STABILITY AND BIFURCATION ANALYSIS OF A PIPE CONVEYING PULSATING FLUID WITH COMBINATION PARAMETRIC AND INTERNAL RESONANCES

Transcription

1 Mthemticl nd Computtionl Applictions Vol. 0 No. pp STABILITY AND BIFURCATION ANALYSIS OF A PIPE CONVEYING PULSATING FLUID WITH COMBINATION PARAMETRIC AND INTERNAL RESONANCES Lingqing Zhou Fngqi Chen Yushu Chen Deprtment of Mthemtics Nnjing University of Aeronutics nd Astronutics Nnjing 006 PR Chin Deprtment of Mechnics Tinjin University Tinjin 0007 PR Chin zlqrex@sin.comfngqichen@nu.edu.cn Abstrct- The stbility nd bifurctions of hinged-hinged pipe conveying pulsting fluid with combintion prmetric nd internl resonnces re studied with both nlyticl nd numericl methods. The system hs geometric cubic nonlinerity. Three types of criticl points for the bifurction response equtions re considered. These points re chrcterized by double zero nd two negtive eigenvlues double zero nd pir of purely imginry eigenvlues nd two pirs of purely imginry eigenvlues respectively. With the id of norml form theory the expressions for the criticl bifurction lines leding to incipient nd secondry bifurctions re obtined. Possible bifurctions leding to -D tori re lso investigted. Numericl simultions confirm the nlyticl results. Keywords- Pipe with Pulsting Fluid Nonliner Vibrtion Perturbtion Methods Prmetric Resonnces Stbility Bifurction;. INTRODUCTION The liner nd nonliner dynmics of pipes conveying fluid hs been studied widely during the lst decdes. Detiled review nd extensive bibliogrphy on this flowinduced vibrtions nd instbilities of piping nd cylindricl structures were provided by Pdoussis et l [-]. The prmetric instbilities depending on the mplitude nd frequency of flow fluctution will occur when the flow velocity hs hrmoniclly fluctuting component over men vlue. A lot of investigtions bsed on linerized nlyticl models of these prmetric instbility problems for simply supported pipes were done by Chen [4] Pdoussis nd Issid[5] Pdoussis nd Sundrrjn [6] Ginsberg [7] nd Arirtnm nd Nmchchivy [8] yrmn nd Tien [9]. They studied the prmetric nd combintion resonnces nd evluted instbility with numericl methods. In [0] Pnd nd Kr studied the nonliner dynmics of hingedhinged pipe conveying pulsting fluid subjected to combintion nd principle prmetric resonnce in the presence of internl resonnce with the method of multiple scles nd numericl methods. Using the method of multiple scles Pnd nd Kr [] studied the nonliner plnr vibrtion of pipe conveying pulsting fluid subjected to principl prmetric resonnce in the presence of internl resonnce. By considering the effect of motion constrints modeled s cubic springs the nonliner dynmics of simply

