Research on experimental measurement of acoustic resistance and major accuracy influencing factors...

Journal of Mechanical Science and Technology 28 (4) (2014) 1219~1227

www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-014-0302-1

Research on experimental measurement of acoustic resistance and major accuracy

influencing factors analysis†

Wang Xiaoqing1, Xiang Yang1,* , Guo Zhiyong2, Xia Xuebao1, Shi Yuxiao1, Xue Peng1 and Wu Shaowei1

1School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, 430063, China 2CCCC Tianjin Port & Waterway Prospection & Design Research Institute CO.,LTD., Tianjin, 300461, China

(Manuscript Received July 5, 2013; Revised December 28, 2013; Accepted February 6, 2014)

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Abstract

An experimental method of measuring acoustic surface radiation resistance is developed. The principle of the method is based on ob-

taining source velocity and field pressure generated by the source. To measure surface radiation resistance, measuring probe was devel-

oped and measuring system was set up. Then, major factors that affect measurement accuracy is discussed and great improvements are

got. After that, experiments of measuring baffled circular piston were conducted to analyze its applicable frequency range. To verify

actual application effect, experiments of measuring the tube and cylinder heads of a diesel engine were performed. The results show that

this measuring system can obtain resistance values in the frequency range from 460 to 1900 Hz with high precision. The measuring sys-

tem has features of simple operation, convenient use, and high accuracy. Therefore, it can be used to determine surface resistance matrix

of various structures.

Keywords: Acoustic resistance; Resistance matrix; Lumped parameter model; Acoustic radiation

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1. Introduction

The essential method in noise control area is quiet structure

designing achieved by acoustic optimization design which

take into account of vibration and noise characteristic in de-

signing stage of product development. Thus, to assess the

effect of optimization method and compare advantages and

disadvantages of different designing methods, an objective

function is needed, and sound power radiated from the surface

of the structure is just in case.

Though the sound powder radiated from structures can be

calculated by numerical integration or by means of finite and

boundary element methods, these methods are usually time

consuming, especially those with complex surface geometries

[1, 2].

In lumped parameter model [3], the sound power radiated

from the surface of structure can be written in volume velocity

and resistance matrix. Thus, the noise of the structure can be

lower by optimization of the resistance matrix. In other words,

resistance matrix can be used to evaluate the noise level ra-

diated from structures.

The resistance matrix can be solved with acoustic finite

element method and acoustic boundary method, which are

suite to structure with simple boundary conditions. However,

the experimental method will not be affected by surface boun-

dary condition and it can measure structure with complex

surface shape. By measuring several typical acoustic struc-

tures, Koopmann [1, 4, 5] made the conclusion that the expe-

rimental method can give a precise value of the resistance in a

certain frequencies. Then, Arenas [6] do some research work

on the experimental method. However, the measured value

they measured has big error with analytical values. Thus,

based on previous research [7-9], this paper is to design a

measuring probe, find out its major affecting factors and then

use the probe in practical application.

2. Principle of resistance measurement

In lumped parameter model [1], the acoustic power output

of a vibrating of structure can be written in the resistance ma-

trix with the volume velocities of the surface elements. Each

of the surface elements was assumed to be vibrating as a pis-

ton. The equation of the power output was determined as:

,

T

a v

*1W = U U

2R (1)

where, 1 T

NU U U is the volume velocities of the sur-

*Corresponding author. Tel.: +86 27 50853170, Fax.: +86 27 86581997

E-mail address: [email protected] † Recommended by Associate Editor Cheolung Cheong

© KSME & Springer 2014

mailto:[email protected]

1220 Wang X. et al. / Journal of Mechanical Science and Technology 28 (4) (2014) 1219~1227

face elements,

11 1

1

N

N NN

R R

R

R R

is the acoustic surface

radiation resistance matrix.

