Experimental study and acoustic characteristics analysis of defective-state Helmholtz-ring phononic crystal muffler

Experimental study and acoustic characteristics analysis of defective-state Helmholtz-ring phononic crystal muffler | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Experimental study and acoustic characteristics analysis of defective-state Helmholtz-ring phononic crystal muffler Yang Bai, Yuehua Chen, Jiahui Zheng This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4963361/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 22 Nov, 2024 Read the published version in Scientific Reports → Version 1 posted 10 You are reading this latest preprint version Abstract To improve the sound absorption performance of the expansion chamber muffler at low frequencies, a Helmholtz-ring phononic crystal muffler is designed based on the local resonance mechanism. The phononic crystal muffler exhibits strong sound attenuation performance at deep sub-wavelength scales. Firstly, the phononic crystal scatterer is designed as a ring-type Helmholtz resonant chamber, and a certain amount of cell units is periodically arranged inside an expansion chamber muffler. Secondly, the effects of the dimension parameters of scatterers on the bandgaps are studied. The transmission loss of the phononic crystal muffler, together with the pressure loss at low Mach numbers, is investigated. Subsequent focus is devoted to analyzing the effects of point and linear defective states on the acoustic transmission characteristics of the phononic crystal muffler. The results show that a significant improvement in both transmission loss and aerodynamic performance of the proposed phononic crystal muffler is observed when compared to the original expansion chamber muffler. Additionally, the transmission loss within the bandgap can be further enhanced when the phononic crystal muffler is in a defective state. Finally, experimental investigations were conducted to validate the effectiveness of the phononic crystal muffler within its bandgap range. Physical sciences/Physics/Applied physics/Acoustics Physical sciences/Engineering/Mechanical engineering Phononic crystal muffler defect state analysis local resonance transmission loss pressure loss Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1 Introduction A muffler is a device designed to reduce noise by absorbing, reflecting sound waves, or altering their propagation paths through specific structural designs. Mufflers have been widely used for reducing the flow noise inside ducts, such as in a vehicle exhaust system or the ventilation system of home appliances [1].The expansion chamber muffler [2]-[4]. is a common type of muffler, its working principle involves the use of abrupt changes in the cross-sectional area of the duct to reflect sound waves. By incorporating expansion sections or resonant chambers within the duct, standing waves are produced, which dissipate sound energy. Traditional muffler has a good performance on middle and high frequency noise. But it has a poor performance on low-frequency noise [5]. Mid and low-frequency noises are common in industrial and living environments, such as those produced by mechanical equipment, motor vehicles, and air conditioning systems. Low-frequency noise, due to its long wavelengths, can easily penetrate barriers and is also readily perceptible to the human ear [6]. Low-frequency sound absorption is still a challenge for most passive noise control applications due to the space requirements to absorb large acoustic wavelengths [7]. Therefore, enhancing the performance of expansion chamber mufflers in the mid and low-frequency range is of significant research interest. Phononic crystals have unique advantages in controlling low-frequency noise. Phononic crystals are materials with periodic structures [8], and the concept of phononic crystals originated from the development of photonic crystals independently proposed by Yablonovitch [9] and John [10] in 1987. The initial interest in phononic crystals arose due to the existence of phononic Bragg bandgaps where the propagation of sound waves is prohibited [11]. In the year 2000, Liu et al. [12][13] discovered another mechanism for generating elastic wave bandgaps, namely, the local resonance mechanism, by studying three-dimensional phononic crystals composed of rubber, lead, and epoxy resin, going beyond the Bragg scattering mechanism. The introduction of the concept of local resonance-type phononic crystals broadened the scope of phononic crystal research [14]-[16]. Based on the local resonance mechanism, it is generally possible to achieve control of large wavelengths with small-sized structures. Due to the significant advantages of phononic crystal technology in noise control, the application of this technology in the research of pipeline mufflers has gradually attracted more attention. Zhang et al. [17] have developed a new type of metamaterial-based silencer that utilizes the principle of anomalous reflection to achieve efficient sound isolation in hollow pipes with sub-wavelength thickness. Shi [18], applying Bloch wave theory, studied the wave propagation in periodic microperforated pipe mufflers and examined the dispersion characteristics of periodic microperforated mufflers. Liu [19] proposed a compact hybrid muffler phononic crystal, calculated the transmission loss of a Unit cell using a two-dimensional transfer matrix method, and analyzed the bandgap characteristics as well as the noise reduction mechanism in the hybrid muffler. Liu et al. [20] integrated a labyrinthine metasurface array into the muffler design, thereby achieving high transmission loss and broad bandwidth acoustic insulation within the low-frequency spectrum. Almeida [21] reengineered the existing intake muffler by utilizing the concept of wave attenuation bands induced by Bragg scattering, which has led to an improvement in the transmission loss of mufflers employed in refrigerant compressors. Liu [22] proposed a periodic waveguide made of axially mounted expansion muffler arrays on a pipeline, generating bandgaps at low frequencies and effectively attenuating the acoustic noise transmission in the pipeline system. An [23] proposed and optimized a universal unit for metamaterial mufflers, which can be installed in limited space without increasing its size, reducing low-frequency and mid-high-frequency noise in pipeline systems. Kheybari [24] applied locally resonant Acoustic Metamaterial Baffles (AMB) to improve the design of internal baffles in mufflers. The research results indicate that AMB significantly increases the transmission loss of the muffler. This paper focuses on the design of a phononic crystal muffler, examining its bandgap characteristics and acoustic transmission properties. It particularly investigates how defect states impact the acoustic transmission and fluid dynamics of the muffler. The study offers a comprehensive understanding of phononic crystal mufflers, detailing their characteristics and performance across various aspects. 2 Theoretical formulas 2.1 Finite element method and the Bloch's theorem The Helmholtz resonator is a common acoustic resonance system, usually composed of a relatively large main cavity and a relatively small neck. The main cavity stores gas, while the neck connects the main cavity to the external environment. When the frequency of sound waves matches the resonant frequency of the Helmholtz resonator, gas vibrates back and forth inside the cavity, creating a resonance effect [25]. The scatterer of the ring phononic crystal is designed to resemble the form of a Helmholtz resonator, as shown in Fig. 1, and the unit cells of the ring phononic crystal are periodically arranged along the axial direction. In the figure, the green region represents air, and the yellow region represents the rigid solid structure. In the modeling process, the boundary between the air domain and solid domain is established as rigid, with the structure of the scatterer being neglected. Therefore, only the propagation of sound waves in air needs to be considered. Calculating the band structure is fundamental in phononic crystal research as it reflects the dispersion relationship of elastic waves in an infinite periodic phononic crystal. In classical acoustics, the wave equation for sound waves typically describes the pressure wave \(\:p(r,\theta\:,z)\) and the particle velocity potential, known as the Helmholtz equation or the wave equation. In cylindrical coordinates, the wave equation for sound waves is [26]-[28]: $$\:{\nabla\:}^{2}p(r,\theta\:,z)-{k}^{2}p(r,\theta\:,z)=0$$ 1 Where, \(\:r\) , \(\:\theta\:\) , and \(\:z\) represent the radial, azimuthal, and axial coordinates, respectively. \(\:k=\omega\:∕c\) , representing the wave number, \(\:\omega\:\) represents the angular frequency. \(\:c\) is the speed of sound. \(\:{\nabla\:}^{2}\) denotes the Laplacian operator. At a temperature of 20 degrees Celsius, the density of air is 1.2044 kg/m 3 , and the speed of sound is 343.2 m/s. The weak form of the wave equation can be derived by introducing a test function \(\:v\) and applying Green's formula, which can be expressed as: $$\:{\int\:}_{V}\:\left[\frac{1}{r}\frac{\partial\:}{\partial\:r}\left(r\frac{\partial\:p}{\partial\:r}\right)v+\frac{1}{{r}^{2}}\frac{{\partial\:}^{2}p}{\partial\:{\theta\:}^{2}}v+\frac{{\partial\:}^{2}p}{\partial\:{z}^{2}}v\right]rdrd\theta\:dz-{k}^{2}{\int\:}_{V}\:pvrdrd\theta\:dz=0$$ 2 where, \(\:v\) is the test function. Typically, boundary terms can be naturally eliminated or simplified by appropriately choosing the test function or applying boundary conditions, so that they do not need to be explicitly included in the equation. Bloch [29] demonstrated that electron waves propagate without scattering in a three-dimensional periodic medium, taking the form of Bloch waves expressed as a product of a periodic function and a plane wave. According to Bloch's theorem, in periodic lattice media, sound waves propagate in the form of plane waves modulated by lattice periods, and can be expressed as: $$\:p\left(\varvec{r}\right)={p}_{k}\left(\varvec{r}\right){e}^{i(\varvec{k}\cdot\:\varvec{r})}$$ 3 In the