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Departamento de
Engenharia Mecânica
Sound radiation and transmission
Professor Phil Joseph
SOUND RADIATION BY A PISTON
• The piston generates plane waves in the tube with particle velocity
equal to its own.
• The sound wave generated by the piston has pressure
amplitude
p = 0c0 v
where 0c0 is the acoustic impedance of air and v is the piston
velocity amplitude.
• The sound radiated by the piston can be quantified using sound
power. This is given by
where S is the surface area of the piston,
is the mean-square velocity.
We can assess the sound radiation of other sources by comparison
with this simple result.
2002
1 . vScSupWrad
2v
SOUND RADIATION BY A VIBRATING PISTON
A dipole is typical of an oscillating source – it is less efficient than a monopole
SOUND RADIATION BY A MONOPOLE SOURCE
Quadrupoles are less efficient than monopoles or dipoles.
SOUND RADIATION BY A QUADRUPOLE SOURCE
Quadrupoles are less efficient than monopoles or dipoles.
SOUND RADIATION BY A QUADRUPOLE SOURCE
RADIATION OF SOUND BY A VIBRATING
STRUCTURE
• Vibrations of most mechanical systems exhibit complex frequency-
dependent spatial distributions of amplitude and phase. But the sound
power can be written simply as
where S is the total surface area, 0c0 is the acoustic impedance of air,
is called the radiation ratio or radiation efficiency
L = 10 log10 is called the radiation index (dB re 1)
200 vScWrad
2v is the surface-averaged mean square normal velocity
(perpendicular to surface) (over time) (over space)
When = 1 a structure radiates as efficiently as the 1-D piston.
At low frequencies, << 1 (structure is small compared with acoustic
wavelength)
At high frequencies, 1 (structure is large compared with acoustic
wavelength)
NB > 1 is possible ( is not a true efficiency).
200 vScWrad
RADIATION OF SOUND BY A VIBRATING
STRUCTURE
EXAMPLE OF RADIATION RATIOS OF SIMPLE SOURCES
• Analytical solutions exist for a pulsating sphere or an oscillating sphere.
• These correspond to a monopole and a dipole source at low frequencies.
• Real structures can be represented by these simple models when they are small compared with the wavelength.
100 1000 10
-5
10 -4
10 -3
10 -2
10 -1
10 0
Frequency [Hz]
Transition frequency
a=/3
f = 1.1 kHz
Dipole: Oscillating
sphere f 4
Sphere radius 0.1 m
Monopole: Pulsating
sphere f 2
Ra
dia
tio
n r
atio
Structure large
compared with
Structure small
compared with
• e.g. An engine
0.60.350.2 m,
surface area Srad =
0.8 m2. Radius of
equivalent sphere
(S=4a2) is 0.25 m.
• 3a= f = c/ = c/3a
= 450 Hz
100 1000 10
-5
10 -4
10 -3
10 -2
10 -1
10 0
Frequency [Hz]
Transition frequency
a=/3
f = 450 Hz
Dipole: Oscillating
sphere f 4
Sphere radius 0.25 m
Ra
dia
tio
n r
atio
Structure large
compared with
Structure small
compared with
EXAMPLE OF RADIATION RATIOS OF SIMPLE SOURCES
MEASURED ‘RADIATION EFFICIENCY’ OF 24 DIESEL
ENGINES
• ‘Radiation efficiency’ obtained from measurements of engine block vibration and 1 m Lp. Similar to a dipole with roll-off around 400 Hz. Constant level of +8 dB: noise from sump and valve cover, effects of the test cell and near field effects.
100 1k 10k
-20
-10
0
10
20
Frequency, Hz
Radia
tion index, dB
mean of 14 engines
dipole a = 0.25 m
(+8 dB)
+/- standard deviation
SOUND RADIATION FROM AN OPEN PIPE
• At low frequencies (ka < 1) most sound is reflected back by an open
end and only a small amount of sound is radiated.
• At high frequencies (ka > 1) much more sound is radiated by the
open end.
