Muffler and Silencer DesignVibro-Acoustics Consortium Web Meeting

University of Kentucky

Vibro-Acoustics Consortium

June 25, 2020

Vibro-Acoustics Consortium

Overview

2

• Fundamental Concepts

• Illustrative Examples

• Conceptual Design

• Detailed Design

• Prototyping / Testing

Vibro-Acoustics Consortium

Performance Measures Transmission Loss

3

Transmission loss (𝑇𝐿) of the muffler:

𝑊

𝑊𝑊 Anechoic

Termination

1 2

𝑇𝐿 10 log𝑊𝑊

Vibro-Acoustics Consortium

Performance Measures Insertion Loss

4

𝐼𝐿 𝑑𝐵 𝐿 𝐿

Insertion loss depends on : • Muffler• Lengths of pipes• Termination (baffled vs. unbaffled)• Source impedance

Muffler

𝐿

𝐿

Note: TL is a property of the muffler; IL is a “system” performance measure.

𝐿

Vibro-Acoustics Consortium

Example Transmission and Insertion Loss

5

2412

122 6Source

-50

-40

-30

-20

-10

0

10

20

0 200 400 600 800 1000

Frequency (Hz)

TL a

nd IL

(dB

)

Insertion LossTransmission Loss

Pipe resonances

Inlet Pipe Outlet Pipe

Expansion Chamber Muffler

Dimensions in Inches

𝜆𝑐𝑓

343 m/s𝑓

Acoustic Wavelength

Vibro-Acoustics Consortium

Reactive Muffler

6

Helmholtz Resonators

Quarter wave tube

Vibro-Acoustics Consortium

Reactive and Dissipative Muffler

7

Sound absorbing material

Perforated tubes

Neck

Volume

Vibro-Acoustics Consortium

The Helmholtz Resonator

8

𝐹 𝑝𝑆

𝑥

𝑉

𝑆

𝐿 𝐿 is the equivalent length of the neck (some air on either end also moves).

Damping due to viscosity in the neck are neglected

(resonance frequency of the Helmholtz resonator)

𝑘𝜌 𝑐 𝑆𝑉

𝑚 𝜌 𝑆 𝐿

𝑚𝑥 𝑘𝑥 𝑝𝑆 𝑥 𝑗𝜔𝑢 𝑥𝑢𝑗𝜔

𝑗 𝜔𝑚𝑘𝜔 𝑢 𝑝𝑆

𝑧𝑝

𝑆 𝑢 𝑗1𝑆

𝜔𝑚𝑘𝜔

𝑧 → 𝜔𝑘𝑚 𝑐

𝑆𝐿 𝑉

Vibro-Acoustics Consortium

The Helmholtz Resonator

9

𝑉 0.001 m3

𝐿 25 mm𝑆

2 x 10-4 m2

𝑆 8 x 10-4 m2

Anechoic termination

0

5

10

15

20

0 50 100 150 200 250 300

Frequency (Hz)

TL (d

B)

35 Hz

𝑇𝐿 10 log 1𝑐 2𝑆⁄

𝜔𝐿 𝑆⁄ 𝑐 𝜔𝑉⁄

Vibro-Acoustics Consortium

Quarter Wave Resonator

10

𝑧

𝐿

𝑆

𝑆 𝑇𝐿 10 logtan 𝑘𝑙 4 𝑆 𝑆⁄

4 𝑆 𝑆⁄

𝑧𝑗𝜌 𝑐𝑆 cot

𝜔𝐿𝑐 0

𝜔𝑛𝜋𝑐2𝐿 𝑛 1,3,5, …

𝑓𝑛𝑐4𝐿

𝐿𝑛𝑐4𝑓 𝑛

𝜆4 𝑛 1,3,5, …or

𝜔𝐿𝑐

𝑛𝜋2 𝑛 1,3,5, …and

Vibro-Acoustics Consortium

Parallel Impedance

11

Side Branch

𝑧

𝑧𝑧

𝑢 𝑝𝑧 𝑧

𝑧𝑝𝑆𝑢

𝑧 𝑧𝑧 𝑧

𝑧 0

𝑧 0

Vibro-Acoustics Consortium 12

𝑧Incident Wave

Reflected Wave

Transmitted Wave

𝑝 𝑝

𝑢 𝑢

𝑝 𝑝

𝑢

𝑧𝑝 𝑝𝑆𝑢

Series (Transfer) Impedance

Vibro-Acoustics Consortium

Muffler Elements

13

Straight Pipe𝐿

S

𝑆

LB

Quarter Wave TubeCone

𝑧

𝑧

𝑉

𝐿𝑆

Helmholtz Resonator

Perforate

Vibro-Acoustics Consortium

Overview

14

• Fundamental Concepts

• Illustrative Examples

• Conceptual Design

• Detailed Design

• Prototyping / Testing

Vibro-Acoustics Consortium

Muffler Elements Simple Expansion Chamber

15

18

2 26

0

5

10

15

20

25

30

0 100 200 300 400 500 600 700 800

Frequency (Hz)

TL (d

B)

Dimensions in inches

𝑇𝐿 10 log14 4 cos 𝑘𝐿 𝑚

1𝑚 sin 𝑘𝐿

where 𝑚 is the expansion ratio (chamber area/pipe area) = 9 in this example and L is the length of the chamber.

