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Sound radiation from structures

• Prof. dr. Ines Lopez Arteaga

Structural Acoustics

Department of Mechanical Engineering

What do they have in common?

What do they have in common?

Source Transmission Receiver Soundboard

What do they have in common

Courtesy of Goodyear S.A.

Source Transmission Receiver Soundboard

/ Mechanical Engineering

PAGE 18

Wave types

• Longitudinal
• Transverse
• Bending

C= constant C=C(f) Acoustic / Structural Structural

Longitudinal Transverse Bending

/ Mechanical Engineering

PAGE 20

Structural wave types

• Pure / corrected bending wave
• Mainly transverse vibrations

Most important for acoustic radiation

Corrected bending wave includes effects

• f rotary inertia and

shear deformation

• most strongly excited (lowest mechanical impedance in audio

freq.range)

• radiates most effectively (as compared to other wave types)

/ Mechanical Engineering

PAGE 22

Transverse vibrations an infinite string

2 2

t u dx T dx x T

L

                 x u    

2 2 2 2 2

1 t u c x u

s 

   

L s

T c  

where is the phase velocity [m/s]

/ Mechanical Engineering

PAGE 23

Transverse bending of a beam

Fourth order partial differential equation:

4 4 2 2

      x u EI t u

L



Consider the solution

 

 

kx t i

Ae t x u







,

2 4

  EI k

L



4 2

  EI k

L B

 

4 L B B

EI k c     

L



where is the mass per unit length.

fh c c

L B

8 . 1 

with

 E cL 

and

  2  f

,

h is the beam height.

/ Mechanical Engineering

PAGE 24

Bending waves in plates

Fourth order partial differential equation:

 

2 1 12

4 4 2 2 4 4 4 2 3 2 2

                    y u y x u x u Et t u

s

 

y x B

k k k     

Vectorial sum in x- and y-direction:

2 2 2 y x B

k k k  

s



where is the mass per unit area.

fh c c

L B

8 . 1 

with

 

2

1     E cL

and

  2  f

,

h is the plate thickness.

/ Mechanical Engineering

PAGE 25

Dispersion

Beam: Plate: In both cases:

f h fh f c

B

/ ) (    

!

fh c c

L B

8 . 1 

with

 

2

1     E cL

and

  2  f

,

h is the plate thickness.

fh c c

L B

8 . 1 

with

 E cL 

and

  2  f

,

h is the beam height.

https://www.youtube.com/watch?v=dwMIaDg4Zeg

/ Mechanical Engineering

PAGE 27

Phase velocity, group velocity

k c  

Phase velocity:

k cg    

Group velocity:

/ Mechanical Engineering

PAGE 28

Critical frequency plates

At the critical frequency (coincidence frequency) the acoustic wave velocity and the bending wave velocity are equal: This leads to: h f c c c

c L B

8 . 1   with

 

2

1     E cL h : plate thickness.

h c c f

L c

8 . 1

2



The critical frequency of a plate only depends on the material properties and plate thickness

/ Mechanical Engineering

PAGE 29

Sound radiation from infinite plates

 

k kB   sin t c f k

L B B

8 1 2 2 .     

c f k    2 2  

/ Mechanical Engineering

PAGE 31

Sound radiation from infinite plates

max

p

x ik yp yp

B

e u u





max

Solving for gives: With: Note that:

t i k k iy x ik B yp

e e e k k u c t y x p

B B





2 2

2 2

1

  

 

max

) , , (

2 2 B y

k k k   And that the pressure increases rapidly as

1

2 2

 k kB

/ Mechanical Engineering

PAGE 32

Sound radiation from infinite plates

“hydrodynamic shortcut” Near field

f<fcrit (B< air)

Far field

f>fcrit (B> air) B 

 

B air

    sin f

air

1   f h

B 



/ Mechanical Engineering

PAGE 34

Radiation ratio (Radiation efficiency)

Radiation ratio: Sound power radiated by the plate divided by the sound power radiated by a large rigid piston with the same surface area and same r.m.s. vibration velocity piston plate plate

   

Piston

/ Mechanical Engineering

PAGE 35

Sound power radiated by a vibrating piston

Acoustic radiation of a large (compared to the acoustic wavelength) and rigid piston. rms rmsu

p r2   

For plane waves the velocity and the pressure are related through the specific acoustic impedance .

p c u 1  

Therefore: 2 rms

cSu   

/ Mechanical Engineering

PAGE 37

Radiation ratio infinite plate

2 2

1 1 k kB

piston plate plate

     

Plate

2 2 2

1 k k u cS

B prms

   

Piston 2 prms

cSu   

/ Mechanical Engineering

PAGE 38

Radiation ratio infinite plate

f f k k

c B piston plate plate

       1 1 1 1

2 2



Radiation from finite plates

/ Mechanical Engineering

PAGE 41

Below the critical frequency (subsonic modes) radiation efficiency depends on modeshape

+

• +

+

• +

+

• At low frequencies odd modes are better radiators (higher radiation

efficiency) than even modes Odd mode Even mode

Radiation from finite plates

/ Mechanical Engineering

PAGE 42

+

• +

+

• +

+

• Odd mode=

Edge radiation Even mode= Corner radiation Below critical frequency dipole and quadrupole cancellations

Dipole Dipole Dipole Dipole Dipole Dipole Dipole

+ + + +

Radiation from finite plates

/ Mechanical Engineering

PAGE 43

Above the critical frequency efficient radiation, each part of the plate radiates independently as a monopole

• +

+

• +

+

• Odd mode

Even mode

+ + + + +

/ Mechanical Engineering

PAGE 49

Radiation ratio finite plates

1 1 1    f fc

plate



for

f f

c



Above the critical frequency approximately similar to infinite plates

/ Mechanical Engineering

PAGE 50

Radiation ratio finite plates

Design curve for broadband mechanical excitation of flat plates

/ Mechanical Engineering

PAGE 51

Conclusions

• Infinite plates can only radiate sound above the

critical frequency.

• Real (finite) plates can radiate sound at all

frequencies, but below the critical frequency they are inefficient radiators.

/ Mechanical Engineering

PAGE 52

Recommended books

• Structure-Borne Sound: Structural Vibrations and

Sound Radiation at Audio Frequencies (3rd Edition),

• L. Cremer, M. Heckl, and B. A. T. Petersson, Springer

Berlin, 2005.

• Fourier acoustics: Sound radiation and nearfield

acoustic holography, E.G. Williams, Academic Press, London, 1999.

• Fundamentals of noise and vibration analysis for

engineers, M. Norton, d. Karckub, Cambridge University Press, 2003.

Sound radiation from structures

• Prof. dr. Ines Lopez Arteaga

Structural Acoustics

Department of Mechanical Engineering