Analysis of the Nonlinear Structural-Acoustic Resonant Frequencies of a Rectangular Tube with a Flexible End Using Harmonic Balance and Homotopy Perturbation Methods

Abstract

The structural acoustic problem considered in this study is the nonlinear resonant frequencies of a rectangular tube with one open end, one flexible end, and four rigid side walls A multiacoustic single structural modal formulation is derived from two coupled partial differential equations which represent the large amplitude structural vibration of the flexible end and acoustic pressure induced within the tube. The results obtained from the harmonic balance and homotopy perturbation approaches verified each other. The effects of vibration amplitude, aspect ratio, the numbers of acoustic modes and harmonic terms, and so forth, on the first two resonant natural frequencies, are examined.

1. Introduction

Over the past decades, many researchers worked on linear structural-acoustic research works (e.g., [1–7]) and nonlninear structural vibration problems (e.g., [8–15]), separately. The structural-acoustic problem of rectangular tube (or similar problems) has been studied for many years in various studies. So far, only few research works about structural-acoustics have adopted the assumption of large amplitude vibration [16–20]. Few classical solutions for nonlinear structural-acoustic problems have been developed to date, although there are many approaches available for solving nonlinear governing differential equation (e.g., [21–30]. In the study reported in this paper, the homotopy perturbation and harmonic balance methods are used and assessed. It is because these two methods were employed to determine the large amplitude free vibration of beams and nonlinear oscillators in previous studies and agreed well with the other published results [31]. The results obtained from these two methods verified each other. In finite element and other numerical approaches for solving the problems of nonlinear structural vibrations (e.g., [32–35]), it is necessary for setting a set of residual equations or global matrix equations and then solving them for the eigenvalue solutions. All these approaches require a significant effort as an eigenvalue problem. The present study uses the multiacoustic and single structural mode approach to develop the classical solutions which do not require a significant amount of computational effort and preprocessing inputs.

2. Theory

2.1. Governing Equations

In Figure 1, the acoustic pressure within the rectangular tube induced by the flexible end is given by the following homogeneous wave equation [1]: ∇²1𝑃−𝐶²_𝑎𝜕²𝑃𝜕𝜏²=0, (2.1) where P = the pressure within the tube; τ = time; 𝐶_𝑎 = sound speed.

Figure 1 

The rectangular tube with one open end, one flexible end, and four rigid side walls.

The boundary conditions are given by𝜕𝑃𝜕𝑥=0,at𝑥=0and𝑎,𝜕𝑃𝜕𝑦=0,at𝑦=0and𝑏,𝑃=0,at𝑧=0,𝜕𝑃𝜕𝑧=−𝜌_𝑎𝜕²𝑤_𝑙(𝑡)𝜕𝜏²𝜑𝑥,𝑦,at𝑧=𝑙, (2.2a)(2.2b)(2.2c)(2.2d)where 𝑤_𝑙(𝜏)=𝐴cos(𝜔𝜏)=flexible plate vibration response; 𝜑(𝑥,𝑦)=vibration mode shape=sin(𝜋/𝑎)sin(𝜋/𝑏); 𝜌_𝑎=airdensity;𝐴=displacement amplitude (or displacement at time=0); 𝜔=vibration frequency.

According to [1], the general multiacoustic mode solution of (2.1) is 𝑃=^𝑈_𝑢^𝑉_𝑣𝐿_𝑢𝑣sinh𝜇_𝑢𝑣𝑧+𝑁_𝑢𝑣cosh𝜇_𝑢𝑣𝑧cos𝑢𝜋𝑥𝑎cos𝑣𝜋𝑦𝑏cos(𝜔𝜏), (2.3) where 𝜇_𝑢𝑣=(𝑢𝜋/𝑎)²+(𝑣𝜋/𝑏)²−(𝜔/𝐶_𝑎)²;𝑢 and 𝑣=0,2,4,… are the acoustic mode numbers; 𝑈 and 𝑉 are the last acoustic mode numbers; 𝐿_𝑢𝑣 and 𝑁_𝑢𝑣 are unknown coefficients to be determined by the boundary conditions.

