326. Dynamic synchronization of the unbalanced rotors for the excitation of longitudinal traveling waves

Transcript

1 . Dynamic synchonizaion of he unbalanced oos fo he exciaion of longiudinal aveling waves. Saseeyeva K. Ragulsis Z. Navicas Kazah Naional Pedagogical Univesiy named afe bay Tole bi s. 8 lmay Kazahsan aigula@mail.u Kaunas Univesiy of Technology Sudenu s. T-8 Kaunas ihuania zenonas.navicas@u.l (Received Januay 8; acceped 7 Febuay 8 bsac. The poblem abou he synchonizaion of he unbalanced oos fo he exciaion of he longiudinal aveling waves in he elasic sysem of a ba ype is examined. The seady-sae egimes of moion ae invesigaed he condiions of hei exisence and sabiliy ae obained. Keywods: synchonizaion elasic sysem longiudinal aveling waves.. Inoducion The sudy of he exciaion mehods of pescibed vibaions of a ceain elasic sysem pesens one of he impoan poblems of he heoy of vibaion machines and devices. In his case he poblem abou he synchonous opeaion of seveal viboexcies conneced wih he unied oscillaoy sysem aises []. In he pesen wo he poblem abou he synchonizaion of seveal vibaion excies of he elasic sysem fo he exciaion of he longiudinal aveling waves is invesigaed. The equaions of he poblem ae nonlinea. Fo he soluion he small paamee which enables o use mehods of he heoy of peiodic soluions of nonlinea diffeenial equaions is inoduced.. The fomulaion of he poblem The sysem consising of a semi-infinie ba wih he elasically conneced end and n vibaion excies conneced o i is analysed. x Y x Fig.. Model of he sysem i m u m m B B X The body of he vibaion excie is aached o he ba a he poin whee - he cene of he axis of oaion of he oos of viboexcies. The exciing masses of vibaion excie ae locaed a poins B and B and hey oae synchonously in opposie diecions so ha he exciaion along he ba is ceaed []. I is designaed B B n. ccoding o Fig. he poins B B have he coodinaes ( x u B ( x u cos sin B ( x u cos sin x cons. VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87 Kineic enegy of he -h viboexcie is equal o T ( m m( u ( J m m( sin ( u whee m is he mass concenaed a he poin and m ae he masses concenaed a poins B and B J is he momen of ineia of he oo of he -h vibaion excie wih espec o he cene of he axis of oaion.

2 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS The equaions of moion of his sysem have he fom ( J m m( x u sin H M ( ( m m u m F ( ( ( sin cos whee H ae dissipaive foces H - coefficiens of viscous ficion fo he oaion of he case of he -h vibaion excie wih espec o he axis M ae he momens of exenal foces F in ae ineial foces. The longiudinal vibaions of a ba ae descibed by he equaion [] u u ξ ρf δ ( x x F in ( x whee u ( x is he displacemen of he coss secion wih he abscissa x ρ is he mass of he uni of volume E - he modulus of elasiciy of he maeial F - cosssecional aea ξ - coefficien chaaceizing exenal damping δ - Diac's dela funcion. Fo he elasically fixed end of he ba he bounday condiion has he following fom c x ( x in u u ε u ε u ε ε ( F F in in in in ε F ε F u Fin ae peiodic funcions of. Subsiuing ( ino he equaions (8 ( ( and equalizing coefficiens wih he idenical degees of ε in he igh and lef sides of he equaliy he diffeenial equaions fo he deeminaion of u Fin ae obained. Subsiuion of ( ino equaion (8 and equaing of coefficiens a ε and ε gives he equaions of he zeo and fis appoximaion ( J m ( ( ( J m Φ whee Φ Φ m( u sin H M. u u I follows ha can be epesened in he following fom whee c is he coefficien of siffness of he sping. x By inoducing he dimensionless coodinae he equaion ( and he condiion ( ae he fom u u ξ ρf δ ( F in ( c x. (7 Thus equaions ( ( ( wih he condiion (7 ae he diffeenial equaions of moion fo his sysem.. Seady-sae egimes of moion The mehod of small paamee is used fo invesigaion of he seady-sae egimes of moion deemined by equaions ( ( (. Then he equaion ( aes he fom ( J m εφ (8 whee Φ m( x u sin H M (9 ε is he small paamee a he end of he calculaions assumed equal o one. The seady-sae egimes of moion ae epesened in he fom ω ( whee cons. Equaion ( is he diffeenial equaion fo he deeminaion of. Peiodiciy condiion of accoding o he equaion ( has he fom m( u sin( ω H M ( ω VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87 Φ he uppe dash indicaes aveaging wih espec o. Consans ω and n ae deemined fom he condiion ( bu befoe i is necessay o find he funcions u n fom ( (. Subsiuing ( wih ω in ( and ( and equaing he coefficiens a ε i is obained u u ξ ρf δ ( F in ( F m m( u mω cos( ω. ( ( in. Soluion of he poblem descibing he foced longiudinal vibaions of he ba Fuhe he deeminaion of he soluions u( of he equaion ( wih he bounday condiion (7 is

