sound transmission loss of a panel backed by a small enclosure
TRANSCRIPT
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Sound Transmission Loss of a Panel Backed by a Small Enclosure
by
Karel Ruber, Sangarapillai Kanapathipillai and RobertRandall
reprinted from
Journal of LOW FREQUENCYNOISE, VIBRATION
AND ACTIVE CONTROL VOLUME 34 NUMBER 4 2015
MULTI-SCIENCE PUBLISHING COMPANY LTD.
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Sound Transmission Loss of a Panel Backed by
a Small Enclosure
Karel Ruber *, Sangarapillai Kanapathipillai and Robert Randall
School of Mechanical and Manufacturing Engineering, University of New South Wales,
Australia
First submitted: 11 June 2014/Revised: 20 April 2015, 10 August 2015 and 02
September 2015/Accepted: 02 September 2015
ABSTRACT
Sound transmission loss (STL) tests of acoustic insulation panels are commonly
performed in large reverberation rooms. Large size rooms are required by
acoustic standards to ensure that a large number of modes can be excited in thefrequency range of interest, to create a diffuse sound field. However for STL
measurements in low frequency range small enclosures should be able to
provide adequate homogenous sound fields, namely ̀ pressure sound fields’. The
expected effect of the air sealed in an enclosure backing a panel, is to increase
the stiffness of the panel artificially raising the first natural frequency of the
panel, which corresponds to a minimum value in the STL spectrum. In this
paper the influence of the air cavity’s added stiffness on the panel STL is
investigated in detail. As expected the effect of the sealed air is to increase the
plate stiffness and as a result to increase the frequency of its first natural mode,
however the effect on the STL in this frequency region is unexpectedly
insignificant which removes the need for correcting STL measurements using
small enclosures in low frequency range-around their first natural frequency of the panels.
Keywords: Sound transmission loss, insertion loss, natural frequency
1. INTRODUCTION
The effect on the vibration and the STL of flexible panels backed by air enclosures
has been investigated extensively by many researchers. Dowell and Voss [1]
established that the first panel natural frequency increases as the size of the
enclosure is reduced due to the stiffening effect of the air in the enclosure. The
second panel mode is affected by the mass loading of the air which reduces its
natural frequency. Lyon [2] identifies three frequency ranges for the Noise
Reduction (NR) of panels where the thickness and material of the panel and the boxdepth is such as the panel’s first natural frequency is below the first natural
frequency of the air cavity. In the lowest frequency range where the panel size is
much smaller than the excitation wave length, the sound pressure is considered
uniform regardless of the incident angle and the air in the box is modelled as a
simple spring adding its stiffness to the bending stiffness of the panel. His model is
based on equating the volume displacement of the air cavity and the panel. The
result is a NR independent of the frequency of excitation in this frequency range.
For the middle frequency range dominated by the first panel resonance and a several
higher panel modes a formula is developed for calculating the envelope of the
minimum NR.
For shallow cavities backing relatively flexible panels Pretlove [3, 4] concludes
that calculations of the first mode natural frequency and modal shape are inaccuratewithout including the coupling effect of higher panel modes. For cases where the
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Vol. 34 No. 4 2015
*Corresponding author e-mail: [email protected] (K Ruber)
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Sound Transmission Loss of a Panel Backed by a Small Enclosure
stiffness of the air is lower than the stiffness of the panel only the first panel mode
needs to be included. The noise reduction spectrum has a dip (the first minimum) at
the natural frequency of the panel stiffened by the enclosure air support, which is
higher than the natural frequency of the panel in vacuo (216.4 Hz vs. 209.3 Hz
respectively for the structural example that was used). In all calculations the
radiation impedance of the external fluid is assumed to be negligible.
Oldham and Hillarby [5, 6] use similar rationale regarding the uniformity of the
pressure field in the enclosure at low frequencies and the adequacy of using only the
first panel mode for calculating the Insertion Loss (IL) of a panel backed by an
enclosure. Their research also considers the effect of vibrating panel sound sources.
The sound radiation of several panel modes was simulated with an array of speakers.
The results show that the depth of the enclosure changes the frequency of the first
dip of the IL similarly to the changes in the NR described in previous researches.