Applying the boundary condition (2.2c) to (2.3) gives 𝑁_𝑢𝑣=0. (2.4) Then applying the boundary condition (2.2d) to (2.3) gives 𝜕𝑃=𝜕𝑧^𝑈_𝑢^𝑉_𝑣𝐿_𝑢𝑣𝜇_𝑢𝑣cosh𝜇_𝑢𝑣𝑙cos𝑢𝜋𝑥𝑎cos𝑣𝜋𝑦𝑏cos(𝜔𝜏)at𝑧=𝑙=−𝜌_𝑎𝜕²𝐴cos(𝜔𝜏)𝜕𝜏²𝜑𝑥,𝑦=𝜌_𝑎𝜔²𝐴cos(𝜔𝜏)𝜑𝑥,𝑦⇒𝐿_𝑢𝑣=𝜌_𝑎𝜔²𝛼_𝑢𝑣𝛼_cos𝐴𝜇_𝑢𝑣cosh𝜇_𝑢𝑣𝑙, (2.5) where 𝛼_𝑢𝑣=∫^𝑏₀∫^𝑎₀𝜑(𝑥,𝑦)cos(𝑢𝜋𝑥/𝑎)cos(𝑣𝜋𝑦/𝑏)𝑑𝑥𝑑𝑦; 𝛼_cos=^𝑏₀^𝑎₀cos𝑢𝜋𝑥𝑎cos𝑣𝜋𝑦𝑏²𝑑𝑥𝑑𝑦. (2.6) Therefore, the acoustic pressure and modal acoustic pressure force acting on the flexible end at 𝑧=𝑙 are given by𝑃_𝑙=𝜌_𝑎𝜔^2𝑈_𝑢^𝑉_𝑣𝛼_𝑢𝑣𝛼_costanh𝜇_𝑢𝑣𝑙𝜇_𝑢𝑣cos𝑢𝜋𝑥𝑎cos𝑣𝜋𝑦𝑏(),𝐹𝐴cos𝜔𝜏_𝑙=∫^𝑏₀∫^𝑎₀𝑃_𝑙𝜑𝑥,𝑦𝑑𝑥𝑑𝑦𝛼_𝜑=𝜌_𝑎𝜔^2𝑈_𝑢^𝑉_𝑣𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣𝑙𝜇_𝑢𝑣𝐴cos(𝜔𝜏), (2.7a)(2.7b)where 𝛼_𝜑=∫^𝑏₀∫^𝑎₀𝜑(𝑥,𝑦)²𝑑𝑥𝑑𝑦.

According to the approach from Chu and Herrmann [15], the governing equation for the large amplitude vibration of a flexible plate is given by𝜌𝑑²𝑤_𝑙𝑑𝜏²+𝜌𝜔²_𝑠𝑤_𝑙+𝛽𝑤³_𝑙=0, (2.8a) where 𝜌 = the panel surface density; 𝜔_𝑠=𝐸𝑡²/12𝜌(1−𝜈²)[(𝜋/𝑎)²+(𝜋/𝑏)²] = the fundamental linear resonant frequency of the plate; 𝛽=(𝐸ℎ/12(1−𝜈²))(𝛾/𝑎⁴) is the nonlinear stiffness coefficient that is due to the large amplitude vibration; E is the Young’s modulus of the plate; 𝛾=3𝜋⁴[((3/4)−(𝜈²/4))(1+𝑟⁴)+𝜈𝑟²];𝑟=𝑎/𝑏 is the aspect ratio; 𝜈 is Poisson’s ratio; and 𝑡 = the plate thickness.

Consider the modal acoustic pressure in (2.7b) on the plate. Equation (2.8a) is modified and given by 𝜌𝑑²𝑤_𝑙𝑑𝜏²+𝜌𝜔²_𝑠𝑤_𝑙+𝛽𝑤³_𝑙=𝐹_𝑙. (2.8b)

2.2. Harmonic Balance Method

Consider the structural vibration response in terms of harmonic terms [20]:𝑤_𝑙=^𝐻_ℎ𝐴_ℎcos(ℎ𝜔𝜏), (2.9a) where 𝐴_ℎ is the amplitude of the ℎth harmonic response; ℎ=1,3,…𝐻; and 𝐻 is the last harmonic order number; 𝐴=∑^𝐻_ℎ𝐴_ℎ.

