Vibro-acoustic coupling analysis of dynamic performance of the double panel system with mechanical links

The theoretical prediction model for the dynamic performance of a double panel system that takes into account the effect of mechanical links is developed in this paper. The double panel system is established considering both the vibro-acoustic coupling and the effect of the mechanical links between two flexible panels. Firstly, the modal characteristics of the double panel system are analyzed and compared with the results calculated by the finite element method. The accuracy and efficiency of the proposed model have been validated. Subsequently, the forced response of the double panel system under different external excitations is studied. It is shown that the mechanical link resulted in less response level of the double panel system in in low frequency range due to the structural path in the energy transmission of the double panel system. Finally, the effect of the mechanical link parameters on the dynamic performance of the double panel system is discussed. The results reveal that the stiffness coefficients, location distributions, and number of mechanical links can significantly influence the dynamic behavior of the double panel system. Therefore, the theoretical model can be used in optimization techniques to improve the sound transmission characteristics of the double panel system.


Introduction
A double panel system usually consists of two flexible panels and a middle cavity.Compared with single panel structures, there are many advantages of double panel systems, such as lightweight, good mechanical, and sound insulation properties.Double panel systems can also be favorably altered to meet specific sound insulation requirements.However, it is challenging to investigate the dynamic performance of double panel systems because of the complex vibro-acoustic coupling phenomena.
2][3][4][5] The modeling approaches were either in analytical or numerical forms.For example, Yairi et al. 6 analyzed the sound radiation of an infinite double-leaf panel by a Helmholtz integral for acoustic sound field coupled with the classical thin plate theory.Anto´nio et al. 7 proposed an analytical model to calculate the acoustic insulation of infinite double panel walls considering the full interaction between the acoustic medium and the structures.Chazot and Guyader 8 predicted the sound transmission loss through double panels by a patch-mobility approach and compared the results with experimental measurements.Xin et al. 9,10 studied the vibro-acoustic performance of a double panel partition with classical boundary conditions by employing the modal superposition method.They explored the influence of system parameters on the sound insulation capability of the double-panel partitions.Craik and Smith 11 used Statistical energy analysis (SEA) to predict sound trans-mission through double leaf lightweight partitions.Sgard et al. 12 discussed the prediction of the low frequency diffuse field transmission loss through double-wall sound barriers based on a finite element model.Sahu et al. 13 developed a coupled finite element-boundary element technique to study the energy transmission characteristics and the sound radiation behavior of a finite double-leaf structure placed in an infinite baffle subjected to external excitation.The previous investigations showed that the sound insulation capacities of the double panel structures would deteriorate at low frequencies because of the mass-air-mass resonance.][16] Most often, the mechanical links connecting two flexible panels of double panel systems are a necessity in some practical applications to support structural loads.The structural transmission path provided by mechanical links will introduce significant alterations to the natural modal and forced response characteristics of the vibro-acoustic coupling system.There are numerous of literature dealing with the double panel system with mechanical links to investigate the effect of the structural energy transmitting path.Lin and Garrelick 17 studied the sound transmission properties of an infinite double panel partition just considering the translational stiffness of the structural connections.Takahashi 18 investigated the sound radiation characteristics of periodically connected infinite double-panel structures in which the acoustic resonances in the middle cavity were neglected.Bao and Pan 19 examined the effect of structural paths on the active control of double panel partitions through experiments.Brunskog 20 presented a prediction model for transmission loss of double-panel structures including the effects of the periodicity studs using the spatial transform technique.Cheng et al. 21studied the energy transmission and noise insulation properties of a mechanically linked doublewall structure, taking into account the vibro-acoustic coupling effect of the middle cavity and mechanical links.Legault and Atalla 22 investigated the sound transmission through an aircraft sidewall representative double panel structure with periodic mechanical links by employing a four-pole formulation to model the links.Takahashi et al. 23 established an analytical model to analyze the double-glazed windows with viscoelastic connectors between the two glass panels and discuss the effects of spacing, contact area, and viscoelastic properties of the connectors on the performance in terms of sound insulation.Yang et al. 24 studied the sound transmission mechanism of a flexibly-linked finite length double-wall structure by using the modal superposition method in conjunction with the envelope rectangular technique.Oyelade 25 proposed a theoretical model for the sound transmission of a double panel clamp mounted on an infinite rigid baffle with magnetically connected stiffness.Wrona et al. 26 presented a novel semi-active control approach for a double-panel system with a bistable link mounted between the panels, in which the acoustic radiation of the double panel system for narrow-band noise can be effectively controlled with the semi-active link.These studies suggest that the structural path is important to be considered in the energy transmission analysis of the double panel system.The transmission loss can be increased significantly when the mechanical links are properly designed.
For the double panel system with mechanical links, it is rather difficult to set up an efficient theoretical model due to the energy transmission mechanisms and the vibro-acoustic coupling effects along with many variables and the consequent error propagation.The numerical methods also cannot provide reliable results in wide frequency range calculations for the high computation costs.In particular, there is still a lack of detailed parameters analysis of the double panel system with mechanical links.Therefore, it is necessary to build theoretical simulation models to evaluate the influence of the parameters on the dynamic behavior of the double panel system.
In the previous work of the Zhang and Du, 27 the panel system composed of two panels connected by mechanical links is examined while ignoring the energy transmission of the bounded fluid between two panels.The results show that different parameters of the mechanical links have significant influences on the structural energy transmission path of the double panel system.The aim of this research is to further compare the strengths of the structural path through the mechanical links and the acoustic path through the middle cavity on the dynamic performance of the double panel system.A concise overview of the prediction model for the dynamic performance of the double panel system with mechanical links is provided in section 2. The improved Fourier series method is employed to describe the field functions of the flexible panels and the middle cavity.The dynamic equations of the double panel system were established by using the energy formulation with the Rayleigh-Ritz method.In section 3, the numerical calculation examples are first conducted to predict the natural modal characteristics of the double panel system.The results are compared to FEM ones to validate the accuracy of the proposed theoretical model.Then the forced response of the double panel system under different external excitations is analyzed.At last, the effects of the mechanical link parameters on the dynamic behavior of the double panel system are discussed in detail.The conclusions and final remarks are listed in section 4.