For their case studies the estimated IL is negative at the first panel resonance
(modified by the cavity stiffness) which can be explained by the increase in the
sound power output at resonance due to a “negative loading” of the sound source
which was experimentally confirmed measuring the cone velocity and the current
supplied to the speakers. It is worth noting that unlike the IL the STL cannot become
negative at any frequency for a passive system because the transmitted sound powercannot be larger than the incident sound power for the same system. Lau and
Tang [7] visualised the sound field inside the enclosure for a panel with various edge
flexibilities at frequencies above one tenth of the uncoupled first natural frequency
of the enclosure. Below this frequency the sound fields inside the enclosure are
uniform regardless of the edge flexibility and the degree of structural-acoustic
coupling.
Xin et al. [8–15] have investigated extensively the effects of various factors on
the STL of double panel structures separated by an air cavity. In the papers which
investigate the effect of the cavity depth on the STL [8, 11], the results show that the
frequency of the STL dip at the first panel resonance is not affected by the depth of
the cavity (the distance between the panels) for double panels with clamped or
simply supported edges, while the frequency of the dip caused by thepanel–cavity–panel resonance is significantly affected by the distance between
the panels. Although their double panel model has an air enclosure the behaviour of
the air backed by a flexible panel is different to the case when the second panel is
rigid as in our model.
A step is taken in this paper to explore the influence of the air cavity’s added
stiffness on the panel STL. The feasibility of low frequency STL measurements of
panels with one of the reverberation rooms replaced by a small enclosure is
investigated.
2. SOUND TRANSMISSION LOSS AND OTHER SOUND INSULATION
MEASUREMENTS
Airborne sound insulation performance of acoustic partitions is frequently
quantified by the sound transmission loss (STL or TL) defined as the ratio of the
incident sound power to the sound power transmitted by the acoustic partition. The
STL is the most common laboratory measurement of the sound attenuation
capabilities of structural partitions such as windows, doors and walls. International
and national standards describe methods of measuring the STL of partitions between
rooms or for facades [16–21].
In practice the STL of a partition is usually measured from the spatially and time
averaged SPL differences in two reverberation rooms separated by the panel and
correcting for the absorption in the receiver side rooms [18]. The standard requires
that volume of the rooms must be at least 50 m3 each. Large reverberation rooms
for testing STL are expensive and becoming more difficult to access.
An alternative method for measuring the STL of panels, requiring only one
reverberation room for the noise source side, is achieved by measuring the soundpower radiated by the panel in the receiver room with a sound intensity (SI) probe
[19]. The receiver room can be a regular room with moderate to high sound
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Karel Ruber, Sangarapillai Kanapathipillai and Robert Randall
Vol. 34 No. 4 2015
absorption. SI measurement systems are significantly cheaper than having an
additional reverberation room however they are still relatively expensive compared
to a regular sound level meter and the STL measurement requires adequate technical
skills to obtain valid results.
The sound transmission loss is defined as the power ratio of the incident to
transmitted sound expressed in decibels (dB) [22]:
(1)
where, W iis the incident sound power and W
t is the transmitted sound power.
In a free field the power based STL can be expressed in terms of a ratio of
squared pressures:
(2)
Piis the incident sound pressure which is the difference between the total pressure
on the noise source side of the panel (Ps) and the reflected sound pressure from thepanel on that side (P
r) which is difficult to measure.
Other methods to quantify the insulation of performance of partitions and
enclosures are the sound level difference also called noise reduction and the
insertion loss. The sound level difference and NR ignore the effect of the reflected
sound from the partition is generally used as a first estimate of the STL of partitions
in situ conditions (low accuracy).
The insertion loss is more commonly used for evaluating sound reduction of
enclosures is calculated from the difference between the sound pressure levels at the
same location, with and without the acoustic element in place
(3)
where, SPLwo
is the SPL without the acoustic element and SPLw
is the SPL with the
acoustic element.
It can be shown that the STL is virtually the same as the IL in free fields when
the impedance of the noise source is large enough so its power output is not affected
by the load [23]. When the acoustic element is in place the sound pressure on the
receiving side Pw
is the same as Pt
in Equation (2) and Pwo
is approximately the
same as Piwithout the acoustic element in place, therefore under this condition the
IL and STL become the same. Some correction for different distances to the sound
source may be needed for free field test conditions to make Piand P
woidentical. For
measurements in reverberant fields the effect of the second reverberant room needs
to be considered.
