Acoustic modeling of honeycomb structures for second load direction with damping consideration

Abstract

The control of low-frequency noise in buildings has become a critical issue because of the inadequate sound insulation in low frequencies (<200 Hz) for most partitions and doors. In the present study, an analytical model has been proposed to specify the relationship between the acoustic parameters such as acoustic impedance of honeycomb and the honeycomb geometry. In this regards, three materials (Polycarbonate, Polypropylene, and Polyurethane), seven angles (45°, 30°, 15°, 0°, −15°, −30°, −45°), and three cell arrangement (1 × 1, 2 × 2, and 3 × 3) were considered to find out the effect of these parameters on the sound absorption coefficient. Sound absorption in the frequency range of 1–500 Hz for angles of 45°, 30°, 15°, 0°, −15°, −30°, −45° was demonstrated using MATLAB software. A comparison of the results showed that the greater the angle’s absolute value, the more the sound absorption coefficient. By comparing all three materials’ absorption coefficient values, it was concluded that the 1 × 1 arrangement had greater absorption than the 2 × 2 and 3 × 3 arrangements. Also, the number of resonance peaks of polyurethane material for all angles was greater than the others in the same frequency range. Then, the panel with the honeycomb core of Polyurethane and 1 × 1 arrangement was recommended for cells with an internal angle of −45° in the present research case study for having more sound absorption.

Keywords

Sandwich structures honeycomb panels acoustic impedance sound absorption coefficient stiffness sound transmission loss

Introduction

Honeycomb structures are widely used in various engineering applications, such as civil, aerospace, and automotive, because of their superior mechanical performance.

The multifunctional properties of honeycomb structures have generated tremendous interest in their application in ultra-light structures. The most attractive properties are those that govern the use of cellular solids as cores for panels and shells having lower weight than competing materials and potentially superior heat dissipation, vibration control, and energy dissipation characteristics.^1–3 Since the late 1950s, many papers have been published on the vibration of sandwich structures. El-Raheb investigated the structural-acoustic performance of periodic truss core panels.⁴ The frequency response of two-dimensional panels has been studied by using the transfer matrix method. The research has then been extended to evaluate double truss-like panels’ behavior and discuss the disappearance phenomenon for this configuration, of the mass–spring-mass resonance, and its dependence on panel aspect ratio.⁵

However, while composite structures combine high stiffness and strength at a low weight, they offer poor acoustic properties by radiating noise at low vibrational frequencies, particularly in sandwich composite structures due to their high stiffness-to-weight ratios. Therefore, considerable interest has been taken in studying sandwich structures’ acoustic properties to improve their performance in the engineering applications mentioned above.^6–12 The critical current issue is improving the acoustic performance includes adding additional sound-absorbing material to the structures. In many applications such as aerospace and automotive increase in weight leads to an increase in cost. Consequently, improving acoustic performance without sacrificing structural performance or weight is considered a technical challenge these days.

Investigations have been conducted regarding the understanding of wave-speed characteristics and transmission loss for sandwich panels. Rajaram et al.¹³ studied the influence of different panel design parameters, such as core density, core material, cell size, and cell structure, on the transmission loss of honeycomb sandwich panels. They presented transmission loss (TL) results of panels in three classes of core shear wave speeds—subsonic, transonic, and supersonic. Peters and Nutt experimentally studied the influence of different design parameters, such as core density, core material, and cell size on honeycomb sandwich structures’ wave speeds. They also measured bending and shear wave speeds related to the transmission loss performance for various material configurations.¹⁴ Grosveld et al. have improved transmission loss of honeycomb panel configurations by using the Finite element method.¹⁵ Backström and Nilsson proposed a vibration theory for the sandwich beam model based on Bernoulli–Euler and Timoshenko models. They showed that by implementing frequency-dependent parameters, sandwich composite beams’ vibration could be approximated using simple fourth-order beam theory.¹⁶ Some researchers worked on wave propagation in a sandwich structure, measurement, and prediction of wave speeds.^17–20 Mohamed et al. studied glass fiber-reinforced two-part Polyurethane composites and low-density polyurethane foam utilized to design and manufacture composite structural insulation panels using vacuum-assisted resin transfer molding process for temporary housing applications. Using these types of composite panels in building construction will result in cost-efficient, high-performance products due to inherent advantages in design flexibility. The use of core-filled composite structures offers additional benefits such as high strength, stiffness, lower structural weight, ease of installation and structure replacement, and higher buckling resistance than the conventional panels.²¹ The effort of Ahmed et al. utilizes finite element computations to assess the low-frequency wave propagation characteristics (i.e., phase velocity and dispersive properties) in commercially available aluminum honeycombs made by bonding thin corrugated sheets. Their results illustrate that the dispersive behavior and acoustic anisotropy of the studied honeycombs are more significant at higher porosities and high frequencies and identify the frequencies below which honeycombs exhibit their least dispersive acoustic behavior.²² Liu et al.²³ conducted an experimental and numerical investigation to reduce vibration and noise from a steel-concrete composite. Wang et al. presented the analytical model for the sandwich plate with pyramidal truss cores to investigate the acoustic property of transmission loss. The two-dimensional periodic model was established based on the assumption that the trusses were regarded as Euler-Bernoulli beams.²⁴ Hasheminejad et al.²⁵ proposed an exact model for sound transmission through a cylindrical sandwich shell of the infinite extent that includes a tunable electrorheological fluid core and is obliquely notified by a plane progressive acoustic wave. Sharma et al.²⁶ investigated the acoustic radiation responses of laminated sandwich baffled flat panels subjected to harmonic loading in an elevated thermal environment using a novel coupled finite and boundary element formulation based on the higher-order shear deformation shell theory.

Essassi et al. presented the experimental and numerical analysis results of the damping properties of a bio-based sandwich with an auxetic core. They showed that the resonance frequencies and the loss factor obtained from experimental and numerical analysis were in close agreement.²⁷ One of the essential mechanical properties of materials is Poisson’s ratio, which is positive for most materials. However, certain materials exhibit “auxetic” properties, have a negative Poisson’s ratio. Thus auxetic and non-auxetic materials exhibit different deformation mechanisms. In this regard, Duc et al.^28–35 studied the mechanical property of auxetic structure under different situations like static, vibration, and dynamic loading. They showed that auxetics could be useful in applications such as knee and elbow pads, body armor, robust shock, packing material, absorbing material, and sponge mops.

