## 3 Simulation Results and Discussion
### Dynamic Behavior of TB-TSS with Fixed-Length Tethers
As shown in Equations (19) and (20), tension is related to centripetal acceleration, which is relevant to \(\alpha_{i}^{\prime 2}+\cos^{2}(\alpha_{i})(\omega+\theta_{i}^{\prime})^{2}=( \omega+\theta_{i}^{\prime})^{2}\) when \(a_{i}^{\prime}=0\). Consequently, tether tension decreases with decreasing roll angular velocity when it corresponds to the orbital angular velocity. When the roll angular velocity is opposite the orbital angular velocity, the tension remains small as the two angular velocities almost cancel each other out. In case 2, the initial attitude angles were set as \(\theta_{1}(0)=\theta_{2}(0)=\alpha_{1}(0)=\alpha_{2}(0)=20^{\circ}\), indicating that the initial in-plane and out-of-plane angles were equal and the two tethers were collinear. Figure 6a shows the dynamic response of in-plane libration angles \(\theta_{1}\) and \(\theta_{2}\), and it can be seen that the two curves oscillate periodically and overlap. The out-of-plane pitch motions are illustrated in Figure 6b, where it can be seen that the motions of pitch angles \(\alpha_{1}\) and \(\alpha_{2}\) are identical. The tension in this case is depicted in Figure 6c, where it can be seen that tension T\({}_{1}\) is larger than tension T\({}_{2}\) and the two curves exhibit quasiperiodic variation. The amplitude of tension in Figure 6c is almost equal to the maximum value of tension in Figure 5c, despite the addition of out-of-plane pitch motion. Figure 7 presents the dynamic characteristics of the TB-TSS with fixed-length tethers in case 3. In this simulation, the initial attitude angles were set as \(\theta_{1}(0)=\alpha_{1}(0)=20^{\circ}\), \(\theta_{2}(0)=\alpha_{2}(0)=-20^{\circ}\). As seen in Figure 7a,b, the oscillations of libration angles do not exhibit a purely cosine variation; instead, they show quasiperiodic oscillation. Unlike in Figures 5 and 6, here the oscillations of the two in-plane and two out-of-plane angles do not overlap and do not have higher frequencies. In this case, the oscillation amplitude of \(\theta_{1}\) and \(\theta_{2}\) is larger than the initial attitude angles, while the oscillation amplitude of \(\alpha_{1}\) and \(\alpha_{2}\) never exceeds \(20^{\circ}\). As shown in Figure 7a, the maximum value of \(\theta_{1}-\theta_{2}\) is nearly \(60^{\circ}\), which is greater than the initial difference of \(40^{\circ}\). This indicates that the initial attitude angles may increase the response amplitude of roll angles. The maximum value of roll angles is nearly \(30^{\circ}\), and the increase is nearly half of the initial roll angles. This also suggests that libration energy can be delivered from out-of-plane to in-plane. Figure 7c shows that tether tension T\({}_{1}\) and T\({}_{2}\) almost overlap each other, although the amplitude of T\({}_{1}\) is slightly larger than T\({}_{2}\). Moreover, the peak value of each tether tension occurs when the libration angle changes from zero to the maximum (or minimum) value. Comparing  
with Figures 5c and 6c, it can be observed that, in this case, the peak values of tether tension are significantly higher than in the previous two cases. This indicates a close relationship between peak tether tension and initial libration angle. Figure 8 displays the libration responses and tether tensions in case 4, where the initial attitude angles were determined as \(\theta_{1}=\theta_{2}=0\), \(\alpha_{1}=\alpha_{2}=20^{\circ}\). As shown in Figure 8a,b, the in-plane and out-of-plane libration motion overlap each other. Moreover, the out-of-plane motion oscillates sinusoidally, while the in-plane motion exhibits random oscillation. A comparison of Figure 8 with Figure 5 leads to a definitive conclusion: libration energy can be transferred from out-of-plane motion to in-plane motion, but not vice versa. In case 4, the amplitudes of tether tension in Figure 8c are smaller than those in the previous cases. This phenomenon occurs because the dominant centripetal acceleration is located in the orbit plane.