3. STL AND IMPEDANCE
The spectrum of the STL of a typical structural element, varies considerably over
frequency ranges [22]. At the lower end of the audio frequency range the STL is
controlled mainly by the panel stiffness while at higher frequencies it is the mass of
the panel which controls the STL level. Between these frequency regions the first
panel resonance reduces the STL to a level which is mainly controlled by the
structural damping of the panel. It is useful to express the effect of the stiffness,
mass and damping of a panel in a frequency range as impedances.
The effect of the panel impedance on the STL is expressed by the following
alternative formula where the panel STL is calculated from the panel acoustic
separation impedance Z s
[24]
(4)
= − =
IL SPL SPL log P
P20wo w
wo
w
=
=
R log
P
Plog
P
P10 20i
t
i
t
2
2
=
R log
W
W 10 i
t
ρ = + R log
Z
c10 1
2
s
2
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(5)
where, Ps
is the sum of incident and reflected waves and this formula avoids the
need to separate Ps
in its components. The units of the acoustic separation
impedance are the same as the units of the acoustic specific impedance, namelyN × s/m3.
It is important to notice that the STL (and IL) have minima at the first panel
resonance (the fundamental mode) where the panel impedance is also minimal. In
addition the fundamental plate mode is the most efficient radiation mode in the low
frequency range (below the critical frequency) [25].
3.1. Natural frequencies, mode shapes and impedances of simply supported
plates
For a simply supported panel in vacuo the natural frequencies are given by the
following formula [26]:
(6)
where, m is mass of the panel per unit area, r1
and r2
are the modal indices of the rth
mode and D is the bending stiffness:
(7)
where, E is Young Modulus, h is the thickness of the panel and ν is Poisson’s ratio.A panel vibrating close to its first natural frequency has a mode shape given by:
(8)
where, x, y are in plane Cartesian coordinates and r = 1 for the first natural frequencyof the panel.
The transverse panel displacement ξ in z direction at point 2 ( x2, y
2) on the panel,
due to a harmonic transverse force F z
at position 1 ( x1, y
1) at any frequency ω , can
be calculated:
(9)
The transverse velocity u at any point on the plate ( x2, y
2) is obtained by
differentiating the displacement ξ with respect to time, which is equivalent to a
multiplication by j ω in frequency domain.
(10)
The point mobility Y of the glass panel, for a point force can be derived from the
above formula by dividing the velocity by the force F z.
(11)
The driving point mobility at the centre of the panel ( xc, y
c) is then given by:
(12)
∑ξ ω ϕ ϕ
ω η ω ω ( )
( ) ( )( )
( )=
+ −
∞
= x y x y x y
M j F x y, ,
, ,
1, ,
r r
r r
zr2 2
2 2 1 1
2 21 11
ϕ ( ) =
x y sin π x
L
sin π y
L
, 2r
x y
ν ( )=
− D
Eh
12 1
3
2
ω
π π
=
+
mr π
l
r π
l
Dr
x y
1
2
2
2
= −
Z P P
us
s t
panel
∑ω ω ξ ω ω ϕ ϕ
ω η ω ( ) ( )
( ) ( )( )
( )= =
+ −
∞
=u x y j x y j x y x y
M j F x y, , , ,
, ,
1
,r r
r r
zr2 2 2 2
2 2 1 1
2 2 2 21
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Hence the driving point impedance at the centre of the panel ( xc, y
c) can be
calculated from:
(13)
Inserting the material properties for silica glass given in Table 1 into equations
(6)-(13) for a 3 mm thick (h), by 0.6 m wide (lx) and 0.33 m long (ly) glass panel
the natural frequencies listed in Table 2 and the impedance spectrum in Figure 1 are
obtained.
The minimum value of the plate impedance is at the resonance frequency of
95.1 Hz which corresponds to the first natural frequency calculated by equation (6).
Our current research is interest is focused on the first natural frequency of the glass
panel at 95.1 Hz.
ω ω
( )( )
= Z x yY x y
, , 1
, ,Plate c c
c c
1
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Table 1.
Silica glass mechanical properties (from Comsol)
Density ρ 2203 Kg/m3
Young’s Modulus E 73.1 GPa
Poisson’s ratio ν 0.17
Loss factor η 0.01
Table 2.
