\Big{(}\tau_{f2}\Big{)}=X_{f2},\dot{X}_{3}\Big{(}\tau_{f2}\Big{)}=V_{f2}\) to Equation (26), we can obtain the coefficients:  
\[