The resistance matrix could be determined by numerical

method, however, due to its huge amount of preparation work

and time-consuming calculation, experimental method has

some advantage in determining the values of the resistance

matrix. The single term of resistance matrix is defined as:

, Re

m

a mn

n

p r

u r

R (2)

where mp r

is the sound pressure of an arbitrarily point m

on the structure surface, and nu r

is the volume velocity of

an arbitrarily point n on the structure surface. It is called

self-resistance when m= n and cross-resistance when m n≠ .

Thus, a small source with known volume velocity is needed to

measure radiation resistance, and it must meet two conditions

to simulate a point simple source. First, the size of the source

should be small enough, so that its influence on the shape can

be ignored when it is installed on the structure surface. Second,

it must be able to radiate a sound pressure that strong enough

to be measured in a certain frequency range, so that every

point on the structure can be measured. These two require-

ments are conflicting, because a source with smaller size

means it will has lower power output at low frequency.

In the research, due to flat diaphragm and moving like pis-

ton motion [10], a flat panel loudspeaker is used to simulate a

point simple source, and it was enclosed to guarantee its work-

ing as a point simple source, otherwise, it is more likely work-

ing as a point double source and its radiation output ability

will be very weak.

According to the acoustic theory, the sound pressure inside

the small volume cavity is determined by the harmonic oscil-

lation velocity of the cavity wall and it is defined as:

2

0 0 ˆjρ c

p uωV

(3)

where 0 is the density of the medium,

0c is sound velocity

in the medium, is the circular frequency ( 2 f ),V is the

cavity volume, and u is the harmonic oscillation velocity of

the cavity wall.

The volume velocity of the chamber wall can be calculated

from formula Eq. (3) if ignoring the influence of standing

wave in the cavity and given the condition that the volume of

the cavity is known and the response of the microphone mea-

suring sound pressure have no phase error. In fact, however, it

is difficult to measure the volume of the cavity accurately due

to the irregular shape of the loudspeaker. Besides, the phase

error of microphone response cannot be ignored [11]. There-

fore, it’s necessary to calibrate the probe before experiment.

The calibration is conducted with a cavity (called calibra-

tion cavity) whose volume is known fixed on the source cavity

and the microphone fixed on the top of the calibration is used

to measure the sound pressure in it. Schematic diagram of the

calibration is shown in Fig. 1(a). Since the air compression in

the source cavity is corresponding with the air expansion in

the calibration cavity, which means ˆ ˆs cu u . So the transfer

function cF , referred as sound pressure ratio between the two

cavities, is determined by:

2 2

0 0 0 0ˆ ˆ ˆ

=ˆ

/c c sc

s c s

p jρ c u jρ c u

p ωV ωF

V

s

c

V

V (4)

where cV and

sV is the volume of the calibration cavity and

source cavity, respectively. Thus,sV can be calculated by mea-

suring the sound pressure of the two cavities. So the formula

Eq. (3) can be written as:

2

0 0

ˆ ˆ .cs c

Vu j p F

c

(5)

The volume velocity of the arbitrary point k on the sur-

face of the structure can be calculated with

2

0 0

ˆ ˆ .ck s k c

Vu r j p r F

c

(6)

When the sound pressure of the point i on the surface of the

structure is measured, then the acoustic radiation resistance

between two arbitrary points can be calculated with

210 0

, .k i

a ik c

k c s k

p r p rρ cR F

u r jωV p r

R R (7)

Then the formula can be further simplified as:

2 21 10 0 0 0

, Im Imi

a ik c m c

c s k c

p rρ c ρ cR F F F

ωV p r ωV

(8)

where m i s kF p r p r

. As complex number, cF and

mF

can be expressed as:

1 0j ψ ψ jψ

c c cF A e A e

(9)

1 0j j

m m mF A e A e

(10)

where 1 0ψ ψ ψ and 1 0 . 1

cF and 0ψ are the

phase angle of the microphones in calibration cavity and

source cavity during calibration measurement. 1 and 0 are

the phase angle of the microphones at field point and source

cavity during experimental measurement. Thus, formula Eq.

