## 3 Simulation Results and Discussion
### Dynamic Behavior of TB-TSS with Fixed-Length Tethers  
In this section, we consider a fully deployed two-piece dumbbell TB-TSS model with a constant tether length. The libration characteristics of the TB-TSS are investigated by taking four degrees of freedom. Specifically, dynamic Equations (15)-(18) describe the nonlinear model of the TB-TSS with fixed-length tethers, while Equations (19) and (20) are utilized to solve the tether tension. The numerical simulation of the libration characteristics of the TB-TSS with fixed-length tethers was performed using the MATLAB ODE45 solver. The physical parameters of the system were the same as those in [22]. The orbital altitude of the main satellite was \(r=7100\) km. The mass of the three satellites was \(m_{0}=10^{4}\) kg, \(m_{1}=100\) kg, and \(m_{2}=500\) kg. The two fixed-length tethers were set at \(l_{1}=l_{2}=10\) km. To investigate the dynamic behavior of this system, five cases of initial attitude were set, and the relevant initial angles and their specific values are listed in Table 1. The initial attitude angles were set to ignore the initial out-of-plane pitch angles, as in case 1, and gradually change to only the initial pitch angles. Cases 5 and 4 were set as the controls to study the effect of initial attitude on system dynamic characteristics. In order to avoid the impact of the initial angular velocity on the system characteristics, other initial variables were given as zero: \(\theta_{1}^{\prime}(0)=\theta_{2}^{\prime}(0)=\alpha_{1}^{\prime}(0)=\alpha_{2 }^{\prime}(0)=0\). Figure 5 shows the dynamic responses of the TB-TSS with fixed length following initial attitude angles \(\theta_{1}(0)=\theta_{2}(0)=20^{\circ}\), \(\alpha_{1}(0)=\alpha_{2}(0)=0^{\circ}\). In this case, the initial in-plane angles are equal to \(20^{\circ}\) and the initial out-of-plane angles are zero, indicating the absence of initial disturbance in the out-of-plane direction. In Figure 5a, the lines for \(\theta_{1}\) and \(\theta_{2}\) overlap, resembling a cosine function graph, while the line for \(\theta_{1}-\theta_{2}\) is zero. Meanwhile, Figure 5b shows three zero lines representing libration angles \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{1}-\alpha_{2}\), indicating that the initial in-plane angles do not affect the out-of-plane motion when the TB-TSS is fully deployed. Additionally, Figure 5c shows the tether tension of the two tethers, with slightly higher tension for L\({}_{1}\) than L\({}_{2}\) due to the gravitational effect of subsatellite 2. Furthermore, by comparing Figure 5a,c, it is observed that the period of in-plane roll motion \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**n** & **\(\theta_{1}\)(0)** (\({}^{\circ}\)) & **\(\theta_{2}\)(0)** (\({}^{\circ}\)) & **\(\alpha_{1}\)(0)** (\({}^{\circ}\)) & **\(\alpha_{2}\)(0)** (\({}^{\circ}\)) \\ \hline Case 1 & 20 & 20 & 0 & 0 \\ Case 2 & 20 & 20 & 20 & 20 \\ Case 3 & 20 & \(-\)20 & 20 & \(-\)20 \\ Case 4 & 0 & 0 & 20 & 20 \\ Case 5 & 0 & 0 & 20 & \(-\)20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Five test cases for initial attitude angles and their specific values. Figure 4: Comparison of libration motion and tether tension when the tether deploying velocity is 1 m/s: **(a)** libration motion in present study (squares) and [22] (solid line); **(b)** tether tension in present study (dotted line) and [22] (solid line). is equal to the period of tension change. The maximum tension occurs when the in-plane libration angles pass through zero, implying that the two tethers are aligned along the local vertical. Combining the trends from both figures, it is evident that the tension changes rapidly from peak to bottom as the libration angle changes from zero to its maximum value, while tension exhibits only slight variation as the libration angle changes from the maximum to the minimum point.