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Study On Mesh Power Losses In High ContactRatio (HCR) Gear Drives
Rama ThirumuruganAssociate Professor
Center for Design Analysis and TestingDepartment of Mechanical Engineering
Dr Mahalingam College of Engineering and Technology, PollachiTamilnadu
G.MuthuveerappanProfessor
Machine Design SectionDepartment of Mechanical Engineering
Indian Institute of Technology Madras, ChennaiTamilnadu
Abstract—Continuous demands for higher efficiency geardrives need an understanding on the role of power loss thatdepends on the frictional forces. The load dependent toothmesh power losses (i.e., sliding and rolling power losses) arethe dominant power loss components at heavy loads withlow or moderate pitch line velocities. Hence, these two powerlosses are given more importance in this study to evaluate theperformance of the HCR gear drive. The calculation of slidingpower loss is carried out based on the Elastohydrodynamiclubrication (EHL) model proposed by Xu et al.[1], but with theconsideration of load sharing ratio(LSR). Higher contact ratiois achieved through addendum enlargement in this work.Thecomparative study of normal and high contact ratio gears isdiscussed for better understanding on gear mesh losses on HCRgear drive.
Keywords – Power loss; Efficiency; Finite element analysi;,High contact ratio; Spur gear
I. INTRODUCTION
The efficiency of power transmission system is con-sidered as one of the important design factors due to thefollowing reasons ([2]),
1) Efficient power transmission systems ensure fueleconomy.
2) With less fuel consumption, less pollutant gasesand particulate are emitted to the environment.
3) Since power losses amount to heat generationwithin the gearbox, which leads to several gearfailure modes such as scoring and fatigue.
4) Improved efficiency of a gearing system can reducethe requirements on the capacity of the lubricationsystem and the gearbox lubricant and thereby re-ducing the operation costs of the system.
5) Efficiency prediction can assist in estimating thepower requirements during the design stage of amachine and thus ensures reliable operation of thesystem. It can also assist in estimating the poweroutput for a given power input.
The total power loss of the gearbox is attributable tosliding and rolling frictional losses between the gear teeth,windage losses due to complex interactions with the airsurrounding the gears and oil splashing and churning lossesinside the gearbox as well as the losses associated with thebearings and seals. While churning and windage losses aremostly geometry and speed related, friction losses are mainlyassociated with sliding velocities and load. Gears are usuallyoperated under mixed EHL condition, where the lubricantfilm thickness is comparable to the surface asperity heightssuch that actual metal-to-metal contact are possible.
Friction can be stated as the resistance to motion be-tween two surfaces in relative sliding and rolling underdry or lubricated contact conditions. Lubricant applied to acontact significantly alters the contact conditions and hencereduces the friction. Applied load, speed, parameters relatedto contact geometry, surface roughness and lubricant as awhole helps to define the lubrication conditions. As thesefactors are changing instantaneously during a mesh cycle,the friction will also change accordingly.
Friction at a gear mesh has two components: slidingfriction and rolling friction. Sliding friction is a direct resultof the relative sliding between the two contacting surfaces.The magnitude of the sliding frictional force depends on thecoefficient of friction between the contacting surfaces.
Where as the hydrodynamic rolling (or pumping) loss isthe power required to entrain and compress the lubricant toform a pressurized oil film, which separates the gear teeth.Based on disk machine data, Crook [3] found that the rollingloss was simply a constant value multiplied by the EHLcentral film thickness. Most of the literature uses the centralfilm thickness model proposed by Hamrock and Dowson[4] adjusted with for thermal effects using Cheng’s thermalreduction factor ([5]) to estimate rolling power loss.
Martin [6] has provided an extensive review of frictionpredictions in gear teeth published in boundary lubrication,mixed and EHL regimes.
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In some of the earlier works, the sliding frictional losswas calculated by using average frictional coefficient valuein a mesh cycle ([7], [8],[9],[10] and [11]).
The differences between the sliding force treatment ofBuckingham’s and Merritt’s equations were explained byYada [12].
In many of the works, the variation in the coefficient offriction during a mesh cycle was calculated based on theempirical equations derived by several authors like Benedictand Kelly [13],ODonoghue and cameron [14], Kelly andLemanski [15], Drozdov and Gavrikov[16], Xu et al.[1] etc.
