Farfield analysis of noise sources

• US 7,330,396 B2
• Filed: 10/13/2004
• Issued: 02/12/2008
• Est. Priority Date: 10/13/2004
• Status: Active Grant

• Current Assignee: Wayne State University
• Original Assignee: Wayne State University
• Inventors: Wu, Sean F.
• Primary Examiner(s): PIHULIC, DANIEL T
• Application Number:
  US10/964,393
• Publication Number:
  US 20060080418A1
• Time in Patent Office: 1,217 Days
• Field of Search: 367/13
• US Class Current: 367/13
• CPC Class Codes: G01H 17/00 Measuring mechanical vibrations or ultrasonic, sonic or infrasonic waves, not provided for in the preceding groups see provisionally also G01H1/00
• CPC Class Codes: G01H 3/125 for representing acoustic field distribution using optical means G01H9/002; sonar systems for imaging G01S7/56, G01S15/89; acoustic holography G03H3/00

Abstract

An algorithm diagnoses the noise source strength distribution on an arbitrarily shaped object based on acoustic pressures measured in the far field, known as FANS. FANS enable one to acquire a quick estimate of the acoustic pressure at locations that are off limit to traditional measurement microphones. Generally, in the method of the present invention, the noise source is modeled in terms a plurality of virtual spherical wave sources distributed on an auxiliary surface conformal to a source boundary from the inside, but not on the source boundary itself. Sound is then measured at a plurality of measurement points external to the source boundary. The sound field is reconstructed on the source boundary surface itself.

Claims

1. A method for diagnosing a noise source including the steps of:

a) modeling the noise source strength in terms of a distribution of plurality of virtual spherical wave sources conformal to a source boundary from the inside, but not on the source boundary;

b) measuring sound at a plurality of points external to the source boundary; and

c) reconstructing sound field on the source boundary surface other than the plurality of points based upon said steps a) and b).

2. The method of claim 1 wherein said step c) further includes the step of reconstructing sound at the plurality of virtual sources.

3. The method of claim 1 further including the step of distributing the plurality of virtual spherical wave sources on an auxiliary surface conformal to a boundary of the source.

4. The method of claim 1 wherein the number of the plurality of points is approximately four times the number of virtual spherical wave sources.

5. The method of claim 1 wherein the reconstructed sound is given by:

$\hat{p}<br><br>\left(x;<br><br>\mathrm{\omega <br><br>}\right)=\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br><br><br>\left\{C_n<br><br>h_0^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)+h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>\left[D_n<br><br>Y_1^0<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)+E_n<br><br>Y_1^{-1}<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)+F_n<br><br>Y_1^1<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)\right]\right\},$

where h₀⁽¹⁾(kr_n)and h₁⁽¹⁾(kr_n) are the first kind of the spherical Hankel functions of order 0 and 1, respectively, and Y₁⁰(θ

_n,φ

_n), Y₁<sup>−

1</sup>(θ

_n,φ

_n) and Y₁¹(θ

_n,φ

_n), are the spherical harmonics.

6. The method of claim 1 wherein the expansion coefficients are obtained by

$\begin{array}{c}C=\arg \end{array}$

⁢

min ⁢

{ 

W Γ

⁡

( Φ

Γ

⁢

ξ

- p ^ Γ

) 

2 2 + λ

2 ⁢



W S ⁡

[ Φ

S Ψ

S ] ⁢

ξ



2 2 } , ( 10 )

where W_S denotes a H×

H non-singular diagonal weight matrix whose elements depend on the quadrature formula and the mesh on the target source surface S, Ψ

_S is the normal derivative of Φ

_S, which is given by [Ψ

_S]_m=∂

_vΦ

_m(x^S;

ω

), =1to H and m=1 to M, and x^S represent the points on the target surface S, and Φ

_Γ

is given by

${\mathrm{\Phi <br><br>}}_{\mathrm{\Gamma <br><br>}}={\left[\begin{array}{cccccc}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \\ \mathrm{\cdots <br><br>}& h_0^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)& h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^0<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)& h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^{-1}<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)& h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^1<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)& \mathrm{\cdots <br><br>}\\ & \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \end{array}\right]}_{M\times <br><br>4<br><br>N}.$

7. The method of claim 1 wherein the number of the plurality of points is approximately two times the number of virtual spherical wave sources.

