d e t e r m n n g ower Losses etra Hengartner s a Textron Sx Sgma black belt for flud and power at Textron Flud & ower, located n Huddersfeld, England. She leads cross-functonal teams to mprove work processes throughout the busness. revously, she was a senor desgn engneer and a senor applcatons engneer of Textron s custom-made heavy ndustral gear unts. Dr. Davd Mba s senor lecturer and postgraduate course drector n the automotve, mechancal and structures engneerng department at Cranfeld Unversty, located n Cranfeld, England. He s drector of the master s course Desgn of otatng Machnes and s responsble for ts ndustral collaboratons and s drector of the contnued-professonal-development course Gear Desgn. He also leads the department s machne dagnostcs group, whch s nvolved wth consultng and doctoral research. Mba specalzes n machne dagnostcs, rotor dynamcs and machne desgn. Also, he s developng a research and expermental faclty to predct nonload gear losses n worm gearboxes. n t h e h e l c a l g e a r m e s h
A Case Study etra Hengartner and Davd Mba Management Summary Currently, legslaton s n place n the Unted Kngdom to encourage a reducton n energy usage. As such, there s an ncreased demand for machnery wth hgher effcences, not only to reduce the operatonal costs of the machnery, but also to cut captal expendture. The power losses assocated wth the gear mesh can be dvded nto speed- and load-dependent losses. Ths artcle revews some of the mathematcal models proposed for the ndvdual components assocated wth these losses, such as wndage, churnng, sldng and rollng frcton losses. A mathematcal model s proposed to predct the power losses on helcal gears hghlghtng the major contrbutor to losses n the gear mesh. Furthermore, the mathematcal model s valdated wth a case study. Nomenclature A g Arrangement constant A g = 0. C 1 Constant, 9.66 C Constant, 9 10 7 D Element dameter F F S L c CL S ollng force Sldng force Contact lne length Churnng power loss ollng power loss Sldng power loss WL Wndage power loss Introducton To meet the ncreased demand for machnery wth hgher effcences, supplers must desgn equpment that reduces the operatonal costs of the machnery and cuts captal expendture. In the past, gears have been consdered as hghly effcent n transmttng loads, but the requrements from the customer to acheve a mnmum effcency target and penaltes for noncomplance are becomng more and more strngent. A reducton n the power loss of a gearbox wll cut the runnng costs of the equpment as t becomes more effcent and also uses less lubrcant to cool the gear teeth. Ths n turn wll reduce the sze of the auxlary equpment, such as the lubrcaton pump, and wll also lead to a reducton n the heat exchanger capacty. All ths wll contrbute to an overall smaller footprnt of the equpment, whch saves space that can be at a premum n some applcatons. The power losses consst of speed- and load-dependent losses. Speed-dependent losses can be dvded nto wndage losses, churnng losses, bearng churnng losses and seal losses. The load-dependent losses are made up of sldng frcton loss, rollng frcton loss and bearng loss. Speed-Dependent Losses Wndage losses. As gears rotate, lubrcant s flung off the gear teeth n small ol droplets due to the centrfugal force actng on the lubrcant. These lubrcant droplets create a fne mst of ol that s suspended nsde the gear housng/case. The effect of ths ol mst s an ncrease n wndage frctonal resstance on the gears and hence an ncrease n the power consumpton. In addton, the expulson of the oly atmosphere from the tooth spaces as the gear teeth come nto engagement creates turbulence wthn the gearbox and ncreases the power consumpton. The combnaton of these factors, as well as the losses at the sde faces of the gears, contrbute to the total wndage losses. Factors that nfluence the magntude of the f V S V T b d f d k f g h m n n w w n β λ oughness factor Sldng