First 3 natural modes of the glass panel.
r 1 r 2 f [ Hz]
1 1 95.1
2 1 161.4
3 1 271.8
Figure 1. Driving point impedance magnitude and phase of the simply supported glass panel.
Impedance magnitude
Impedance phase
60 70 80
100
50
0
−50
−100
90 100
Frequency (Hz)
110 120 130 140
60 70 80 90 100
Frequency (Hz)
110 120 130 140
M a g n i t u d e ( N ∗ s e c / m )
P h a s e ( d e g )
103
102
10
1
100
Zplate at the center
Zplate at the center
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When a harmonic uniform pressure P z
is impinging on the plate the normal
displacements of the plate are given by Fahy [6]:
(14)
The modal force F r
is:
(15)
For our plate dimensions and P z =1 Pa, F
ris calculated as 0.16 N.
3.2. Natural frequency and impedance of a plate with air support
The effect of the air in the sealed box behind the panel is to create an elastic support
increasing the stiffness of the panel. The volume stiffness of sealed air cavity of volume
V is defined as the pressure increase P required to reduce the volume by ∆V [2]:
(16)
where, ρ is the air density and c is the speed of sound. For our plate backed by a0.25 m deep air enclosure, k
air= 2,875,864 [N/m5]. The volume stiffness of a simply
supported panel is given by [2, 24]:
(17)
k _plate = P/∆V = 1000 D/( A^3 F (α )) (17)
where A is the panel area and α is the aspect ratio. The function F (α ) is presentedas a graph in the reference however the volume stiffness can be also calculated from
the expression of static deflection under a static uniform pressure P [27].
(18)
(19)
(20)
The calculations above indicate that the sealed air in the box backing the plate
increases the plate stiffness by 15% and therefore the panel’s first natural frequency
will increase by a factor equal to the square root of the stiffness increase. In our case
the natural frequency is expected to be 102 Hz.
On the other side of the panel the surrounding air impedance has inertial and
damping characteristics. A decrease in the first natural frequency will be expected
because of the mass loading of the plate by the air, as described in Fahy [28] for a
baffled circular piston.
(21)
However this formula applies to a disc of radius “a” and because only half of
the plate is loaded by the air and the plate movement is not rigid, an approximatevalue was calculated as 92.9 Hz. The equivalent radius of the plate was
approximated to be half the diagonal dimension of the plate. Below the first
=∆
=
k P
V
18, 973, 285 N / m plate5
∫ ∫ ξ )(∆ =V x y dx dy,lx ly
0,0
,
( )=∆
=α
k P
V
D
A F
1000 plate 3
ρ =∆
=k PV
cV
air
2
ρ =m a8 / 3air3
∫ ∫ ϕ ( ) ( )=F P x y x y dydx, ,rlx ly
z r
0 0
∑ξ ω ϕ
ω η ω
( )( )
( )=
+ −
∞
= x y x y
M j F , ,
,
1
r
r r
rr2 2
1
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acoustic resonance of the enclosure the acoustic field inside the enclosure is
expected to be uniform [4, 5]. In this frequency range the real part of the
radiation impedance is low and its loading effect on the panel can be ignored.
When the mass loading effect is added together with the spring effect of the air
volume, to the simply supported model of the panel the first natural frequency iscalculated as 99.7 Hz. As expected in this case the impedance of the panel has a
minimum value at the modified natural frequency of the panel. The impedances
for both cases are presented in Figure 2.
4. EXPERIMENTAL TEST SYSTEM FOR MEASURING STL AND IL
In the proposed method for measuring the STL of small panels at low frequency
range, the panel forms the lid of a rigid enclosure consisting of walls with very high
STL compared to the panel under test. This construction ensures that the transmitted
sound through the panel is the only significant transmission path from the source.
A test system consisting of a glass panel 3 mm thick (h), by 610mm long and
340 mm wide was attached to an open box of 600 mm (l x) × 330 mm (l
y) internal
dimensions and a depth of 250 mm. The box was built from 24 mm thick plywood
panels to provide suitable acoustic insulation and an enclosed speaker was installed
in one of its bottom corners (Figure 3). The lowest acoustic natural frequency of the
box defined by its largest wall length is 285.8 Hz, which is well above the first
natural frequency of the simply supported panel.