Anderson and Loewenthal [17] proposed a method forpredicting the power loss and efficiency of a steel spurgear set of arbitrary geometry supported by ball bearings.The method algebraically accounts for losses due to gearsliding, rolling, and windage and incorporates an expressionfor ball-bearing power loss. This method provides an ac-curate estimate of spur-gear-system efficiency at part loadas well as full load. They have used Benedict and Kelly[13] frictional coefficient equation to calculate the slidingfrictional loss. They have extended their work to explorethe effect of parameters on HCR gear performance with theassumption that the teeth are rigid and equal load sharingfor simultaneously contacting pairs ([18]).
Heingartner and Mba [19] modified the above approachto calculate the sliding and rolling friction losses in helicalgear.
Arto and Asko [20] used the Benedict and Kelly model toestimate the sliding frictional loss. They have also includedthe load dependent bearing power loss to calculate the totalpower loss.
Benedict-Kelley’s formula was evaluated by consideringthe oil sump temperature and the ambient pressure, butseveral parameters such as piezoviscosity or limiting shearstress are ignored. The influence of surface roughness wasalso discarded in their equation, which is realistic for fullyflooded conditions, but certainly questionable when mixedlubrication prevails and direct asperity interactions cannotbe avoided. Another shortcoming of their equation is dueto the log10 in the analytical expression, which (a) leads toinfinite friction near pitch points and (b) for high slide-to-rollconditions, underestimates friction forces.
Due to the above shortcomings, Dib et al. [21] proposeda new traction law based on measurements from a two-discmachine which accounts for lubricant properties and surfacefinish and integrated in a 3-D dynamic model of gearswith consideration of tooth friction. They have simulatedthe model with no friction, constant frictional coefficient,friction based on Benedict and Kelly [13]equation as well asKelly and Lemanski [15]equation. They claimed that theirmodel yields best results for both low and higher speeds.They have extended their study to predict the power lossesin high- speed gears ([22]).
Xu at al.[1] proposed a model to evaluate the frictionrelated mechanical efficiency losses for parallel-axis gearpairs. Their model combined a gear load distribution model,a friction coefficient model and a mechanical efficiency for-mulation to predict the instantaneous mechanical efficiencyof a gear pair under typical operating, surface and lubricationconditions. They have also derived new friction coefficientformula by performing a multiple linear regression analysisto a large number of EHL model simulations representingvarious combinations of all key parameters influencing thefriction coefficient. The new friction coefficient formula wasshown to agree with the measured traction data. They havealso highlighted the influence of key basic gear geometricparameters, tooth modifications, operating conditions, sur-face finish and lubricant properties on mechanical efficiency.
Kuria and Kihiu [2] used the Xu et al. [1] model fordetermining the overall efficiency of a multistage tractorgearbox including all gear pairs, lubricant, surface finishrelated parameters and operating conditions. Rolling frictionand windage losses were also included in their study andfound that the overall efficiency varies over the path ofcontact of the gear meshes ranging between 94% and 99.5%.
Even though the availability of literatures on the per-formance of the normal contact ratio (NCR) gear drives isabundant, the performance study of the HCR gear drives hasbeen rarely found. Also normally the higher contact ratiosare achieved by way of enlarging the addendum beyondthe standard values, this increases the sliding power losses.Hence a thorough efficiency analysis has to be made forHCR gear drives.
This work explores the performances of both the NCRand HCR spur gears in terms of load dependent mesh powerlosses by considering the LSR between the simultaneouslymeshing teeth pairs.The typical contact points along thepath of contact and load sharing between simultaneouslyengaging pairs of NCR and HCR gear drives are evaluatedbased on the author’s previous works([23] and [24]).
II. POWER LOSS CALCULATION
The different kinds of power losses that occur withinthe gear box can be grouped into two categories, one asload dependent power loss (Pload) and the other as loadindependent power loss (Pno load). Hence, the total powerloss is considered as
PTotal = Pload + Pno load (1)
Load dependent power loss has got the contributionfrom tooth mesh power loss (PMesh) and bearing powerloss (Pb,load). Tooth mesh power loss consists of slidingfrictional power loss components (Ps) and rolling power losscomponents (PR).
The load independent power loss has got the contributionfrom gear windage loss (Pw), oil churning loss (Pc), load
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independent bearing loss (Pb,no load) and seal loss (Pseal).Thus the total power loss is restated as
PTotal = (PMesh+Pb,load)+(Pw+Pc+Pb,no load+Pseal)(2)
where,
PMesh = Ps + PR (3)
As the load acting on the gear tooth varies continuouslyduring a mesh cycle, the load sharing based load dependentpower loss calculation has to be done to estimate the geardrive efficiency to a reasonable accuracy. The load sharingratio by a pair in the point i ((LSR)i) is evaluated usingmulti pair contact model(MPCM)([24]).
