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1 Univerity of Southmpton Reerch Repoitory eprint Soton Copyright nd Morl Right for thi thei re retined by the uthor nd/or other copyright owner. A copy cn be downloded for peronl non-commercil reerch or tudy, without prior permiion or chrge. Thi thei cnnot be reproduced or quoted extenively from without firt obtining permiion in writing from the copyright holder/. The content mut not be chnged in ny wy or old commercilly in ny formt or medium without the forml permiion of the copyright holder. When referring to thi work, full bibliogrphic detil including the uthor, title, wrding intitution nd dte of the thei mut be given e.g. AUTHOR (yer of ubmiion) "Full thei title", Univerity of Southmpton, nme of the Univerity School or Deprtment, PhD Thei, pgintion

2 UNIVERSITY OF SOUTHAMPTON FACUTY OF ENGINEERING AND APPIED SCIENCE INSTITUTE OF SOUND AND VIBRATION RESEARCH THE INFUENCE OF STRUCTURA-ACOUSTIC COUPING ON THE DYNAMIC BEHAVIOUR OF A ONE-DIMENSIONA VIBRO-ACOUSTIC SYSTEM by Gihwn Kim A thei ubmitted for the degree of Doctor of Philoophy Jnury 009

3 UNIVERSITY OF SOUTHAMPTON ABSTRACT Fculty of Engineering nd Applied Science Intitute of Sound nd Vibrtion Reerch Doctor of Philoophy THE INFUENCE OF STRUCTURA-ACOUSTIC COUPING ON THE DYNAMIC BEHAVIOUR OF A ONE-DIMENSIONA VIBRO- ACOUSTIC SYSTEM by Gihwn Kim The im of thi thei i to invetigte the tructurl-coutic coupling effect on the dynmic behviour of vibro-coutic ytem under pive/ctive control. The implet model of vibro-coutic ytem one cn conider i one-dimenionl coutic cvity driven by ingle-degree-of-freedom (SDOF) tructure. Thi imple model i ued to demontrte the phyicl chrcteritic of the coupling phenomenon. Thi imple nlyticl model cn provide vriou degree of tructurl-coutic coupling, which re dependent upon (i) the tructurl-coutic tiffne rtio, (ii) tructurl-coutic nturl frequency rtio, (iii) tructurl dmping, nd (iv) coutic dmping. In thi ce, lthough the geometric coupling fctor i not included becue the SDOF tructure h ingle mode, 80 percent of the fctor tht determine the degree of coupling cn be ccounted for by the imple nlyticl model. The coupling mechnim, in the imple vibro-coutic ytem, i invetigted uing the mobilityimpednce pproch. In order to provide the threhold of the degree of coupling, coupling fctor i clculted in term of non-dimenionl tructurl-coutic prmeter. Vibrocoutic repone re repreented by the coutic potentil energy in the cvity nd the kinetic energy of the tructure coupled to the coutic cvity. The vibro-coutic repone re invetigted for vriou coupled ce. The principle re demontrted by controlling the coutic potentil energy in the one-dimenionl finite coutic tube driven by the SDOF tructure. Three control trtegie re pplied; pive control, ctive feedforwrd control nd decentrlied velocity feedbck control. Pive control i invetigted to chieve phyicl inight into the reltive benefit of pive control i

4 tretment. In the more trongly coupled ce, couticl modifiction were preferble for the reduction of the coutic potentil energy. On the other hnd, in the more wekly coupled ce, tructurl modifiction were more effective. For hrmonic diturbnce, n ctive feedforwrd control trtegy i conidered for the control of the coutic potentil energy in the cvity driven by the tructure under externl hrmonic excittion. For the ctive feedforwrd control ytem, thi tudy ue the concept of optiml impednce, which i defined the rtio of the control force to the velocity of econdry ource when the coutic potentil energy i minimied. In the more trongly coupled ce, ll the coutic mode were effectively uppreed t the reonnce frequencie. On the other hnd, in the more wekly coupled ce, ll the coutic mode were controllble in the more trongly coupled ce. However, the tructurl mode w generlly uncontrollble. For brodbnd diturbnce, decentrlied velocity feedbck control i formulted to invetigte the reltive control effectivene of tructurl nd coutic ctutor for the control of the coutic potentil energy. In the more trongly coupled ce, the control configurtion of uing the coutic ctutor w preferble. On the other hnd, in the more wekly coupled ce, the decentrlied velocity feedbck control trtegy uing both the ctutor w deirble. ii

5 CONTENTS Chpter 1. Introduction Bckground 1 1. iterture review Modelling of vibro-coutic ytem Pive control Active control Aim nd objective Contribution Thei tructure 16 Chpter. Anlyticl Model of Vibro-coutic Sytem 19.1 Introduction 19. Arbitrry impednce terminted ytem 1..1 Acoutic input impednce uing the impednce pproch 1.. Non-dimenionl tructurl-coutic prmeter 7..3 Acoutic preure nd prticle velocity 8.3 Structurl-coutic coupling in vibro-coutic ytem Conceptul repreenttion of vibro-coutic ytem Coupling fctor 33.4 Vibro-coutic repone in vriou coupled ce Acoutic potentil energy in n coutic cvity Kinetic energy of tructure coupled with n coutic cvity Simultion reult on vibro-coutic repone 4.5 Concluion 49 iii

6 Chpter 3. Pive Control of Acoutic Potentil Energy in Vibro-coutic Sytem Introduction Pive control of coutic potentil energy Effect of chnging tructurl-coutic tiffne rtio - modifying tructurl tiffne Effect of chnging tructurl-coutic nturl frequency rtio ω / ω - modifying tructurl m Effect of chnging tructurl lo fctor η - modifying tructurl dmping Effect of chnging n coutic lo fctor η - modifying coutic dmping Summry of pive tretment on coutic potentil energy Pive control of coutic potentil energy uing n borptive medium Experimentl invetigtion on vibro-coutic ytem Experimentl etup Structurl modifiction of loudpeker Meuring coutic potentil energy Experimentl reult Concluion 91 K / K Chpter 4. Active Feedforwrd Control of Acoutic Potentil Energy in Vibro-coutic Sytem Introduction 9 4. Anlyticl model Acoutic potentil energy under feedforwrd control Primry ource contribution Secondry ource contribution Minimition of coutic potentil energy 97 iv

7 4.4 Optiml impednce of feedforwrd control ytem Feedforwrd control effect on tructurl kinetic energy Cumultive um of coutic potentil energy Experimentl vlidtion on coutic potentil energy under feedforwrd control Experimentl etup Experimentl reult Concluion 135 Chpter 5. Decentrlied Velocity Feedbck Control of Acoutic Potentil Energy in Vibro-coutic Sytem Introduction Velocity feedbck control uing n coutic ctutor Acoutic repone due to n coutic ctutor Optiml feedbck gin for vriou coupling condition Control of coutic potentil energy uing n coutic ctutor Velocity feedbck control uing tructurl ctutor Acoutic repone due to tructurl ctutor Feedbck gin for vriou coupling condition Control of coutic potentil energy uing tructurl ctutor Decentrlied velocity feedbck control uing both ctutor Acoutic repone due to decentrlied controller Feedbck gin for vriou coupling condition Control of coutic potentil energy uing decentrlied controller Velocity feedbck control effect on tructurl kinetic energy Cumultive um of coutic potentil energy Concluion 185 Chpter 6. Concluion Concluion Recommendtion for future work 191 v

8 Appendice 19 A Anlyi of low-frequency pproximte one-dimenionl vibro-coutic ytem 19 B Sound preure t rbitrry poition in tube with two different medi uing the trnfer mtrix method 03 C Integrl of the modulu qured frequency repone function 09 Reference 11 vi

9 IST OF FIGURES Figure1.1 Figure 1. Combined flexible tructure 3D rbitrry encloure ytem where the flexible tructure i under n externl excittion force F( x, ω ), which i cting in the x direction t driving frequency ω Combined ingle-degree-of-freedom (SDOF) tructurl driver 1D (onedimenionl) finite cloed tube ytem where the SDOF tructure i under n externl excittion force F( x, ω ) in the x direction t the frequencyω nd the 1D finite cloed tube i urrounded by rigid wll Figure.1 Acoutic preure nd prticle velocity repreenttion of combined tructure coutic tube ytem which i under the externl time hrmonic force on the j t tructure, f () t = Fe ω 0 t x = 0 nd i terminted by rbitrry impednce Z t x =. M nd K re tructurl m nd tiffne of pring with contnt tructurl lo fctor η repectively. Alo, j = 1 nd ω i driving frequency Figure. () coupled ytem repreented by uncoupled tructurl impednce Z S nd uncoupled coutic impednce Z A0 rection force F, where the driving force 0 ; (b) tructurl force F nd coutic F = F + F Figure.3 A block digrm repreenttion of eqution (.39) where F 0 i the input force pplied to the tructure, U S i output velocity t the input poition ( x = 0 ), YS i the uncoupled tructurl mobility nd Z A0 i the uncoupled coutic impednce Figure.4 Modulu of coupling fctor for vriou tructurl-coutic tiffne rtio K / K where tructurl lo fctor = 10 nd n coutic lo 3 fctor = 10 (olid line: K / K = 10, dhed line: K η dotted line: K / K = 1) η / K 10 3 = nd Figure.5 () coutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in ech coupled ce (b) tructurl kinetic energy rbitrrily normlied by tht t / λ = where the tructurl nturl frequency i t / λ = 0.1(ω / ω = 5), the contnt tructurl nd coutic lo fctor repectively (olid line: trongly coupled ce with wekly coupled ce with with K / K = 1) K η = 10 K / K = 10 nd η = 10, dhed line 3 3 = nd dotted line: intermedite ce / K 10 Figure.6 Opertionl deflection hpe of the coutic preure in the cvity normlied 3 by mximum modulu: () trongly coupled ce with K / K = 10, (b) wekly vii

10 coupled ce with K 3 = nd (c) intermedite ce with K / K = 1 / K 10 where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5 ), the contnt tructurl nd coutic lo fctor repectively η = 10 nd η = 10 Figure.7 () coutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in ech coupled ce (b) tructurl kinetic energy rbitrrily normlied by tht t / λ = where the tructurl nturl frequency i t / λ = 0.8 (ω / ω = 0.6), the contnt tructurl nd coutic lo fctor repectively (olid line: trongly coupled ce with wekly coupled ce with with K / K = 1) K η = 10 K / K = 10 nd η = 10, dhed line 3 3 = nd dotted line: intermedite ce / K 10 Figure.8 Figure 3.1 Opertionl deflection hpe of the coutic preure in the cvity normlied 3 by mximum modulu: () trongly coupled ce with K / K = 10, (b) wekly 3 coupled ce with K / K = 10 nd (c) intermedite ce with K / K = 1 where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6), the contnt tructurl nd coutic lo fctor η = 10 nd η = 10 repectively. Combined SDOF tructure finite cloed tube ytem under the externl time j t hrmonic force on the tructure, f () t = Fe ω 0 t x = 0. The tube h n infinitely lrge impednce Z in the nlyticl model depicted in figure.1. M nd K re tructurl m nd tiffne of pring with contnt tructurl lo fctor η repectively. Figure 3. Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl tiffne i increed by fctor of 5 where the tructurl nd coutic lo fctor η = η = 10 (olid line: before increing the tructurl tiffne where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5), nd dhed line: fter increing the tructurl tiffne where the tructurl nturl frequency i t / λ = 0. ( ω / ω = 1.1)) Figure 3.3 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl tiffne i increed by fctor of 5 where the tructurl nd coutic lo fctor η = η = 10 (olid line: before increing the tructurl tiffne where the tructurl nturl frequency i t / λ = 0.8 (ω / ω = 0.6), nd dhed line: fter increing the tructurl tiffne where the tructurl nturl frequency i t / λ = 1. 8 ( ω / ω = 0. 3)) viii

11 Figure 3.4 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl m i increed by fctor of 5 where the tructurl nd coutic lo fctor η = η = 10 (olid line: before increing the tructurl m where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5), nd dhed line: fter increing the tructurl m where the tructurl nturl frequency i t / λ = 0.05 ( ω / ω = 11. )) Figure 3.5 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl m i increed by fctor of 5 where the tructurl nd coutic lo fctor η = η = 10 (olid line: before increing the tructurl m where the tructurl nturl frequency i t / λ = 0.8(ω / ω = 0.6), nd dhed line: fter increing the tructurl m where the tructurl nturl frequency i t / λ = 0.36 ( ω / ω = 1.4 )) Figure 3.6 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl lo fctor i increed by fctor of 5 where the tructurl nturl frequency i t / λ = 0.1(ω / ω = 5) (olid line: before increing the tructurl lo fctor where the tructurl nd coutic lo fctor η = η = 10, nd dhed line: fter increing the tructurl lo fctor where the tructurl lo fctor coutic lo fctor η = 10 ) η = 5 10 nd the Figure 3.7 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl lo fctor i increed by fctor of 5 where the tructurl nturl frequency i t / λ = 0.8(ω / ω = 0.6 ) (olid line: before increing the tructurl lo fctor where the tructurl nd coutic lo fctor η = η = 10, nd dhed line: fter increing the tructurl lo fctor where the tructurl lo fctor coutic lo fctor η = 10 ) η = 5 10 nd the Figure 3.8 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the coutic lo fctor i increed by fctor of 5 where the tructurl nturl frequency i t / λ = 0.1(ω / ω = 5) (olid line: before increing the coutic lo fctor where the tructurl nd coutic lo fctor η = η = 10, nd dhed line: fter increing the coutic lo fctor where the coutic lo fctor tructurl lo fctor η = 10 ) η = 5 10 nd the Figure 3.9 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the coutic lo fctor i increed by fctor of 5 where the tructurl nturl frequency i t / λ = 0.8(ω / ω = 0.6 ) (olid line: before increing the coutic lo fctor where the tructurl nd ix

12 coutic lo fctor η = η = 10, nd dhed line: fter increing the coutic lo fctor where the coutic lo fctor tructurl lo fctor η = 10 ) η = 5 10 nd the Figure 3.10 Combined SDOF tructure finite cloed tube ytem, which h the borptive medium t the rigid end urfce of the cloed tube in the region of 0 x Figure 3.11 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) without the borptive medium for given tiffne rtio in the ce when the borptive medium i pplied in the region 0 x where the tructurl nturl frequency i t / λ = 0.1 ( ω / ω = 5) nd the tructurl nd coutic lo fctor η = η = 10 (olid line: without the borptive medium, nd dhed line: with the borptive medium / 0 = 0.7, b / = 0.3 nd the lo fctor η = 0. ) b Figure 3.1 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) without the borptive medium for given tiffne rtio in the ce when the borptive medium i pplied in the region 0 x where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6) nd the tructurl nd coutic lo fctor η = η = 10 (olid line: without the borptive medium, nd dhed line: with the borptive medium / 0 = 0.7, b / = 0.3 nd the lo fctor η = 0. ) b Figure 3.13 Experimentl etup of one-dimenionl coutic tube driven by loudpeker: () chemtic digrm (b) experimentl etup Figure 3.14 oudpeker ued to excite the coutic tube: () tndrd loudpeker nd (b) modified loudpeker Figure 3.15 Structurl velocity with repect to the input voltge to the loudpeker nd phge ngle of the tndrd loudpeker where the known dummy m m = 10g, f 1 = 190Hz, f = 130Hz nd the reference vlue for the mplitude of the FRF i 1 V / V in db cle (olid line: without the dummy m nd dotted line: with the dummy m) Figure 3.16 Structurl velocity with repect to the input voltge to the loudpeker nd phge ngle of the modified loudpeker where the known dummy m m = 100g, f 1 = 96Hz, f = 69Hz nd the reference vlue for the mplitude of the FRF i 1 V / V in db cle (olid line: without the dummy m nd dotted line: with the dummy m) Figure 3.17 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for theory nd tht t 50Hz for experiment in the more trongly coupled ce where the tiffne rtio K / K = 0.1, the tructurl nturl frequency i x

13 190Hz ( ω / ω = 0.9 ) nd the lo fctor η = 0.16, η = 0.01: () theory nd (b) experiment Figure 3.18 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for theory nd by tht t 50Hz for experiment in the more wekly coupled ce where the tiffne rtio K / K = 0.1, the tructurl nturl frequency i 96Hz ( ω / ω = 1.8 ), nd the lo fctor η = 0., η = 0.01 : () theory nd (b) experiment Figure 4.1 A combined SDOF tructure one dimenionl coutic tube ytem controlled by n coutic piton in feedforwrd control trtegy Figure 4. Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in the bence of control for given tiffne rtio when the tructurl nturl 3 frequency i t / λ = 0.1(ω / ω = 5) where the tiffne rtio K / K = 10 nd the lo fctor η = η = η = 10 (olid line: without control nd dhed line: with feedforwrd control) Figure 4.3 Modulu nd phe ngle of the optiml feedforwrd controller given in eqution (4.17) for given tiffne rtio in the ce when the tructurl nturl 3 frequency i t / λ = 0.1(ω / ω = 5) where the tiffne rtio K / K = 10 nd the lo fctor η = η = η = 10 (olid line: trongly coupled ce with K / K = , dhed line: wekly coupled ce with K / K = 10 nd dotted line: intermedite ce with K / K = 1) Figure 4.4 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in the bence of control for given tiffne rtio when the tructurl nturl 3 frequency i t / λ = 0.8(ω / ω = 0.6) where the tiffne rtio K / K = 10 nd the lo fctor η = η = η = 10 (olid line: without control nd dhed line: with feedforwrd control) Figure 4.5 Figure 4.6 Modulu nd phe ngle of the optiml feedforwrd controller given in eqution (4.17) for given tiffne rtio in the ce when the tructurl nturl 3 frequency i t / λ = 0.8(ω / ω = 0.6) where the tiffne rtio K / K = 10 nd the lo fctor η = η = η = 10 (olid line: trongly coupled ce with 3 3 K / K = 10, dhed line: wekly coupled ce with K / K = 10 nd dotted line: intermedite ce with K / K = 1) Conceptul repreenttion of the vibro-coutic ytem depicted in figure 4.1 in term of the optiml impednce Z opt nd the impednce of the econdry ource Z Figure 4.7 Strongly coupled ce: impednce t the econdry ource poition (t x = ) decribed in figure 4.5 with rbitrry normlition where the tructurl xi

14 nturl frequency i t / λ = 0.1( ω / ω = 5) in figure (), (c) nd (e), nd t / λ = 0.8( ω / ω = 0.6) in figure (b), (d) nd (f) repectively, tiffne 3 rtio K / K = 10 nd K / K 10 3 =, lo fctor η = η = η = 10 (olid line: normlied impednce of the econdry ource normlied coutic input impednce t the econdry ource poition dotted line: normlied optiml impednce Z opt ) Z, dhed line: Z A nd Figure 4.8 Wekly coupled ce: impednce t the econdry ource poition (t x = ) decribed in figure 4.5 with rbitrry normlition where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5) in figure (), (c) nd (e), nd t / λ = 0.8(ω / ω = 0.6) in figure (b), (d) nd (f) repectively, tiffne rtio 3 3 K / K = 10 nd K / K = 10, lo fctor η = η = η = 10 (olid line: normlied impednce of the econdry ource Z, dhed line: normlied coutic input impednce t the econdry ource poition Z A nd dotted line: normlied optiml impednce Z opt ) Figure 4.9 Intermedite ce: impednce t the econdry ource poition (t x = ) decribed in figure 4.5 with rbitrry normlition where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5 ) in figure (), (c) nd (e), nd t / λ = 0.8 (ω / ω = 0.6 ) in figure (b), (d) nd (f) repectively, tiffne 3 rtio K / K = 1 nd K / K = 10, lo fctor η = η = η = 10 (olid line: normlied impednce of the econdry ource Z, dhed line: normlied coutic input impednce t the econdry ource poition Z A nd dotted line: normlied optiml impednce Z opt ) Figure 4.10 Structurl kinetic energy rbitrrily normlied by tht t / λ = in the bence of control for given tiffne rtio where the tructurl nturl 3 frequency i t / λ = 0.1( ω / ω = 5 ), tiffne rtio K / K = 10, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with feedforwrd control) Figure 4.11 Structurl kinetic energy rbitrrily normlied by tht t / λ = in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.8 ( / ω ω = ), tiffne rtio K / K = 10, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with feedforwrd control) Figure 4.1 Cumultive um of coutic potentil energy normlied by the ummed coutic potentil energy in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5), tiffne 3 rtio K / K = 10, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with feedforwrd control) xii

