Natural Frequency Analysis of Two Nonlinear Panels Coupled with a Cavity Using the Approximate Elliptic Integral Solution and the Method of Harmonic Residual Minimization

Abstract

The nonlinear structural acoustic problem considered in this study is the nonlinear natural frequency analysis of flexible double panels using the elliptic integral solution method. There are very limited studies for this nonlinear structural-acoustic problem, although many nonlinear plate or linear double panel problems have been tackled and solved. A multistructural/acoustic modal formulation is derived from two coupled partial differential equations which represent the large amplitude structural vibrations of the flexible panels and acoustic pressure induced within the air gap. One is the von Karman's plate equation and the other is the homogeneous wave equation. The results obtained from the proposed method approach are verified with those from a numerical method. The effects of vibration amplitude, gap width, aspect ratio, the numbers of acoustic modes and harmonic terms, and so forth on the resonant frequencies of the in-phase and out of phase modes are examined.

Academic Editor:Zengji Du

Department of Civil and Architectural Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong

Received 8 April 2014; Revised 21 October 2014; Accepted 3 November 2014; 28 January 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In practice, thin double panels are very common for sound reduction. As it is easy to get thin panels excited nonlinearly, the nonlinear effect on the resonant frequencies of double panel is studied in this paper. However, there are very limited studies for this nonlinear structural-acoustic problem, although many nonlinear plate or linear double panel problems have been tackled and solved. Thus, the following paragraph would be the reviews about linear double panel and nonlinear panel vibration.

Over the past decades, many researchers such as London [1], Cummings and Mulholland [2], and Price and Crocker [3] studied double panels and adopted the multiple-reflection theory and statistical energy analysis to develop various formulas for prediction. In their works, the panel sizes were assumed very large so that the first few structural resonant frequencies were far below 50 Hz and not concerned. Thus, the concept of the mass-cavity-mass, which ignored the panel stiffness, was valid for common audible frequency range. The statistical energy method was suitable for cases of high modal density or high frequency. If a smaller panel is considered, the first few fundamental resonant frequencies would move to a higher frequency range (say 80-150 Hz). Under this condition, the bending stiffness of the panel cannot be ignored, and the statistical energy method would not be suitable. In the works of Xin and Lu [4-7] and Xin et al. [8-10], the detailed theoretical and experimental investigations about the sound absorptions, radiations, and vibrations of various sandwich panels and double panels were included. In some of their works, the solution method adopted was suitable for double panel systems of finite or infinite extent and applicable for both low- and high-frequency ranges with these merits; the proposed method compares favorably with a number of other approaches. Besides, it was pointed out that the structural-acoustic coupling affected significantly the sound and vibration behaviors at low frequencies and could not be ignored. Pan and Bao [11] and Kaiser et al. [12] adopted the modal analysis to develop their theoretical models for studying a linear double panel problem at low frequencies. Langer and Antes [13] studied double walls with different pane configurations using the finite and boundary element methods. In [14, 15], the works considered the resonant frequencies of the first two modes of the structural-acoustic problem. It was proven experimentally and theoretically in these works that the first two modes could seriously deteriorate the noise reduction performance. That was an important finding. Hence, the in-phase and out of phase modes of double panels are the focus of this paper. The structural-acoustic models in the aforementioned works were capable of predicting multimode responses, but not the large amplitude vibration effect, and the modal coupling effects on the resonant frequencies were not investigated. There are very limited studies for nonlinear double panel problems. Some studies (e.g., [16-24]) are related to the nonlinear structural vibrations, but their focuses are different from that in this paper. In fact, there are numerous theories used for solving various nonlinear vibration problems (e.g., [25-35]). In some approaches for solving the problems of large amplitude structure or nonlinear oscillation (e.g., [36-39]), they require a significant effort in the tedious eigenvalue solution steps. The present study uses the proposed elliptical integral method to develop a concise elliptical integral solution for the large amplitude free vibration analysis.

2. Theoretical Formulation

According to the formulation of Chu and Herrmann [25], the governing equation for the large amplitude vibration of a rectangular plate is given by [figure omitted; refer to PDF] Note that in the studies of [23-25], the panel vibration amplitude ranges are zero to [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] = panel thickness. Within this vibration amplitude range, no panel buckling occurs.

