Study the effect of thickness on the natural and resonant frequencies of polyurethane foams

Abstract

In this paper, the effects of thickness variations on the natural and resonant frequencies of polyurethane foams have been studied. The governing Biot’s equations are solved using an analytical method, and the natural and resonant frequencies are presented alongside the variations in solid, fluid displacement, and pressure. By selecting the characteristics of three different foam samples from various references, the variations in the frequency characteristics of the foam with changes in thickness are investigated over a broad range, and specific results are provided. Based on the presented results, it is observed that as thickness increases, the solid and fluid natural frequencies, along with the resonant frequencies, shift to lower frequencies. Interestingly, the values of the natural frequencies at lower thicknesses are repeated at higher thicknesses, maintaining their values and nature while following a specific repetition rule.

Keywords

Resonant frequency natural frequency thickness polyurethane foam potential function method

1. Introduction

Natural and damped frequencies are fundamental characteristics of each structure and medium. Natural frequencies are inherently altered by changing the attributes of any medium, and this relationship is consensual. In other words, by recognizing the natural frequencies in the system, the behavior and output of the system can be predicted. Natural and damping frequencies are part of the inherent characteristics of the system, and recognizing their relationship with variable changes can be effective in designing and identifying the behavioral characteristics of a class of materials.

Recognizing the position of natural frequencies in any dynamic system is one of the most basic measurements performed in any system’s vibration analysis. In other words, by identifying the position of natural frequencies, it is possible to comment on the behavior of free and forced vibrations of the system. The detection of the natural and resonant frequencies in porous materials due to their two-phase characteristics has already been considered.

In 2016, Hasani Baferani et al. (Hasani Baferani et al., 2016) presented a method for obtaining the natural and resonant frequencies in porous media, considering Biot’s equations. This research demonstrates the relationship between natural and resonant frequencies with the acoustic absorption behavior of this class of materials and yields significant results. Based on the offered results, it was observed that the maximum-minimum variations in the acoustic absorption coefficient occur at the location of the solid natural frequencies of the porous media. In other words, by recognizing the position of the structure’s solid and fluid natural frequencies, it is possible to make an acceptable prediction of the acoustic absorption coefficient behavior of the porous media.

Yamaguchi et al. (Yamaguchi et al., 2009) studied the damped vibration analysis of automotive double walls with a porous material. The damped vibration of the double walls with a porous layer is computed considering the coupling in damping between layers. Additionally, the modal damping between the layers has been discussed. Kinkelaar et al. (Kinkelaar et al., 1998) investigated the vibrational properties of polyurethane foam to ensure a comfortable ride and attenuation at the appropriate frequency ranges. Changing the foam formulation and processing variables could alter the foam’s vibrational characteristics. Jaouen et al. (Jaouen et al., 2008) presented the elastic and damping characterizations of acoustical porous materials such as melamine and polymeric foam. Based on the presented experimental results, the influence of viscoelasticity and anisotropy is often observed in porous materials.

Zhang and Dupuls (Zhang and Dupuis, 2011) presented the experimental techniques and identification of the dynamic properties of flexible polyurethane foam used in automotive seats. Mozafari and Najafian (Mozafari and Najafian, 2017) investigated the vibration frequencies and mode shapes of honeycomb sandwich panels with different cores using numerical methods. Ezzatnwshan and Sadraei (Ezzatneshan and Sadraei, 2023) presented the effects of vibration on droplet dynamics inside a three-dimensional (3D) porous medium, focusing on frequency, amplitude, and surface wettability. A lattice Boltzmann method based on the Allen–Cahn equation (A-C LBM) is used for finite element analysis.

This paper studied the effect of thickness on the resonant and natural frequencies of porous media. The governing equations of Biot’s equations have been solved analytically by the potential function method. The resonant and natural frequencies of polyurethane foams have been obtained, and the mentioned parameters have been studied for actual samples. Different characteristics of porous media, such as fluid and solid displacements and pressure, have been plotted to obtain selected resonant and natural frequencies of foams. Also, the variation of absorption coefficients of the mentioned foams has been compared with the thickness variation.

