A Particle Reinforced Gradient Honeycomb Sandwich Panel for Broadband Sound Insulation

Abstract

The sound insulation capacity of traditional sound insulation boards is limited by the law of mass, and any improvement in sound insulation capacity contradicts the demand for light weight. In order to overcome this shortcoming, a lightweight particle reinforced gradient honeycomb sandwich panel is proposed to achieve lightweight sound insulation. The sound insulation of the particle reinforced honeycomb sandwich panel is calculated based on the transfer matrix method. The accuracy of the theory is verified through finite element simulation, and the influence of material and structural parameters on the sound insulation performance of the sandwich panel is analyzed. The results show that the sound insulation of the honeycomb sandwich panel can be significantly improved by adding reinforcement particles to the aluminum matrix, and the sound insulation also increases as the particle mass fraction of the reinforcement increases. In addition, the valley value of the sound insulation curve moves towards the low frequency direction, which indicates that the sound insulation performance of the sandwich plate at low frequencies is effectively improved.

1. Introduction

The sound insulation capacity of traditional sound insulation panels is limited by the law of mass, so the requirements of improving the sound insulation capacity of panels and reducing weight cannot be met at the same time. Zhang Zhifu et al. [1] proposed an acoustic insulation metamaterial with distributed piezoelectric resonators based on the problem of unsatisfactory sound transmission loss (STL) when using thin-plate structures in the low–mid frequency range. The results show that sound insulation in the low–mid frequency range is improved by more than 5 dB compared with that of the substrate, even up to 44.49 dB. Hicham Kaidouchi et al. [2] used finite element analysis software to study the acoustic properties of sandwich plates with different materials, and the results showed that glass fiber reinforced polymer (GFRP) cores with fiber reinforced plastic (FRP) finishing materials had better acoustic properties. GFRP cores with FRP surfaces can be used instead of aluminum cores, thus reducing the quality of the panels. Lee [3] analyzed the forced vibration and transmission loss of a nonlinear vibrating panel supported by a partition cavity using the elliptic integration method; the results showed that nonlinear vibration would make overall transmission loss worse for wideband excitation, and linear vibration would make overall transmission loss worse for narrowband excitation. Compared with other sandwich panel structures, the lightness of a honeycomb core sandwich panel is particularly prominent in stiffness, vibration reduction, energy absorption, impact resistance, heat insulation and other properties, and has been widely studied and applied in the fields of aerospace, ships and railways [4].

The sound insulation performance of a honeycomb sandwich panel is related to the structure of the honeycomb core layer. At first, research on the sound insulation performance of honeycomb core layers depended on the parameter equivalent theory of the honeycomb core layer. Relevant scholars deduced the stress deformation of the cell structure based on the beam bending model in material mechanics. Gibson [5] deduced the Young’s modulus and Poisson’s ratio of the honeycomb core layer based on the traditional beam model theory, obtained the expression of the above parameters, and analyzed the sound insulation performance of the honeycomb core layer. Kumar et al. [6] equated the traditional honeycomb core layer to an orthotropic plate based on the structural bending wave method and homogenization equivalent theory, and solved the acoustic propagation loss of a honeycomb core sandwich panel structure. Wang [7] established a theoretical prediction model for the sound insulation of honeycomb core sandwich panels based on Biot theory and transfer matrix method, and verified the effectiveness of the transfer matrix method in calculating the sound insulation of a honeycomb core sandwich panel by comparing the theoretical prediction results with the experimental results. In addition, a small number of scholars began to pay attention to honeycomb core sandwich panels with structural gradients [8,9]. Mazloomi et al. [10] designed a concave honeycomb core sandwich panel structure with internal angle gradient change and solved its natural frequency and radiated sound power. Li et al. [11] studied the acoustic characteristics of a honeycomb core sandwich panel with gradient internal angles based on the spectral element method. The above research on sound insulation performance plays a theoretical support and guidance role in the design of new lightweight sound insulation structures.