Then, the modal acoustic pressure force at 𝑧=𝑙 in (2.7b) is revised and also consists of higher harmonic terms. 𝐹_𝑙(𝑡)=𝜌_𝑎^𝐻_ℎ^𝑈_𝑢^𝑉_𝑣(ℎ𝜔)²𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣,ℎ𝑙𝜇_𝑢𝑣,ℎ𝐴_ℎcos(ℎ𝜔𝜏), (2.9b)where 𝜇_𝑢𝑣,ℎ=(𝑢𝜋/𝑎)²+(𝑣𝜋/𝑏)²−(ℎ𝜔/𝐶_𝑎)².

Similarly, (2.8a) is revised and given by 𝜌𝑑²𝑤_𝑙𝑑𝜏²+𝜌𝜔²_𝑠𝑤_𝑙+𝛽𝑤³_𝑙=𝜌_𝑎^𝐻_ℎ^𝑈_𝑢^𝑉_𝑣(ℎ𝜔)²𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣,ℎ𝑙𝜇_𝑢𝑣,ℎ𝐴_ℎ()⇒()cosℎ𝜔𝜏−𝜌ℎ𝜔²𝐴_ℎ+𝜌𝜔²_𝑠𝐴_ℎ+𝛽𝐺_ℎ𝐴₁,𝐴₃…𝐴_𝐻=𝜌_𝑎(ℎ𝜔)^2𝑈_𝑢^𝑉_𝑣𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣,ℎ𝑙𝜇_𝑢𝑣,ℎ𝐴_ℎ, (2.10) where 𝐺_ℎ is a set of functions which contains 𝐴₁,𝐴₃,…𝐴_𝐻.

If 𝐻=1, then 𝐺₁𝐴₁=34𝐴³₁. (2.11) Consider the harmonic balance for 𝐻=1 and ignore the higher harmonic terms in (2.10). Then, the following equation is obtained: −𝜌(𝜔)²𝐴₁+𝜌𝜔²_𝑠𝐴₁3+𝛽4𝐴³₁=𝜌_𝑎(𝜔)^2𝑈_𝑢^𝑉_𝑣𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣,1𝑙𝜇_𝑢𝑣,1𝐴₁. (2.12) If 𝐻=5, then𝐺₁𝐴₁,𝐴₃,𝐴₅=34𝐴³₁+34𝐴²₁𝐴₃+32𝐴²₃𝐴₁+32𝐴²₅𝐴₁+34𝐴²₃𝐴₅+32𝐴₁𝐴₃𝐴₅,𝐺₃𝐴₁,𝐴₃,𝐴₅=34𝐴³₃+14𝐴³₁+32𝐴²₁𝐴₃+32𝐴²₅𝐴₃+34𝐴²₁𝐴₅+32𝐴₁𝐴₃𝐴₅,𝐺₅𝐴₁,𝐴₃,𝐴₅=34𝐴³₅+32𝐴²₁𝐴₅+34𝐴²₁𝐴₃+34𝐴²₃𝐴₁+32𝐴²₃𝐴₅. (2.13a)(2.13b)(2.13c)Consider the harmonic balance for 𝐻=1,3,5 and ignore the higher harmonic terms in (2.10). Then, the following three equations are obtained:−𝜌(𝜔)²𝐴₁+𝜌𝜔²_𝑠𝐴₁+𝛽𝐺₁𝐴₁,𝐴₃…𝐴_𝐻=𝜌_𝑎(𝜔)^2𝑈_𝑢^𝑉_𝑣𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣,1𝑙𝜇_𝑢𝑣,1𝐴₁,−𝜌(3𝜔)²𝐴₃+𝜌𝜔²_𝑠𝐴₃+𝛽𝐺₃𝐴₁,𝐴₃…𝐴_𝐻=𝜌_𝑎(3𝜔)^2𝑈_𝑢^𝑉_𝑣𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣,3𝑙𝜇_𝑢𝑣,3𝐴₃,−𝜌(5𝜔)²𝐴₅+𝜌𝜔²_𝑠𝐴₅+𝛽𝐺₅𝐴₁,𝐴₃…𝐴_𝐻=𝜌_𝑎(5𝜔)^2𝑈_𝑢^𝑉_𝑣𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣,5𝑙𝜇_𝑢𝑣,5𝐴₅. (2.14a)(2.14b)(2.14c) According to (2.9a), one more equation is obtained: 𝐴=𝐴₁+𝐴₃+𝐴₅. (2.14d)Note that 𝐴 is the initial modal displacement which is a known parameter. Thus, there are four unknowns 𝐴₁,𝐴₃,𝐴₅, and 𝜔 in (2.14a)–(2.14d). The resonant frequency ω is obtained by solving the four equations.