To approximate theoretical simply supported boundary conditions, 1 mm thick
aluminium strips, were fixed to the inner edges of the box and also on the clamping
frame. The strips were fixed to protrude above the surfaces of the plywood and
provide the only contact between the glass and the test box. This configuration
allows panel edges rotations while constraining all edge translations [29]. The
clamping force was adjusted by turning the screws of ten adjustable clamps.
The effect of varying the clamping force was tested and observed to change the
frequency of the minimum STL in a range of 90 Hz to 110 Hz. The frequency range
corresponds to a change in boundary conditions between simply supported and fixedconditions [10]. Theoretically the increase in the clamping force should not change
the boundary condition but in reality small misalignments between the supporting
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555
Figure 2. Driving point impedance of the glass panel.
Impedance magnitude
60 70 80 90 100 110 120 130 140
M a g n i t u d e ( N
∗ s e c / m )
104
103
102
101
Zplate on air boxZplate
Zplate on air boxZplate
Impedance phase
100
50
0
−50
−10060 70 80 90 100
Frequency (Hz)
Frequency (Hz)
110 120 130 140
P h a s
e ( d e g )
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strips will constrain rotations (as well as transverse displacements) at the paneledges and this effect increases with the clamping force. In practical applications the
boundary conditions are somewhere between simply supported and fixed
conditions.
The other source of error in the experimental boundary condition compared to
the theoretical simply supported panel is that the size of the experimental panel
(610 mm by 340 mm) is slightly larger than that modelled between the supports (by
5 mm on all sides).
The outside and inside SPL were measured with two B&K 4189-L (Type 1)
microphones and a B&K Pulse analyser while the output channel analyser produced
a white noise band limited from 20 to 200 Hz. The outside acoustic field (in a large
room) approaches a free field and the transmitted (receiver) sound was measured
200 mm above the centre of the glass plate. The inside sound was measured close
to the opposite bottom corner of the speaker location [30]. All measurements where
performed with a frequency resolution of 0.5 Hz using a Hanning window.
5. FINITE ELEMENT ANALYSIS
5.1. Modal analysis of a plate with and without air support
A model of the plate was created and analysed in Comsol FEA using shell elements
and eigenvalue analysis. It should be noted here that simple structure such as plates
can be also effectively modelled with the finite difference method [31]. The FEA
model of the test system for STL measurement consists of the air domain inside the
box enclosed by rigid acoustic walls boundary conditions on all sides, except on the
side of the panel. The panel is constrained on all edges with pinned (simply
supported) boundary conditions.
The air on the outside is modelled by a hemispherical domain with a sphericalradiation condition on its outer boundary which can simulate an anechoic or infinite
acoustic field, as long as the acoustic waves from the sound source approach a
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Figure 3. Sound transmission loss test box.
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spherical wave pattern close to the outer boundary (Figure 4). The glass panel was
meshed with 434 triangular shell elements and the air domain was meshed with
10721 tetrahedral elements with a maximum size of 150 mm which provided more
than the normally required 7 elements per wavelength up to a frequency of 320 Hz.
For the frequency analysis a sound power point source was located in one of the
corners of the box simulating a corner speaker. A perfectly matched layer (PML)
was added to the bottom of the box to simulate the anechoic termination of the open
box case study.
The results for the FEA analysis for the simply supported plate in vacuo indicate
that its first natural frequency is 94.9 Hz which is close to the theoretical calculated
value of 95.1 Hz (Figure 5). The effect of the material damping has no significant
effect on the natural frequency of the panel.
In the frequency range around first panel resonance for the case the sound source
is inside the box, the sound transmitted through the panel is mainly radiated from an
area in the centre of the panel and the wave propagation pattern is indeed close to
spherical (Figures 6 and 7).The modal analysis results are close to the theoretical results for both cases
(Table 3).
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0
0.5
–0.4
–0.2
0
0.2
zy
x
Figure 4. FEA model of the simply supported panel (blue mesh), the air in the box and the free field above the panel
(the hemisphere).
y
z
x y
z
x
Eigenfrequency = 94.919884Surface: total displacement (m)
(b)
Eigenfrequency = 94.92107 + 0.474593iSurface: total displacement (m)
(a)
Figure 5. First natural frequency and mode shape of the simply supported glass panel in vacuo, with and without
material damping.
