## 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
Calculation of the Internal Force of Space Flexible Web Rope Segments  
In general, because it is difficult to obtain the dynamic stress-strain relationship of the web rope, the static stress-strain relationship curve of the web rope is often used in engineering. Using segmented linearization, the constitutive relationship of the web rope is simplified into the sum of linear elasticity and linear velocity damping. That is, the tensile force of the m space flexible web rope segment is obtained by the following formula:  
\[H_{m}=\left\{ \begin{array}{cc}\mu(\dot{\delta}_{m})+p_{m}(\delta_{m})&\delta _{m}>0\\ 0&\delta_{m}\leq 0\\ \end{array} \right. \tag{9}\]  
In the above formula:  
\(p_{m}(\delta_{m})\)--Linear tensile function of the webbing rope in section m;  
\(\mu\)--Tension damping coefficient of rope segment of webbing unit;  
strain \(\varepsilon_{m}\) is obtained by the following formula:  
\[\varepsilon_{m}=\frac{L_{m}-L_{m,0}}{L_{m,0}} \tag{10}\]  
\(L_{m,0}\)--Original length of web rope; \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols (Variables)** & **Meaning** \\ \hline \(K_{m}\) & Spring stiffness of spring damping element \\ \hline \(\zeta_{m}\) & Damping coefficient of spring damping element \\ \hline \(Ox_{m}y_{m}z_{m}\) & The jointed coordinate system of the flexible rope segment in the first space \\ \hline \(Ox_{x}y_{x}z_{a}\) & Inertial coordinate system of the flexible rope segment in space \\ \hline \(\alpha_{m},\ \beta_{m},\ \chi_{m}\) & Three independent angular parameters representing the position of a fixed point rotating rigid body are the rotation Angle, nutation Angle and precession Angle. \\ \hline \(p_{m}(\delta_{m})\) & Linear tension function of the nth segment web rope \\ \hline \(\mu\) & Tension damping coefficient of web rope \\ \hline \(\varepsilon_{m}\) & Strain of web rope \\ \hline \(L_{m,0}\) & The original length of the web rope \\ \hline \(L_{m}\) & Length of web rope after deformation \\ \hline \(\dot{\varepsilon}_{m}\) & Strain differentiation of net rope \\ \hline \(g_{m}\) & Mass of flexible web rope segment \\ \hline \(G_{b,n}\) & Normal mass matrix of rope segment \\ \hline \(F_{b,m}\) & Normal fluid resistance of rope segment \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Explanation of symbols and variable meanings. \(L_{m}\)--The length of the deformation is as follows:  
\[L_{m}=\sqrt{\left(x_{m}-x_{m-1}\right)^{2}+\left(y_{m}-y_{m-1}\right)^{2}+\left(z _{m}-z_{m-1}\right)^{2}} \tag{11}\]  
The strain differential of the web cord unit can be obtained by differentiating the above:  
\[\dot{e}_{m}=\frac{(x_{m}-x_{m-1})(\dot{x}_{m}-\dot{x}_{m-1})+\left(y_{m}-y_{m-1 }\right)(\dot{y}_{m}-\dot{y}_{m-1})+(z_{m}-z_{m-1})(\dot{z}_{m}-\dot{z}_{m-1})} {L_{m,0}L_{m}} \tag{12}\]  
Dynamical equation of spatial flexible web expansion system  
If the mass of the space flexible rope segment is recorded as \(g_{m}\), the mass matrix of the m space flexible rope segment is shown in the conjoined coordinate system as follows:  
\[G_{b.m}= \begin{bmatrix}g_{m}+g_{a.m}&0&0\\ 0&g_{m}+g_{a.m}&0\end{bmatrix}  \tag{13}\]  
The mass of the nth space flexible rope segment and \((m+1)\) space flexible rope segment were divided in half and concentrated in the m space elastic rope segment by engineering approximation.