## Abstract

The authors present numerical results for a systematic parametric study of the effect of honeycomb core geometry on the sound transmission and vibration properties of in-plane loaded honeycomb core sandwich panels using structural acoustic finite element analysis (FEA). Honeycomb cellular structures offer many distinct advantages over homogenous materials because their effective material properties depend on both their constituent material properties and their geometric cell configuration. From these structures, a wide range of targeted effective material properties can be achieved thus supporting forward design-by-tailoring honeycomb cellular structures for specific applications. One area that has not been fully explored is the set of acoustic properties of honeycomb and understanding of how designers can effectively tune designs in different frequency ranges. One such example is the insulation of target sound frequencies to prevent sound transmission through a panel. This work explored the effect of geometry of in-plane honeycomb cores in sandwich panels on the acoustic properties the panel. The two acoustic responses of interest are the general level of sound transmission loss (STL) of the panel and the location of the resonance frequencies that exhibit high levels of sound transmission, or low sound pressure transmission loss. Constant mass honeycomb core models were studied with internal cell angles ranging in increments from −45 deg to +45 deg. Effective honeycomb moduli based on static analysis of honeycomb unit cells are calculated and correlated to the shift in resonance frequencies for the different geometries, with all panels having the same total mass. This helps explain the direction of resonance frequency shift found in the panel natural frequency solutions. Results show an interesting trend of the first resonance frequencies in relation to effective structural properties. Honeycomb geometries with smaller core internal cell angles, under constant mass constraints, shifted natural frequencies lower, and had more resonances in the 1–1000 Hz range, but exhibited a higher sound pressure transmission loss between resonant frequencies.

## Introduction and Motivation

Cellular materials have become increasingly popular in research and design due to their macro material properties that substantially differ from their micro, or host material properties [1–7]. One specific group of cellular materials, hexagonal honeycomb structures are frequently used in applications requiring a high out-of-plane stiffness to weight ratio [4,8] resulting in a lightweight core material of a sandwich construction. In addition to their generally good lightweight stiffness properties, they are also useful in impact absorption [3], low energy loss elastomeric materials [1,2,9], and thermal management [10,11].

Cellular honeycomb meta-materials also offer a major advantage in that they can be tailored with material properties that are specific to an application. If conventional homogenous materials are used, then the designer is limited to a specific set of fixed material properties. However, since cellular materials depend on both the fixed constituent material properties and the geometry of the structure, a broad range of effective properties can be attained by modifying the geometry of the cells. Through such modification, materials are designed for certain applications and their varying set of geometric parameters also provides a good platform for optimization [3,12].

In addition to the applications mentioned, there may also be acoustic applications where this same type of design flexibility and tailoring of properties would be desirable. Some applications include the design of vehicles and building walls to attain targeted acoustic signatures. The target acoustic signature in this study is the level of sound transmission across a range of frequencies through a panel, also known as the STL curve. For a panel, it may be possible to control which incident frequencies have reduced magnitudes of sound pressure transmission and which ones have high magnitudes of sound pressure transmission. The STL is determined by the mechanical properties of the partition's material and with honeycomb cellular materials, it is possible to vary these mechanical properties through the control of cell wall geometry, varying the acoustic response.

The acoustic properties of sandwich panels have been the subject of intense study, with most being analytically expressed for use in studying the behavior of sandwich structures. In one of the first studies on the STL of sandwich panels, Kurtze and Watters [13] elucidated some distinct advantages over single layer panels. Ford et al. [14] improved upon their model by incorporating a compressible core, allowing for flexural and dilatational motions, to analyze foam core panels. That analytical model has since been refined and improved by other researchers [15–18]. The study of multilayer sandwich panels eventually led into the study of alternative anisotropic core materials. Moore and Lyon [19] studied the effects of using orthotropic cores on the panel's performance using both analytical expressions and experimental validation. One of the most common orthotropic core materials used in this and similar studies is the honeycomb core [12,20,21] and similar trusslike periodic panels [22,23]. Specifically, Ruzzene [7], employed a spectral finite element (FE) model to study honeycomb truss core panels over a broad range of frequencies. Efforts to efficiently model the out-of-plane STL behavior of honeycomb sandwich panels using hybrid analytical and FE methods [24], and approximations of orthotropic stiffness properties for finite panels [25] have also recently been developed.

In most of the previous endeavors to study the effects of honeycomb cores, only certain core configurations have been tested with acoustic responses not related directly to the geometric and structural properties. These factors require further exploration for use in effectively designing structures to achieve targeted acoustic properties while satisfying other mechanical properties. We analyzed which acoustic properties can be controlled through variation of the honeycomb core geometry and the significant parameters that control those properties in an effort to develop a design method. In addition, a FE model was employed which, unlike custom analysis codes, is available through commercial software and can be easily recreated. Unlike most studies that use out-of-plane honeycomb cores, this study focuses on the use of honeycomb cores that are in-plane with the loading.