In addition to the core structure, the materials of the sandwich panel and core also have a significant impact on the sound insulation characteristics of sandwich panel. A new type of composite material, particle reinforced composite materials can be obtained by combining reinforcement particles with a matrix through physical or chemical methods. In addition to the preparation process, the parameters and properties of reinforcement particles also have a significant impact on the mechanical properties of particle reinforced composites [12,13,14,15]. Mori T [16] proposed the Mori Tanaka model to calculate the properties of particle reinforced composite materials through the properties of matrix materials and reinforcement particle materials. Xu [17] predicted the mechanical parameters of functionally graded nano-reinforced plates under thermal load based on the Halpin Tsai micromechanical model, and studied the acoustic and vibration characteristics of functionally graded nano-reinforced plates based on high-order shear deformation plate theory. Compared with traditional composite structures, a particle reinforced gradient composite has the advantages of good mechanical properties, strong designability and versatility, and it can go beyond the potential of the material itself to develop new lightweight structures and new acoustic materials [18,19].

Based on the above research, a particle reinforced composite was applied to a gradient honeycomb core sandwich plate for the first time. By adding particle reinforcement and developing honeycomb sandwich plates that account for the gradient effect of the structure and material, the Young’s modulus of the whole structure is improved and the mass of the core structure is reduced, achieving light sound insulation, which breaks with the traditional design idea that improving overall sound insulation performance mainly depends on improving the quality of the sandwich plates, and ensures a light weight of the structure.

2. Theory

A gradient honeycomb core sandwich panel is composed of a panel and a gradient honeycomb core layer, as shown in Figure 1. The thickness, Young’s modulus, density, Poisson’s ratio and area of the m-th layer can be denoted h_m, E_m, ρ_m. v_m and S_m, respectively.

2.1. Equivalent Parameters of Particle Reinforced Composites

The material parameters of particle reinforced composites can be solved by the material parameters of matrix and reinforcement. Based on Mori-Tanaka equivalent theory [16], the Young’s modulus E and Poisson’s ratio v of the above materials can be expressed as:

$$E=\frac{9KG}{3K+G}$$

(1)

$$v=\frac{3K-2G}{2(3K+G)}$$

(2)

where K is the bulk modulus of the composite and G is the shear modulus of the composite. The expressions are:

$$K=K_m+\frac{V_{\mathrm{r}}\left(K_r-K_m\right)}{1+\left(1-V_r\right)\left[3\left(K_r-K_m\right)/\left(3K_{\mathrm{m}}+4G_m\right)\right]}$$

(3)

$$G=G_m+\frac{V_r\left(G_r-G_m\right)}{1+\left(1-V_r\right)\left[\left(G_r-G_m\right)/\left(G_r+r_m\right)\right]}$$

(4)

where K_m is the bulk modulus of the matrix, K_r is the bulk modulus of the reinforcement, G_m is the shear modulus of the matrix, G_r is the shear modulus of the reinforcement, r and m are the intermediate parameters of the reinforcement, and V_r is the volume fraction of the reinforcement.

According to the relationship between mass, volume and density, the total volume fraction of reinforcement particles has the following relationship with the total mass fraction:

$$V_r=\frac{W_r}{W_r+\left({\rho }_r/{\rho }_m\right)\left(1-W_{\mathrm{r}}\right)}$$

(5)

where ρ_r is the density of the reinforcement, ρ_m is the matrix density and W_r is the mass fraction of the reinforcement particles. According to the gradient change in the volume fraction of each sublayer of the honeycomb core, the following particle distribution modes of reinforcement are given. Depending on its symmetry or asymmetry and linearity or nonlinearity, the gradient distribution mode of the reinforcement particles is categorized as a V-type distribution, X-type distribution, O-type distribution or C-type distribution. The schematic diagram is shown in Figure 2.

