## 4 Numerical Modeling  
Let us conduct numerical modeling and consider the small spacecraft model, the scheme of which is shown in Figure 1, and the main characteristics are given in Table 1[3]. This allows us to compare the simulation results with the symmetric formulation in [3]. If the initial deflection is represented by the expression in Table 1, the results shown in Figure 2 are obtained. \(\mathrm{u}_{z0}\) in Figure 2 refers to the maximum deviation of the free edge of the flexible element at the initial deflection. More significant values of \(u_{z0}\) than shown in Figure 2 can cause a stability loss of the flexible element [34; 53] and have not been considered in this work. The analysis of Figure 2 shows its fundamental similarity with the results obtained in [3] without taking into account the initial deflection (Figure 9 in [3]). In a wider range of initial deflections, the dependence of the inertia force on the initial deflection would probably be more noticeable. However, in this case, there is a stability loss of the flexible element. Therefore, this comparison is incorrect. It should also be noted that with the linear dependence of \(u_{z0}\) on \(x\), the pattern described in [3] is observed because the dependence of the inertia force on the initial deflection is viewed if the second partial derivative of \(u_{z0}\) with respect to \(x\) is different from zero. Let us consider another spacecraft, the simulation of which was performed in [34]. However, in [34], a scheme with one flexible element was considered. The design of this spacecraft provides four flexible elements (Figure 3) [3]. The main characteristics of the simulated spacecraft are given in Table 2[34]. This allows us to compare the simulation results with the results presented in [34]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Designation** & **Value** & **Dimension** \\ \hline Spacecraft prototype & – & Aist–2D [3] & – \\ Solar panel frame material & – & MA2 & – \\ Mass of the spacecraft & \(m\) & 530 & kg \\ Coefficient of thermal & \(\lambda\) & 96.3 & \(W/(\mathrm{m}\times\mathrm{K})\) \\ conductivity & \(\sigma\) & \(5.67\times 10^{-8}\) & \(W/(m^{2}\times\mathrm{K}^{4})\) \\ Stefan–Boltzmann constant & \(\sigma\) & \(2.6\times 10^{-6}\) & \(\mathrm{K}^{-1}\) \\ Linear expansion coefficient & \(\alpha\) & \(2.6\times 10^{-6}\) & \(\mathrm{K}^{-1}\) \\ External heat flux & \(Q_{0}\) & 1400 & \(W/m^{2}\) \\ Vacuum temperature & \(T_{\mathrm{C}}\) & 3 & \(\mathrm{K}\) \\ Initial temperature of the & \(T(x,\ y,\ z,\ 0)\) & 200 & \(\mathrm{K}\) \\ solar panel frame & & \(c\) & 1130.4 & \(J/(kg\times K)\) \\ Specific heat & \(\rho\) & 1780 & \(kg/m^{3}\) \\ Density & \(\rho\) & \(1780\) & \(kg/m^{3}\) \\ Lame coefficient \(\Lambda\) & \(\Lambda\) & \(3\times 10^{10}\) & \(\mathrm{Pa}\) \\ Lame coefficient \(\
u\) & \(\
u\) & \(1.6\times 10^{10}\) & \(\mathrm{Pa}\) \\ Number of plate layers & \(N\) & 3 & – \\ Layer thickness & \(\Delta z\) & 2 & mm \\ Solar panel length & \(l\) & 2.5 & \(\mathrm{m}\) \\ Poisson’s coefficient & \(\mu\) & 0.3 & – \\ Number of flexible elements & \(k\) & 2 & – \\  & \(a\) & 1 & \(\mathrm{s}\) \\ Model parameters & \(A\) & \(10^{-4}\) & \(\mathrm{m}^{-3}\) \\  & \(C\) & 200 & \(\mathrm{K}/\mathrm{m}\) \\  & \(M\) & 3 & \(\mathrm{K}/\mathrm{m}\) \\ Initial deflection & \(u_{z0}\) & \(u_{z0\times\ 2}/l^{2}\) & \(\mathrm{m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values of the small spacecraft used in numerical simulation [3]. Figure 3: The scheme of the “Vozvrat-MKA” spacecraft [3].