By. Satya Seetharaman, B.E., M.S.. Graduate Program in Mechanical Engineering. The Ohio State University. 2009. Dissertation Committee: Ahmet Kahraman ...
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
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 jetlubrication 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 ii
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 jetlubricated 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.
iii
Dedicated to Amma, Appa & Chithra, Sujatha, Kavita
iv
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 one’s 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 PetryJohnson, 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. vi
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, Ohio Graduate Research Associate, Gear and Power Transmission, Research Laboratory, Dept. of Mechanical Engineering, The Ohio State University, Columbus, Ohio
Mar. 2005 – present………………
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 vii
TABLE OF CONTENTS
Abstract……………………………………………………………………………….
Page ii
Dedication…………………………………………………………………………….
iv
Acknowledgments…………………………………………………………………....
v
Vita……………………………………………………………………………….…..
vii
List of Tables..………………………………………………………………...……...
xii
List of Figures...……………………………………………………………...……….
xiii
Nomenclature…………………………………………………………………………
xvi
CHAPTERS:
1
Introduction…………………………………………..…………………………...
1
1.1 Research Background and Motivation.………………………………….……
1
1.2 Literature Review…………….……………..….……………………………..
3
1.3 Scope and Objectives……………..……...…………......…………………….
6
1.4 Dissertation Outline……………………...…………......…………………….
9
References for Chapter 1……………..……...…………......…………………….
12
viii
2
Oil Churning Power Losses of a Gear Pair: Model Formulation....……………...
16
2.1 Introduction.....………………….……………….……………………………
16
2.2 Gear Pair Oil Churning Power Loss Model....………………………………..
20
2.3 Drag 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
2.3.2.1 Laminar Flow Conditions..…………………………….…………
30
2.3.2.2 Turbulent Flow Conditions……………………………………….
33
2.4 Power Loss due to Root Filling………………………...…………………….
35
2.5 Oil Pocketing Power Loss…………………………………………………….
40
2.5.1
Backlash Flow Area………………...……………….………………..
44
2.5.2
End Flow Area…………………..……………………………………
50
2.5.2.1 Calculation of Area Qc 2 ……………………………….…………
51
2.5.2.2 Calculation of Area Qt(,1mj) ……………..………………………….
54
2.5.2.3 Calculation of Area Qb( m,1 )j ……………..………………………….
56
2.5.3
3
Power Loss due to Oil Pocketing..……………………………………
58
2.6 Example Oil Churning Analysis………………………..…………………….
62
2.7 Concluding Remarks………………………………………………………….
68
References for Chapter 2...……………………………………………………….
70
Windage Power Losses of a Gear Pair: Model Formulation…...………………...
75
3.1 Introduction.....………………….……………….……………………………
75
ix
3.2 Windage Pocketing Power Loss Model..…....………………………………..
78
3.3 Windage Drag Power Losses….....………………...…………………………
89
3.4 Example Windage Power Loss Analysis and Parametric Studies………..…..
91
3.5 Concluding Remarks….…………………………………………………….
105
References for Chapter 3...………………………………………………………. 107 4
Validation of Oil Churning and Windage Power Loss Models……...…………... 108 4.1 Introduction.....………………….……………….…………………………… 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
4.2.3
Power Loss due to Drag on the Faces of a Gear...…………………… 118
4.3 Validation of Windage Power Loss Model…………......……………………. 126 4.4 Concluding Remarks…………………………………………………………. 130 References for Chapter 4...………………………………………………………. 131 5
Application of Oil Churning Power Loss Model to an Automotive transmission.. 132 5.1 Introduction.....………………….……………….…………………………… 132 5.2 Validation of Gear Pair Oil Churning Model….…………………………….. 136 5.2.1
Test Machine and Oil Churning Test Procedure….....…..…………… 138
5.2.2
Gear Specimens and Parameters Studied……......…………………… 138
5.2.3
Power Loss due to Drag on the Faces of a Gear...…………………… 141
5.3 Validation of Windage Power Loss Model…………......……………………. 142 5.4 Concluding Remarks…………………………………………………………. 147 x
5.4.1
Test Machine and Oil Churning Test Procedure….....…..…………… 147
5.4.2
Gear Specimens and Parameters Studied……......…………………… 149
5.5 Concluding Remarks…………………………………………………………. 153 References for Chapter 5...………………………………………………………. 154 6
Conclusions and Recommendations for Future Work.…………………………... 156 6.1 Summary….....………………….……………….…………………………… 156 6.2 Conclusions and Contributions……………..….…………………………….. 157 6.3 Recommendations for Future Work……...…………......……………………. 160
Bibliography …………………………..………………...…………………………... 166
xi
LIST OF TABLES Table
2.1
Page The design parameters of the example spur gears …………………..…….....
xii
63
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 0 < m1 < m2 < m3 ………………….
2.4
41
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 Qc 2 ,(b) the overlap area Qt(,1mj) and (c) the excluded area Qb( m,1 )j , all at an arbitrary position
m ………………………………………………………………...…. 2.8
Effect of (a) temperature T , (b) oil level parameter h , (c) face width b and (d) gear module m% on total spin power loss PT …………..………………….
2.9
65
Components of PT for a gear pair having m% = 2.32 mm and b = 26.7, 19.5 and 14.7 mm at h = 1.0 and 80 ºC; (a) Pdp , (b) Pdf , (c) Prf , and (d) Pp .....
3.1
52
67
Effect of (a) temperature T , (b) face width b and (c) gear module m% on
Pw ………………………………………………………………………….....
xiii
93
3.2
Components of Pw for the gear pairs having m% = 2.32 mm and varying b at
80o C ; (a) windage pocketing power loss Pwp (b) windage drag power losses Pwd .………………………………………………………………...…. 3.3
96
Windage power loss Pwp and its components, Pwp ,e and Pwp ,b , for gear pairs having m% = 2.32 mm at 80o C ; (a) b = 14.7 mm (b) b = 19.5 mm (c)
b = 26.7 mm. ……………………………………………………………...…. 3.4
97
Variation of the end flow area Ae(,1m )j with rotational position for the gear pairs having (a) m% = 2.32 mm , and (b) m% = 3.95 mm……………………….. 100
3.5
Variation of (a) density, (b) pressure and (c) velocity of control volume ( m) Η11 of the gear pair having m% = 2.32 mm with rotational position for (a1,
b1,
c1)
the
end
area
and
(a2,
b2,
c2)
the
backlash
area
o
( b = 19.5 mm , 80 C )……………………………………………………….... 3.6
102
Variation of (a) density, (b) pressure and (c) velocity of control volume ( m) Η11 of the gear pair having m% = 3.95 mm with rotational position for (a1,
b1,
c1)
the
end
area
and
(a2,
b2,
c2)
the
backlash
area
( b = 19.5 mm , 80o 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
m% = 3.95 mm and
b = 19.5 mm, and (b) 40-tooth gear with m% = 2.32 mm and b = 19.5 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] PsT for a gear pair (a) 30o C , (b) 50o C , (c) 70o C , and (d) 90o C ; m% = 2.32 mm, b = 19.5 mm and
h = 1.0 ……………………………………………………………………….. 120
xiv
4.6
Comparison of predicted vs. measured PsT for a gear pair at (a) h = 0.05 , (b) h = 0.5 , (c) h = 1.0 , and (d) h = 1.5 ; oil at 80o C (up-in-mesh),
m% = 2.32 mm, and b = 19.5 mm…………………………………………….... 122 4.7
Comparison of predicted to the measured [4.1] PsT for a gear pair having
m% = 2.32 mm and face width values (a) b = 14.7 mm, (b) b = 19.5 mm, and (c) b = 26.7 mm; oil at 80o C and h = 1.0 ………………………………...... 4.8
124
Comparison of predicted to the measured [4.2] PsT for a gear pair of face width b = 19.5 mm and modules (a) m% = 2.32 mm and (b) m% = 3.95 mm; oil at 80o C and h = 1.0 ………….…………………………………………….... 125
4.9
Comparison of predicted and measured [4.2] Pw for (a) the 40-tooth gear pair having m% = 2.32 mm and b = 26.7 mm and (b) 23-tooth gear pair having m% = 3.95 mm and b = 19.5 mm ……………………...……………....
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) 1st and 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) 2nd gear, (b) 3rd gear, (c) 4th gear, (d) 5th gear, and (e) 6th gear….. 152
xv
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
M
Total number of gear rotational increments
m
Gear rotational position index m ∈ [1, M ]
m%
Gear module
N
Number of teeth of gear
n
Unit normal vector
n
Average number of cavities
P
Power loss
p
Pressure xvi
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
x
Axis parallel to gear face
y
Axis perpendicular to gear face
δ
Boundary layer thickness
ε
Energy per unit mass
φ
Immersion angle
γ
Ratio of specific heats
κ
Momentum thickness
µ
Viscosity
θ
Angle, tangential direction
ρ
Density xvii
τ
Shear stress
Ω, ω
Rotational speed in rpm and rad/s, respectively. ω =
ξ
Flow factor
ψ
Stream function
ζ
Displacement thickness
Subscripts b
Base or backlash
c
Cavity
d
Drag
e
End
f
Face
i = 1, 2
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
xviii
2π Ω 60
T
Total
v
Constant Volume
w
Windage
Superscripts
m
Rotational position
L
Laminar
T
Turbulent
r
Radial
θ
Tangential
w
Wall
xix
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
1
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 2
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 multi3
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 Karman’s [1.8] equations and a modified Reynolds number, Bones 4
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 5
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/CFDbased methodology. The dimensional analysis approach deduces a lot from experimental 6
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.
7
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 multimesh 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 systemlevel 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 8
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
9
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 10
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.
11
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, 217–232.
[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.
12
[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, 337–346. [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, 1266–1273. [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, 335–342. [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
13
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), 51–59. [1.17] Diab, Y., Ville, F., and Velex, P., 2006, “Investigations on Power Losses in High Speed Gears,” Journal of Engineering Tribology, 220, 191–298. [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, 116–122. [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), 331–341. [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. 14
[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.
15
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 churning of the lubricant if the rotating components of the gearbox are immersed in an oil bath (dip-lubricated) or due to windage if 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 16
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 Karman’s [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, 17
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. 18
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 Höhn et al [2.15] also stresses this apparent lack of dependence of oil type on load independent losses. In their experiments, Höhn 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 19
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:
Pdi = Pdpi + Pdfi + Prfi ,
i = 1, 2,
(2.1)
where Pdpi is the power loss due to oil/air drag on the periphery (circumference) of a gear, Pdfi is the power loss due to oil/air drag on the faces (sides) of a gear, and Prfi 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 Pd = Pd 1 + Pd 2 .
20
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
PT = Pd + Pp .
(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.
21
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 roi (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 ∂v ∂t = 0 and
∂p ∂t = 0 .
The oil pressure is assumed to vary only in the radial direction, i.e.
pi = pi (r ) and ∂pi ∂θ = 0 . 22
Gear 1
ω1 Gear 2
l1
φ1
O1
Oil Level
l2
ro1
ro 2
h1
ω2
O2
φ2
Figure 2.1 Definition of oil churning parameters for a gear pair immersed in oil.
23
h2
(r ) Further, the radial component of the oil velocity vdpi 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, 24
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. ( θ) of the oil along the Based on the above detailed assumptions, the velocity vdpi
tangential direction on the periphery of the gear and the pressure distribution around the periphery are defined by solving the continuity equation [2.17]
(r ) ∂vdpi
(r ) vdpi
(θ)
1 ∂vdpi + + [ ]= 0, ∂r r r ∂θ
(2.3a)
25
and the Navier-Stokes equations of motion [2.17] along r and θ directions
(r ) ∂vdpi
∂t
( θ) ∂vdpi
∂t
r) + v (dpi
( θ) + vdpi
(r ) ∂vdpi
∂r
( θ) ∂vdpi
∂r −
+
+
(θ) (r ) vdpi ∂vdpi
∂θ
r
( r ) (θ) vdpi vdpi
r
+
1 ∂pi µ + 2 [r rρ ∂θ r ρ
−
(θ) 2 [vdpi ]
= r (θ) 2 (r ) ∂vdpi 1 ∂pi µ ∂ vdpi (r ) [ ], − + − vdpi − 2 ρ ∂r ρr 2 ∂θ2 ∂θ
( θ) ( θ) vdpi ∂vdpi
∂θ
r
( θ) ∂ 2vdpi 2
∂r
2
+
(2.3b)
= (θ) ∂ 2vdpi
∂θ
2
+r
( θ) ∂vdpi
∂r
+2
(r ) ∂vdpi
∂θ
(2.3c) ( θ) − vdp ].
where ρ and µ are the density and the dynamic viscosity of the lubricant at a given operating temperature. Implementing the assumptions stated earlier, one obtains
(θ) ∂vdpi
= 0,
(2.3d)
ρ ( θ) 2 ∂pi [v ] = , r dpi ∂r
(2.3e)
∂θ
(θ) d 2vdpi
dr 2
(θ) ( θ) 1 dvdpi vdpi + − 2 = 0. r dr 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 ( θ) ( θ) = ωi roi at r = roi and vdpi = 0 as r → ∞ , where ωi is the depth) are given as vdpi
26
rotational speed of the gear and r ∈ [roi , ∞) . Solving Eq. (2.3f) and applying these boundary conditions, the fluid velocity along the periphery of the gear is found as
( θ) vdpi
ωi roi2 = . r
(2.4)
Given the dynamic viscosity µ 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
(r ) ∂vdpi
rr ) τ(dpi
= 2µ
=0,
(2.5a)
θθ) τ(dpi
(θ) (r ) 1 ∂vdpi vdpi = 2µ [ + ] = 0, r ∂θ r
(2.5b)
r θ) τ(dpi
2µωi roi2 ∂ v dpi 1 ∂v dpi = µ[r ( )+ ]=− . ∂r r r ∂θ r2
∂r
(θ)
(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 w) = 2µωi . outside radius of gear i, r = roi (tangential wall shear stress), is found as τ(dpi w) Defining the friction drag coefficient as Cdpi = 2τ(dpi ρ U i2 , where U i = ωi roi is the free-
27
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
1 w) Fdpi = ρU i2 Adpi Cdpi = Adpi τ(dpi . 2
(2.6)
Here, the wetted surface area of the periphery is given as Adpi = 2φi roi bi , where bi is face width of gear i and φi = cos −1[1 − hi ] . A dimensionless immersion parameter is defined from Figure 2.1 as hi = hi roi , where hi is the immersion depth of gear i. Accordingly, hi ≤ 0 represents the case of no oil-gear interactions and hi ≥ 2 corresponds to the case of a gear that is submerged in oil. For hi ≥ 2 , hi = 2 is used in the above formulation to obtain φi = π such that the entire periphery of the gear is subject to drag, i.e.
Adpi = 2πroi bi . For hi ≤ 0 , it is understood that the gear is submerged in air. Then, the parameters hi = 2 and φi = π 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
28
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 Adpi w) and τ(dpi known, the drag force is then evaluated as
Fdpi = 4µ bi roi ωi φi .
(2.7a)
Finally, the power loss due to drag on the periphery of a single gear is given as the product of Fdpi and the tangential velocity at the periphery as
Pdpi = 4µ bi roi2 ωi2 φi .
(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 Re = 2ρωi roi2 µ within the range (10)5 to (10)6 [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 29
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 ( L) profile vdfi is considered to be linear, i.e.
( L) vdfi = U i y δi( L ) ,
(2.8a)
where δi( L ) is the laminar boundary layer thickness, U i is the free-stream velocity of the lubricant flowing across the faces of the gear i as defined previously, and y 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
30
δdfi
∫
ζi =
[1 −
0 δdfi
κi =
∫ 0
vdfi Ui
vdfi Ui
[1 −
] dy = 12 δi ,
vdfi Ui
(2.8b)
] dy = 16 δi .
(2.8c)
Here, ζi 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 κi 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 µ k being the kinematic viscosity of the oil, the skin friction coefficient, Ci( L ) must be taken into account and is defined as
Ci( L ) =
( L) 2µ dvdfi ρU i2 dy
= y =0
2µ k ( L) δi U i
.
(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 U i is constant, the rate of increase of momentum thickness is directly proportional to the wall shear stress, i.e.
31
∂κi 1 = Ci . ∂x 2
(2.9)
By setting Ci = Ci( L ) and κi = κi( L ) 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
δi( L ) = 3.46
µk x , Ui
(2.10)
where x is the length parameter, as defined in Figure 2.1. At x = l i = 2roi sin φi , the skin friction coefficient is obtained by substituting Eq. (2.10) into Eq. (2.8d) as
Cl( L ) = 0.578 i
µk . li Ui
(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 x = l i as
1 ( L) Fdfi = ρU i2 Adfi Cl( L ) . i 2
(2.12a)
The wetted area of the face is
32
Adfi = roi2 [ π2 − sin −1 (1 − hi ) − (1 − hi ) hi (2 − hi )] .
(2.12b)
For the case of a fully submerged gear (or for air windage), hi = 2 and the wetted surface area of the gear reduces to Adfi = πroi2 while the length parameter is x = l i = 2roi . 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
( L) Fdfi
=
1.5 0.41ρµ0.5 k U i Adfi
roi sin φi
.
(2.12c)
With U i = ωi roi , 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
( L) Pdfi
=
2.5 2 0.41 ρµ0.5 k ωi roi Adfi
sin φi
.
(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]
33
(T ) vdfi = Ui [
y δi(T )
]1/ 7 ,
(2.14a)
where the superscript T indicates turbulent flow, with δi(T ) 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
ζi(T ) = 18 δi(T ) and
7 δ(T ) . Defining the skin friction coefficient for flow over a κi(T ) = 72 i
rough flat plate that is turbulent from the leading edge as [2.17]
Ci(T ) =0.02[
µk U i δi(T )
]0.167 ,
(2.14b)
and substituting the expressions for Ci(T ) , ζi = ζ i(T ) and κi = κi(T ) into the momentum integral equation, the boundary layer thickness is derived as
T) δ(dfi = 0.142 x6 / 7 (
µ k 1/ 7 ) . Ui
(2.15a)
As in the laminar flow formulation, the value of Ci(T ) at x = l i , as shown in Figure 2.1, is derived from Eq. 2.14(b) as
Cl(T ) = 0.0276 [ i
µ k 1/ 7 ] , l iU i
(2.15b)
34
and the drag force on the faces of the gear under turbulent flow conditions is given as
(T ) Fdfi = 12 ρU i2 Adfi Cl(T ) .
(2.16a)
i
Substituting Eq. (2.15b) and accounting for the both faces of the gear, the drag force on the faces of the gear i in the turbulent flow regime is written as
(T ) Fdfi =
1.86 1.72 0.025ρµ0.14 roi Adfi k ωi
(sin φi )0.14
.
(2.16b)
Finally, the power loss due to drag on the faces of the gear in the turbulent flow regime is found as
(T ) Pdfi =
2.86 2.72 0.025ρµ 0.14 roi Adfi k ωi
(sin φi )0.14
.
(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 35
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 ψ(r , θ) , the following assumptions are made: (i) Flow is assumed to be radial with all ∂ ( ) ∂θ = 0 and ψi (r , θ) = ψi (r ) .