The volume fractions of reinforcement particles with the above gradient distributions are expressed as follows [20]:

$$V_{\mathrm{V}}=V_{\mathrm{r}}\times \frac{\left(2k-1\right)}{N}$$

(6)

$$V_{\mathrm{X}}=2V_{\mathrm{r}}\times \frac{\left|2k-1-N\right|}{N}$$

(7)

$$V_{\mathrm{O}}=2V_{\mathrm{r}}\times \left(1-\frac{\left|2k-1-N\right|}{N}\right)$$

(8)

$$V_{\mathrm{C}}=3V_{\mathrm{r}}\times {\left(\frac{2k-1-N}{N}\right)}^2$$

(9)

2.2. Sound Insulation Theory of Honeycomb Sandwich Panel

The key to calculate the sound insulation of a honeycomb sandwich panel is to analyze the propagation law of sound waves in a solid. By calculating the transmission relationship between sound pressure and vibration velocity on both sides of the medium, the transmission matrix and sound insulation loss of the sandwich plate are obtained. Firstly, the scalar displacement potential function and vector displacement potential function are introduced to describe the P-wave and S-wave components of the sound wave in a solid, respectively [21,22]. The displacement potential function of a sound wave can be written as:

$$\begin{array}{l}\varphi \left(x,z,t\right)=\left(a_{+}^{\mathrm{L}}{e}^{i{k}_z^{\mathrm{L}}z}+a_{{}_{-}}^{\mathrm{L}}{e}^{-i{k}_z^{\mathrm{L}}z}\right)e^{i\left(Kx-\omega t\right)}\\ \phi \left(x,z,t\right)=\left(a_{{}_{+}}^{\mathrm{T}}{e}^{i{k}_z^{\mathrm{T}}z}+a_{{}_{-}}^{\mathrm{T}}{e}^{-i{k}_z^{\mathrm{T}}z}\right)e^{i\left(Kx-\omega t\right)}\end{array}$$

(10)

where the amplitudes of forward and backward-propagating sound waves are denoted as a₊ and a₋, respectively, and L and T represent P-waves and S-waves, respectively. k_z^T and k_z^L are, respectively, the S-wave and P-wave components of the wave vector k_z in the z direction, which can be solved by the following equation [21]:

$$\left\{\begin{array}{l}{\overrightarrow{k^{\mathrm{L}}}}^2={\overrightarrow{k_z^{\mathrm{L}}}}^2+{\overrightarrow{k_x}}^2\\ {\overrightarrow{k^{\mathrm{T}}}}^2={\overrightarrow{k_z^{\mathrm{T}}}}^2+{\overrightarrow{k_x}}^2\end{array}\right.$$

(11)

$${\left(\overrightarrow{k^{\mathrm{L}}}\right)}^2=\frac{\rho {\omega }^2}{\lambda +2\mu },{\left(\overrightarrow{k^{\mathrm{T}}}\right)}^2=\frac{\rho {\omega }^2}{\mu }$$

(12)

The vibration displacement function in the x and z directions can be solved by the potential functions [23]:

$$w_x\left(x,z,t\right)=\frac{\partial \varphi }{\partial x}-\frac{\partial \phi }{\partial z},w_z\left(x,z,t\right)=\frac{\partial \varphi }{\partial z}+\frac{\partial \phi }{\partial x}$$

(13)

The vibration velocity and stress in the x and z directions can be solved from the vibration displacement [23]:

$$v_x=\frac{\partial w_x}{\partial t},v_z=\frac{\partial w_z}{\partial t},T_x=\mu \left(\frac{\partial w_x}{\partial x}+\frac{\partial w_z}{\partial z}\right),T_z=\lambda \left(\frac{\partial w_x}{\partial x}+\frac{\partial w_z}{\partial z}\right)+2\mu \frac{\partial w_z}{\partial z}$$

(14)

In order to facilitate the calculation of the transfer matrix of each layer, the intermediate quantity is defined by the sum difference expression of the displacement potential function:

$$\begin{array}{l}{C}^{\mathrm{L}}\left(z\right)=a_{+}^{\mathrm{L}}{e}^{i{k}_z^{\mathrm{L}}z}+a_{-}^{\mathrm{L}}{e}^{-i{k}_z^{\mathrm{L}}z},D^{\mathrm{L}}\left(z\right)=a_{+}^{\mathrm{L}}{e}^{i{k}_z^{\mathrm{L}}z}-a_{-}^{\mathrm{L}}{e}^{-i{k}_z^{\mathrm{L}}z}\\ C^{\mathrm{T}}\left(z\right)=a_{+}^{\mathrm{T}}{e}^{i{k}_z^{\mathrm{T}}z}+a_{-}^{\mathrm{T}}{e}^{-i{k}_z^{\mathrm{T}}z},D^{\mathrm{T}}\left(z\right)=a_{+}^{\mathrm{T}}{e}^{i{k}_z^{\mathrm{T}}z}-a_{-}^{\mathrm{T}}{e}^{-i{k}_z^{\mathrm{T}}z}\end{array}$$

