Study on Nonlinear Dynamic Characteristics of 2K-H Planetary Gear System

Transcription

1 16 Internatonal Conference on Materals, Informaton, Mechancal, Electronc and Computer Engneerng (MIMECE 16) ISBN: Study on Nonlnear Dynamc Characterstcs of K-H Planetary Gear System Yaoman Zhang, Xanzhan Zhu and Shuxan Ja ABSTRACT A K-H type planetary gear reducer wth nvolute spur gear s adopted as the research object. Consderng the tme-varyng meshng stffness, the tme-varyng backlash of the gear meshng and the tme varyng pressure angle and other nonlnear factors, the lateral-torsonal-swng coupled nonlnear dynamc model of the planetary gear transmsson system s establshed. Accordng to the equatons of Lagrange, the dynamc dfferental equatons are derved. By ntroducng the state varable, the second-order dfferental equaton of the coupled system s reduced to the frst-order dfferental equaton. The ffth-order step-varyng adaptve Runge-Kutta numercal ntegraton method s used to solve the dynamc equaton. Compared wth the prevous dynamc models, ths nonlnear dynamc model for the planetary gear transmsson system s more closely to the practcal operatng condtons of the transmsson system, whch has lad a foundaton for further evaluatng the dynamc characterstcs of the planetary gear transmsson system n the future. INTRODUCTION The planetary gear system s not only wdely appled to the automotve and vessels feld, but also more and more popular among the CNC machne tools due to ts excellent propertes ncludng compact structure, large transmsson power and hgh bearng capacty[1]. A large number of meanngful researches on the dynamc characterstcs of the planetary gear transmsson system are carred out by many domestc and foregn scholars. Consderng the meshng stffness fluctuaton and the meshng frequency errors, the dynamc model for K-H planetary gear reducer system was establshed and solutons of the tme and frequency domans was obtaned by Zong-de Fang[]. The pure torson model for compound planetary gear system on the bass of the lumped mass method was frstly establshed and the natural characterstcs of the system was studed by Kahraman A[]. 1 The adaptve step-varyng gll ntegral Yaoman Zhang, Xanzhan Zhu, Shuxan Ja, School of Mechancal Engneerng & Automaton, Northeastern Unversty, Shenyang, Laonng, 11819, P.R. Chna 7

2 method was used to solve several steady-state forced responses ncludng harmonc and nonharmonc sngle cycle, subharmonc, quas-perodcty and chaos of the planetary gear system under vares parameters by Zhmn Sun[4]. Based on torsonalvbratory models, nfluences of the rotate speed and gear clearance on the bfurcaton characterstcs of the planetary gear system was studed wth the numercal ntegral method by Tongje L[5]. Consderng several factors ncludng the frcton, tmevaryng mesh stffness, gear clearance and comprehensve meshng error, the dynamc model was establshed and solved by step-varyng Gll method and the dynamc responses of frcton between gear tooth surfaces was obtaned by Enyong Zhu[6]. Consderng the effects of gear translaton movement on the pressure angle and contact rato, the lateral-torsonal dynamc model to study the dynamc propertes of the gear system was establshed by Woohyung Km[7]. Nonlnear propertes of the planetary gear transmsson system such as chaos and bfurcaton were studed by and dynamc responses was obtaned wth the lumped mass and fnte element methods by Robert G. Parker[8]. In summary, a seres of studes on the planetary gear transmsson system dynamcs have been conducted by a number of scholars at home and abroad whch has formed a relatvely complete system of the dynamc theory. However, most of models establshed by scholars are purely torsonal or translatonal-torsonal coupled dynamc models and the uneven stress along the tooth wdth was gnored. Based on prevous studes, consderng the tme-varyng meshng stffness, tme-varyng gear clearance, tme-varyng pressure angels and the swng of gears, the lateral-torsonal-swng coupled nonlnear dynamc model was bult. In addton, the effects of dfferent supportng stffness on dynamc responses of the system have been studed under the gven workng condtons. THE PLANETARY GEAR SYSTEM AND ITS SOLID MODEL The research object-k-h type planetary gear reducer wth nvolute spur gear conssts of a sun gear, a rng gear, a carrer and three planetary gears. The geometrc parameters of the system are shown n Table I. The three-dmensonal model s establshed accordng to these geometrc parameters n Table I, and ts sold model was shown n the Fgure 1. 8