(ii) Steady Stokes flow is assumed, i.e. ∇ 4ψi (r ) = 0 . (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, 2θci , is defined by the angle of the cavity at
36
Oi (a)
rbi
θbi Oil
roi
rri
ωi n =1
n = ni n=2
n = ni −1
(b)
dr
dθ ci
2θci
roi
rpi rri Figure 2.2 Definition of geometric parameters associated with root filling power losses. 37
the pitch radius, rpi , and the outer and inner boundaries are formed by the outside diameter, roi , and the root radius, rri , 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
∇ 4ψi =
d 4ψ i dr 4
+
2 d 3ψ i 1 d 2 ψ i 1 d ψ i − + =0, r dr 3 r 2 dr 2 r 3 dr
(2.18a)
whose solution has the form
ψi = D1i + D2i r 2 + D3i log r + D4i r 2 log r .
(2.18b)
Given the boundary conditions
0 at r = roi ψi = 0 at r = rri d ψ (i ) −ω i roi = dr −ωi rri
θ ≤ θci ,
(2.19a)
θ ≤ θci ,
at r = roi at r = rri
θ ≤ θci , θ ≤ θci ,
one obtains from Eq. (2.18b)
38
(2.19b)
D1i + roi2 D2i + log roi D3i + roi2 log roi D4i = 0 , 2roi D2i +
1 D3i + roi (1 + 2 log roi ) D4i = −ωi roi , roi
D1i + rri2 D2i + log rri D3i + rri2 log rri D4i = 0 , 2rri D2i +
1 D3i + rri (1 + 2 log rri ) D4i = −ωi rri . rri
(2.20a) (2.20b)
(2.20c) (2.20d)
These equations are solved to determine D1i , D2i , D3i and D4i . With these coefficients, the tangential velocity inside the annular cavity is written as
v ( θ) ( r ) = −
D ∂ψ = −[2 D2i r + 3i + D4i r + 2 D4i r log r ] . r ∂r
(2.21a)
The tangential shear stress on the fluid inside the cavity between adjacent teeth is defined as
τ(rfirθ) (r ) = µ r
D D ∂ v ( θ) ( r ) [ ] = 2µr[ 33i − 4i ] . r r ∂r r
(2.21b)
The force acting on the fluid at (r , θ) inside the cavity between adjacent teeth becomes
Frfi (r , θ) = Ac τ(rfirθ) (r ) = 2µAc [
D3i r2
− D4i ] ,
39
(2.22a)
where Ac 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
roi
Trfi =
D
∫ Frfi (r , θ)dr =2µAc (roi − rri )[ roi 3riri − D4i ] .
(2.22b)
rri
Finally, the power loss due to root filling inside one tooth cavity can be calculated as the product of Trfi and ωi . If there are an average of ni cavities below the oil level where eddies take place, the power loss of gear i due to root filling is then given as
Prfi = 2ni µAc ωi (roi − rri )(
D3i − D4i ) . roi rri
(2.23)
The power loss of the gear pair due to root filling becomes Prf = Prf 1 + Prf 2 .
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 40
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 threedimension 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. 41
ω2 (0) H12
(0) H11
(a)
m=0 (0) H 21
ω1
(m )
H12 1
(m )
H11 1
(b) m = m1 (m )
H 21 1
(m )
H11 2
(c) m = m2
(m )
H11 3
(d) m = m3
Figure 2.3 Illustration of a side view of fluid control volumes of the gear mesh interface at different rotational positions 0 < m1 < m2 < m3 .
42
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 H ij(0) , the j-th control volume of the i-th gear at position m = 0 , is associated with the corresponding backlash flow area Ab(0) ,ij and end (0) flow Ae(0) ,ij areas. These leakage areas, and subsequently the volume of H ij , change
with rotation of the gears. The first task involves the computation of backlash and end flow areas at a rotational incremental angle of θim = θi 0 + mθi M ( m ∈ [0, M − 1] ) where
θi 0 = [(rsi2 rbi2 ) − 1]1/ 2 and θi = [(roi2 rbi2 ) − 1]1/ 2 − [(rsi2 rbi2 ) − 1]1/ 2 . Here, it is noted that the flow areas of a control volume at m = M are equal to those of the control volume (M ) (0) (M ) preceding it at m = 0 , i.e. Ae(0) ,i ( j +1) = Ae,ij and Ab ,i ( j +1) = Ab ,ij .
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 (0) in Figure 2.3(a). The centers of the gears are first cavity of gear 1 marked as H11
marked as O1 and O2 , with the operating center distance O1O2 = rp1 + rp 2 = e , where
rpi 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 43
Ae(,mij) Ae(,mij)
) Ab( m ,ij
Figure 2.4 Three-dimensional representation of a control volume showing backlash and end flow areas.
44
that O1B (0) = rs1 , where rs1 is the radius at the SAP for the driving gear. Points D and E are 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 O 1 D = O1E = ro1 remain unchanged with rotational position
m , where ro1 is the outside radius of gear 1. Using basic geometry, the angles to be defined at m = 0 are given as
ϕ1(0) = cos −1[
e2 + rs21 − ro22 ], 2e rs1
(2.24a)
−1 ϕ(0) 2 = cos [
e2 + ro22 − rs21 ], 2e ro 2
(2.24b)
φ1(0)
φ(0) 2
=(
ro21 rb21
− 1)
−1
= cos [
β(0) =
1/ 2
−1 rb1
− cos [
ro1
]− (
e2 + (O2 E (0) )2 − ro21 2e O2 E (0)
rs21 rb21
− 1)1/ 2 + cos −1[
rb1 DE ]+ , rs1 ro1
],
(2.24d)
π − ϕ1(0) − φ1(0) , 2
(2.24e)
where DE is the tooth thickness of gear 1 at its tip. calculated, angles CO2 J (0) = (
(2.24c)
Once the initial angles are
2π DE − ) and CO2G (0) = CO2 J (0) − JO2G (0) are defined n2 ro 2
by letting O2 J = ro 2 and assuming more over that O2 E (0) ≈ O2G (0) , where
O2 E (0) = (e2 + ro21 − 2ero1 sin[β(0) ])1/ 2 , 45
(2.24f)
O2
ω2
ϕ2( m)
φ2( m)
E C
φ1( m)
ϕ1( m)
ω1 β (m) O1
Figure 2.5 Geometry of two gears in mesh at an arbitrary position m . 46
P M N G E
D
m) Ae(,1 j
) Qb( m ,1 j
C J
K
F B
Qt(,1mj)
Figure 2.6 Definition of the end area at an arbitrary position m .
47
JO2G (0) = (
ro22 rb22
− 1)1/ 2 − cos −1[
rb 2 r O E (0) 2 ] − [( 2 ) − 1]1/ 2 + cos −1[ b 2(0) ] . (2.24g) ro 2 rb 2 O2 E
(0) Accordingly, EO2G (0) = CO2 J (0) − JO2G (0) − φ(0) 2 and the initial backlash length EG
(the shortest distance between point D on gear 1 and the involute profile of gear 2) is then given as EG (0) = O2 E (0) ⋅ EO2G (0) . Thus, the backlash flow area for the first control (0) volume H11 at the initial position can then be calculated as
Ab(0) ,11 = b EG
(0)
,
(2.25)
where b is the effective face width of the gear pair. With the same gear pair now at an arbitrary rotational position m , the required angles at that position are given as
β( m ) = β(0) + (m − 1) ∆θ1m ,
(2.26a)
φ1( m)
r DE ro21 O1C ( m) 2 1/ 2 −1 rb1 = + ( 2 − 1) − cos [ ] − [( ) − 1]1/ 2 + cos −1[ b1( m) ] , (2.26b) ro1 ro1 rb1 rb1 O1C
ϕ1( m )
π ( m) ( m) 2 − β − φ1 , = ( m) ( m) π β + φ1 − 2 ,
β( m ) ≤ β
(m)
π 2
π > 2
48
,
(2.26c)
ϕ(2m ) = sin −1[
O1C ( m ) sin(ϕ1( m ) ) O2C ( m )
],
(2.26d)
where distance O2C ( m ) = e2 + (O1C ( m ) )2 − 2eO1C ( m ) cos[ϕ1( m ) ]
is calculated from
known
defines
angles
and
lengths.
One
further
distance
O2 E ( m ) = e2 + ro21 − 2e ro1 cos[ϕ1( m ) + φ1( m ) ] and the angle
φ(2m )
( m) ( m) −1 ro1 sin[φ1 + ϕ1 ] = sin [ ] − ϕ(2m ) . ( m) O2 E
(2.26e)
With O2 J ( m ) = O2 J = ro 2 , CO2G ( m ) = CO2 J ( m ) − JO2G ( m ) , assuming O2 E (0) ≈ O2G (0) , and further calculating
CO2 J ( m) = CO2 J (0) − (
JO2G (0) = (
ro22 rb22
ro22 rb22
− 1) 2 + cos −1[ 1
− 1)1/ 2 − cos −1[
1 rb 2 r O C ( m) 2 ] + [( 1 ) − 1] 2 − cos −1[ b 2( m) ] , (2.26f) ro 2 rb1 O1C
rb 2 r O E (m) 2 ] − [( 2 ) − 1]1/ 2 + cos −1[ b 2( m) ] , ro 2 rb 2 O2 E
(2.26g)
the angle EO2G ( m ) = CO2 J ( m ) − JO2G ( m ) − φ(2m ) is calculated and the backlash length EG ( m ) is estimated as
EG ( m ) = O2 E ( m ) ⋅ EO2G ( m ) . It is noted here that EG ( m ) is
defined by drawing a line from the tip of the trailing edge of the driving gear to the involute surface of the driven gear along the direction tangent to the major diameter of 49
gear 2. The error associated with this estimation is minimal while the computational ( m) at benefits are quite significant. The backlash flow area for the control volume H11
position m is then calculated as
) ( m) Ab( m , ,11 = b EG
(2.27)
where b is the effective face width of the gear pair. The same procedure is applied to the ( m) other control volumes as well. In Figure 2.3, for instance, calculations for H12 are also
) ( m) ( M + m) repeated the same way to determine Ab( m , calculations of all ,12 . Since Ab ,i ( j +1) = Ab ,ij
control volumes at M discrete positions covering one base pitch are sufficient to determine the change of backlash areas of all cavities throughout the gear mesh. ) Likewise, the same formulation is also applied to the backlash areas Ab( m ,2 j of control
volumes H 2( mj) of gear 2.
2.5.2 End Flow Area ) Focusing on a control volume H1( m in Figure 2.6, the end area Ae(,1m)j can be j
defined as
) ( m) Ae(,1m)j = Qc 2 − Qt(,1mj) − Qb( m ,1 j − QBCK ,
(2.28)
50
where Qc 2 is the total tooth cavity area bounded by the major diameter and two adjacent teeth of gear 2 shown in Figure 2.7(a), Qt(,1mj) is the portion of Qc 2 overlapped by the tooth of gear 1 (area defined by points B, D, E and F) as represented in Figure 2.7(b), ) Qb( m ,1 j is the excluded area of the segment along the trailing profile (defined by the points ( m) E, G, J and F) as seen in Figure 2.7(c) and QBCK is the small portion of the area shown
in Figure 2.6, as bounded by the involute segment BC of gear 1 and CK of gear 2. At (0) ( m) the initial position, QBCK = 0 . As the gears rotate, area QBCK increases while remaining
typically very small compared to the total area of the cavity Qc 2 . For this reason, it is neglected here while it can be included, if desired.
2.5.2.1
Calculation of Area Qc 2 : From Fig. 2.7(a), the transverse area Qc 2
between the two adjacent gear teeth can be written as
Qc 2 = QJO2 K − QJO2M − QKO2 N − QMO2 N + QPMN .
(2.29)
Here, QJO2 K is the area of the segment defined by lines O2 J , O2 K (defined from the center of gear 2 to the tips of two adjacent teeth), area QKO2 N represents the region between the involute profile segment KN and the segment defined by the angle KO2 N , and area QJO2 M is the region between the involute profile segment JM and the segment defined the angle JO2 M , where M and N are points where the base circle of gear 2
51
O2 (m) QJO M
E
2
( m) QPMN
rrf
M
D
G
( m) QMO 2N (m) QKO 2N
P
ri 2
N
O2
E
C J
F
B
F (m) QGO E 1
(m) QDO 1B
ro 2 52
( m) QJO F 1
(m) QEO D
J
1
K
(m) QFO E 1
(b)
(a)
(c)
O1
Figure 2.7 Parameters used in calculation of (a) the total tooth cavity area Qc 2 ,(b) the overlap area Qt(,1mj) and (c) the excluded ) area Qb( m ,1 j , all at an arbitrary position m .
52
intersects the involute profiles of gear 2, as shown in Figure 2.7(a). Area QMO2 N is defined by the segment MN and the angle MO2 N . Finally, the area bounded by points M, N and P is represented by QPMN where P is a point on the root circle. Assuming that QPMN is the area of a semi-circle at the root fillet radius, rrf 2 , and by using area ( m) ( m) integrals, while at the same time recognizing that QJO M = QKO N for teeth having 2
2
symmetric involute profiles in both drive and coast sides, area Qc 2 can be written from Eq. (2.29) as
Qc 2 = 12 ro 2 (
2πro 2 − To 2 ) − 2 ∫∫ I JM (θ)drd θ N2 JO2 M , 2πrb 2 2 1 1 − 2 rb 2 ( − Tb 2 ) + 2 πrrf 2 N2
(2.30)
where radius I JM (θ) represents the radius of a point along the involute profile at a given angle θ , To 2 and Tb 2 are circular tooth thicknesses of gear 2 at the tip and base circle, and N 2 is the number of teeth in gear 2. A simplified version I JM (θ) = rb 2 [1 + (3θ) 2 / 3 ] is used here by expressing the involute profile equation in the Taylor series form and retaining the first order terms [2.12]. With this, the area integral bounded by the involute profile segment JM is given by
∫∫ JO2 M
2.33 I JM (θ)drd θ = 12 rb22 [(θM − θ J ) + 0.46(θ2.33 M − θJ )
+ 1.25(θ1.67 M
− θ1.67 J )] 53
,
(2.31)
where, θ J and θM denote the angles measured from the horizontal line X 2O2 to the lines O2 J and O2 M , respectively. The value of area Qc 2 is obtained in closed-form by substituting Eq. (2.31) into Eq. (2.30).
2.5.2.2 Calculation of Area Qt(,1mj) : In order to calculate the overlap area Qt(,1mj) ) associated with the control volume H1( m j shown in Figure 2.6, the intersection points B
and F of the outside circle of the driven gear with the involute surfaces of the driving gear must be defined.
For this purpose, distances O1F ( m ) and O1B ( m ) are determined
numerically such that O2 B ( m ) = O2 F ( m ) = ro 2 in Figure 2.7(b). The overlap area Qt(,1mj) is then given as
(m) (m) (m ) Qt(,1mj) = QDO B + QEO1D + QFO E − QFO B . 1
1
1
(2.32)
( m) Here, QDO B is the area of the region bounded by the involute profile segment BD , lines 1
(m) O1B , O1D and angle BO1D , area QFO is the area of the region bounded by the 1E (m) involute profile segment FE , lines O1F , O1E and angle FO1E , area QFO is the area 1B
of the region bounded by the arc BF , lines O1B , O1F and angle BO1F and, finally, area QEO1D is the area of the region bounded by the lines ED , O1D , O1E and the
54
angle DO1E . With this nomenclature, Figure 2.7(b) shows all of the four areas used in Eq. (2.32) to define Qt(,1mj) . Equation (2.32) can then be written in integral form as
Qt(,1mj) =
1 I BD (θ)drd θ + ro1To1 + ∫∫ I FE (θ)drd θ − ∫∫ I BF (θ)drd θ . 2 DO B FO E FO B
∫∫
1
1
(2.33)
1
Here, functions I BD (θ) and I FE (θ) define the involute profiles on the drive and coast sides, both approximated by I (θ) = rb1[1 + 0.5(3θ)2 / 3 ] , so that their respective integral terms can be evaluated as
∫∫ BO1D
∫∫ FO1E
I BD (θ)drd θ = 12 rb21[(θ B − θ D ) + 0.46{(θ R − θ D )2.33 − (θR − θ B ) 2.33} 1.67
+ 1.25{(θ R − θ D )
1.67
− (θ R − θ B )
}]
I FE (θ)drd θ = 12 rb21[(θ E − θ F ) + 0.46{(θ E − θQ )2.33 − (θ F − θQ ) 2.33} 1.67
+ 1.25{(θ E − θQ )
1.67
− (θ F − θQ )
, (2.34a)
, (2.34b)
}]
where, angles θ B , θ D , θ E and θ F are measured from a horizontal line X1O1 , which is perpendicular to the lines joining the gear centers, to the lines O1B , O1D , O1E and O1F , (m) respectively, as shown in Figure 2.7(b). For evaluating the area QFO B , the radius r of 1
any point on the arc BF can be expressed in terms of the gear angle θ as I BF (θ) = e sin θ − ro22 − e2 sin 2 θ . With this, one obtains 55
∫∫
I BF (θ)drd θ = 12 ro22 (θ B − θ F ) − 14 e2 [sin(2θ B ) − sin(2θ F )]
BO1F
+ 12 ro22 [sin −1 (
e cos θ B e cos θ F ) − sin −1 ( )] ro 2 ro 2
+ 12 ero 2 [cos θ B 1 − (
(2.34c)
e cos θ B 2 e cos θ F 2 ) − cos θF 1 − ( ) ], ro 2 ro 2
with angles θ B and θ F defined in Figure 2.7(b). Area Qt(,1mj) is then determined by substituting Eqs. (2.34a) - (2.34e) into Eq. (2.33).