2 Stbility nd Bifurction Anlysis of Pipe Conveying Pulsting Fluid 0 supported pipes conveying pulsting fluid ws further investigted nd some new dynmicl behviors including qusi-periodic nd chotic motions were obtined []. in nd Song [] investigted the stbility nd prmetric resonnces of supported pipes conveying pulsting with numericl methods. The post-divergence behvior of extensible fluid-conveying pipes supported t both ends ws studied by Modrres- Sdeghi nd Pdoussis [4] nd supercriticl pitchfork bifurction ws obtined. Using numericl methods Wng nd Ni [5] investigted the stbility nd chotic motions of stnding pipe conveying fluid. A spectrl element model ws developed for the uniform stright pipelines conveying internl unstedy fluid [6]. By using the Melnikov method the globl dynmics of prmetriclly excited conveying fluid ner 0: resonnce ws studied nd chotic dynmics my exist in the system [7]. The stbility nd dynmics of cntilevered pipe conveying fluid with motion-limiting constrints nd liner spring support were investigted [8]. Using the Euler-Bernoulli bem theory nd nonliner Lgrnge strin theory new nonliner model of stright pipe conveying fluid ws presented [9]. The vibrtion ws nlyzed with the Glerkin method. Using Hmilton's principle nd Glerkin method Sin [0] investigted the non-liner vibrtions of slightly curved pipes conveying fluid with constnt velocity. Periodic nd chotic motions were observed in the trnsverse vibrtions. In this pper the stbility nd bifurctions of hinged-hinged pipe conveying pulsting fluid with combintion prmetric nd internl resonnces re studied both nlyticlly nd numericlly. Three types of criticl points for the bifurction response equtions re discussed. These points re chrcterized by double zero nd two negtive eigenvlues double zero nd pir of purely imginry eigenvlues nd two pirs of purely imginry eigenvlues respectively. With the id of norml form theory the expressions for the criticl bifurction lines leding to incipient nd secondry bifurctions re obtined. Possible bifurction solutions nd their stbility re investigted. Numericl simultions re lso given which verify the nlyticl results.. FORMULATION OF THE PROBLEM Consider uniform horizontl pipe hinged t both ends conveying fluid with flow-velocity hving hrmoniclly pulsting component superimposed over stedy one (Fig ). Assume tht the motion is plnr nd the uniform cross-section remins plne during the motion nd the tube behves like n Euler-Bernoulli bem in trnsverse vibrtion. It is lso ssumed tht the fluid is incompressible nd hs plug flow conditions. The eqution of trnsverse motion of the pipe including the nonlinerity due to midline stretching is 4 5 y * y y y y EI E I MU ( M m) c 4 4 x x t xt t t * [ U ( ) EA L ( ') d E A L ( ' ')d ] y MU T M L x y x y y x t L L () x with the boundry conditions y y y(0 t) y( L t) (0 t) ( L t) 0 () x x where x is the longitudinl coordinte y is the trnsverse deflection T is the

3 0 L. Zhou F. Chen nd Y. Chen externlly imposed xil tension m nd M re the mss per unit length of pipe nd fluid mterils respectively A is the cross sectionl re of the pipe L is the length * EI is the flexurl stiffness of the pipe mteril E is the coefficient of internl dissiption of the pipe mteril which is ssumed to be viscoelstic nd of the Kelvin- Voigt type nd c is the externl dmping fctor U is the fluid velocity which hs the following form U U ( sin t ) () 0 whereu 0 is the men flow velocity nd re the mplitude nd frequency of the flow-velocity fluctution which my led to prmetric instbilities. Figure. Schemtics of the model. Introducing the following dimensionless quntities x L w y EI ( ) / M / M TL t t u ( ) UL L L M m n EI M m EI * * cl * E / EI( M m) { I L ( M m) E } AL k I * EA * (4) EI( M m) the eqution of motion becomes [0] * / / * w'''' w'''' { u u( )} w'' uw' w w * (5) k ( w') d w'' w' w'd w'' u u 0 ( sin ) (6) The primes nd dots represent differentition with respect to non-dimensionl longitudinl coordinte nd non-dimensionl time. To express the smllness of the / mplitude of motion w we scle it with the fctor s in [0] where the smll prmeter is mesure of mplitude nd is lso used s book keeping device in the subsequent perturbtion nlysis. Introducing this scling fctor nd using Eq.(6) for pulsting flow velocity the non-dimensionl eqution of motion cn be written s w'''' ( u u u sin ) w'' u cos ( ) w'' ( u u sin ) w' w w'''' w k ( w') d w'' O( ) 0 (7) with the ssocited boundry conditions w(0 ) w( ) w''(0 ) w''( ) 0 (8) n

