Pres Css Ifs 3maggio2011

Radiated Acoustic Panel by a Panel

Vincenzo D’Alessandro

[email protected]

CSS/IFS Lesson – 3 May 2011

ælab ‐Vibrations and Acoustics Laboratory Department of Aerospace Engineering

Università degli Studi di Napoli “Federico II” Via Claudio 21, 80125, Napoli, Italy

www.dias.unina.it

Outline I Section – Introduction to Acoustic

What is Sound? Frequency and Wavelength Examples The dB Acoustic Power, Intensity and

Impedance What can we hear? Octave Bands

Radiated Acoustic Power by a Panel - 3 May 2011 2

What is Sound?Sound waves are compressional oscillatory disturbance (longitudinal waves) that propagate in a fluid. These waves involve molecules of the fluid moving back and forth in the direction of propagation (with no net flow), accompanied by chances in pressure, density and temperature. These phenomena requires the presence of a sound source and an elastic medium which allows the propagation and for the latter reason, the sound can spread in a vacuum. The sound sourceconsists of a vibrating element which transmits its movement to the particles of the surrounding medium, which oscillate around their equilibrium position. Sound: any pressure variation (in air, water or other medium) that the human ear can detect. We define the sound pressure, that is the difference between the instantaneous value of the total pressure and the static pressure, as the quantity that can be heard. The number of pressure variation per second is called frequency of the sound, and is measured in Hertz (Hz). The frequency of a sound produces it’s distinctive tone. The normal range of hearing for a healthy young person extends approximately 20 Hz up to 20 KHz.

3Radiated Acoustic Power by a Panel - 3 May 2011

Frequency and WavelengthThese pressure variations travel through any elastic medium (such as air) from the source of the sound to the listener's ears. The speed of sound c is 344m/s (1238km/h) at room temperature. Knowing the speed and frequency of a sound, we can calculate the wavelength l — that is, the distance from one wave top or pressure peak to the next. l = c/f .OSS. Air: c = 331.4 + 0.6 t c depends on temperature! We can see high frequency sounds have short wavelengths and low frequency sounds have long wavelengths. A sound which has only one frequency is known as a pure tone. In practice pure tones are seldom encountered and most sounds are made up of different frequencies. Even a single note on a piano has a complex waveform.


5

Examples

Pure Tone

Bi-tone

Impulse

Tone with Random base

Radiated Acoustic Power by a Panel - 3 May 2011

6

Examples


7

Examples


The dBThe second main quantity used to describe a sound is the size or amplitude of the pressure fluctuations. The pressure changes associated with a sound wave can be very small if compared with ambient pressure. The human hear can perceive as sound pressure variation in the range 20 mPa – 104 Pa. The ambient pressure at sea level is about 1 atm = 1.013 x 105 Pa.It’s obviously that, if we measured sound in Pa, we would end up with some quite large, unmanageable numbers. To avoid this, another scale is used — the decibel or dB scale. The decibel is not an absolute unit of measurement. It is a ratio between a measured quantity and an agreed reference level. The dB scale is logarithmic and uses the hearing threshold of 20 mPa as the reference level. This is defined as 0 dB. When we multiply the sound pressure in Pa by 10, we add 20 dB to the dB level. So 200 mPa corresponds to 20 dB, 2000 mPa to 40 dB and so on. Thus, the dB scale compresses a range of a million into a range of only 120 dB.


The dBThe Sound Pressure Level (SPL) is defined as:

where:

100

[ ] 20log rmspSPL dB

p

12

2

0

1lim

T

rms Tp p t dt

T


Acoustic Power, Intensity, ImpendanceAcoustic Power W: acoustic energy produced by a source in the time unit [W]

Acoustic Intensity I: power per area unit [W/m2] depends on source and field

For spherical source:

Acoustic Impedance Z.The relation between sound pressure and velocity of particles is:

where Z = ρ0c0 is called acoustic impedance.