(8) can be written with

Wang X. et al. / Journal of Mechanical Science and Technology 28 (4) (2014) 1219~1227 1221

2 2

-0 0 0 0, Im Im

jj ψm m

a ik j

c

ψ

c c c

ρ c A e ρ c Ae

ωV A e ωV A

R (11)

Then, it can be simplified as:

2 2

0 0 0 0, sin - sinm m

c

k

c

a i

c c

ρ c A A ρ cψ θ

ωV A A ωV R (12)

where -θ ψ .

Thus, the acoustic radiation resistance can be calculated

with formula Eq. (12) by first amend the phase error of the

microphones with 1

cF calculated by the calibration mea-

surement, then by measure the sound pressure of arbitrary

points on the surface of the structure and the pressure inside of

the source cavity with the designed probe.

3. Measuring system

A schematic of the probe designed is shown in Fig. 1 and

the measuring system is shown in Fig. 2, which including the

designed experimental device, NI 9263 Analog voltage output

module, NI 9234 dynamic signal acquisition modules, NI

cDAQ-9172 chassis, and the computer for data acquisition

and post-processing. The loudspeakers are flat panel type with

rated power output of 2 W and diameter of 57 mm and 50 mm.

The size of omnidirectional electret microphones, named

DGO9767CD, is 9.7 mm6.7 mm with sensitivity of -24B, -

30 dB, -32 dB. Data acquisition program which was pro-

gramed with LabVIEW can gather data automatically after the

setup of initial parameters. The data post-processing program

is programmed with MATLAB.

In each of the experiments, the input volume of loudspeaker

was kept constant; the loudspeaker was driven with sinusoidal

signal and the frequency range from 20 to 4000 Hz with fre-

quency increment of 20 Hz. The sampling rate is 25.6 KHz

and the sampling number is 40000. The acoustic radiation

resistance can be calculated with the actual experimental data

andcF calculated by the calibration measurement.

4. Major influencing factors

Besides the rated power and diameter of the loudspeaker,

the air tightness of the source cavity, the sensitivity of the

microphone and the matching of the source and field micro-

phones are also important factors that influencing the measur-

ing results. Thus, experiments of baffled circular piston self-

resistance were performed to analyze these factors.

To simulate measuring the radiation resistance of a piston in

an infinite baffled plate, the experimental device was placed

on the floor of the hemi-anechoic chamber with a distance of 2

m to its surrounding sound-absorbent material, and the field

microphone was placed directly above the center of the louds-

peaker diaphragm, as illustrated in Fig. 3.

Calling back Eq. (12), the radiation resistance can be calcu-

lated with the experimental and calibration data. To the con-

venience of comparison, the experiment solutions were

changed to a dimensionless quantity by multiplied by the area

of the speaker diaphragm iS and divided by the characteris-

tic impedance of the acoustic medium c . To the convenience

of explain the degree of the factors, loudspeaker with diameter

of 57 mm is used in the following experiments due to its sig-

nificant improvements.

By assuming the diaphragm of the loudspeaker moves as a

piston, the self-resistance ii of the baffled circular piston

can be calculated using Eq. (13) [12]

1

0

21iii

J krcS

kr

(13)

where 1J is the first order Bessel function and r is the radius

of the loudspeaker’s diagram.

(a) Schematic diagram of the calibration

(b) Schematic of the probe

Fig. 1. Resistance probe.

Fig. 2. Schematic of experimental set-up.

Fig. 3. Self-resistance measurement (baffled circular piston).