4 Stbility nd Bifurction Anlysis of Pipe Conveying Pulsting Fluid 0 where * / * / u 0 u (9) Using the method of multiple scles introducing the time scle T n n 0 d nd the time derivtives D0 D d d D D 0 0D d Dn Tn n 0 we write the expnsion of w( ) in the form w( ) w0 ( T0 T ) w ( T0 T ) (0) Substituting Eq.(0) into (7) nd (8) nd equting coefficients of like powers of on both sides one cn obtin 0 ' '' '''' O( ) : D w u D w ( u ) w w w (0 ) w ( ) w (0 ) w ( ) 0 () '' '' O( ) : D w u D w ( u ) w w ' '' '''' D D w D w D w u D w u sin T D w '''' ' ' '' ' '' ' u cos T ( ) w u u sin T D w kw w dx '' '' w (0 ) w ( ) w (0 ) w ( ) 0 () According to () we cn write i 0 i 0 ( ) ( ) ( ) T T w T T A T e A ( T ) ( ) e cc () 0 0 where the complicte expressions of m( )( m ) re given in reference [0]. Substituting () into () considering the cse of the internl resonnce nd combintion prmetric resonnce i.e. (4) the modultion equtions cn be written s A C A e A 8S A A 8S A A A 8G A A e H A e 0 (5) ' i T it 4 A C A e A 8S A A 8S A A A 8G A e H Ae 0 (6) ' -i T it 4 5 The coefficients which re very complicted nd cn be seen in the ppendix of reference [0] re omitted here. ( ) Letting [ ( ) i ( )] i n T An pn T qn T e ( n ) (7) substituting it into Eqs.(5) nd (6) crrying out lgebric mnipultions nd seprting rel nd imginry prts we cn obtin the normlized reduced equtions s follows [0] p C p C q e p e q S ( p p q ) S ( p q q ) ' R I R I R I S ( p p p q ) S ( q p q q ) q H p H q R I 4R 4I G ( p p p q p q q ) G ( p q p p q q q ) (8) R I q C q C p e q e p S ( p p q ) S ( p q q ) ' R I R I I R S ( q p q q ) S ( p p p q ) p H q H p R I 4R 4I G ( p q q q p q p ) G ( p q q p p p q ) (8b) R I n

5 04 L. Zhou F. Chen nd Y. Chen p C p C q e p e q S ( p p q ) S ( q p q ) ' R I R I 4R 4I S ( p p p q ) S ( p q q q ) q H p H q R I 5R 5I G ( p p q ) G ( q p q ) (8c) R I q C q C p e q e p S ( q p q ) S ( p p q ) ' R I R I 4R 4I S ( p q q q ) S ( p p q q ) p H q H p R I 6R 6I G ( q p q ) G ( p p q ) (8d) R I where ( ) / 4 ( ) / 4. (9) The chrcteristic eqution of the cobi mtrix evluted t the initil equilibrium point ( p q p q) (0000) for Eq.(8) is 4 R R R R4 0 (0) where R R R R 4 re complicted functions of the prmeters nd omitted here. According to the Routh-Hurwitz criterion the initil equilibrium point ( p q p q ) (0000) is stble if the following conditions re stisfied. R R R R 0 R ( R R R ) R R 0 R 4 0. () 0 4. BIFURCATION ANALYSIS Conditions () imply tht ll the eigenvlues of the cobi mtrix hve negtive rel prts. When () re not stisfied this is not the cse. Three cses will be discussed here... Cse : Double zero nd two negtive eigenvlues Tking R 4 R R R 4 0 Eq.(0) hs double zero nd two negtive eigenvlues 4.One choice of prmeters tht stisfy these conditions is C R 0 C R C I C I H 4 I 4 0 H5I 0 0 H 4 R H R H 6 R H 6 I 0. Let us consider s perturbtion prmeters. Using the prmeter trnsformtions nd the stte vrible trnsformtion p 4 x q 4 x () p 4 x q x4 where ij ( i j 4) re given in the ppendix (A.) one my trnsformtion Eq.(8) into new system s follows dx x x x 4 x4 Nf ()