2. Governing equations

Biot’s equations, which are the most complete governing equations for porous materials in terms of solid and fluid displacements, are as follows (Biot, 1956a, 1956b):

P \frac{\partial}{\partial x} [\frac{\partial u_{s}^{1}}{\partial x} + \frac{\partial u_{s}^{2}}{\partial y}] + N (\frac{\partial}{\partial y} [\frac{\partial u_{s}^{1}}{\partial y} - \frac{\partial u_{s}^{2}}{\partial x}]) + Q \frac{\partial}{\partial x} [\frac{\partial u_{f}^{1}}{\partial x} + \frac{\partial u_{f}^{2}}{\partial y}] = - ω^{2} {\tilde{ρ}}_{11} u_{s}^{1} - ω^{2} {\tilde{ρ}}_{12} u_{f}^{1}

(1)

P \frac{\partial}{\partial y} [\frac{\partial u_{s}^{1}}{\partial x} + \frac{\partial u_{s}^{2}}{\partial y}] - N (\frac{\partial}{\partial x} [\frac{\partial u_{s}^{1}}{\partial y} - \frac{\partial u_{s}^{2}}{\partial x}]) + Q \frac{\partial}{\partial y} [\frac{\partial u_{f}^{1}}{\partial x} + \frac{\partial u_{f}^{2}}{\partial y}] = - ω^{2} {\tilde{ρ}}_{11} u_{s}^{2} - ω^{2} {\tilde{ρ}}_{12} u_{f}^{2}

(2)

R \frac{\partial}{\partial x} [\frac{\partial u_{f}^{1}}{\partial x} + \frac{\partial u_{f}^{2}}{\partial y}] + Q \frac{\partial}{\partial x} [\frac{\partial u_{s}^{1}}{\partial x} + \frac{\partial u_{s}^{2}}{\partial y}] = - ω^{2} {\tilde{ρ}}_{12} u_{s}^{1} - ω^{2} {\tilde{ρ}}_{22} u_{f}^{1}

(3)

R \frac{\partial}{\partial y} [\frac{\partial u_{f}^{1}}{\partial x} + \frac{\partial u_{f}^{2}}{\partial y}] + Q \frac{\partial}{\partial y} [\frac{\partial u_{s}^{1}}{\partial x} + \frac{\partial u_{s}^{2}}{\partial y}] = - ω^{2} {\tilde{ρ}}_{12} u_{s}^{2} - ω^{2} {\tilde{ρ}}_{22} u_{f}^{2}

(4)

where

u_{s}^{i} (i = 1, 2)

and

u_{f}^{i} (i = 1, 2)

are the solid and fluid displacements in different directions, respectively. Also,

ω

represents the angular frequency. In addition, elastic coefficients

P, R, Q

, and

N

and densities

{\tilde{ρ}}_{11}, {\tilde{ρ}}_{12}

, and

{\tilde{ρ}}_{22}

have been investigated by Biot (Biot and Willis, 1957) as follows:

P = \frac{(1 - ϕ) [1 - ϕ - \frac{K_{b}}{K_{s}}] K_{s} + ϕ \frac{K_{s}}{K_{f}} K_{b}}{1 - ϕ - \frac{K_{b}}{K_{s}} + ϕ \frac{K_{s}}{K_{f}}} + \frac{4}{3} N Q = \frac{[1 - ϕ - \frac{K_{b}}{K_{s}}] ϕ K_{s}}{1 - ϕ - \frac{K_{b}}{K_{s}} + ϕ \frac{K_{s}}{K_{f}}} R = \frac{ϕ^{2} K_{s}}{1 - ϕ - \frac{K_{b}}{K_{s}} + ϕ \frac{K_{s}}{K_{f}}} {\tilde{ρ}}_{11} = (1 - ϕ) ρ_{s} - {\tilde{ρ}}_{12} {\tilde{ρ}}_{22} = ϕ ρ_{f} - {\tilde{ρ}}_{12} {\tilde{ρ}}_{12} = (1 - α_{\infty}^{'}) ϕ ρ_{f}

