# Dynamic Analysis of Tethered Satellites with a Payload Moving Along a Flexible Tether
## 4 Dynamic Response Analysis
The direction of payload movement is clockwise when transferring from satellite A to satellite B, and counterclockwise when transferring from satellite B to satellite A, relative to the straight line connecting the two satellites. This phenomenon is attributed to the Coriolis effect. As the payload speed increased, the tether underwent greater deformation (Figures 7c-f and 8c-f). Finally, an analysis was conducted on the axial force exerted on the tether as the payload moved between satellites A and B. In a tethered satellite system where the two satellites were connected by a tether, differing orbital radii resulted in variations in gravitational and centrifugal forces. Consequently, the tether typically experienced a positive axial force; that is, a tensile force. As the payload moved along the tether, the Coriolis force, as previously described, acted on the payload, causing the lateral deformation of the tether. Therefore, the axial force was generated on the tether due to the combined effects  
Figure 8: Attitudes of tethered satellites for different payload speeds as the payload moves from satellite A to satellite B: (**a**) 10 m/s, (**b**) 20 m/s, (**c**) 30 m/s, (**d**) 40 m/s, (**e**) 50 m/s, and (**f**) 60 m/s. The green circle, red square, and blue triangle represent satellite A, satellite B, and the payload, respectively. Figure 9: Attitudes of tethered satellites for different payload speeds as the payload moves from satellite B to satellite A: (**a**) 10 m/s, (**b**) 20 m/s, (**c**) 30 m/s, (**d**) 40 m/s, (**e**) 50 m/s, and (**f**) 60 m/s. The green circle, red square, and blue triangle represent satellite A, satellite B, and the payload, respectively. of the gravitational, centrifugal, and Coriolis forces. This axial force can be computed by multiplying the Lagrange strain, expressed by Equation (6), by Young's modulus \(E\) and the cross-sectional area \(A\). The changes in the axial force along the tether length were investigated as the payload, with a mass of 100 kg and speed of 20 m/s, moved between the two satellites at various attitudes. Figures 10 and 11 illustrate the axial forces as functions of tether length when the payload travels from satellites A to B and B to A, respectively. In these figures, a position value of 0 km represents the location of satellite A, whereas a position  
Figure 10 shows the axial forces along the tether length for the 10 attitudes depicted in Figure 6b as the payload moves from satellite A to satellite B. At attitude 1 in Figure 6b, where the payload was located at satellite A, a smooth and positive axial force is observed throughout the entire length of the tether, as shown in Figure 10a. This indicates that the entire tether was under tension. At attitude 2 in Figure 6b, where the payload has moved slightly towards satellite B, fluctuations in the axial force were observed near the payload position (approximately 2 km), as shown in Figure 10b. As the payload moved further towards satellite B, the magnitude of these fluctuations increased, causing the axial force to alternate between positive (tensile) and negative (compressive) forces (Figure 10c-j). Sections of the tether with a negative axial force, that is, a compressive force, were compressed axially, which can result in slack in the tether owing to lateral deformation. Another notable observation from Figure 10 is that the amplitude of the  
Figure 11: Axial forces in the tether for various attitudes shown in Figure 7b: (**a**) attitude 1, (**b**) attitude 2, (**c**) attitude 3, (**d**) attitude 4, (**e**) attitude 5, (**f**) attitude 6, (**g**) attitude 7, (**h**) attitude 8, (**i**) attitude 9, and (**j**) attitude 10.  
axial force fluctuations increased as the payload moved from satellite A to satellite B. The maximum axial force observed in Figure 10 is approximately 400 N.  
Figure 11 shows the axial forces along the tether length for the 10 attitudes depicted in Figure 7b as the payload moved from satellite B to satellite A.