What can we hear?We have already defined sound as any pressure variation which can be heard by a human ear. This means a range of frequencies from 20 Hz to 20 kHz for a young, healthy human ear. In terms of sound pressure level, audible sounds range from the threshold of hearing at 0 dB to the threshold of pain which can be over 130 dB. Although an increase of 6 dB represents a doubling of the sound pressure, an increase of about 10 dB is required before the sound subjectively appears to be twice as loud. (The smallest change we can hear is about 3 dB). The subjective or perceived loudness of a sound is determined by several complex factors. One such factor is that the human ear is not equally sensitive at all frequencies (equal loudness countour). It is most sensitive to sounds between 2 kHz and 5 kHz, and less sensitive at higher and lower frequencies.


Octave BandsThe human hearing mechanism is more sensitive to frequency ratios rather than actual frequencies:

The frequency of a sound determines its pitch as perceived by a listener a frequency ratio of two is a perceived pitch change of one octave, no matter

what the actual frequencies are

This phenomenon can be summarized by saying that the pitch perception of the ear is proportional to the logarithm of frequency rather than to frequency itself.

The octave is such an important frequency interval to the ear that so-called octave band analysis has been defined as a standard for acoustic analysis. Each octave band has a bandwidth equal to about 70% of its centre frequency. This type of spectrum is called constant percentage bandwidth (CPB) because each frequency band has a width that is a constant percentage of its centre frequency. It is possible to define constant percentage band analysis with frequency bands of narrower width. A common example of this is the one-third-octave spectrum, whose filter bandwidths are about 27% of their centre frequencies


Octave Bands

octave bands

one-third octave bands

, 1 ,2c i c iff

1/ 3, 1 ,2c i c iff

lower band limit

upper band limit

1/ 22 xlow cff

1/ 22 xup cff

where x=1 for octaves, x=3 for 1/3 octaves


Acoustical Analysis of a Panel II Section – Acoustical Analysis of a Panel

• Radiated Acoustic Power• Incident Acoustic Power• Transmission Loss TL• Radiation Efficiency• Acoustical Analysis in Discrete Formulation


Radiated Acoustic PowerThe power radiated from a vibrating surface is defined as:

1

2*Real , ,rad S Sp Q V Q d

, ,S SV Q j w Q

thus, considering that the displacement is related to the normal velocity and multiplying and dividing for the wavenumber k:

The effective harmonic acoustic power radiated by a vibrating surface, Πrad, can be evaluated by using the Rayleigh Integral formulation for plane radiators. The Rayleigh integral, in fact, expresses the sound pressure p radiated from a vibrating surface Σ in a field point Q, as function of the surface normal displacements, w.

where QS is a generic point on Σ and R=|Q-QS|.

By substituting the pressure expressed by Rayleigh in the power expression:

2

*01Re , ,

2 2

jkR

rad s

ew Q d V Q d

R

2

0

2, ,

jkR

s

ep Q w Q d

R


Radiated Acoustic Power

2

*0 Re , ,4

jkR

rad s

ej V Q V Q d d

c kR

where c is the speed of the sound and k=ω/c. By applying the Euler’s formula for complex analysis and considering the real part, the previous expression becomes:

In the above equation, it is possible introducing the radiation function

that is symmetricaland that presents a “removable” singularity (hence also in the expression of Πrad )

2

*0sin

, ,4rad s

kRV Q V Q d d

c kR

expjkRs

s

jk Q QeR

kR kQ Q

, , , ,s sR Q Q R Q Q

0

sinlim 1

s

s

Q Qs

k Q Q

kQ Q


Radiated Acoustic Power

20

4

*

*

sin, , if

, , if

S S

rad

S S

kRV Q V Q d d Q Q

kRc V Q V Q d d Q Q

The power radiated can be expressed in its final formulation by using complex number properties as:


Incident Acoustic PowerThe incident acoustic power on the give elastic surface due to the impinging pressure distribution is given by:

In the hypothesis of a generic plane wave impinging on a plate with an angle ϑi , the incident power can be expressed in term of pressure pi as:

where a and b are plate dimensions.

