# Modeling and experimental verification of a new muffler based on the theory of quarter-wavelength tube and the Helmholtz muffler

- Can Wu
^{1}Email author, - Lei Chen
^{1}, - Jing Ni
^{1}and - Jing Xu
^{1}

**Received: **28 April 2016

**Accepted: **12 August 2016

**Published: **19 August 2016

## Abstract

To address the problem of low frequency noise of the internal combustion engine, several existing muffler design methods, such as the theory of the quarter-wavelength tube and the Helmholtz muffler, were examined and compared. This paper proposes a new type of muffler design method, which has the advantages of both the quarter-wavelength tube and the Helmholtz muffler. An example is carried out to illustrate the analysis of original signal, the design of the new muffler and the improvement of the in-car noise. The transmission loss of the new muffler is studied by theoretical method and finite element method. The vehicle test of the new muffler demonstrates excellent performance with a wider noise elimination frequency band and smaller radial size.

## Keywords

## Background

The internal combustion (IC) engine is a major source of automobile noise (Sanjid et al. 2014), and as the standards for automobile noise are becoming increasingly strict, there is greater demand for quieter engines. The process of induction plays a large part in creating engine noise. For some small high-speed machines and some other large turbochargers, the induction noise can reach up to 5 dB (A) higher than the engine noise sometimes (Chiatti et al. 2015). Therefore, it is of great significance to find an approach to control induction noise (Mondal et al. 2014).

Intake noise control begins with the air intake system design, the structure of the valve, cam curve shape, and many other factors, while at the same time, noise control is restrained by power performance and economic efficiency (Boutin and Becot 2015). At present, the actual application of active noise control technology remains in its infancy (Zhou et al. 2009). Therefore, passive noise control is currently the most commonly applied method in the intake noise control, and its core principle is based in muffler design. Mofakhami et al. (2008) studied sound transmission through multilayered viscoelastic air filled cylinders. His result showed that using constant and frequency-dependent viscoelastic material, with high loss factor, leads to uniform noise reduction in the frequency domain. Miandoab et al. (2015) analyzed the nonlinear dynamics and chaotic behavior of nanoresonators with electrostatic forces on both sides, which indicated that the necessary condition for the creation of chaos in the resonator is intersection of the system steady state response with the homoclinic orbit. Ning et al. (2015) used a quasi-one-dimensional model based on the compressible Navier–Stokes equations and a finite volume method to investigate the transient motion of a fluid inside oscillating axisymmetric tubes. Li (2010) developed an objective function to optimize parameters (number and location) of resonator-like cavities, which is based on the tuned weighting coefficient and the acoustic potential energy. Unnikrishnan et al. (2010) presents a simplified modeling approach for numerical simulation of a coupled cavity-resonator system, which was validated by experimentation. It is shown that the resonator volume fraction required to significantly (more than 5 dB) suppress the cavity axial mode. Otherwise, the PZT piezoelectric plates (Li et al. 2013a) and multiple PVDF beam arrays (Li et al. 2013b) are used in the design of the muffler to achieve the purpose of energy collection and recycling.

