AN INVESTIGATION OF LOAD-INDEPENDENT POWER LOSSES OF GEAR.pdf

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AN INVESTIGATION OF LOAD-INDEPENDENT POWER LOSSES OF GEAR

SYSTEMS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

By

Satya Seetharaman, B.E., M.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2009

Dissertation Committee:

Ahmet Kahraman, Advisor

Vish Subramaniam

Gary L. Kinzel

Robert A. Siston


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ABSTRACT

Physics-based fluid mechanics models are proposed to predict load-independent

(spin) power losses of gear pairs due to oil churning and windage. The oil churning

power loss model is intended to simulate spin losses in dip-lubricated conditions while

the windage power loss model is intended to simulate spin power losses under jet-

lubrication conditions. The total spin power loss, in either case, is defined as the sum of

(i) power losses associated with the interactions of individual gears with the environment

surrounding the gears, and (ii) power losses due to pumping of the oil or air-oil mixture at

the gear mesh. Power losses in the first group are modeled through individual

formulations for drag forces induced by the fluid, which is the lubricant in the case of oil

churning power losses and air or air-oil mixture in the case of windage power losses, on a

rotating gear body along its periphery and faces, as well as for eddies formed in the

cavities between adjacent teeth. Gear mesh pocketing/pumping losses are predicted

analytically as the power loss due to squeezing of the fluid as a consequence of volume

contraction of the mesh space between mating gears as they rotate. The pocketing losses

are modeled through means of an incompressible fluid flow approach in the case of oil

churning power losses. When the gear pairs rotate under windage conditions, a

compressible fluid flow methodology is considered for predicting the pocketing losses.

The power loss models are applied to a family of unity-ratio spur gear pairs to quantify


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the individual contributions of each power loss component to the total spin power loss.

The influence of operating conditions, gear geometry parameters and lubricant properties

on spin power loss are also quantified.

The oil churning and windage power loss models are validated through

comparisons to extensive experiments performed on spur gear pairs under dip- and jet-

lubricated conditions, over wide ranges of gear parameters and operating conditions.

The direct comparisons between model predictions and measurements demonstrate that

the model is indeed capable of predicting the measured spin power loss values as well as

the measured parameter sensitivities reasonably well, reinforcing the possibility of

utilizing the proposed model as a computationally effective design tool for predicting

power losses in geared systems. The spin power loss model is further generalized to

handle the several complex and varying gear configurations and operating conditions

present in an actual manual transmission in order to come up with a transmission spin

power loss model, which when coupled with a transmission mechanical power loss model

and existing bearing power loss prediction methodologies, can predict the total power

loss in a transmission. This transmission power loss model formed by these three power

loss components is validated through comparison to actual power loss measurements

from a six-speed example manual transmission, indicating that the transmission power

loss model can indeed be used for design and product improvement activities.


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Dedicated to

Amma, Appa

&

Chithra, Sujatha, Kavita


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ACKNOWLEDGEMENTS

I would like to express my heart-felt gratitude to my advisor, Prof. Ahmet

Kahraman, for guiding me throughout this fruitful journey, leading to the completion of

my dissertation. But for his invaluable foresight, discretion and plenteous

encouragement, it would be hard to fathom the maturity of my dissertation being beyond

just nascent, and in turn, my own growth onto an enriching career in research and letters.

It is often said that one need only put in the effort; one need not worry about the fruits of

ones labors. From him, I have learned to conduct research in an honest and dedicated

manner; I have learned to put in the hard yards without expectations. The fruition of such

an effort has been my development as an individual and engineer, en route to a more

complete being. My genuine appreciation is also due towards the wisdom and guidance

of Prof. Vish Subramaniam, all along the very many stages of my dissertation research.

His patience in listening to my research problem provided me with a platform to voice

my ideas and queries when the roads were hard to navigate. Many thanks are due

towards Prof. Gary Kinzel and Dr. Robert Siston for agreeing to serve on my doctoral

committee and for their searching inquiries and critical views of my dissertation research.

I am grateful to the financial support extended by General Motors Powertrain, Europe,

for sponsoring my dissertation work.


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I would like to thank Jonny Harianto and Sam Shon for their tremendous help and

technical expertise, in the areas of programming and experimentation. I have spent many

hours in their office, and my admiration for the nature of their job has only grown tenfold

over these years. Sincere thanks also go to Gary Gardner for putting up with my requests

in correcting uncorrectable holes in test rigs; most appreciation to David Talbot for

stimulating discussions on power losses and assistance with Visual Basic programming.

Thanks are also owed to all my lab mates for putting up with me. To Travis Petry-

Johnson, Mike Moorhead, Tim Szweda and Hai Xu, many thanks towards your help in

gear pair and transmission testing, and friction power loss calculations. To my friends,

particularly, Vijay Kanagala, Sachit Rao and Satyajit Ambike, while those endless hours

of arguments and articulations over coffee and similar such substances have detracted me

from my immediate goals, I have only one thing to say: you all rock! Live on!

I stand as who I am today because of the love, trust and support of my parents.

My words go only a small way in conveying my feelings towards them; so does this

dissertation, which is dedicated to them. I spend my wakeful hours in eternal debt to

them. Amma and Appa, you never stopped believing in me even when I did: look, I have

come home now. My three sisters, for whom this dissertation is dedicated to as well,

have nursed and guided me towards the twin pedestal of dignity and strength. No matter

what, my love and gratitude towards my sisters can never be broken. These have been

several wonderful years, between my transition from callow youth to patient adult. I

have lost a lot, but in the process, I have gained more than what I bargained for. Here is

to hoping my legs will keep running till they no longer can. At least, till one Boston.


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VITA

Feb. 18, 1981 ... Born Tiruchirapalli, India

Sep. 1998 Jun. 2002 . B.E., Mechanical Engineering, Regional

Engineering College, Tiruchirapalli, India

Aug. 2002 Jun. 2004 . M.S., Mechanical Engineering, Applied

Mechanics, Iowa State University, Ames, Iowa

Sep. 2004 Mar. 2005. Graduate Teaching Associate, Dept. of

Mechanical Engineering, The Ohio State

University, Columbus, OhioMar. 2005 present Graduate Research Associate, Gear and Power

Transmission, Research Laboratory, Dept. of

Mechanical Engineering, The Ohio State

University, Columbus, Ohio

PUBLICATIONS

1. Seetharaman, S. and Kahraman, A., 2009, Oil Churning Power Losses of a Gear

Pair: Model Formulation,ASME Journal of Tribology, 131(2), 022201 (11 pages).

2. Seetharaman, S., Kahraman, A., Moorhead, M. D., Petry-Johnson, T., T., 2009, Oil

Churning Power Losses of a Gear Pair Experiments and Model Validation,ASME

Journal of Tribology, 131(2), 022202 (10 pages).

FIELDS OF STUDY

Major Field: Mechanical Engineering


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TABLE OF CONTENTS

Page

Abstract. ii

Dedication. iv

Acknowledgments.... v

Vita... vii

List of Tables........ xii

List of Figures....... xiii

Nomenclature xvi

CHAPTERS:

1 Introduction..... 1

1.1Research Background and Motivation.. 1

1.2Literature Review...... 3

1.3Scope and Objectives............ 6

1.4Dissertation Outline.......... 9

References for Chapter 1............ 12


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2 Oil Churning Power Losses of a Gear Pair: Model Formulation....... 16

2.1Introduction....... 16

2.2

Gear Pair Oil Churning Power Loss Model...... 20

2.3Drag Power Losses....... ... 22

2.3.1 Power Loss due to Drag on the Periphery of a Gear..... 22

2.3.2 Power Loss due to Drag on the Faces of a Gear... 29

2.3.2.1Laminar Flow Conditions... 30

2.3.2.2

Turbulent Flow Conditions. 33

2.4

Power Loss due to Root Filling.... 35

2.5Oil Pocketing Power Loss. 40

2.5.1 Backlash Flow Area...... 44

2.5.2 End Flow Area.. 50

2.5.2.1Calculation of Area 2cQ . 51

2.5.2.2

Calculation of Area( ),1m

t jQ ... 54

2.5.2.3Calculation of Area ( ),1m

b jQ ... 56

2.5.3 Power Loss due to Oil Pocketing.. 58

2.6Example Oil Churning Analysis... 62

2.7

Concluding Remarks. 68

References for Chapter 2.... 70

3 Windage Power Losses of a Gear Pair: Model Formulation...... 75



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3.2Windage Pocketing Power Loss Model........ 78

