Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

4.8 TRANSMISSION LOSS FOR WALLS

Back

One procedure for noise control is to provide an acoustic barrier or wall to

reduce the transmission of sound. For design purposes, one must be able to

predict the transmission loss for the wall over a wide range of frequencies. In

this section, we will examine the more general case of transmission of sound

through a panel or partition.

Transmission of Sound 107

The general variation of the transmission loss with frequency for a

homogeneous wall is shown in Fig. 4-10. We note that there are three

general regions of behavior for the wall or panel:

(a) Region I: stiffness-controlled region

(b) Region II: mass-controlled region

Techniques for prediction of the transmission loss for each of these regions

are given in the following material.

4.8.1 Region I: Sti!ness-Controlled Region

At low frequencies, the wall or panel vibrates as a whole, and sound transmission

through the panel is determined primarily by the stiffness of the

panel. Let us consider a panel, as shown in Fig. 4-11, in which the mediumis

the same on both sides of the panel, and the panel is very thin. The expressions

for the acoustic pressure and particle velocity on each side of the panel

may be written as follows:

p1рx; tЮ ј A1 ejр!t_kxЮ ю B1 ejр!tюkxЮ (4-132)

(Incident wave) ю (Reflected wave)

p2рx; tЮ ј A2 ejр!t_kxЮ (4-133)

(Transmitted wave)

108 Chapter 4

FIGURE 4-10 General variation of the transmission loss with frequency for a

homogeneous wall or panel.

u1рx; tЮ ј р1=_ocЮЅA1 ejр!t_kxЮ _ B1 ejр!tюkxЮ_ (4-134)

u2рx; tЮ ј р1=_ocЮA2 ejр!t_kxЮ (4-135)

At the surface of the panel (for a very thin panel), the particle velocities

are both equal to the instantaneous velocity of the panel, VрtЮ. We may

write the following expressions from Eqs (4-134) and (4-135) for x ј 0:

A1 _ B1 ј A2 (4-136)

VрtЮ ј

A2 ej!t

_oc

(4-137)

If the panel has a finite stiffness, the net force acting on the panel is

equal to the ‘‘spring-force’’ of the panel. The specific mechanical compliance

or mechanical compliance per unit area will be denoted by the symbol CS.

The compliance is the reciprocal of the spring constant. If we make a force

balance at the surface of the thin panel, we obtain the following expression:

p1р0; tЮ _ p2р0; tЮ ј _

VрtЮ dt ј _

A2 ej!t

j!c_oCS

(4-138)

Transmission of Sound 109

FIGURE 4-11 Vibration of a panel in the stiffness-controlled region, Region I. VрtЮ is the vibrational velocity of the panel.

Making the substitutions from Eqs (4-132) and (4-133) for the acoustic

pressure force, we obtain the following expression for the coefficients:

A1 ю B1 _ A2 ј ю

jA2

!_ocCS

(4-139)

Combining Eqs (4-136), (4-137), and (4-139), we obtain the following

expression for the ratio of coefficients:

A1 ј

1 _

2!_ocCS

1 ю р1=2!_ocCSЮ2 (4-140)

The sound power transmission coefficient for normal incidence may be

determined from Eq. (4-140):

atn ј

Itr

Iin ј j ptrj2

j pinj2 ј

____

1 ю р1=2!_ocCSЮ2 (4-141)

Substituting for the frequency, ! ј 2_f , we obtain an alternative form of

Eq. (4-141):

1=atn ј 1 ю р4_f _ocCSЮ_2 ј 1 ю рKSЮ_2 (4-142)

where:

KS ј 4_f _ocCS (4-143)

If we repeat the development for the case of oblique incidence of the

sound wave, we obtain the following expression for the sound power transmission

for an angle of incidence _:

atр_Ю ј

1 ю рcos _=KSЮ2 (4-144)

In many situations in noise control work, the sound waves strike the surface

at all angles of incidence (random incidence). The average sound power

transmission coefficient for random incidence of the sound waves is given

by:

at ј 2

р_=2

atр_Ю cos _ sin _ d_ (4-145)

If we use the expression for atр_Ю from Eq. (4-144) in Eq. (4-145), we obtain

the following expression for the sound power transmission coefficient in the

stiffness-controlled region, Region I:

at ј K2

S lnр1 ю K_2

S Ю ј K2

S lnр1=atnЮ (4-146)