3 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS descibed. he poins wih he abscissa x (o in he dimensionless coodinaes of he ba he foces F in ( ae applied. Using he condiion (7 he equaion ( is ewien in he following fom u u u ξ ρf c δ F in. (7 ( The case n is examined. The soluion of his poblem is educed o he inegaion of he diffeenial equaion of he vibaions of he ba u u ξ ρf c u (8 valid eveywhee excep a he poins and a which he foces ae applied ( m m( u mω cos( ω F in and ( m m( u mω cos( ω F in especively. Equaion (8 mus be inegaed wih he following condiions: u when ( ( u u ( ( Fin and ( ( u u ( ( Fin (9 ( whee u ( u when u ( u when u ( u when <. The soluion of equaion (8 is sough in he fom u( θ cosω ψ sin ω. ( Designaing ξ ω ρ ω β ρfω µ β E c c and subsiuing he expession ( ino he equaion (8 he funcions θ ( and ψ ( ae found ( ( θ e e e e e ( ( ψ ie e e e e ( whee i i ae posiive eal numbes ( ( β µ ( β µ ( β β. ( ( The funcion u ( θ cosω ψ sin ω is deemined fo [ ] heefoe he funcions θ ψ ( mus saisfy he condiions θ θ ( ψ ( ψ ( ( ( β β obained fom (7; ( he funcion u ( θ cosω ψ sin ω appoaches zeo when i. e. θ ψ when. ( ( ( Besides he funcions u ( u ( and ( u ( θ cosω ψ sin ω mus saisfy he condiions (9 and (. Subsiuing condiions ( in ( i is obained / ( β θ e e e / β / β e e / β / ( β ψ ie e e / β / β e e. / β ( VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87

4 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS I is assumed ha ( ψ acg. θ ( ( θ e e e e e ( Pesening u ( in he fom (9 i is obained ψ ( ( ( ( u ( cos( ω ie e e e e. ( ( cos( ω (7 ( ( Wih he saisfacion of condiions ( when ( cos( ω i is obained ( b ( ( ( whee e e M ( ( θ e e e (8 b ( ( ( ( e e M ψ ie e e. ( ( ab ( Using he connecion condiions obained fom e e M M ( β bounday condiions (9 and ( i follows ha ( ( acg Λ acgλ ( θ θ ψ ψ acg Λ acgλ θ θ θ ( a a θ ( ( bcos ( acg Λ ( Λ acgλ ; β λ ch λ sh λ cos λ sin M ψ ψ ψ a λ λ aψ ( ( bsin β M µ ch µ sh µ cos µ sin ( λ ( / β λ ( / β θ θ ψ ψ ( λ ( / β λ ( / β θ θ θ a a θ ( ( bcos ( µ ( / β µ ( / β β ( µ ( / β µ ( / β ψ ψ ψ a aψ ( ( bsin ν ( β Λ ( ν m m m m whee a ω aβ a ρ F m m b ω bβ b ρf he inegaion consans in ( (8 ae deemined. ( Funcions u ( ae epesened in he fom ( ( ( u ( cos( ω (9 whee he ampliude of vibaions of he poins of a ba and he iniial phase ( θ ψ VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87 ν ν ( ( λ λ e λ e g λ e ( λ λ e λ e λ e g ( h ( / β g ( g ( / β h Λ ( BB CC D ( BC CB D B C C B D B B C C D Λ acg a ae B µ [ λ cos λ sin ] ( λ λ a ae C [ λ cos λ sin ] ( λ λ a ae B µ [ µ cos sin ] a ae C [ cos µ sin ]