Talebitooti et al. presented an analytical model based on multi-objective optimization by applying a Genetic Algorithm. The summation of sound transmission loss and transverse displacement and the structure’s weight was regarded as two cost functions, optimized in a diffuse field. Their work’s significant achievement was to design an optimization algorithm to improve the vibro-acoustic fitness and weight of the sandwich doubly curved shells.³⁶ Fu et al. proposed a theoretical model with vibro-acoustic behavior analyses of laminated functionally graded carbon nanotube-reinforced composite plates based on Reddy’s higher-order shear deformation theory. Additionally, they calculated the sound pressure and radiation efficiency and carried out a numerical comparison with available literature results to confirm the presented model’s validity.³⁷ Wen et al.³⁸ proposed an ultra-light-weight sandwich panel with a closed octahedral core, which demonstrated the excellent acoustic property of sound transmission loss as well as outstanding mechanical performance. Hasheminejad and Jamalpoor proposed a three-dimensional analytical model developed to control sound transmission through an acoustically baffled supported hybrid smart double sandwich panel partition of rectangular planform. They verified their model by using the finite element model.³⁹ Guo et al. investigated a supported sandwich plate’s sound insulation performance with an hourglass lattice core. The governing equations of the lattice core sandwich plate were established using the Reissner sandwich plate theory. The solutions were derived using the Fourier series expansion. The developed model was verified by comparing the results of the models from the existing literature and the finite element method (FEM).⁴⁰ Marx and Rabiei⁴¹ proposed an experimental setup to determine the mechanical properties of a steel-steel composite metal foam (SS-CMF) and composite metal foam core sandwich panels (SS-CMF-CSP) manufactured and tested under quasi-static tension. Ye et al. presented a new higher-order refined analysis model to analyze a sandwich plate’s static and free vibration with a soft functionally graded material core. The governing equations were derived from equilibrium differential equations of motions. The accuracy and convergence of the presented model and numerical solution were validated versus available results.⁴² Fiber-metal laminates are hybrid sandwich composite structures made of thin metallic sheets and layers of fiber-reinforced plastics. Beylergil et al.⁴³ for the first time, experimentally evaluated the effects of polyamide 66 nonwoven interlayers on the tensile, three-point bending, interlaminar shear strength, and low-velocity impact responses of fiber-metal laminates were investigated by coupling acoustic emission, thermography, and microscopy techniques. Arani et al. worked on the vibration response of a porous sandwich composite beam. They presented some experimental and theoretical efforts to model the vibration caused by the electromagnetic field in sandwich beam structures.^44–46

Consequently, applying the appropriate mechanical modeling of the honeycomb structure is the fundamental calculation of acoustic parameters such as acoustic impedance. Understanding of correlation of acoustic impedance and geometrical and mechanical properties of structure (such as stiffness) is fundamental for acousticians to design more applicable sound isolations concerning honeycomb structures. Since low-frequency noise control in buildings is difficult due to the inadequate sound insulation between 50 and 200 Hz for most partitions and doors, the study of low-frequency noise reduction is critical for building acoustics performance.⁴⁷ Stiff and light panels instead of heavy panels can improve the sound insulation, but very few test data are available. Recent research by Wen-chao and Chung-fai⁴⁸ shows that a cement surface sheet with Nomex paper honeycomb core design can also provide additional noise control performance. It was understood that the honeycomb core could enhance the damping of the structure and consequently improve the sound insulation at frequencies between 50 and 100 Hz. However, sound insulation has reduced because of vibration resonance at 125 Hz, and the acoustic insulation performance in the medium frequency region between 250 and 1000 Hz was less than those of thin and heavy metal plates due to the existence of local honeycomb core shear resonance between 600 and 2000 Hz.⁴⁹ A double-layered honeycomb design has been employed to increase the local resonance frequency, while the static strength remained the same.⁵⁰

The present work’s novelty is presenting a simple analytical model for the sound absorption coefficient calculation of honeycomb structure. In this case, it is possible to acoustically design suitable honeycomb structure only by changing honeycombs’ geometries. The sound absorption coefficient and acoustic impedances can alter by changing honeycomb characteristics (material and geometry).

This paper is organized into five sections. Section 2 is dedicated to the theoretical determination of honeycomb’s equivalent density and stiffness by theoretical solutions. Section 3 deals with the theoretical framework of sound transmission through multilayer systems with considering attenuation. Before the conclusion section, section 4 presents the main contribution of this work, namely, the effect of the honeycomb structure’s geometrical parameters to calculate the acoustic impedance of the matching layers for increasing the design of sound absorber by changing the geometry of honeycomb.

Mechanical modeling of honeycomb and assumptions consideration

Geometrical characteristic of a honeycomb structure

The honeycomb is a network of six-celled cells with a certain number of cells in both the y and x directions arranged in an array. Specifications for each cell of the honeycomb structure are defined, which determine the geometry of each cell, including internal angle $(θ)$ , side length (h), arm length (L), arm width (H), and thickness (t).

In Figure 1, the properties shown on the cell are presented for two negative and positive internal angles. The cell structure for negative angles is called the auxetic structure.

Figure 1.

Geometrical properties of the honeycomb structure cell: (a) positive internal angle and (b) negative internal angle.

From Figure 1, it is possible to obtain the following geometrical equations for both positive and negative interior angles.

H = 2 h + 2 l \sin θ

(1)

L = 2 l \cos θ

(2)

A limitation of interior angles for both positive and negative (auxetic) interior angles is considered to prevent changing the internal angle of the cell structure. This limitation is within an interval of $+ 45^{°}$ and $- 45^{°} .$ ⁵¹

Speed of sound in a solid structure

The speed of sound for pressure waves in stiff materials such as metals is sometimes given for “long rods” of the material in question, in which the speed is more comfortable to measure. In rods where their diameter is shorter than a wavelength, pure pressure waves’ speed may be simplified and given by.⁵²

C = \sqrt{\frac{E}{ρ}}

(3)

Where E is Young’s modulus.