110 Chapter 4

The transmission loss for the stiffness-controlled region is given by the

following:

TL ј 10log10р1=atЮ ј 10 log10р1=K2 SЮ_10log10Ѕlnр1юK_2

S Ю_ (4-147)

The transmission loss for normal incidence may be written as follows:

TLn ј 10log10р1=atnЮ ј 10 log10р1юK_2

S Ю (4-148)

TLn ј р10Юрlog10 eЮ lnр1юK_2

S Ю ј 4:3429 lnр1юK_2

SЮ (4-149)

lnр1юK_2

S Ю ј 0:23026 TLn (4-150)

If we substitute the expression from Eq. (4-150) into Eq. (4-147), we obtain

the final expression for the transmission loss for Region I, the stiffnesscontrolled

region:

TL ј 20log10р1=KSЮ_10 log10р0:23026 TLnЮ (4-151)

For a rectangular panel, the expression for the specific mechanical

compliance is given by the following:

CS ј

768р1_2Ю

_8Eh3р1=a2 ю1=b2Ю2 (4-152)

The quantities a and b are the width and height of the panel; h is the

thickness of the panel; and E and are the Young’s modulus and

Poisson’s ratio for the panel material, respectively. For a circular panel

with a diameter D and thickness h, the specific mechanical compliance is

given by:

CS ј

3D4р1_2Ю

256Eh3 (4-153)

Some properties of various panel materials are given in Appendix C.

4.8.2 Resonant Frequency

As the frequency of the incident wave is increased, the plate will resonate at

a series of frequencies, called the resonant frequencies. The lowest resonant

frequency marks the transition between Region I and Region II behavior.

The resonant frequencies are a function of the plate dimensions. For a

rectangular plate having dimensions a _ b _ h thick, the resonant frequencies

are given by the following expression (Roark and Young, 1975):

fmn ј р_=4

ffiffiffi

3 p

ЮcLhЅрm=aЮ2 ю рn=bЮ2_ (4-155)

Transmission of Sound 111

The factors m and n are integers, 1; 2; 3; . . . . The quantity cL is the speed of

longitudinal sound waves in the solid panel material:

cL ј

_wр1 _ 2Ю

_ _1=2

(4-156)

The quantity _w is the density of the panel material. Usually, the lowest

resonant frequency (the fundamental frequency) is the most predominant

frequency. This frequency corresponds to m ј n ј 1 in Eq. (4-155):

f11 ј р_=4

ffiffiffi

3 p

ЮcLhЅр1=aЮ2 ю р1=bЮ2_ (4-157)

The magnitude of the transmission loss at the first few resonant frequencies

is strongly dependent on the damping at the edges of the panel.

The fundamental resonant frequency for a circular plate is given by the

following expressions. For a circular plate of diameter D and thickness h

clamped at the edge (Roark and Young, 1975):

f11 ј

10:2cLh

ffiffiffi3p D2

(4-158)

For a circular plate with a simple supported edge, the fundamental resonant

frequency is given by a similar equation:

f11 ј

5:25cLh

ffiffiffi3p D2

(4-159)

4.8.3 Region II: Mass-Controlled Region

For frequencies higher than the first resonant frequency, the transmission

loss of the panel is controlled by the mass of the panel and is independent of

the stiffness of the panel. In this region, some acoustic energy is transmitted

through the panel and the remainder is reflected at the panel surfaces. This is

the physical situation analysed in Sec. 4.7.

The sound power transmission coefficient for normal incidence is

given by Eq. (4-128):

atn ј 1 ю

_f _wh

_1c1

_ _2

ј 1 ю

_fMS

_1c1

_ _2

(4-160)

The quantity MS is called the surface mass, or the panel mass per unit

surface area:

MS ј _wh (4-161)

The quantity _w is the density of the wall or panel, and _1 and c1 are the

density and speed of sound in the air around the panel, respectively.

112 Chapter 4

The transmission loss for normal incidence is related to the sound

power transmission coefficient for normal incidence:

TLn ј 10 log10р1=atnЮ (4-162)

For random incidence (field incidence), it has been found experimentally

that the transmission loss for the mass-controlled region is related to TLn by

the following expression (Beranek, 1971):

TL ј TLn _ 5 (4-163)

In many cases, the second term is Eq. (4-160) is much larger than 1. In

these cases, the reciprocal of the sound power transmission coefficient for

normal incident is proportional to f 2. The transmission loss is proportional

to 20 log10р f Ю, so that if the frequency is doubled the transmission loss will

be increased by 20 log10р2Ю or 6 dB/octave for the mass-controlled region.