velocty ollng velocty Face wdth oot dameter Outsde dameter Gear dp factor (ato of dppng depth to element outer dameter) f g = 1 element fully submerged Isothermal central flm thckness Normal module otatonal speed Load parameter Normal gear contact load Helx angle Gearbox space functon µ Coeffcent of frcton µ 0 Ambent vscosty at ambent temperature ν φ φ t Subscrpts Knematc vscosty at operatng temperature Ol mxture functon, φ = 1 ol-free atmosphere Thermal reducton factor Element under consderaton w w w. p o w e r t r a n s m s s o n. c o m w w w. g e a r t e c h n o l o g y. c o m G E A T E C H N O L O G Y S E T E M B E / O C T O B E 0 0 5
wndage loss nclude the rotatonal speed of the gear because power losses rse wth an ncrease n perpheral velocty. Other factors are the tooth module, the amount of ol mst present nsde the casng and the dameter of the gears. A mathematcal model to predct wndage loss was proposed by Anderson et al. (efs. 1 ). However, Anderson s wndage loss equaton accounted for nether the tooth module nor the helx angle. Townsend detaled a wndage loss equaton whch ncluded an ol mxture functon φ and a gearbox space functon λ and s presented here as Equaton 1 (ef. ). The ol mxture functon φ ndcates the state/type of atmosphere nsde the gear unt wth φ = 1 ndcatng an ol-free atmosphere. The gearbox space functon λ s set at 1 for free space and reduces to a value of 0.5 for a closely fttng enclosure,.e. the fttng of baffles or shrouds around the gear. WL.9 f.9 f.9 0.75 : = n (0.16 d + d b m Ol Churnng Loss. Townsend defned churnng losses as the acton of the gears movng the lubrcant nsde the gear case and referred n partcular to the losses due to entrapment of the lubrcant n the gear mesh, whch s more applcable to spur gears than to helcal gears (ef. ). Factors nfluencng the ol churnng loss are the vscosty of the ol, as ths ressts the moton of the gears; perpheral velocty; operatng temperature; the tooth module; the helx angle; and the submerged depth of the gears. All rotatng components that are n drect contact wth the lubrcant,.e. dpped nto the ol, contrbute to the churnng losses, and the deeper the components are submerged, the hgher the losses. Wth larger helx angles, the power losses are lower as the gear teeth slce through the lubrcant rather than dsplacng the lubrcant along the whole gear face wdth. Other expressons for determnng churnng losses have been proposed (ef. 4). The Brtsh Standard BS ISO/T 14179 art 1 detals churnng loss equatons whch had been modfed for the effect of lubrcant vscosty, element dameter, the gear dp factor and the arrangement constant (ef. 5). These churnng loss expressons were splt nto three dfferent sectons, whch are detaled n Equatons 4. Churnng Loss for Smooth Outsde Dameters (.e. shafts): CL 7.7 f g v n D : = 6 A 10 φ λ Churnng Loss for Smooth Sdes of Dscs (.e. gear sde faces, both faces) CL g g 4.7 1.474 f g v n D : = 6 A 10 1.15 L 5.7 )10 0 (1) () () Churnng Loss for Tooth Surfaces: Load-Dependent Losses Sldng frcton loss. rncpally, the nstantaneous sldng frcton loss s a functon of the nstantaneous sldng velocty and the frcton force, whch tself s a functon of the nstantaneous normal tooth load and the nstantaneous coeffcent of frcton. The magntude of sldng velocty depends on the poston of contact along the contact path wth a peak velocty at the start of the approach. The velocty reduces to 0 at the ptch pont of the two matng gears and rses agan to a peak value at the end of the recess. The effect of the sldng frcton loss s an ncrease n power consumpton, where the magntude depends on the pont of contact. It s nfluenced by the angular velocty of the gears, the rato of the rollng veloctes, the pont of contact, the contact rato and the lubrcant propertes. Anderson et al. analyzed the sldng frcton losses along the path of contact and postulated expressons for the nstantaneous sldng velocty and