3.3Windage Drag Power Losses........ 89

3.4Example Windage Power Loss Analysis and Parametric Studies.... 91

3.5

Concluding Remarks.. 105


4 Validation of Oil Churning and Windage Power Loss Models...... 108


4.2

Validation of Gear Pair Oil Churning Model... 110

4.2.1

Test Machine and Oil Churning Test Procedure....... 110

4.2.2 Gear Specimens and Parameters Studied...... 114


4.3Validation of Windage Power Loss Model....... 126

4.4Concluding Remarks. 130


5 Application of Oil Churning Power Loss Model to an Automotive transmission.. 132


5.2Validation of Gear Pair Oil Churning Model... 136

5.2.1 Test Machine and Oil Churning Test Procedure....... 138



5.3Validation of Windage Power Loss Model....... 142



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5.4.1 Test Machine and Oil Churning Test Procedure....... 147




6 Conclusions and Recommendations for Future Work.... 156

6.1Summary....... 156

6.2Conclusions and Contributions..... 157

6.3

Recommendations for Future Work.......... 160

Bibliography ........ 166


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LIST OF TABLES

Table Page

2.1 The design parameters of the example spur gears ....... 63


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LIST OF FIGURES

Figure Page

2.1 Definition of oil churning parameters for a gear pair immersed in oil..... 23

2.2 Definition of geometric parameters associated with root filling power losses. 36

2.3 Illustration of a side view of fluid control volumes of the gear mesh

interface at different rotational positions 1 2 30 m m m< < < . 41

2.4 Three-dimensional representation of a control volume showing backlash and

end flow areas .. 43

2.5 Geometry of two gears in mesh at an arbitrary position m . 45

2.6 Definition of the end area at an arbitrary position m ....... 46

2.7 Parameters used in calculation of (a) the total tooth cavity area 2cQ ,(b) the

overlap area

( )

,1

m

t jQ and (c) the excluded area

( )

,1

m

b jQ , all at an arbitrary positionm .... 52

2.8 Effect of (a) temperature T, (b) oil level parameter h , (c) face width band

(d) gear module m% on total spin power loss TP ... 65

2.9 Components of TP for a gear pair having 2.32m =% mm and 26.7,b = 19.5

and 14.7 mm at 1.0h = and 80 C; (a) dpP , (b) dfP , (c) rfP , and (d) pP ..... 67

3.1 Effect of (a) temperature T, (b) face width b and (c) gear module m% on

wP ..... 93


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3.2 Components of wP for the gear pairs having 2.32m =% mm and varying bat

o80 C; (a) windage pocketing power loss wpP (b) windage drag power

losses wdP ..... 96

3.3

Windage power loss wpP and its components, ,wp eP and ,wp bP , for gear pairs

having 2.32m =% mm ato80 C; (a) 14.7 mmb = (b) 19.5 mmb = (c)

26.7 mm.b = .... 97

3.4 Variation of the end flow area ( ),1m

e jA with rotational position for the gear

pairs having (a) 2.32m =% mm , and (b) 3.95m =% mm.. 100

3.5 Variation of (a) density, (b) pressure and (c) velocity of control volume

( )11

m of the gear pair having 2.32 mmm =% with rotational position for (a1,

b1, c1) the end area and (a2, b2, c2) the backlash area

( 19.5 mmb = , o80 C).... 102

3.6 Variation of (a) density, (b) pressure and (c) velocity of control volume

( )11

m of the gear pair having 3.95 mmm =% with rotational position for (a1,

b1, c1) the end area and (a2, b2, c2) the backlash area

( 19.5 mmb = , o80 C)........ 103

4.1 (a) A view and (b) the layout of the gear efficiency test machine [4.1, 4.2].... 111

4.2 One of the test gear boxes shown in dip-lubrication arrangement [4.1]....... 113

4.3 Two examples of test gears (a) 23-tooth gear with 3.95m =% mm and

19.5b = mm, and (b) 40-tooth gear with 2.32m =% mm and 19.5b = mm

[4.1, 4.2].... 115

4.4 Illustration of oil level parameters [4.2]... 117

4.5 Comparison of predicted to the measured [4.2] sTP for a gear pair (a) 30Co ,

(b) 50Co , (c) 70Co , and (d) 90Co ; 2.32m =% mm, 19.5b = mm and

1.0h = .. 120


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4.6 Comparison of predicted vs. measured sTP for a gear pair at (a) 0.05h = ,

(b) 0.5h = , (c) 1.0h = , and (d) 1.5h = ; oil at 80Co (up-in-mesh),

2.32m =% mm, and 19.5b = mm.... 122

4.7

Comparison of predicted to the measured [4.1] sTP for a gear pair having

2.32m =% mm and face width values (a) 14.7b = mm, (b) 19.5b = mm, and

(c) 26.7b = mm; oil at 80 Co and 1.0h = ...... 124

4.8 Comparison of predicted to the measured [4.2] sTP for a gear pair of face

width 19.5b = mm and modules (a) 2.32m =% mm and (b) 3.95m =% mm; oil

at 80Co and 1.0h = ..... 125

4.9 Comparison of predicted and measured [4.2] wP for (a) the 40-tooth gear

pair having 2.32 mmm =% and 26.7 mmb = and (b) 23-tooth gear pair

having 3.95 mmm =% and 19.5 mmb = ....... 127

5.1 Flowchart of the transmission power loss computation methodology..... 137

5.2 Example system: A 6-speed manual transmission ....... 143

5.3 Rotating gears on the planes of (a) 6th

gear pair, (b) 4th

gear pair, (c) 3rd

and

5th

gear pairs, (d) 2nd

gear pair, (e) 1stand reverse gear pairs, and (f) final

drive gear pair .......... 144

5.4 Experimental test setup for measuring transmission power losses of the

example 6-speed transmission. Safety guards are removed for

demonstration purposes............ 148

5.5 Comparison of measured [5.11] and predicted transmission mechanical

power loss values...... 150

5.6 Comparison of measured [5.11] and predicted transmission spin power loss

values: (a) 2ndgear, (b) 3rdgear, (c) 4thgear, (d) 5thgear, and (e) 6th gear.. 152


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NOMENCLATURE

A Area

b Gear facewidth

C Friction drag coefficient

c Specific heat

F Drag force

H Control volume

h Immersion depth

h Dimensionless immersion parameter

l Length parameter along gear faces

Total number of gear rotational increments

m Gear rotational position index [1, ]m M

m% Gear module

Number of teeth of gear

n Unit normal vector

n Average number of cavities

P Power loss

p Pressure


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Q Cavity area, Heat

R Universal gas constant

Re Reynolds number

r Gear radius

T Torque, temperature or tooth thickness

t Time

U Free-stream velocity

u Internal energy

V Volume

v Velocity vector

v Velocity

W Work

Axis parallel to gear face

Axis perpendicular to gear face

Boundary layer thickness

Energy per unit mass

Immersion angle

Ratio of specific heats

Momentum thickness

Viscosity

Angle, tangential direction

Density


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Shear stress

, Rotational speed in rpm and rad/s, respectively. 260

=

Flow factor

Stream function

Displacement thickness

Subscripts

b Base or backlash

c Cavity

d Drag

e End

f Face

1,2i= Gear index

j Index for control volumes

h Adiabatic

k Kinematic

m Rotational position

o Outside

p Periphery, pitch, pocketing

rf Root filling

s Start of active profile, shaft


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T Total

v Constant Volume

w Windage

Superscripts

m Rotational position

L Laminar

T Turbulent

r Radial

Tangential

w Wall


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CHAPTER 1

INTRODUCTION

1.1 Research Background and Motivation

Gears have found prevalent use in automotive industry, turbines and compressors,

gear pumps and most commonly in vehicle drive trains. Since gears transmit power

through rotational motion at different speeds, torques and direction, power is lost through

dissipation taking place due to friction between geared elements and spin losses

accounted for by the environment surrounding the system.

Power losses experienced by drive trains of passenger vehicles have been one of

the major concerns in automotive powertrain engineering over the past few decades.

Such losses directly impact fuel consumption of the vehicle, helping define how good a

vehicle is in terms of its fuel economy and gas/particulate emission levels. Recent

environmental regulations have made it imperative to look into the emission levels and

fuel economy of geared transmissions, and to seek improvements that might enhance

efficiency and reduce power losses. In tow with this line of thought, the study of power

losses and efficiency of geared transmissions has become an important area of interest, as


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dwindling fossil fuel resources has stressed the urgent need to come up with means of

improving efficiency of transmissions.

