A novel gradient multi-Helmholtz cavities muffler for low-frequency broad band gaps

doi:10.21203/rs.3.rs-3335898/v1

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A novel gradient multi-Helmholtz cavities muffler for low-frequency broad band gaps

https://doi.org/10.21203/rs.3.rs-3335898/v1

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With the development of local resonance theory, more and more researchers use Helmholtz cavity structures to reduce low-frequency noise. However, the absorption frequency is very narrow in the traditional structure. It is very inflexible in practical applications. To address this challenge, we first proposed a gradient multi-Helmholtz cavities muffler for low-frequency broad band gaps. By simplifying the structure into a neck-cavity model, we derived the resonance frequency of a single Helmholtz cavity. Next, we use COMSOL software to calculate the transmission loss (TL). The results of our analysis confirmed that the structure can reduce noise in its resonance frequency, and the range of absorption frequency is enlarged with the number of the cavities increases, almost 24.75 times that of one cavity. The transmission loss around the center frequency also increases from 45dB to near 100dB. In addition, the relationship between noise attenuation characteristics and the distance of the cavities is also studied. Results show that a deeper valley appears in the transmission loss curve with the increase of the distance, which greatly affects the sound absorption performance. The gradient multi-Helmholtz cavities designed in our paper enlarges the range of absorption frequency and improve the ability of noise reduction. Our findings can be widely applied to guide the design and optimization of metamaterial mufflers.

Physical sciences/Engineering/Mechanical engineering

Physical sciences/Materials science/Condensed matter physics

acoustic metamaterial

gradient

Helmholtz cavities

noise reduction

The interior of a ship is a very complicated piping arrangement. Due to imperfect working process, mechanical equipment, pumps, valves would inevitably produce vibration and noise during system load shock, and then forming a plurality of noise sources [1–2]. Seawater pipeline system is connected to the hull, it is difficult to reduce noise because the large size of the hull can create a large radiator. In addition, strong vibration will make the pipeline accessories and the connection loosening to rupture. Therefore, Noise control in pipeline system is always the focus of academic debate [3–7]. How to improve noise absorption performance in pipeline system is an urgent requirement to maintain the vitality of the ship.

The traditional way to improve the noise absorption performance in pipeline system is to choose low noise pump, set up elastic joints, bellows and other components, install muffler elbow, water muffler and so on [8–9]. To some extent, these measures can avoid valve suddenly open and close, reduce pipeline elbow. However, the control of broad-frequency noise reduction by traditional way is severely restricted, it is difficult to meet the target requirements of ship quietness design. How to design a structure in pipeline system with low-frequency and broad-frequency sound absorption is a challenge problem to many researchers.

Traditional materials have a lot of limitations in the practical engineering application, so, artificial materials with special physical characteristics gradually come into the sight of scientific researchers, such as metamaterial and phononic crystals [10–14]. Over the last couple of decades, artificial metamaterial have expanded into the field of acoustics and vibration, used for noise reduction. Then, as industrial society develops towards miniaturization and integration, the new type of artificial materials with microstructure feature size of subwavelength has been made further development. Which is later known as acoustic metamaterial [15–18]. The most typical physical properties in acoustic metamaterials is ‘single negative’ or ‘double negative’ properties [19], In the range of negative properties, the structure tend to absorb low frequency or control sound wave propagation. Guided by this theory, a lot of acoustic devices were designed to control sound wave or reduce noise in low frequency.

Helmholtz cavity is one of the most typical acoustic metamaterial which is widely studied in recent years [20–22].

The reason is Helmholtz cavity can reduce noise in very low frequency, and the absorption coefficient of it is very considerable. However, the disadvantage of this material is the range of frequency is very narrow. Ding and Zhao [23] designed a novel one-dimensional Helmholtz cavity structure and carried out corresponding experiment. They proved that the absorption coefficient can reach over 0.85 in its resonance frequency. Lee et al. [24] took advantage of the “double negative” properties and designed an acoustic metamaterial structure. They successfully add the thin membrane into the Helmholtz cavity. The result is also very good. For the seawater pipe system, Shen [25] designed a Helmholtz muffler based on phononic crystal theory, the Helmholtz cavity is placed periodically in the seawater pipe. In order to further optimize the range or bandgap, he adjusted the structural parameters of the Helmholtz cavity and obtained good results. This design has important application value to the sound absorption and vibration reduction of seawater pipeline.

Influenced by this Helmholtz cavity theory, this paper proposes a new concept of gradient Helmholtz cavity. This design is more applicable because it can reach a very wide range of absorption frequency within 500Hz. The range of the noise-absorption frequency can also be enlarged as the number of cavities increases, limit can be reached close to 300Hz, near 24.75 times that of one cavity.