Consider the acoustic pressures within the gap acting on the two panels (see Figure 1). Equation (1a) is modified and given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the modal displacements of the two panels at [figure omitted; refer to PDF] = 0 and [figure omitted; refer to PDF] . The mode shapes of the two panels are [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] is time; [figure omitted; refer to PDF] is the panel surface density; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the modal pressure force induced by the panel vibration; [figure omitted; refer to PDF] are the fundamental linear natural frequencies of the panels; [figure omitted; refer to PDF] is the nonlinear stiffness coefficient that is due to the large amplitude vibration; [figure omitted; refer to PDF] is Young's modulus of the panels; [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] is the aspect ratio; [figure omitted; refer to PDF] is Poisson's ratio; and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the panel length and width.

Figure 1: A double panel and its mode shapes.

[figure omitted; refer to PDF]

[figure omitted; refer to PDF] and [figure omitted; refer to PDF] are periodic and considered to be a summation of harmonic terms and are given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the [figure omitted; refer to PDF] th-order harmonic responses and [figure omitted; refer to PDF] = the number of harmonic terms used.

Similarly, the total modal acoustic pressure forces acting on the panel surfaces are given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the acoustic pressure forces induced by the [figure omitted; refer to PDF] th-order harmonic components of the two nonlinear panel vibrations, respectively.

In Figure 1, the acoustic pressure within the gap induced by the flexible enclosure panel and panel absorber is given by the following homogeneous wave equation [14, 15]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the acoustic pressure within the gap induced by the [figure omitted; refer to PDF] th harmonic component of the nonlinear panel vibration and [figure omitted; refer to PDF] is the speed of sound.

The boundary conditions of the air gap are [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the panel width and length, respectively; [figure omitted; refer to PDF] is the gap width; [figure omitted; refer to PDF] = air density, [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the [figure omitted; refer to PDF] th harmonic components of the nonlinear panel displacements at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, which can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the [figure omitted; refer to PDF] th harmonic components of the nonlinear panel displacements at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. [figure omitted; refer to PDF] is the panel mode shape.

According to [14, 15], the general multiacoustic mode solution of (1a)-(1c) is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] is the acoustic mode; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the acoustic mode numbers; [figure omitted; refer to PDF] is the excitation frequency. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are coefficients that depend on the boundary conditions at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the numbers of acoustic mode numbers used. [figure omitted; refer to PDF] is sound speed. [figure omitted; refer to PDF] is the time function.

By applying the boundary conditions in (2c) and (2d) to (5a) and (5b), the unknown coefficients, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , can be found, and thus the [figure omitted; refer to PDF] th harmonic component of the modal internal pressure forces within the gap at [figure omitted; refer to PDF] and 0 (i.e., [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ) is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the maximum acoustic mode numbers.

Then, the total modal pressure forces within the gap at [figure omitted; refer to PDF] and 0 and the external pressure force acting on the panel absorber ( [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ) are given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the number of harmonic terms used.

Putting (10a) and (10b) into (1a)-(1b) yields [figure omitted; refer to PDF] Note that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the displacement responses of the two panels and given by [figure omitted; refer to PDF] Figure 1 shows the two modes of the double panel. Then set [figure omitted; refer to PDF] for the in-phase mode and [figure omitted; refer to PDF] for the out of phase mode in (8a) and (8b). Hence, we get the following.

For the in-phase mode [figure omitted; refer to PDF] For the out of phase mode [figure omitted; refer to PDF] Consider the dummy term [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in (13a) and (13b): [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are unknowns to be determined.

Then, consider the elliptical integral solutions in [23] for the following equations: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the elliptic integrals; [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the modulus of [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] is the elliptic cosine; and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the modal displacements at [figure omitted; refer to PDF] or the vibration amplitudes.

The periods of the nonlinear modal vibrations are given by [figure omitted; refer to PDF] Replacing [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] in (14a) and (14b) gives [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the harmonic residuals, as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are not the exact solution to (14a) and (14b). [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the harmonic components of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Using (15a) and (15b), the summation of the first three terms in (17a) and (17b) are set to zero, as follows: [figure omitted; refer to PDF] According to (15a) and (15b), [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] depend on the unknowns, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. Hence, (18a) and (18b) can be rewritten as [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be the optimum values for the minimization of the overall harmonic residual squares in (19a) and (19b). Hence, the periods of the large amplitude modal vibrations are given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the natural frequencies of the two nonlinear modes.

