Sound Transmission Loss of a Panel Backed by a Small Enclosure

8/19/2019 Sound Transmission Loss of a Panel Backed by a Small Enclosure

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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

549

JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL Pages 549 – 568

Vol. 34 No. 4 2015

*Corresponding author e-mail: [email protected] (K Ruber)


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550 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL


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

551


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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


JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL552


6/21

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

Vol. 34 No. 4 2015


553

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




8/21

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

Vol. 34 No. 4 2015


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 )


9/21

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



Figure 3. Sound transmission loss test box.


10/21

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).

Vol. 34 No. 4 2015


557

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.


11/21



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.


12/21

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

Vol. 34 No. 4 2015


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


13/21

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



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


Figure 9. Average sound transmission coefficient of the glass panel τ average for the open and closed bottom cases.


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Vol. 34 No. 4 2015


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


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


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



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.

Vol. 34 No. 4 2015


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



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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Documents

Sound Transmission Loss of a Panel Backed by a Small Enclosure