RESEARCH ON THE THEORIES OF BEM BASED CYCLOSTATIONARY NEAR FIELD ACOUSTIC HOLOGRAPHY. Haibin Zhang, Weikang Jiang and Quan Wan

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1 ICV14 Cairn Autralia 9-12 July, 2007 REEARCH ON THE THEORIE OF BEM BAED CYCLOTATIONARY NEAR FIELD ACOUTIC HOLOGRAPHY Haibin Zhang, Weikang Jiang and Quan Wan tate Key Laboratory of Mechanical ytem and Vibration, hanghai Jiaotong Univerity No.800, Dongchuan Road, hanghai , P. R. China, Abtract The cyclo-tationary vibrational ignal modulated by other ignal i an example of a non-tationary ignal found frequently in internal combution engine and rotating machinery. The modulation frequency component may be extracted from the total ignal uing econd order cyclic tatitic. In eviou reearch, the author ued the cyclo-tationary near field acoutic holography (CYNAH) technique in which the cyclic pectral denity function were ued to recontruct phyical quantitie. In thi paper the CYNAH method i combined with the boundary element method to overcome the limitation of the planar CYNAH method. The method ued in the conventional BEM baed NAH to overcome non-uniquene and the ill-poed nature of the recontruction are alo implemented here. Reult of a imulation how atifactory agreement between computed and analytical value. 1. INTRODUCTION The nearfield acoutic holography (NAH) [1 wa developed quickly to recontruct the ound field in lat twenty year. The NAH algorithm were developed in ome different way by combining with the different algorithm. Fat Fourier tranfer (FFT) apoach i implemented for radiating ource with imple ofile [1-2 which require the ource boundary to match the eparable geometrie of the acoutic wave equation uch a plana cylindrical and pherical geometrie. The boundary element method (BEM) and ingular value decompoition (VD) were applied to recontruct the ound field radiated from the ource with irregular ofile [3-4. The Helmholtz equation leat quare (HEL) [5 and wave uperpoition algorithm (WA) [6 could be conidered a the equivalent ource method, which were alo illutrated ueful for 3-dimenional complicated ource. All thee NAH method recontructed the ound field in the frequency or wave-number domain. ome reearcher recontructed the ound field in a range of the continuou frequency on the ource urface and then obtained the tranient ound information in time domain via invere FFT [7. Thi method wa utilized to locate the brake queal noie ource and engine knocking noie. The cyclo-tationary ignal modulated eriouly by other ignal i a pecial example of the non-tationary ignal, which i found frequently in the rotating machinery. The modulating wave component are uually reulted from periodic pule vibration, while the carrier wave

2 ICV July 2007 Cairn Autralia component are reulted from the free ocillation or other random diturbance. The cyclic tatitic of the cyclotationary ignal [8-9 uch a cyclic pectral denity (CD) are applied for diagnoing ome machinery fault. The type and mechanim of mechanical fault can be diagnoed by analyzing the ingle channel of cyclo-tationary ignal. Recontructing the cyclotationary ound field on the machine urface will upply much information for diagnoing the fault uch a location and type of mechanical fault. An algorithm named cyclo-tationary nearfield acoutic holography (CYNAH) [10-11 wa uggeted to recontruct the cyclotationary ound field on a planar ource urface. The CD hologram by CYNAH how the feature and location of modulating wave, ince the CD function i able to decompoe the modulating wave and carrier wave. In order to recontruct the cyclotationary ound field for irregularly haped ource, the CYNAH baed on BEM i eented here. 2. THE DERIVATION OF BEM-BAED CYNAH THEORY 2.1 NAH baed on BEM in the tationary ound field Helmholtz integral equation i the foundation of the NAH algorithm baed on BEM for tationary ound field, which can be evaluated