## Acoustic Properties of Sandwich Panels

It is important to make a clear distinction between two different concepts, sound insulation and sound absorption. Sound absorption is the conversion of sound energy into heat energy [26]. Adding materials with high absorption coefficients can enhance the dampening of vibrations in a partition, which though beneficial for reducing the level of sound within a space is not so for reducing the sound between adjacent spaces. Alternatively, we focused on sound insulation, which is useful for attenuating the sound pressure levels between spaces through the “blocking” of sound by a partition between a sound source and the listener [26]. While sound absorption can sometimes relate to the sound insulation, a distinction is made between the two concepts.

### STL Curve.

A standard performance metric for the sound insulation of a material is the sound pressure transmission loss [8,26,27], which describes the ratio between the incident and transmitted sound pressure across the surface of a partition or panel. The STL of a panel is frequency dependent and is influenced by the material and geometry of the panel [8,26]. The controlling factor for the STL depends on the specific frequency range being analyzed. The STL characteristics can be divided into four distinct regions based on the frequency range under analysis, as shown in the schematic illustrated in Fig. 1.

Fig. 1
Fig. 1
Close modal

At lower frequencies, the STL is controlled by the panel's stiffness; damping and mass have little effect [8,27]. For constant frequency, the STL in this region increases with a higher first resonance frequency, and the range ends with the appearance of the first resonance frequency [28].

At slightly higher frequencies, the STL is controlled by the natural resonance of the finite panel. At these driving frequencies, the high level of displacement causes a transfer of large amounts of sound energy to the transmitted side, resulting in noticeable discrete valleys in the STL response curve [8,26].

After the resonance range, the sound pressure transmission loss of the panel is primarily governed by its mass in what is known as the mass controlled region. In this range the STL can be predicted by the mass law
$STL=10log[1+(mωcosθ2ρc)2]$
(1)
where m is the mass per unit surface area of the panel, ω is the angular frequency, θ is the angle of incidence, ρ is the density of the acoustic medium, and c is the speed of sound in the acoustic medium. For normally incident waves, the incidence angle is 0 deg. For normal incidence, and assuming /(2ρc) ≫ 1, the mass law simplifies to the normal incidence mass law
$STL=20log(fm)-42 dB$
(2)

where f = ω/2π, is the cyclic frequency in Hertz. It should be noted that this law assumes high frequencies (short wavelengths) and an infinitely long panel [26–28].

At even higher frequencies, bending waves can result in the coincidence effect, which leads to the coincidence region. The coincidence effect occurs at a critical frequency of the panel when the freely propagating bending waves in the panel match the wavelength of the incident sound [26,29]. The resulting coincidence effect is an efficient transfer of sound pressure and a noticeable dip in the STL at the coincident frequency [8,29].

Stiffness, mass, damping, and resonant properties are the major components affecting the sound transmission capabilities of a panel. Honeycomb cellular materials, if used as the core material in a three layer sandwich construction, offer many different possibilities. Due to the variable range of effective properties possible with honeycomb cellular materials, these properties can be changed while keeping other crucial properties constant. In this research, we focused on elucidating the stiffness and resonance regions of the STL response caused by the variability within this region.

## Geometry and Effective Moduli of Honeycomb Structures

The sandwich panels under study consist of three layers, with two face sheets and a honeycomb core in between. The material for both the face sheets and the core is aluminum, and all panels have a core height, H, of 8.66 cm and face sheet thicknesses of 2.5 mm. The length of the panel, L, depends on the number of periodic unit cells in the honeycomb core. The thickness of the core cell walls varies and is adjusted with geometry in order to keep each honeycomb sandwich panel with the same total mass.

### Geometric Parameters.

The unit cell is highlighted for two honeycomb panels in Fig. 2. In the design of honeycomb materials, conventional unit cell geometric parameters are used to define the topology, shown in Fig. 3; cell angle (θ), vertical member height (h), angled member length (l), and cell wall thickness (t). The unit cells in the figure shown are of a standard hexagonal model (θ = 30 deg, h = l) and a frequently used auxetic model (θ = − 30 deg, h = 2l). To study a full range of geometric parameters on STL, the cell angle is varied from −45 deg to +45 deg in 5 deg increments, and the h, l, and t values are adjusted to maintain a specific unit cell size and mass.

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal
The relationship between the unit cell size and geometric parameters is given by
$Lx=2lcosθ$
(3)
$Ly=2(h+lsinθ)$
(4)

For this study, unit cell sizes are determined to fit a cell configuration of either a 1 × 40 or 2 × 80 in the specified panel size. The total length, L, of the panel is the number of longitudinal cells (40 or 80) multiplied by Lx. Any constraints on the overall size of the structure, which is the case with this design problem, require a change in one of the conventional geometric parameters accompanied by a modification of other parameters to adhere to the design size constraints. To account for this modification, the authors developed a method to ensure constant unit cell size between varying configurations. The reference size is determined by standard hexagonal geometry (θ = 30 deg, h = l). Using these parameters in Eq. (4), and equating Ly=H=8.66 cm for the case of one cell in the height direction; solve for h = l = 2.8867 cm. Equation (3) is then used to define Lx=5 cm, for a total panel length of 40(5 cm) = 200 cm. For the case of two cells in the height direction, equate Ly = H/2; solve for h = l = 1.4433 cm, then Lx = 2.5 cm, for a total panel length of L=80(2.5 cm) = 200 cm. Using these reference values for Lx and Ly, for the other cell wall angle geometries, use Eq. (3) to solve for l, then Eq. (4) to solve for h. These calculations are summarized in Tables 1 and 2. Figure 4 shows the honeycomb core for a few of the geometric configurations studied; each model fits in an overall panel length of L = 2 m.