equation, the function \(\:{p}_{k}\left(\varvec{r}\right)\) is a periodic function, satisfying the condition \(\:{p}_{k}\left(\varvec{r}\right)={p}_{k}\left(\varvec{r}+{\varvec{R}}_{n}\right)\) , \(\:\varvec{k}\) is the wave vector. When employing the finite element method to solve for the band structure of a ring phononic crystal, it is necessary to discretize the continuous physical domain Ω within the unit cell of the phononic crystal into a finite number of subdomains or elements. Using shape functions to approximate the sound pressure \(\:p(r,\theta\:,z)\) , it can be expressed as: $$\:p(r,\theta\:,z)=\sum\:_{i=1}^{n}{N}_{i}(r,\theta\:,z){p}_{i}$$ 4 Where, \(\:{p}_{i}\) is the sound pressure at node 𝑖, and 𝑛 is the total number of nodes. Based on the weak form of the wave equation for sound waves, shape functions are used to construct the local matrices for each element: $$\:{\varvec{M}}_{e}=\frac{1}{{c}^{2}}{\int\:}_{{{\Omega\:}}_{e}}{\varvec{N}}_{e}{\varvec{N}}_{e}^{T}d{\Omega\:}$$ 5 $$\:{\varvec{K}}_{e}={\int\:}_{{{\Omega\:}}_{e}}(\nabla\:{\varvec{N}}_{e})(\nabla\:{\varvec{N}}_{e}^{T})d{\Omega\:}$$ 6 Where, \(\:{\varvec{M}}_{e}\) is the mass matrix, \(\:{\varvec{K}}_{e}\) is the stiffness matrix, \(\:{{\Omega\:}}_{e}\) is the element domain, \(\:{\varvec{N}}_{e}\) is the vector of shape functions within the element domain, \(\:\nabla\:\) is the gradient operator in cylindrical coordinates. According to the global numbering of each node of elements, the local mass and stiffness matrices are assembled into global matrices, resulting in the following formula: $$\:(\varvec{K}-{\omega\:}^{2}\varvec{M})\varvec{P}=0$$ 7 where, P is the vector of sound pressures. Bloch-Floquet boundary [30][31] conditions are the manifestation of Bloch's theorem in wave equations, particularly applicable to the solution of linear wave equations in periodic media. By applying Bloch-Floquet boundary conditions (as shown in Eq. 8 ) on the axial boundaries of the cylindrical phononic crystal unit cell, it is feasible to achieve Bloch waves with periodic amplitude modulation along the axis. $$\:-{\varvec{n}}_{\text{d}\text{s}\text{t}}\cdot\:\left(-\frac{1}{{\rho\:}_{C}}\nabla\:{p}_{\text{d}\text{s}\text{t}}\right)={\varvec{n}}_{\text{s}\text{r}\text{c}}\cdot\:\left(-\frac{1}{{\rho\:}_{C}}\nabla\:{p}_{\text{s}\text{r}\text{c}}\right){\text{e}}^{-i{\varvec{k}}_{\text{F}}a}$$ 8 In the equation, \(\:{p}_{\text{d}\text{s}\text{t}}={p}_{\text{s}\text{r}\text{c}}{e}^{-i{\varvec{k}}_{\text{F}}a}\) , \(\:\varvec{n}\) is the outward normal vector of the periodic boundary. \(\:{p}_{\text{s}\text{r}\text{c}}\) is the sound pressure at the source boundary, and \(\:{p}_{\text{d}\text{s}\text{t}}\) is the sound pressure at the destination boundary. According boundary conditions to modify the global matrices, thus obtaining a new eigenvalue problem: $$\:\left(\stackrel{-}{\varvec{K}}\right(\varvec{k})-{\omega\:}^{2}\stackrel{-}{\varvec{M}}(\varvec{k}\left)\right)\stackrel{-}{\varvec{P}}=0$$ 9 When \(\:\varvec{k}\) varies along the boundary of the irreducible Brillouin zone, the dispersion relation \(\:\omega\:\left(\varvec{k}\right)\) obtained from Eq. (14) describes the band structure of the phononic crystal [32]. The modeling and computation of phononic crystals are conducted using the finite element software COMSOL [33]-[35]. Due to the circular symmetry, the phononic crystals be simplified into two-dimensional models, where the variation of physical quantities is only related to the axial and radial positions. 2.2 Transmission characteristic Based on the infinite periodic structure, ideal phononic crystals exhibit perfect elastic wave shielding within the bandgap range. However, in practical applications, the periodic structure of phononic crystals is inevitably finite. Consequently, some sound waves within the bandgap frequency range may not be effectively attenuated and can pass through the finite structure. Therefore, for finite periodic phononic crystals, there is a need for metrics that can reflect the characteristics of sound wave transmission. The transmission loss can effectively reflect the transmission characteristics of sound waves within the acoustic field of mufflers. Axial plane wave theory is most commonly used to analyze a typical muffler system, wherein it is assumed that the plane wave propagates along the axis of the cylindrical chamber muffler [36]. At the entrance boundary of the acoustic field within muffler, an incident plane wave sound source is set, and the exit boundary is designated as a non-reflecting boundary. With the given entrance boundary conditions, the total sound pressure \(\:{p}_{t}\) is calculated by the following expression: $$\:{p}_{t}=\sum\:_{i\in\:\text{b}\text{n}\text{d}}\:{A}_{\text{i}\text{n}}{e}^{i\varphi\:}({S}_{ij}+{\delta\:}_{ij}){p}_{i}$$ 10 In the equation, \(\:{A}_{\text{i}\text{n}}\) is the amplitude of the incident wave sound pressure, \(\:{e}^{i\varphi\:}\) represents the phase of the incident wave, where 𝜙 is the phase angle, \(\:{S}_{ij}\) is a component of the scattering matrix, \(\:{\delta\:}_{ij}\) is the Kronecker delta function, and \(\:{p}_{i}\) is the port mode shape. The matrix form of the linear equation set for solving the transmission loss of the acoustic field within the muffler can be expressed as: $$\:(\varvec{K}-{\omega\:}^{2}\varvec{M})\varvec{P}=\varvec{Q}$$ 11 where, and \(\:\varvec{Q}\) is the sound source vector. Once \(\:\varvec{P}\) is solved, the input and output acoustic power at the entrance and exit of the acoustic field within muffler can be obtained. The definition of transmission loss is the difference between the input incident sound power level and the output radiated sound power level, calculated by the following formula [37]: $$\:TL=10\times\:\text{l}\text{g}\left(\frac{{W}_{\text{i}\text{n}}}{{W}_{\text{o}\text{u}\text{t}}}\right)$$ 12 Here, \(\:{W}_{\text{i}\text{n}}\) is the input sound power, and \(\:{W}_{\text{o}\text{u}\text{t}}\) is the output sound power. As depicted in Fig. 2(a), the phononic crystal muffler is shown, which contains 2 × 6 ring scatterers within its expansion chamber. The two-dimensional simplified model of the air domain of the muffler is established and discretized using the COMSOL. The Fig. 2(b) shows a two-dimensional mesh model of the muffler, which includes a total of 14,983 elements, with the maximum mesh cell length being 8 mm. 3 Results analysis 3.1 Bandgap characteristics As shown in Fig. 3(a), the band structure of the ring phononic crystal is depicted. The dimension parameters of ring phononic crystal are as follows: lattice constant a 1 = 50 mm, side length of resonator a 2 = 42 mm, resonator neck width l = 4mm, resonator neck length h = 10 mm, and resonator rigid body thickness b = 2 mm. The distance of the model from the symmetry axis is set to R = 50 mm. The range of the first bandgap for ring phononic crystal is from 409 Hz to 707 Hz. The central frequency of the first bandgap is 550 Hz, which is significantly lower than the Bragg gap observed in lattices of comparable size, suggesting that it arises from local resonance phenomena. The second bandgap extends from 1897 Hz to 2514 Hz. It is widely accepted that the position and width of local resonance bandgaps are closely related to the dimensions of the scatterer structures rather than the specific crystal configuration. Consequently, to obtain a lower central frequency and a wider bandgap, it is necessary to investigate the impact of the dimension parameters of scatterer on the band structure. To examine the effects of parameter variations on the first bandgap of ring phononic crystal, the model parameters previously described were established as initial parameters. Subsequently, the side length of resonator ( a 2 ), neck width of resonator ( l ), neck length of resonator ( h ), and the radius ( R ) of the unit cell were individually adjusted while maintaining the constancy of other parameters. For each set of parameters, the band structure of ring phononic crystal was computationally determined. The influence of the above four parameters on the starting frequency (the lowest frequency corresponding of a certain bandgap), the ending frequency (the highest frequency corresponding of a certain bandgap) and the bandwidth of the first bandgap of the phonon crystal are shown in Fig. 3(b). The results indicate that an increment in a 2 results in a progressive broadening of the first bandgap, coupled with a shift to lower frequencies. As h increases, the first bandgap progressively shifts to lower frequencies while simultaneously becoming narrower. An increase in l causes the first bandgap to gradually move towards higher frequencies, along with an expansion in width. When the radius of the unit cell slightly increases (within the range of the lattice constant), the starting and ending frequencies of the first bandgap, as well as the bandwidth, all tend to increase, but the variation is relatively minor. In summary, to achieve a bandgap at lower frequencies with sufficient width for the ring phononic crystal, it is necessary to increase the side length of the resonator in the radial direction and the length of the neck. Additionally, it is advisable to appropriately reduce the width of the neck. 3.2 Acoustic transmission characteristics The following analysis will delve deeper into the characteristics of the phononic crystal muffler at low frequencies. Figure 4(a) displays the band structure of the ring phononic crystal with adjusted scatterer parameters: a 2 is now 46 mm, h is 40 mm, and b is reduced to 1 mm. These modifications result in a first bandgap frequency range of 270–579 Hz. As previously mentioned, the bandgap of the ring phononic crystal is affected by the radius, but the first bandgap does not vary significantly with minor changes in the radius. Consequently, ring phononic crystals with two different radii within the muffler can be considered to have similar first bandgaps. For subsequent analysis, the bandgap of the phononic crystal with the smaller radius will serve as the reference point. Figure 4(b) compares the transmission loss of the phononic crystal muffler with that of an expansion chamber muffler. It is evident that within the bandgap range, the phononic crystal muffler exhibits a significantly higher transmission loss compared to the expansion chamber muffler, with the maximum transmission loss reaching up to 130 dB, an increase from the original 13 dB. Figures 4(c) and 4(d) respectively illustrate the sound pressure level and particle acceleration distributions within the expansion chamber muffler and the phononic crystal muffler at a frequency of 360 Hz. It is evident from these figures that the localized resonance occurring within the resonators of the phononic crystal muffler leads to a rapid attenuation of sound pressure as the acoustic wave propagates. This localized resonance is a result of the specific design of the resonators, which enhances the interaction with the acoustic waves, thereby increasing the energy dissipation. In contrast, the expansion chamber muffler, lacking such resonant structures, shows a significantly smaller attenuation amplitude, indicating less effective sound wave energy reduction. 