4.8.4 Critical Frequency

As the frequency of the impinging sound wave increases in the masscontrolled

region, the wavelength of bending waves in the material, which are

frequency-dependent, approaches the wavelength of the sound waves in the

air. Coincidence (equality of the wavelengths) first occurs at grazing incidence,

or for an angle of incidence of 908. When this condition happens,

the incident sound waves and the bending waves in the panel reinforce

each other. The resulting panel vibration causes a sharp decrease in the

panel transmission loss. This point corresponds to the transition from

Region II behavior to Region III behavior.

The critical frequency (or wave coincidence frequency) is given by the

following expression (Reynolds, 1981):

fc ј

ffiffiffi3p c2

_cLh

(4-164)

If we combine Eqs (4-161) and (4-164), we find that the product рMS fcЮ is a

function of the physical properties of the panel and the sonic velocity (c) in

the air around the panel:

MS fc ј

ffiffiffi

p3 c2_w

_cL

(4-165)

4.8.5 Region III: Damping-Controlled Region

For frequencies above the critical frequency, the transmission loss is

strongly dependent on the frequency of the incident sound waves and the

internal damping of the panel material.

Transmission of Sound 113

For sound waves striking the panel at all angles (random incidence) at

frequencies greater than the critical frequency, the following empirical fieldincidence

expression applies for the transmission loss in the dampingcontrolled

region (Beranek, 1971):

TL ј TLnр fcЮю10 log10р_Юю33:22log10р f =fcЮ_5:7 (4-166)

The quantity TLnр fcЮ is the transmission loss for normal incidence at the

critical frequency:

TLnр fcЮ ј 10log10 1 ю

_MS fc

_1c1

_ _2

" #

(4-167)

The quantity _ is the damping coefficient for the panel material. Some

numerical values for the damping coefficient for various materials are

given in Appendix C.

For the damping-controlled region, the transmission loss is proportional

to 33:22 log10р f Ю. If the frequency is doubled, the transmission loss is

increased by 33:22 log10 р2Ю ј 10 dB/octave.

Example 4-4. An oak door has dimensions of 0.900m (35.4 in) wide by

1.800m (70.9 in) high by 35mm (1.38 in) thick. The air on both sides of the

door has a temperature of 208C (688F), for which c ј 343:2 m/s (1126 ft/

sec), _ ј 1:204 kg=m3 (0.0752 lbm=ft3), and zo ј 413:3 rayl. Determine the

transmission loss for the following frequencies: (a) 63 Hz, (b) 250 Hz, and (c)

2000 Hz.

We find the following properties for the oak door from Appendix C:

Longitudinal sound wave wave speed cL ј 3860 m/s (12,700 ft/sec)

Density _w ј 770 kg=m3 (48.1 lbm=ft3Ю Critical frequency product

MS fc ј р11,900 Hz-kg/m2) (343.2/346.1)2

ј 11,700 Hz-kg/m2 (2397 Hz-lbm/ft2)

Damping factor _ ј 0:008

Young’s modulus E ј 11:2 GPa р1:62 _ 106 psi)

Poisson’s ratio ј 0:15

The first resonant frequency is found from Eq. (4-157):

f11 ј 0:4534cLhр1=a2 ю 1=b2Ю

f11 ј р0:4534Юр3860Юр0:035ЮЅр1=0:902Ю ю р1=1:802Ю_ ј 94:5Hz

The specific mass is:

MS ј _wh ј р770Юр0:035Ю ј 26:95 kg=m2 р5:52 lbm=ft2Ю

114 Chapter 4

The critical or wave coincidence frequency is found from the ratio MS fc:

fc ј

MS fc

MS ј р11,700Ю

р26:95Ю ј 434:1Hz

(a) For f ј 63 Hz.