15 Figure 4.13 Cumultive um of coutic potentil energy normlied by the ummed coutic potentil energy in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6), tiffne 3 rtio K / K = 10, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with feedforwrd control) Figure 4.14 Schemtic digrm of the experimentl etup for the predicted coutic potentil energy in tructurl-coutic coupled ytem under feedforwrd control Figure 4.15 Strongly coupled ce: coutic potentil energy normlied by tht t the ttic tte ( f = 0) for theory nd tht t 50Hz for experiment where tiffne rtio K / K = 0.1, K / K = 0. 1, uncoupled tructurl reonnce i t 190Hz, nd lo fctor η = η = 0.16, η = 0.01 (olid line: before control nd dhed line: fter control) Figure 4.16 Strongly coupled ce: cumultive um of coutic potentil energy by the ummed coutic potentil energy over the frequency rnge ( 0 f 500Hz) in the bence of control where tiffne rtio K / K = 0.1, K / K = 0. 1, uncoupled tructurl reonnce i t 190Hz, nd lo fctor η = η = 0.16, η = 0.01(olid line: before control nd dhed line: fter control) Figure 4.17 Wekly coupled ce: coutic potentil energy normlied by tht t the ttic tte ( f = 0) for theory nd tht t 50Hz for experiment where tiffne rtio K / K = 0.03, K / K = 0. 1, uncoupled tructurl reonnce i t 96Hz, nd lo fctor η = 0., η = 0.16, η = 0.01 (olid line: before control nd dhed line: fter control) Figure 4.18 Wekly coupled ce: cumultive um of coutic potentil energy by the ummed coutic potentil energy over the frequency rnge ( 0 f 500Hz) in the bence of control where tiffne rtio K / K = 0.03, K / K = 0. 1, uncoupled tructurl reonnce i t 96Hz, nd lo fctor η = 0., η = 0.16, η = 0.01 (olid line: before control nd dhed line: fter control) Figure 5.1 Figure 5. Model of combined SDOF tructure one-dimenionl coutic tube ytem uing n coutic ctutor driven by velocity feedbck controller t x = Summed coutic potentil energy function of feedbck gin rtio H / ρ c S normlied by tht in the bence of control for given tiffne A 0 0 rtio when the tructurl nturl frequency i t / λ = 0.1 ( ω / ω = 5) nd t / λ = 0.8(ω / ω = 0.6) repectively where tiffne rtio K / K = 0.1 nd lo fctor K / K = 10 3 = = = 10 (olid line: trongly coupled ce with η η η 3, dhed line: wekly coupled ce with K / K = 10 nd dotted line: intermedite ce with K / K = 1) xiii

16 Figure 5.3 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.1(ω / ω = 5), tiffne rtio K / K = 0.1, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with velocity feedbck control uing the coutic ctutor) Figure 5.4 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0. 8 ( ω / ω = 0. ), tiffne rtio K / K = 0.1, lo 6 fctor η = η = η = 10 (olid line: without control nd dhed line: with velocity feedbck control uing the coutic ctutor) Figure 5.5 Model of combined SDOF tructure one dimenionl coutic tube ytem uing tructurl ctutor driven by velocity feedbck controller t x = 0 Figure 5.6 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.1(ω / ω = 5), tiffne rtio K / K = 0.1, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with velocity feedbck control uing the tructurl ctutor) Figure 5.7 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6 ), tiffne rtio K / K = 0.1, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with velocity feedbck control uing the tructurl ctutor) Figure 5.8 Figure 5.9 Model of combined SDOF tructure one dimenionl coutic tube ytem uing tructurl nd coutic ctutor driven by velocity feedbck controller t x = 0 nd t x = repectively Summed coutic potentil energy function of feedbck gin rtio HA / ρ 0c0S normlied by tht under control uing only the tructurl ctutor for given tiffne rtio when the tructurl nturl frequency i t / λ = 0. 1 ( ω / ω = ) nd t / λ = 0.8( ω / ω = 0.6) repectively where tiffne rtio 5 K / K = 0.1 nd lo fctor η = η = η = 10 (olid line: trongly coupled ce with K / K = 10 3, dhed line: wekly coupled ce with nd dotted line: intermedite ce with K / K = 1) K / K = 10 3 Figure 5.10 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.1(ω / ω = 5), tiffne rtio K / K = 0.1, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with decentrlied velocity feedbck control uing both the ctutor) xiv

17 Figure 5.11 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6 ), tiffne rtio K / K = 0.1, lo fctor η = η = η = 10 (olid line: without control nd dhed line: with decentrlied velocity feedbck control uing both the ctutor) Figure 5.1 Structurl kinetic energy rbitrrily normlied by tht t / λ = in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0. 1 ( ω / ω = 5 ), tiffne rtio K / K = 0.1, lo fctor η = η = η = 10 (olid line: without control, dhed line: with velocity control uing the coutic ctutor, dhed-dotted line: with velocity control uing the tructurl ctutor nd dotted line: with decentrlied velocity feedbck control uing both the ctutor) Figure 5.13 Structurl kinetic energy rbitrrily normlied by tht t / λ = in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0. 8 ( ω / 0. ω = 6), tiffne rtio K / K = 0.1, lo fctor η = η = η = 10 (olid line: without control, dhed line: with velocity control uing the coutic ctutor, dhed-dotted line: with velocity control uing the tructurl ctutor nd dotted line: with decentrlied velocity feedbck control uing both the ctutor) Figure 5.14 Cumultive um of coutic potentil energy normlied by the ummed coutic potentil energy in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5), tiffne 3 rtio K / K = 10, lo fctor η = η = η = 10 (olid line: without control, dhed line: with control uing the coutic ctutor, dhed-dotted line: with control uing the tructurl ctutor, dotted line: with control uing both the ctutor) Figure 5.15 Cumultive um of coutic potentil energy normlied by the ummed coutic potentil energy in the bence of control for given tiffne rtio where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6), tiffne 3 rtio K / K = 10, lo fctor η = η = η = 10 (olid line: without control, dhed line: with control uing the coutic ctutor, dhed-dotted line: with control uing the tructurl ctutor, dotted line: with control uing both the ctutor) xv

18 IST OF TABES Tble 3.1 Tble 3. Comprion of normlied ummed coutic potentil energy over the frequency rnge ( 0 / λ ) ccording to the pive tretment in the ce when the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5 ) nd t / λ = 0.8( ω / ω = 0.6 ) Mechnicl component of tndrd nd modified loudpeker Tble 4.1 Summed tructurl kinetic energy, over the frequency rnge of ( 0 / λ ), normlied by tht before control when the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5) nd t / λ = 0.8( ω / ω = 0.6 ) repectively Tble 4. Summed coutic potentil energy, over the frequency rnge of ( 0 / λ ), normlied by tht before control when the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5) nd t / λ = 0.8( ω / ω = 0.6 ) repectively Tble 5.1 Optiml feedbck gin rtio ( HA / ρ 0c0S ) nd ummed coutic potentil energy over the frequency rnge ( 0 / λ ) normlied by tht in the bence of control for given tiffne rtio in the brcket when the tructurl nturl frequency i t / λ = 0.1 ( ω / ω = 5 ) nd t / λ = 0. 8 ( ω / 0.6 ω = ) repectively Tble 5. Feedbck gin rtio ( HS / ρ 0c0S) nd ummed coutic potentil energy over the frequency rnge ( 0 / λ ) normlied by tht in the bence of control for given tiffne rtio in the brcket when the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5) nd t / λ = 0.8 ( ω / ω = 0.6 ) repectively Tble 5.3 Feedbck gin rtio for the coutic ctutor ( HA / ρ 0c0S) nd ummed coutic potentil energy over the frequency rnge ( 0 / λ ) normlied by tht under the velocity feedbck control implemented by the tructurl ctutor for given tiffne rtio in the brcket when the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5) nd t / λ = 0.8 ( ω / ω = 0.6 ) repectively Tble 5.4 Summed coutic potentil energy, over the frequency rnge ( 0 / λ ), normlied by tht before control when the tructurl nturl frequency i t / λ = 0.1 ( ω / ω = 5) nd t / λ = 0.8( ω / ω = 0.6 ) repectively Tble 5.5 Summed tructurl kinetic energy over the frequency rnge ( 0 / λ ) ccording to the ctutor type, normlied by tht in the bence of control, for given tiffne rtio when the tructurl nturl frequency i t / λ = 0. 1 ( ω / ω = 5) nd t / λ = 0.8( ω / ω = 0.6 ) repectively xvi

19 GOSSARY OF TERMS A, B B c c c 0 b Complex wve mplitude Bulk modulu Complex ound peed Sound peed in the lole medium Complex ound peed in the borptive medium E ( ) K ω Structurl kinetic energy Eˆ ( ˆ) Normlied tructurl kinetic energy k E ( ) P ω Acoutic potentil energy Eˆ ( ˆ P Acoutic potentil energy normlied by tht t the ttic tte E p FRF ( ω ) Meured pproximte coutic potentil energy F( x, ω ) Externl excittion force F CA, F CS Velocity feedbck control force F Force F F F 0 P S S opt F Acoutic rection force Primry force Secondry control force F Optiml econdry control force f () t Structurl force Externl time hrmonic fore f () P t Primry time force fs () t Secondry time force f 1, f Frequencie t hlf-power point f Nturl frequency f 1 Nturl frequency of the tndrd or the modified loudpeker f Nturl frequency of the loudpeker with the known dummy m G( jω ) Feedforwrd controller G Optiml feedforwrd controller opt H A, h S H S Velocity feedbck gin Trnfer function vector j 1 J P Approximte coutic potentil energy Jˆ ˆ P min ( ) Normlied minimum level of the coutic potentil energy k Complex coutic wvenumber k Complex coutic wvenumber in the borptive medium b K K Acoutic bulk tiffne Structurl tiffne xvii

20 ˆK Structurl-coutic tiffne rtio Tube length ˆ / λ Acoutic medium length 0 b M m m n Pxω (, ) Px ˆ( ˆ, ˆ) P (, ) x ω Pˆ ( ˆ, ˆ ) x PS ( x, ω ) Pˆ ( xˆ, ˆ) S Aborptive medium length Structurl m Known dummy m Number of meuring point Integer Acoutic preure in the coutic tube Normlied coutic preure Acoutic preure due to primry force Normlied coutic preure due to primry force Acoutic preure due to econdry control force Normlied coutic preure due to econdry control force PFRF ( xi, ω ) Trnfer frequency repone function of the ound preure with repect to the input voltge P Acoutic preure t x = 0 1 P Reulting coutic preure t x = P P P p S p P p S FRF Acoutic preure on the tructure Acoutic preure on the rbitrry impednce Non-dimenionliing fctor for the coutic preure m -length vector of the normlied totl ound preure m -length vector of normlied ound preure due to the primry ource m -length vector of normlied ound preure due to the econdry ource p Meured frequency repone function vector H p FRF Hermitin trnpoe of the vector pfrf S Cro-ectionl re U( x, ω ) Prticle velocity in the coutic tube Uˆ( xˆ, ˆ) Normlied prticle velocity U (, ) P x ω Prticle velocity due to primry force Uˆ ( ˆ, ˆ ) P x Normlied prticle velocity due to primry force US ( x, ω ) Prticle velocity due to econdry control force Uˆ ( ˆ, ˆ S x ) Normlied prticle velocity due to econdry control force U Non-dimenionliing fctor for the prticle velocity U Prticle velocity t x = 0 1 U Reulting prticle velocity t x = U opt Prticle velocity on the rbitrry impednce U Velocity due to both the primry force nd econdry control force t x = xviii

21 U P Velocity due to the primry ource t x = U Velocity due to optiml econdry control force t x = U S U x ˆx x Y i S S opt CS Prticle velocity on the tructure Structurl velocity Co-ordinte for the coutic field in the cvity Any normlied poition long the tube th i meuring point Uncoupled tructurl mobility Y Coupled tructurl mobility Z A0 Acoutic input impednce t x = 0 Z ˆ A0 Normlied coutic input impednce t x = 0 Z A Acoutic input impednce t x = Z ˆ A Normlied coutic input impednce t x = Arbitrry impednce Z Z ˆ Normlied rbitrry impednce Z opt Optiml impednce Z S Z ˆS Uncoupled tructurl impednce Normlied tructurl impednce Z VA, Z VS Controlled impednce vi velocity feedbck controller Z ˆVA, Z ˆVS Normlied controlled impednce vi velocity feedbck controller Z 11, Z Point impednce Z 1, Z 1 Trnfer impednce η Acoutic lo fctor η ρ 0 Structurl lo fctor Ambient denity ρ b Ambient denity in the borptive medium λ Acoutic wvelength ω Excittion frequency ω Fundmentl nturl frequency of cloed-cloed tube when / λ = 1/ ω Uncoupled tructurl nturl frequency ˆω Structurl-coutic nturl frequency rtio ( = ω / ω ) xix

22 Chpter 1. Introduction CHAPTER 1 INTRODUCTION 1.1 Bckground Within the context of vibro-coutic, there re two min nlyticl ytem involving wve field, which re purely tructurl nd purely couticl ytem. When thee two uncoupled ubytem combine, more intriguing phyicl chrcteritic pper. The interction between tructurl vibrtion nd coutic wve propgtion h been the ubject of much reerch. The mutul tructurl-coutic interction i dependent upon the propertie of both uncoupled tructurl nd couticl ytem [Kinler et l (198) nd Fhy (001)]. Eltic tructure in free-field tend to vibrte in direction norml to the urfce t reonnce frequencie [Cremer nd Heckl (1973)]. The tructurl vibrtion cue ound rdition to n coutic field diplcing nd compreing urrounding fluid in contct with the tructurl urfce [Fhy (1985)]. When the coutic field i limited to n coutic cvity encloed by n rbitrry tructure under tructurl excittion, it i more chllenging to invetigte the phyicl behviour of the vibro-coutic ytem. For n encloed ir cvity intercting with plne tructure, modlinterction model h been derived by Fhy (1985), which decribe the behviour of the 1

23 Chpter 1. Introduction encloed fluid nd tructure in term of the uncoupled nturl mode. Anlyticl pproche bed on modl repreenttion hve been ued in the nlyi of tructurl-coutic interction in rectngulr cvitie with five couticlly rigid wll nd one flexible wll. The effect of n underlying coutic cvity on the flexible plte vibrtion w tudied by Dowell nd Vo (1963). They howed tht only the fundmentl plte mode w trongly ffected by the cvity. Pretlove (1965, 1966) preented the concept of cvity- nd plte-controlled mode in the rectngulr cvity under flexible tructurl excittion, which depend on the reltive energy contribution of ech ubytem. The tructurl-coutic interction w dicued in term of the reltive plte nd coutic cvity tiffne in which the coupled mode were obtined from uncoupled in vcuo plte nd cvity mode. Dowell et l (1977) generlied the concept bed on plte nd cvity mode by dicuing the coupling between the tructurl plte nd coutic cvity in the vibro-coutic ytem. In the ce when tructurl nturl frequency i unchnged by the coutic preure in the cvity uch tht it behve like n in vcuo tructurl plte, the coutic preure loding on the plte cn be neglected. In nother ce when there i ignificnt chnge in tructurl reonnce frequency due to the coutic preure in the cvity, the effect of the cvity modifie the effective tiffne. The former i clled wekly coupled ce nd the ltter i clled trongly coupled ce in thi thei. The time-verged coutic potentil energy i conidered meure of the globl coutic environment in n encloure [Bullmore et l (1987)]. It i ueful to e the effectivene of pive or ctive control by invetigting the control effect on the coutic potentil energy in the encloure of interet. Sound nd vibrtion problem cn be olved by pive or ctive control method. Pive control involve modifiction of the tiffne, m or dmping of the vibrting ytem o tht it i le reponive to it excittion ource [Med (1999)]. The modifiction my tke the form of bic tructurl chnge [Wng et l (198) nd Olhoff (1976)] or dding pive element uch me [McMilln nd Kene (1996, 1997)], vibrtion ioltor [Crede (1965) nd Nhif et l (1985)] nd vriou dmper [zn (1959), Med (1960) nd Hendy (1986)]. On the other hnd, ctive control ytem require ctutor driven by control input fed through controller uing ignl from enor on the vibrting ytem. The ctive control of ound nd vibrtion h been invetigted in ingle-input ingle-output (SISO) ytem nd in multiple-input multiple-output (MIMO) ytem under feedforwrd control or feedbck control [Nelon nd Elliott (199) nd Fuller et l (1996)].