2.3. Homotopy Perturbation Method

Consider the free large amplitude vibration of a flexible panel and the corresponding governing equation 𝑑²𝑤_𝑙𝑑𝜏²+𝜔²_𝑠𝑤_𝑙+𝛽^′𝑤_𝑙³=0, (2.15) where 𝛽^′=𝛽/𝜌.

Using the homotopy perturbation approach [29, 30], equation (2.15) can be linearized and construct the following homotopy (note that there are some alternative ways of constructing the homotopy equation, e.g., [36]):𝑑²𝑤_𝑙𝑑𝜏²+𝜔²𝑤_𝑙+𝑞𝜔²_𝑠𝑤_𝑙−𝜔²𝑤_𝑙+𝛽^′𝑤³_𝑙𝑤=0,_𝑙=𝑤_𝑙,0+𝑞𝑤_𝑙,1+⋯, (2.16a)(2.16b)where 𝑞∈[0,1].𝑤_𝑙,0 and 𝑤_𝑙,1 are the linear and first order approximate terms. Their initial conditions and approximate forms are given by𝑤_𝑙,0=𝐴;𝑑𝑤_𝑙,0𝑤𝑑𝜏=0,at𝜏=0,_𝑙,1=0;𝑑𝑤_𝑙,1𝑤𝑑𝜏=0,at𝜏=0,_𝑙,0(),𝑤=𝐴cos𝜔𝑡_𝑙,1()−1=𝐵cos𝜔𝜏3cos(3𝜔𝜏), (2.17a)(2.17b)(2.17c)(2.17d)where 𝜔 is the approximate natural frequency of the nonlinear system. 𝐴 and 𝐸 and vibration amplitudes of the 𝑤_𝑙,0 and 𝑤_𝑙,1, respectively.

Substituting (2.16b) into (2.16a), collecting terms of the same power of 𝑞, gives𝑑²𝑤_𝑙,0𝑑𝜏²+𝜔²𝑤_𝑙,0𝑑=0,₂𝑤_𝑙,1𝑑𝜏²+𝜔²𝑤_𝑙,1+𝛽^′𝑤³_𝑙,0+𝜔²_𝑠−𝜔²𝑤_𝑙,0=0. (2.18a)(2.18b)According to [37], the variational formulation is given by 𝐽𝑤_𝑙,1=₀^2𝜋/𝜔−12𝑑𝑤_𝑙,1𝑑𝜏²+𝜔²𝑤²_𝑙,1+𝜔²_𝑠−𝜔²𝑤_𝑙,0𝑤_𝑙,1+𝛽^′𝑤³_𝑙,0𝑤_𝑙,1𝑑𝜏=0. (2.19) Consider 𝜕𝐽/𝜕𝐵=0 and 𝜕𝐽/𝜕𝜔=0. Then, the resonant frequency is given by 𝜔²_𝑜−𝜔²_𝑠−34𝛽^′𝐴²=0⇒𝜔_𝑜=𝜔²_𝑠+34𝛽^′𝐴², (2.20) where 𝜔_𝑜 is the resonant frequency of the large amplitude vibration.

Now consider the modal acoustic pressure force acting on the panel in (2.7b). Equation (2.18a) can be rewritten and given by 𝑑²𝑤_𝑙,0𝑑𝜏²+𝜔²_𝑜𝑤_𝑙,0=𝐹_𝑙⇒−𝜌𝜔²𝐴+𝜌𝜔²_𝑠3𝐴+𝛽4𝐴³=𝜌_𝑎(𝜔)^2𝑈_𝑢^𝑉_𝑣𝛼²_𝑢𝑣𝛼_cos𝛼_𝜑tanh𝜇_𝑢𝑣𝑙𝜇_𝑢𝑣𝐴. (2.21) Equation (2.21) is exactly the same as (2.12), which is developed from the harmonic balance method.