3. Results and Discussions

Using (20a) and (20b), the resonant frequencies of the in-phase and out of phase modes of simply supported double panels can be obtained for various vibration amplitude ratios. Each double panel is made of two aluminum panels measuring 0.3048 m × 0.3048 m × 1.2192 mm. The material properties are Young's modulus [figure omitted; refer to PDF] N/m² , Poisson's ratio [figure omitted; refer to PDF] = 0.3, and mass density [figure omitted; refer to PDF] kg/m³ . In Tables 1(a) and 1(b), the first 25 symmetrical acoustic modes and the first four harmonic terms are used for the convergence checks of the resonant frequencies of the in-phase and out of phase modes. The frequency ratio is defined as [figure omitted; refer to PDF] . It is shown that the nine acoustic modes and two harmonic terms approach is good enough for a converged and accurate frequency solution. Figures 2(a) and 2(b) show the comparisons between the free vibration time histories of the in-phase and out of phase mode vibrations obtained from the present elliptical integral method and numerical integration method [36]. It can be seen that the two sets of time histories are well agreed with each other.

Table 1: (a) Acoustic mode and harmonic convergences of the frequency ratio of an aluminum plate (in-phase mode, vibration amplitude/thickness = 1). (b) Acoustic mode and harmonic convergences of the frequency ratio of an aluminum plate (out of phase mode, vibration amplitude/thickness = 1).

(a)

[figure omitted; refer to PDF] / [figure omitted; refer to PDF]	Number of harmonic terms = 1	2	3	4
Number of acoustic modes = 1	1.3981	1.3981	1.3981	1.3981
4	1.3961	1.3962	1.3962	1.3962
9	1.3961	1.3961	1.3961	1.3961
16	1.3961	1.3961	1.3961	1.3961
25	1.3961	1.3961	1.3961	1.3961

(b)

[figure omitted; refer to PDF] / [figure omitted; refer to PDF]	Number of harmonic terms = 1	2	3	4
Number of acoustic modes = 1	2.9676	2.9676	2.9676	2.9676
4	2.9505	2.9505	2.9505	2.9505
9	2.9502	2.9502	2.9502	2.9502
16	2.9502	2.9502	2.9502	2.9502
25	2.9502	2.9502	2.9502	2.9502

(a) Free vibration time histories obtained from the two solution methods (in-phase mode, initial amplitude ratio = 1.4, gap width/panel width = 0.5). (b) Free vibration time histories obtained from the two solution methods (out of phase mode, initial amplitude ratio = 0.4, gap width/panel width = 0.5).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

In Figures 3-6, the cases of vibration amplitude = 0 represent the results of linear vibration. It is because the nonlinear stiffness term is zero (in the linear vibration theory, there is no nonlinear term in the governing equation). In Figures 3(a) and 3(b), the vibration amplitude ratios are plotted against the frequency ratios of the in-phase and out of phase mode vibrations for various gap widths. The aspect ratio is 1. In Figure 3(a), when the vibration amplitude = 0, the frequency ratio = 1 (it is purely linear vibration. In the linear vibration, it is assumed that the vibration amplitude is very small); when the vibration amplitude = 1.4, the frequency ratio is about 1.7 (it is highly nonlinear vibration). The natural frequency increases by 70% due to the nonlinear stiffness caused by the axial deformation along the panel (see Figure 3(c) for further explanation). The frequency ratios of the in-phase mode vibration are not very sensitive to the gap width (the three curves almost exactly coincide) while they are quite sensitive to and monotonically increasing with the vibration amplitude. When the amplitude is small, the frequency ratios increase very slowly. When the amplitude is large, the increase rates of the frequency ratios are almost constant. Note that the cases of infinite gap width in Figures 3(a) and 3(b) can be considered as a single panel without any gap effect. When compared with those in Figure 3(a), the frequency ratios of the out of phase mode vibration show higher degree of sensitivity to the gap width and lower degree of sensitivity to the vibration amplitude, respectively.