nstantaneous frcton force, where the frcton force was a functon of the nstantaneous coeffcent of frcton and gear load. The sldng frcton loss s dependent on the poston of contact durng the engagement cycle (efs. 1 ). Whle the model proposed was specfc for spur gears, the authors have modfed the expresson for the nstantaneous coeffcent of frcton to nclude helcal gearng. Ths was accomplshed by modfyng the expresson for nstantaneous coeffcent of frcton to take nto account the helcal gear contact length. The nstantaneous sldng power loss s gven as: where CL : = 7.7 f g v n : = 10 S F S D The postulated expresson proposed by Anderson et al. for the nstantaneous coeffcent of frcton for spur gears s detaled n Equaton 7 and was employed by the authors for ths nvestgaton (efs. 1 ). The coeffcent of frcton s gven as: w( C1 (7) µ : = 0.017 log b µ 0 S T The contact length for helcal gears, L c, has been detaled and was substtuted by the authors nto Equaton 7 (ef. 7). Therefore, the modfed equaton for the coeffcent of frcton for helcal gears g 4.7 A 10 f b tan( ) β 6 F : = µ w( S S (4) (5) (6) 4 S E T E M B E / O C T O B E 0 0 5 G E A T E C H N O L O G Y w w w. g e a r t e c h n o l o g y. c o m w w w. p o w e r t r a n s m s s o n. c o m
F ( : = C h( φ b t s gven as: : = 0.017 log µ The coeffcent of frcton used n ths analyss s ndependent of the gear surface temperature, whch strctly speakng s naccurate. The expresson n Equaton 8 was substtuted nto Equaton 6 to obtan the sldng power losses. ollng frcton loss. The rollng frcton loss s dependent on the nstantaneous rollng velocty and the nstantaneous lubrcant flm thckness. As the gear teeth come nto mesh, an elastohydrodynamc lubrcant flm s developed between the teeth n contact. The acton of the gear teeth durng the engagement draws the lubrcant nto the contact zone. The parameters that nfluence the rollng frcton loss are the lubrcant flm thckness, the angular velocty of the gears, the workng pressure angle and the pont of contact along ts contact path. The lubrcant propertes nfluence the buldup of the lubrcant flm, ts shear values and ts thermal behavor. In addton, the gear materal and the normal tooth load also nfluence the flm thckness. In eferences 1, Anderson et al. postulated the nstantaneous rollng frcton force as: where the rollng power loss s gven as: wn C1 Lc µ 0 S T F ( : = C h( φ b : = 10 F (8) (9) (10) Ths expresson for nstantaneous rollng force ncludes a thermal reducton factor that accounts for the decrease n ol flm thckness as the ptch-lne velocty ncreases (ef. 6). A relatonshp between thermal loadng factor and reducton factor was presented by Anderson et al. and employed by the authors of ths nvestgaton (ef. 1). The paper by Anderson et al. mpled that pror to computng the thermal reducton factor, the thermal loadng factor must be determned. To account for helcal gears, the expresson of Equaton 1 was modfed by the authors, as the contact lne length n helcal gears s not synonymous wth the face wdth. The modfed nstantaneous rollng force s gven as: F : C (11) T = h( φ t t L c Fgure 1 Schematc of the back-to-back arrangement for the hgh-speed gear unt. where L c, has been defned as the contact length for helcal gears. The expresson was substtuted nto Equaton 10 to obtan the sldng power losses. Mathematcal Model The mathematcal model employed for ths nvestgaton conssted of the followng expressons: 1. The wndage loss equaton as employed by Townsend (ef. ).. Churnng loss expressons as detaled n BS ISO/T 14179, art 1 (ef. 5).. The rollng and sldng frcton losses as postulated by Anderson et al. (ef. 1). Model Valdaton The gearbox used to valdate the model was a sngle-stage, double-helcal, speed-ncreasng gear unt wth ol flm bearngs of the crcular type. The nlet ol temperature was 49 C, and the maxmum bearng temperature dd not exceed 87 C. The nput shaft and the output shaft end were sealed by means of