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z y
x
Eigenfrequency = 99.809997 + 0.751575i
Surface: Velocity, x components (m/ s)
Isosurface: Sound pressure level (dB)
(a)
z y
x
Eigen frequency = 93.101133 + 3.910204Surface: Velocity, x components (m/ s)
Isosurface: Sound pressure level (dB)
(b)
Figure 6. Modal analysis of the simply supported plate with surrounding air inside the box and free field outside. Only
half model is shown. On the left the bottom of the box is open and the natural frequency of the
plate is lower than on the right where the bottom of the box is closed.
freq(1) = 70 Hz Surface: velocity, x component (m/ s)
Isosurface: sound pressure level (dB) arrow volume: + intensity (RMS)
121
121
117
–10
–20
–0.02
113
109
104
100
96.1
92
87.9
83.7
83.7
z y
x
2×10 –20
×10 –3
(a)
freq(1) = 70 Hz Surface: velocity, x component (m/ s)
Isosurface: sound pressure level (dB) arrow volume: + intensity (RMS)
z y
x
129
129
12512
10
8
6
4
2
0
–1.17×10 –21
121
117
113
108
104
100
96
91.8
91.8
1.4×10 –3
×10 –4
(b)
Figure 7. Equal pressure surfaces of SPL, plate velocity contours and sound intensity arrows with a sound power point
source excitation in the bottom, front, right side corner of box, at 70 Hz.
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The results confirm the stiffening effect of the air cavity for the first panel mode.
This however is not true for the second panel mode which is not a volume
displacement mode and the air inside the cavity is pushed from side to side
presenting a mass like (inertial) impedance and as a result lowering the second
panel natural frequency. Both type of behaviours are presented by Fahy and
Gardonio [32].
5.2. FEA frequency response–plate impedance and STL
The basic outputs from the FEA program are the acoustic pressure and velocity
fields which are calculated from the geometry, material properties and excitation
sources of the model. The frequency analysis can be performed by directly solving
the motion equations/matrices or by modal superposition which is faster. The modal
superposition was found to be not working properly therefore the direct method was
used.
It was found that the power output of the sound power point source was varying
considerably over the frequency range and the direct method to calculate the STL
based on equation (1) gave negative STL for the closed box case study, therefore this
method was not used and the alternative equation based on the separation
impedance presented earlier in the paper was used instead.
The separation impedance can be evaluated at each point on the plate while the
STL is a property of the whole panel. On the edges of the plate the velocity is zerotherefore the local separation impedance is infinite. Averaging the separation
impedance over the plate to obtain an average STL for the plate will result in an
infinite value. To avoid this problem the local sound transmission coefficient of
the plate τ i
is calculated first for each point (i) on the plate using the following
equation [24].
(22)
Because τ i is not constant over the panel the average (composite) soundtransmission coefficient of the plate τ average is obtained by integrating the values
over the panel and dividing the result to its area as shown in [22].
(23)
The averaged sound transmission coefficient of the plate τ average is then used forcalculating the STL of the plate.
(24)
The results show the expected shapes of the STL curves for both cases with a
minimum at the first panel resonance for the case the box is opened at the bottom
(Figure 8).However contrary to the expected changes in the STL, when the box is closed the
minimum STL does not occur at the natural frequency of the plate stiffened by the
ρ τ = +
c
Z 1
2i
s
2
∑∑ ∫∫
τ τ
τ = =S
S S dS
1i i
i total
average
τ =
R log101
10
average
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559
Table 3.
Analytical and FEA natural frequencies of the panel backed by an open or closed box
Plate with air mass Plate with air mass loading and
Analysis/Calculation loading (Open Box) air stiffness (Closed Box)
Analytical 92.9 Hz 99.7 Hz
FEA 93.1 Hz 99.8 Hz
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air backing. Instead it occurs at the same natural frequency of the plate in the open
box. A similar behaviour was observed by Xin et al [8], when their calculated STL
of a double panel structure was found to be insensitive to the size of the air cavity
(depth) between them. Inspection of the total sound transmission coefficient of the
plate and the separation impedance curves shows that there is no significant
difference for the open and closed boxes (Figure 9).
Examining the separation impedance for the open and closed box shows them to
be approximately the same (Figure 10).