A. Sliding Power Loss Calculation
1) Sliding Force: Sliding frictional force is recognizedas one of the vital sources of power loss in gear drive at anycontact point (i), which is a function of normal load (Fn) atthat contact point and it is given as
(Fs)i = µi(LSR)i(Fn)i (4)
where µi is the instantaneous coefficient of friction.
A large number of empirical formulae found in theliterature to determine µi were obtained by curve fittingof measured data collected from twin-disk type tests. Asthe selection of model that calculates µi can significantlyaffects the system losses, a suitable model has to be assumed.The formula developed by Benedict and Kelley [13] isconveniently modified in this work to incorporate the loadsharing effect and the modified equation is given as
µi = 0.0127log1029.66(LSR)i(Fn)iυo(vs)i(vT )2i
(5)
where,υo is the absolute viscosity in cPs and the rolling and slidingvelocities are respectively given by vT and vS in m/sec as
(vT )i = [(vp)i + (vg)i] (6)
(vs)i = [(vp)i − (vg)i] (7)
where (vp)i and (vg)i are the sliding velocity of pinionand gear respectively at the contact point i. The Eq. 5shows higher values near the pitch point for µi, but it isexperimentally proved by Xu (2005) [25] that this value iszero at pitch point (Figure 1), where slide to roll ratio (SR)is zero.
The model proposed by Xu et al. [1] includes the keyparameters like sliding velocity, contact pressure, the surfaceroughness, lubricant dynamic viscosity, radius of curvatureand entrainment velocity that are influencing friction be-tween the contact surfaces of the gear and it is given as,
µi = ef(SRi,(PH)i,νo,s)(PH)b2i |SRi|b3 (Ve)
b6i νo
b7Rb8i (8)
in which
f(SRi, (PH)i, ν0, s) =
(b1 + b4 |SRi| (PH)iLog(ν0)+b5e−|SRi|(PH)iLog(ν0) + b9e
s
)(9)
and PH is maximum Hertzian contact pressure, which isevaluated by using MPCM in this work.The entraining velocity (ve)i in m/sec is given by
(ve)i =1
2[(vp)i + (vg)i] (10)
The slide- to -roll ratio (SR) is given by
(SR)i =(vS)i(ve)i
(11)
in whichs is the RMS composite surface roughness in µm andbi = - 8.92, 1.03, 1.04, - 0.35, 2.81, - 0.10, 0.75, - 0.39, and0.62 for i = 1 to 9, respectively.
The coefficient of friction as per Eqs. 5 and 8 simulatedalong the path of contact for NCR gear using MATLABfor an input speed (Nin) as1500 rpm, power (Pin) as 2.5kW and for the lubricants of specified properties (Table I)is plotted in Figure 1 for comparison.
TABLE I. PROPERTIES OF THE LUBRICANT ([25])
Properties ValueInlet temperature (K) 373.15Viscosity (Pa s) 0.0065Density (kg/m3) 813Pressure viscosity coefficient (1/Pa) 1.2773×10−8
Temperature viscosity coefficient (1/K) 0.0217Thermal conductivity (W/mK) 0.1176Coefficient of thermal expansion (1/K) 6.53×10−4
Specific heat (J/kgK) 2000
−1 −0.5 0 0.5 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
µ
X/pb
Xu, et al. (2007)
Benedict and Kelley (1961)
HPTCLPTC
Fig. 1. Comparison of friction models(number of teeth=50, module=1, pressure angle =20o,
cutter tip radius=0.3m, backup ratio=2.2, gear ratio=1 andcontact ratio=1.755)
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2) Sliding Power Loss: The instantaneous sliding powerloss in kW is given by
(Ps)i = 10−3(FS)i(vS)i (12)
The instantaneous sliding force and power loss calculatedthrough the respective formulae are shown in Figures 2(a)and 2(b). It is observed that the model proposed by Benedictand Kelly [13] predicts more sliding frictional force at pitchpoint, where there is no sliding which is unreasonable,However, it correctly predicts zero power loss at this point.