Honeycomb cell direction

Different types of loading influence the honeycomb panels’ mechanical response significantly; therefore, the load type should be considered in the modeling. Two types of typical loading, in-plane and out-plane, are shown in Figure 2. Furthermore, types of loading influence the sound absorption because of the panel’s different stiffness due to the different loading directions. The present study concentrates on the out-plane loading, which is called the second direction.

Figure 2.

Two different cell directions: (a) in-plane and (b) out-plane loading.

Effective properties of honeycomb sandwich panels

The effective mechanical properties of the honeycomb core can be obtained by mathematical analysis of one cell. It is required to calculate the equivalent structure’s density and structure Young’s modulus to obtain the sound absorption coefficient.

Effective density

Effective density can be obtained by examining the repeating member of the structure, a cell, the mass of the air confined to each cell, plus the mass of the constituent material of each cell, which is divided by the volume of a cell. Then, the effective density can be written as follows.

ρ_{e} = \frac{m_{e}}{v_{e}} = \frac{m + m_{0}}{v + v_{0}}

(4)

Where $ρ_{e}$ is the effective cell density, m is the mass of hexagonal cell constituent, m_o is the mass of air inside the cell, m_e is the cell effective mass, v is the volume of hexagonal cell constituent, v_o is the volume of air inside the cell, and v_e is the effective cell volume.

Since the mass of air confined to the cell can be neglected against the mass of the cell’s constituent material $(m > > m_{0})$ and the volume of the cell can also be neglected against the volume of the confined air $(v_{0} > > v),$ it can be rewritten as follows:

ρ_{e} = \frac{m}{v_{0}}

(5)

For the constituent of the cell we have the equation (6):

ρ = \frac{m}{v}

(6)

By combining equations (5) and (6), the following equation is obtained.

ρ_{e} = ρ \frac{v}{v_{0}} = ρ \frac{A \times b}{A_{0} \times b} = ρ \frac{A}{A_{0}}

(7)

Where A, is the cross-section area of the three sides specified is equivalent to the cell’s constituent and A₀ is the cross-section area of air confined inside the cell. Also, equations (8) and (9) represent the levels A and A₀, respectively.

A = 2 \times t \times l + t \times h

(8)

A_{0} = 2 \times l \cos θ \times h + 2 \times l \cos θ \times l \sin θ

(9)

Now, by replacing equations (8) and (9) in equation (7) and simplifying, the effective density is obtained as follows:

ρ_{e} = ρ \frac{\frac{t}{l} (\frac{h}{l} + 2)}{2 \cos θ (\frac{h}{l} + \sin θ)}

(10)

Effective Young’s modulus in the second orientation

The effective Young’s modulus of the honeycomb cell indicates network stiffness in the second orientation. According to Young’s modulus, which is the stress-strain ratio, we need to obtain these two quantities in this cell orientation. Because of the symmetry in each cell’s hexagonal structure, it is sufficient to obtain the stress and strain for two of the six sides. These two sides are illustrated in Figure 3.

Figure 3.

The stress applied to the two sides of the cell in the second orientation.

In terms of Figure 3 and the definitions of the stress and strain correlations, we obtain:

σ_{2} = \frac{F}{b (l c o s θ)}

(11)

In the above equation, F is the force of the sound wave.

Due to the non-occurrence of the buckling phenomenon, the strain correlation is obtained by neglecting the displacement of the side length h.⁵³

ε_{2} = \frac{δ_{2} \cos θ}{h + l \sin θ}

(12)

$δ_{2}$ is the deflection related to the side with length $ℓ .$

In equation (12), $δ_{2} \cos θ$ is the amount of side-to-length, $ℓ$ displacement is in line with the stress, and $ℓ \cos θ$ (the initial length) is in this direction. The deflection of a beam with length $ℓ$ and cross-section area $t \times b$ is influenced by the force p, whose force is perpendicular to its axis define as below⁵³:

δ_{2} = \frac{p l^{3}}{12 E I}

(13)

In equation (13), p is the force caused by the sound wave, $ℓ$ is the side length, E is the Young modulus of the beam, and I is the second torque of the surface. In Figure 4, if the beam’s neutral axis is assumed to correspond to the y-axis, and if the stress is applied to the z-axis, then the beam will have torque around the x-axis.

Figure 4.

A loading beam along the z-axis with a torque around the x-axis.

In this case, the second torque of the surface around the x-axis is obtained from the following correlation.⁵⁴

I_{x} = \int_{A} z^{2} d A

(14)

In equation (14), the surface element is located on the xz plane and is expressed as $d A = d z . d x .$

By replacing into equation (14) and considering the dimensions of the beam in Figure 6, we have:

I_{x} = \int_{- b / 2}^{b / 2} \int_{- t / 2}^{t / 2} z^{2} d z d x

(15)

In Figure 4, we show the vertical component of the force $F \sin θ$ applied to the side with length $ℓ$ and by replacing it in equation (13):

δ_{2} = \frac{F \cos θ l^{3}}{12 E I}

(16)

Now by combining 12 to 16, the following equation for the strain can be obtained:

ε_{2} = F {(\frac{l}{t})}^{3} \frac{\cos^{2} θ}{E b (h + l \sin θ)}

(17)

Finally, by applying the equations (11) and (17) and replacing them in the main correlation of Young’s modulus based on stress and strain, the effective Young’s modulus in the second orientation is obtained as follows:

E_{e, 2} = \frac{σ_{2}}{ε_{2}} = E {(\frac{t}{l})}^{3} \frac{(\frac{h}{l} + \sin θ)}{\cos^{3} θ}

(18)

Since Young’s modulus convert to the complex form, $E^{*} = E_{a} + E_{2} j$ , while considering the hysterics loss, $E_{a}$ the real part is the Young’s modulus and $E_{2}$ is the imaginary part that considers the loss effect. Therefore, viscous losses can defined by $η$ with $η = \frac{E_{2}}{E_{a}}$ correlation. Then, the complex Young’s modulus and equation (18) can be written as follows:

E^{*} = E_{a} (1 + η j)

E_{e, 2}^{*} = E_{a} (1 + i η) {(\frac{t}{l})}^{3} \frac{(\frac{h}{l} + \sin θ)}{\cos^{3} θ}

(19)

Also, it can be mentioned that if the number of cells increases by multiply n in constant thickness, the spring constant in its corresponding mechanical model decreases n times, and finally, the following relationships can be obtained.