In the second section, a concise derivation of resonant frequency of multi-cavity Helmholtz muffler is given. In the third section, its noise reduction characteristics are verified by COMSOL software. The simulation and variation trend of transmission loss in gradient-change Helmholtz cavity is also clarified in section 3. Our analysis and simulation provide new insights into a better understanding of gradient-change Helmholtz cavity and can be generally used to guide the simulation and optimization of certain acoustic devices.

In this article, the main contributions and applications include:

By simulating and comparing the physical parameters of a single Helmholtz cavity, we get a very low frequency of sound absorption, and the result has a good agreement with the theoretical data.
A novel gradient-change Helmholtz cavity is designed, low frequency band gap is enlarged with the number of the cavities increase, and the maximum range can reach 300Hz, near 24.75 times that of one cavity. In addition, the transmission loss around the center frequency is also increases from 45dB to near 100dB.
The relationship between noise attenuation characteristics and the distance of the cavities is further study. Results show that a deeper valley appears in the transmission loss curve with the increase of the distance, the valley splits the formerly wide muffler band in two, which greatly affects the sound absorption performance.

The structure of a typical Helmholtz resonator is shown in Fig. 1. The structure consists of a slender neck and cavity. In the slender neck, the fluid Euler equation can be approximated as:

$$\rho \frac{{\partial v}}{{\partial t}}+\nabla p=0$$

Take the line integral over the slender neck, Eq. (1) can be expressed as:

$$\int_{{{L_0}}}^{0} {\left( {\rho \frac{{\partial v}}{{\partial t}}+\nabla p} \right)dx} =0$$

For plane wave approximation, the wavelength linearity satisfies:$\lambda \gg {L_0}$;$\lambda \gg \sqrt {{S_0}}$, and$v=v{e^{jwt}}$, Eq. (2) can be expressed as:

$$- j\omega \rho {L_0}v+{p_0} - {p_{{L_0}}}$$

So, the expression of acoustic impedance and equivalent acoustic quality is obtained:

$$X=\frac{P}{{v{S_0}}}=j\omega \frac{{P{L_0}}}{{{S_0}}} \Rightarrow Ma=\frac{{\rho {L_0}}}{{{S_0}}}$$

In the cavity, by conflating conservation of mass with the equation of state, we can get:

$$\frac{{\partial p}}{{\partial t}}+\rho {c^2}\nabla \cdot v=0$$

Take the volume integral over the cavity, Eq. (5) can be expressed as:

$$\int {\left( {\frac{{\partial p}}{{\partial t}}+\rho {c^2}\nabla \cdot v} \right)} dV=0$$

If$\lambda \gg \sqrt {{S_0}}$;$\lambda \gg \sqrt[3]{{{V_0}}}$, So the first term of Eq. (6) can be linearized, and the second term becomes a surface integral and still linearized as:

$$- j\omega p{V_0}+p{c^2}v{S_0}=0$$

So the reactance and the volume can be expressed as:

$$X=\frac{P}{{v{S_0}}}=\frac{{p{c^2}}}{{j\omega {V_0}}} \Rightarrow {C_a}=\frac{{{V_0}}}{{\rho {c^2}}}$$

Combining slender neck and cavity in the Helmholtz resonator, the acoustic reactance can be expressed as:

$$X=j\omega \frac{{\rho {L_0}}}{{{S_0}}}+\frac{{p{c^2}}}{{j\omega {V_0}}}$$

Set acoustic reactance to zero, resonant frequency can be obtained:

$\omega =c\sqrt {\frac{{{S_0}}}{{{V_0}{L_0}}}}$ , $f=\frac{\omega }{{2\pi }}$ (10)

Where c is the speed of sound. ${S_0},{L_0},{V_0}$are the area of slender neck, the length of slender neck and the volume of the cavity, respectively.

The pipe with one Helmholtz cavity is first studied, simulation diagram is shown in Fig. 2. The pipe consists of three parts, Perfect matching layer, background pressure field layer and main pipe layer, length of each part are${{\text{L}}_{{\text{Pef}}}}=5{\text{ cm}}$, ${{\text{L}}_{{\text{bac}}}}=12{\text{ cm}}$, ${{\text{L}}_{{\text{pipe}}}}=50{\text{ cm}}$, respectively. The diameter of the pipe is${{\text{D}}_{{\text{Pef}}}}=5{\text{ cm}}$. For the single Helmholtz cavity, the geometric Parameters are${{\text{D}}_{{\text{cav}}}}=8{\text{ cm}}$,${{\text{h}}_{{\text{cav}}}}=3{\text{ cm}}$,${{\text{D}}_{{\text{nec}}}}=3{\text{ cm}}$,${{\text{h}}_{{\text{nec}}}}=6{\text{ cm}}$.