5 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS ae D [ µ cos µ sin ] 7 8 ( a e [ λ cos ( λ sin ( 7 8 ( λ λ ae D [ µ cos µ sin ] 8 7 ( a e [ λ cos ( λ sin ( 8 7 ( λ λ µ / β / / β λ λ λ ( / β µ ] ] λ λ ( β λ ( / ( / µ ( β λ λ µ λ µ µ β λ λ µ λ µ u ( ( Now u ( is epesened in he fom (9 ( ( ( cos( ω ( ( cos( ω ( ( cos( ω ( ( ( cos( ω ( b ( ( ( whee e e M ( ( ( ( b ( e e M ( ab ( e e M ( M ( β ab ( ( e e M ( ch cos β ( ( acg Λ acgλ ( ( ( acg Λ acgλ ( ( ( acg Λ ( Λ acgλ ( ( ( g ( acg Λ acgλ ( ; h M λ µ λ µ λ µ λ µ χ ch χ sh χ cos χ sin λ λ χ χ λ µ λ µ χ λ µ λ µ χ Λ ν ν ( / β ( / β (( λ λ e λ e g λ e ( λ λ e λ e λ e g. ( Repesening he funcion u ( in he fom (9 i is obained u ( ( ( ( cos( ω ( ( cos( ω ( ( ( cos( ω < ( b ( ( whee e e M ( ( ( b a ( ( e e M ( β ab ( e e M ( ch cos β ( ( acg Λ acgλ ( ( ( acg Λ acgλ ( ( g ( acg Λ acgλ (. h ( Diffeeniaing he funcion u ( wih espec o ( he value of he funcion x ( ( a β is found hen adding x ( and ( ( u ( u ( u ( i is obained ( ( u ( x ( cos( ω ( ( ( ( cos( ω cos( ω ( b ( ( whee e e M ( ( ( ( ( ( b e M ( ab ( e M ( M ( ( acg Λ acgλ ( acg Λ acgλ ( acg Λ ( Λ acgλ (. ( ( In he same way diffeeniaing he funcion u ( wih espec o he value of he funcion VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87

6 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS ( x( ( a is found hen by adding β x ( and ( ( ( ( u u u ( i is obained u ( ( ( x ( cos( ω ( ( ( ( cos( ω cos( ω ( b ( ( whee e M ( b ( ( e e µ µ M ( ( ( acg Λ acgλ ( ( ( acg Λ acgλ ( acg. µ. The exisence and sabiliy of he soluions Fom he peiodiciy condiions Φ m( u sin( ω H M ω whee he funcions u x ( and u x ( ae now aleady nown he consans ω and ae deemined. H M Designaing h M hese condiions m m can be wien in he fom F ( h ω M / ω sin sin ( ( ( (. ( ccoding o (7 he condiion fo exisence of he soluions is expessed by he inequaliy ( F sin F sin ( ( sin( ( <. (8 Values depend on he fequencies of exciaion ω which ae deemined fom he equaion ( ( ( ( F F F F cos( sin (. (9 I follows fom (9 ha F < F <. Only hose values of ae sable which saisfy he inequaliy ( ( ( ( cos( cos( <. (. Vibaions of a ba in he absence of damping If he viscous damping is disegaded ha is ξ hen he expessions ( fo ae epesened in he fom a when b when < β : > β : β ( β ( Φ ω ( x u sin( ω h ω M. ( Equaions ( by aing ino accoun ( ( lead o he following expessions Φ Φ ( ( sin( ω hω M ( ( ( ( sin sin ( ( sin ( ( ω sin( hω M ( ( sin. ( Fom ( he expession fo he deeminaion of is obained: ( F sin F sin sin( ( ( sin( ( (7 ( ( ( ( ( hω M / ω sin sin whee F ( since µ - eal posiive numbes. In he case a he funcions u x ( and u x ( ae ansfomed o he fom u x e cos( ω β cos( ω β cos( ω β ( u x cos( ω β VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87 e β sin( ω β cos( ω ( µ whee b cos( ( µ a b cos( cos( ( µ b cos( ( ( µ b sin( (

