Dynamic Analysis of Single-Stage Planetary Gearings by the FE Approach

Transcription

1 Dynamic Analysis o Single-Stage Planetary Gearings by the FE Aroach Kuo Jao Huang a, Shou Ren Zhang b, Jui Tang Tseng c a,b Deartment o Mechanical Engineering, Chung Hua University c Wind Energy Equiment Technology Det. Mechanical and System Research Lab., TR a kjhuang@chu.edu.tw Abstract. This study rooses an aroach to analying modal behavior o single-stage lanetary gear systems using a general-urose inite element (FE) ackage. At irst, using the derived tooth roile equations o gears generated by a rack cutter, high quality gear elements are arametrically and automatically created. Then, the dynamic models o the gear systems can be constructed ater assigning boundary and gear contact conditions. Next, the structural natural requencies and modal shaes are obtained by the ackage LS-DYNA. Besides, dimensionless sloe is used to evaluate the eect o systematical suorting bearing stinesses on gearing modal characteristics. Both the modal roerties o lanetary gearings with and without meshing hase dierences are included. The roosal aroach may beneit dynamic investigation on wide tyes o lanetary gearings. Keywords: Modal Analysis, Natural Frequency, Finite Element, Planetary Gearing, Stiness, Young s Modulus. ntroduction Planetary gearings have broadly alied in wide industrial ields owing to their excellent eatures including high ower density and low noise and vibration esecially in the vehicle, aircrat, wind turbine, robot industries. n order to urther enhance gear erormance to meet severer alications, lanetary gearing dynamics has become essential research toics or amore than three decades [-3]. Using a three dimensional discrete model, Kahraman [4] abundantly investigated on helical lanetary gear dynamics in which the author categoried lanet hasing conditions and calculated modal shaes and dynamic orces caused by static undulation transmission errors in lanetary gearings. However, the author ointed out the categoriation is rimarily or comatibility and it may be misleading in 3D models. According to the later work o Velex and Flamand [5], they resented the stiness o meshing gear airs aect lanetary gear dynamics than the stinesses o shats, sun and ring gears, and bearings. Recently, lentiul dynamic results relevant to lanetary gear were contributed by Parker el al [6-7] which including investigation on lanet meshing hase and its inluence on liner or nonlinear gear dynamics or three- and our-lanets gearings o equally sacing and diametrically symmetrical with or without lanet meshing hase dierence (MPD). For examles, Lin and Parker [6] calculated natural requencies o lanetary gear systems. Also, the nonlinearity due to meshing stiness discontinuity o gear airs was discussed. Furthermore, in the study o [7], the authors discussed the natural requencies o lanetary gearing and their reetition number. The vibration modes were classiied into three tyes including the rotational, translational and lanet modes. Beside o single-stage lanetary gearings, aarently the dynamic investigation o multistage lanetary gearings is much more comlicated. The works [9, 0] might be the only two about that. Using the equivalent discrete models to gear dynamics may beneit model simliication. However, due to comlexities o structure and tooth roiles, erhas, it may state that discrete models can only be alied in very limited tyes o lanetary gearings under over simliied design and oerating considerations. With advancement o comuter and comuting technologies, the continuum aroach using the FE analysis was adoting in gear dynamics. Two recent studies [, ], the FE continuum aroach was alied in analying gear dynamic analyses. However, using commercialied FE ackages to analye modal behaviors o lanetary gearings is still very limited. Thereore, in this study, using a general uroses FE ackage LS-DYNA, a dynamic aroach to analying modal characteristics o general lanetary gearings is resented. n addition, inluences o bearing stinesses are also discussed.. Equations o Rack Cutter The roile o involute gear teeth is tyically comosed o three arts which are addendum chamers, working involutes, and illets, those can resectively be generated by the two straight lines and an addendum circular curve o a rack, a hob, or a grinding disk. Firstly, as deicted in Fig., deine two coordinate systems S n and S, which resectively arallel to a 7