The resonance-type resistance silencer is based on the principle of the Helmholtz resonator (HR). It has the advantages of simple structure, high amount of noise elimination, and small pressure loss, among other attributes. It is widely used in automotive engine air intake noise control (Sohn and Park 2011). Arefi et al. (2014) was able to improve reverberation time in a conference room using HRs with defined dimensions, diffusers, and sound absorbers. Mao and Pietrzko (2010) carried out an experimental investigation of passive control of sound transmission through a double-glazed window by using an arrangement of HRs. It was shown that a considerable reduction of the transmitted sound pressure levels has been achieved around the mass-air-mass resonance frequency (50–120 Hz). Lee et al. (2013) studied the effect of leakage on the acoustic performance of reactive silencers, such as expansion chambers, HRs, and quarter-wave resonators. Sanada and Tanaka (2013) used two degree of freedom Helmholtz-based resonators with a flexible panel to extend the frequency range of resonant sound absorbers. Yasuda et al. (2013), based on the typical structure, designed a muffler with an interconnecting hole on the tail tube, which was proposed to improve its acoustic performance. Park (2013) introduced a micro-perforated panel of absorbers backed by HRs to improve sound absorption in the low-frequency region, where conventional micro-perforated panel absorbers cannot provide sufficient absorption. Singh and Rienstra (2014) presented a systematic derivation of a solution of the nonlinear HR equation, in order to obtain analytically expressions for impedances close to resonance, while including nonlinear effects. Atak et al. (2014) combined two concepts to design acoustic lenses that are based on HRs. It was shown that using HR-based sonic crystals leads to better acoustic lens designs, especially at the low frequencies, where the local resonances are pronounced. At present, research focuses primarily on designing or analyzing HR, which has a good effects on the air inlet system’s low-frequency noise elimination; however, a larger proportion of intake system noise is high-frequency, especially during rapid acceleration. In order to improve an automobile’s riding comfort, it is necessary to establish a method to suppress the engine’s air inlet high-frequency noise during rapid acceleration.

The air inlet system of traditional engines mainly relies on absorption by the quarter-wavelength straight-tube to suppress high-frequency noise, but two major problems exist. First, the quarter-wavelength straight-tube has a narrow denoising band and cannot cover the low-frequency noise band completely, which results in incomplete noise reduction. Second, due to the greatly restricted radial size of the air inlet tube in the engine’s inlet system, the longer length of the traditional quarter-wavelength straight-tube will be difficult to arrange in a compact car air inlet system. Therefore, in this paper, we propose a new type of muffler design, which has the advantages of both the quarter-wavelength straight-tube and Helmholtz muffler, with a wider noise elimination frequency band and smaller radial size. When used in a practical automobile test, the muffler showed excellent performance in high frequency noise reduction of the air inlet system, and this performance was supported by the experimental data.

## Methods

The principle of the resonance muffler is based on a hole in the tube that connects with the resonance cavity. When the sound wave comes to the resonance structure, the gas will flow back and forth in the hole like a piston reciprocating motion under the influence of the acoustic pressure. Part of the acoustic energy can be consumed into heat energy by the aperture wall friction and damping effects. At present, the main types of resonant silencer are the quarter-wavelength tube and the Helmholtz muffler.

### Quarter-wavelength tube model

*TL*) is as follows (Pang et al. 2006):

*TL*tends to be infinity, which means the

*TL*reaches the maximum, the bypass tube length is:

*λ*/4, 3

*λ*/4, 5

*λ*/4, etc.,

*TL*reaches the maximum. For the first value being selected (n = 1), the bypass tube’s length is:

From the Eq. 4, we know the resonance frequency of the quarter-wavelength tube merely depends on the length of the tube. The longer the tube, the lower frequency.

*r*is the quarter-wavelength tube’s radius (m).

In spite of the above model as the simplified approximate model of ideal quarter wave tube model, the resonant frequency of the quarter-wavelength tube is closely relevant to the length of the wavelength tube and the radius of the connecting tube from Eq. 6. And the shape, direction or volume of the wavelength tube has less impact on the resonant frequency.

### Helmholtz muffler model

*m*is the mass of air in the connected tube, and it can be expressed as \(m = A_{c} l_{c} \rho_{0}\),

*k*is the elastic coefficient of the resonance cavity, and it can be expressed as \(k = \rho_{0} c_{0}^{2} A_{c}^{2} /V\). \(A_{c}\) is the section area (m

^{2}) of the connecting tube, \(l_{c}\) is the length of the connecting tube (m), \(\rho_{0}\) is the gas density (kg/m

^{3}), and \(V\) is the volume of resonance cavity (m

^{3}).

In spite of the above model as the simplified approximate model of the ideal Helmholtz muffler, the resonance frequency of Helmholtz muffler is closely related to the cross-sectional area and length of connection tube, as well as the volume of the resonance cavity from Eq. 8, while the overall length of the muffler has little influence on the resonance frequency.

