VIBRATION OF MULTI-STAGE GEAR DRIVES INFLUENCED BY NONLINEAR COUPLINGS

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1 ωt + ϕ ωt + ϕ ENOC-8, Sant Petersburg, Russa, June, July, 4 8 VIBRATION OF MULTI-STAGE GEAR DRIVES INFLUENCED BY NONLINEAR COUPLINGS Mroslav Byrtus Department of Mechancs Faculty of Appled Scences Unversty of West Bohema Czech Republc byrtus@kme.zcu.cz Vladmír Zeman Department of Mechancs Faculty of Appled Scences Unversty of West Bohema Czech Republc zemanv@kme.zcu.cz Abstract Ths paper deals wth mathematcal modellng of nonlnear vbraton of mult-stage large rotatng shaft systems wth gears and rollng-element bearngs. Gearng and bearng couplngs brng nto the system nonlnear phenomena lke mpact motons due to the possblty of the mesh nterrupton. The moton of the system s then nfluenced by the nternal knematc exctaton n gearng, by the parametrc exctaton caused by perodc change of number of teeth n gear meshng and can be further nfluenced by mpacts of freely rotatng gears whch transmt no statc torque. The developed methodology s tested for gear drve vbraton of a twostage gearbox. s = B B 4 k Key words Gear drves, vbraton, modal synthess method, nonlnear systems, bfurcaton, chaos. Introducton Large rotatng systems, especally gear drves and gearboxes occur as parts of many mechancal devces transmttng the torque wth relatvely small loss of power. But they represent man nternal exctaton sources of these systems. Ther vbraton analyses are commonly performed wth the assumpton of the small deformatons and lnearzed couplng forces. But ths assumpton s not correct for certan operatonal states when the nfluence of couplng nonlneartes s domnant. Therefore, the nonlnear models of gear and bearng couplngs are developed and the nfluence of couplng nonlneartes s nvestgated. In last years the subject of large rotatng system modellng has been studed n our department. The result of ths effort s a creaton of modal synthess method allowng to model systems wth complcated structures. The advantage of ths method s the separated modellng of subsystems usng dfferent modellng tools and separated modellng of dscrete couplngs. Each couplng s descrbed s = G G B B k Fgure. s = Scheme of two-stage gearbox. by geometrcal and materal propertes and by correspondng couplng force characterstcs. The man contrbuton of the modal synthess method conssts n the degrees of freedom number (DOF reducton, whch s based on the transformaton descrbed by a transformaton matrx assembled from chosen egenmode shapes of subsystems. The selecton of sutable egenmode shapes s mostly governed by the range of exctng force frequency spectrum. The fnal condensed mathematcal model has then sgnfcantly lower number of DOF than the orgnal mathematcal model and can be advantageously used for numercal nvestgaton of nonlnear effects caused by nonlnear couplng characterstcs or by parametrc exctaton n couplngs. Ths contrbuton s focused on modellng of a model

2 gearbox vbraton (fgure whch s supposed to be ncluded nto a drve between two wheels by means of dscrete torsonal couplngs characterzed by torsonal stffnesses k. The drvng and drven wheels rotate by the constant angular speeds ω and ω and the statc loadng of rotatng parts of the gearbox s defned by the ntal statc torsonal drve deformaton ϕ between wheels. The gearbox vbratons are excted by knematc transmsson errors of gear pars transmttng the power and by ther tme dependent meshng stffnesses.. Bearng model The bearng model, used here, respects real number of rollng elements unformly dstrbuted between the nner and outer race (see fgure. Then the vector fs B n equaton ( (model of rotatng parts can be expressed n followng form [Byrtus and Zeman, 5] f B s =,j ( t,j F,j + t ax,jf ax,j, ( Modellng of gearboxes Presented approach to the gearbox modellng uses the modal synthess method based on sutable system decomposton nto subsystems and on separate modellng of couplngs among subsystems.. Modellng of subsystems Generally, the subsystems rotatng wth angular velocty ω s are descrbed n local generalzed coordnates q s (t wth system of n s ordnary dfferental equatons n matrx form [Zeman and Hlavac, 995] M s q s (t + (B s + ω s G s q s (t + K s q s (t = = f E s (t + f B s + f G s, s =,,..., S, ( where M s, B s and K s are symmetrcal mass, dampng and stffness matrces of the uncoupled subsystems of order n s and G s s skew symmetrcal matrx of the gyroscopc effects of the same order for s =,,...,S only. These matrces are usually created by means of fnte element method combned wth dscrete parameters representng masses of rgd gear dscs. External forced exctaton s descrbed by vector fs E(t. Vector fb s expresses the couplng forces n rollng-element bearngs and vector fs G represents the forces n spur helcal gear couplngs. All force effects descrbed n vectors above are actng on the subsystem s. δ e, j y H, j ax ax F e, j H, j η F r, j ϑ ϑ v v z w u ψ S F S ϕ ψ ζ δ, j Fgure. Scheme of a bearng couplng y, ax ϕ x where vectors t,j, t ax,j descrbe the bearng geometry. Among bearng ndces belong only these whch correspond to bearngs coupled wth the subsystem s, s =,...,S. Smlarly, the vector fs B n equaton ( (model of a stator housng can be expressed n followng form fs B = (ẽ,j F,j + ẽ ax,j F,j ax, (,j where vectors ẽ,j, ẽ ax,j descrbe the geometry of contact ponts at the stator. Bearng ndces are governed by the same condtons as above. In the general coordnate space S q = [q T, qt,..., qt S ]T R n, n = n s (4 s= of the whole system the bearng model s descrbed by the global couplng vector n the form f B f B f B =. =,j f B S (c,j F,j + c ax,jf ax,j. (5 Vectors c,j and c ax,j descrbe global geometrcal propertes of each bearng contact pont j n bearng and have ths structure c,j = [ t T,j ẽt,j ]T, c ax,j = [ ( t ax,j T (ẽ ax,j T ]T (6. To stablze the numercal smulaton of the nonlnear model t s effcent to separate the lnear part of nonlnear bearng force characterstc. The lnear part s descrbed by stffness and dampng matrces n the general coordnate space (4 and the bearng force vector (5 can be then rewrtten n followng form f B = K B q B B q + (c,j f,j + c ax,jf,j ax, (7,j where K B and B B are global stffness and dampng bearng matrces. Ther structure depends on the number of rollng elements and on the nodal ponts to whch they are fxed on the shafts (for detals see [Zeman and Hajzman, 5]. The bearng dampng matrx s supposed to be proportonal to the stffness matrx B B = β B K B (8

3 and functons f,j (,j and f,j ax( ax,j respect the possblty of contact loss between the rollng-element j and the outer race n the bearng and depend on lnearzed rollng-elements stffness that s calculated n dependence on an external statc torsonal load of drvng and drven shafts [Byrtus and Zeman, 5].. Gearng model The force effect of the spur helcal gear couplng G z (fgure n ( s expressed by vector fs G = ± δ z, F z (t, d z, d z, (9 z where sgn - (mnus corresponds to drvng gear and sgn + (plus corresponds to drven gear (fgure. Vector δ z, = [..., δz, T,... ]T s the n s -dmensonal extended vector gven by geometrcal parameters α (normal pressure angle, β (angle of nclnaton of the teeth, γ (angle of poston vector, r (rollng radus of the gear. The drvng (drven gear s fxed on the shaft at the nodal pont (j. Detals are shown n [Zeman and Hlavac, 995]. The resultant force F z transmtted by gearng z can be approxmately expressed n the form F z (t, d z, d z = k z (td z + b z d z + f z (t, d z, ( where k z (t s tme dependent meshng stffness and b z s coeffcent of vscous dampng of gearng on gear mesh lne. Nonlnear functon f z (t, d z of gearng deformaton d z corrects the lnear elastc part of the force F z n the phases of the gear mesh nterrupton. z y j x Fgure. Gearng deformaton drven gear drvng gear A v w j T w j v j r R j j u r j u j Scheme of a gearng couplng d z (t = δ T z, q (t + δ z,j q j (t + z (t ( of gears n mesh fxed on shafts at nodal ponts and j expresses the relatve moton of theoretcal j contact pont of teeth on the gear mesh lne. Vectors q (t = [... u v w ϕ ϑ ψ... ] T and q j (t havng smlar form descrbe dsplacements of nodal ponts and j. The functon z (t, defnng knematc transmsson error of gearng z, can be expressed by Fourer seres K z (t = ( C z,k coskω zt + S z,k snkω zt, ( k= where ω z are meshng frequences. Analogous to the bearng model, we can express the global gear couplng vector n general coordnate space (4 n followng way f G f G f G =. Z. = c z F z (t,q, q + f G (t, ( f S G z= where