On Acoustic Radiation by Rotating Dipole Sources in Frequency Domain

Transcription

1 Moder Applied Sciece Vol. 5, No. 5; Ocober 211 O Acousic Radiaio by Roaig Dipole Sources i Frequecy Doai Zhihog Liu Ceer of Eergy ad Eviroe, QigDao Techological Uiversiy 11 FuShu Road, QigDao 26633, Chia Tel: E-ail: lzhqigdao@163.co Chuijie Yi Ceer of Eergy ad Eviroe, QigDao Techological Uiversiy E-ail: Chuijieyi@vip.163.co Received: May 18, 211 Acceped: July 23, 211 doi:1.5539/as.v55p8 Absrac I order o deal wih he acousic radiaio fro roaig sources, his paper describes a frequecy-doai uerical ehod for predicig soud radiaio. The ehod is based o he aalyical Gree s fucio of roaig oopole ad dipole source i free space. Soud radiaio odel is esablished ad characerisics of soud field are discussed by uerical siulaio ad he relaioship bewee radiaed soud frequecies ad acousic aure frequecy of source, agular frequecy ad is haroics ca be revealed.the radiaed soud field has a srog direciviy, fudaeal frequecy rasiig i he roary shaf direcio ad haroics spreadig alog radial direcio ad frequecy shif pheoea appearig clearly i higher roaig speed of source. The ehod has a heoreical sigificace for explorig he low-oise roaig achiery. Keywords: Acousic radiaio, Roaig dipole sources, Aalyical gree s fucio 1. Iroducio Due o he relaive oio bewee ediu ad soud source, he soud radiaio of ovig source has becoe he difficuly i acousic doai. More aeio o he coplexiy of soud geeraio of ovig soud sources has bee paid by soe auhors. P.M.Morse ad K.U.Zgard have bee discussed soud pressure of liear oio poi source usig he coordiae rasfor. This research is oly available for siple ovig sources ad for he coplex oio source he coordiae rasfor is difficul or ipossible. M.V.Lowso derived firsly acousic equaios fro roaig poi force.the equaios is applied o predic he soud radiaio of roor ad he predicio pressure is exac, bu he equaio is oly used o roor i free field ad i fac he roor is ofe i case or i box, he effec fro he boudaries is eglec. This will give ore iaccurae for he predicio pressure. I order o overcoe he disadvaage, K.D.Had D.J.Leeh as give soud field equaios of roaig poi force wih cylidrical boudary usig direc boudary elee ehod, Helholz equaio ad is oral derivaive ad W.H.Jeo ad D.J.Lee exeded his research, discussed cerifugal fa feaures of he soud source ad forecasig ehod ad Ha-Li Choid ad D. J. Lee bega o sudy roaig acousic source radiaio i ope hi wall pipe. These ehods have esablished he foudaio for soud radiaio of ovig source, bu here are soe probles eeded o overcoe furher. The assupio of poi source ad plae wave siplify he fac bu he ore iaccurae for he predicio will brig. Moreover, he soud pressure equaios are ofe give i ie doai. The ie odel is difficul o calculae ad obai he uerical soluio. I order o deal wih he probles above, i he prese paper he aalyical expressios of he pressure of roaig dipole i frequecy doai based o uerical Gree s fucio is derived i secio2. Characerisics of Soud field are give by uerical siulaio i secio3. The resuls are preseed i secio4.the aalyical ehod provided ipora heoreical value for he research of soud radiaio ad oise corol of ovig source. 2. Aalyical forulae of roaig dipole sources I his secio, he aalyical Gree s fucio ad soud pressure odel of roaig dipole sources is derived i he frequecy doai. Roaig dipole source s aalyical Gree's fucio ad soud field expressios i frequecy doai based o he roaig poi source Gree's fucio is derived. Two dipole for, he horizoal 8 ISSN E-ISSN