2.5.2.3
) Calculation of Area Qb( m ,1 j :
) Next, the excluded area Qb( m ,1 j of the
segment along the trailing profile (defined by the points E, G, J and F in Figure 2.6) of ) the control volume H1( m j is written as
) ( m) ( m) ( m) ( m) Qb( m ,1 j = QGO E + QJO G − QJO F − QFO E , 1
1
1
(2.35)
1
(m) where any area QXO Y is defined as the area of the segment between the line XY and the 1
angle XO1Y , as shown in Figure 2.7(c). Expressing Eq. (2.35) in terms of the area integrals, one writes
) ( m) Qb( m ,1 j = QGO E + 1
∫∫ JO1G
I JG (θ)drd θ −
∫∫ JO1F
56
I JF (θ)drd θ −
∫∫ FO1E
I FE (θ)drd θ ,
(2.36)
where functions I JG (θ) and I FE (θ) define the involute profiles on the drive side of gear 2 and the coast side of gear 1, respectively, whereas function I JF (θ) defines the segment JF that is a portion of the outside circle of gear 2 intersecting gear 1 profile at points F ( m) and B , respectively. Area QGO E is approximated by a triangle with known dimensions 1
for its edges EG ( m) , O1E ( m) and O1G ( m) . It is calculated as
( m) ( m) QGO )( s − O1E ( m) )( s − O1G ( m ) ) , E = s ( s − EG 1
(2.37a)
where s = 12 ( EG ( m ) + O1E ( m ) + O1G ( m ) ) . Given I JF (θ) = e sin θ − ro22 − e2 sin 2 θ , area ( m) QJO F is calculated as 1
∫∫
I JF (θ)drd θ = 12 ro22 (θ F − θ J ) − 14 e 2 [sin(2θ F ) − sin(2θ J )]
JO1F
+ 12 ro22 [sin −1 (
e cos θ J e cos θ F ) − sin −1 ( )] ro 2 ro 2
+ 12 ero 2 [cos θ F 1 − (
(2.37b)
e cos θ J 2 e cos θ F 2 ) − cos θ J 1 − ( ) ], ro 2 ro 2
where angles θ F and θ J are defined from the horizontal line X1O1 to the lines O1F and (m) O1 J , respectively. Area QFO E is calculated from Eq. (2.34b), with angles θ E and θ F , 1
defined from the horizontal line X1O1 to the lines O1E and O1F , respectively, taken as
57
the upper and lower limits of integration, while the area
( m) QJO G 1
with
2 I JG (θ) = e sin θ − rinv − e 2 sin 2 θ is obtained as
∫∫
2 I JG (θ)drd θ = 12 rinv (θG − θ J ) − 14 e2 [sin(2θG ) − sin(2θ J )]
JO1G 2 + 12 rinv [sin −1 (
e cos θG e cos θ J ) − sin −1 ( )] ro 2 ro 2
+ 12 erinv [cos θG 1 − (
(2.37c)
e cos θG 2 e cos θ J 2 ) − cos θ J 1 − ( ) ], ro 2 ro 2
where rinv is a length that varies as a function of the rotational angle from O1 J to O1G . Substituting Eqs (2.37a) to (2.37c) into Eq. (2.36) will give a closed form expression for ) excluded area Qb( m ,1 j .
2.5.3 Power Loss due to Oil Pocketing ) ( m) With Ae(,mij) (one at each side) and Ab( m at ,ij defined, the volume of the cavity H ij
a rotational position m is given by Vij( m) = bAe(,mij) . Next, the velocity of the oil squeezed out through the backlash and end flow areas are determined by using the integral form of the continuity equation [2.17]
d ρ dV = − ∫ ρ v ⋅ n dS , dt V∫ S
(2.38a)
58
where V is an arbitrary volume, ρ is the density of the oil, v is the velocity vector and n is the outward unit vector normal to the surface element S of the arbitrary volume V.
Integrating the above equation and assuming one-dimensional flow, the general flow velocity escaping through a discharge area can be written in terms of a flow area A , rotational speed ω and the rate of change of flow volume as
v=−
1 dV 1 dV =− ω. A dt A dθ
(2.38b)
The negative sign here vanishes since dV d θ is negative during pocketing, as each successive volume is smaller than the previous, until the minimum position is reached. Thus, for the first control volume, the velocity of the lubricant escaping through the backlash flow area and end flow area can be written, respectively, as
) vb( m ,ij =
(m) ξ( m) dVij ωi , ) dθ Ab( m ,ij
(2.39a)
( m)
ve( ,mij)
(1 − ξ( m) ) dVij = ωi , dθ 2 Ae(,mij)
(2.39b)
) ( m) ( m) where ξ( m) = Ab( m ,ij (2 Ae,ij + Ab,ij ) is a flow factor defined as ratio of the backlash flow
area to the total flow area. Since there are two areas of end leakage and also one area of backlash leakage, a three-dimensional analysis would be required in calculating the flow 59
velocity along each flow area. A portion of the oil is squeezed out through the end leakage area while the rest of it is discharged through the backlash leakage area. This flow factor ξ( m ) is introduced in order to ease the computational demand associated with such a three-dimensional computation, which is beyond the scope of this dissertation. Here, including ξ( m ) in Eq. (2.39a, b), in essence, captures the portion of lubricant escaping through the backlash and end leakage areas, respectively, due to the pumping action of the gears. For a gear pair with a very large face width relative to its outside diameter, ξ( m) ≈ 1 , as the backlash flow area dominates over the end leakage area, while the end leakage area dominates over the backlash flow area for gears with narrow face width, and hence, ξ( m) ≈ 0 .
The pressure of the control volume H ij( m) at the backlash and end flow areas can be estimated by using Bernoulli’s principle [2.17], respectively, as
) ( m −1) 1 −1) 2 ( m) 2 pb( m + 2 ρ[(vb( m ,ij = pb ,ij ,ij ) − (vb,ij ) ] ,
(2.40a)
) ( m −1) 1 −1) 2 ( m) 2 pe( m + 2 ρ[(ve( m ,ij = pe,ij ,ij ) − (ve,ij ) ] .
(2.40b)
The conservation of momentum principle [2.17] gives the forces acting on the oil escaping through the backlash and end areas of H ij( m) as
60
) ( m) ( m) ( m) Fb(,mij ) = − ∫ pb( m ,ij dAb ,ij = − pb ,ij Ab ,ij ,
(2.41a)
S ) ( m) ( m) ( m) ( m) Fe(,m ij = − ∫ pe,ij dAe,ij = − pe,ij Ae,ij .
(2.41b)
S
The power loss due to pocketing at H ij( m) is equal to the product of the discharge velocity and the pocketing force:
) ( m) (m ) ( m) ( m) Pp( m ,ij = vb ,ij Fb ,ij + 2ve,ij Fe,ij .
(2.42a)
This represents only the power loss for the j-th control volume at the m-th rotational position for gear i. The same procedure is applied to all J1( m ) control volumes of gear 1 and J 2( m ) control volumes of gear 2 to determine the total instantaneous power loss of a gear pair due to pocketing that is then averaged over the total number of rotational increments, M, to find the average power loss as
1 Pp = M
J 2( m ) J1( m ) m ( ) ( m) ∑ ∑ Pp,1 j + ∑ Pp,2 j . m =1 j =1 j =1 M
(2.42b)
Additional power losses can take place in the meshing zone due to secondary effects loss such as lift-off of the lubricant and acceleration of the lubricant trapped in the 61
meshing zone in the circumferential direction. Such effects were not included in this paper since they were found to be negligibly small when compared to the Pp values predicted above.
2.6 Example Oil Churning Analysis
The drag and pocketing components of the total gear pair spin loss PT defined in Eq. (2.2) are determined by using Eqs. (2.7), (2.13) (or Eq. (2.17) if the regime is turbulent), (2.23), and (2.42b), respectively. These equations, all given in closed-form, provide the total spin loss as a function of gear parameters, lubricant properties, and operating conditions. In order to exercise the proposed gear pair spin loss formulation, two unity-ratio spur gear pairs having different modules are considered: a 40-teeth pair at a module m% = 2.32 mm and a 23-teeth gear pair having m% = 3.95 mm. The 40-teeth gear pair is considered to have three face width values of b = 14.7, 19.5 and 26.7 mm, whereas the 23-teeth gear pair has a facewidth of b = 19.5 mm. The gears are operated at a center distance of 91.5 mm, under no load (i.e. zero transmitted torque).
Also, since the
example analysis is concerned with computing power losses involved in the case of unloaded conditions, the surface roughness of the gears does not play any role in the subsequent analyses. Table 2.1 lists all essential parameters of these two example gear pairs. In addition to b and m% , rotational speed Ω (in rpm), oil temperature T (in ºC), and the oil immersion depth parameter h are also varied to quantify their influence on the 62
_______________________________________________________________ Parameter
23T gears
40T gear
_______________________________________________________________ Number of teeth
23
40
Module, mm
3.95
2.32
Pressure angle, deg
25.0
28.0
Base diameter, mm
82.34
81.89
Outside diameter, mm
100.34
95.95
Circular tooth thickness, mm
6.435
2.925
Root fillet radius, mm
1.63
0.83
Face width, mm
19.5
14.7, 19.5, 26.7
Center Distance, mm
91.5
91.5
_______________________________________________________________
Table 2.1 The design parameters of the example spur gears.
63
resultant PT . The rotational speed ranged from Ω = 1000 rpm to Ω = 4000 rpm , with oil immersion level ranging from h = 0.5 , which refers to an oil immersion level of 25 % of the pitch radius of the gear to h = 1.5 , which refers to an oil immersion level of 75 % of the pitch radius of the gear. Also, the operating oil temperature ranged from 20 ºC to 100 ºC. It is to be noted that under the oil immersion depths considered in the example analysis, the lubricant properties are that of the oil itself with air not taking any part in the formulations. It is useful to mention here that each analysis shown in the following figures took less than a second of CPU time on a personal computer, indicating that the proposed model is indeed suitable as a practical design tool. In Figure 2.8(a), the influence of T on PT is demonstrated for m% = 2.32 mm, b = 19.5 mm and h = 1.0 . Here, a transmission fluid ( µ = 0.217, 0.035 and 0.011 Pa-s and ρ = 847.3, 823.3 and 799.3 kg / m3 at T = 20, 60 and100 ºC, respectively) is considered as the lubricant. As shown in Figure 2.8(a), PT increases with reduction in T (increased µ ) regardless of the value of Ω . For example, spin loss at T = 20 ºC and Ω = 4, 000 rpm is about 0.3kW higher than PT value at the same Ω and at T = 100 ºC. This difference is due to significant changes in viscosity of the oil (about 15 times higher viscosity at 20 ºC than at 100 ºC, which changes Pdpi and Prfi linearly (Eqs. (2.7) and (2.23)), while it has a more complex influence on Pdfi and Pp . This predicted effect of viscosity conflicts with the experimental observations of Luke and Olver [2.8] and that of 64
(a)
1.0
(b)
20 o C
h = 1.5
o
0.8
60 C
h = 1.0
100 o C
h = 0.5
0.6 PT [kW]
0.4
0.2
0.0 (c)
1.0
0.8
(d)
b = 26.7 mm
% = 3.95 mm m
b = 19.5 mm
m% = 2.32 mm
b = 14.7 mm
0.6 PT [kW]
0.4
0.2
0.0 0
1000
2000
3000
4000 0
1000
2000
3000
4000
Speed [rpm]
Figure 2.8 Effect of (a) temperature T , (b) oil level parameter h , (c) face width b and (d) gear module m% on total spin power loss PT .
65
Changenet and Velex [2.14] while agreeing well with experiments described in Chapter 4, which talks about the validation of the oil churning and windage power loss models. In Figure 2.8(b), the influence of the quasi-static oil levels on PT is demonstrated for a gear pair having m% = 2.32 mm and b = 19.5 mm, operated in lubricant at a temperature of 80 ºC ( µ = 0.0185 Pa-s and ρ = 811.3 kg / m3 ). Here, the oil levels h = 0.5 , 1.0 and 1.5 represent 25, 50 and 75% immersion of the gears in oil, according to Figure 2.1. As all components of PT are dependent on h , significant increases in PT are observed with increased immersion depth, with almost 2.5 times larger PT values at h = 1.5 compared to those at h = 0.5 . Figure 2.8(c) shows the influence of b on PT for a gear pair having m% = 2.32 mm operating at h = 1.0 and 80 ºC. Here, the gear pair having b = 26.7 mm experiences about 2.8 times higher PT than the gear pair with b = 14.7 mm. The components of PT for the gear pair with m% = 2.32 mm and varying face width values are shown in Figure 2.9. Here, in line with the formulation given above, the major changes with b take place in Pdp and Pp components of the power loss as shown in Figures 2.9(a) and 2.9(d). More importantly, Figure 2.9 indicates that Pp and Pdf are the two most significant components of oil churning power loss of this gear pair while contributions of Pdp and
Prf to PT are both quite negligible. For instance, the power loss values of the gear pair having b = 19.5 mm at Ω = 4, 000 rpm are such that Pp is about 74.4% of PT followed by 25.1% for Pdf , while the other two components Pdp and Prf are 0.48% and 0.02% 66
(a) Pdp
0.0030
(b) Pdf
0.20
b = 26.7 mm b = 19.5 mm
0.0024
0.15
Power Loss [kW]
b = 14.7 mm
0.0018
0.10 0.0012
0.05
0.0006
0.00
0.0000
(c) Prf
0.00015
0.00012 Power Loss [kW]
(d) Pp
0.8
0.6
0.00009 0.4 0.00006 0.2 0.00003
0.0
0.00000 0
1000
2000
3000
4000
0
1000
2000
3000
4000
Speed [rpm]
Figure 2.9 Components of PT for a gear pair having m% = 2.32 mm and b = 26.7, 19.5
and 14.7 mm at h = 1.0 and 80 ºC; (a) Pdp , (b) Pdf , (c) Prf , and (d) Pp .
67
of PT , respectively.
The last comparison in Figure 2.8(d) illustrates the influence of module m% on PT for b = 19.5 mm, h = 1.0 and oil at an operating temperature of 80 ºC. Here, the coarser pitch gear pair with m% = 3.95 mm is predicted to have more spin power loss than the gear pair having m% = 2.32 mm. For instance, at Ω = 4, 000 rpm, PT for m% = 3.95 mm is about 0.1 kW higher than that of the gear pair having m% = 2.32 mm.
As Pdf and Pdp
components are not expected to be influenced significantly by tooth size according to the formulations presented, and Prf is rather negligible, this difference in PT values with module can be attributed primarily to the pocketing losses. As one last observation from Figures 2.8 and 2.9, the value of PT changes with Ω ( ω = 2πΩ 60 ) in a non-linear fashion. For instance, in Figure 2.8(c) for b = 19.5 mm, PT ∝ ω2.11 while the corresponding components of the power loss in Figure 2.9 reveal the dependency of the individual components of power loss on speed as
Pdf ∝ ω2.5 , Pdp ∝ ω2 , Prf ∝ ω and Pp ∝ ω1.8 .
2.7 Concluding Remarks
A physics-based fluid mechanics model was proposed in this chapter to predict power losses of gear pairs due to oil churning. Individual formulations were proposed for evaluating drag, root filling and pumping components of the gear pair spin losses by treating power losses associated with pumping of the oil from the gear mesh, drag forces 68
induced by the oil on a rotating gear body along its periphery and faces, and finally, eddies formed in the cavities between adjacent teeth. The model was applied to a unityratio spur gear pair to quantify the individual contributions of each component to the total oil churning power loss. At the end, the influence of operating conditions, selected gear geometry parameters, module and face width, and lubricant properties on oil churning power losses were quantified for the example gear pair.
69
References for Chapter 2
[2.1]
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.
[2.2]
Martin, K. F., 1978, “A Review of Friction Predictions in Gear Teeth,” Wear, 49, 201–238.
[2.3]
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, 217–232.
[2.4]
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.
[2.5]
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.
[2.6]
Von Karman, T., 1921, “On Laminar and Turbulent Friction,” Z. Angew. Math. Mech., 1, 235-236.
70
[2.7]
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.
[2.8]
Luke, P., and Olver, A., 1999, “A Study of Churning Losses in Dip-Lubricated Spur Gears,” Proceedings of the Institute of Mechanical Engineers, Part G, Journal of Aerospace Engineering, 213, 337–346.
[2.9]
Ariura, Y., Ueno, T., and Sunamoto, S., 1973, “The lubricant churning loss in spur gear systems,” Bulletin of the JSME, 16, 881-890.
[2.10] 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, 1266–1273. [2.11] 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. [2.12] 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, 335–342. [2.13] Diab, Y., Ville, F., Houjoh, H., Sainsot, P., and Velex, P., 2005, “Experimental and Numerical Investigations on the Air-Pumping Phenomenon in High-Speed Spur and Helical Gears,” Proceedings of the Institution of Mechanical Engineers, Part C: J. Mechanical Engineering Science, 219, 785-800. [2.14] Changenet, C., and Velex, P., 2007, “A Model for the Prediction of Churning Losses in Geared Transmissions – Preliminary Results“, ASME Journal of Mechanical Design, 129(1), pp. 128-133. 71
[2.15] Höhn, B.R., Michaelis, K., Völlmer, T., 1996, “Thermal Rating of Gear Drives: Balance between Power Loss and Heat Dissipation, “AGMA, Fall Technical Meeting, 96FTM8, pp. 1-12. [2.16] Changenet, C., and Velex, P., 2008, “Housing Influences on Churning Losses in Geared Transmissions“, ASME Journal of Mechanical Design, 130 (6), 062603, 6 pages. [2.17] Streeter, L. V., and Wylie, E. B., 1985, Fluid Mechanics, Eighth edition, McGraw-Hill, New York. [2.18] Schlichting, H., 1955, Boundary-Layer Theory, McGraw-Hill Book Company, New York. [2.19] 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. [2.20] 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. [2.21] Martin, K. F., 1978, “A Review of Friction Predictions in Gear Teeth,” Wear, 49, 201–238. [2.22] 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, 217–232. [2.23] 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 72
Blade Aspect Ratio and Solidity,” ASME Journal of Basic Engineering, 83 (4), 719-723. [2.24] 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. [2.25] Von Karman, T., 1921, “On Laminar and Turbulent Friction,” Z. Angew. Math. Mech., 1, 235-236. [2.26] 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.
[2.27] Luke, P., and Olver, A., 1999, “A Study of Churning Losses in Dip-Lubricated Spur Gears,” Proceedings of the Institute of Mechanical Engineers, Part G, Journal of Aerospace Engineering, 213, 337–346. [2.28] Ariura, Y., Ueno, T., and Sunamoto, S., 1973, “The lubricant churning loss in spur gear systems,” Bulletin of the JSME, 16, 881-890. [2.29] 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, 1266–1273. [2.30] 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. [2.31] 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, 335–342. 73
[2.32] Diab, Y., Ville, F., Houjoh, H., Sainsot, P., and Velex, P., 2005, “Experimental and Numerical Investigations on the Air-Pumping Phenomenon in High-Speed Spur and Helical Gears,” Proceedings of the Institution of Mechanical Engineers, Part C: J. Mechanical Engineering Science, 219, 785-800. [2.33] Changenet, C., and Velex, P., 2007, “A Model for the Prediction of Churning Losses in Geared Transmissions – Preliminary Results“, ASME Journal of Mechanical Design, 129(1), pp. 128-133. [2.34] Höhn, B.R., Michaelis, K., Völlmer, T., 1996, “Thermal Rating of Gear Drives: Balance between Power Loss and Heat Dissipation, “AGMA, Fall Technical Meeting, 96FTM8, pp. 1-12. [2.35] Changenet, C., and Velex, P., 2008, “Housing Influences on Churning Losses in Geared Transmissions“, ASME Journal of Mechanical Design, 130 (6), 062603, 6 pages. [2.36] Streeter, L. V., and Wylie, E. B., 1985, Fluid Mechanics, Eighth edition, McGraw-Hill, New York. [2.37] Schlichting, H., 1955, Boundary-Layer Theory, McGraw-Hill Book Company, New York. [2.38] 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.