24 Chpter 1. Introduction Active noie control (ANC) nd ctive noie nd vibrtion control (ANVC) ytem hve given ucceful reult for the control of tonl nd ttionry rndom diturbnce. Nelon nd Elliott (199) hve implemented control ytem for the control of tonl diturbnce in propeller ircrft or engine noie in cr. ANC or ANVC ytem hve lo been pplied to the control of ttionry rndom noie in ircrft [Mthur et l (1997) nd Guigou nd Fuller (1999)] nd in cr [Sutton et l (1994) nd Prk et l (00)]. Thee control ytem require MIMO dptive feedforwrd or feedbck controller with lrge number of enor nd ctutor, which re reltively bulky, hevy nd cotly ytem. The prcticl problem encountered in the control ytem uing lrge-re ditributed trnducer motivted the deire to find decentrlied controller. A decentrlied control ytem i compoed of multiple independent mll loclied control unit tht hve ingle-input ingle-output (SISO) feedbck controller. SISO feedbck control cn provide ctive dmping, which i effective in reducing the repone t reonnce frequencie by implementing direct velocity feedbck. Petitjen et l (00) invetigted low-frequency wide-bnd ound rdition control of pnel incorported with lrge number of control unit providing dmping to low-frequency reonnce mode. They found imilr vibrtion reduction on the pnel whether it w under decentrlied velocity feedbck control or under MIMO optiml feedbck control. Elliott et l (00) propoed new control configurtion bed on decentrlied velocity feedbck control unit on pnel uing piezocermic ptch ctutor with mll ccelerometer on the centre. They noticed tht the ctive dmping effect on the totl kinetic energy nd totl ound rdition grow the control gin i grdully increed. However, thi control behviour i vlid up to n optiml feedbck gin, bove which the control effect diminihe nd the kinetic energy nd the ound rdition incree gin. The control mechnim i tht the velocity feedbck control unit work ky-hook dmper which borb the tructurl energy. The phyicl behviour of the velocity feedbck control with lrge control gin i to pin the pnel t the enor poition. The ubject of thi thei i to invetigte the influence of tructurl-coutic coupling on the dynmic behviour of one-dimenionl vibro-coutic ytem under pive/ctive control under vriou tructurl-coutic coupling condition: with trong, wek nd intermedite coupling. Previou literture in thi re re firt reviewed before the detil objective re outlined. 3

25 Chpter 1. Introduction 1. iterture review 1..1 Modelling of vibro-coutic ytem The control of ound nd vibrtion in vibro-coutic ytem tretche cro wide rnge of ppliction uch irplne fuelge [Elliott et l (1990), Bullmore et l (1990) nd Guigou nd Fuller (1999)], cr comprtment [Sutton et l (1994) nd Nefke et l (198)], etc. Elliott et l (1990) preented erie of in-flight experiment on the ctive control of propellerinduced penger cbin noie in n ircrft. They implemented locl control ytem (twoloudpeker nd two-microphone) nd globl control ytem (16-loudpeker nd 3- microphone) inide the cbin. Bullmore et l (1990) crried out theoreticl tudie nd compred the reult to the previou experimentl work [Elliott et l (1990)]. They modelled the tructurl repone of the ircrft fuelge finite, iotropic thin cylindricl hell, nd the cbin coutic repone cylindricl room. The theoreticl reult howed good greement with the previou experimentl reult provided tht the theoreticl externl coutic preure forcing of the hell i repreenttive of the meured propeller preure field on the ircrft fuelge. Guigou nd Fuller (1999) invetigted the control of ircrft interior noie by pplying fom-pvdf mrt kin mounted in the cockpit. They implemented feedforwrd control ytem to reduce ound field interior noie in the fuelge diturbed by the crown pnel of n ircrft, which w excited by loudpeker locted outide of the cockpit nd driven by bnd-limited rndom excittion. Sutton et l (1994) developed n ctive control ytem for the control of interior noie in utomobile by countercting the lowfrequency rumble noie with loudpeker intlled inide the comprtment. Nefke et l (198) reviewed the formultion of the finite element method for tructurl-coutic nlyi of n encloed cvity uing n coutic model of penger comprtment under tructurl excittion. Prcticl vibro-coutic ytem cn be depicted imple model, which include key fctor uch coutic fluid in n encloure, tructurl ytem under externl excittion being coupled with the coutic fluid nd tructurl-coutic coupling [Dowell et l (1977)]. The tructurl-coutic coupling w decribed by compct mtrix formultion for the tedytte nlyi of tructurl-coutic ytem by Kim nd Brennn (1998). They invetigted tructurl-coutic coupling theory in modl coordinte uing the impednce-mobility 4

26 Chpter 1. Introduction pproch in the combined flexible tructure 3D (three-dimenionl) rbitrry encloure ytem. Alo, they howed tht the degree of coupling i dependent upon five fctor: (i) the rtio of the coutic bulk tiffne to tructure tiffne; (ii) the coincidence of the coutic nd tructurl uncoupled nturl frequencie; (iii) geometric coupling between the coutic nd tructurl mode; (iv) tructurl dmping in the flexible tructure; nd (v) coutic dmping in the cvity. Figure 1.1 depict combined flexible tructure 3D rbitrry encloure ytem where the flexible tructure, under n externl force, excite the interior ound field in the rbitrry encloure. Thi implified tructurl-coutic model cn be ued to invetigte the interior ound field in vriou utomobile nd flight vehicle excited by vibrting wll. The tructurl-coutic interction in the coupled ytem reult in coupled tructurl mode in the flexible tructure nd coupled coutic mode in the ound field of the encloure. The uncoupled tructurl nd coutic mode re tructurl norml mode in-vcuo nd coutic norml mode of the cvity urrounded by rigid wll repectively. F( x, ω ) flexible tructure x z couticlly rigid wll y Figure1.1 Combined flexible tructure 3D rbitrry encloure ytem where the flexible tructure i under n externl excittion force F( x, ω ), which i cting in the negtive x direction t driving frequency ω 5

27 Chpter 1. Introduction 1.. Pive control Pive control of vibrtion cn be implemented by modifying the tiffne, m or dmping of the vibrting ytem. The modifiction cn be chieved by bic tructurl chnge or the ddition of pive element uch me, pring, fluid dmper or dmped rubber [Med (1999)]. yon (1963) computed the ound preure in the rectngulr prllelepiped encloure with one flexible wll under incident ound excittion in vriou frequency rnge. He howed the poibility of etimting the ound preure inide the encloure combined with vriou pnel. The coupled frequencie nd mode in rectngulr cvity bcked by imply upported plte hve been preented function of the thickne of the plte by Scrp (000). Vriou tructurl nd couticl pive control tretment hve been pplied to vibrting plte coupled with n coutic cvity in n encloure. Nrynn et l (1981) invetigted the reduction of trnmitted ound through vicoeltic ndwich pnel into rectngulr encloure. The reult were tht ignificnt noie reduction could be obtined by the contrined dmping lyer tretment t the fundmentl tructurl reonnce of the ndwich pnel. Mhiro et l (007) invetigted the ttenution effect on the rdited coutic power of the vibrting pnel which incorported honeycomb tructure, which llow Helmholtz reontor effect. They preented the poibility of uing thi pive control model in vriou field including floor impct inultion to chieve the ttenution t rbitrry frequencie. iu et l (006) dicued pive nd ctive vibro-coutic noie control method for ttenuting interior noie in box tructure, which cn be cbin of vehicle nd ircrft. They dopted the tructurl intenity method to predict the poible loction of pive dmper on the tructure nd concluded tht the dmper hould be put t the loction which re nerby the energy ource poition. Ro nd Burdio (1999) propoed the concept of wek ound rditing cell nd pplied the cell on the vibrting piton coupled with cvity under be tructurl excittion. The cell mounted on the vibrting tructure ct nerly out-of-phe nd nerly of the me trength over wide frequency rnge providing the control of low-frequency tructurlly rdited noie. Eteve nd Johnon (005) preented n dptive-pive olution to control the brodbnd ound trnmiion into imply upported cylinder excited by n externl plne wve. They pplied pive ditributed vibrtion borber for tructurl mode nd dptive Helmholtz 6

28 Chpter 1. Introduction reontor for coutic mode in the cvity. Their numericl imultion demontrted tht optimum noie reduction required the dptive Helmholtz reontor for coutic mode nd the ditributed vibrtion borber for the tructurl reonnce tht mnifet themelve in the coutic pectrum. Morelnd (1984) dded lyer of dmping mteril to the encloure wll nd put borptive lining on the interior wll for noie reduction in the encloure in vriou frequency rnge. He concluded tht extenionl dmping lyer or porou borbent lining on the interior wll provided little noie reduction t low frequencie. Oh et l (1999) identified the interior ound field chrcteritic of cvity with luminium fom lining on the wll of rectngulr encloure. They determined the uitble thickne of the ound borber which mximie the ound borption effect in the encloure. More recently, vibro-coutic problem hve been delt with in one-dimenionl cvity driven by ingle-degree-of-freedom (SDOF) tructurl driver to invetigte the influence of phyicl prmeter of couticl nd tructurl ytem on the ound field in the cvity. Hong nd Kim (1996) hve developed the nlyi method of generl vibro-coutic problem in one-dimenionl model incorporting tructurl dmping nd borbing mteril on the tructurl driver. Cur et l (1995) invetigted tructurl-coutic interction in uni-dimenionl coutic cvity coupled to SDOF ytem. They concluded tht the reonnce frequencie of the coupled model were modified by the effect of the interction between fluid nd tructure. The reonnce frequencie of the coupled model hd ignificnt chnge only when n couticl reonnce frequency mtched tructurl reonnce frequency. The reonnce repone of the coutic cvity to the tructurl excittion hd ignificnt vrition when there w reltively low vrition of the vicou dmping of the tructurl or coutic ytem. Previou pive control trtegie in vibro-coutic ytem hve been implemented either tructurlly or couticlly by mny reercher. They hve invetigted ound reduction in encloure coupled with tructurl ytem incorporting tructurl or couticl tretment. However, there i gp in the knowledge on the reltive benefit of pive control tretment to minimie the ound preure of the cvity in vriou tructurl-coutic coupling ce ince the coutic repone in the cvity h quite different chrcteritic depending on the degree of coupling [Dowell et l (1977)]. 7

29 Chpter 1. Introduction 1..3 Active control Since the firt ctive control concept w ptented by ueg (1936), vriou nlyticl tudie of ctive control hve been crried out with the im of determining the phyicl guideline for effective control deign. The fundmentl theory of the ctive control of the ound field in encloure h been developed by Nelon et l (1987) nd Elliott et l (1987). Nelon et l (1987) preented n nlyi of ctive method to produce globl ound reduction in hrmoniclly excited rectngulr encloure. They demontrted tht ubtntil reduction in the totl coutic potentil energy in the encloure were poible uing econdry ound ource locted t ditnce from the primry ource which i le thn hlf wvelength t the frequency of interet. Elliott et l (1987) crried out comprtive invetigtion on the meured reult of ctive minimition experiment nd thoe predicted from theory [Nelon et l (1987) ] in the lightly dmped rectngulr encloure, which w two dimenionl over the frequency rnge of interet. The condition under which ignificnt reduction in the totl coutic potentil energy in the encloure could be chieved were experimentlly invetigted uing primry nd three econdry loudpeker. The ctive control of hrmonic ound trnmiion into vriou coutic cvitie h been of interet in recent yer. Fuller nd Jone (1987) invetigted the feibility of uing ctive vibrtion control of ircrft fuelge to reduce the interior noie level in finite luminium cylinder model. The cylinder model w excited by one monopole ource, repreenttive of ingle propeller, nd controlled by mini-hker ttched to the exterior of the cylinder. They found tht the ctive vibrtion control ytem provided reonbly good reduction on the interior noie level t reonnce nd off-reonnce frequencie of the cylinder model. Elliott et l (1990) preented erie of in-flight experiment on the ctive control of propellerinduced penger cbin noie in n ircrft uing two loudpeker. They invetigted t the firt three hrmonic of the blde pge frequency nd imultneouly controlled three hrmonic with effective noie reduction t ome et loction. Snyder nd Tnk (1993) invetigted the minimition of rdited coutic power nd coutic potentil energy in rigid wlled rectngulr cvity coupled with flexible pnel. The dptive feedforwrd control ytem were implemented uing vibrtion-bed error enor uch hped piezoelectric polymer film enor which were n lterntive to the 8

30 Chpter 1. Introduction ue of coutic error enor uch microphone. They concluded tht the ue of vibrtion error ignl in n dptive feedforwrd ctive control ytem w prcticl lterntive to ttenute coutic rdition of the pnel vi vibrtion control input. Blchndrn et l (1996) developed nlyticl nd experimentl tudie for controlling the interior noie in rectngulr encloure with flexible wll excited by n externl peker. The ctive control w implemented by uing led zirconte titnte piezoelectric (PZT) ctutor bonded to the flexible wll. They demontrted tht ignificnt noie reduction cn be relied by uing ctive control cheme. Mohmmd nd Elliott (005) invetigted the ctive control of ound trnmiion into rectngulr encloure coupled with flexible tructurl pnel driven by unit point force uing econdry coutic ource. Their interet w the effect of full tructurl-coutic coupling between the vibrting pnel nd the interior coutic field in the cvity. They demontrted tht better reduction in the interior coutic field i obtined when the econdry ource i cloe to the pnel rther thn when it i remote. The mjority of previou reerch h been crried out in coupled tructurl-coutic ytem with one type of tructurl-coutic coupling, which i the encloure coupled with prticulr pnel. They demontrted coniderble ound reduction in the encloure under feedforwrd control. However there exit need to invetigte the control of the ound field in the coupled ytem with vriou tructurl-coutic coupling, nd the phyicl behviour of the feedforwrd controller in ech coupling ce. The ctive control of the ound field in tructurl-coutic coupled ytem ubject to rndom diturbnce cn lo be implemented uing feedbck controller. The ctive feedbck control trtegy h been invetigted in the minimition of ound trnmiion nd rdition from flexible plte. Fuller (1990) invetigted the ctive control of ound trnmiion from clmped eltic circulr thin plte, under plne coutic wve incident, uing point force. The optiml control gin to minimie cot function proportionl to the rdited coutic power w clculted bed on qudrtic optimiztion. Hi reult demontrted tht globl ttenution of brodbnd rdited ound level cn be chieved with one or two control force t reonnce nd off-reonnce frequencie with control efficiency determined by the nture of the coupling between the plte mode of repone nd the trnmitted field. Meirovitch nd Thngjithm (1990) tudied the problem of uppreing the coutic rdition preure generted by the vibrtion of imply-upported rectngulr eltic plte. The influence on the control effectivene of vriou deign fctor, uch controlled mode nd ctutor, 9

31 Chpter 1. Introduction w invetigted. They concluded tht tifctory control cn be chieved by ufficient number of ctutor nd by the choice of reltively lrge number of mode. The ctive control of ound trnmiion through pnel cn be implemented by uing piezoelectric control ctutor. Wng et l (1991) demontrted the reltive benefit of piezoelectric control ctutor nd point force ctutor to reduce ound trnmiion through thin rectngulr plte mounted in n infinite bffle under n incident hrmonic plne wve. The reult howed tht point force ctutor provided more effective control of the ound trnmiion thn piezoelectric ctutor. Johnon nd Elliott (1995) invetigted the ctive control of ound power rdition from rectngulr pnel mounted in bffle excited by ingle incident hrmonic plne wve uing econdry piezoelectric ctutor to determine the effect of cncelling the net volume velocity of the pnel. They howed tht the firt rdition mode w the dominnt rditor of ound power t low frequencie nd the cncelltion of the volume velocity w good trtegy for the reduction of ound power trnmiion t low frequencie. Alo, they uggeted tht volume velocity enor, uch polyvinylidene fluoride (PVDF) film, nd uniform-force ctutor cn be ued mtched ctutor/enor pir in feedbck control ytem to chieve the me ttenution poible with feedforwrd control ytem. When flexible plte under externl excittion combine with n coutic cvity, noie i trnmitted into the encloure. The ctive control of ound trnmiion into encloure h been invetigted by mny reercher. Pn et l (1989) developed the ctive control of ound trnmiion into rectngulr cvity coupled with tet pnel, which h reonnce frequencie well eprted from the cvity reonnce frequencie, uing point force ctutor. They demontrted two different control mechnim for minimiing the ound trnmiion through pnel into cvity. The ytem repone dominted by pnel-controlled mode or by cvity controlled mode cn be minimied by uppreing the pnel-vibrtion or by djuting the pnel-velocity ditribution repectively. Griffin et l (1999) developed n pproch whereby the feedbck control of tructurl nd coutic problem in cylinder cn be decribed flexible tructure repone. The cylinder, repreenttive of pce lunch vehicle, w umed in which meure of the diturbnce nd direct meurement of the ound preure were not vilble. They concluded tht it i poible to ctively uppre rdited ound in n coutic cvity ubject to brodbnd diturbnce through flexible tructure uing tructurl ening nd feedbck control. Al-Byiouni nd Blchndrn 10

32 Chpter 1. Introduction (005) invetigted the ctive control of ound trnmiion through flexible pnel into n encloure ubject to n incident phericl wve uing piezoelectric ctutor bonded on the pnel. They highlighted the generl model decription uing phericl wve incidence inted of plne wve incidence. Recent development in enor nd ctutor technologie opened new reerch direction to decentrlied control. Centrlied control ytem operte with lrge number of error enor nd ctutor vi multi-input multi-output (MIMO) controller, which i centrlied controller connecting ll the enor nd ctutor for pecific mode. Decentrlied control overcome the hortcoming of centrlied control which i tht they re complex, cotly nd hevy. Decentrlied control ytem cn be implemented by multiple ingle-input ingleoutput (SISO) loclied independent control unit tht my ignificntly mitigte computtion lod. Control of the tructure my involve the modifiction of m, tiffne nd dmping effect on the vibrtion of the tructure. However for the pecific ce of the control of the tructure the mot uitble trtegy i ctive dmping, which cn be obtined by feeding bck from velocity enor on the vibrting tructure to control ctutor [Preumont (00)]. Elliott et l (00) invetigted reduction in both the kinetic energy nd rdited ound power from pnel excited by plne coutic wve, which incorported n rry of collocted 4 4 force ctutor nd velocity enor in decentrlied velocity feedbck control cheme. They hve hown tht both the kinetic energy of the pnel nd it trnmitted ound power cn be ignificntly reduced for n optiml vlue of feedbck gin under the decentrlied control cheme which i conditionlly tble. Alo, they implemented decentrlied control uing 16 piezoelectric ctutor nd 16 velocity enor for prcticl purpoe nd compred the reult with force ctutor nd velocity enor rry. The reulting reduction in the kinetic energy nd ound power were not gret with the force ctutor but were till worthwhile. Hung et l (003) invetigted n ctive vibrtion ioltion ytem which involve electromgnetic ctutor intlled in prllel with ech of four pive mount between flexible equipment tructure nd be tructure. Decentrlied velocity feedbck control w experimentlly implemented nd howed good control performnce in the reduction of vibrtion of the equipment tructure over wide frequency rnge. Decentrlied control cheme hve been lo ued to reduce ound trnmiion through pnel coupled with n coutic cvity. Grdonio et l (004) theoreticlly nd experimentlly invetigted ound rdition/trnmiion through mrt pnel, incorporting 16 cloely 11

33 Chpter 1. Introduction pced ccelerometer enor nd piezocermic ctutor trnducer pir connected by inglechnnel velocity feedbck controller, well coupled with rectngulr encloure. The pnel w excited by monopole ource in the cvity or by trnvere point force. The theoreticl nd experimentl reult demontrted tht for both the coutic nd force ource, good reduction of the kinetic energy of the pnel nd it totl ound power rdition cn be chieved t low frequencie for optiml control gin. Kim nd Brennn (1999) invetigted the feedforwrd control of hrmonic nd rndom ound trnmiion into rectngulr cvity excited by plne coutic wve nlyticlly nd experimentlly. They configured the ctive control in three wy with (i) ue of ingle point-force ctutor, (ii) ue of ingle coutic piton ource nd (iii) imultneou ue of both point-force ctutor nd n coutic piton ource. Both the coutic nd tructurl ctutor were driven vi independent controller with the me reference ignl. Alo, they howed tht the configurtion of both coutic nd tructurl ctutor w deirble for the ctive control of hrmonic nd rndom ound trnmiion into coupled tructurl-coutic ytem whoe repone w governed by plte nd cvity-controlled mode. The literture review h been preented in the re of the ctive control of ound rdition nd trnmiion through flexible tructure into n coutic cvity. Previou reerch h hown tht it i poible to chieve good reduction in the tructurl kinetic energy of the vibrting tructure nd it ound rdition/trnmiion. However, the influence on the control effectivene w not tckled veru key fctor chrcteriing vibro-coutic ytem uch the degree of tructurl-coutic coupling. Vibro-coutic problem, where the tructurlcoutic coupling i key prmeter, my require robut control trtegy uing tructurl nd coutic ctutor. The chrcteritic of the ound-preure repone in the cvity depend on the coupling mechnim nd cn be djuted by vrying the tructurl modl propertie [Pn (199)] which cn be chnged by cvity mode in trongly coupled ce nd unchnged in wekly coupled ce [Dowell et l (1977)]. The previou ucceful control performnce of decentrlied control motivte the invetigtion on the ctive control of the interior ound field in vibro-coutic ytem. The vibro-coutic ytem my be effectively controlled in vriou coupled ce by uing tructurl nd coutic ctutor driven by decentrlied velocity feedbck controller. 1