4.1 Selection and match of microphones

The two microphones used in the experiments have impor-

tant influence on the measuring results. To improve the signal

to noise ratio (SNR) at low frequencies, the sensitivity of the

field microphone should be as high as possible. In addition,

the sound volume of the loudspeaker should be increased as

much as possible. However, due to the nature of the louds-

peaker, the amplitude of the sound pressure radiated from it as

a function of frequency is shown in Fig. 4(a). It can be seen

that the microphone will give a distorted response if the vo-

lume of the loudspeaker too high. So the SNR should increase

under the condition that the response has no distortion.

It is known that the volume of calibration cavity is about

three times of that of source cavity. Recalling formula Eq. (4),

it can be seen that the frequency response of source micro-

phone is about three times of that of calibration microphone.

Thus, the source microphone will come to the point of distor-

tion first when the source microphone have same sensitivity.

Then, the sound volume should not turn up again. However,

the volume can be increased if the source microphone has

almost the same response with field microphone, which can be

achieved by adjusting the sensitivity of the source microphone.

The self-resistance of the baffled circular piston as a function

of frequency of the two cases can be seen in Figs. 4 and 5.

The comparison between Figs. 4 and 5 shows that the

measured results can be improved by increasing of SNR

achieved by turning up the volume of the loudspeaker under

the case that the two microphones have normal response.

4.2 The air tightness of the source cavity

In the experiment, the air tightness has a very significant in-

fluence on the measured results, which will improve a lot if

the source cavity has a good airtightness. The Blu Tack was

used to achieve a well seal in the source cavity due to its good

sealing effect, convenient installation, good reutilization and

price is low.

As is shown in Fig. 6, with the airtightness of the source

cavity enhanced, the measured results improved a lot com-

pared to Fig. 5.

(a) Amplitude of sound pressure

(b) Transfer function

(c) Self-resistance of a baffled circular piston as a function of frequen-

cy

Fig. 4. Microphones with same sensitivity: 0 2

24 dBS S (Loud-

speaker: D = 57 mm, P = 2W; Vol = 19/%max

Vol ).

(a) Amplitude of sound pressure(Vol. = 19/%max

Vol )



cy

Fig. 5. Microphones with match sensitivities: 0

30 dB,S 2

S

24 dB ; Vol. = 22/%max

Vol (Loudspeaker: D = 57 mm, P = 2W).


5. Applicable frequency range of the probe

To get the frequency range applied and its accuracy of the

probe, four experiments of baffled circular piston were con-

ducted. The first one was self-resistance measurement carried

out as mentioned above, and the results were shown in Fig.

8(a).

The other three ones were cross-resistance measurements.

Similar to the self-resistance measurements, the probe was

placed on the floor of the hemi-chamber, and the field point

was measured with the microphone at one, two and three ikd

(50 mm) from the center of the loudspeaker. The cross-

resistance in situ measurement of baffled circular piston is

show in Fig. 7. The measured cross-resistances are compared

with the analytical values in Figs. 8(b)-(d), in which the ana-

lytical solutions of the cross-resistance between the i th ele-

ment and the k th element were calculated with formula Eq.

(14), presented by Hashimoto (2001) [13]

21 i 1 k0 i k ik

ik

i k ik

J kr J kr2ρ ck S S sinkd=

π kr kr kd

(14)

where iS and kS are the areas of the i th and k th elements

respectively, /i ir S and /k kr S are the equiva-

lent radii of the i th and k th elements respectively, ikd is the

distance between the centers of the two circular vibrating pis-

(a) Amplitude of sound pressure (Vol = 19/%max

Vol )



cy

Fig. 6. Airtightness (Loudspeaker: D = 57 mm, P = 2 W; Microphone

030 dB,S

224 dB;S Vol. = 22/%

maxVol ).

Fig. 7. Cross-resistance measurement of a baffled circular piston.

(a) Self-resistance

(b) Cross-resistance (d = 50 mm)

(c) Cross-resistance (d = 100 mm)

(d) Cross-resistance(d = 150 mm)

Fig. 8. Radiation resistance of baffled circular piston as a function of

frequency.


ton elements.

