Active reduction of identified machine drive system vibrations in the form of multi-stage gear units

Transcription

1 83 ISSN 397. MECHANIKA. 4 Volume (): 8389 Active reductio of idetified machie drive ytem vibratio i the form of multi-tage gear uit T. Dzitowi*, A. Dymare** *Sileia Uiverity of Techology, Koariego 8a, 44 Gliwice, Polad, tomaz.dzitowi@poll.pl **Sileia Uiverity of Techology, Koariego 8a, 44 Gliwice, Polad, adrzej.dymare@poll.pl Itroductio The dyamical procee occurrig i machie drive ytem are caued by vibratio occurrig i the ytem. They have a direct impact o oie emiio, fatigue tregth, cotrollality ad tality. Therefore, oe of the baic criteria ued i deigig cotemporary mechaical tructure are the dyamical propertie of the tructure decribed i the form of frequecy trafer fuctio. By owig them, it i poible to avoid operatig the ytem withi reoace rage which ca compromie the machie life ad reliality [-7]. The tructural ad parametric idetificatio of ytem coiderig the dyamical machie propertie i oe of the method for extedig it life ad improvig reliality. Thi ta ca be performed by uig the algorithm of paive ad active ythei [ 8,, 6]. The paper preet the method for determiig the tructural ad dyamical parameter of machie drive ytem by uig the algorithm of active ythei. Thi method mae it poible to chooe the parameter ad the cotrollig force of the ytem to maitai the load at pecified worig coditio withi rated limit. The active ythei of mechaical ytem i thi paper i udertood a a calculatio method ued to idetify the tructure of a mechaical ytem, icludig it parameter ad active force, meetig the deired characteritic i the form of elected reoace ad ati-reoace frequecie. The characteritic feature of the obtaied active mechaical ytem i the fact that the determied parameter do ot chage over time ad the fact that obtaiig the aumed propertie require deliverig a exteral ource of eergy. The reearch o the paive ad active ythei of torioally ocillatig mechaical drive ytem i developed by the author of thi paper. The itroductio to the ythei of machie drive ytem, actig a torioally ocillatig ytem, i preeted i []. Thi ectio preet the method of ytheizig the ditributio of characteritic o a cotiued fractio maig it poible to obtai the parameter ad the model of a igle-axi ytem. I additio, paper [] preet the ythei of drive ytem uig the method of characteritic ditributio o partial fractio (brached ytem), the mixed method (cacade-brached ytem). The, i the cae of dyamical model of the whole drive ytem obtaied, the author determie the cotrollig force applied o the ytem that couteract the ocillatio-iducig dyamical load active ythei [3, 4, 6, 8]. Afterward, i [7] the author preeted a attempt to exted the exitig method of paive ythei, aimed at obtaiig the model of drive ytem icludig multi-tage gear uit. Thi paper cotitute a exteio of the latter iue, by the problem of vibratio reductio i idetified machie drive ytem. The reductio method wa baed o uig the active ythei maig it poible to obtai the deired mechaical effect by properly electig the dyamical propertie of the ytem, icludig the calculatio of active force a a fuctio of ytem couplig.. Idea of active ythei of o-reduced machie drive ytem The firt tage of ytheizig the mechaical ytem cotitute creatig mathematical fuctio that, o oe had, meet the requiremet et for thoe ytem, while o the other, ca be preciely implemeted i real ytem. The problem i about creatig ratioal fuctio meetig the feaility requiremet ad the deired dyamical requiremet. The method for determiig the aalytic form of the dyamical characteritic coit i aumig the equece of reoace ad ati-reoace frequecie (pole ad zeroe of the earched dyamical characteritic Fig. ). The ta formulated i the above-metioed way, aimed at determiig the form of dyamical characteritic, doe ot require performig the approximatio proce, but itead it mae it poible to tart the very ythei at oce. If the fuctio decrig the dyamical propertie of the dicrete ocillatig ytem i aumed a dyamic tiffe or the dyamical flexility, the the followig traform are ued for idividual characteritic: V Y, () V i molity (mechaical admittace), where Y i dyamical flexility o the plae of complex variable,, where U Z, () U i immolity (mechaical impedace), Z i dyamical tiffe o the plae of complex variable. By uig the Eq. () ad () it i poible to traform the ytem motio decriptio from geeralized diplacemet ad force geeralized ito a motio decriptio by uig geeralized velocity ad force value. Thi i the way to traform the dyamical flexility fuctio of