To further investigate the reasons that the frequency of the minimum STL in the
closed box case, is not the same as the natural frequency of the plate with the air
stiffening effect (100 Hz), the components of the separation impedance were
examined (Figure 11 and Figure 12).
Figure 11 shows that the inside pressure for the closed bottom box has an
unexpected shape – a minimum at the open box natural frequency of the panel
followed by a maximum at the natural frequency of panel stiffened by the air
support. Figure 12 shows that the average panel velocity magnitude has the
expected behaviour of maxima at the panel resonances for the open and closed box
case studies.
As a result of the particular shape of the inside pressure spectrum (Figure 11) forthe closed box, the maximum value (at approx. 100 Hz) is cancelled by the maxima
in both the outside pressure and velocity curves which also occur at the same
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Figure 8. STL calculation of the glass plate from the total sound transmission coefficient of the plate for open and
closed bottom box case studies. The curves are almost identical.
Open bottom boxClosed box
STL from averaged τ (10*log10(1/ τ_avg))
2
3
4
5
6
7
8
9
10
11
12
13
14
S T L ( d B )
70 75 80 85 90 95 100 105 110 115
Freq (Hz)
Freq (Hz)
Average sound transmission coefficient (τ) of the glass panel
70
τ a
v e r a g e
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.10.05
75 80 85 90 95 100 105 110 115
Open bottom boxClosed box
Figure 9. Average sound transmission coefficient of the glass panel τ average for the open and closed bottom cases.
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561
Figure 10. The separation impedance of the glass panel for the open and closed bottom cases.
Figure 11. Average pressure magnitude on each side of panel for open bottom and closed box cases. The pressure on
the inside the box side of the panel in the closed box case has a minimum at the same frequency
and similar magnitude as in the open box case.
Figure 12. Averaged panel velocity for the first mode (the middle of the plate).
Z_int - Averaged separation impedance - magnitude
Open bottom boxClosed box
Freq (Hz)
70
2000
1000
500
200
100
75
S e p a r a t i o n i m p e d a n c e ( N ⋅ s / m 3 )
80 85 90 95 100 105 110 115
70 75
S o u n d p r e s s u r e l e v e l ( d B )
80
Sound pressure levels on the plate -Inside andoutside the box -Box closed and open
85 90 95
Freq (Hz)
100 105 110 1159095
100105110
115120125130135140145150155
160165170
Open bottom box -Outside sound pressure
Open bottom box -Inside sound pressure
Closed box -Outside sound pressure
Closed box -Inside sound pressure
Open bottom boxClosed box
0.01
0.02
0.05
0.1
0.2
0.5
A v e r a g e d p a n e l v e l o c i t y ( m / s )
70 75 80 85 90 95 100 105 110 115
Freq (Hz)
Averaged panel velocity-magnitude, for the open and closed box
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frequency. Hence the separation impedance does not have a minimum at the
natural frequency of the air stiffened panel but instead its minimum is close to the
natural frequency of the panel without the air support. This results in almost
identical STL curves for the closed and open box cases, as shown before (Figure 8).
To validate the FEA results with experimental results the panel driving point
impedance at its centre point has been also simulated in FEA and experimental
results were obtained with an impact hammer (see section 6.2). A harmonic force
of 1 [N] was applied to the middle of the plate of the FEA model and the simulation
calculated the resulting panel velocity at that point (Figure 13).
The driving point minimum impedance frequency corresponds to the natural
frequency of the panel for both the open and closed box cases respectively, asexpected, unlike the separation impedance (Figure 10).
6. EXPERIMENTAL RESULTS
6.1. SPL measurements
The closed box case was tested while the open box case was not considered given
that opening the bottom of the box would allow the noise from the sound source side
to bypass the glass panel. Comparing the inside and outside sound pressure spectra
with the FEA results, it is evident that the shapes are similar (Figure 14). In
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JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL562
102
101
10072 76 80 84 88 92 96
Freq (Hz)
Driving point impedance of the panel
D r i v i n g p o i n t i m p e d a n
c e ( N ∗ s / m )
100 104 108 112
Open bottom box
Closed box
Figure 13. Driving point impedance for the middle of the glass plate for the open bottom box and the closed box. The
closed box impedance is minimal at the natural frequency of the panel with the air support
backing.