6 Stbility nd Bifurction Anlysis of Pipe Conveying Pulsting Fluid 05 dx x x x 4x4 Nf (b) dx x x x 4x4 Nf (c) dx4 4x 4x 4x 44x4 Nf4 (d) where ij ( i j 4) re given in the ppendix (A.) Nfi ( i 4) re third order nonliner terms whose expressions re very complicted nd omitted here. The cobi mtrix of Eq.() evluted on the initil equilibrium solution ( x x x x ) (0000) t the criticl point 4 Pc ( c c 0) is now in the cnonicl form ( x i 0) (4) The locl dynmicl behvior of system () is chrcterized by the criticl vribles x nd x. Further more the bifurction solutions for the non-criticl vribles x nd x 4 my be determined from Eq.() up to leding order terms []. Therefore one my verify tht neglecting x nd x 4 (i.e. setting x = x 4 = 0) in the first two equtions of Eq.() does not effect the results of the bifurction solution ( x x ) nd their stbility conditions up to leding order terms. So in order to consider the bifurction nd stbility properties of system () in the vicinity of the criticl point P c one only needs to nlyze the following two-dimensionl system: dx x x Nff dx x x Nff (5) where Nff i re complicted third order nonliner terms nd omitted. Now bsed on the reduced system (5) the results nd formule obtined in [] cn be pplied here. Using these methods we cn study the stbility nd bifurctions of this model nlyticlly. Applying the generl formul yields the following results. The stbility conditions for the initil equilibrium solution xi 0 re described by nd (6) which leds to two criticl lines. One of these is L: ( ) (7) long which sttic bifurction solution tkes plce from the initil equilibrium solution nd the solution is expressed by x ( ) x ( ) x (8)

7 06 L. Zhou F. Chen nd Y. Chen It is clled pitchfork bifurction. On the other hnd the second criticl line L : ( ) (9) describes dynmic boundry where the initil equilibrium solution loses its stbility nd bifurctes fmily of limit cycles. Agin from Tble of reference [] one my find the stbility condition for the fmily of limit cycles given by. 0 (0) so the fmily of limit cycles bifurcting from the initil equilibrium solution is stble. The sttic bifurction solution (8) becomes unstble on the third criticl line L :..0 0 () from which nother fmily of limit cycles which is usully clled secondry Hopf bifurctions occurs. The frequency of this fmily of limit cycles is ( ) 0 () c where since the secondry Hopf bifurction solution exists in the region locted on the right of the criticl line L (see Fig.). The stbility condition for this fmily of limit cycles is given s follows: () c Therefore the secondry Hopf bifurction solution is stble. The criticl bifurction lines re illustrted in the prmeter spce in Fig. From Fig it is seen tht there my exist stble trivil nd non-trivil equilibrium solutions periodic motions in this cse. Figure. The bifurction digrm in the cse of double zero nd two negtive eigenvlues... Cse :Double zero nd pir of purely imginry eigenvlues Tking the prmeters C R C R 0 C I C I H 4 I H 5 I 0 0 H 4 H H 6 H 6 0 R R R I in Eq.(0) yields R R R4 0 R so the eigenvlues of the cobin re

8 Stbility nd Bifurction Anlysis of Pipe Conveying Pulsting Fluid 07 0 nd 4 i where i. Choosing nd s prmeters nd using the prmeter trnsformtion introducing the following stte vrible trnsformtion p 5 x 0 q x (4) p x q x one my obtin the following equtions dx ( ) x ( ) x 5( ) x x4 Nm dx x 5( ) x ( ) x ( ) x4 Nm dx ( ) x x ( ) x ( ) x4 Nm 6 6 dx ( ) x 5( ) x ( ) x ( ) x4 Nm4 (5) where the third order nonliner terms Nm i (i = 4) re omitted here. Using n intrinsic method of hrmonic nlysis [] we obtin the norml form of Eq.(5) s follows dy y dy y ( ) y ( ) y ( ) y d 5 5 ( ) ( ) ( ) y 5 d ( ) ( ) y (6) where y y re the vribles tht re trnsformtionl systems which re topology equivlent to the originl systems. The trnsformtionl systems cn disply the dynmicl behviors of the originl ones. System (6) hs the following equilibrium solutions. (i) The initil equilibrium solution (E.S.) y y 0. Evluting the cobin t the initil equilibrium solution we obtin the stbility conditions s follows