2 84.4 Y(ω), rad/nm.3 Y ( ) ω, rad/ 5 Fig. Example of dyamical characteritic the ytem ito the molity fuctio or the dyamical tiffe fuctio of the ytem ito the immolity fuctio, ued both to ytheize mechaical ytem. Becaue of the cla of mechaical ytem uder coideratio, the ythei ta will be limited to the cae of emi-defiite ytem, whe the umber of the obtaied ytem elemet i odd. The dyamical fuctio meetig thee aumptio i the immolity fuctio decribed by the followig formula: U H i j b z, (3) where: H i ay poitive umber, equece of reoace frequecie, b, b,, b are z, z,, z are equece of ati-reoace frequecie. At the firt tep it i eceary to ubject the characteritic (3) to ythei by uig the method of ditributio o a cotiued fractio (ued for the firt time to ytheize electrical circuit the Cauer method). Fially, it i poible to obtai the immolity value i the form of a cotiued fractio havig the followig form: U. c z c z z c z z (4) A a reult of ytheizig the immolity characteritic (3) we ca obtai a dicrete ocillatig ytem preeted i Fig., havig the aumed propertie i the form of reoace ad ati-reoace frequecie. c c z c -z z -z z Fig. Cacade model of a drive ytem The ytem obtaied cotitute a tart poit for determiig the followig: the cotrollig reducig force ad the force elimiatig vibratio ad the o-reduced drive ytem parameter i the form of a multi-tage gear uit. Oce the parameter ad tructure of the paive ytem are determied, it i poible to commece determiig the value of force maig it poible to talize, reduce or elimiate ytem vibratio ear the reoace ytem focu. The force excitatio, a a ettig value, ca be performed i the ytem o ay idetified iertial ele-

3 85 met. The paper limit the earch for the cotrollig force to the firt iertial elemet. To do thi, the dyamical characteritic of flexility () are modified by itroducig: vibratio frequecy drop parameter h for the elected reoace frequecie (a example of the vibratio reductio defied i thi way i preeted i Fig. 3), i the followig form: Y H hb hb b i j z, (5) the parameter of hiftig the vibratio frequecie Δω for the elected reoace frequecie (a example of the vibratio elimiatio defied i thi way i preeted i Fig. 3), i the followig form: Y H b b i j z. (6) the parameter of reductio ad hiftig for the elected reoace frequecie of the ytem, by itroducig parameter h ad Δω (a example of the characteritic i preeted i Fig. 3), i the followig form: Y3 H hb hb b b i j z. (7) Itroducig thoe characteritic modificatio Eq. () mae it poible to reduce ytem vibratio ear the reoace frequecy rage characteritic Eq. (5), reoace frequecy hift characteritic Eq. (6) ad hybrid of the above-metioed cae characteritic Eq. (7)..4 Y(ω), rad/nm.3 YY(ω ( ) ) Y(ω) ( ). Y(ω ( ) ) Y3(ω ( ) ) ω, rad/ 5 Fig. 3 Example of dyamical characteritic: Y(ω) - created by taig ito accout the free vibratio frequecy drop parameter, Y(ω) - by taig ito accout the free vibratio frequecy hift parameter, Y3(ω) - created by taig ito accout the free vibratio frequecy drop ad hift parameter To obtai a vibratio drop ear the reoace frequecy rage of the ytem uder aalyi (Fig. ), we adopt the rule of cotrol [9, 3], maig it poible to calculate the force excitatio a a feedbac fuctio i the followig form: where p v p v p v M, (8) p, p, p;v,v,, v are amplificatio coefficiet of the cotrol ytem depedig o the poitio ad velocity of the iertial elemet of the ytem aalyzed. The, the paper preet the method for determiig the above-lited coefficiet, which, coequetly, will mae it poible to defie the cotrollig force. By uig the idetified tructure parameter (Fig. ) it i poible to cotruct a dyamic tiffe matrix i the followig form: Z c c c c c z z z, c z c c z c z c z (9)