34 Chpter 1. Introduction 1.3 Aim nd objective The implet model of vibro-coutic ytem i one-dimenionl coutic cvity, cloed t one end by hrd wll nd t the other by ingle-degree-of-freedom (SDOF) tructure. Thi imple model h been ued to demontrte the phyicl chrcteritic of the coupling phenomenon. The eqution of motion of m-pring ytem coupled with onedimenionl coutic cvity cn be found in mny textbook [Richrd nd Med (1968) nd Fhy (001)]. The dvntge of uing the geometriclly imple model to develop the nlyi of vibro-coutic problem re: (i) to verify the nlyi procedure in the cvity where plne wve propgte compred to the vilble exct olution nd (ii) to give better phyicl inight into the nture of vibro-coutic problem, which cn be trightforwrdly extended to three-dimenionl ytem [Hong nd Kim (1995)]. The imple model w ued to evlute the ound trnmiion through mll cvity-bcked pnel t the fundmentl reonnce frequency [Guy nd Pretlove (1973)]. Crgg nd Ayorinde (1990) ued the imple model to clculte the reonnce frequencie of tructurlcoutic coupled ytem introducing the concept of iochronim between uncoupled tructurl nd coutic nturl frequencie. Hong nd Kim (1996) hve developed the nlyi method of generl vibro-coutic problem uing the imple model incorporting tructurl dmping nd borbing mteril on the tructurl driver. They demontrted tht the effect of coutic borbing mteril pplied on ll or prt of the tructure well tructurl dmping element cn be hndled uing the imple model. Cur et l (1995) invetigted tructurlcoutic interction in the imple model. They dicued the influence on the tructurlcoutic interction for the different prmeter: the geometricl chrcteritic of the cvity (re of the cro-ection nd length), the phyicl quntitie chrcteriing the fluid (denity nd vicou dmping) nd the tructurl driver (m, vicou dmping nd tiffne). Alo, they concluded tht the reonnce frequencie of the coupled model were modified by the effect of the interction between fluid nd tructure, with repect to thoe of the uncoupled ytem. cour et l (000) conducted experiment on the ctive control of encloed ound field in the imple model vi wll impednce chnge. The imple model involved primry ource, vibrting hrmoniclly with fixed velocity mplitude, t the left end nd controlled impednce t the right end. The controlled coutic impednce w implemented two wy: the firt i direct feedforwrd control nd the econd i hybrid pive/ctive feedbck 13

35 Chpter 1. Introduction control with borbing mteril. They concluded tht the one-dimenionl ound field cn be uccefully controlled for brodbnd excittion by uing hybrid feedbck control method when feedforwrd control cnnot be pplied. Thi thei i minly concerned with the pive/ctive control of the coutic potentil energy in tructurl-coutic coupled ytem under vriou coupled condition: with trong, intermedite nd wek coupling. The principle re demontrted by controlling the coutic potentil energy in one-dimenionl finite coutic tube driven by ingle-degree-offreedom (SDOF) tructure in the three coupled ce nd when the tructurl nturl frequency i below nd bove the fundmentl coutic mode of cloed-cloed tube repectively. Figure 1. how combined SDOF tructurl driver 1D (one-dimenionl) finite cloed tube ytem where the tructurl driver, under n externl force, excite the interior ound field in the finite cloed tube urrounded by rigid wll. Compred to generl 3D (three-dimenionl) model hown in figure 1.1, the SDOF tructure repreent the flexible tructure being compoed of rigid m with tructurl dmping where M nd complex pring K (1 + jη ) incorported K i tructurl tiffne nd η i contnt tructurl lo fctor. In ddition, the 1D finite cloed tube replce the 3D rbitrry encloure. F( x, ω ) K (1 + jη ) M SDOF tructure x 1D coutic tube couticlly rigid Figure 1. Combined ingle-degree-of-freedom (SDOF) tructurl driver 1D (onedimenionl) finite cloed tube ytem where the SDOF tructure i under n externl excittion force F( x, ω ) in the x direction t frequencyω nd the 1D finite cloed tube i urrounded by rigid wll Thi imple nlyticl model cn provide vriou degree of tructurl-coutic coupling, which re dependent upon (i) the tructurl-coutic tiffne rtio; (ii) tructurl-coutic 14

36 Chpter 1. Introduction nturl frequency rtio; (iii) tructurl dmping; nd (iv) coutic dmping. In thi ce, lthough the geometric coupling fctor i not included becue the SDOF tructure h ingle mode, 80 percent of the fctor to determine the degree of coupling cn be ccounted for by the imple nlyticl model. The effect of the tructurl-coutic coupling on the following three control trtegie re invetigted. pive control trtegy - to chieve the phyicl inight into the reltive benefit of pive control tretment uch tiffne, m, tructurl dmping, coutic dmping nd borptive medium ctive feedforwrd control - to invetigte the phyicl behviour of the feedforwrd controller minimiing the coutic potentil energy in the vibro-coutic ytem under hrmonic excittion of the tructure decentrlied velocity feedbck control - to invetigte the reltive control effectivene of tructurl nd coutic ctutor on the reduction of the coutic potentil energy in the vibro-coutic ytem under brodbnd excittion of the tructure 1.4 Contribution The originl contribution of the work reported in thi thei re 1. Proviion of non-dimenionl coupling fctor to determine the threhold of the degree of coupling for the dynmic behviour of vibro-coutic ytem (Chpter ). Determintion of the reltive benefit of pive control tretment in vriou tructurl coutic coupled ce through prmetric tudy of tructurl-coutic non-dimenionl prmeter (Chpter 3) 3. Phyicl interprettion on the effectivene of feedforwrd control ytem dicuing the control mechnim for minimiing the coutic potentil energy in the vibro-coutic ytem under hrmonic excittion of the tructure in vriou tructurl-coutic coupled ce (Chpter 4) 15

37 Chpter 1. Introduction 4. Proviion of novel control trtegy uing tructurl nd coutic ctutor in decentrlied velocity feedbck control cheme for the control of the coutic potentil energy in the vibro-coutic ytem under rndom excittion of the tructure in vriou tructurl-coutic coupled ce (Chpter 5) 1.5 Thei tructure Thi thei invetigte three control trtegie for the control of the coutic potentil energy in combined ingle-degree-of-freedom (SDOF) tructure 1D (one-dimenionl) tube ytem under vriou tructurl-coutic coupled condition: pive control (Chpter 3), feedforwrd control (Chpter 4 ) nd decentrlied velocity feedbck control (Chpter 5). For pive control, the reltive control effect of pive tretment on the coutic potentil energy re invetigted bed on non-dimenionl tructurl-coutic prmeter in chpter 3. The effectivene of feedforwrd control ytem nd decentrlied velocity feedbck control ytem i invetigted under hrmonic excittion in chpter 4 nd under rndom excittion in chpter 5 repectively for the ctive control of the coutic potentil energy. In Chpter, the dynmic behviour of the 1D vibro-coutic ytem nd tructurl-coutic coupling mechnim re invetigted bed on the mobility-impednce pproch. The dynmic behviour i dicued for vriou vibro-coutic ytem uch : emi-infinite tube, finite-open tube nd finite-cloed tube driven by SDOF tructure t one end. An rbitrry-impednce terminted tube i invetigted by deriving coutic input impednce uing n impednce pproch. Alo, the coutic preure nd prticle velocity in the nlyticl model re preented in term of non-dimenionl tructurl-coutic prmeter. The coupling mechnim i invetigted bed on the mobility-impednce pproch. In order to provide the threhold of the degree of coupling, coupling fctor i clculted in combined SDOF tructure finite cloed tube ytem. The vibro-coutic repone, in the cloed tube ytem, i repreented by the coutic potentil energy in the cvity nd the kinetic energy of the tructure coupled to the coutic cvity. The vibro-coutic repone re dicued in vriou coupled ce determined by the threhold of the degree of coupling. 16

38 Chpter 1. Introduction In Chpter 3, the effect of pive tretment on the reduction of the coutic potentil energy in imple vibro-coutic ytem re invetigted in vriou coupling ce: trong, intermedite nd wek coupling. The imple vibro-coutic model i configured by onedimenionl finite-cloed tube driven by ingle-degree-of-freedom (SDOF) tructure. The pive control of the coutic potentil energy i invetigted involving tructurl nd couticl modifiction bed on tructurl-coutic non-dimenionl prmeter: tructurlcoutic tiffne rtio (tiffne), tructurl-coutic nturl frequency rtio (m), tructurl lo fctor (tructurl dmping) nd coutic lo fctor (coutic dmping). Alo, the effect of borptive medium on the reduction of the coutic potentil energy i invetigted by plcing it t the rigid end urfce of the cvity. Experimentl invetigtion on the coutic potentil energy in vibro-coutic ytem i crried out bed on the non-dimenionl tructurl-coutic propertie in the more trongly coupled ce nd in the more wekly coupled ce. In Chpter 4, the min concern i to invetigte the performnce of feedforwrd control of the coutic potentil energy in combined SDOF tructure one dimenionl coutic tube ytem under the three coupled condition dicued in chpter 3. An nlyticl model of the vibro-coutic ytem, driven by SDOF tructure nd controlled by n coutic piton in feedforwrd control cheme, i decribed. The control performnce on the coutic potentil energy i tudied by invetigting the optiml feedforwrd controller. In order to invetigte the phyicl mechnim of the feedforwrd control on the coutic potentil energy, the phyicl chrcteritic of the optiml impednce, preented by the econdry ource, re dicued. When the coutic potentil energy i minimied, the feedforwrd control effect on the dynmic behviour of the primry tructure re dicued in term of the kinetic energy of the tructure. The quntittive feedforwrd control effect on the coutic potentil energy i invetigted by preenting cumultive um of the coutic potentil energy over the frequency rnge of interet. Experimentl vlidtion on the control performnce of the feedforwrd controller i crried out to vlidte the theoreticl reult. In Chpter 5, the ctive velocity feedbck control of the coutic potentil energy, in the imple vibro-coutic model of interet in thi thei, i invetigted under brodbnd diturbnce in the three coupled ce dicued in chpter 3 nd 4. The ctive velocity feedbck control ytem i configured in three wy: uing (i) n coutic ctutor, (ii) tructurl ctutor nd (iii) both the ctutor. Reltive control effectivene of the coutic 17

39 Chpter 1. Introduction ctutor nd the tructurl ctutor, driven by velocity feedbck controller, i invetigted. When the ctive velocity feedbck control i implemented uing the coutic ctutor, the optiml gin of the velocity feedbck controller i determined in ech coupled ce when the ummed coutic potentil energy, over the frequency rnge of interet, i minimied. When the ctive velocity feedbck control i implemented uing the tructurl ctutor, the criticl dmping of the SDOF tructure i ued the gin of the velocity feedbck controller driving the econdry tructurl ctutor, ince the velocity feedbck unit work kyhook dmper nd obviouly top the tructure t the optiml condition. The dynmic coupling between the tructurl ctutor nd the coutic cvity i conidered to invetigte the control effectivene of the coutic potentil energy. When the decentrlied velocity feedbck control i implemented uing both the ctutor, the ctive dmping of the coutic ctutor i optimied under the velocity feedbck control implemented by the tructurl ctutor. The velocity feedbck control effect on the dynmic behviour of the primry tructure i dicued in term of the kinetic energy of the SDOF tructure coupled to the coutic cvity. The reltive control performnce of the velocity feedbck controller i demontrted in term of the cumultive um of the coutic potentil energy over the frequency rnge of interet. The bet control configurtion i uggeted for the control of the coutic potentil energy in the vriou coupled ce. 18

40 Chpter. Anlyticl model CHAPTER ANAYTICA MODE OF A ONE-DIMENSIONA VIBRO-ACOUSTIC SYSTEM.1 Introduction In thi chpter imple vibro-coutic model, which i ued extenively in thi thei, i decribed. The imple vibro-coutic ytem conit of finite one-dimenionl coutic tube excited by ingle-degree-of-freedom (SDOF) tructure t one end with rbitrry impednce t the other end. The chpter minly concern n invetigtion into the tructurlcoutic coupling effect on the dynmic behviour of the imple vibro-coutic ytem bed on the mobility-impednce pproch. A non-dimenionl coupling fctor i provided to determine the threhold of the degree of tructurl-coutic coupling. Structurl vibrtion induce coutic wve propgtion in the cvity to which the tructure i connected. The nlyi of the vibro-coutic ytem trt from the invetigtion of purely tructurl nd coutic ytem. Acoutic wve propgtion in the cvity h been tudied 19

41 Chpter. Anlyticl model extenively nd the relted wve eqution hve been preented in mny text book for exmple [Kinler et l (198) nd Fhy (001)]. They dicu the mutul tructurl-coutic interction in vibro-coutic ytem nd preent the phyicl chrcteritic of the ytem which re dependent upon the propertie of both the purely tructurl nd coutic ytem. The nlyi of vibro-coutic ytem cn be implified by uing the mobility-impednce pproch [Hixon (1977) nd Kim nd Brennn (1999)]. The concept of mobility nd impednce re exploited for the nlyi of the tructurl-coutic coupling in vibro-coutic ytem. The degree of coupling in vibro-coutic ytem i one of key fctor to chrcterie the vibro-coutic repone [Dowell et l (1977)]. In the ce when tructurl nturl frequency i unchnged by the coutic preure in the cvity uch tht it behve like n in-vcuo tructure, the coutic preure loding on the tructure cn be neglected. In nother ce when there i ignificnt chnge in tructurl reonnce frequency due to the coutic preure in the cvity, the dynmic behviour of the tructure i ubject to the coutic preure loding. The former i clled wekly coupled ce nd the ltter i clled trongly coupled ce in thi thei. The phyicl chrcteritic of the coupling phenomenon cn be chieved by uing imple vibro-coutic model. The imple model mke it poible to verify the nlyi procedure, in the cvity where plne wve propgte, compred to the vilble exct olution. Alo, the imple geometry my give better phyicl inight into the nture of vibro-coutic problem, which cn be trightforwrdly extended to threedimenionl ytem. In thi chpter, the dynmic behviour of the imple vibro-coutic ytem i invetigted in the vriou coupled ce nd when the tructurl nturl frequency i below nd bove the fundmentl coutic mode of cloed-cloed tube repectively. In ection., the coutic input impednce in imple vibro-coutic model, under SDOF tructurl excittion t one end nd terminted by rbitrry impednce t the other end, i derived uing n impednce pproch. Alo, the coutic preure nd prticle velocity in the nlyticl model re preented in term of non-dimenionl tructurl-coutic prmeter. In ection.3, the coupling mechnim i invetigted bed on the mobility-impednce pproch. In order to provide the threhold of the degree of coupling, coupling fctor i clculted in combined SDOF tructure finite cloed tube ytem. In ection.4, the vibro-coutic repone, in the cloed tube ytem, re repreented by the coutic potentil energy in the cvity nd the kinetic energy of the tructure coupled to the coutic cvity. The vibro-coutic repone re dicued in the vriou coupled ce determined by the threhold of the degree of 0

42 Chpter. Anlyticl model coupling with correponding ODS (opertionl deflection hpe). Thi chpter i cloed in ection.5 with ome concluion.. Arbitrry impednce terminted ytem The mutul tructurl-coutic interction in vibro-coutic ytem i determined by repective uncoupled tructurl nd coutic impednce. In the ce when one-dimenionl tube, under tructurl excittion t one end, i terminted by n rbitrry impednce t the other end, the uncoupled coutic impednce cn be imply derived by uing the impednce pproch. The coutic input impednce of the one-dimenionl coutic tube, terminted by n rbitrry impednce t the other end, i derived t both the end of the tube uing the impednce pproch. For the comprtive nlyi of the vibro-coutic ytem, nondimenionlied tructurl-coutic prmeter re introduced. Alo, the coutic preure nd the prticle velocity in the vibro-coutic ytem re derived in term of the non-dimenionl prmeter...1 Acoutic input impednce uing the impednce pproch Figure.1 depict n coutic preure nd prticle velocity repreenttion of combined ingle-degree-of-freedom (SDOF) tructure coutic tube ytem, which i ubjected to n j t externl time hrmonic force on the tructure f () t = Fe ω 0 t x = 0 nd i terminted by n rbitrry impednce Z t x =. The coutic preure nd the prticle velocity t x = 0 in the tube re denoted by nd U repectively, nd the reulting coutic preure nd P1 1 prticle velocity t x = re denoted by nd U repectively. Alo, the coutic preure P nd the prticle velocity on the tructure re denoted by P nd U repectively, nd on the rbitrry impednce re denoted by P nd U repectively. S S 1

43 Chpter. Anlyticl model f () t = Fe ω j t 0 M P S P 1 P P Z K (1 + jη ) U S U1 U U x = 0 x = Figure.1 Acoutic preure nd prticle velocity repreenttion of combined tructure coutic tube ytem which i under the externl time hrmonic force on the tructure, j t f () t = Fe ω 0 t x = 0 nd i terminted by rbitrry impednce Z t x =. M nd K re tructurl m nd tiffne of pring with contnt tructurl lo fctor η repectively. Alo, j = 1 nd ω i driving frequency The SDOF tructure i compoed of m nd pring with contnt tructurl lo fctor. The tructurl dmping model i dopted rther thn vicou dmping for convenience being conitent n coutic model. The uncoupled tructurl impednce cn be clculted by dding up the mechnicl impednce of the m nd the tiffne connected in prllel [Hixon (1977)], which i given by Z S K (1 + jη ) = jωm + (.1) jω where nd Z S i the uncoupled tructurl impednce of the SDOF tructure, M i tructurl m K i tiffne of pring with contnt tructurl lo fctor η repectively. When conidering the chrcteritic of the tructurl impednce with frequency, pring nd m effect re dominnt t low frequency nd t high frequency repectively. The tructurl ytem h reonnce t the frequency which the rective prt of the uncoupled tructurl impednce um to zero. The coutic input impednce i defined by the rtio of the coutic force to the coutic prticle velocity t the input poition ( x = 0 ), which i

44 Chpter. Anlyticl model Z A0 SP U 1 = (.) 1 where Z A0 i the coutic input impednce, nd P1 nd U1 re coutic preure nd prticle velocity t the input poition ( x = 0 ) in the tube with cro-ectionl re S repectively. The coutic input impednce defined in eqution (.) i the uncoupled coutic impednce of the coutic tube depicted in figure.1. The coutic preure t the end point of the tube cn be decribed in term of impednce nd prticle velocitie, nd i given by SP S = (.3) ZSU S SP1 Z11 Z1 U1 SP = Z Z U 1 (.4) SP = Z U (.5) where Z 11, Z, Z 1 nd Z 1 repreent point nd trnfer impednce of the coutic tube with cro-ectionl re S. When the coutic tube, depicted in figure.1, i terminted by the rbitrry impednce Z t x =, P = P by equilibrium of force, nd U = U by continuity of motion. The prticle velocity U cn be clculted by combining eqution (.4) nd (.5), nd i given by U Z1 = Z + Z U 1 (.6) The coutic input impednce Z A0 t x = 0 cn be clculted from the coutic preure nd the prticle velocity reltion, given in eqution (.4) nd (.6), uing the definition of the coutic input impednce given in eqution (.) to give Z Z Z = Z Z + Z A (.7) 3

45 Chpter. Anlyticl model The coutic preure nd the prticle velocity in the open tube, hown in figure.1, tke the form of two trvelling wve in oppoite direction, which re given by jkx jkx P( x, ω) = Ae + Be (.8) 1 jkx jkx U ( x, ω) = ( Ae Be ) (.9) ρ c 0 where A nd B re complex wve mplitude. Alo the prmeter ρ 0 i mbient denity nd the prmeter c i complex ound peed in the loy coutic medium with contnt coutic lo fctor η. The lo in the borptive medium cn be repreented by complex coutic wvenumber which i defined by [Brennn nd To (001)] k = ω 1 ρ 0 B(1 + jη ) (.10) where k i complex coutic wvenumber nd B i the bulk modulu, which i preure incree needed to cue given reltive decree in volume under uniform compreion. For mll coutic lo fctor, the complex coutic wvenumber given in eqution (.10) cn be rewritten ω ω 1 k = (1 j η ) (.11) c c 0 where c i the complex ound peed in the loy coutic medium, which i pproximtely 1 c c0 / 1 j η (.1) where c0 = B / ρ0. The rel prt of the complex wve number relte to the wve propgtion nd the imginry prt govern the wve ttenution in the coutic medium. 4