^{2}) of the main tube.

### New muffler design

The new muffler consisted of a connecting tube and cavity resonance. It exhibits the functions of both the quarter-wavelength tube and the Helmholtz muffler. The design steps are as follows:

Step 1: Try to determine the value of the connecting tube’s radius \(r\). Considering the acoustic radiation effect in the nozzle and convenience to manufacture, the connecting tube’s radius \(r\) is recommended for 6-8 mm (The diameter of main tube is 54-96 mm for most of IC engines).

Step 3: Try to determine the value of the connecting tube’s length \(l_{c}\). Since the air in the connecting tube is considered as a lumped mass, the connecting tube’s length \(l_{c}\) is not more than half of the wavelength. Taking into account the influence of the tube wall, the connecting tube’s length \(l_{c}\) should not be significantly less than its diameter. So the connecting tube’s length \(l_{c}\) is recommended for \(r \le l_{c} \le 4r\).

Equation 6 is deduced under the assumptions that the geometry of the resonance cavity is significantly less than the acoustic wavelength. It was also assumed that the wave motion and mass distribution conditions in the connection tube and the resonant cavity can be ignored. These hypotheses are hard to meet if the ratios of height and width are too large or too small. In addition, when the ratios of height and width become too small, the transverse wave is dominant in the resonant cavity, and the intersection area of the cavity and connecting tube becomes larger. This leads to strong three-dimensional effects in the intersection area and therefore reduces the accuracy of the calculation.

## Experiment details

Vehicle testing conditions

Testing condition | Testing state |
---|---|

The normal temperature | 15 °C |

The starting speed | 15 km/h |

Full-throttle acceleration | Engine speed 1000–5000 rpm full-throttle acceleration in second gear |

The test distance | 200 m |

## Results and discussion

### Experiment analysis

The resulting curve of the vehicle prototype’s in-car noise analysis is shown below.

*z*is the number of engine cylinders,

*i*is the engine stroke coefficient (for four stroke is 2), \(k_{o}\) is the harmonic number (order number), and \(f_{e}\) stands for the noise frequency.

### Examples and solutions

Because the noise energy is too high at the range of 470–570 Hz, we designed a new muffler with a wide noise suppression frequency band.

Size: \(a = 25\;{\text{mm}}\), \(b = 20{\kern 1pt} \;{\text{mm}}\)

### Validation and discussion

The frequency spectrum of the new muffler is difficult to obtain by the theoretical calculation. So the Finite Element Method (FEM) was used to calculate the transmission loss. The details of the model are as follows:

## Conclusion

- 1.
This new silencer consists of a connecting tube and a unidirectional asymmetric resonance cavity. The muffler exhibits the advantages of both the quarter-wavelength tube and the Helmholtz muffler. The new muffler has a wider frequency band of noise elimination than the quarter-wavelength tube, and it is similar to the Helmholtz muffler. However, the radial size of the new muffler is significantly less than the regular Helmholtz muffler.

- 2.
The new muffler’s noise elimination frequency has been derived via a calculation method and FEM method proposed in this study. The size of the muffler’s connection tube was selected based on the diameter of the intake tube, and then according to the frequency calculation, we derived the specific optimal geometric dimensions of the cavity resonance muffler.

- 3.
The experimental data shows that there is a wide noise elimination frequency band around the calculated noise frequency. The new type of muffler design presented here may be used in the air intake system of some types of cars, thus improving the driving experience of both driver and passengers.

## Declarations

### Authors’ contributions

CW carried out the muffler model studies, participated in experiment design. LC manufactured the new muffler and carried out the experiment verification. JN participated in experiment data analysis. JX carried out the Helmholtz muffler studies. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Funding

The work was supported by National Natural Science Foundation of China (Grant No. 51405118); Zhejiang Provincial Natural Science Foundation (Grant No. LQ14E050011) and Research Project of Zhejiang Provincial Department of Education (Grant No. Y201120741).

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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