## 3 Non-Bracket Construction Method
### Analysis Method
#### 3.1.1 Nonlinear Dynamic Equilibrium Iteration Dynamic Balance Equation  
The dynamic balance equation (Equation (1)) adopts the Rayleigh damping matrix (Equation (2)), in which the natural vibration circle frequency and damping ratio can be set virtually:  
\[[\mathrm{M}]\{\dot{\mathrm{U}}\}+[\mathrm{C}]\{\dot{\mathrm{U}}\}+[\mathrm{K}] \{\mathrm{U}\}=\{\mathrm{F}_{(\mathrm{i})}\} \tag{1}\]  
\[[\mathrm{C}]=\alpha[\mathrm{M}]+\beta[\mathrm{K}] \tag{2}\]  
\[\alpha=\frac{2\omega_{\mathrm{i}}\omega_{\mathrm{j}}(\xi_{\mathrm{i}}\omega_{ \mathrm{j}}-\xi_{\mathrm{j}}\omega_{\mathrm{i}})}{\omega_{\mathrm{j}}^{2}- \omega_{\mathrm{i}}^{2}} \tag{3}\]  
\[\beta=\frac{2(\xi_{\mathrm{j}}\omega_{\mathrm{j}}-\xi_{\mathrm{i}}\omega_{ \mathrm{i}})}{\omega_{\mathrm{j}}^{2}-\omega_{\mathrm{i}}^{2}} \tag{4}\]  
where \(\{\mathrm{U}\}\), \(\{\dot{\mathrm{U}}\}\), and \(\{\dot{\mathrm{U}}\}\) are deformation, velocity, and acceleration vectors, respectively; \(\{\mathrm{F}(\mathrm{t})\}\) is the load time history vector; \([\mathrm{C}]\) is the Rayleigh damping matrix; \([\mathrm{M}]\) is the quality matrix; \([\mathrm{K}]\) is the stiffness matrix; \(\alpha\) and \(\beta\) are the Rayleigh damping coefficients; \(\omega_{\mathrm{i}}\) and \(\omega_{\mathrm{i}}\) are the \(i\)th and \(j\)th natural vibration circle frequencies, respectively; and \(\xi_{\mathrm{i}}\) and \(\xi_{\mathrm{j}}\) are the damping ratios corresponding to \(\omega_{\mathrm{i}}\) and \(\omega_{\mathrm{j}}\), respectively. If \(\xi_{\mathrm{i}}=\xi_{\mathrm{j}}=\xi\), Equations (3) and (4) can be simplified to Equations (5) and (6):  
\[\alpha=\frac{2\omega_{\mathrm{i}}\omega_{\mathrm{j}}\xi_{\mathrm{i}}}{\omega_ {\mathrm{j}}+\omega_{\mathrm{i}}} \tag{5}\]  
\[\beta=\frac{2\xi}{\omega_{\mathrm{j}}+\omega_{\mathrm{i}}} \tag{6}\]  
Time step \(\Delta\mathrm{T}_{\mathrm{s}}\) is one of the key factors determining the convergence speed of NDFEM form finding analysis. The shorter \(\Delta\mathrm{T}_{\mathrm{s}}\) is, the easier the convergence of dynamic analysis will be, while the sum of total time steps to reach static equilibrium \(\Sigma\mathrm{N}_{\mathrm{ls}}\) will be large and the analysis will not be efficient enough. In a dynamic analysis, reasonable \(\Delta\mathrm{T}_{\mathrm{s}}\) should ensure that total kinetic energy reaches the peak in fewer time steps based on the convergence of dynamic analysis. Considering all above, the form finding analysis of NDFEM is divided into three stages: In the initial stage, the cable strut system moves violently, and a short time step can be set for dynamic analysis to facilitate the iterative convergence of dynamic balance equation. In the middle stage, the main deformation direction of the cable strut system is clear and the whole system tends to reach the static equilibrium configuration. At this time, a large time step should be set to quickly approach the static equilibrium state with fewer time steps and configuration updating times. In the third stage, the cable strut system vibrates near the static equilibrium state. At this time, a larger time step should be set to make the configuration iteration converge as soon as possible and the whole cable strut system finally reach the static equilibrium state. In view of the important influence of the time step on dynamic balance iteration and analysis efficiency, it is proposed to automatically adjust the time step of each dynamic analysis by using the adjustment coefficient \(C_{\text{bs}}\) in the analysis process. The adjustment strategy is as follows:  
1. The initial time step \(\Delta T_{\text{s}\,(1)}\) is used in the first configuration iteration;
2. If the time steps of the (m \(-\) 1)th dynamic analysis \(N_{\text{bs(m}-1)}\) = [\(N_{\text{bs}}\)] and the total kinetic energy still do not decrease, then the time step of the \(m\)th dynamic analysis is \(\Delta T_{\text{s(m)}}\) = \(\Delta T_{\text{s(m-1)}}\)\(\times\)\(C_{\text{bs}}\);
3.