2 normal and a transverse cross sections o the rack cutter. Then, the transverse cross section roile reresented in coordinate system S is illustrated in Fig. in which x is collinear to the middle line o the rack cutter and y is coincident to the standard itch line o the cutter. As shown in Fig., the transverse cross section roile is comosed o two straight lines, and, and an addendum ellitic curve. M 0 is the intersection o the straight and the ellitic curve, and M is the intersection o the two straight lines. Accordingly, the osition vector o a transverse cross section roile or the rack cutter in a homogenous coordinate orm is written as T x y R () Thus, the transverse cross section o the rack in S is derived as ollows: (a) Proile equations o straight line backlash. Finally, E c, used to incororating the crowning modiication. (b) Proile equations o straight line cosar cos r g x y ( sin a tan b sin ) S tan (5) As shown in Fig., is distance MM 0, distance rom M to M 3, which is an arbitrary oint on straight line. The transverse cuter angle is determined by the chamering arameters h j and b j as illustrated in Fig.. (c) Proile equations o addendum ellitic curve x ar sin rsin y ( ar tanbcosrcos ) Sg tan (6) cos x r g y ( sin a tan b) S tan a r () where is the illet radius o the rack cutter, the angle to designate the illet ellitic coordinates, and r the radial distance. where is a distance between the starting oint M 0 and M which is an arbitrary oint on line ; is a the distance along the gear ace width, and transverse ressure angle, and the helical angle. Sg is a sign symbol which is ositive when deriving a right hand helical gear and minus when deriving a let hand one. Besides, is or the let side roile o rack cutter and is or the let side. a r and b, which resectively are the distance rom the lower starting oint M 0 and the hal tooth sace at centrode line, are exressed in Eqs. () and (3) as a m ce m (3) r d n c n m b t ct / ( Ec mn) tan (4) 4 Transverse section Fig.. The rack cutter and coordinate systems Straight line Normal section Straight line Addendum ellitic curve mn where d is the dedendum coeicient o the gear, the normal module, m the transverse module, c the t distance o the addendum circular arc, the addendum correction actor, and c t the transverse 8

3 Fig.. The roile o a transverse cross section o the rack cutter and coordinate system S.. Equations o Generated Gears Then, introducing homogenous transormation matrix (HTM) and the equation o meshing or gears [3], the roile equations o involutes, illets, and addendum chamers or the gear teeth are obtained. As shown in Fig. 3, coordinate systems S, S, and S which are resectively rigidly connected to the rack cutter, the generated gear, and the ixed rame. r is the radius o circular centrode. The osition vector R o the generated gear in S can be obtained by erorming coordinate transormation matrix rom S to S as R = MR, (7) where M is the HTM rom S to S. Finally, both erorming the coordinate transormation in Eq. (7) and using the equation o meshing to the rack roile equations, the three arts o generated gear tooth are resectively derived as x ( cosar cos r)cos ( sin artansin bsgtan r)sin y [( cosar cos r)sin ( sin artansin bsgtan r)cos ] (c) Proile equations o illet curve x ( ar sinnrsin r)cos ( artant b cost rcosrsg tan )sin y [( ar sinnrsin r)sin ( artant b cost rcosrsg tan )cos ] V. Constraints and Planet Phasing (9) (0) Three ollowing geometry constraints need being satisied while constitution o lanetary gear systems. (i) Center distance constraint The constraint o center distances o external and internal gear airs holds that Fig. 3. Coordinate systems or gear generation. (a) Proile equations o involute x ( cos a r )cos ( sin a tan r r r r brs tan )sin g y [( cos a r )sin ( sin a tan brs tan )cos ] g (b) nvolute roile equations o addendum chamer (8) s r r r r r () s where r, r, and r r are the oerating itch radii o sun, lanet, and ring gears, resectively. (ii) Gear meshing constraint Only when the distributed angles o lanet gears satisy Eq. (), the gearing is assemble. s r k / nt () c s r where and are the teeth numbers o sun and ring k gears, resectively. Also, is the distributed angle o c the kth lanet gear ositioning around carrier as illustrated in Fig.4 and nt denotes integer. Secially, or an equally sacing gearing, sum o tooth numbers o lanet and ring gears must satisy Eq. (3), s r / n nt (3) where n is the number o lanet gears. More seciied tye o equally sacing gearings in which there are no MPD among lanet gears. t is required s r / n and / n nt (4) (iii) Addendum intererence constraint n order to avoid addendum intererence between adjacent lanet gears, the distributed angle intervals between them are restricted as given in Eq. (5) in 9