9 08 L. Zhou F. Chen nd Y. Chen 0 0 ( ) 0 (7) So the initil equilibrium solution is unstble. (ii) The sttic bifurction solution (S.B.) y y ( 5 ) / 4 0 (8) The stbility conditions for this solution re s follows 0 0 ( ) 0 (9) (iii) Hopf bifurction solution (H.B.) y y 0 5 5( ) /(5 5) 0.8( ) (40) The stbility conditions for this solution re s follows (4) (iv) Hopf bifurction solution (H.B.) y ( ) y (4) Obviously when nd (4) there exists H.B. solution. The stbility conditions for this solution re ( )( ) 0 (44) so this bifurction solution is unstble. The bifurction digrm is shown s in Fig. Here the dshed lines just mrk the regions of different bifurction solutions. From Fig we cn see tht there exist stle non-trivil equilibrium solution nd periodic motion under certin conditions.

10 Stbility nd Bifurction Anlysis of Pipe Conveying Pulsting Fluid 09 Figure. The bifurction digrm in the cse of double zero nd pir of purely imginry eigenvlues.. Cse : two pirs of purely imginry eigenvlues If the vlues of prmeters re chosen s / 4 ( ) C R C R C I C I H 4 I H 5 I 0 0 H 4 H H 6 H 6 0 R R R I then we hve R R 0 in Eq.(0) nd the eigenvlues re i 4 i (45) Considering s prmeters using the prmeter trnsformtion nd the stte vrible trnsformtion p 0 0 x q 0 0 x p / 4 / 4 / 4 / 4 x q x / 4 / 4 / 4 / 4 4 Eq.(8) becomes dx 9 ( ) x ( ( ) ) x ( ) x 4 4 / 4 x Nh / 4 5/ 4 4 dx x ( ) x ( ) x4 Nh 8 dx 9 / 4 ( ) x x ( ) x 4 4 / 4 / 4 ( ( ) x Nh 4 dx4 ( ) x x ( ) x4 Nh4 (47) 4 The third order nonliner terms Nhi ( i 4) re lso omitted here. (46)

11 0 L. Zhou F. Chen nd Y. Chen With the method of multiple scles nd computer lgebr [4] we get the norml form of Eq.(47) in polr coordintes s follows: r r ( r r ) 0 0 r r ( b r b r ) 0 0 c c r c r (48) 0 c d0r d0r 5/ 4 / 4 where / 4 / 4 / 4 / 4 / 4 / / 4 / 4 / 4 / 4 b0 7 b / 4 / 4 / 4 / 4 c0 c / 4 99 d re the vribles tht re trnsformtionl systems which re / 4 / 4 d0 nd r r nd topology equivlent to the originl systems. We now discuss the min types of equilibrium sttes nd their stbility for system (48). These fll into four ctegories. (i) Trivil stte: The initil equilibrium solution (E.S.): r r 0. Evluting the cobi mtrix t the initil equilibrium solution results in the stbility conditions for the E.S. s 0 nd 0 i.e. nd 0 so it is unstble. 0 (ii) Pure mode : Hopf bifurction solution (i.e. self-sustined oscilltion H.B.(I) with frequency ): r ( ) r c c0r.0.6 (49) Obviously when 0 there exists H.B.(I) solution. Evluting the cobi mtrix t the H.B.(I) solution results in the following b 0 stbility conditions 0 nd ( ).The 0 condition 0 implies tht 0 So H.B.(I) solution is unstble. (iii) Pure mode : Hopf bifurction solution (H.B.(II) with frequency ) r 0 0