While fuel consumption alone is a sufficient reason for seeking reduced drivetrain

power losses, there are other supplementary reasons as well. Excessive power losses

within the transmission amount to additional heat generation and higher temperatures,

thus, adversely impacting gear contact fatigue and scuffing failure modes [1.1]. A gear

pair that is more efficient will result in lesser heat generation, and hence translates into

better performance. In addition, the design of the lubrication system as well as quantity

of lubricant within the transmission is also related to the amount of heat generated. A

more efficient transmission will free up the demands on the capacity and the size of the

lubrication system; consequently, the amount and quality of the gearbox lubricant are

also eased with improved efficiency. This in turn reduces the overall weight of the unit,

contributing to further efficiency enhancements and reduction in power loss.

In a geared system, the total power loss is comprised of two groups of losses: (i)

load-dependent (friction induced) mechanical power losses and (ii) load-independent

(viscous) spin losses. Sliding and rolling friction losses at the loaded gear meshes and at

the bearings largely define the load-dependent mechanical power losses. The total

mechanical loss is then given as the sum of losses from all gear meshes and bearings.

The sliding friction losses are related to the coefficient of friction, normal load and

sliding velocity on the contact surfaces while the rolling friction losses occur due to the

formation of an elastohydrodynamic (EHL) film [1.1]. Meanwhile, load-independent

spin losses are caused by a host of factors including viscous dissipation of bearings and


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gear oil churning and windage that are present as a result of oil/air drag on the face and

sides of the gears as well as pocketing/squeezing of lubricant in the meshing zone.

Moreover, oil shearing taking place in the synchronizers and in seals are also possible

sources of frictional and viscous dissipation, and can be added to the total transmission

power loss.

Spin loss in geared transmissions are of primary importance, as experimental

results have shown them to account for the bulk of the total power loss at higher speeds

of operation and at light loads. The spin power loss can be broken down into several

components as mentioned above. Drag losses, defined as losses taking place on the

periphery and faces of a gear, are computed separately for each gear of the gear pair and

then summed up to give the total drag loss of a gear pair. Power losses can also take

place due to swirling motion of lubricant in the cavity between adjacent teeth through the

mechanism of root filling and subsequent transportation into the meshing zone.

Pocketing/squeezing losses are defined as losses taking place in the meshing zone when

lubricant is squeezed out of the gear meshes due to the pumping action of gear pairs.

When gear pairs rotate in free air, windage loss takes over as the dominant mode of

power loss, requiring a compressible fluid flow approach to compute such losses.

1.2 Literature Review

While there many published studies on prediction of mechanical power losses of a

single gear pair (see ref. [1.1] for a detailed review of literature on such studies), there are

only a few published studies available on modeling spin losses in gear pairs and in multi-


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mesh gearboxes and transmissions. At low speeds of rotation, power losses mostly stem

from friction between gear teeth and oil/air drag acting on the faces and periphery of gear

pairs, resulting in friction and oil churning losses. At higher speeds, spin losses due to

pumping action in the meshing zone and windage also come into prominence. Several

seminal studies [1.2-1.4] focused on the drag torque and heat transfer associated with

circulation and secondary flows induced by rotation of discs submerged in a fluid. The

induced flows were dependent on the geometries of the bladed rotating elements and its

enclosure [1.2]. Subsequent methodologies for estimating churning and windage losses

in gears borrowed heavily from the fundamental principles espoused by the above

referenced studies, i.e. [1.2-1.4], and were mostly empirical adaptations of the same. The

environment inside the gearbox, which can at best be described as a pseudo-single phase

mixture of oil and vapor, and also the difficulties in measurement of gear teeth

temperature, has made the study of spin losses very challenging. As computational fluid

dynamics (CFD) tools are not very practical and not readily available to model such a

complex application, most of the published models or formulae to predict windage and/or

churning losses are based on dimensional analysis of experimental data or approximate

hydrodynamic formulations to characterize the flow of lubricant around rotating gears.

Following the works of Blok [1.5] and Niemann and Lechner [1.6], the first in

situ temperature measurements were made to calculate the real surface temperature at all

points along the line of action of mating gears [1.5], for a heat transfer approach towards

computing power losses. Bones [1.7] carried out churning torque loss measurements for

discs of varying geometries at different immersion depths for various fluids. Using a

combination of von Karmans [1.8] equations and a modified Reynolds number, Bones


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proposed a methodology to calculate churning losses in gears. Moreover, he also

proposed the existence of three different regimes, which the churning losses fall under:

laminar, transition and turbulent. Terekhov [1.9] proposed a similar methodology for

single and meshing gears by testing several gear pairs of different geometries under

highly viscous lubricant conditions and low rotational speeds [1.10]. Both Bones and

Terekhov expressed churning loss in terms of a dimensionless churning torque. Luke and

Olver [1.10] carried out a number of experiments to determine churning loss in single and

meshed spur gear pairs. Experimental conditions were similar to that proposed by Bones,

and geometry of the gear pairs and operating conditions were varied to study their

influence on churning torque.

Ariura et al. [1.11] measured losses from jet-lubricated spur gear systems

experimentally. They proposed an analysis of the power required to pump the oil trapped

between mating gears. Akin et al. [1.12, 1.13] analyzed the effect of rotationally induced

windage on the lubricating oil distribution in the space between adjacent gear teeth in

spur gears. The purpose of their study was to provide formulations to study lubricant

fling-off cooling. They proposed that impingement depth of the oil into the space

between adjacent gear teeth and the point of initial contact was an important aspect in

determining cooling effectiveness. Pechersky and Wittbrodt [1.14] analyzed fluid flow in

the meshing zone between spur gear pairs to assess the magnitude of the fluid velocity,

temperature and pressures that result from meshing gear teeth. By far, this study can be

considered as one of the few computational models available to predict the pumping

action of lubricant trapped in the meshing zone, while it does not extend the formulation

to computation of the power loss due to the pumping action in the meshing zone. Diab et


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al [1.15] came up with an approximate hydrodynamic model of oil-air squeezing in the

meshing zone of spur and helical gear pairs. They inferred this phenomenon of pumping

the lubricant from the gear mesh to be a substantial source of power loss, especially under

high operating speeds [1.15].

Meanwhile, most of the studies on gear windage have been experimental,

focusing on losses of gears or disks of various sizes rotating in free air and surrounded by

different enclosures. Dawson [1.16] studied the problem of gear windage extensively and

investigated windage losses from isolated spur/helical gears rotating in free air. Several

empirical formulae resulted from this study. Diab et al [1.17] carried out windage-related

experiments with disks and gears of various shapes and sizes and subsequently came up

with predictions based on fluid flow around a rotating gear and also through dimensional

analysis, to characterize windage loss in pinion-gear pairs. Eastwick and Johnson [1.18]

provided an extensive review of studies on gear windage to conclude that the general

solution for reducing power loss due to windage has not yet been well-established. Wild

et al. [1.19] studied the flow between a rotating cylinder and a fixed enclosure by using a

CFD model and through experiments. Similarly, Al-Shibl et al. [1.20] proposed a CFD

model of windage power loss from an enclosed spur gear pair.

1.3 Scope and Objectives

As seen from the literature review, spin power loss models mostly incorporate

either a dimensional analysis based approach or an approximate hydrodynamic/CFD-

based methodology. The dimensional analysis approach deduces a lot from experimental


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observations that are particular to the environment whereas the approximate

computational formulations are intensive in effort, and also do not shed sufficient light

into the physics of spin power losses. When the majority of the factors involved in

computing spin losses are taken into account, the nature of the environment surrounding

the gear pairs in a transmission or gearbox makes it extremely difficult to formulate a

fluid mechanics-based approach. Also, to perform extensive studies based on oil level,

gear geometry parameters, oil inlet temperature, lubricant parameters and operating

conditions requires a spin loss model that will be able to throw insight into the

contributions of operating parameters without compromising on computational

efficiency.

Accordingly, the first objective of this dissertation is to develop a physics-based,

analytical, fluid mechanics model of gear pair spin power losses, which is capable of

quantifying the impact of key system parameters with minimal computational effort.

Specific objectives in this regard are as follows:

Develop physics-based fluid mechanics models for prediction of spin power

losses of single gear pairs that incorporates the effect of key parameters such as

gear geometry, operating conditions and lubricant properties on spin power loss.

These models will account for losses associated with oil churning and air

windage.

Perform detailed parametric studies to identify and rank-order the key

parameters influencing spin power losses of a gear pair.


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Since gear pair spin loss models were not previously formulated in an analytical,

physics-based manner to accommodate the effect of the various parameters that influence

them, the modeling of power losses of an entire transmission has also been based mostly

on measurements and empirical formulations. Accordingly, the second objective of this

dissertation is to develop a methodology for predicting the overall power loss of a multi-

mesh gear transmission. Specifically, this dissertation will accomplish the following:

Develop a physics-based transmission mechanical power loss model by

incorporating a generalized power flow formulation, which combines a gear

pair mechanical efficiency formulation proposed by Xu et al. [1.1] and the

bearing mechanical power loss model proposed by Harris [1.21].