Fig. 4
Fig. 4
Close modal
Table 1

Geometric and effective properties of single row honeycomb cores

Cell angle (deg)Unit cellsl (mm)h (mm)t (mm)G12*  (MPa)ρ* (kg/m3)E11*  (MPa)E22*  (MPa)
−451 × 4035.3668.301.560.5792707.0121.0
−401 × 4032.6464.281.670.86227013.328.1
−351 × 4030.5260.811.781.2327024.636.2
−301 × 4028.8757.731.881.6827044.944.9
−251 × 4027.5854.971.972.2527083.154.2
−201 × 4026.6052.402.052.9427016063.7
−151 × 4025.8850.002.133.7627034073.2
−101 × 4025.3947.712.204.7527088382.4
−51 × 4025.1045.492.265.93270396090.8
01 × 4025.0043.302.327.3427098.3
51 × 4025.1041.112.379.042704550105
101 × 4025.3938.892.4111.12701170109
151 × 4025.8836.602.4513.6270519112
201 × 4026.6034.202.4816.8270283112
251 × 4027.5831.642.4921.0270170111
301 × 4028.8728.872.5026.6270107107
351 × 4030.5225.802.4934.927067.999.9
401 × 4032.6422.322.4748.227043.191.0
451 × 4035.3618.302.4373.427026.780.1
Cell angle (deg)Unit cellsl (mm)h (mm)t (mm)G12*  (MPa)ρ* (kg/m3)E11*  (MPa)E22*  (MPa)
−451 × 4035.3668.301.560.5792707.0121.0
−401 × 4032.6464.281.670.86227013.328.1
−351 × 4030.5260.811.781.2327024.636.2
−301 × 4028.8757.731.881.6827044.944.9
−251 × 4027.5854.971.972.2527083.154.2
−201 × 4026.6052.402.052.9427016063.7
−151 × 4025.8850.002.133.7627034073.2
−101 × 4025.3947.712.204.7527088382.4
−51 × 4025.1045.492.265.93270396090.8
01 × 4025.0043.302.327.3427098.3
51 × 4025.1041.112.379.042704550105
101 × 4025.3938.892.4111.12701170109
151 × 4025.8836.602.4513.6270519112
201 × 4026.6034.202.4816.8270283112
251 × 4027.5831.642.4921.0270170111
301 × 4028.8728.872.5026.6270107107
351 × 4030.5225.802.4934.927067.999.9
401 × 4032.6422.322.4748.227043.191.0
451 × 4035.3618.302.4373.427026.780.1
Table 2

Geometric and effective properties of double row honeycomb cores

Cell angle (deg)Unit cellsl (mm)h (mm)t (mm)G12*   (MPa)ρ* (kg/m3)E11*  (MPa)E22*   (MPa)
−452 × 8017.6834.150.780.5792707.0121.0
−302 × 8014.4328.870.941.6827044.944.9
−152 × 8012.9425.001.063.7627034073.2
152 × 8012.9418.301.2213.6270519112
302 × 8014.4314.431.2526.6270107107
452 × 8017.689.1511.2273.427026.780.1
Cell angle (deg)Unit cellsl (mm)h (mm)t (mm)G12*   (MPa)ρ* (kg/m3)E11*  (MPa)E22*   (MPa)
−452 × 8017.6834.150.780.5792707.0121.0
−302 × 8014.4328.870.941.6827044.944.9
−152 × 8012.9425.001.063.7627034073.2
152 × 8012.9418.301.2213.6270519112
302 × 8014.4314.431.2526.6270107107
452 × 8017.689.1511.2273.427026.780.1