SOUND RADIATION FROM AN OPEN PIPE
DISPERSIVE AND NON-DISPERSIVE WAVES
• Dispersive waves have a wavespeed that depends on frequency.
EFFECT OF BLADE BLENDING
• For thin structures we need
to consider the effect of
bending modes in plates or
beams.
• Acoustic waves in air have a
wavelength air = c0/f where
c0 is the constant wavespeed
(343 m/s).
• But bending vibrations in
plates and beams have
wavelengths B 1/f.
• There is therefore a
frequency at which their
wavelengths are equal, B =
air. This is called the critical
frequency.
0.01
0.1
1
10
Frequency Hz
Wave
len
gth
m
100 1k 10k
air
critical frequency
1 mm 2 mm
5 mm
10 mm
steel
CRITICAL FREQUENCY
The critical frequency is important for the radiation of sound from panels and the transmission of sound through partitions.
e.g. Steel plate 1 mm thick: h = 0.001, fc = (12.4/0.001) Hz = 12.4 kHz
Note that hfc is similar for steel, aluminium or glass (because cL is similar).
Material Young's
modulus
E, N/m2
Poisson's
ratio,
Density
m, kg/m3
hfc, m/s at
20C in air
Steel 2.0 x 1011 0.28 7800 12.4
Aluminium 7.4 x 1010 0.33 2770 11.9
Magnesium 4.5 x 1010 0.30 1800 13.5
Plastic 1.5 x 1010 0.35 1410 19.1
Glass 6.0 x 1010 0.24 2400 12.7
L
mc
hc
c
Eh
cf
8.1
)1(12
2
20
2/122
0
RADIATION FROM PLATES IN BENDING
• At low frequencies f << fc: << 1
• For an infinite plate with a free bending wave, the sound radiation is zero. This is because of acoustic short-circuiting between the radiation from maxima and minima of the vibration pattern.
• For a finite plate, the edges and corners result in a net radiation of sound:
+
+ +
+ +
baffle residual
cancellation
+ +
+
SOUND RADIATION BY A VIBRATING PLATE
• Below the critical frequency an evanescent field is created.
RADIATION FROM PLATES IN BENDING
• At high frequencies f > fc: 1
B
air
radiated sound
Sound is radiated in a direction at an angle to the plate normal:
air = B sin
At the critical frequency, sin = 1, i.e. sound is radiated parallel
to the plate.
SOUND RADIATION BY A VIBRATING PLATE
• Above the critical frequency sound is radiated to the far-field.
EXAMPLE OF RADIATION RATIOS OF THIN PLATES
31.5 63 125 250 500 1k 2k 4k 8k -40
-30
-20
-10
0
10
Frequency [Hz]
plate 2.5 mm x 1 m x 0.3 m
plate 5 mm x 1 m x 0.3 m
plate 10 mm x 1 m x 0.3 m
infinite plate 2.5 mm
10
lo
g
, d
B r
e 1
critical frequency decreases as h increases.
below the critical frequency acoustic short-circuiting reduces .
thick (or stiff) plates radiate more noise than thin ones for a
given vibration level.
Sound transmission
SOUND TRANSMISSION • We often use walls, barriers or partitions
to block out sound.
• Consider a sound field incident on a panel.
• Most of the sound is reflected, but due to motion of the panel, a small part is transmitted to the other side.
• The Sound Reduction Index, R (or transmission loss TL) is defined as
•
R = 10 log10 ( Ii / It ) [dB]
where Ii is the incident intensity and It is the transmitted intensity.
NB a large sound reduction index implies a good acoustic
performance.
Usually the sound reduction index is used for a diffuse incident sound
field (equal intensity from all directions).
Transmitted
Reflected
Incident V
TYPICAL SOUND REDUCTION INDEX CURVE
fc
NB ‘Mass law’ region applies over much of the frequency range of
interest for automotive structures.
‘MASS LAW’
For a plane wave normally incident on a panel of mass per unit area
in air:
where 0c0 is the characteristic specific impedance of air (= 415 rayls).