f G (t s vector of nternal knematc exctaton generated n gear meshng that can be expressed n form Z ( f G (t = k z (t z (t + b z z (t c z. (4 z= The global vector of geometrcal parameters of the gearng z n general coordnate space (4 has followng structure c z = [ δ T z, δt z,j T ] T. (5 For the same reason as above, t s effcent to separate up the lnear part of nonlnear gearng force characterstc. Equaton ( can be then rewrtten to ths form Z f G = K G q B G q + c z f z (t,q + f G (t, (6 z= where K G and B G are global lnearzed stffness and dampng matrces of gear couplngs whose structure s descrbed n detal n [Zeman and Hajzman, 5]..4 Parametrc exctaton n gear couplng The behavour of the gearng s moreover nfluenced by the perodc tme-varyng meshng stffness whose mathematcal model has come through a longtme progress. In the frst contrbutons publshed n the fftes the meshng stffness was consdered to be constant equal to the mean meshng stffness. Because of the change of tooth par stffness durng the mesh, dfferent gearng models were developed. The stffness of a partcular tooth par s consdered to be perodc. The perod depends on the duraton of one tooth par mesh. It s nfluenced by the tooth profle, profle error, gear contact rato and lubrcant propertes n gearng. Partcular courses of the mesh stffness can be expressed n a analytcal way. Authors [Ca and Hayash, 994] proposed the gear mesh stffness k p (t for a tooth par p n the form k p (t = k m K p (t for t t p, t p + ε γ T, k p (t = otherwse (7

4 kz/km Fgure 4. rato ε γ. ε γ =.8 ε γ =. ε γ =.8 k z,.5.5 ξ (path of contact pont.5.5 ε γ = Relatve gear mesh stffness for dfferent values of contact where (.8 K p (t = (ε γ T (t t p +.8 ε γ T (t t p +.55 (8 dependng on the contact rato ε γ and on the perod of the gear mesh. The parameter k m represents maxmal value of the gear mesh stffness of one tooth par on the assumpton that at tme t = t p teeth enter nto mesh and at t = t p + ε γ T get out of the mesh. The gear mesh perod s then equal to T = (π/(p z ω, where parameter p z ndcates the number of the teeth of the drve gear mounted on a shaft rotatng wth angular velocty ω. Fgure 4 shows the dependance of the relatve meshng stffness k z /k m on the path of the contact pont n the front plane of the gearng for dfferent values of contact rato ε γ. The parameter k z, corresponds to the mean meshng stffness. The bold lnes dsplay the resultng gear mesh stffness, whch s gven as a sum of the stffnesses of teeth beng n the gear mesh for a gven tme. The resultng meshng stffness can be expressed n followng form k z (t = k p (t, (9 p where ndex p s restrcted to the tooth pars beng n gear mesh for the gven tme t..5 Condensed mathematcal model It s advantageous to assemble condensed mathematcal model of the system wth reduced degrees of freedom (DOF number, because, n partcular, the housng subsystem could have too large DOF number and ths can hnder from consecutve performng of varous dynamcal analyses. The modal transformatons q s (t = m V s x s (t, s =,,..., S, ( are ntroduced for ths purpose. Matrces m V s R ns,ms are modal submatrces obtaned from modal analyss of the mutually uncoupled, undamped and non-rotatng subsystems, whereas m s (m s n s s the number of the chosen master modes of vbraton. The new confguraton space of the dmenson m s defned by vector S x = [x T, xt,..., xt S ]T, m = m s. ( s= The model ( can be then rewrtten usng terms (7 and (6 n the global condensed form ( ẍ(t + B + ω G + V T (B B + B G V ẋ(t+ ( + Λ + V T (K B + K G V x(t = = V T( (c,j f,j (q + c ax,jf,j ax (q+ j Z + c z f z (t,q + f G (t + f E (t, z= ( wheref E (t = [ (f E (t T, (f E (t T,..., (f E S (tt ] T s the global vector of external exctaton, B = dag ( m Vs T B m s V s, ( ωs G = dag m Vs T ω G s m V s, V = dag ( m V s ( are block dagonal matrces (ω S = holds for the stator subsystem and Λ = dag ( m Λ s s dagonal matrx assembled of spectral submatrces m Λ s R ms,ms of the subsystems. Analyss of two-stage gearbox vbraton The mathematcal model of gear drve s strongly nonlnear due to the possblty of gear mesh nterrupton and n consequence of nonlnear bearng couplngs respectng loss of contact n some contact ponts n dependance on poston of journal centre. To perform the dynamcal analyss the condensed mathematcal model ( has to be transformed nto the state space to use the tme ntegraton method. The tme ntegraton s started from the ntal state x( = (Λ + V T (K B + K G V V T f E (, ẋ( = ω (4 to mnmze the startup transent motons. Vector ω expresses the ntal angular speed of shafts. In general, the vector f E ( can descrbe an arbtrary external exctaton at the start of numercal ntegraton. The presented methodology was appled to the twostage gearbox (fgure whch can be dsassembled nto three subsystems drve shaft wth gears (s =,