3.3 Defect state characteristics In phononic crystals, the removal of scatterers introduces defect states, which can significantly alter the propagation of elastic waves. Specifically, the elimination of a single scatterer results in a point defect, which may manifest as a single point or multiple points. The removal of an entire row of scatterers generates a line defect, potentially appearing as a straight or broken line. In phononic crystals, the intentional introduction of defects can lead to the localization of elastic waves around these defects. This phenomenon is particularly beneficial for controlling long-wavelength sound waves using small structures. To facilitate the description of defect positions within the muffler, we assign labels to internal scatterers in the muffler in the format shown in Fig. 5(c) (e.g., i4 represents the scatterer in the fourth row of the i-th column). Due to the structural symmetry of the phononic crystal muffler, only variations in the first three columns need to be considered. Figure 5(a) shows that the impact of point defects on transmission loss is concentrated in the frequency ranges of 200–340 Hz, 350–500 Hz, and 510–820 Hz. The removal of scatterer j1 leads to an increase in transmission loss across the frequency span of 350–820 Hz, and the elimination of scatterer i2 increases transmission loss within the frequency band of 200–340 Hz. When scatterers j1 and i2 are removed together, transmission loss is significantly enhanced in the ranges of 200–340 Hz and 500–820 Hz, with new peaks at frequencies of 270 Hz, 610 Hz, and 820 Hz. Figure 5(b) illustrates the effect of a linear defect on transmission loss for the phononic crystal muffler, primarily affecting the frequency ranges of 200–340 Hz, 350–500 Hz, and 510–1000 Hz. After removing scatterers in columns 1, 2, and 3, the muffler's transmission loss is enhanced in the ranges of 200–340 Hz and 510–1000 Hz, and shows a decrease in the 350–500 Hz range. These results demonstrate that by accurately manipulating the defect state, it is possible to effectively suppress sound waves at specific frequencies. 3.4 Fluid dynamic characteristics The aerodynamic performance is a crucial aspect of assessing the quality of a muffler [38]-[40], reflecting the resistance offered by the muffler to airflow, or the impact of installing the muffler on the flow characteristics of the original system. Pressure loss, calculated as \(\:PL={P}_{\text{i}\text{n}}-{P}_{\text{o}\text{u}\text{t}}\) [38], is an evaluation metric for this performance, where \(\:{P}_{\text{i}\text{n}}\) is the total pressure at the inlet and \(\:{P}_{out}\) at the outlet. A two-dimensional simplified model of the muffler air domain was established, similar to the approach used in acoustic calculations. When the air velocity Mach number is less than 0.3, the fluid can be considered incompressible; therefore, the air within the muffler is assumed an ideal fluid with no density variation or heat exchange. The standard k - ε turbulence model [42][43] was employed to calculate the pressure loss of the muffler. The muffler inlet boundary was set as a velocity inlet, defining the inlet velocity, while the outlet boundary was set as a pressure outlet with atmospheric pressure, defined as 0 Pa. Figure 6 illustrates the pressure loss of the phononic crystal muffler and the simple expansion chamber muffler at different inlet velocities. It is observed that the pressure loss escalates with an increase in the inlet velocity. The phononic crystal muffler, devoid of defect states, exhibits a significantly slower rate of pressure loss increase with rising inlet velocity in comparison to the expansion chamber muffler. At identical inlet velocities, the pressure loss for the phononic crystal muffler with defect states exhibits a marginal increment relative to the pressure loss of the phononic crystal muffler without defect states. Figure 7 shows the pressure distributions and turbulent kinetic energy distributions of the phononic crystal muffler and the expansion chamber muffler at an inlet velocity of 30 m/s. Compared to the expansion chamber muffler, the inner ring resonator walls of the phononic crystal muffler obstruct the radial flow of the airflow, resulting in weaker turbulent intensity inside and smoother airflow. When the phononic crystal muffler is in a defective state, although there is some enhancement of turbulence near the defect region close to the pipeline, the overall impact of fewer defects on the internal flow field of the phononic crystal muffler is not significant. These results indicate that the phononic crystal muffler possesses excellent fluid dynamic performance. 4 Experiment and validation The phononic crystal model used for finite element calculations and experiments is shown in Fig. 8(a), with specific geometric parameters listed in Table 1 . The expansion chamber of the muffler contains three ring resonators and features a point defect. This experiment utilized the four-microphone method with an impedance tube to measure the transmission loss for the phononic crystal muffler, based on the theoretical foundation of the standing wave separation method. As shown in Fig. 8(c), a loudspeaker emitted white noise into the impedance tube, generating a planar incident wave M 1 upon encountering the phononic crystal muffler sample. A portion of the sound waves were reflected, forming the planar reflected sound wave N 1 . Another portion passed through the sample, entered the transmission tube, and generated the planar transmitted sound wave M 2 . The planar transmitted sound wave encountered the absorbent termination, where some was absorbed, and some was reflected, forming the transmitted reflected wave N 2 . The sound pressure values measured by microphones at different positions were then used to calculate the self-power spectra W in and W pro of the incident and transmitted waves, respectively. Finally, the transmission loss of the test sample equal to \(\:10\times\:\text{lg}\left({W}_{\text{i}\text{n}}∕{W}_{\text{p}\text{r}\text{o}}\right)\) . Table 1 The parameters of the phononic crystal muffler sample Symbol H 1 H 2 R 1 R 2 a 1 a 2 l b h Geometrical parameters (mm) 98.0 52.0 52.0 14.5 26.0 23.0 3.0 3.0 8.0 Figure 8(a) displays the band structure of the phononic crystal in the experiment, which shows the first bandgap ranging from 1084 to 1683 Hz. Figure 8(b) illustrates the transmission loss curve of the phononic crystal muffler obtained from experimental testing, compared with the results calculated by the finite element method. It shows that, the experimental results are in good agreement with those calculated by the finite element method, and a significant enhancement in transmission loss is observed at frequencies within the bandgap. This confirms the accuracy of the finite element method in calculating band structures and transmission loss values, and demonstrates the effectiveness of using a phononic crystal muffler. At the same time, this demonstrates that, through the judicious design of bandgap properties, the phononic crystal muffler can effectively attenuate noise within specific frequency ranges, proving its potential as a highly efficient acoustic control component. 5 Conclusions This study introduces the engineering of a Helmholtz-like ring resonator phononic crystal muffler, with its acoustic characteristics numerically analyzed using finite element method. In the study, the effects of the dimension parameters of ring scatterers on the bandgap were investigated, revealing that increasing the side length of the resonator lowers the central frequency and broadens the bandgap. Experimental validations of the acoustic transmission characteristics for a phononic crystal muffler affirmed the accuracy of the calculations and the efficacy of the muffler within the bandgap range. The study further investigated the effects of point and linear defects on transmission loss, showing that strategically positioned defects can significantly enhance noise reduction performance and broaden the bandwidth. Fluid dynamics simulations revealed that scatterers within the phononic crystal muffler impede radial airflow and reduce turbulence. Consequently, compared to a conventional expansion chamber muffler, the phononic crystal muffler incurs lower pressure loss at equivalent inlet velocities, highlighting superior aerodynamic performance. Integrating phononic crystal technology into muffler design, this study introduces a muffler with robust low-frequency absorption and commendable aerodynamics, charting a novel pathway for controlling low-frequency noise in muffler technology. Declarations Competing Interests Statement 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. Author Contribution Y.B. and Y.C. initiated the research; Y.B. conducted the theoretical and numerical simulation work; J.Z and Y.B. designed and carried out the experiments; Y.B. wrote and revised the manuscript; Y.C. guided and supervised this study. Acknowledgements The authors thank the financial supports from the Natural Science Foundation of Zhejiang Province and Ningbo City, China (Nos. LY20E050006, 2021Z098). Data Availability The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy concerns. References Lee, J. K., Oh, K. S. & Lee, J. W. Methods for evaluating in-duct noise attenuation performance in a muffler design problem. J. 