46 Chpter. Anlyticl model The coutic preure nd the prticle velocity t x = 0 cn be obtined by etting x to zero in eqution (.8)~(.9) to give P1 = A+ B (.13) U 1 1 = ( A B ) (.14) ρ c 0 nd the reulting coutic preure nd prticle velocity t x mnner by = re given in the me P = ( A+ B)co k j( A B)ink (.15) 1 U = [( A B)co k j( A B)in ] ρ c + k 0 (.16) Combining eqution (.13)~(.16) give the trnfer mtrix of the coutic tube, which i given by [ Munjl (1987) ] P co k jρ0cin k P1 U = jin k/ ρ0c co k U 1 (.17) Rerrnging eqution (.17) give the impednce mtrix of the coutic tube, which i co k 1 ρ0c ρ0c P1 jin k jin k U1 P = 1 cok (.18) ρ0c ρ0c U jin k jin k Eqution (.4) cn thu be written 5

47 Chpter. Anlyticl model co k 1 ρ0cs ρ0cs SP1 jin k jin k U1 SP = 1 cok U ρ0cs ρ0cs jin k jin k (.19) Subtituting the correponding impednce in eqution (.19) into eqution (.7) give the coutic input impednce Z A0 t x = 0, which i given by Z Z co k + j in k ρ0cs = ρ cs Z co k + j in k ρ cs A0 0 0 (.0) The generl coutic input impednce given in eqution (.0) h pecific form ccording to the impednce rtio Z / ρ 0cS. When the impednce rtio Z / ρ 0cS = 1, the coutic input impednce Z A0 i tht of emi-infinite tube [Kinler et l (198)], which i Z = ρ cs emi-infinite tube (.1) A0 0 When the impednce rtio Z / ρ 0cS = 0, the coutic tube h n open condition t x = nd the coutic input impednce i in k ZA0 = jρ0cs open tube (.) co k Alo, when the impednce rtio Z / ρ 0cS 0 i infinitely lrge, the coutic input impednce Z A0 i tht of rigidly cloed tube, which i co k ZA0 = jρ0cs cloed tube (.3) in k The coutic input impednce t x impednce Z with the uncoupled tructurl impednce =, Z A cn be obtined imply by replcing the rbitrry Z S, nd i given by 6

48 Chpter. Anlyticl model Z A ZS co k + j in k ρ0cs = ρ0cs ZS co k + j in k ρ cs 0 (.4) It cn be een tht the behviour of the impednce Z A i ffected by the impednce rtio ZS / ρ cs 0. The impednce Z A i tht of n open tube given in eqution (.) in the ce when the impednce rtio ZS / ρ 0cS i extremely mll, nd i tht of cloed tube given in eqution (.3) in the ce when the impednce rtio Z / ρ 0cS 0 i infinitely lrge. S.. Non-dimenionl tructurl-coutic prmeter For frequency dependent tructurl nd coutic chrcteritic in the combined ytem depicted in figure.1, the prmeter / λ i defined ω = (.5) λ c π 0 where i the tube length, λ i coutic wvelength, ω i the driving frequency nd ound peed in the lole coutic medium. c 0 i One of the non-dimenionl prmeter ued to chrcterie the vibro-coutic ytem i the tructurl-coutic tiffne rtio given by K K BS / = (.6) K where K i coutic bulk tiffne, which i the coutic tiffne of the cloed tube when the tructurl motion i ttic. In ddition, ρ 0 i the mbient denity of the coutic medium in the tube with cro-ectionl re S. 7

49 Chpter. Anlyticl model Another importnt prmeter i the proximity of the uncoupled tructurl nturl frequency nd the uncoupled fundmentl coutic nturl frequency. The tructurl-coutic nturl frequency rtio i defined by ω πc M 0 ω = K (.7) where ω ( = π c0 / ) i the uncoupled fundmentl coutic nturl frequency of cloedcloed tube when / λ = 1/ in eqution (.5). Alo, ω ( = K / M ) i the uncoupled tructurl nturl frequency...3 Acoutic preure nd prticle velocity Acoutic preure nd prticle velocity in the coutic tube re the combintion of poitive going wve nd negtive going wve, which re given in eqution (.8)~(.9). The complex wve mplitude cn be clculted by pplying correponding boundry condition. Since the tructure nd the coutic tube hre the me velocity t the input poition ( x = 0 ), the coutic prticle velocity t the point cn be repreented by the rtio of the force F 0 to the um of the uncoupled tructurl impednce Z S given in eqution (.1), nd the uncoupled coutic input impednce Z A0 given in eqution (.0). The prticle velocity t x = 0, U (0, ω ) i given by U (0, ω ) = Z S F0 + Z A0 (.8) On the other hnd, the coutic tube i terminted by the rbitrry impednce the prticle velocity t x =, U(, ω ) i given by Z t x =. So, SP(, ω) U(, ω ) = (.9) Z 8

50 Chpter. Anlyticl model where Pω (, ) i the coutic preure t x = due to the force F 0. Subtituting boundry condition given in eqution (.8)~(.9) into eqution (.8)~(.9) give complex wve mplitude A nd B, which re Z co k j in k ρ0c F + 0 ρ0cs A = 1 + Z Z S + ZA0 co k + j in k ρ0cs (.30) Z co k j in k ρ0c F + 0 ρ0cs B = 1+ Z Z S + ZA0 co k + j in k ρ0cs (.31) Subtituting the complex wve mplitude given in eqution (.30)~(.31) into eqution (.8)~(.9) give the coutic preure nd the prticle velocity in the vibro-coutic ytem depicted in figure.1, which re Px (, ω) = ρ0c Z S F + Z 0 0 A0 Z co k ( x) + jin k ( x) ρ cs Z co k + j in k ρ cs 0 (.3) U( x, ω) = Z S F + Z 0 0 A0 Z co k ( x) + j in k ( x) ρ cs Z co k + j in k ρ cs 0 (.33) For generl comprtive nlyi between tructurl propertie nd correponding coutic propertie, it i convenient to non-dimenionlie the primry coutic preure nd the prticle velocity follow: 9

51 Chpter. Anlyticl model Px ˆ( ˆ, ˆ) = Zˆ co ( j )(1 x) j in ( j )(1 x) 1 jη Zˆ + Zˆ jη S A0 co ( π( jη ) ˆ) + jzˆ in ( π( jη ) ˆ ) jη ( π η ˆ ˆ) + ( π η ˆ ˆ) (.34) Uˆ( xˆ, ˆ) = Z 1 co ( )(1 ) ˆ ˆ in ( )(1 ) ˆ jη ( π jη xˆ ) + jz ( π jη xˆ ) ˆ + Zˆ S A0 ˆ ˆ ˆ ( π jη ) + jz ( π jη ) co ( ) in ( ) jη (.35) where Px ˆ( ˆ, ˆ) = Px ( ˆ, ˆ)/ P, Uˆ( xˆ, ˆ) U( xˆ, ˆ)/ U, = S S 0 0 Z ˆ = Z / ρ c S, Z ˆ / A0 = ZA0 ρ0c0s, Zˆ = Z / ρ0c0s nd ˆ = / λ. xˆ ( = x/ ) i ny normlied poition long the tube. Alo, P nd U re non-dimenionliing fctor for the coutic preure nd the prticle velocity, which re defined by F / S 0 nd F0 / ρ 0c0S repectively. The normlied tructurl impednce Zˆ S ( = ZS / ρ0c0s) nd the normlied coutic impednce Z ˆ ( / A0 = ZA 0 ρ0c0s ) in eqution (.34)~(.35) cn be written in non-dimenionl form uing non-dimenionl prmeter given in eqution (.5)~(.7) Zˆ S ˆ 1 ( ˆ ω) + jη = j (.36) π K ˆˆ Zˆ A0 ( π η ˆ ) + ( π η ˆ ) Zˆ co ( j ) j in ( j ) jη = jη co ( ) ˆ + ˆ in ( ) ˆ jη ( π jη ) jz ( π jη ) (.37) where Kˆ ( = K / K ) i the tructurl-coutic tiffne rtio nd ˆ ω ( = ω / ω ) i the tructurl-coutic nturl frequency rtio repectively..3 Structurl-coutic coupling in vibro-coutic ytem 30

52 Chpter. Anlyticl model The vibro-coutic repone cn be generlly chrcteried by the degree of the tructurlcoutic coupling. A coupling mechnim i invetigted bed on conceptul repreenttion of vibro-coutic ytem uing the mobility-impednce pproch. Alo, coupling fctor i ued to invetigte the effect of the tructurl-coutic prmeter on the degree of coupling in vibro-coutic ytem..3.1 Conceptul repreenttion of vibro-coutic ytem The velocity of the tructure, in the combined ytem depicted in figure.1, t the input poition ( x = 0 ), U i equl to the prticle velocity t the point given in eqution (.8) by S continuity of motion, which cn be rewritten U S = Z S F0 + Z A0 (.38) where Z i the uncoupled tructurl impednce nd Z 0 i the uncoupled coutic S impednce of the tube repectively. A Figure.() nd (b) decribe conceptul tructurl-coutic coupled repreenttion of the combined ytem, depicted in figure.1, t the input poition ( x = 0 ) in term of uncoupled tructurl nd coutic impednce. The uncoupled tructurl impednce Z S nd uncoupled coutic impednce Z A0 re connected in prllel hown in figure.() hring the me velocity. The driving force F 0 i ditributed between the tructure nd the coutic cvity ccording to their impednce. The uncoupled tructurl impednce effective force pplied to the tructure F to the velocity U S Z S i the rtio of the. The uncoupled coutic impednce velocity U S Z repreent the rtio of the effective force pplied to the cvity to the A0 hown in figure.(b). F 31

53 Chpter. Anlyticl model F 0 Z S F Z S Z A0 F U S Z A0 U S U S Figure.() coupled ytem repreented by uncoupled tructurl impednce nd uncoupled coutic impednce Z A0 force F0 = F + F Figure.(b) tructurl force F nd Z S coutic force F, where the driving The tructurl velocity t the input poition ( x = 0 ), U given in eqution S (.38) cn be rewritten in term of the uncoupled tructurl mobility impednce Z A0, which i Y S nd the uncoupled coutic U S YS = F 0 (.39) 1 + YZ S A0 Eqution (.39) cn be repreented by block digrm which h ingle input force F 0 pplied to the tructure nd ingle output velocity U t the input poition ( x = 0 ) with S cloed loop trnfer function Y /(1 + Y Z ) S S A0 hown in figure.3. The tructurl velocity U S i ffected by both the uncoupled tructurl mobility YS nd the uncoupled coutic impednce Z. A0 F0 + Y S U S Z A0 Figure.3 A block digrm repreenttion of eqution (.39) where F 0 i the input force pplied to the tructure, U S i output velocity t the input poition ( x = 0 ), YS i the uncoupled tructurl mobility nd Z A0 i the uncoupled coutic impednce 3

54 Chpter. Anlyticl model Eqution (.39) cn be rewritten by dividing by the force F 0 to give Y CS YS = (.40) 1 + YZ S A0 where Y i the rtio of the tructurl velocity to the force F nd i defined coupled CS tructurl mobility. The degree of the tructurl-coutic coupling in the combined ytem i determined by the mgnitude of YZ S A0 U S 0 defined the coupling fctor. The coupled tructurl mobility Y tend to the uncoupled tructurl mobility Y in the ce CS S when the modulu of the coupling fctor YZ S A0 << 1. Thi ce implie tht the uncoupled coutic impednce Z A0 i much mller thn the uncoupled tructurl impednce Z S. The effect of the coutic impednce on the tructure i negligible nd the tructurl repone i determined only by the tructurl chrcteritic though it i in-vcuo. In thi ce, thi tructurl-coutic coupled ytem i id to be wekly coupled. The coupled ytem, under wekly coupled condition, men phyiclly n coutic tube driven by hevy nd tiff tructure. In the other ce when the modulu of the coupling fctor YZ 0 >> 1, the coupled tructurl mobility i the invere of the uncoupled coutic impednce YCS Z A0 S A. The repone of the tructure i ubject to the coutic chrcteritic of the tube. The tructurl repone h pek nd trough t the frequencie with low nd high coutic impednce repectively. Thi tructurl-coutic coupled ytem i id to be trongly coupled. In thi ce, the phyicl ytem, under trongly coupled condition, i n coutic tube driven by light nd flexible tructure..3. Coupling fctor The degree of tructurl-coutic coupling in vibro-coutic ytem cn be determined by the modulu of coupling fctor dicued in ection.3.1. The coupling fctor of vibro- 33

55 Chpter. Anlyticl model coutic ytem i defined by the product of the uncoupled tructurl mobility uncoupled coutic impednce Z. A0 Y S nd The coupling fctor of the combined SDOF tructure finite cloed tube ytem cn be obtined from the uncoupled tructurl nd coutic impednce given in eqution (.1) nd (.3), which i ω co k YZ S = ρ cs K (1 j ) M in k A0 0 + η ω (.41) Alterntively, it cn be written in term of the non-dimenionl prmeter given in eqution (.5)~(.7) YZ S A0 ( π jη ˆ ) ( ˆ ) ˆ co ( ) ˆ π = K 1 ( ˆ ˆ (.4) ω) + jη jη in π( jη ) The coupling fctor, given in eqution (.4), i the product of three term repreenting: (i) the tructurl-coutic tiffne rtio, (ii) the uncoupled tructurl mode nd (iii) the uncoupled coutic mode in the vibro-coutic ytem. The coupling fctor YZ S A0 i proportionl to the tructurl-coutic tiffne rtio. Phyiclly, lrge tiffne rtio repreent cloed tube driven by flexible tructure. In thi ce the tructurl repone in the vibro-coutic ytem i ubject to coutic chrcteritic of the cloed tube under more trongly coupled condition. On the other hnd, mll tiffne rtio repreent cloed tube driven by tiff tructure. In thi ce the tructurl repone in the vibro-coutic ytem i determined only by the tructurl chrcteritic under more wekly coupled condition. The modulu of the uncoupled tructurl mode relted term given in eqution (.4) cn be written 34

56 Chpter. Anlyticl model πˆ πˆ = 1 ( ˆ ωˆ ) + jη ˆ ( 1 ( ˆ ω) ) + η (.43) The term ˆ ω ˆ in the denomintor i equl to ω / ω, o thi term depend on how cloe the frequency i to the tructurl nturl frequency. The modulu in eqution (.43) h mximum vlue t the tructurl nturl frequency nd thu the vibro-coutic ytem become more trongly coupled. At the tructurl nturl frequency, the modulu decree with the tructurl lo fctor η. When the tructurl lo fctor η i mller, the vibrocoutic ytem i under more trongly coupled condition. Otherwie, the vibro-coutic ytem i under more wekly coupled condition. The cotngent term repreenting the uncoupled coutic mode given in eqution (.4) cn be expnded uing trigonometric identitie ( ˆ ) ( π jη ) co π( jη ) co( πˆ)coh( πη ˆ) in( ˆ + j π)inh( πη ˆ ) = in ( ) ˆ in( πˆ)coh( πη ˆ) jco( πˆ)inh( πη ˆ) (.44) The mximum modulu of the uncoupled coutic mode relted term given in eqution (.4) cn be pproximtely determined t / λ = n/ for mll coutic lo fctor η, which i ( π η ˆ ) ( π jη ˆ ) mx co ( j ) jη in ( ) πη n (.45) where in( π ˆ ) = 0, co( π ˆ ) =± 1, inh( πη ˆ) πη ˆ nd coh( πη ˆ) 1. It cn be een tht the mximum modulu decree with the coutic lo fctor η nd t higher frequencie. The vibro-coutic ytem i more trongly coupled for mller coutic lo fctor η nd t lower frequencie. Otherwie, the tructurl-coutic interction in the vibrocoutic ytem become more wekly coupled. 35

57 Chpter. Anlyticl model Figure.4() nd (b) how the modulu of the coupling fctor YZ S A0, given in eqution (.4), for vriou vlue of tructurl-coutic tiffne rtio K / nd nturl frequency rtio ω / ω. The tructurl nturl frequency i t / λ = 0. 1 in figure.4() nd t / λ = 0. 8 in figure.4(b). Alo, the uncoupled coutic mode re t / λ = n/ in figure.4() nd (b) where n i n integer. K In one extreme ce when the tiffne rtio K / K = 10, the modulu of the coupling fctor i much lrger thn the threhold of 1 hown in figure.4() nd (b) (olid line). In thi ce, the vibro-coutic ytem behve like trongly coupled ytem. When the tructurl nturl frequency i below the fundmentl coutic mode, the degree of coupling, in figure.4(), i le with frequency due to the mobility of the tructurl m reducing 1/ ω. Alo, when the tructurl nturl frequency i bove the fundmentl coutic mode, the degree of coupling, in figure.4(b), i more trongly coupled with frequency up to the tructurl nturl frequency due to the mobility of the tructurl tiffne increing ω. The degree of coupling i reduced t frequencie higher thn the tructurl nturl frequency. 3 In nother extreme ce when the tiffne rtio K / K 10 3 =, the modulu of the normlied coupling fctor i much mller thn the threhold of 1 hown in figure.4() nd (b) (dhed line). In thi ce, the vibro-coutic ytem behve like wekly coupled ytem. The degree of coupling how imilr behviour with frequency for the chnge of the tructurl nturl frequency. In the intermedite ce when the tiffne rtio K / K = 1, the vibro-coutic ytem how compounded behviour of previou two extreme ce hown in figure.4() nd (b) (dotted line). When the tructurl nturl frequency i below the fundmentl coutic mode, the vibro-coutic ytem h more trongly coupled condition t the firt three pek, which re the uncoupled tructurl mode t / λ = 0. 1 nd the uncoupled coutic mode t / λ = 0. 5 nd t / λ = 1 hown in figure.4(). The degree of coupling i more wekly coupled with frequency due to the mobility of the tructurl m. Alo, when the tructurl nturl frequency i bove the fundmentl coutic mode, the vibro-coutic ytem i more trongly coupled t ll the four pek hown in figure.4(b). 36

58 Chpter. Anlyticl model In ummry, the degree of coupling in vibro-coutic ytem i minly determined by the tructurl-coutic tiffne rtio K / K in the two extreme ce. However, in the intermedite ce, the degree of coupling i dependent on coincidence of uncoupled tructurl nd coutic reonnce. When the tructurl nturl frequency ω i cloe to multiple of the coutic fundmentl nturl frequency ω, the vibro-coutic ytem become more trongly coupled. Otherwie, the vibro-coutic ytem become more wekly coupled. 37

59 Chpter. Anlyticl model / λ () Structurl nturl frequency i t / λ = 0.1 ( ω / ω = 5) / λ (b) Structurl nturl frequency i t / λ = 0.8 ( ω / ω = 0.6 ) Figure.4 Modulu of coupling fctor for vriou tructurl-coutic tiffne rtio K / K where tructurl lo fctor = 10 nd n coutic lo fctor = 10 (olid line: K / K = 10 3, dhed line: K η 3 = nd dotted line: / 1 / K 10 K K = ) η 38