(5)

where

K_{s}

denotes the bulk modulus of the elastic solid,

K_{b}

is the bulk modulus of the frame in vacuum, and

K_{f}

represents the bulk modulus of the fluid. Also,

ϕ

denotes the porosity of porous media. Notably, the dynamic tortuosity,

α_{\infty}^{'}

, is calculated based on the Johnson et al. model (Johnson et al., 1987), and the bulk modulus of the fluid is attained from the Champoux-Allard model (Champoux and Allard, 1991). All assumption is considered based on the statement presented in Ref. (Hasani Baferani et al., 2016). By considering the solution procedure presented in Ref. (Hasani Baferani et al., 2016), the governing equations are decoupled as:

ξ_{1} \frac{d^{4} ψ_{1}}{d x^{4}} + ξ_{4} \frac{d^{2} ψ_{1}}{d x^{2}} + ξ_{5} ψ_{1} = 0

(6)

N \frac{d^{2} ψ_{2}}{d x^{2}} + [ω^{2} ({\tilde{ρ}}_{11} - \frac{{\tilde{ρ}}_{12}^{2}}{{\tilde{ρ}}_{22}}) - N k^{2} \sin^{2} θ] ψ_{2} (x) = 0

(7)

In addition, the coefficients are

ξ_{1} = - \frac{Q λ_{1}}{λ_{3}}, ξ_{2} = P - \frac{{Q λ}_{2}}{λ_{3}} - \frac{ω^{2} {\tilde{ρ}}_{12} λ_{1}}{λ_{3}}, ξ_{3} = ω^{2} ({\tilde{ρ}}_{11} - \frac{{\tilde{ρ}}_{12} λ_{2}}{λ_{3}}), ξ_{4} = ξ_{2} - 2 k^{2} ξ_{1} \sin^{2} θ, ξ_{5} = ξ_{3} - k^{2} ξ_{2} \sin^{2} θ + k^{4} ξ_{1} \sin^{4} θ, λ_{1} = P R - Q^{2}, λ_{2} = R ω^{2} {\tilde{ρ}}_{11} - Q ω^{2} {\tilde{ρ}}_{12}, λ_{3} = R ω^{2} {\tilde{ρ}}_{12} - Q ω^{2} {\tilde{ρ}}_{22}

(8)

The partial differential equations (6) and (7) are based on potential functions, that is, $ψ_{1}$ and $ψ_{2}$ . By solving the decoupled equations, as obtained:

ψ_{1} (x) = C_{1} e^{η_{1} x} + C_{2} e^{- η_{1} x} + C_{3} e^{η_{2} x} + C_{4} e^{- η_{2} x}

(9)

ψ_{2} (x) = C_{5} e^{η_{3} x} + C_{6} e^{- η_{3} x}

(10)

where

\begin{array}{l} η_{1} = \frac{\sqrt{- 2 ξ_{1} (ξ_{4} + \sqrt{ξ_{4}^{2} - 4 ξ_{1} ξ_{5}})}}{2 ξ_{1}} \\ η_{2} = \frac{\sqrt{2 ξ_{1} (- ξ_{4} + \sqrt{ξ_{4}^{2} - 4 ξ_{1} ξ_{5}})}}{2 ξ_{1}} \\ η_{3} = \sqrt{\frac{ω^{2}}{N} ({\tilde{ρ}}_{11} - \frac{{\tilde{ρ}}_{12}^{2}}{{\tilde{ρ}}_{22}}) - k^{2} \sin^{2} θ} \end{array}

(11)

By using Equations (9) and (10), all field variables can be obtained based on six unknown parameters, that is, $C_{1}$ through $C_{6}$ , which can be determined by applying boundary conditions. These boundary conditions are related to the air-porous medium interface and porous medium-hard wall interface (Allard and Atalla, 2009; Hasani Baferani et al., 2016). By splitting the impedance difference, which represents the ratio of pressure to velocity, the absorption coefficient can be determined at each frequency. The mentioned reference provides additional information on how to obtain the absorption coefficient.

3. Verification

In this section, to ensure the validity of the simulation results, the variations in the absorption coefficient of a foam sample with two different tortuosity coefficients, as presented in a Ref. (Asadi et al., 2015) based on the Transfer Matrix Method, are compared with the results obtained for the same foam sample using the present method, and the comparison is illustrated in Figure 1. Based on the presented results, it can be observed that there is a good agreement between the presented results, and the proposed method is validated.