a follow: C p( r) = p( r ) G( r ) iρω vn( r ) G( r ) d (1) n with 1 r V f, 0 r V, C = 1/ 2 r, i a mooth urface Ω / 4π r, i a nonmooth urface where, p(r) mean the complex eure in the field, p(r ) and v n (r) mean the eure and the normal velocity on the urface, repectively. V and Vf imply the interior volume and the jk r r G( r ) 4 exterior volume, repectively. = e π r r denote the boundary. i the free-pace Green function with wave number k=ω/c, where ω and c reeent the angular frequency and wave peed, repectively. n denote the outward normal direction to the ource urface. The coefficient C depend on the boundary moothne. Ω denote the olid angle. Equation (1) can be olved numerically by dicretizing the boundary into element and employing the hape function. Two matrix equation are formed including urface olution and field olution a follow: CP = DP M Vn (2) Ph = DhP M hvn (3) where, P h and P are the vector correponding to the meaured ound eure on the hologram and the unknown eure on the urface, repectively. V n mean the normal velocity on the urface. The ubcript h and denote the hologram and the ource urface, repectively. D, M, D h and M h are the tranfer matrixe. The relation between the hologram eure and the urface eure or the normal velocity can be built a follow: 1 P = [ D D M M V (4) h h h h h 1 h n P = [ D M M D P (5)

3 ICV July 2007 Cairn Autralia where, D = ( D C I ) with I meaning the identity matrix. Then, the eure and the normal velocity on the urface can be obtained in an inverely oce. 2.2 Cyclo-tationary : definition and opertie The definition of cyclo-tationary can be found in many paper and book about ignal oceing. Uually, a random ignal u(t) i conidered a n th order cyclo-tationary if it time-domain n th order moment i a periodical function of the time, t [8. The econd-order cyclo-tationarity, which i recently applied in mechanical fault diagnoi, i dicued a follow. The econd-order cyclic tatitical quantitie, cro-cd function, are ued a the tranform variable in CYNAH method. The CD function can be evaluated in temporal or frequency viewpoint. The baic equation of econd-order cro-cd in the frequency domain i introduced a follow: ( f ) = U ( f + 2) V ( f 2) (6) uv where the operator t and upercript denote the time-average and the conjugate operato repectively. U(f+/2) and V(f-/2) reeent the pectral component of cyclo-tationary ignal u(t) and v(t) at frequencie f+/2 and f /2, repectively. The frequency i the cyclic frequency. According to the filter operty of CD, CD can be obtained by a pair of cro-cd a follow: ( f ) = ( f ) ( f ) ( f ) (7) 2.3 CYNAH baed on BEM for cyclo-tationary ound field uu uv If the radiating ound ignal i cyclo-tationary, the pectrum of modulating wave and carrier wave are mixed up in the PD figure, and they can not be decompoed in the recontructed ound field by current NAH method. The cyclic tatitic of cyclo-tationary ignal i changing periodically and containing the information of the modulating wave, which i helpful to diagnoe ome mechanical fault. Recontructing the CD function on the urface from the CD function on the hologram can not be implemented directly becaue of the lack of phae information of CD function. Howeve the cro-cd function contain the phae information between urface and hologram. Theore, uing equation (7) to obtain CD after recontructing the cro-cd function on the urface firtly will be a good choice. uppoing a channel of eure ignal on ome field point a the erence, it pectrum p(r, f, t) depend on time becaue of the non-tationary nature. The equation (1) can be written a follow by multiplying p(r, f, t) to both ide: C p( f + p ( r, f = p( r, f + p ( r vn ( r, f + p ( r, f [ j2πρ( f + 2) G( r, f + 2) d where the upercript denote the conjugate operato and mean the cyclic frequency. Although the hidden parameter of frequency f in free-pace Green function i exhibited in equation (8), the actual meaning and exeion of Green function i the ame a that in equation (1). Temporal factor come up in the exeion of the pectral component of the eure and the normal velocity. It tree the non-tationary feature of cyclo-tationary ignal and indicate the temporal variation of pectral component. According to the vu, f n vv G( r, f + 2) t (8)