(a) Vibration amplitude ratio versus frequency ratio (in-phase mode, nine acoustic modes, two harmonic terms). (b) Vibration amplitude ratio versus frequency ratio (out of phase mode, nine acoustic modes, two harmonic terms). (c) Axial forces and moments acting on a plate element (side view). When the vibration amplitude is large, [figure omitted; refer to PDF] is large, the axial force acting on the panel is significant, and the nonlinear stiffness due to the axial deformation must be considered. When the vibration amplitude is small, [figure omitted; refer to PDF] and the axial force is neglectable (the linear vibration theory is valid for small deflection), where [figure omitted; refer to PDF] is the axial force; [figure omitted; refer to PDF] is the bending moment; [figure omitted; refer to PDF] is the slope.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(a) (Gap width/panel width)^-1 versus frequency ratio (in-phase mode, nine acoustic modes, two harmonic terms). (b) (Cavity depth/width)^-1 versus frequency ratio (out of phase mode, nine acoustic modes, two harmonic terms). (c) Volume changes of the in-phase and out of phase modes.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(a) Aspect ratio versus frequency ratio (in-phase mode, nine acoustic modes, two harmonic terms). (b) Aspect ratio versus frequency ratio (out of phase mode, nine acoustic modes, two harmonic terms).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(a) Vibration amplitude ratio versus acoustic stiffness (in-phase mode, nine acoustic modes, two harmonic terms). (b) Vibration amplitude ratio versus acoustic stiffness (gap width = 0.3048 m, out of phase mode, nine acoustic modes, two harmonic terms). (c) Vibration amplitude ratio versus acoustic stiffness (gap width = 0.1524 m, out of phase mode, nine acoustic modes, two harmonic terms). (d) Vibration amplitude ratio versus acoustic stiffness (gap width = 0.0508 m, out of phase mode, nine acoustic modes, two harmonic terms).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

In Figures 4(a) and 4(b), the inverses of the gap width/panel width, [figure omitted; refer to PDF] , are plotted against the frequency ratios of the in-phase and out of phase mode vibrations for various vibration amplitude ratios. The aspect ratio is 1. In Figure 4(a), the frequency ratios of the in-phase mode vibration is almost unchanged for [figure omitted; refer to PDF] . It is implied that the effect of [figure omitted; refer to PDF] is not significant on the frequency ratios for [figure omitted; refer to PDF] . In the case of amplitude ratio = 1.4, the frequency ratio significantly increases when [figure omitted; refer to PDF] . Relatively, the frequency ratio of amplitude ratio = 1.4 is slightly more sensitive than that of amplitude ratio = 0. In Figure 4(b), the frequency ratios of the out of phase mode vibration are sensitive to [figure omitted; refer to PDF] when compared with those in Figure 4(a). Again, it can be seen that the higher the vibration amplitude ratio is, the higher the frequency ratio is. Besides, the slopes of the three curves, which represent the cases of amplitude ratio = 0, 1, 1.4, are almost the same. In Figure 4(c), it can be seen that the "net cavity volume change" of the in-phase mode vibration is smaller than that of the out of phase mode vibration. The amount of the volume change is larger and the acoustic stiffness is higher when the vibration amplitude is larger. Relatively, the effect of vibration amplitude on the natural frequency is relatively smaller when the acoustic stiffness is higher. That is why the frequency ratio of the in-phase mode vibration is less sensitive.

In Figures 5(a) and 5(b), the aspect ratios, [figure omitted; refer to PDF] , are plotted against the frequency ratios of the in-phase and out of phase mode vibrations for various vibration amplitude ratios. The gap width is 0.0508 m. In Figure 5(a), the frequency ratios of the in-phase mode vibration are almost constant for the zero vibration amplitude or [figure omitted; refer to PDF] in the cases of amplitude ratio = 1 and 1.4. They are almost linearly decreasing with the aspect ratio for [figure omitted; refer to PDF] . The effect of vibration amplitude is larger when the aspect ratio is smaller. In Figure 5(b), the frequency ratio of the out of phase mode is almost constant for [figure omitted; refer to PDF] . They are increasing with the aspect ratio for the cases of [figure omitted; refer to PDF] . Relatively, they are more sensitive to the aspect ratio than those of the in-phase mode vibration. When the aspect ratio is larger, the differences between the three frequency ratio curves are smaller (i.e., the effect of vibration amplitude is smaller).

