# Integration of Freeplay-Induced Limit Cycles Based On a State Space Iterating Scheme
## 5 LCO Results of Time Integrations with the H\(\!\)non-RK45 Method  
An I.C. case \(\mathbf{x}_{0}=\mathbf{x}(t)\) is specified as a case where only the pitch angle \(\alpha(t)\) starts with a non-zero value \(\alpha_{0}\), while the rest of the system states are zero--that is, \(\mathbf{x}_{0}=\left[0,\alpha_{0},0,0,0\right]^{T}\). The amount of freeplay is \(2\delta=0.1^{\circ}\). We first carried out time integrations with the H\(\!\)non-RK45 method at an airspeed of \(U=20\) m/s and various \(\alpha_{0}\) values. The results show that the system behaviors depend highly on \(\alpha_{0}\). For example, when the initial non-dimensional pitch angle, \(\alpha_{0}/\delta\), equals \(5.00\), a three-domain LCO (3-D LCO) is found in which the phase trajectory of \(\alpha(t)\) between 13 and 15 s passes all three subdomains defined in Section 2.1, as illustrated in Figure 13b. However, when \(\alpha_{0}/\delta=1.00\), a two-domain LCO (2-D LCO)  
Figure 11: Eigenvalues of the underlying linear system (ULS) of the wing section model varying with the flow velocity.  
Figure 12: Eigenvalues of the overlying linear system (OLS) of the wing section model varying with the flow velocity.  
emerges, and its phase trajectory passes only two subdomains, \(S_{0}\) and \(S_{2}\), as presented in Figure 14.  
To determine the effect of \(\alpha_{0}/\delta\) on LCO behaviors, 100 cases of \(\alpha_{0}/\delta\) that vary from 0.1 to 100 subjects with a 1-cos function and 73 cases of airspeed \(U\) that increases from 12 to 20 m/s following a piecewise linear function were considered, as illustrated in Figure 15. Therefore, 7300 cases featuring a combination of \(\alpha_{0}/\delta\) and \(U\) were specified, and the results derived from the time integrations are presented in Figure 16.  
Figure 14: A two-domain LCO (2-D LCO) emerges when the flow velocity is 20 m/s and the initial non-dimensional pitch angle is 1.00, where the response of the freeplay state is presented by (**a**) a time history of 0–15 s and (**b**) a phase trajectory of 13–15 s.  
Figure 13: A three-domain LCO (3-D LCO) emerges when the flow velocity is 20 m/s and the initial non-dimensional pitch angle is 5.00, where the response of the freeplay state is presented by (**a**) a time history of 0–15 s and (**b**) a phase trajectory of 13–15 s.  
It was found that the 3-D LCO exists in a continuous region with a clear boundary, while the 2-D LCOs are distributed in numerous separated regions, as shown in Figure 16. A close-up of the boundary of the 3-D and the 2-D LCO regions is provided in Figure 17. Since Figures 16 and 17 can only be obtained under various \(\alpha_{0}\) and \(U\) values, we are still  
Figure 16: System behaviors derived from the 7300 cases, where a 3-D LCO region with a clear boundary and a 2-D LCO region mingled with the converging region are found.  
Figure 15: Calculation cases with 73 airspeeds and 100 initial pitch angles for each airspeed.  
unaware about how other states will affect the two LCO regions if they have non-zero values in the I.C. cases.  
The time integrations for the 7300 cases consume a total central processing unit (CPU) time of 4687 s (about 1.3 h); however, only the effect of \(\alpha_{0}\) is investigated among all system states. All calculations in this paper were carried out via the software package MATLAB r2014 running on a computer with a quad-core and an eight-thread processor (Intel Core i5-10210U CPU @ 1.60 GHz and 2.11 GHz) and 16.0 GB of random access memory.