3. Results and Discussions

In this numerical study, the first two resonant frequencies of the rectangular tube with a flexible end are considered and obtained by solving (2.10) and (2.21). The material properties of the flexible end or (flexible panel at 𝑧=𝑙) are as follows: Young’s modulus = 7.1 × 10¹⁰ N/m², Poisson’s ratio = 0.3, and mass density = 2700 kg/m³. The dimensions of the tube are 0.2 m × 0.2 m × 1.0 m. The panel thickness is 1 mm. The linear 1st structural resonant frequencies of the panel (not mounted to the tube end), 𝜔_𝑠 = 121.878 Hz. The first nine acoustic modes (i.e., 𝑢=0,2,4;𝑣=0,2,4) and first three harmonic terms (i.e., 𝐻=1,3,5) are employed in the convergence study. Tables 1(a) and 1(b) the harmonic term convergence for various amplitudes. The four acoustic modes (i.e., 𝑢=0,2;𝑣=0,2) are used in the cases in Tables 1(a) and 1(b). As aforementioned in (2.12) and (2.21), the linear resonant frequencies obtained from the homotopy perturbation method and the harmonic balance method with one harmonic term are exactly the same. It can be seen that the effect of the 3rd harmonic term on the first two resonant frequencies can be ignored. The solutions with the first two harmonic terms can achieve 3-digit accuracy. Tables 2(a) and 2(b) present the results of the acoustic mode convergence studies for various vibration amplitudes. One harmonic term (i.e., 𝐻=1) is used in the cases in Tables 2(a) and 2(b). It can be seen that the effect of the higher acoustic modes (i.e., 𝑢=4,6…;𝑣=4,6…) on the first two resonant frequencies can be ignored. The solutions with the four acoustic modes can achieve 3-digit accuracy.

Table 1 

(a) Harmonic term convergence for various amplitudes (1st resonant frequency). (b) Harmonic term convergence for various amplitudes (2nd resonant frequency).

Table 2 

(a) Acoustic mode convergence for various amplitudes (1st resonant frequency). (b) Acoustic mode convergence for various amplitudes (2nd resonant frequency).

In Figures 2(a) and 2(b), the vibration amplitude ratio, 𝐴/ℎ, is plotted against the frequency ratio for various panel thicknesses, 𝑡=0.5, 0.6, and 0.7 mm, and first two resonant frequencies.

(a)

(b)

Figure 2 

(a) The vibration amplitude ratio versus the 1st nonlinear resonant frequency (𝑡=0.5, 0.6, 0.7 mm). (b) The vibration amplitude ratio versus the 2nd nonlinear resonant frequency (𝑡=0.5, 0.6, 0.7 mm).

The material properties and the other dimensions are the same as those considered in Tables 1(a) and 1(b). The linear 1st structural resonant frequencies of the three panels (not mounted to the tube end) in Figures 2(a) and 2(b) are 𝜔_𝑠=60.939, 73.127, and 85.315 Hz for 𝑡=0.5, 0.6, and 0.7 mm, respectively. Note that these three linear structural resonant frequencies are lower than the linear 1st acoustic resonant frequency of the tube with one rigid end and one open end (i.e., 𝜔_𝑎 = 85.75 Hz). Generally, the resonant frequencies in all cases are monotonically increasing with the amplitude ratio. In Figure 2(a), the 1st nonlinear resonant frequencies are much smaller than the corresponding linear 1st structural resonant frequencies when the amplitude ratio is small. In Figure 2(b), the 2nd nonlinear resonant frequencies are always higher than the 1st linear acoustic resonant frequency of the tube with one rigid end and one open end. According to a comparison between the three curves, the differences between the 1st nonlinear resonant frequencies of the three cases in Figure 2(a) are getting small for large amplitude ratio. On the contrary, the differences between the 2nd nonlinear resonant frequencies of the three cases in Figure 2(b) are getting large for large amplitude ratio.

In Figures 3(a) and 3(b), the vibration amplitude ratio, 𝐴/ℎ, is plotted against the frequency ratio for various panel thicknesses, 𝑡=0.8, 0.9, and 1.0 mm, and first two resonant frequencies. The linear 1st structural resonant frequencies of the three panels (not mounted to the tube end) in Figures 3(a) and 3(b) are 𝜔_𝑠=97.503, 109.69, and 121.878 Hz for 𝑡=0.8,0.9, and 1.0 mm respectively. Note that these three linear structural resonant frequencies are higher than the 1st linear acoustic resonant frequency of the tube with one rigid end and one open end. Although the curves in Figures 3(a) and 3(b) show similar trends in Figures 2(a) and 2(b), there are some other observations found. According to a comparison between the curves in Figures 2(a) and 3(a), the differences between the 1st nonlinear resonant frequencies in Figure 3(a) are obviously smaller than those in Figure 2(a) for all amplitude ratios. In Figure 3(b), the differences between the 2nd nonlinear resonant frequencies are quite constant for all amplitude ratios, while the differences between the 2nd nonlinear resonant frequencies in Figure 2(b) are getting large for large amplitude ratio.