If 𝑚 10

𝑇𝐿 10 log14 𝑚 sin 𝑘𝐿

𝑚 𝑑 𝑑⁄

Vibro-Acoustics Consortium 16

Muffler Elements Quarter Wave Tube

29 18

2 26

Dimensions in inches

0

5

10

15

20

25

30

0 100 200 300 400 500 600 700 800

Frequency (Hz)TL

(dB)

Vibro-Acoustics Consortium

A Note on Quarter Wave Tubes

17

𝑧

𝑧 𝑧

Vibro-Acoustics Consortium 18

Muffler Elements Extended Inlet

18

2 269

0

5

10

15

20

25

30

0 100 200 300 400 500 600 700 800

Frequency (Hz)

TL (d

B)

Dimensions in inches

(similar for extended outlet)

Vibro-Acoustics Consortium 19

Muffler Elements Extended Inlet and Outlet

9 9

4 6

0

10

20

30

40

50

0 100 200 300 400 500 600 700 800

Frequency (Hz)

TL (d

B)

Dimensions in inches

Vibro-Acoustics Consortium 20

Muffler Elements Extended Inlet and OutletDimensions in inches

0

10

20

30

40

50

60

0 500 1000 1500 2000

Tran

smis

sion

Los

s (d

B)

Frequency (Hz)

𝐿

𝐿/2

𝐿/4

Simple expansion chamberSide branch 1Side branch 2

Expansion Chamber

Side Branch 1

Side Branch 2

Vibro-Acoustics Consortium 21

Munjal, 2011

Double Tuned Expansion Chamber

Vibro-Acoustics Consortium

Overview

22

• Fundamental Concepts

• Illustrative Examples

• Conceptual Design

• Detailed Design

• Prototyping / Testing

Vibro-Acoustics Consortium 23

Plane Wave Modeling

𝑝𝑆 𝑢

𝑝𝑆 𝑢

𝑝𝑆 𝑢

cos 𝑘𝐿𝑗𝜌 𝑐𝑆 sin 𝑘𝐿

𝑗𝑆𝜌 𝑐 sin 𝑘𝐿

𝑆𝑆 cos 𝑘𝐿

𝑝𝑆 𝑢

Vibro-Acoustics Consortium 24

𝑆

𝑆

1 2

𝑝 𝑝 𝑝

𝑆𝑢 𝑆 𝑢 𝑆𝑢

𝑧𝑝𝑆 𝑢

𝑝𝑆 𝑢

𝑆𝑢𝑝𝑧 𝑆𝑢

𝑝𝑆 𝑢

1 01 𝑧⁄ 1

𝑝𝑆 𝑢

Parallel Impedance

Transfer Matrix

Vibro-Acoustics Consortium 25

𝑧Incident Wave

Reflected Wave

Transmitted Wave

𝑝 𝑝

𝑢 𝑢

𝑝 𝑝

𝑢

𝑧𝑝 𝑝𝑆𝑢

Series (Transfer) Impedance

𝑝𝑆 𝑢

1 𝑧0 1

𝑝𝑆 𝑢

Transfer Matrix

Vibro-Acoustics Consortium

Empirical Equations Perforated Panel

Sullivan and Crocker (1978)

𝑧 1 𝜌𝑐𝜎⁄ 2.4 𝑗0.02𝑓

𝑧 1 𝜎⁄ 0.006 𝑗𝑘 𝑡 0.75𝑑

Sullivan and Crocker (1978))

Rao and Munjal (1986)

26

𝑧 1 𝜎⁄ 7.337 10 1 72.23𝑀 𝑗2.2245 10 𝑓 1 51𝑡 1 204𝑑

𝑓

𝑀

frequency

flow Mach number

𝜌 density of air𝑐 speed of sound in air

𝜎 porosity𝑡 panel thickness 𝑑 hole diameter

𝑘 wave number ( ⁄ )

Vibro-Acoustics Consortium 27

For each element:

Overall:

𝑝𝑆 𝑢

𝑇 𝑇𝑇 𝑇

𝑝𝑆 𝑢

𝑇 𝑇 𝑇 𝑇 ⋅⋅⋅ 𝑇 𝑇 𝑇𝑇 𝑇

Combining Elements

𝑇𝐿 10 log𝑆

4𝑆 𝑇𝑆 𝑇𝜌𝑐

𝜌𝑐𝑇𝑆

𝑆𝑆 𝑇

Transmission Loss

Vibro-Acoustics Consortium 28

Plane Wave Modeling Virtual DesignPlane Wave Modeling (Virtual Design)

Source

In course work

A real muffler

Vibro-Acoustics Consortium

Plane Wave Modeling Tips

29

1. Examine the muffler to be modeled and determine a direction of sound propagation. This will normally be the direction having the longest cross-dimension and is also usually the same as the flow path.

2. Details like welds, seams, and fillets that are small compared to an acoustic wavelength can be ignored. However, leaks should be included.

3. Identify and include quarter wave tubes. Often in the case of right-angle turns, quarter wave tubes are unintentionally included. Additionally, be sure to include mass effects in the case of extended inlets or outlets.

Vibro-Acoustics Consortium

Plane Wave Modeling Tips

30

4. The shape of the cross-section, whether circular, elliptical, or rectangular will have no effect.

5. Sound will be reflected due to changes in the cross-sectional area. Significant area changes should be accounted for. Gradual changes in cross-sectional area can be modeled as cones.

6. If the porosity (perforation rate) exceeds 30%, a perforate can generally be ignored and treated as open.

Vibro-Acoustics Consortium

Example Small Engine Muffler

31

Perforated

Vibro-Acoustics Consortium

Example Small Engine Muffler

32

Flow Path

Vibro-Acoustics Consortium

Example Small Engine Muffler

33

Quarter Wave TubesQuarter Wave Tubes

Vibro-Acoustics Consortium

Example Small Engine Muffler

34

SIDLAB Model

Vibro-Acoustics Consortium

Example Small Engine Muffler

35

Vibro-Acoustics Consortium 36

A Note on Perforates and Flow

Frequency (Hz)1000 2000 3000 40000

Frequency (Hz)1000 2000 3000 40000

Frequency (Hz)1000 2000 3000 40000

Frequency (Hz)1000 2000 3000 40000

40

30

10

20

Tran

smis

sion

loss

(dB)

40

30

10

20

Tran

smis

sion

loss

(dB)

30

10

20

Tran

smis

sion

loss

(dB)

40

30

10

20

Tran

smis

sion

loss

(dB)

40

No flow M=0.01

M=0.03 M=0.04

Vibro-Acoustics Consortium

Example Cross Flow Muffler

37

InletOutlet

: Propagation: Quarter-wavelength tube: Cross-flow: Helmholtz resonator

Vibro-Acoustics Consortium

Example Cross Flow Muffler

38

1st chamber2nd chamber3rd chamber

Vibro-Acoustics Consortium

Example Cross Flow Muffler

39

0

20

40

60

80

100

120

0 100 200 300 400 500 600

Tran

smis

sion

Los

s (d

B)

Frequency (Hz)

Measurement

SIDLAB

MAP

Vibro-Acoustics Consortium

Overview

40

• Fundamental Concepts

• Illustrative Examples

• Conceptual Design

• Detailed Design

• Prototyping / Testing

Vibro-Acoustics Consortium 41

Determination of Transfer Matrix

3-D Waves

Plane Waves

𝑝𝑆 𝑢

𝑇 𝑇𝑇 𝑇

𝑝𝑆 𝑢𝑝

𝑆 𝑢

𝑝𝑆 𝑢

Vibro-Acoustics Consortium 42

Determination of Transfer Matrix

𝑝𝑆 𝑢

𝑇 𝑇𝑇 𝑇

𝑝𝑆 𝑢

3-D Waves

Plane Waves

𝑝𝑆 𝑢

𝑝𝑆 𝑢

𝐴 𝑝,

𝐴 𝑝,

𝐴 𝑝,

𝐴 𝑝,

𝑇 𝐴 /𝐴 𝑇1𝑆 𝐴

𝐴 𝐴𝐴

𝑇 𝑆1𝐴 𝑇

𝑆𝑆

𝐴𝐴

Vibro-Acoustics Consortium 43

Example Cross Flow Muffler

Virtual prototyping using acoustic BEM (MAP software)

Vibro-Acoustics Consortium

Overview

44

• Fundamental Concepts

• Illustrative Examples

• Conceptual Design

• Detailed Design

• Prototyping / Testing

Vibro-Acoustics Consortium 45

Transmission Loss Standards

ASTM E2611-09, “Standard Test Method for Measurement of Normal Incidence Sound Transmission of Acoustical Materials Based on the Transfer Matrix Method,” American Society of Testing and Materials, Philadelphia, 2009.