(15)

The relationship between stress, vibration velocity and intermediate quantity is as follows:

$$\left[\begin{array}{l}{v}_x\\ v_z\\ T_z\\ T_x\end{array}\right]=\chi \left[\begin{array}{l}{C}^{\mathrm{L}}\\ D^{\mathrm{L}}\\ D^{\mathrm{T}}\\ C^{\mathrm{T}}\end{array}\right]i{e}^{i\left(Kx-\omega t\right)}$$

(16)

and

$$\chi =\left[\begin{array}{cccc}\omega K& 0& -\omega k_z^{\mathrm{T}}& 0\\ 0& \omega k_z^{\mathrm{L}}& 0& \omega K\\ -\rho {\omega }^2\varsigma & 0& -2\rho {\omega }^2{k}_z^{\mathrm{T}}\gamma /k^{\mathrm{T}}& 0\\ 0& -2\rho {\omega }^2{k}_z^{\mathrm{L}}\gamma /k^{\mathrm{T}}& 0& \rho {\omega }^2\varsigma \end{array}\right]$$

(17)

where the matrix element contains an intermediate quantity.

Based on the sum difference expression of the displacement potential function in Equation (15), the transfer matrix of the sum difference expression between the interfaces on both sides of each layer of the medium can be solved via:

$${\left[\begin{array}{c}{C}^{\mathrm{L}}\\ D^{\mathrm{L}}\\ D^{\mathrm{T}}\\ C^{\mathrm{T}}\end{array}\right]}_{z+h}=P\left(h\right){\left[\begin{array}{c}{C}^{\mathrm{L}}\\ D^{\mathrm{L}}\\ D^{\mathrm{T}}\\ C^{\mathrm{T}}\end{array}\right]}_z$$

(18)

where P(h) is the transfer matrix of the medium layer:

$$P\left(h\right)=\left[\begin{array}{cccc}\cos (k_z^{\mathrm{L}}h)& i\sin (k_z^{\mathrm{L}}h)& 0& 0\\ i\sin (k_z^{\mathrm{L}}h)& \cos (k_z^{\mathrm{L}}h)& 0& 0\\ 0& 0& \cos (k_z^{\mathrm{T}}h)& i\sin (k_z^{\mathrm{T}}h)\\ 0& 0& i\sin (k_z^{\mathrm{T}}h)& \cos (k_z^{\mathrm{T}}h)\end{array}\right]$$

(19)

In combination with Equations (17) and (19), the transfer matrix of the vibration velocity and stress between the interfaces on both sides of each layer of the medium can be deduced [24]:

$${\left[\begin{array}{c}{v}_x\\ v_z\\ T_z\\ T_x\end{array}\right]}_{z+h}=\chi P\left(h\right){\chi }^{-1}{\left[\begin{array}{c}{v}_x\\ v_z\\ T_z\\ T_x\end{array}\right]}_z=A\left(h\right){\left[\begin{array}{c}{v}_x\\ v_z\\ T_z\\ T_x\end{array}\right]}_z$$

(20)

For a honeycomb core layer with a gradient structure, the honeycomb is divided into multiple sub layers along the direction perpendicular to the panel, as shown in Figure 3. Calculate the area of each sublayer, construct the area matrix, and modify the transfer matrix of the sound insulation theory through the area matrix. The expression of the area matrix is as follows:

If the thickness of each sublayer is small enough, the cross-sectional area of the sublayer can be regarded as unchanged. The relationship between the vibration velocity component and the stress component between upper and lower interfaces in each sublayer is as follows:

$${\left[\begin{array}{c}{v}_x^j\\ v_z^j\\ F_z^j\\ F_x^j\end{array}\right]}_{z_j}=S_c^j\circ A\left(\frac{h}{n}\right){\left[\begin{array}{c}{v}_x^j\\ v_z^j\\ F_z^j\\ F_x^j\end{array}\right]}_{z_{j-1}}$$