74
CHAPTER 3
WINDAGE POWER LOSSES OF A GEAR PAIR: MODEL FORMULATION
3.1 Introduction In Chapter 2, a fluid mechanics based oil churning power loss model for a spur gear pair was proposed. As depicted in Figure 2.1, the spur gear pair was immersed in a liquid medium (lubricating oil bath). In such a case of gears operating while immersed in an oil bath, with this method of lubrication often called dip lubrication, the oil churning losses under dip-lubrication conditions were represented by the sum of (i) the losses due to oil drag on the faces and periphery of the gears, (ii) the losses associated with eddies formed during the filling of lubricant in the cavities between adjacent gear teeth and (iii) losses caused by the displacement of the oil trapped within the gear mesh due to squeezing and pumping. Examples of geared systems operating under dip-lubrication condition are ample in low to medium speed automotive transmissions and in industrial gearbox applications. When the operating gear speeds are relatively high, the jet-lubrication method is preferred instead of dip-lubrication, as is the case with most aerospace powertrain and
75
turbine applications. In this method of lubrication, a high-pressure jet of oil is directed towards the gear mesh interface via a nozzle, and oil is not allowed to accumulate within the gearbox to cause any oil churning losses. Hence, power losses under jet-lubrication conditions can mostly be attributed to windage losses due to the interaction between air or air-oil mixture and the rotating gears. Pocketing power loss due to squeezing of the compressible air/air-oil mixture from the meshing zone forms the main component of windage power loss. In addition to pocketing loss, windage drag power loss due to air drag along the periphery and surfaces of the rotating gears can also be expected to contribute additionally to the total windage power loss. It was shown experimentally by Petry-Johnson et al. [3.1] that power losses in dip-lubricated gearboxes with very low static oil levels can also be dominated by air windage. With the assumption that the oil jet directed at the gear mesh has a negligible influence, the windage power loss Pw of a gear pair is given as
Pw = Pwp + Pwd ,
(3.1)
where Pwp is the air pumping/pocketing power loss and Pwd = Pwd1 + Pwd 2 is the total drag power loss of a gear pair. Meanwhile, the drag power loss of each gear i ( i = 1, 2 ) is given as the sum of drag losses at its periphery, Pwdpi , and its faces (sides), Pwdfi , as
Pwdi = Pwdpi + Pwdfi .
(3.2)
76
Most of the investigations on gear windage power losses have been experimental in nature, focusing on losses of gears or disks of various sizes rotating in free air and surrounded by different enclosures. In one such study, Dawson [3.2] performed a large number of gear windage experiments using spur and helical gears rotating in free air and derived empirical formulae to describe windage power losses as a function of gear geometry, lubricant (air) and speed parameters. Diab et al [3.3] carried out windage experiments on disks and gears of an assortment of shapes and sizes, and subsequently came up with quasi-analytical predictions based on fluid flow around a rotating gear and also through dimensional analysis, to characterize windage loss in pinion-gear pairs. Their model does not consider the presence of the mating gear and hence, the major component of windage power loss in the form of fluid compression and expansion taking place in the meshing zone between gear teeth is not considered in the model. Eastwick and Johnson [3.4] have recently provided an extensive review of studies on gear windage to conclude that a physics-based, generalized solution for reducing power loss due to windage has not yet been well established. Wild et al [3.5] studied the flow between a rotating cylinder and a fixed enclosure by using a computational fluid dynamics (CFD) model and accompanying experiments. Similarly, Al-Shibl et al [3.6] proposed a CFD model of windage power loss from an enclosed spur gear pair. Review of these studies indicates that there is a need for an accurate and computationally efficient windage power loss model for predicting power losses of jetlubricated, high-speed spur gear pairs. Accordingly, the main objective of this chapter is to develop a fluid mechanics-based model for predicting windage losses of spur gear pairs. This model will consist of a compressible fluid flow gear mesh pocketing power 77
loss model, and an air drag formulation through a modification of the drag power loss formulations from the oil churning model presented in Chapter 2.
Following the
formulation of the windage power loss model, the validation of the model with respect to actual windage loss experiments [3.1] and empirical results [3.2] will ensue in Chapter 4. At the end of this chapter, the model will be exercised for a pair of example spur gear sets to quantify the contributions of the major components of windage power loss and to highlight the computational ease of use of the proposed windage power loss model through a parametric study. In the subsequent sections, formulations will be proposed to compute Pwp , Pwdpi and Pwdfi , so that the total windage power loss of a spur gear pair can be determined according to Eq. (3.1).
3.2 Windage Pocketing Power Loss Model Formulation of the model for power loss due to pumping of air or air-oil mixture from the gear mesh is identical to that of oil pumping power loss proposed in Chapter 2, in terms of the changes in discharge areas and control volumes formed between the mating teeth of the gears as they rotate. The main difference between the windage pocketing power loss model and the oil churning pocketing model is that the medium considered here is air or a mixture of air and oil, depending on the aeration ratio, and hence, the assumption made in Chapter 2 that the fluid is incompressible is no longer valid: variations in density as the pocket volume changes with rotational position has to be considered in case of compressible flow, and corresponding variations in pressure, temperature and velocity of the fluid ejected through the respective discharge areas will 78
follow. The windage pocketing phenomenon is defined through the squeezing action of any control volume formed between a gear tooth and the mating tooth cavity over successive rotational positions, as illustrated in Figure 2.3. Looking at Figure 2.3, it is necessary to conceive the pockets to be filled by air or a mixture of air and oil, instead of the incompressible lubricant. As the air or air-oil mixture gets squeezed out from the openings at the ends and at the backlash (as seen in Figure 2.4) due to successive compression of the control volume till the volume is minimized at a certain rotational gear position, a pressure differential is created between the ambient atmosphere outside the control volume and the air-oil mixture or air inside the control volume. In the process, the mixture is squeezed out at typically high velocities through the discharge areas. The high velocity of compressible fluid ejected through the flow areas will result in power losses. The formulation defined below for the windage pocketing power loss in terms of compressible fluid flow will first evaluate the discharge areas formed between the involute profiles of the mating gears. This evaluation procedure for the discharge areas follows very much on the lines for computing the flow areas in case of the meshing zone filled with lubricant as laid out in Chapter 2. Next, a methodology to compute the velocity of the air or air-oil mixture being squeezed out through the end and backlash flow areas will be proposed through means of the conservation of mass and conservation of energy principles. The pressure inside the control volume will then be computed using two different methodologies, which depart from each other at higher operating speeds of rotation (Mach numbers higher than 0.3): (i) the compressible Bernoulli formulation and, (ii) isentropic relationship between pressure and density inside control volume. The 79
conservation of momentum principle will be used to compute the force acting on the atmosphere inside the control volume through the product of the mass flow rate and the velocity of the fluid escaping through the discharge areas.
Finally, the windage
pocketing power loss will be then be defined as the product of the force acting on the fluid inside the control volume and the velocity of the compressible fluid being ejected out of the control volume. It has to be noted that computation of velocity of the fluid escaping through the backlash discharge area can be handled in two different ways. One line of thought is to treat the pockets (control volumes) formed between the mating gear teeth as separate entities, with no pressure differential being communicated between opposing pockets or (0) , which is the first cavity of gear 2 and control volumes (an example labeled as H 21 (0) H12 , which is the second cavity of gear 1, both at the beginning of contact, seen in
Figure 2.3(a)). Another way of approach is to consider the pressure differential between the opposing control volumes, as a result of which, not all of the fluid lying in the backlash discharge region gets ejected through the backlash discharge area. Some of the fluid will in fact be transported inside the opposing control volume having lesser pressure and forcefully ejected through the end discharge area corresponding to that control volume, reinforcing the velocity of the compressible mixture already being ejected through those particular end flow areas, by augmenting the mass flow rate of the fluid ejected through the end area.
In order to alleviate concerns about the physics of
broaching the problem through either means of thought, in this work, the control volumes
80
or pockets will be considered as independent entities, with power loss taking place through both the discharge areas. Referring to Figure 2.4, areas of the openings of a given cavity H ij( m) (j-th cavity ) of gear i) at a rotational position m along its backlash and end (side) openings, Ab( m ,ij and
Ae(,mij) , are defined as in Chapter 2, respectively, as
) ( m) Ab( m , ,ij = b EG
(3.3)
) ( m) Ae(,mij) = Qc 2 − Qt(,mij ) − Qb( m ,ij − QBCK ,
(3.4)
where b is the effective face width of the gear pair and EG ( m) is the backlash length (the shortest distance between a point on the coast side of gear 1 and the involute profile of gear 2), seen in Figure 2.6. As defined in Figure 2.7(a), Qc 2 is the total tooth cavity area bounded by the major diameter and two adjacent teeth of gear 2, Qt(,mij ) is the portion of ) Qc 2 overlapped by the tooth of gear I, as shown in Figure 2.7(b), Qb( m ,ij is the excluded ( m) area of the segment along the trailing profile, depicted in Figure 2.7(c), and QBCK is the
small portion of the area bounded by the involute segment BC of gear 1 and CK of gear 2. ) ( m) Closed form expressions for Ab( m ,ij and Ae,ij were derived in Chapter 2 based on
gear tooth geometry and gear kinematics. The very same expressions will be used here as 81
well, with the only different that the medium inside the pockets is air or a mixture of air and oil, and hence, the assumption made in Chapter 2 that the fluid is incompressible is no longer valid. Meanwhile, as a considerable departure from the incompressible fluid flow formulation of the earlier oil pocketing model of Chapter 2, the windage pocketing formulations must use a compressible flow approach to compute the flow velocities through the end and backlash flow areas, as reiterated in preceding paragraphs. Applying the principle of conservation of mass [3.7] to an arbitrary control volume, the total rate of change of mass in a control volume is equal to the sum of the rate of change of mass accumulated inside the control volume and the total mass exiting (or entering) the control volume such that
d ρdV + ∫ ρ v ⋅ n dS = 0 . dt V∫ S
(3.5)
Here V is an arbitrary volume, ρ is the density of the squeezed air (or air-oil mixture), v is the velocity vector of the air/mixture escaping through the control volume, and n is the outward unit vector normal to the surface element S of the arbitrary volume V. Applying Eq. (3.5) to sample control volumes, H ij( m) and H ij( m+1) between successive rotational positions m and (m+1), the discretized equation for conservation of mass at the end and backlash flow areas between time steps t and t + ∆t can be written as
1 ∆t
) ( m) ρ(em,ij+1)Vij( m+1) − ρ(em,ij)Vij( m) + 2ρ∞ ve( m ,ij Ae,ij = 0,
82
(3.6a)
1 ∆t
ρ( m+1)Vij( m+1) − ρ( m)Vij( m) + ρ∞ v ( m) A( m) = 0. b,ij b ,ij b ,ij b,ij
(3.6b)
(m) m) , ρ(e,ij and Vij( m) are the backlash area discharge In the above set of equations, ρb,ij
density, the end area discharge density and the volume of the fluid inside control volume ( m+1) m +1) H ij( m) , which is at an incremental rotational position m, ρb,ij and ρ(e,ij and Vij( m) are
the physical quantities for the same control volume H ij( m+1) , but which is now at an ) ( m) incremental rotational position (m+1), ρ∞ is the ambient air density, and ve( m ,ij and vb,ij
represent the end flow and backlash flow velocities at rotational position m. Simplifying this further, with ∆t = ∆θi ωi , where ∆θi is the incremental rotation angle between positions m and (m+1) of gear i and ωi is the rotational velocity of gear i, expressions for the backlash and end flow velocities can be obtained as
) vb( m ,ij
ωi ρb( m,ij)Vij( m) − ρb( m,ij+1)Vij( m+1) , = ( m) ρ∞ Ab,ij ∆θi
(3.7a)
) ve( m ,ij
ωi ρ(em,ij)Vij( m) − ρ(em,ij+1)Vij( m+1) . = ( m) 2ρ∞ Ae,ij ∆θi
(3.7b)
Denoting the energy per unit mass inside the control volume at the end and backlash flow areas during the rotational increment m by ε(em,ij) and εb( m,ij) , and energy per unit mass of the same control volume at location (m+1) by ε(em,ij+1) and εb( m,ij+1) , and 83
denoting the energy per unit mass at ambient conditions as ε∞ , the discretized form of the second law of thermodynamics [3.7] can be applied to the control volumes between time instants t and t + ∆t (i.e. between positions m and (m+1)) at the end and backlash flow areas, respectively, as
( m +1) ( m+1) ( m+1) − ρ(em,ij) ε(em,ij)Vij( m ) δQh δWs ρe,ij εe,ij Vij ) ( m) − = + 2ρ∞ ε∞ ve( m ,ij Ae,ij = 0 , δt δt ∆t
(3.8a)
( m +1) ( m+1) ( m+1) − ρb( m,ij) εb( m,ij)Vij( m ) δQh δWs ρb,ij εb,ij Vij ) ( m) − = + ρ∞ ε∞ vb( m ,ij Ab ,ij = 0 . δt δt ∆t
(3.8b)
In the above equation, the energy per unit mass is given by the sum of the internal and kinetic energies. The thermodynamic process is assumed to be adiabatic ( δQh δt = 0 ), and the shaft work is deemed to be negligible ( δWs δt = 0 ). In addition, potential energy and surface tension effects as well as electrical and magnetic effects are neglected. Given the fact that there exists no kinetic energy inside the control volume at a given time instant because of zero velocity inside the control volume, the energy per unit mass inside the control volume at the end and backlash flow areas during the rotational ) ( m) increment m is written in terms of the internal energy ue( m ,ij and ub,ij , with the internal +1) +1) energy of the same control volume at position (m+1) given by ue( m and ub( m for the ,ij ,ij
respective discharge areas. Denoting the energy per unit mass at ambient conditions as the sum of the internal energy at ambient conditions, u∞ , and the kinetic energy of the 84
fluid escaping through the end and backlash flow areas, Eqs. (3.8a) and (3.8b) can be written based on these assumptions in the following form:
+1) ( m +1) ) ( m) ρe( m,ij+1)ue( m − ρ(em,ij)ue( m ,ij Vij ,ij Vij
∆t +1) ( m +1) ρb( m,ij+1)ub( m − ρb( m,ij)ub( m,ij)Vij( m ) ,ij Vij
∆t
) 2 (ve( m ,ij ) ) ( m) + 2ρ∞ + u∞ ve( m ,ij Ae,ij = 0 , 2
(3.9a)
(v ( m ) ) 2 b ,ij ) ( m) + ρ∞ + u∞ vb( m ,ij Ab ,ij = 0. 2
(3.9b)
) ( m) ( m +1) +1) ) ( m) ( m +1) Noting that ue( m = cvTe(,m , ub( m = cvTb(,mij +1) and ,ij = cvTe,ij , ue,ij ij ,ij = cvTb ,ij , ub,ij ) ( m +1) u∞ = cvT∞ , where cv is the specific heat at constant volume, and Te(,m , Tb(,mij ) and ij , Te,ij
Tb(,mij +1) refer to the temperatures corresponding to successive rotational increments, m and m + 1 , along the end and backlash flow areas, Eqs. (3.9a) and (3.9b) can be written as
+1) ( m +1) ) ( m) cv [ρ(em,ij+1)Te(,m − ρ(em,ij)Te(,m ij Vij ij Vij ]
∆t cv [ρb( m,ij+1)Tb(,mij +1)Vij( m +1) − ρb( m,ij)Tb(,mij )Vij( m ) ] ∆t
) 2 (ve( m ,ij ) + 2ρ∞ + cvT∞ ve( ,mij) Ae(,mij) = 0 , (3.10a) 2
(v ( m ) ) 2 b ,ij ) ( m) + ρ∞ + cvT∞ vb( m ,ij Ab,ij = 0 . (3.10b) 2
Further, assuming that the thermodynamic process is isentropic, the following relation can be written for the temperature, pressure and density between two successive rotational increments [3.7]:
85
+1) Pe(,m ij ) Pe(,m ij
γ
γ
+1) γ−1 ( m +1) Te(,m ρe,ij ij = ( m) = ( m) . Te,ij ρe,ij
(3.11)
Here γ = c p cv is the compressibility factor. With this, Eq. (3.10a) and (3.10b) can be written in terms of the end and backlash flow velocities and the end and backlash densities, respectively. Substituting for the flow velocity along backlash and end flow areas into Eq. (3.10a) and (3.10b), two non-linear algebraic equations are obtained, one for each discharge density, which can be solved using the Gauss-Newton method with a suitably defined step size. Once the density at the backlash and end flow areas are determined through successive rotational increments m ∈ [1, M ] , the pressure at the backlash and end flow areas can be calculated by two different ways.
The first method uses the
compressible version of the Bernoulli equation [3.7] as
Pb(,mij +1)
ρ( m +1) 1 γ − 1 ( m +1) ( m ) 2 b ,ij = ( m ) Pb(,mij ) + ρb,ij (vb,ij ) − (vb( m,ij+1) )2 , ρb,ij 2 γ
(3.12a)
+1) Pe(,m ij
ρe( m,ij+1) ) 1 γ − 1 ( m +1) ( m ) 2 ( m +1) 2 = ( m) Pe(,m ij + ρe,ij (ve,ij ) − (ve,ij ) . ρe,ij 2 γ
(3.12b)
The second way is to use Eq. (3.11) directly to write
86
γ
Pb(,mij +1)
+1) Pe(,m ij
ρ( m+1) ( m ) b ,ij = Pb,ij , ρb( m,ij1 )
=
( m+1) ( m ) ρe,ij Pe,ij ( m) ρe,ij
(3.13a)
.
(3.13b)
At low Mach numbers, the compressible Bernoulli formulation for pressure calculations is compatible with the isentropic relationship for pressure computation. The simulations carried out in Section 3.4 as well as the simulations used in Chapter 4 for the validation of the windage pocketing power loss model will make use of the isentropic relationship between density and pressure by computing the pressure inside the control volume for the backlash and end flow areas using Eq. (3.13a) and (3.13b), respectively. The rest of the windage pocketing power loss formulation follows the same procedure as laid out for incompressible fluid flow pocketing formulation of Chapter 2. The conservation of momentum [3.7] gives the forces acting on the air or air-oil mixture exiting through the backlash and end flow areas, respectively, as the product of the mass flow rate times the velocity of fluid ejected through the corresponding discharge areas, i.e.,
Fb(,mij +1)
+1) ( m+1) Pb(,mij +1) Ab( m v ,ij b,ij = RTb(,mij )
2
P (m) b,ij Pb(,mij +1)
87
(
γ−1 ) γ
,
(3.14a)
+1) Fe(,m = ij
2
( +1) ve( m (m) ,ij Pe,ij +1) ) Pe(,m RTe(,m ij ij
+1) ( m+1) Pe(,m Ae,ij ij
γ−1 ) γ
,
(3.14b)
where R = 8.314 J/kg/mol is the universal gas constant and P∞ is the ambient pressure. Finally, the power loss due to pocketing of control volume H ij( m) at a given incremental rotational position m is equal to the product of the discharge velocity and the pocketing force, with the factor of two accounting for both the end flow areas:
( m) ( m) ( m) (m ) ( m) Pwp ,ij = vb,ij Fb,ij + 2ve,ij Fe,ij .
(3.15)
This represents the power loss for the j-th control volume of gear i at the m-th rotational position. The same procedure is applied to all J1( m ) control volumes of gear 1 and J 2( m ) control volumes of gear 2 shown in Figure 2.3 to determine the total instantaneous power loss of a gear pair due to pocketing, which is averaged over the entire gear mesh cycle to find the average windage pocketing power loss, very much similar to the way it is defined in Chapter 2 for the incompressible fluid flow analysis as
Pwp
1 = M
J 2( m ) J1( m ) ( ) ( m) m P + P ∑ ∑ wp,1 j ∑ wp,2 j . m =1 j =1 j =1 M
88
(3.16)
In the formulation presented above, it is assumed that opposing control volumes linked to each other through the backlash discharge areas are independent entities, with a pressure differential existing only between the inside of the control volume and the ambient conditions. In reality, as explained in the introductory portion of the chapter, there is the possibility of a pressure differential existing between opposing control volumes themselves, with the backlash discharge area acting as the bridge in communicating this pressure differential between the opposing control volumes. The involved task of considering the pressure differential between opposing control volumes as dependent entities falls beyond the scope of this dissertation.