### Effective Properties.

The effective static mechanical properties of the honeycomb core can be approximated by Gibson and Ashby's cellular materials theory (CMT) [4], which is described in this section. The unique advantage that honeycomb cellular materials offer is that the conventional geometric parameters can be changed in order to achieve varying overall properties that the host material lacks. For low deformation, before a critical stress is reached, the main deformation in honeycombs is the bending of the cell walls, with the honeycomb exhibiting behaviors that are elastically linear. The CMT developed by Gibson and Ashby (CMT) [4] uses a linear static analysis of the unit cell wall geometry based on Bernoulli–Euler beam theory to develop formulas for the static effective elastic properties for honeycombs, that use the geometric cellular parameters as variables. The derived formulas have been shown to give good agreement with experimental and FE models [1,4,30–32]. The static effective elastic properties studied here include in-plane elastic modulus in the longitudinal, x and transverse, y directions and in-plane shear modulus
$E11*=E(tl)3cosθ(hl+sinθ)sin2θ$
(5)
$E22*=E(tl)2(hl+sinθ)cos3θ$
(6)
$G12*=E(tl)3(hl+sinθ)cos3θ$
(7)
where E is the elastic modulus of the constituent material, and h, l, t, and θ are the geometric unit cell parameters. Since the accuracy of the Gibson–Ashby effective structural moduli in terms of unit cell geometric parameters has been validated in the open literature both experimentally and using FEA, an analysis was not repeated in the present work. Based on the unit cell geometry, the effective density of the honeycomb structure can be calculated from
$ρ*=ρtl(hl+2)2cosθ(hl+sinθ)$
(8)

where ρ is the density of the constituent material. In this study, the effective density is kept constant so that each honeycomb sandwich panel has the same total mass. Constant effective density ensures that all models (shown in Tables 1 and 2) have the same material volume and consequently the same overall mass. Constant mass was selected as this property governs the overall magnitude of the sound transmission response, with some deviations. To achieve constant effective mass for the various unit cell geometries in Tables 1 and 2, Eq. (8) was used to solve for the required cell wall thickness t.

In the present work, the Gibson–Ashby effective honeycomb moduli based on static analysis of honeycomb unit cells are calculated for each of the considered geometries in Tables 1 and 2.

These properties are not used to model for the sound transmission properties of the honeycomb sandwich panels, but instead are used to relate the shift in resonance frequencies for the different geometries, with all panels having the same total mass. The use of these effective properties helps explain the direction of resonance frequency shift found in the panel natural frequency solutions.

The effective elastic moduli and mass density could theoretically be used in a layer-wise composite sandwich panel dynamic model with the discrete honeycomb core cell geometry replaced by an effective homogeneous material with orthotropic properties. However, if the effective material properties for the honeycomb core obtained from the Gibson–Ashby model were to be used as a homogenized effective core medium, since the moduli are constant and frequency independent, such a model would be inaccurate for frequencies beyond the first few natural frequencies of the panel [33]. Another approach to extend model accuracy to higher frequencies is to develop dynamic effective properties of lattice structures from the detailed microstructure in the framework of a continuum medium, see for example Refs. [34] and [35]. In the present work, we use a direct approach and model the detailed cell walls of the unit cell with FEs and assemble the honeycomb structure for the sandwich panel using FEA. Other approaches include the indirect [36], and direct assembly of spectral elements [7].

## Finite Element Model

To obtain accurate numerical solutions for the honeycomb sandwich panel vibration and STL properties over the frequency band from 0 to 1000 Hz, corresponding to the stiffness and resonance driven frequency ranges, a detailed structural-acoustic FEA model with abaqus/CAE v6.10 was developed. The FEA model, shown in Fig. 5, consists of a honeycomb sandwich panel coupled with an acoustic fluid domain attached to the top of the panel. Sound pressure on the coupled surface is used to compute the transmitted pressure due to incident propagating plane acoustic waves on the bottom side of the panel. The panel was loaded on the bottom side to mimic an incoming plane sound pressure wave of controlled frequency. Since the acoustic medium is modeled as air, the radiated acoustic pressure due to elastic vibration of the bottom face sheet is small in comparison with the incident sound pressure. The acoustic resonances of the air internal cavities within the gaps in the honeycomb cell walls lie beyond the 1000 Hz frequency range of interest, and the effects of air within the cavities are neglected.

Fig. 5
Fig. 5
Close modal

### Model Setup.

The two-dimensional (2D) model represents in-plane loading of the sandwich panel. In contrast to homogenized models with effective material properties, the honeycomb core unit cell geometry and face sheets are modeled in detail using an assembled mesh of beam elements. The beam elements are modeled with rectangular profile sections defining the cell wall thickness and out-of-plane unit depth for the 2D planar model. The thickness of the profile in the xy-plane is varied to achieve constant mass between the various models with geometric cell properties in Tables 1 and 2. All sections are assigned the material properties of aluminum, with ρ = 2700 kg/m3, E = 71.9 GPa, and ν = 0.3. The air medium is modeled with a semicircular domain to simulate the effects of air on the surface of the honeycomb structure embedded in a rigid baffle. On the exterior circular boundary, absorbing impedance boundary conditions are applied to model outgoing propagating acoustic waves radiating from the panel to the far-field. The radial distance from of the circular boundary was increased until all model solutions were nonreflecting. The effect of the air medium coupled with the structural panel produces mechanical energy loss that manifests as a small damping effect for the elastic panel. The acoustic fluid medium is given the acoustic properties of air, with mass density ρ = 1.2 kg/m3 and speed of sound c = 343 m/s, and bulk modulus κ = 141,179 N/m2.