3 Fgure 1. The three-dmensonal model of K-H planetary gear system. Table I. Geometrc parameters of the planetary gear transmsson system. Modulus(mm) Number of teeth Tooth wdth(mm) Pressure angle( ) Sun gear Planetary gear Rng gear THE ESTABLISHMENT OF NONLINEAR DYNAMIC MODEL The rng gear belongs to the fxed part whch has no freedom, whle the other components n the system all have fve degrees of freedom ncludng rotatonal, translatonal and swng freedom. Consderng the deformaton of bearngs and the uneven stress along the tooth wdth, the smplfed dynamc model was shown n Fgure. WhereOxyz was a fxed coordnate wth the center of planetary gear transmsson system as the orgn and at the same tme, O was a follow-up coordnate wth the orgn concded wth the axes of the th planetary gear and fxed wth the carrer. The axes of three follow-up coordnates always stay parallel wth the correspondng axes of fxed coordnates. Wth the fve degrees of freedom of the sun gear and planetary gear represented by symbols x,y, k k kx, ky, kz ( k c,s), respectvely and the fve degrees of freedom of the th planetary gear represented by symbols,, p, p, p ( 1,, ),the generalzed dsplacement array q for the system can be expressed as q xs, ys, sx, sy, sz, xc, yc, cx, cy, cz, 1, 1, p 1c, p 1c, p 1c T,, pc, pc, pc,,, pc, pc, pc 9

4 Fgure. The smplfed dynamc model of K-H planetary gear system. The Establshment of Dynamc Equatons The dynamc equatons of the system can be derved from the Lagrange equatons wth the knetc and potental energy expressed by the generalzed coordnate q. The dynamc equatons of the sun gear, the th planetary gear and carrer are derved n the same way, so takng the carrer as an example shown as equatons (1) to (5). mc mp sn xc m p r c cz cz cz cz 1 1 cz rc cos cz cz cz kcuxxc (1) f s b f s b ksp f s sp b krp f s rp b 1 xc xc r cos cz mc mp yc mp cz r c cz cz cz 1 1 sn k y c cz cz cz cu c f ssp b f srp b ksp f s sp b krp f s rp b 1 yc y c cx cx cz c cy pu pc 1 I I k () cy cy cz c cx pu pc 1 I I k (4) ()

5 I I m r m r ( r ) cz p p c cz p c c cz sn cos r x y x r y I c c c cz c c c cz p pc 1 k f s b m c, p f s b 1 cz cz Where respectvely, I j x, y, z cj f s b k f s b stands for the mass of carrer and th planetary gear, are the rotatonal nerta of carrer, k xcu (5), k ycu, k xcu, k ycu and kpu, k,, pu k pu k purepresent brace stffness of the component drecton, k and k representng the meshng stffness between the sun, rng and the th planetary s t s, r represents the dsplacement along the mesh drecton gear,respectvely, tp between the sun gear, rng gear and the th planetary gear, respectvely,, tp t s r stands for the tme-varyng pressure angle between the sun gear, rng gear and the th planetary gear, respectvely and b, b are the tme-varyng gear clearance between the th planetary gear and the sun gear, rng gear respectvely. The Determnaton of Nonlnear Parameters As shown n Fgure (a), the tme-varyng pressure angle and poston angle derved from the geometrc relatonshp between the sun and the th planetary gear can be expressed as equatons (6) to (7). (a)sun gear and the th planetary gear (b)rng and the th planetary gear Fgure. Gear mesh model. rbs rbp arccos 1,, xp xs yp ys yp y s arctan 1,, x x As well, the tme-varyng pressure angle and poston angle between the sun gear and the th planetary gear are depcted n Fgure (b). p s (6) (7) 1