Generalize the gear pair spin loss model developed for single gear pairs so as to

handle any gear pair configuration observed in a transmission and incorporate

the same with a bearing viscous loss model [1.21] to predict the total spin loss

of the transmission.

Perform detailed parametric studies to identify and rank-order the key system-

level parameters influencing total power loss of a transmission.

As is the case with any model, these formulations cannot be used in confidence

unless validated through comparisons to actual experiments. The last objective of this

dissertation is the validation of the above proposed models. Specifically:

Validation of the gear pair oil churning power loss model by comparing

predictions to measurements from another sister study on oil churning power


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losses, over a wide range of input speed, oil inlet temperature, geometric

parameters, oil level and lubricant conditions [1.22].

Validation of the gear pair windage power loss model by comparing predictions

to measurements from the gear windage experiments of Petry-Johnson et al

[1.23] and to the windage experiments of Dawson [1.16].

Validation of both the spin and mechanical power loss predictions of the

transmission power loss model through comparisons to measurements from a

companion study [1.24] on a six-speed sample manual transmission.

1.4 Dissertation Outline

Chapter 2 details the methodology of oil churning power losses in gear pairs, with

subsections focusing on the physics behind drag loss on the periphery and on the faces of

the gears in the laminar and turbulent regime, root filling mode of power loss due to the

formation of eddies in the cavity between adjacent gear teeth, and, finally, power loss due

to pocketing/squeezing taking place in the meshing zone as a consequence of successive

compression/expansion of tooth cavity volume. The oil churning power loss

formulations assume the fluid surrounding the gears as well as the fluid inside the

meshing zone to be incompressible in nature. Detailed formulations are presented for

each component of oil churning power loss, with the chapter winding up with an example

oil churning analysis that includes extensive parametric studies on the influence of


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lubricant parameters, gear geometry parameters and operating conditions on oil churning

power losses.

Chapter 3 is devoted to the formulation of a gear pair windage power loss model.

The nature of the compressible air-oil mixture in the meshing zone under jet-lubrication

conditions affords for a departure from the incompressible pocketing power loss

formulation developed as a part of the oil churning power loss model. A modeling

strategy based on the conservation laws is proposed to calculate the power loss due to the

compressible mixture being squeezed out through the meshing zone; drag loss

formulations from the oil churning power loss model are adapted to include the effect of

windage conditions. Chapter 3 also presents an example windage power loss analysis,

with adequate light thrown on the influence of system parameters on the total windage

power loss, which will be given by the sum of the windage drag and windage pocketing

power losses.

In Chapter 4, the gear pair oil churning and windage power loss models delineated

in Chapters 2 and 3 are validated through comparisons to measurements conducted on a

family of unity-ratio spur gear pairs, under dip- and jet-lubricated conditions, so as to

simulate oil churning and windage modes of operation. This extensive validation effort

includes measurements conducted at different speed and temperature conditions, over

different gear design variations as well as with the gear pairs being operated under

different lubricant conditions.

Chapter 5 deals with the proposition and validation of the transmission power loss

model, which incorporates the generalized transmission spin and mechanical power loss


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models, along with an existing bearing power loss formulation, to compute the total

power loss in an automotive transmission. Validation efforts are carried out through

comparisons to measurement from a sample six-speed manual transmission operated over

a wide range of parameters and under several gear configurations.

Literature review pertaining to particular parts of the formulation proposed in this

dissertation is made available at the beginning of each chapter towards an easy

understanding of the body of work lying behind the relevant models. The repetition of

such critical reviews on initial and existing methodologies between the introductory

chapter and the rest of this dissertation is intentional.

Finally, Chapter 6 provides an extended summary of the entire work while

outlining the major conclusions and contributions of this dissertation. Also, a detailed list

of recommendations for future work is presented at the end of Chapter 6.


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References for Chapter 1

[1.1] Xu, H., Kahraman, A., Anderson, N.E., Maddock, D.G., 2007, Prediction of

Mechanical Efficiency of Parallel-Axis Gear pairs, ASME Journal of Mechanical

Design, 129, 58-68.

[1.2] Daily, J. W., and Nece, R. E., 1960, Chamber Dimensional Effects on Induced

Flow and Frictional Resistance of Enclosed Rotating Disks, ASME Journal of

Basic Engineering, 82, 217232.

[1.3] Mann, R.W., and Marston, C.H., 1961, Friction Drag on Bladed disks in

Housings as a Function of Reynolds Number, Axial and Radial Clearance and

Blade Aspect Ratio and Solidity, ASME Journal of Basic Engineering, 83 (4),

719-723.

[1.4] Soo, S. L., and Princeton, N. J., Laminar Flow Over an Enclosed Rotating Disc,

Transactions of the American Society of Mechanical Engineers, 80, 287-296.

[1.5] Blok, H., 1957, Measurement of Temperature Flashes on Gear Teeth under

Extreme Pressure Conditions, Proceeding of The Institution of Mechanical

Engineers, 2, 222-235.

[1.6] Niemann G., and Lechner G., 1965, The Measurement of Surface Temperature

on Gear Teeth,ASME Journal of Basic Engineering, 11, 641-651.


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[1.7] Bones, R.J., 1989, Churning Losses of Discs and Gears Running Partially

Submerged in Oil, Proceedings of the ASME 5th International Power

Transmission and Gearing Conference, Chicago, 355-359.

[1.8]

Von Karman, T., 1921, On Laminar and Turbulent Friction, Z. Angew. Math.

Mech., 1, 235-236.

[1.9] Terekhov, A.S., 1975, Basic Problems of Heat Calculation of Gear Reducers,

JSME International Conference on Motion and Powertransmissions, Nov. 23-26,

1991, 490-495.

[1.10]

Luke, P., and Olver, A., 1999, A Study of Churning Losses in Dip-Lubricated

Spur Gears, Proc. Inst. Mech. Eng.: J. Aerospace Eng., Part G, 213, 337346.

[1.11] Ariura, Y., Ueno, T., and Sunamoto, S., 1973, The lubricant churning loss in

spur gear systems,Bulletin of the JSME, 16, 881-890.

[1.12] Akin, L. S., and Mross, J. J., 1975, Theory for the Effect of Windage on the

Lubricant Flow in the Tooth Spaces of Spur Gears, ASME Journal of

Engineering for Industry, 97, 12661273.

[1.13] Akin, L. S., Townsend, J. P., and Mross, J. J., 1975, Study of lubricant jet flow

phenomenon in spur gears,Journal of Lubrication Technology, 97, 288-295.

[1.14] Pechersky, M. J., and Wittbrodt, M. J., 1989, An analysis of fluid flow between

meshing spur gear teeth, Proceedings of the ASME 5th International Power

Transmission and Gearing Conference, Chicago, 335342.

[1.15] Diab, Y., Ville, F., Houjoh, H., Sainsot, P., and Velex, P., 2005, Experimental

and Numerical Investigations on the Air-Pumping Phenomenon in High-Speed


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Spur and Helical Gears,Proceedings of the Institution of Mechanical Engineers,

Part C: J. Mechanical Engineering Science, 219, 785-800.

[1.16] Dawson, P. H., 1984, Windage Loss in Larger High-Speed Gears, Proceedings

of the Institution of Mechanical Engineers, Part A: Power and Process

Engineering, 198(1), 5159.

[1.17] Diab, Y., Ville, F., and Velex, P., 2006, Investigations on Power Losses in High

Speed Gears,Journal of Engineering Tribology, 220, 191298.

[1.18] Eastwick, C. N., and Johnson, G., 2008, Gear Windage: A Review, ASME

Journal of Mechanical Design, 130, 034001, 6 pages.

[1.19] Wild, P. M., Dijlali, N., and Vickers, G. W., 1996, Experimental and

Computational Assessment of Windage Losses in Rotating Machinery, ASME

Trans. J. Fluids Eng., 118, 116122.

[1.20] Al-Shibl, K., Simmons, K., and Eastwick, C. N., 2007 Modeling Gear Windage

Power Loss From an Enclosed Spur Gears, Proceedings of the Institution of

Mechanical Engineers, Part A, 221(3), 331341.

[1.21] Harris, A. T., 2001,Rolling Bearing Analysis, Fourth Edition, Wiley & Sons, Inc.,

New York.

[1.22] Moorhead, M., 2007, Experimental Investigation of Spur Gear Efficiency and

the Development of a Helical Gear Efficiency Test Machine, M.S. Thesis, The

Ohio State University, Columbus, Ohio.