For the honeycomb sandwich panels, a connected mesh of B22 planar beam elements with quadratic order were used for the face skins and the core honeycomb section. These elements were chosen in order to reduce computational effort while keeping a high level of accuracy compared to linear beam elements [1,2,5,37]. Using previously established mesh convergence results, the elements were sized to have at least four elements per the smallest edge in the cell walls of the honeycomb core [1,37]. The air domain uses AC2D3 elements, which are three-node 2D acoustic triangular elements, with the mesh seed size of the elements in close proximity of the honeycomb structure at 0.012 m. By using a mesh bias from the center of the semicircular domain to the edge, the element size gradually increases to 0.08 m at the circular edge of the air domain; see Fig. 6. The reason for the changing mesh size is that the air directly in contact with the honeycomb panel is of more concern for the data post processing with the remainder of the air domain only used for visualization. Though this mesh size exceeds software documentation2 recommendations based on frequency (wavelength), mesh convergence was nonetheless verified.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

A unit amplitude time-harmonic pressure load was applied to the bottom face sheet of the structure to simulate the incident sound wave loading. For all analyses, direct incidence (α = 0 deg) is specified for simplicity, representing a worst-case head on transmission. The ends of the structure, both ends of face sheets and honeycomb core cell walls adjacent to the ends, have pinned boundary conditions, constrained in the x and y directions. A surface based tie constraint was used to maintain interaction between the structure and acoustic medium, specifying the honeycomb as the parent surface and the air as the child surface. This coupling of the structural field with the acoustic field ensures their deformity as a single component. As discussed earlier, is also important to ensure that sound does not reflect from the circular absorbing surface of the air region into the area of data collection. To model the acoustic region as a semi-infinite, an absorbing interaction via surface impedance is defined along the circular surface of the air region. For each honeycomb sandwich panel, a circular absorbing boundary with a radius of 2 m was verified to produce nonreflecting numerical solutions.

### Analysis Procedure.

Before performing a direct steady-state frequency response analysis for the structural acoustic model, a natural frequency extraction was used to determine the undamped modal natural frequencies and corresponding mode shapes of the honeycomb sandwich panels in vacuo. Since the damping effects of the air are small, these modal frequencies, which correlate to the resonating frequencies of the structure, closely determine the location of peak vibration amplitude levels [26]. The frequency step for the direct steady-state analysis for the coupled structural acoustics model was conducted over a range of 1–1000 Hz, ensuring that analysis pervaded into the stiffness region and well into the resonance region. Each interval between natural frequencies contained seven evaluation frequencies with a bias parameter of two toward the natural frequencies, generating a refined response around those natural frequencies.

### Response Collection.

The STL of the panel over a range of frequencies is a common metric for describing the acoustic performance of a panel. STL is a measure of the ratio of incident power to transmitted power across the panel surfaces. For a structural panel coupled with air on both sides, the ratio of power corresponds to the ratio of sound pressure across the panel surfaces. To simplify the analysis, we compute the sound pressure transmission loss (STLp), which is defined as the ratio between the incident and transmitted acoustic pressure, in decibels, through the panel
$STLp=10log10|pi2pt2|$
(9)
where pi and pt are the root-mean-square value of pressure on the incident and transmitted sides, respectively. These values are computed numerically from
$pi2=pi·pi, pt2=pt·pt$
(10)
where
$p=〈p1,p2,...,pn〉$
(11)

is a vector of the pressure values at the air FE nodes along the face sheets of the honeycomb.

During the steady state analysis using abaqus, a history output is specified for the acoustic pressure of all the air nodes in direct contact with the transmitted (top) side of the sandwich panel. The acoustic pressure results assume the form of complex numbers so that the magnitude is recorded. Since the loading is specified as a unit pressure wave, the incident pressure on all the nodes on the incident side of the panel is one. These results were used to calculate the sound pressure transmission loss of each of the panels using Eq. (9).

### Model Validation.

As discussed earlier, a convergence study with mesh refinement and absorbing boundary placement was used to verify accuracy of the structural acoustic FE model. To further ensure consistency in the results, the FE modeling procedure was cross-checked with results from other square truss core sandwich panel models [7,22], which use custom spectral element analysis codes for the lattice panel with numerical post processing of an analytical spatial transform acoustic response solution. Results for the sound pressure transmission loss in the 1–1000 Hz frequency range for a square truss core sandwich panel, which was only used for trends and location of resonance frequencies, and was not the subject of further study, were compared [38]. It was found that the FE model followed the same trends in the STL response as the spectral element models, especially for the resonant frequencies, which was the main concern in this study [38].

## Natural Frequency Extraction Results

The natural frequency extraction was performed on the entire honeycomb sandwich panel models presented for both the 1 × 40 models and the 2 × 80 unit cell honeycomb models. As discussed earlier, each model has the same total mass. Though all the results are recorded for use in the division of frequency increments in the steady state analysis, we report only the first ten here. For the models with a single row (1 by 40) of periodic unit cells, the first ten frequencies for the honeycomb sandwich panels are shown in Table 3 (positive angles) and Table 4 (negative angles). It should be noted that the + 45 deg model only has nine modes within the specified range. The mode shapes for the honeycomb sandwich panels corresponding to the first ten natural frequencies exhibit a classical global flexural shape superimposed with local modes within the unit cells as the mode number increases. At some higher modes, especially for the negative angle honeycomb core geometries, nonflexural dilatational mode shapes are exhibited; examples are shown in [38].