(a)

(b)

Figure 3 

(a) The vibration amplitude ratio versus the 1st nonlinear resonant frequency (𝑡=0.8, 0.9, 1 mm). (b) The vibration amplitude ratio versus the 2nd nonlinear resonant frequency (𝑡=0.8, 0.9, 1 mm).

In Figures 4(a) and 4(b), the 1st and 2nd nonlinear resonant frequencies are plotted against 𝜔_𝑠/𝜔_𝑎, for various vibration amplitude ratios (where 𝜔_𝑠 = the linear 1st structural resonant frequency of the panel, not mounted to the tube end; 𝜔_𝑎 = the linear 1st acoustic resonant frequency of the tube with one rigid end and one open end). In Figure 4(a), it can be seen that the 1st nonlinear resonant frequencies of all cases are always below 𝜔_𝑎 and getting close to it, when 𝜔_𝑠/𝜔_𝑎 is increasing. Similar to the observation in Figure 4(a), the 2nd nonlinear resonant frequencies of all cases in Figure 4(b) are always below 3𝜔_𝑎 and getting close to it, when 𝜔_𝑠/𝜔_𝑎 is increasing. Besides, it can be seen that the 2nd nonlinear resonant of all cases always higher than a certain frequency and converge to it when 𝜔_𝑎 is getting small.

(a)

(b)

Figure 4 

(a) The 1st nonlinear resonant frequency versus 𝜔𝑠/𝜔𝑎 (𝐴/𝑡=0.4, 0.8, 1.2 mm). (b) The 2nd nonlinear resonant frequency versus 𝜔𝑠/𝜔𝑎 (𝐴/𝑡=0.4, 0.8, 1.2 mm).

In Figures 5(a) and 5(b), the vibration amplitude ratio, 𝐴/ℎ, is plotted against the frequency ratio for various aspect ratios, 𝑎/𝑏=1, 1.5, and 2 mm, and first two resonant frequencies. The 1st linear structural resonant frequencies of the three cases are kept the same. It can be seen that the differences between the three curves are very small; and thus the effect of aspect ratio is very minimal on the nonlinear resonant frequencies of the structural acoustic system.

(a)

(b)

Figure 5 

(a) The vibration amplitude ratio versus the 1st nonlinear resonant frequency (𝑎/𝑏=1.0, 1.5, 2.0). (b) The vibration amplitude ratio versus the 2nd nonlinear resonant frequency (𝑎/𝑏=1.0, 1.5, 2.0).

4. Conclusions

This paper presents a multimode formulation, based on the classical nonlinear panel and homogeneous wave equations, for the nonlinear vibrations of a flexible panel mounted to an end of a rectangular tube. The first two resonant frequencies are obtained by solving the multimode differential equations and using the harmonic balance method and homotopy perturbation method. The solutions from the two methods are found to agree well with each other. The convergence study shows the number of acoustic modes and harmonic terms needed for an accurate result. The effects of vibration amplitude, panel thickness, aspect ratio, and so forth have also been investigated. The main findings include the following: (1) if the linear 1st structural resonant frequency of the panel is higher than the linear 1st acoustic resonant frequency of the tube with one rigid end and one open end, the 1st nonlinear resonant frequency of the structural-acoustic system is less sensitive to the panel thickness than that of the rectangular tube, which the linear 1st structural resonant frequency of the panel is lower than the linear 1st acoustic resonant frequency; (2) if the linear 1st structural resonant frequency of the panel is lower than the linear 1st acoustic resonant frequency of the tube with one rigid end, the 2nd nonlinear resonant frequency of the structural-acoustic system is less sensitive to the panel thickness for small amplitude ratio, and more sensitive for large amplitude ratio; and (3) in each case considered, the 1st nonlinear resonant frequency is always lower than the linear 1st acoustic resonant frequency, and the 2nd nonlinear resonant frequency is always higher than the 1st linear acoustic resonant frequency and lower than 2nd linear acoustic resonant frequency, respectively.

Acknowledgment

The work described in this paper was fully supported by a Grant from the Research Grants Council of the Hong Kong, Hong Kong (Project no. 9041496 (CityU 116209)).