60 Chpter. Anlyticl model.4 Vibro-coutic repone in vriou coupled ce In thi ection, the dynmic behviour of the coupled ytem, depicted in figure.1, i dicued under vriou coupling condition demontrted in ection.3 for the pecific ce of n infinitely lrge impednce t x =. The vibro-coutic repone i repreented by the coutic potentil energy in the cvity under tructurl excittion nd the kinetic energy of the tructure coupled to the cvity..4.1 Acoutic potentil energy in n coutic cvity The totl time-verged coutic potentil energy in one-dimenionl coutic cvity i given by integrting the relevnt energy denity over the entire volume conidered [Nelon nd Elliott (199)], which i given for the complex ound peed c by S = ω (.46) EP ( ω) Px (, ) dx 4 ρ 0 0 c where EP ( ω ) i the coutic potentil energy nd Pxω (, ) i the coutic preure t ny poition in the cvity. For the pecific ce of cloed tube, the coutic preure Pxω (, ) cn be written by etting the impednce rtio Z / ρ 0cS to infinity in eqution (.3) to give = F co k ( x) 0 Px (, ω) jρ0c Z S + Z A0 in k (.47) where the uncoupled coutic impednce Z A0 i tht of the cloed tube given in eqution (.3). The coutic preure given in eqution (.47) cn lo be rewritten in term of nondimenionl prmeter. Setting Z ˆ to infinity in eqution (.34) give the non-dimenionl coutic preure in the cvity to be 39

61 Chpter. Anlyticl model ( π jη ˆ ˆ x) ( ) 1 co ( )(1 ) Px ˆ( ˆ, ˆ) = j Zˆ + Zˆ jη in π( jη ) ˆ S A0 (.48) The normlied tructurl impednce Z ˆS i defined in eqution (.36) nd the normlied coutic input impednce Z ˆ A 0 cn be obtined by etting Z ˆ to infinity in eqution (.37) to give Zˆ A0 ( π jη ˆ ) ( ˆ ) co ( ) = j jη in π( jη ) (.49) Hence, the coutic potentil energy normlied by tht t the ttic tte ( ˆ = 0 ) cn be written in term of non-dimenionl prmeter Eˆ p ( ˆ) = ˆ ˆ Px ( ˆ, ) dxˆ ˆ Px ( ˆ,0) dxˆ (.50) where Eˆ ˆ ˆ P( ) = EP( )/ EP(0). The coutic potentil energie E ( ˆ P ) nd the coutic potentil energy t the ttic tte ( ˆ = 0 ), E P (0) re defined repectively by S P 1 ˆ E ˆ ˆ P ( ) = P( x, ) dx 4 ρ 0 0 c ˆ ˆ (.51) P S P 1 ˆ = P ˆ 4 ρ 0 0 c ˆ E (0) ( x,0) dx (.5) The coutic potentil energy t the ttic tte, given in eqution (.5), cn be pproximtely clculted by etting ˆ 0 into the coutic preure given in eqution (.48). The coutic potentil energy t the ttic tte for mll tructurl nd coutic lo fctor i S P ˆ K EP(0) 4 ρ ˆ 0 c 1 + K (.53) 40

62 Chpter. Anlyticl model The coutic potentil energy t the ttic tte i determined by tructurl-coutic tiffne rtio K / K in the cvity with pecific cro-ectionl re S, mbient denity ρ 0, complex ound peed c nd the non-dimenionliing fctor P ( = F0 /S)..4. Kinetic energy of tructure coupled with n coutic cvity The tructurl velocity t the input poition ( x = 0 ) cn be ffected by the coutic preure in vibro-coutic ytem depending on the degree of tructurl-coutic coupling. The tructurl kinetic energy i defined by [Meirovitch (1986)], which i 1 EK( ) M U ω = (.54) where EK ( ω ) i the tructurl kinetic energy, M i the tructurl m nd U i the tructurl velocity. The tructurl kinetic energy cn be rewritten by ubtituting the tructurl velocity given in eqution (.38) into eqution (.54) 1 EK( ω ) = M F0 Z S 1 + Z A0 (.55) where F0 i the excittion force, Z S i the uncoupled tructurl impednce, nd A0 Z i the coutic input impednce of cloed tube given in eqution (.3). The tructurl kinetic energy, given in eqution (.55), cn be written in non-dimenionl form Eˆ K ( ˆ) = Zˆ S 1 + Zˆ A0 (.56) where ˆ ˆ ˆ 1 EK( ) = EK( )/ M F0 / ρ0c0s Z ˆ ( = Z / ρ c S i the normlied nd A0 A ) coutic input impednce of cloed tube given in eqution (.49). 41

63 Chpter. Anlyticl model.4.3 Simultion reult of the vibro-coutic repone In thi ection, ome imultion reult on the coutic potentil energy, given in eqution (.50), nd on the tructurl kinetic energy, given in eqution (.56), re preented in vriou coupled ce with phyicl interprettion. Alo, the tructurl nd coutic ODS (Opertionl Deflection Shpe) re demontrted in order to invetigte their contribution to the coutic potentil energy nd ptil ditribution in the cvity. Figure.5(), (b) nd.7(), (b) how the coutic potentil energy nd the tructurl kinetic energy in combined tructure finite cloed tube in vriou tructurl-coutic coupled ce when the tructurl nturl frequency i t / λ = 0. 1 nd t / λ = 0. 8 repectively. The coutic potentil energy i normlied by tht t the ttic tte ( / λ = 0) in ech coupled ce nd the tructurl kinetic energy i rbitrrily normlied by tht t / λ =. Alo, figure.6(), (b), (c) nd.8(), (b), (c) demontrte the ODS, normlied by mximum modulu, with repect to the normlied rbitrry poition x / in vriou coupled ce when the tructurl nturl frequency i t / λ = 0. 1 nd t / λ = 0. 8 repectively. The tructurl nd coutic ODS re clculted by uing the rel prt of the coutic preure in the cvity. In the trongly coupled ce with tructurl-coutic tiffne rtio K / K = 10 3, the nturl frequency rtio ω / ω h negligible effect on the vibro-coutic repone due to inignificnt tructurl impednce. The coutic potentil energy in the cvity h coutic mode of n open-cloed tube t / λ = (n 1)/ 4 hown in figure.5() nd.7() (olid line) where n i n integer. The minimum of the coutic potentil energy i roughly contnt with frequency due to the velocity of the tructurl m being controlled by the coutic preure in the cvity. In thi ce, the tructure h negligible effect on the coutic potentil energy providing n open-tube condition t x = 0. The kinetic energy of the tructure h reonnce t / λ = (n 1)/ 4 nd i ubject to the coutic preure in the cvity hown in figure.5(b) nd.7(b) (olid line). Alo, the tructurl kinetic energy h nti-reonnce t / λ = n/ due to the dominnt coutic input impednce in eqution (.55), which h reonnce t / λ = n/. The coutic potentil energy, hown in figure.5() nd.7() (olid line), i contributed by ech coutic mode. The coutic potentil energy i dominted by the fundmentl coutic ODS hown in figure.6() nd.8(). 4

64 Chpter. Anlyticl model Alo, the coutic ODS in the coutic potentil energy hve low mplitude t the input poition ( x = 0 ) nd high mplitude t the end of the cloed tube. In the wekly coupled ce with tructurl-coutic tiffne rtio K / K = 10 3, the vibrocoutic repone re enitive to the nturl frequency rtio ω / ω. If the tructurl nturl frequency i t / λ = 0. 1 below the fundmentl coutic mode, the coutic potentil energy h dominnt tructurl mode t / λ = 0. 1 nd coutic mode of cloed-cloed tube t / λ = n/ hown in figure.5() (dhed line) where n i n integer. The minimum of the coutic potentil energy in the cvity reduce with frequency due to the velocity of the tructurl m reducing 1/ω. The kinetic energy of the tructure i not ffected by the coutic preure in the cvity in-vcuo nd h reonnce t / λ = 0. 1 hown in figure.5(b) (dhed line). In thi ce, the tructurl ODS dominte over the coutic potentil energy but the coutic ODS i inignificnt hown in figure.6(b). Alo, the coutic ODS h generlly the me mplitude t both the end of the cvity but the tructurl ODS h lower mplitude t the input poition ( x = 0 ) due to the rigid boundry condition t the other end. In the wekly coupled ce, when the tructurl nturl frequency i t / λ = 0. 8 bove the fundmentl coutic mode, the coutic potentil energy in the cvity h tructurl mode t / λ = 0. 8 nd coutic mode of cloed-cloed tube t / λ = n/ hown in figure.7() (dhed line) where n i n integer. Alo, the minimum of the coutic potentil energy incree due to the tructurl tiffne effect nd reduce due to the tructurl m effect t frequencie below nd bove the tructurl nturl frequency repectively. The kinetic energy of the tructure i generlly determined by only the tructurl chrcteritic. However, the tructurl kinetic energy i ffected by the coutic preure t / λ = 0. 5 nd t / λ = 1 hown in figure.7(b) (dhed line) due to the more trongly coupled tructure into the cvity cued by the le tructurl m with the me tiffne compred to the ce of the tructurl nturl frequency i t / λ = In thi ce, the tructurl nd coutic ODS hve more or le equivlent contribution to the coutic potentil energy hown in figure.8(b). The ptil ditribution of the ODS t both the end of the cvity h imilr trend to the ce of the tructurl nturl frequency t / λ =

65 Chpter. Anlyticl model In the intermedite ce with tructurl-coutic tiffne rtio K / K = 1, the vibrocoutic repone how different behviour ccording to the nturl frequency rtio ω / ω in the wekly coupled ce. When the tructurl nturl frequency i t / λ = 0. 1, the coutic potentil energy h dominnt tructurl mode t / λ = 0.13 nd coutic mode of cloed-cloed tube motly t / λ = n/ hown in figure.5() (dotted line) where n i n integer. The tructurl mode i t the higher frequency thn tht in the wekly coupled ce, which i due to the coutic bulk tiffne dded into the tructurl tiffne. The minimum of the coutic potentil energy reduce with frequency for the me reon in the wekly coupled ce. The tructurl kinetic energy h reonnce t / λ = 0.13 nd t / λ = n/ being ubject to the coutic preure in the cvity hown in figure.5(b) (dotted line). Alo, the tructurl kinetic energy h nti-reonnce t / λ = n/ for the me reon in the trongly coupled ce. In thi ce, the tructurl nd coutic ODS in the coutic potentil energy hve imilr ptil ditribution to thoe in the wekly coupled ce hown in figure.6(c). In the intermedite ce, when the tructurl nturl frequency i t / λ = 0. 8, the coutic potentil energy h only coutic mode hown in figure.7() (dotted line) due to more trongly coupled tructure. Compred to the ce of the tructurl nturl frequency t / λ = 0. 1, the tructurl mode h n inignificnt contribution to the coutic potentil energy. The minimum of the coutic potentil energy i roughly contnt for the me reon in the trongly coupled ce. The kinetic energy of the tructure h reonnce t the reonnce frequencie of the coutic preure in the cvity hown in figure.7(b). Alo, the nti-reonnce occur t / λ = n/ for the me reon in the trongly coupled ce. In thi ce, the coutic ODS h more or le equivlent contribution to the coutic potentil energy hown in figure.8(c). The coutic ODS h low mplitude t the input poition ( x = 0 ) nd high mplitude t the other end in the trongly coupled ce. 44

66 Chpter. Anlyticl model / λ () Normlied coutic potentil energy / λ (b) Normlied tructurl kinetic energy Figure.5 () coutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in ech coupled ce (b) tructurl kinetic energy rbitrrily normlied by tht t / λ = where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5), the contnt tructurl nd coutic lo fctor η = 10 nd η = 10 repectively (olid line: trongly coupled 3 3 ce with K / K = 10, dhed line wekly coupled ce with K / K = 10 nd dotted line: intermedite ce with K / K = 1) 45

67 Chpter. Anlyticl model : / λ = 1/4 : / λ = 3/ 4 : / λ = 5/ 4 x / () Strongly coupled ce : / λ = 0.1 : / λ = 1/ : / λ = 1 : / λ = 3/ x / (b) Wekly coupled ce : / λ = 0.13 : / λ = 1/ : / λ = 1 : / λ = 3/ x / (c) Intermedite ce Figure.6 Opertionl deflection hpe of the coutic preure in the cvity normlied by 3 mximum modulu: () trongly coupled ce with K / K = 10, (b) wekly coupled ce with K 3 / K = 10 nd (c) intermedite ce with K / K = 1 where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5), the contnt tructurl nd coutic lo fctor η = 10 nd η = 10 repectively 46

68 Chpter. Anlyticl model / λ () Normlied coutic potentil energy / λ (b) Normlied tructurl kinetic energy Figure.7 () coutic potentil energy normlied by tht t the ttic tte ( / λ = 0) in ech coupled ce (b) tructurl kinetic energy rbitrrily normlied by tht t / λ = where the tructurl nturl frequency i t / λ = 0.8 ( ω / ω = 0.6), the contnt tructurl nd coutic lo fctor η = 10 nd η = 10 repectively (olid line: trongly coupled 3 3 ce with K / K = 1, dhed line wekly coupled ce with K / K = 10 nd dotted line: 0 intermedite ce with K / K = 1) 47

69 Chpter. Anlyticl model : / λ = 1/4 : / λ = 3/4 : / λ = 5/4 x / () Strongly coupled ce : / λ = 0.8 : / λ = 1/ : / λ = 1 : / λ = 3/ x / (b) Wekly coupled ce : / λ = 0.3 : / λ = 3/4 : / λ = 1. x / (c) Intermedite ce Figure.8 Opertionl deflection hpe of the coutic preure in the cvity normlied by 3 mximum modulu: () trongly coupled ce with K / K = 10, (b) wekly coupled ce with K 3 / K = 10 nd (c) intermedite ce with K / K = 1 where the tructurl nturl frequency i t / λ = 0.8(ω / ω = 0.6), the contnt tructurl nd coutic lo fctor η = 10 nd η = 10 repectively 48

70 Chpter. Anlyticl model.5 Concluion The dynmic behviour of imple vibro-coutic ytem h been invetigted in vriou coupled ce. The imple vibro-coutic ytem conit of finite one-dimenionl coutic tube excited by ingle-degree-of-freedom (SDOF) tructure t one end nd terminted by n rbitrry impednce t the other end. In order to invetigte the mutul tructurl-coutic interction in the vibro-coutic ytem, coupling fctor h been derived for thi ytem uing the mobility-impednce pproch. The dynmic behviour of the vibro-coutic ytem h been dicued by invetigting the coutic potentil energy nd the tructurl kinetic energy. The vibro-coutic repone cn be chrcteried by the degree of the tructurl-coutic coupling in the imple vibro-coutic ytem. In the trongly coupled ce, the coutic potentil energy h mjor contribution from the coutic mode nd the tructurl kinetic energy i ubject to the coutic loding in the cvity. On the other hnd, in the wekly coupled ce, the coutic potentil energy i dominted by the tructurl mode nd the tructurl kinetic energy i generlly determined by only the tructurl chrcteritic though it i in-vcuo. In thi ce, the tructurl nturl frequency below the fundmentl coutic mode mke the vibro-coutic ytem more wekly coupled. Alo, in the intermedite ce, the vibro-coutic repone re frequency dependent nd how compound behviour of the previou two extreme ce depending on the nturl frequency rtio. The degree of tructurl-coutic coupling in the imple vibro-coutic ytem will be exploited for the nlyi of vibro-coutic repone in pive nd ctive control cheme in the following chpter. 49

71 Chpter 3. Pive Control CHAPTER 3 PASSIVE CONTRO OF ACOUSTIC POTENTIA ENERGY IN A VIBRO-ACOUSTIC SYSTEM 3.1 Introduction Thi chpter conider the effect of pive control tretment on the reduction of the coutic potentil energy in combined SDOF (ingle-degree-of-freedom) tructure - 1D (one-dimenionl) finite cloed tube ytem. The three coupled ce, dicued in chpter, re tudied when the tructurl nturl frequency i below nd bove the fundmentl coutic mode of cloed-cloed tube repectively. The pive control tretment involve tructurl nd couticl modifiction. The tructurl modifiction re implemented by chnging tructurl tiffne, tructurl m nd tructurl dmping. Alo, the couticl modifiction re implemented by chnging coutic dmping nd plcing borptive medium in the cvity. When pplying different medi in vibro-coutic ytem, the trnfer mtrix reltion cn be ued to relte the ound preure nd the prticle velocity t one end of n coutic element to thoe t the other end [Munjl (1987) nd Song et l (1999)]. In ection 3. i n invetigtion into the pive control of the coutic potentil energy involving tructurl nd couticl modifiction bed on tructurl-coutic nondimenionl prmeter: tructurl-coutic tiffne rtio (tiffne), tructurl-coutic nturl frequency rtio (m), tructurl lo fctor (tructurl dmping) nd coutic lo 51

72 Chpter 3. Pive Control fctor (coutic dmping). In ection 3.3, the effect of n borptive medium on the reduction of the coutic potentil energy i invetigted by plcing it t the rigid end urfce of the cvity. In ection 3.4, n experimentl invetigtion on the coutic potentil energy i crried out bed on the non-dimenionl tructurl-coutic propertie in the more trongly coupled ce nd in the more wekly coupled ce. Thi chpter i cloed in ection 3.5 with ome generl concluion bout the effect of the pive control tretment on the reduction of the coutic potentil energy in vriou coupled ce. 3. Pive control of coutic potentil energy Pive control tretment cn be implemented by modifying tiffne, m nd dmping. In thi ection, pive control of the coutic potentil energy, in combined SDOF tructure finite cloed tube ytem depicted in figure 3.1, i invetigted by wy of prmetric tudy: the effect of chnging the tructurl-coutic tiffne rtio (tiffne), tructurlcoutic nturl frequency rtio ω / ω (m), tructurl lo fctor η (tructurl dmping) nd coutic lo fctor η (coutic dmping). The cloed tube ytem i depicted in figure K / K.1 but with n infinitely lrge impednce Z. The effect of the tructurl-coutic prmeter on the coutic potentil energy re invetigted for two nturl frequency rtio ce, nmely tht the tructurl nturl frequency i below or bove the fundmentl coutic mode of cloed-cloed tube. The pive control performnce i evluted by exmining the ummed coutic potentil energy over the frequency rnge ( 0 / λ ) for the brodbnd control. j t f () t = Fe ω 0 K (1 + jη ) M x = 0 x = Figure 3.1 Combined SDOF tructure finite cloed tube ytem under the externl time j t hrmonic force on the tructure, f () t = Fe ω 0 t x = 0. The tube h n infinitely lrge impednce Z in the nlyticl model depicted in figure.1. M nd K re tructurl m nd tiffne of pring with contnt tructurl lo fctor η repectively. 5

73 Chpter 3. Pive Control 3..1 Effect of chnging tructurl-coutic tiffne rtio K / K - modifying tructurl tiffne The modifiction of the tructurl-coutic tiffne rtio i crried out by increing the tructurl tiffne in the combined SDOF tructure finite cloed tube ytem depicted in figure 3.1. The coutic potentil energy in the cvity, defined in eqution (.46), i clculted for given tiffne rtio nd in the ce when the tructurl tiffne i increed by fctor of 5. Figure 3.(), (b), (c) nd figure 3.3(), (b), (c) how the coutic potentil energy, in the imple vibro-coutic ytem, for given tiffne rtio when the tructurl nturl frequency i t / λ = 0.1 nd t / λ = 0. 8 repectively. The coutic potentil energy i normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio before increing the tructurl tiffne. K / K In the trongly coupled ce with tructurl-coutic tiffne rtio K / K = 10, the effect of chnging the tructurl tiffne by fctor of 5 on the coutic potentil energy i negligible for the two nturl frequency rtio ω / ω hown in figure 3.() nd 3.3(). In thi ce, the coutic potentil energie for the two vlue of the tructurl tiffne overlp due to inignificnt tructurl impednce. 3 In the wekly coupled ce with tructurl-coutic tiffne rtio K / K = 10 3 nd the tructurl nturl frequency t / λ = 0. 1, the tructurl mode t / λ = 0. 1 i effectively hifted to / λ = 0. by the tiffne chnge nd the mplitude i reduced hown in figure 3.(b). Alo, the mplitude of the fundmentl coutic mode t / λ = 0. 5 i increed due to the proximity of the tructurl mode. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decee by bout 7dB due to the hifted tructurl mode with mller mplitude. In the wekly coupled ce, when the tructurl nturl frequency i t / λ = 0. 8, the tructurl mode t / λ = 0. 8 i hifted to / λ = 1. 8 with mller mplitude by the tiffne chnge hown in figure 3.3(b). The fundmentl nd econd coutic mode re decreed due to the increed tructurl tiffne effect. However, the pek t the third nd fourth coutic mode re increed due to the proximity of the tructurl mode. In thi ce, the 53