The results shows that the probe can get resistance values

with higher precision in frequency between 460~1900 Hz, but

there are errors in low and high frequencies results from rea-

sons listed below:

(1) At frequencies below 460 Hz, The errors of the meas-

ured results is caused by lower SNR, which results from the

lower power radiated from the diaphragm of the loudspeaker.

(2) The radiation resistance measurement is limited by the

noise floor of this hemi-anechoic chamber, which is at approx-

imately 2/ 1 10S c R , so it can be seen that the experi-

mental results will not show good agreement with analytical

values below 0.01.

(3) At frequencies above 2000 Hz, the measuring probe

should not be viewed as a point simple source, because the

wavelength is shorter and the ratio of it to the size of the probe

is not big enough, and thus the probe is not correspondence

with the condition of the lumped parameter model. Besides,

the performance of the loudspeaker is also decreased.

6. Measuring examples

6.1 Tube

The tube with one end open and the other closed is taken as

an example to test the probe in measuring the resonance fre-

quencies, for only the circularly resonances can be excited in

the tube due to the circular symmetry of the boundary surface.

The tube is made of 2 mm polyvinyl chloride (PVC) with a

length of 85.7 cm and an inner diameter of 7.2 cm. In experi-

ments, the tube was placed on the floor of a hemi-anechoic

chamber and the gap between the tube and the floor was

sealed by BLU Tack. The measuring probe was placed at the

center of the tube, as illustrated in Fig. 9.

To verify accuracy of the measured results, it was compared

to numerical values which were calculated with Vir-

tual.Lab.Acoustics using BEM [14]. In the process of numeri-

cal calculation, the air is set as homogeneous medium with

density of 3

0 1.21 kg / mρ and the velocity of the sound

0 343 m / sc . The boundary condition of velocity is exerted

at the center of the tube’s bottom in a circular area with radius

46 mmr , 1 m / sv . The acoustic grid model, field point

and boundary condition of velocity is shown in Fig. 10. The

comparison between the measured results and numerical val-

ues of the radiation resistance as a function of frequency is

shown in Fig. 11, in which the frequency increment in both

case is 2 Hz.

As shown in Fig. 11, the measured results and numerical

values agree very well in heights and widths of the resonance,

as well as the depths of the valleys, which suggest that the

probe can measure the resistance of the tube in frequencies

with high-precision, especially in the frequencies range from

260 Hz to 1700 Hz. Besides, the depth of the valleys is deter-

mined by the airtight seal of the probe, so the quality of the

airtight seal is very high. Certainly, the measured results are

not very good at frequencies below 400 Hz or higher frequen-

cies up 1700 Hz, which has the same reason listed above.

During the period of numerical calculation, the scattering,

refraction and diffraction have been taken in account, so the

inaccuracy of the numerical values compared to real values

mainly come from errors in geometric modeling, material

property and boundary conditions defining and so on. In the

case, the shape of circular tube is simple, and the precision of

its model is high, so the numerical values are of high approx-

imation accuracy to its real values. Therefore, the conclusion

can be made that the probe can measure out resistance values

with higher precision in actual conditions.

6.2 Cylinder heads of a diesel engine

Take cylinder head of a diesel engine produced by Weichai

Power Company as an example. The diesel engine was placed

at the center of the hemi-anechoic chamber, and the number of

the cylinder head is 1, 2, 3, 4, 5 and 6 from left to right, as

illustrated in Fig. 12(a). As in the baffled circular piston ex-

Fig. 9. In-situ measurement of tube.

Fig. 10. Acoustic grid model of tube.

Fig. 11. Acoustic resistance of tube as a function of frequency.


ample, the probe was placed at the center of the cylinder heads

with field microphone on the holder above the diaphragm to

measure self-resistance of the cylinder head. Then the probe

was moved from number one to number 6 one by one to com-

plete self-resistance experiments of cylinder head. Similarly,

the cross-resistance was measured with the probe at the center

of the cylinder head and the field microphone placed at the

center of other cylinder head, as illustrated in Fig. 12(b).