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Kabir, M., Mostavi, A. & Ozevin, D. Noise isolation with phononic crystals to enhance fatigue crack growth detection using acoustic emission. J. Civ. Struct. Health Monit. 8 (3), 529–542. https://doi.org/10.1007/s13349-018-0291-6 (2018). Mimani, A. & Munjal, M. L. Transverse plane wave analysis of short elliptical chamber mufflers: An analytical approach. J. Sound Vib. 330, 1472–1489. https://doi.org/10.1016/j.jsv.2010.09.035 (2011). Chu, C. I., Hua, H. T. & Liao, I. C. Effects of three-dimensional modes on acoustic performance of reversal flow mufflers with rectangular cross-section. Comput. Struct . 79 (8), 883–890. https://doi.org/10.1016/S0045-7949(00)00184-X (2001). Liu, C., Ji, Z. L. & Fang, Z. Numerical analysis of acoustic attenuation and flow resistance characteristics of double expansion chamber silencers. Noise Control Eng. J. 61 (5), 487–499. https://doi.org/10.1016/j.apacoust.2018.07.021 (2013). Xue, F. & Sun, B. B. Experimental study on the comprehensive performance of the application of U-shaped corrugated pipes into reactive mufflers. Appl. Acoust. 141 , 362–370. https://doi.org/10.1016/j.apacoust.2018.07.021 (2018). Chen, Y. & Lv, L. Design and evaluation of an Integrated SCR and exhaust muffler from marine diesels. J. Mar. Sci. Technol. 20 (3), 505–519. https://doi.org/10.1007/s00773-014-0302-1 (2015). Guo, R., Huang, Z., Sun, Z. Z. & Luo, R. Research on flow characteristics of the irregular multi-chamber perforated resonator. Appl. Acoust. 184 , 108351. https://doi.org/10.1016/j.apacoust.2021.108351 (2021). Fu, J., Zhang, Z. F., Chen, W., Mao, H. & Li, J. X. Computational fluid dynamics simulations of the flow field characteristics in a novel exhaust purification muffler of diesel engine. J. Low Freq. Noise Vib. Act. Control 37 (4), 816–833. https://doi.org/10.1177/1461348418790488 (2018). Wang, T., Gao, J. R. & Bu, Y. S. Performance Analysis of Improved Vehicle Muffler. Mechanika 24 (5), 751–756. https://doi.org/10.5755/j01.mech.24.5.17784 (2018). Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 22 Nov, 2024 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 19 Sep, 2024 Reviews received at journal 18 Sep, 2024 Reviews received at journal 07 Sep, 2024 Reviewers agreed at journal 07 Sep, 2024 Reviewers agreed at journal 07 Sep, 2024 Reviewers invited by journal 07 Sep, 2024 Editor assigned by journal 03 Sep, 2024 Editor invited by journal 03 Sep, 2024 Submission checks completed at journal 30 Aug, 2024 First submitted to journal 23 Aug, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4963361","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":356257818,"identity":"40416c74-95ac-40b5-9657-8d7b932c117c","order_by":0,"name":"Yang Bai","email":"","orcid":"","institution":"Ningbo University","correspondingAuthor":false,"prefix":"","firstName":"Yang","middleName":"","lastName":"Bai","suffix":""},{"id":356257819,"identity":"38935891-3e8a-47de-a975-f53913a2ff75","order_by":1,"name":"Yuehua Chen","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4klEQVRIiWNgGAWjYDACCQiVwMDAfIAZzDxAvBa2BJK18BgQp4V/dvOxx7xtNnkGx898e1zYxiDHdyOB8XMBPkvuHEs35m1LKzY4k7vdeGYbg7HkjQRm6Rl4tBhI5JhJ87YdTtxwIHcbkMGQuOFGAhszD14t+d+AKv8nbjj/5hlISz0RWnLYgCoPAA0HMxgSDAhpkbiRZiY551xy4swbz8ykZ5yTMJx55mGzND4t/DOSn0m8KbNL7Duf/Ey6oMxGnu948sHP+LSAABOSAlA0MTYQ0ABU8oOgklEwCkbBKBjRAADGZUtNLrG2bwAAAABJRU5ErkJggg==","orcid":"","institution":"Ningbo University","correspondingAuthor":true,"prefix":"","firstName":"Yuehua","middleName":"","lastName":"Chen","suffix":""},{"id":356257820,"identity":"2769557d-c672-4876-b9a2-3befbd2b3e05","order_by":2,"name":"Jiahui Zheng","email":"","orcid":"","institution":"Ningbo University","correspondingAuthor":false,"prefix":"","firstName":"Jiahui","middleName":"","lastName":"Zheng","suffix":""}],"badges":[],"createdAt":"2024-08-23 09:59:24","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4963361/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4963361/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-024-79762-9","type":"published","date":"2024-11-22T15:57:54+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":65716674,"identity":"a224766a-1389-4c86-bfc2-74b21ae06a2d","added_by":"auto","created_at":"2024-10-01 15:51:34","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":266504,"visible":true,"origin":"","legend":"\u003cp\u003e3D Model of the ring phononic Crystal.\u003c/p\u003e","description":"","filename":"Fig1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/49da594e869bda4f17167b0b.jpg"},{"id":65717223,"identity":"37ac632a-5f53-4fb2-ba82-e3aa0088b8af","added_by":"auto","created_at":"2024-10-01 15:59:33","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":480257,"visible":true,"origin":"","legend":"\u003cp\u003e(a)\u003cstrong\u003e \u003c/strong\u003e3-D model of the phononic crystal muffler. (b) 2-D mesh model of the acoustic field within the phononic crystal muffler.\u003c/p\u003e","description":"","filename":"Fig2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/9b86f2d57f7a34e2ce26d8a1.jpg"},{"id":65716667,"identity":"fd9f8e71-4f23-4e2f-aa2e-52c28ee49e20","added_by":"auto","created_at":"2024-10-01 15:51:33","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":1013262,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Band structure of the Helmholtz-ring phononic crystal. (b) Influence of different parameters on the first bandgap of the Helmholtz-ring phononic crystal.\u003c/p\u003e","description":"","filename":"Fig3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/5b0f908a6c37e7575d66bef3.jpg"},{"id":65716670,"identity":"cd981773-7304-4e73-91d7-e9fa117bb088","added_by":"auto","created_at":"2024-10-01 15:51:33","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":882010,"visible":true,"origin":"","legend":"\u003cp\u003e(a)\u003cstrong\u003e \u003c/strong\u003eBand structure of the phononic crystal. (b) Comparison of transmission loss. (c) Comparison of sound pressure level distributions at 360 Hz for phononic crystal muffler and expansion chamber muffler. (d) Comparison of particle acceleration distributions at 360 Hz for phononic crystal muffler and expansion chamber muffler.\u003c/p\u003e","description":"","filename":"Fig4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/27e1d2941f8964be2a768145.jpg"},{"id":65717224,"identity":"11674245-9b1e-4e0f-a53a-99834c50e3bb","added_by":"auto","created_at":"2024-10-01 15:59:33","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":810573,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Influence of various point defects on the transmission loss for the phononic crystal muffler. (b) Influence of various linear defects on the transmission loss for the phononic crystal muffler. (c) Schematic diagram illustrating the numbering of scatterers in the phononic crystal muffler.\u003c/p\u003e","description":"","filename":"Fig5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/fc6855572dcb0c95a5880ebd.jpg"},{"id":65716671,"identity":"a305cfc3-327b-4860-8279-9f0df1ee9377","added_by":"auto","created_at":"2024-10-01 15:51:33","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":439764,"visible":true,"origin":"","legend":"\u003cp\u003ePressure loss curves for the expansion chamber muffler, the phononic crystal muffler, and the defective phononic crystal mufflers.\u003c/p\u003e","description":"","filename":"Fig6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/d0f1e61ac9326897ff64d8b7.jpg"},{"id":65717225,"identity":"b44e79bf-1ba9-471c-866a-08567c99496a","added_by":"auto","created_at":"2024-10-01 15:59:33","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":518063,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Internal pressure distributions of the mufflers. (b) Internal turbulent kinetic energy distributions of the mufflers.\u003c/p\u003e","description":"","filename":"Fig7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/c8269ef6438bd7ca7e91f79d.jpg"},{"id":65717609,"identity":"8abbfd99-2d74-4782-82ed-f33b58f12e97","added_by":"auto","created_at":"2024-10-01 16:07:33","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":1137797,"visible":true,"origin":"","legend":"\u003cp\u003e(a) 3-D model of the phononic crystal muffler sample. (b) The overall experiment setup. (c) Schematic diagram of the four-microphone method for measuring transmission loss. (d) Band structure of the phononic crystal sample. (e) Comparison of transmission loss between experimental result and current result.\u003c/p\u003e","description":"","filename":"Fig8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/386fc07ac150b5551b6114c4.jpg"},{"id":69834974,"identity":"51862a07-0559-46fb-b80d-f8a8bf38b541","added_by":"auto","created_at":"2024-11-25 16:11:04","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":6085937,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4963361/v1/8f487ccd-0ed0-42a2-bff7-06f600ca1300.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Experimental study and acoustic characteristics analysis of defective-state Helmholtz-ring phononic crystal muffler","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eA muffler is a device designed to reduce noise by absorbing, reflecting sound waves, or altering their propagation paths through specific structural designs. Mufflers have been widely used for reducing the flow noise inside ducts, such as in a vehicle exhaust system or the ventilation system of home appliances [1].The expansion chamber muffler [2]-[4]. is a common type of muffler, its working principle involves the use of abrupt changes in the cross-sectional area of the duct to reflect sound waves. By incorporating expansion sections or resonant chambers within the duct, standing waves are produced, which dissipate sound energy. Traditional muffler has a good performance on middle and high frequency noise. But it has a poor performance on low-frequency noise [5]. Mid and low-frequency noises are common in industrial and living environments, such as those produced by mechanical equipment, motor vehicles, and air conditioning systems. Low-frequency noise, due to its long wavelengths, can easily penetrate barriers and is also readily perceptible to the human ear [6]. Low-frequency sound absorption is still a challenge for most passive noise control applications due to the space requirements to absorb large acoustic wavelengths [7]. Therefore, enhancing the performance of expansion chamber mufflers in the mid and low-frequency range is of significant research interest.