## 3 Application of a Tailored Tether to AWES
### Approximate Response Modeling of Simplified AWES  
In the previous section, the computation was executed whenever any elements failed, and the failure time and criteria were calculated at each step. For a very long tether, however, this requires expensive computational efforts. This section describes the approximate modeling of an AWES following the hypothesis that the mechanical work of an approximate response model should be close to one of the original models, as shown in Figure 5.  
For an approximate response model, the parameters at each point, such as stiffness, force, and elongation, should be obtained. The stiffnesses of the tether at the first stage (I) and the last stage (III) are given by:  
\[K_{1}=\frac{1}{\frac{1}{k_{\text{cls}}}+\frac{1}{k_{s}}+\frac{1}{k_{\text{sys} }}+\frac{n-1}{k_{p}}+\frac{n-2}{k_{\text{s}rp}}+\frac{1}{k_{\text{s}rp}}},\ \ \ K_{3}=\frac{1}{\frac{2}{k_{\text{cls}}}+\frac{n}{k_{s}}+\frac{n-1}{k_{\text{ c}rp}}} \tag{13}\]  
Fromthe approximate force-displacement relationship, the corresponding load values are expressed as:  
\[F_{1}=K_{1}\delta_{1},\ F_{2}=K_{2}\big{(}\delta_{2}-(n+1-1)\big{(}l_{s}-l_{p} \big{)}\big{)},\ \ F_{3}=K_{3}\big{(}\delta_{3}-(n+1-1)\big{(}l_{s}-l_{p}\big{)}\big{)} \tag{14}\]  
Next, for the numerical computation, the responses are considered in detail for each stage. Response at stage I  
The equations of motion can be obtained according to the Euler-Lagrange equation \(\frac{\partial}{\partial t}\Big{(}\frac{\partial T}{\partial l_{i}}\Big{)}+ \frac{\partial V}{\partial q_{i}}=Q_{i}\). Equation (15) expresses the kinetic and potential energy and the final equations are expressed in Equation (16) in matrix form. \[T=\frac{1}{2}m\dot{x}_{1}^{2}+\frac{1}{2}M\dot{X}_{1}^{2} \tag{15}\] \[V=\frac{1}{2}K_{1}\delta^{2}=\frac{1}{2}K_{1}(\dot{x}_{1}-X_{1}- L)^{2} \tag{16}\]  
Next, the forced vibration response of the tether at the first stage can be obtained as follows. \[x_{1}(t)=q_{m}(t)+q_{M}(t)=A_{1}(t-\tau_{0})+B_{1}+C_{1}cos\omega_{1}(t-\tau_{ 0})+D_{1}sin\omega_{1}(t-\tau_{0})+\frac{1}{m}\frac{K_{1}L}{\omega_{1}^{2}} \tag{17}\] \[X_{1}(t)=q_{m}(t)-\frac{m}{M}q_{M}(t)=A_{1}(t-\tau_{0})+B_{1}- \frac{m}{M}C_{1}cos\omega_{1}(t-\tau_{0})-\frac{m}{M}D_{1}sin\omega_{1}(t-\tau _{0})-\frac{1}{M}\frac{K_{1}L}{\omega_{1}^{2}}\]  
Figure 5: Approximation technique for tether response of an AWES. where \({\omega_{1}}^{2}=\frac{K_{1}}{m}\big{(}1+\frac{m}{M}\big{)}\). By applying the initial conditions \(x(\tau_{0})=x_{0},\ \dot{x}(\tau_{0})=v_{0},X(\tau_{0})=X_{0}\), \(\dot{X}(\tau_{0})=V_{0}\) to Equation (17), we can obtain the coefficients. \[A_{1}=\frac{mv_{0}+MV_{0}}{m+M},\ B_{1}=\frac{mx_{0}+MX_{0}}{m+M},\ C_{1}=- \frac{K_{1}L(m+M)-mM\omega^{2}(x_{0}-X_{0})}{m(m+M)\omega_{1}^{2}},\ \ D_{1}=\frac{M(v_{0}-V_{0})}{(m+M)\omega_{1}} \tag{18}\]  
_Response at stage II_  
The tether response at stage II is shown in Figure 6.  
In Figure 6,  
\[K_{1}=\frac{1}{\frac{1}{k_{\text{cls}}}+\frac{1}{k_{s}}+\frac{1}{k_{\text{sys} }}+\frac{n-1}{k_{p}}+\frac{n-2}{k_{\text{s}rp}}+\frac{1}{k_{\text{s}rp}}},\ \ \ K_{3}=\frac{1}{\frac{2}{k_{\text{cls}}}+\frac{n}{k_{s}}+\frac{n-1}{k_{\text{ c}rp}}} \tag{13}\]0  
Since the response of the second stage (II) is inelastic, the virtual work resulting from nonconservative force needs to be plugged into the Lagrange equation. \[ \begin{array}{ll}\delta W&=F_{\text{ext}}\overset{\rightarrow}{i}\times \delta(\Delta)=-[F_{1}-K_{d}(\Delta-\delta_{1})]\overset{\rightarrow}{i}\times \delta(\Delta)=-[F_{1}-K_{d}(x-X-L-\delta_{1})]\overset{\rightarrow}{i}\times (\delta x_{2}-\delta X_{2})\overset{\rightarrow}{i}\\ &=-[F_{1}-K_{d}(x_{2}-X_{2}-L-\delta_{1})]\delta x_{2}+[F_{1}-K_{d}(x_{2}-X_{ 2}-L-\delta_{1})]\delta X_{2}=Q_{1}\delta x_{2}+Q_{2}\delta X_{2}\end{array}  \tag{20}\]  
Based on the rule of [PERSON]'s equation, \(\frac{\partial T}{\partial t}\left(\frac{\partial T}{\partial\dot{q}_{i}} \right)+\frac{\partial V}{\partial\dot{q}_{i}}=Q_{i}\), the equations of motion are given by:  
\[\left[ \begin{array}{cc}m&0\\ 0&M\end{array} \right]\left\{