7 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS ( β ( Λ β β β β acg / β / β acg ( ( Λ acg a a µ a a β sin λ a µ λ a µ λ λ cos λ sin aχ aχ λ cos ( λ sin ( a a a µ a β cos λ a µ λ a µ λ λ cos λ sin aχ aχ λ cos ( λ sin ( ( / β λ ( / β ( / β λ λ ( / β λ ( / β µ ( / β ( / β λ λ λ ( / λ µ χ λ µ λ µ λ µ χ Taing ino accoun ha β λ µ λ µ. < β i is obained ha c c c β ( c ω. ( ρf When / β aes he maximum value c c c c β ω ; ( ( ρf c when / β < : c β c ( c ω ; ρf (7 c when / β > : c c β c c ( ( c c ω. (8 ρf ρf Peiodiciy condiions lead o he expessions Φ sin( ω e β h ω M sin( β sin β sin β (9 Φ cos( ω e β sin β hω M. Fom (9 i is obained F cos( β F sin β sin( ( cos(β F F F F sin(β cos (β ( whee F (( h ω M / ω sin( β sin β e ( ( h ω M / ω ( sin β e F moeove one is o ae ino accoun ha accoding o ( c ω. In ( F < F <. ρf ccoding o ( he condiion of exisence is expessed by he inequaliy F cos( β F sin β cos(β <. ( VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN

8 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS The sabiliy condiion of hose egimes of moion which saisfy he condiion of exisence is he inequaliy e β cos( > e sin( β. ( In he case b he funcions u x ( and u x ( ae he fom moeove one is o ae ino accoun ha c ω ρf. The expessions fo deeminaion of he values ae found: sin( h ω M h ω M (8 ω ω hence i follows ha he inequaliy u x cos( ω ( cos( ω u x cos( ω cos( ω ( ( h ω M h ω M ω ω < (9 b whee e s ( s s e ( b s ( s s e ( ab s ( s s e ( s e ( b ( s ( s s e ( b e s ( s e s / β ( / β / β s s / β / β a s ( s s e a ( s ( e s e a s ( s e (. By aing ino accoun ha c c ( > β i is obained ha β ( c ω. ( ρf The aveaged values Φ i i he fom Φ Φ ω sin( hω M ω sin( h ω M. ae epesened in ( Fom ( he equaliy fo deeminaion of he fequency ω is obained: ( h ω M ( h ω M (7 is he condiion of exisence of he seady-sae egimes of moion. The sabiliy condiion of he obained egimes of moion is deemined by he inequaliy cos(. ( ( < e and h ω M h ω M hen sin( fom whee i is found ; π. The soluion accoding o ( is sable when < and π is sable when >. In he case when h ω M h ω M and h ω M hen sin( whee ω - ω ( ω he soluion of he equaion. If he condiion of h ω M exisence is saisfied i. e. < hen he ω ( ω soluion will be sable when he inequaliy cos( < is sisfied: if he soluion ( π / / hen i is sable when if π ( π / π / > hen i will be sable when. In he case when h ω M h ω M and hose soluions (8 saisfying he condiion of exisence will be sable which belong o ( π / if < and hω M ω hω M ω ( π / π if he inequaliies saisfied ( π π / if > and < > ( > and ( ae 8 VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87