\u003c/p\u003e \u003cp\u003ePhononic crystals have unique advantages in controlling low-frequency noise. Phononic crystals are materials with periodic structures [8], and the concept of phononic crystals originated from the development of photonic crystals independently proposed by Yablonovitch [9] and John [10] in 1987. The initial interest in phononic crystals arose due to the existence of phononic Bragg bandgaps where the propagation of sound waves is prohibited [11]. In the year 2000, Liu et al. [12][13] discovered another mechanism for generating elastic wave bandgaps, namely, the local resonance mechanism, by studying three-dimensional phononic crystals composed of rubber, lead, and epoxy resin, going beyond the Bragg scattering mechanism. The introduction of the concept of local resonance-type phononic crystals broadened the scope of phononic crystal research [14]-[16]. Based on the local resonance mechanism, it is generally possible to achieve control of large wavelengths with small-sized structures.\u003c/p\u003e \u003cp\u003eDue to the significant advantages of phononic crystal technology in noise control, the application of this technology in the research of pipeline mufflers has gradually attracted more attention. Zhang et al. [17] have developed a new type of metamaterial-based silencer that utilizes the principle of anomalous reflection to achieve efficient sound isolation in hollow pipes with sub-wavelength thickness. Shi [18], applying Bloch wave theory, studied the wave propagation in periodic microperforated pipe mufflers and examined the dispersion characteristics of periodic microperforated mufflers. Liu [19] proposed a compact hybrid muffler phononic crystal, calculated the transmission loss of a Unit cell using a two-dimensional transfer matrix method, and analyzed the bandgap characteristics as well as the noise reduction mechanism in the hybrid muffler. Liu et al. [20] integrated a labyrinthine metasurface array into the muffler design, thereby achieving high transmission loss and broad bandwidth acoustic insulation within the low-frequency spectrum. Almeida [21] reengineered the existing intake muffler by utilizing the concept of wave attenuation bands induced by Bragg scattering, which has led to an improvement in the transmission loss of mufflers employed in refrigerant compressors. Liu [22] proposed a periodic waveguide made of axially mounted expansion muffler arrays on a pipeline, generating bandgaps at low frequencies and effectively attenuating the acoustic noise transmission in the pipeline system. An [23] proposed and optimized a universal unit for metamaterial mufflers, which can be installed in limited space without increasing its size, reducing low-frequency and mid-high-frequency noise in pipeline systems. Kheybari [24] applied locally resonant Acoustic Metamaterial Baffles (AMB) to improve the design of internal baffles in mufflers. The research results indicate that AMB significantly increases the transmission loss of the muffler.\u003c/p\u003e \u003cp\u003eThis paper focuses on the design of a phononic crystal muffler, examining its bandgap characteristics and acoustic transmission properties. It particularly investigates how defect states impact the acoustic transmission and fluid dynamics of the muffler. The study offers a comprehensive understanding of phononic crystal mufflers, detailing their characteristics and performance across various aspects.\u003c/p\u003e"},{"header":"2 Theoretical formulas","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e\n\u003ch2\u003e2.1 Finite element method and the Bloch's theorem\u003c/h2\u003e\n\u003cp\u003eThe Helmholtz resonator is a common acoustic resonance system, usually composed of a relatively large main cavity and a relatively small neck. The main cavity stores gas, while the neck connects the main cavity to the external environment. When the frequency of sound waves matches the resonant frequency of the Helmholtz resonator, gas vibrates back and forth inside the cavity, creating a resonance effect [25]. The scatterer of the ring phononic crystal is designed to resemble the form of a Helmholtz resonator, as shown in Fig.\u0026nbsp;1, and the unit cells of the ring phononic crystal are periodically arranged along the axial direction. In the figure, the green region represents air, and the yellow region represents the rigid solid structure. In the modeling process, the boundary between the air domain and solid domain is established as rigid, with the structure of the scatterer being neglected. Therefore, only the propagation of sound waves in air needs to be considered.\u003c/p\u003e\n\u003cp\u003eCalculating the band structure is fundamental in phononic crystal research as it reflects the dispersion relationship of elastic waves in an infinite periodic phononic crystal. In classical acoustics, the wave equation for sound waves typically describes the pressure wave \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:p(r,\\theta\\:,z)\\)\u003c/span\u003e\u003c/span\u003e and the particle velocity potential, known as the Helmholtz equation or the wave equation. In cylindrical coordinates, the wave equation for sound waves is [26]-[28]:\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ1\" class=\"mathdisplay\"\u003e$$\\:{\\nabla\\:}^{2}p(r,\\theta\\:,z)-{k}^{2}p(r,\\theta\\:,z)=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:r\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\theta\\:\\)\u003c/span\u003e\u003c/span\u003e, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:z\\)\u003c/span\u003e\u003c/span\u003e represent the radial, azimuthal, and axial coordinates, respectively. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:k=\\omega\\:∕c\\)\u003c/span\u003e\u003c/span\u003e, representing the wave number, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\omega\\:\\)\u003c/span\u003e\u003c/span\u003e represents the angular frequency. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:c\\)\u003c/span\u003e\u003c/span\u003e is the speed of sound. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\nabla\\:}^{2}\\)\u003c/span\u003e\u003c/span\u003e denotes the Laplacian operator. At a temperature of 20 degrees Celsius, the density of air is 1.2044 kg/m\u003csup\u003e3\u003c/sup\u003e, and the speed of sound is 343.2 m/s. The weak form of the wave equation can be derived by introducing a test function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:v\\)\u003c/span\u003e\u003c/span\u003e and applying Green's formula, which can be expressed as:\u003c/p\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ2\" class=\"mathdisplay\"\u003e$$\\:{\\int\\:}_{V}\\:\\left[\\frac{1}{r}\\frac{\\partial\\:}{\\partial\\:r}\\left(r\\frac{\\partial\\:p}{\\partial\\:r}\\right)v+\\frac{1}{{r}^{2}}\\frac{{\\partial\\:}^{2}p}{\\partial\\:{\\theta\\:}^{2}}v+\\frac{{\\partial\\:}^{2}p}{\\partial\\:{z}^{2}}v\\right]rdrd\\theta\\:dz-{k}^{2}{\\int\\:}_{V}\\:pvrdrd\\theta\\:dz=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:v\\)\u003c/span\u003e\u003c/span\u003e is the test function. Typically, boundary terms can be naturally eliminated or simplified by appropriately choosing the test function or applying boundary conditions, so that they do not need to be explicitly included in the equation.\u003c/p\u003e\n\u003cp\u003eBloch [29] demonstrated that electron waves propagate without scattering in a three-dimensional periodic medium, taking the form of Bloch waves expressed as a product of a periodic function and a plane wave. According to Bloch's theorem, in periodic lattice media, sound waves propagate in the form of plane waves modulated by lattice periods, and can be expressed as:\u003c/p\u003e\n\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ3\" class=\"mathdisplay\"\u003e$$\\:p\\left(\\varvec{r}\\right)={p}_{k}\\left(\\varvec{r}\\right){e}^{i(\\varvec{k}\\cdot\\:\\varvec{r})}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn the equation, the function \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{k}\\left(\\varvec{r}\\right)\\)\u003c/span\u003e\u003c/span\u003e is a periodic function, satisfying the condition \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{k}\\left(\\varvec{r}\\right)={p}_{k}\\left(\\varvec{r}+{\\varvec{R}}_{n}\\right)\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{k}\\)\u003c/span\u003e\u003c/span\u003e is the wave vector.