E_{e, 2}^{*} = \frac{E_{a}}{2 n} (1 + i η) {(\frac{t}{l})}^{3} \frac{(\frac{h}{l} + \sin θ)}{\cos^{3} θ}

(20)

Where n denotes the number of cells in constant thickness.

As shown in equation (20), the relationship between the effective Young’s modulus in honeycomb and the geometry and material of the panel is obtained. This relationship shows that the effective Young’s modulus of the honeycomb is a function of the geometry and material of the honeycomb panel. In other words, it means that by changing the geometry of the panel, different coefficients of effective Young’s modulus can be created.

Effective acoustic impedance of honeycomb sandwich panels

Effective sound speed

To obtain the sound speed in the core of the panel, we use equation (3). However, instead of Young’s modulus and density, the complex effective Young’s modulus and the effective density are set, respectively. The output is called complex effective sound speed. By combining equations (3), (10), and (20), the effective sound speed in the panel’s core is obtained for the honeycomb structure by second orienting the cells as follows.

\begin{array}{l} C_{1}^{*} = \sqrt{\frac{E_{e, 1}^{*}}{ρ_{e}}} = {C^{'}}_{1} + i {C^{″}}_{1} \\ {C^{'}}_{1} = 2 \frac{t \cos^{2} θ}{\sin θ} \sqrt{\frac{E_{a} (\sqrt{η^{2} + 1} + 1)}{n ρ x (y \cos θ - x \sin θ + 2 x)}} \\ {C^{″}}_{1} = 2 \frac{t \cos^{2} θ}{\sin θ} \sqrt{\frac{E_{a} (\sqrt{η^{2} + 1} - 1)}{n ρ x (y \cos θ - x \sin θ + 2 x)}} \end{array}

(21)

Where y and x are the cell height and the cell thickness, $θ,$ $ρ,$ E_a, and t are the inner angle, the density of the constituent material, Young’s modulus of the constituent material, and the thickness of the cell walls, respectively.

Effective wavenumber

Equation (22) would be used to obtain the complex wavenumber, and complex effective sound speed must be used instead of sound speed.

k = \frac{ω}{C}, (\frac{r a d}{m})

(22)

By inserting equation (21) into equation (22), a complex wavenumber can be obtained for the nucleus with the honeycomb structure in the second type of cell orientation.

\begin{array}{l} k_{2}^{*} = \frac{ω}{C_{2}^{*}} = {k_{2}}^{'} + i {k_{2}}^{''} \\ {k_{2}}^{'} = \frac{1}{4} \frac{n y^{2} ρ ω (x \cos θ - y \sin θ + 2 y)}{t x E_{a} \sqrt{η^{2} + 1} \cos θ} \sqrt{\frac{E_{a} (\sqrt{η^{2} + 1} + 1)}{n y ρ (x \cos θ - y \sin θ + 2 y)}} \\ {k_{2}}^{''} = - \frac{1}{4} \frac{n y^{2} ρ ω (x \cos θ - y \sin θ + 2 y)}{t x E_{a} \sqrt{η^{2} + 1} \cos θ} \sqrt{\frac{E_{a} (\sqrt{η^{2} + 1} - 1)}{n y ρ (x \cos θ - y \sin θ + 2 y)}} \end{array}

(23)

Effective acoustic impedances

According to the definition given of the acoustic impedance $(r = ρ c),$ the complex acoustic impedance of the honeycomb can be obtained as follows:

\begin{array}{l} r_{2}^{*} = ρ_{e} C_{2}^{*} = {r_{2}}^{'} + i {r_{2}}^{''} \\ {r_{1}}^{'} = \frac{2 t^{2}}{y^{2}} \sqrt{\frac{ρ (x \cos θ - y \sin θ + 2 y) E_{a} (\sqrt{η^{2} + 1} + 1)}{n y}} \\ {r_{2}}^{''} = \frac{2 t^{2}}{y^{2}} \sqrt{\frac{ρ (x \cos θ - y \sin θ + 2 y) E_{a} (\sqrt{η^{2} + 1} - 1)}{n y}} \end{array}

(24)

Sound wave transmission through multilayer systems considering attenuation

The multilayer transmission situation is shown in Figure 5. A plane- attenuated longitudinal sound wave is propagating from left to right (the positive $x$ direction) through a series of $n$ layers of material of differing specific acoustic impedance. The incident wave travels through medium 1 and undergoes a series of reflections and transmissions in the subsequent layers until a transmitted wave emerges into medium n. The acoustic impedance and length of the kth layer ( $1 \leq k \leq n$ ) are denoted $z_{k}^{*},$ and $l_{k} .$ The nth medium has an infinite length, that is, $l_{n} = \infty .$

Figure 5.

A plane longitudinal sound wave transmission.

Subsequently, the following assumptions will be considered:

The reflected wave in the medium $n$ is assumed to be non-existent as this layer is assumed of infinite length;

Although the bounded media may produce multiple reflections and transmissions in each layer, it is sufficient to consider the steady-state condition.

No energy loss is assumed while the incident wave is generated.

We consider the wave attenuation only due to the constant loss factor ( $η$ ).

