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VTAF 01 – Sound in Buildings and Environment

10. Building Acoustics 2 - Airborne sound insulation models

NIKOLAS  VARDAXIS

DIVISION  OF  ENGINEERING  ACOUSTICS,  LTH,  LUND  UNIVERSITY

2021.05.11

(2)

… recap from last lecture (I)

• Airborne sound insulation measurements (ISO standards)

⎟⎟⎠

⎞

⎜⎜⎝

+ ⎛

= ( ) ( ) 10log ( )

)

( A f

f S L f

L f

R S R

Statement of results:

• R’w(C50-3150 ; Ctr)

• Rw(C50-3150 ; Ctr)

63 125 250 500 1000 2000 4000 15

20 25 30 35 40 45 50 55 60

Frequency [Hz]

R [dB]

(3)

… recap from last lecture (II)

• Impact sound pressure level measurements (ISO standards)

⎟⎠

⎜ ⎞

⎝

+ ⎛

= 10

) log (

10 ) ( )

( A f

f L f

Ln R

0 10 20 30 40 50 60 70

100 160 250 400 630 1000 1600 2500

Ln[dB]

Frequency  [Hz]

Statement of results:

• L’n,w(Cl,50-2500 )

• Ln,w(Cl,50-2500 )

(4)

Spectrum adaptation terms

Rw=36  dB C  -­1

Ctr -­4

Rw=36  (-­1,-­4)

Rw=36  dB C  -­3

Ctr -­11

Rw=36  (-­3,-­11)

• Example: Airborne sound insulation measurement

(5)

Physics of sound transmission

• A sound field in a room (or a source on a floor) excites waves in the adjacent wall or on the floor.

• The structure vibrates and produced waves are transmitted through the structure. Eventually this event excites either other structures or the air in another room.

• Thinking about our SDOF, as a homogeneous solution, described in terms of natural frequencies (Eigen-).

• Eigenfrequencies and eigenmodes depend on boundary and coupling conditions. How a wall is mounted and the surrounding elements are key to determining final properties of a wall

• Field measurements (in-situ) and laboratory data differ (Rw vs. R’w).

(6)

Physics of sound transmission

• The relationship between incident and transmitted waves depends on speed of sound in the two media (stiffness and density) and angle of incidence (we did not see that really, but it is so).

• Sound insulation of a material is an interplay between mass, stiffness and damping.

• Diffuse sound fields are typically assumed – but it is not the case at low frequencies, remember (Schröder frequency)!

(7)

Outline

Introduction

Examples

Impact:

SS-EN 12354-2

Flanking transmission

Summary

Airborne:

SS-EN 12354-1

Analytical models for calculation of R(f) in

single and double-leaf walls

(8)

DEF: Coincidence – Critical frequency (I)

• The wavelength of a bending wave λB is dependent on frequency, bending stiffness and mass density.

• When the wavelength of sound in air coincides with the structural wavelength à Coincidence phenomena

‒ Radiation efficiency becomes very high

‒ Poor insulation properties

Impinging wave

(9)

DEF: Coincidence – Critical frequency (II)

• Bending wave velocity in a plate

• If f = f

c

thus c

B

= c

o

= 340 m/s (f

c

= critical frequency)

• Or expressed as a function of the coincidence number

2 4

m f B cB

= π ʹ′ʹ′

B m fc c ʹ′ʹ′

= 2π

2 0

h fc = K

NOTE: The condition for coincidence is that λB=λsin(φ). Therefore, if the incidence angle φ decreases, the coincidence frequency fc increases according to fc(φ)=fc/sin2(φ). The lowest frequency at which coincidence occurs (critical frequency) is at the incidence angle φ=90º.