12 Stbility nd Bifurction Anlysis of Pipe Conveying Pulsting Fluid r ( ) b 0 c d0r (50) Obviously when 0 there exists H.B.(II) solution. Evluting the cobi mtrix t the H.B.(II) solution results the following 0 stbility conditions 0 nd ( ) 0 i.e. b nd 0so H.B.(II) solution is lso unstble. 0 (iv) Mixed modes: qusi-periodic solution (D tori with frequency ): 0( ) b0( ) r b0 0b0 b0( ) 0( ) r b b c c0r c0r c d0r d0r (5) Obviously when 0 nd there exists D tori. The stbility conditions for the qusi-periodic solutions re obtined from the trce nd determinnt of the cobin given by Tr ( r b r ) 0 0 [ 0( 0 b0)( ) b0( 0 b0)( )] 0 b b Det 4( b b ) r r 0 (5) i.e The criticl lines re illustrted in Fig 4. Here the dshed lines lso just mrk the regions of different bifurction solutions. From Fig 4 one cn see tht there exist stble mixed modes qusi-periodic motions under certin conditions in this cse. 0 Figure 4. The bifurction digrm in the cse of two pirs of purely imginry eigenvlues.

13 L. Zhou F. Chen nd Y. Chen 4. NUMERICAL SIMULATIONS In this section with the fourth-order Runge-Kutt method the phse portrits of system (4) re obtined for different vlues nd. For cse choosing ( ) ( ) ( ) ( ) ( ) ( ) respectively we obtin the phse portrits of system (.4) s in Fig 5-Fig 7. Notice tht these prmeters of ( ) re in the stble regions of the initil equilibrium solution sttic bifurction solution nd limit cycle respectively numericl results gree with the nlyticl ones. For cse choosing ( ) ( ) (in the stble region of the sttic bifurction solution) ( ) ( ) (in the stble region of the Hopf bifurction solution) respectively we obtin the phse portrits of system (.4) s in Fig 8-Fig 9. For cse choosing ( ) ( ) (in the stble region of -D tori) the phse portrits re shown s in Fig 0. () Figure 5. Trjectory strting from ( p q q q ) ( ) converges to the E.S. for cse (b) () Figure 6. Trjectory strting from ( p q q q ) ( ) converges to the S.B. for cse (b)

14 Stbility nd Bifurction Anlysis of Pipe Conveying Pulsting Fluid () Figure 7. Trjectory strting from ( p q q q ) ( ) converges to the L.C. for cse (b) () Figure 8. Trjectory strting from ( p q q q ) ( ) converges to the S.B. for cse (b) () Figure 9. Trjectory strting from ( p q q q ) ( ) converges to the H.B. for cse (b) () Figure 0. Trjectory strting from( p q q q ) ( ) converges to the -D tori for cse (b)

15 4 L. Zhou F. Chen nd Y. Chen 5. CONCLUSIONS With the method of norml forms the bifurction solutions nd their stbility of hinged-hinged pipe conveying pulsting fluid with combintion prmetric nd internl resonnces re studied in detil. When the stbility conditions for the initil equilibrium solutions re not stisfied bifurctions including pitchfork bifurction Hopf bifurction -D tori my occur. Complicted dynmicl phenomen of this model re presented here. Numericl simultions gree with the nlyticl results. 6. ACKNOWLEDGMENTS This reserch ws supported by the Ntionl Nturl Science Foundtion of Chin (Nos ) Chin Postdoctorl Science Foundtion (No. 0T605) nd Ntionl Reserch Foundtion for the Doctorl Progrm of Higher Eduction of Chin (08005). 7. APPENDIX ( 7) ( ) ( ) ( ) ( 5)(4 ) (A.) 6 (5 ) (5 07) 4 6 6( 5) (7 ) (6 9 ) ( 5) (6 55) ( 7) 4 6 6( 5) (7 5) (95 46) 6 6( 5) 4 [( 7) (5 ) ] 6 6( 5) [( 5) (7 ) ] ( 5) [( ) ( ) ] 8( 5) ( ) 6 6( 5) [(7 ) ( 9) ] 4 6 6( 5) [( ) (7 ) ] 6 6( 5) 4 9[( ) ( ) ] 6( 5) [( 4) (7 6) ] 4( 5) 4