In Figures 6(a)-6(d), the vibration amplitudes are plotted against the acoustic stiffness ratios of the in-phase and out of phase modes for various gap widths. The aspect ratio is 1. Note that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are defined as [figure omitted; refer to PDF] in Figures 6(a) and 6(b). In Figure 6(a), the acoustic stiffness ratio of the in-phase mode is almost constant for [figure omitted; refer to PDF] , while it is nonlinearly decreasing with the vibration amplitude for [figure omitted; refer to PDF] . The physical meaning of negative acoustic stiffness is that the acoustic force direction is the same as the panel vibration direction. Besides, when the gap is small, the acoustic stiffness is strong (it can be imagined that a larger force is required to compress a smaller cavity). Thus, the effect of vibration amplitude on the natural frequency is relatively smaller when the gap width is smaller (or the acoustic stiffness is higher). That is why the acoustic stiffness of large gap width in Figure 6(a) is more sensitive than that of small gap width. Unlike those in Figure 6(a), the three curves in Figures 6(b)-6(d) are far from each other (that is why they are plotted in three graphs). The slopes of the three curves are similar to each other, and the acoustic stiffness vales are nonlinearly decreasing with the vibration amplitude. Generally, the acoustic stiffness of the out of phase mode vibration is less sensitive to the vibration amplitude than that of the in-phase mode vibration. It is because the acoustic stiffness value of the out of phase mode vibration, [figure omitted; refer to PDF] , is higher than that of in-phase mode vibration, [figure omitted; refer to PDF] . As aforementioned, the effect of vibration amplitude on the natural frequency is relatively smaller when the acoustic stiffness is higher.

From Figure 6(a), it is known that the acoustic stiffness values would be negative and the frequency ratios in some cases could be less than 1 (see Figures 5(a) and 5(b)). Unlike the acoustic stiffness, the nonlinear structural stiffness is always positive and nonlinearly increasing with the vibration amplitude (see Figure 7). It is noted that the nonlinear stiffness is zero when the vibration amplitude is zero (in the linear vibration, it is assumed that the vibration amplitude is very small). When the vibration amplitude is small, the slope of the curve representing the nonlinear stiffness is deep (it means that the nonlinear stiffness increases quickly). When the vibration amplitude is large, the slope is almost constant. The nonlinear stiffness constantly increases with the vibration amplitude.

Figure 7: Vibration amplitude ratio versus structural nonlinear stiffness.

[figure omitted; refer to PDF]

4. Conclusions

A multistructural/acoustic mode formulation, which is based on the classical nonlinear plate equation and homogeneous wave equation, has been presented for the large amplitude free vibrations of a double panel. The present study proposes the elliptical integral method to obtain the natural frequencies of various nonlinear double panels. The effects of the gap width, aspect ratio, and vibration amplitude ratio on the resonant frequencies of the in-phase and out of phase modes have also been investigated. It has been found that (1) the present elliptic integral solutions agree reasonably well with those obtained from the numerical integration method, and they are convergent with increasing the numbers of acoustic modes and harmonic terms; (2) the frequency ratio of the in-phase mode of a double panel is more like that of a single panel, and the frequency ratio of the out of phase mode is more sensitive to the gap width and aspect ratio; (3) the acoustic stiffness of the in-phase mode is more sensitive to the vibration amplitude than that of the out of phase mode; and (4) unlike the structural stiffness, the acoustic stiffness of a double panel may be negative and depends on the vibration amplitude.

In practice, when acoustic engineers design a double panel used for noise reduction, they would set the natural frequency not overlapping with the dominant noise frequency bandwidth. Thus, the natural frequency results in this study can be used as reference in noise reduction panel design.

Acknowledgments

The work described in this paper was fully supported by a grant from the Research Grants Council of Hong Kong (project title: A New Solution Method to Nonlinear Panel-Cavity Coupling Problems: Multilevel Residue Harmonic Balance Method ). The author would like to express his gratitude and appreciation to Dr. W. Y Poon's advices and contributions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

[1] A. London, "Transmission of reverberant sound through double walls," The Journal of the Acoustical Society of America , vol. 22, no. 2, pp. 270-279, 1950.

[2] A. Cummings, K. A. Mulholland, "The transmission loss of finite sized double panels in a random incidence sound field," Journal of Sound and Vibration , vol. 8, no. 1, pp. 126-133, 1968.

