## 3 Results and Discussion
### Multiple Freeplay Nonlinearities
#### 3.2.1 Discrete Scanning Method
The relevant results are presented in \[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]6  
Later, with the increase of \(\delta_{\rm B}/\delta_{\rm T}\), branch B moves downward and eventually disappears in Figure 20, while branches A and C move toward the upper dotted line. The system exhibits LCO-i characteristics similar to Figure 5a, determined by the bending freeplay. Comparing the trend of \(A_{\rm B}/\delta_{\rm B}\) at \(\delta_{\rm B}/\delta_{\rm T}=5.0\) in Figure 20 with the trend of LCO-i in Figure 5a, it can be seen that branch A corresponds to the unstable LCO, and branch C corresponds to the stable LCO. The corresponding time-domain bifurcation diagram in Figure 20 also agrees well with the calculation results of branch C.  
Next, we specifically look at the time-domain bifurcations of the system under different freeplay size ratios in \[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]7 At \(\delta_{\rm B}/\delta_{\rm T}=0.1\), the system response decays and eventually reaches a static equilibrium position when the dynamic pressure is lower than the linear flutter boundary. Due to the existence of the unstable LCO branch C, when the dynamic pressure is slightly higher than the linear flutter dynamic pressure, the system response does not diverge. This is because the initial conditions at this time are not of great enough amplitude to excite the system into a divergent state. The time-response results given in Figure 21 further verify the calculation results of the LCO branch C. However, the other two LCO branches A and B in the frequency-domain calculation results are not reflected in the corresponding time-domain bifurcation diagrams. Considering that the current initial conditions might not excite the system to enter the LCO, we increased the amplitude of the initial conditions. The corresponding time-domain bifurcation diagrams are shown in \[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]8 When the initial conditions increase from 10 times the freeplay size to 100 times, quasi-periodic or chaotic motions appear between dynamic pressures of 25.4 and 31.3 kPa. However, the existence of such vibrations with a non-zero amplitude is uncertain because the system enters a decaying motion under some dynamic pressures between 25.4 and 31.3 kPa. In addition, a quasi-periodic motion occurs alone at 35.7 kPa. In general, the existence of LCO branches A and B theoretically causes the system to exhibit more quasi-periodic or chaotic motions, but such vibrations are uncertain and have great dependencies on the initial conditions. For the case of \(\delta_{\rm B}/\delta_{\rm T}\geq 0.25\), the system mainly exhibits quasi-periodic or chaotic motions when the dynamic pressure is less than about 38 kPa. The nondimensional vibration amplitude also becomes larger as the dynamic pressure increases. As \(\delta_{\rm B}/\delta_{\rm T}\) increases, the nondimensional vibration amplitude becomes smaller, but it changes more dramatically at around 38 kPa. Finally, the jumping phenomenon of the LCO appears at \(\delta_{\rm B}/\delta_{\rm T}=5.0\). When the dynamic pressure is higher than about 38 kPa but below the linear flutter boundary, the system exhibits LCO motions except for within the dynamic pressure region in which the response decays at \(\delta_{\rm B}/\delta_{\rm T}=0.25\). Considering that \(\delta_{\rm B}/\delta_{\rm T}\geq 0.5\), the nondimensional LCO amplitude increases with increasing dynamic pressure in the bending DOF, but it first decreases and then increases in the torsional DOF. There is a maximum value of the nondimensional LCO amplitude in the torsional DOF at around 38.0 kPa; this should be carefully considered because structural damage can occur here.