16 Stbility nd Bifurction Anlysis of Pipe Conveying Pulsting Fluid ( ) ( ) 6 6 6( 5) ( ) ( 6) 4( 5) [( ) ( ) ] 8 6 6( 5) 4 ( 8) (9 4 ) 6 6( 5) 44 (A.) 8. REFERENCES.M. P. Pidoussis Flow-induced instbilities of cylindricl structures Applied Mechnics Re-view M. P. Pidoussis nd D. X. Li Pipes conveying uid: model dynmicl problem ournl of Fluids nd Structures M. P. Pidoussis Fluid-Structure Interctions Slender Structure nd Axil Flow Acdemic Press London S. S. Chen Dynmic stbility of tube conveying uid ASCE ournl of Engineering Mechnics M. P. Pidoussis nd N. T. Issid "Dynmic stbility of pipes conveying fluid" ournl of Sound nd Vibrtion M. P. Pidoussis nd C. Sundrrjn Prmetric nd combintion resonnces of pipe conveying pulsting fluid ASME ournl of Applied Mechnics Ginsberg The dynmic stbility of pipe conveying pulstile flow Interntionl ournl of Engineering Science S. T. Arirtnm nd N.S. Nmchchivy Dynmic stbility of pipes conveying pulsting fluid ournl of Sound nd Vibrtion K. yrmn nd W. M. Tien Chotic oscilltors in pipes conveying pulsting fluid Nonliner Dynmics L. N. Pnd nd R. C. Kr Nonliner dynmics of pipe conveying pulsting fluid with combintion principle prmetric nd internl resonnces ournl of Sound nd Vibrtion L. N. Pnd nd R. C. Kr Nonliner dynmics of pipe conveying pulsting fluid with prmetric nd internl resonnces Nonliner Dynmics L. Wng A further study on the non-liner dynmics of simply supported pipes conveying pulsting fluid Interntionl ournl of Non-Liner Mechnics D. in nd Z.Y. Song Prmetric resonnces of supported pipes conveying pulsting fluid ournl of Fluids nd Structures Y. Modrres-Sdeghi nd M.P.Pldoussis Nonliner dynmics of extensible fluidconveying pipes supported t both ends ournl of Fluids nd Structures L. Wng nd Q. Ni A note on the stbility nd chotic motions of restrined pipe conveying fluid ournl of Sound nd Vibrtion U. Lee nd. Prk Spectrl element modeling nd nlysis of pipeline conveying internl unstedy fluid ournl of Fluids nd Structures R.. McDonld nd N. Sri Nmchchivy Pipes conveying pulsting fluid ner

17 6 L. Zhou F. Chen nd Y. Chen 0: resonnce: Globl bifurctions ournl of Fluids nd Structures D. in nd G.S. Zou Bifurctions nd chotic motions in the utonomous system of restrined pipe conveying fluid ournl of Sound nd Vibrtion S.I. Lee nd. Chung New nonliner modeling for vibrtion nlysis of stright pipe conveying fluid ournl of Sound nd Vibrtion B.G.Sinir Bifurction nd chos of slightly curved pipes Mthemticl nd Computtionl Applictions P.Yu nd Q. S. Bi Anlysis of non-liner dynmics nd bifurctions of double pendulum ournl of Sound nd Vibrtion P. Yu nd K. Huseyin Sttic nd dynmic bifurctions ssocited with doublezero eigenvlue Dynmics nd Stbility of Systems P. Yu nd R. Huseyin Bifurctions ssocited with double zero nd pir of pure imginry eigenvlues SIAM ournl on Applied Mthemtics P. Yu Anlysis on double Hopf bifurction using computer lgebr with the id of multiple scles Nonliner Dynmics