The frequency, f ј 63 Hz < 94:5Hz ј f11; therefore, this case lies in Region

I, the stiffness-controlled region. The specific mechanical compliance may be

evaluated from Eq. (4-152):

CS ј р768Юр1 _ 0:152Ю

р_8Юр11:2Юр109Юр0:035Ю3Ѕр1=0:90Ю2 ю р1=1:80Ю2_2

ј 70:81 _ 10_9 m3=N

CS ј 70:81 nm=Pa

The value of the parameter defined by Eq. (4-143) is as follows:

KS ј 4_ fZ1CS ј р4_Юр63Юр413:3Юр70:81Юр10_9Ю ј 0:02317

The sound power transmission coefficient may be calculated from Eq.

(4-146):

at ј K2

S lnр1 ю K_2

S Ю ј р0:02317Ю2 lnЅ1 ю р0:02317Ю_2_ ј 0:004042

The transmission loss for a frequency of 64 Hz is as follows:

TL ј 10 log10р1=0:004042Ю ј 23:9dB

(b) For f ј 250 Hz.

For this case, f11 ј 94:5Hz < 250 Hz < 434:1Hz ј fc; therefore, the operating

region is Region II, the mass-controlled region. The sound power

transmission coefficient for normal incidence is found from Eq. (4-160):

atn ј 1 ю

_fMS

_ _2

ј 1 ю р_Юр250Юр26:95Ю

р413:3Ю

_ _2

ј 1 ю р51:21Ю2

1=atn ј 2623:8

The transmission loss for normal incidence is found from Eq. (4-162):

TLn ј 10 log10р1=atnЮ ј 10 log10р2623:8Ю ј 34:2dB

The transmission loss with random incidence for a frequency of 250 Hz is

found from Eq. (4-163):

TL ј 34:2 _ 5 ј 29:2dB

Transmission of Sound 115

The frequency, f ј 2000Hz > 434:1Hz ј fc; therefore, this case lies in

Region III, the damping-controlled region. The transmission loss for normal

incidence at the critical frequency is found from Eq. (4-167):

TLnр fcЮ ј 10log10 1 ю р_Юр11,700Ю

р413:3Ю

_ _2

( )

ј 10log10р1ю7909Ю

ј 39:0dB

The transmission loss for a frequency of 2000Hz is found from Eq. (4-166):

TL ј 39:0ю10log10р0:008Юю33:22log10р2000=434:1Ю_5:7

TL ј 39:0юр_21:0Юю22:0_5:7 ј 34:3dB

Example 4-5. A steel plate (density 7700 kg/m3) has dimensions of 0.900m

(35.4in) by 1.800m (70.9in). The air on both sides of the plate has a characteristic

impedance of 413.3 rayl (at 208C) and sonic velocity of 343.3 m/s.

At a frequency of 500 Hz, it is desired to have a transmission loss of 30dB.

Determine the required thickness of the plate.

This problem involves iteration, because we do not know the region

for the transmission loss. Let us begin by trying Region II, the masscontrolled

region. The required transmission loss for normal incidence is

given by Eq. (4-163):

TLn ј TLю5 ј 30ю5 ј 35dB

We may use Eqs (4-160) and (4-162) to determine the surface mass:

TLn ј 10log10Ѕ1юр_MS f =Z1Ю2_ ј 35dB

р_MS f =Z1Ю2 ј 1035=10 _1 ј 3161:3

The surface mass is:

MS ј р3161:3Ю1=2р413:3Ю

р_Юр500Ю ј 14:79kg=m2 ј _wh

The required thickness (if the TL region is Region II) is as follows:

h ј

14:79

7700 ј 0:00192m ј 1:92mm р0:076 inЮ

Now, let us check the assumption of Region II behavior. The critical

frequency is found from the MS fc product, obtained from Appendix C for

steel:

MS fc ј р99,700Юр343:2=346:1Ю2 ј 98,040 Hz-kg/m2

116 Chapter 4

The critical or wave-coincidence frequency is:

fc ј

MS fc

MS ј р98,040Ю

р14:79Ю ј 6630 Hz > 500 Hz ј f

The first resonant frequency for the panel is found from Eq. (4-157):

f11 ј р0:4534Юр5100Юр0:00192ЮЅр1=0:900Ю2 ю р1=1:800Ю2_

f11 ј 6:85Hz < 500 Hz ј f

The frequency f ј 500 Hz lies in Region II, because f11 < f < fc, and

the required panel thickness is:

h ј 1:92mm р0:076 inЮ

Пресс-релиз популярных книг

4.8 TRANSMISSION LOSS FOR WALLS

Популярные книги

Популярные статьи

Навигация