6 rbr rbp arccos 1,, x y p p y p rp arctan 1,, (9) x Dsplacement along the lne of acton between the sun gear and the th planetary gear can be derved from the equatons of the tme-varyng pressure angle and the poston angle, shown as ssp r ( sz cz ) rbp cos sn bs p c xs xp r c cz sp (1) ys yp r cos cos c cz sp Smlarly, the dsplacement along the lne of acton between the rng gear and the th planetary gear can also be gven as srp r rbp cos sn brcz p c xp r c cz rp (11) yp r sn cos c cz rp f s b tp tp t s, r p s the nonlnear functon related to the clearance, shown as stp b tp stp >b tp f s b s b t s r stp b - tp stp < b tp tp tp tp < tp, In whch, b b rbs rbp nv nv b b rbr rbp nv nv SOLUTION AND RESULT ANALYSIS OF THE DYNAMIC EQUATIONS The dynamc equatons for the system are solved by the ffth-order step-varyng adaptve Runge-Kutta numercal ntegraton method whch s usually appled to solve the frst order dfferental equatons. However, the dynamc equatons are the secondorder equatons, so t s necessary to transform them nto the frst order dfferental equatons va state varables X expressed as T X x, x, x, x... x, x, x, x x, x, y, y,,,,,,, x, x, y, y,,,,,,, s s s s sx sx sy sy sz sz c c c c cx cx cy cy cz cz,,,,,,,,,,,,,,, pc 1 p1c p1c pc 1 pc 1 pc 1 pc pc,,,,,,,,,,,,, pc pc pc pc pc pc pc pc pc pc Expressons for the second dervatve of all the varables are derved va smultaneous equatons and reduced the order wth state varables ntroduced. Durng the process of the smulaton, key parameters n the planetary gear system are gven, such as base radus of the sun, rng and planetary gear can be expressed as r 8.46mm, r 5.74mm, r 1.14mm, respectvely, the radus of the carrer s bs br bp rc 1.5mm, the mass of all of the components can be expressed as ms.kg, m.6kg, m.4kg, the ntal gear clearance can be gven as b b 5μm, c p, T (8) (1)

7 5 and all of the moment of nertas can be ndcated as =8.9 1, I I cz = , Icx = I =.5 1 cy I sz 4 5, I p.7 1 and I p sx = I = sy = I = kg m p 5. 4, The ffth-order step-varyng adaptve Runge-Kutta numercal ntegraton method s adopted to solve the dynamc equatons and obtan the steady responses for the system wth the moment M 4Nm appled on the nput shaft of the sun gear. The lateral degrees of freedom s nduced by the deformaton of bearngs, so that the values of dynamc responses wth dfferent supportng stffness can be obtaned to study the effects of the supportng stffness on dynamc responses. Only the vbraton along the y drecton of the sun gear and carrer and the vbraton along the drecton of the frst planetary as well as ts correspondng phase plane portrat are shown when the value of supportng stffness for the sun gear, 8 planetary gear and carrer s. 1 N/m.The steady responses for the system are gven from Fgure 4 to Fgure 6. As shown from Fgure 4 to Fgure 6, compared wth the sun gear and carrer, the ampltude of the planetary s much larger and vbrates more serously. The ampltude of the planetary decreased slghtly wth the supportng stffness added up to 8. 1 N/m, whch s helpful to mprove the propertes of even loads. In practce, the value of the stffness for the planetary s not allowed too small, for the power s passed to the carrer va the planetary. As far as the sun gear and the carrer, the ncrease of the supportng stffness has lttle effect on the ampltude along y drecton, whle the perodcty of dynamc responses for the system s more obvous. The phase plane portrats of the sun and planetary gear are expressed as closed and roughly ellptcal curve bands wth some wdth, whch demonstrates the responses are 8 quas-perodc, when the supportng stffness s. 1 N/m. Compared wth the sun and planetary, phase plane portrat of the carrer s less regular, but the response s stll quas-perodc wth the loop curve and propertes of the tme-doman fgure. 1 x y s -. _ys(m=s) tme (s) x y s x 1-6 (a) Dsplacement-tme responses of sun gear (b) Velocty-dsplacement phase plane of sun gear Fgure 4. Smulaton results of sun n y drecton.

