## 3 Results and Discussion
In the practical structural design of a similar structural-acoustic system, the disappearance of the linear solution (or small-amplitude solution), which is replaced by the nonlinear solution (or large-amplitude solution), implies that a stronger structural design is needed to overcome the larger vibration amplitude. Figures 8-10 show the vibration amplitude plotted against the excitation frequency for various tube lengths. The first peak value due to the acoustic resonance and first jump-down frequency increase with the tube length, while the first peak frequency and the first trough frequency decrease with the tube length. In the case of \(l=10a\), the tube length is the longest among the three cases. Thus, the second peak occurs around \(\omega/\omega_{l}=2.1\), which is lower than the first jump-down frequency, and there are more acoustic resonant peaks on the solution line. In the case of \(l=3a\), the tube length is the shortest among the three cases. The trough occurs out of the frequency range (i.e. \(\omega/\omega_{l}>6\)). Figure 7: Comparison between the solutions obtained from the proposed method and numerical integration method [6] (number of structural modes \(=2\), number of acoustic modes \(=4\), \(l=5a\), \(\gamma=8\), \(\theta=0\); the orange arrow represents the “jump phenomenon”; “**C**” represents the particular point on the solution curve). Figure 8: Vibration amplitude ratio versus normalized excitation frequency for various duct lengths (number of structural modes = 2, number of acoustic modes = 4, \(\gamma=2\), \(\theta=0\), \(l=3a\); the orange arrow represents the “jump phenomenon”). Figure 9: Vibration amplitude ratio versus normalized excitation frequency for various duct lengths (number of structural modes = 2, number of acoustic modes = 4, \(\gamma=2\), \(\theta=0\), \(l=5a\); the orange arrows represent the “jump phenomenon”). Figures 11-13 show the vibration amplitude plotted against the excitation frequency for various phase shift parameters. The first peak frequency due to the acoustic resonance, the first jump-down frequency, and the trough frequency increase with the phase shift parameter, while the first peak value decreases with the tube length. In the cases of \(\theta=5\) and \(10\), it is implied that the tube boundaries are not perfectly rigid and the corresponding air particle velocities are non-zero. It is noted that the first peak is quite small, marginally detectable when the phase shift is only \(8\) degrees. Figure 11: Vibration amplitude ratio versus normalized excitation frequency for various acoustic boundary conditions (number of structural modes = 2, number of acoustic modes = 4, \(\gamma=2\), \(\theta=0\), \(l=5a\); the orange arrows represent the “jump phenomenon”). Figure 10: Vibration amplitude ratio versus normalized excitation frequency for various duct lengths (number of structural modes = 2, number of acoustic modes = 4, \(\gamma=2\), \(\theta=0\), \(l=10a\); the orange arrows represent the “jump phenomenon”). Moreover, the phase shift does not significantly affect the higher acoustic resonances. Figure 14 shows the vibration amplitudes of the tubes filled with air and CO\({}_{2}\) plotted against the corresponding excitation frequency. The density of CO\({}_{2}\) and the corresponding sound speed are about 50% higher than those in the case of air. There is a big difference between the two cases. In the case of CO\({}_{2}\), the first peak is much smaller and the jump phenomenon disappears. The peaks due to higher acoustic resonances are out of the frequency range. In the practical structural design of a similar structural-acoustic system, the disappearance of the nonlinear solution (or large-amplitude solution), which is replaced by the  
Figure 12: Vibration amplitude ratio versus normalized excitation frequency for various acoustic boundary conditions (number of structural modes = 2, number of acoustic modes = 4, \(\gamma\) = 2, \(\theta\) = 5, \(l\) = 5\(a\); the orange arrows represent the “jump phenomenon”).