4 ad the active force matrix, becaue of the fact that the earched active force act o the firt iertial elemet oly, i the followig form: p v p v p3 v3 p v M. () M ad matrix Uig the matrix determie the followig polyomial: det Z M A A Z allow u to A A A. () To calculate the value of active force, the obtaied polyomial hould be divided by the coefficiet A, ad the compared to the polyomial characterizig the adopted dyamical propertie of the reoace ad free vibratio drop coefficiet for elected frequecie. The preeted equatio tae the followig form depedig o the cae uder coideratio, whe a active force i applied: det det det Z M h h A, () i Z M A Z i M A i, (3) h h. (4) Oce the coefficiet foud at the ame polyomial expoet Eq. ()-(4) are compared, we ca determie the parameter of the active force reducig the idetified ytem vibratio. The earlier paper of the author [4-6] preeted umerical calculatio of the active force applied i the torioally ad logitudially ocillatig idetified mechaical ytem. The obtaied ytem preeted i Fig. i a model of the reduced ytem for the firt haft axi ad cotitute a tartig poit for creatig a drive ytem with couplig toothed gear (primary ytem). The ue of the revere reductio rule, ayig that the ietic ad potetial eergy value of the reduced ad primary ytem are the ame, doe ot provide a uamguou olutio. To do thi, we itroduce a additioal coditio ayig that all ubytem of the primary ytem hould meet the deired dyamical requiremet i the form of elected reoace value. The ythei of the primary ytem begi from buildig a tramittace matrix: Y Y Y Y Y 3 Y Y3 Y Y Y3 Y Y3 Y33 Y3 Y Y Y 3, (5) 86 where Y X F, i,, ; j,, ij are i j operator tramittace betwee i iput ad j iput ad are equal to the proportio of the i repoe traform to the j forcig traform to be ued to determie zeroe for the dyamical characteritic decrig the dyamical propertie of primary ytem. The the ytem i diegaged to create two ubytem i relatio to the firt reduced elemet z ad the reoace frequecie of the diegaged ubytem are determied i relatio to the firt reduced ma. By taig the highet value from the equece of the pre-et reoace frequecy value ad from the equece of the ati-reoace value determied, it i poible to cotruct the dyamical characteritic decrig the propertie of the earched primary ubytem at the firt axi. Such a fuctio i ubject to ythei by uig the method of characteritic ditributio o a cotiued fractio to determie the ytem parameter before performig the reductio at it haft with tiffe c, i the followig form: U b H H Hc z H, (6) where i max ati-reoace frequecy determied for z ytem i relatio to z,, are the value of iertial elemet of the ubytem obtaied, c i the value of elatic elemet. For the value obtaied it i eceary to elect the cotat of proportioality H to meet the followig requiremet: H H. (7) The ythei performed reult i obtaiig a ytem tructure taig ito accout the firt diegagemet preeted i Fig. 4. Fig. 4 Model of a drive ytem taig ito accout the firt diegagemet Further ythei ued the ubytem of the earched drive ytem reduced at the haft with tiffe of c, obtaied a a reult of deigig the tructure ad parameter of the eparated primary ubytem with tiffe of c (Fig. 4). The parameter of the ubytem uder aalyi tae the followig value: c c z H, c,,c,. (8) z At the later tage of the ythei, the obtaied ubytem i diegaged i relatio to the ubequet iertial elemet reduced ad the ubytem ati-reoace frequecie are determied. After aumig the dyamical z c -z -z z

5 87 propertie of the ytem i the ame way a for the egagemet z i relatio to the iertial elemet decribed above, the dyamical characteritic ditributed o a cotiued equatio i created, taig the followig form: where U b H H 3 Hc H H i the cotat of proportioality,, (9) i max ati-reoace frequecy determied for ubytem i relatio to, 3, are the value of iertial elemet of the ubytem obtaied, c i the value of elatic elemet. Thi tep of the ythei reult i obtaiig the ytem tructure by taig ito accout the firt ad ecod diegagemet preeted i Fig. 5. elatic elemet: c ; c r c r cz i ;, c r c z i. Fially, we obtai the drive ytem before reductio preeted i Fig. 6, meetig the deired propertie i the form of reoace frequecie aumed i advace ad ratio for idividual drive ytem tage. r c r r c r r 3 r c 3 r - r c -r c - r r 3 c -z Fig. 6 Model of a drive ytem taig ito accout all couplig Fig. 5 Model of a drive ytem taig ito accout the firt ad ecod diegagemet O the other had, the ubytem of the drive ytem reduced at the haft with tiffe of c 3 tae the followig value: H,c,,c,. () z The above-decribed proce cotiue util performig the diegagemet i relatio to the -z elemet. Whe ytheizig drive ytem with couplig, it i eceary to aume the requiremet regardig the total ytem tramiio ratio. Thi ratio i the product of elected igle tramiio ratio at each gear uit tage expreed by the followig formula: i i,i i i, () where i i total tramiio ratio, i i, i+ i ratio for ubequet gear uit tage. To obtai the real parameter of the ytheized toothed gear it i eceary to tae ito accout (3). It ca be doe by multiplyig the drive ytem parameter obtaied by the elected ratio at idividual tage (it refer to the cae, whe the ytem i reduced at the drivehaft), i the followig form: iertial elemet: r ; r 3 r 3 i3 r z ; i. ; ; r i ; r z i, 3 -z z 3 r 3 i ; r z i, The ythei reult i obtaiig a machie drive ytem taig ito accout couplig i the form of a multi-tage gear uit. The baic property of the ytem model created ad it ubytem i arrowig dow the critical (reoace) frequecie to the umber aumed at the begiig of the ta. 3. Numerical example Numerical calculatio ued to determie the parameter ad tructure of the ytem model i the form of two-tage gear uit are preeted. The iput value, i order to obtai the tructure ad parameter of the ytem, coited of: the reoat ad ati-reoat frequecie (pole ad zero of the deired dyamic characteritic) i the form of: b rad/, b 5 rad/, b3 rad/, b4 5 rad/, z 5 rad/, z 75 rad/, z3 5 rad/ ; vibratio frequecy drop parameter h for the elected reoace frequecie i the form of: h b rad/, h b3 rad/, h b4 rad/ ; the parameter of hiftig the vibratio frequecie Δω for the elected reoace frequecie i the form of: b 5 rad/, b3 5 rad/, b4 5 rad/ ; which were ued to decribe the dyamic propertie of the dicrete vibratig ytem i the form of dyamical flexility: Y, ()

6 Y Y, (3) Y, (4), (5) ad the total gear ratio of the ytheied gear: 4 i 3 i i3. (6) The characteritic fuctio Eq. ()-(5) were ued to determie the tructure ad et of parameter i accordace with the formal arragemet preeted i the previou chapter. The reult of the variou tage of drive ytem ythei are ummaried i Table ad Table. 3 The reult of the variou tage of drive ytem ythei Cacade model of a drive ytem Model of a drive ytem taig ito accout the firt diegagemet, gm c,, gm c, c 35 c 35 z..4 c z 8 c z z 4.6 c z 4.6 c 866 Model of a drive ytem taig ito accout the Model of a drive ytem taig ito accout all firt ad ecod diegagemet couplig, gm c,, gm c, c 35 r.4 4 r.4 c r 35.7 c z 8 r r.3 c r c r.4 4z 4.6 4r.5 c 3r Table The value of cotrollig force parameter value p, v, p, v, p3, v3, p4, v4, Y() Y() Y3() Table 4. Cocluio Thi paper preet the problem of idetifyig the torioally vibratig dicrete ytem, coidered a machie ubaemblie featurig deired mechaical propertie. The preeted idetificatio method mae it poible to obtai the parameter ad tructure of o-reduced machie drive ytem o accout of the aumed reoace ad ati-reoace value ad the tramiio ratio. I additio, the paper preet the method of active vibratio reductio of the idetified machie drive ytem. The aim of idetifyig a active ytem i to eure that the ytem meet the ey operatio requiremet. The baic criterio here i the reductio ad elimiatio of vibratio ear the locatio i the ytem where the reoace origiate. The idetified tructure of multi-tage toothed gear uit i ot the oly model featurig the pre-et dyamical propertie. Uig the preeted method mae it poible to obtai ot oly the cacade-tructured ytem, but alo the brached ad brached-cacade tructure. The tructure obtaied i uch a way ad parameter value ca be ued to elect a optimum drive ytem. Further reearch o ytheizig drive ytem will focu o fidig the criterio for chooig appropriate drive ytem obtaied a a reult of the ythei. Referece. Dzitowi, T.; Dymare, A. 5. The ythei of machie drivig ytem. th Iteratioal Scietific ad Egieerig Coferece: Machie-buildig ad techophere o the border of the XXI Cetury 5: 66-7.