Model description and formulation
The studied double panel system is illustrated in Figure 1, consisting of two flexible panels mounted on the infinite rigid baffle with the dimensions of a 3 b 3 d 1 and a 3 b 3 d 2 , respectively.The two panels, namely Panel 1 and Panel 2, are connected by a mechanical link, forming a tightly coupled structure.The equivalent forces and moments of the mechanical link connecting two panels have been considered by assuming the translational and rotational stiffnesses of linear springs.Between the two panels, a middle cavity is surrounded by rigid acoustic boundaries.The height of the cavity is h.The vibroacoustic coupling between the panels and the middle cavity led to a complex energy transmission mechanism.
For the bending vibration of the panel structures, the two-dimensional improved Fourier series method is used to describe the displacement function, which can overcome the boundary differential discontinuities of panel structures.The displacement function of the panels is as follows: where , and the expressions of four supplemental functions are given by The acoustic field bounded between the two panels makes up a middle cavity with two flexible walls and four rigid walls.Under the three-dimensional improved Fourier series framework, two supplementary functions will be introduced in the sound pressure function to correct the discontinuity in the first-order spatial derivative on unrigid acoustic boundaries.The sound pressure of the acoustic field in the middle cavity are as follows: where l m c x = m c x p=a, l m c y = m c y p=b, l m c z = m c z p=h.The expressions of the supplemental functions are given by In this work, the energy formulation is employed to drive the governing equations for the double panel system.The Lagrange functions of the panels and the middle cavity can be respectively represented by where U 1 and U 2 are the elastic strain energies of Panel 1 and Panel 2, respectively.T 1 and T 2 are the kinetic energies of the two panels.U link is the potential energy stored in the mechanical link.U c and T c are the acoustic potential and kinetic energy of the middle cavity, respectively.W ext is the work done by external excitations.W cp and W pc are the work done by sound pressure of the middle cavity and structure vibration of the panels on each other which represent the vibro-acoustic interaction between the subsystems.According to energy conservation, it can be derived that The total potential energy and total kinetic energy of the panels can be described by the following expressions: # where represents the bending stiffness of the panels, m is the Poisson's ratio, k i x 0 and K i x 0 are the translational and rotational boundary restraint stiffness coefficients on the structural edge x i = 0, respectively.Similarly, the meanings of k i x a , K i x a , k i y 0 , K i y 0 , k i y a , K i y a can be derived.r i and d i are the mass density and thickness of the panels, respectively.
The potential energy stored in the mechanical link is where k l is the translational stiffness coefficient of the mechanical link.K lx and K ly are the rotational stiffness coefficient of the mechanical link around the x-axis and y-axis, respectively.(x l , y l ) represents the coordinates of the join point between Panel 1 and Panel 2.
The potential energy and kinetic energy of the middle cavity can be described by the following expressions: where r 0 and c 0 are the density and sound speed of the acoustic medium inside the middle cavity.
Considering an external point force acts at the point of (x e , y e ) on Panel 1, the work done by the force is expressed as: where F 0 represents the amplitude of the external force.Substituting the field functions in equations ( 1) and (2) into the Lagrangian functions of the panels and the middle cavity, and minimizing equations will lead to the following matrix equations of the double panel system: With the solved expansion coefficients in equation ( 14), the bending displacement and sound pressure of the double panel system can be directly obtained.Additionally, the modal characteristics of the double panel system can be given by solving the eigenvalue problem of the matrix equations.

