## 3 Results and Discussion  
In this section, the material properties in the numerical cases are as follows: [PERSON]'s modulus = \(7.1\times 10^{10}\) N/m\({}^{2}\), Poisson's ratio = 0.3, and mass density = 2700 kg/m\({}^{3}\); panel dimensions = 0.4 m \(\times\) 0.4 m \(\times\) 2 m; damping ratio \(\xi\) = 0.02. Tables 2-4 show the convergence studies of normalized panel vibration amplitude for various excitation magnitudes and frequencies. The nine acoustic mode solutions, three structural mode solutions, and 2 nd level solutions are normalized as one hundred. The acoustic boundaries are rigid (i.e., 0 = 0). The zero, 1 st and 2 nd level solutions contain 1, 2 and 3 harmonic components, respectively (i.e., \(\sin(\omega t)\), \(\sin(3\omega t)\), and \(\sin(5\omega t)\)). It is shown that the 1 st level 4 acoustic mode and two structural mode approach is good enough for convergent and accurate vibration amplitude solution. numerical method, which enables the capture of unstable solutions (e.g., [21; 22; 23]). According to [14], the solutions from the proposed method come from the cubic algebraic equations transformed from the Duffing differential equations representing the nonlinear panel vibration. The unstable solutions occur when the cubic algebraic equations have three real solutions. Generally, the unstable solution is in between the nonlinear and nearly linear solutions. It is slightly lower than the real nonlinear solution around the peak frequency and slightly higher than the nearly linear solution when the excitation frequency is far from the peak frequency. Figure 5: Comparison between the solutions obtained from the proposed method and numerical integration method [6] (number of structural modes = 2, number of acoustic modes = 4, \(l=5a\), \(\gamma=2\), \(\theta=0\); the orange arrows represent the ”jump phenomenon”; “A”, “B”, and “C” represent the particular points on the solution curve). Figure 6: Comparison between the solutions obtained from the proposed method and numerical integration method [6] (number of structural modes = 2, number of acoustic modes = 4, \(l=5a\), \(\gamma=4\), \(\theta=0\); the orange arrows represent the ”jump phenomenon”; “A”, “B”, and “C” represent the particular points on the solution curve). Relatively, the agreement in Figure 5 is the best because of smaller nonlinearity. In Figures 6 and 7, some deviations are observed around the major peak values. Note that there are some minor peaks around \(\omega/\omega_{l}=2.3\) and \(3.4\) in Figure 7. In these three cases, the first acoustic resonances occur around \(\omega/\omega_{l}=1.4\). In the case of \(\gamma=2\), the well-known jump phenomena, which are due to the first structural resonance and higher acoustic resonances, occurs around \(\omega/\omega_{l}=2.3\), \(3.9\) and \(4.5\) (see Points A, B and C). The nonlinear solution line discontinues there. Generally, the nearly linear solution lines are much lower than the nonlinear solution lines. In the cases of \(\gamma=4\) and \(8\), the jump phenomena due to the first structural resonance and higher acoustic resonances occur at the higher frequencies. Unlike the trough on the solution line in the case of \(\gamma=2\), a small peak and a large peak occur on the nonlinear solution lines of \(\gamma=4\) and \(8\) around \(\omega/\omega_{l}=4.2\), respectively. Besides, Points A and B in the case of \(\gamma=4\) are very close. There are no Points A and B in the case of \(\gamma=8\) (i.e., no discontinuous portion). In Figure 7, due to the large excitation level, the vibration responses are highly nonlinear. The nonlinear solution lines further extend so that Points A and B are joined and the linear solution line disappears. In this case, there is only one real solution obtained from the cubic algebraic equation transformed from the Duffing differential equations representing the nonlinear panel vibration.