3.3 Windage Drag Power Losses Drag loss formulations of the oil churning model proposed in Chapter 2 are applicable to the windage power loss model as well, though with some minor changes to accommodate for the presence of air or air-oil mixture surrounding the gears. The drag power loss is broken down into power loss taking place on the periphery (circumference) of the gears and power loss taking place along the faces (sides) of the gears. The formulations for drag loss on the periphery and faces of the gears are applied to both gears in the pair, and the total losses are summed up to find the total drag power loss of the gear pair, in the manner as laid out in Chapter 2. In applying the periphery drag loss formulation, laminar flow between concentric rotating disks is considered to be the flow methodology. This represents the scenario of the outer disk placed at infinity, aping the presence of the flanges/casing, and the inner 89
disc placed at the center mimicking the presence of the gear. Such an approach was used to come up with an expression for the shear stress built up due to resistance to fluid flow along the periphery in Chapter 2.
This resistance to fluid flow takes place in the
tangential direction along the periphery of the gears. The drag loss on the periphery is then given by the product of the built up shear force and the velocity of the fluid along the periphery of the gear in the tangential direction. Referring to Chapter 2 for equations concerning peripheral drag due to oil churning, as well as setting the immersion depth parameter hi = hi roi = 2 where hi is the static depth of the fluid around the gear i and
roi is the outside (major) radius of gear i (i.e. gear is fully submerged in air or air-oil mixture), the windage drag power loss of gear i along its periphery is derived as
Pwdpi = 4 πµ bi roi2 ωi2 .
(3.17)
Here, µ is the dynamic viscosity of air (or a homogeneous mixture of air and oil) and bi is the face width of gear i. Similarly, the power losses due to air drag along the faces of gear i can be derived from the same oil churning formulations of Chapter 2. In formulating the drag loss along the faces of the gears, a boundary layer approach was used in Chapter 2, with the laminar and turbulent regimes handled separately, based on the Reynolds number for fluid flow across the faces. Also, in Chapter 2, a linear velocity profile was considered for fluid flow across the faces of the gears to represent laminar conditions, and the boundary layer thickness and skin friction coefficient were calculated through the application of the von 90
Karman momentum integral equation [3.7]. Finally, the power loss due to drag on the faces of gear i was written as the product of the shear stress built up across the faces of the gear due to boundary layer formation and the velocity of the fluid flowing across the faces of the gear. With h = 2 , windage drag power loss of gear i along its faces in the laminar regime is hence given as
( L) 2.5 4 Pwdfi = 0.41 π ρµ0.5 k ωi roi ,
(3.18)
where superscript L denotes the laminar regime, and µ k and ρ are the kinematic viscosity and density of air (or air-oil mixture). Likewise, the oil churning power loss model of Chapter 2 used the Prandtl-one-seventh power law for the velocity profile in turbulent regime, from which the boundary layer thickness and skin friction coefficient were computed again, using von Karman’s equation [3.7]. The turbulent flow face drag formula is adopted here for computing the windage power loss across both the faces of the gear, with flow being turbulent and by setting h = 2 , as
(T ) Pwdfi = 0.025πρµ0.14 ωi2.86 roi4.72 . k
(3.19)
3.4 Example Windage Power Loss Analyses and Parametric Studies
In order to investigate the influence of various operating parameters and also to quantify the contributions of individual components on the total windage power loss, two 91
unity-ratio example spur gear pairs, whose gear geometry parameters are listed in Table 2.1 (for the sake of consistency, the same family of spur gear pairs used in the oil churning analysis of Chapter 2 is also used for the example analysis presented in this chapter), are considered for this purpose. It is to be noted that the backlash flow area is given by the product of the face width of the gears and the backlash length at all points over one mesh cycle, keeping the variations in backlash flow area consistently the same throughout the mesh cycle. This is considered for the sake of simplicity of analysis. The central theme of this section is then to (i) demonstrate the effective contributions of drag and pocketing power loss components to the total windage power loss Pw and (ii) investigate the influence of various gear pair parameters on Pw . The first example gear pair is a 40-teeth design with a module of m% = 2.32 mm and three different face width values of b = 14.7, 19.5 and 26.7 mm , respectively. The second example spur gear pair is a 23-teeth design, with m% = 3.95 mm and b = 19.5 mm . Both gear pairs are operated at a center distance of 91.5 mm, under zero transmitted torque and under jet lubrication conditions. In these example analyses, the rotational speed Ω = Ω1 =
60 ω 2π 1
is varied within a range of up to 10,000 rpm (corresponding to a
maximum gear pitch line velocity of about 48 m/s). Figure 3.1(a) shows the influence of temperature on Pw for the first example gear pair (40-teeth) with m% = 2.32 mm and b = 19.5 mm. The operating gearbox temperature ranges from 30o C to 100o C .
For computing the windage drag power losses, the
properties
respective
of
air
at
those
92
temperature
values
were
considered.
1.6 (a)
1.2
(b)
T = 30 o C
b = 26.7 mm
T = 50 o C
b = 19.5 mm
T = 70 C
b = 14.7 mm
o
Pw
T = 100 C o
0.8
[kW]
0.4
0.0 0
2000
4000
6000
8000
10000 0
2000
4000
6000
8000
10000
1.6 (c)
m% = 3.95 mm 1.2
m% = 2.32 mm
Pw [kW]
0.8
0.4
0.0 0
2000
4000
6000
8000
10000
Ω [rpm]
Figure 3.1 Effect of (a) temperature T , (b) face width b and (c) gear module m% on Pw .
93
Windage pocketing power loss was computed by evaluating the density at the backlash and end flow areas at a particular operating temperature value. As seen in Figure 3.1(a), Pw increases with reduction in temperature, regardless of the operating speed, due to a
significantly non-linear influence of the temperature on Pwp and a weakly non-linear effect of temperature on the drag power losses Pwd . As a sample quantification of the effect of temperature on the windage power losses, Pw at 30o C and 7,000 rpm is predicted to be about 0.36 kW higher that Pw value at the same speed and at 100o C . This is a direct result of the influence of temperature on both viscosity and density of the medium surrounding the gears as well as inside the meshing zone. As the operating temperature increases, the air/air-oil mixture inside the gearbox, being compressible in nature, sees an increase in the operating kinematic viscosity value and a decrease in operating density. This combined effect in turn causes a decrease in the windage power loss value, as evinced by the curves in Figure 3.1(a), with density having a stronger influence in reducing the windage power loss than kinematic viscosity. Figure 3.1(b) shows the influence of face width b on Pw for a gear pair having m% = 2.32 mm operating at 80o C . The gear pair having b = 26.7 mm is predicted to
experience about 1.5 times the total windage power loss as compared to a gear pair with b = 19.5 mm, and about 2.5 times the total windage power loss as experienced by the
gear pair at b = 14.7 mm, all at the same operating speed and temperature conditions. For instance, Pw for the gear pair having wide face width of b = 26.7 mm at 80o C and 5,000 rpm is 0.22 kW, as compared to Pw of 0.14 kW for the gear pair with medium face 94
width b = 19.5 mm, and Pw of 0.09 kW for the gear pair with narrow face width b = 14.7 mm, all at the same operating temperature and speed values. An increase in face width opens up more of the control volume for ejection of the fluid and as a result, a greater velocity profile can be predicted, leading to higher windage pocketing power losses and hence, total windage power loss. In order to throw more light on the influence of face width on windage power loss, windage pocketing and drag power loss for the gear pair with m% = 2.32 mm and varying face width values are shown in Figures 3.2(a) and 3.2(b), respectively, at the same operating conditions. Corresponding to the formulation laid out earlier in Chapter 2, the influence of b on windage pocketing power loss is significant, very much more so than it is on the windage drag power loss. It is also seen in Figure 3.2 that the pocketing loss of this particular gear pair constitutes a very large portion of Pw (in this example, more than 98%) with drag losses more or less staying as a tertiary influence under these operating conditions. Hence, the variation of the total windage power loss with face width is primarily due to the windage pocketing power loss, which is strongly affected by changes in face width. Figure 3.3 shows the individual components of windage pocketing power loss, in terms of power loss through the backlash and end flow areas, for varying face widths and at an operating temperature of 80o C over the same speed range. As seen in this figure, the value of the face width has a strong influence on both the end and backlash components of windage pocketing power loss, with an almost two and a half fold increase taking place between windage pocketing power loss over narrow and wide face 95
1.0 (a)
b = 26.7 mm
0.8
b = 19.5 mm Pwp
b = 14.7 mm
0.6
[kW]
0.4
0.2
0.0 0
2000
4000
6000
8000
10000
6000
8000
10000
0.008 (b)
0.006
Pwd
0.004
[kW]
0.002
0.000 0
2000
4000
Ω [rpm]
Figure 3.2 Components of Pw for the gear pairs having m% = 2.32 mm and varying b at
80o C ; (a) windage pocketing power loss Pwp (b) windage drag power losses Pwd . 96
1.0 (a) b = 14.7 mm
(b) b = 19.5 mm
Pwp
0.8
Pwp, e Pwp,b Pwp
0.6
[kW]
0.4
0.2
0.0 0
2000
4000
6000
8000
10000 0
2000
4000
6000
8000
10000
1.0 (c) b = 26.7 mm
0.8
Pwp
0.6
[kW]
0.4
0.2
0.0 0
2000
4000
6000
8000
10000
Ω [rpm]
Figure 3.3 Windage power loss Pwp and its components, Pwp,e and Pwp,b , for gear pairs
having m% = 2.32 mm at 80o C ; (a) b = 14.7 mm (b) b = 19.5 mm (c) b = 26.7 mm. 97
width values.
Further portrayed in the figures is the constitution of the windage
pocketing power loss in terms of its individual components. At narrow face width values ( b = 14.7 mm), the backlash and end power loss components of the windage pocketing power loss contribute more or less equally to the total windage pocketing power loss, as seen in Figure 3.3(a). As the face width is increased, the end pocketing power loss is the more dominant component of the total windage pocketing power loss, with pocketing power loss at the ends constituting over 65 % of the total pocketing power loss over the entire operating speed range at wide face width values, seen in Figure 3.3(c). The final comparison in Figure 3.1(c) portrays the influence of m% on Pw for b = 19.5 mm at 80o C .
The 23-teeth coarse pitch gear pair, with m% = 3.95 mm, is
predicted to have more windage power loss than the 40-teeth gear pair having m% = 2.32 mm. As an example, at 5,000 rpm, Pw for m% = 3.95 mm is about 0.15 kW greater than that of the 40-tooth gear pair having m% = 2.32 mm. As Pwdf and Pwdp components are not expected to be influenced significantly by tooth size according to the formulations presented, this difference in Pw values with module can be attributed primarily to Pwp . This is due to the fact that volume change for the coarser pitch gear pair is greater than for the fine pitch 40-teeth gear pair. Since volume change depends on the size of the mating teeth falling into place in the tooth space cavity as the gears rotate, a larger tooth can cause greater volume displacement as tracked by the 23-teeth coarse pitch gear pair and hence, a greater pocketing power loss is attributed to this fact, both under compressible and incompressible flow conditions.
98
The variation of windage power loss with operating speed is non-linear in nature, as seen in Figures 3.1 - 3.3. For instance in Figure 3.2, for b = 19.5 mm, Pw ∝ ω12.1 while the power loss-speed relationships for individual components are Pwdf ∝ ω2.5 (or
Pwdf ∝ ω2.86
for turbulence), Pwdp ∝ ω2 and Pwp ∝ ω1.8 .
The trend of windage
pocketing power loss variation with speed is mixed and non-linear in nature. At lower operating speed values, the windage pocketing power loss varies with speed in a quadratic fashion, similar to the one depicted in incompressible flow conditions as seen in Chapter 2. At higher operating speeds, the windage pocketing power loss shows an almost linear behavior, with the steeper rise in the velocity of fluid being ejected contributing as a direct effect towards this trend. Figures 3.4, 3.5 and 3.6 are included here in order to throw more light on the physics of the phenomenon of windage pocketing power losses. In Figure 3.4(a) and 3.4(b), the variation of the end flow area of a single control volume with gear rotation is shown for the gears having m% = 2.32 mm and m% = 3.95 mm, respectively. The range of end area values shown for the fine pitch and coarse pitch gear pair has to be understood from the point of view of the total cavity area available for the pocketing purposes. As an average, the total cavity area between adjacent teeth for the 40-teeth gear is about 11.5 mm 2 (roughly, the product of the working tooth height times the average space between adjacent teeth, which is given by the difference between the circular pitch and the tooth thickness at pitch point), and for the 23-teeth gear pair, the total cavity area afforded for pocketing purposes is approximately 50.6 mm 2 . 99
30
(a) m% = 2.32 mm
25
( m) H11
20
H12
( m)
m) Ae(,1 j
( m) H10
15 [mm2 ]
10
5 0
30
(b) m% = 3.95 mm
25
20 m) Ae(,1 j
15
[mm2 ]
10
5
0 0
Rotational Position (m)
1
Figure 3.4 Variation of the end flow area Ae(,1m)j with rotational position for the gear pairs
having (a) m% = 2.32 mm , and (b) m% = 3.95 mm. 100
From Figure 3.4(a), the variation of the end flow area corresponding to the pocket (m) H11 shows that the end area decreases as the gears enter into mesh with each other,
reaching a minimum position at the mating pitch point and then expanding beyond that (m) point. At the same time as the control volume H11 undergoes compression/expansion, (m) depending on the contact ratio, there will be other volumes, for example H12 and (m) H10 in Figure 3.4(a), which will be undergoing compression and expansion,
simultaneously, with consecutive contact points separated by one base pitch distance. In this way, over one base pitch rotation of the gears, all the pockets undergoing compression and expansion have to be considered for computing the average pocketing power loss, as explained in the methodology for power loss computation. Similar such variations of end flow area is encountered in case of the other example gear set, with compression/expansion taking place over different intervals for the 23-teeth gear pair due to differences in module, contact ratio and number of teeth, and hence, cavity areas. As explained earlier, the backlash discharge area is given by the product of the backlash and the face width values for the respective sample gear pairs, and hence, variations in backlash area is neglected for the sake of simplicity. Figure 3.5 shows the variation of density, pressure and velocity along the end and (m) backlash flow areas for control volume H11 of the 40-teeth gear pair ( m% = 2.32 mm,
b = 19.5 mm) at Ω = 1,000, 5,000 and 10,000 rpm. Figures 3.5(a1), 3.5(b1) and 3.5(c1)
show the variation of density, pressure and velocity along the end flow area with respect (m) while Figures 3.5(a2), 3.5(b2) and 3.5(c2) show the same to rotational position for H11
101
6
6
(a1)
1,000 rpm
5
5
5,000 rpm
( m) ρe,11
( m) ρb,11
10,000 rpm
4 [kg/m3 ]
(a2)
4
[kg/m3 ]
3
3
2
2
1
1 1.0
1.0
(b1)
0.7
(b2)
0.7
) pe(m ,11
( m) pb,11
[GPa]
[GPa] 0.4
0.4
0.1
0.1 100
100
(c1)
(c2)
50
50
) ve(m ,11
(m)
vb,11
[m/s]
0
0
[m/s]
-50
-50
-100
-100
0
Rotational Position (m)
0
1
Rotational Position (m)
1
(m) Figure 3.5 Variation of (a) density, (b) pressure and (c) velocity of control volume Η11 of the gear pair having m% = 2.32 mm with rotational position for (a1, b1, c1)
the end area and (a2, b2, c2) the backlash area ( b = 19.5 mm , 80o C ).
102
6
6
(a1)
(a2)
1,000 rpm
5
5
5,000 rpm
) ρe(m ,11
) ρb(m ,11
10,000 rpm
4 [kg/m3 ]
4
[kg/m3 ]
3
3
2
2
1 1.0
1 1.0
(b1)
0.7
(b2)
0.7
( m) pe,11
) pb( m ,11
[GPa] 0.4
[GPa] 0.4
0.1 100
0.1 100
(c1)
50
(c2)
50
) ve( m ,11
) vb( m ,11
[m/s]
0
[m/s]
-50
0
-50
-100
-100 0
Rotational Position (m)
1
0
Rotational Position (m)
1
(m) Figure 3.6 Variation of (a) density, (b) pressure and (c) velocity of control volume Η11 of the gear pair having m% = 3.95 mm with rotational position for (a1, b1, c1)
the end area and (a2, b2, c2) the backlash area ( b = 19.5 mm , 80o C ).
103
along the backlash flow area. As seen from the figures, the variation of the governing parameters over one mesh cycle follows the compression/expansion process for the respective pocket and flow areas over the same mesh cycle. The velocity of the fluid ejected through the end and backlash flow areas, as seen in Figures 3.5(c1) and 3.5(c2), respectively, shows negative values for velocity just beyond the point in the mesh cycle where expansion has started taking place, whereby fluid is sucked into the pocket due to the loss of pressure differential created by the expansion of the control volume. Variation of density and pressure with rotational position for the backlash flow area shows a slightly higher value than those for the end flow area. This is due to the fact that the backlash flow area has been assumed, for the sake of computational simplicity, to be the product of the face width and the backlash length, while the end flow area has been allowed to vary, as seen in Figure 3.4(a) for the 40-teeth gear pair. Over the entire mesh cycle, the average end flow area is larger in value than the backlash flow area, resulting in lower density values for the end flow area versus the backlash area. Correspondingly, the pressure changes with rotational position, for the end and backlash flow areas, in Figures 3.5(b1) and 3.5(b2), respectively, show very similar trends and within the same operating range, for both the flow areas. Also noted in Figure 3.5 is the variation of these parameters with speed, validating the physical notion that an increase in speed corresponds to an increase in the density and pressure and hence, an increase in the velocity of the fluid being squeezed out. The variation of velocity with respective flow areas do not show a similar trend with respect to one another because of the way the velocity is computed. The difference between product of volume and density changes between rotational positions are 104
considered in the computation of fluid velocity for the end and backlash flow areas, as given by Eq. (3.7), and hence, steeper changes in such differential products between mesh positions will produce steeper velocity gradations, as seen in Figures 3.5(c1) and 3.5(c2). As reported in Figure 3.5, similar such trends are noted in Figure 3.6, where the same parameters are presented for the 23-teeth coarse pitch gear pair ( m% = 3.95 mm, b = 19.5 mm). Evidently, the coarser pitch gear pair has a higher velocity, density and
pressure differential than the fine pitch 40-teeth gear pair, due to larger changes taking place with respect to volume variation within mesh position over each incremental step, hence leading to greater windage pocketing power losses. It is also worthwhile to note that each analysis performed and outlined in this study took less than a second of CPU time, corroborating the fact that the proposed windage power loss model eschews the computational intensity of CFD type formulations, making it viable as an effective and practical design tool.