4. Results and discussion

In this section, three foam samples have been examined based on different references to investigate the effect of thickness on natural and resonant frequencies, in addition to the behavior of the absorption coefficient at different thicknesses, whose characteristics are presented in Table 1. The considered foams were selected from various references and included a wide range of other characteristics to study the behavior of natural frequencies by changes in thickness.

Table 1.

Non-acoustical and mechanical characteristics of different used polyurethane foams.

Foam number	Porosity	Flow resistivity	Tortuosity	Viscous length	Thermal length	Density	Young modulus	Poisson ratio	Structural damping
1 (Allard and Atalla, 2009)	0.98	6600	1.03	200	380	11.2	2.93E+05	0.2	0.06
2 (Dauchez et al., 2001)	0.98	13,500	1.7	80	160	30	270,000	0.3	0.1
3 (Atalla et al., 1998)	0.8	3,200,000	2.56	6	24	125	1,000,000	0.3	0.1

The study of the changes in natural and resonance frequencies is a practical and comprehensive approach to gain a deeper understanding of different systems, due to variations in their characteristics. This paper focuses on how thickness variations affect changes in the natural frequencies and resonance of porous materials. Through this behavioral analysis, a better understanding of these systems can be achieved. Exploring the behavioral changes of natural frequencies, which are intrinsic characteristics of various systems, can aid in a more comprehensive understanding of porous material systems. Engineers and designers can benefit from behavioral studies of porous materials as sound-absorbing materials, which can help them use them more effectively and completely.

The process of obtaining natural and resonant frequencies is based on the method presented in Ref. (Hasani Baferani et al., 2016), which is derived from examining the fluid and solid parts of displacement and pressure in the frequency range of the mentioned frequencies. In each case, the variation of absorption coefficient of the desired foam is also provided. It should be noted that these results are for deformable foams at a temperature of 27°C, an air pressure of 87,800 Pa, and a humidity of 34%. In these figures, two Damping Gain (DG) values are considered, which will have a value between 0 and 1. To investigate the natural frequencies at DG = 0, three pressure graphs and the displacement of the fluid and solid parts are plotted on the vertical axis compared to the frequency on the horizontal axis, and the resonant frequencies at DG = 1 have been considered.

To better illustrate the subject, the specifications of various real polyurethane foam samples were selected from scientific references, and the change in behavior of these foams with varying thicknesses was investigated. Due to limitations in presenting results, only three foam samples were chosen for the article, but the established behavioral patterns also apply to all other foam samples.

4.1. Natural and resonance frequencies of foam 1

This foam has minimum flow resistivity, tortuosity, density, and structural damping compared to other selected foams. On the other hand, it has the highest viscous and thermal characteristic lengths, as well as the percentage of porosity among the selected foams. As presented in Ref. (Hasani Baferani et al., 2016), the way to detect the natural frequencies of the solid compared to the fluid natural frequencies is by observing significant changes in the minimum or maximum displacement values of the fluid displacement and pressure.

Figure 2 presents the solid and fluid displacements, pressure, and absorption coefficients of foam 1 with a thickness of 0.05 m. As can be seen, at a thickness of 0.05 m, a significant variation in pressure and fluid displacement is observed only at the frequency of 1553 Hz. Therefore, the frequency of 1553 Hz is considered the fluid natural frequency, while the other two peaks in the solid phase displacement diagram, which are 845 and 2536 Hz, are regarded as the solid natural frequencies. The value of the resonant frequency in DG = 1 is well represented in the solid displacement, with a value of 871 Hz. Additionally, the maximum absorption at this thickness occurs at the frequency of 1995 Hz. To investigate the effect of thickness on the mentioned frequencies, Figure 3 is examined mentioned parameter for 0.1 m thicknesses.

It can be observed that by increasing the thickness, the values of natural and resonant frequencies have shifted to lower frequencies. For foam sample 1 at a thickness of 0.1 m, the fluid natural frequencies are 765 and 2347 Hz, compared to the thickness of 0.05 m, which had a peak of 1553 Hz. This phenomenon indicates that the increase in thickness caused the frequencies to move to lower values. Additionally, the solid natural frequencies are 442, 1267, 2113, and 2958 Hz for 0.1 m foam 1. In comparison to the two frequencies at 0.05 m thickness, it is evident that the transition to lower frequencies has occurred at this stage. The resonant frequency is also at 464 Hz, and a transition to lower frequencies has happened at this frequency as well. The maximum absorption coefficient occurred at the frequency of 874 Hz.