Numerical simulation and discussion
In this section, several numerical examples will be performed to validate the proposed theoretical model of the double panel system and analyze the dynamic performance of the double panel system influenced by the parameters of mechanical links including the variation of the stiffness coefficients, the location distributions, and the number.
The numerical simulation parameters are as follows: The dimensions of two flexible panels are 0.

Free modal analysis
The free modal characteristics of the double panel system are first investigated from the previously described theoretical model.The comparison results are calculated by the finite element method (FEM).The double panel system for the eigenvalue analysis is shown in Figure 1: two panels are connected at the midpoint through a single-point mechanical link.As mentioned before, the mechanical link can be assumed as elastic springs to consider the equivalent forces and moments.The stiffness of the translational spring is denoted as k l = 10 9 N/m, and the stiffness of the rotational spring is denoted as K lx = K ly = 0 N/rad in this numerical example.
The first 10 natural frequencies of the double panel system under the classical boundary conditions are indicated in Table 1.The present analytical and FEM results show good agreement in the natural frequencies of the double panel system, which demonstrates the high feasibility and accuracy of the proposed theoretical model.Further improvements in the details of modeling could minimize the differences.
Figures 2 and 3 display the mode shapes of the simply supported double panel system obtained from the present model and FEM, respectively.The vibroacoustic coupling of the double panel system can be performed visually through the figures.It can be observed that some of the modes are generated according to the effects of the single-point mechanical link.

Forced response analysis
To analyze the forced response characteristics of the double panel system with the single-point mechanical link, three kinds of external excitations as shown in Figure 4 are considered.These are as follows: The vibration acceleration response curves of the simply supported double panel system under the excitation of the point and line forces are illustrated in Figures 5 and 6, respectively.The comparison between Table 1.Natural frequency of the double panel system with a single-point mechanical link located at (x l , y l ) = (0.      the present model and FEM further validates the feasibility and accuracy of the theoretical model in the forced response analysis of the double panel system. For the surface force applied to the double panel system, the average quadratic acceleration of panel structures, as defined in equation ( 16), is calculated to measure the response characteristics of the double panel system.
where A i is the surface area of the panel.The average acceleration level of the simply supported double panel system excited by the surface force is plotted in Figure 7.To indicate the structural path of the energy transmission of the double panel system, the results of the double panel system with and without the single-point mechanical link are also presented.
As can be seen in Figure 7, the resonance peaks of the curves are moved to the higher frequencies because the mechanical link forms a structural path of energy transmission in the double panel system.The forced response of the double panel system with mechanical links at a certain frequency could be less than a double panel system without the structural transmission path.This is a result of reducing the additional mass of the middle cavity in traditional double panel systems in low frequency.