[3] A. J. Price, M. J. Crocker, "Sound transmission through double panels using statistical energy analysis," The Journal of the Acoustical Society of America , vol. 47, no. 3, pp. 683-693, 1970.

[4] F. X. Xin, T. J. Lu, "Analytical and experimental investigation on transmission loss of clamped double panels: implication of boundary effects," Journal of the Acoustical Society of America , vol. 125, no. 3, pp. 1506-1517, 2009.

[5] F. X. Xin, T. J. Lu, "Sound radiation of orthogonally rib-stiffened sandwich structures with cavity absorption," Composites Science and Technology , vol. 70, no. 15, pp. 2198-2206, 2010.

[6] F. X. Xin, T. J. Lu, "Transmission loss of orthogonally rib-stiffened double-panel structures with cavity absorption," Journal of the Acoustical Society of America , vol. 129, no. 4, pp. 1919-1934, 2011.

[7] F. X. Xin, T. J. Lu, "Analytical modeling of sound transmission through clamped triple-panel partition separated by enclosed air cavities," European Journal of Mechanics, A/Solids , vol. 30, no. 6, pp. 770-782, 2011.

[8] F. X. Xin, T. J. Lu, C. Q. Chen, "Vibroacoustic behavior of clamp mounted double-panel partition with enclosure air cavity," Journal of the Acoustical Society of America , vol. 124, no. 6, pp. 3604-3612, 2009.

[9] F. X. Xin, T. J. Lu, C. Q. Chen, "Sound transmission through simply supported finite double-panel partitions with enclosed air cavity," Journal of Vibration and Acoustics, Transactions of the ASME , vol. 132, no. 1, 2010.

[10] F. X. Xin, T. J. Lu, C. Q. Chen, "External mean flow influence on noise transmission through double-leaf aeroelastic plates," AIAA Journal , vol. 47, no. 8, pp. 1939-1951, 2009.

[11] J. Pan, C. Bao, "Analytical study of different approaches for active control of sound transmission through double walls," The Journal of the Acoustical Society of America , vol. 103, no. 4, pp. 1916-1922, 1998.

[12] O. E. Kaiser, S. J. Pietrzko, M. Morari, "Feedback control of sound transmission through a double glazed window," Journal of Sound and Vibration , vol. 263, no. 4, pp. 775-795, 2003.

[13] S. Langer, H. Antes, "Analyses of sound transmission through windows by coupled finite and boundary element methods," Acta Acustica , vol. 89, no. 1, pp. 78-85, 2003.

[14] Y. Y. Lee, C. F. Ng, "Sound insertion loss of stiffened enclosure plates using the finite element method and the classical approach," Journal of Sound and Vibration , vol. 217, no. 2, pp. 239-260, 1998.

[15] Y. Y. Lee, E. W. M. Lee, "Widening the sound absorption bandwidths of flexible micro-perforated curved absorbers using structural and acoustic resonances," International Journal of Mechanical Sciences , vol. 49, no. 8, pp. 925-934, 2007.

[16] F. Alijani, M. Amabili, "Chaotic vibrations in functionally graded doubly curved shells with internal resonance," International Journal of Structural Stability and Dynamics , vol. 12, no. 6, 2012.

[17] C. S. Chen, C. W. Chen, W. R. Chen, "Dynamic stability characteristics of functionally graded plates under arbitrary periodic loads," International Journal of Structural Stability and Dynamics , vol. 13, no. 6, 2013.

[18] D. K. Nguyen, "A Timoshenko beam element for large displacement analysis of planar beams and frames," International Journal of Structural Stability and Dynamics , vol. 12, no. 6, 2012.

[19] V. Dogan, R. Vaicaitis, "Nonlinear response of double-wall cylindrical shell vibrations under random excitation," Journal of Aerospace Engineering , vol. 19, no. 1, pp. 46-54, 2006.

[20] Y. Wei, R. Vaicaitis, "Nonlinear models for double-wall systems for vibrations and noise control," Journal of Aircraft , vol. 34, no. 6, pp. 802-810, 1997.

[21] Y. Y. Lee, "Structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one flexible plate," Applied Acoustics , vol. 63, no. 11, pp. 1157-1175, 2002.