\u003c/p\u003e\n\u003cp\u003eWhen employing the finite element method to solve for the band structure of a ring phononic crystal, it is necessary to discretize the continuous physical domain Ω within the unit cell of the phononic crystal into a finite number of subdomains or elements. Using shape functions to approximate the sound pressure \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:p(r,\\theta\\:,z)\\)\u003c/span\u003e\u003c/span\u003e, it can be expressed as:\u003c/p\u003e\n\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ4\" class=\"mathdisplay\"\u003e$$\\:p(r,\\theta\\:,z)=\\sum\\:_{i=1}^{n}{N}_{i}(r,\\theta\\:,z){p}_{i}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the sound pressure at node 𝑖, and 𝑛 is the total number of nodes. Based on the weak form of the wave equation for sound waves, shape functions are used to construct the local matrices for each element:\u003c/p\u003e\n\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ5\" class=\"mathdisplay\"\u003e$$\\:{\\varvec{M}}_{e}=\\frac{1}{{c}^{2}}{\\int\\:}_{{{\\Omega\\:}}_{e}}{\\varvec{N}}_{e}{\\varvec{N}}_{e}^{T}d{\\Omega\\:}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ6\" class=\"mathdisplay\"\u003e$$\\:{\\varvec{K}}_{e}={\\int\\:}_{{{\\Omega\\:}}_{e}}(\\nabla\\:{\\varvec{N}}_{e})(\\nabla\\:{\\varvec{N}}_{e}^{T})d{\\Omega\\:}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{M}}_{e}\\)\u003c/span\u003e\u003c/span\u003e is the mass matrix, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{K}}_{e}\\)\u003c/span\u003e\u003c/span\u003e is the stiffness matrix, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{{\\Omega\\:}}_{e}\\)\u003c/span\u003e\u003c/span\u003e is the element domain, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\varvec{N}}_{e}\\)\u003c/span\u003e\u003c/span\u003e is the vector of shape functions within the element domain, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\nabla\\:\\)\u003c/span\u003e\u003c/span\u003e is the gradient operator in cylindrical coordinates. According to the global numbering of each node of elements, the local mass and stiffness matrices are assembled into global matrices, resulting in the following formula:\u003c/p\u003e\n\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ7\" class=\"mathdisplay\"\u003e$$\\:(\\varvec{K}-{\\omega\\:}^{2}\\varvec{M})\\varvec{P}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere, \u003cstrong\u003eP\u003c/strong\u003e is the vector of sound pressures. Bloch-Floquet boundary [30][31] conditions are the manifestation of Bloch's theorem in wave equations, particularly applicable to the solution of linear wave equations in periodic media. By applying Bloch-Floquet boundary conditions (as shown in Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e) on the axial boundaries of the cylindrical phononic crystal unit cell, it is feasible to achieve Bloch waves with periodic amplitude modulation along the axis.\u003c/p\u003e\n\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ8\" class=\"mathdisplay\"\u003e$$\\:-{\\varvec{n}}_{\\text{d}\\text{s}\\text{t}}\\cdot\\:\\left(-\\frac{1}{{\\rho\\:}_{C}}\\nabla\\:{p}_{\\text{d}\\text{s}\\text{t}}\\right)={\\varvec{n}}_{\\text{s}\\text{r}\\text{c}}\\cdot\\:\\left(-\\frac{1}{{\\rho\\:}_{C}}\\nabla\\:{p}_{\\text{s}\\text{r}\\text{c}}\\right){\\text{e}}^{-i{\\varvec{k}}_{\\text{F}}a}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn the equation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{\\text{d}\\text{s}\\text{t}}={p}_{\\text{s}\\text{r}\\text{c}}{e}^{-i{\\varvec{k}}_{\\text{F}}a}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{n}\\)\u003c/span\u003e\u003c/span\u003e is the outward normal vector of the periodic boundary. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{\\text{s}\\text{r}\\text{c}}\\)\u003c/span\u003e\u003c/span\u003e is the sound pressure at the source boundary, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{\\text{d}\\text{s}\\text{t}}\\)\u003c/span\u003e\u003c/span\u003e is the sound pressure at the destination boundary. According boundary conditions to modify the global matrices, thus obtaining a new eigenvalue problem:\u003c/p\u003e\n\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ9\" class=\"mathdisplay\"\u003e$$\\:\\left(\\stackrel{-}{\\varvec{K}}\\right(\\varvec{k})-{\\omega\\:}^{2}\\stackrel{-}{\\varvec{M}}(\\varvec{k}\\left)\\right)\\stackrel{-}{\\varvec{P}}=0$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhen \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{k}\\)\u003c/span\u003e\u003c/span\u003e varies along the boundary of the irreducible Brillouin zone, the dispersion relation \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\omega\\:\\left(\\varvec{k}\\right)\\)\u003c/span\u003e\u003c/span\u003e obtained from Eq.\u0026nbsp;(14) describes the band structure of the phononic crystal [32]. The modeling and computation of phononic crystals are conducted using the finite element software COMSOL [33]-[35]. Due to the circular symmetry, the phononic crystals be simplified into two-dimensional models, where the variation of physical quantities is only related to the axial and radial positions.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n\u003ch2\u003e2.2 Transmission characteristic\u003c/h2\u003e\n\u003cp\u003eBased on the infinite periodic structure, ideal phononic crystals exhibit perfect elastic wave shielding within the bandgap range. However, in practical applications, the periodic structure of phononic crystals is inevitably finite. Consequently, some sound waves within the bandgap frequency range may not be effectively attenuated and can pass through the finite structure. Therefore, for finite periodic phononic crystals, there is a need for metrics that can reflect the characteristics of sound wave transmission. The transmission loss can effectively reflect the transmission characteristics of sound waves within the acoustic field of mufflers.\u003c/p\u003e\n\u003cp\u003eAxial plane wave theory is most commonly used to analyze a typical muffler system, wherein it is assumed that the plane wave propagates along the axis of the cylindrical chamber muffler [36]. At the entrance boundary of the acoustic field within muffler, an incident plane wave sound source is set, and the exit boundary is designated as a non-reflecting boundary. With the given entrance boundary conditions, the total sound pressure \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{t}\\)\u003c/span\u003e\u003c/span\u003e is calculated by the following expression:\u003c/p\u003e\n\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ10\" class=\"mathdisplay\"\u003e$$\\:{p}_{t}=\\sum\\:_{i\\in\\:\\text{b}\\text{n}\\text{d}}\\:{A}_{\\text{i}\\text{n}}{e}^{i\\varphi\\:}({S}_{ij}+{\\delta\\:}_{ij}){p}_{i}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn the equation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{A}_{\\text{i}\\text{n}}\\)\u003c/span\u003e\u003c/span\u003e is the amplitude of the incident wave sound pressure, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{e}^{i\\varphi\\:}\\)\u003c/span\u003e\u003c/span\u003e represents the phase of the incident wave, where 𝜙 is the phase angle, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{S}_{ij}\\)\u003c/span\u003e\u003c/span\u003e is a component of the scattering matrix, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\delta\\:}_{ij}\\)\u003c/span\u003e\u003c/span\u003e is the Kronecker delta function, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{p}_{i}\\)\u003c/span\u003e\u003c/span\u003e is the port mode shape. The matrix form of the linear equation set for solving the transmission loss of the acoustic field within the muffler can be expressed as:\u003c/p\u003e\n\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ11\" class=\"mathdisplay\"\u003e$$\\:(\\varvec{K}-{\\omega\\:}^{2}\\varvec{M})\\varvec{P}=\\varvec{Q}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{Q}\\)\u003c/span\u003e\u003c/span\u003e is the sound source vector. Once \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varvec{P}\\)\u003c/span\u003e\u003c/span\u003e is solved, the input and output acoustic power at the entrance and exit of the acoustic field within muffler can be obtained. The definition of transmission loss is the difference between the input incident sound power level and the output radiated sound power level, calculated by the following formula [37]:\u003c/p\u003e\n\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ12\" class=\"mathdisplay\"\u003e$$\\:TL=10\\times\\:\\text{l}\\text{g}\\left(\\frac{{W}_{\\text{i}\\text{n}}}{{W}_{\\text{o}\\text{u}\\text{t}}}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eHere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{W}_{\\text{i}\\text{n}}\\)\u003c/span\u003e\u003c/span\u003e is the input sound power, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{W}_{\\text{o}\\text{u}\\text{t}}\\)\u003c/span\u003e\u003c/span\u003e is the output sound power.\u003c/p\u003e\n\u003cp\u003eAs depicted in Fig.\u0026nbsp;2(a), the phononic crystal muffler is shown, which contains 2 \u0026times; 6 ring scatterers within its expansion chamber. The two-dimensional simplified model of the air domain of the muffler is established and discretized using the COMSOL. The Fig.\u0026nbsp;2(b) shows a two-dimensional mesh model of the muffler, which includes a total of 14,983 elements, with the maximum mesh cell length being 8 mm.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e"},{"header":"3 Results analysis","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n\u003ch2\u003e3.1 Bandgap characteristics\u003c/h2\u003e\n\u003cp\u003eAs shown in Fig.\u0026nbsp;3(a), the band structure of the ring phononic crystal is depicted. The dimension parameters of ring phononic crystal are as follows: lattice constant \u003cem\u003ea\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;50 mm, side length of resonator \u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;42 mm, resonator neck width \u003cem\u003el\u003c/em\u003e\u0026thinsp;=\u0026thinsp;4mm, resonator neck length \u003cem\u003eh\u003c/em\u003e\u0026thinsp;=\u0026thinsp;10 mm, and resonator rigid body thickness \u003cem\u003eb\u003c/em\u003e\u0026thinsp;=\u0026thinsp;2 mm. The distance of the model from the symmetry axis is set to \u003cem\u003eR\u003c/em\u003e\u0026thinsp;=\u0026thinsp;50 mm. The range of the first bandgap for ring phononic crystal is from 409 Hz to 707 Hz. The central frequency of the first bandgap is 550 Hz, which is significantly lower than the Bragg gap observed in lattices of comparable size, suggesting that it arises from local resonance phenomena. The second bandgap extends from 1897 Hz to 2514 Hz. It is widely accepted that the position and width of local resonance bandgaps are closely related to the dimensions of the scatterer structures rather than the specific crystal configuration. Consequently, to obtain a lower central frequency and a wider bandgap, it is necessary to investigate the impact of the dimension parameters of scatterer on the band structure.