# Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
## 6 LCO Analysis of Time Integrations with the SSI Scheme
### LCO Results under an Airspeed of 20 m/s
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Iteration-\(n\)** & \(x_{j}\) & \(x_{1}\)**, m** & \(x_{3}\)**, m/s** & \(x_{4}\)**, \({}^{\circ}\)/s** & \(x_{5}\) & \(x_{6}\) & **CPU time** \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{\(X_{0,j}^{\{1\}}\)} & min. & \(-0.1\) & \(-0.1\) & \(0.00\) & \(-0.1\) & \(-0.1\) & \multirow{2}{*}{—} \\ \cline{2-5}  & & max. & \(+0.1\) & \(+0.1\) & \(30.00\) & \(+0.1\) & \(+0.1\) & \\ \hline \multirow{2}{*}{2} & \multirow{2}{*}{\(X_{0,j}^{\{2\}}\)} & min. & \(-7.34\times 10^{-4}\) & \(-6.14\times 10^{-2}\) & \(0.00\) & \(0.00\) & \(-3.43\times 10^{-3}\) & \\ \cline{2-5}  & & max. & \(+5.42\times 10^{-3}\) & \(+5.60\times 10^{-2}\) & \(29.98\) & \(+4.79\times 10^{-1}\) & \(0.00\) & \multirow{2}{*}{82.82 s} \\ \cline{2-5}  & & \(\Delta\Big{|}X^{\{1,2\}}\Big{|}\Big{/}\Big{|}X^{\{1\}}\Big{|}\) & \(-96.92\%\) & \(-41.23\%\) & \(0.06\%\) & \(-8.12\%\) & \(-98.29\%\) & \\ \hline \multirow{3}{*}{3} & \multirow{2}{*}{\(X_{0,j}^{\{3\}}\)} & min. & \(-6.83\times 10^{-4}\) & \(-2.49\times 10^{-2}\) & \(0.02\) & \(0.00\) & \(-7.44\times 10^{-4}\) & \\ \cline{2-5}  & & max. & \(+5.91\times 10^{-5}\) & \(+1.57\times 10^{-3}\) & \(27.42\) & \(+2.11\times 10^{-1}\) & \(+1.11\times 10^{-5}\) & \\ \cline{2-5}  & & \(\Delta\Big{|}X^{\{2,3\}}\Big{|}\Big{/}\Big{|}X^{\{2\}}\Big{|}\) & \(-87.93\%\) & \(-77.46\%\) & \(8.61\%\) & \(-55.81\%\) & \(-77.97\%\) & \\ \hline \multirow{3}{*}{4} & \multirow{2}{*}{\(X_{0,j}^{\{4\}}\)} & min. & \(-5.35\times 10^{-4}\) & \(-2.50\times 10^{-2}\) & \(0.00\) & \(0.00\) & \(-7.44\times 10^{-4}\) & \\ \cline{2-5}  & & max. & \(+4.97\times 10^{-5}\) & \(+1.48\times 10^{-3}\) & \(12.09\) & \(+2.12\times 10^{-1}\) & \(0.00\) & \multirow{2}{*}{78.66 s} \\ \cline{2-5}  & & \(\Delta\Big{|}X^{\{3,4\}}\Big{|}\Big{/}\Big{|}X^{\{3\}}\Big{|}\) & \(-21.20\%\) & \(0.23\%\) & \(55.88\%\) & \(0.77\%\) & \(-1.39\%\) & \\ \hline \multirow{2}{*}{5} & \multirow{2}{*}{\(X_{0,j}^{\{5\}}\)} & min. & \(-5.04\times 10^{-4}\) & \(-2.52\times 10^{-2}\) & \(0.00\) & \(0.00\) & \(-7.47\times 10^{-4}\) & \\ \cline{2-5}  & & max. & \(+6.10\times 10^{-5}\) & \(+1.46\times 10^{-3}\) & \(12.14\) & \(+2.11\times 10^{-1}\) & \(+1.10\times 10^{-5}\) & \\ \cline{2-5}  & & \(\Delta\Big{|}X^{\{4,5\}}\Big{|}\Big{/}\Big{|}X^{\{4\}}\Big{|}\) & \(-3.32\%\) & \(0.81\%\) & \(-0.41\%\) & \(-0.30\%\) & \(1.86\%\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ranges of states, relative changes in ranges, and the central processing unit (CPU) time consumed in each iteration. The airspeed is 20 m/s. An iteration will terminate if all ranges vary within 5% compared to those in the last iteration. Figure 21: Iteration plots for each state in each iterating process, presenting the procedure for the range \(X_{0,j}^{\{n\}}\) being contracted to \(X_{0,j}^{\{n+1\}}\). Each one of the subplots in this figure is similar to that in Figure 20.  
After Iteration-4, clear spatial patterns emerge on each of the five iteration plots, as shown in the rightmost column of Figure 21. For example, we can focus on the iteration plot for \(x_{0,4}\), which is located at the third row and fourth column in Figure 21, and re-plot it in Figure 22. Three fixed points can be easily distinguished: one is at \(x_{0,4}\approx 10.6\) deg/s, one is at \(x_{0,4}\approx 0.45\) deg/s, and the last one is at \(x_{0,4}=0\). The first two fixed points represent two LCO solutions, while the last fixed point indicates a convergence of the system responses. The amplitudes and frequencies are derived from the time integrations of Iteration-4; then, a plot of the LCO amplitude/LCO frequency versus \(x_{0,4}\) is given in Figure 23. It is found that the amplitude of the LCO arising at \(x_{0,4}\approx 10.6\) deg/s is larger than 1.0, which indicates that this is a 3-D LCO, and the other LCO is determined as a 2-D LCO, as its relative amplitudes are less than or around 1.0.