(21)

where n is the equivalent number of layers of the gradient honeycomb layer, j is the serial number of the sublayer, h is the thickness of the gradient honeycomb layer, A is the transfer matrix of the medium of each sublayer, S_c is the area matrix of each sublayer, and the specific expression of S_c is as follows [21,24]:

$$S_c^j=\left[\begin{array}{cccc}1& 1& 1/S^j& 1/S^j\\ 1& 1& 1/S^j& 1/S^j\\ S^j& S^j& 1& 1\\ S^j& S^j& 1& 1\end{array}\right]$$

(22)

Recursively multiply the transfer matrix of each sublayer in Equation (21) to obtain the total transfer matrix B(h) considering the change in cross-sectional area:

$${\left[\begin{array}{c}{v}_x^j\\ v_z^j\\ F_z^j\\ F_x^j\end{array}\right]}_{z_n}=B\left(h\right){\left[\begin{array}{c}{v}_x^j\\ v_z^j\\ F_z^j\\ F_x^j\end{array}\right]}_{z_0}={\displaystyle \prod _{j=1}^n\left(S_c^j\circ A\left(h/n\right)\right)}{\left[\begin{array}{c}{v}_x^j\\ v_z^j\\ F_z^j\\ F_x^j\end{array}\right]}_{z_0}$$

(23)

The relationship between the total transfer matrix of the structure and the transfer matrix of each layer:

$$T=B_n\left(h_n\right)\times B_{n-1}\left(h_{n-1}\right)\times \cdots \times B_1\left(h_1\right)$$

(24)

Both sides of the sandwich plate structure were set to be in a fluid medium environment. Under this condition, the vibration velocity component in the x direction is discontinuous, and the stress component in the x direction is 0. Therefore, the relationship between the total transfer matrix and the vibration velocity and stress on both sides of the sandwich plate structure can be simplified as follows [24,25]:

$${\left[\begin{array}{c}{v}_z\\ T_z\end{array}\right]}_{z_n}=T^{\prime }{\left[\begin{array}{c}{v}_z\\ T_z\end{array}\right]}_{z_0}$$

(25)

And

$$T^{\prime }=\left[\begin{array}{cc}{T}_{22}-\frac{T_{21}{T}_{42}}{T_{41}}& T_{23}-\frac{T_{21}{T}_{43}}{T_{41}}\\ T_{32}-\frac{T_{31}{T}_{42}}{T_{41}}& T_{33}-\frac{T_{31}{T}_{43}}{T_{41}}\end{array}\right]$$

(26)

Assuming that both sides of the sandwich plate structure are infinite spaces, the boundary continuity conditions in this case are [24,25]:

$$\tau \left[\begin{array}{c}1\\ Z_n\end{array}\right]=T^{\prime }\left[\begin{array}{c}1+r\\ Z_0\left(1-r\right)\end{array}\right]$$

(27)

where τ Is the transmission coefficient, r is the reflection coefficient, Z₀ is the dielectric impedance on one side of the composite plate structure, Z_n is the dielectric impedance on the other side, and the impedances can be expressed as follows:

$$Z_0=\frac{{\rho }_0{c}_0}{\cos {\theta }_0},Z_n=\frac{{\rho }_n{c}_n}{\cos {\theta }_n}$$

(28)

The expression of the sound pressure transmission coefficient can be obtained by solving Equation (27):

$$\tau =\frac{2\left({T^{\prime }}_{12}{T^{\prime }}_{21}-{T^{\prime }}_{11}{T^{\prime }}_{22}\right)Z_0}{{T^{\prime }}_{21}-Z_0{T^{\prime }}_{22}-\left({T^{\prime }}_{11}-Z_0{T^{\prime }}_{12}\right)Z_n},$$

(29)

The sound intensity transmission coefficient can be solved using the sound pressure transmission coefficient:

$${\tau }^{\prime }=1-\tau \cdot {\tau }^{\ast }$$

(30)