Considering the first layer, we denote $p_{i 1}$ and $v_{i 1},$ the incident wave pressure and particle velocity, respectively. Similarly, the reflected wave pressure and particle velocity are denoted $p_{r 1},$ and $v_{r 1} .$ Now, it is time to focus on an intermediate medium, the kth medium (1 < k < n). Two waves are propagating in opposite directions, the transmitted and reflected waves. The transmitted wave has pressure $p_{t k}$ and particle velocity $v_{t k},$ and the reflected wave has $p_{r k},$ and particle velocity $v_{r k} .$ The nth layer has only one propagating wave as it is semi-infinite, the transmitted wave, of which the pressure and particle velocity are denoted $p_{t n},$ and $v_{t n},$ respectively.

The mathematical introduction method applied in Saffar and Abdullah⁵⁵ published by the same author which yields to the following equation to link the pressures and particle velocities in both the first and last media:

[\begin{matrix} P_{n} (0) \\ V_{n} (0) \end{matrix}] = [\begin{matrix} P_{n - 1} (l_{n - 1}) \\ V_{n - 1} (l_{n - 1}) \end{matrix}] = [T_{n - 1}] [T_{n - 2}] \dots [T_{3}] [T_{2}] [\begin{matrix} P_{1} (l_{1}) \\ V_{1} (l_{1}) \end{matrix}],

(25)

where

\begin{array}{l} [T_{k}] = \frac{1}{2} [\begin{matrix} \cos k_{1 k} l_{k} g + j \sin k_{1 k} l_{k} h & \frac{b \times z_{k}}{k_{k}} (\cos k_{1 k} l_{k} h + j \sin k_{1 k} l_{k} g) \\ \frac{k_{k}}{b \times z_{k}} (\cos k_{1 k} l_{k} h + j \sin k_{1 k} l_{k} g) & \cos k_{1 k} l_{k} g + j \sin k_{1 k} l_{k} h \end{matrix}], (1 < k < n) \\ g = (e^{k_{2 k} l_{k}} + e^{- k_{2 k} l_{k}}); \\ h = (e^{- k_{2 k} l_{k}} - e^{k_{2 k} l_{k}}); \\ b = [k_{1 k} - η_{k} k_{2 k} + j (k_{2 k} + η_{k} k_{1 k})] . \end{array}

(26)

is the transfer matrix of the kth layer.

In this equation, $z_{k},$ $l_{k},$ and $η_{k}$ are the acoustic impedance, the thickness of the layer and the loss factor of the kth layer, respectively. Additionally, $k_{1 k},$ and $k_{2 k}$ are the real and imaginary parts of the complex wavenumber. $k_{k}^{*}$ , the kth layer, can be calculated from the equations mentioned in section 2. By considering in equation (25), the expressions of the pressure and the particle velocity given in equation (26), we obtain:

[\begin{matrix} 1 & 1 \\ \frac{1}{z_{n}^{*}} & - \frac{1}{z_{n}^{*}} \end{matrix}] [\begin{matrix} A_{n}^{*} \\ B_{n}^{*} \end{matrix}] = [T_{n - 1}] [T_{n - 2}] \dots [T_{3}] [T_{2}] [\begin{matrix} e^{- j k_{1}^{*} l_{1}} & e^{j k_{1}^{*} l_{1}} \\ \frac{1}{z_{1}^{*}} e^{- j k_{1}^{*} l_{1}} & - \frac{1}{z_{1}^{*}} e^{j k_{1}^{*} l_{1}} \end{matrix}] [\begin{matrix} A_{1}^{*} \\ B_{1}^{*} \end{matrix}],

(27)

or equivalently:

[\begin{matrix} A_{n}^{*} \\ B_{n}^{*} \end{matrix}] = {[\begin{matrix} 1 & 1 \\ \frac{1}{z_{n}^{*}} & - \frac{1}{z_{n}^{*}} \end{matrix}]}^{- 1} [T_{n - 1}] [T_{n - 2}] \dots [T_{3}] [T_{2}] [\begin{matrix} e^{- j k_{1}^{*} l_{1}} & e^{j k_{1}^{*} l_{1}} \\ \frac{1}{z_{1}^{*}} e^{- j k_{1}^{*} l_{1}} & - \frac{1}{z_{1}^{*}} e^{j k_{1}^{*} l_{1}} \end{matrix}] [\begin{matrix} A_{1}^{*} \\ B_{1}^{*} \end{matrix}] .

(28)

where $A_{1}^{*},$ and $A_{n}^{*}$ are the complex amplitude of the incident pressure wave from the first and nth layer, respectively. Additionally, $B_{1}^{*},$ and $B_{n}^{*}$ are the complex amplitudes of the reflected pressure wave from the first and nth layer, respectively. As $A_{1}^{*}$ is known, and $B_{n}^{*} = 0,$ equation (5) is a two-equation linear system with two unknown variables, and can be solved to obtain $A_{n}^{*}$ and $B_{1}^{*} .$ Subsequently, we can deduce the sound-pressure transmission coefficient (T) for a system of $n$ layers $T .$ This expression is given by

T = | \frac{A_{n}^{*}}{A_{1}^{*}} | . and R = 1 - T

(29)

The above equation’s physical meaning is that the transmission coefficient is a function of the acoustic impedance of the matching layers, their number, their internal losses, and the matching layer thicknesses. Therefore, increasing the transmission coefficient is possible by choosing the optimal parameters and the optimal combination of these parameters. Now, we write the sound-power transmission ( $T_{Π}$ ) and sound-power reflection ( $R_{Π}$ ) coefficients and, finally, the sound absorption coefficient $α$ for a honeycomb structure.

Now assuming three environments (Figure 6), which the first and third environments (the two sides of the panel) are the same, it is possible to calculate both R and T values as follows:

R = \frac{i [2 \frac{ω ρ_{s}}{r_{1}} \cos (k_{2} L) + (\frac{r_{2}}{r_{1}} - \frac{ω^{2} {ρ_{s}}^{2}}{r_{2} r_{1}} - \frac{r_{1}}{r_{2}}) \sin (k_{2} L)]}{[2 \cos (k_{2} L) - 2 \frac{ω ρ_{s}}{r_{2}} \sin (k_{2} L)] + i [2 \frac{ω ρ_{s}}{r_{1}} \cos (k_{2} L) + (\frac{r_{2}}{r_{1}} - \frac{ω^{2} {ρ_{s}}^{2}}{r_{2} r_{1}} + \frac{r_{1}}{r_{2}}) \sin (k_{2} L)]}

(30)

Figure 6.