Table 3

Natural frequencies of positive angle constant mass honeycomb core sandwich panels

Mode #5 deg 10 deg 15 deg 20 deg 25 deg 30 deg 35 deg 40 deg 45 deg
143.046.149.353.557.962.868.575.383.8
288.294.4100.7109.4118.7129.7143.2160.3183.6
3138.3148.3158.4172.3187.6205.9228.6258.1299.5
4189.5203.3217.0236.0257.2283.0315.5358.6420.5
5241.4259.3276.8301.0327.8360.8403.0459.9542.9
6293.6316.0337.4366.7399.0439.0490.5560.8664.8
7346.2373.2398.8433.3471.0517.8578.4661.6762.8
8399.2430.8460.8500.6543.8597.2666.5761.2786.0
9452.7488.9523.4568.6617.4677.3755.2762.2906.2
10506.7547.6586.6637.4691.7758.2760.3862.9
Mode #5 deg 10 deg 15 deg 20 deg 25 deg 30 deg 35 deg 40 deg 45 deg
143.046.149.353.557.962.868.575.383.8
288.294.4100.7109.4118.7129.7143.2160.3183.6
3138.3148.3158.4172.3187.6205.9228.6258.1299.5
4189.5203.3217.0236.0257.2283.0315.5358.6420.5
5241.4259.3276.8301.0327.8360.8403.0459.9542.9
6293.6316.0337.4366.7399.0439.0490.5560.8664.8
7346.2373.2398.8433.3471.0517.8578.4661.6762.8
8399.2430.8460.8500.6543.8597.2666.5761.2786.0
9452.7488.9523.4568.6617.4677.3755.2762.2906.2
10506.7547.6586.6637.4691.7758.2760.3862.9
Table 4

Natural frequencies of negative angle (auxetic) constant mass core sandwich panels

Mode #−45 deg −40 deg −35 deg −30 deg −25 deg −20 deg −15 deg −10 deg −5 deg
111.714.216.819.622.625.729.033.436.3
223.628.734.039.745.752.059.068.174.2
336.343.952.261.070.580.691.7106.4116.2
449.860.071.283.396.3110.3125.6145.8159.0
564.377.191.3106.7123.3141.2160.8186.2202.5
679.995.4112.5131.2151.5173.2196.7227.2246.4
796.8114.8134.9156.9180.7206.2233.5268.7290.7
8114.9135.5158.4183.7210.9240.1270.9310.8335.4
9134.4157.4183.2211.6242.2274.8309.1353.5380.7
10155.2180.6209.2240.6274.4310.4348.0396.7426.6
Mode #−45 deg −40 deg −35 deg −30 deg −25 deg −20 deg −15 deg −10 deg −5 deg
111.714.216.819.622.625.729.033.436.3
223.628.734.039.745.752.059.068.174.2
336.343.952.261.070.580.691.7106.4116.2
449.860.071.283.396.3110.3125.6145.8159.0
564.377.191.3106.7123.3141.2160.8186.2202.5
679.995.4112.5131.2151.5173.2196.7227.2246.4
796.8114.8134.9156.9180.7206.2233.5268.7290.7
8114.9135.5158.4183.7210.9240.1270.9310.8335.4
9134.4157.4183.2211.6242.2274.8309.1353.5380.7
10155.2180.6209.2240.6274.4310.4348.0396.7426.6

The lowest first modal frequency for the regular honeycomb core sandwich panels occurred at 43.0 Hz (+5 deg model) after which the first natural frequency increased as the cell angle increased (becoming more positive) for each of the positive angle models. The lowest first modal frequency for the auxetic core sandwich panels occurred at 11.7 Hz (−45 deg model). The first natural frequencies again occurred for each auxetic core as the cell angle increased (becoming less negative). In general, we observed that the lower angle models, meaning less positive or more negative angles, exhibited lower overall natural frequencies at each of the mode numbers. The increase of the cell angle caused an increase in the first natural frequency. In addition, those models with lower first natural frequencies exhibited a smaller spacing between subsequent natural frequencies.

It is important to emphasize that the Gibson–Ashby formulas for effective static material properties are based on an infinite number of unit cells in orthogonal directions such that distortions in cell wall members due to boundary effects are not accounted for. When the number of cell rows is small, as with our simulations, the boundary affects may alter the effective elastic moduli as predicted by Eqs. (5)(7). The result is that two different models can have identical Gibson–Ashby predictive effective stiffness properties, while still exhibiting slightly differing responses due to boundary effects. Note this effect with our 2 × 80 configuration models. All have the exact same effective properties as their 1 × 40 counterparts, but the unit cell size is scaled down by half. The natural frequency results for the 1 × 80 unit cell models in 15 deg increments are shown in Table 5.