74 Chpter 3. Pive Control ummed coutic potentil energy over the frequency rnge decree by bout 4dB. The reltively mll reduction in the ummed coutic potentil energy i due to the more dominting tructurl mode in the ce when the tructurl nturl frequency i t / λ = 0. 1 decribed in the ODS (opertionl deflection hpe) of the coutic preure hown in figure.6(b) nd.8(b). In the intermedite ce with tructurl-coutic tiffne rtio nturl frequency t K / K = 1 nd the tructurl / λ = 0. 1, the tructurl mode t / λ = 0.13 i hifted to / λ = 0. with mller mplitude by the tiffne chnge hown in figure 3.(c). The hifted tructurl mode incree the fundmentl coutic mode t / λ = 0. 5 for the me reon in the wekly coupled ce. In thi ce, the ummed coutic potentil energy over the frequency rnge decree by bout 5dB. In the intermedite ce, when the tructurl nturl frequency i t / λ = 0. 8, the coutic mode in the coutic potentil energy re hifted to higher frequencie due to the increed tructurl tiffne hown in figure 3.3(c). In thi ce, the ummed coutic potentil energy over the frequency rnge decree by bout 3dB. Compred to the ce of the tructurl nturl frequency t / λ = 0. 1, the reltively mll reduction on the ummed coutic potentil energy i due to the more trongly coupled tructure, which cue only coutic mode to feture ignificntly in the coutic potentil energy. To ummrie the vriou ce conidered, the chnge of the tructurl-coutic tiffne K / K i more effective on reducing the coutic potentil energy in the more wekly coupled ce nd when the tructurl nturl frequency i below the fundmentl coutic mode. 54

75 Chpter 3. Pive Control / λ 3 () Strongly coupled ce ( K / K = 10 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3. Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl tiffne i increed by fctor of 5 where the tructurl nd coutic lo fctor η = η = 10 (olid line: before increing the tructurl tiffne where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5), nd dhed line: fter increing the tructurl tiffne where the tructurl nturl frequency i t / λ = 0.( ω / ω = 1.1)) 55

76 Chpter 3. Pive Control / λ 3 () Strongly coupled ce ( K / K = 10 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.3 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl tiffne i increed by fctor of 5 where the tructurl nd coutic lo fctor η = η = 10 (olid line: before increing the tructurl tiffne where the tructurl nturl frequency i t / λ = 0.8 ( ω / ω = 0.6 ), nd dhed line: fter increing the tructurl tiffne where the tructurl nturl frequency i t / λ = 1.8 ( ω / ω = 0.3 )) 56

77 Chpter 3. Pive Control 3.. Effect of chnging tructurl-coutic nturl frequency rtio ω / ω - modifying tructurl m The modifiction of the tructurl-coutic nturl frequency rtio ω / ω i crried out by increing the tructurl m in the combined SDOF tructure finite cloed tube ytem depicted in figure 3.1. The coutic potentil energy in the cvity i clculted for given tiffne rtio nd in the ce when the tructurl m i increed by fctor of 5. Figure 3.4(), (b), (c) nd figure 3.5(), (b), (c) how the coutic potentil energy for given tiffne rtio when the tructurl nturl frequency i t / λ = 0. 1 nd t / λ = 0. 8 repectively. The coutic potentil energy i normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio before increing the tructurl m. In the trongly coupled ce with tructurl-coutic tiffne rtio K / K = 10, the effect of chnging the tructurl m by fctor of 5 on the coutic potentil energy i generlly negligible for the two nturl frequency rtio ω / ω hown in figure 3.4() nd 3.5(). When the tructurl nturl frequency i t / λ = 0. 1, the coutic mode in the coutic potentil energy hve minor hift to lower frequencie due to the increed tructurl m. The increed tructurl m provide n increed tructurl impednce t the input poition ( x = 0 ). Hence, the coutic mode of n open-cloed tube t / λ = (n 1)/ 4 hift towrd thoe of cloed-cloed tube t / λ = n/ where n i n integer. On the other hnd, when the tructurl nturl frequency i t / λ = 0. 8, the coutic potentil energie for the two vlue of the tructurl m overlp due to more trongly coupled tructure. Under the more trongly coupled condition, the m chnge of the tructurl impednce h le effect on the coutic potentil energy. 3 In the wekly coupled ce with tructurl-coutic tiffne rtio K / K = 10 3 nd the tructurl nturl frequency t / λ = 0. 1, the tructurl mode t / λ = 0. 1 i hifted to / λ = 0.05 with mller mplitude by the m chnge hown in figure 3.4(b). Alo, the mplitude of the coutic mode t / λ = n/ re reduced due to the impednce of the tructurl m increing ω where n i n integer. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decee by bout 7dB due to the reduced tructurl nd coutic mode. 57

78 Chpter 3. Pive Control In the wekly coupled ce, when the tructurl nturl frequency i t / λ = 0. 8, the tructurl mode t / λ = 0. 8 i effectively hifted to / λ = 0.36 with mller mplitude by the m chnge hown in figure 3.5(b). The fundmentl coutic mode t / λ = 0. 5 h more or le the me mplitude due to the proximity of the tructurl mode. Alo, the mplitude of the ret of the coutic mode re decreed for the me reon in the ce when the tructurl nturl frequency i t / λ = In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decree by bout 9dB. The reltively lrge reduction in the ummed coutic potentil energy i due to the more effective hift of the tructurl mode. In the intermedite ce with tructurl-coutic tiffne rtio nturl frequency t K / K = 1 nd the tructurl / λ = 0. 1, the tructurl mode t / λ = 0.13 i hifted to / λ = 0.06 with mller mplitude by the m chnge hown in figure 3.4(c). The coutic mode re hifted to thoe of cloed-cloed tube t / λ = n/ due to the increed tructurl impednce where n i n integer. Alo, the mplitude of the coutic mode re reduced for the me reon in the wekly coupled ce. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decree by bout 6dB. In the intermedite ce, when the tructurl nturl frequency i t / λ = 0. 8, the coutic mode in the coutic potentil energy re hifted to lower frequencie due to the increed tructurl m hown in figure 3.5(c). The increed tructurl m cue the coutic mode to hift towrd thoe of cloed-cloed tube t / λ = n/ where n i n integer. Alo, the further hift of higher coutic mode i due to the impednce of the tructurl m increing ω. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decree by bout 1dB. Compred to the ce of the tructurl nturl frequency t / λ = 0. 1, the reltively mll reduction in the ummed coutic potentil energy i due to the more trongly coupled tructure, which cue only coutic mode to feture in the coutic potentil energy. To ummrie the vriou ce conidered, the chnge of the tructurl-coutic nturl frequency rtio ω / ω i more effective on reducing the coutic potentil energy in the more wekly coupled ce nd when the tructurl nturl frequency i bove the fundmentl coutic mode. 58

79 Chpter 3. Pive Control / λ () Strongly coupled ce ( K / K 1 3 = 0 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.4 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl m i increed by fctor of 5 where the tructurl nd coutic lo fctor η = η = 10 (olid line: before increing the tructurl m where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5), nd dhed line: fter increing the tructurl m where the tructurl nturl frequency i t / λ = 0.05( ω / ω = 11.)) 59

80 Chpter 3. Pive Control / λ 3 () Strongly coupled ce ( K / K = 10 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.5 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl m i increed by fctor of 5 where the tructurl nd coutic lo fctor η = η = 10 (olid line: before increing the tructurl m where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6), nd dhed line: fter increing the tructurl m where the tructurl nturl frequency i t / λ = 0.36( ω / ω = 1.4 )) 60

81 Chpter 3. Pive Control 3..3 Effect of chnging tructurl lo fctor η - modifying tructurl dmping The pive tretment of tructurl dmping i implemented by increing the tructurl lo fctor η in the combined SDOF tructure finite cloed tube ytem depicted in figure 3.1. The coutic potentil energy in the cvity i clculted for given tiffne rtio nd in the ce when the tructurl lo fctor i increed by fctor of 5. Figure 3.6(), (b), (c) nd figure 3.7(), (b), (c) how the coutic potentil energy for given tiffne rtio when the tructurl nturl frequency i t / λ = 0. 1 nd t / λ = 0. 8 repectively. The coutic potentil energy i normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio before increing the tructurl lo fctor. In the trongly coupled ce with tructurl-coutic tiffne rtio K / K = 10, the effect of increing the tructurl lo fctor by fctor of 5 on the coutic potentil energy i negligible for the two nturl frequency rtio ω / ω hown in figure 3.6() nd 3.7(). In thi ce, the coutic potentil energie for the two vlue of the tructurl lo fctor overlp ince the effect of the tructurl impednce i negligible. 3 In the wekly coupled ce with tructurl-coutic tiffne rtio / K = 10 nd the tructurl nturl frequency t / λ = 0. 1, the tructurl lo fctor chnge i effective in reducing the mplitude of the tructurl mode t / λ = 0. 1 hown in figure 3.6(b). However, the coutic mode t / λ = n/ re not ffected by the tructurl lo fctor chnge where n i n integer. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decee by bout 9dB. The vibro-coutic repone, t low frequencie below the firt coutic reonnce, cn be explined by wy of the low-frequency pproximted model decribed in Appendix. A. The low-frequency coutic potentil energy in the cvity cn be decribed by the tored trin energy in the coutic pring depicted in figure A.1, which i the low-frequency pproximtion of the one-dimenionl coutic tube. In the ce when the tructurl-coutic tiffne rtio K 3 i much mller thn 1, the mplitude of the trin energy i minly ubject to the tructurl lo fctor t the reonnce frequency. K / K 61

82 Chpter 3. Pive Control In the wekly coupled ce, when the tructurl nturl frequency i t / λ = 0. 8, the tructurl mode t / λ = 0. 8 i effectively reduced hown in figure 3.7(b). However, the tructurl lo fctor chnge h inignificnt effect on the coutic mode t / λ = n/ where n i n integer. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decee by bout 4dB. Compred to the ce of the tructurl nturl frequency t / λ = 0. 1, the tructurl mode i more effectively reduced due to decreed criticl dmping of the tructure. The criticl dmping i proportionl to qure root of the product of the tructurl tiffne nd the tructurl m. In thi ce, the tructure h le tructurl m with given tructurl-coutic tiffne rtio for the higher tructurl nturl frequency. Alo, the reltively mll reduction in the ummed coutic potentil energy i due to the le dominnt tructurl mode over the coutic potentil energy decribed in the ODS of the coutic preure hown in figure.6(b) nd.8(b). In the intermedite ce with tructurl-coutic tiffne rtio K / K = 1 nd the tructurl nturl frequency t / λ = 0. 1, the tructurl lo fctor chnge i effective on reducing the mplitude of the tructurl mode t / λ = 0.13 hown in figure 3.6(c). However, the coutic mode t / λ = n/ re not ffected by the tructurl lo fctor chnge for the me reon in the wekly coupled ce where n i n integer. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decree by bout 7dB. In the intermedite ce, when the tructurl nturl frequency i t / λ = 0. 8, the mplitude of the coutic mode in the coutic potentil energy re reduced t reonnce frequencie due to the tructurl lo fctor chnge hown in figure 3.7(c). The mller reduction of the mplitude t higher reonnce frequencie i due to the tructurl dmping decreing ω. Compred to the ce of the tructurl nturl frequency t / λ = 0. 1, the coutic preure in the cvity i more effectively ubject to the tructurl dmping due to the more trongly coupled tructure. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decree by bout 4dB. The reltively mll reduction in the ummed coutic potentil energy i due to the more trongly coupled tructure, which cue only coutic mode in the coutic potentil energy. To ummrie the vriou ce conidered, the chnge of the tructurl lo fctor η i generlly effective t the tructurl mode, but i ineffective t the coutic mode on reduction 6

83 Chpter 3. Pive Control of the coutic potentil energy in the cvity. The ummed coutic potentil energy, over the frequency rnge of interet, decree more effectively in the more wekly coupled ce nd in the ce when the tructurl nturl frequency i below the fundmentl coutic mode. 63

84 Chpter 3. Pive Control / λ () Strongly coupled ce ( K / K 1 3 = 0 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.6 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl lo fctor i increed by fctor of 5 where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5 ) (olid line: before increing the tructurl lo fctor where the tructurl nd coutic lo fctor η = η = 10, nd dhed line: fter increing the tructurl lo fctor where the tructurl lo fctor η = 5 10 nd the coutic lo fctor η = 10 ) 64

85 Chpter 3. Pive Control / λ 3 () Strongly coupled ce ( K / K = 10 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.7 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the tructurl lo fctor i increed by fctor of 5 where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6 ) (olid line: before increing the tructurl lo fctor where the tructurl nd coutic lo fctor η = η = 10, nd dhed line: fter increing the tructurl lo fctor where the tructurl lo fctor η = 5 10 nd the coutic lo fctor η = 10 ) 65

86 Chpter 3. Pive Control 3..4 Effect of chnging n coutic lo fctor η - modifying coutic dmping The pive tretment of coutic dmping i implemented by increing the lo fctor η of the coutic medium in the combined SDOF tructure finite cloed tube ytem depicted in figure 3.1. The coutic potentil energy in the cvity i clculted for given tiffne rtio nd in the ce when the coutic lo fctor i increed by fctor of 5. Figure 3.8(), (b), (c) nd figure 3.9(), (b), (c) how the coutic potentil energy for given tiffne rtio when the tructurl nturl frequency i t / λ = 0. 1 nd t / λ = 0. 8 repectively. The coutic potentil energy i normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio before increing the coutic lo fctor. In the trongly coupled ce with tructurl-coutic tiffne rtio K / K = 10, the effect of chnging the coutic lo fctor on the coutic potentil energy i ignificnt for the two nturl frequency rtio ω / ω hown in figure 3.8() nd 3.9(). All the coutic mode in the coutic potentil energy re effectively reduced t the reonnce frequencie. The ummed coutic potentil energy over the frequency rnge of interet decree by bout 1dB for both the nturl frequency rtio. 3 In the wekly coupled ce with tructurl-coutic tiffne rtio K / K = 10 3, the coutic lo fctor chnge i effective only t the coutic mode t / λ = n/, where n i n integer, on the reduction of the coutic potentil energy for the two nturl frequency rtio ω / ω hown in figure 3.8(b) nd 3.9(b). In the ce when the tructurl nturl frequency i t / λ = 0. 1, the inignificnt effect t the tructurl mode i due to the mplitude of the trin energy minly ubject to the tructurl lo fctor t the reonnce frequency decribed in Appendix. A. The ummed coutic potentil energy, over the frequency rnge of interet, decree by bout 0.dB due to the inignificnt effect t the dominting tructurl mode. Alo, in the ce when the tructurl nturl frequency i t / λ = 0. 8, the ummed coutic potentil energy decree by bout 4dB. The reltively lrge reduction in the coutic potentil energy i due to the more trongly coupled tructure, which cue more dominnt coutic mode over the coutic potentil energy. 66

87 Chpter 3. Pive Control In the intermedite ce with tructurl-coutic tiffne rtio nturl frequency t K / K = 1 nd the tructurl / λ = 0. 1, the coutic potentil energy i reduced effectively t the coutic mode, i.e. t / λ = n/ where n i n integer hown in figure 3.8(c). Alo, the mplitude of the tructurl mode t / λ = 0.13 cn be reduced to ome degree by the coutic lo fctor chnge. A dicued in the low-frequency pproximted model decribed in Appendix. A, the mplitude of the trin energy i ffected by both the tructurl nd coutic lo fctor t the reonnce frequency when the tructurl-coutic tiffne rtio K / K i equl to 1. In thi ce, the ummed coutic potentil energy over the frequency rnge of interet decree by bout 8dB. Alo, in the ce when the tructurl nturl frequency i t / λ = 0.8, the mplitude of the ll the coutic mode re effectively reduced hown in figure 3.9(c). In thi ce, the ummed coutic potentil energy decree by bout 11dB. Compred to the ce of the tructurl nturl frequency t / λ = 0. 1, the ummed coutic potentil energy i more effectively reduced for the me reon in the wekly coupled ce. To ummrie the vriou ce conidered, the chnge of the coutic lo fctor η i generlly effective t the coutic mode, but i ineffective t the tructurl mode on reduction of the coutic potentil energy. The ummed coutic potentil energy, over the frequency rnge of interet, decree more effectively in the more trongly coupled ce nd in the ce when the tructurl nturl frequency i bove the fundmentl coutic mode.. 67

88 Chpter 3. Pive Control / λ () Strongly coupled ce ( K / K 1 3 = 0 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.8 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the coutic lo fctor i increed by fctor of 5 where the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5 ) (olid line: before increing the coutic lo fctor where the tructurl nd coutic lo fctor η = η = 10, nd dhed line: fter increing the coutic lo fctor where the coutic lo fctor η = 5 10 nd the tructurl lo fctor η = 10 ) 68

89 Chpter 3. Pive Control / λ 3 () Strongly coupled ce ( K / K = 10 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.9 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio in the ce when the coutic lo fctor i increed by fctor of 5 where the tructurl nturl frequency i t / λ = 0.8( ω / ω = 0.6) (olid line: before increing the coutic lo fctor where the tructurl nd coutic lo fctor η = η = 10, nd dhed line: fter increing the coutic lo fctor where the coutic lo fctor η = 5 10 nd the tructurl lo fctor η = 10 ) 69

90 Chpter 3. Pive Control 3..5 Summry of pive tretment on coutic potentil energy In thi ection, the pive control of the coutic potentil energy, in the imple vibrocoutic ytem depicted in figure 3.1, i ummried ccording to the pive tretment conidered. For two vlue of the tructurl nturl frequency, the comprion of the ummed coutic potentil energy over the frequency rnge ( 0 / λ ), normlied by tht before increing the correponding prmeter by fctor of 5, i hown in tble 3.1. The tructurl nturl frequency i t / λ = 0.1 (ω / ω = 5), which i below the firt coutic reonnce, or i t / λ = 0. 8 ( ω / ω = 0. ), which i bove the firt coutic reonnce for given tiffne rtio. 6 In the trongly coupled ce, the couticl modifiction, involving the chnge of the coutic lo fctor η, i preferble for the reduction of the coutic potentil energy in both the ce of two vlue of the tructurl nturl frequency. In the wekly coupled ce, the tructurl modifiction, involving the chnge of the tiffne rtio K / K, the nturl frequency rtio ω / ω or the tructurl lo fctor η, i preferble for the reduction of the coutic potentil energy. In the ce when the tructurl nturl frequency i t / λ = 0. 1, the tructurl lo fctor η i more effective in reducing the coutic potentil energy. On the other hnd, in the ce when the tructurl nturl frequency i t / λ = 0. 8, the chnge of the nturl frequency rtio ω / ω i more effective in reducing the coutic potentil energy. In the intermedite ce, the couticl modifiction, involving the chnge of the coutic lo fctor η, i preferble for the reduction of the coutic potentil energy. Alo, in the ce when the tructurl nturl frequency i t / λ = 0. 8, the coutic lo fctor chnge h more ignificnt effect on the reduction. 70