Parameter analysis of mechanical links
The effect of the mechanical link parameters, the stiffness coefficients, the location distributions, and the number, on the dynamic performance of the double panel system, is studied in this section.All simulations assume the surface force excitation applied on Panel 1 of the simply supported double panel system.
The first parameter to be studied is the stiffness of mechanical links.The effect of the translational and rotational stiffness of the single-point mechanical link on the increase in the averaged acceleration level of the double panel system is presented in Figures 8 and 9, respectively.
It can be found that the effect of the mechanical link stiffness on the averaged acceleration level of the double panel system depends on the energy transmission paths.The higher stiffness decreases the response of Panel 2 in the low frequency range and increases it in the high frequency range, and there is no significant change in the response of Panel 1.
Then the effect of location distributions of the single-point mechanical link on the double panel system dynamic performance is presented.Four locations of the mechanical link are considered as shown in Figure 10.The link stiffness coefficients of the adopted singlepoint mechanical link are considered as k l = 10 4 N/m and K lx = K ly = 10 2 N/rad.
The average acceleration level of Panel 2 with different mechanical link locations is plotted in Figure 11.The results show that the locations of the mechanical link significantly affected the response of the double panel system at the specific frequencies while the vibration mode shapes of Panel 1 and Panel 2 present opposite directions due to better impedance matching.In contrast, there will not be an obvious impact of the mechanical link position on the response of the double panel system if the vibration mode shapes of Panel 1 and Panel 2 present the same direction.The next mechanical link parameter analyses use four distributions illustrated in Figure 12 to study the influence of the number of mechanical links on the response characteristics of the double panel system.
The average acceleration level of Panel 2 under the variation of the number of mechanical links is given in Figure 13.In the low-frequency region, the response of the double panel system decreases and the resonance frequencies move to higher frequency with increasing the number of mechanical links.In the high-frequency region, the response of the double panel system increases as the number of mechanical links increases because of the better energy transitivity of structural paths.The results provide further evidence of the above analysis.

Conclusion
In this study, a theoretical model considering the fully vibro-acoustic coupling interactions is established to predict the dynamic performance of the double panel system with mechanical links.The structural and acoustic field functions of the double panel system are described by adopting the improved Fourier series method, and the governing equations of the system are obtained based on the energy principle.The validity of the proposed model is verified by comparing the present results with the numerical results calculated by FEM firstly.From the results of the simulations, it can be found that the free modal and forced response characteristics of the double panel system are significantly influenced by the mechanical links.Consequently, the effect of the mechanical links on the dynamic performance of the double panel system is further analyzed and discussed.The parameter analysis results have shown that the additional mass effect of the middle cavity in the double panel system can be decreased in the low frequency range by the structural energy transmission path   of mechanical links.This investigation can be applied to the vibration and noise control of the double panel system by taking advantage of the stiffness and distribution of the mechanical links between the panel structures without significantly changing the structures.

Figure 1 .
Figure 1.Schematic diagram of the double panel system with a mechanical link.

where K 1 ,
K 2 , and K c are the stiffness matrices of Panel 1, Panel 2, and the middle cavity, respectively.M 1 , M 2 , and M c are the mass matrices of Panel 1, Panel 2, and the middle cavity, respectively.C 11 , C 12 , C 21 , and C 22 are the contribution matrices of the mechanical link.C 1c , C 2c , C c1 , and C c2 are the contribution matrices of the vibro-acoustic coupling.F 1 is the external load vector caused by the point force acted on Panel 1. W 1 , P and W 2 are the vectors that contain all the unknown Fourier expansion coefficients in the field functions of the double panel system.And 5 3 0.4 3 0.003 m 3 and 0.5 3 0.4 3 0.002 m 3 , respectively.Both panels are assumed to be made of aluminum with Young's modulus of 71 GPa, Poisson's ratio of 0.3, and a density of 2750 kg/m 3 .The height of the middle cavity h = 0.3 m.The air in the middle cavity is assumed to have parameter values of c 0 = 340 m/s and r 0 = 1.21 kg/m.
(a) Panel 1 excited by the unit point force on (0.05, 0.04) m, (b) Panel 1 excited by the unit line force along the diagonals, and (c) Panel 1 excited by the unit surface force.

Figure 2 .
Figure 2. Mode distribution of the simply supported double panel system calculated by the present model.

Figure 3 .
Figure 3. Mode distribution of the simply supported double panel system calculated by FEM.

Figure 4 .
Figure 4. External excitation of the double panel system: (a) point force, (b) line force, and (c) surface force.

Figure 5 .
Figure 5.The acceleration response of the simply supported double panel system excited by the point force: (a) (0.05, 0.04) m on Panel 1 and (b) (0.45, 0.36) m on Panel 2.

Figure 6 .
Figure 6.The acceleration response of the simply supported double panel system excited by the line force: (a) (0.05, 0.04) m on Panel 1 and (b) (0.45, 0.36) m on Panel 2.

Figure 7 .
Figure 7.The average acceleration level of the simply supported double panel system excited by the surface force: (a) Panel 1 and (b) Panel 2.

Figure 8 .
Figure 8.The average acceleration level of the double panel system with respect to the translational stiffness of the mechanical link: (a) Panel 1 and (b) Panel 2.

Figure 9 .
Figure 9.The average acceleration level of the double panel system with respect to the rotational stiffness of the mechanical link: (a) Panel 1 and (b) Panel 2.