a shaft-mounted ol flnger and non-contactng baffle rngs n the housng. A lubrcaton pump was drven va a set of reducton gears from the nput shaft. The pump suppled the lubrcant for the gear sprayers and the forcefed bearng. The gear unt was tested n a back-to-back arrangement (see Fg. 1). A torque loader was ftted between the output shaft couplng end of the slave unt and the output shaft tal end of the test unt. The torque loader used for the experment employed a pressurzed ol system. The ol was suppled va a rotary unon (see Fg. 1). The torque loader conssts of an nner rotor and outer rotor, whch are supported n bearngs. Ol s fed nto the space between the two rotors, creatng a torsonal load n the test rg. The bearng losses n the torque loader were calculated separately beforehand and subtracted from the motor nput power pror to calculatng the gearbox effcency. The load condtons for the experment ncluded 5%, 50%, 75%, and 100% of full load (8.95 MWatts), at 100% nput speed (1,460 rpm). Expermental torque readngs were taken wth a telemetrc system from the low speed shaft va stran gauges. Appendx A detals some gear data. It must be noted that the rg was run at full load and maxmum speed (1,460 rpm) for a perod of four hours. w w w. p o w e r t r a n s m s s o n. c o m w w w. g e a r t e c h n o l o g y. c o m G E A T E C H N O L O G Y S E T E M B E / O C T O B E 0 0 5 5
Table 1 Expermental and Theoretcal esults of Experment I. ower nput (kw) 8,95 6,714 4,476,8 % load 100% 75% 50% 5% Expermental total loss (kw) 15. 116.8 11.5 106.75 Expermental total loss (kw) for gear and wndage only 55.08 50.15 50.16 49.4 redcted total loss (kw) 57.41 5. 49.86 47.4 Breakdown of Losses Sldng frcton loss (kw) 10.88 6.67.05 0.4 ollng frcton loss (kw) 5.71 5.81 5.98 6.7 Churnng loss (kw) 0 0 0 0 Wndage loss (kw) 40.8 40.8 40.8 40.8 gearbox space functon were assumed. The total predcted power losses were the sum of the losses from the lubrcaton pump and the gearbox. esults are detaled n Table 1. A comparson between predcted and expermentally determned power losses s shown n Fgure. The model predcts a steady declne n power loss correspondng to a reducton n load. It was observed that by ncreasng the power nput at a fxed rotatonal speed, the wndage losses remaned the same and the rollng frcton power losses decreased whle the sldng loss ncreased. Dscusson and Concluson The mathematcal model detaled n ths paper has shown to be vald, provdng an ndcaton of the contrbuton of each element wthn a gearbox wth helcal gearng to the total power loss. The predctons and the expermental results show a good correlaton, although the expermental results do not provde a breakdown of the varous power losses. In the breakdown of losses n Experment I, t can clearly be seen that the sldng frcton losses are heavly load dependent, ncreasng wth load. However, the rollng frcton losses decreased slghtly wth an ncreased load, and ths s due to a decrease n ol flm thckness. As the speed was constant durng the experment, wndage losses remaned constant throughout the tests. Ths nvestgaton dd not revew or nclude new mathematcal models for load-dependent bearng losses and speed-dependent bearng churnng and seal losses. Acknowledgments The authors wsh to express ther grattude to Davd Brown, Textron ower Transmsson for allowng the publcaton of ths nvestgaton, undertaken as a degree project for a master s of scence n Desgn of otatng Machnes, Cranfeld Unversty. Fgure Comparson between expermental gear and wndage losses and predcted power losses. Ths paper was presented at the ASME/AGMA 00 Internatonal ower Transmsson and Gearng Conference, held Sept. 5, 00, n Chcago, IL, and was publshed n roceedngs of the 00 ASME Desgn Engneerng Techncal Conferences & Computers and Informaton n Engneerng Conference. It s republshed here wth permsson from ASME. ror to undertakng ths test, the gear unt was vsually nspected