The relationship between sound insulation loss and the sound intensity transmission coefficient is:

$$TL=-10\lg \left({\tau }^{\prime }\right)$$

(31)

3. Foundation and Validation of Theoretical Analysis

3.1. Parameter Characteristics and Preconditions of Model Application

For the relevant parameters of the particle reinforced gradient honeycomb sandwich plate given by the above model, the incident medium is air, ρ₀ = 1.21 kg/m³, sound velocity c₀ = 344 m/s. The materials of the panel and core layer are composite materials with aluminum alloy as the base material and carbon nanoparticles as the reinforcement particles. The density of the aluminum alloy is ρ_m = 2700 kg/m³, the Young’s modulus is E_m = 70 Gpa, and Poisson’s ratio is v_m = 0.33; the density of the carbon nanoparticles is ρ_r = 1500 kg/m³, the Young’s modulus E_r = 3700 Gpa, and Poisson’s ratio v_r = 0.304; the mass fraction of reinforcement particles is 10%, and their distribution mode is a C-type distribution. The surface thickness of the gradient honeycomb sandwich panel is d = 0.5 mm, the core height is h = 9 mm, the honeycomb wall thickness is t = 0.1 mm, the minimum honeycomb side length is a₁ = 3 mm, and the maximum honeycomb side length is a₂ = 4 mm. The geometric parameters of the honeycomb core layer are shown in Figure 4.

In order to calculate and analyze the sound insulation characteristics of the whole frequency band to the greatest extent possible, the sound insulation loss in the 10–3000 Hz frequency band of the particle reinforced gradient honeycomb sandwich panel is calculated and verified in this paper.

3.2. Model Validation

In this section, simulation analysis is carried out using the finite element method to verify the reliability of the model. The basic idea of this method is to discretize the structure and use a finite number of elements to represent the composite matching layer. The elements are connected by a limited number of nodes, solved according to their deformation coordination conditions, and then verified using the theoretical method in this paper. The finite element model is shown in Figure 5. The background pressure field is applied to the vertical incident air field. The calculation model is divided into five layers from top to bottom. The top layer is a perfect matching layer (PML), the thickness of which is 2 mm; the second layer is an air domain, the thickness of which is 2 mm; the third layer is the sandwich plate structure; the fourth layer is also an air domain, set in the model as 2 mm; the bottom layer is a perfect matching layer (PML) with a thickness of 2 mm. The air domain represents the background pressure field, and the solid domains represent the panel and core, respectively. Taking a gradient honeycomb cell as a periodic element, each side boundary of the hexagonal prism is set as a periodic Floquet boundary, which is used to represent the infinite sandwich plate structure. Above the air domain, the perfectly matched layer (PML) represents a semi-infinite air domain. The PML layer and air domain at the top and bottom were used to construct an 8-layer mapping grid, the air domain and gradient honeycomb structure inside the cellular core were used to construct a sweeping grid, and the air domain between cells was used to construct a free quadrilateral grid. The mesh size is determined by the maximum frequency according to analysis and calculation. Refining the mesh of the finite element model changes the convergence of the mesh. If two consecutive mesh refinements do not fundamentally change the results, the mesh can be considered to have converged and no further refinement is required. The convergence of the grid greatly affects calculation accuracy. In this paper, the grid size is defined as 0.1 mm by adjusting the grid density, so that the results of sound insulation calculation can converge.

Compare the sound insulation loss calculated via the theoretical model and the finite element model under the same parameters; the curve is shown in Figure 6. The theoretical solution and the simulation solution agree well in most frequency bands, which proves the correctness of the proposed theoretical model. In addition, through comparison, it can be seen that the two curves only have a certain amount of error in the medium and high frequency regions around 3000 Hz. This is because, in order to improve computational efficiency, mesh generation is not strictly in accordance with the requirements for acoustic mesh.