R increases at 20 dB/decade (6 dB/octave). For air, the above
expression reduces to:
R(0) 20 log10(f) 42
e.g. for 1 mm steel at 1 kHz, R(0) = 36 dB.
(between first panel resonance and critical frequency)
00
102
log20)0(c
R
For a plane wave normally incident on a panel of mass per unit area in air:
where 0c0 is the characteristic specific impedance of air (= 415 rayls). R increases at 20 dB/decade (6 dB/octave). For air, the above expression reduces to:
R(0) 20 log10(f) 42
e.g. for 1 mm steel at 1 kHz, R(0) = 36 dB.
For a diffuse field (plane waves assumed to propagate in all directions with equal probability), many measurements of real partitions suggest the following empirical formula applies in much of the frequency range:
Rd = R(0) 5
This approximation is sometimes called the ‘field incidence sound reduction index’. NB Rd cannot be less than 0. The above formula is only valid for R(0) >> 0.
(between first panel resonance and critical frequency)
00
102
log20)0(c
R
‘MASS LAW’
SUMMARY: SOUND REDUCTION THROUGH
PARITIONS
For practical automotive structures at most frequencies ‘mass law’ applies. So
• Adding stiffening has no effect over much of the frequency range (it can make matters worse by lowering the critical frequency).
• Adding damping to the structure is only effective at high frequencies.
• Doubling the mass gives only a 6 dB increase in transmission loss. Reducing vehicle mass will tend to worsen airborne noise transmission.
• Attention should be paid to weak points, close up gaps, etc.
DOUBLE SKIN PARTITIONS
• Double panels can give much increased transmission loss at high frequencies without significant increase in mass: R = R1 + R2.
21
212
000
)(8.1
2
1
d
cf
But only effective above:
where d is the gap width and i is mass per unit area
of panel i.
e.g. two 1 mm steel panels 1 cm apart are more effective than a
single 2 mm panel above about f0=400 Hz. Below this, they
behave like a single 2 mm partition. Also double windows.
Vehicle trim materials often have a heavy barrier layer
as well as foam layers to introduce this effect.
EFFECTS OF PLATE THICKNESS
negative effect of increased thickness
radiation efficiency is increased in the acoustic short-circuiting region.
overall effect also depends on vibration level.
positive effect of increased thickness
sound reduction index increased (mass law)
But not at all frequencies!
31.5 63 125 250 500 1k 2k 4k 8k -40
-30
-20
-10
0
10
Frequency [Hz]
10 log10
1 mm
2 mm
5 mm
1m x 1m steel plate
31.5 63 125 250 500 1k 2k 4k 8k 0
10
20
30
40
50
Frequency [Hz]
R dB
1 mm
2 mm
5 mm
EFFECTS OF STIFFERNING RIBS
Adding stiffeners to a plate increases its impedance. This is beneficial if the stiffeners are added where a force excitation acts (e.g. where a resilient mount is attached).
Adding stiffeners to a plate reduces the acoustic short-circuiting effect (lengthening the perimeter), hence increasing the radiation efficiency.
31.5 63 125 250 500 1k 2k 4k 8k -40
-30
-20
-10
0
10
Frequency [Hz]
1m x 1m x 1mm steel plate
no ribs
4 ribs
4x4 ribs
INSERTION LOSS
When a sound reducing measure of any kind (e.g. an enclosure, a barrier, a
resilient mount) is introduced, its effectiveness can be measured by its insertion loss. This is defined as the difference in sound pressure levels with and without the measure:
IL = Lp,without Lp,with [dB]
If a sound source is surrounded by an enclosure, two opposite effects occur:
The enclosure introduces attenuation, measured by TL.
A reverberant field builds up inside the enclosure, increasing the level of the incident sound field by some amount L.
Moreover for close fitting enclosures the sound power radiated by the source can be increased, also contributing to L.
Then the insertion loss is given by:
IL = TL L < TL
To maximize the effect of an enclosure, absorption should be added to reduce the reverberant field and hence L. If no absorption or damping is present, IL 0.
EXAMPLE OF ENGINE ENCLOSURE