7 89. Dzitowi, T.; Dymare, A. 9. Sythei ad eitivity of multiaxial drive ytem, Acta Mechaica et Automatica 3(4): Dzitowi, T.; Dymare, A.. Active ythei of multiaxial drive ytem uig a comparative method, oural of Achievemet i Material ad Maufacturig Egieerig 49(): Dzitowi, T.; Dymare, A.. Active ythei of machie drive ytem uig a comparative method, oural of Vibroegieerig 4(): Dymare, A.; Dzitowi, T. 3. Paive reductio of ytem vibratio to the deired amplitude value, oural of Vibroegieerig 5(3): Dymare, A.; Dzitowi, T. 3. Active ythei of dicrete ytem a a tool for reductio vibratio, Solid State Pheomea 98: Dzitowi, T.; Dymare, A.. Sythei of machie drive ytem a a tool of deigig multitage gear with required frequecy pectrum. [w:] Proceed. of the 9th Iteratioal Coferece o Vibratio Egieerig ad Techology of Machiery: Dzitowi, T.; Dymare, A. 3. Active ythei of machie drive ytem, Applied Mechaic ad Material 43: Craig,.. 5. Itroductio to Robotic: Mechaic ad Cotrol. Pearo Educatio. 4p.. Marchele, K. 99. Dyamic of Machie Tool. Warzawa: WNT. 53p (i Polih).. Męży, A.. Optimizatio of Dyamic Propertie of Machie Drivig Sytem. Gliwice: Sileia Uiverity of Techology Pre. 3p (i Polih).. Par,.S.; Kim,.S Dyamic ytem ythei i term of bod graph prototype, KSME Iteratioal oural. (3): Rao,. S.. Hitory of Rotatig Machiery Dyamic. Spriger Netherlad. 358p Świtońi, E.; et al. 4. The Modellig of Mechatroic Driver Sytem, Gliwice: Sileia Uiverity of Techology Pre. 6p (i Polih). 5. Świtońi, E.; Męży, A. 8. Selectio of optimum dyamic feature for mechatroic drive ytem, Automatio ad Cotructio 7(3): Redfield, R.C.; Kriha, S Dyamic ytem ythei with a bod graph approach, part I: Sythei of oe-port impedace,. Dy. Sy., Mea. Cotrol 5(3): Wu,.S.; Che, D.W.. Torioal vibratio aalyi of gear-brached ytem by Fiite Elemet Method, oural of Soud ad Vibratio 4(): T. Dzitowi, A. Dymare MAŠINŲ PAVAROE NUSTATYTŲ MECHANINIŲ VIRPESIŲ AKTYVUS SLOPINIMAS DAUGIAPAKOPĖSE KRUMPLIARATINĖSE PAVAROSE R e z i u m ė Straipyje aalizuojama mašių pavaroje utatytų mechaiių virpeių atyvu lopiima daugiapaopėe rumpliaratiėe pavaroe. Slopiimo metoda pagrįta atyvia iteze leidžiačia paieti orimą mechaiį efetą teiigai pareat itemo diamie avybe. Norimo diamiė avybė upratamo, aip rezoaų ir atirezoaų dažių ea diamiių charateritių formoje. Toiu būdu parito avybė yra išeitie taša diretiių virpačių itemų parametriiam ir trutūriiam idetifiavimui. T. Dzitowi, A. Dymare ACTIVE REDUCTION OF IDENTIFIED MACHINE DRIVE SYSTEM VIBRATIONS IN THE FORM OF MULTI-STAGE GEAR UNITS S u m m a r y The paper preet the problem of vibratio reductio i the idetified machie drive ytem i the form of igle or multi-tage gear uit. The reductio method wa baed o uig a active ythei maig it poible to obtai a deired mechaical effect by properly electig the dyamical ytem propertie. The deired dyamical propertie are udertood throughout the paper a a equece of reoace ad ati-reoace frequecie i the form of dyamical characteritic. The propertie aumed i uch a way cotitute a tartig poit for the parametric ad tructural idetificatio of dicrete ocillatig ytem. Keyword: mechaical impedace, mechaical admittace, reoat ad ati-reoat frequecie. Received Mai 9, 3 Accepted March, 4