5 drven shaft wth gears (s = and housng (s = wred n selected nodal ponts wth the fxed frame. Subsystems are joned by dscrete couplngs gear meshngs G z (z =, and rollng-element bearngs (B to B 4 consderng twenty rollng-elements n each bearng. The ntal number of DOF of the uncoupled subsystems after dscretzaton by FEM usng MAT- LAB code for rotatng subsystems and software package ANSYS for housng was n = 9 (drve shaft, n = 9 (drven shaft and n 5 (housng. Numercal experments show that the reduced (condensed model ( of the complete system of the order m = 6 (m =, m =, m = s acceptable n the frequency range up to 5 Hz. The gearbox vbraton was nvestgated for the frst transmsson degree, when gearng G transmts the power. The second gear par rotates freely about the deformed shaft axles. The vbraton s caused by the nternal knematc exctaton n the form (4 for z = and by the tme-varyng change of meshng stffness k (t n the form (8 and (9. Only three frst ampltudes of the Fourer seres ( were taken nto account and have followng values S, = 5 6 m, S, = S,, S, = S,, C, = C, = C, =. (5. Constant gear mesh The lnearzed condensed model ( for zero nonlnear functons (f,j (q, f,j ax(q and f z(t,q was used for determnaton of the constant gear mesh regons. Usng the lnearzaton we neglect the possblty of gear mesh nterrupton and we consder constant stffness gven by statc loadng of each rollngelement n all bearngs. Supposng the nternal knematc exctaton n gearng defned by (, (4 and (5, we can nvestgate for whch operatonal parameters revolutons per mnute and statc load the gear mesh s constant and we can fnd a boundary of nterrupted gear mesh. From knowledge of the tme response of the system n confguraton space x(t we can formulate the condton of constant gear mesh z mn d z(t = mn t T t T { ct z Vx(t + z(t} >. (6 Ths condton generally holds for both lnearzed and full nonlnear model of the gearbox. Fgure 5 shows area of nterrupted gear mesh (whte area and of constant gear mesh (grey area of the frst gear mesh transmttng the power n dependence on two operatonal parameters on revolutons per mnute of the drvng shaft and on external statc preload actng on rotatng dsks mounted to drve and drven shafts. Results were ganed from the lnearzed model for steady state moton. The steady gearng deformatons are then expressed by Fourer seres wth above mentoned harmonc components wth the fundamental frequency equal to the meshng frequency. There are statc deformaton ϕ [rad] e 6 e 5 e 5 e 5 5e 6 e 5 e 5 e 5 5e 6 5e 6 e 5 e 5e 6 e 5 5e 6 5e 6 5e 6 e 5 5e e 6 5e 6 revolutons per mnute 5e 6 5e 6 Fgure 5. Map of constant gear mesh of frst gearng G. many peaks correspondng to resonance states of the gearbox on the boundary between constant and nterrupted gear mesh. The gear box shows greater dsplacements and thence greater gearng deformatons and greater tendency towards gear mesh nterrupton n all resonance states.. Nonlnear two-stage gearbox vbraton Further, we are concerned wth the qualtatve analyss of the two-stage gearbox nonlnear vbraton, whch s nfluenced by gear mesh nterrupton and tme-varyng meshng stffness. The nonlnear model ncludes such nonlnear phenomena lke mpacts n gearng and nonlnear contact forces transmtted by rollng-elements of bearngs. These phenomena, whch are shown for systems wth several DOF number n [Thomson, ], are sources of nonlnear effects n soluton of the model. The tme response of such a system s accompaned by bfurcaton of soluton n dependence on chosen operatonal parameters. Vbro-mpact systems are characterzed by perod doublng scenaro, when the perod number of the tme response ncreases unexpectedly twce for a certan values of operatonal parameters. Ths