8. The method of claim 1 wherein the reconstructed sound can alternatively be determined by:

$\hat{p}<br><br>\left(x;<br><br>\mathrm{\omega <br><br>}\right)=\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br><br><br>\left[C_n<br><br>h_0^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)+D_n<br><br>h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^0<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)\right],$

9. The method of claim 1 wherein the expansion coefficients are obtained by

$\begin{array}{c}C=\arg \end{array}$

⁢

min ⁢

{ 

W Γ

⁡

( Φ

Γ

⁢

ξ

- p ^ Γ

) 

2 2 + λ

2 ⁢



W S ⁡

[ Φ

S Ψ

S ] ⁢

ξ



2 2 } , ( 10 )

where W_S denotes a H×

H non-singular diagonal weight matrix whose elements depend on the quadrature formula and the mesh on the target source surface S, Ψ

_S is the normal derivative of Φ

_S, which is given by [Ψ

_S]_m=∂

_vΦ

_m(x^S;

ω

), =1 to H and m=1 to M, and x^S represent the points on the target surface S, and Φ

_Γ

is given by

${\mathrm{\Phi <br><br>}}_{\mathrm{\Gamma <br><br>}}={\left[\begin{array}{ccc}& \mathrm{\cdots <br><br>}& \\ \mathrm{\cdots <br><br>}& h_0^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br><br><br>h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^0<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)& \mathrm{\cdots <br><br>}\\ & \mathrm{\cdots <br><br>}& \end{array}\right]}_{M\times <br><br>2<br><br>N}.$

10. A system for diagnosing a noise source comprising:

a plurality of virtual spherical wave noise sources;

a plurality of transducers for measuring sound at a plurality of measurement points; and

a computer diagnosing the noise source based upon the sound measured at the plurality of measurement points and the based upon the virtual spherical wave noise sources.

11. The system of claim 10 wherein the plurality of virtual spherical wave sources are distributed on an auxiliary surface conformal to a boundary of the noise source.

12. The system of claim 10 wherein the computer reconstructs sound at a plurality of points other than the plurality of measurement points.

13. The system of claim 10 wherein the computer reconstructs sound at the plurality of virtual spherical wave noise sources.

14. The system of claim 10 wherein the number of the plurality of measurement points is approximately four times the number of virtual spherical wave sources.

15. The system of claim 10 wherein the number of the plurality of measurement points is approximately two times the number of virtual spherical wave sources.

16. The system of claim 10 wherein the reconstructed sound is given by:

$\hat{p}<br><br>\left(x;<br><br>\mathrm{\omega <br><br>}\right)=\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br><br><br>\left\{C_n<br><br>h_0^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)+h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>\left[D_n<br><br>Y_1^0<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)+E_n<br><br>Y_1^{-1}<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)+F_n<br><br>Y_1^1<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)\right]\right\},$

where h₀⁽¹⁾(kr_n) and h₁⁽¹⁾(kr_n) are the first kind of the spherical Hankel functions of order 0 and 1, respectively, Y₁⁰(θ

_n,φ

_n), Y₁<sup>−

1</sup>(θ

_n,φ

_n) and Y₁¹(θ

_n,φ

_n), are the spherical harmonics, and

${\mathrm{\Phi <br><br>}}_{\mathrm{\Gamma <br><br>}}={\left[\begin{array}{cccccc}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \\ \mathrm{\cdots <br><br>}& h_0^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)& h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^0<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)& h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^{-1}<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)& h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^1<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)& \mathrm{\cdots <br><br>}\\ & \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \mathrm{\cdots <br><br>}& \end{array}\right]}_{M\times <br><br>4<br><br>N}.$

17. The system of claim 10 wherein the reconstructed sound is determined by:

$\hat{p}<br><br>\left(x;<br><br>\mathrm{\omega <br><br>}\right)=\underset{n=1}{\overset{N}{\sum <br><br>}}<br><br><br><br>\left[C_n<br><br>h_0^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)+D_n<br><br>h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^0<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)\right],$

where C_n and D_n is solved by minimizing the least-square errors between an approximated solution {circumflex over (p)} and the acoustic pressure {circumflex over (p)}_Γ

measured on Γ

, where Φ

_Γ

is a matrix containing h₀⁽¹⁾(kr_n), h₁⁽¹⁾(kr_n) and Y₁⁰(θ

_n,φ

_n) only,

${\mathrm{\Phi <br><br>}}_{\mathrm{\Gamma <br><br>}}={\left[\begin{array}{ccc}& \mathrm{\cdots <br><br>}& \\ \mathrm{\cdots <br><br>}& h_0^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br><br><br>h_1^{\left(1\right)}<br><br>\left({\mathrm{kr}}_n\right)<br><br>Y_1^0<br><br>\left({\mathrm{\theta <br><br>}}_n,{\mathrm{\varphi <br><br>}}_n\right)& \mathrm{\cdots <br><br>}\\ & \mathrm{\cdots <br><br>}& \end{array}\right]}_{M\times <br><br>2<br><br>N}.$