[1.23] Petry-Johnson, T. T., Kahraman, A., Anderson, N.E., and Chase, D. R., 2008, An

Experimental Investigation of Spur Gear Efficiency, ASME Journal of

Mechanical Design,130 (6), 062601, 10 pages.


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[1.24] Szweda, T, A., An Experimental Study of Power Loss of an Automotive Manual

Transmission, MS thesis, The Ohio State University, Columbus, Ohio, 2008.


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CHAPTER 2

OIL CHURNING POWER LOSSES OF A GEAR PAIR: MODEL

FORMULATION

2.1 Introduction

Power losses in gear pairs and transmissions can be broadly classified into two

groups as detailed in the first chapter: load-dependent (friction induced or mechanical)

power losses and load-independent (spin) power losses due to viscous dissipation. In

automotive applications, geared components of systems such as manual transmissions,

transfer cases and front or rear axles might rotate at reasonably high speeds (say, gear

pitch-line velocities in excess of 20 to 30 m/s) to cause significant amounts of spin power

losses. While load-dependent and spin power losses can be comparable in magnitude

under high-load and low operating speed conditions, the spin losses typically dominate

the overall power losses at these higher operating speeds.

Spin power losses of a gearbox are either due to churningof the lubricant if the

rotating components of the gearbox are immersed in an oil bath (dip-lubricated) or due to

windageif the lubrication method is jet-type and the surrounding medium is air or a fine

mist of air and oil. Focusing on the most fundamental component of the gearbox, i.e. a

gear pair in mesh, this chapter aims at developing a novel physics-based fluid


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mechanics model of oil churning losses due to interactions of the gears, both as a pair and

as individual entities, with the surrounding lubricant medium. A companion model to

handle windage losses in a jet-lubrication scenario will be proposed in Chapter 3.

While there has been a large body of work dealing with load-dependent power

losses ([2.1, 2.2]), there are only a few published studies on modeling spin losses.

Several studies (e.g. [2.3-2.4]) proposed formulations for the drag torque associated with

circulation and secondary flows induced by rotation of a disc submerged in a fluid.

Subsequent studies on churning and windage losses in gears relied heavily on these

fundamental studies, and they can mostly be characterized as empirical adaptations of the

same. The convolution of the environment inside the gearbox, which can at best be

described as a pseudo-single phase mixture of oil and vapor, as well as the complexity of

gears rotating in mesh have made the study of spin losses a very challenging one. As

computational fluid dynamics (CFD) tools are not very practical and not readily available

to model such a complex application, most of the published models or formulae to predict

windage and/or churning losses are based on dimensional analysis of experiments or

approximate hydrodynamic formulations to characterize the flow of lubricant around

rotating gears.

Taking a critical view of existing methodologies, Bones [2.5] carried out churning

torque measurements for discs of varying geometries at different immersion depths using

three different fluids. Using a combination of von Karmans [2.6] equations and a

Reynolds number based on the disc chord length as the characteristic length, he proposed

a methodology to calculate churning losses in gears in three different regimes: laminar,


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transition and turbulent. Terekhov [2.7] proposed a similar methodology for single and

meshing gears based on gear experiments under highly viscous lubricant conditions and

low rotational speeds. Both Bones and Terekhov expressed churning loss in terms of a

dimensionless churning torque. Luke and Olver [2.8] performed a number of

experiments to determine churning loss in single and meshed spur gear pairs. They

compared their experimental observations on spin power losses with the empirical

formulations of Bones [2.5] and Terekhov [2.7] and found that contrary to what Bones

had predicted, the spin power losses were not strongly affected by the viscosity of the

lubricant. Furthermore, their observations called into question the attempt used to

characterize spin power loss based on a Reynolds number dependent on lubricant

viscosity. Ariura et al. [2.9] measured losses from jet-lubricated spur gear systems

experimentally. They proposed an analysis of the power required to pump the oil trapped

between mating gears. Akin et al. [2.10, 2.11] analyzed the effect of rotationally induced

windage on the lubricating oil distribution in the space between adjacent gear teeth in

spur gears. The purpose of their study was to provide formulations to study lubricant

fling-off cooling. They proposed that impingement depth of the oil into the space

between adjacent gear teeth and the point of initial contact was an important aspect in

determining cooling effectiveness.

Pechersky and Wittbrodt [2.12] analyzed fluid flow in the meshing zone between

spur gear pairs to assess the magnitude of the fluid velocity, temperature and pressures

that result from meshing gear teeth. This work can be considered as one of the few

computational models available to study the pumping action of lubricant trapped in the

meshing zone, while it does not extend the formulation to computation of the power loss.


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Diab et al [2.13] came up with an approximate hydrodynamic model of oil-air squeezing

in the meshing zone of spur and helical gear pairs. They inferred this phenomenon of

pumping the lubricant from the gear mesh to be a substantial source of power loss,

especially under high operating speeds [2.13]. A more recent study by Changenet and

Velex [2.14] investigated the influence of meshing gear on oil churning power losses by

performing a number of gear oil churning experiments to come up with empirical

formulae for power losses. Parameters included were gear module, diameter and face

width, speed and lubricant viscosity. Their empirical formulae suggested that the

influence of viscosity on oil churning loses was insignificant at high speeds of rotation

for single gears, corroborating similar findings from the experimental observations of

Luke and Olver [2.8]. Another relevant work by Hhn et al [2.15] also stresses this

apparent lack of dependence of oil type on load independent losses. In their experiments,

Hhn et al [2.15] measured gear and bearing power losses and forged a balance between

generated heat in the gearbox due to gears and bearings and the dissipated heat in the

form of free and forced convection and through radiation as well, from housing and

rotating parts, to calculate mean lubricant temperature.

Examination of the above body of literature reveals that gear pair spin power loss

models were either based on dimensional analysis of controlled experimental data or a

CFD-based computational methodology, with unconvincing light thrown on the influence

of key system parameters on the viscous power losses. Both approaches have major

limitations in terms of their practicality and computational demand, and hence, come

short of providing a better understanding of gear spin power losses, including the

significance of key system parameters. Accordingly, this chapter aims at developing a


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physics-based, analytical, fluid mechanics model of gear pair spin power losses, which is

capable of quantifying the impact of key system parameters with minimal computational

effort. Load-independent windage power losses will be handled separately in Chapter 3

by taking into account the compressible nature of air or air-oil mixture surrounding the

gears and in the meshing zone, with validation of both the models presented in Chapter 4

through comparisons to tightly controlled experiments and empirical studies.

2.2 Gear Pair Oil Churning Power Loss Model

The churning power losses of a gear pair can be grouped into two categories. The

first category is comprised of drag losses associated with the interactions of each

individual gear with the surrounding medium. For a gear i that is partially or fully

immersed in oil, the drag power loss will be modeled as the sum of three individual

components:

di dpi dfi rfiP P P P= + + , 1, 2,i= (2.1)

where dpiP is the power loss due to oil/air drag on the periphery (circumference) of a

gear, dfiP is the power loss due to oil/air drag on the faces (sides) of a gear, and rfiP is

the power loss that occurs during the filling of the cavity between adjacent teeth with oil.

With these components determined for each gear, the total drag power loss of the gear

pair is then found as 1 2d d dP P P= + .


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The second category of churning power losses consists of losses due to

interactions of the gear pair with surrounding medium (oil) at the gear mesh interface,

with squeezing/pocketing losses being the dominant mode of power loss. With the gear

mesh pocketing power losses pP and drag power losses predicted, the total churning

power losses of a gear pair is then found as

T d pP P P= + . (2.2)

The following sections describe the details of the oil churning power loss model.

Specifically, Section 2.3 provides the drag power loss formulations along the periphery

and faces of each gear and Section 2.4 details the power loss due to pocketing,

establishing the framework for computing the backlash and end flow areas, as well as

computing the power loss due to pocketing. Many other sources of churning power

losses can also be identified, including those associated with lift-off of the leftover oil

within the mesh at the exit of the mesh and power loss due to the transport or acceleration

of the lubricant swirling inside each tooth cavity as it is carried into the meshing zone.

These effects are not included in the proposed model as preliminary formulations of these

effects suggest that they are secondary. Similarly, the effect of enclosures in the form of

flanges or shrouds in the near vicinity of rotating gears as reported experimentally by

Changenet and Velex [2.16] will not be included as such effects are beyond the scope of

this dissertation.