4 ICV July 2007 Cairn Autralia cyclo-tationary theory, equation (8) will be exeed by cro-cd function by time-averaging the both ide with time t a following: C ( f ) = r ( r, f ) G( r, f + 2) n (9) V ( r, f ) j2πρ( f + 2) G( r, f + 2) d r where, r V and r denote the cro-cd function at the frequency f with the cyclic frequency, repectively. In the ame way, the other derivation among comparative erie of cro-cd function can be decribed a follow: C p ( r, f + p( f = p ( r p ( r, f + p( r, f [ n n G( r, f, f + v ( r, f [ j2πρ( f 2) G( r, f 2) d 2) Time-averaging the both ide of equation (10), equation (11) can be obtained below: C rp ( f ) = r ( r, f ) [ G( r, f 2) n (11) Vr ( r, f ) [ j2πρ( f 2) G( r, f 2) d Comparing the equation (9) and (1), it can be found that the two equation are imilar to each other except that the econd-order tatitical quantitie take the place of the eure and the normal velocity. Thu, all ocedure applied to BEM-baed NAH, including dicretizing the boundary urface, utilizing the ioparametric quadratic hape function and aembling erie of linear equation into matrix mode, can alo be implemented to equation (9). Thee ocedure will not be introduced in detail here. The relationhip between a erie of cro-cd function vecto involved equation (9), will be given directly. The relationhip among the cro-cd function, urface-to-hologram and urface-to-urface, can be determined by two equation a follow: [ = [ Dh ( f + 2) [ r [ M h ( f + 2) [ Vr (12) C [ r = [ D ( f + 2) [ r [ M ( f + 2) [ Vr (13) [ [ where,, r [ V and r are the cro-cd function vector at the frequency f with the cyclic frequency between field eure, urface eure and urface normal velocity to the erence ignal, repectively. [ D h ( f + 2) [ M ( f + 2) and h are dipole and monopole tranfer matrixe between the hologram and the ource urface, repectively. [ D ( f + 2) and [ M ( f + 2) are the dipole and the monopole tranfer matrice from ource urface to itelf analogouly. It can be found that the tranfer matrice are imilar between the CYNAH and NAH except for the frequency hift. The uccinct exeion with concealed frequency (f+/2) for tranfer matrixe are obtained a follow: [ [ 1 [ [ [ = Dh M hm D r = G p r 1 [ = [ D D M M [ V = [ G [ V h h r v r (15) D = ( D CI ) where with I being an identity matrix. Thi pair of reultant ytem [ [ equation build the relationhip between and r [ V or r [. r [ V and r can be obtained inverely via equation (16) and (17) a follow: (10) (14)

5 ICV July 2007 Cairn Autralia 1 [ r [ G p [ 1 [ V [ G [ r = (16) = v (17) Another pair of matrix equation can be derived in the imilar way, and it uccinct exeion with concealed frequency (f-/2) for tranfer matrixe are a follow: [ rp [ [ 1 [ = [ [ r G p rp 1 [ V = [ G [ r v where,, r V and r are the comparative cro-cd function vector at frequency [ [ r [ V r f with cyclic frequency relative to, and, repectively. The correponding frequencie in different tranfer matrixe and conjugate operator are fallible if carelene. The CD vector of the eure on the urface can be obtained after the harvet of a pair of cro-cd vector of between the eure on the urface and the erence. o the CD vector of the eure on the urface doe. The CD function on the i th urface node, at the frequency f with the cyclic frequency, can be evaluated by equation (20) and (21) a follow: rr V i i ri ri rp rr (18) (19) = (20) = V V (21) ri ri where, i the CD function of the erence ignal, ubcript i mean the i th element of vector correponding to the i th node. Thu, the urface CD function of either eure