Source Visualization by Using Statistically Optimized Near-Field Acoustical Holography in Conical Coordinates

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1 Purdue University Purdue e-pubs Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering 8-01 Source Visualiation by Using Statistically Optimied Near-Field Acoustical Holography in Conical Coordinates J Stuart Bolton Purdue University, bolton@purdue.edu Yong T. Cho Kyungil University Follow this and additional works at: Bolton, J Stuart and Cho, Yong T., "Source Visualiation by Using Statistically Optimied Near-Field Acoustical Holography in Conical Coordinates" (01). Publications of the Ray W. Herrick Laboratories. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information.

2 SOURCE VISUALIZATION BY USING STATISTICALLY OPTIMIZED NEAR-FIELD ACOUSTICAL HOLOGRAPHY IN CONICAL COORDINATES Yong T. Cho* and J. Stuart Bolton Ray W. Herrick Laboratories Purdue University *Currently working at School of Mechanical and Automotive Engineering, Kyungil University, Korea 1

3 INTRODUCTION NAH is a useful tool for visualiing noise sources throughout a 3D space. - Fast since implemented using spatial Fourier transform. - Needs ero padding of measurement results to avoid wrap-around error. - Meaningless velocity results close to measurement edge due to discontinuity. Statistically Optimied Nearfield Acoustical Hologrpahy - First introduced by Jørgen Hald in planar coordinates - No spatial Fourier transform involved. - More accurate result over entire measurement area.

4 Aeroacoustic sources are more closely defined in conical geometry NAH: non-regular geometries X SONAH: conical geometry O Wave functions in conical geometry are formulated by modifying cylindrical wave functions 3

5 Conical geometry definition Conical array parallel to surface of diverging flow Reconstruction surface hologram r= r o + tan α r α r s Cylindrical array intersects flow r o r ho Rigid Boundary, =0 y ø r x r o 4

6 The sound pressure, p(r), can be expressed as linear combination of the measured sound pressure p(r n ), N n= 1 p() r c () r p( r ) n n If a good representation of the sound field can be obtained by using a finite subset of wave functions, the coefficients c n can be determined. Φ SONAH in Conical Coordinates - Conical SONAH formulation (1) N ( r) c ( r) Φ ( r ), m = 1... M Km n Km n n= 1 5

7 - Conical SONAH formulation () 1 H ( kr) imφ ik p(, r φ, ) = P ( r, k ) e e dk ( ) m= () 1 m r () 1 m π = Hm kr r s m s Defining wave function Φ k, m ( r) in conical coordinates, Φ () 1 m k, m ( ) () 1 Hm H ( kr) e e ( kr) r imφ ik r, r s k r = i k k k k for for k k < k k H ( k ( r + tanα )) Φ (r) Φ (,, ),. () 1 m r o imφ ik k, m = k, mr φ e e r r () 1 s Hm( kr r s) 6

8 - Conical SONAH formulation (3) + * A A = Φk, (r, ) Φ, (r, ) q m h j kq m h i ji SONAH in Conical Coordinates m= q= = m= q= H ( k ( r + tanα)) H ( k ( r + tanα)) ( ( ) ( ) ) } imφhi, φh, j + kq hi, h, j e + * A α = Φ (r ) Φ (r) q,, q, *() 1 () 1 m r ho h, j m r ho h, i () 1 Hm( kr r s) j k m h j k m m= q= *() 1 () 1 Hm kr rho + h, jtanα Hm kr ro + i = () 1 m= q= Hm( kr r s) ( ( )) ( ( tanα)) ( ( ) ( ) ) } imφi φh, j + kq i h, j e 7

9 Estimated pressure p(r) is, r N n r αr rn = T pcr = T p ( A A+ I) 1 A n= 1 p() c () p( ) () θ () where, p T is measured pressure vector at r n Estimated radial particle velocity u r (r) is, ur ( θ ) + T + r p A A () βr I 1 + A () where, SONAH in Conical Coordinates - Conical SONAH formulation (4) A + β(r) pressure and particle velocity. is a correlation vector that relates measured 8

10 Dipole numerical simulation r ho = cm r o = 9 cm N Φ = 3 N = 17 α = 15 o 9

11 - Dipole numerical simulation (1000 H) Directly measured and backward projected pressure Directly measured p r ho = cm Directly measured p r ho = 9 cm Back projected p r o = 9 cm (MSE : % ) 10

12 - Dipole numerical simulation (1000 H) Directly measured and backward projected particle velocity Directly measured u r r ho = 9 cm Back projected u r r o = 9 cm (MSE : 0.17 % ) 11

13 - Conical SONAH measurement Microphone arrays and loudspeakers Spatially averaged pressure (r o =10.6 cm, 15.6 cm, α =15 ) (r o =10.6 cm) p [db] frequency [H] inc = cos α cm, N Φ = 3, N = 5 1

14 - Conical SONAH measurement result (1), 684 H Measured p, r o = 10.6 cm Measured p, r o = 15.6 cm Back projected p Back projected u r Back Projected p, cylindrical Back Projected u r, cylindrical 13

15 - Conical SONAH measurement result (), 648 H Measured p, r o = 10.6 cm SONAH in Conical Coordinates Measured p, r o = 15.6 cm Back projected p Back projected u r Back Projected p, cylindrical Back Projected u r, cylindrical 14

16 - Conical SONAH mean square error Directly measured and back projected pressure Two loudspeakers r o =15.6 cm to 10.6 cm (α =15 ) Frequency (H) MSE (%) Directly measured and back projected pressure, velocity Numerical dipole simulation r o =14.15 cm to 9 cm (α =15, 1000 H) Pressure Velocity MSE (%)

17 - Computation effort required + + Estimate N Ø N by N Ø N square matrices AA α A ji, j number of elements in SONAH matrices Hologram Square Cylindrical Conical Arbitrary Elements N Ø N N Ø N 4N Ø N 4N Ø N N Ø =3, N =17 544* N Ø =3, N =5 800* N Ø =3, N = * (0.5) 176 (1) (68) (176) 16

18 - Conclusions SONAH in Conical Coordinates Conical SONAH accuracy confirmed Numerical simulation and loudspeaker measurement Reasonable to use cylindrical wave functions for conical geometry More detailed visualiation of sources by back projection from conical to cylindrical surfaces Cylindrical and conical SONAH matrix quite different computation time conical SONAH >> cylindrical SONAH Finite difference calculation would be required to calculate particle velocity normal to conical surface (and hence intensity and sound power) 17