There are usually two types of cylinder head used in diesel

engine, integral type and single type. To explain the difference

between the two types of cylinder head, resistance of them

will be compared in the experiments, in which the integral

cylinder head can be seen as baffled plate in a sense. Thus, the

resistance values of integral cylinder head can be replaced by

baffled circular piston. The self-resistance and cross-resistance

of the cylinder head are shown in Figs. 13 and 14, in which

the analytical values was resistance solutions of baffled circu-

lar piston.

The results in Fig. 13 show that the self-resistance of cy-

linder head decreases a lot compared to baffled circular piston,

which suggest that the single type cylinder head can lower the

noise power level radiated than integral ones. In other words,

the quiet structure designing have been achieved in a sense by

replace integral cylinder head to single ones.

At low frequencies, it can be find that the resistance has a

minor variation with the cylinder head changed, which sug-

gests that the resistance of the cylinder head will be affected

by its nearby boundary conditions. On the other hand, the

resistance changes very little at higher frequencies above 1200

Hz, which suggest that the boundary condition has little influ-

ence on the resistance. Thus, the conclusion can be get that

shape optimization is suitable for problems of structural noise

(a) No.1 cylinder head

(b) No.2 cylinder head

(c) No.3 cylinder head

(d) No.4 cylinder head

(e) No.5 cylinder head

(f) No.6 cylinder head

Fig. 13. Self-resistance of cylinder head as a function of frequency.

(a) Diesel engine

(b) In-situ measurement of cylinder head

Fig. 12. Cylinder heads.


control at low frequencies.

In the Fig. 14, the cross-resistance values of the cylinder

heads are smaller comparing to self-resistance at frequencies

above 1200 Hz. Only the cross-resistance between #1cylinder

and #2, #3 are above 210 at some frequencies, which sug-

gests that the cross-resistance have little contributions in the

sound power radiated from the diesel engine, especially the

cross-resistance between two cylinder head with far distance.

7. Conclusions

A method for measuring acoustic surface radiation resis-

tance has been presented. In this study, measuring probes was

designed and measuring system was constructed. By measur-

ing resistance of baffled circular piston, the influencing factors

were found and weakened. In this way, the probe can measure

out the resistance with higher precision in a certain frequen-

cies. Then, the probe was applied to actual problems of tube

and cylinder head. The results show that:

(1) This measuring probe can obtain resistance values with

higher precision in the frequency range from 460 to 1900 Hz.

(2) The match up of the source and field microphones, to-

gether with sensitivity of them is critical to the measured re-

sults.

(3) The airtightness of the source cavity is very important in

the experiments. A good airtightness of the source cavity will

improve the experimental results a lot.

(4) The measuring system has features of simple operation,

convenient use, and high accuracy.

Acknowledgement

This work was supported by the National Natural Science

Foundation of China (Grant No. 51079118, 51279148).

Reference

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(a) Cylinder head #1: #2

(b) Cylinder head #1: #2

(c) Cylinder head #1: #4

(d) Cylinder head #1: #5

(e) Cylinder head #1: #6

Fig. 14. Cross-resistance of cylinder head as a function of frequency.


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Wang Xiaoqing received his B.S. and

M.S. degrees in Marine Engineering

from Chongqing Jiaotong University

and Wuhan University of Technology,

China, in 2010 and 2013. He is currently

a doctoral student in The University of

Alabama. His research interest include

vibration and noise control, and electron

beam additive fabrication.

Xiang Yang, Ph.D., is currently a Pro-

fessor at the School of Energy and Pow-

er Engineering, Wuhan University of

Technology. Prof. Xiang’s research

interests include vibration and noise

control, optimal design, condition moni-

toring and fault diagnosis technology,

and signal processing technology.

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