or normal velocity are obtained and CD at any point on the urface can be evaluated via the hape function. By comparing the BEM-baed CYNAH and the BEM-baed NAH, it can be found that the ill-poed nature and non-uniquene oblem alo occur in the CYNAH algorithm. Theore, the truncated VD (ingular value decompoition) ocedure [12 i adopted to overcome the ill-poed nature in CYNAH. The combined Helmholtz integral equation formulation (CHIEF) method [13 i implemented to eliminate the non-uniquene oblem happening on the oblematic frequency. rr 3. IMULATION TUDY OF CYNAH ALGORITHM BAED ON BEM 3.1 imulation A numeric imulation cae i tudied in thi ection to verify CYNAH algorithm baed on BEM. It i a cylinder with two pherical end-cap encloing a finite length line ource inide, which determine the normal velocity and eure on the cylinder urface (ee figure1). The pulating ignal of the line ource are cyclotationary. The reaon of chooing thi numeric cae i that the facility of ynthetically generating eure and normal velocity for location collocated on urface and hologram. Figure 2 i the diagram of the portion of the cylinder with two pherical end-cap, including 40 element and 122 node. The input parameter are elected a r 0 =0.3m, a=0.01m, and length L=0.5m. The vibrating velocity ignal on the finite-length line ource in temporal mode i evaluated by equation (22) a follow: v( t) = a( t) co(2πf t) (22) a

6 ICV July 2007 Cairn Autralia X 2a r o Z Y L Figure 1. Cylinder radiator with pherical end-cap pherical encloing linear cyclotationary vibrator. Figure2. The partition plot of cylinder with two end-cap, 40 element and 122 node where a(t) i a purely tationary random ignal with zero mean and f a =200 Hz. The velocity ignal i a cyclotationary amplitude-modulation (AM) ignal with the unique cyclic frequency 400 Hz. The carrier wave co(2πf a t) i modulated by a(t). It i eay to ee that the pectrum of modulation ignal a(t) become two ideband around f a under the effect of carrier wave by mean of comparing the PD drawing of v(t) and a(t) a hown in figure 3(a) and (b). In figure 3(c), the CD function of v(t) remove the diturbance of carrier wave and extract the characteritic of modulation wave a(t) olely. The normal velocity and the eure on the urface and hologram can be evaluated by equation (23) and (24) a follow: ρ / 2 0cU ( k) ka L 1 jk r z p( k) = j 2 e dz L / 2 r z (23) U ( k) a L / 2 jk r z 1 jk r z r z vn ( k) = e dz / L r z n (24). where, r mean the coordinate of urface point or field point, z mean the coordinate of point on line ource, a mean the radiu of the line ource with the finite length L, n i the outward normal direction. U(k) in the equation (23) and (24) i the normal velocity on the line ource urface. The PD of the ound eure radiated from the ource i hown in figure 4(a), which i a little different from the PD of v(t) in figure 3(b) becaue of the influence of tranfer function. Howeve the ound eure i till cyclotationary. The CD of the ound eure at =400 Hz i hown in figure 4(b), and the pectral characteritic of the modulating component are hown clearly. (a). The PD plot of a(t) (b) The PD plot of v(t) (c) The CD plot of v(t), =400 Hz Figure 3. Analyi of the imulation velocity ignal

7 ICV July 2007 Cairn Autralia (a) PD of the eure ignal. (b) CD of eure ignal, = 400 Hz Figure 4. Analyi of the imulation acoutic eure ignal for erence point 3.2 The CYNAH recontruction of ound field A cylindrical hologram i et to meaure the ynthetically generating eure in the ound field, which i 1.6m long with the radiu 0.4m. The hologram i compoed of 224 point with angle pace Δθ=π/8 radian and height pace Δz=0.12m between the neighbouring point. The hologram i around the ource urface axially. The erence point i aumed to be et at coordinate (0.35, 0, 0). The urface element and node are et a hown in figure 2. The reult of contruction plotted in figure 5 are