\u003c/p\u003e\n\u003cp\u003eTo examine the effects of parameter variations on the first bandgap of ring phononic crystal, the model parameters previously described were established as initial parameters. Subsequently, the side length of resonator (\u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e), neck width of resonator (\u003cem\u003el\u003c/em\u003e), neck length of resonator (\u003cem\u003eh\u003c/em\u003e), and the radius (\u003cem\u003eR\u003c/em\u003e) of the unit cell were individually adjusted while maintaining the constancy of other parameters. For each set of parameters, the band structure of ring phononic crystal was computationally determined. The influence of the above four parameters on the starting frequency (the lowest frequency corresponding of a certain bandgap), the ending frequency (the highest frequency corresponding of a certain bandgap) and the bandwidth of the first bandgap of the phonon crystal are shown in Fig.\u0026nbsp;3(b). The results indicate that an increment in \u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e results in a progressive broadening of the first bandgap, coupled with a shift to lower frequencies. As \u003cem\u003eh\u003c/em\u003e increases, the first bandgap progressively shifts to lower frequencies while simultaneously becoming narrower. An increase in \u003cem\u003el\u003c/em\u003e causes the first bandgap to gradually move towards higher frequencies, along with an expansion in width. When the radius of the unit cell slightly increases (within the range of the lattice constant), the starting and ending frequencies of the first bandgap, as well as the bandwidth, all tend to increase, but the variation is relatively minor.\u003c/p\u003e\n\u003cp\u003eIn summary, to achieve a bandgap at lower frequencies with sufficient width for the ring phononic crystal, it is necessary to increase the side length of the resonator in the radial direction and the length of the neck. Additionally, it is advisable to appropriately reduce the width of the neck.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n\u003ch2\u003e3.2 Acoustic transmission characteristics\u003c/h2\u003e\n\u003cp\u003eThe following analysis will delve deeper into the characteristics of the phononic crystal muffler at low frequencies. Figure\u0026nbsp;4(a) displays the band structure of the ring phononic crystal with adjusted scatterer parameters: \u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e is now 46 mm, \u003cem\u003eh\u003c/em\u003e is 40 mm, and \u003cem\u003eb\u003c/em\u003e is reduced to 1 mm. These modifications result in a first bandgap frequency range of 270\u0026ndash;579 Hz. As previously mentioned, the bandgap of the ring phononic crystal is affected by the radius, but the first bandgap does not vary significantly with minor changes in the radius. Consequently, ring phononic crystals with two different radii within the muffler can be considered to have similar first bandgaps. For subsequent analysis, the bandgap of the phononic crystal with the smaller radius will serve as the reference point. Figure\u0026nbsp;4(b) compares the transmission loss of the phononic crystal muffler with that of an expansion chamber muffler. It is evident that within the bandgap range, the phononic crystal muffler exhibits a significantly higher transmission loss compared to the expansion chamber muffler, with the maximum transmission loss reaching up to 130 dB, an increase from the original 13 dB.\u003c/p\u003e\n\u003cp\u003eFigures 4(c) and 4(d) respectively illustrate the sound pressure level and particle acceleration distributions within the expansion chamber muffler and the phononic crystal muffler at a frequency of 360 Hz. It is evident from these figures that the localized resonance occurring within the resonators of the phononic crystal muffler leads to a rapid attenuation of sound pressure as the acoustic wave propagates. This localized resonance is a result of the specific design of the resonators, which enhances the interaction with the acoustic waves, thereby increasing the energy dissipation. In contrast, the expansion chamber muffler, lacking such resonant structures, shows a significantly smaller attenuation amplitude, indicating less effective sound wave energy reduction.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n\u003ch2\u003e3.3 Defect state characteristics\u003c/h2\u003e\n\u003cp\u003eIn phononic crystals, the removal of scatterers introduces defect states, which can significantly alter the propagation of elastic waves. Specifically, the elimination of a single scatterer results in a point defect, which may manifest as a single point or multiple points. The removal of an entire row of scatterers generates a line defect, potentially appearing as a straight or broken line. In phononic crystals, the intentional introduction of defects can lead to the localization of elastic waves around these defects. This phenomenon is particularly beneficial for controlling long-wavelength sound waves using small structures. To facilitate the description of defect positions within the muffler, we assign labels to internal scatterers in the muffler in the format shown in Fig.\u0026nbsp;5(c) (e.g., i4 represents the scatterer in the fourth row of the i-th column). Due to the structural symmetry of the phononic crystal muffler, only variations in the first three columns need to be considered.\u003c/p\u003e\n\u003cp\u003eFigure 5(a) shows that the impact of point defects on transmission loss is concentrated in the frequency ranges of 200\u0026ndash;340 Hz, 350\u0026ndash;500 Hz, and 510\u0026ndash;820 Hz. The removal of scatterer j1 leads to an increase in transmission loss across the frequency span of 350\u0026ndash;820 Hz, and the elimination of scatterer i2 increases transmission loss within the frequency band of 200\u0026ndash;340 Hz. When scatterers j1 and i2 are removed together, transmission loss is significantly enhanced in the ranges of 200\u0026ndash;340 Hz and 500\u0026ndash;820 Hz, with new peaks at frequencies of 270 Hz, 610 Hz, and 820 Hz. Figure\u0026nbsp;5(b) illustrates the effect of a linear defect on transmission loss for the phononic crystal muffler, primarily affecting the frequency ranges of 200\u0026ndash;340 Hz, 350\u0026ndash;500 Hz, and 510\u0026ndash;1000 Hz. After removing scatterers in columns 1, 2, and 3, the muffler's transmission loss is enhanced in the ranges of 200\u0026ndash;340 Hz and 510\u0026ndash;1000 Hz, and shows a decrease in the 350\u0026ndash;500 Hz range. These results demonstrate that by accurately manipulating the defect state, it is possible to effectively suppress sound waves at specific frequencies.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n\u003ch2\u003e3.4 Fluid dynamic characteristics\u003c/h2\u003e\n\u003cp\u003eThe aerodynamic performance is a crucial aspect of assessing the quality of a muffler [38]-[40], reflecting the resistance offered by the muffler to airflow, or the impact of installing the muffler on the flow characteristics of the original system. Pressure loss, calculated as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:PL={P}_{\\text{i}\\text{n}}-{P}_{\\text{o}\\text{u}\\text{t}}\\)\u003c/span\u003e\u003c/span\u003e [38], is an evaluation metric for this performance, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{P}_{\\text{i}\\text{n}}\\)\u003c/span\u003e\u003c/span\u003e is the total pressure at the inlet and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{P}_{out}\\)\u003c/span\u003e\u003c/span\u003e at the outlet. A two-dimensional simplified model of the muffler air domain was established, similar to the approach used in acoustic calculations. When the air velocity Mach number is less than 0.3, the fluid can be considered incompressible; therefore, the air within the muffler is assumed an ideal fluid with no density variation or heat exchange. The standard \u003cstrong\u003ek\u003c/strong\u003e-\u003cstrong\u003e\u0026epsilon;\u003c/strong\u003e turbulence model [42][43] was employed to calculate the pressure loss of the muffler. The muffler inlet boundary was set as a velocity inlet, defining the inlet velocity, while the outlet boundary was set as a pressure outlet with atmospheric pressure, defined as 0 Pa.\u003c/p\u003e\n\u003cp\u003eFigure 6 illustrates the pressure loss of the phononic crystal muffler and the simple expansion chamber muffler at different inlet velocities. It is observed that the pressure loss escalates with an increase in the inlet velocity. The phononic crystal muffler, devoid of defect states, exhibits a significantly slower rate of pressure loss increase with rising inlet velocity in comparison to the expansion chamber muffler. At identical inlet velocities, the pressure loss for the phononic crystal muffler with defect states exhibits a marginal increment relative to the pressure loss of the phononic crystal muffler without defect states.\u003c/p\u003e\n\u003cp\u003eFigure 7 shows the pressure distributions and turbulent kinetic energy distributions of the phononic crystal muffler and the expansion chamber muffler at an inlet velocity of 30 m/s. Compared to the expansion chamber muffler, the inner ring resonator walls of the phononic crystal muffler obstruct the radial flow of the airflow, resulting in weaker turbulent intensity inside and smoother airflow. When the phononic crystal muffler is in a defective state, although there is some enhancement of turbulence near the defect region close to the pipeline, the overall impact of fewer defects on the internal flow field of the phononic crystal muffler is not significant. These results indicate that the phononic crystal muffler possesses excellent fluid dynamic performance.