scenaro could repeat tll the moton becomes chaotc. Moreover, the tme response of nonlnear model s nfluenced by mpacts n gear meshngs wth one freely rotatng gear. Here, we nvestgate the nfluence of the phase shft between the knematc and parametrc exctaton n gearng G. All results ntroduced here are ganed for the condensaton level of the gearbox and for knematc transmsson errors mentoned n the chapter before. Fgure 6 shows bfurcaton dagrams of gearng deformaton of gearng G for a chosen revoluton range of the gearbox, whch s nscrbed wth the bold black lne n fgure 5. The correspondng statcal external preload s characterzed by torsonal doformaton ϕ =. rad. In each bfurcaton dagram a lne desgnatng the zero gearng deformaton s plotted. Each dot plotted under ths lne corresponds to gear mesh nterrupton durng one perod of moton. Grey dots agree wth local maxma of gearng deformaton and the black ones agree wth local mnma, respectvely. In the chosen operatonal area exst perodcal solutons

6 gearng deformaton [m] Fgure 6. Bfurcaton dagrams of the gearng deformaton G. gearng deformaton [m].5.5 x 4 constant gear mesh varyng gear mesh no shft varyng gear mesh shft π/ Fgure 7. x 5 mesh G. tme [s] constant gear mesh varyng gear mesh no shft varyng gear mesh shft π/ tme [s] Gearng deformaton of the freely rotatng gear n gear whch follow the perod doublng scenaro. Ths nonlnear phenomenon mostly accompanes the crossng of the area of constant gear mesh, as [Byrtus and Zeman, 5] shows. Snce the tme course of the gearng deformaton has a quas-perodcal structure, t s very dffcult to detect ponts of perod doublng n the dagram 6. The three dagrams showed here, correspond to constant meshng stffness (above, tme-varyng meshng stffness wth no shft between the parametrc and knematc exctaton (n the mddle and tme-varyng meshng stffness wth a phase shft equal to π/ between the parametrc and knematc exctaton (below. Fgure 7 shows gearng deformaton n gear mesh G wth the freely rotatng gear for the three above mentoned states and for rpm (above, characterzed by perodcal response and rpm (below, characterzed by chaotc response. The deformatons are bounded by the gearng clearance showed by the two bold lnes. 4 Concluson The paper descrbes the methodology of the large coupled rotatng systems modellng and the analyss of ther nonlnear vbratons. The models of these systems suppose a flexble shafts and stator and nonlnear gear couplngs between rotor subsystems and nonlnear rollng-element bearngs. The whole system model s created by means of the modal synthess method whch allows to reduce number of degrees of freedom of the mathematcal model. The methodology s appled to the two-stage gearbox to smulate the dynamc response caused by knematc transmsson error of gears and by tme-varyng meshng stffnesses. Vbraton s accompaned by nonlnear phenomena lke bfurcaton, perodc and quas-perodc solutons and chaos. These modes of motons are very nterestng from the theoretcal pont of vew and for gearbox desgn from the dynamc loadng and nose pont of vew. Ths work was supported by the research project MSM of the Mnstry of Educaton, Youth and Sports of the Czech Republc and by the Research Project number A-TP/9 of Mnstry of Industry and Trade of the Czech Republc. References Byrtus, M. and Zeman, V. (5 Modellng and analyses of gear drve nonlnear vbraton. In Proc. of the Ffth EUROMECH Nonlnear Dynamcs Conference, Endhoven, August 7-, pp (full text on cdrom. Ca, Y. and Hayash, T. (994 The lnear approxmated equaton of vbraton of a par of spur gears (Theory and experment. In Journal of Mechancal Desgn, 6, pp Thompson, J. M. and Stewart, H. B. ( Nonlnear dynamcs and chaos. John Wley & Sons, Chchester. Zeman, V. (994 Dynamcs of rotatng machnes by the modal synthess method. In Proceedngs of the Tenth World Congress IFToMM, Oulu, Fnland, June 4, pp Zeman, V. and Hajžman, M. (5 Modellng of shaft system vbraton wth gears and rollng element bearngs. In Proc. of Colloquum Dynamcs of Machnes 5, Prague, February 8-9, pp Zeman, V. and Hlaváč, Z. (995 Mathematcal modellng of vbraton of gear transmssons by modal synthess method. In Proceedngs of the Nnth World Congress on the Theory of Machnes and Mechansms, Vol., pp. 97-4, Poltechnco d Mlano, Italy.