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2.3 Drag Power Losses

2.3.1 Power Loss Due to Drag on the Periphery of a Gear

A rotating gear pair that is fully or partially immersed in oil, as shown in Figure

2.1, is subjected to drag forces that are induced along the direction of flow on the

periphery (circumference) and faces (sides) of the gears, thus, contributing to churning

power losses. A gear pair rotating in free air experiences similar drag forces in the form

of air windage, as detailed in Chapter 3. In formulating the drag forces and drag power

losses due to oil churning, each gear is modeled as an equivalent circular cylinder of

radius oir (the outside radius of the gear). This employs the assumption that at medium

to high speeds of rotation, the behavior of a gear immersed in oil follows that of a

cylindrical disk, as the oil swirling around the gear will not feel the effects of the tooth

cavities. The implication of the above assumption can be extended to gears with

increased tooth height as well, because the Reynolds number defined below in the

following section takes into account the characteristic length scale as the outside diameter

of the gears and an increase in tooth height will be reflected in an increase in the outside

diameter and hence can be accommodated in the drag loss formulation. It has to be noted

though that at low speeds of rotation, substantial changes in tooth thickness will change

the nature of the boundary layer from that seen for flow around a circular disk.

Steady-flow conditions are assumed in this formulation such that 0v t = and

0p t = . The oil pressure is assumed to vary only in the radial direction, i.e.

( )i ip p r= and 0ip = .


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Figure 2.1 Definition of oil churning parameters for a gear pair immersed in oil.

1

1

1or 1O

1l

1h 2h 2

2l

2O

2

2or

Gear 1

Gear 2

Oil Level


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Further, the radial component of the oil velocity ( )rdpiv is considered to be zero. In

addition, flow conditions are assumed to be incompressible, with density remaining

unaffected by changes in pressure, while the viscosity of the lubricant has a strong

exponential dependence on the operating temperature. It is further assumed that under

incompressible flow conditions, changes in pressure in the meshing zone have a

negligible effect on the operating viscosity and density of the lubricant. Also, changes in

pressure close to the contact zone are not relevant to the load-independent power losses,

and as a result, overall changes in pressure are not reflected in the viscosity

characteristics of the lubricant, either in the meshing zone or the contact region.

Meanwhile, the bulk temperature of oil was assumed to be known in this model.

From the standpoint of the analysis, the drag along the periphery reflects the case

of flow in the annulus between two rotating disks, with the outside disk remaining

stationary and placed at infinity to simulate the casing and the inner disk rotating at the

operating speed of the gears in consideration. Regarding the presence of any flanges

enveloping the gears, the drag formulations are valid for as long as the distance between

the flanges or shrouds and the gear surfaces is greater than the total boundary layer

thickness of both the surfaces, i.e. from the gears and the flanges, and as such, the current

formulation cannot, in its entirety, be extended to include the effect of flanges. A more

qualitative analysis, taking into account the interactions between boundary layers arising

from the surfaces of the gear as well as that of the flanges, once the flanges are close

enough to the gear surfaces, must be taken into account to model the presence of flanges.

This boundary layer could be a combined boundary layer or a separate boundary layer,


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under laminar or turbulent conditions, based on the rotational speed. Future formulations

and extensions to the drag power losses will need to focus on the same. Also, surface

roughness effects are not taken into consideration while formulating the drag power

losses. Surface roughness might play a tertiary role in case of gears with very rough

surfaces, potentially changing the behavior of drag forces, especially in the turbulent

regime. The drag formulations consider only friction drag and the effects of form drag

has not been characterized in this dissertation. The inclusion of form drag will need more

information on flow separation and behavior of the lubricant beyond the wake region and

as has been noted earlier, it is assumed that there is a negligible wake region on

separation, which gives enough reason to neglect form drag in the current formulation.

Also to be noted is that the effects of air ingestion at medium speeds and low oil levels

will change the nature of the analysis from incompressible to compressible flow and

hence, the dependence of density on temperature and pressure would change, bordering

the analysis on two-phase flow. Imposing single-phase conditions on the drag

formulations can be considered as a gross simplification that avoids the computational

difficulties involved in a two-phase mixture flow.

Based on the above detailed assumptions, the velocity ( )dpiv

of the oil along the

tangential direction on the periphery of the gear and the pressure distribution around the

periphery are defined by solving the continuity equation [2.17]

( ) ( ) ( )1

[ ] 0

r rdpi dpi dpiv v v

r r r

+ + =

, (2.3a)


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and the Navier-Stokes equations of motion [2.17] along r and directions

( ) ( ) ( ) ( ) ( ) 2( )

( ) ( )2( )

2 2

[ ]

v

1 [ 2 ],

r r rdpi dpi dpi dpi dpir

dpi

rdpi dpiri

dpi

v v v v v

t r r r v vp

vr r

+ + =

+

(2.3b)

( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( )2 2( )2

2 2 2

1 [ 2 ].

rdpi dpi dpi dpi dpi dpi

dpi

rdpi dpi dpi dpii

dp

v v v v v vv

t r r r

v v v vpr r v

r rr r

+ + + =

+ + + +

(2.3c)

where and are the density and the dynamic viscosity of the lubricant at a given

operating temperature. Implementing the assumptions stated earlier, one obtains

( )

0dpiv

=

, (2.3d)

( ) 2[v ] idpip

r r

=

, (2.3e)

( ) ( ) ( )2

2 2

10

dpi dpi dpid v dv v

r drdr r

+ = . (2.3f)

The boundary conditions for the oil circulating along the periphery of the gear (and also

at the end of the enclosure, assuming the gear to be immersed in an enclosure of infinite

depth) are given as ( ) i oidpiv r = at oir r= and

( ) 0dpiv = as r , where i is the


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rotational speed of the gear and [ , )oir r . Solving Eq. (2.3f) and applying these

boundary conditions, the fluid velocity along the periphery of the gear is found as

2( ) i oidpi

rv

r

= . (2.4)

Given the dynamic viscosity of the oil, the radial, axial and tangential

components of the shear stress built up on the periphery due to the rotational motion are

evaluated next as

( )( )

2 0

rdpirr

dpi

v

r

= =

, (2.5a)

( ) ( )( ) 1

2 [ ] 0

rdpi dpi

dpi

v v

r r

= + =

, (2.5b)

( ) ( ) 2( )

2

v v 21[ ( ) ]

rdpi dpir i oi

dpi

rr

r r r r

= + =

. (2.5c)

The above set of equations indicates that only the tangential shear stress acting on the

periphery of the gear is nonzero. From Eq. (2.5c), the tangential shear stress at the

outside radius of gear i, oir r= (tangential wall shear stress), is found as( )

2

w

idpi = .

Defining the friction drag coefficient as ( ) 22 wdpi idpiC U= , where i i oiU r= is the free-


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stream velocity of the lubricant near the periphery of the gear i, the drag force acting on

the periphery of the gear is found as

( )21

2

wdpi i dpi dpi dpi dpiF U A C A= = . (2.6)

Here, the wetted surface area of the periphery is given as 2dpi i oi iA r b= , where ib is face

width of gear iand 1cos [1 ]i ih = . A dimensionless immersion parameter is defined

from Figure 2.1 as i i oih h r= , where ih is the immersion depth of gear i. Accordingly,

0ih represents the case of no oil-gear interactions and 2ih corresponds to the case

of a gear that is submerged in oil. For 2ih , 2ih = is used in the above formulation to

obtain i = such that the entire periphery of the gear is subject to drag, i.e.

2dpi oi iA r b= . For 0ih , it is understood that the gear is submerged in air. Then, the

parameters 2ih = and i = are used together with the properties of air to calculate air

windage loss at the periphery of the gear, as proposed in Chapter 3.

It has to be noted that the oil levels considered are static oil levels, though in

reality, oil levels will not remain static with gear rotations, resulting in dynamically

changing oil levels. Inclusion of these dynamic oil levels that would require a more

advanced computational model is beyond the scope of the present formulation whose

main aim is to provide a simple, physics-based explanation of the major phenomena

pertinent to oil churning losses. Also, to a certain extent, the assumption of a static oil


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level is still relevant based on the fact that a reduction of oil level at one chamber or

location will average off the increase in dynamic oil levels at some other location within

the transmission/gearbox. As a result, the averaging effect of dynamic oil level plays into

the hands of assuming static oil levels to circumvent modeling difficulties. With dpiA

and( )wdpi known, the drag force is then evaluated as

4dpi i oi i iF b r= . (2.7a)

Finally, the power loss due to drag on the periphery of a single gear is given as the

product of dpiF and the tangential velocity at the periphery as

2 24dpi i oi i iP b r= . (2.7b)

2.3.2 Power Loss Due to Drag on the Faces of a Gear

In modeling the power loss due to drag on the faces of a gear, laminar and

turbulent flow regimes will be handled separately. At low to medium speeds, depending

on the size of the individual gears, the flow can be assumed to be laminar, with transition

to turbulence taking place at higher rotational speeds corresponding to a Reynolds

number defined as 2Re 2 i oir= within the range5(10) to 6(10) [2.17]. Because of a

large velocity gradient between outer and inner flow due to the no-slip boundary

conditions on the surface of the gear, viscous effects are no longer negligible and a


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boundary layer is formed along the faces of the gear, in which viscous effects become

predominant. Assuming that the boundary layer remains close to the surface of the gear,

with a negligible wake region formed on separation, the flow across the faces of a gear

can be modeled as flow across a flat plate, under laminar or turbulent flow conditions. In

cases when the gear is fully submerged in oil, an alternate method of flow near a rotating

disk [2.18] can be used, which provides the same dependence of power loss on the faces

with the operating parameters as detailed below in the present formulation. However, the

approach of modeling the flow across the faces as flat plate flow is more advantageous

since it can handle partial immersion cases and turbulence more conveniently.

