## 4 Theoretical Model of the Space Flexible Web Expansion Dynamics
### Geometric Nonlinear Characteristics of the Flexible Web  
Because of the particular \"tensile and unpressurized\" mechanical characteristics of the flexible web, the mechanical response of the wire rope is geometrically nonlinear even if the deformation is small under linear elasticity. In the simple flexible rope system shown in Figure 4 below, the two sides of the rope are fixed at points A and B, the distance between them is \(L\), the stiffness of the flexible rope is EA, and the length of the rope is \(L_{0}\), the initial strain is \(\varepsilon_{0}\), the initial internal tension is \(\sigma_{0}\), and the vertical displacement of the rope is 1 at the midpoint (\(M\)-point) under the action of external force \(F_{0}\), and the degree of elongation of the rope is \(\Delta L\). At this point, the strain increases to \(\varepsilon\), the stress increases to \(\sigma\), and the M-point moves to \(M_{1}\).  
As shown in diagram 3, under the action of external force \(F_{0}\), the rope strain \(\varepsilon\) is solved as follows:  
\[\begin{array}{ll}\varepsilon=&\frac{\Delta L}{L_{0}}=\frac{2\left[\frac{1}{L _{0}\times(1+\varepsilon_{0})}-\frac{1}{2}L_{0}\right]}{\cos\theta_{A}-\frac{ 1}{2}L_{0}}\\ &=\frac{L_{0}(1+\varepsilon_{0})\sec L_{0}-L_{0}}{L_{0}}\\ &=(1+\varepsilon_{0})\sec\theta_{A}-1\\ &=(1+\varepsilon_{0})\sqrt{\frac{4\pi^{2}}{L^{2}}+1}-1\end{array} \tag{1}\]  
Figure 4: Simple flexible rope system diagram.  
Under the assumption of small elastic deformation on linear elasticity, the following relationships can be used to approximate:  
\[\sin\theta_{A}\approx\frac{l}{\frac{1}{2}L}=\frac{2l}{L} \tag{2}\]  
\[\begin{array}{c}\varepsilon=(1+\varepsilon_{0})\sqrt{\frac{(2l)^{2}}{L^{2}}+1 }-1\\ \lim\limits_{\frac{\alpha}{L}\to 0}=(1+\varepsilon_{0})\left(\frac{2l^{22}}{L^{2}}+1 \right)-1\end{array} \tag{3}\]  
From the static equilibrium conditions at the midpoint \(M_{1}\) of the rope:  
\[F_{0}=2\sigma\sin\theta_{A} \tag{4}\]  
The constitutive model of the simple rope system shown in Figure 3 is available under the linear elasticity hypothesis:  
\[\sigma=EA\varepsilon\;\varepsilon_{0}=\frac{\sigma_{0}}{EA} \tag{5}\]  
The force control equations of simple flexible rope systems (6) can be obtained by combining Formulas (3)-(5):  
\[F_{0}=(EA+\sigma_{0})\frac{8}{L^{3}}l^{3}+\frac{4\sigma_{0}}{L}l \tag{6}\]  
As can be seen from Formula (6), even under the assumption of linear elasticity with small deformation, there is no linear relationship between the external force and displacement of a simple flexible rope. Refer to the flexible rope system in Figure 3, assuming EA = 100,000 N for the stiffness of the rope and \(L_{0}\) = 2.5 m for the original length of the rope, the relationship between external force and vertical displacement at the midpoint of the web rope is shown in Figure 5 below when the initial internal tension of the rope is 0 N, 200 N, 400 N, 600 N, 800 N, and 1000 N, respectively.  
Figure 5: Relationship between external force of flexible rope and vertical displacement of midpoint of rope.  
Analyzing Figure 4 above, when the initial internal tension \(\sigma_{0}\) of the rope segment is larger, the curve changes faster, and the slope is more extensive, which means the flexible rope is better able to resist external forces. The above examples show that the dynamic problem of webbing rope is different from that of linear system analysis and has prominent nonlinear dynamic characteristics, especially the flexible webbing system in the following chapter.