Schematic of sound transmission from a honeycomb structure.

Instead of the acoustic impedance and wavenumber in this environment, a complex equivalent can be utilized using the second environment’s damping (panel core).

We use the following relationships, regardless of the orientation of the cells.

r^{*} = r^{'} + {ir}^{″}

(31)

k^{*} = k^{'} + i k^{″}

Now, by placing the equation (31) in equation (30) instead of the acoustic impedance and the wavenumber of the second environment and multiplying the resultant in its conjugate, the correlation of the wave power reflection in the first environment is obtained.

R_{Π} = {| R |}^{2} = \frac{Ψ (ω, θ)}{Γ (ω, θ)} = \frac{A (ω, θ) \sinh^{2} (k^{''} L) + B (ω, θ) \sinh (k^{''} L) + C (ω, θ)}{D (ω, θ) \sinh^{2} (k^{''} L) + E (ω, θ) \sinh (k^{''} L) + F (ω, θ)}

(32)

Expansion of coefficients $F (ω, θ)$ , $E (ω, θ)$ , $D (ω, θ)$ , $C (ω, θ)$ , $B (ω, θ)$ , and $A (ω, θ)$ is given in the Appendix.

T_{Π} = 1 - {| R |}^{2} = \frac{Θ (ω, θ)}{Γ (ω, θ)} = \frac{A^{'} (ω, θ) \sinh^{2} (k^{''} L) + B^{'} (ω, θ) \sinh (k^{''} L) + C^{'} (ω, θ)}{D (ω, θ) \sinh^{2} (k^{''} L) + E (ω, θ) \sinh (k^{''} L) + F (ω, θ)}

(33)

Expansion of coefficients $C^{'} (ω, θ)$ , $B^{'} (ω, θ)$ , and $A^{'} (ω, θ)$ is given in the Appendix.

The power transmitting through the boundaries of the first and second environments is divided into two parts: the first power consumed in the second environment (panel core) and the second power imported into the third environment.

T_{Π} = α (ω, θ) + {T_{Π}}^{'} = \frac{A^{'} (ω, θ) \sinh^{2} (k^{''} L) + B^{'} (ω, θ) \sinh (k^{''} L)}{Γ (ω, θ)} + \frac{C^{'} (ω, θ)}{Γ (ω, θ)}

(34)

The following two equations can be extracted from equation (34)

α (ω, θ) = \frac{A^{'} (ω, θ) \sinh^{2} (k^{''} L) + B^{'} (ω, θ) \sinh (k^{''} L)}{Γ (ω, θ)}

(35)

{T_{Π}}^{'} = \frac{C^{'} (ω, θ)}{Γ (ω, θ)}

(36)

Equation (35) indicates the core of the panel’s absorption coefficient, and equation (36) represents the transmitted power of the panel (input power to the third environment).

It should be noted that if there is no damping in the second environment, the imaginary part of the wavenumber will also be zero, and eventually, the sound absorption coefficient $α$ will be zero for all angles and frequencies.

Results

In this section, the sound absorption coefficient of sandwich panels with honeycomb cores has been investigated at different angles with various constituents. The quantity of absorption coefficient based on equations was mentioned in the previous section. The physical meaning of that explains more here. In general, sound waves striking an arbitrary surface are transmitted, reflected, or absorbed; the amount of energy going into a transmission, reflection, or absorption depends on the acoustic properties of the surface. The absorbed sound may either be transmitted or dissipated. Sound energy is dissipated by simultaneous actions of viscous and thermal mechanisms. Sound absorbers are used to dissipate sound energy and to minimize its reflection. The absorption coefficient, $α,$ is a typical quantity used for measuring the sound absorption of a material and is known to be the function of the incident wave frequency. It is defined as the ratio of energy absorbed by a material to the energy incident upon its surface. Sound absorption coefficient in the frequency range of 1–500 Hz for angles of 45°, 30°, 15°, 0°, −15°, −30°, −45° will be demonstrated using MATLAB software.

A specific dimension (popular in the industry) of the panel is selected (Figure 7).

Figure 7.

A piece of panel in which the absorption of acoustics is investigated with the second type orientation.

In this research, 0.7 mm is considered the thickness of the layers, and 1 mm is chosen for the thickness of the cells’ walls which are popular in the industry (http://www.monopan.de/en/lieferspezifikationen.html).

The constituents of the honeycomb core and their physical properties

Three types of polymeric materials are selected as the honeycomb core. Their properties are presented in Table 1.

Table 1.

Material properties of honeycomb core.⁵⁶

Material	$η$	$E (G p a)$	$ρ (k g / m^{3})$
Polycarbonate	0.03	2.7	1170
Polypropylene	0.016	1.76	932
Polyurethane	0.025	0.167	1160

Panel sound absorption coefficient in the second orientation

In this section, we assume the cells of the panel core to be oriented in the second direction concerning the sound pressure wave, by the previous section to maintain the hexagonal shape of the cell, the interior angle being in degrees −45 < $θ$ < 45 and this limitation also allows us to consider two modes in the second orientation for aligning the cells in the segment shown in Figure 8.

Figure 8.

1 × 1 arrangement for organizing cells in a segment with an orientation of the second type.

1—First arrangement (1 × 1)

In this case, as shown in Figure 8, the honeycomb thickness consists of one cell, and its height comprises one cell.

2—Second arrangement (2 × 2)

In this case, according to Figure 9, two cells are included in the core thickness, and the height is two cells.

Figure 9.

2 × 2arrangement for aligning cells in a slice with a second orientation.

3—Third arrangement (3 × 3)

In this case, according to Figure 10, three cells are included in the core thickness, and the height is three cells.

Figure 10.

3 × 3 arrangement for aligning cells in a slice with a second orientation.