4 which h is the addendum coeicient and number o lanet gears s th lanet gear is the tooth h sin (5) Beside o the above assembly constraints, the MPD between arbitrary two lanet gears is introduced. The MPD k between the kth and st external gear airs is exressed as nt (6) k s s k k k c c c V. Modal Shae Categoriation According to the categoriation in the researches [4, 6] which are in essence basing on discrete models o mass-sring systems, vibration modes o lanetary gearings are categoried as translation, rotation, and lanet modes. n this aer, or identiication, the ith mode o them is denoted as Ti, Ri, and Pi, resectively. The reetition number o translation modes is two. Also, lanet mode exists only as the number o lanet gears n is larger than our with a reetition o n-3. kth lanet gear Sun gear (a) Ti (b) Ri (c) Pi Fig. 4. Phase dierence between the kth and the th lanet gears o with a distributed angle interval V. Gearing Element Creation k c Firstly, not via CAD model, the analyed elements o gears are created using a C code directly. Through that high quality meshing elements o gears are arametrically built using the derived roiles. Sequentially, mesh elements o the sun, lanet, and ring gears are successively created. Next, incororating comonent models including driving and driven shats, and carrier and also their bearings using 4 sets equivalent srings, the D/3D FE models o a comlete gearing are built as shown in Fig 5. Perorming LS-DYNA to modal analyses, modal characteristics o the lanetary are solved ater assigning suitable materials, boundary conditions, and other required settings. (a) (b) nut shat Fig. 5. D/3D element models o lanetary gearing Fig.6. llustration o the three modal shaes categories Analying Case V. Results o Gearing with MPD An examle o lanetary gearing ( ) o n=3 lanet gears with MPD is analyed. ts gear data is given as: module=6mm, ressure angle= 0, teeth numbers o gears are s =9, =34, and r =89, resectively. The D/3D FE models o the gearing have been created in Fig. 5. Besides, Young s modulus o steel is assigned as E= MPa and bearing stinesses k B =0 8 N/m at all. Using D FE model o Fig.5(a) and single tooth air in contact external or internal gear meshes, 9 calculated structural modal requencies are deicted in Tab.. There are no twin translation modes or this kind o gearing with MPD and no lanet mode or n=3. Ones o two tyes o obtained modes are shown in Fig. 7. The stinesses and result o this gearing will be used as the reerence base to comare with the results as investigating the inluences o varieties o bearing and material stinesses aterwards. To exlain inluence o the number o meshing tooth airs in mating gears, the structural modal requencies using double meshing tooth airs are also calculated. Figure 8 comares the resulting structural modal requencies using single tooth air and double tooth airs in which shows that the number o tooth airs in contact does not signiicantly aected the lower requency mode o T-!4 and R. However, it aect most o the requencies o the other modes. n this study, only the setting o the single tooth 30