It can be seen from Figure 6 that at 10–1000 Hz, the sound insulation loss of the sandwich plate first increases and then decreases as frequency increases, and the peak value at 580 Hz is about 18 dB. In the frequency band above 1000 Hz, the sound insulation loss increases as frequency increases. The above rules are in accordance with the law of sound insulation quality of the panel. With a change in frequency, the sound insulation loss will successively produce a stiffness control zone, a structural resonance zone, a quality control zone and a coincidence effect zone. Coincidence frequency is an important sign of the sound insulation curve, and its expression is as follows:

$$f=\frac{c_0^2}{2\pi h{\sin }^2\phi }\sqrt{\frac{12\rho \left(1-{\nu }^2\right)}{E}}$$

(32)

where c₀ is the speed of sound in the air, h is the thickness of the plate, ρ is the plate density, φ is angle of incidence of the sound waves, ν is the Poisson’s ratio of the material, and E is the Young’s modulus of the material.

4. Parameter Research and Discussion

4.1. Influence of Geometric Parameters of Honeycomb

Gradient honeycomb contains many geometric parameters, such as panel thickness, core height, honeycomb side length, honeycomb wall thickness, and so on. In this section, the influence of the above parameters on sound insulation in the gradient honeycomb sandwich panel will be analyzed in turn.

4.1.1. Effects of Panel Thickness and Core Height

The panel is very thin compared to the sandwich, but it is stiff and dense and takes most of the bending load. The panel thickness was changed to d = (0.5 mm, 1 mm, 1.5 mm), and the influence of sandwich panel thickness on sound insulation was compared and analyzed. According to Figure 7a, sound insulation loss improves and the coincidence frequency of the sandwich panel slightly decreases as the panel thickness increases. The bending stiffness and surface density of sandwich panels increase with the increase of panel thickness. According to the sound insulation characteristic curve, the sound transmission loss in the stiffness control area in the low frequency range and the quality control area in the middle and high frequency range are both improved. At the same time, since the influence of panel thickness change on bending stiffness is more significant, the wavelength of the bending wave of the sandwich panel structure increases; thus, the coincidence frequency decreases slightly with increasing panel thickness at the coincidence. As a conclusion, within the design requirements for sandwich panel quality, the panel thickness can be increased to improve sound insulation loss.

The core layer height was changed to h = (9 mm, 12 mm, 15 mm), and the effect of honeycomb core layer height on sound insulation was compared and analyzed. According to Figure 7b, sound insulation increases with increasing core layer height, since the increase in core layer thickness can effectively improve the bending stiffness of the sandwich panel, and the surface density of the sandwich panel also increases. However, as the height of the core layer increases, its effect on improvement in sound insulation gradually decreases. Meanwhile, with increasing core layer height, the coincidence frequency of the sandwich panel declines significantly. As the core layer thickness increases, the bending stiffness of the sandwich panel is enhanced because the core layer keeps it away from the middle surface. At the same time, since the core layer’s density is much smaller than that of the panel, a change in its height has little effect on the quality of the sandwich panel. As a result, the coincidence frequency declines significantly with increasing core layer height. In conclusion, within the stiffness design requirements for the sandwich panel, the coincidence frequency can be effectively adjusted by appropriately increasing or decreasing the core layer thickness.

4.1.2. Effects of Honeycomb Side Length

First, keep the maximum side length a₂ = 4 mm unchanged, change the minimum side length of the gradient honeycomb core to a₁ = (3 mm, 3.4 mm, 3.8 mm), and analyze the influence of the minimum side length of the gradient honeycomb core on sound insulation, as shown in Figure 8a. Secondly, keep the minimum side length a₁ = 3 mm unchanged, change the maximum side length of the gradient honeycomb core to a₂ = (3.2 mm, 3.6 mm, 4 mm), and analyze the influence of the maximum side length of the gradient honeycomb core on sound insulation, as shown in Figure 8b.

It can be seen from Figure 8 that sound insulation loss increases with increasing minimum side length of the gradient honeycomb and decreases with increasing maximum side length of the gradient honeycomb. This is because change in side length affects the equivalent density of the honeycomb core, and the number of honeycomb cells per unit area within the sandwich panel and the degree of filling density also change. It can be seen that the sound insulation loss in the gradient honeycomb is related not only to the size of both ends of the core layer, but also to the size of periodic elements used for analysis. In the analysis in this paper, the maximum side length of a single gradient cell is usually taken to be the periodic cell size of the cell. However, according to a comparison of Figure 8 and Figure 9, an increase in honeycomb side length has a weaker effect on improvement of sound insulation than changes in panel thickness and core height.