to ensure gear contact markngs were satsfactory full face and full depth. The mathematcal model for the helcal gears n ths nstance was accomplshed by doublng the face wdth. The ol mx functon was assumed to be φ = and the gearbox space functon was taken eferences as λ = 1 as the gear case walls were suffcently far away from the 1. Anderson, N.E., and S.H. Loewenthal. Spur gear system effcency at part and full load, Techncal eport 79-46. NASA Techncal aper 16, 1980. gears to be consdered as free space. No ol churnng took place, as. Anderson, N.E., and S.H. Loewenthal. Effcency of nonstandard and hgh contact ths was a spray-lubrcated arrangement. The expermental results rato nvolute spur gears, Journal of Mechansms, Transmssons, and Automaton n Desgn, Vol. 108, March 1986, pp. 119 16. provded the effcency for the complete gear unt; therefore, the. Townsend, D.. Lubrcaton and coolng for hgh speed gears, Orgnal Equpment bearng losses, seal losses and absorbed power for the lubrcaton Manufacturng Conference, Sept. 9 11, 1985, hladelpha, A. 4. Luke,. and A.V. Olver. A study of churnng losses n dp-lubrcated spur gears, pump had to be calculated from manufacturer s nformaton (see roceedngs of the Insttuton of Mechancal Engneers, art G, Vol. 1, 1999, pp. Appendx B). The power loss calculatons for the bearngs assumed 7 46. 5. Brtsh Standards Insttuton, BS ISO/T 14179:001(E), Gears Thermal Capacty, the maxmum clearance condton. As non-contactng seals were BSI, London, Unted Kngdom, 001. employed, no power losses from the seals were assumed. 6. Wu, S., and H.S. Cheng, A frcton model of partal-ehl contacts and ts applcaton to power loss n spur gears, Trbology Transacton, Vol. 4, art, 1991, pp. As the lubrcaton pump reducton gears were not separately 98 407. sprayed, the ol mst present nsde the gear case was assumed suffcent to provde lubrcaton. Agan the same ol mx functon and Alexandra, VA, December 198. 7. Amercan Gear Manufacturers Assocaton, AGMA Standard 18.01, atng the pttng resstance and bendng strength of spur and helcal nvolute gear teeth, AGMA, 6 S E T E M B E / O C T O B E 0 0 5 G E A T E C H N O L O G Y w w w. g e a r t e c h n o l o g y. c o m w w w. p o w e r t r a n s m s s o n. c o m
Appendx A Gear Data. Unts non teeth number z 1 115 Wheel teeth number z 1 Center dstance a 609.6 mm Normal module m 8 mm Normal pressure angle α 0 deg. Helx angle β 5 deg. Face wdth b 85.75 mm non shft coeffcent Wheel shft coeffcent non, Young s modulus x 1 0.989 x 0.181 E 1 07,000 N/mm Double helcal Appendx A Calculated Gear Data. Contact lne length L c mm 499.57 Operatng dameter speed Gear contact tangental load Gear contact normal load Equvalent Young s modulus v mm/s 78,811 W pt N 11,588.59 W pn N 1,74.6 E eq N/mm 7,47.5 non torque T 1 Nm 58,551.58 Appendx B Bearng, Lubrcaton ump and Seal Losses. Wheel, Young s modulus non, osson s rato Wheel, osson s rato E 07,000 N/mm ny 1 0. ny 0. ower nput (kw) 8,95 6,714 4,476,8 % load 100% 75% 50% 5% Expermental total loss (kw) 15. 116.8 11.5 106.75 Specfc heat C 1 544 J/kgK Specfc heat C 544 J/kgK Thermal conductvty K 1 46 W/mK Thermal conductvty K 46 W/mK Applcaton non speed n 1 1,460 rpm Transmtted power 8,95 kw ower Loss n Each Bearng non couplng bearng (kw) non tal bearng (kw) Wheel couplng bearng (kw) Wheel tal bearng (kw).9 1.7 19.7 17.8.9 1.7 19.7 17.8.9.67.4.09.9.67.4.09 Lubrcant Lubrcant factor 1 for mneral ol Lubrcaton ump Vscosty ny 46 mm /s at 1 deg. K ny 1 mm /s at deg. K Specfc gravty ρ 87 kg/m Dynamc vscosty η 0.018 a*s Absorbed power (kw) 16.59 16.59 16.59 16.59 Seal Losses at Each Shaft (kw) non couplng (kw) 0 0 0 0 Vscosty-pressure coeffcent α.0e 08 m /N Wheel couplng (kw) 0 0 0 0 Thermal conductvty K f 1.5E 01 W/(m*K) Total loss (kw) 70.5 66.67 6.19 57. w w w. p o w e r t r a n s m s s o n. c o m w w w. g e a r t e c h n o l o g y. c o m G E A T E C H N O L O G Y S E T E M B E / O C T O B E 0 0 5 7