Figure 4 investigated the solid and fluid displacement, pressure, and absorption coefficient of polyurethane foam 1 in 0.15 m thicknesses. As previously reported, by investigating the behavior of foam at a thickness of 0.15 m, it was found that with the increase in thickness, the values of natural and resonant frequencies shifted to lower frequencies. At a thickness of 0.15 m, the fluid natural frequencies are 503, 1553, and 2613 Hz. In comparison, at a thickness of 0.1 m, the frequencies were 765 and 2347 Hz, and at 0.05 m, the frequency was 1553 Hz. It can be observed that the frequency of 1553 Hz is repeated at both thicknesses of 0.05 m and 0.15 m. A similar trend is noted for the solid natural frequencies, which have values of 282, 845, 1408, 1972, and 2536 Hz, with the frequencies of 845 and 2536 Hz being repeated at the two thicknesses of 0.05 m and 0.15 m. Initial observations of variations in natural frequency in thickness indicate the repetition of natural frequency values at particular thicknesses.

Regarding the resonant frequency, it is transferred to lower frequencies and is located at 309 Hz. The remarkable point about the resonant frequency compared to the natural frequencies in this foam sample is that with the increase in thickness, the number of resonant frequencies has not changed, and only its value has been shifted to lower frequencies with the increase in thickness. The value of the maximum absorption frequency is 2727 Hz, and the value is 0.99. The variation of the acoustic absorption coefficient is shown in Figure 5 for three different thicknesses.

By considering the variation of absorption coefficients presented in Figure 5, it can be seen that with the increase in thickness, the maximum absorption coefficient has also been shifted to lower frequencies. In other words, one of the reasons for improving absorption behavior with increasing thickness is the transfer of fluid and solid natural frequencies to lower frequencies. The maximum-minimum behavior in this sample of foams occurred around the solid natural frequency and led to the improvement of the acoustic absorption behavior at lower frequencies. According to the results obtained from the investigation of the behavior of foam 1, in order to understand more about the reported process, the changes in foam 1 have been examined at greater thicknesses in Table 2.

Table 2.

The first 20 natural frequencies of solid (blue) and fluid (red) of polyurethane foam 1 for different thicknesses for below 3000 Hz Frequency.

Table 2 presents the first 20 natural frequencies of foam 1 at different thicknesses below 3000 Hz. In other words, the natural frequencies at a thickness of 0.05 m are repeated at 0.15, 0.25, 0.35, 0.45, and so on up to 0.95 m. It is observed that for this thickness of 0.05, with an increase in thickness of 0.1 m, there is again a repetition in the natural and resonant frequencies.

Regarding the thickness of 0.1 m, it is observed that the natural and resonant frequencies have been repeated at thicknesses of 0.3, 0.5, 0.7, and 0.9 m. For this thickness, with an increase of 0.2 m in each thickness, the frequencies have been repeated again. Regarding the thickness of 0.15 m, natural and resonant frequencies have been repeated at thicknesses of 0.45 and 0.75 m, which differ in height by 0.3 m. Similarly, for a thickness of 0.2 m, natural and resonant frequencies have been repeated at thicknesses of 0.6 and 1 m. As can be seen, the thickness increment factor at this stage is 0.4 m. The same trend of increase is observed for a thickness of 0.25 m, which has been repeated at 0.75 m, and for 0.3 m, which has been repeated at 0.9 m, with a difference of 0.6 m.

Regarding the repetition of frequencies within specific thickness ranges, it can be noted that the characteristic natural frequency presented is independent of thickness. By increasing thickness, each of frequencies are repeated again at specified thicknesses. This phenomenon is not dependent on specific solid and fluid frequencies, and repetition has occurred in both categories of frequencies. Considering the concept that the natural frequency is an inherent characteristic of the system, the reason for the repetition can be attributed to the constancy of specific intrinsic characteristics of the system, which remain unchanged with increasing thickness, resulting in the natural frequencies of both solid and fluid remaining stable and unchanged. In order to further investigate the behavior of natural frequency changes with variations in thickness, foam sample number 2 will also be examined.