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e"},{"header":"4 Experiment and validation","content":"\u003cp\u003eThe phononic crystal model used for finite element calculations and experiments is shown in Fig. 8(a), with specific geometric parameters listed in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. The expansion chamber of the muffler contains three ring resonators and features a point defect. This experiment utilized the four-microphone method with an impedance tube to measure the transmission loss for the phononic crystal muffler, based on the theoretical foundation of the standing wave separation method. As shown in Fig. 8(c), a loudspeaker emitted white noise into the impedance tube, generating a planar incident wave \u003cem\u003eM\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e upon encountering the phononic crystal muffler sample. A portion of the sound waves were reflected, forming the planar reflected sound wave \u003cem\u003eN\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e. Another portion passed through the sample, entered the transmission tube, and generated the planar transmitted sound wave \u003cem\u003eM\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e. The planar transmitted sound wave encountered the absorbent termination, where some was absorbed, and some was reflected, forming the transmitted reflected wave \u003cem\u003eN\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e. The sound pressure values measured by microphones at different positions were then used to calculate the self-power spectra \u003cem\u003eW\u003c/em\u003e\u003csub\u003ein\u003c/sub\u003e and \u003cem\u003eW\u003c/em\u003e\u003csub\u003epro\u003c/sub\u003e of the incident and transmitted waves, respectively. Finally, the transmission loss of the test sample equal to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:10\\times\\:\\text{lg}\\left({W}_{\\text{i}\\text{n}}∕{W}_{\\text{p}\\text{r}\\text{o}}\\right)\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n \u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eThe parameters of the phononic crystal muffler sample\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSymbol\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eH\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eH\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eR\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eR\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003ea\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eb\u003c/em\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e\u003cem\u003eh\u003c/em\u003e\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eGeometrical parameters (mm)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e98.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e52.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e52.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e14.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e26.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e23.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eFigure 8(a) displays the band structure of the phononic crystal in the experiment, which shows the first bandgap ranging from 1084 to 1683 Hz. Figure\u0026nbsp;8(b) illustrates the transmission loss curve of the phononic crystal muffler obtained from experimental testing, compared with the results calculated by the finite element method. It shows that, the experimental results are in good agreement with those calculated by the finite element method, and a significant enhancement in transmission loss is observed at frequencies within the bandgap. This confirms the accuracy of the finite element method in calculating band structures and transmission loss values, and demonstrates the effectiveness of using a phononic crystal muffler. At the same time, this demonstrates that, through the judicious design of bandgap properties, the phononic crystal muffler can effectively attenuate noise within specific frequency ranges, proving its potential as a highly efficient acoustic control component.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n \u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003cp\u003eThis study introduces the engineering of a Helmholtz-like ring resonator phononic crystal muffler, with its acoustic characteristics numerically analyzed using finite element method. In the study, the effects of the dimension parameters of ring scatterers on the bandgap were investigated, revealing that increasing the side length of the resonator lowers the central frequency and broadens the bandgap. Experimental validations of the acoustic transmission characteristics for a phononic crystal muffler affirmed the accuracy of the calculations and the efficacy of the muffler within the bandgap range. The study further investigated the effects of point and linear defects on transmission loss, showing that strategically positioned defects can significantly enhance noise reduction performance and broaden the bandwidth. Fluid dynamics simulations revealed that scatterers within the phononic crystal muffler impede radial airflow and reduce turbulence. Consequently, compared to a conventional expansion chamber muffler, the phononic crystal muffler incurs lower pressure loss at equivalent inlet velocities, highlighting superior aerodynamic performance. Integrating phononic crystal technology into muffler design, this study introduces a muffler with robust low-frequency absorption and commendable aerodynamics, charting a novel pathway for controlling low-frequency noise in muffler technology.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eCompeting Interests Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe 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.\u003c/p\u003e\n\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\n\u003cp\u003eY.B. and Y.C. initiated the research; Y.B. conducted the theoretical and numerical simulation work; J.Z and Y.B. designed and carried out the experiments; Y.B. wrote and revised the manuscript; Y.C. guided and supervised this study.\u003c/p\u003e\n\u003ch2\u003eAcknowledgements\u003c/h2\u003e\n\u003cp\u003eThe authors thank the financial supports from the Natural Science Foundation of Zhejiang Province and Ningbo City, China (Nos. LY20E050006, 2021Z098).\u003c/p\u003e\n\u003ch2\u003eData Availability\u003c/h2\u003e\n\u003cp\u003eThe data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy concerns.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eLee, J. K., Oh, K. S. \u0026amp; Lee, J. W. Methods for evaluating in-duct noise attenuation performance in a muffler design problem. \u003cem\u003eJ. Sound Vib\u003c/em\u003e. \u003cstrong\u003e464\u003c/strong\u003e, 114982. https://doi.org/10.1016/j.jsv.2019.114982 (2020).\u003c/li\u003e\n\u003cli\u003eHe, Z. R., Ji, Z. L. \u0026amp; Huang, H. P. Acoustic attenuation analysis of expansion chamber mufflers with non-uniform cold and hot flow. \u003cem\u003eJ. Sound Vib\u003c/em\u003e. \u003cstrong\u003e568\u003c/strong\u003e, 118062. https://doi.org/10.1016/j.jsv.2023.118062 (2024).\u003c/li\u003e\n\u003cli\u003eZhao, S. 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Performance Analysis of Improved Vehicle Muffler. \u003cem\u003eMechanika\u003c/em\u003e \u003cstrong\u003e24\u003c/strong\u003e (5), 751\u0026ndash;756. https://doi.org/10.5755/j01.mech.24.5.17784 (2018).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Phononic crystal muffler, defect state analysis, local resonance, transmission loss, pressure loss","lastPublishedDoi":"10.21203/rs.3.rs-4963361/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4963361/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTo improve the sound absorption performance of the expansion chamber muffler at low frequencies, a Helmholtz-ring phononic crystal muffler is designed based on the local resonance mechanism. The phononic crystal muffler exhibits strong sound attenuation performance at deep sub-wavelength scales. Firstly, the phononic crystal scatterer is designed as a ring-type Helmholtz resonant chamber, and a certain amount of cell units is periodically arranged inside an expansion chamber muffler. Secondly, the effects of the dimension parameters of scatterers on the bandgaps are studied. The transmission loss of the phononic crystal muffler, together with the pressure loss at low Mach numbers, is investigated. Subsequent focus is devoted to analyzing the effects of point and linear defective states on the acoustic transmission characteristics of the phononic crystal muffler. The results show that a significant improvement in both transmission loss and aerodynamic performance of the proposed phononic crystal muffler is observed when compared to the original expansion chamber muffler. Additionally, the transmission loss within the bandgap can be further enhanced when the phononic crystal muffler is in a defective state. Finally, experimental investigations were conducted to validate the effectiveness of the phononic crystal muffler within its bandgap range.\u003c/p\u003e","manuscriptTitle":"Experimental study and acoustic characteristics analysis of defective-state Helmholtz-ring phononic crystal muffler","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-10-01 15:51:29","doi":"10.21203/rs.3.rs-4963361/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-09-19T10:55:34+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-09-18T17:21:23+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-09-07T19:06:01+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"33710899036691200085709231609841269072","date":"2024-09-07T18:25:37+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"320431754093211377508697842308805002512","date":"2024-09-07T17:51:38+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-09-07T17:46:04+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-09-03T05:36:25+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2024-09-03T05:32:32+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-08-30T14:51:09+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2024-08-23T09:57:04+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"a28f9323-c6ff-41b4-9652-a1bdf59f5558","owner":[],"postedDate":"October 1st, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":37881376,"name":"Physical sciences/Physics/Applied physics/Acoustics"},{"id":37881377,"name":"Physical sciences/Engineering/Mechanical engineering"}],"tags":[],"updatedAt":"2024-11-25T16:03:53+00:00","versionOfRecord":{"articleIdentity":"rs-4963361","link":"https://doi.org/10.1038/s41598-024-79762-9","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2024-11-22 15:57:54","publishedOnDateReadable":"November 22nd, 2024"},"versionCreatedAt":"2024-10-01 15:51:29","video":"","vorDoi":"10.1038/s41598-024-79762-9","vorDoiUrl":"https://doi.org/10.1038/s41598-024-79762-9","workflowStages":[]},"version":"v1","identity":"rs-4963361","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4963361","identity":"rs-4963361","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}