Impinging wave

(10)

Critical frequency for common materials

Material Coincidence number (K) Thickness [mm] fc [Hz]

Concrete 18 160 110

Light concrete 38 70 540

Gypsum 32 10 3200

Steel 12-13 1 12000

Glass 18 3 6000

K 60000 Eρ

=

• For a homogeneous isotropic plate of uniform thickness, the coincidence number is:

(11)

Outline

Double-leaf wall

Exact method

Analytical models for calculation of R in single and double-leaf walls

Single-leaf wall

Approximate method

(12)

Wall types

(13)

Sound reduction index of single-leaf partitions

• Exact method

‒ Region I: Stiffness-controlled region (f < f11)

‒ Region II: Mass-controlled region (f11 < f < fc)

‒ Region III: Damping-controlled region (fc< f)

(14)

Sound reduction index of single-leaf partitions

• Region I: Stiffness-controlled region (f < f11)

‒ Panel vibrates as a whole (considered thin)

( )

( )

2 2 2

3 8

2

2 2

1 1

) 1

( 768

4 ) (

1 ln log 1 10

log 10 )

(

⎟⎠

⎜ ⎞

⎝

⎛ +

=

=

+

⎟⎟

⎠

⎞

⎜⎜⎝

= ⎛

b Eh a

C

C c f f

K

K K f

R

s

s F F S

S S

π

υ ρ π

Cs: mechanical compliance for a rectangular plate E: Young’s modulus of the material the wall is made of h: wall thickness

a, b: plate dimensions : Poisson’s ratio of the wall

: density of the surrounding fluid (F), i.e. air cF: wave propagation speed in the fluid (F), i.e. air

cLplatewave propagation speed in the plate (longitudinal wave)

υ ρF

⎟⎠

⎜ ⎞

⎝

⎛ +

= 2 2

11

1 1

3

4 c h a b

f π Lplate

For a simply supported plate of dimensions a x b

(15)

Sound reduction index of single-leaf partitions

• Region II: Mass-controlled region (f11 < f < fc)

‒ Sound reduction independent of stiffness (controlled by mass inertia)

‒ Some energy transmitted and part reflected at panel surface

c dB f fm

R

F F

'' 5 1

log 10 )

(

2

⎟−

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞

⎜⎜⎝

+⎛

= ρ

π

m’’= h is the surface mass of the panelρ

NOTE: Although the above equation is valid for frequencies up to fc, it yields only accurate results for f

≤ 0.5fc. The mathematical expresion around fc is mathematically cumbersome and rarely used, so approximate methods were developed.

Mass law >>1

Random incidence correction

(16)

Sound reduction index of single-leaf partitions

• Region III: Damping-controlled region (f

c

< f)

‒ Curve “dip” controlled by internal material damping

Important for design (low insulation)

‒ Contribution from plate reverberant field dominates (only above fc )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞

⎜⎜⎝

+⎛

=

⎟⎟

⎠

⎞

⎜⎜⎝

+ ⎛

+

=

'' 2

1 log 10 ) (

7 . 5 log

22 . 33 ) log(

10 ) ( ) (

F F

c c

c c

c m f f

R

f dB f f

R f

R

ρ π

η

is the total loss factor or damping of the panel η

Lplate F

c hc

f c π

2 3

=

(17)

Notes: Sound reduction index of single-leaf partitions

• As the frequency of the incident sound wave increases, the wavelength of the bending-waves in the plate approaches the wavelength of sound waves in air (remember that bending waves are dispersive).

• When the wavelengths match, the panel offer little resistance and vibrates with same velocity as incident sound wave

‒ Curve “dip” controlled by internal material damping

‒ Important for design (low insulation)

Lplate F

c hc

f c π

2 3

=

(18)

Notes: Sound reduction index of single-leaf partitions

• Design in respect to coincidence frequency

Make coincidence either very low (100-125 Hz) or very high (3150-4000 Hz). 100 Hz – 4000 Hz is important for human hearing.

Very low: thick wall, low density, high Young’s modulus: e.g. 15 cm concrete.

Very high: thin wall, high density, low Young’s modulus: e.g. 13 mm plasterboard.

Both these construction elements are typical indeed!