Figure 14. Closed box SPL measurements: upper curve - inside SPL measured near the bottom corner of the box; lower
curve - outside SPL 20 cm above the middle of the panel.
110
100
70
80
90
60
40
50
300 20 40 60 80 100 120
(Hz)
( d B / 2 0 . 0
J P a )
140 160 180 200
Inside SPL
Outside SPL
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particular, the sound pressure measured inside the box has a similar shape, aminimum followed by a maximum – as the FEA inside pressure results.
When the outside sound pressure levels are subtracted from the inside sound
pressure levels the NR has a minimum at the open box natural frequency of the
panel (Figure 15).
Variation of as much as +/− 5 Hz in the location of the peaks in the measured SPLand panel velocity were observed between some tests with identical settings, which
were most likely caused by slight variability in the panel location in the fixture as
well as the clamping force variability
6.2. Glass panel driving point impedance from hammer test
The driving point impedance was measured with a modal hammer and an
accelerometer for the closed box and the results were compared to the Comsol
results, see Figure 16.
Some of the differences between the impedances can be attributed to the real
panel edge constraints and damping being different from the theoretical values used
in the FEA and the analytical calculations.
6.3. Lumped element model of the glass panel and air stiffness- point force
impedance
In order to investigate the behaviour of the inside pressure observed in Figure 11 and
Figure 14 simple equivalent lumped mass, stiffness and damping model simulating
the first panel resonance was considered.
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563
Figure 15. Sound level differences between inside and outside microphones.
25.0
20.0
15.0
10.0
5.0
0.070 75 80 85 90 95
0.8
2.8
100
Frequency (Hz)
Measured SPL and ILs
S P L a n d I L ( d B )
105 110 115 120
Mics close to each other
Open lid
3 mm Glass panel (lid) or
3 mm Glass insertion loss
Figure 16. Measured and FEA simulated panel driving point impedances for the open bottom box.
80.0 90.0 100.0 110.0 120.0
Driving impedance at the centre of the glass panel
I m p e d a n c e m a g n i t u d e [ N * s / m ]
Frequency (Hz)
100.0
10.0
1.0
Measured impedance
FEA(comsol) calculated
impedance
Calculated impedance(using matlab)
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The effect of the sealed box was simulated by adding the “air spring” between
the excitation force and the glass panel (Figure 17).
The equivalent panel mass was set to 1 kg and stiffness of the air in the sealed
box Kb
is 50 kN/m while the stiffness of the 3 mm glass panel for point force in its
centre - K1
(exciting first natural mode) is about 450 kN/m (nine times stiffer).
The results of a Matlab simulation are given in Figure 18.
It is worth noting that when the air stiffness is included the impedance is minimal
at the first natural frequency of the panel without air stiffness effect, while the
maximum is at the system natural frequency which includes the air spring effect.
7. CONCLUSIONS
This study shows that when a panel is backed by a sealed air enclosure the first
natural frequency of the panel is raised by the stiffening effect of the air spring in
the closed cavity. Theoretical models based on modal superposition show a change
of the panel impedance and STL minimum to the new higher natural frequency.
However FEA analysis and measurements do not indicate this change. At closer
examination the sound pressure inside the enclosure has minimum at the original
free field panel natural frequency and a maximum at the natural frequency of the
panel with the air support of the closed enclosure. The maximum internal soundpressure annuls the maximum in the transmitted sound pressure at the resonance,
while the minimum in the internal sound pressure at the panel free field natural
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K b-air box
stiffness
F for Z1b case
F for Z1 case
Panel mass
K 1-pannel
stiffness
C1-panel
damping Panel
boundary
constraints
Figure 17. Glass panel driving point impedance simulated with lumped elements for the case the sound source is inside
the box and the box is closed (orange) and the box is open (green).
60 70 80 90 100 110 120 130100
101
102
103Impedance
Frequency (Hz)
Z1
Z1b
I m p e d a n c e ( N * s e c / m )
Figure 18. Lumped elements simulation of the glass panel driving point impedance with nd without the effect of the
sealed air in the box.
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frequency results in an unchanged panel impedance and STL when compared to the
free field case similar to the results provided by the double panel model developed
by Xin et al [8, 11].
It is concluded that these findings could lead to performing STL measurements
of panels installed on small enclosures without any corrections, although this is
limited to the low frequency range.
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