2.3.2.1 Laminar Flow Conditions: In the laminar regime, the flow velocity

profile( )Ldfiv is considered to be linear, i.e.

( ) ( )L Li idfiv U y= , (2.8a)

where( )Li is the laminar boundary layer thickness, iU is the free-stream velocity of the

lubricant flowing across the faces of the gear ias defined previously, and is the axial

direction perpendicular to the faces (coming out of the paper) along which the velocity

profile varies. The displacement thickness and momentum thickness of the boundary

layer are defined, respectively, as


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12

0

[1 ]

dfidfi

i ii

vdy

U

= = , (2.8b)

16

0

= [1 ]

dfi

dfi dfii ii i

v vdy

U U

= . (2.8c)

Here, i gives the distance that the outer streamlines are shifted or displaced outward of

the surface of the gear as a result of retarded flow in the boundary layer, whereas i

represents the reduced momentum flux in the boundary layer as a result of shear force on

the surface of the gear [2.17]. Due to the mostly viscous flow across the gear surface,

with k being the kinematic viscosity of the oil, the skin friction coefficient,( )LiC must

be taken into account and is defined as

( )( )

2 ( )0

22=

LdfiL k

i Li iiy

dvC

dyU U=

=

. (2.8d)

The boundary layer momentum integral equation of von Karman [2.17] is applied to

solve for the boundary layer thickness. Since flow across the faces of the gear is taken to

be the case as flow across a flat plate for which the free-stream velocity iU is constant,

the rate of increase of momentum thickness is directly proportional to the wall shear

stress, i.e.


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12

iiC

x

=

. (2.9)

By setting ( )Li iC C= and ( )Li i = to represent the laminar regime and incorporating

Eqs. (2.8b) and (2.8c), Eq. (2.9) is solved to obtain the boundary layer thickness as

( )3.46

L ki

iU

= , (2.10)

wherexis the length parameter, as defined in Figure 2.1. At 2 sini oi ix r= = l , the skin

friction coefficient is obtained by substituting Eq. (2.10) into Eq. (2.8d) as

( )0.578

i

L k

i i

CU

=

ll

. (2.11)

The drag force on the face of the gear in the laminar regime is then given in terms of the

skin friction coefficient at ix= l as

( ) ( )21

2 iL L

i dfidfiF U A C= l . (2.12a)

The wetted area of the face is


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2 12

[ sin (1 ) (1 ) (2 )]dfi oi i i i iA r h h h h= . (2.12b)

For the case of a fully submerged gear (or for air windage), 2ih = and the wetted surface

area of the gear reduces to 2dfi oiA r= while the length parameter is 2i oir= =l . Eq.

(2.12a) takes into account only one face of the gear while calculating the drag force.

Hence, in order to include both faces of the gear, Eq. (2.12a) is multiplied by a factor of 2

and further simplified to obtain

0.5 1.5( )

0.41

sin

k i dfiLdfi

oi i

U AF

r

=

. (2.12c)

With i i oiU r= , the power loss due to drag on the faces of the gear in the laminar regime

is given as the product of the drag force on the faces of the gear and the tangential

velocity of the gear as

0.5 2.5 2( )

0.41

sin

k i oi dfiLdfi

i

r AP

=

. (2.13)

2.3.2.2 Turbulent Flow Conditions: For turbulent flow over a flat plate, the

velocity profile obeys the Prandtl one-seventh power law [2.17]


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( ) 1/ 7

( )[ ]

Tidfi T

i

yv U=

, (2.14a)

where the superscript T indicates turbulent flow, with ( )Ti being the boundary layer

thickness in the turbulent regime. The displacement and momentum thicknesses are

found from Eqs. (2.8b) and (2.8c), with the velocity profile defined in Eq. (2.14a) as

( ) ( )18

T Ti i = and

( ) ( )772

=T Ti i . Defining the skin friction coefficient for flow over a

rough flat plate that is turbulent from the leading edge as [2.17]

( ) 0.167

( )=0.02[ ]T ki T

i i

CU

, (2.14b)

and substituting the expressions for( )TiC ,

( )Ti i = and

( )Ti i = into the momentum

integral equation, the boundary layer thickness is derived as

( ) 6 / 7 1/ 70.142 ( )T kdfii

xU

= . (2.15a)

As in the laminar flow formulation, the value of ( )TiC at ix=l , as shown in Figure 2.1,

is derived from Eq. 2.14(b) as

( ) 1/ 70.0276[ ]i

T k

i i

CU

=

ll

, (2.15b)


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and the drag force on the faces of the gear under turbulent flow conditions is given as

( ) ( )212 i

T Ti dfidfiF U A C= l . (2.16a)

Substituting Eq. (2.15b) and accounting for the both faces of the gear, the drag force on

the faces of the gear iin the turbulent flow regime is written as

0.14 1.86 1.72( )

0.14

0.025

(sin )

k i oi dfiTdfi

i

r AF

=

. (2.16b)

Finally, the power loss due to drag on the faces of the gear in the turbulent flow regime is

found as

0.14 2.86 2.72

( )0.14

0.025

(sin )k i oi dfiTdfi

i

r AP

=

. (2.17)

2.4 Power Loss due to Root Filling

When a gear pair is partially immersed in oil, the space (cavity) between adjacent

teeth is filled with oil as it enters the oil bath, with the number of filled cavities

depending on the immersion depth of the gears. The rate of filling of the cavities

depends on speed of rotation and the operating temperature. The cavities will be filled

completely with lubricant when the operating temperature is high (i.e. the oil viscosity is


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low) while partial filling could take place at lower operating temperatures. When the oil

swirls past the edges of a tooth cavity, it can be modeled as flow across an annular cavity.

This swirling action of the oil, caused by the tooth surfaces acting as the sidewalls,

creates eddies within the cavity. The flow direction of eddies differ from that of the

general flow, causing energy dissipation and power loss.

Figure 2.2 shows a schematic of flow across an annular cavity. The sidewalls

representing the inner sides of the gear teeth are considered stationary for this model. It is

assumed that the gear is stationary and the lubricant is swirling across the adjacent

cavities. Further, the flow is assumed to be a creeping flow of an incompressible viscous

liquid (oil). When the flow is relatively slow, the terms involving squares of velocity in

the Navier-Stokes equation can be neglected, allowing for an analytical solution.

Denoting the stream function by ( , )r , the following assumptions are made:

(i) Flow is assumed to be radial with all ( ) 0 = and ( , ) ( )i ir r = .

(ii) Steady Stokes flow is assumed, i.e. 4 ( ) 0i r = .

(iii)Only radial solutions are considered for the bi-harmonic equation,

(iv) Any effect of gravity is neglected.

In addition, as the exact geometry of the cavity between two teeth shown in

Figure 2.2(b) is rather complex, an approximate annular shape is considered in its place.

The angle of this approximate annular shape, 2 ci , is defined by the angle of the cavity at


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Figure 2.2 Definition of geometric parameters associated with root filling power losses.

bir

oir

2n=

rir

1in n =

1n=

bi

i

in n=

iO

Oil

oir

pir

rir

cid

dr

2 ci

(a)

(b)


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the pitch radius, pir , and the outer and inner boundaries are formed by the outside

diameter, oir , and the root radius, rir , respectively, as shown in Figure 2.2(b).