Lplate F

c hc

f c π

2 3

=

(19)

Notes: Mass law simplification

• Simplification:

⎟⎟⎠

⎞

⎜⎜⎝

≈ ⎛

F Fc f fm

R ρ

π 2 log '' 20 )

(

c dB f fm

R

F F

'' 5 1

log 10 )

(

2

⎟−

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞

⎜⎜⎝

+⎛

= ρ

π

Mass law

Random incidence correction

0 10 20 30 40 50 60 70

Sound reduction(dB)

Frequency (Hz)

concrete leight-weight wall mineral wool

Mass law (6 dB/octave)

Doubled weight leads to 6 dB increased insulation

>>1

(20)

Sound reduction index of a single leaf wall – Summary

nollmods område

fåmods område R [dB]

frekvens [Hz]

+6 dB/oktav masslagen R0

mindre dämpning mångmods

område

kritisk frekvens, fc

(21)

Outline

Double-leaf wall

Exact method

Analytical models for calculation of R in single and double-leaf walls

Single-leaf wall

Approximate method

(22)

Sound reduction index of single-leaf partitions (I)

• Approximate method – Just a review (not used)

Region I: Mass-controlled region (f < f1)

Region II: “Plateau” (f1 < f < f2)

Region III: Stiffness-controlled region (f2< f)

Hyphotesis: Infinite panel and diffuse field excitation

NOTE: f1and f2are not the resonance and coincident frequency explained in the exact method!

(23)

Sound reduction index of single-leaf partitions (II)

• Region I: Mass-controlled region (f < f1)

‒ Transmission independent of panel stiffness

( )

m

( )

f c dB

f

R( ) 20log ´´ 20log 20log F F ⎟5

⎠

⎜ ⎞

⎝

⎛

+

= π

ρ

(24)

Sound reduction index of single-leaf partitions (III)

• Region II: “Plateau” (f1 < f < f2)

‒ Governed by internal damping

‒ Height of the plateau depends on material

‒ f1 and f2 are the lower and upper limits of the plateau

» Calculated with expresions of adjoining regions

(25)

Sound reduction index of single-leaf partitions (IV)

• Region III: Mass-controlled region (f2 < f )

‒ Governed by stiffness of the panel

( )

2

log 22 . 33 )

( )

( f R f2 ff

R = +

NOTE: The slope of the expression (10 dB/octave) should just be used only for the 2 octaves above f2. For the following octaves, one should use a slope equal to 6 dB/octave,

i.e. “20log(f/f2oct)” instead of “33.22log(f/f2)”, where f2oct is the frequency where the 3rd octave above f2 starts.

(26)

Example – Approximate method

1. A glass window has a thickness of 8.3 mm. Using the “approximate”

method, determine the sound reduction index as a function of

frequency in octaves over the range from 63 Hz to 8000 Hz.

(27)

Outline

Double-leaf wall

Exact method

Analytical models for calculation of R in single and double-leaf walls

Single-leaf wall

Approximate method

(28)

Introduction

• How to improve single-wall? Add a second wall!

Double-leaf wall literature is extensive

Theoretical analysis, less developed due to complexity

Analyses often carried out using advanced methods in software:

FEM (Finite Elements Method), SEA (Statistical Energy Analysis)…

• Several theoretical derivations of sound transmission

Double-leaf wall without mechanical coupling Double walls with structural connections

(29)

Sound reduction index of double-leaf walls

• Approximate empirical model for a double leaf wall without structural connections, with cavity filled with porous absorber (Sharp 1978)

⎟⎠

⎜ ⎞

⎝

+ ⎛

+

=

⎟⎟

⎠

⎞

⎜⎜⎝

+ ⎛

=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎟⎟⎠

⎞

⎜⎜⎝

+ ⎛

=

⎟⎟⎠

⎞

⎜⎜⎝

+ ⎛

=

S R A

R R

A L S

L R

A L S

L R

A L S

L R

DoubleWall DoubleWall

2 2 1

3 3

1

3 3

2 2

2 2

1 1

log 10 log 10 log

10 log 10

⎪⎩

⎪⎨

⎧

>

+ +

<

<

+

+

<

=

d

d M

f f dB

R R

f f f

dB d

f R

R

f f R

R

; 6

; 29

) log(

20

;