Table 5

Natural frequencies for selected double row core sandwich plates

Mode #−45 deg −30 deg −15 deg 15 deg 30 deg 45 deg
110.317.425.845.559.381.9
220.935.051.991.8121.2177.7
332.053.479.9143.0191.2288.6
443.972.38108.6194.6261.4403.9
556.692.1138.3247.0331.9520.6
670.4112.7169.1300.1402.1636.7
785.4134.3200.8354.3472.5752.2
8101.6157.0233.6409.4543.1767.7
9119.3180.8267.4465.4614.0866.8
10138.3205.8302.0522.4685.2980.7
Mode #−45 deg −30 deg −15 deg 15 deg 30 deg 45 deg
110.317.425.845.559.381.9
220.935.051.991.8121.2177.7
332.053.479.9143.0191.2288.6
443.972.38108.6194.6261.4403.9
556.692.1138.3247.0331.9520.6
670.4112.7169.1300.1402.1636.7
785.4134.3200.8354.3472.5752.2
8101.6157.0233.6409.4543.1767.7
9119.3180.8267.4465.4614.0866.8
10138.3205.8302.0522.4685.2980.7

It can be observed that the natural frequencies differ from the frequencies of the 1 × 40 models with identical effective stiffness properties further emphasizing that while Gibson–Ashby CMT theory provides a good reference point for comparing models, further interpretation is necessary. Consequently, these effective properties must be experimentally determined to account for this discrepancy. The trends of the 2 × 80 models were consistent with the 1 × 40 models, however. The models with the greater internal cell angles have first natural frequencies that were higher with a greater spacing between modes.

### Trends and Observations.

The natural frequency (vibrational mode) of a structure depends on its mass, shape, and stiffness properties. Since all of the modes are directly related to the first mode, the location and trend of the natural frequencies can be understood by studying the occurrence of the first natural frequency for the models. In addition to describing the behavior of the subsequent vibrational modes, the first natural frequency also describes the extent of the stiffness region. Even though the overall stiffness of a complex structure depends on several interrelated parameters, we attempted to determine the significance of any single property. Scatter plots were created to visualize the potential effects that each of the effective stiffness properties based on the Gibson–Ashby CMT, including effective shear modulus ($G12*$) and effective Young's modulus in the x ($E11*$) and y ($E22*$) directions, have on the first natural frequency value. The plots are shown in Figs. 7–9.

In viewing the results, the shear modulus and Young's modulus in the x-direction showed logarithmic relationships with the first natural frequency. For the Young's modulus in the x and y directions, the positive angle models behave differently than that with negative angles. Indeed, for the $E11*$, the negative and positive angled models mirrored each other in their behavior. We next performed a sensitivity analysis to study the influence of the effective stiffness properties to determine if any could be eliminated from future evaluations. Based on the results, none were deemed suitable for elimination in that all properties were highly interrelated in their effect on the overall structural stiffness. The shear modulus, however, was the most promising for potentially predicting the behavior based upon a single parameter. We will explore this modulus in future research. Since the natural frequencies were also adequate indicators of dips in the STL, we hypothesize that the models with greater internal cell angles (and greater shear moduli) should have sound pressure transmission loss dips that are spaced farther apart from each other.

## Sound Pressure Transmission Loss Results

We next ran all of the steady-state simulations and calculated the STLp for each of the panels. As predicted, for the panels interacting with air, damping is small, and the dips in the transmission curve did correspond with the natural frequencies of the panel in vacuo, specifically aligning with the odd mode numbers from the natural frequencies as shown in Fig. 10. At these frequencies, the structure vibrated at higher amplitudes, thus transmitting more acoustic pressure. The reason the dips correspond to odd modes is due to the normal incident wave, and symmetry of the panel flexural modes about the centerline.

Fig. 10
Fig. 10
Close modal

This transmission resulted for some, but not all, of the panels are shown below. Though some of our results were omitted for clarity and brevity purposes, the general trends are clearly indicated. In Fig. 10, we provide the analysis of 15 deg increments of the positive angle cores. For the positive angle models, the behavior of the sound pressure transmission loss exhibited trends similar to that of the natural frequencies. The first dip occurred at higher frequencies for the models with greater internal cell angle (more positive), thus shifting up the resonance controlled region and causing greater spacing between dips as predicted.

The results of the negative angle cores (Fig. 11) also exhibited the same behavior. As the angle increased (to become less negative), the dips in the STLp became fewer and less frequent. The −45 deg exhibited the lowest first frequency dip and consequently the smallest spacing between dips. For the negative angles, we also observed that the other dips began occurring at the higher frequencies. These correspond to the dilatational vibration modes, which began occurring at earlier frequencies for models with smaller internal cell angles, which was most particularly prevalent for the −45 deg model beginning at 500 Hz.

Fig. 11
Fig. 11
Close modal

As predicted, the mass does appear to be the governing factor in determining the magnitude of the sound pressure transmission loss. It can be observed in both the regular honeycombs and the auxetics that the magnitude of the curve, not at the resonances, remains relatively constant between all the models due to their same mass.