3.5 Concluding Remarks
A physics-based fluid mechanics power loss model was proposed to predict power losses for gear pairs operating under windage conditions. The framework of the model included individual formulations for windage losses on the periphery and faces of the gears as well as a compressible fluid model for power loss due to pocketing taking place in the meshing zone.
The windage conditions simulate jet-lubricated operating 105
conditions or low oil-level dip-lubricated conditions. As the first stab, the windage power loss model was applied to two example unity-ratio gear sets with varying gear geometry parameters, to quantify the contributions of each of the components of the total windage power loss. In both example cases, the windage pocketing losses were shown to dominate the total gear pair windage losses, as was the case in incompressible flow computations performed in Chapter 2. Also, the influence of operating conditions, gear geometry parameters and lubricant properties on windage power loss was quantified for the gear pairs in consideration.
106
References for Chapter 3
[3.1]
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.
[3.2]
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), 51–59. [3.3]
Diab, Y., Ville, F., and Velex, P., 2006, “Investigations on Power Losses in High Speed Gears,” Journal of Engineering Tribology, 220, 191–298.
[3.4]
Eastwick, C. N., and Johnson, G., 2008, “Gear Windage: A Review,” ASME Journal of Mechanical Design, 130, 034001 (6 pages).
[3.5]
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, 116–122.
[3.6]
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), 331–341.
[3.7]
Streeter, L. V., and Wylie, E. B., 1985, Fluid Mechanics, Eighth edition, McGraw-Hill, New York. 107
CHAPTER 4
VALIDATION OF OIL CHURNING AND WINDAGE POWER LOSS MODELS
4.1 Introduction In the previous two chapters, two novel models, one for predicting oil churning power losses, and the other for predicting windage power losses of spur gear pairs, were proposed. Both the models employed fundamental principles of fluid mechanics and a host of critical assumptions, so that the formulations were reduced to closed-form expressions, making them practical for application towards design purposes. In cases where a numerical computational methodology has to be used, care was taken to rein in the computational effort, as is the case for the compressible fluid pocketing power loss component of the windage power loss model. While these two models are desirable in the sense that they can be used conveniently and efficiently to predict power losses associated with gear pairs, they would not be viable tools for actual product development unless they are validated through real-world experiments conducted on gear pairs. This chapter takes on the onerous task of such a validation of both the power loss models.
108
It has to be noted upfront that the idea of such a validation process is not to provide for accurate point-to-point comparison with experiments; such a comparison is not possible when experimental uncertainties and modeling assumptions are taken into account. The broader objective is to provide for a comparison between predictions and experiments, wherein the predictions follow the experimental trend, within limits of experimental scatter, over wide ranges of operating parameters. Provided the models can produce such predictions, they will be deemed validated as practical and computationally efficient tools, which can provide a reasonable estimate of load-independent power losses occurring in gear pairs. The most basic gear system, a spur gear pair, will be used in the experimental comparisons in this chapter. Another system-level validation will be provided in Chapter 5, where the proposed spin loss models will be applied to a six-speed automotive transmission and its predictions will be compared to transmission power loss measurements. As a part of this study on load-independent losses, a companion experimental study was performed by Moorhead [4.1] to measure oil churning power losses of unityratio spur gear pairs having different design parameters.
These experiments were
performed at different static oil levels and operating oil temperature, as well as over a wide range of operating speed. The oil churning power loss model of Chapter 2 will be used to simulate the experiments of Moorhead [4.1] for verifying the trends exhibited by the predictions as well as for quantifying the accuracy of the predicted power loss values
109
through a direct comparison to experimental results. Section 4.2 will briefly describe the experimental set-up of Moorhead [4.1] and provide comparisons between his experiments and model predictions for validating the oil churning power loss model. For validation of the windage power loss model of Chapter 4, the experimental database of Petry-Johnson et al [4.2] will be used. This body of experimental data was generated by performing load-independent tests on jet-lubricated spur gear pairs, with no accumulation of the oil within the gearbox, so as to simulate windage conditions at high operating speeds. These experiments will be simulated in Section 4.3 to determine the accuracy of the windage power loss model predictions. In addition, a direct comparison of the windage model formulations to the empirical formula of Dawson [4.3] will also be carried out as a part of this validation effort.
4.2 Validation of the Gear Pair Oil Churning Power Loss Model 4.2.1 Test Machine and Oil Churning Test Procedure A gear efficiency test machine used previously by Petry-Johnson et al. [4.2] to study power losses of jet-lubricated gearboxes, as shown in Figure 4.1(a), was used by Moorhead [4.1], with some modifications, to perform gear pair oil churning experiments. Only the most relevant details of the test set-up are provided here. The test machine consists of two identical gearboxes, each containing one gear pair and four identical cylindrical roller bearings. In this power circulatory arrangement, one gear from each gearbox is connected to the corresponding gear of the other gearbox, as shown 110
test gearboxes
torque meter
belt drive
Ω1 high-speed spindle
Ω2
AC motor
Split Load Coupling
Figure 4.1 (a) A view and (b) the layout of the gear efficiency test machine [4.1, 4.2].
111
schematically in Fig. 4.1(b) [4.1]. The input shaft of the reaction gearbox is connected to a high-speed spindle through a flexible coupling and a precision non-contact type torquemeter. The high-speed spindle is driven by a belt with a 3:1 ratio speed increase from a variable speed AC motor. Figure 4.2 shows a picture of one of the gearboxes that is configured to perform dip-lubricated efficiency tests [4.1].
The other gearbox has the exactly the same
arrangement. Valves of the oil return were closed in order to maintain an oil bath within each gearbox. Equal static oil levels were ensured in both the gearboxes via a flow path between them. Sheet metal baffle plates were added inside the gearboxes in front of the oil drainage ports to minimize oil circulation in the drainage lines. Large diameter graduated site glasses were used to monitor the static oil level within each gearbox. A ball valve placed between each gearbox and its site glass was closed during the tests to ensure that the site glasses did not act as fluid capacitors. The power circulation loop can be activated by applying a torque at the split coupling to load the gear meshes. Here, only a very little amount of such a torque was applied so that any tooth separations or rattle can be prevented. In addition, all the test gears were super-finished through a chemical polishing process to minimize the surface roughness amplitudes.
This way, the load-dependent (mechanical) losses in these
experiments can be deemed negligible and the total torque provided to the loop, TT , as measured by the torque-meter, represents the total spin losses from both gearboxes. Since both gearboxes are identical, the spin power loss of an individual gearbox is
112
Figure 4.2 One of the test gear boxes shown in dip-lubrication arrangement [4.1].
113
determined as PsT = 12 ωTT , where ω is the rotational speed of the unity-ratio test spur gears. Torque TT was measured with a non-contact digital torque-meter. Moorhead [4.1] reported that the torque-meter has a full-scale range of 50 N-m, and overall accuracy of 0.03% of full scale for the installed system. It was also reported that the rotational speed Ω = Ω1 = Ω 2 = 60ω 2π (in rpm) was typically maintained within 0.20.3% of the set (intended) speed, and exhibited a negligible ( ±2 rpm) fluctuation about this value, which was averaged out during data collection [4.1]. 4.2.2 Gear Specimens and Parameters Studied Two basic unity-ratio spur gear designs were implemented in Moorhead’s study [4.1]. The design parameters of these test gears were provided earlier in Table 2.1. Figure 4.3 shows a picture of a pair of these test gears [4.1]. Moorhead’s study [4.1] provided data for investigating the influence of the following parameters: (i)
Oil Viscosity – Oil churning tests were performed by using a typical manual transmission fluid. The lubricant temperature was not controlled, and was allowed to vary while being recorded continuously [4.1].
Each test was started with
lubricant temperature at 30ºC, and spin losses were measured as oil temperature rose gradually to its steady-state value. Then, another test was started at the same test conditions, but now with the oil heated to an elevated temperature (say 100ºC) and the test continued until the gearbox was cooled down to the same steady-state
114
Figure 4.3 Two examples of test gears (a) 23-tooth gear with m% = 3.95 mm and b = 19.5 mm, and (b) 40-tooth gear with m% = 2.32 mm and b = 19.5 mm [4.1, 4.2].
115
temperature. Combining these two tests provided continuous spin loss data for each condition within a wide range of oil temperature from 30ºC ( µ k = 131.2 cSt) and 100ºC ( µ k = 13.1 cSt). This allowed for a complete evaluation of the influence of lubricant temperature and viscosity for every test. (ii)
Rotational Speed – The rotational speed of gears at each test was varied up to 6,000 rpm (corresponding to a pitch-line velocity of 29 m/s).
Either 6-point
( Ω = 1, 000 rpm to 6,000 rpm with an increment of 1,000 rpm) or 3-point ( Ω = 2, 000 , 4,000 and 6,000) tests were performed for each test to generate sufficient data for quantifying the impact of the rotational speed on spin power losses. (iii) Gear Module –Two different test gear designs, a 40-teeth gear pair having a module
of m% = 2.32 mm and a 23-teeth gear pair having a module of m% = 3.95 mm were included in the test matrix to collect data on the influence of gear module. (iv) Gear Face Width – Tests on gears of module m% = 2.32 mm were performed with three different face width values to investigate the impact of the face width on spin power loss. The face width values used are b = 14.2, 19.5 and 26.7 mm.
(v)
Immersion Depth – Figure 4.4 shows various static oil levels considered for the actual test gearbox with the side cover removed. According to Figure 4.4(a), these unity ratio gear pairs with ro1 = ro 2 have the same immersion depth h1 = h2 = h
116
(a)
ro r
h1 = h2 r
pitch circles (b)
h = 1.5 r
h = 1.0 r
h = 0.5 r
h = 0.05 r
Figure 4.4 Illustration of oil level parameters [4.2].
117
since gear centers O1 and O2 are on the same horizontal plane.
Under this
configuration,
given
the
dimensionless
immersion
parameter
is
as
h = h1 ro1 = h2 ro 2 . The test matrix included tests conducted at values of h = 0.5, 1.0 and 1.5, as shown schematically in Figure 4.4(b). Here, h = 1.0 indicates an oil level at the centers of the gears (half immersion), and h = 0.5 and 1.5 represent 25% and 75% immersion, respectively.
4.2.3 Comparison of Oil Churning Power Loss Model to Experiments The predictions of the spur gear oil churning power loss model formulated in Chapter 2 will be compared to the experimental data of Moorhead [4.1] to determine the accuracy of the oil churning power loss model; all of the comparisons will be done for an up-in-mesh rotational condition. The comparisons to predictions will be presented as a function of operating speed, Ω , thus allowing for a better quantitative comparison. Since the measured total spin power loss, PsT , represents the gearbox spin power losses that include bearing losses as well, the viscous bearing torque losses for the four cylindrical element bearings are calculated in mW as [4.4]
Pb,spin
4(10−7 ) f L µ k 2 / 3Ω5 / 3d m3 ω = 6.4(10−5 ) f L d m3 ω
118
µ k Ω > 2000 µ k Ω < 2000
.
(4.1)
where d m is the mean bearing diameter in millimeters ( d m = 60 mm in the experiments), f L is a bearing viscous friction constant depending on the type of bearing and lubrication method ( f L =2.2 - 4 for oil bath type lubrication; in this study, f L = 4 ), Ω is the shaft speed in rpm, ω is the shaft speed in rad/s and µ k is the kinematic viscosity of the lubricant in mm2/s, and added to the spin power loss values for the gear pair to obtain the total PsT of the gearbox. One should note that Eq. (4.1) is rather simplistic and does not include h as a parameter for oil churning effects at the bearing locations. It was used in this study due to the lack of a more sophisticated published bearing model. The shaft seal losses were reported to be negligible as compared to bearing power losses [4.2], and hence, were not included in the total spin power loss predictions. In Figure 4.5, the comparisons between measured and predicted PsT values of a gear pair having m% = 2.32 mm and b = 19.5 mm are presented at four different oil temperature values when h = 1.0 . A wide range of inlet temperatures ranging from 30 to 100°C is chosen for this comparison. The total spin power loss decreases with an increase in oil inlet temperature for both measurements and predictions since the viscosity reduces with an increase in oil temperature. The decrease in measured power loss is not as steep as it is for the predictions between temperature ranges of 70°C to 100°C. It is also seen that the comparison between the measured and predicted PsT values is reasonably good, with predictions well within 0.1 kW of measurements, with the exception of the data point corresponding to Ω = 4, 000 rpm at 100°C where the difference is about 0.2 kW. Such deviations at higher speeds and higher operating 119
1.2
(a) µ k = 131.2 cSt ( 30 o C )
(b) µ k = 54.3 cSt ( 50 o C )
Experiment Prediction - Total Prediction - Bearings
0.8
0.4
0.0 PsT [kW] 1.2
(d) µ k = 13.1 cSt (100 o C )
(c) µ k = 27.8 cSt ( 70 o C )
0.8
0.4
0.0 0
1000
2000
3000
4000 0
1000
2000
3000
4000
Ω [rpm]
Figure 4.5 Comparison of predicted to the measured [4.2] PsT for a gear pair (a) 30o C , (b) 50o C , (c) 70o C , and (d) 90o C ; m% = 2.32 mm, b = 19.5 mm and h = 1.0 .
120
temperatures can potentially be attributed to the simplicity of the bearing viscous power loss model, which does not specifically take into account variations in oil viscosity, as it considers only the range afforded by the product of oil viscosity and operating speed, together. It is also observed from Figure 4.5 that the predicted variation of PsT with Ω also agrees reasonably well with the measured trends. Figure 4.5 also shows the bearing power loss component of the total predicted spin power loss at the different temperature levels. The bearing power losses (from all four rolling element bearings) account for about one-third of the total gearbox losses, indicating that the remaining two-third comes from the gears. Hence, while bearing power losses are not negligible, they fail to overshadow the pocketing and drag components of the gear churning power losses.
Next, the predicted and measured influences of immersion depth parameter h on PsT are compared in Figure 4.6 for four different oil levels. These comparisons are for a gear pair with b = 19.5 mm and m% = 2.32 mm at 80°C.
The predicted and measured
PsT shown in Figure 4.6 also compare reasonably well with differences being within 0.1 kW, with the exception of the high speed portion of the h = 1.5 curve. At high speeds and higher oil level immersion, assumptions of static oil level as considered while modeling the oil churning power loss underestimates the total power loss, shown through the discrepancy in Figure 4.6(d); swell effects of the oil need to be considered while modeling the churning losses. Overall, it can be stated that the model captures the
121
1.2
(a) h = 0.05
(b) h = 0.5
Experiment 0.8
Prediction
0.4
0.0 PsT [kW] 1.2
(d) h = 1.5
(c) h = 1.0
0.8
0.4
0.0 0
1000
2000
3000
4000 0
1000
2000
3000
4000
Ω [rpm]
Figure 4.6 Comparison of predicted vs. measured PsT for a gear pair at (a) h = 0.05 , (b) h = 0.5 , (c) h = 1.0 , and (d) h = 1.5 ; oil at 80o C m% = 2.32 mm, and b = 19.5 mm.
122
(up-in-mesh),
influence of h on PsT sufficiently. Figure 4.6 also establishes the significance of h on PsT clearly, with both measured and predicted PsT values increasing with an increase in immersion depth. As an example, at an operating speed of Ω = 3, 000 rpm, predicted PsT is 0.21 kW at h = 0.5 , which is almost doubled to 0.41 kW when h = 1.5 . Figure 4.7 provides comparisons between measured and predicted PsT values at three different face width values of b = 14.7 , 19.5 and 26.7 mm, for gears having m% = 2.32 mm, operated at h = 1.0 and oil temperature of 80°C. The measured and predicted PsT values again agree quite well for all three b values, as maximum difference is less than 0.1 kW.
This figure shows sizable increase in PsT as the face width
increases. For instance, measured and predicted PsT values are 1.01 kW and 0.98 kW, respectively, for the gear pair having b = 26.7 mm in Figure 4.7(c) at Ω = 4, 000 rpm, whereas corresponding values for b = 19.5 mm are 0.72 kW and 0.64 kW in Figure 4.7(b) that are about 0.3 kW less than those for b = 26.7 mm. Likewise, the measured and predicted PsT of the pair having b = 14.7 mm at Ω = 4, 000 rpm are 0.53 kW and 0.49 kW representing about half of the losses at b = 26.7 mm. Figure 4.7 also shows clearly that the model is capable of capturing the influence of b on PsT accurately. As a final comparison, Figure 4.8 shows the influence of module m% on measured and predicted PsT . Two separate gear pairs, one with m% = 2.32 mm and the other with m% = 3.95 mm (both having b = 19.5 mm) are considered for this comparison at h = 1.0
123
1.2
1.2
(b) b = 19.5 mm
(a) b = 14.7 mm
Experiment 0.8
0.8
Prediction
PsT [kW] 0.4
0.4
0.0 0
1000
2000
3000
0.0 4000 0
1000
2000
3000
4000
1.2
(c) b = 26.7 mm
0.8
PsT [kW] 0.4
0.0 0
1000
2000
3000
4000
Ω [rpm]
Figure 4.7 Comparison of predicted to the measured [4.1] PsT for a gear pair having m% = 2.32 mm and face width values (a) b = 14.7 mm, (b) b = 19.5 mm, and (c) b = 26.7 mm; oil at 80o C and h = 1.0 .
124
1.2 (b) m% = 3.95 mm
(a) m% = 2.32 mm
Experiment 0.8
Prediction
PsT [kW]
0.4
0.0 0
1000
2000
3000
4000 0
1000
2000
3000
4000
Ω [rpm]
Figure 4.8 Comparison of predicted to the measured [4.2] PsT for a gear pair of face width b = 19.5 mm and modules (a) m% = 2.32 mm and (b) m% = 3.95 mm; oil at 80o C and h = 1.0 .
125
(up-in-mesh) and oil temperature of 80°C.
The agreement between the model and
experiments are again well within 0.1 kW in both the cases. Gear module comes across as an important parameter, influencing both the spin and mechanical power loss and hence, is a key operating parameter in ensuring the reduction of power losses with lower module.