2 1

0 2

1

0 ⎟⎟

⎠

⎞

⎜⎜⎝

⎛

+ ʹ′ʹ′

= ʹ′ʹ′

2 1

0

1 1

2 d m m

f c ρF π

fd 55d

=

RM denotes the mass law with M=m1+m2

R1 and R2 denote the individual sound reduction index for each leaf d: distance between the two leaves i.e. (cavity thickness)

NOTE: Diffuse field assumed in both rooms

(30)

• Improvement in the sound reduction index of a double-leaf wall respect to a single wall, and also when including insulation in the cavity.

Examples (i)

(31)

Examples (ii)

0 10 20 30 40 50 60 70 80

50 80 125 200 315 500 800 1250 2000 3150

frekvens [Hz]

Reduktionstal [dB]

R3 R2 R1

135 mm

Gipsplatta, 13 mm Mineralull

Stålreglar c/c 600 R1= tomt hålrum mm

R3 = 140 mm mineralull R2 = 30 mm mineralull Rw [dB]

R3: 55 R2: 49 R1: 43

• Variation in the sound reduction index of a double-leaf wall when varying parameters in the cavity (inclusion of insulation and its thickness).

(32)

Examples (iii)

(33)

Examples (iv)

(34)

Examples (v)

(35)

Examples (vi)

“Rule of thumb”: decoupled structures perform much better à

acoustic bridges eliminated

(36)

Outline

Introduction

Examples

Impact:

SS-EN 12354-2

Flanking transmission

Summary

Airborne:

SS-EN 12354-1

Analytical models for calculation of R(f) in

single and double-leaf walls

(37)

Review of flanking transmission treatments

• Sound transmission

– Airborne

– Structure-borne

• Transmission paths

– Direct transmission (D) – Flanking paths (Fi)

– In total: 13 paths (1 direct / 12 flanking)

• Flanking: cause of problems related with acoustic comfort

– Difference between lab and in-situ measurements ~4 dB

»Estimation methods described in SS-EN 12354:2000

Acoustic performance as sum of individual contributions

(38)

Remember…

… Laboratory vs. Field situation (flanking transmission comes into play)

[REF] Vigran(2008)

ISO 717-1:2013 ISO 10140-2:2010

Rw

ISO 717-1:2013 ISO 16283-1:2014

R’w

SS-EN12354-1:2000

Prediction of R’wfrom the individual acoustic performances (Rw) of the elements involved in the junction, as sum of individual contributions

(39)

Design example – Decoupling structural elements

(40)

Design example: timber volume elements

(41)

Design example: elastic interlayers

(42)

Summary

• Analytical calculation methods of reduction sound index – Single-leaf wall

» Exact method

» Approximate method (not used anymore)

– Double-leaf wall

– There is more complex and effective solutions

(43)

References: SS-EN 12354:2000 series

• SS-EN12354-1:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 1: Airborne sound insulation between rooms (2000).

• SS-EN12354-2:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 2: Impact sound insulation between rooms (2000).

• SS-EN12354-3:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 3: Airborne sound insulation against outdoor sound (2000).

• SS-EN12354-4:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 4: Transmission of indoor sound to the outside (2000).

• SS-EN12354-5:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 5: Sound levels due to service equipment (2000).

• SS-EN12354-3:2000, Building Acoustics– Estimation of acoustic performance of buildings from the performance of elements – Part 6: Sound absorption in enclosed spaces (2000).

(44)

Thank you for your attention!

nikolas.vardaxis@construction.lth.se

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(mineral wool products, porous fibreboard products, foam plastic, fabric, felt etc). 2.