### Comparison and/or Discussion.

The models with increased cellular angles (and higher shear modulus) did exhibit first natural frequencies and larger spacing between dips. We provide the range of spacing between resonances, plots quantifying the spacing in Fig. 12, with the spacing between the first and second sound pressure transmission loss dips measured for each of the models. The spacing ranged from 24.7 Hz (−45 deg model) to 215.7 Hz (+45 deg model). Should designers wish to isolate a particular frequency and create a large range around that frequency, larger spacing may be more desirable. This large spacing range variability with models of the same mass clearly indicates the flexibility of these honeycomb panels.

Fig. 12
Fig. 12
Close modal

Though we did not present the results of our 2 × 80 model steady-state simulations in their entirety here, we did make comparisons between their 1 × 40 counterparts, the results of which are in Fig. 13. Here, the +30 deg single row model and the +30 deg double row model are identical in all of the calculated effective stiffness properties analyzed in this paper. Though they did exhibit similar sound pressure transmission loss curves, they did deviate after the first resonance. The general shape of the resonance dips was identical but the spacing differed. We also noticed a similar trend in our comparison of the models in the −30 deg configuration (Fig. 14). As we emphasize in earlier discussion on the natural frequency, the CMT equations are inadequate for fully defining the acoustic behavior due to the neglect of boundary effects.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

In this study of the sound pressure transmission loss, we also noted dips in the shape of the resonance in the sound pressure transmission loss curve. These negative angle models, while having more resonance dips, do have dips that are sharper, indicating a high sound pressure transmission loss remains for the range between these frequencies. Compared to the positive angle models, a better overall performance was evident across all frequencies, as there are more frequencies with a high STLp, which is desirable. To consider both the quantity and width of the dips, we calculated the area under the curve as a metric to compare the panels, the results of which are shown in Table 6.

Table 6

Values for area under the sound pressure transmission loss curve for the constant mass panels

Area under curve (dB Hz)
Model (deg)Stiffness regionResonance regionTotal
−45346.744,081.844,428.5
−40441.644,841.645,283.0
−35543.444,526.545,070.0
−30646.343,574.144,220.4
−25768.944,092.944,862.0
−20892.743,606.344,499.0
−151019.142,331.543,350.5
−101196.642,738.543,935.0
−51337.141,602.142,939.0
51626.540,401.842,028.0
101762.339,342.141,104.0
151891.239,044.340,935.5
202090.136,859.238,949.0
252286.335,355.837,642.0
302649.437,642.940,292.3
352774.835,874.738,650.0
403095.637,227.940,324.0
453434.336,218.339,652.6

Area under curve (dB Hz)
Model (deg)Stiffness regionResonance regionTotal
−45346.744,081.844,428.5
−40441.644,841.645,283.0
−35543.444,526.545,070.0
−30646.343,574.144,220.4
−25768.944,092.944,862.0
−20892.743,606.344,499.0
−151019.142,331.543,350.5
−101196.642,738.543,935.0
−51337.141,602.142,939.0
51626.540,401.842,028.0
101762.339,342.141,104.0
151891.239,044.340,935.5
202090.136,859.238,949.0
252286.335,355.837,642.0
302649.437,642.940,292.3
352774.835,874.738,650.0
403095.637,227.940,324.0
453434.336,218.339,652.6

While the positive angle models perform better in the stiffness region and have fewer sound pressure transmission resonances in the resonance region, their total values for area under the curve were consistently less than the negative angle models, presenting a tradeoff.

## Conclusions

In our analysis of the acoustic transmission loss properties of in-plane honeycomb sandwich panels, we developed a detailed finite element model with commercial software that provided consist results with the literature. By studying the acoustic response of varying honeycomb cellular core geometries in a sandwich construction, several interesting findings were elucidated that will be of both use to designers and will be the subject of future research. First, for the normal incident pressure simulated and symmetric flexural modes, the odd numbered natural frequencies aligned with the honeycomb panel resonances, resulting in large dips in the sound pressure transmission loss. For plane waves incident at angles other than 0 deg normal, we anticipate that both even and odd mode numbers are active resulting in dips in the response at all panel natural frequencies. Second, the natural frequencies and resulting resonances could be shifted, without compromising mass, by varying the effective stiffness properties of the core. Third, under constant mass constraints, the general magnitude of the STL at nonresonance frequencies was observed as constant, suggesting that mass was the main controlling magnitude factor. Fourth, the constant mass models with auxetic cores exhibited a greater number of resonances compared to the positive angle models, while maintaining a consistently higher STLp between those resonances as evidenced by the area under the curve values. We will undertake future study to further elucidate each of these discoveries. The influence of the geometric and effective structural parameters on the acoustic response will continue to be an area of focus as all are easily controlled by the designer. We also intend to study other honeycomb core configurations and topologies. Finally, as with any computational model, experimental studies are needed to further validate this developed FE model.

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