## 3 Results and Discussion
### Multiple Freeplay Nonlinearities
#### 3.2.1 Discrete Scanning Method
However, if the amplitude of the initial condition becomes 1.02 times its original value, the system will eventually enter a stable LCO. This comparison indicates that the LCO of branch B is unstable at this dynamic pressure, and with sufficient initial amplitude, the system will be attracted to the stable LCO of branch A. In addition, we can find that the system at 45.26 kPa with larger initial conditions has strong amplitude fluctuations at the beginning, but as time goes on, the fluctuations become progressively smaller, which is the complete opposite of the phenomenon at 42.27 kPa. The above interesting phenomena reflect the weakening of the coupling effect of the two LCO branches and the expansion of the attraction domain of the stable LCO during the transformation process of the system from quasi-periodic motion to the LCO. Figure 17 shows variations of the Poincare section with dynamic pressure at \(\delta_{\rm B}/\delta_{\rm T}=0.1\) and \(\delta_{\rm B}/\delta_{\rm T}=0.5\). Figure 18 shows the variation of the largest Lyapunov exponent with time [43; 44], and it finally converges to a positive value with increasing time step \(N\), indicating  
\[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]4 Time responses of systems at different dynamic pressures and initial conditions. (\(Q=42.27\) kPa: harmonic initial conditions of branch A (**right**); \(Q=43.53\) kPa: harmonic initial conditions of branch A (**right**); \(Q=45.26\) kPa: harmonic initial conditions of branch B (**left**), 1.02 times the left initial conditions (**right**). \(\delta_{\rm B}/\delta_{\rm T}=0.1\). The phase parameter \(\phi_{\rm c}\) of the harmonic initial condition is set to \(\pi/2\).) According to the solving interval and the continuity of the LCO solutions with the parameters, three LCO branches are defined here. In Figure 20, when \(\delta_{\text{B}}/\delta_{\text{T}}\leq 0.25\), LCO branches A and B are located on either side of the yellow dotted line (two solution intervals) and converge at the top right. In contrast, branch C is relatively independent. With the  
\[A_{\rm B}/A_{\rm T}=f_{\rm R}[f_{\rm DE}(A_{\rm B}/\delta_{\rm B}),f_{\rm DF}(A_{ \rm T}/\delta_{\rm T})] \tag{24}\]5 Frequency-domain LCO solutions and corresponding time-domain bifurcation diagrams in bending and torsional DOFs (\(K_{\text{B}}=5.0\times 10^{3}\) N \(\cdot\) m/rad, \(K_{\text{T}}=1.5\times 10^{3}\) N \(\cdot\) m/rad, and \(\delta_{\text{B}}/\delta_{\text{T}}\) is 0.1, 0.25, 0.5, 1.0, and 5.0 from top to bottom). decrease of \(\delta_{\rm B}/\delta_{\rm T}\), Branch C moves toward the right dotted line, and branches A and B move toward the yellow dotted line. The intersection point of branches A and B moves down and left, which means the right boundary of the dynamic pressure region of LCO branches A and B in Figure 19 is shifted to the left. If \(\delta_{\rm B}/\delta_{\rm T}\) decreases to 0, LCO branches A and B would disappear and the system would exhibit LCO-ii characteristics similar to Figure 5b, as determined by the torsion freeplay. From the frequency-domain LCO solutions in Figure 19, there is no matching LCO solution in a certain dynamic pressure region when \(\delta_{\rm B}/\delta_{\rm T}\leq 0.25\). The corresponding time-marching results show that the system response decays in this dynamic pressure region. Branch C is in the right of the region. This dynamic pressure region keeps shrinking with increasing \(\delta_{\rm B}/\delta_{\rm T}\), and it disappears completely at \(\delta_{\rm B}/\delta_{\rm T}=0.5\). Through further computational analysis, it was found that branches B and C meet and undergo recombination when \(\delta_{\rm B}/\delta_{\rm T}\) is approximately 0.2685.