4.3 Validation of the Gear Pair Windage Power Loss Model For validation of the windage power loss model, measurements of Petry-Johnson et al. [4.2] will be compared against the predictions of the windage power loss model detailed in Chapter 4. Petry-Johnson et al [4.2] used jet-lubrication for most of the experiments. The test machine set-up and methodology for measuring spin power losses under jet-lubrication conditions are described in detail in ref. [4.2, 4.5]. The same test machine as explained in ref. [4.1] was employed in this case as well with the same test gear pairs whose parameters listed in Table 2.1. The spin power loss measurements were carried out at rotational speeds of Ω = 2, 000 rpm to Ω = 10, 000 rpm in increments of 2,000 rpm. The 40-teeth and 23-teeth test gears have different face width values, apart from having different modules. The operating temperature was maintained at 110°C. Figure 4.9 provides for direct comparisons between predictions of the windage power loss model of Chapter 3, and measured data from Ref. [4.2]. First, predicted and measured windage power loss values, Pw , for the 40-teeth gear pair having m% = 2.32 mm and b = 26.7 mm are compared in Figure 4.9(a). Here, the gearbox bearing losses are accounted for by using the same viscous bearing power loss formula [4.4], 126
1.6 Experiment
Prediction 1.2
Pw
0.8
[kW]
0.4
0.0 0
2000
4000 6000 Ω [rpm]
8000
10000
0
2000
4000
8000
10000
1.6
1.2
Pw
0.8
[kW]
0.4
0.0 6000
Ω [rpm]
Figure 4.9 Comparison of predicted and measured [4.2] Pw for (a) the 40-tooth gear pair having m% = 2.32 mm and b = 26.7 mm and (b) 23-tooth gear pair having m% = 3.95 mm and b = 19.5 mm . 127
taking note of the fact that even though the gear pair rotates in windage conditions, the bearings are lubricated separately to function properly. As seen in Figure 4.9(a), the predictions match well with the measured Pw values, with the maximum deviation between measurements and prediction being well within 0.2 kW over the operating speed range, with the maximum deviation occurring at the highest operating speed value of Ω = 10, 000 rpm. Deviations at higher Ω values are potentially caused by the simplicity of bearing power loss formula [4.4] and as well as some of the critical assumptions of the windage model. The predicted and measured [4.2] Pw values are compared in Figure 4.9(b) for the 23-teeth gear pair having m% = 3.95 mm and b = 19.5 mm. The agreement between the two is again acceptable and mostly within the bounds of experimental scatter, with maximum deviations being well within 0.2 kW. The comparison provided in Figures 4.9(a) and 4.9(b) for the two different gear designs demonstrates the accuracy of the windage model.
In spite of the fact that the proposed model was a reduced-order
analytical model that employed many simplifying assumptions, the observed level of accuracy is rather encouraging, given the fact that computational effort required is negligible. While gear pair spin loss experiments like the one presented in ref. [4.2] are sparse, literature includes several experimental studies on windage losses of a single gear. In such cases, the windage losses are limited to air drag as no pocketing losses can occur. In one such study, Dawson [4.3] performed a large number of spur and helical gear windage loss measurements. These measurements were curve-fit to derive an empirical 128
power loss formula for a single gear operating in free air that takes into account the operating speed ω , gear outside radius ro and air properties (density ρ and kinematic viscosity µ k ) as [4.3]
2.85 4.7 Pw = aρµ0.15 ro . k ω
(4.2)
where a is an empirical constant. Meanwhile, Eq. (3.19) from Chapter 3 computes the drag losses on faces of a gear in the turbulent regime as
2.86 4.72 Pw = 0.025πρµ0.14 ro . k ω
(4.3)
This empirical formula, Eq. 4.2, of Dawson is very much identical to the analytical formula of Eq. (4.3) for the face drag power loss in turbulent regime, not only in terms of the parameters involved but also in terms of the values of the exponents. Dawson’s empirical formula and Eq. (4.3) from the windage power loss model both have ρ , µ k , ωi and roi as the only parameters influencing drag power loss. In addition, Dawson’s exponents for ρ , µ k , ωi and roi are 1.0, 0.15, 2.85, and 4.7, as seen in Eq. (4.2) compared to 1.0, 0.14, 2.86 and 4.72 in Eq. (4.3), respectively. This indicates that the proposed physics-based drag windage power loss formulation is indeed valid within the range of experimental scatter as seen through Dawson’s measurements.
129
4.4 Concluding Remarks In this chapter, oil churning and windage power loss models proposed in earlier chapters were used to simulate the experiments of two companion studies [4.1, 4.2] on load-independent spin power losses of spur gear pairs operating under dip- and jetlubricated conditions. These experiments were performed over a wide range of operating speed, temperature, oil levels and key gear design parameters, to allow for quantification of their influence on spin power losses. Through comparisons to these measurements, the physics-based oil churning and windage power loss models were shown to be capable of predicting the measured spin power loss values under both oil churning and windage conditions, within a reasonable degree of accuracy. It was also shown that the models predict the parameter sensitivities and experimental trends accurately. In addition, a close agreement between the proposed windage power loss model formulations and a published empirical formula [4.3] was reported for further validation of the models. In view of these comparisons, it can be concluded that the models are realistically accurate, and hence, can be used for larger scale transmission power loss simulations.
130
References for Chapter 4 [4.1]
Moorhead, M, D., “Experimental investigation of spur gear efficiency and the development of a helical gear efficiency test machine,” MS thesis, The Ohio State University, Columbus, Ohio, 2007.
[4.2]
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.
[4.3]
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), 51–59.
[4.4]
Harris, A. T., 2001, Rolling Bearing Analysis, Fourth Edition, Wiley & Sons, Inc., New York.
[4.5]
Petry-Johnson, T., “Experimental Investigation of Spur Gear Efficiency,” MS thesis, The Ohio State University, Columbus, Ohio, 2007.
131
CHAPTER 5
APPLICATION OF OIL CHURNING POWER LOSS MODEL TO AN AUTOMOTIVE TRANSMISSION
5.1 Introduction With the oil churning power loss model of a gear pair, which was proposed in Chapter 2, having been validated through comparisons to experiments in Chapter 4, the last remaining objective of this dissertation is the expansion of the formulations to predict spin power losses of an entire transmission. Given the motivations listed in Chapter 1 in regards to fuel economy and emissions goals imposed on automotive drive trains and transmissions, achieving a spin loss model would be valuable in quantifying and minimizing such losses. The literature on power losses of geared transmissions is very sparse, with most studies adapting existing empirical and dimensional-analysis based formulations to predict friction and spin losses in gear pairs in transmissions. Changenet and Velex [5.1] proposed a power loss model for a 6-speed manual transmission using thermal networks. The model predicts temperature distribution and efficiency by considering power loss due to rolling and sliding friction, rolling bearing elements and oil shearing taking place in
132
the synchronizers. Heingartner and Mba [5.2] proposed a methodology for estimating power losses in high-speed gear units by combining an empirical power loss expression for oil churning and windage [5.3], along with a rolling and sliding friction loss model [5.4]. The predictions of the combined empirical models for friction and viscous losses in the gear unit were compared with experiments conducted by the same authors on an example single-stage double-helical speed-increasing unit.
This study lead them to
conclude that increasing the load at a constant speed increased the sliding losses and slightly decreased the rolling friction losses, while the windage losses remained constant. While their study is a useful addition to the literature on transmission power loss, no insights were thrown on the physics behind the mechanism of transmission power losses. Changenet and Velex [5.5] continued on the same vein of thought by providing for an experimental versus theoretical comparison of power losses in a six-speed manual transmission.
The theoretical estimates were again carried out through empirical
formulations for friction and oil churning power losses [5.5], and therefore, they were not physics-based. From their comparisons, they inferred the transmission efficiency to be higher at higher operating temperature and higher load values. Handschuh and Kilmain [5.6] carried out experiments on characterizing the efficiency of high-speed helical gear trains that find extensive use in tilt-rotor aircrafts. The results of the experimental study were compared to empirical estimates for power losses due to friction [5.4] and viscous dissipation [5.7] in the gear meshes and bearings. They concluded that power losses increased with speed, with torque transmitted by the gear train having a negligible effect on the power losses. van Dongen [5.8] quantified the influence of speed and load on transmission efficiency in manual and automatic transmissions through dimensional 133
analysis, leading to the conclusion that transmission efficiency increased as load increased, with oil churning acting as the dominant source of power loss in transmnissions. In general, the complexity of the environment inside the gearbox as well as difficulties in measuring gear tooth temperature has made the study of transmission power losses very challenging. A critical review of the above studies only goes on to show that transmission power losses were tackled from the point of view of approximate analyses, with major limitations existing in the application of such empirical formulae in estimating power losses in transmissions, as experimental conditions do not remain consistent.
Also,
constraints on computational time and effort prevent the seeking of solutions through a CFD-based methodology.
Hence, applicability of such models to larger multi-gear
transmissions has been cumbersome. With the above thoughts in mind, this chapter will focus on the prediction of the total transmission power loss of a manual transmission, which is given by the sum of the power losses at the gear mesh due to friction, and the spin power losses, as well as the power losses due to mechanical and viscous dissipation in bearings. As the centerpiece of this work, the fluid mechanics-based model of oil churning losses of a gear pair as formulated in Chapter 2 will be utilized here and applied to the transmission as a whole. This model will then be combined with a transmission mechanical power loss model, based on a gear pair friction power loss model [5.9] and bearing power loss models [5.10] to predict the total power loss of a manual transmission. The transmission power loss predictions will be validated through comparisons to measurements [5.11] from an 134
example 6-speed manual transmission, preformed under tightly controlled test conditions. Application of the fluid-dynamics based oil churning power loss model and also the mechanical power loss model of Xu et al [5.9] will provide insight into the physics behind the transmission power losses. In general, the total transmission power loss PT is comprised of two groups of losses, as explained in the introductory chapter: (i) load-dependent (friction induced) mechanical power losses Pm and (ii) load-independent (spin) losses Ps , i.e. PT = Pm + Ps . Friction-induced losses at the loaded gear meshes, Pmg , and at the bearings, Pmb , define the load-dependent mechanical power losses. The total mechanical loss is the sum of losses from all loaded gear meshes and bearings: Pm = Pmg + Pmb .
Load-independent (spin) losses are caused by a host of factors, as detailed in earlier chapters, including viscous dissipation of bearings, Psb , churning of oil and windage by gears, Psg , and are given as Ps = Psg + Psb ; the physics of the loadindependent power losses has been laid out in Chapters 2 and 3 of this dissertation, in sufficient detail. Other secondary losses in components such as synchronizers and oil seals can also be added to this total. As stated in preceding chapters, while losses from these two groups might be comparable under high-load and low-speed conditions, the spin losses tend to dominate over the load-dependent power losses at higher speeds and lighter loads.
Section 5.2 will provide an introduction to the methodology of manual
transmission power loss prediction, with sub-sections outlining gear pair spin and mechanical power loss models in brief. Finally, the transmission power loss model will be 135
applied to an actual manual transmission and compared to transmission power loss measurements from a companion study on an example six-speed manual transmission for validation purposes.
5.2 Manual Transmission Power Loss Prediction Methodology The transmission power loss prediction methodology comprises of three stages of power loss computations as illustrated in the flowchart of Figure 5.1. The first task in this methodology is to determine (i) the load transmitted by each gear pair, (ii) forces acting on each bearing (including preloads), (iii) speed of each gear, and (iv) oil level at each gear location. These calculations, represented by “Kinematics and Power Flow” in Figure 5.1, are repeated at each duty cycle increment j, characterized by the transmission input speed
ω j , input torque T j , gear state g j , and a user defined operating temperature θ j and static oil level (or oil volume). These operating conditions and corresponding bearing, gear and transmission geometric data are then used to compute (1) load-independent spin losses of losses of each rotating gear pair and bearings and (2) load-dependent (mechanical) losses of every load carrying gear mesh and bearings. All these components of power losses are added to represent the losses at a transmission cycle increment j and the same computations are repeated for all discrete increments to determine the variation of the power losses with a particular fuel economy cycle.
136
Lubricant Conditions θ j , Oil Level
Gear Parameters Bearing Parameters Transmission Geometry
Duty Cycle Tj , ωj , g j
Kinematics & Powerflow Analysis
Bearing Mech. Loss Model
Gear Mech. Loss Model
Gear Spin Loss Model
Pmb
Pmg
Psg
Σ Pm
Bearing Spin Loss Model
Psb
Σ Ps
Σ PT Total Transmission Power Loss
Figure 5.1 Flowchart of the transmission power loss computation methodology.
137
5.2.1 Gear Pair Spin Power Loss Model. The gear pair spin power loss model employed in the computation of loadindependent power losses in the transmission has been described in detail in Chapter 2. Accordingly, the total gear pair spin power loss will be given as the sum of the drag loss on the periphery of the gears, the drag loss on the faces of the gears, power loss due to root filling in the cavity between adjacent gear teeth, and finally, the power loss due to pocketing/squeezing of the lubricant in the meshing zone.
With drag loss on the
periphery, faces and power loss due to root filling of gear i denoted as Pdpi , Pdfi and Prfi , respectively, and pocketing power loss in the meshing zone given as Pp , and expressions for the same given by Eqs. (2.7b), (2.13), (2.23) and 2.42(b), respectively, for the laminar regime (with Eq. (2.17) replacing Eq. (2.13) if flow along the faces is turbulent), the total gear pair spin power loss is given as
Psg = Pdp1 + Pdp 2 + Pdf 1 + Pdf 2 + Prf 1 + Prf 2 + Pp .
(5.1)
5.2.2 Gear Pair Mechanical Power Loss Model. As illustrated in the flowchart of Figure 5.1, the gear mesh mechanical power loss model combines a gear load distribution model, a gear contact friction model and a gear mechanical efficiency formulation [5.9]. The gear load distribution model predicts the load and contact pressure distribution at every contact point over the tooth surface of the gear, as the mesh position is incremented. The corresponding load distribution along with 138
geometric parameters is input to the gear contact friction model, which determines the distribution of friction coefficient µ along the contacting surfaces. This friction coefficient value is then input to the mechanical efficiency formulation to determine the instantaneous power loss at an incremental rotational position m ( m ∈ [1, M ] ). The same procedure is repeated at other discrete rotational positions until m = M , when a complete gear mesh cycle of rotation is achieved. The instantaneous mechanical efficiency values over the entire mesh cycle are then averaged to obtain the average mechanical power loss of the gear pair at the given instantaneous duty cycle condition. Xu et al [5.9] used a thermal EHL model as the basis to predict µ distribution along the gear contacts. In order to overcome the high computational demand of the EHL model, they performed a large number of EHL analyses for the lubricant considered, within typical ranges of all key contact parameters including surface characteristics, operating conditions and contact pressure distribution. Then a multiple linear regression analysis of the analyzed EHL results was performed to reduce these simulations into a single friction coefficient formula given as [5.9]
µ
b1 +b4 SR Ph log10 (ν 0 )+b5 exp( − SR Ph log10 ν 0 )+b9e S Pb2 SR b3 Veb6 νb7 Rb8 0 h = e
.
(5.2)
This formula contains most of the key parameters including the absolute viscosity of the lubricant ν 0 (in cPs), the RMS composite surface roughness S (in µm), the effective radius of curvature R (in meters) and the contact pressure Ph (in Pa). SR is the sliding ratio SR = 2 (u1 − u2 ) (u1 + u2 ) where u1 and u2 are the velocities of the contacting
139
surfaces in the sliding direction.
In Eq. (5.2), bi (i = 1, 2,..., 9) are constant coefficients
that are established for the manual transmission fluid of interest. This formula was shown to agree well with both actual EHL analyses and traction measurements [5.9]. After finding the friction coefficient µ( m) ( z , Θ) at each contact segment centered by the coordinates ( z , Θ) at each incremental rotational position m = [1, M ] , the friction force is calculated as [5.9]
F ( m ) ( z , Θ) = µ( m ) ( z , Θ)W ( m ) ( z , Θ) ,
(5.3a)
where W ( m) ( z , Θ) is the normal force at the center of the contact segment. After the friction forces at each loaded contact point q ∈ [1, Q ] are computed, the instantaneous gear mesh mechanical power is obtained as [5.9]
Q
( m) Pmg = Pin − ∑ F ( m) (u1 − u2 ) .
(5.3b)
q =1
Finally, average value of mechanical power loss over a mesh cycle is computed as [5.9]
Pmg =
M
1 M
∑ Pmg(m) .
(5.3c)
m =1
140
5.2.3 Bearing Power Loss Model The total bearing power loss is given by the sum of the bearing mechanical power loss and bearing viscous (spin) power loss. For the computation of the bearing mechanical power losses, the axial and radial components of forces acting on bearings placed at either ends of a shaft s are computed at each duty cycle increment, for a given input torque, input speed and average bulk temperature value. The preload force, if any, is included in the computation of the axial component of the bearing force. Once the effective bearing force is obtained from force and moment balance, the friction moment on a bearing i located on shaft s is computed as [5.10]
(i , s ) M mb = Fˆ λ d ,
(5.4a)
where λ is a friction factor, d = 0.5(d o + di ) is the mean bearing diameter (in mm), and Fˆ is the effective bearing force. The value of Fˆ is taken to be the greater of the radial
bearing force or the combined product of
4 cot β 5
times the axial bearing force, where β is
the bearing contact angle. With a rotational shaft speed of ωs , the power loss due to (i ,s ) (i ,s ) = M bm ωs . Similarly, the friction at i bearings of a given shaft s is computed as Pbm
spin torque loss for a bearing i on a shaft s is given by: [5.10]
(i , s ) M sb
(10−7 ) f L µ k 2 / 3Ω 2 / 3d m3 = 160(10−7 ) f L d m3
µ k Ω > 2000 µ k Ω < 2000 141
,
(5.4b)
where f fl is a bearing viscous friction constant, ns is the shaft speed in rpm and ν k is the kinematic viscosity of the lubricant in mm2/s. With this, the spin power loss of the (i ,s ) ωs . same bearing is found as Psb(i ,s ) = M sb
5.3 Application of the Power Loss Models to an Actual Transmission In this section, the manual transmission power loss methodology shown in the flowchart of Figure 5.1 will be applied to a production manual transmission. This 6-speed manual transmission shown in Figure 5.2 will be considered as the example transmission. This transmission consists of four shafts, input, upper output, lower output and differential. All the gears on the input shaft are fixed to it so that they rotate at the input speed. As all speed gears on both output shafts are engaged with their counterparts on the input shaft at all instances, they also rotate at speeds proportional to the input speed according to their gear ratios. The speeds of the final drive gears, on the other hand are found from the speed gear and final drive ratios. With this and the duty cycle data ( T j ,
ω j , g j ( g j ∈ [1, 6] ), rotational speeds of all of the gears are calculated in the ‘Kinematics and Powerflow’ routine. Likewise, individual immersion depths of each gear pair are determined based on the volume of oil in the transmission. In addition, bulk transmission temperature θ j at a discrete time instant j ( j ∈ [1, J ] ) is input to the transmission power loss model. Figure 5.3 shows the orientation of a total of 9 gear pairs (15 gears) of the 6-speed example manual transmission for given arbitrary static oil level. With this oil level, 142
Figure 5.2 Example system: A 6-speed manual transmission.
143
Figure 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) 1st and reverse gear pairs, and (f) final drive gear pair. 144
immersion parameter h of each gear is determined and drag spin loss calculation are carried out using the model of Section 5.2.1 for all 15 gears. Likewise, pocketing losses for each of the 9 gear meshes are also computed using the pocketing loss model of the same section. If a gear is not immersed in oil, its oil churning drag loss is assumed to be zero and the windage loss is computed instead by considering the properties of air replacing that of oil. Similarly, the pocketing loss of any gear mesh above the static oil level is assumed to be negligible. For the 6-speed manual transmission, the total gear spin loss at a given duty cycle instance j is then found as
15
9
i =1
r =1
Psg = ∑ [ Pd ]i + ∑ Pp . r
(5.5a)
As each shaft in this transmission is supported by a pair of tapered roller bearings, the spin power loss due to bearings is given as
4
2
Psb = ∑∑ Psb(i , s ) .
(5.5b)
s =1 i =1
With this, the total spin power loss of the transmission due to gears and bearings is Ps = Psm + Psb . Next, following the mechanical loss branch of Figure 5.1 and employing the model presented in Section 5.2.2, the mechanical gear mesh losses are computed.