−1 −0.5 0 0.5 10
10
20
30
40
50
60
70
Fs(N
)
X/pb
Xu, et al. (2007)
Benedict and Kelley (1961)
LPTC HPTC
(a) Sliding force
−1 −0.5 0 0.5 10
0.002
0.004
0.006
0.008
0.01
0.012
0.014
Ps(k
W)
X/pb
Xu, et al. (2007)
Benedict and Kelley (1961)
LPTC HPTC
(b) Sliding power loss
Fig. 2. Sliding frictional force and power loss
B. Rolling Power Loss in NCR Gears
1) Rolling force (FR): As the EHL film thickness (hL)developed between the gear teeth surfaces at any contactpoint in a mesh cycles controls the rolling force (FR), thehL and FR are calculated at all contact points along thepath of contact using the formula referred by Anderson andLoewenthal [18]. The instantaneous rolling force at any pointi due to build up of EHL film is given by
Rolling force(FR)i = 9X107(φt)i(hL)ib (13)
where φt is the thermal reduction factor to account for theeffect of temperature rise at high speed conditions and b is
the face width of the gear tooth the film thickness (hL) usedin the Eq.13 is expressed as
(hL)i = 2.69U0.67i G0.53W−0.067i (1− 0.61e−0.73Ki)Ri
(14)where, Ri is the effective radius in the direction of rollingand Ki is the ellipticity parameterU is the dimensionless speed parameter and it is given as
Ui =(ve)iυoE′Ri
(15)
in which,
E′ =2(
1−ν2p
Ep+
1−ν2g
Eg
) (16)
the material parameter(G) is expressed as a product of E’and the pressure viscosity coefficient (α)
G = E′α (17)
the load parameter Wi is given as
Wi =F
E′R2i
(18)
The calculated instantaneous EHL film thickness is plot-ted in Figure 3. An abrupt drop in the lubricant film thicknessobserved near the pitch point of the NCR gear is due to asudden increase in the load at this region.
−1 −0.5 0 0.5 15.6
5.65
5.7
5.75
5.8
x 10−8
hL
(m
)
X/pb HPTCLPTC
Fig. 3. EHL lubricant film thickness in NCR gears
2) Rolling Power Loss: The instantaneous rolling powerloss in kW is given by
(PR)i = 10−3(FR)i(vT )i (19)
The calculated rolling force and the power loss are shownin Figure 4
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172
−1 −0.5 0 0.5 1
0.0405
0.041
0.0415
0.042
FR
(N
)
X/pbLPTC HPTC
(a) Rolling force
−1 −0.5 0 0.5 10.000108
0.000109
0.00011
0.000111
0.000112
0.000113
PR
(k
W)
X/pbLPTC HPTC
(b) Rolling power loss
Fig. 4. Rolling force and power loss
C. Total Mesh Power Losses
The total instantaneous tooth mesh loss at any contactpoint on the path of contact is the summation of the rollingand sliding power losses at that particular instant. As contactalternates between double pairs and single pair, in NCRgears, the contribution from both teeth pairs, which are insimultaneous contact, has to be considered for calculatingthe total mesh power losses along the path of contact. Thetotal instantaneous power loss ((TPint)i) for a teeth pair isgiven as
(TPint)i = (Ps)i + (PR)i (20)
Total power losses (TP) due to sliding and rolling for NCRgear for the simultaneous contact is given as
TP =
2B′∑A′
(TPint)idx+C′∑B′
(TPint)idx+
2D′∑C′
(TPint)idx
(21)
The contact points A’ to D’ along the path contact are givenin [24]
and same for HCR gear is given as
TP =
3B∑A
(TPint)idx+ 2C∑B
(TPint)idx+
3D∑C
(TPint)idx+ 2E∑D
(TPint)idx+
3F∑E
(TPint)idx
(22)
The contact points A to F along the path contact are givenin [24]Thus, the average total power loss for a mesh cycle is givenas
(TP )ave =TP
A′D′for NCR gear (23)
(TP )ave =TP
AFfor HCR gear (24)
where A’D’ and AF are the length of path of contact of NCRand HCR gear respectivelyThe efficiency is given by
η = 100
(Pin − (TP )ave
Pin
)(25)
The instantaneous total power loss and efficiency calculatedat every discrete point along the path of contact are shownin Figure 5. It is observed that the Benedict and Kelly [13]model predicts more power loss when compared to that ofXu et al.[1]. The Xu el al.[1] model has been used in thiswork for further parametric analysis on HCR gear drives.
III. EFFECT OF ADDENDUM
The instantaneous values of µ and hL obtained for twodifferent values of addendum (1m for NCR gear and 1.29mof HCR gear) are shown in Figure 6. Simulated results showhigher µ at the tip and root for HCR gear pair becauseof higher sliding velocities at these points due to enlargedaddendum. There is a sudden rise in the value of µ at highestpoint of single tooth contact (HPSTC) and lowest point ofsingle tooth contact (LPSTC) due to single pair contact forNCR gear pair. The hL is less in single pair contact regiondue to high load acting at this region for NCR gear. Whereas,hL is more near the pitch point region due to the reducedshared load caused by triple pair contact for HCR gear pair.