91 Chpter 3. Pive Control Tble 3.1 Comprion of normlied ummed coutic potentil energy over the frequency rnge ( 0 / λ ) ccording to the pive tretment in the ce when the tructurl nturl frequency i t / λ = 0.1( ω / ω = 5) nd t / λ = 0.8( ω / ω = 0.6) degree Nturl frequency rtio ω / ω = 5 / ω / ω = 0.6 tiffne rtio nturl frequency rtio tructurl lo fctor coutic lo fctor of coupling K / K ω / ω η η Strong 0 / 0 0 / 0 0 / 0-1dB / -1dB Wek -7dB / -4dB -7dB / -9dB -9dB / -4dB -0.dB / -4dB Intermedite -5dB / -3dB -6dB / -1dB -7dB / -4dB -8dB / -11dB 3.3 Pive control of coutic potentil energy uing n borptive medium In thi ection, n borptive medium i ued one of pive control tretment for the reduction of the coutic potentil energy in the imple vibro-coutic ytem depicted in figure 3.1. The pive control effect i invetigted for given tiffne rtio when the tructurl nturl frequency i t / λ = 0. 1 nd t / λ = 0. 8 repectively. The borptive medium i plced t the rigid end urfce of the cloed tube hown in figure The cloed tube i compoed of the coutic medium in the region of 0 x 0, which h mbient denity ρ 0 nd complex ound peed c, nd the borptive medium in the region of x, which h mbient denity ρ b nd complex ound peed c b. 0 j t f () t = Fe ω 0 K (1 + jη ) M x = 0 x = 0 x = Figure 3.10 Combined SDOF tructure finite cloed tube ytem, which h the borptive medium t the rigid end urfce of the cloed tube in the region of 0 x 71

92 Chpter 3. Pive Control Sound propgtion in rigid-frme porou mteril i governed by the effective denity nd the effective bulk modulu of the fluid in the pore pce. Thee quntitie re frequencydependent, complex nd non-liner. Due to the complexity of thee quntitie, it i difficult to obtin phyicl inight into the coutic behviour. Brennn nd To (001) derived very imple expreion of the chrcteritic impednce nd wvenumber for ound propgtion in the rigid-frme porou mteril. The lo in the borptive medium cn be repreented by complex coutic wvenumber tht given in eqution (.11) k b ω ω 1 = (1 j ηb) (3.1) c c b 0 where k b i the complex wvenumber in the borptive medium. The complex ound peed in the borptive medium c b cn be defined for mll lo fctor by / 1 1 cb c0 j ηb (3.) where c0 i the ound peed in lole medium nd η b i the contnt lo fctor in the borptive medium. The contnt lo fctor η b in the borptive medium i frequencydependent in rigid-porou frme mteril but i conidered contnt vlue in the very imple model dopted here for convenience. The coutic potentil energy, in the imple vibro-coutic ytem with two different medi depicted in figure 3.10, cn be clculted by umming up the coutic potentil energy in ech medium, which i defined in eqution (.46), to give S S E Px dx Px dx 0 p ( ω) = (, ω) + (, ) 0 4 ρ 0 0 c 4 ρ b cb ω (3.3) When pplying different medi in vibro-coutic ytem, the trnfer mtrix reltionhip cn be ued to relte the ound preure nd the prticle velocity t one end of n coutic element to thoe t the other end. The trnfer mtrix reltionhip of the imple vibro-coutic ytem 7

93 Chpter 3. Pive Control with borptive medium hown in figure 3.10 cn be written uing eqution (B.18) in Appendix. B P1 co k0 jρ 0cin k0 co kbb jρbcbin kbb P3 U = 1 jin k0/ ρ 0c co k0 jin kbb / ρ bcb co kbb U3 (3.4) where nd U re the ound preure nd the prticle velocity t x = 0 repectively, nd P1 1 P3 nd U3 re the ound preure nd the prticle velocity t x = repectively. Alo, k ( = ω / c) nd k ( = ω / c ) re complex wve number in the coutic medium with length b b 0 nd in the borptive medium with length = ) repectively. b ( 0 The prticle velocity U i zero becue the cloed tube i rigidly terminted t x =. In thi 3 ce the uncoupled coutic impednce t the input poition ( x = 0 ) i given by P = (3.5) 1 ZA0 S U 1 U3 = 0 The uncoupled coutic impednce cn be rewritten by combining eqution (3.4) with eqution (3.5) to give ρ c co k co k ρ cin k in k ZA0 = jρ0cs ρ c in k co k + ρ cco k in k b b 0 b b 0 0 b b b b 0 b b 0 0 b b (3.6) If nd re et to nd zero repectively, then the uncoupled coutic impednce 0 b become tht of the cloed tube without the borptive medium given in eqution (.3). The normlied uncoupled coutic impednce i given in term of non-dimenionl prmeter repectively by Zˆ A0 ρb( jη)co k0co kbb ρ0( jηb)in k0in kb = j b jη ρ ( jη )in k co k + ρ ( jη )co k in k b 0 b b 0 b 0 b b (3.7) 73

94 Chpter 3. Pive Control where k0 = π( jη ) ˆˆ 0, k ˆˆ b b = π( jηb) b, ˆ 0 = 0/ nd ˆ b b /. Alo, η nd = η b re the contnt coutic lo fctor in the coutic medium ( 0 x 0 borptive medium ( x ) repectively. 0 ) nd in the The trnfer mtrix reltionhip for the ound preure P nd the prticle velocity U t n rbitrry poition in the coutic medium ( 0 x 0 ) in figure 3.10 i given uing eqution (B.0) in Appendix. B to give P co kx jρ cin kx P 0 1 U = jin kx/ ρ 0c co kx U 1 (3.8) The ound preure t n rbitrry poition in the region of 0 x 0 cn be written by expnding the trnfer mtrix in eqution (3.8) to give ( ) 1 ρ 0 Px (, ω) = co kxp ( j cin kxu ) (3.9) 1 The prticle velocity t x = 0, U1 cn be written in term of the uncoupled tructurl nd coutic impednce which i given in eqution (.8) nd repeted here for convenience U 1 = Z F0 + Z S A0 (3.10) Similrly, the ound preure t x = 0, P 1 cn be written uing eqution (3.5) nd (3.10) P = 1 Z A0 F 0 Z + Z S S A0 (3.11) Subtituting the prticle velocity nd ound preure P given in eqution (3.10)~(3.11) U1 1 into eqution (3.9) give the ound preure t n rbitrry poition in the coutic medium ( 0 x 0 ), which i 74

95 Chpter 3. Pive Control F 1 P x = Z kx j cs kx x (3.1) ( ) 0 (, ω) A0co ρ 0 in (0 0) S ZS + ZA0 The ound preure given in eqution (3.1) cn be written in non-dimenionl form ˆ ˆ 1 ˆ Px ( ˆ, ) = ZA0 cokx j inkx ˆ Z jη CS (3.13) where Px ˆ( ˆ, ˆ) = Px ( ˆ, ˆ)/ P, kx = π( jη ) ˆ ˆ x, ˆ = / λ nd xˆ( = x/ ) i ny normlied poition long the tube. The ttic preure P i defined the rtio of the ttic force F 0 to the cro-ectionl re S of the cloed tube. The trnfer mtrix reltionhip for the ound preure nd the prticle velocity t n rbitrry poition in the borptive medium ( 0 x ) in figure 3.10 i given uing eqution (B.) in Appendix. B to give P co k x jρ c in k x co k jρ cin k P b b b b U = jin kbx/ ρ bcb co kbx jin k0/ ρ 0c co k 0 U 1 (3.14) The ound preure t n rbitrry poition in the region of 0 x cn be written by expnding the trnfer mtrix in eqution (3.14) to give ( ) Px (, ω) cokxco k ( ρ c/ ρ c)inkxink P = b 0 b b 0 b 0 1 ( ρ0 b 0 ρb b b j cco k xin k + c in k xco k U (3.15) ) 0 1 Subtituting the prticle velocity nd the ound preure t x = 0 given in eqution (3.10)~ (3.11) into eqution (3.15) give the ound preure t n rbitrry poition in the borptive medium ( x ), which i 0 F 1 Px = Z kx k c c kx k { ρ ρ } 0 (, ω) A0 co b co 0 ( b b / 0 )in b in S ZS + Z A0 0 75

96 Chpter 3. Pive Control { ρ ρ } j cs co k xin k + c S in k x co k ( x ) (3.16) 0 b 0 b b b 0 0 The ound preure given in eqution (3.16) cn be rewritten in non-dimenionl form 1 ˆ ˆ ˆ ρb jη Px ( ˆ, ) = ZA0 cokx b cok0 inkx b ink Zˆ 0 ρ0 jη CS b ρ b j co kbxin k0 + in kbxco k 0 (3.17) jη ρ0 jηb where, kx= π( jη ) x ˆ ˆ. b b The coutic potentil energy given in eqution (3.3) cn be written in non-dimenionl form Eˆ P ( ˆ) = ρ 4 + η Px dx Px dxˆ ˆ 0 1 ˆ ˆ 0 b ( ˆ, ) ˆ+ ˆ( ˆ, ˆ) 0 ˆ ρ 4 0 b + η 1 0 Px ˆ( ˆ,0) dxˆ (3.18) where Eˆ ( ˆ) = E ( ˆ)/ E (0). (0) i the coutic potentil energy in the bence of control P P P E P defined in eqution (.5) nd the coutic potentil energy E ( ˆ P ) i defined by ˆ S P ˆ ˆ S P E ( ) (, ) ˆ P = P x dx+ P( x, ) dxˆ 4 ρ c 4 0 ˆ 0 1 ˆ ˆ ˆ ˆ 0 ˆ ρ 0 b c b (3.19) Figure 3.11(), (b), (c) nd figure 3.1(), (b), (c) how the coutic potentil energy for given tiffne rtio when the tructurl nturl frequency i t / λ = 0. 1 nd t / λ = 0. 8 repectively. The coutic potentil energy i normlied by tht t the ttic tte ( / λ = 0) for given tiffne rtio without the borptive medium. 76

97 Chpter 3. Pive Control In the trongly coupled ce with tructurl-coutic tiffne rtio K / K = 10, the effect of the borptive medium on the coutic potentil energy i ignificnt for the two nturl frequency rtio ω / ω hown in figure 3.11() nd 3.1(). All the coutic mode re effectively reduced t the reonnce frequencie. The ummed coutic potentil energy over the frequency rnge of interet decree by bout 9dB in the ce of both the nturl frequency rtio. 3 In the wekly coupled ce with tructurl-coutic tiffne rtio borptive medium i effective only t the coutic mode t K / K = 10 3, the / λ = n/, where n i n integer, for both the nturl frequency rtio ω / ω hown in figure 3.11(b) nd 3.1(b). In the ce when the tructurl nturl frequency i t / λ = 0. 1, the tructurl mode t / λ = 0. 1 i not ffected by the borptive medium in the cvity. The ummed coutic potentil energy, over the frequency rnge of interet, decree by bout 0.1dB due to the inignificnt effect t the dominting tructurl mode. Alo, in the ce when the tructurl nturl frequency i t / λ = 0. 8, the ummed coutic potentil energy decree by bout 3dB. The reltively lrge reduction on the ummed coutic potentil energy i due to the more dominting tructurl mode in the ce when the tructurl nturl frequency i t / λ = 0. 1 dicued in ection 3.. In the intermedite ce with tructurl-coutic tiffne rtio K / K = 1 nd the tructurl nturl frequency t / λ = 0. 1, the coutic potentil energy i reduced effectively t the coutic mode t / λ = n/ where n i n integer hown in figure 3.11(c). Alo, the mplitude of the tructurl mode t / λ = 0.13 cn be reduced to ome degree by the borptive medium in the cvity. In thi ce, the ummed coutic potentil energy, over the frequency rnge of interet, decree by bout 5dB. Alo, in the ce when the tructurl nturl frequency i t / λ = 0. 8, the mplitude of the ll the coutic mode re effectively reduced by the borptive medium hown in figure 3.1(c). In thi ce, the ummed coutic potentil energy decree by bout 8dB. The reltively lrge reduction on the ummed coutic potentil energy i due to the more trongly coupled tructure dicued in ection

98 Chpter 3. Pive Control To ummrie the vriou ce conidered, the effect of the borptive medium, plced t the rigid end urfce of the cloed tube, on reduction of the coutic potentil energy i imilr to tht of chnging the coutic lo fctor dicued in ection The borptive medium in the cvity i generlly effective t the frequencie where the coutic mode dominte the coutic repone, but i ineffective t the tructurl mode for the reduction of the coutic potentil energy. The ummed coutic potentil energy, over the frequency rnge of interet, decree more in the more trongly coupled ce nd in the ce when the tructurl nturl frequency i bove the fundmentl coutic mode. 78

99 Chpter 3. Pive Control / λ 3 () Strongly coupled ce ( K / K = 10 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.11 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) without the borptive medium for given tiffne rtio in the ce when the borptive medium i pplied in the region x where the tructurl nturl frequency i t / λ = ( ω / ω = ) nd the tructurl nd coutic lo fctor η = η = 10 (olid line: without the 5 borptive medium, nd dhed line: with the borptive medium / 0 = 0.7, / 0.3 b = nd the lo fctor η b = 0. ) 79

100 Chpter 3. Pive Control / λ 3 () Strongly coupled ce ( K / K = 10 ) / λ (b) Wekly coupled ce ( K / K 10 3 = ) / λ (c) Intermedite ce ( K / K = 1) Figure 3.1 Acoutic potentil energy normlied by tht t the ttic tte ( / λ = 0) without the borptive medium for given tiffne rtio in the ce when the borptive medium i pplied in the region x where the tructurl nturl frequency i t / λ = ( ω / ω = 0 ) nd the tructurl nd coutic lo fctor η = η = 10 (olid line: without.6 the borptive medium, nd dhed line: with the borptive medium / =, b / = 0.3 nd the lo fctor η b = 0. ) 80

101 Chpter 3. Pive Control 3.4 Experimentl invetigtion on vibro-coutic ytem Some experimentl work w crried out in order to vlidte the nlyticl model hown in figure 3.1 nd to upport the imultion reult. The experimentl rig w configured with loudpeker, finite wter pipe, microphone nd ccelerometer for the SDOF tructure, finite coutic tube, coutic nd tructurl enor repectively. The loudpeker w modified for different tructurl chrcteritic under different tructurl-coutic coupling condition. The tructurl modifiction of the loudpeker w bed on non-dimenionl tructurl-coutic prmeter, defined in chpter, dominting over the degree of tructurlcoutic coupling. The finite coutic tube driven by the tndrd or modified peker w invetigted by meuring the frequency repone function of the ound preure in the vibro-coutic ytem with repect to the input voltge to the loudpeker. The coutic potentil energy in the vibro-coutic ytem w pproximtely clculted by umming up the qured mgnitude of the meured frequency repone function t every meurement point long the coutic tube Experimentl etup A tet-rig w deigned to behve in the imilr wy to the nlyticl model, depicted in figure 3.1, hown in figure The loudpeker hd diphrgm of rdiu 46 mm nd w excited by the dynmic ignl nlyer (Dt Phyic) through the A1 coutic mplifier (Cmbridge Audio A1 V.0). The other end ( x = ) of the tube w terminted by thick wooden plte (10 mm ) to give cloed tube. Alo, the coutic tube hd dimenion of length = 1m, dimeter φ = 0.1m nd wll thickne t = 4mm. The frequency repone function of the ound preure in the tube were meured with repect to the input voltge to the loudpeker by uing even omni-directionl ub-miniture microphone (type EM-60B). The microphone were plced equiditntly long the centre line of the tube with pce of d = 0.16m, nd were eled by ilicone. The meured ound preure t every meurement point w ped to the dynmic ignl nlyer through n ISVR 8-chnnel coutic mplifier. The frequency repone function of the ound preure t ech meurement point with repect to the input voltge to the 81

102 Chpter 3. Pive Control loudpeker w clculted. The firt microphone w plced t x = 0.0m on the ide of the loudpeker diphrgm nd the eventh one w t x = 0.98m. The ditnce between the microphone w determined o tht the ditnce between them w le thn qurter of wve length t the mximum frequency (500Hz). The coutic field inide the tube hd plne wve propgtion within the frequency rnge of interet ( 0 ~ 500 Hz): the firt reonnce frequency in the rdil direction w 1.7 khz. The frequency rnge w ufficient to invetigte one tructurl mode nd three coutic mode. 8-Chnnel Acoutic Amplifier Dynmic Signl Anlyer A1 Acoutic Amplifier Input Voltge oudpeker x = 0 t Microphone φ d x = 1m Wooden Plte () Schemtic digrm Mic.7 Mic.6 Wooden Plte Mic.5 Mic.4 Mic.3 Mic. Mic.1 oudpeker (b) Experimentl etup Figure 3.13 Experimentl etup of one-dimenionl coutic tube driven by loudpeker: () chemtic digrm (b) experimentl etup 8

103 Chpter 3. Pive Control 3.4. Structurl modifiction of loudpeker The min concern of thi experiment w to invetigte the coutic potentil energy in vibro-coutic ytem in different coupled ce: more trongly nd more wekly coupled. For the more wekly coupled ce, it w required to modify the tructurl chrcteritic of tndrd loudpeker. The tructurl modifiction of the tndrd loudpeker w informed by the non-dimenionl tructurl-coutic prmeter etblihed in chpter. The degree of tructurl-coutic coupling i determined by the coupling fctor given in eqution (.4). If ll the coutic chrcteritic nd the tructurl lo fctor in the coupling fctor t certin frequency re fixed, the coupling fctor i proportionl to the tructurl-coutic tiffne rtio ˆK nd the invere of the tructurl-coutic nturl frequency rtio ˆω. The coupling fctor cn be rewritten YZ S A0 1 (3.0) KM Hence more wekly coupled ce cn be chieved by increing the tructurl m nd tiffne. Figure 3.14 how the tndrd nd modified loudpeker for the more trongly nd more wekly coupled ce repectively. The tndrd loudpeker w compoed of pper cone with rubber upenion t it perimeter nd i eled by metl cover on the bck. The tndrd loudpeker w modified by dding m of 100g compoed of led hot nd increing tiffne with 3 luminium bem mounted to the loudpeker metl ce inted of the rubber upenion. When it come to the dded m on the loudpeker, the m need to be determined in prcticl wy becue the tndrd loudpeker h uch light moving m. Even though more m cn produce more wekly coupled condition, the dded m w limited o tht the ttic diplcement, due to weight, w mll. The perimeter of the modified loudpeker cone w eled by uing ilicone to void ir lekge between the luminium bem nd the wooden flnge of the coutic tube. 83

104 Chpter 3. Pive Control Aluminium bem Silicone ed hot () Stndrd loudpeker (b) Modified loudpeker Figure 3.14 oudpeker ued to excite the coutic tube: () tndrd loudpeker nd (b) modified loudpeker The mechnicl component of the tndrd nd modified loudpeker cn be clculted from the meured frequency repone function with nd without known dummy m. The nturl frequency of the tndrd or the modified loudpeker cn be written bed on the SDOF behviour chrcteritic of loudpeker f 1 K = 1 π M f 1 = π K M + m (3.1) where f 1 nd f re the nturl frequency of the tndrd or modified loudpeker without nd with known dummy m m repectively. The tructurl moving m tiffne M nd K of the loudpeker cn be clculted from eqution (3.1), where the moving m include the m of n ccelerometer (B&K 4375) of. 4g. The tructurl lo fctor η cn be clculted from qulity fctor nd i given [Meirovitch (1986)] by 84