Sound absorption coefficient polycarbonate in the second orientation

To investigation the amount of sound absorption of the polycarbonate by using data in Table 1 and replacing them into the sound absorption equation, it is possible to plot sound absorption curves at a frequency range of 1–500 Hz for angles: 45°, 30°, 15°, 0°, −15°, −30°, −45°.

Figure 11 shows the sound absorption amount of the panels for negative and positive angles, for 1 × 1 and second load orientation of cells.

Figure 11.

Polycarbonate honeycomb panel sound absorption curves for negative and positive angles in 1 × 1 mode and the second type orientation for cells.

As shown in Figure 11, two phenomena occur; first, the resonance frequency shifts to the lower frequency range (shift to the left side of the curve). Consequently, more sound absorption can be reached at larger angles. Second, the value of each resonance frequency peak is larger in negative angles. Therefore, negative angles are better for sound absorption.

According to the correlation $ω = \sqrt{\frac{K}{m}},$ for the smaller K, the resonant frequency of the panel is shifted to lower values, and therefore, to have better absorption at lower frequencies, we must choose the size of the inner angle of the cells as large as possible. However, as shown in Figure 11, the absorption at −45° has more peaks than that at 45°. From the above explanation, it can be concluded that for a sandwich panel with a honeycomb core with one row of cells with an orientation of the second type in its thickness, the −45° angle is the best.

Now, sound absorption is considered for the 2 × 2 arrangement; Figure 12 shows the panel sound absorption curves for negative and positive angles.

Figure 12.

Polycarbonate honeycomb panel absorption curves for negative and positive internal angles at 2 × 2 mode and in the second load orientation for cells.

The same fact, as shown in Figure 11, can be observed in Figure 12. A comparison between Figures 11 and 12 demonstrate a little bit better situation regarding more sound absorption when 1 × 3 arrangement is employed. However, working frequency shifts to a higher frequency regarding more sound absorption.

The last case is the 3 × 3 arrangement illustrated in Figure 13.

Figure 13.

Polycarbonate honeycomb panel absorption curves for negative and positive internal angles at 3 × 3 and in the second load orientation for cells.

It can be concluded from the comparison between Figures 11 to 13 that the number of peaks in the 1 × 1 mode is more than the 2 × 2 and 3 × 3 modes because the arm length $l$ decreases with the increasing number of cells at a constant thickness (based on presented theory) and at a constant angle, concerning appearing at the power of three at the denominator increases the effective complex Young’s modulus and transfers the frequencies to larger values. In other words, with the increase in the number of cells at the constant thickness, the number of peaks in the frequency range is reduced, and eventually, we will have less sound absorption—furthermore, the working frequency of the panel shifts to the higher one.

Sound absorption coefficient for polypropylene and polyurethane in the second orientation

Table 1 is employed to investigate the sound absorption of polypropylene and Polyurethane panels. The panel sound absorption can be illustrated for angles of 45, 30, 15, 0, −15, −30, −45, and in the frequency range of 1–500 Hz for 1 × 1, 2 × 2, and 3 × 3 modes in Figures 14 to 19.

Figure 14.

Polypropylene honeycomb panel absorption curves for negative and positive internal angles in 1 × 1 mode and the second type orientation for cells.

Figure 15.

Polypropylene honeycomb panel absorption curves for negative and positive internal angles in 2 × 2 mode and the second orientation for the cells.

Figure 16.

Polypropylene honeycomb panel absorption curves for negative and positive internal angles in 3 × 3 mode and the second orientation for the cells.

Figure 17.

Absorption curves of honeycomb core polyurethane with negative and positive internal angles in 1 × 1 mode and second type orientation for cells.

Figure 18.

Panel absorbance curves of honeycomb core with polyurethane with negative and positive internal angles in 2 × 2 mode and second type orientation for cells.

Figure 19.

Panel absorbance curves of honeycomb core with polyurethane with negative and positive internal angles in 3 × 3 mode and second type orientation for cells.

The same results were obtained as the previous section. A little more shift frequency to the lower frequency has been observed when polypropylene is used rather than polycarbonate. Polypropylene is a better absorber than Polycarbonate, regarding more sound absorption. Also, the case of 1 × 1 mode is much better than 2 × 2 and 3 × 3 modes if more sound absorption is needed in lower frequencies.

Figures 17 to 19 show the panel sound absorption curves with negative and positive angles for 1 × 1, 2 × 2, and 3 × 3 modes, respectively, when Polyurethane is employed. As shown in Figures 17 to 19, the same result as the previous one was observed. The only difference is the high impact of material effect to increase sound frequency band absorption. They demonstrate that Polyurethane is superior to the other two substances, ultimately resulting in less effective Young’s modulus.

Consequently, by comparing the sound absorption curves of all three materials, it can be concluded that the 1 × 1 arrangement has more sound absorption than the 2 × 2 and 3 × 3 ones. Also, the number of Polyurethane material peaks for all angles is more than the other two materials, so in the second load type orientation for more sound absorption, the panel with the honeycomb core of Polyurethane and 1 × 1 arrangement is recommended for cells with an internal angle of −45°.

Conclusion

Today, noise pollution has become a serious issue, especially in metropolises. Hence, the suitable absorbents for absorbing noise at a wide bandwidth, especially at lower frequencies, and occupying the least space in the structure, are of great interest.

The results showed that the sandwich structure allows the acoustic function to be controlled by changing the cells’ geometrical and physical properties. By modifying the single cell’s geometrical and physical properties, it can be obtained the effective and desirable acoustic properties of the honeycomb panel.

In this paper, the effective complex Young’s modulus for the honeycomb structure was obtained considering attenuation. Furthermore, the complex acoustic impedance and the effective sound velocity were calculated. Finally, we investigated the core sandwich panel containing the honeycomb grid, and it was observed that a lower Young’s modulus material such as Polyurethane should be selected for higher absorption. It was also observed that the 1 × 1 mode is a good cell arrangement to maintain the honeycomb structure regarding the more sound absorption.