Substitute the data of Helmholtz cavity into Eq. 10 and let$c=340m/s$, the resonant frequency is$f=478.29{\text{Hz}}$.

In order to verify the sound absorption effect of Helmholtz cavity at the resonant frequency, the commercially software, COMSOL Multiphysics 5.5, was used in the analyses.

The plane wave radiation is applied on one side and transmission loss of sound propagation is measured on the other side. The schematic diagram is shown in Fig. 3. Transmission loss is defined bellow:

$$TL{\text{=}}{L_i} - {L_t}=10\lg \frac{{{I_i}}}{{{I_t}}}$$

Where${L_i}$ and ${L_t}$ are the power of incident sound and transmission sound, and ${I_i}$,${I_t}$ are the power of the incident sound and transmission sound, which are defined as follows:

$$\begin{gathered} {I_i}=\int_{{{S_P}}} {\frac{{{{\left| {{P_i}} \right|}^2}}}{{2\rho {c_0}}}ds} \hfill \\ {I_t}=\int_{{{S_P}}} {\frac{{{{\left| {{P_t}} \right|}^2}}}{{2\rho {c_0}}}ds} \hfill \\ \end{gathered}$$

Where${P_i}$ and ${P_t}$ are the pressure of incident sound and transmission sound, ${S_P}$ is the area of the inlet and outlet. The transmission coefficient of the sound intensity is defined below:

$$T=\frac{{{I_t}}}{{{I_i}}}=\frac{{{{\left| {{P_t}} \right|}^2}/2\rho {c_0}}}{{{{\left| {{P_i}} \right|}^2}/2\rho {c_0}}}$$

If the inlet and outlet areas of the pipeline are the same, the relationship between the loss and transmission coefficient can be defined as follows:

$$TL=10\lg \left( {{t^{ - 1}}} \right)$$

The transmission loss is calculated using COMSOL, which is shown in Fig. 4. The muffling frequency is around$480{\text{Hz}}$, which has a good agreement with the resonant frequency, also, the transmission loss reaches nearly 45dB. It is proved that the Helmholtz cavity can truly have good silencing properties near its resonant frequency.

However, if we want to get a transmission loss over 30dB, the range of absorption frequency is only 12.12Hz. The reason is Helmholtz cavity can only get a very narrow muffling peak in its resonant frequency. Obviously, the result does not satisfy the need of broadband noise reduction. Through the derivation of the theory in the previous section, it can be concluded that if more different resonance frequencies can be obtained, and these frequencies are close enough, the range of frequency can definitely increases.

In this paper, we propose the concept of a gradient resonance frequency, the resonant frequency position is controlled by adjusting the geometric parameters inside the Helmholtz cavity, which are defined as follows:

$$\begin{gathered} {{\text{D}}_n}={{\text{D}}_1}\cdot {\text{(0}}{\text{.9+0}}{\text{.1n) }} \hfill \\ {{\text{h}}_n}={{\text{h}}_1}\cdot {\text{(0}}{\text{.9+0}}{\text{.1n) }} \hfill \\ \end{gathered}$$

Where ${{\text{D}}_n}$and ${{\text{h}}_n}$are the diameter and height of cylindrical cavity, n represents the number of Helmholtz cavities, and${{\text{D}}_1}{\text{=}}8{\text{cm}}, {{\text{h}}_1}{\text{=}}3{\text{cm}}$.

The first case to be considered is n = 2, which is depicted in Fig. 5(a). The transmission loss is also calculated using COMSOL. As depicted in Fig. 5(b), it can been seen that there are two distinct resonance frequency peaks. Also, the range of frequency increases to 112.13Hz when the transmission loss is over 30dB. That's a tenfold increase compared to n = 1. So, the width of reduction frequency can be greatly broadened by arranging Helmholtz cavities in a gradient way.