P 𝒊𝒏𝒄 (𝝎 )=𝟏𝟐

Real{ ∑❑

❑

𝒑𝒊 (𝝎 ,𝑸𝒊 ) ∙𝑽 ∗ (𝝎 ,𝑸𝒊 )𝒅S}2

0

cos2i

i i

p abc


Transmission Loss TLWhen sound wave interacts with an infinite barrier, part of the wave is absorbed, part is redirect and the rest is transmitted through the surface.

By writing the energy balance:

By dividing both members for the power incident:

The TL is the fraction of the sound energy incident on a structure that is transmitted through it

10 10

110log 10log inc

rad

TL


Transmission Loss TL


Transmission Loss TLMass Law. The panel stiffness and damping have no effect and the TL depends on the surface density of the panel and it increases by 6 dB per doubling of mass.

diffuse incidence

normal incidence in air

2

0

2

02

0

0.9782

10log

12

ln

1 0.2082

s

s

s

cTL

c

c


𝑇𝐿 (ω )=20 𝑙𝑜𝑔10(ωρ 𝑠𝑐𝑜𝑠θρ0𝑐 0 )Incidence θ≠ 0

- 42.5

Transmission Loss TL


Transmission Loss TLCoincidence effect. The panel is transparent to the acoustic radiation, almost all the incident sound is transmitted through the panel.

1,2

2

,2s

cx y

cf

D

Coincidence frequency. The frequency at which the bending wavelength λb in the panel equals the wave sound projected wavelength λ/sinϑ: a high degree of coupling between panel and air is achieved.

Critical frequency: lowest possible value of the coincidence frequency.For homogeneous panel and for θ=90°:

For metallic panel the critical frequency is obtained by dividing 12000 for the thickness expressed in mm.Radiated Acoustic Power by a Panel - 3 May 2011 23

Radiation Efficiency σThe radiation efficiency of a vibrating surface is defined as the ratio between the acoustic power radiated and vibrational energy of the panel. It is also defined as the ratio between the acoustic power radiated by the panel and the power radiated by an infinitely rigid piston with the same area and the same mean square displacement.

In fact, if an infinite rigid surface such as a piston vibrates at a frequency at which the surface’s dimension are considerably greater that the acoustic wavelength in the medium, the air cannot move out of the way laterally, and the particle velocity of the air must be equal to the velocity of the surface, and so σ=1: the piston is a perfect radiator. In most practical case, the radiation efficiency is either below or very close to unity, but can also exceed the unity. For this reason often another parameter is reported: the radiation resistance.


Radiation Efficiency σWallace (1972) produced an analytic solution to calculate the modal radiation efficiency, that is to say the radiation efficiency of every single flecural mode of a plate.

By considering a panel simply supported on its four edges, we assume this velocity distribution over the surface:

Considering acoustical symmetry, for diffuse incidence can be written


cos(α/2) if m is odd integer, else sin(α/2) if m even integer cos(β/2) if n is odd integer, else sin(β /2) if n even integer

Radiation Efficiency σ


Acoustical Analysis in Discrete FormulationBy considering the expression so far calculated, we can analyze in discrete coordinates the acoustical behaviour of a panel.

Displacement in NG discrete coordinate:

where H is the transfer matrix, F the modalmatrix and F the force vector.

Since we suppose a pressure load, the force vector is given by

where p(w) is the acting pressure and A the nodal equivalent area matrix. A is a diagonal matrix, and for the i-th grid point it is


Acoustical Analysis in Discrete FormulationRadiated acoustic power

where


20

4

*

*

sin, , if

, , if

S S

rad

S S

kRV Q V Q d d Q Q

kRc V Q V Q d d Q Q

2

0 ( ) ( ) ( ) ( ) 4

H

rad V A R A Vc

Acoustical Analysis in Discrete FormulationIncident acoustic power

Radiation efficiency

where


P 𝒊𝒏𝒄 (𝝎 )=𝟏𝟐

Real{ ∑❑

❑

𝒑𝒊 (𝝎 ,𝑸𝒊 ) ∙𝑽 ∗ (𝝎 ,𝑸𝒊 )𝒅S}

Acoustical Analysis in Discrete Formulation


Modal radiation efficiency: analytical and FEA calculation

Modal radiation efficiency

Acoustical Analysis in Discrete Formulation


Transmission Loss

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