The sliding and rolling forces(Figure 7(a) to 7(b)), re-spective power losses(Figure 8(a) to 8(b)), total power lossand efficiency(Figure 9 to 10) are compared between NCRand HCR gears. It is observed that the total power lossis higher in HCR gear drive compared to that of NCRgear. This is attributed to the increase in mesh durationfor HCR gear drive compared to NCR gear drive thatultimately increases the power loss. The total power lossis the area under the total power loss curve and this hasbeen calculated using Simpson’s one third rule. The averagetotal power losses calculated for all the six cases (addendum=1m,1.093m,1.125m,1.158m,1.223m,1.291m) are shown inFigure 11. From the results, it is observed that an increase in
Proceedings of the 1st International and 16th National Conference on Machines and Mechanisms (iNaCoMM2013), IIT Roorkee, India, Dec 18-20 2013
173
−1 −0.5 0 0.5 10
0.005
0.01
0.015
0.02
0.025
TP
(k
W)
X/pb
Xu, et al. (2007)
Benedict and Kelley (1961)
LPTC HPTC
(a) Total mesh power loss
−1 −0.5 0 0.5 199
99.1
99.2
99.3
99.4
99.5
99.6
99.7
99.8
99.9
100
η
X/pb
Xu, et al. (2007)
Benedict and Kelley (1961)
LPTC HPTC
(b) Efficiency
Fig. 5. Instantaneous total mesh power loss and efficiency.
the addendum always increases both the sliding and rollingpower losses, but the amount of increase is more in slidingpower loss when compared to rolling power loss.
IV. CONCLUSIONS
Analytical study on instantaneous power loss in NCRand HCR gear drives and the effects of gear parameters onload dependent total mesh power loss on HCR gear driveshave been studied and discussed in this work.The simulatedresults reveal that an increase in addendum to increases thecontact ratio always increases the power loss in a similarway a decrease in the pressure angle to increase the contactratio also increases the power loss.
REFERENCES
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−1.5 −1 −0.5 0 0.5 1 1.50
0.005
0.01
0.015
0.02
0.025
µ
X/pb
ha =1 (ε=1.755)
ha =1.291 (ε=2.2)
HPTCLPTC
(a) Coefficient of friction
−1.5 −1 −0.5 0 0.5 1 1.55.6
5.65
5.7
5.75
5.8
5.85
5.9
5.95x 10
−8
hL
(m
)
X/pb
ha =1 (ε=1.755)
ha =1.291 (ε=2.2)
LPTC HPTC
(b) EHL film thickness
Fig. 6. Coefficient of friction and film thickness for NCR and HCR geardrive
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−1.5 −1 −0.5 0 0.5 1 1.50
1
2
3
4
5
6
7
8
Fs(N
)
X/pb
ha =1 (ε=1.755)
ha =1.291 (ε=2.2)
HPTCLPTC
(a) Sliding frictional force
−1.5 −1 −0.5 0 0.5 1 1.50.04
0.0405
0.041
0.0415
0.042
0.0425
0.043
0.0435
FR
(N)
X/pb
ha =1 (ε=1.755)
ha =1.291 (ε=2.2)
HPTCLPTC
(b) Rolling force
Fig. 7. Sliding and Rolling force
−1.5 −1 −0.5 0 0.5 1 1.50
0.001
0.002
0.003
0.004
0.005
0.006
Ps(k
W)
X/pb
ha =1 (ε=1.755)
ha =1.291 (ε=2.2)
LPTC HPTC
(a) Sliding power loss
−1.5 −1 −0.5 0 0.5 1 1.50.000108
0.000109
0.00011
0.000111
0.000112
0.000113
0.000114
0.000115
PR
(k
W)
X/pb
ha =1 (ε=1.755)
ha =1.291 (ε=2.2)
LPTC HPTC
(b) Rolling power loss
Fig. 8. Sliding and Rolling power loss
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−1.5 −1 −0.5 0 0.5 1 1.50
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
TP
(k
W)
X/pb
ha =1 (ε=1.755)
ha =1.291 (ε=2.2)
HPTCLPTC
Fig. 9. Total mesh power loss
−1.5 −1 −0.5 0 0.5 1 1.599.65
99.7
99.75
99.8
99.85
99.9
99.95
100
ηX/p
b
ha =1 (ε=1.755)
ha =1.291 (ε=2.2)
LPTC HPTC
Fig. 10. Efficiency
0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.350
1
2
3
4
5
6
7
Rolling power loss
Sliding power lossTotal power loss
Po
we
r lo
ss
(W
)
ha (mm)
Fig. 11. Power loss with respect to addendum.
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