2 Moder Applied Sciece Vol. 5, No. 5; Ocober 211 (alog he orieaio) ad verical direcios (alog he z axis), are discussed. Two poi sources of opposie phase, as show i Figure.1 ad Figure. 2. We ca ge ulipole field by he parial derivaives of free-field roaig poi source s Gree s fucio ha are ake wih respec o he source coordiaes. By akig he Gree forula as a coo sarig poi. 2.1 Free-field aalyical Gree s fucio The Gree s fucio G(x, y,τ) is defied as he respose of he flow o a ipulsive poi source represeed by dela fucios of space ad ie: 2 1 G 2 2 c Where x y x y x y x y Where x, G x y , whe,he G. 1 x y 1 r s Gx, y,, 4 x y c 4rs c (2) are for he observer s posiio x a ie, y, are for he source s locaio ad ie r x y, c are for acousic velociy. Ĝ is he Fourier rasfored Gree s fucio defied by ˆ 1 G Gx, / y, i e d 2 Takig he Fourier rasfor o boh sides of he forula (2) ca ge Gree's fucio i frequecy-doai ˆ 1 i i r Gx, y,, e e s 2 8 rs Where is aure frequecy of source, / c is wave uber. Expadig forula (3) by Legedre ad Spherical haroic fucios i he spherical coordiaes: ikrs e i i b Y Y 4rs 4 b P By he Euler's forula ad addiio heore Where b (, ) (, ) 2 1 (cos ) (4) i( ) ( ) (cos ) i i i i i P a e e a e e e e j( r ) h r r r j( r) h r r r, 1/2, 1 1! a P P! cos cos is he agle bewee he vecor r ad r. Takig forula (4) io (3), Gree s fucio of roaig poi source i free space is give by ˆ i i i i i Gx, y,, 2 21bae e e e 8 Verical Gree's fucio of roaig dipole source is defied he derivaives of Ĝ ha is ake wih respec o he source coordiaes of he direcio of Z( ),which is show as below: Gˆ Gˆ / z ( Gˆ / r)cos (6) z 2 (2 1) i i i i 2 8 (7) Gˆ r/ r ( i ) cos Z b a e e e e j( r) h r r r Where b. j ( r) hr r r s (1) (3) (5) Published by Caadia Ceer of Sciece ad Educaio 81

3 Moder Applied Sciece Vol. 5, No. 5; Ocober 211 Horizoal Gree's fucio of dipole source is he derivaives of Ĝ alog he orieaio 1 ˆ ˆ G G r ˆ (2 1) G r/ r ( i / r)[ ] b 2 8 (8)! i i i i PP P P ie e e e 1! 2.2 Soud pressure odel ad Direciviy Soud pressure of roaig dipole source O he basis of he assupio, he acousic field is for a give ie-haroic source disribuio i Qs Qe. Soud pressure is give by 1 R pr, Qs Gd Qs d 4 R c (9) Takig he Fourier rasfor o boh sides of he forula (9) ca ge soud pressure i frequecy-doai 1 1 i i p r, Q ˆ s Ge dd Q e Gd 2 2 (1) Where Q, he Q is poi source sregh, Assupio he A is he Fourier rasfor of 1 i A Qe d 2 (11) For a give ie-haroic source, forula (11) is rewrie as Where is he source aural frequecy. A Q (12) The soud pressure of roaig dipole sources are preseed 2 (2 1)! i pz Ds( i ) cos 2 B P P cos e 8! i e i (21) B! p Ds 2 P Pcosi P P cos 8 r 1! i i e e We defied he dipole sregh D s Qd s, ad assue ha d is vaishigly sall Direciviy of roaig dipole source i r The far-field direciviy paer defied by reovig he e r defied by i r (13) (14) facor so ha he direciviy paer D, is li p r,, x ( e r) D, (15) far-field direciviy of roaig dipole source i r,, D, re G z z i r D re G (16) 82 ISSN E-ISSN