Since the flow across the cavity is assumed to be a creeping flow caused by

tangential velocities on the top and bottom curved boundaries, the bi-harmonic equation

for radial flow can be written as

4 3 24

4 3 2 2 3

2 1 10i i i ii

d d d d

r drdr dr r dr r

= + + = , (2.18a)

whose solution has the form

2 21 2 3 4log logi i i i iD D r D r D r r = + + + . (2.18b)

Given the boundary conditions

0 at ,

0 at ,

oi cii

ri ci

r r

r r

= =

= (2.19a)

( ) at ,

at ,

i i oi oi ci

i ri ri ci

r r rd

dr r r r

= =

=

(2.19b)

one obtains from Eq. (2.18b)


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2 21 2 3 4log log 0i oi i oi i oi oi iD r D r D r r D+ + + = , (2.20a)

2 3 4

12 (1 2log )oi i i oi oi i i oi

oi

r D D r r D r r

+ + + = , (2.20b)

2 21 2 3 4log log 0i ri i ri i ri ri iD r D r D r r D+ + + = , (2.20c)

2 3 4

12 (1 2log )ri i i ri ri i i ri

ri

r D D r r D r r

+ + + = . (2.20d)

These equations are solved to determine 1iD , 2iD , 3iD and 4iD . With these coefficients,

the tangential velocity inside the annular cavity is written as

( ) 32 4 4( ) [2 2 log ]

ii i i

Dv r D r D r D r r

r r

= = + + +

. (2.21a)

The tangential shear stress on the fluid inside the cavity between adjacent teeth is defined

as

( )( ) 3 4

3

( )( ) [ ] 2 [ ]

r i irfi

D Dv rr r r

r r rr

= =

. (2.21b)

The force acting on the fluid at ( , )r inside the cavity between adjacent teeth becomes

( ) 342

( , ) ( ) 2 [ ]r irfi c c irfiD

F r A r A Dr

= = , (2.22a)


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where cA is the area of the cavity shown in Fig. 2.2(b). The torque loss due to the

swirling flow of the lubricant inside the cavity is then found as

34( , ) 2 ( )[ ]

oi

ri

r

irfi rfi c oi ri i

oi rir

DT F r dr A r r D

r r= = . (2.22b)

Finally, the power loss due to root filling inside one tooth cavity can be calculated as the

product of rfiT and i . If there are an average of in cavities below the oil level where

eddies take place, the power loss of gear idue to root filling is then given as

342 ( )( )

irfi i c i oi ri i

oi ri

DP n A r r D

r r= . (2.23)

The power loss of the gear pair due to root filling becomes 1 2rf rf rf P P P= + .

2.5 Oil Pocketing Power Loss

As two adjacent teeth of a gear approach the interface of the mating gear, the

cavity between them is intruded upon by a tooth of the mating gear. This results in a

swift reduction in the volume of the cavity, forcing the lubricant out through the openings

at the ends and the backlash. Figures 2.3(a) to 2.3(d) illustrate the side view of a pair of

gears at a sequence of four discrete rotational positions. Here, a cavity of gear 1 (the

driving gear), reduces in area as it moves to the center of the gear mesh, reaching its


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minimum in Figure 2.3(d). Similarly, an example cavity of the mating gear is shown to

experience the same continuous contraction before it reaches a minimum at the position

shown in Figure 2.3(b). In general, there are multiple cavities that are squeezed

simultaneously, exactly the same way, phase-shifted at the base pitch of the gear pair.

This squeezing action of the oil through the respective flow areas results in power loss,

which adds to the total spin loss experienced by the gear pair.

In the meshing zone, control volumes are defined by involute surfaces and root

profiles of the mating gear teeth, with the number of control volumes depending on the

involute contact ratio of the gear pair. Viewing one of the control volumes in three-

dimension in Figure 2.4, the end flow areas are defined by the tooth height and several

involute and root profile parameters of the respective gears, whereas the backlash flow

area is defined by the shortest chord length that joins the trailing edge of the tooth on the

driving gear and the involute surface of the corresponding mating tooth on the driven

gear. The first task here is to calculate the velocity of the lubricant escaping through the

backlash and end flow areas. This will be done by means of the continuity equation,

assuming an incompressible fluid. Since each control volume varies with rotation of the

gears, it is necessary to determine the flow areas as a function of angular position of the

mating gears. After the velocity of the lubricant squeezed out through the respective flow

areas is calculated, the power loss due to pocketing is determined by using the principle

of conservation of momentum. The above approach does not consider the effects of air

ingestion, which can take place at higher rotational speeds. In such cases, a compressible

fluid formulation is required that is a steep departure from the incompressible flow

formulation presented here. This problem will be tackled separately in Chapter 3.


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Figure 2.3 Illustration of a side view of fluid control volumes of the gear mesh interface

at different rotational positions 1 2 30 m m m< < < .

1( )11

mH

1( )21m

H

1( )12

mH

2( )11

mH

3( )11

mH

(0)11H

(0)21H

(0)12H

1

2

0m=

1m m=

2m m=

3m m=


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The initial mesh position for this pocketing formulation is taken as the one when the tip

of a tooth of the driven gear first comes in contact with the involute surface of tooth of

the driving gear at its start of active profile (SAP) as shown in Figure 2.3(a). At this

initial position, a given control volume(0)ijH , thej-th control volume of the i-th gear at

position 0m= , is associated with the corresponding backlash flow area (0),b ijA and end

flow(0),e ijA areas. These leakage areas, and subsequently the volume of

(0)ijH , change

with rotation of the gears. The first task involves the computation of backlash and end

flow areas at a rotational incremental angle of 0im i im M = + ( [0, 1]m M ) where

2 2 1/ 20 [( ) 1]i si bir r = and

2 2 1/ 2 2 2 1/ 2[( ) 1] [( ) 1]i oi bi si bir r r r = . Here, it is noted that

the flow areas of a control volume at m M= are equal to those of the control volume

preceding it at 0m= , i.e. (0) ( ),, ( 1)M

e ije i jA A+ = and(0) ( ), ( 1) ,

Mb i j b ijA A+ = .

2.5.1 Backlash Flow Area

Figures 2.5 and 2.6 show the transverse geometry of two spur gears in mesh at an

arbitrary rotational position m . With gear 1 designated as the driving gear, the first step

is to establish the necessary angles and distances at this initial position, focusing on the

first cavity of gear 1 marked as (0)11H in Figure 2.3(a). The centers of the gears are

marked as1

O and2

O , with the operating center distance1 2 1 2p p

O O r r e= + = , where

pir is the pitch radius of gear i. At this starting position, point B , which is the

intersection of the leading profile of driving gear 1 with the outside (major) circle of gear

2, coincides with the contact point C. This point also represents the SAP of gear 1 so


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Figure 2.4 Three-dimensional representation of a control volume showing backlash andend flow areas.

( ),m

e ijA

( )

,

m

e ijA

( ),m

b ijA


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that (0)1 1sO B r= , where 1sr is the radius at the SAP for the driving gear. Points D and

Eare located at the tip of the leading and trailing profiles of gear 1, whereas point F

marks the intersection of the major circle of the driven gear 2 with the trailing tooth

profile of gear 1. Distances 1 1 1oO D O E r = = remain unchanged with rotational position

m , where 1or is the outside radius of gear 1. Using basic geometry, the angles to be

defined at 0m= are given as

2 2 2(0) 1 1 2

1 1cos [ ]2

s o

s

e r r

e r

+

= , (2.24a)

2 2 2(0) 1 2 12

2

cos [ ]2

o s

o

e r r

e r

+ = , (2.24b)

2 2(0) 1/ 2 1 1/ 2 11 1 1 11 2 2

1 1 11 1

( 1) cos [ ] ( 1) cos [ ]o b s b

o s ob b

r r r r DE

r r rr r

= + + , (2.24c)

2 (0) 2 2(0) 1 2 1

2 (0)2

( )

cos [ ]2

oe O E r

e O E

+

= , (2.24d)

(0) (0)(0)1 1

2

= , (2.24e)

where DE is the tooth thickness of gear 1 at its tip. Once the initial angles are

calculated, angles (0)22 2

2( )

o

DECO J

n r

= and (0) (0) (0)2 2 2CO G CO J JO G= are defined

by letting 2 2oO J r= and assuming more over that(0) (0)

2 2O E O G , where

(0) 2 2 (0) 1/ 22 1 1( 2 sin[ ])o oO E e r er = + , (2.24f)


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Figure 2.5 Geometry of two gears in mesh at an arbitrary position m .

E

C

( )m

( )

1

m

( )1

m

( )2m

( )2m

1O

2O

1

2


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Figure 2.6 Definition of the end area at an arbitrary position m .

M

P

N

G E D

C

KFB

( ),1m

b jQ

( )

,1

m

e jA

( ),1m

t jQ

AN INVESTIGATION OF LOAD-INDEPENDENT POWER LOSSES OF GEAR.pdf

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