5 air in contact is used in the ollowing discussion on the eect o bearing stinesses. Table Structural modal requencies o gearing ( ) Mode tye Translation mode, Ti Rotation mode, Ri Planet mode, Pi Natural requency (H) 47.36, 48.54, 8.93, 97.36, 37.98, 45.98, 89.5, 06.8, , 99.89, 38.03, , 696.8, 39, 39.3, , 47., Nil (No lanet mode or the gearing o n=3) increased with the increase o systematical bearing stinesses. Moreover, using the curve o R to be a low and high boundary line, thus, all 9 natural requencies can be clearly categoried into two grous: the 0 lower requency modes o T, T, T3, T4, R, T5, T6, T7, T8, and T9 are the lower grou, and the other 9 higher ones o R, R3, T0-T5, and R4 are assigned as the higher grou. Besides, an index, i terming as dimensionless sloe, is used or revealing the change tendency o structural natural requencies under various systematical bearing stinesses and is deined as (a) (b) i - i i i k -k k Bi, Bi, - Bi, - (7) Fig.7. Two examles to the resulted translation and rotation mode shaes: (a) T5 (37.98 H); (b) R4 (349.4 H) Fig.8. Comaring the resulting structural modal requencies by using the settings o single tooth air and double tooth airs in contact Eect o Suorting Bearing Stinesses At irst, basing on the above reerence gearing o k B =0 8 N/m together with its gear data and material roerties, eect o systematical bearing stiness is discussed which means that all comonent stinesses are simultaneously changed. The systematical bearing stiness is assigned rom 0. times o 0 7 N/m to 00 times o the maximum value o 0 0 N/m. Fig. 9(a) obviously shows the calculated structural natural requencies o the gearings under various stiness assignments that shows all structural requencies are where i is calculated natural requency under assigning bearing stinesses k B,i. Then, the dimensionless sloes rom Fig. 9(a) are deicted in Fig. 9(b) which shows dimensionless sloes o T, T, T3, and T4 in the lower grou are always large esecially at lower bearing stinesses, which imlies that the natural requencies o those modes are increased more evidently than that o the others. These lower modal requency modes are more deending on bearing stiness bearing when its meshing (material) stiness is comaratively high. Besides, among the stiness interval, s o R, T5, T6, T7, T8, and T9 are increased with increase o bearing stinesses which exhibits the eect o these requencies due to stiness increase becomes signiicant as bearings becomes more sti. When bearing stinesses are 0 times increased to 0 9 N/m, most s achieve their maxima which means gear modal behavior near that they is more sensitive to stiness. Ater that, s become smaller. The modal requencies increased less signiicantly are those to the higher grou including R, R3, T0-T3. However, as the bearing stinesses are assigned larger than N/m, the increasing tendency o s o the higher grou is also becoming signiicantly. Noticeably, T4, T5, and R4 are almost not aected at all through the interval. 3

6 Nature requency (H) one rotation mode vanishes when comaring with the requencies o no hase dierence. The unequal gear meshing stiness not only diverts the reetitive translation modes o gearing ( ) to two distinct translation modes but also causes a rotation mode turning into a translation one since the sun gear under this vibration mode has linear dislacement. Dimensionless sloe Systematical bearing stiness Kb ( N/m) (a) Systematical bearing stiness Kb ( N/m) (b) Fig. 9. s and s to systematical bearing stiness V. Results o Gearing no MPD n addition to the lanetary gearing ( ) with MPD among lanet gears, an equally sacing gearing o no MPD is used to contrast their modal characteristics. Thereore, modal requencies o no MPD gearing are also calculated. The gear data o the gearing ( ) which is designed very similar to the gearing ( ) are given as the ollows: module=6mm, ressure angle= 0, teeth numbers o gears are s =8, =34, r and =87. n a similar manner, the FE model o the gearing ( ) are built and shown in Fig. 0. As the categoriation in Re. [4, 6], modal shaes o no MPD gearing can be comletely divided to three tyes as illustration in Fig. 6. Using the same reerence bearing stiness o 0 8 N/m, but only the case o simultaneously changing systematical bearing stinesses k B rom 0 7 to 0 0 N/m is discussed. The analyed result is shown in Fig.. Comaring with the results o ( ) in Fig. 9, the no MPD gearing ( ) likewise has 9 structural modal requencies as deicted in Tab.. Since, the meshing stiness o external gear airs and internal gear airs or gearing ( ) are identical. The reetition number o its translation modes is two. Almost erectly these requencies are twice aearing. Noticeably, gearing ( ) exhibits one additional translation mode but Fig.0. D FE model o no MPD gearing ( ) Nature requency (H) Dimensionless sloe Systematical bearing stiness Kb ( N/m) (a) Systematical bearing stiness Kb ( N/m) (b) Fig.. s and s o gearing ( ) or assigning various systematical bearing stiness 3