9 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS hω M ω hω M ω < (. w.9 φ -φ u x u x ( π / π if he inequaliies saisfied. < and ( ae w.9 φ -φ -. In all he analyzed cases c ω ρf. - u x u x Noe: When > and c ω ρf : c < e < e < e c c < e < e <. e wih he gowh of he degee appoaches zeo i. e. beginning fom a ceain value of he fequency ω he value of he funcion by a small amoun fom zeo. e will diffe x x Fig.. The dependence of funcions u and u on (unsable and sable soluion The hee-dimensional image of he funcion ( u ( ( u ( u ( ( ( u (.. wih he given paamees (in he c case ω > is epesened in Fig.. ρf 7. The esuls of numeical calculaions a w.9 φ -φ -. s iniial daa fo he case ξ i is acceped: a b ρ n h. h. M. M... u(. -. m m m whee a b ρ F ρf ρ ρ E n c H M x h M. m m c In his case ω o ω /. ρf The equaion fo deeminaion of he fequency ω has wo soluions: ω. 9 ω.. Fequency ω does no saisfy he condiion of exisence b w.9 φ -φ -. 8 Since ω > / hus he values ae found fom he equaion (8. When ω. 9 hen sin(. 97 fom whee. 778 and.. < consequenly he value. coesponds o he sable soluion. he poin. he funcion u x is defined a he poin. - he funcion u x. The dependence of he pesened funcions on is epesened in Fig.. VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87 u( ( Fig.. The image of he funcion u( : u ( ( (.; u (.. ; u (. 9

10 . DYNMIC SYNCHRONIZTION OF THE UNBNCED ROTORS FOR THE EXCITTION OF ONGITUDIN TRVEING WVES.. SRSEKEYEV K. RGUSKIS Z. NVICKS a w.7 φ -φ.87 If o accep as iniial he following daa in he case ξ : a b ρ. n h. h. 7 M M. hen wo values of fequency will be obained ω.7 and ω c s ω o ω and ρf ω < and ω < hen he values ae found fom he equaion (. Wih ω he equaion ( has soluions 79 and. 87 and only he soluion. 87 saisfies he sabiliy condiion (. Wih ω fom he wo soluions. 8 and.97 sable posiion will be he second one. The dependence of funcions u x ( and u ( fom is epesened in Fig.. x w.7 φ -φ.79 u( u(. -.. b w.7 φ -φ u x u x w.7 φ -φ u x u x w.777 φ -φ w.777 φ -φ.97 u x u x u x u x x x Fig.. Dependence of funcions u and u fom (unsable and sable soluion The hee-dimensional image of he funcion u( ( wih he given paamees (in he case c ω < is epesened in Fig.. ρf -. ( Fig.. Image of he funcion u( : u ( ; ( ( u ( ; u ( 8. Conclusions ppoximae analyical soluion of he examined poblem abou he synchonizaion of viboexcies fo he geneaion of longiudinal avelling waves in a ba can be disseminaed o he moe geneal case fo any numbe of vibaion excies in he sysem and fo moe complicaed sysems. On he basis of esuls of invesigaions some qualiies of he sysem ae evealed he obained inequaliies and equaions ae suiable fo pacical applicaion. Refeences. Blechman I. I. Synchonizaion of Dynamic Sysems. Moscow p.. Ragulsis. K. Ragulsis K. M. Vibaing Sysems wih he Dynamically Dieced Vibaion Excie. eningad Machine Building 987 p. (Vibaion echnology. Issue.. Vibaions in Engineeing. Vibaions of inea Sysems. Vol.. Moscow Machine Building 978 p.. Filipov. P. Vibaions of Defomable Sysems. Moscow Machine Building 97 7 p. VIBROMECHNIK. JOURN OF VIBROENGINEERING. 8 JNURY/MRCH VOUME ISSUE ISSN 9-87