[22] Y. Y. Lee, X. Guo, E. W. M. Lee, "Effect of the large amplitude vibration of a finite flexible micro-perforated panel absorber on sound absorption," International Journal of Nonlinear Sciences and Numerical Simulation , vol. 8, no. 1, pp. 41-44, 2007.

[23] C. K. Hui, Y. Y. Lee, J. N. Reddy, "Approximate elliptical integral solution for the large amplitude free vibration of a rectangular single mode plate backed by a multi-acoustic mode cavity," Thin-Walled Structures , vol. 49, no. 9, pp. 1191-1194, 2011.

[24] Y. Y. Lee, Q. S. Li, A. Y. T. Leung, R. K. L. Su, "The jump phenomenon effect on the sound absorption of a nonlinear panel absorber and sound transmission loss of a nonlinear panel backed by a cavity," Nonlinear Dynamics , vol. 69, no. 1-2, pp. 99-116, 2012.

[25] H. N. Chu, G. Herrmann, "Influence of large amplitudes on free flexural vibrations of rectangular elastic plates," Journal of Applied Mechanics , vol. 23, pp. 532-540, 1956.

[26] C. Yang, "Approximate super- and sub-harmonic response of a multi-DOFs system with local cubic nonlinearities under resonance," Journal of Applied Mathematics , vol. 2012, 2012.

[27] Z. Zhou-Lian, L. Chang-Jiang, H. Xiao-Ting, C. Shan-Lin, "Free vibration analysis of rectangular orthotropic membranes in large deflection," Mathematical Problems in Engineering , vol. 2009, 2009.

[28] R. A. Jafari-Talookolaei, M. H. Kargarnovin, M. T. Ahmadian, M. Abedi, "An investigation on the nonlinear free vibration analysis of beams with simply supported boundary conditions using four engineering theories," Journal of Applied Mathematics , vol. 2011, 2011.

[29] Y. Y. Lee, W. Y. Poon, C. F. Ng, "Anti-symmetric mode vibration of a curved beam subject to autoparametric excitation," Journal of Sound and Vibration , vol. 290, no. 1-2, pp. 48-64, 2006.

[30] Y. Y. Lee, J. L. Huang, C. K. Hui, C. F. Ng, "Sound absorption of a quadratic and cubic nonlinearly vibrating curved panel absorber," Applied Mathematical Modelling , vol. 36, no. 11, pp. 5574-5588, 2012.

[31] W. Y. Poon, C. F. Ng, Y. Y. Lee, "Dynamic stability of a curved beam under sinusoidal loading," Proceedings of the Institution of Mechanical Engineers G: Journal of Aerospace Engineering , vol. 216, no. 4, pp. 209-217, 2002.

[32] R. Soroush, A. Koochi, A. S. Kazemi, M. Abadyan, "Modeling the effect of van der waals attraction on the instability of electrostatic cantilever and doubly-supported nano -beams using modified a domian method," International Journal of Structural Stability and Dynamics , vol. 12, no. 5, 2012.

[33] I. V. Andrianov, V. I. Olevs'kyy, J. Awrejcewicz, "Analytical perturbation method for calculation of shells based on 2D Pade approximants," International Journal of Structural Stability and Dynamics , vol. 13, no. 7, 2013.

[34] M. Gu, S.-Y. Ren, "Parametric vibration of stay cables under axial narrow-band stochastic excitation," International Journal of Structural Stability and Dynamics , vol. 13, no. 8, 2013.

[35] S.-H. Yin, "A new explicit time integration method for structural dynamics," International Journal of Structural Stability and Dynamics , vol. 13, no. 3, 2013.

[36] Y. Y. Lee, H. Y. Sun, J. N. Reddy, "Non-linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates," International Journal for Numerical Methods in Engineering , vol. 65, no. 1, pp. 45-61, 2006.

[37] R. Ma, J. Li, C. Gao, "Existence of positive solutions of a discrete elastic beam equation," Discrete Dynamics in Nature and Society , vol. 2010, 2010.

[38] S. L. Zhu, C. Sheng, "Oscillation and nonoscillation criteria for nonlinear dynamic systems on time scales," Discrete Dynamics in Nature and Society , vol. 2012, 2012.

[39] W. Chen, Z. Han, S. Sun, T. Li, "Oscillation behavior of a class of second-order dynamic equations with damping on time scales," Discrete Dynamics in Nature and Society , vol. 2010, 2010.