4 Moder Applied Sciece Vol. 5, No. 5; Ocober Nuerical Aalysis of Soud field Acousic field characerisics are give i he codiio of ie-haroic source. Siulaio paraeers: Aziuh of observaio poi /4,source aure frequecies are respecively23hz ad 68Hz,roaig agular frequecies are 112 rad/s ad 56 rad/s, acousic field frequecy is show k, k is haroic orders for arbirary ieger ha ses k 5,,5.For sybol cocisio i figures usig subsiue for ady subsiue for ha are evaluaed wih, /18, / 6, / 3, / 2 respecively. For far-field r 2ad r.3.siulaio resuls are deosraed i Figure. 3 ad Figure.4. 1I Figure.3, wih he icreasig of aure frequecies, he far-field soud pressure apliude of verical dipole icreases, he scope of haroic disribuio expad ad fudaeal frequecy copoe reduce as he observaio agle augeaio. Haroic disribuio is he os abuda i he observaio agle of /2.2There is oly fudaeal frequecy i he roaig axis direcio.3wih he icreasig of agular frequecies, he disribuio of haroics ad soud pressure apliude icrease. 4I Figure.4 soud pressure apliude is ipaced grealy by he aure frequecies. he scope of haroic disribuio expads wih he icreasig of view agle.5 Fudaeal frequecy is evidece i he sall observaio agle as ad /18.here is o haroic orders i he observaio agle of /2.6The aure ad agular frequecies chagig have grea effec o he haroic disribuio. Roaig dipole sources have a iesive space direciviy ha are show i Figure.5 ad Figure.6.Wih he icreasig of haroic orders, he direciviy of verical dipole becoes iesive. Especially, soud power focuses o he rage fro /6o /3 i Figure.5. I Figure.6 wih he icreasig of haroic orders, he direciviy of horizoal dipole becoes iesive. Especially, soud power focuses o he rage fro /3o /2.The disribuio fro horizoal ad verical roaig dipole source a differe observaio agle is show by Figure7.The oal soud pressure is aily fro horizoal roaig dipole source. 4. Coclusios A frequecy-doai aalyical ehod has bee developed for prediciog soud field of roaig diople source.the ehod is based o aalyical Gree s fucio i freqecy-doai.soud pressure odel ad he characerisics of soud field of ovig source are discussed. By uerical siulaios we ca ge ha he soud field frequecy ivolves aure frequecy,agular frequecy ad is haroics,he disribuio fro horizoal ad verical roaig dipole source, ad The iesive direciviy.the ehod ad resuls have ipora heoreical sigificace o he ovig source soud field characerisic aalysis ad explorig he low-oise desig of roary achie. Refereces Ha-Li Choi,D. J. Lee. (26). Develope of he uerical ehod for calculaig soud radiaio fro a roaig dipole source i a opeed hi duc. Joural of Soud ad Vibraio,295: doi:1.116/j.jsv , hp://dx.doi.org/1.116/j.jsv K.D. Ih, D.J. Lee. (1997). Develope of he direc boudary elee ehod for hi bodies wih geeral boudary codiios. Joural of Soud ad Vibraio, 22: LiuQiuhog,QiDaog. (24). A uberical ehod ad a aeroacosical odel of he cerifugal fa. Joural of XiA Jiao Tog Uiversiy,138(3): Lowso M V. (1965). The soud field for sigulariies i oio. Proceedig of he Royal Sociey i Lodo, Series A,286: P. M.Morse,K. U. Igard. (1968). Theoreical Acousics, McGraw-Hil. New York. W. H. Jeo, D. J. Lee. (2). A aalysis of geeraio ad radiaio of soud for a cerifugal fa. i: Seveh ICSV,121(3): W.H. Jeo, D.J. Lee. (2). A aalysis of he flow ad aerodyaic acousic sources of a cerifugal ipeller. Joural of Soud ad Vibraio, 222: doi:1.16/jsvi , hp://dx.doi.org/1.16/jsvi W.H. Jeo, D.J. Lee. (23). A uerical sudy o he flow ad soud fields of cerifugal ipeller locaed ear a wedge. Joural of Soud ad Vibraio, 266: WuJiuhui,CheHualig. (2). Acousic Soluio of Roaig Poi Source i Frequecy Doai. Joural of XiA Jiao Tog Uiversiy, 34(1): Published by Caadia Ceer of Sciece ad Educaio 83

5 Moder Applied Sciece Vol. 5, No. 5; Ocober 211 Zhihogliu, Chuijieyi,e al. (29). Research for Characerisics of Roaig Dipole Acousic Source i Spaial Acousic Field. Moder Applied Sciece,3(6): Figure 1. Horizoal roaig dipole source Figure 2. Verical roaig dipole source (a) =68Hz, =56rad/s (b) =68Hz, =112rad/s (c) =23Hz, =56rad/s (d) =23Hz, =112rad/s ) Figure 3. Soud pressure apliude disribuio of verical dipole wih observaio agle a differe aure ad agular frequecies 84 ISSN E-ISSN

6 Moder Applied Sciece Vol l. 5, No. 5; Ocober 211 (a =68Hz, =56rad/s ) (b) =68Hz, =112rad/s (c) =23Hz, =56rad/s (d) =23Hz, =112rad/s Figure 4. Soud pressure apliude disribuio of horizoal dipole wih observaio agle a differe aure ad agular frequecies (a)k=1 (b)k= =2 (c) k=3 (d)k= =5 Figure 5. The direciviy of verical dipole wih differe haroic orders a sae aure ad fudaeal frequecy Published by Caadia Ceer of Sciece ad Educaio 85

7 Moder Applied Sciece Vol. 5, No. 5; Ocober 211 (a)k=1 (b)k=2 (c)k=3 (d)k=5 Figure 6. The direciviy of horizoal dipole wih differe haroic orders a sae aure ad fudaeal frequecy (a) (b) /2 Figure 7. The disribuio fro horizoal ad verical dipole a differe observaio agle 86 ISSN E-ISSN