7 Table Structural requencies o gearing ( ) Mode tye Translation mode, Ti Rotation mode, Ri Planet mode, Pi Natural requency (H) (47.6, 47.6), (85.63, 85.63), (44.4, 44.4), (.5,.5), (33.59, 33.59), (807.78, ), (38.99, 38.99).85, 90.90, 899.5, 67., Nil (No lanet mode or the gearing o n=3) X. Conclusions This study roosed an aroach to analying dynamic behavior o single-stage lanetary gear systems using a general-urose FE ackage. Through which, the modal roerties o two kinds o single-stage lanetary gear systems which are with ( ) and without ( ) MPD among lanet gears were analyed. Using this continuous geometric gear model, inluences o suorting bearing stinesses, structural modal characteristics o gearings can be discussed. Using the roosal model, wide tyes o lanetary gearing dynamics can be investigated. n uture, studies on modal and dynamic analyses o single- and multile-stage helical lanetary gears using D/3D FE model o variant conigurations are also attemted. Exectedly, the roosal aroach may beneit gear dynamics including wide sorts o lanetary gearings. Acknowledgements The authors would like to thank the unding suorts (i) rom National Science Council under the grant o NSC-95--E-6-008, Taiwan, R.O.C., and (ii) rom Chu Hua University under the grant o CHU-97--M-00. The sotware suort rom National Center or High Perormance Comuting (NCHPC) is also acknowledged. Reerences. Hidaka, T., Terauchi Y., and Nagamura K., Dynamic Behavior o Planetary Gear-5th Reort, Dynamic ncrement o Torque, Bulletin o the JSME, Vol., (979), Botman, M., Eicyclic Gear Vibration, ASME J. Eng. nd, Vol. 7, (976), Cunlie, F., Smith J. D. and Welbourn D. B., Dynamic Tooth Loads in Eicyclic Gears, ASME J. Eng. nd., (974), Kahraman A. Planetary Gear Train Dynamics, ASME J. Mech. Des., Vol. 6, (994), Velex, P., and Famand, L., Dynamic Resonse o Planetary Trains to Mesh Parametric Excitations, ASME J. Mech. Des., Vol. 8, (996), Lin, J. and Parker R. G., Analytical Characteriation o the Unique Proerties o Planetary Gear Free Vibration, ASME J. Vib. Acoust., Vol., (999),. 36-3,. 7. Lin, J. and Parker R. G., Sensitivity o Planetary Gear Natural Frequencies and Vibration Modes to Model Parameters, J. Sound Vib., Vol. 8, (999),. 09-8, 8. Kahraman, A., Free torsional Vibration Characteristics o Comound Planetary Gear Sets, Mech. Mach. Theory, Vol. 36, (00), Kiracote, D. R. and Parker R. B., Structured Vibration Modes o General comound Planetary Gear Systems, ASME J. Vib. Acoust., Vol. 9, (007), Parker, R. G., Agashe V., and Vijayakar S. M., Dynamic Resonse o a Planetary Gear System using a Finite Element/Contact Mechanics Model, ASME J. Mech. Des., Vol., (000), Yuksel, C., and Kahraman, A., Dynamic Tooth Loads o Planetary Gear Sets Having Tooth Proile wear, Mech. Mach. Theory, Vol. 39, (004), Abousleiman, V, and Velex, P., Ahybrid 3D Finite Element/lumed Parameter Model or Quasi-static and dynamic analyses o Planetary/Eicyclic Gear Sets, Vol. 4, (006), Litvin F. L., Gear Geometry and Alied Theory, (004) New York: Cambridge U. ublished. 4. LSTC, LS-DYNA Hel Manual, (003). 33

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