To make the results more obviously, three cases n = 4, 6, and 8 are considered, respectively. Based on the transmission loss of 30 dB, the frequency range of sound absorption in different numbers of cavities was measured. In Fig. 6(a), four gradient cavities are added in the pipe, result is shown in Fig. 6(b), width of the frequency to reduce noise reach up to 203.28Hz. Results show that the width of silencing frequency increases as the number of cavities increases, and the transmission loss at the peak is increasing. The same phenomenon appears in six cavities and eight cavities. The schematic diagram is shown in Fig. 7(a) and Fig. 7(c) and transmission loss is shown in Fig. 7(b) and Fig. 7(d), the width can reach up to 230.86Hz and 273.54Hz, almost 22.5 times that of one cavity. This result is quite striking and perfect. This is a big improvement compared to only one Helmholtz cavity.

The Helmholtz cavity theory is used to reduce the noise of the structure. Helmholtz cavity structure is composed of a large cavity and a small neck. Since the volume of the neck is much smaller than that of the cavity, the fluid in the neck can be considered approximated incompressible. However, the fluid in the neck is considered compressible under the motion of the neck fluid. As the fluid in the cavity is compressed, the pressure increases naturally, providing a reverse restoring force. According to the previous description, the fluid in the neck can be regarded as a mass and the fluid in the lumen can be regarded as a spring. On this basis, in order to increase the number of resonance frequency of the entire structure, we adjust the geometrical parameters of the cavities so that they can varies with gradient. When the excitation frequency is close to the resonant frequency, the Helmholtz cavities can absorb sound waves. Figure 8 shows the acoustic pressure of this muffler in frequency f = 400Hz when there is four Helmholtz cavities, the result shows that most of the sound pressure in the pipe is absorbed into the cavity. Because the gradient variation of Helmholtz cavity has affected the resonance frequency, the original resonant frequencies is overlapped. Although 400Hz is not a resonant frequency, the whole structure can still achieve a good sound attenuation performance at 400Hz

When noise enters the muffler, the Helmholtz cavities act as a filter, preventing sound waves of a certain width of frequency from passing through. However, when there is only one or two cavities, sound waves cannot be fully absorbed, and some will escape from the Helmholtz cavity. By comparing different cases, it is easy to find the width of the absorption frequency increases by the number of cavities gradient-increasing, and as the number increases, the lifting speed gradually slows down, eventually close to 300Hz, almost 24.75times that of one cavity, which is shown in Fig. 9. In addition, the transmission loss around the center frequency also increases from 45dB to near 100dB. So, as the number of cavities increases, not only the range of band gap is broadened, but also the ability to absorb sound waves is improved.

In order to further study the relationship between noise attenuation characteristics and the distance of the cavities, another simulation is made by changing their distance from 15 cm to 37.5 cm. We can notice from the transmission loss in Fig. 10. (a), a valley gradually appears at 420Hz, and the valley becomes more and more obvious when the distance of the cavities increases. The valley splits the formerly wide muffler band in two, makes the resonant frequencies of the two cavities independent of each other. The phenomenon results in the structure not being able to absorb sound in wide range of silence frequency, and reduces the muffler ability to some extent. Figure 10(b) and Fig. 10 (c) shows the acoustic pressure of two cavities with the distance 15cm and 37.5cm, respectively. The frequency of 420Hz at the valley is selected, it is easy to find that most of the sound pressure in the pipe is absorbed into the cavity when the distance is 15cm, prove that the valley has little effect on the noise-reduction feature. However, when the distance is 37.5cm, most sound pressure appears in the end of the pipe, as a result, the Helmholtz cavity basically no longer has sound absorption ability in the valley.

In our work, a novel gradient multi-Helmholtz muffler is used to reduce low frequency noise. Through theoretical modeling, finite element simulation and distance comparison analysis, the structural design and noise reduction performance of the muffler are studied.

The corresponding simulation results agree with the theory derivation, which verify that the multi-cavity Helmholtz muffler can reduce noise in its resonant frequency, and the resonant frequency is below 500Hz, which can meet the requirements of low frequency sound absorption. Our study show that the width of the absorption frequency can be enlarged by increasing the number of cavities gradiently. This performance is better than some traditional Helmholtz cavities that have only a narrow fixed noise-reduction frequency, the maximum range can reach near 24.75 times that of one cavity. In addition, the transmission loss around the center frequency is also increases from 45dB to near 100dB. This superior quality has a very good effect on broadband and low frequency noise reduction. The relationship between noise attenuation characteristics and the distance of the cavities is also further study in our paper. Results show that a deeper valley appears in the transmission loss curve with the increase of the distance, the valley splits the formerly wide muffler band in two, which greatly affects the sound absorption performance. The theory of resonant frequency, corresponding simulation in this paper provide new significance for research into acoustic mufflers.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

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No competing interests reported.

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A novel gradient multi-Helmholtz cavities muffler for low-frequency broad band gaps

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1. Introduction

2. Theory and model

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