Vibro-Acoustics Consortium 46

Example Cross Flow Muffler

0

20

40

60

80

100

120

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Tran

smis

sion

Los

s (d

B)

Frequency (Hz)

Experiment

SIDLAB

3D MAP

Vibro-Acoustics Consortium 47

Example Cross Flow Muffler

InletOutlet

: Propagation: Quarter-wavelength tube: Cross-flow: Helmholtz resonator

123

Vibro-Acoustics Consortium

VAC Meeting May 15-16, 2014

Muffler Design

48

Example Helmholtz Resonator Included

Vibro-Acoustics Consortium

Summary

49

• Muffler fundamentals introduced.

• Recommend the use of plane wave simulation in the early design stages.

• Numerical simulation is useful for finalizing the geometry and details.

• Prototyping using 3D printing is recommended for an initial test.

Vibro-Acoustics Consortium

References• ASTM E2611-19, “Standard Test Method for Measurement of Normal Incidence

Sound Transmission of Acoustical Materials Based on the Transfer Matrix Method,” Philadelphia, 2019.

• D. A. Bies, C. H. Hansen, and C. Q. Howard, Engineering Noise Control, 5th Edition, CRC Press, 2018.

• H. Bodén, U. Carlsson, R. Glav, H-P Wallin, and M. Åbom, Sound and Vibration, The Marcus Wallenberg Laboratory, KTH, Stockholm, 2011.

• D. W. Herrin, X. Hua, Y. Zhang, and T. Elnady, “The Proper Use of Plane Wave Models for Muffler Design,” SAE International Journal of Passenger Cars –Mechanical Systems, Vol. 7, No. 3 (2014).

• M. L. Munjal, Acoustics of Ducts and Mufflers, 2nd Edition, Wiley-Interscience, United Kingdom, 2014.

50

Vibro-Acoustics Consortium 51

List of Variables

𝑓

𝑀

frequency

flow Mach number

𝜌 density of air𝑐 speed of sound in air

𝜔 angular frequency

𝜎 porosity𝑡 panel thickness 𝑑 hole diameter𝜐 air kinematic viscosity

𝑘 wave number

𝐶 end correction coefficient𝐽 Bessel function of the first kind of order 1𝑏 the distance between two neighboring holes in axial directionsℎ the distance between two neighboring holes in circumferential directions

If either 𝑚 or 𝑛 is equal to zero, 𝜀 1; otherwise, 𝜀 0.5; 𝜀

Vibro-Acoustics Consortium

Eq. 5 (Ji, 2010)

𝑍 1 𝜎⁄ 1 𝑡 𝑑 8𝑘𝜐 𝜌𝑐⁄⁄ 𝑗𝑘 𝑡 𝐶 𝑑

𝐶4𝜋

1𝜁𝜂 ⁄ 𝜀

𝐽 𝜋 𝑚𝜁 𝑛𝜂

𝑚 ℎ 𝑏⁄ 𝑛 𝑏 ℎ⁄⁄

𝜁 𝑑 𝑏⁄ ,     𝜂 𝑑 ℎ⁄

Eq. 4 (Rao and Munjal, 1986)

Empirical Equations Perforated Panel

52

𝑍 1 𝜎⁄ 7.337 10 1 72.23𝑀 𝑗2.2245 10 𝑓 1 51𝑡 1 204𝑑

Vibro-Acoustics Consortium53

Eq. 6 (Allam and Åbom, 2011)

the absolute value or the peak particle velocity inside the holes

𝑟 Re𝑗𝜔𝑡𝜎𝑐 1

2𝜅 𝑗

𝐽 𝜅 𝑗𝐽 𝜅 𝑗

2𝛽𝑅𝜎𝜌𝑐

𝑢𝜎𝑐

𝑥 Im𝑗𝜔𝑡𝜎𝑐 1

2𝜅 𝑗

𝐽 𝜅 𝑗𝐽 𝜅 𝑗

𝛿 𝜔 1 𝑢𝜎𝑐

𝜎𝑐

Surface Resistance 𝑅12 2𝜂𝜌𝜔β is equal to 2 for holes with rounded edges and 4

for holes with sharp edges

Dimensionless shear wave-number𝜅 𝑑 𝜔 4𝜐⁄

𝑍 𝑟 𝑗𝑥

End correction𝛿 0.85𝑑

Empirical Equations Perforated Panel