only for the node howed in figure 2, which locate on a generatrix of cylindrical urface, becaue the acoutic field i axially ymmetrical. The recontructed frequency i 50Hz under the cyclic frequency 400Hz. The recontruction error i ε m = a a 100% defined a, where and a are recontructed CD vector and analytical CD vecto repectively, i the norm operator. The recontructed CD function without any noie diturbance i hown in figure 5(a). The error i equal to 1.91% which i o accurate becaue of the lack of noie diturbance. The recontruction error increae when the noie i added to the ynthetically generating eure in the ound field. A zero mean tationary random white-noie i added to the ound eure on the hologram, which i independent of the ource and 10dB lower than the mallet ound eure level on the hologram. The recontructed CD function on the urface i plotted in the figure 5(b). The error increae to 3.4% and the reult i till atifactory..(a) without diturbance of noie (b) diturbed by noie :computed, : analytical Figure 5. Recontruction of CD for cylindrical radiator at f=50hz, with cyclic frequency = 400 Hz

8 ICV July 2007 Cairn Autralia 4. CONCLUION CYNAH baed on BEM i developed for tudying the information of acoutic field radiated from cyclo-tationary ource with irregular ofile. The BEM-baed CYNAH algorithm extend the application of NAH method into cyclo-tationary ound field, epecially for the acoutic ource with complicated ofile found frequently in internal combution engine and rotating machinery. It i hown in imulation that the recontructed ound filed agree with the analytical one with quite well accuracy. The CD hologram by CYNAH how the location and feature of the modulating wave, which i helpful for undertanding the modulating mechanim. ACKNOWLEDGEMENT Thi work i aided by the Natural cience Fund of China (No ). REFERENCE [1 J. D. Maynard, E. G. William, Y. Lee, Nearfield acoutic holograhy: I. Theory of generalized holography and the development of NAH, Journal of the Acoutical ociety of America 78, (1985). [2 E. G. William, H. D. Dardy, Generalized nearfield acoutic holography for cylinderal geometry: theory and experiment, Journal of the Acoutical ociety of America 81, (1987). [3 W. A. Veronei, J. D. Maynard, Digital Holographic Recontruction of ource with Arbitrarily haped urface, Journal of the Acoutical ociety of America 85, (1989). [4 M. Bai, Application of BEM(boundary element method)-baed acoutic holography to radiation analyi of ound ource with arbitrarily haped geometrie, Journal of the Acoutical ociety of America 92, (1992). [5 Z. Wang,. F. Wu, Helmholtz equation leat quare method for recontructing the acoutic eure field, Journal of the Acoutical ociety of America 102, (1997). [6 L. ong, G. Koopmann, J. Fahnline, Numerical error aociated with the method of uperpoition for computing acoutic field, Journal of the Acoutical ociety of America 89, (1991). [7 J. Hald, Time domain acoutical holography and it application, Journal of ound and Vibration 35, 16-25(2001). [8 W.A. Gardne Exploitation of pectral redundancy in cyclotationary ignal, IEEE ignal Proceing Magazine 8, 14 36(1991). [9 G.K. Yeung, W.A. Gardne earch efficient method of detection of cyclotationary ignal, IEEE Tranaction on ignal Proceing 44, (1996). [10 Q. Wan, W.K. Jiang, Near field acoutic holography (NAH) theory for cyclotationary ound field and it application, Journal of ound and Vibration 290, (2006). [11 Q. Wan, W.K. Jiang, Near field acoutic holography for cyclotationary ound field and it partial ource decompoition ocedure, Journal of Vibration and Acoutic, Tranaction of the AME 127, (2005). [12 E.G. William, et al., Interior near-field acoutical holography in flight, Journal of the Acoutical ociety of America 108, (2000). [13 H. A. chenck, Imoved integral formulation for acoutic radiation oblem, Journal of the Acoutical ociety of America 44, 42 58(1968).