4.2. Natural and resonant frequencies of foam 2

Now, the variety of natural and resonant frequencies of foam sample 2 has been examined. As can be seen from the results provided in Table 1, the characteristics of tortuosity, flow resistivity, and density have increased in foam sample 2. In contrast, the thermal and viscous characteristic lengths, as well as Young’s modulus, have decreased. Similar to the previous section, a thickness of 0.05 m will be examined first and then compared with the subsequent thickness data.

Figure 6 presents the solid and fluid displacement, pressure, and absorption coefficient of polyurethane foam 2 in 0.05 m thicknesses. The fluid natural frequency is located at 944 Hz, while the solid natural frequencies are found at 527, 1592, and 2665 Hz. The resonant frequency is also 641 Hz. The maximum absorption coefficient occurs at a frequency of 1005 Hz. Considering the changes in the characteristics of foam 2 compared to foam 1, an increase in the number of natural frequencies of the solid phase is observed. As noted, the characteristics of the solid phase, including Young’s modulus, have decreased, while structural damping, Poisson’s ratio, and density have increased. Additionally, changes in the intrinsic properties have caused the natural frequencies at the same thickness to shift to lower frequencies compared to foam 1. Specifically, the fluid natural frequency has changed from 1553 Hz in foam 1 to 944 Hz in foam 2.

Naturally, various non-acoustical and mechanical characteristics of the foam influence the behavior of its natural frequency positions. A comparison of the changes in acoustic absorption behavior in the two foam samples shows that the improvement in acoustic behavior has occurred at lower frequencies. The behavior of foam sample 2 will be further examined at a thickness of 10 cm.

Figure 7 investigated the solid and fluid displacement, pressure, and absorption coefficient of polyurethane foam 2 in 0.1 m thicknesses. In foam 2, the fluid natural frequencies are 452, 1478, and 2606 Hz, which have shifted to lower frequencies compared to the 0.05 m thickness. For the solid natural frequencies, there are six frequencies at 263, 792, 1325, 1860, 2397, and 2934 Hz, where the transition to lower frequencies has occurred with an increase in thickness. The resonant frequency has also shifted to a lower frequency of 304 Hz compared to the previous thickness. The maximum absorption coefficient occurred at this thickness is 362 Hz. With the increase in thickness, similar to the last sample, the transition of natural and resonant frequencies to lower frequencies is also observed in this foam sample.

Figure 8 presents the solid and fluid displacement, pressure, and absorption coefficient of polyurethane foam 2 in 0.15 m thicknesses. In this thickness, the fluid natural frequencies occurred at frequencies of 298, 944, 1662, and 2415 Hz, which transition to lower frequencies can be realized. As observed, the frequency of 944 Hz related to a thickness of 0.05 is repeated here. The same trend is seen for the solid natural frequencies, with values of 175, 527, 881, 1236, 1592, 1949, 2307, and 2665 Hz, where the values of 527, 1592, and 2665 are repeated from the thickness of 0.05 m. The resonance frequency has also shifted to lower frequencies and is in the range of 202 Hz. The absorption coefficient is located at a frequency of 219 Hz. The phenomenon of repeating natural and resonant frequencies, similar to foam sample 1, is also observed in this foam at a thickness of 0.15 m.

As observed from the results presented in Table 3, the trend reported in the previous foam sample is similarly repeated in this foam with different specifications. It’s interesting that this foam also adheres to the rule of repetition in thicknesses with a specified incremental rate using the same previous procedure. A closer examination reveals that the new frequencies generated with increasing thickness overlap with the repetitive frequencies of previous thicknesses in some cases. Additionally, the nature of the frequencies in the repetition in subsequent thicknesses does not change, and the fluid and solid natural frequencies remain unchanged. Considering the recurrence of the mentioned phenomena regarding foam sample number 2, this examination will also be conducted for another foam sample.

Table 3.

The first 20 natural frequencies of solid (blue) and fluid (red) of polyurethane foam 2 for different thicknesses for below 3000 Hz Frequency.

4.3. Natural and resonant frequencies of foam 3

Foam 3 has the highest values of tortuosity, flow resistivity, and density compared to previous samples, while its viscous and thermal characteristic lengths and porosity are lower than those of other samples.