These
computations are done only for the gear pairs that carry load. In any given forward gear 145
state, g j ∈ [1, 6] , this transmission has only two loaded gear pairs. For instance, the speed gear pair (sg) in Figure 5.3(e) and the lower final drive (fd) gear pair of Figure 5.3(f) are the only load carrying gear meshes that must be considered in mechanical loss computations. Accordingly, total mechanical gear mesh loss of the transmission at the same duty cycle instance is given as
( sg ) ( fd ) Pmg = Pmg + Pmg ,
(5.6a)
Likewise, for each bearing with its load computed (including preload), the mechanical power loss from all the eight bearings of the 6-speed sample manual transmission is given as
4
2
(i ,s ) Pmb = ∑∑ Pmb ,
(5.6b)
s =1 i =1
and the total mechanical power loss of the transmission due to gears and bearings is found as Pm = Pmg + Pmb .
Finally, the total power loss of the transmission is given as
PT = Ps + Pm .
146
5.4 Transmission Power Loss Experiments and Model Validation Chapter 4 details the validation of the gear pair spin power loss model. Hence, the validation activity in this chapter will be performed at the transmission level. For this purpose, an experimental setup was designed to measure the mechanical and spin power losses of the 6-speed manual transmission over a wide range of input torque, operating speed and bulk transmission temperatures [5.11]. In this section, a brief description of the experimental procedure will be provided and direct comparisons between the power loss measurements and predictions will be made to assess the accuracy of the proposed model.
5.4.1 Test Set-up and Testing Methodology Figure 5.4 shows the dynamometer setup that was designed and implemented for measurement of power losses of the 6-speed manual transmission through tests carried out by Szweda [5.11]. A more detailed description of the experimental setup is provided in ref. [5.11]. Describing only the necessary details, a 110 kW DC drive motor is connected to the example manual transmission through a slender shaft, a pair of flexible couplings and a precision torque-meter at the input side. A support bracket and an adapter plate are used to hold the transmission in place at its intended position. On the output side, a precision torque-meter is connected to the output shaft by a flexible coupling. On the other side of the output torque-meter, a long shaft and another flexible coupling is used to connect to a 1.6 ratio speed-increaser that is connected to an eddy-current load brake dynamometer (400 kW). A thermocouple inserted from the bottom drain hole measures the bulk transmission temperature.
147
Figure 5.4 Experimental test setup for measuring transmission power losses of the example 6-speed transmission. Safety guards are removed for demonstration purposes.
148
Measurements were repeated at various input speeds ( ω j = ωin ) and gear stages,
g j , for both the loaded and unloaded test cases. The input speed was maintained within ±20 rpm and the input torque in the loaded test case was maintained within ±2 Nm. First, unloaded tests were conducted with output shaft disconnected. The torque, Tin , measured by the input-side torque-meter was used to compute the transmission spin power loss Ps = ω jTin . Repeating the same test with the output side connected and applying a nominal input torque value of T j , the power loss of the transmission was found as
PT = ω j (Tin − Tout Γ j ) where Γ j is the input-to-output speed ratio and Tout is the torque value measured by the output-side torque-meter.
5.4.2 Model Validation Figure 5.5 provides for comparison between the measured and predicted dimensionless transmission mechanical power loss values, Pm = Pm Pref , where Pm is the transmission mechanical power loss value and Pref is a reference power loss value for the example 6-speed manual transmission. The comparisons are done an input operating speed range of 1,000 to 4,000 rpm and for an input torque of 100 Nm at 80o C with the production oil volume. The composite root-mean-square surface roughness of each gear pair was measured to be S = 0.56 µ m. As is apparent from the figure, the predicted and measured dimensionless transmission mechanical power loss values show good agreement over the entire range of operating conditions and gear stages, with maximum deviations
149
0.5
Experiment
(a) 1000 rpm
(b) 2000 rpm
(c) 3000 rpm
(d) 4000 rpm
Prediction
0.4
Mechanical Power Loss Pm
0.3 0.2
0.1 0.0 0.5 0.4
0.3 0.2
0.1 0.0 1
2
3
4
5
6
1
2
3
4
5
6
Gear Range
Figure 5.5
Comparison of measured [5.11] and predicted transmission mechanical power loss values.
150
between the measured and predicted Pm values being well within 0.1 kW. This suggests that the transmission mechanical power loss model elucidated in Section 5.3 is sufficiently accurate for studies to characterize friction losses in gear trains and transmissions and can also be used to study methods for reduction of the load-dependent friction power losses. Figure 5.6 provides for comparisons between the measured and predicted dimensionless spin power loss values, Ps = Ps Pref , where Ps is the transmission spin power loss and Pref is the same reference power loss value, for gear stages 2 to 6 and for a range of operating speed, Ωin = 1,000 to 4,000 rpm. Here, the same volume of oil at 80o C was used for these tests. As shown in Figure 5.6, the predicted and measured spin power loss values agree quite well over the entire operating range, with the predictions remaining consistently lower than the measured spin power loss values, though following the same trend as the measurements. The maximum deviation between the predicted and measured Ps values are within about 0.3 kW. Several factors might be the reason for this difference between the measurements and predictions, the most important of them being the fact that the spin loss model captures only the gear and bearing related losses. Inclusion of the other sources of spin power losses such as synchronizers and seals should bring the predictions even closer to the measurements. Also, the oil churning model uses static oil levels to compute the immersion depths of gears in churning. Under dynamic conditions, effective oil levels should be expected to be higher, especially for the upper chamber of the transmission, increasing the predicted losses further. It is also important to note that the bearing viscous power loss model used in this study is rather simplistic as it 151
2.0
Experiment
(a) 2nd Gear
(b) 3rd Gear
(c) 4th Gear
(d) 5th Gear
Prediction 1.6
1.2
Spin Power Loss Ps
0.8
0.4
0.0 2.0
1.6
1.2
0.8
0.4
0.0 0
1000
2000
2.0
3000
4000
0
Experiment
1000
2000
3000
4000
(e) 6th Gear
Spin Power Loss Ps
Prediction 1.6
1.2
0.8
0.4
0.0 0
1000
2000
3000
4000
Input Speed [rpm]
Figure 5.6 Comparison of measured [5.11] and predicted transmission spin power loss values: (a) 2nd gear, (b) 3rd gear, (c) 4th gear, (d) 5th gear, and (e) 6th gear. 152
cannot account for several key parameters, most importantly the oil immersion depth. Therefore, bearing viscous power loss predictions are expected to be less accurate. On the whole, these comparisons indicate that the transmission spin power loss model formulated and developed is indeed effective in predicting the spin power loss of gear pairs under different configurations in a transmission in a computationally effective way, thus providing confidence to end users.
5.5 Concluding Remarks In this chapter, a physics-based methodology was proposed to predict the mechanical and spin power losses of geared transmissions. The transmission power loss model incorporates a transmission spin power loss model and a transmission mechanical power loss model in conjunction with bearing mechanical and viscous power loss models in order to predict the load-dependent and load-independent power losses of all the gear pairs in a transmission. Comparisons of the predictions to tightly controlled transmission power loss measurements demonstrate good correlation over wide ranges of operating speed, input torque and gear range, indicating that the model can be used for design and product improvement activities.
153
References for Chapter 5 [5.1]
Changenet, C. and Velex, P., 2005, “Power Loss Predictions in a 6-speed Manual Gearbox Using Thermal Networks”, ASME Design Engineering Technical
Conferences and Computers and Information in Engineering Conference, DETC2005-84290, Long Beach, CA.. [5.2]
Heingartner, P., and Mba, D., 2003, “Determining Power Losses in The Helical Gear Mesh; Case Study,” Proceeding of DETC’3, ASME 2003 Design
Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, I [5.3]
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, 337–346
[5.4]
Anderson, N. E., and Loewenthal, S. H., 1982, “Design of Spur Gears for Improved Efficiency,” ASME Journal of Mechanical Design, 104, pp. 767–774..
[5.5]
Changenet, C., and Velex, P., 2007, “A Model for the Prediction of Churning Losses in Geared Transmissions – Preliminary Results“, ASME Journal of
Mechanical Design, 129(1), pp. 128-133. [5.6]
Handschuh, R. F., Kilmain, C. J., 2003, “Efficiency of High-Speed Helical Gear Trains,” 59th Annual Forum and Technology Display sponsored by the American Helicopter Society, Phoenix, AZ.
154
[5.7]
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), 51–59. [5.8]
van Dongen, L. A. M., 1982, “Efficiency Characteristics of Manual and Automatic Passenger Car Transaxles,” SAE Technical Paper Series, 820741, Warrendale, PA: Society of Automotive Engineers.
[5.9]
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. [5.10] Harris, A. T., 2001, Rolling Bearing Analysis, Fourth Edition, Wiley & Sons, Inc., New York. [5.11] Szweda, T, A., “An Experimental Study of Power Loss of an Automotive Manual Transmission,” MS thesis, The Ohio State University, Columbus, Ohio, 2008.
155
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK
6.1 Summary This dissertation lays the framework for a physics-based fluid mechanics model to predict oil churning and windage power losses in gear pairs. The oil churning and windage power loss models, both developed for the case of a spur gear pair, have been validated through comparisons to power loss measurements conducted on a family of unity-ratio spur gear pairs, under dip- and jet-lubricated conditions. Validations were carried out over a wide range of operating conditions, different oil immersion depths, lubrication conditions and through variations in gear geometry parameters, to enhance the fidelity of the proposed models. The influence of key system parameters were also quantified through detailed example analyses of the respective power loss models. Further, the oil churning power loss model, in conjunction with a mechanical power loss model and existing bearing power loss models, has been generalized, in the form of a transmission power loss model, to predict the total power loss in automotive manual transmissions.
Power loss measurements on a sample six-speed manual
transmission corroborate the predictions of the proposed transmission power loss model. 156
The power loss computation methodologies proposed in this dissertation have the potential to be generalized so as to form the foundation for future formulations that can handle other types of complex gear drives, beyond just spur and helical. It cannot be more highly stressed that in lieu of the lack of physics-based models that exist in transmission power loss literature, the models developed in this dissertation, with all their limitations, present a detailed first-stab towards the prediction of power losses in gears without compromising on computational efficiency in providing a clear window into the mechanism of such power losses in geared systems.
6.2 Conclusions and Contributions The comprehensive modeling efforts to predict oil churning and windage power losses in gear pairs, backed by experimental validations on both the gear pair and the transmission levels, have resulted in the following general conclusions:
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Predictions of the oil churning power loss model compare reasonably well with dip-lubricated spin loss measurements. Concurrently, predictions of the windage power loss model show a good correlation with jet-lubricated experiments. This proves the effectiveness of both of the model prediction methodologies. The transmission power loss model also compares well with measurements from a sample six-speed manual transmission, which makes the transmission power loss model suitable as a design tool.
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Oil immersion depth has maximal influence on power loss due to oil churning, with a substantial increase seen in oil churning losses as the oil immersion depth increases. In the case of transmission spin losses, the influence of oil immersion depth is also as significant, with greater spin losses predicted at higher oil fill levels.
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Power loss due to oil churning and windage decrease with an increase in operating temperature, regardless of operating speed. Correspondingly, oil churning and windage power losses are influenced heavily by operating speed, registering substantial increase with an increase in gear rotational speed. In the case of power losses in a transmission, the influence of operating speed is more pronounced at higher gear stages.
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Face width, gear module and outside diameter influence the total spin power loss to a much lesser extent than lubricant parameters and operating conditions, with comparatively minor increases seen in the model predictions with an increase in gear geometry parameters. Surface roughness plays no part in influencing either the oil churning or windage modes of power losses.
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Power loss due to oil churning is several times higher than power losses loss due to windage, under similar operating conditions.
Also, drag and pocketing
components of load-independent power losses overshadow the power loss due to root filling.
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This dissertation research provides a fundamental understanding of the mechanisms of spin power losses of gear pairs as well as of larger transmission systems. The following can be listed as the main contributions of this study to the sate-of-the-art.
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This study has provided the first physics-based analytical model for oil churning and windage losses of a spur gear pair. As previous efforts relied on empirical or CFD type approaches, their capabilities were rather limited, both computationally and methodologically. The analytical and closed-form nature of the proposed oil churning and windage power loss models makes possible for computationally effective power loss estimations under various operating conditions, even during the early stages of product design, allowing for discrimination of various design concepts based on their efficiency attributes.
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The spin power loss model developed in this study represents a general formulation, which has the potential to form a foundation for similar such models for hypoid gears in automotive axles (a significant source of power losses) and planetary gear sets in automatic transmissions.
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This study also constitutes the first theoretical model for predicting both loaddependent and load-independent power losses of multi-mesh gear systems such as manual transmissions and gearboxes. While being limited in its contributions to the fundamental aspect of this topic, the proposed model has the potential to impact upon the intended goals of this study in terms of fuel economy of vehicles with manual transmissions. The model proposed will include all key system
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parameters so that cost effective strategies to lower transmission power loss can be developed.
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The proposed gear pair and transmission power loss models will also distinguish themselves as the first validated power loss models.
Comparisons to
measurements from tightly controlled gear pair experiments as well as experiments conducted on a six-speed manual transmission make for a complete multi-parameter validation of the models at both gear pair and transmission levels. This provides added confidence to end users.
6.3 Recommendations for Future Work Although the proposed models are exhaustive in their approach, they are by no means capable of addressing all the functional issues that come into play when geared systems are looked into in their full glory. As the topic of efficiency of automotive applications is becoming more crucial, especially in high power applications, and with increasing demands placed on conservation of fossil-based resources, it is vital to expand on some of the stated objectives of this dissertation, as a means to enhance the accuracy of the current model predictions. The following are some recommendations for future work:
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Effect of enclosures: Drag power losses in the case of oil churning and windage modes of power loss did not include the effects of enclosures. The presence of peripheral shrouds, baffles and flanges will certainly change the nature of the 160
external flow or boundary layer between the enclosure and the gear surfaces. As such, the current formulations are valid for as long as the distance between the enclosures and the gear surfaces is greater than the combined thickness of boundary layers from both the surfaces. Future modifications of the formulations should look into the effects of peripheral clearance on churning and windage losses. This can have potential impact on noise and vibrations induced through pocketing power losses as well, with an increase in the intensity of noise produced upon impact of the ejected jet of lubricant from the meshing zone as enclosures wrap gear pairs more tightly.
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Presence of form drag: The drag formulations detailed in Chapters 2 and 3 take into account only the skin friction drag while calculating the total drag loss on the periphery and faces of the gears, before boundary layer separation has taken place.
The effects of pressure or form drag are not included based on the
assumption that the wake region formed on boundary layer separation is negligible. This has to be looked into at greater depth due to the fact that form drag can potentially increase churning power loss at low viscosity values and higher speeds of operation. The inclusion of form drag will need more details on the nature of flow separation and adverse pressure gradient leading to separation.
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Dynamic oil levels: All through the power loss formulations, it has been assumed that oil immersion depths are quasi-static in nature, with a broad based supposition that a reduction in oil level in one part of the chamber encasing the gear pairs will result in an increase in oil level at another part of the chamber, 161
leading to an averaging effect, which aids in the assumption. In reality, oil levels are highly dynamic in nature, with oil swirling all around the gear faces and periphery, making it very difficult to peg onto a particular oil level value for computation purposes. The inclusion of dynamic oil levels, through means of a fundamental relationship between oil level, operating speed, enclosure/gear size and lubricant conditions, can greatly aid in improving the accuracy of power loss predictions.
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Improvements to the pocketing power loss formulation: It has been consistently shown that pocketing power loss is the major component of the total gear pair spin power loss. Future revisions to the pocketing component of the spin power loss model must include the effects of a moving control volume approach, and the motion of the control surface. Further more, flow through the pockets is threedimensional, with a flow factor in the current formulation taking this into account. A more detailed three-dimensional analysis can account for such assumptions of the current work. Also, during the modeling of pocketing power loss, it has been assumed that there exists no pressure differential between opposing control volumes thereby leading to the consideration of the control volumes as separate entities in time.
A renewed and in depth approach will consider the
communication between opposing control volumes as a continua, which will augment the mass flow rate of the fluid ejected through the end flow areas.
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Contribution of helix angle: The spin power loss model has been developed for the simplest case of spur gears. In most transmission applications, the gears used 162
are helical in nature, for obtaining higher speeds and smoother progression of mating, leading to a reduction in operating noise. When the spin power loss model is applied to the scenario of a pair of helical gears, changes in the flow areas due to helix angle has to be considered when applying the pocketing power loss formulation. On the whole, power losses are expected to be lesser for helical gears than in the case of spur gears; the formulations have to be subtended to take this into account.
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Surface roughness of gears: In the turbulent regime, while modeling drag losses on the faces and periphery, surface roughness might potentially change the behavior of drag forces, leading to earlier flow separation by enhancing the intense mixing taking place in the boundary layer. This effect can be considered while modifications to the drag losses are being made. Also, at lower speeds of rotation, the boundary layer characteristics might change from that of a growth around a disk. Future revisions to the drag loss formulations should take this into account.
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Effects of air-ingestion: Air-ingestion, which can take place for medium oil levels and at high speeds, will change the atmosphere prevailing in the meshing zone, thereby changing the nature of dependence of density on temperature and pressure.
Hence, the analysis itself will then borders on a two-phase flow
approach, increasing the complexity of the problem but also making it a more complete approach through such an inclusion. Partial pressures, densities and viscosities can be utilized in such a case, depending on the aeration ratio. 163
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Effect of rotational direction: Experimental evidence has shown that a change in the direction of gear pair rotation from up-in-mesh to down-in-mesh reduces the total spin power losses of a gear pair. A down-in-mesh rotational arrangement will result in lesser amount of lubricant squeezed through the meshing zone, but greater amounts of air ingestion, leading to a methodology, which will need a combined compressible-incompressible flow approach. This approach, though very obvious, is not easy, as it is extremely hard to deduce the fraction of air to oil ratio in a gear mesh (aeration ratio), at a given rotational position. This works in superposition with the effects of air-ingestion as explained in the previous point.
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Application to hypoid and planetary gear sets:
Subtle changes in the
formulations, in terms of tooth cavity volumes and immersion areas have to be made to accommodate other gear geometries, such as cross-axis gears and planetary gear sets. Future formulations will focus on such geometrical aspects. On a final note, while the intended goal of this dissertation is to come up with a physics-based methodology for predicting spin power losses in gear pairs and transmissions, along the way, broad-based assumptions have been impressed upon the formulation to eschew computational difficulties, with proper justifications. Based on these assumptions, which certainly do not detract from the ease of use of the proposed models, it has to be noted that a point-to-point comparison between measurements and model predictions is certainly not the aim of this dissertation; the main focus of coming up with such a methodology is to predict the trends of load-independent power losses in gear pairs from a simplistic but effective engineering viewpoint. In that sense, this 164
dissertation achieves what it set out to accomplish. It is hoped that future users of the models formulated in this dissertation will understand and appreciate the limitations of this work in terms of the complexity of the problem handled and the apparent sensitivity of the power loss predictions to a whole host of parameters other than the most important ones considered in this work.
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