Presentation of Fluid-Structure Interaction: radiated acoustic power from a panel

1. CSS/IFS Lesson – 3 May 2011 RadiatedAcoustic Panel by a Panel Vincenzo D’Alessandro vin.dalessandro@studenti.unina.it ælab‐Vibrations and AcousticsLaboratory Departmentof AerospaceEngineering Università degli Studi di Napoli “Federico II” Via Claudio 21, 80125, Napoli, Italy www.dias.unina.it

3. What is Sound?

4. Frequency and Wavelength

5. Examples

6. The dB

7. Acoustic Power, Intensity and Impedance

8. What can we hear?

9. Octave BandsRadiatedAcousticPower by a Panel - 3 May2011 2

10. 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 source consists 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. 3 RadiatedAcousticPower by a Panel - 3 May2011

11. Frequency and Wavelength These 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. 4 RadiatedAcousticPower by a Panel - 3 May2011

12. Examples Pure Tone Bi-tone Impulse Tone with Random base 5 RadiatedAcousticPower by a Panel - 3 May2011

13. Examples 6 RadiatedAcousticPower by a Panel - 3 May2011

14. Examples 7 RadiatedAcousticPower by a Panel - 3 May2011

15. The dB The 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, 2000mPa to 40 dB and so on. Thus, the dB scale compresses a range of a million into a range of only 120 dB. 8 RadiatedAcousticPower by a Panel - 3 May2011

16. The dB The Sound Pressure Level (SPL) isdefinedas: where: RadiatedAcousticPower by a Panel - 3May2011 9

17. Acoustic Power, Intensity, Impendance AcousticPower W: acousticenergyproducedby a source in the time unit[W] AcousticIntensity I: power per area unit [W/m2]  depends on source and field For spherical source: AcousticImpedance Z. The relation between sound pressure and velocity of particlesis: where Z = ρ0c0iscalledacousticimpedance. RadiatedAcousticPower by a Panel - 3May2011 10

18. 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 about3 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. RadiatedAcousticPower by a Panel - 3May2011 11

20. a frequency ratio of two is a perceived pitch change of one octave, no matter what the actual frequencies areThis 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 RadiatedAcousticPowerby a Panel - 3 May 2011 12

21. Octave Bands octave bands one-third octave bands lower band limit upper band limit where x=1 for octaves, x=3 for 1/3 octaves RadiatedAcousticPower by a Panel - 3May2011 13

23. Radiated Acoustic Power

24. Incident Acoustic Power

25. Transmission Loss TL

26. Radiation Efficiency

27. Acoustical Analysis in Discrete FormulationRadiatedAcousticPower by a Panel - 3 May2011 14

28. Radiated Acoustic Power The power radiated from a vibrating surface is defined as: 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: thus, considering that the displacement is related to the normal velocity and multiplying and dividing for the wavenumber k: RadiatedAcousticPower by a Panel - 3 May2011 15

29. Radiated Acoustic Power 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 symmetrical and that presents a “removable” singularity (hence also in the expression of Πrad) RadiatedAcousticPower by a Panel - 3 May2011 16

30. Radiated Acoustic Power The power radiated can be expressed in its final formulation by using complex number properties as: RadiatedAcousticPower by a Panel - 3May2011 17

31. Incident Acoustic Power The 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. P𝒊𝒏𝒄𝝎=𝟏𝟐Real𝒑𝒊𝝎, 𝑸𝒊∙𝑽∗𝝎, 𝑸𝒊𝒅S RadiatedAcousticPower by a Panel - 3May2011 18

32. Transmission Loss TL When 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: П𝑖𝑛𝑐=П𝑟+П𝑎+П𝑡 By dividing both members for the power incident: 1=ρ+α+τ The TL is the fraction of the sound energy incident on a structure that is transmitted through it RadiatedAcousticPower by a Panel - 3May2011 19

33. Transmission Loss TL RadiatedAcousticPower by a Panel - 3 May2011 20

34. Transmission Loss TL Mass Law. The panel stiffness and damping have no effect and the TL depends on the surfacedensity of the panel and itincreases by 6 dB per doubling of mass. Incidence θ≠ 0 diffuse incidence 𝑇𝐿ω=20𝑙𝑜𝑔10ωρ𝑠𝑐𝑜𝑠θρ0𝑐0 normal incidence in air 𝑇𝐿ω=20𝑙𝑜𝑔10𝑓 ρ𝑠 - 42.5 RadiatedAcousticPower by a Panel - 3May2011 21

35. Transmission Loss TL RadiatedAcousticPower by a Panel - 3May2011 22

36. Transmission Loss TL Coincidenceeffect. The panel is transparent to the acoustic radiation, almost all the incident sound is transmitted through the panel. Coincidencefrequency. 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. RadiatedAcousticPower by a Panel - 3May2011 23

37. Radiation Efficiency σ The radiationefficiencyof a vibratingsurfaceisdefinedas the ratio between the acousticpowerradiated and vibrationalenergy of the panel. Itisalsodefinedas the ratio between the acousticpowerradiated by the panel and the powerradiated by an infinitelyrigidpiston with the same area and the samemeansquaredisplacement. 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 pistonis a perfectradiator. In mostpractical case, the radiationefficiencyiseitherbelow or veryclose to unity, but can alsoexceed the unity. For thisreasonoftenanotherparameterisreported: the radiationresistance. RadiatedAcousticPower by a Panel - 3 May2011 24

38. Radiation Efficiency σ Wallace (1972) produced an analyticsolution to calculate the modalradiationefficiency, thatis to say the radiationefficiency of every single flecural mode of a plate. By considering a panel simplysupported on itsfouredges, we assume thisvelocitydistribution over the surface: Consideringacousticalsymmetry, for diffuse incidence can be written cos(α/2)if m isoddinteger, else sin(α/2) if m eveninteger cos(β/2)ifn isoddinteger, else sin(β /2) ifn eveninteger RadiatedAcousticPower by a Panel - 3May2011 25

39. Radiation Efficiency σ RadiatedAcousticPower by a Panel - 3May2011 26

40. Acoustical Analysis in Discrete Formulation By considering the expression so far calculated, we can analyze in discrete coordinates the acousticalbehaviour of a panel. Displacement in NG discrete coordinate: where H is the transfer matrix, F the modal matrix and F the force vector. Sincewe suppose a pressure load, the force vector isgiven by where p(w) is the acting pressure and A the nodalequivalent area matrix. A is a diagonalmatrix, and for the i-thgridpointitis RadiatedAcousticPower by a Panel - 3May2011 27

41. Acoustical Analysis in Discrete Formulation Radiatedacousticpower where RadiatedAcousticPower by a Panel - 3May2011 28

42. Acoustical Analysis in Discrete Formulation Incidentacousticpower Radiationefficiency where P𝒊𝒏𝒄𝝎=𝟏𝟐Real𝒑𝒊𝝎, 𝑸𝒊∙𝑽∗𝝎, 𝑸𝒊𝒅S RadiatedAcousticPower by a Panel - 3May2011 29

43. Acoustical Analysis in Discrete Formulation Modalradiationefficiency Modalradiationefficiency: analytical and FEA calculation RadiatedAcousticPower by a Panel - 3May2011 30

44. Acoustical Analysis in Discrete Formulation TransmissionLoss RadiatedAcousticPower by a Panel - 3 May2011 31