According to the results presented in Figure 9, the fluid natural frequencies of this foam are 674 Hz and 2023 Hz. On the other hand, there are three solid natural frequencies, which are 495 Hz, 1486 Hz, and 2477 Hz, respectively. The resonant frequency is also 538 Hz. The maximum absorption coefficients occurred at a frequency of 539 Hz.

Figure 10 presents the solid and fluid displacement, pressure, and absorption coefficient of polyurethane foam 3 in 0.1 m thickness. As previously mentioned, the number of frequencies has increased. Based on the presented results, the fluid natural frequencies have been occurred at frequencies of 337, 1012, 1686, and 2361 Hz. On the other hand, for the solid natural frequencies at this thickness, the frequencies are 248, 743, 1239, 1734, 2229, and 2725 Hz. The resonant frequency occurs at 269 Hz, which, similar to previous foams, decreases with increasing thickness, and the peak of absorption coefficient has also reached at the frequency of 269 Hz. The repeated trend observed in the two previous foam samples is also noted in this foam sample.

Figure 11 investigated the solid and fluid displacement, pressure, and absorption coefficient of polyurethane foam 3 in 0.15 m thicknesses. Similar to the previous foams, the increase in thickness has led to a rise in the number of natural frequencies in this range. Additionally, the frequencies at a thickness of 0.05 m have been repeated at a thickness of 0.15 m. At this thickness, the fluid natural frequencies are observed at 225, 674, 1124, 1574, 2023, 2473, and 2923 Hz, while the solid frequencies are noted at 165, 495, 826, 1156, 1486, 1817, 2147, 2477, and 2807 Hz. Furthermore, the resonance frequency has decreased to 179 Hz. The maximum absorption coefficient occurs at a frequency of 180 Hz.

Table 4 presents the first 20 natural frequencies of solid and fluid natural frequencies of polyurethane foam 3 for different thicknesses. Upon examining foam sample number 3, it is evident that the phenomena mentioned in the two previous foam samples have occurred in this foam regarding its natural frequencies for both solid and fluid sections. Changes in structural characteristics naturally lead to an increase or decrease in frequencies at a specific thickness, and with the reinforcement of the structural section, there is an increase in the characteristics of solid natural frequency. Additionally, changes in the non-acoustical properties of the foam have resulted in variations in the number of fluid natural frequencies.

Table 4.

The first 20 natural frequencies of solid (blue) and fluid (red) of polyurethane foam 3 for different thicknesses for below 3000 Hz Frequency.

Regarding foam sample 3, a process entirely similar for the two previous foam samples is observed, with an increase in thickness leading to the repetition of natural frequencies at higher thicknesses for this foam as well. Considering the analysis conducted on three foam samples with entirely different specifications and the occurrence of natural frequency repetition at subsequent thicknesses, this phenomenon is repetitive. It is an inherent characteristic of the system.

Regarding all foam samples, it has been observed that after damping the system, there is only one resonant frequency present, and this frequency is close to the first solid natural frequency of the sample. The rate of decrease with increasing thickness for both the resonant frequency and the first solid natural frequency is nearly the same, and they exhibit a similar decreasing trend with increasing thickness.

5. Conclusions

In this paper, the variation of solid and fluid natural frequencies and resonant frequencies of three different polyurethane foams have been studied by changes in the thickness of samples. Biot’s governing equations have been solved analytically by using the potential function method. Various properties such as fluid and solid displacements and pressure presented for obtaining the fluid and solid natural frequencies have been presented. Based on the results presented in this paper, the following notes can be concluded:

1. By increasing the thickness, the natural and resonant frequencies of porous material shifted to lower frequencies.

2. The values of solid and fluid natural frequencies in lower thickness have been repeated at higher frequencies.

3. The nature of the frequencies in the repetition in subsequent thicknesses does not change, and the fluid and solid natural frequencies remain unchanged.

4. By examining several different foam samples, the behavior noted was replicated in foams with various specifications.

5. It has been observed that after damping the system, there is only one resonant frequency in all foam samples, and this frequency is close to the first solid natural frequency of polyurethane foam.

Footnotes

Declaration of Conflicting Interests

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

Funding

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

ORCID iDs

Mehran Heidari

Abolfazl Hasani Baferani

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