## 3 Results and Discussion
### Multiple Freeplay Nonlinearities
#### 3.2.1 Discrete Scanning Method
It can also be seen in Figure 11 that the two LCO branches start from one point and then gradually move away from each other when \(\dot{\delta}_{\rm B}/\delta_{\rm T}=0.01\). Therefore, the appearances of such quasi-periodic motions are probably caused by the interaction between the two LCO branches, which should have similar amplitudes and irrational frequency ratios [42]. Once the two LCO branches move far away from each other, these quasi-periodic motions will be destroyed, and the response trajectories will be attracted by one of the LCO branches. Figure 14 shows plots of the variations of the LCO frequency \(\omega\) and the phase difference \(\Delta\phi\) with dynamic pressure. The frequency of the quasi-periodic motion refers to the frequency of the component with the highest proportion. The frequency-domain calculation results show that the frequencies of the two LCO branches are relatively close until the turning point of the torsional LCO frequency at about 46.0 kPa; however, the phase differ  
\[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]0 Poincaré section variation with dynamic pressure at \(\dot{\delta}_{\rm B}/  
ences between the responses in the bending and torsional DOFs of the two branches have completely opposing trends. After the system enters the LCO at 46.2 kPa, the frequency and phase of the time-domain LCO response are consistent with the LCO solutions of branch A. Figure 15 shows the spectral-analysis results of the time-domain response in the yellow area. Not surprisingly, we find that the LCO response at 47 kPa mainly contains the components of one frequency and its multiples, while the quasi-periodic motion at 45 kPa contains some irreducible frequencies. The frequency of the component with the highest proportion is consistent with the frequencies of the two LCO branches (Figure 14). However, some other frequency components cannot be obtained by the method in this paper; these may be generated by the coupling effects of LCO branches with similar amplitudes and frequencies but different phase differences. Another interesting phenomenon is that when the two LCO branches are very close, this quasi-periodic motion disappears, such as when the dynamic pressure is less than 44 kPa at \(\delta_{\rm B}/\delta_{\rm T}=0.01\) or less than 28 kPa at \(\delta_{\rm B}/\delta_{\rm T}=0.1\). \[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]1 LCO frequency and phase-difference variations with dynamic pressure at \(\delta_{\rm B}/\delta_{\rm T}=0.01\). \[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]2 Time responses and spectra for 45 and 47 kPa and \(\delta_{\rm B}/\delta_{\rm T}=0.01\). To further study the transformation process of this quasi-periodic motion to the LCO, the time-domain response of systems with different dynamic pressures and initial conditions at \(\delta_{\rm B}/\delta_{\rm T}=0.1\) was calculated, and the results are shown in \[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]3 When the default initial conditions \([10\delta_{\rm B},10\delta_{\rm T},0.0,0.0]^{T}\) are given, a quasi-periodic motion occurs at 42.27 and 43.53 kPa. If a harmonic initial condition corresponding to branch A is given, the response trajectory is immediately attracted to the LCO at 43.53 kPa. However, at 42.27 kPa, even given similar harmonic initial conditions, the amplitude of the system fluctuates increasingly as time goes on, and it eventually enters quasi-periodic motion. This phenomenon shows that with increasing dynamic pressure, the torsional LCO amplitude difference between the two branches expands, and this leads to the weakening of the coupling effect between the LCO branches. At 45.26 kPa, when a harmonic initial condition corresponding to branch B is given, the system response decays.