Footnotes

Appendix

\begin{array}{l} A (ω, θ) = [{(\frac{2 ω ρ_{s}}{r_{1}} \tan (k^{'} L) + \frac{r^{'} ({ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} - {| r |}^{2})}{r_{1} {| r |}^{2}})}^{2} \\ + {(\frac{r^{''} ({ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})}^{2} c o s^{2} (k^{'} L)] \end{array}

B (ω, θ) = - 4 ρ_{s} ω \frac{r^{''} ({ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{{r_{1}}^{2} {| r |}^{2}} \cosh (k^{″} L)

\begin{array}{l} C (ω, θ) = [{(\frac{r^{''} ({ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})}^{2} \\ + {(\frac{2 ρ_{s} ω}{r_{1}} c o t (k^{'} L) - \frac{r^{'} ({ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} - {| r |}^{2})}{r_{1} {| r |}^{2}})}^{2}] \sin^{2} (k^{'} L) \cosh^{2} (k^{″} L) \end{array}

\begin{array}{l} D (ω, θ) = [{(- \frac{2 ρ_{s} ω r^{''}}{{| r |}^{2}} + \frac{2 ρ_{s} ω}{r_{1}} \tan (k_{1} L) - \frac{r^{'} (- {ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})}^{2} \\ + (\frac{2 ρ_{s} ω r^{'}}{{| r |}^{2}} + 2 \tan (k_{1} L) + \frac{r^{''} ({ρ_{s}}^{2} ω^{2} - {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})] c o s^{2} (k^{'} L) \end{array}

\begin{array}{l} E (ω, θ) = [(2 c o s (k^{'} L) - \frac{2 ρ_{s} ω r^{'}}{{| r |}^{2}} \sin (k^{'} L) - \frac{r^{''} ({ρ_{s}}^{2} ω^{2} - {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}} \sin (k^{'} L)) \times \\ (- \frac{2 ρ_{s} ω r^{''}}{{| r |}^{2}} + \frac{2 ρ_{s} ω}{r_{1}} \tan (k^{'} L) - \frac{r^{'} (- {ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})] \\ + [(\frac{2 ρ_{s} ω}{r_{1}} c o s (k^{'} L) + \frac{2 ρ_{s} ω r^{''}}{{| r |}^{2}} \sin (k^{'} L) + \frac{r^{'} (- {ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}} \sin (k^{'} L)) \times \\ (- \frac{2 ρ_{s} ω r^{'}}{{| r |}^{2}} - 2 \tan (k^{'} L) - \frac{r^{''} ({ρ_{s}}^{2} ω^{2} - {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})] \end{array}

\begin{array}{l} F (ω, θ) = [{(2 c o t (k^{'} L) - \frac{2 ρ_{s} ω r^{'}}{{| r |}^{2}} - \frac{r^{''} ({ρ_{s}}^{2} ω^{2} - {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})}^{2} \\ + {(\frac{2 ρ_{s} ω}{r_{1}} c o t (k^{'} L) + \frac{2 ρ_{s} ω r^{''}}{{| r |}^{2}} + \frac{r^{'} (- {ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})}^{2}] \sin^{2} (k^{'} L) \cosh^{2} (k^{″} L) \end{array}

\begin{array}{l} A^{'} (ω, θ) = 4 [(\frac{r^{'} ρ_{s} ω - r^{''} r_{1}}{{| r |}^{2}} + \tan (k^{'} L)) (\frac{r^{'} ρ_{s} ω}{{| r |}^{2}} + \frac{r^{''} ({ρ_{s}}^{2} ω^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}} + \tan (k^{'} L)) \\ + (\frac{r^{''} ρ_{s} ω + r^{'} r_{1}}{{| r |}^{2}}) (\frac{r^{''} ρ_{s} ω}{{| r |}^{2}} - \frac{r^{'} ({ρ_{s}}^{2} ω^{2} - {| r |}^{2})}{r_{1} {| r |}^{2}} - \frac{2 ρ_{s} ω}{r_{1}} \tan (k^{'} L))] c o s^{2} (k^{'} L) \end{array}

\begin{array}{l} B^{'} (ω, θ) = 2 [[(\frac{2 ρ_{s} ω}{r_{1}} c o t (k^{'} L) + \frac{2 ρ_{s} ω r^{''}}{{| r |}^{2}} + \frac{r^{'} (- {ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}}) \\ (- \frac{2 ρ_{s} ω r^{'}}{{| r |}^{2}} - 2 \tan (k^{'} L) - \frac{r^{''} ({ρ_{s}}^{2} ω^{2} - {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})] + \\ [(2 \cot (k^{'} L) - \frac{2 ρ_{s} ω r^{'}}{{| r |}^{2}} - \frac{r^{''} ({ρ_{s}}^{2} ω^{2} - {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}}) \\ (- \frac{2 ρ_{s} ω r^{''}}{{| r |}^{2}} + \frac{2 ρ_{s} ω}{r_{1}} \tan (k^{'} L) - \frac{r^{'} (- {ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})]] \sin (k^{'} L) c o s (k^{'} L) \cosh (k^{″} L) + \\ \frac{4 ρ_{s} ω}{r_{1}} \frac{r^{''} ({ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}} \cosh (k^{″} L) \end{array}

\begin{array}{l} C^{'} (ω, θ) = 4 [(\cot (k^{'} L) - \frac{ρ_{s} ω r^{'}}{{| r |}^{2}} - \frac{r^{''} ({ρ_{s}}^{2} ω^{2} + {r_{1}}^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}}) (\cot (k^{'} L) + \frac{r^{''} r_{1} - r^{'} ρ_{s} ω}{{| r |}^{2}}) \\ + (\frac{r^{''} ρ_{s} ω + r^{'} r_{1}}{{| r |}^{2}}) (\frac{2 ρ_{s} ω}{r_{1}} \cot (k^{'} L) + \frac{ρ_{s} ω r^{''}}{{| r |}^{2}} + \frac{r^{'} (- {ρ_{s}}^{2} ω^{2} + {| r |}^{2})}{r_{1} {| r |}^{2}})] \sin^{2} (k^{'} L) \cosh^{2} (k^{″} L) \end{array}

Declaration of conflicting interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Saber Saffar

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