4.1.3. Effects of Honeycomb Wall Thickness

The honeycomb core wall thickness was changed to t = (0.1 mm, 0.2 mm, 0.3 mm), and the influence of honeycomb core wall thickness on sound insulation was compared and analyzed. It can be seen from Figure 9 that sound insulation loss increases with increasing wall thickness of the honeycomb core layer. Due to the increase in honeycomb wall thickness, the bending stiffness and surface density of the sandwich panel improve, thus enhancing its sound insulation loss. At the same time, the sound insulation valley frequency starts to move towards the high frequency direction, which is due to the fact that the effect of honeycomb wall thickness change on the surface density of the sandwich panel structure is greater than its effect on the bending stiffness of the sandwich panel structure. This indicates that increasing the honeycomb wall thickness is also an effective way to optimize sound insulation in the studied test band. However, increasing the honeycomb wall thickness, as well as increasing the panel thickness and core height, significantly changes the surface density of the sandwich panel and impacts the overall structural quality.

4.2. Influence of Particle Mass Fraction of Reinforcement

Carbon nanoparticles with mass fractions of W_r = (0, 5%, 10%) were added to the sandwich plate as reinforcement, and the influence of the particle mass fraction of reinforcement on sound insulation was compared and analyzed. Assuming that the reinforcement particle gradient distribution mode is a C-shaped distribution, using Equations (1)–(5), the Young’s moduli can be calculated for the three mass fractions, which are 70 GPa, 182 GPa and 278 GPa, respectively. The calculation results show that the Young’s modulus gradually increases as the mass fraction of reinforcement particles increases. According to Figure 10, sound insulation loss improves and the sound insulation valley value moves to lower frequencies as the mass fraction increases. Based on the sound insulation’s characteristic curve, the surface density of the sandwich panel increases due to increasing mass fraction, which enhances the sound transmission loss in the mass control area, and the coincidence frequency shifts forward with an increase in Young’s modulus, which makes the sound insulation in the coincidence effect area improve significantly. In addition, by comparing Figure 7, Figure 8, Figure 9 and Figure 10, the effect of reinforcement particle mass fraction on sound insulation, as well as coincidence frequency range, is more significant compared to the parameters of honeycomb geometry.

4.3. Sound Insulation and Light Weight

The sound insulation loss of a traditional honeycomb sandwich panel and the particle reinforced gradient honeycomb sandwich panel with W_r = 10% are calculated respectively, and the comparison results are shown in Figure 11. The results show that the gradient honeycomb sandwich plate structure proposed in this paper has higher sound insulation in most frequency bands than the traditional honeycomb sandwich plate under the same quality conditions. In addition, the calculation shows that the area density of the gradient honeycomb sandwich panel is 3.20 kg/m³, and that of the traditional honeycomb sandwich panel is 4.12 kg/m³. Therefore, the gradient honeycomb is more suitable for light sound insulation.

5. Conclusions

In this paper, particle reinforcement and a gradient honeycomb core sandwich plate are combined, and using the transfer matrix method to calculate the sound insulation of the particle reinforced gradient honeycomb core sandwich plate, the accuracy of the theory is verified via finite element simulation. It is found that the height of the core layer, thickness of the panel and thickness of the honeycomb wall affect the quantity of sound insulation and the fitting frequency, mainly by changing the stiffness and surface density of the structure, while changing the honeycomb side length mainly leads to changes in the density of the periodic structure, which provides a theoretical basis for the geometric design of the core layer. When reinforced particles are added to the aluminum matrix, the sound insulation capacity of honeycomb core sandwich panels is significantly improved. With increases in the mass fraction of reinforced particles, the Young’s modulus of the structure increases, as does the sound insulation capacity. The valley value of the sound insulation curve moves towards the low-frequency direction, which improves sound insulation performance in the low frequency band. In addition, the surface density of the gradient honeycomb decreases compared with traditional honeycomb, which provides new theoretical guidance for the design of lightweight sound insulation structures.