## 4 Theoretical Model of the Space Flexible Web Expansion Dynamics
### Theoretical Model of Traction Expansion Dynamics of Space Flexible Web
#### 4.2.2 Theoretical Model of Traction Expansion Dynamics  
Compared with rigid systems, a new space flexible web is a typical multi-body system with significant motion and large deformation coupling. Establishing an accurate mathematical analysis equation is complex, and its relatively reasonable dynamic analysis model can be built according to the following discrete ideas. Disperse the web rope into finite segments and then the mass of each new spatial flexible strip element is concentrated at the two endpoints, that is, the nodes of each rope segment. Because the cable is very soft, it can only bear the tension along the cable direction and cannot withstand the pressure. Therefore, it can be assumed that the new type of flexible cable node is connected by a virtual \"spring,\" which has the force characteristics of being pulled without pressure. Based on considering the damping effect of the new type of space flexible web, the new type of space flexible mesh segment unit can be treated as a \"concentrated mass damping spring model\" [17], as shown in Figure 6a below:  
In the dynamic model shown in Figure 5 above, the nth new space flexible web segment element is equivalent to a spring damping element whose mass is concentrated at the endpoint, where \(m\leq 90\) (a total of 90 rope segment elements), and node n connects node n and node (\(m+1\)) new flexible space bar segment units. Assuming the spring stiffness \(K_{m}\) and damping coefficient \(\zeta_{m}\) in the spring damping element, the coordinate components of the node of the nth new space flexible web rope segment are \(x_{m},y_{m},z_{m}\), as shown in Figure 6b above. Figure 7 below shows that the conjoined coordinate system of the nth space flexible web rope segment is \(Ox_{m}y_{m}z_{m}\), and the \(x\)-, \(y\)-, and \(z\)-axis of the coordinate system corresponds to the primary, secondary, and tangent vectors of the rope curve, respectively. The inertial coordinate system \(Ox_{x}y_{x}z_{a}\) and the conjoined coordinate system can be determined by three Euler angles: \(\alpha_{m}\), \(\beta_{m}\), and \(\chi_{m}\). The inertial coordinate system can be rotated three times to match the conjoined coordinate system. First, go around the axis \(Ox_{a}\) to the angle \(\beta_{m}\), from \(Ox_{a}y_{a}z_{a}\) to \(OMy_{1}z_{a}\); then, turn the \(\alpha_{m}\) corner around ON and go from \(OMy_{1}z_{a}\) to \(OMy_{2}z_{m}\). Finally, round the \(Ox_{m}\) axis to the \(\chi_{m}\) angle, from \(OMy_{2}z_{m}\) to \(Ox_{m}y_{m}z_{m}\). The transformation matrix of the connective coordinate \(Ox_{m}y_{m}z_{m}\) and inertial coordinate system \(Ox_{a}y_{a}z_{a}\) of the space flexible web rope segment is shown in the following Formula (7):  
\[B_{n}=\begin{bmatrix}\cos\chi_{m}\cos\beta_{m}-\sin\beta_{m}\sin\chi_{m}\cos \alpha_{m}&\sin\beta_{m}\cos\chi_{m}+\sin\chi_{m}\cos\beta_{m}\cos\alpha_{m} &\sin\alpha_{m}\sin\chi_{m}\\ -\cos\beta_{m}\sin\chi_{m}-\sin\beta_{m}\cos\alpha_{m}\cos\chi_{m}&-\sin\beta _{m}\sin\chi_{m}+\cos\alpha_{m}\cos\beta_{m}\cos\chi_Because the web rope is soft, it can not bear ordinary bending moments and shearing force but can only bear tangential pulling force. Therefore, ignore Euler Angle \(\chi_{m}\), or  
Figure 6: Diagram of rope segment units. Figure 7: New spatial flexible web rope segment conversion coordinate system. assume \(\chi_{m}=0\) and simplify the transformation matrix between the two coordinate systems as follows:  
\[A_{m}=\left[ \begin{array}{ccc}\cos\beta_{m}&\sin\beta_{m}&0\\ -\cos\alpha_{m}\sin\beta_{m}&\cos\alpha_{m}\cos\beta_{m}&\sin\alpha_{m}\\ \sin\alpha_{m}\sin\beta_{m}&-\sin\alpha_{m}\cos\beta_{m}&\cos\alpha_{m}\\ \end{array} \right] \tag{8}\]  
In order to understand the whole calculation process more clearly, the variables and symbols involved in the calculation are summarized as shown in Table 1 below.