8 1.5 x y c _yc(m=s) tme (s) x 1 - y c x 1-6 (a) Dsplacement-tme responses of carrer (b) Velocty- dsplacement phase plane of carrer Fgure 5. Smulaton results of carrer n y drecton. x _1(m=s) tme (s) x x 1-6 (a) Dsplacement-tme responses (b) Velocty-dsplacement phase plane of of the frst planetary gear the frst planetary gear Fgure 6. Smulaton results of the frst planetary gear n drecton. 9 However, wth the supportng stffness ncreasng to. 1 N/m, the phase plane portrats of the planetary gear and carrer become more regular as quas-perodc responses, whle quas-perodc response of the sun gear changes nto the sngle-cycle smple harmonc vbraton response. The results above have shown a trend of the dynamc responses changng from the quas-perodcty to the perodcty for the system wth the ncreased stffness properly. As far as the feed system of the machne tool, although the quas-perodc vbraton can be defned, but t s not lkely easy to fnd the regular of the quas-perodc vbraton lke the perodc vbraton, whch means t s more helpful to elmnate the vbraton wth the quas-perodc vbraton converted nto the perodc vbraton n order to mprove the machnng precson of the machne tool. CONCLUSIONS In ths study, consderng the nonlnear factors ncludng the tme-varyng meshng stffness, tme-varyng gear clearance and tme-varyng pressure angles, etc, the lateral-torsonal-swng couplng nonlnear dynamc model for K-H type planetary gear reducer s establshed by the lumped mass method. Also, the effects of support stffness on dfferent dynamc responses wth Runge-Kutta numercal ntegraton method solved s explored, whch has lad a foundaton for further researchng the dynamc propertes n the future. 4

9 (1) Based on the prevous models, the model bult wth swng freedom nduced by the uneven stress along the tooth wdth, tme-varyng pressure angles and tme-varyng gear clearance nduced by lateral movement consdered, s more closely to practcal operatng condtons of transmsson system, whch are meanngful to further analyze dynamc characterstcs of planetary gear transmsson system n the future. () Dynamc equatons for the system are solved by the ffth-order step-varyng adaptve Runge-Kutta numercal ntegraton method and dynamc responses of dfferent supportng stffness for the system are compared wth each other both have a gudng sgnfcance to choose the proper bearng. ACKNOWLEDGEMENT Ths work was fnancally supported by the Natonal Natural Scence Foundaton of Chna(515759) and the Fundamental Research Funds for the Central Unverstes(N14141) and Project of Educaton Department of Laonng Provnce(L15184). REFERENCES 1. Ynmn Song, Ce Zhang. 6. Modfed Torsonal Model Development and Natural Characterstcs Analyss of K-H Epcyclc Gearng, J. Journal of Mechancal Engneerng, 5: Zongde Fang, Yunwen Shen, Zhendong Huang Analyss of Dynamc Behavour of K-H Planetary Gearng, J. Journal of Northwestern Polytechncal Unversty, 4: Kahraman A. 1. Free Torsonal Vbraton Characterstcs of Compound Planetary Gear Sets, J. Mechansm and Machne Theory, 6: Zhmn Sun, Lnhong J, Yunwen Shen.. Nonlnear Dynamcs of K-H Planetary Gear Tran, J. Journal of Tsnghua Unversty, 5: Tongje L, Rupeng Zhu. 11. Nonlnear Torsonal Vbraton Modelng and Bfurcaton Characterstc Study of a Planetary Gear Tran, J. Journal of Mechancal Engneerng, 1: Enyong Zhu, Shjng Wu, Mngxng Deng. 1. Wth the Frcton Nonlnear Dynamc Model of Planetary Gear Transmsson System, J. Journal of Vbraton and Shock, 8: Km W, Hong Hee Yoo. 1. Dynamc Analyss for a Par of Spur Gears Wth Translatonal Moton Due to Bearng Deformaton, J. Journal of Sound and Vbraton, 9: Vjaya Kumar Ambarsha, Robert G. Parker. 7. Nonlnear Dynamcs of Planetary Gears Usng Analytcal and Fnte Element Models, J. Journal of Sound and Vbraton, :